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author:
- 'O. Guibert and T. Mansour'
title: ' [restricted $132$-involutions and Chebyshev polynomials]{}'
---
[LABRI (UMR 5800), Université Bordeaux 1, 351 cours de la Libération, 33405 Talence Cedex, France]{}\
[guibert@labri.fr]{}, [toufik@labri.fr]{}
Abstract {#abstract .unnumbered}
========
We study generating functions for the number of involutions in $S_n$ avoiding (or containing once) $132$, and avoiding (or containing once) an arbitrary permutation $\tau$ on $k$ letters. In several interesting cases the generating function depends only on $k$ and is expressed via Chebyshev polynomials of the second kind. In particular, we establish that involutions avoiding both $132$ and $12\dots k$ have the same enumerative formula according to the length than involutions avoiding both $132$ and any [*double-wedge pattern*]{} possibly followed by fixed points of total length $k$. Many results are also shown with a combinatorial point of view.
Introduction
============
A permutation is a bijection from $[n]=\{1,2,\ldots,n\}$ to $[n]$. Let $S_n$ be the set of permutations of length $n$.
Let $\alpha\in S_n$ and $\tau\in S_k$ be two permutations. We say that $\alpha$ [*contains*]{} $\tau$ if there exists a subsequence $1\leq i_1<i_2<\dots<i_k\leq n$ such that $(\alpha_{i_1},\dots,\alpha_{i_k})$ is order-isomorphic to $\tau$; in such a context $\tau$ is usually called a [*pattern*]{}. We say that $\alpha$ [*avoids*]{} $\tau$, or is $\tau$-[*avoiding*]{}, if such a subsequence does not exist. The set of all $\tau$-avoiding permutations in $S_n$ is denoted $S_n(\tau)$. For an arbitrary finite collection of patterns $T$, we say that $\alpha$ avoids $T$ if $\alpha$ avoids any $\tau\in T$; the corresponding subset of $S_n$ is denoted $S_n(T)$.
While the case of permutations avoiding a single pattern has attracted much attention, the case of multiple pattern avoidance remains less investigated. In particular, it is natural, as the next step, to consider permutations avoiding pairs of patterns $\tau_1$, $\tau_2$. This problem was solved completely for $\tau_1,\tau_2\in S_3$ (see [@SS]), for $\tau_1\in S_3$ and $\tau_2\in S_4$ (see [@W]), and for $\tau_1,\tau_2\in S_4$ (see [@B1; @Km] and references therein). Several recent papers [@CW; @MV1; @Kr; @MV2] deal with the case $\tau_1\in S_3$, $\tau_2\in S_k$ for various pairs $\tau_1,\tau_2$. Another natural question is to study permutations avoiding $\tau_1$ and containing $\tau_2$ exactly $t$ times. Such a problem for certain $\tau_1,\tau_2\in S_3$ and $t=1$ was investigated in [@R], and for certain $\tau_1\in S_3$, $\tau_2\in S_k$ in [@RWZ; @MV1; @Kr]. The tools involved in these papers include continued fractions, Chebyshev polynomials, and Dyck words.
[*Chebyshev polynomials of the second kind*]{} (in what follows just Chebyshev polynomials) are defined by $$U_r(\cos\theta)=\frac{\sin(r+1)\theta}{\sin\theta}$$ for $r\geq0$. Evidently, $U_r(x)$ is a polynomial of degree $r$ in $x$ with integer coefficients. Chebyshev polynomials were invented for the needs of approximation theory, but are also widely used in various other branches of mathematics, including algebra, combinatorics, and number theory (see [@Ri]).
[*Dyck words*]{} are words $w$ of $\{x,\overline{x}\}^*$ verifying that $|w|_x = |w|_{\overline{x}}$ and that for all $w = w' w''$, $|w'|_x \geq |w'|_{\overline{x}}$. Dyck words of length $2n$ are enumerated by the $n$th Catalan number $C_n=\frac{1}{n+1}\binom{2n}{n}$ whose generating function is $C(x)=\frac{1-\sqrt{1-4x}}{2x}$.
We also consider words of $\{a,b^2\}^*$ of length $n$ enumerated by the $n$th Fibonacci number $F_n$ with $F_0=F_1=1$ and $F_{n}=F_{n-1}+F_{n-2}$ whose generating function is $F(x)=\frac{1}{1-x-x^2}$.
Apparently, for the first time the relation between restricted permutations and Chebyshev polynomials was discovered by Chow and West in [@CW], and later by Mansour and Vainshtein [@MV1; @MV2; @MV3; @MV4], Krattenthaler [@Kr]. These results related to a rational function $$R_k(x)=\frac{2t U_{k-1}(t)}{U_k(t)}, \qquad t=\frac{1}{2\sqrt{x}}$$ for all $k\geq 1$.\
An involution is a permutation such that its cycles are of length $1$ or $2$ that is $\alpha\in S_n$ is an involution if and only if $\alpha(\alpha_i)=i$ for all $i\in [n]$.
Some authors considered involutions with forbidden patterns.\
Regev in [@Regev] provided asymptotic formula for $12\cdots
k$-avoiding involutions of length $n$ and he also established that $1234$-avoiding involutions of length $n$ are enumerated by Motzkin numbers. Gessel [@Gessel] exhibited the enumeration of such $12\cdots k$-avoiding involutions of length $n$. Moreover, Gouyou–Beauchamps [@GouyouBYoung] obtained by an entirely bijective proof very nice exact formulas for the number of $12345$-avoiding and $123456$-avoiding involutions of length $n$.\
Gire [@GireThese] studied some permutations with forbidden subsequences and established a one-to-one correspondence between 1-2 trees having $n$ edges and permutations of length $n$ avoiding patterns $321$ and $231$, the latter being allowed in the case where it is itself a subsequence of the pattern $3142$. Guibert [@GuibertThese] also established bijections between 1-2 trees having $n$ edges and another set of permutations with forbidden subsequences and $3412$-avoiding involutions of length $n$ and $4321$-avoiding involutions of length $n$ (and so with $1234$-avoiding involutions of length $n$ by transposing the corresponding Young tableaux obtained by applying the Robinson-Schensted algorithm). He also shown [@GuibertThese] a bijection between vexillary involutions (that is $2143$-avoiding involutions) and $1243$-avoiding involutions. More recently, Guibert, Pergola and Pinzani [@GPP] established a one-to-one correspondence between 1-2 trees having $n$ edges and vexillary involutions of length $n$. So all these sets are enumerated by the $n$th Motzkin number $\sum_{i=0}^{\lfloor\frac{n}{2}\rfloor}\binom{n}{2i}C_{i}$. It remains a connected open problem: in [@GuibertThese] conjectures that $1432$-avoiding involutions of length $n$ are also enumerated by the $n$th Motzkin number.
In this paper we present a general approach to the study of involutions in $S_n$ avoiding $132$ (or containing $132$ exactly once), and avoiding (or containing exactly once) an arbitrary pattern $\tau\in
S_k$. As a consequence, we derive all the previously known results for this kind of problems, as well as many new results. Some results are also established by bijections as for an example a bijection between $132$-avoiding involutions and primitive Dyck words.
The paper is organized as follows. The case of involutions avoiding both $132$ and $\tau$ is treated in Section $2$. We present an explicit expression in terms of Chebyshev polynomials for several interesting cases. The case of involutions avoiding $132$ and containing $\tau$ exactly once is treated in section $3$. Here again we present an explicit expression in terms of Chebyshev polynomials for several interesting cases. Finally, the cases of involutions containing $132$ exactly once and either avoiding or containing exactly once an arbitrary pattern $\tau$ is treated in sections $4$ and $5$; respectively.
Avoiding $132$ and another pattern {#sec2}
==================================
Let $I_T(n)$ denote the number of involutions in $S_n(132)$ avoiding $T$, and let $I_T(x)=\sum_{n\geq 0}I_T(n)x^n$ be the corresponding generating function. The following proposition is the base of all the other results in this section, which holds immediately from definitions.
\[prom\] For any involution $\pi\in S_n(132)$ such that $\pi_j=n$ holds either,
1. for $1\leq j\leq [n/2]$, $\pi=(\beta,n,\gamma,\delta,j)$, where
- $\beta$ is a $132$-avoiding permutation of the numbers $n-j+1,\dots,n-2,n-1$,
- $\delta$ is a $132$-avoiding permutation of the numbers $1,\dots,j-2,j-1$ such that $\delta\cdot\beta$ is the identity permutation of $S_{j-1}$,
- $\gamma$ is a $132$-avoiding involution of the numbers $j+1,j+2,\dots,n-j-1,n-j$;
2. for $j=n$, $\pi=(\beta,n)$ where $\beta$ is an involution in $S_{n-1}(132)$.
As a corollary of Proposition \[prom\] we get the generating function for the number of involutions in $S_n(132)$ as follows.
[(see [@SS Prop. 5])]{} Let $C(t)$ be the generating function for the Catalan numbers; then $$I_\varnothing(x)=\frac{1}{1-x-x^2C(x^2)}.$$
Proposition \[prom\] yields for all $n\geq 1$, $$I_\varnothing(n)=\sum_{j=1}^{[n/2]} C_{j-1}I_\varnothing(n-2j)+I_\varnothing(n-1),$$ where $C_{j-1}$ is the $(j-1)$th Catalan number. Besides $I_\emptyset(0)=1$, therefore in terms of generating function we get that $$I_\varnothing(x)=1+x^2C(x^2)I_\varnothing(x)+xI_\varnothing(x).$$
We can also prove this result by a bijective point of view.
Let $P_{x,\overline{x}} = \{ w \in \{x,\overline{x}\}^* :
\mbox{ for all } w = w' w'', |w'|_x \geq |w'|_{\overline{x}} \}$ be the language of primitive Dyck words.\
The number of such words of $P_{x,\overline{x}}$ of length $n$ is the central binomial coefficient $\binom{n}{\lfloor\frac{n}{2}\rfloor}$ with $n \geq 0$. Indeed, any primitive Dyck word $w$ of $P_{x,\overline{x}}$ can be uniquely written as $w_0 x w_1 x \ldots x w_p$ where $w_i$ is a Dyck word (that is $w_i \in P_{x,\overline{x}}$ and $|w_i|_x =
|w_i|_{\overline{x}}$) for all $0 \leq i \leq p$, but $w$ can also be uniquely written as $w_0 \overline{x} w_1 \overline{x} \dots
\overline{x} w_{\lceil \frac{p}{2} \rceil - 1} \overline{x}
w_{\lceil \frac{p}{2} \rceil} x w_{\lceil \frac{p}{2} \rceil + 1}
x \ldots x w_p$. So primitive Dyck words $w$ of $P_{x,\overline{x}}$ of length $n$ are in bijection with bilateral words of $\{ w \in \{x,\overline{x}\}^* : |w|_x =
|w|_{\overline{x}} \mbox{ or } |w|_x = |w|_{\overline{x}} - 1 \}$ of length $n$ trivially enumerated by $\binom{n}{\lfloor\frac{n}{2}\rfloor}$.
\[Phi\] There is a bijection $\Phi$ between involutions in $S_n(132)$ and primitive Dyck words of $P_{x,\overline{x}}$ of length $n$. Moreover, the number of fixed points of the involution corresponds to the difference between the number of letters $x$ and $\overline{x}$ into the primitive Dyck word.
Let $\pi$ be a $132$-avoiding involution on $[n]$ having $p$ fixed points. According to Proposition \[prom\] we have $\pi=\pi'\pi''x\pi'''$ with $|\pi'|=\frac{n-p}{2}$ ($\pi'$ has no fixed points and constitutes cycles with $\pi''$ or $\pi'''$), $\pi''$ does not contain fixed point and $\pi(x)=x$ ($x$ is the first fixed point). We obtain two $132$-avoiding involutions on $[n+1]$ from $\pi$: the first one is given by inserting a fixed point between $\pi'$ and $\pi''$, and the second one (iff $\pi$ has at least one fixed point) is given by modifying the first fixed point $x$ by a cycle starting between $\pi'$ and $\pi''$. All $132$-avoiding involutions can be obtained (and only once) by applying this rule, starting from the empty involution of length $0$.\
Let $w$ be a primitive Dyck word of $P_{x,\overline{x}}$ of length $n$ such that $|w|_x-|w|_{\overline{x}}=p$. So we have $w = w_0 x
w_1 x \ldots x w_p$ where $w_i$ are Dyck words for all $0 \leq i
\leq p$. We obtain two primitive Dyck words of length $n+1$ from $w$: $x w$ and $x w_0 \overline{x} w_1 x \cdots x w_p$ (iff $p>0$). All primitive Dyck words can be obtained (and only once) by applying this rule, starting from the empty word of length $0$.
Clearly, these two generating trees for the $132$-avoiding involutions and the primitive Dyck words can be characterized by the following succession system:\
$\left\{\begin{array}{lcll}
\multicolumn{4}{l}{(0)} \\
(0) & \leadsto & (1) & \\
(p) & \leadsto & (p+1) , (p-1) & \mbox{if } p \geq 1
\end{array}\right.$
Figure \[fig-ag\] shows the bijection $\Phi$ between $132$-avoiding involutions and primitive Dyck words (and the labels of the succession system which characterizes them) for the first values.
=590.0pt
\[nb132ptfix\] The number of $132$-avoiding involutions of length $n$ is $\binom{n}{\lfloor \frac{n}{2} \rfloor}$. Moreover, the number of $132$-avoiding involutions of length $n$ having $p$ fixed points with $0 \leq p \leq n$ (and $p$ is odd iff $n$ is odd) is the ballot number $\binom{n}{\frac{n+p}{2}} -
\binom{n}{\frac{n+p}{2}+1}$.
Indeed, the number of primitive Dyck words $w$ of $P_{x,\overline{x}}$ according to the length and $|w|_x-|w|_{\overline{x}}$ is given by the ballot number (or Delannoy number [@Errera] or distribution $\alpha$ of the Catalan number [@Kre]).
In particular, the number of $132$-avoiding involutions of length $2n$ without fixed points is $C_n$ the $n$th Catalan number. The following theorem is the base of all the other results in this section.
\[thg\] Let $T$ set of patterns, $T'=\{(\tau,|\tau|+1) : \tau\in T\}$, and let $S_T(x)$ be the generating function for the number of $T$-avoiding permutations in $S_n(132)$. Then $$I_{T'}(x)=\frac{1}{1-x^2S_T(x^2)}+\frac{x}{1-x^2S_T(x^2)}I_T(x).$$
Proposition \[prom\] with definitions of $T'$ yields for $n\geq
1$, $$I_{T'}(n)=I_T(n-1)+\sum_{j=1}^{[n/2]} s_T(j-1)I_{T'}(n-2j),$$ where $s_T(j-1)$ is the number of permutations in $S_{j-1}(132,T)$. Hence, in terms of generating functions we have $$I_{T'}(x)-1=x\cdot I_T(x)+x^2 S_T(x^2)\cdot\frac{I_{T'}(x)+I_{T'}(-x)}{2}+x^2S_T(x^2)\cdot\frac{I_{T'}(x)-I_{T'}(-x)}{2},$$ so the theorem holds.
Avoiding $132$ and $12\dots k$ {#subsec21}
------------------------------
\[ex1\] [(see [@SS])]{} Let us find $I_{123}(x)$; let $T'=\{123\}$ and $T=\{12\}$, so Theorem \[thg\] gives $$I_{123}(x)=\frac{1}{1-x^2S_{12}(x^2)}+\frac{x}{1-x^2S_{12}(x^2)}I_{12}(x),$$ where by definitions $S_{12}(x)=I_{12}(x)=\frac{1}{1-x}$, hence $$I_{123}(x)=\frac{1+x}{1-2x^2},$$ which means the number of involutions $I_{123}(n)$ is given by $2^{[n/2]}$ for all $n\geq 0$.
Similarly, $I_{1234}(x)=\frac{1}{1-x-x^2}$, so the number of involutions $I_{1234}(n)$ is given by $F_n$ the $n$th Fibonacci number.
The case of varying $k$ is more interesting. As an extension of Example \[ex1\] let us consider the case $T=\{12\dots k\}$.
\[th12k\] For all $k\geq 1$, $$I_{12\dots k}(x)=\frac{1}{x\cdot U_k\left( \frac{1}{2x} \right)}
\sum_{j=0}^{k-1} U_j\left( \frac{1}{2x} \right).$$
Immediately, the theorem holds for $k=1$. Let $k\geq 2$; Theorem \[thg\] gives $$I_{12\dots k}(x)=\frac{1}{1-x^2S_{12\dots (k-1)}(x^2)}+\frac{x}{1-x^2S_{12\dots (k-1)}(x^2)}I_{12\dots (k-1)}(x).$$ On the other hand, the generating function for the sequence $S_n(132,12\dots(k-1))$ is given by $R_{k-1}(x)$ (see [@CW Th. 1]) with $R_k(x)=\frac{1}{1-xR_{k-1}(x)}$ (see [@MV1]) we get that $$I_{12\dots k}(x)=R_k(x^2)+xR_k(x^2)I_{12\dots (k-1)}(x).$$ Besides $I_{1}(x)=R_0(x)=1$, hence by use induction on $k$ and definitions of $R_k(x)$ the theorem holds.
We consider now a combinatorial point of view for this result.
Let $\pi$ be a $132$-avoiding involution. Clearly, if $\pi$ avoids $12\dots k$ then $\pi$ has less than $k$ fixed points. Moreover, if $\pi$ of length $n$ having less than $k$ fixed points is obtained from an $132$-avoiding involution $\sigma$ of length less than $n$ having $k$ fixed points (and take $\sigma$ as big as possible) by applying the rules described for bijection $\Phi$ given by Theorem \[Phi\], then $\pi$ contains a subsequence $1 2
\ldots k$ because the first fixed points of $\sigma$ become cycles into $\pi$ such that the beginning of these cycles and the last remaining fixed points of $\sigma$ into $\pi$ constitute a subsequence of type $1 2 \ldots k$. So the succession system $(*)$\
$\left\{\begin{array}{lcll}
(0) \\
(0) & \leadsto & (1) \\
(p) & \leadsto & (p+1) , (p-1) & 1 \leq p \leq k-2 \\
(k-1) & \leadsto & (k-2)
\end{array}\right.$\
characterizes the generating tree of the involutions avoiding both $132$ and $1 2 \ldots k$.
It is easy to see that for $k$ odd, the number of involutions of length $2m$ avoiding both $132$ and $12\dots k$ is the twice of the number of involutions of length $2m-1$ avoiding both $132$ and $12\dots k$.
Moreover, the reader can note that the set of labels of this succession system is finite and so the corresponding generating function is rational. More precisely, we immediately deduce from the previous succession system that the number of involutions of length $n$ avoiding both $132$ and $12\dots k$ and having $p$ fixed points is given by the $(p+1)$th component of the vector given by $V_k . M_k^n$ where $V_k = ( \ 1 \ 0 \ 0 \ \dots \ 0 \ )$ is a vector of $k$ elements and $M_k = \left(
\begin{array}{ccccccccc}
0 & 1 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & \cdots & 0 & 0 & 0 \\
0 & 1 & 0 & 1 & 0 & \cdots & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 1 & \cdots & 0 & 0 & 0 \\
& & & & & \ddots & & & \\
0 & 0 & 0 & 0 & 0 & \cdots & 1 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & \cdots & 0 & 1 & 0 \\
\end{array} \right)$ is a $k \times k$ matrix.\
In another way, we can see that as an automaton where the states are $0, 1, \dots, k-1$ and the transitions are arrows from $i$ to $i+1$ for $0 \leq i < k-1$ and from $i$ to $i-1$ for $0<i\leq
k-1$.
The bijection $\Phi$ establishes a one-to-one correspondence between involutions of length $n$ avoiding both $132$ and $12\dots
k$ and having $p$ fixed points, and primitive Dyck words $w = w_0
x w_1 x \ldots x w_p$ of $P_{x,\overline{x}}$ of length $n$ such that $w_i$ is a Dyck word of height less than $k-p+i$ (that is $w_i \in P_{x,\overline{x}}$ and $|w_i|_x = |w_i|_{\overline{x}}$ and for all $w_i = w' w''$, $|w'|_x-|w'|_{\overline{x}}<k-p+i$) for all $0 \leq i \leq p$.\
In particular involutions of length $2n$ avoiding both $132$ and $12\dots k$ without fixed points are in bijection by $\Phi$ with Dyck words of length $2n$ of height less than $k$.
These Dyck words of bounded height was considered by Kreweras [@Kre] and Viennot [@V]. In particular, Dyck words of length $2n$ of height less than $1$, $2$, $3$, $4$, $5$ are respectively enumerated by $0$, $1$, $2^{n-1}$, $F_{n-2}$, $\frac{3^{n-1}+1}{2}$ for all $n \geq 1$.\
We provide some simple bijections for special cases $k=3,4,5$ (related to Example \[ex1\]) by generating some well known words in the same way as involutions avoiding both $132$ and $1 2 \ldots k$.
Fist of all, we consider the case $k=3$ and the words of $\{a,b\}^*$ or $a\{a,b\}^*$ enumerated by the powers of $2$ we can generate from the empty word labeled $(0)$ by the rules:\
$\left\{\begin{array}{lcl}
w (0) & \leadsto & a w (1) \\
a w (1) & \leadsto & a w (2) , b w (0) \\
w (2) & \leadsto & a w (1)
\end{array}\right.$\
such that the words labeled $(0)$ start by $b$ whereas the words labeled $(1)$ or $(2)$ start by $a$.\
So, words of $\{a,b\}^n$ (respectively $a\{a,b\}^n$) are in bijection with involutions avoiding both $132$ and $123$ of length $2n$ (respectively $2n+1$) enumerated by $2^n$ (respectively $2^n$).
Next we consider the case $k=4$ and the words of $\{a,b^2\}^\ast$ enumerated by the Fibonacci numbers we can generate from the empty word labeled $(0)$ by the rules:\
$\left\{\begin{array}{lcl}
w (0) & \leadsto & a w (1) \\
a w (1) & \leadsto & a a w (2) , b^2 w (0) \\
a w (2) & \leadsto & b^2 w (3) , a a w (1) \\
w (3) & \leadsto & a w (2)
\end{array}\right.$\
such that the words labeled $(0)$ or $(3)$ start by $b^2$ whereas the words labeled $(1)$ or $(2)$ start by $a$.\
So, words of $\{a,b^2\}^\ast$ of length $n$ are in bijection with involutions in $S_n(132,1234)$ enumerated by $F_n$ the $n$th Fibonacci number.
Now we consider the case $k=5$ and the words of $\{a,b,c\}^\ast a$ or $\{a,b,c\}^\ast a \cup b \{a,b,c\}^\ast a$ enumerated by the powers of $3$ we can generate from the empty word labeled $(0)$ by the rules:\
$\left\{\begin{array}{lcl}
w (0) & \leadsto & a w (1) \\
w (1) & \leadsto & b w (2) , w (0) \\
w = b w' = b b^* c w (2) & \leadsto & w (3) , a w' (1) \\
w = b w' = b b^* a w (2) & \leadsto & c w' (3) , w (1) \\
w (3) & \leadsto & w (4) , b w (2) \\
w (4) & \leadsto & c w (3)
\end{array}\right.$\
such that the words labeled $(0)$ or $(1)$ start by $b^\ast a$, the words labeled $(3)$ or $(4)$ start by $b^\ast c$, and the words labeled $(2)$ start by $b$ (and have one letter more than words labeled $(0)$ or $(4)$ at the same level).\
So, words of $\{a,b,c\}^n a$ (respectively $\{a,b,c\}^n a \cup b \{a,b,c\}^n a$) are in bijection with involutions avoiding both $132$ and $12345$ of length $2n+1$ (respectively $2n+2$) enumerated by $3^n$ (respectively $2.3^n$).
Figure \[fig-forbid\] (an output of the software `forbid` [@GuibertThese]) shows the first values for the number of involutions avoiding both $132$ and $12\dots k$ for $3 \leq k \leq
5$ according to the number of fixed points.
Involutions $\pi \in S_n(132,123)$ according to $|\{\pi(x)=x\}|$ for $1 \leq n \leq 15$
=1 =2 =2 =4 =4 =8 =8 =16 =16 =32 =32 =64 =64 =128 =128
2: - 1 0 2 0 4 0 8 0 16 0 32 0 64 0
1: 1 0 2 0 4 0 8 0 16 0 32 0 64 0 128
0: 0 1 0 2 0 4 0 8 0 16 0 32 0 64 0
1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: [n]
Involutions $\pi \in S_n(132,1234)$ according to $|\{\pi(x)=x\}|$ for $1 \leq n \leq 15$
=1 =2 =3 =5 =8 =13 =21 =34 =55 =89 =144 =233 =377 =610 =987
3: - - 1 0 3 0 8 0 21 0 55 0 144 0 377
2: - 1 0 3 0 8 0 21 0 55 0 144 0 377 0
1: 1 0 2 0 5 0 13 0 34 0 89 0 233 0 610
0: 0 1 0 2 0 5 0 13 0 34 0 89 0 233 0
1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: [n]
Involutions $\pi \in S_n(132,12345)$ according to $|\{\pi(x)=x\}|$ for $1 \leq n \leq 15$
=1 =2 =3 =6 =9 =18 =27 =54 =81 =162 =243 =486 =729 =1458 =2187
4: - - - 1 0 4 0 13 0 40 0 121 0 364 0
3: - - 1 0 4 0 13 0 40 0 121 0 364 0 1093
2: - 1 0 3 0 9 0 27 0 81 0 243 0 729 0
1: 1 0 2 0 5 0 14 0 41 0 122 0 365 0 1094
0: 0 1 0 2 0 5 0 14 0 41 0 122 0 365 0
1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: [n]
Avoiding $132$ and $213\dots k$
-------------------------------
\[ex2\] Let us find $I_{213}(x)$; let $T'=\{213\}$ and $T=\{21\}$, so Theorem \[thg\] gives $$I_{213}(x)=\frac{1}{1-x^2S_{21}(x^2)}+\frac{x}{1-x^2S_{21}(x^2)}I_{21}(x),$$ where by definitions $S_{21}(x)=I_{21}(x)=\frac{1}{1-x}$, hence $$I_{213}(x)=\frac{1+x}{1-2x^2},$$ which means the number of involutions $I_{213}(n)$ is given by $2^{[n/2]}$ for all $n\geq 0$.
Similarly, $I_{2134}(x)=\frac{1}{1-x-x^2}$, so the number of involutions $I_{2134}(n)$ is given by $F_n$ the $n$th Fibonacci number.
We can easily prove by a combinatorial way that $I_{213}(n)$ is given by $2^{[n/2]}$.\
An involution $\pi$ avoiding both $132$ and $213$ of length $n$ can be either $(n+1-i)(n+2-i)\dots n \pi' 12\dots i$ with $i\geq1$ or $12\dots n$ such that $\pi'$ is also an involution avoiding both $132$ and $213$ (of length $n-2i$, and if we subtract $i$ to each element). We code this recursive decomposition of an involution $\pi$ avoiding both $132$ and $213$ by a word of nonnegative integers formed by the successive positive numbers $i$ and whose last nonnegative integer is (the smallest integer of) the number of the fixed points in $\pi$ divided by $2$. This coding is clearly bijective.\
For example, the involutions $\epsilon$, $1$, $12$, $21$, $123$, $321$, $1234$, $4231$, $3412$ and $4321$ are respectively coded by $0$, $0$, $1$, $10$, $1$, $10$, $2$, $11$, $20$ and $110$. Moreover, the involution $21 \linebreak[0] \ 19 \linebreak[0] \ 20
\linebreak[0] \ 16 \linebreak[0] \ 17 \linebreak[0] \ 18
\linebreak[0] \ 15 \linebreak[0] \ 14 \linebreak[0] \ 9
\linebreak[0] \ 10 \linebreak[0] \ 11 \linebreak[0] \ 12
\linebreak[0] \ 13 \linebreak[0] \ 8 \linebreak[0] \ 7
\linebreak[0] \ 4 \linebreak[0] \ 5 \linebreak[0] \ 6
\linebreak[0] \ 2 \linebreak[0] \ 3 \linebreak[0] \ 1$ in $S_{21}(132,213)$ is coded by $123112$.\
Thus, involutions avoiding both $132$ and $213$ of length $n$ are coded by words $w = w_1 w_2 \ldots w_{l-1} w_l$ with $l \geq 1$, $w_j \geq 1$ for all $1 \leq j < l$, $w_l \geq 0$ and $\sum_{j=1}^{l} w_j = \lfloor \frac{n}{2} \rfloor$. Trivially, words $w$ are in bijection with words $w_1 w_2 \ldots w_{l-1}
(w_l+1)$ which are compositions of $\lfloor \frac{n}{2} \rfloor +
1$ into $l$ positive parts enumerated by $2^{\lfloor \frac{n}{2}
\rfloor}$.
The case of varying $k$ is more interesting. As an extension of Example \[ex2\] let us consider the case $T=\{213\dots k\}$. Similarly as Theorem \[th12k\] we have
\[th21k\] For all $k\geq 1$, $$I_{213\dots k}(x)=\frac{1}{x\cdot U_k\left( \frac{1}{2x} \right)}
\sum_{j=0}^{k-1} U_j\left( \frac{1}{2x} \right).$$
Therefore, Theorem \[th12k\] and Theorem \[th21k\] yields $I_{123\dots k}(n)=I_{213\dots k}(n)$. We establish a bijection for this result.
\[12k=2134k\] There is a bijection between involutions avoiding both $132$ and $12\ldots k$ of length $n$ and involutions avoiding both $132$ and $2134\ldots k$ of length $n$, for any $k \geq 3$.\
Moreover, two involutions in bijection have the same number of fixed points $p$ for all $0 \leq p \leq k-3$ whereas the involutions avoiding both $132$ and $12\ldots k$ having $k-2$ or $k-1$ fixed points correspond to the involutions avoiding both $132$ and $2134\ldots k$ having $k-2$ or more fixed points.
In order to establish this result we consider a generating tree for the involutions avoiding both $132$ and $2134\ldots k$ which is characterized by the same succession system $(*)$ given in Subsection \[subsec21\] characterizing a generating tree for the involutions avoiding both $132$ and $12\ldots k$.\
So let $\pi$ be an involution avoiding both $132$ and $2134\ldots k$ of length $n$ and let $q=|\{\pi(x)=x\}|$ be the number of fixed points of $\pi$. The label $(p)$ of $\pi$ is defined by $p=q$ if $q \leq k-3$ or by $p=k-2$ if $q \geq k-2$ and $(n+k) \bmod 2 = 0$ or by $p=k-1$ if $q \geq k-2$ and $(n+k) \bmod 2 = 1$. Of course, the empty involution of length $0$ has label $(0)$. We obtain $\sigma$ an involution avoiding both $132$ and $2134\ldots k$ of length $n+1$ by applying the following rules:
- If $p \in [0,k-3]$, we have $\pi=\pi'\pi''$ with $|\pi'|=\frac{n-p}{2}$, and then $\sigma$ obtained by inserting a fixed point between $\pi'$ and $\pi''$ has label $(p+1)$.
- If $p=k-2$, we have $\pi=\pi'x\pi''$ with $\pi(x)=x=\frac{n+4-k}{2}$, and then $\sigma$ obtained by inserting a fixed point between $\pi'$ and $x$ has label $(k-1)$.
- If $p \in [1,k-3]$, we have $\pi=\pi'\pi''x\pi'''$ with $|\pi'|=\frac{n-p}{2}$, $\pi(x)=x$ and $\pi(y) \neq y$ for all $1 \leq y < x$. Then $\sigma$ obtained by modifying the first fixed point $x$ by a cycle starting between $\pi'$ and $\pi''$ (and ending in $x$) has label $(p-1)$.
- If $p=k-1$, we have $\pi=\pi'x(x+1)\pi''$ with $\pi(x)=x=\frac{n+3-k}{2}$, and then $\sigma$ obtained by inserting a fixed point between $\pi'$ and $x$ has label $(k-2)$.
- If $p=k-2$, we have $\pi=\pi'(x-j)(x-j+1)\dots(x+j)\pi''$ with $j
\geq 0$, $\pi(x)=x=\frac{n+4-k}{2}$, $|\pi'|=\frac{n-k}{2}+1-j$ and $e>x$ for all $e \in \pi'$. Then $\sigma$ obtained by modifying the $2j+1$ fixed points between $\pi'$ and $\pi''$ by $j+1$ consecutive cycles each of difference (between the index and the value) $j+1$ that is $(\pi_1'+1)(\pi_2'+1)\dots(\pi_{\frac{n-k}{2}+1-j}'+1)
(\frac{n-k}{2}+3)(\frac{n-k}{2}+4)\dots(\frac{n-k}{2}+3+j)
(\frac{n-k}{2}-j+2)(\frac{n-k}{2}-j+3)\dots(\frac{n-k}{2}+2)
\pi''$ has label $(k-3)$.
There is a bijection between permutations avoiding both $132$ and $12\ldots k$ of length $n$ and permutations avoiding both $132$ and $2134\ldots k$ of length $n$, for any $k \geq 3$.
By Proposition \[prom\], we deduce that permutations $\pi$ avoiding both $132$ and $12\ldots k$ (respectively $2134\ldots k$) of length $n$ are in bijection with involutions without fixed points $(\pi^{-1}+n) \pi$ avoiding both $132$ and $12\ldots k$ (respectively $2134\ldots k$) of length $2n$. Moreover, a particular case of Theorem \[12k=2134k\] establishes a one-to-one correspondence between involutions avoiding both $132$ and $12\ldots k$ without fixed points and involutions avoiding both $132$ and $2134\ldots k$ without fixed points.
Avoiding $132$ and $(d+1(d+2)\dots k12\dots d$
----------------------------------------------
\[ex22\] By Proposition \[prom\] it is easy to obtain for $n\geq1$, $$I_{231}=n;\quad I_{321}=[n/2]+1.$$
We consider a combinatorial approach to show Example \[ex22\]. Clearly, we have that involutions avoiding both $132$ and $231$ of length $n$ are $i (i-1) \ldots 1 (i+1) (i+2) \ldots n$ for all $1 \leq i \leq n$ and that involutions avoiding both $132$ and $321$ of length $n$ are $(i+1) (i+2) \ldots (2i) 1 2 \ldots i (2i+1) (2i+2) \ldots n$ for all $0 \leq i \leq \lfloor \frac{n}{2} \rfloor$.
As an extension of Example \[ex22\] let us consider the case $T=\{[k,d]\}$, where $[k,d]=(d+1,d+2,\dots,k,1,2,\dots,d)$.
\[thkd\] For any $k\geq 2$, $k/2\geq d\geq 1$, $$I_{[k,d]}=\frac{1}{x(U_d(t)-U_{d-1}(t))}\left[
U_{d-1}(t)+\frac{U_{k-2d-1}(t)}{U_{k-d}(t)U_{k-d-1}(t)}\sum_{j=0}^{k-d-1} U_j(t)
\right],\quad t=\frac{1}{2x}.$$
Proposition \[prom\] yields, in the second case the generating function for the number of involutions $[k,d]$-avoiding permutations is $xI_{[k,d]}(x)$. In the first case, we assume that $\gamma$ either $(1)$ avoiding $12\dots (k-d)$, or $(2)$ containing $12\dots (k-d)$. In $(1)$, $\beta$ and $\delta$ avoiding $12\dots (k-d-1)$, so the generating function for these number of involutions is $x^2R_{k-d-1}(x^2)I_{12\dots(k-d)}(x)$ (similarly Theorem \[th12k\]). In $(2)$, $\beta$ and $\delta$ avoiding $12\dots (d-1)$, so the generating function for these number of involutions is $x^2R_{d-1}(x^2)(I_{[k,d]}(x)-I_{12\dots(k-d)}(x))$ (the generating function for the number of involutions in $S_n(132,[k,d])$ such containing $12\dots(k-d)$ is given $I_{[k,d]}(x)-I_{12\dots(k-d)}(x)$). Therefore $$\begin{array}{ll}
I_{[k,d]}(x)&=1+xI_{[k,d]}(x)+x^2R_{k-d-1}(x^2)I_{12\dots(k-d)}(x)+\\
&+x^2R_{d-1}(x^2)(I_{[k,d]}(x)-I_{12\dots(k-d)}(x)),
\end{array}$$ which means that $$I_{[k,d]}(x)=\frac{1}{1-x-x^2R_{k-d-1}(x^2)}\cdot(1+x^2I_{12\dots (k-d)}(x)(R_{k-d-1}(x^2)-R_{d-1}(x^2))).$$ Hence, by use the identities $R_k(x)=\frac{1}{1-xR_{k-1}(x)}$ and $R_a(x)-R_b(x)=\frac{U_{a-b-1}(t)}{\sqrt{x}U_a(t)U_b(t)}$, the theorem holds.
\[ex3412\] Theorem \[thkd\] yields for $k=4$ and $d=2$, the number of involutions $I_{3412}(n)$ is given by $F_n$ the $n$th Fibonacci number.
We consider a combinatorial approach to show Example \[ex3412\]. An involution $\pi$ avoiding both $132$ and $3412$ of length $n$ can be written $i\pi' 1(i+1)(i+2)\cdots n$ with $1\leq i\leq n$ such that $\pi'$ is also an involution avoiding both $132$ and $3412$ (of length $i-2$, and if we subtract $1$ to each element). We code $\pi$ by a word of $\{a, b^2\}^\ast$ of length $n$ in that way: $a$ if $\pi_i=i$, $b^2$ if $\pi_i<i$ and nothing if $\pi_i>i$ for all $1\leq i\leq n$. This coding is clearly bijective.\
Following [@MV2] we say that $\tau\in S_k$ is a [*wedge*]{} pattern if it can be represented as $\tau=(\tau^1,\rho^1,\dots,\tau^r,\rho^r)$ so that each of $\tau^i$ is nonempty, $(\rho^1,\rho^2,\dots,\rho^r)$ is a layered permutation of $1,\dots,s$ for some $s$, and $(\tau^1,\tau^2,\dots,\tau^r)=(s+1,s+2,\dots,k)$. For example, $645783912$ and $456378129$ are wedge patterns.
For a further generalization of Theorem \[th12k\], Theorem \[th21k\] and [@MV2 Th. 2.6], consider the following definition. We say that $\tau\in S_{2l}$ is a [*double-wedge*]{} pattern if there exist a wedge pattern $\sigma\in S_{l-1}$ such that $$\tau=(\sigma^{-1}+l,2l,\sigma,l)\ \mbox{or}\ \tau=(\sigma+l,2l,\sigma^{-1},l).$$ For example, the double-wedge patterns of length $10$ are $6789(10)1234\linebreak[0] 5$, $7689(10)\linebreak[0]
213\linebreak[0] 4\linebreak[0] 5\linebreak[0]$, $7869(10)\linebreak[0] 31245\linebreak[0]$, $7896(10)\linebreak[0]
41235\linebreak[0]$, $8679(10)\linebreak[0] 23145\linebreak[0]$, $8796(10)\linebreak[0] 42135\linebreak[0]$, $8967(10)\linebreak[0]
34125\linebreak[0]$, $9678(10)\linebreak[0] 23415\linebreak[0]$ and $9768(10)\linebreak[0] 32415\linebreak[0]$.
\[dwedge\] For any double-wedge pattern $\tau\in S_{2l}(132)$ $$I_\tau(x)=I_{12\dots(2l)}(x)=\frac{R_l(x^2)}{1-xR_l(x^2)}.$$
First of all, let us find the generating function $I_\rho(x)$ where $\rho=(\sigma^{-1},2l,\sigma,l)$. By use Proposition \[prom\] we obtain in the first case $xI_\rho(x)$, and in the second case $x^2S_\sigma(x^2)I_\rho(x)$ where $S_\sigma(x^2)$ is the generating function for the number of permutations in $S_n(132,\sigma)$, therefore ($1$ for the empty permutation) $$I_\rho(x)=1+xI_\rho(x)+x^2S_\sigma(x^2)I_\rho(x).$$ On the other hand, Mansour and Vainshtein proved $S_\sigma(x)=R_{l-1}(x)$ for any wedge pattern $\sigma$, so $$I_\rho(x)=\frac{1}{1-x-x^2R_{l-1}(x^2)}.$$ By use the identity $R_l(x)=\frac{1}{1-xR_{l-1}(x)}$ we have $$I_\rho(x)=\frac{R_l(x)}{1-xR_l(x^2)}.$$ Now, let us find $I_{12\dots 2l}(x)$ in terms of $R_j(x)$. So, by use the identity $$\sum_{j=0}^{2l}U_j(t)=\frac{U_{2l}(t)U_{l-1}(t)}{U_l(t)-U_{l-1}(t)}$$ and use the symmetric inverse operation, the first part of the theorem holds.
\[extwedge\] For any wedge pattern $\sigma\in S_{l-1}$ the generating function for the number of permutations in $S_n(132,(\sigma^{-1}+l,2l,\sigma,l,2l+1,\dots,k))$ (or $S_n(132,(\sigma+l,2l,\sigma^{-1},l,2l+1,\dots,k))$, or $S_n(132,(\sigma+l,2l,\sigma,l,2l+1,\dots,k))$) is given by $R_{k}(x)$, for all $k\geq 2l$.
Let $\tau=(\sigma^{-1}+l,2l,\sigma,l)$ and let $S_\tau(x)$ be the generating function for the number of permutations in $S_n(132,\tau)$. By [@MV2 Th. 1] we have $$S_\tau(x)=1+x(S_\tau(x)-S_\sigma^{-1}(x))S_\sigma(x)+xS_\sigma^{-1}(x)S_\tau(x).$$ On the other hand, by [@MV2 Th. 2.6] and $\sigma$ a wedge pattern in $S_{l-1}(132)$ we have $S_{\sigma^{-1}}(x)=S_\sigma(x)=R_{l-1}(x)$. Therefore, by use the identity $R_l(x)=\frac{1}{1-xR_{l-1}(x)}$ we get $$S_\tau(x)=\frac{R_l(x)(1-xR_{l-1}(x)R_l(x))}{1-xR_l^2(x)}.$$ By use the definitions of Chebyshev polynomials of the second kind it is easy to see $$\frac{R_l(x)(1-xR_{l-1}(x)R_l(x))}{1-xR_l^2(x)}=R_{2l}(x),$$ hence by use again Theorem [@MV2 Th. 1] we have $S_{(\tau,2l+1,\dots,k)}(x)=R_k(x)$. Similarly we obtain the other cases.
As a corollary of Theorem \[dwedge\] we have
\[idwedge\] For any double wedge pattern $\tau\in I_{2l}(132)$ $$I_{(\tau,2l+1,2l+2,\dots,k)}(x)=I_{12\dots k}(x).$$
Since, if $S_\beta(x)=S_\gamma(x)$ and $I_\beta(x)=I_\gamma(x)$, then Theorem \[thg\] yields $I_{\tau'}(x)=I_{\beta'}(x)$, and by use [@MV2 Th. 1] we have $S_{\tau'}(x)=S_{\rho'}(x)$, where $\tau'=(\tau_1,\dots,\tau_p,p+1)$ and $\rho'=(\rho_1,\dots,\rho_p,p+1)$ two patterns in $S_{p+1}$. Hence, the theorem holds by use Theorem \[dwedge\], Theorem \[extwedge\], and induction on $p$.
In view of Theorem \[dwedge\] and Theorem \[idwedge\] it is a challenge to find a bijective proof.
Avoiding $132$ and two other patterns
-------------------------------------
Now, let us restrict more than two patterns ($132$ and two other patterns).
\[ex3\] Let us find $I_{123,213}(x)$; let $T'=\{123,213\}$ and $T=\{12,21\}$, so Theorem \[thg\] gives $$I_{123,213}(x)=\frac{1}{1-x^2S_{12,21}(x^2)}+\frac{x}{1-x^2S_{12,21}(x^2)}I_{12,21}(x),$$ where by definitions $S_{12,21}(x)=I_{12,21}(x)=1+x$, hence $$I_{123,213}(x)=\frac{1+x+x^2}{1-x^2-x^4},$$ which means the number of involutions $I_{123,213}(2n)$ is given by $F_{n+1}$, and $I_{123,213}(2n+1)$ is given by $F_{n}$ for all $n\geq 0$, where $F_m$ is the $m$th Fibonacci number.
We consider a combinatorial approach to show Example \[ex3\] by distinguishing the cases of odd and even length.\
An involution $\pi$ avoiding $132$, $123$ and $213$ of length $2n+1$ can be written either $(2n+1) \pi' 1$ or $(2n) (2n+1) \pi''
2 1$ or $1$ (if $n=0$) such that $\pi'$ and $\pi''$ are also involutions avoiding $132$, $123$ and $213$ (of length $2n-1$ for $\pi'$ if we subtract $1$ to each element, of length $2n-3$ for $\pi''$ if we subtract $2$ to each element). We code $\pi$ by a word of $\{a,b^2\}^*$ of length $n$ in that way: $a$ if $\pi_i=2n+2-i$, $b^2$ if $\pi_i=2n+1-i$ and nothing if $\pi_i=2n+3-i$ for all $1\leq i\leq n$. This coding is clearly bijective.\
An involution $\pi$ avoiding $132$, $123$ and $213$ of length $2n$ can be written either $(2n) \pi' 1$ (that includes $2 1$ if $n=1$) or $(2n-1) (2n) \pi'' 2 1$ or $1 2$ (if $n=1$) or the empty involution (if $n=0$) such that $\pi'$ and $\pi''$ are also involutions avoiding $132$, $123$ and $213$ (of length $2n-2$ for $\pi'$ if we subtract $1$ to each element, of length $2n-4$ for $\pi''$ if we subtract $2$ to each element). We code $\pi$ by a word of $\{a,b^2\}^*$ of length $n+1$ in that way: $a$ if $\pi_i=2n+1-i$ for all $1\leq i\leq n-1$, $b^2$ if $\pi_n=n+1$, $b^2$ if $\pi_i=2n-i$ for all $1\leq i\leq n-2$, $b^2a$ if $\pi_{n-1}=n+1$ and $aa$ if $\pi_n=n$. Moreover, the empty involution is coded by $a$. This coding is clearly bijective.
Using definitions and Theorem \[thg\] it is easy to see the following.
\[th12k213\] For all $k\geq 1$, $$I_{123\dots k, 213}(x)=I_{(k-1)\dots 21k, 123}(x)=\frac{1+x+x^2+\dots+x^{k-1}}{1-x^2-x^4-\dots-x^{2(k-1)}}.$$
\[ex312decr\] Using Proposition \[prom\] it is easy to see for $n\geq 1$, $$I_{213,321}(n)=\frac{1}{2}((-1)^n+3),\quad
I_{213,4321}(n)=[n/2]+1.$$
We consider a combinatorial approach to show Example \[ex312decr\]. Clearly, we have that involutions avoiding $132$, $213$ and $321$ of length $n$ are $1 2 \ldots n$ and also $(m+1) (m+2) \ldots n 1 2 \ldots m$ if $n=2m$ with $m \geq 1$. We also have that involutions avoiding $132$, $213$ and $4321$ of length $n$ are $(n+1-i) (n+2-i) \ldots n (i+1) (i+2) \ldots (n-i) 1 2 \ldots i$ for all $0 \leq i \leq \lfloor \frac{n}{2} \rfloor$.
Avoiding $132$ and containing another pattern
=============================================
Let $I_\tau^r(n)$ denote the number of involutions in $S_n(132)$ containing $\tau$ exactly $r$ times, and let $I_\tau^r(x)=\sum_{n\geq 0}I_\tau^r(n)x^n$ be the corresponding generating function. Let us start by the following example.
\[exbb0\] By Proposition \[prom\] it is easy to see $$I_{12}^1(x)=xI_{1}^1(x)+x^2I_{12}^1(x),$$ which means $I_{12}^1(x)=\frac{x^2}{1-x^2}$.
As extension of Example \[exbb0\] let us consider the case $\tau=12\dots k$.
\[th1\_12\] For all $k\geq 1$; $$I_{12\dots k}^{1}=\frac{1}{U_k\left( \frac{1}{2x} \right)}.$$
By Proposition \[prom\] we have for $n\geq k$, $$I_{12\dots k}^{1}(n)=I_{12\dots (k-1)}^{1}(n-1)+\sum_{j=1}^{[n/2]} s_{12\dots (k-1)}(j-1)I_{12\dots k}^{1}(n-2j),$$ where $s_{12\dots k}(j-1)$ is the number of $12\dots k$-avoiding permutations in $S_{j-1}(132)$. Besides $I_{12\dots k}^{1}(n)=0$ for all $n\leq k-1$, and $I_{12\dots k}^{1}(k)=1$. Similarly as proof of Theorem \[th12k\] we have $$I_{12\dots k}^{1}(x)=xR_{k}(x^2)I_{12\dots (k-1)}^{1}(x).$$ Hence, by induction on $k$ with initial condition $I_1^{1}=x$, the theorem holds.
Similarly as Theorem \[th1\_12\] we have an explicit formula when $\tau=213\dots k$ or $\tau=23\dots k1$.
For all $k\geq 2$; $$I_{213\dots k}^{1}=\frac{1-x^2}{U_k\left( \frac{1}{2x} \right)},
\quad
I_{23\dots k1}^1(x)=\frac{x^3}{(1-x)U_{k-2}\left( \frac{1}{2x} \right)}.$$
More generally, by Proposition \[prom\] and the argument proof of Theorem \[th12k\] we get
\[th12kr\] For any $k,r\geq 1$ $$I_{12\dots k}^{r}(x)=xI_{12\dots (k-1)}^{r}(x)+x^2\sum_{2a+b=r} S_{12\dots (k-1)}^{a}(x^2)I_{12\dots k}^{b}(x),$$ where $S_{12\dots (k-1)}^{a}(x)$ is the generating function for the number of permutations in $S_n$ containing $12\dots (k-1)$ exactly $a$ times.
In [@Kr] found an explicit formula for $S_{12\dots k}^{r}(x)$, so Theorem \[th12kr\] yields a recurrence for $I_{12\dots
k}^{r}(x)$. For example, following [@Kr] ([@MV1 Th. 3.1]) we have a recurrence for $I_{12\dots k}^{r}(x)$ where $r=1,2,\dots,2k$.
Let $k\geq 1$; for $r=1,2,\dots,2k$ $$I_{12\dots k}^{r}(x)=xI_{12\dots (k-1)}^r(x)+x^2R_{k-1}(x^2)I_{12\dots k}^{r}(x)
+x^2\sum_{2a+b=r,\ a>0} x^{a-1}I_{12\dots k}^{b}(x)\frac{U_{k-1}^{a-1}\left( \frac{1}{2x} \right)}{U_k^{a+1}\left( \frac{1}{2x} \right)}.$$
The above Theorem yields for $r=2$ an explicit formula for $I_{12\dots k}^{2}(x)$.
For all $k\geq 1$, $$I_{12\dots k}^{2}(x)=\frac{1}{U_k\left( \frac{1}{2x} \right)}\sum_{i=1}^k
\frac{\sum_{j=0}^{k-i} U_j\left( \frac{1}{2x} \right)}{U_{k+1-i}\left( \frac{1}{2x} \right) U_{k-i}\left( \frac{1}{2x} \right)}.$$
Containing $132$ once and avoiding another pattern
==================================================
We first relate involutions containing $132$ once to $132$-avoiding involutions.
\[bij132once-132\] There is a bijection $\Psi$ between involutions containing $132$ exactly once of length $n$ having $p$ fixed points with $1\leq p\leq n$ and $132$-avoiding involutions of length $n-2$ having also $p$ fixed points.
Let $\pi=\pi'xz\pi''y\pi'''$ with $\pi(x)=x$, $\pi(y)=z$ and $1+x=y<z$ be an involution containing $132$ once (that is subsequence $xzy$) of length $n$ having $p$ fixed points. We replace the subsequence $xzy$ by a fixed point between $\pi''$ and $\pi'''$ in order to obtain an $132$-avoiding involution of length $n-2$ having $p$ fixed points. Note that the only possibility to have exactly once $132$ subsequence is a cycle with a fixed point just to its left. Moreover, $y=x+1$ in order to forbid another $132$ subsequence and cycles are only allowed from $\pi'$ to $\pi''$ and from $\pi'$ to $\pi'''$ (and not from $\pi''$, $\pi'$, $\pi''$, $\pi'''$ respectively to $\pi'''$, $\pi'$, $\pi''$, $\pi'''$) whereas fixed points can uniquely be into $\pi'''$. Clearly the involution we obtain avoids $132$ and in particular, the fixed point $z-2$ cannot be a part of an $132$-subsequence because it cannot be the $3$ or $2$ (all the elements on its left are greater than it) and it cannot be the $1$ (there is no cycle starting on its right).\
Let $\sigma = \sigma' \sigma'' \sigma''' t \sigma''''$ with $\sigma(t) = t$ and $\sigma(i) \neq i$ for all $1 \leq i < t$ (that is $t$ is the first fixed point), $\sigma'(i) > t$ for all $1 \leq i \leq |\sigma'|$ (all the elements of $\sigma'$ are cycles ending into $\sigma''''$), $\sigma''(i) \in
[|\sigma'\sigma''|+1,t-1]$ for all $1 \leq i \leq |\sigma''|$ and $\sigma'''(i) \in [|\sigma'|+1,|\sigma'\sigma''|]$ for all $1 \leq
i \leq |\sigma'''|$ ($\sigma''\sigma'''$ is entirely constituted by cycles from $\sigma''$ to $\sigma'''$) be an $132$-avoiding involution of length $n-2$ having $p$ fixed points. We modify the fixed point $t$ by a cycle starting between $\sigma''$ and $\sigma'''$ (and ending between $\sigma'''$ and $\sigma''''$) and by adding a fixed point just to the right of $\sigma''$ in order to obtain an involution containing $132$ once of length $n$ having $p$ fixed points. Proposition \[prom\] leads immediately to the decomposition of $\sigma$. The involution we obtain contains $132$ exactly once that is the subsequence we modify and insert. There is no other $132$-subsequence and in particular, the fixed point inserted and the start of the new cycle cannot be the $3$ or $2$ of another $132$-subsequence (all the elements on their left are greater than them), the fixed point inserted and the start of the new cycle and the end of the new cycle cannot be the $1$ of another $132$-subsequence (there is no cycle starting on their right), the end of the new cycle cannot be the $3$ of another $132$-subsequence (because in that case the $2$ must be connected to $\sigma'$ and the $1$ must be the fixed point inserted or an element of $\sigma'''$ that forms an $231$-subsequence), and the end of the new cycle cannot be the $2$ of another $132$-subsequence (because in that case the $1$ must be an element of $\sigma'\sigma''$ or the start of the new cycle and the $3$ must be the fixed point inserted or an element of $\sigma'''$ that forms an $312$-subsequence excepted for the fixed point inserted and the new cycle).\
So we have established a bijection between $\pi$ an involution containing $132$ once and $\sigma$ an $132$-avoiding involution where $t=z-2$, $\pi'$ corresponds to $\sigma'\sigma''$, $\pi''=\sigma'''$ and $\pi'''$ corresponds to $\sigma''''$.
For example, the involution $22 \linebreak[0] \ 19 \linebreak[0] \
17 \linebreak[0] \ 18 \linebreak[0] \ 16 \linebreak[0] \ 12
\linebreak[0] \ 11 \linebreak[0] \ 13 \linebreak[0] \ {\bf 9}
\linebreak[0] \ {\bf 14} \linebreak[0] \ 7 \linebreak[0] \ 6
\linebreak[0] \ 8 \linebreak[0] \ {\bf 10} \linebreak[0] \ 15
\linebreak[0] \ 5 \linebreak[0] \ 3 \linebreak[0] \ 4
\linebreak[0] \ 2 \linebreak[0] \ 20 \linebreak[0] \ 21
\linebreak[0] \ 1 \linebreak[0] \ 23$ containing $132$ once (the subsequence $9$ $14$ $10$) corresponds to the $132$-avoiding involution $20 \linebreak[0] \ 17 \linebreak[0] \ 15 \linebreak[0]
\ 16 \linebreak[0] \ 14 \linebreak[0] \ 10 \linebreak[0] \ 9
\linebreak[0] \ 11 \linebreak[0] \ 7 \linebreak[0] \ 6
\linebreak[0] \ 8 \linebreak[0] \ {\bf 12} \linebreak[0] \ 13
\linebreak[0] \ 5 \linebreak[0] \ 3 \linebreak[0] \ 4
\linebreak[0] \ 2 \linebreak[0] \ 18 \linebreak[0] \ 19
\linebreak[0] \ 1 \linebreak[0] \ 21$.
The number of involutions containing $132$ exactly once of length $n$ having $p$ fixed points with $1 \leq p \leq n$ is the ballot number $\binom{n-2}{\frac{n+p}{2}-1}-\binom{n-2}{\frac{n+p}{2}}$. Moreover, the number of involutions containing $132$ exactly once of length $n$ is $\binom{n-2}{\lfloor \frac{n-3}{2} \rfloor}$.
We immediately deduce this result from bijection $\Psi$ of Theorem \[bij132once-132\] and Corollary \[nb132ptfix\]. In fact, the number of involutions containing $132$ once of length $n$ is either the number of $132$-avoiding involutions of length $n-2$ if $n$ is odd or the number of $132$-avoiding involutions of length $n-2$ having more than one fixed point if $n$ is even.
Of course, some of the following results can immediately be obtained from bijection $\Psi$ of Theorem \[bij132once-132\] and results of Section \[sec2\].
Let $J_\tau(n)$ denote the number of involutions in $S_n(\tau)$ such containing $132$ exactly once, and let $J_\tau(x)=\sum_{n\geq
0} J_\tau(n)x^n$ be the corresponding generating function. The following proposition is the base of all the other results in this section, which holds immediately from definitions.
\[prom2\] Let $\pi$ an involution in $S_n$ such that contains $132$ exactly once, and let $\pi_j=n$. Then holds either
1. or $\pi_n=n$;
2. or $\pi=(\pi',n,\pi'',\pi''',j)$ where $1\leq j\leq n/2$, $\pi'''={\pi'}^{-1}$ and $\pi'$ avoids $132$.
3. or $\pi=(\pi',m,2m+1,\pi'',m+1)$ where $n=2m+1$, $\pi''={\pi'}^{-1}$ and $\pi'\in S_{m-1}(132)$.
Another approach to find the generating function of involutions in $S_n$ containing $132$ exactly once is by use Proposition \[prom2\].
\[thcc1\] Let $C(t)$ be the generating function for the Catalan numbers; then $$J_\varnothing(x)=\frac{x^3C(x^2)}{1-x-x^2C(x^2)}.$$
According to Proposition \[prom2\] with terms of generating functions we get the following: the first part of the proposition yields $xJ_\varnothing(x)$, the second part of the proposition yields $x^2C(x^2)J_\varnothing(x)$, and the third part of the proposition gives $x^3C(x^2)$. Hence $$J_\varnothing(x)=xJ_\varnothing(x)+x^2C(x^2)J_\varnothing(x)+x^3C(x^2).$$
\[excc1\] From Proposition \[prom2\] it is easy to see that, the number of the involutions in $S_n(123)$ and containing $132$ exactly once is $2^{(n-3)/2}$ for $n$ odd, otherwise is $0$. Also, $J_{1234}(n)=F_{n-3}$ the $(n-3)$th Fibonacci number, $J_{12345}(n)=3^{[(n-3)/2]}$.
Again, the case of varying $k$ is more interesting. As an extension of Example \[excc1\] let us consider the case $\tau=12\dots k$.
\[thcc2\] For all $k\geq 1$, $$J_{12\dots k}(x)=\frac{x}{U_k\left( \frac{1}{2x} \right)} \sum_{j=1}^{k-2} U_{j}\left( \frac{1}{2x} \right).$$
Proposition \[prom2\] with use the generating function of permutations in $S_n(132,12\dots k)$ given by $R_k(x)$, yields $$J_{12\dots k}(x)=xJ_{12\dots(k-1)}+x^2R_{k-1}(x^2)J_{12\dots k}(x)+x^3R_{k-1}(x^2).$$ By use the relation $R_k(y)=1/(1-yR_{k-1}(y))$ we get that $$J_{12\dots k}(x)=xR_k(x^2)J_{12\dots(k-1)}(x)+x^3R_{k-1}(x^2)R_k(x^2),$$ so induction on $k$ with Example \[excc1\] gives the theorem.
Similarly, we obtain another case $\tau=213\dots k$.
\[thcc22\] For all $k\geq 3$, $$J_{213\dots k}(x)=\frac{x}{U_k\left( \frac{1}{2x} \right)}
\left[
xU_2\left(\frac{1}{2x}\right) +\sum_{j=2}^{k-2} U_{j}\left( \frac{1}{2x} \right)
\right].$$
Similarly as proof of Theorem \[thcc2\] with use the generating function for the number of $213\dots k$-avoiding permutations in $S_n(132)$ is given by $R_k(x)$ (see [@MV2]), we obtain that $$J_{213\dots k}(x)=xR_k(x^2)J_{213\dots(k-1)}(x)+x^3R_{k-1}(x^2)R_k(x^2),$$ and by induction with $J_{213}(x)=x^4R_3(x^2)$ (it is easy to see) the theorem holds.
Theorem \[thcc22\] yields $J_{2134}(2n+3)=J_{2134}(2n+4)=F_{2n}$ the $(2n)$th Fibonacci number for all $n \geq 0$.
\[excc2\] Proposition \[prom2\] yields, $J_{231}(n)=1$ for all $n\geq 1$, and $J_{2341}(n)=2^{[(n-1)/2]}-1$ for all $n\geq 1$.
Once again, the case of varying $k$ is more interesting. As an extension of Example \[excc2\] let us consider the case $\tau=23\dots k1$.
\[thcc3\] For all $k\geq 3$, $$J_{23\dots k1}(x)=\frac{x^2U_{k-3}\left( \frac{1}{2x} \right)}{(1-x)U_{k-2}\left( \frac{1}{2x} \right)}
\left[ 1+\frac{1}{U_{k-1}\left( \frac{1}{2x} \right)}\sum_{j=1}^{k-3} U_j\left( \frac{1}{2x} \right) \right].$$
Similarly as proof Theorem \[thcc2\] we have that $$J_{23\dots k1}(x)=xJ_{23\dots k1}(x)+x^2R_{k-2}(x^2)J_{12\dots(k-1)}(x)+x^3R_{k-2}(x^2),$$ so by using Theorem \[thcc2\] the theorem holds.
More generally, we present an explicit expression when $\tau=[k,d]$ as follows.
\[thcc4\] For $k\geq 4$, $2\leq d\leq k/2$, $$J_{[k,d]}(x)=\frac{R_d(x^2)}{1-xR_d(x^2)}\left[
x^2R_{k-d-1}(x^2)+\frac{x^2(R_{k-d-1}(x^2)-R_{d-1}(x^2))}{U_{k-d}\left(\frac{1}{2x}\right)}\sum_{j=1}^{k-d-2} U_j\left(\frac{1}{2x}\right)
\right].$$
According to Proposition \[prom2\] in terms of generating functions we get the following. In first case $xJ_{[k,d]}(x)$. In the third case, if $\pi'$ contains $12\dots(k-d-1)$ then $\pi$ contains $[k,d]$ which is a contradiction, we get that $\pi'$ avoids $12\dots(k-d-1)$, hence $x^3R_{k-d-1}(x^2)$. Finally, in the second case, let us observe two subcases $\pi''$ contains $12\dots (k-d)$ or avoids $12\dots (k-d)$; so by use the same argument of the third case we get $$x^2R_{k-d-1}(x)J_{12\dots (k-d)}(x)+x^2R_{d-1}(x^2)(J_{[k,d]}(x)-J_{12\dots(k-d)}(x)).$$ Therefore, if we add all these cases we get $J_{[k,d]}(x)$. Hence, by Theorem \[thcc2\] this theorem holds.
Containing $132$ once and containing another pattern
====================================================
Let $J_\tau^r(n)$ denote the number of involutions in $S_n$ such containing $132$ exactly once and containing $\tau$ exactly $r$ times. Let $J_\tau^r(x)=\sum_{n\geq 0} J_\tau^r(n)x^n$ be the corresponding generating function. Let us start be the following result.
\[thd1\] For all $k\geq 1$, $$J_{12\dots k}^1(x)=0.$$
By Proposition \[prom2\] it is easy to see $$J_{12\dots k}^1(x)=xJ_{12\dots (k-1)}^1(x)+x^2R_{k-1}(x^2)J_{12\dots k}(x).$$ with $J_{12}^1(x)=0$, hence induction on $k$ gives the theorem.
Similarly as Theorem \[thd1\] we have another case where $\tau=23\dots k1$.
\[thd2\] For all $k\geq 1$, $$J_{23\dots k1}^1(x)=0.$$
\[exdd1\] Proposition \[prom2\] yields the following. The number of involutions $J_{21}^1(n)=1$ for $n\geq 3$, and $J_{213}(n)=2^{(n-8)/2}(1+(-1)^n)$.
Once again, the case of varying $k$ is more interesting. As an extension of Example \[exdd1\] let us consider the case $\tau=213\dots k$.
\[thd3\] For all $k\geq 3$, $J_{213\dots
k}^1(x)=\frac{x(1-x^2)}{U_k\left(\frac{1}{2x}\right)}$.
By Proposition \[prom2\] it is easy to obtain $$J_{213\dots k}^1(x)=xJ_{213\dots (k-1)}^1(x)+x^2R_{k-1}(x^2)J_{213\dots k}(x).$$ with $J_{21}^1(x)=x^3/(1-x)$ (which is yield directly from definitions), hence induction on $k$ gives the theorem.
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| {
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---
abstract: |
Let $N$ and $P$ be smooth manifolds of dimensions $n$ and $p$ ($n\geq p\geq2$) respectively. Let $\Omega^{I}(N,P)$ denote an open subspace of $J^{\infty
}(N,P)$ which consists of all Boardman submanifolds $\Sigma^{J}(N,P)$ of symbols $J$ with $J\leq I$. An $\Omega^{I}$-regular map $f:N\rightarrow P$ refers to a smooth map having only singularities in $\Omega^{I}(N,P)$ and satisfying transversality condition. We will prove what is called the homotopy principle for $\Omega^{I}$-regular maps in the existence level. Namely, a continuous section $s$ of $\Omega^{I}(N,P)$ over $N$ has an $\Omega^{I}$-regular map $f$ such that $s$ and $j^{\infty}f$ are homotopic as sections.
address: 'Department of Mathematical Sciences, Faculty of Science, Yamaguchi University, Yamaguchi 753-8512, Japan'
author:
- YOSHIFUMI ANDO
title: 'A homotopy principle for maps with prescribed Thom-Boardman singularities'
---
[^1]
Introduction {#introduction .unnumbered}
============
Let $N$ and $P$ be smooth ($C^{\infty}$) manifolds of dimensions $n$ and $p$ respectively with $n\geq p\geq2$. In \[B\] there have been defined what are called the Boardman manifolds $\Sigma^{I}(N,P)$ in $J^{\infty}(N,P)$ for the symbol $I=(i_{1},i_{2},\cdots,i_{r})$, where $i_{1},i_{2},\cdots,i_{r}$ are a finite number of integers with $i_{1}\geq i_{2}\geq\cdots\geq i_{r}\geq0$. We say that a smooth map germ $f:(N,x)\rightarrow(P,y)$ has $x$ as a Thom-Boardman singularity of the symbol $I$ if and only if $j_{x}^{\infty}f\in\Sigma^{I}(N,P)$. Let $\Omega^{I}(N,P)$ denote an open subset of $J^{\infty}(N,P)$ which consists of all Boardman manifolds $\Sigma^{J}(N,P)$ with symbols $J$ of length $r$ satisfying $J\leq I$ in the lexicographic order. It is known that $\Omega^{I}(N,P)$ is an open subbundle of $J^{\infty
}(N,P)$ with the projection $\pi_{N}^{\infty}\times\pi_{P}^{\infty}$, whose fiber is denoted by $\Omega^{I}(n,p)$. A smooth map $f:N\rightarrow P$ is called an $\Omega^{I}$-*regular map* if and only if (i) $j^{\infty
}f(N)\subset\Omega^{I}(N,P)$ and (ii) $j^{\infty}f$ is transverse to all $\Sigma^{J}(N,P)$.
We will study a homotopy theoretic condition for a given continuous map to be homotopic to an $\Omega^{I}$-regular map. Let $C_{\Omega^{I}}^{\infty}(N,P)$ denote the space consisting of all $\Omega^{I}$-regular maps equipped with the $C^{\infty}$-topology. Let $\Gamma_{\Omega^{I}}(N,P)$ denote the space consisting of all continuous sections of the fiber bundle $\pi_{N}^{\infty
}|\Omega^{I}(N,P):\Omega^{I}(N,P)\rightarrow N$ equipped with the compact-open topology. Then there exists a continuous map $$j_{\Omega^{I}}:C_{\Omega^{I}}^{\infty}(N,P)\rightarrow\Gamma_{\Omega^{I}}(N,P)$$ defined by $j_{\Omega^{I}}(f)=j^{\infty}f$. It follows from the well-known theorem due to Gromov\[G1\] that if $N$ is a connected open manifold, then $j_{\Omega^{I}}$ is a weak homotopy equivalence. This property is called the homotopy principle (the terminology used in \[G2\]). If $N$ is a closed manifold, then it becomes a hard problem for us to prove the homotopy principle. As the primary investigation preceding \[G1\], we must refer to the Smale-Hirsch Immersion Theorem (\[H\]), $k$-mersion Theorem due to \[F\] and the Phillips Submersion Theorem for open manifolds (\[P\]). In \[E1\] and \[E2\], Èliašberg has proved the well-known homotopy principle in the $1$-jet level for $\Omega^{n-p+1,0}$-regular maps, say fold-maps. As for the Thom-Boardman singularities, du Plessis\[duP\] has proved that if $i_{r}>n-p-d^{I}$, where $d^{I}$ is the sum of $\alpha_{1},\cdots,\alpha_{r-1}$ with $\alpha_{\ell}$ being $1$ or $0$ depending on $i_{\ell}-i_{\ell+1}>1$ or otherwise, then $j_{\Omega^{I}}$ is a weak homotopy equivalence.
In this paper we prove the following homotopy principle in the existence level for closed manifolds.
Let $n\geq p\geq2$. Let $N$ and $P$ be connected manifolds of dimensions $n$ and $p$ respectively with $\partial N=\emptyset$. Assume that $\Omega
^{I}(N,P)$ contains $\Sigma^{n-p+1,0}(N,P)$ at least. Let $C$ be a closed subset of $N$. Let $s$ be a section of $\Gamma_{\Omega^{I}}(N,P)$ which has an $\Omega^{I}$-regular map $g$ defined on a neighborhood of $C$ into $P$, where $j^{\infty}g=s$.
Then there exists an $\Omega^{I}$-regular map $f:N\rightarrow P$ such that $j^{\infty}f$ is homotopic to $s$ relative to a neighborhood of $C$ by a homotopy $s_{\lambda}$ in $\Gamma_{\Omega^{I}}(N,P)$ with $s_{0}=s$ and $s_{1}=j^{\infty}f$.
In \[A1\] we have given Theorem 0.1 for the symbol $I=(n-p+1,\overbrace
{1,\cdots,1}^{r-1},0)$ with a partially sketchy proof using the results in \[E1\] and \[E2\]. The singularities of this symbol $I$ are often called $A_{r}$-singularities or Morin singularities. The detailed proof are given in \[An4, Theorem 4.1\] and \[An6, Theorem 0.5\] for the symbol $I=(n-p+1,0)$. We will use these two theorems in the proof of Theorem 0.1 in this paper.
Recently it turns out that this kind of the homotopy principle has many applications. Theorem 0.1 is very important even for fold-maps in proving the relations between fold-maps, surgery theory and stable homotopy groups (\[An4, Theorem 1\] and \[An5, Theorems 0.2 and 0.3\]). The homotopy type of $\Omega^{n-p+1,0}$ determined in \[An3\] and \[An5\] has played an important role. We can now readily deduce the famous theorem about the elimination of cusps in \[L1\] and \[E1\] (see also \[T\]) from these theorems.
The homotopy principle in the existence level for maps and singular foliations having only what are called $A$, $D$ and $E$ singularities are proved in \[An2\] and \[An7\].
In \[Sady\] Sadykov has applied \[An1, Theorem 1\] to the elimination of higher $A_{r}$ singularities ($r\geq3$) for Morin maps when $n-p$ is odd. This result is a strengthened version of the Chess conjecture proposed in \[C\].
As an application of Theorem 0.1 we prove the following theorem. We note that the simplest case is also a little stronger form of the Chess conjecture.
Let $n\geq p\geq2,$ and $N$ and $P$ be connected manifolds of dimensions $n$ and $p$ respectively. Let $I=(n-p+1,i_{2},\cdots,i_{r-1},1,1)$ and $J=(n-p+1,i_{2},\cdots,i_{r-1},1,0)$ such that $n-p+1-i_{2}$ and $r$ $(r\geq3)$ are odd integers. Then if $f:N\rightarrow P$ is an $\Omega^{I}$-regular map, then $f$ is homotopic to an $\Omega^{J}$-regular map $g:N\rightarrow P$ such that $j^{\infty}f$ and $j^{\infty}g$ are homotopic in $\Gamma_{\Omega^{I}}(N,P)$.
In Section 1 we explain notations which are used in this paper. In Section 2 we review the definitions and the fundamental properties of the Boardman manifolds, from which we deduce several further results about higher intrinsic derivatives in Section 3. In Section 4 we reduce the proof of Theorem 0.1 to the proof of Theorem 4.1 by the induction, and prepare a certain rotation of the tangent spaces defined around the singularities of given symbol in $N$ to deform the section $s$. In Section 5 we prepare several lemmas which are used in the deformation of the section $s$ in the proof of Theorem 4.1. We prove Theorem 4.1 in Section 6 and prove Theorem 0.2 in Section 7.
Notations
=========
Throughout the paper all manifolds are Hausdorff, paracompact and smooth of class $C^{\infty}$. Maps are basically continuous, but may be smooth (of class $C^{\infty}$) if necessary. Given a fiber bundle $\pi:E\rightarrow X$ and a subset $C$ in $X,$ we denote $\pi^{-1}(C)$ by $E|_{C}.$ Let $\pi^{\prime
}:F\rightarrow Y$ be another fiber bundle. A map $\tilde{b}:E\rightarrow F$ is called a fiber map over a map $b:X\rightarrow Y$ if $\pi^{\prime}\circ
\tilde{b}=b\circ\pi$ holds. The restriction $\tilde{b}|(E|_{C}):E|_{C}\rightarrow F$ (or $F|_{b(C)}$) is denoted by $\tilde{b}|_{C}$. In particular, for a point $x\in X,$ $E|_{x}$ and $\tilde{b}|_{x}$ are simply denoted by $E_{x}$ and $\tilde{b}_{x}:E_{x}\rightarrow F_{b(x)}$ respectively. We denote, by $b^{F}$, the induced fiber map $b^{\ast}(F)\rightarrow F$ covering $b$. For a map $j:W\rightarrow X$, let $j^{\ast}(\tilde{b}):j^{\ast}E\rightarrow(b\circ
j)^{\ast}F$ over $W$ be the fiber map canonically induced from $b$ and $j$. A fiberwise homomorphism $E\rightarrow F$ is simply called a homomorphism. For a vector bundle $E$ with a metric and a positive function $\delta$ on $X$, let $D_{\delta}(E)$ be the associated disk bundle of $E$ with radius $\delta$. If there is a canonical isomorphism between two vector bundles $E$ and $F$ over $X=Y,$ then we write $E\cong F$.
When $E$ and $F$ are smooth vector bundles over $X=Y$, Hom$(E,F)$ denotes the smooth vector bundle over $X$ with fiber Hom$(E_{x},F_{x})$, $x\in X$ which consists of all homomorphisms $E_{x}\rightarrow F_{x}$.
Let $J^{k}(N,P)$ denote the $k$-jet space of manifolds $N$ and $P$. Let $\pi_{N}^{k}$ and $\pi_{P}^{k}$ be the projections mapping a jet to its source and target respectively. The map $\pi_{N}^{k}\times\pi_{P}^{k}:J^{k}(N,P)\rightarrow N\times P$ induces a structure of a fiber bundle with structure group $L^{k}(p)\times L^{k}(n)$, where $L^{k}(m)$ denotes the group of all $k$-jets of local diffeomorphisms of $(\mathbf{R}^{m},0)$. The fiber $(\pi_{N}^{k}\times\pi_{P}^{k})^{-1}(x,y)$ is denoted by $J_{x,y}^{k}(N,P)$.
Let $\pi_{N}$ and $\pi_{P}$ be the projections of $N\times P$ onto $N$ and $P$ respectively. We set $$J^{k}(TN,TP)=\bigoplus_{i=1}^{k}\text{\textrm{Hom}}(S^{i}(\pi_{N}^{\ast
}(TN)),\pi_{P}^{\ast}(TP))$$ over $N\times P$. Here, for a vector bundle $E$ over $X$, let $S^{i}(E)$ be the vector bundle $\cup_{x\in X}S^{i}(E_{x})$ over $X$, where $S^{i}(E_{x})$ denotes the $i$-fold symmetric product of $E_{x}$. If we provide $N$ and $P$ with Riemannian metrics, then the Levi-Civita connections induce the exponential maps $\exp_{N,x}:T_{x}N\rightarrow N$ and $\exp_{P,y}:T_{y}P\rightarrow P$. In dealing with the exponential maps we always consider the convex neighborhoods (\[K-N\]). We define the smooth bundle map $$J^{k}(N,P)\mathbf{\rightarrow}J^{k}(TN,TP)\text{ \ \ \ over }N\times P$$ by sending $z=j_{x}^{k}f\in J_{x,y}^{k}(N,P)$ to the $k$-jet of $(\exp
_{P,y})^{-1}\circ f\circ\exp_{N,x}$ at $\mathbf{0}\in T_{x}N$, which is regarded as an element of $J^{k}(T_{x}N,T_{y}P)(=J_{x,y}^{k}(TN,TP))$ (see \[K-N, Proposition 8.1\] for the smoothness of exponential maps). More strictly, (1.2) gives a smooth equivalence of the fiber bundles under the structure group $L^{k}(p)\times L^{k}(n)$. Namely, it gives a smooth reduction of the structure group $L^{k}(p)\times L^{k}(n)$ of $J^{k}(N,P)$ to $O(p)\times
O(n)$, which is the structure group of $J^{k}(TN,TP)$.
Recall that $S^{i}(E)\ $has the inclusion $S^{i}(E)\rightarrow\otimes^{i}E$ and the canonical projection $\otimes^{i}E\rightarrow S^{i}(E)$ (see \[B, Section 4\] and \[Mats, Ch. III, Section 2\]). Let $E_{j}$ be subbundles of $E$ $(j=1,\cdots,i)$. We define $E_{1}\bigcirc\cdots\bigcirc E_{i}=\bigcirc
_{j=1}^{i}E_{j}$ to be the image of $E_{1}\otimes\cdots\otimes E_{i}=\otimes_{j=1}^{i}E_{j}\rightarrow\otimes^{i}E\rightarrow S^{i}(E)$. When $E_{j+1}=\cdots=E_{j+\ell}$, we often write $E_{1}\bigcirc\cdots\bigcirc
E_{j}\bigcirc^{\ell}E_{j+1}\bigcirc E_{j+\ell+1}\bigcirc\cdots\bigcirc E_{i}$ in place of $\bigcirc_{j=1}^{i}E_{j}.$
Boardman manifolds
==================
We review well-known results about Boardman manifolds in $J^{\infty}(N,P)$ (\[B\], \[L2\] and \[Math2\]). Let $I=(i_{1},\cdots,i_{r})$ be a Boardman symbol with $i_{1}\geq\cdots\geq i_{r}\geq0$. For $k\leq r$, set $I_{k}=(i_{1},i_{2},\cdots,i_{k})$ and $(I_{k},0)=(i_{1},i_{2},\cdots,i_{k},0)$. In the infinite jet space $J^{\infty}(N,P)$, there have been defined a sequence of the submanifolds $\Sigma^{I_{1}}(N,P)\supseteq\cdots\supseteq\Sigma^{I_{r}}(N,P)$ with the following properties. In this paper we often write $\Sigma^{I_{r}}$ for $\Sigma^{I_{r}}(N,P)$ if there is no confusion.
Let $\mathbf{P}=(\pi_{P}^{\infty})^{\ast}(TP)$ and $\mathbf{D}$ be the total tangent bundle defined over $J^{\infty}(N,P)$. We explain important properties of the total tangent bundle $\mathbf{D}$, which are often used in this paper. Let $f:(N,x)\rightarrow(P,y)$ be a germ and $\digamma$ be a smooth function in the sense of \[B, Definition 1.4\] defined on a neighborhood of $j_{x}^{\infty}f$. Given a vector field $v$ defined on a neighborhood of $x$ in $N$, there is a total vector field $D$ defined on a neighborhood of $j_{x}^{\infty}f$ such that $D\digamma\circ j^{\infty}f=v(\digamma\circ
j^{\infty}f)$. It follows that $d(j^{\infty}f)(v)(\digamma)=D\digamma
(j^{\infty}f)$ for $d(j^{\infty}f):TN\rightarrow T(J^{\infty}(N,P))$ around $x$. This implies $d(j^{\infty}f)(v)=D.$ Hence, we have $\mathbf{D\cong(}\pi_{N}^{\infty})^{\ast}(TN)$.
First we have the first derivative $\mathbf{d}_{1}:\mathbf{D}\rightarrow
\mathbf{P}$ over $J^{\infty}(N,P)$. We define $\Sigma^{I_{1}}(N,P)$ to be the submanifold of $J^{\infty}(N,P)$ which consists of all jets $z$ such that the kernel rank of $\mathbf{d}_{1,z}$ is $i_{1}$. Since $\mathbf{d}_{1}|_{\Sigma^{I_{1}}(N,P)}$ is of constant rank $n-i_{1}$, we set $\mathbf{K}_{1}=$Ker$(\mathbf{d}_{1})$ and $\mathbf{Q}_{1}=$Cok$(\mathbf{d}_{1})$, which are vector bundles over $\Sigma^{I_{1}}(N,P)$. Set $\mathbf{K}_{0}=\mathbf{D}$, $\mathbf{P}_{0}=\mathbf{P}$ and $\Sigma^{I_{0}}(N,P)=J^{\infty}(N,P)$. We can inductively define $\Sigma^{I_{k}}(N,P)$ and the bundles $\mathbf{K}_{k}$ and $\mathbf{P}_{k}$ over $\Sigma^{I_{k}}(N,P)$ ($k\geq1$) with the properties:
\(1) $\mathbf{K}_{k-1}|_{\Sigma^{I_{k}}(N,P)}\supseteq\mathbf{K}_{k}$ over $\Sigma^{I_{k}}(N,P)$.
\(2) $\mathbf{K}_{k}$ is an $i_{k}$-dimensional subbundle of $T(\Sigma
^{I_{k-1}}(N,P))|_{\Sigma^{I_{k}}(N,P)}.$
\(3) There exists the $(k+1)$-th intrinsic derivative $\mathbf{d}_{k+1}:T(\Sigma^{I_{k-1}}(N,P))|_{\Sigma^{I_{k}}(N,P)}\linebreak
\rightarrow\mathbf{P}_{k}$ over $\Sigma^{I_{k}}(N,P),$ so that it induces the exact sequence over $\Sigma^{I_{k}}(N,P):$$$\begin{array}
[c]{l}\mathbf{0\rightarrow}T(\Sigma^{I_{k}}(N,P))\overset{\text{inclusion}}{\hookrightarrow}T(\Sigma^{I_{k-1}}(N,P))|_{\Sigma^{I_{k}}(N,P)}\overset{\mathbf{d}_{k+1}}{\longrightarrow}\mathbf{P}_{k}\text{ }\rightarrow\mathbf{0.}\end{array}$$ Namely, $\mathbf{d}_{k+1}$ induces the isomorphism of the normal bundle$$\nu(I_{k}\subset I_{k-1})=(T(\Sigma^{I_{k-1}}(N,P))|_{\Sigma^{I_{k}}(N,P)})/T(\Sigma^{I_{k}}(N,P))$$ of $\Sigma^{I_{k}}(N,P)$ in $\Sigma^{I_{k-1}}(N,P)$ onto $\mathbf{P}_{k}$.
\(4) $\Sigma^{I_{k+1}}(N,P)$ is defined to be the submanifold of $\Sigma
^{I_{k}}(N,P)$ which consists of all jets $z$ with dim(Ker$(\mathbf{d}_{k+1,z}|\mathbf{K}_{k,z}))=i_{k+1}$. In particular, $\Sigma^{I_{k}}(N,P)$ is the disjoint union $\cup_{j=0}^{i_{k}}\Sigma^{(I_{k},j)}(N,P).$
\(5) Set $\mathbf{K}_{k+1}=$Ker$(\mathbf{d}_{k+1}|\mathbf{K}_{k})$ and $\mathbf{Q}_{k+1}\mathbf{=}$Cok$(\mathbf{d}_{k+1}|\mathbf{K}_{k})$ over $\Sigma^{I_{k+1}}(N,P)$. Then it follows that $(\mathbf{K}_{k}|_{\Sigma
^{I_{k+1}}(N,P)})\cap T(\Sigma^{I_{k}}(N,P))|_{\Sigma^{I_{k+1}}(N,P)}=\mathbf{K}_{k+1}$. We have the canonical projection $\mathbf{e}_{k}:\mathbf{P}_{k}|_{\Sigma^{I_{k+1}}(N,P)}\rightarrow\mathbf{Q}_{k+1}$.
\(6) The intrinsic derivative$$d(\mathbf{d}_{k+1}|\mathbf{K}_{k}):T(\Sigma^{I_{k}}(N,P))|_{\Sigma^{I_{k+1}}(N,P)}\rightarrow\text{Hom}(\mathbf{K}_{k+1},\mathbf{Q}_{k+1})\text{ \ \ over
}\Sigma^{I_{k+1}}(N,P)$$ of $\mathbf{d}_{k+1}|\mathbf{K}_{k}$ is of constant rank $\dim(\Sigma^{I_{k}}(N,P))-\dim(\Sigma^{I_{k+1}}(N,P))$. We set $\mathbf{P}_{k+1}=\operatorname{Im}(d(\mathbf{d}_{k+1}|\mathbf{K}_{k}))$ and define $\mathbf{d}_{k+2}$ to be$$\mathbf{d}_{k+2}=d(\mathbf{d}_{k+1}|\mathbf{K}_{k}):T(\Sigma^{I_{k}}(N,P))|_{\Sigma^{I_{k+1}}(N,P)}\rightarrow\mathbf{P}_{k+1}$$ as the epimorphism.
\(7) There exists the bundle homomorphism of constant rank$$\mathbf{u}_{k}:\text{Hom}(\mathbf{K}_{k}\bigcirc\mathbf{K}_{k-1}\bigcirc
\cdots\bigcirc\mathbf{K}_{1},\mathbf{P})\rightarrow\text{Hom}(\mathbf{K}_{k},\mathbf{Q}_{k})\text{ \ \ over }\Sigma^{I_{k}}(N,P)$$ such that the image of $\mathbf{u}_{k}$ coincides with $\mathbf{P}_{k}$. We denote, by $\mathbf{c}_{k}$, the map $\mathbf{u}_{k}$ as the epimorphism onto $\mathbf{P}_{k}$. Furthermore, $\mathbf{u}_{k}$ is defined as the composition$$\begin{aligned}
& \text{Hom}(\mathbf{K}_{k}\bigcirc\mathbf{K}_{k-1}\bigcirc\cdots
\bigcirc\mathbf{K}_{1},\mathbf{P})\overset{\text{inclusion}}{\hookrightarrow
}\text{Hom}(\mathbf{K}_{k}\otimes\mathbf{K}_{k-1}\bigcirc\cdots\bigcirc
\mathbf{K}_{1},\mathbf{P})\\
& \cong\text{Hom}(\mathbf{K}_{k},\text{Hom}(\mathbf{K}_{k-1}\bigcirc
\cdots\bigcirc\mathbf{K}_{1},\mathbf{P}))\overset{\underrightarrow
{\text{Hom}(id_{\mathbf{K}_{k}},\mathbf{c}_{k-1})}}{}\text{Hom}(\mathbf{K}_{k},\mathbf{P}_{k-1})\nonumber\\
& \overset{\underrightarrow{\text{Hom}(id_{\mathbf{K}_{k}},\mathbf{e}_{k})}}{}\text{Hom}(\mathbf{K}_{k},\mathbf{Q}_{k})\nonumber\end{aligned}$$ (\[B, Theorem 7.14\]).
\(8) For a smooth map germ $f:N\rightarrow P$ such that $j^{\infty}f$ is transverse to $\Sigma^{I_{k}}(N,P)$, let $S^{I_{k}}(j^{\infty}f)$ denote $(j^{\infty}f)^{-1}(\Sigma^{I_{k}}(N,P))$. If $f|S^{I_{k}}(j^{\infty
}f):S^{I_{k}}(j^{\infty}f)\rightarrow P$ is of kernel rank $i_{k+1}$ at $x$, then $j_{x}^{\infty}f\in\Sigma^{I_{k+1}}(N,P)$.
\(9) The submanifold $\Sigma^{I_{k}}(N,P)$ is actually defined so that it coincides with the inverse image of the submanifold $\widetilde{\Sigma}^{I_{k}}(N,P)$ in $J^{k}(N,P)$ by $\pi_{k}^{\infty}$. The codimension of $\Sigma^{I_{k}}(N,P)$ in $J^{\infty}(N,P)$ is described in \[B, Theorem 6.1\].
\(1) It is known that $\Omega^{I}(N,P)$ is an open subset of $J^{\infty}(N,P)$: Let $I=(i_{1},i_{2},\cdots,i_{r})$. We prove that the closure of $\Sigma
^{I}(N,P)$ is contained in the subset which consists of all submanifolds $\Sigma^{J}(N,P)$ of the symbol $J$ of length $r$ with $J\geq I$. Let $z\in
J^{\infty}(N,P)$ lies in the closure of $\Sigma^{I}(N,P)$. By definition, we first have $\dim((\mathrm{Ker}(\mathbf{d}_{1,z}))\geq i_{1}$. If the symbol of $z$ is $J$ with $J\neq I$, then we can inductively prove that $z$ has a number $k$ such that $\mathrm{\dim}(\mathrm{Ker}(\mathbf{d}_{j,z}|\mathbf{K}_{j-1,z}))=i_{j}$ for $1\leq j\leq k<r$ and $\dim(\mathrm{Ker}(\mathbf{d}_{k+1,z}|\mathbf{K}_{k,z}))>i_{k+1}$. This implies the assertion.
\(2) If a symbol $J$ is an infinite series $(j_{1},j_{2},\cdots,j_{k},\cdots)$ and $\mathrm{codim}\Sigma^{J}(N,P)\leq n$, then $j_{1},j_{2},\cdots
,j_{k},\cdots$ are equal to $0$ except for a finite number of $j_{k}$’s.
Polynomials
===========
Let $V$ and $\ W$ be vector spaces with inner product of dimensions $v$ and $w$ respectively. Let $e_{1},e_{2},\cdots,e_{v}$ and $d_{1},d_{2},\cdots
,d_{w}$ be orthogonal basis of $V$ and $W$ respectively. We introduce the inner product in Hom$(\otimes^{\ell}V,W)$ as follows. Let $h_{i}\in\mathrm{Hom}(\otimes^{\ell}V,W)$ ($i=1,2$) and let$$h_{1}(e_{i_{1}}\otimes\ldots\otimes e_{i_{\ell}})=\sum_{j=1}^{w}a_{i_{1}i_{2}\cdots i_{\ell}}^{j}d_{j}\text{ \ \ and \ \ }h_{2}(e_{i_{1}}\otimes
\ldots\otimes e_{i_{\ell}})=\sum_{j=1}^{w}b_{i_{1}i_{2}\cdots i_{\ell}}^{j}d_{j}.$$ Then we define the inner product by$$\langle h_{1},h_{2}\rangle=\sum_{j=1}^{w}\sum_{i_{1}i_{2}\cdots i_{\ell}}a_{i_{1}i_{2}\cdots i_{\ell}}^{j}b_{i_{1}i_{2}\cdots i_{\ell}}^{j}.$$ Let $S$ and $T$ be isomorphisms of $V$ and $W$ which preserve the inner products respectively. We define the action of $(T,S)$ on Hom$(\otimes^{\ell
}V,W)$ by $(T,S)h=T\circ h\circ(\otimes^{\ell}S^{-1})$. We show by induction on $\ell$ that this inner product is invariant with respect to this action. We represent $S^{-1}$ by the matrix $(s_{ij})$ under the basis $e_{1},e_{2},\cdots,e_{w}$.
The assertion for $\ell=1$ is well known. Assume that the assertion holds for $\ell-1$. Under the canonical isomorphism Hom$(\otimes^{\ell}V,W)\cong
\mathrm{Hom}(V,\mathrm{Hom}(\otimes^{\ell-1}V,W))$ we let $h\in\mathrm{Hom}(\otimes^{\ell}V,W)$ correspond to $\overline{h}$, which satisfies $\overline{h}(e_{i_{1}})(e_{i_{2}}\otimes\ldots\otimes e_{i_{\ell}})=h(e_{i_{1}}\otimes e_{i_{2}}\otimes\ldots\otimes e_{i_{\ell}})$. Then we have that $\langle h_{1},h_{2}\rangle=\Sigma_{j=1}^{v}\langle\overline{h_{1}}(e_{j}),\overline{h_{2}}(e_{j})\rangle$. Hence, we have that $$\begin{aligned}
\langle(T,S)h_{1},(T,S)h_{2}\rangle & =\sum_{i=1}^{v}\langle((T,S)\overline
{h_{1}})(S^{-1}(e_{i})),((T,S)\overline{h_{2}})(S^{-1}(e_{i}))\rangle\\
& =\sum_{i=1}^{v}\langle\overline{h_{1}}(S^{-1}(e_{i})),\overline{h_{2}}(S^{-1}(e_{i}))\rangle\\
& =\sum_{i=1}^{v}\langle\overline{h_{1}}(\Sigma_{j=1}^{v}s_{ij}e_{j}),\overline{h_{2}}(\Sigma_{k=1}^{v}s_{ik}e_{k})\rangle\\
& =\sum_{i=1}^{v}(\Sigma_{j=1}^{v}s_{ij}(\Sigma_{k=1}^{v}s_{ik}\langle\overline{h_{1}}(e_{j}),\overline{h_{2}}(e_{k})))\rangle\\
& =\sum_{j=1}^{v}\sum_{k=1}^{v}(\Sigma_{i=1}^{v}s_{ij}s_{ik})\langle
\overline{h_{1}}(e_{j}),\overline{h_{2}}(e_{k})\rangle\\
& =\sum_{j=1}^{v}\sum_{k=1}^{v}\delta_{jk}\langle\overline{h_{1}}(e_{j}),\overline{h_{2}}(e_{k})\rangle\\
& =\sum_{j=1}^{v}\langle\overline{h_{1}}(e_{j}),\overline{h_{2}}(e_{j})\rangle\\
& =\langle h_{1},h_{2}\rangle.\end{aligned}$$
We recall that Hom$(\Sigma_{j=1}^{\ell}\bigcirc^{j}V,W)$ is identified with the set of polynomials of degree $\leq\ell$ having the constant $0$ (see \[Mats, Ch. III, Section 2\]). Let $\mathbf{V}$ and $\mathbf{W}$ be smooth vector bundles with metric over a manifold $S$ with fibers $V$ and $W$ respectively. Then Hom$(\bigcirc^{\ell}\mathbf{V},\mathbf{W})$ is also a vector bundle with metric. For a point $c\in S$, take an open neighborhood $U$ around $c$ such that $\mathbf{V}|_{U}$ and $\mathbf{W}|_{U}$ are the trivial bundles, say $U\times V$ and $U\times W$ respectively. Then an element of Hom$(\bigcirc^{\ell}\mathbf{V},\mathbf{W})|_{U}$ is identified with a polynomial $\Sigma_{j=1}^{w}(\Sigma_{|\omega|=\ell}A_{\omega}^{j}(c)x_{1}^{\omega_{1}}x_{2}^{\omega_{2}}\cdots x_{v}^{\omega_{v}})d_{j}$, $c\in
U$, where $\omega=(\omega_{1},\omega_{2},\cdots,\omega_{v})$, $\omega_{i}\geq0$ ($i=1,\cdots,v$), and $|\omega|=\omega_{1}+\cdots+\omega_{v}$ and $A_{\omega}^{j}(c)$ is a real number. If $A_{\omega}^{j}(c)$ are smooth functions of $c,$ then $\{A_{\omega}^{j}(c)\}$ defines a smooth section of Hom$(\bigcirc^{\ell}\mathbf{V},\mathbf{W})|_{U}$ over $U$.
We now provide $N$ and $P$ with Riemannian metrics respectively. Then they induce the metrics on $\mathbf{D}$ and $\mathbf{P}$, and hence induces the metric on Hom$(\mathbf{K}_{k}\bigcirc\mathbf{K}_{k-1}\bigcirc\cdots
\bigcirc\mathbf{K}_{1},\mathbf{P})$. Furthermore, we can prove inductively that $\mathbf{P}_{k},$ and also $\mathbf{Q}_{k+1}$ as the orthogonal complement of Im$(d_{k+1}|\mathbf{K}_{k})$ inherit the induced metrics by (5) and (6) in Section 2 respectively. Consequently we have the induced metric on Hom$(\mathbf{K}_{k+1},$ $\mathbf{Q}_{k+1})$.
Let us recall $\mathbf{d}_{k}|\mathbf{K}_{k-1}:\mathbf{K}_{k-1}\rightarrow
\mathbf{P}_{k-1}$ and $\mathbf{e}_{k-1}:\mathbf{P}_{k-1}\rightarrow
\mathbf{Q}_{k}$ over $\Sigma^{I_{k}}(N,P),$ which induces the commutative diagram$$\begin{array}
[c]{lllllll}\mathbf{K}_{k} & \rightarrow & \mathbf{K}_{k-1} & \rightarrow & \text{
\ \ \ \ \ \ \ }\mathbf{P}_{k-1} & \rightarrow & \mathbf{Q}_{k}\\
& & \downarrow & & \text{ \ \ \ \ \ \ \ }\curvearrowright & \swarrow
\mathbf{j}_{\mathbf{Q}_{k}} & \\
\mathbf{0} & \rightarrow & \mathbf{K}_{k-1}/\mathbf{K}_{k} & \rightarrow &
\text{Hom}(\mathbf{K}_{k-1},\mathbf{Q}_{k-1}). & &
\end{array}$$ Since $\mathbf{Q}_{k}$ is the cokernel of $\mathbf{d}_{k}|\mathbf{K}_{k-1}$, we obtain the canonical isomorphism$$\mathbf{j}_{\mathbf{Q}_{k}}:\mathbf{Q}_{k}\rightarrow\mathrm{\operatorname{Im}}(\mathbf{d}_{k}|\mathbf{K}_{k-1})^{\bot}\text{ \ \ \ \ over \ }\Sigma^{I_{k}}(N,P),$$ where the symbol $\bot$ refers to the orthogonal complement. We also use the notation $\mathbf{j}_{\mathbf{Q}_{k}}:\mathbf{Q}_{k}\rightarrow
\mathrm{Hom}(\mathbf{K}_{k-1},\mathbf{Q}_{k-1})$.
Let $k\geq2$. We now construct the homomorphism, for $1\leq i\leq k$,$$\mathbf{q}(k)^{i+1,i+1}:T(\Sigma^{I_{i-1}}(N,P))|_{\Sigma^{I_{k}}(N,P)}\bigcirc\mathbf{K}_{i}\bigcirc\mathbf{K}_{i-1}\bigcirc\cdots
\bigcirc\mathbf{K}_{1}\rightarrow\mathbf{Q}_{1}$$ over $\Sigma^{I_{k}}(N,P)$ inductively by using $\mathbf{d}_{i+1}|_{\Sigma^{I_{k}}(N,P)}:T(\Sigma^{I_{i-1}}(N,P))|_{\Sigma^{I_{k}}(N,P)}\rightarrow\mathbf{P}_{i}|_{\Sigma^{I_{k}}(N,P)}$ as follows. By the inclusion $\mathbf{P}_{i}|_{\Sigma^{I_{k}}(N,P)}\subset\mathrm{Hom}(\mathbf{K}_{i},\mathbf{Q}_{i})|_{\Sigma^{I_{k}}(N,P)}$ we have the homomorphism$$\mathbf{q}(k)_{\otimes}^{i+1,2}:(T(\Sigma^{I_{i-1}}(N,P))|_{\Sigma^{I_{k}}(N,P)})\otimes\mathbf{K}_{i}\rightarrow\mathbf{Q}_{i}\text{ \ \ \ \ over
}\Sigma^{I_{k}}(N,P).$$ Suppose that we have constructed the homomorphism, for $j\leq i$,$$\mathbf{q}(k)_{\otimes}^{i+1,i-j+2}:T(\Sigma^{I_{i-1}}(N,P))|_{\Sigma^{I_{k}}(N,P)}\otimes\mathbf{K}_{i}\otimes\mathbf{K}_{i-1}\otimes\cdots
\otimes\mathbf{K}_{j}\rightarrow\mathbf{Q}_{j}$$ over $\Sigma^{I_{k}}(N,P)$. By using $\mathbf{j}_{\mathbf{Q}_{j}}:\mathbf{Q}_{j}\rightarrow$Hom$(\mathbf{K}_{j-1},\mathbf{Q}_{j-1})$ over $\Sigma^{I_{k}}(N,P)$, we obtain the homomorphism$$\mathbf{q}(k)_{\otimes}^{i+1,i-j+3}:T(\Sigma^{I_{i-1}}(N,P))|_{\Sigma^{I_{k}}(N,P)}\otimes\mathbf{K}_{i}\otimes\mathbf{K}_{i-1}\otimes\cdots
\otimes\mathbf{K}_{j-1}\rightarrow\mathbf{Q}_{j-1}$$ over $\Sigma^{I_{k}}(N,P)$. By setting $j=2$, we obtain $\mathbf{q}(k)_{\otimes}^{i+1,i+1}$. It remains to prove that $\mathbf{q}(k)_{\otimes
}^{i+1,i+1}$ is symmetric. This fact has been esssentially stated in \[B, Section 7, p.413\] without proof. Following the proof of \[B, Theorem 4.1\] we briefly prove it.
Let $z\in\Sigma^{I_{k}}(N,P)$. By the Riemannian metric of $P$, we consider the convex neighborhood of $P$ around $\pi_{P}^{\infty}(z)=y$. Let us canonically identify $\mathbf{Q}_{1,z}$ with a subspace of $T_{y}P$ by the isomorphism $\mathbf{P}_{z}\rightarrow T_{y}P$. By taking a basis of $\mathbf{Q}_{1,z}$ and projecting it by the exponential map, we have the local coordinates $y_{1},y_{2},\cdots,y_{p-n+i_{1}}$ on the convex neighborhood of $y$. Then we identify $\mathbf{Q}_{1,z}$ with Hom$(\frak{m}_{y}^{\mathbf{Q}}/(\frak{m}_{y}^{\mathbf{Q}})^{2},\mathbf{R})$, where $\frak{m}_{y}^{\mathbf{Q}}$ is the ideal generated by $y_{1},y_{2},\cdots,y_{p-n+i_{1}}$. Let $D$ and $D_{j}$ be sections of $T(\Sigma^{I_{i-1}}(N,P))|_{\Sigma^{I_{k}}(N,P)}$ and $\mathbf{K}_{j}$ defined around $z$ and let $\alpha\in
\frak{m}_{y}^{\mathbf{Q}}$. Then (3.1) is regarded as the homomorphism induced from$$T(\Sigma^{I_{i-1}}(N,P))_{z}\otimes\mathbf{K}_{i,z}\otimes\mathbf{K}_{i-1,z}\otimes\cdots\otimes\mathbf{K}_{1,z}\otimes\frak{m}_{y}^{\mathbf{Q}}/(\frak{m}_{y}^{\mathbf{Q}})^{2}\longrightarrow\mathbf{R}$$ which maps $D\otimes D_{i}\otimes\cdots\otimes D_{1}\otimes\alpha$ to $(DD_{i}\cdots D_{1}\alpha)(z)$ (see (a) and (b) in the proof of \[B, Theorem 4.1\]). We have to show the following for the symmetry (consult Remark 3.1 below to avoid the infinity of the dimensions of the tangent spaces). In the expression with $[D_{j},D_{j-1}]=D_{j}D_{j-1}-D_{j-1}D_{j}$$$DD_{i}\cdots D_{j}D_{j-1}\cdots D_{1}\alpha-DD_{i}\cdots D_{j-1}D_{j}\cdots
D_{1}\alpha=DD_{i}\cdots\lbrack D_{j},D_{j-1}]\cdots D_{1}\alpha$$ for some $j$ with $1<j\leq i+1$ ($D_{i+1}=D$), we have that $[D_{j},D_{j-1}]$ is the section of $\mathbf{K}_{j-1}$ for $j\leq i$ and of $T(\Sigma^{I_{i-1}}(N,P))_{z}$ for $j=i+1$ by \[B, Lemma 3.2\]. Since $\mathbf{K}_{j}|_{\Sigma^{I_{k}}(N,P)}\subset\mathbf{K}_{j-1}|_{\Sigma^{I_{k}}(N,P)}$, the length of $DD_{i}\cdots\lbrack D_{j},D_{j-1}]\cdots D_{1}$ is $i$, $D$ and $\ [D,D_{i}]$ lie in $T(\Sigma^{I_{i-1}}(N,P))_{z}$ and since $T(\Sigma
^{I_{i-1}}(N,P))_{z}\subset T(\Sigma^{I_{i-2}}(N,P))_{z}$, we have that $(DD_{i}\cdots\lbrack D_{j},D_{j-1}]\cdots D_{1}\alpha)(z)=0$ by Ker$(\mathbf{d}_{i,z})=T(\Sigma^{I_{i-1}}(N,P))_{z}$ in (2.1). This is what we want.
In particular, if $i=1$ and we restrict $T(\Sigma^{I_{i-1}}(N,P))|_{\Sigma
^{I_{k}}(N,P)}$ to $\mathbf{K}_{1},$ then we have the homomorphism $\mathbf{q}(k)^{2,2}|(\mathbf{K}_{1}\bigcirc\mathbf{K}_{1}):\mathbf{K}_{1}\bigcirc\mathbf{K}_{1}\rightarrow\mathbf{Q}_{1}$ over $\Sigma^{I_{k}}(N,P)$, which induces the nonsingular quadratic form $(\mathbf{K}_{1}/\mathbf{K}_{2})\bigcirc(\mathbf{K}_{1}/\mathbf{K}_{2})\rightarrow
\mathbf{Q}_{1}$ on each fiber.
We can entirely do the arguments in Sections 2 and 3 on $J^{\ell}(N,P)$ for a large $\ell$. We provide $N$ and $P$ with Riemannian metrics. For any points $x\in N$ and $y\in P$, we have the local coordinates $(x_{1},...,x_{n})$ and $(y_{1},...,y_{n})$ on convex neighbourhoods of $x$ and $y$ associated to orthonormal basis of $T_{x}N$ and $T_{y}P$ respectively. Let us define the canonical embedding $\mu_{\infty}^{\ell}:J^{\ell}(TN,TP)\rightarrow J^{\infty
}(TN,TP)$ such that $\pi_{\ell}^{\infty}\circ\mu_{\infty}^{\ell}=id_{J^{\ell
}(TN,TP)}$ and that the $i$-th components for $i>\ell$ of elements of the image $\mu_{\infty}^{\ell}$ are the null homomorphisms of $\mathrm{Hom}(S^{i}(\pi_{N}^{\ast}(TN)),\pi_{P}^{\ast}(TP))$. We regard $\mu_{\infty}^{\ell}$ as the map to $J^{\infty}(N,P)$ under the identification (1.2). Any element $z\in\mu_{\infty}^{\ell}(J^{\ell}(TN,TP))$ is represented by a $C^{\infty}$ map germ $f:(N,x)\rightarrow(P,y)$ such that any $i$-th derivative of $f$ with $i>\ell$ vanishes under these coordinates. It is clear that We can prove that $\mathbf{D}|_{\mu_{\infty}^{\ell}(J^{\ell}(TN,TP))}$ is tangent to $\mu_{\infty}^{\ell}(J^{\ell}(TN,TP))$. Indeed, for $\sigma
=(\sigma_{1},...,\sigma_{n})$ with non-negative integers $\sigma_{i}$, let us recall the functions $X_{i}$ and $Z_{j,\sigma}$ with $1\leq i\leq n$ and $1\leq j\leq p$ defined locally on a neighbourhood of $J^{\infty}(N,P)$ by, for $z=j_{x}^{\infty}f,$ $$X_{i}(z)=x_{i}\text{ \ and \ }Z_{j,\sigma}(z)=\frac{\partial^{|\sigma|}(y_{j}\circ f)}{\partial x_{1}^{\sigma_{1}}\cdots\partial x_{n}^{\sigma^{n}}}(x),$$ which constitute the local coordinates on $J^{\infty}(N,P)$ as described in \[B, Section 1\]. Let $\Phi$ be a smooth function defined locally on $\mu_{\infty}^{\ell}(J^{\ell}(TN,TP))$ and let $D_{i}\in\mathbf{D}$ be the total tangent vector corresponding to $\partial/\partial x_{i}$ by the canonical identification of $\mathbf{D}$ and $(\pi_{N}^{\infty})^{\ast}(TN)$. Let $\sigma^{\prime}=(\sigma_{1},...,\sigma_{i-1,}\sigma_{i}+1,\sigma
_{i+1},...,\sigma_{n})$. Then we have $$D_{i}(\Phi)(z)=\frac{\partial(\Phi\circ j^{\infty}f)}{\partial x_{i}}(x)=\frac{\partial\Phi}{\partial X_{i}}(z)+\underset{j,\sigma}{\sum}\frac{\partial\Phi}{\partial Z_{j,\sigma}}(z)Z_{j,\sigma^{\prime}}(z)$$ by \[B, (1.8)\]. If $z\in\mu_{\infty}^{\ell}(J^{\ell}(TN,TP))$, then $Z_{j,\sigma}(z)$ vanishes for $|\sigma|>\ell$. Hence, $D_{i}(\Phi)$ is a smooth function defined locally on $\mu_{\infty}^{\ell}(J^{\ell}(TN,TP))$. This implies that $D_{i}$ is tangent to $\mu_{\infty}^{\ell}(J^{\ell}(TN,TP))$. Since $\mathbf{D}_{z}$ consists of all linear combinations of $D_{1},...,D_{n}$, we have that $\mathbf{D}_{z}\subset T_{z}(\mu_{\infty
}^{\ell}(J^{\ell}(TN,TP)))$. Therefore, we can do the required arguments on $\mu_{\infty}^{\ell}(J^{\ell}(TN,TP))$, namely also on $J^{\ell}(TN,TP)$.
Primary obstruction
===================
Let $\Gamma_{\Omega^{I}}^{tr}(N,P)$ denote the subspace of $\Gamma_{\Omega
^{I}}(N,P)$ consisting of all continuous sections of $\pi_{N}^{\infty}|\Omega^{I}(N,P):\Omega^{I}(N,P)\rightarrow N$ which are transverse to each $\Sigma^{J}(N,P)$. For $\frak{s}\in\Gamma_{\Omega^{I}}^{tr}(N,P)$, we set $S^{I_{j}}(\frak{s})=\frak{s}^{-1}(\Sigma^{I_{j}}(N,P))$, $S^{I_{j},0}(\frak{s})=\frak{s}^{-1}(\Sigma^{I_{j},0}(N,P))$, $(\frak{s}|S^{I}(\frak{s}))^{\ast}(\mathbf{K}_{j})=K_{j}(S^{I}(\frak{s}))$ and $(\frak{s}|S^{I}(\frak{s}))^{-1}\mathbf{Q}_{1}=Q(S^{I}(\frak{s}))$. We often write $S^{I}(\frak{s})$ as $S^{I}$ if there is no confusion.
Let $L=(\ell_{1},\ell_{2},\cdots,\ell_{r})$ and $I=(i_{1},i_{2},\cdots
,i_{k},0,\cdots,0,\cdots)$ such that $I_{r}\leq L$, codim $\Sigma^{L}(n,p)\leq
n$ and codim $\Sigma^{I}(n,p)\leq n$, where $k$ may be larger than $r$. Let $C(I^{+})$ (resp. $C(I)$) refer to the union $C\cup(\cup_{J>I}S^{J}(s))$ (resp. $C\cup(\cup_{J\geq I}S^{J}(s))$), where $J$ are symbols of infinite length and $C$ is a closed subset of $N$.
We show in this section that it is enough for the proof of Theorem 0.1 to prove Theorem 4.1.
Let $n\geq p\geq2$. Let $N$ and $P$ be connected manifolds of dimensions $n$ and $p$ respectively with $\partial N=\emptyset$. Let $L$ and $I=(i_{1},i_{2},\cdots,i_{k},0)$ be as above. Assume that $\Omega^{L}(N,P)$ contains $\Sigma^{n-p+1,0}(N,P)$ at least. Let $s$ be a section of $\Gamma_{\Omega^{L}}^{tr}(N,P)$ which has an $\Omega^{L}$-regular map $g(I^{+})$ defined on a neighborhood of $C(I^{+})$ into $P$, where $j^{\infty}g(I^{+})=s$. Then there exists a homotopy $s_{\lambda}\in\Gamma_{\Omega^{L}}^{tr}(N,P)$ relative to a neighborhood of $C(I^{+})$ with the following properties.
$(1)$ $s_{0}=s$ and $s_{1}\in\Gamma_{\Omega^{L}}^{tr}(N,P)$.
$(2)$ There exists an $\Omega^{L}$-regular map $g_{I}$ defined on a neighborhood of $C(I)$, where $j^{\infty}g_{I}=s_{1}$ holds.
$(3)$ $s^{-1}(\Sigma^{I}(N,P))=(j^{\infty}g_{I})^{-1}(\Sigma^{I}(N,P))$.
The case $I=(n-p+1,0)$ of Theorem 4.1 follows from Theorem 1 of \[An1\], where a partially sketchy proof was given and the detailed proof was given in \[An4, Theorem 4.1\] and \[An6, Theorem 0.5\]. Let us explain how it follows. In fact, we have $\Omega^{n-p+1,0}(N,P)=\Sigma^{n-p}(N,P)\cup\Sigma^{n-p+1,0}(N,P)$, and if we set $N_{0}=S^{n-p}(s)\cup S^{n-p+1,0}(s)$, then $C((n-p+1,0)^{+})=C\cup(N\setminus N_{0})$. Let $U$ and $U^{\prime}$ be closed neighborhoods of $C\cup(N\setminus N_{0})$ with $U\subset$Int$U^{\prime}$, where $g((n-p+1,0)^{+})$ is defined. Since $s\in\Gamma_{\Omega^{L}}^{tr}(N,P)$, $s|N_{0}$ is a section of $\Gamma_{\Omega^{n-p+1,0}}^{tr}(N_{0},P)$ and $g((n-p+1,0)^{+})|(U^{\prime}\cap N_{0})$ is an $\Omega^{n-p+1,0}$-regular map. Hence, we obtain a homotopy $u_{\lambda}\in\Gamma_{\Omega^{n-p+1,0}}^{tr}(N_{0},P)$ relative to a neighborhood of $U\cap N_{0}$ and an $\Omega^{n-p+1,0}$-regular map $f_{0}:N_{0}\rightarrow P$ such that $s_{0}|N_{0}=u_{0}$ and $u_{1}=j^{\infty}f_{0}$. Then we obtain a required homotopy $s_{\lambda}$ by defining $s_{\lambda}|N_{0}=u_{\lambda}$ and $s_{\lambda}|U=j^{\infty}g((n-p+1,0)^{+})$.
We will prove Theorem 4.1 for $I>(n-p+1,0)$ in Section 6.
We now prove Theorem 0.1 for $\Omega^{L}$ for this symbol $L=(\ell_{1},\ell_{2},\cdots,\ell_{r})$ in place of $\Omega^{I}$ by using Theorem 4.1. In Sections 4, 5 and 6 we use the notation $\Omega$ for $\Omega^{L}$.
\[Proof of Theorem 0.1\]Suppose that the section $s$ given in Theorem 0.1 lies in $\Gamma_{\Omega}^{tr}(N,P)$. Let $I=(i_{1},i_{2},\cdots,i_{k},0,\cdots,0,\cdots)$ be the largest symbol such that $I_{\ell}\leq L$ and codim $\Sigma^{I}(n,p)\leq n$. We can choose such a symbol $I$ by using Section 2 (4) and Remark 2.1 (2). Then we first set $C(I^{+})=C$ and $g(I^{+})=g$. By Theorem 4.1 there exists an $\Omega$-regular map $g_{I}$ defined on a neighborhood of $C(I)$, where $j^{\infty}g_{I}=s$ holds. If we note Remark 2.1 (2), then we can prove Theorem 0.1 by the downward induction on the symbols $I$ in the lexicographic order.
We begin by preparing several notions and results, which are necessary for the proof of Theorem 4.1. For the map $g(I^{+})$ and the closed subset $C(I^{+})$, we take an open neighborhood $U(C(I^{+}))^{\prime}$ of $C(I^{+})$, where $j^{\infty}g(I^{+})=s$. Without loss of generality we may assume that $N\setminus U(C(I^{+}))^{\prime}$ is nonempty. Take a smooth function $h_{C(I^{+})}:N\rightarrow\lbrack0,1]$ such that $$\left\{
\begin{array}
[c]{ll}h_{C(I^{+})}(x)=1 & \text{for }x\in C(I^{+}),\\
h_{C(I^{+})}(x)=0 & \text{for }x\in N\setminus U(C(I^{+}))^{\prime},\\
0<h_{C(I^{+})}(x)<1 & \text{for }x\in U(C(I^{+}))^{\prime}\setminus C(I^{+}).
\end{array}
\right.$$ By the Sard Theorem (\[H2\]) there is a regular value $r$ of $h_{C(I^{+})}$ with $0<r<1$. Then $h_{C(I^{+})}^{-1}(r)$ is a submanifold and we set $U(C(I^{+}))=h_{C(I^{+})}^{-1}([r,1])$. We decompose $N\setminus
\mathrm{Int}U(C(I^{+}))$ into the connected components, say $L_{1},\cdots,L_{j},\cdots$. It suffices to prove Theorem 4.1 for each $L_{j}\cup
$Int$U(C(I^{+}))$. Since $\partial N=\emptyset$, we have that $N\setminus
U(C(I^{+}))$ has empty boundary. If $L_{j}$ is not compact, then Theorem 4.1 holds for $L_{j}\cup$Int$U(C(I^{+}))$ by Gromov’s theorem (\[G1, Theorem 4.1.1\]). Therefore, it suffices to consider the special case where
(C1) $N\setminus\mathrm{Int}U(C(I^{+}))$ is compact, connected and nonempty,
(C2) $\partial U(C(I^{+}))$ is a submanifold of dimension $n-1$,
(C3) for the smooth function $h_{C(I^{+})}:N\rightarrow\lbrack0,1]$ satisfying (4.1) there is a sufficiently small positive real number $\varepsilon$ with $r-2\varepsilon>0$ such that $r-t\varepsilon$ ($0\leq t\leq2$) are all regular values of $h_{C(I^{+})}$. We have that $h_{C(I^{+})}^{-1}([r-2\varepsilon,1])$ is contained in $U(C(I^{+}))^{\prime}$.
We set $U(C(I^{+}))_{t}=h_{C(I^{+})}^{-1}([r-(2-t)\varepsilon,1])$. In particular, we have $U(C(I^{+}))_{2}=U(C(I^{+}))$. Furthermore, we may assume that
(C4) $s\in\Gamma_{\Omega}^{tr}(N,P)$ and $S^{I}(s)$ is transverse to $\partial
U(C(I^{+}))_{0}$ and $\partial U(C(I^{+}))_{2}$.
In what follows we choose and fix a Riemannian metric of $N$, which satisfies
**Orthogonality Condition**; *for the symbol* $I$*,* $K_{j-1}(S^{I}(s))/K_{j}(S^{I}(s))$ *is orthogonal to* $S^{I_{j}}(s)$ *in* $S^{I_{j-1}}(s)$ *for* $k\leq j\leq1$ *on* $S^{I}(s)$ ** $(S^{I_{0}}(s)=N)$.
Let $\nu(\Sigma^{I})$ be the normal bundle $(T(J^{\infty}(N,P))|_{\Sigma^{I}})/T(\Sigma^{I}(N,P))$ and let $c(I)=\dim\nu(\Sigma^{I})$. Let us fix a direct sum decomposition$$\nu(\Sigma^{I}){{{{=\oplus}}}}_{j=1}^{k}\nu(I_{j}\subset I_{j-1})|_{\Sigma
^{I}(N,P)},$$ and the direct sum decomposition $\mathbf{K}_{1}={{{{\oplus}}}}_{j=1}^{k-1}(\mathbf{K}_{j}\mathbf{/K}_{j+1})\oplus\mathbf{K}_{k}$ over $\Sigma
^{I}(N,P)$. Let $\mathbf{j}_{\mathbf{K}}:\mathbf{K}_{1}\rightarrow\nu
(\Sigma^{I})$ over $\Sigma^{I}(N,P)$ be the composition of the inclusion $\mathbf{K}_{1}\rightarrow T(J^{\infty}(N,P))$ and the projection $T(J^{\infty}(N,P))|_{\Sigma^{I}(N,P)}\rightarrow\nu(\Sigma^{I})$. We have the monomorphism$$\mathbf{j}_{\mathbf{K}}\circ(s|S^{I})^{\mathbf{K}_{1}}:K_{1}(S^{I}(s))\rightarrow\mathbf{K}_{1}|_{\Sigma^{I}(N,P)}\rightarrow\nu(\Sigma^{I}).$$
For $s\in\Gamma_{\Omega}^{tr}(N,P)$, let $\frak{n}(s,I)$ or simply $\frak{n}(I)$ be the orthogonal normal bundle of $S^{I}(s)$ in $N$. Let $\frak{n}(s,I_{j}\subset I_{j-1})$ be the orthogonal normal bundle of $S^{I_{j}}(s)$ in $S^{I_{j-1}}(s)$ over $S^{I}(s)$. Then we have the canonical direct sum decomposition such as$$\frak{n}(s,{{{{I)=\oplus}}}}_{j=1}^{k}\frak{n}(s,I_{j}\subset I_{j-1}),$$ Furthermore, we obtain the bundle map $$ds|\frak{n}(s,I):\frak{n}(s,I)\rightarrow\nu(\Sigma^{I})$$ covering $s|S^{I}:S^{I}(s)\rightarrow\Sigma^{I}(N,P)$. Let $\mathbf{i}_{\frak{n}(s,I)}:\frak{n}(s,I)\subset TN|_{S^{I}}$ denote the inclusion. We define $\Psi(s,I):K_{1}(S^{I}(s))\rightarrow\frak{n}(s,I)\subset TN|_{S^{I}}$ to be the composition$$\begin{aligned}
& \mathbf{i}_{\frak{n}(s,I)}\circ((s|S^{I})^{\ast}(ds|\frak{n}(s,I)))^{-1}\circ((s|S^{I})^{\ast}(\mathbf{j}_{\mathbf{K}}\circ(s|S^{I})^{\mathbf{K}_{1}}))\\
\text{ \ \ \ \ \ \ } & :K_{1}(S^{I}(s))\rightarrow(s|S^{I})^{\ast}\nu
(\Sigma^{I})\rightarrow\frak{n}(s,I)\hookrightarrow TN|_{S^{I}}.\end{aligned}$$ We note that this homomorphism does not use the decomposition in (4.2) and we can take the direct sum decompositions in (4.2) to be compatible with those in (4.3). Let $i_{K_{1}(S^{I}(s))}:K_{1}(S^{I}(s))\rightarrow TN|_{S^{I}}$ be the inclusion.
If $f$ is an $\Omega$-regular map, then it follows from the definition of $\mathbf{D}$ that $\mathbf{i}_{K_{1}(S^{I}(j^{\infty}f))}=\Psi(j^{\infty}f,I)$.
In what follows let $M=S^{I}(s)\setminus$Int$(U(C(I^{+})))$. Let Mono$(K_{1}(S^{I}(s))|_{M},TN|_{M})$ denote the subset of Hom$(K_{1}(S^{I}(s))|_{M},TN|_{M})$ which consists of all monomorphisms $K_{1}(S^{I}(s))_{c}\rightarrow T_{c}N$, $c\in M$. We denote the bundle of the local coefficients $\mathcal{B}(\pi_{j}(\mathrm{Mono}(K_{1}(S^{I}(s))_{c},T_{c}N))),$ $c\in M,$ by $\mathcal{B}(\pi_{j})$, which is a covering space over $M$ with fiber $\pi_{j}(\mathrm{Mono}(K_{1}(S^{I}(s))_{c},T_{c}N))$ defined in \[Ste, 30.1\]. From the obstruction theory due to \[Ste, 36.3\], it follows that the obstructions for $i_{K_{1}(S^{I}(s))}|_{M}$ and $\Psi(s,I)|_{M}$ to be homotopic are the primary differences $d(i_{K_{1}(S^{I}(s))}|_{M},\Psi(s,I)|_{M})$, which are defined in $H^{j}(M,\partial
M;\mathcal{B}(\pi_{j}))$ with the local coefficients . We show that all of them vanish. In fact, note $I>(n-p+1,0)$. If $i_{1}=n-p+1$, then we have$$\dim M<\dim S^{i_{1}}=n-i_{1}(p-n+i_{1})=n-i_{1}.$$ If $i_{1}>n-p+1$, then $$\dim M\leq\dim S^{i_{1}}=n-i_{1}(p-n+i_{1})<n-i_{1}.$$ Since $(\mathbf{R}^{i_{1}},\mathbf{R}^{n})$ is identified with $GL(n)/GL(n-i_{1})$, it follows from \[Ste, 25.6 38.2\] that $\pi_{j}($$(\mathbf{R}^{i_{1}},\mathbf{R}^{n}))\cong\{\mathbf{0}\}$ for $j<n-i_{1}(\leq p-1)$. Hence, there exists a homotopy $\psi^{M}(s,I)_{\lambda
}:K_{1}(S^{I}(s))|_{M}\rightarrow TN|_{M}$ relative to $M\cap U(C(I^{+}))_{1}$ in $(K_{1}(S^{I}(s))|_{M},TN|_{M})$ such that $\psi^{M}(s,I)_{0}=i_{K_{1}(S^{I}(s))}|_{M}$ and $\psi^{M}(s,I)_{1}$$=\Psi(s,I)|_{M}$. Let $\mathrm{Iso}(TN|_{M},TN|_{M})$ denote the subspace of $\mathrm{Hom}(TN|_{M},TN|_{M})$ which consists of all isomorphisms of $T_{c}N$, $c\in
M$. The restriction map $$r_{M}:\mathrm{Iso}(TN|_{M},TN|_{M})\rightarrow\mathrm{Mono}(K_{1}(S^{I}(s))|_{M},TN|_{M})$$ defined by $r_{M}(h)=h|(K_{1}(S^{I}(s))_{c})$, for $h\in\mathrm{Iso}(T_{c}N,T_{c}N)$, induces a structure of a fiber bundle with fiber Iso$(\mathbf{R}^{n-i_{1}},\mathbf{R}^{n-i_{1}})\times{\mathrm{Hom}}(\mathbf{R}^{n-i_{1}},\mathbf{R}^{i_{1}})$. By applying the covering homotopy property of the fiber bundle $r_{M}$ to the sections $id_{TN|_{M}}$ and the homotopy $\psi^{M}(s,I)_{\lambda},$ we obtain a homotopy $\Psi^{M}(s,I)_{\lambda}:$ $TN|_{M}\rightarrow TN|_{M}$ such that $\Psi^{M}(s,I)_{0}=id_{TN|_{M}}$, $\Psi^{M}(s,I)_{\lambda}|_{c}=id_{T_{c}N}$ for all $c\in M\cap U(C(I^{+}))_{1}$ and $r_{M}\circ\Psi^{M}(s,I)_{\lambda}=\psi
^{M}(s,I)_{\lambda}$. We define $\Phi(s,I)_{\lambda}:$ $TN|_{M}\rightarrow
TN|_{M}$ by $\Phi(s,I)_{\lambda}=(\Psi^{M}(s,I)_{\lambda})^{-1}$.
Lemmas
======
Let $I$ be the symbol in Theorem 4.1. In the proof of the following lemma, $\Phi(s,I)_{\lambda}|_{c}$ ($c\in M$) is regarded as a linear isomorphism of $T_{c}N$. Let $r_{0}$ be a small positive real number with $r_{0}<1/10$.
Let $s\in\Gamma_{\Omega}^{tr}(N,P)$ be a section satisfying the hypotheses of Theorem $4.1$. Then there exists a homotopy $s_{\lambda}$ relative to $U(C(I^{+}))_{2-3r_{0}}$ in $\Gamma_{\Omega}^{tr}(N,P)$ with $s_{0}=s$ satisfying
$(1)$ for any $\lambda$, $S^{I}(s_{\lambda})=S^{I}(s)$ and $\pi_{P}^{\infty
}\circ s_{\lambda}|S^{I}(s_{\lambda})=\pi_{P}^{\infty}\circ s|S^{I}(s),$
$(2)$ for any point $c\in S^{I}(s_{1})$, we have $i_{K_{1}(S^{I}(s_{1}))}=\Psi(s_{1},I)$. In particular, $K_{1}(S^{I}(s_{1}))_{c}\subset\frak{n}(s,I)_{c}$.
Recall the exponential map $\exp_{N,x}:T_{x}N\rightarrow N$ defined near $\mathbf{0}\in T_{x}N$. We write an element of $\frak{n}(I)_{c}$ as $\mathbf{v}_{c}$. There exists a small positive number $\delta$ such that the map $$e:D_{\delta}(\frak{n}(I))|_{M}\rightarrow N$$ defined by $e(\mathbf{v}_{c})=\exp_{N,c}(\mathbf{v}_{c})$ is an embedding, where $c\in M$ and $\mathbf{v}_{c}\in D_{\delta}(\frak{n}(I)_{c})$ (note that $e|M$ is the inclusion). Let $\rho:[0,\infty)\rightarrow\mathbf{R}$ be a decreasing smooth function such that $0\leq\rho(t)\leq1$, $\rho(t)=1$ if $t\leq\delta/10$ and $\rho(t)=0$ if $t\geq\delta$.
If we represent $s(x)\in\Omega(N,P)$ by a jet $j_{x}^{\infty}\sigma_{x}$ for a germ $\sigma_{x}:(N,x)\rightarrow(P,\pi_{P}^{\infty}\circ s(x))$, then we define the homotopy $s_{\lambda}$ of $\Gamma_{\Omega}^{tr}(N,P)$ using $\Phi(s,I)_{\lambda}$ by$$\left\{
\begin{array}
[c]{ll}\begin{array}
[c]{l}s_{\lambda}(e(\mathbf{v}_{c}))\\
=j_{e(\mathbf{v}_{c})}^{\infty}(\sigma_{e(\mathbf{v}_{c})}\circ\exp_{N,c}\circ\Phi(s,I)_{\rho(\Vert\mathbf{v}_{c}\Vert)\lambda}|_{c}\circ\exp
_{N,c}^{-1})
\end{array}
& \text{\textrm{if} $c\in M$ \textrm{and} }\Vert\text{$\mathbf{v}_{c}\Vert
\leq\delta,$}\\\begin{array}
[c]{l}s_{\lambda}(x)=s(x)
\end{array}
& \text{\textrm{if} $x\notin\mathrm{Im}(e).$}\end{array}
\right.$$ Here, $\Phi(s,I)_{\rho(\Vert\mathbf{v}_{c}\Vert)\lambda}|_{c}$ refers to $\ell(\mathbf{v}_{c})\circ(\Phi(s,I)_{\rho(\Vert\mathbf{v}_{c}\Vert)\lambda
}|_{c})\circ\ell(-\mathbf{v}_{c})$, where $\ell(\mathbf{v})$ denotes the parallel translation defined by $\ell(\mathbf{v})(\mathbf{a})=\mathbf{a}+\mathbf{v}$. If $\Vert\mathbf{v}_{c}\Vert\geq\delta$, then $\Phi
(s,I)_{\rho(\Vert\mathbf{v}_{c}\Vert)\lambda}|_{c}=\Phi(s,I)_{0}|_{c}$, and if $c\in S^{I}\cap U(C(I^{+}))_{2-3r_{0}}$, then $\Phi(s,I)_{\lambda}|_{c}=\Phi(s,I)_{0}|_{c}$. Hence, $s_{\lambda}$ is well defined. It follows from (5.1) that
\(1) $\pi_{P}^{\infty}\circ s_{\lambda}(x)=\pi_{P}^{\infty}\circ s(x)$,
\(2) $S^{I}(s_{\lambda})=S^{I}(s)$,
\(3) if $c\in S^{I}(s)$, then we have that $\frak{n}(s,I)_{c}\supset
K_{1}(S^{I}(s_{1}))_{c}$ and $i_{K_{1}(S^{I}(s_{1}))}=\Psi(s_{1},I)$.
\(4) $s_{\lambda}$ is transverse to $\Sigma^{I}(N,P)$.
In what follows we set $d_{1}(s,I)=(s|S^{I})^{\ast}(\mathbf{d}_{1})$. We also choose and fix a Riemannian metric of $P$ and identify $Q(S^{I}(s))$ with the orthogonal complement of Im$(d_{1}(s,I))$ in $(\pi_{P}^{\infty}\circ
s|S^{I})^{\ast}(TP)$.
Let $s$ be a section of $\Gamma_{\Omega}^{tr}(N,P)$ satisfying the property $(2)$ for $s$ $($in place of $s_{1}$$)$ of Lemma $5.1$. Then there exists a homotopy $s_{\lambda}$ relative to $U(C(I^{+}))_{2-3r_{0}}$ in $\Gamma
_{\Omega}^{tr}(N,P)$ with $s_{0}=s$ such that
$(1)$ $S^{I}(s_{\lambda})=S^{I}(s)$ for any $\lambda$,
$(2)$ $\pi_{P}^{\infty}\circ s_{1}|S^{I}(s_{1})$ is an immersion into $P$ such that $d(\pi_{P}^{\infty}\circ s_{1}|S^{I}(s_{1})):T(S^{I}(s_{1}))\linebreak
\rightarrow TP$ is equal to $(\pi_{P}^{\infty}\circ s_{1})^{TP}\circ
d_{1}(s_{1},I)|T(S^{I}(s_{1}))$, where $(\pi_{P}^{\infty}\circ s_{1})^{TP}:(\pi_{P}^{\infty}\circ s_{1})^{\ast}(TP)\rightarrow TP$ is the canonical induced bundle map.
Since $\mathbf{K}_{1}\cap T(\Sigma^{I}(N,P))=\{\mathbf{0}\},$ it follows that $(\pi_{P}^{\infty}\circ s)^{TP}\circ d_{1}(s,I)|T(S^{I})$ is a monomorphism. By the Hirsch Immersion Theorem (\[H1, Theorem 5.7\]) there exists a homotopy of monomorphisms $m_{\lambda}^{\prime}:T(S^{I})\rightarrow TP$ covering a homotopy $m_{\lambda}:S^{I}\rightarrow P$ such that $m_{0}^{\prime}=(\pi
_{P}^{\infty}\circ s)^{TP}\circ d_{1}(s,I)|T(S^{I})$ and that $m_{1}$ is an immersion with $d(m_{1})=m_{1}^{\prime}$. Then we can extend $m_{\lambda
}^{\prime}$ to a homotopy $\widetilde{m_{\lambda}^{\prime}}:TN|_{S^{I}}\rightarrow TP$ of homomorphisms of constant rank $n-i_{1}$ relative to $U(C(I^{+}))_{2-3r_{0}}$. In fact, let $m:S^{I}\times\lbrack0,1]\rightarrow
P\times\lbrack0,1]$ and $m^{\prime}:T(S^{I})\times\lbrack0,1]\rightarrow
TP\times\lbrack0,1]$ be the maps defined by $m(c,\lambda)=(m_{\lambda
}(c),\lambda)$ and $m^{\prime}(\mathbf{v},\lambda)=(m_{\lambda}^{\prime
}(\mathbf{v}),\lambda)$ respectively. Let $m^{\ast}(m^{\prime}):T(S^{I})\times\lbrack0,1]\rightarrow m^{\ast}(TP\times\lbrack0,1])$ be the canonical monomorphism induced from $m^{\prime}$ by $m$. Let $\mathcal{F}_{1}=$Im$(m^{\ast}(m^{\prime}))$ and $\mathcal{F}_{2}$ be the orthogonal complement of $\mathcal{F}_{1}$ in $m^{\ast}(TP\times\lbrack0,1])$. Since $\mathcal{F}_{2}$ is isomorphic to $(\mathcal{F}_{2}|_{S^{I}\times0})\times\lbrack0,1]$, we obtain a monomorphism of rank $c(I)-i_{1}$ $$j_{\mathcal{F}}:\text{Im}(d_{1}(s,I)|\frak{n}(I))\times\lbrack0,1]\rightarrow
\mathcal{F}_{2}\text{ \ \ \ \ \ over }S^{I}\times\lbrack0,1]\text{.}$$ Since $d_{1}(s,I)|(TN|_{S^{I}})$ is of constant rank $n-i_{1}$, it induces the homomorphism of kernel rank $i_{1}$$$d:\frak{n}(I)\times\lbrack0,1]\rightarrow\text{Im}(d_{1}(s,I)|\frak{n}(I))\times\lbrack0,1]\overset{j_{\mathcal{F}}}{\rightarrow}\mathcal{F}_{2}\text{.}$$ We define $\widetilde{m^{\prime}}$ to be the composition$$\begin{array}
[c]{c}TN|_{S^{I}}\times\lbrack0,1]\cong(T(S^{I})\oplus\frak{n}(I))\times
\lbrack0,1]\overset{\underrightarrow{\text{ }m^{\ast}(m^{\prime})\oplus
d\text{ \ }}}{}\mathcal{F}_{1}\oplus\mathcal{F}_{2}\\
\rightarrow\text{Im}(m^{\ast}(m^{\prime}))\oplus\text{Cok}(m^{\ast}(m^{\prime
}))\cong m^{\ast}(TP\times\lbrack0,1])\overset{\underrightarrow{\text{
}m^{TP\times\lbrack0,1]}\text{\ }}}{}TP\times\lbrack0,1].
\end{array}$$ We define $\widetilde{m_{\lambda}^{\prime}}$ to be $(\widetilde{m_{\lambda
}^{\prime}}(\mathbf{v}),\lambda)=\widetilde{m^{\prime}}(\mathbf{v},\lambda)$.
Next we construct a homotopy $s_{\lambda}:N\rightarrow\Omega(N,P)$ from $\widetilde{m_{\lambda}^{\prime}}.$ Recall the submanifold $\widetilde{\Sigma
}^{i_{1}}(N,P)$ of $J^{1}(N,P)=J^{1}(TN,TP)$ which corresponds to $\Sigma^{i_{1}}(N,P)$ in Section 2 (9). Then $\pi_{1}^{\infty}|\Sigma
^{I}(N,P):\Sigma^{I}(N,P)\rightarrow\widetilde{\Sigma}^{i_{1}}(N,P)$ becomes a fiber bundle. We regard $\widetilde{m_{\lambda}^{\prime}}$ as a homotopy $S^{I}\rightarrow\widetilde{\Sigma}^{i_{1}}(N,P).$ By the covering homotopy property to $s|S^{I}$and $\widetilde{m_{\lambda}^{\prime}}$, we obtain a homotopy $s_{\lambda}^{\prime}:S^{I}\rightarrow\Sigma^{I}(N,P)$ covering $\widetilde{m_{\lambda}^{\prime}}$ relative to $U(C(I^{+}))_{2-3r_{0}}$ such that $s_{0}^{\prime}=s|S^{I}$.
By using the transversality of $s$ and the homotopy extension property to $s$ and $s_{\lambda}^{\prime}$, we first extend $s_{\lambda}^{\prime}$ to a homotopy defined on a tubular neighborhood of $S^{I}$ and then to a required homotopy $s_{\lambda}\in\Gamma_{\Omega}^{tr}(N,P)$, which satisfies $s_{0}=s$, $s_{\lambda}|S^{I}=s_{\lambda}^{\prime}$ and $\ s_{\lambda}|U(C(I^{+}))_{2-3r_{0}}=s|U(C(I^{+}))_{2-3r_{0}}$.
Here we give two lemmas necessary for the proof of Theorem 4.1. Let $\pi:E\rightarrow S$ be a smooth $c(I)$-dimensional vector bundle with a metric over an $(n-c(I))$-dimensional manifold, where $S$ is identified with the zero-section. Then we can identify $\exp_{E}|D_{\varepsilon}(E)=id_{D_{\varepsilon}(E)}$.
Let $\pi:E\rightarrow S$ be given as above. Let $f_{i}:E\rightarrow P$ $(i=1,2)$ be $\Omega$-regular maps which have singularities of the symbol $I$ exactly on $S$ such that
$(\mathrm{i})$ $f_{1}|S=f_{2}|S$, which are immersions,
$(\mathrm{ii})$ $S=S^{I}(j^{\infty}f_{1})=S^{I}(j^{\infty}f_{2})$,
$(\mathrm{iii})$ $K_{1}(S^{I}(j^{\infty}f_{1}))_{c}=K_{1}(S^{I}(j^{\infty
}f_{2}))_{c}$ are tangent to $\pi^{-1}(c)$,
$(\mathrm{iv})$ $T_{c}(S^{I_{j-1}}(j^{\infty}f_{1}))=T_{c}(S^{I_{j-1}}(j^{\infty}f_{2}))$, $((j^{\infty}f_{1})^{\ast}\mathbf{P}_{j})_{c}=((j^{\infty}f_{2})^{\ast}\mathbf{P}_{j})_{c}$ and $((j^{\infty}f_{1})^{\ast
}(\mathbf{d}_{j+1}\circ d(j^{\infty}f_{1})))_{c}=((j^{\infty}f_{2})^{\ast
}(\mathbf{d}_{j+1}\circ d(j^{\infty}f_{2})))_{c}$ for each number $j$ and any $c\in S$.
Let $\eta:S\rightarrow\lbrack0,1]$ be any smooth function. Let $\varepsilon:S\rightarrow\mathbf{R}$ be a sufficiently small positive smooth function. Let $\mathbf{f}^{\eta}(\mathbf{v}_{c})$ denote $\exp_{P,f_{1}(c)}((1-\eta(c))\exp_{P,f_{1}(c)}^{-1}(f_{1}(\mathbf{v}_{c}))+\eta
(c)\exp_{P,f_{2}(c)}^{-1}(f_{2}(\mathbf{v}_{c})))$ for any $c\in S$ and any $\mathbf{v}_{c}\in\pi^{-1}(c)$ with $\Vert\mathbf{v}_{c}\Vert\leq
\varepsilon(c)$.
Then the map $\mathbf{f}^{\eta}:D_{\varepsilon}(E)\rightarrow P$ is a well-defined $\Omega$-regular map such that
$(1)$ $\mathbf{f}^{\eta}|S=f_{1}|S=f_{2}|S,$
$(2)$ $S=S^{I}(j^{\infty}\mathbf{f}^{\eta})$,
$(3)$ $T_{c}(S^{I_{j-1}}(j^{\infty}\mathbf{f}^{\eta}))=T_{c}(S^{I_{j-1}}(j^{\infty}f_{1})),$ $((j^{\infty}\mathbf{f}^{\eta})^{\ast}\mathbf{P}_{j})_{c}=((j^{\infty}f_{1})^{\ast}\mathbf{P}_{j})_{c}$ and $((j^{\infty
}\mathbf{f}^{\eta})^{\ast}(\mathbf{d}_{j+1}\circ d(j^{\infty}\mathbf{f}^{\eta
})))_{c}=((j^{\infty}f_{1})^{\ast}(\mathbf{d}_{j+1}\circ d(j^{\infty}f_{1})))_{c}$ for each number $j$ and any $c\in S$,
Let us take a Riemannian metric on $E$ which is compatible with the metric of the vector bundle $E$ over $S$. In particular, $S$ is a Riemannian submanifold of $E$. Furthermore, take a Riemannian metric on $P$ such that $f_{i}(S)\cap P$ is a Riemannian submanifold of $P$ around $f(c)$. Then the local coordinates of $\exp_{N,c}(K_{1,c})$ and $\exp_{P,f_{i}(c)}(Q_{c})$ are independent of the coordinates of $S$, where $Q_{c}$ is regarded as a subspace of $T_{f(c)}P$.
We may consider $\eta(c)$ as a constant when dealing with higher intrinsic derivatives in the lemma by the identification (1.2) and the property of the total tangent bundle $\mathbf{D}$ given in the beginning of Section 2. Then the assertion follows from the assumptions and the properties of $\Sigma^{I_{j}}(N,P)$.
The proof of the following lemma is elementary, and so is left to the reader.
Let $\pi:E\rightarrow S$ be given as above. Let $(\Omega,\Sigma)$ be a pair of a smooth manifold and its submanifold of codimension $c(I)$. Let $\varepsilon:S\rightarrow\mathbf{R}$ be a sufficiently small positive smooth function. Let $h:D_{\varepsilon}(E)\rightarrow(\Omega,\Sigma)$ be a smooth map such that $S=h^{-1}(\Sigma)$ and that $h$ is transverse to $\Sigma$. Then there exists a smooth homotopy $h_{\lambda}:(D_{\varepsilon}(E),S)\rightarrow
(\Omega,\Sigma)$ between $h$ and $\exp_{\Omega}\circ dh|D_{\varepsilon}(E)$ such that
$(1)$ $h_{\lambda}|S=h_{0}|S$, $S=h_{\lambda}^{-1}(\Sigma)=h_{0}^{-1}(\Sigma)$ for any $\lambda$,
$(2)$ $h_{\lambda}$ is smooth and is transverse to $\Sigma$ for any $\lambda$,
$(3)$ $h_{0}=h$ and $h_{1}(\mathbf{v}_{c})=\exp_{\Omega,h(c)}\circ
dh(\mathbf{v}_{c})$ for $c\in S$ and $\mathbf{v}_{c}\in D_{\varepsilon}(E_{c}).$
Proof of Theorem 4.1
====================
Consider the bundles $\frak{n}(I)$ ($=\frak{n}(s,I)$) and $Q$ $(=Q(S^{I}(s))$. For a point $c\in S^{I}(s)$, take an open neighborhood $U$ around $c$ such that $\frak{n}(I)|_{U}$ and $Q|_{U}$ are the trivial bundles, say $U\times\mathbf{R}^{c(I)}$ and $U\times\mathbf{R}^{p-n+i_{1}}$ respectively, where $\mathbf{R}^{c(I)}$ has coordinates $(x_{1},\ldots,x_{c(I)})$ and $\mathbf{R}^{p-n+i_{1}}$ has $(y_{1},\ldots,y_{p-n+i_{1}})$. Then we identify an element of $\mathrm{Hom}(\bigcirc^{j}\frak{n}(I),Q)|_{U}$ with polynomials $y_{i}(c)=\Sigma_{|\omega|=j}a_{i}^{\omega}(c)x_{1}^{\omega_{1}}x_{2}^{\omega_{2}}\cdots x_{c(I)}^{\omega_{c(I)}}$, $c\in U$, where $\omega
=(\omega_{1},\omega_{2},\cdots,\omega_{c(I)})$, $\omega_{\ell}\geq0$ ($i=1,\cdots,p-n+i_{1}$), and $|\omega|=\omega_{1}+\cdots+\omega_{c(I)}$ and $a_{i}^{\omega}(c)$ are real numbers. If $a_{i}^{\omega}(c)$ are smooth functions of $c,$ then $\{a_{i}^{\omega}(c)\}$ defines a smooth section of Hom$(\bigcirc^{j}\frak{n}(I),Q)|_{S^{I}}$ over $U$.
We first introduce several homomorphisms between vector bundles over $S^{I}(s)$, which are used for the construction of the required $\Omega
$-regular map in Theorem 4.1.
Let $s\in\Gamma_{\Omega}^{tr}(N,P)$. By deforming $s$ if necessary, we may assume without loss of generality that $s$ satisfies (2) of Lemma 5.1 and (2) of Lemma 5.2, where $s_{1}$ is replaced by $s$.
In the following, let $K_{j}$ ($j\geq1$) refer to $K_{j}(S^{I}(s))$. Let $K_{j}/K_{j+1}$ refer to the orthogonal complement of $K_{j+1}$ in $K_{j}$, $T_{j}^{I}$ refer to the orthogonal complement of $K_{j}/K_{j+1}$ in $\frak{n}(s,I_{j}\subset I_{j-1})$, and $P_{j}^{I}=(s|S^{I})^{\ast}\mathbf{P}_{j}$. Then we have the following isomorphism by (2.2)$$\begin{aligned}
(s|S^{I})^{\ast}(\mathbf{d}_{j+1}\circ ds|\frak{n}(s,I_{j} & \subset
I_{j-1})):\\
\frak{n}(s,I_{j} & \subset I_{j-1})=K_{j}/K_{j+1}\oplus T_{j}^{I}\rightarrow
P_{j}^{I}\text{ }(1\leq j\leq k).\nonumber\end{aligned}$$
We first define the section $\frak{q}^{\prime}(s,I)^{1}$ of $\mathrm{Hom}(\frak{n}(I),\mathrm{\operatorname{Im}}(d_{1}(s,I)))$ over $S^{I}(s)$ defined by $\frak{q}^{\prime}(s,I)^{1}=d_{1}(s,I)|\frak{n}(I)$, which vanishes on $K_{1}|_{S^{I}}$ and gives an isomorphism of $\oplus_{j=1}^{k}T_{j}^{I}$ onto $\mathrm{\operatorname{Im}}(d_{1}(s,I))$.
For $1\leq j\leq k$, $\mathbf{q}(k)^{j+1,j+1}$ in (3.1) induces the homomorphism$$\frak{q}(s,I)^{j+1}:\frak{n}(s,I_{j}\subset I_{j-1})\bigcirc K_{j}\bigcirc
K_{j-1}\bigcirc\cdots\bigcirc K_{1}\rightarrow Q$$ over $S^{I}(s)$ defined as the composition$$((s|S^{I})^{\ast}\mathbf{q}(k)^{j+1,j+1})\circ(((s|S^{I})^{\ast}ds|\frak{n}(s,I_{j}\subset I_{j-1}))\bigcirc id_{K_{j}\bigcirc K_{j-1}\bigcirc\cdots\bigcirc K_{1}}).$$ Furthermore, we define the following section of Hom$(\Sigma_{j=1}^{k+1}\bigcirc^{j}\frak{n}(I),Q)$$$\begin{array}
[c]{ll}\frak{q}^{\prime}(s,I)=\Sigma_{j=1}^{k}\frak{q}(s,I)^{j+1} & \text{over }S^{I}(s).
\end{array}$$ Let us consider the direct sum decompositions$$\begin{aligned}
\frak{n}(s,I) & =\oplus_{j=1}^{k}\frak{n}(s,I_{j}\subset I_{j-1}),\text{
\ \ }\frak{n}(s,I_{j}\subset I_{j-1})=K_{j}/K_{j+1}\oplus T_{j}^{I},\\
K_{1} & =\oplus_{j=1}^{k-1}K_{j}/K_{j+1}\oplus K_{k},\text{ \ \ }(\pi
_{P}^{\infty}\circ s|S^{I})^{\ast}(TP)=Q\oplus Q^{\bot}$$ and the inclusion $i_{Q}:Q\rightarrow(\pi_{P}^{\infty}\circ s|S^{I})^{\ast
}(TP)$. Then we obtain the smooth fiber map $$\frak{q}(s,I)=(\pi_{P}^{\infty}\circ s|S^{I})^{TP}\circ(i_{Q}\circ
\frak{q}^{\prime}(s,I)+\frak{q}^{\prime}(s,I)^{1}):\frak{n}(s,I)\rightarrow TP$$ covering the immersion $\pi_{P}^{\infty}\circ s|S^{I}(s):S^{I}(s)\rightarrow
P$ such that for any $c\in S^{I}(s)$, $\frak{q}(s,I)_{c}$ is regarded as $p-n+c(I)$ polynomials of $c(I)$ variables with constant $0$.
\[Proof of Theorem 4.1\]By Lemmas 5.1 and 5.2 we may assume that $s$ satisfies (2) of Lemma 5.1 and (2) of Lemma 5.2, where $s_{1}$ is replaced by $s$. We define $E(S^{I})$ to be the union of all $\exp_{N}(D_{\delta\circ s}(\frak{n}(I)))$, where $\delta:\Sigma^{I}(N,P)\rightarrow\mathbf{R}$ is a sufficiently small positive function such that $\delta\circ s|(S^{I}\setminus
$Int$U(C(I^{+})){_{2}})$ is constant. This is a tubular neighborhood of $S^{I}$.
It is enough for the proof of Theorem 4.1 to prove the following assertion:
(**A**) [There exists a homotopy $H_{\lambda}$ relative to $U(C(I^{+}))_{2-r_{0}}$ in $\Gamma_{\Omega}^{tr}(N,P)$ with $H_{0}=s$ satisfying the following. ]{}
[$(1)$ $S{^{I}}(H_{\lambda})=S^{I}$ for]{} [any $\lambda$. ]{}
[$(2)$ We have an ]{}$\Omega$-regular[ map $G$ defined on a neighborhood of ${U(}$[$C(I^{+})$]{}${)}_{2-r_{0}}$ to $P$ such that $j^{\infty}G=H_{1}$ on ]{}${U(}$[$C(I^{+})$]{}${)}_{2-r_{0}}\cup E(S^{I}).$
By the Riemannian metric on $P$, we identify $Q$ with the orthogonal $p-n+i_{1}$ dimensional bundle of Im$(d_{1}(s,I))$ in $(\pi_{P}^{\infty}\circ
s|S^{I})^{\ast}(TP)$. Then the map $\exp_{P}\circ(\pi_{P}^{\infty}\circ
s|S^{I})^{TP}|D_{\gamma}(Q)$ is an immersion for some small positive function $\gamma$. In the proof we express a point of $E(S^{I})$ as $\mathbf{v}_{c}$, where $c\in S^{I},$ $\mathbf{v}_{c}\in\frak{n}(I)_{c}$ and $\Vert
\mathbf{v}_{c}\Vert\leq\delta(s(c))$. In the proof we say that a smooth homotopy $$k_{\lambda}:(E(S^{I}),\partial E(S^{I}))\rightarrow(\Omega(N,P),\Omega
(N,P)\setminus\Sigma^{I}(N,P))$$ has the property (C) if it satisfies that for any $\lambda$
(C-1) $k_{\lambda}^{-1}(\Sigma^{I}(N,P))=S^{I}$, and $\pi_{P}^{\infty}\circ
k_{\lambda}|S^{I}=\pi_{P}^{\infty}\circ k_{0}|S^{I\text{ }}$and,
(C-2) $k_{\lambda}$ is smooth and transverse to $\Sigma^{I}(N,P)$.
If we choose $\delta$ sufficiently small compared with $\gamma$, then we can define the $\Omega$-regular map $g_{0}:E(S^{I})\rightarrow P$ by $$g_{0}(\mathbf{v}_{c})=\exp_{P,\pi_{P}^{\infty}\circ s(c)}\circ\frak{q}(s,I)_{c}\circ\exp_{N,c}^{-1}(\mathbf{v}_{c}).$$ It follows from Section 2 that $g_{0}$ has each point $c\in S^{I}$ as a singularity of the symbol $I$ and vice versa. Now we need to modify $g_{0}$ by using Lemma 5.3 so that $g_{0}$ is compatible with $g(I^{+})$. Let $\eta:S^{I}\rightarrow\mathbf{R}$ be a smooth function such that
\(i) $0\leq\eta(c)\leq1$ for $c\in S^{I},$
\(ii) $\eta(c)=0$ for $c\in S^{I}\cap{U(}$[$C(I^{+})$]{}${)}_{2-3r_{0}}$,
\(iii) $\eta(c)=1$ for $c\in S^{I}\setminus{U(}$[$C(I^{+})$]{}${)}_{2-4r_{0}}$.
Then consider the map $G:{U(}$[$C(I^{+})$]{}${)}_{2-3r_{0}}\cup
E(S^{I})\rightarrow P$ defined by $$\left\{
\begin{array}
[c]{ll}G(x)=g(I^{+})(x) & \text{if $x\in{U(}${$C(I^{+})$}${)}_{2-3r_{0}}$},\\
G(\mathbf{v}_{c})=(1-\eta(c))g(I^{+})(\mathbf{v}_{c})+\eta(c)g_{0}(\mathbf{v}_{c}) & \text{if }\mathbf{v}_{c}\text{$\in E(S^{I})$}.
\end{array}
\right.$$ It follows from Lemma 5.3 that $G$ is an $\Omega$-regular map defined ${U(}$[$C(I^{+})$]{}${)}_{2-3r_{0}}\cup E(S^{I})$, that $G|E(S^{I})$ has the singularities of the symbol $I$ exactly on $S^{I}$, and that for any $c\in
S^{I}$, the assumptions (i)-(iv) of Lemma 5.3 holds for $f_{1}=g(I^{+})$ and $f_{2}=g_{0}$.
Set $\exp_{\Omega}=\exp_{\Omega(N,P)}$ for short. Let $h_{1}^{1}$ and $h_{0}^{3}$ be the maps $(E(S^{I}),S^{I})\rightarrow(\Omega(N,P),\Sigma
^{I}(N,P))$ defined by $$\begin{aligned}
h_{1}^{1}(\mathbf{v}_{c}) & =\exp_{\Omega,s(c)}\circ d_{c}s\circ(\exp
_{N,c})^{-1}(\mathbf{v}_{c}),\\
h_{0}^{3}(\mathbf{v}_{c}) & =\exp_{\Omega,j^{\infty}G(c)}\circ
d_{c}(j^{\infty}G)\circ(\exp_{N,c})^{-1}(\mathbf{v}_{c}).\end{aligned}$$ By applying Lemma 5.4 to the section $s$ and $h_{1}^{1}$, we first obtain a homotopy $h_{\lambda}^{1}\in\Gamma_{\Omega}^{tr}(E(S^{I}),P)$ between $h_{0}^{1}=s$ and $h_{1}^{1}$ on $E(S^{I})$ satisfying the properties (1), (2) and (3) of Lemma 5.4. Similarly we obtain a homotopy $h_{\lambda}^{3}\in
\Gamma_{\Omega}^{tr}(E(S^{I}),P)$ between $h_{0}^{3}$ and $h_{1}^{3}=j^{\infty}G$ on $E(S^{I})$ satisfying the properties (1), (2) and (3) of Lemma 5.4.
Next we construct a homotopy of bundle maps $\frak{n}(I)\rightarrow\nu
(\Sigma^{I})$ covering a homotopy $S^{I}\rightarrow\Sigma^{I}(N,P)$ between $ds|\frak{n}(I)$ and $d(j^{\infty}G)|\frak{n}(I)$. Let us recall the additive structure of $J^{\infty}(N,P)$ in (1.2). Then we have the homotopy $\kappa_{\lambda}:S^{I}\rightarrow J^{\infty}(N,P)$ defined by $$\kappa_{\lambda}(c)=(1-\lambda)s(c)+\lambda j^{\infty}G(c)\quad\text{covering
}\pi_{P}^{\infty}\circ s|S^{I}:S^{I}\rightarrow P,$$ where $\pi_{P}^{\infty}\circ s|S^{I}$ is the immersion as in $(2)$ of Lemma 5.2. We show that $\kappa_{\lambda}$ is actually a homotopy of $S^{I}$ into $\Sigma^{I}(N,P)$. Under the identification $(s)^{\ast}\mathbf{P}\cong(\pi
_{P}^{\infty}\circ s)^{\ast}(TP)$ and $s^{\ast}\mathbf{D}\cong TN$, it follows from the decomposition of $\frak{n}(I)$ in (4.3) that$$(s|S^{I})^{\ast}(\mathbf{d}_{j+1}\circ ds|\frak{n}(I_{j}\subset I_{j-1}))=(j^{\infty}G|S^{I})^{\ast}(\mathbf{d}_{j+1}\circ d(j^{\infty}G)|\frak{n}(I_{j}\subset I_{j-1}))$$ over $S^{I}$. These formulas are the direct consequence of the construction of $\frak{q}(s,I)$ used in the definition of $G$ and the definition of the intrinsic derivatives in Sections 2 and 3. By (6.6) we have that $\frak{n}(\kappa_{\lambda},I)_{c}=\frak{n}(I)_{c}$ and $Q(\kappa_{\lambda
})_{c}=Q_{c}$ for any $c\in S^{I}$. Hence, it follows that the equalities of the homomorphisms in (6.6) also hold when $s$ is replaced by $\kappa_{\lambda
}$ $(0\leq\lambda\leq1)$. This implies that $\kappa_{\lambda}$ is a homotopy into $\Sigma^{I}(N,P)$. Hence, the homotopy $(\kappa_{\lambda})^{\nu
(\Sigma^{I})}:\kappa_{\lambda}^{\ast}(\nu(\Sigma^{I}))\rightarrow\nu
(\Sigma^{I})$, $ds$ and $d(j^{\infty}G)$ induce the homotopy of bundle maps $\widetilde{\kappa_{\lambda}}:\frak{n}(I)\rightarrow\nu(\Sigma^{I})$ covering $\kappa_{\lambda}$ such that $\widetilde{\kappa_{0}}=ds$ and $\widetilde
{\kappa_{1}}=d(j^{\infty}G)$. We define the homotopy $h_{\lambda}^{2}:(E(S^{I}),S^{I})\rightarrow(\Omega(N,P),\Sigma^{I}(N,P))$ by$$h_{\lambda}^{2}(\mathbf{v}_{c})=\exp_{\Omega,s(c)}\circ\widetilde
{\kappa_{\lambda}}\circ(\exp_{N,c})^{-1}(\mathbf{v}_{c}).$$ Then we have that $h_{0}^{2}(\mathbf{v}_{c})=h_{1}^{1}(\mathbf{v}_{c})=\exp_{\Omega,s(c)}\circ d_{c}s\circ(\exp_{N,c})^{-1}(\mathbf{v}_{c})$ and $h_{0}^{3}(\mathbf{v}_{c})=h_{1}^{2}(\mathbf{v}_{c})=\exp_{\Omega,j^{\infty
}G(c)}\circ d_{c}(j^{\infty}G)\circ(\exp_{N,c})^{-1}(\mathbf{v}_{c})$ on $E(S^{I})$.
Since $h_{0}^{1}(\mathbf{v}_{c})=h_{1}^{3}(\mathbf{v}_{c})=s(\mathbf{v}_{c})$ for $\mathbf{v}_{c}\in E(S^{I})\cup{U(}$[$C(I^{+})$]{}${)}_{2-3r_{0}}$, we may assume in the construction of $h_{\lambda}^{1}$, $h_{\lambda}^{2}$ and $h_{\lambda}^{3}$ that if $\mathbf{v}_{c}\in E(S^{I})\cup{U(}$[$C(I^{+})$]{}${)}_{2-3r_{0}}$, then $h_{\lambda}^{2}(\mathbf{v}_{c})=h_{0}^{2}(\mathbf{v}_{c})=h_{1}^{2}(\mathbf{v}_{c})$ and $h_{\lambda}^{1}(\mathbf{v}_{c})=h_{1-\lambda}^{3}(\mathbf{v}_{c})$ for any $\lambda$.
Let $\overline{h}_{\lambda}\in\Gamma_{\Omega}^{tr}(E(S^{I})\cup{U(}$[$C(I^{+})$]{}${)}_{2-3r_{0}},P)$ be the homotopy which is obtained by pasting $h_{\lambda}^{1}$, $h_{\lambda}^{2}$ and $h_{\lambda}^{3}$. The homotopies $h_{\lambda}^{1}$ and $h_{\lambda}^{3}$ are not homotopies relative to $E(S^{I})\cap{U(}$[$C(I^{+})$]{}${)}_{2-3r_{0}}$ in general. By using the above properties of $h_{\lambda}^{1}$, $h_{\lambda}^{2}$ and $h_{\lambda}^{3}$, we can modify $\overline{h}_{\lambda}$ to a homotopy $h_{\lambda}\in
\Gamma_{\Omega}^{tr}(E(S^{I}),P)$ satisfying the property (C) such that
\(1) $h_{\lambda}(\mathbf{v}_{c})=h_{0}(\mathbf{v}_{c})=s(\mathbf{v}_{c})$ for any $\lambda$ and any $\mathbf{v}_{c}\in E(S^{I})\cap{U(}$[$C(I^{+})$]{}${)}_{2-2r_{0}}$,
\(2) $h_{0}(\mathbf{v}_{c})=s(\mathbf{v}_{c})$ for any $\mathbf{v}_{c}\in
E(S^{I}),$
\(3) $h_{1}(\mathbf{v}_{c})=j^{\infty}G(\mathbf{v}_{c})$ for any $\mathbf{v}_{c}\in E(S^{I})$.
By (1), we can extend $h_{\lambda}$ to the homotopy $H_{\lambda}^{\prime}\in\Gamma_{\Omega}^{tr}(E(S^{I})\cup{U(}$[$C(I^{+})$]{}${)}_{2-2r_{0}},P)$ defined by $H_{\lambda}^{\prime}|E(S^{I})=h_{\lambda}$ and $H_{\lambda
}^{\prime}|{U(}$[$C(I^{+})$]{}${)}_{2-2r_{0}}=s|{U(}$[$C(I^{+})$]{}${)}_{2-2r_{0}}$.
By the transversality of $H_{\lambda}^{\prime}$ and the homotopy extension property to $s$ and $H_{\lambda}^{\prime}$, we obtain an extended homotopy $$H_{\lambda}:(N,S^{I})\rightarrow(\Omega(N,P),\Sigma^{I}(N,P))$$ relative to ${U(}$[$C(I^{+})$]{}${)}_{2-r_{0}}$ with $H_{0}=s$. Furthermore, we replace $\delta$ and $E(S^{I})$ by smaller ones. Then $H_{\lambda}$ is a required homotopy in $\Gamma_{\Omega}^{tr}(N,P)$ in the assertion (**A**).
Proof of Theorem 0.2
====================
In this section we prove Theorem 0.2 by applying Theorem 0.1.
Under the same assumption of Theorem 0.2, any section $s\in\Gamma
_{\Omega^{I_{r}}}^{tr}(N,P)$ has a homotopy $s_{\lambda}\in\Gamma
_{\Omega^{I_{r}}}^{tr}(N,P)$ such that
$(1)$ $s_{0}=s,$
$(2)$ $s_{1}$ is a section of $\Omega^{J}(N,P)$ over $N$,
$(3)$ $S^{I_{r}}(s_{\lambda})=S^{I_{r}}(s)=S^{I_{r}}(s_{1})$ for any $\lambda$.
We need the following lemma for the proof of Proposition 7.1.
Assume the same assumption of Theorem 0.2. Then, we have
$(1)$ $\mathbf{Q}_{1}\mathbf{|}_{\Sigma^{I_{r}}(N,P)}$, and $\bigcirc
^{2}\mathbf{K}_{r}|_{\Sigma^{I_{r}}(N,P)}$ are trivial line bundles equipped with the canonical orientations respectively,
$(2)$ The homomorphisms $\mathbf{c}_{j}|\mathrm{Hom}(\bigcirc^{j}\mathbf{K}_{r},\mathbf{Q}_{1}):\mathrm{Hom}(\bigcirc^{j}\mathbf{K}_{r},\mathbf{Q}_{1})\rightarrow\mathbf{P}_{j}$ $(1\leq j\leq r)$ and $\mathbf{e}_{j-1}\circ\mathbf{c}_{j-1}:\mathrm{Hom}(\bigcirc^{j-1}\mathbf{K}_{r},\mathbf{Q}_{1})\rightarrow\mathbf{Q}_{j}$ $(1<j\leq r)$ are injective over $\Sigma^{I_{r}}(N,P)$.
\(1) By Section 2 (5), $\mathbf{d}_{2}|\mathbf{K}_{1}:\mathbf{K}_{1}\rightarrow\mathrm{Hom}(\mathbf{K}_{1},\mathbf{Q}_{1})$ induces the isomorphism$$\mathbf{K}_{1}\mathbf{/K}_{2}\rightarrow\text{Hom}(\mathbf{K}_{1}\mathbf{/K}_{2},\mathbf{Q}_{1})\text{ \ \ \ over }\Sigma^{I_{r}}(N,P).$$ This yields $\mathbf{q}:\mathbf{K}_{1}/\mathbf{K}_{2}\bigcirc\mathbf{K}_{1}/\mathbf{K}_{2}\rightarrow\mathbf{Q}_{1}$ over $\Sigma^{I_{r}}(N,P)$, which is a nonsingular quadratic form on each fiber. Since dim$\mathbf{K}_{1}\mathbf{/K}_{2}=n-p+1-i_{2}$ is odd, we choose the unique orientation of $\mathbf{Q}_{1}$, expressed by the unit vector $\mathbf{e}_{p}$, so that the index (the number of the negative eigen values) of $\mathbf{q}_{z}$, $z\in\Sigma^{I_{r}}(N,P)$ is less than $(n-p+1-i_{2})/2$.
Since $\mathbf{K}_{r}|_{\Sigma^{I_{r}}(N,P)}$ is a line bundle, $\bigcirc
^{2}\mathbf{K}_{r}|_{\Sigma^{I_{r}}(N,P)}$ has the canonical orientation.
\(2) We prove the assertion by induction on $j$ ($r\geq3$). Let $j=1$. Since the kernel of $\mathbf{d}_{2}|\mathbf{K}_{1}$ is $\mathbf{K}_{2}$, we have that $\mathbf{c}_{1}=\mathbf{u}_{1}$ induces the inclusion Hom$(\mathbf{K}_{r},\mathbf{Q}_{1})|_{\Sigma^{I_{r}}}\subset\mathrm{Hom}(\mathbf{K}_{1},\mathbf{Q}_{1})|_{\Sigma^{I_{r}}}=\mathbf{P}_{1}|_{\Sigma^{I_{r}}}$ and $\mathbf{e}_{1}|_{\Sigma^{I_{r}}}:\mathbf{P}_{1}|_{\Sigma^{I_{r}}}\rightarrow\mathbf{Q}_{2}|_{\Sigma^{I_{r}}}$ is identified with the restriction Hom$(\mathbf{K}_{1},\mathbf{Q}_{1})|_{\Sigma^{I_{r}}}\rightarrow\mathrm{Hom}(\mathbf{K}_{2},\mathbf{Q}_{1})|_{\Sigma^{I_{r}}}$. Hence, $\mathbf{e}_{1}\circ\mathbf{c}_{1}|($Hom$(\mathbf{K}_{r},\mathbf{Q}_{1})|_{\Sigma^{I_{r}}})$ is injective. Suppose that $\mathbf{e}_{j-2}\circ\mathbf{c}_{j-2}|\mathrm{Hom}(\bigcirc^{j-2}\mathbf{K}_{r},\mathbf{Q}_{1})$ is injective into $\mathbf{Q}_{j-1}$ over $\Sigma^{I_{r}}(N,P)$ for $j-2<r$. Then it follows from the definition of $\mathbf{u}_{j-1}$ that $\mathbf{c}_{j-1}|\mathrm{Hom}(\bigcirc^{j-1}\mathbf{K}_{r},\mathbf{Q}_{1})$ is injective into $\mathrm{Hom}(\mathbf{K}_{j-1},\mathbf{Q}_{j-1})$ over $\Sigma^{I_{r}}(N,P)$. Since the image of $\mathbf{c}_{j-1}$ is $\mathbf{P}_{j-1}$, the map $\mathbf{c}_{j-1}|\mathrm{Hom}(\bigcirc^{j-1}\mathbf{K}_{r},\mathbf{Q}_{1})$ is injective into $\mathbf{P}_{j-1}$. Since $\mathbf{d}_{j}|\mathbf{K}_{r}$ vanishes for $j\leq r$ over $\Sigma^{I_{r}}(N,P)$ and since $\mathbf{d}_{j}|\mathbf{K}_{r}$ is symmetric, the composition $\mathbf{e}_{j-1}\circ\mathbf{c}_{j-1}|\mathrm{Hom}(\bigcirc
^{j-1}\mathbf{K}_{r},\mathbf{Q}_{1})$ is injective into $\mathbf{Q}_{j}$ over $\Sigma^{I_{r}}(N,P)$ for $j\leq r$. Thus the map $\mathbf{c}_{j}|\mathrm{Hom}(\bigcirc^{j}\mathbf{K}_{r},\mathbf{Q}_{1})$ is injective into $\mathbf{P}_{j}$ for $j\leq r$. This proves the lemma.
\[Proof of Proposition 7.1\]In the proof we identify $J^{k}(N,P)$ with $J^{k}(TN,TP)$ by (1.2). By (9) in Section 2, there exists the open subbundles $\widetilde{\Omega}^{L}(N,P)$ of $J^{k}(N,P)$ such that $(\pi_{k}^{\infty
})^{-1}(\widetilde{\Omega}^{L}(N,P))=\Omega^{L}(N,P)$ for $L$ with length $k$. It follows that $(\pi_{r}^{\infty}\circ s)(N\setminus(S^{I_{r}}(s)))\subset
\widetilde{\Omega}^{I_{r-1},0}(N\setminus(S^{I_{r}}(s)),P)$.
We now construct a new section $\widetilde{u}:N\rightarrow\widetilde{\Omega
}^{J}(N,P)$ as follows.
Let $\mathbf{e}_{p}(Q)_{c}$ and $\mathbf{e}(\bigcirc^{r+1}(K_{r})_{c})$ be the oriented vectors induced from $\mathbf{e}_{p}(\mathbf{Q}_{s(c)})$, $\mathbf{e}(\bigcirc^{r+1}\mathbf{K}_{r,s(c)})$ by $s$ respectively. Then we define the section $\phi^{J}:S^{I_{r}}(s)\rightarrow\mathrm{Hom}(\bigcirc^{r+1}K_{r},Q)$ by $\phi^{J}(c)(\mathbf{e}(\bigcirc^{r+1}(K_{r})_{s(c)}))=\mathbf{e}_{p}(Q_{s(c)})$. Then we can extend $\phi^{J}$ to a section $u_{\phi}:S^{I_{r}}(s)\rightarrow\mathrm{Hom}(S^{r+1}((\pi
_{N}^{\infty}\circ s)^{\ast}(TN)),(\pi_{P}^{\infty}\circ s)^{\ast}(TP))$ such that $u_{\phi}(c)|\bigcirc^{r+1}K_{r,c}=\phi^{J}(c)$ for $c\in S^{I_{r}}(s)$. Since $S^{I_{r}}(s)$ is a closed submanifold and since Hom$(S^{r+1}((\pi
_{N}^{\infty}\circ s)^{\ast}(TN)),(\pi_{P}^{\infty}\circ s)^{\ast}(TP))$ is a vector bundle, we extend $u_{\phi}$ arbitrarily to the section $\widetilde
{u_{\phi}}:N\rightarrow\mathrm{Hom}(S^{r+1}((\pi_{N}^{\infty}\circ s)^{\ast
}(TN)),(\pi_{P}^{\infty}\circ s)^{\ast}(TP))$. Then we define $\widetilde{u}$ by $\widetilde{u}=\pi_{r}^{\infty}\circ s\oplus\widetilde{u_{\phi}}$ as the section of $J^{r+1}(N,P)=$ $J^{r+1}(TN,TP)$. We lift $\widetilde{u}$ to the section $s^{J}$ of $J^{\infty}(N,P)$ over $N$. Then we have that $\pi
_{r+1}^{\infty}\circ s^{J}=\widetilde{u}$ and $\pi_{r}^{\infty}\circ s^{J}=\pi_{r}^{\infty}\circ s$. Furthermore, we define the homotopy $s_{\lambda
}\in\Gamma_{\Omega^{I}}(N,P)$ by$$s_{\lambda}=(1-\lambda)s+\lambda s^{J}.$$ It follows from $\pi_{r}^{\infty}\circ s_{\lambda}=\pi_{r}^{\infty}\circ
s=\pi_{r}^{\infty}\circ s^{J}$ that $s_{\lambda}$ is transverse to $\Sigma^{I_{r}}(N,P)$ and $S^{I_{r}}(s_{\lambda})=S^{I_{r}}(s^{J})=S^{I_{r}}(s)$.
We prove that $s^{J}\in\Omega^{J}(N,P)$. For any point $c\in S^{I_{r}}(s)$, let $U_{c}$ be a convex neighborhood of $c$ and let $k$ and $y_{p}\ $be the coordinates of $\exp_{N,c}((K_{r})_{c})$ and $\exp_{P,\pi_{P}^{\infty}\circ
s^{J}(c)}((\pi_{P}^{\infty}\circ s^{J})^{TP}(Q)_{c})$ respectively. Let $D_{k}$ denote the vector of the total tangent bundle $\mathbf{D}$ which corresponds $k$ defined in \[B, definition 1.6\]. It follows from the definition of $\mathbf{D}$ that$$\begin{array}
[c]{ll}(\bigcirc^{r+1}D_{k})y_{p}|_{s^{J}(c)}=\partial^{r+1}y_{p}/\partial
k^{r+1}(c)\neq0 & \text{for }c\in S^{I_{r}}(s).
\end{array}$$ Then it follows from Lemma 7.2 (2) that $$\mathbf{d}_{r+1,s^{J}(c)}|\mathbf{K}_{r,s^{J}(c)}:\mathbf{K}_{r,s^{J}(c)}\rightarrow\mathbf{P}_{r,s^{J}(c)}\supset\text{Hom}(\bigcirc^{r}\mathbf{K}_{r,s^{J}(c)},\mathbf{Q}_{s(c)})$$ is injective. Hence, we have that $s^{J}(S^{I_{r}}(s))\subset\Sigma^{J}(N,P)$. Since $s^{J}(N\setminus(S^{I_{r}}(s))\subset\Omega^{I_{r-1},0}(N,P)$, the assertion is proved. This proves the proposition.
\[Proof of Theorem 0.2\]By the assumption, $j^{\infty}f$ is the section $N\rightarrow\Omega^{I}(N,P)$. By Proposition 7.1, we have the section $s^{J}:N\rightarrow\Omega^{J}(N,P)$ such that $\pi_{P}^{\infty}\circ s$ and $\pi_{P}^{\infty}\circ s^{J}$ are homotopic. By Theorem 0.1 we obtain an $\Omega^{J}$-regular map $g$ such that $j^{\infty}g$ and $s^{J}$ are homotopic. This proves the assertion.
Let $n\geq p\geq2,$ and $N$ and $P$ be as above. Let $I=(n-p+1,1,1,1)$ and $J=(n-p+1,1,1,0)$ such that $n-p$ and $r$ ($r\geq3$) are odd integers. Then if $f:N\rightarrow P$ is an $\Omega^{I}$-regular map with $j^{\infty}f\in
\Gamma_{\Omega^{I}}^{tr}(N,P)$, then $f$ is homotopic to an $\Omega^{J}$-regular map $g:N\rightarrow P$ such that $j^{\infty}f$ and $j^{\infty}g$ are homotopic in $\Gamma_{\Omega^{I}}^{tr}(N,P)$ and that $S^{I}(j^{\infty
}f)=S^{I}(j^{\infty}g)$.
This corollary proves the Chess conjecture (\[C\]). Sadykov\[Sady\] has actually proved this corollary for $J=(n-p+1,1,0)$ in the case of $N$ and $P$ being orientable.
Let $\pi_{0}(X)$ be the arcwise connected components of $X$. Theorem 0.1 asserts that$$(j_{\Omega^{I}})_{\ast}:\pi_{0}(C_{\Omega^{I}}^{\infty}(N,P))\rightarrow
\pi_{0}(\Gamma_{\Omega^{I}}(N,P))$$ is surjective. However, $(j_{\Omega^{I}})_{\ast}$ is not necessarily injective. Let $N=S^{2}$, $P=\mathbf{R}^{2}$ and $I=(1,0)$. Then we have by \[An3\] that $\Omega^{1,0}(2,2)$ is homotopy equivalent to $SO(3)$. it follows from \[Ste, 36.4\] that every two sections of $\Omega^{1,0}(S^{2},\mathbf{R}^{2})$ over $S^{2}$ are mutually homotopic. Namely, $\pi_{0}(\Gamma
_{\Omega^{I}}(N,P))$ consists of a single element. On the other hand, let $f_{\lambda}:S^{2}\rightarrow\mathbf{R}^{2}$ be a homotopy of fold-maps. Define $F:S^{2}\times\lbrack0,1]\rightarrow\mathbf{R}^{2}$ by $F(x,\lambda
)=f_{\lambda}(x)$ so that if $\lambda$ is sufficiently small, then $F(x,\lambda)=f_{0}(x)$ and $F(x,1-\lambda)=f_{1}(x)$. By a very small perturbation of $F$ fixing $f_{0}$ and $f_{1}$, we may assume that $F$ is smooth and $f_{\lambda}$ is still an $\Omega^{1,0}$-regular map for any $\lambda$. Furthermore, the map $F:S^{2}\times\lbrack0,1]\rightarrow
\mathbf{R}^{2}\times\lbrack0,1]$ becomes an $\Omega^{1,0}$-regular map, and $S^{1,0}(F)$ is a submanifold of $S^{2}\times\lbrack0,1]$. By the Jacobian matrix of $F$ we know that the kernel line bundle $K_{1}(j^{\infty}F)$ over $S^{1,0}(F)$ is independent with $\partial/\partial\lambda$, and $T(S^{1,0}(F))\cap K_{1}(j^{\infty}F)=\{\mathbf{0}\}$. This implies that $S^{1,0}(F)$ is regularly projected onto $[0,1]$. Hence, $S^{1,0}(f_{0})$ must be diffeomorphic to $S^{1,0}(f_{1})$. Thus we conclude that $\pi_{0}(C_{\Omega^{I}}^{\infty}(S^{2},\mathbf{R}^{2}))$ is an infinite set.
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[^1]: 2000 *Mathematics Subject Classification.* Primary 58K30; Secondary 57R45, 58A20
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We consider the evolution of a flat, isotropic and homogeneous Friedmann-Robertson-Walker Universe, filled with a causal bulk viscous cosmological fluid, that can be characterized by an ultra-relativistic equation of state and bulk viscosity coefficient obtained from recent lattice QCD calculations. The basic equation for the Hubble parameter is derived under the assumption that the total energy in the Universe is conserved. By assuming a power law dependence of bulk viscosity coefficient, temperature and relaxation time on energy density, an approximate solution of the field equations has been obtained, in which we utilized equations of state from recent lattice QCD simulations QCD and heavy-ion collisions to derive an evolution equation. In this treatment for the viscous cosmology, we found no evidence for singularity. For example, both Hubble parameter and scale factor are finite at $t=0$, $t$ is the comoving time. Furthermore, their time evolution essentially differs from the one associated with non-viscous and ideal gas. Also thermodynamic quantities, like temperature, energy density and bulk pressure remain finite as well. In order to prove that the free parameter in our model does influence the final results, qualitatively, we checked out other particular solutions.'
author:
- |
A. Tawfik$^{1}$, M. Wahba$^1$, H. Mansour$^2$ and T. Harko$^3$\
[$^1$Egyptian Center for Theoretical Physics (ECTP), MTI University, Cairo-Egypt]{}\
[$^2$Department of Physics, Faculty of Science, Cairo University, Giza-Egypt]{}\
[$^3$Department of Physics and Center for Theoretical and Computational Physics,]{}\
[University of Hong Kong, Hong Kong]{}\
title: |
\
Viscous Quark-Gluon Plasma in the Early Universe
---
Introduction {#sec:intro}
============
The dissipative effects, including both bulk and shear viscosity, are supposed to play a very important role in the early evolution of the Universe. The first attempts at creating a theory of relativistic fluids were those of Eckart [@Ec40] and Landau and Lifshitz [@LaLi87]. These theories are now known to be pathological in several respects. Regardless of the choice of equation of state, all equilibrium states in these theories are unstable and in addition signals may be propagated through the fluid at velocities exceeding the speed of light. These problems arise due to the first order nature of the theory, that is, it considers only first-order deviations from the equilibrium leading to parabolic differential equations, hence to infinite speeds of propagation for heat flow and viscosity, in contradiction with the principle of causality. Conventional theory is thus applicable only to phenomena which are quasi-stationary, i.e. slowly varying on space and time scales characterized by mean free path and mean collision time.
A relativistic second-order theory was found by Israel [@Is76] and developed by Israel and Stewart [@IsSt76], Hiscock and Lindblom [HiLi89]{} and Hiscock and Salmonson [@HiSa91] into what is called “transient” or “extended” irreversible thermodynamics. In this model deviations from equilibrium (bulk stress, heat flow and shear stress) are treated as independent dynamical variables, leading to a total of 14 dynamical fluid variables to be determined. For general reviews on causal thermodynamics and its role in relativity see [@Ma95].
Causal bulk viscous thermodynamics has been extensively used for describing the dynamics and evolution of the early Universe or in an astrophysical context. But due to the complicated character of the evolution equations, very few exact cosmological solutions of the gravitational field equations are known in the framework of the full causal theory. For a homogeneous Universe filled with a full causal viscous fluid source obeying the relation $\xi \sim \rho ^{1/2}$, with $\rho $ the energy density of the cosmological fluid, exact general solutions of the field equations have been obtained in [@ChJa97; @MaHa99a; @MaHa99b; @MaHa00a; @MaTr97]. It has also been proposed that causal bulk viscous thermodynamics can model on a phenomenological level matter creation in the early Universe [@ChJa97]. Exact causal viscous cosmologies with $\xi \sim \rho ^{s},s\neq 1/2$ have been considered in Ref. [@MaHa99a].
Because of technical reasons, most investigations of dissipative causal cosmologies have assumed Friedmann-Robertson-Walker (FRW) symmetry (i.e. homogeneity and isotropy) or small perturbations around it [@MaTr97]. The Einstein field equations for homogeneous models with dissipative fluids can be decoupled and therefore are reduced to an autonomous system of first order ordinary differential equations, which can be analyzed qualitatively [@CoHo95].
The role of a transient bulk viscosity in a FRW space-time with decaying vacuum has been discussed in [@AbVi97]. Models with causal bulk viscous cosmological fluid have been considered recently [@ArBe00]. They obtained both power-law and inflationary solutions, with the gravitational constant an increasing function of time. The dynamics of a viscous cosmological fluids in the generalized Randall-Sundrum model for an isotropic brane were considered in [@Chen01]. The renormalization group method was applied to the study of homogeneous and flat FRW Universes, filled with a causal bulk viscous cosmological fluid, in [@Be03]. A generalization of the Chaplygin gas model, by assuming the presence of a bulk viscous type dissipative term in the effective thermodynamic pressure of the gas, was investigated recently in [@Pun08].
Recent RHIC results give a strong indication that in the heavy-ion collisions experiments, a hot dense matter can be formed [@reff1]. Such an experimental evidence might agree with the “new state of matter” as predicted in the Lattice QCD simulations [@reff5]. However, the experimentally observed elliptic flow in peripheral heavy-ion collisions seems to indicate that a thermalized collective QCD matter has been produced. In a addition to that, the success of ideal fluid dynamics in explaining several experimental data e.g. transverse momentum spectra of identified particles, elliptic flow [@reff6], together with the string theory motivated that the shear viscosity $\eta$ to the entropy $s$ would have the lower limit $\approx 1/4\pi$ [@reff7] leading to a paradigm that in heavy- ion collisions, that a [*nearly*]{} perfect fluid likely be created and the quarks and gluons likely go through relatively rapid equilibrium characterized with a thermalization time less than $1$ fm/c [@mueller1].
According to recent lattice QCD simulations [@mueller2], the bulk viscosity $
\xi $ is not negligible near the QCD critical temperature $T_c$. It has been shown that the bulk and shear viscosity at high temperature $T$ and weak coupling $\alpha_s
$, $\xi\sim \alpha_s^2 T^3/\ln \alpha_s^{-1}$ and $\eta\sim T^3/(\alpha_s^2 \ln
\alpha_s^{-1})$ [@mueller3]. Such a behavior obviously reflects the fact that near $T_c$ QCD is far from being conformal. But at high $T$, QCD approaches conformal invariance, which can be indicated by low trace anomaly $(\epsilon-3p)/T^4$ [@karsh09], where $\epsilon$ and $p$ are energy and pressure density, respectively. In the quenched lattice QCD, the ratio $\zeta/s$ seems to diverge near $T_c$ [@meyer08].
To avoid the mathematical difficulties accompanied with the Abel second type non-homogeneous and non-linear differential equations [@TawCosmos], one used to model the cosmological fluid as an ideal (non-viscous) fluid. No doubt that the viscous treatment of the cosmological background should have many essential consequences [@taw08]. The thermodynamical ones, for instance, can profoundly modify the dynamics and configurations of the whole cosmological background [@conseq1]. The reason is obvious. The bulk viscosity is to be expressed as a function of the Universe energy density $\rho$ [@conseq2]. Much progress has been achieved in relativistic thermodynamics of dissipative fluids. The pioneering theories of Eckart [@Ec40] and Landau and Lifshitz [@LaLi87] suffer from lake of causality constrains. The currently used theory is the Israel and Stewart theory [@Is76; @IsSt76], in which the causality is conserved and theory itself seems to be stable [@HiLi89; @Ma95].
In this article, we aim to investigate the effects that bulk viscosity has on the Early Universe. We consider a background corresponding to a FRW model filled with ultra-relativistic viscous matter, whose bulk viscosity and equation of state have been deduced from recent heavy-ion collisions experiments and lattice QCD simulations.
The present paper is organized as follows. The basic equations of the model are written down in Section \[field\]. In Section \[approx\] we present an approximate solution of the evolution equation. Section \[part1\] is devoted to one particular solution, in which we assume that $H=const.$ The results and conclusions are given in Sections \[final\] and \[final2\], respectively.
Evolution equations {#field}
===================
We assume that geometry of the early Universe is filled with a bulk viscous cosmological fluid, which can be described by a spatially flat FRW type metric given by $$\label{1}
ds^{2}=dt^{2}-a^{2}\left( t\right) \left[ dr^{2}+r^{2}\left( d\theta
^{2}+\sin ^{2}\theta d\phi ^{2}\right) \right] .$$ The Einstein gravitational field equations are: $$R_{ik}-\frac{1}{2}g_{ik}R=8\pi GT_{ik}. \label{ein}$$ In rest of this article, we take into consideration natural units, i.e., $c=1$, for instance.
The energy-momentum tensor of the bulk viscous cosmological fluid filling the very early Universe is given by $$T_{i}^{k}=\left( \rho +p+\Pi\right) u_{i}u^{k}-\left( p+\Pi\right)
\delta_{i}^{k},\label{1_a}$$ where $i,k$ takes $0,1,2,3$, $\rho$ is the mass density, $p$ the thermodynamic pressure, $\Pi $ the bulk viscous pressure and $u_{i}$ the four velocity satisfying the condition $u_{i}u^{i}=1$. The particle and entropy fluxes are defined according to $N^{i}=nu^{i}$ and $S^{i}=sN^{i}-\left( \tau\Pi^{2}/2\xi T\right)
u^{i}$, where $n$ is the number density, $s$ the specific entropy, $T\geq0$ the temperature, $\xi$ the bulk viscosity coefficient, and $\tau\geq0$ the relaxation coefficient for transient bulk viscous effect (i.e. the relaxation time), respectively.
The evolution of the cosmological fluid is subject to the dynamical laws of particle number conservation $N_{\text{ };i}^{i}=0$ and Gibbs’ equation $Td\rho=d\left( \rho /n\right) +pd\left( 1/n\right) $. In the following we shall also suppose that the energy-momentum tensor of the cosmological fluid is conserved, that is $T_{i;k}^{k}=0$.
The bulk viscous effects can be generally described by means of an effective pressure $\Pi $, formally included in the effective thermodynamic pressure $p_{eff}=p+\Pi $ [@Ma95]. Then in the comoving frame the energy momentum tensor has the components $T_{0}^{0}=\rho ,T_{1}^{1}=T_{2}^{2}=T_{3}^{3}=-p_{eff}$. For the line element given by Eq. (\[1\]), the Einstein field equations read $$\begin{aligned}
\label{2}
\left( \frac{\dot{a}}{a}\right)^{2} &=& \frac{8\pi}{3}G \;\rho, \\
\frac{\ddot{a}}{a} &=& -\frac{4\pi}{3}G \; \left( 3p_{eff}+\rho \right),
\label{3}\end{aligned}$$ where one dot denotes derivative with respect to the time $t$, $G$ is the gravitational constant and $a$ is the scale factor.
Assuming that the total matter content of the Universe is conserved, $T_{i;j}^j=0$, the energy density of the cosmic matter fulfills the conservation law: $$\label{5}
\dot{\rho}+3H\left( p_{eff}+\rho \right) =0,$$ where we introduced the Hubble parameter $H=\dot{a}/a$. In presence of bulk viscous stress $\Pi $, the effective thermodynamic pressure term becomes $p_{eff}=p+\Pi $. Then Eq. (\[5\]) can be written as $$\label{6}
\dot{\rho}+3H\left( p+\rho \right) =-3\Pi H.$$
For the evolution of the bulk viscous pressure we adopt the causal evolution equation [@Ma95], obtained in the simplest way (linear in $\Pi)$ to satisfy the $H$-theorem (i.e., for the entropy production to be non-negative, $S_{;i}^{i}=\Pi^{2}/\xi T\geq0$ [@Is76; @IsSt76]). According to the causal relativistic Israel-Stewart theory, the evolution equation of the bulk viscous pressure reads [@Ma95] $$\label{8}
\tau \dot{\Pi}+\Pi =-3\xi H-\frac{1}{2}\tau \Pi \left( 3H+\frac{\dot{\tau}}{\tau }-\frac{\dot{\xi}}{\xi }-\frac{\dot{T}}{T}\right).$$ In order to have a closed system from equations (\[2\]), (\[6\]) and (\[8\]) we have to add the equations of state for $p$ and $T$.
As shown in Appendix A, the equation of state, the temperature and the bulk viscosity of the quark-gluon plasma (QGP), can be determined approximately at high temperatures [@karsch07] from recent lattice QCD calculations [@Cheng:2007jq], as $$\label{13}
P = \omega \rho,\hspace*{1cm}T = \beta \rho^r,\hspace*{1cm}\xi = \alpha \rho + \frac{9}{\omega_0} T_c^4,$$ with $\omega = (\gamma-1)$, $\gamma \simeq 1.183$, $r\simeq 0.213$, $\beta\simeq 0.718$, $$\alpha = \frac{1}{9\omega_0} \frac{9\gamma^2-24\gamma+16}{\gamma-1},$$ and $\omega_0 \simeq 0.5-1.5$ GeV. In the following we assume that $\alpha \rho >> 9/\omega_0 T_c^4$, and therefore we take $\xi \simeq \alpha \rho$. In order to close the system of the cosmological equations, we have also to give the expression of the relaxation time $\tau $, for which we adopt the expression [@Ma95], $$\label{tau}
\tau=\xi\rho^{-1}\simeq\alpha .$$
Eqs. (\[13\]) are standard in the study of the viscous cosmological models, whereas the equation for $\tau$ is a simple procedure to ensure that the speed of viscous pulses does not exceed the speed of light. Eq. (\[tau\]) implies that the relaxation time in our treatment is constant but strongly depends on EoS. These equations are without sufficient thermodynamical motivation, but in the absence of better alternatives, we shall follow the practice of adopting them in the hope that they will at least provide some indication of the range of bulk viscous effects. The temperature law is the simplest law guaranteeing positive heat capacity.
With the use of Eqs. (\[8\]), (\[13\]) and (\[tau\]), respectively, we obtain the following equation describing the cosmological evolution of the Hubble function $H$ $$\begin{aligned}
\label{init}
\ddot H + \frac{3}{2} [1+(1-r) \gamma] H\dot H + \frac{1}{\alpha}\dot H - (1+r) H^{-1} \dot H^2 + \frac{9}{4}(\gamma -2) H^3 +
\frac{3}{2}\frac{\gamma}{\alpha} H^2 &=& 0.\end{aligned}$$
An approximate solution {#approx}
=======================
We introduce the transformation $u=\dot{H}$, so that Eq. (\[init\]) is transformed into a first order ordinary differential equation, $$\label{init2}
u\frac{du}{dH}-(1+r)H^{-1}u^{2}+\left(\frac{3}{2}[1+(1-r)\gamma ]H+\alpha
^{-1}\right) u+\frac{9}{4}\frac{1}{(\gamma)}H^{3}+\frac{3}{2}\frac{\gamma}{\alpha}
H^{2}=0.$$ We can rewrite Eq. (\[init2\]) in the form $$\label{OmegH1}
\Omega \frac{d\Omega }{dH} = F_1(H)\Omega + F_0(H),$$ where $$\begin{aligned}
\Omega &=& u \; E \;\; = u\; \exp\left(-\int \frac{1+r}{H} dH\right), \nonumber \\
F_1(H) &=& -\left( \frac{3}{2} [1+(1-r)\gamma]H + \frac{1}{\alpha}\right)E, \nonumber \\
F_0(H) &=& -\left(\frac{9}{4}(\gamma-2) H^3 + \frac{3}{2} \frac{\gamma}{\alpha} H^2\right)E^2. \nonumber\end{aligned}$$ By introducing a new independent variable $z=\int F_1(H)\,dH$, we obtain $$\Omega \frac{d\Omega}{dz} - \Omega = g(z),$$ with $g(z)$ is defined parametrically as, $$\label{gofzz1}
g(z) = \frac{F_0}{F_1}.$$ As shown in Appendix B, $g(z)$ can be approximated as a simple function of $z$ $$g(z)\approx {\cal C}\; z,$$ where ${\cal C}$ is a constant. We proceed with this approximation to get solvable differential equations. Keeping the parametric solution of $g(z)$, Eq. (\[fullgofz\]), results in much more complicated differential equations. This would be the subject of a future work.
From the definitions of $\Omega$ and $z$ we have $$\begin{aligned}
\Omega &=& H^{1+r}\dot H, \label{Eq1} \\
z &=& H^{2+r} \left(\frac{-3[1+(1-r)\gamma]H}{2(1-r)} +\frac
{1}{\alpha r}\right), \label{Eq2}\end{aligned}$$ Analogous to the solution of reduced Abel type canonical equation, $$\label{abel1}
y \frac{dy}{dx} - y = a x$$ (see Appendix C) we obtain the relation $\Omega = z/{\cal P}$. Therefore, from Eqs. (\[Eq1\]) and (\[Eq2\]) we obtain the following first order differential equation for Hubble parameter $H$, $$\label{init-polyn}
{\cal P} \dot H = \frac{-3[1+(1-r)\gamma]}{2(1-r)} H^2 +\frac{1}{\alpha r}H$$ with the solution $$H(t) = \frac{B}{\exp(-Bt/{\cal P})-A} \label{eq:mysolut1}$$ where $$\label{paramsab}
A=\frac{-3[1+(1-r)\gamma]}{2(1-r)}, \hspace*{1cm} B=\frac{1}{\alpha r},$$ and ${\cal P}$ is taken as a free parameter. We can assign any real value to ${\cal P}$. For the results presented in this work, we used a negative value. This negative sign is necessarily to overcome the sign from the integral limits. The geometric and thermodynamic quantities of the Universe read $$\begin{aligned}
a(t) &=& a_0\left(\frac{\exp(-B t/{\cal P})}{\exp(-B
t/{\cal P})-A}\right)^{{\cal P}/A} \label{approx-a}, \\
\rho(t) &=& 3\, H^2= 3 \left(\frac{B}{\exp(-Bt/{\cal P})-A}\right)^2 \label{approx-rho},\\
T(t) &=& \beta \rho^{r}=\beta \left(3 \frac{B^{2}}{[\exp(-Bt/{\cal P})-A]^{2}}\right)^r \label{approx-T},\\
\Pi(t) &=& -2\dot{H}-3\gamma H^{2}=-\frac{B^2}{{\cal P}} \left(\frac{2\exp(-Bt/{\cal P})+3\gamma{\cal P}}{[\exp(-Bt/{\cal P})-A]^2}\right) \label{approx-Pi},\\
q(t)&=&\frac{d}{dt}H^{-1}-1=-\frac{1}{{\cal P}}\exp(-Bt/{\cal P})-1. \label{approx-q}\end{aligned}$$ $a_0$ is an arbitrary constant of the integration. The sign of $q$ indicates whether the Universe decelerates (positive) or accelerates (negative). $q$ can also be given as a function of the thermodynamic, gravitational and cosmological quantities $q(t)=[\rho(t)
+3p(t)+3\Pi(t)]/2\rho(t)$ [@kolbBook].
de Sitter Universe {#part1}
===================
Besides the approximation in $g(z)$, previous solution apparently depends on the free parameter ${\cal P}$. In this section, we suggest a particular solution to overcome ${\cal P}$. Eq. (\[init\]) can easily be obtained by assuming that $H$ doesn’t depend one $t$, i.e, de Sitter Universe. With a simple calculation, we get an estimation for $H$ $$\label{partcH}
H=\frac{4}{9}\frac{\alpha ^{-1}\gamma}{2-\gamma }.$$ The geometric and thermodynamic parameters of the Universe are given by $$\begin{aligned}
a(t) &=& a_{0}\exp \left[ \frac{4\alpha ^{-1}\gamma }{9(2-\gamma )}t\right], \label{partca} \\
\rho(t) &=& 3\left[ \frac{4\alpha ^{-1}\gamma }{9(2-\gamma )}\right] ^{2}, \label{partcrho}\\
T(t) &=& 3^{r}\beta \left[ \frac{4\alpha ^{-1}\gamma }{9(2-\gamma )}\right]
^{2r}, \label{partcT}\\
\Pi(t) &=& -3\gamma \left[ \frac{4\alpha
^{-1}\gamma }{9(2-\gamma )}\right] ^{2}, \label{partcPi}\\
q(t) &=& -1. \label{partcq}\end{aligned}$$ Although we have assumed here that the cosmic background is filled with viscous matter, the assumption that $H=const$ results in an exponential scale parameter, Eq. (\[partca\]). This behavior characterizes the de Sitter space, when $\Lambda=k=0$. $\rho$ and $T$ are finite at small $t$ as given in Fig. \[Figg2\].
Particular Solution {#part2}
===================
Another particular solution for Eq. (\[init\]) can be obtained, when assuming that the dependence of $u$ on $H$ can be given by the polynomial in Eq. (\[init-polyn\]) $$\label{init-partc2}
u=b_{1}H^{2}+b_{2}H,$$ where $b_{1}$ and $b_{2}$ are constants. Some simple calculations show that this form is a solution of the initial equation, Eq. (\[init2\]), if $$\begin{aligned}
b_1 &=&-\frac{3}{2}\frac{1+\gamma }{1-r}, \\
b_2 &=& \frac{1}{r\alpha }.\end{aligned}$$ $b_2$ is identical to $B$ in Eq. (\[paramsab\]). $r$ and $\gamma$ have to satisfy the compatibility relation $$r=\frac{2-\gamma}{2+\gamma^2}.$$ Integrating Eq. (\[init-partc2\]) results in $$\begin{aligned}
\label{Eq:Ht}
H(t) &=& \frac{b_2\exp(-b_2 t)}{1-b_1\exp(-b_2 t)},\end{aligned}$$ where minus sign in the exponential function refers to flipping the integral limits. This was not necessary while deriving the expressions given in Section \[approx\]. The free parameter [P]{} compensates it. The geometric and thermodynamic quantities of the Universe read $$\begin{aligned}
a(t)&=&a_0\left(\frac{\exp(b_2t)-b_1}{\exp(b_2t)}\right)^{1/b_1}, \label{partc2a} \\
\rho(t)&=& 3 \left(\frac{b_2\exp(-b_2 t)}{1-b_1\exp(-b_2 t)}\right)^2, \label{partc2rho}\\
T(t)&=& 3^r\;\beta \left(\frac{b_2\exp(-b_2 t)}{1-b_1\exp(-b_2 t)}\right)^{2r}, \label{partc2T}\\
\Pi(t)&=& \frac{b_2^2 \left[2\exp(b_2t)-3\gamma\right]}{\left[\exp(b_2t)-b_1\right]^2}, \label{partc2Pi}\\
q(t) &=& \exp(b_2t)-1. \label{partc2q}\end{aligned}$$ Obviously , we notice that the scale parameter in Eq. (\[partc2a\]) looks like Eq. (\[approx-a\]), which strongly depends on the free parameter ${\cal
P}$. The other geometric and thermodynamic quantities find similarities in Eq. (\[approx-rho\]) - (\[approx-q\]), respectively. Deceleration parameter $q$ seems to be positive everywhere.
Results {#final}
=======
In present work, we have considered the evolution of a full causal bulk viscous flat, isotropic and homogeneous Universe with bulk viscosity parameters and equation of state taken from recent lattice QCD data and heavy-ion collisions. Three classes of solutions of the evolution equation have been obtained.
In Fig. \[Figg1\], $H(t)$ and $a(t)$ are depicted in dependence on the comoving time $t$. We compare $H(t)$, given by Eq. (\[eq:mysolut1\]), and $a(t)$, given by Eq. (\[approx-a\]), with the counterpart parameters obtained in the case when the background matter is assumed to be an ideal and non-viscous fluid, described by the equations of state of the non-interacting ideal gas, $$\begin{aligned}
H(t) &=& \frac{1}{2t} \label{htideal1}, \\
a(t) &=& \sqrt{t}. \label{atideal1}\end{aligned}$$
In the left panel of Fig. \[Figg1\], $H(t)=\dot a/a$ has an exponential decay, whereas in the non-viscous case, $H(t)$ is decreasing according to Eq. (\[htideal1\]). The latter is much slower than the former, reflecting the nature of the exponential and linear dependencies. The other difference between the two cases is obvious at small $t$. We notice a divergence, or singularity, associated with the ideal non-viscous fluid, Eq. (\[htideal1\]). The viscous fluid results in finite $H$ even at vanishing $t$, as can be seen from Eq. (\[eq:mysolut1\]).
The scale factor $a(t)$ also shows differences in both cases. $a(t)$ in a Universe with an ideal and non-viscous background matter depends on $t$ according to Eq. (\[atideal1\]), which simply implies that $a(t)$ is directly proportional to $t$, and $a(t)$ vanishes at $t=0$, which shows the existence of a singularity of $H$. Assuming that the background matter is described by a viscous fluid results in different $a(t)$-behaviors with increasing $t$. At $t=0$, $a(t)$ remains finite. Correspondingly, $H(t)$ remains also finite. In general, the dependence on $t$ is much more complicated than in Eq. (\[atideal1\]). Here we have an $A/{\cal P}$ root of an exponential function. If $\exp(-Bt/{\cal P})>>A$, $a$ remains constant.
Fig. \[Figg2\] illustrates the dependence of the two thermodynamical quantities, $\rho $ and $T$, on the comoving time. The non-viscous Universe shows a singular behavior in $\rho$ at vanishing $t$, as shown in the left panel of Fig. \[Figg2\]. This is not obvious in the case where we have taken into consideration a finite viscosity coefficient, i.e., $\rho$ is finite at $t=0$. In both cases, $\rho$ is decreasing with increasing $t$, reflecting that the Early Universe was likely expanding. Also the life time of the thermal viscous Universe seems to be shorter than for the non-viscous Universe. Almost the same behavior is observed in the right panel of Fig. \[Figg2\]. The temperature $T$ seems to be finite at vanishing $t$ in the viscous Universe. The $T$-singularity is only present, if we assume that the background matter is non-viscous ideal gas.
In left panel of Fig. \[Figg3\], we show the dependence of the bulk viscous pressure $\Pi$ on $t$. $\Pi$ takes negative values at very small $t$. Then it switches to positive values at some values of $t$. After reaching the maximum value, $\Pi$ decays exponentially with increasing $t$. At larger $t$, $\Pi$ entirely vanishes. The deceleration parameter $q$, given by Eq. (\[approx-q\]), is depicted in the right panel of Fig. \[Figg3\], and it is compared with $q$ for a non-viscous fluid, $q=-3$. The approximate solution, given by Eq. (\[approx-q\]), results in negative $q$ at small $t$, referring to expansion era. $q$ from the particular solution, Eq. (\[partcq\]) is negative everywhere.\
For the particular solution, only the scale factor depends on $t$, Eq. (\[partca\]). The results are given in the right panel of Fig. \[Figg1\]. All cosmological and thermodynamical quantities given by Eq. (\[partcH\]) and Eqs. (\[partcrho\])-(\[partcq\]) are constant in time.
Conclusions {#final2}
===========
It is obvious that the bulk viscosity plays an important role in the evolution of the Early Universe. Despite of the simplicity of our model, it shows that a better understanding of the dynamics of our Universe is only accessible, if we use reliable equation of state to characterize the matter filling out the cosmic background.
We conclude that the causal bulk viscous Universe described by the approximate solution starts its evolution from an initial non-singular state with a non-zero initial value of Hubble parameter $H(t)$ and scale factor $a(t)$, where $t$ is the comoving time. In this treatment, $t$ is given in GeV$^{-1}$. Also the thermodynamical quantities, energy density $\rho$ for instance, are finite at vanishing $t$. Even the temperature $T$ itself shows no singularity at $t=0$. The Hubble parameter $H$ decreases monotonically with $T$ similar to $\rho$. The bulk viscous pressure $\Pi$ likely satisfies the condition that $\Pi<0$ at very small $t$ indicating to inflationary era. Then $\Pi $ switches to positive value. It reaches a maximum value and then decays and vanishes, exponentially, at large $t$. The deceleration parameter $q$ shows an expanding behavior in the case of non-viscous ideal gas and first particular solution. For second particular solution, $q$ starts from zero and increases, exponentially. According to this solution, the Universe was decelerating. The approximate solution shows an interesting behavior in $q(t)$, Eq. (\[approx-q\]). At small $t$, the values of $q$ are negative, i.e. the Universe was accelerating (expansion). At larger $t$, a non-inflationary behavior sets on, $q>0$, i.e., the Universe switched to a decelerating evolution.
In this treatment, we assumed that the Universe is flat, $k=0$, and the background geometry is filled out with QCD matter (QGP) with a finite viscosity coefficient. The resulting Universe is obviously characterized by a shortly increasing and afterward constant scale factor and a fast vanishing Hubble parameter. At $t=0$, both $a(t)$ and $H(t)$ remain finite, i.e., there is no singularity. The validity of our treatment depends on the validity of the equations of states, Eq. \[13\], which we have deduced from the lattice QCD simulations at temperatures larger than $T_c\approx 0.19~$GeV. Below $T_c$, as the Universe cooled down, not only the degrees of freedom suddenly increase [@Tawfik03] but also the equations of state turn to be the ones characterizing the hadronic matter. Such a phase transition - from QGP to hadronic matter - would characterize one end of the validity of our treatment. The other limitation is the very high temperatures (energies), at which the strong coupling $\alpha_s$ entirely vanishes.
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Appendix A: Viscosity coefficient $\xi (T)$ from LQCD {#App:C .unnumbered}
=====================================================
Following the discussion presented in [[@Cheng:2007jq]]{}, the bulk viscosity of QGP can be calculated from the lattice QCD by Eq. (13) in that paper. We assume that the decay factors for pions and kaons are vanishing above the critical temperature of the phase transition QGP-hadrons. The quark-antiquark condensates can be neglected at temperatures higher than the critical one [@TawDom]. Therefore, Eq. (22) of Ref. [[@Cheng:2007jq]]{} would be reduced to $$\label{ze}
9\,\omega_0\,\xi = T\, s\, \left(\frac{1}{c_s^2}-3\right)-4(\rho-3p) +16|\epsilon_v|$$ where $\rho$ is the energy density and $c_s^2=dp/d\rho $ is the square of the speed of sound. The parameter $\omega_o$ is a scale depending on the temperature $T$, and defines the validity of the underlying perturbation theory. In this relation, the viscosity is assumed to have a thermal part which can be determined through lattice calculations, and a vacuum contributing part, which can be fixed using quark and gluon condensates. The vacuum part would take the value $$16 |\epsilon_v| (1 + \frac{3}{8} \cdot 1.6) \simeq (560\ {\rm MeV})^4
\simeq (3 \,T_c)^4 \,$$ Our algorithm is the following. Using lattice QCD results on trace anomaly, $(\epsilon-3p)/T^4$, and other thermodynamical quantities, we can determine the bulk viscosity. To make use of the lattice QCD results, it is useful to make a suitable fit to the data at high temperatures. Then we obtain the following equations of state $$\begin{aligned}
\label{EoS}
p &=&\omega \rho, \hspace*{2cm} T =\beta \rho^r, \hspace*{2cm} c_s^2 = \omega \nonumber\end{aligned}$$ where $\omega=0.319$, $\beta=0.718\pm 0.054$ and $r=0.23\pm 0.196$. Using the equations of state, Eq. (\[EoS\]) in Eq. (\[ze\]), we obtain $$\label{zeta}
\xi(\epsilon)=\frac{1}{9\omega_o}\frac{9\gamma^2-24\gamma+16}{\gamma-1}\rho+\frac{9}{\omega_o}T_c^4.$$
Appendix B: Estimations of $g(z)$ {#App:A .unnumbered}
=================================
For analytical purposes, the function $g(z)$, which is defined in $z$ parameter as $g(z)=F_0/F_1$ in Eq. (\[gofzz1\]), can be numerically estimated depending on the parameter $z$ by using the following procedure. First, we plot it parametrically depending on the parameter $H$, Fig. (\[Figg4\]). Then we fit the resulting curve to various functions. Based on least-square fit, best choice would be a mixture of polynomial and exponential functions, $$\label{fullgofz}
g(z)= a + b\, z + c \frac{\exp(d\, z)+e}{\left[\exp(d\, z)+f\right]^2},$$ where the coefficients read $a=-2.078\pm0.117$, $b=0.091\pm0.007$ and $c=17.332\pm1.553$, $d=0.189\pm0.003$, $e=-0.814\pm0.162$ and $f=2.849\pm0.02$. At small values of $z$, it is clear that the dependence is linear, $$\label{lineargofz2}
g(z) = c + {\cal C} z.$$ Obviously, the intersect $c$ is much smaller than the slope ${\cal C}$. The sign of $g(z)$ can be flipped regarding to the sign of its independent variable $z$. Accordingly, we get $$\label{lineargofz}
g(z)\approx {\cal C} z.$$
To prove this dependence, algebraically, we try to estimate $g(z)$ directly from the division of $F_0$ by $F_1$, which can be approximated by including their first terms only, i.e. $$\label{eq.A1}
g(H)\approx \frac{3(\gamma-2)}{2[1+(1-r)\gamma]}\; H^{1-r},$$ Then, we approximate $z(H)$ to the form, $$\label{eq.A2}
z(H)\approx-\frac{3[1+(1-r)\gamma]}{2(1-r)}\; H^{1-r}.$$ Finally, we now able to derive an approximate estimation for $g(z)$. According to Eq. (\[eq.A1\]) and (\[eq.A2\]), we get $$g(z)\approx \frac{(1-r)(\gamma-2)}{[1+(1-r)\gamma]^2}\; z$$ Amazingly, this expression looks the same as the one we obtained from the numerical approximation with $${\cal C} = \frac{(1-r)(\gamma-2)}{\left[1+(1-r)\gamma\right]^2}.$$
Appendix C: Solution of Abel equation $y\dot y -y = ax$ {#App:B .unnumbered}
=======================================================
To solve Eq. (\[abel1\]) we divide the whole equation by $y^3$ and introduce a new variable $v=1/y$. Then Eq. (\[abel1\]) reads $$\frac{dv}{dx}+v^{2}+axv^{3}=0.$$ We then introduce the function $v=w/x$. $$x\frac{dw}{dx}=w-w^{2}-aw^{3}, \label{abel2}$$ Previous differential equation can be solved by separation of variables $$\int \frac{dw}{w-w^{2}-aw^{3}}=\ln C^{-1}x,$$ where $C$ is an arbitrary constant of integration. To calculate the integral, we write the function to be integrated as $$\frac{1}{w-w^{2}-aw^{3}}=\frac{1}{w}-\frac{aw}{aw^{2}+w-1}-\frac{1}{aw^{2}+w-1}.$$ Let us assume that $\Delta =1+4a>0$ (this implies that $a>0$). $$\int \frac{dw}{w-w^{2}-aw^{3}}=-\frac{1}{2\sqrt{\Delta }}\ln \frac{2aw-\sqrt{\Delta }+1}{2aw+\sqrt{\Delta }+1}-\frac{1}{2}\ln \left( aw^{2}+w-1\right)
+\ln w.$$
Therefore the general solution of Eq. (\[abel2\]) can be written as $$x=C\frac{w}{\sqrt{aw^{2}+w-1}}\left( \frac{2aw+\sqrt{\Delta }+1}{2aw-\sqrt{\Delta }+1}\right) ^{1/2\sqrt{\Delta }},$$ leading to $$y=\frac{1}{v}=\frac{x}{w}=C\frac{1}{\sqrt{aw^{2}+w-1}}\left( \frac{2aw+\sqrt{\Delta }+1}{2aw-\sqrt{\Delta }+1}\right) ^{1/2\sqrt{\Delta }}.$$
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Vrushali A. Bokil [^1]'
- 'Yingda Cheng [^2]'
- 'Yan Jiang [^3]'
- 'Fengyan Li [^4]'
bibliography:
- 'bibfile.bib'
- 'bib\_Li.bib'
- 'Bokil.bib'
- 'cheng.bib'
- 'cheng\_papers.bib'
- 'bokil\_papers.bib'
- 'mimetic.bib'
title: 'Energy Stable Discontinuous Galerkin Methods for Maxwell’s Equations in Nonlinear Optical Media'
---
Maxwell’s equations, nonlinear dispersion, discontinuous Galerkin method, energy stability, error estimates.
Energy relation for the fully discrete schemes with non-periodic boundary conditions in Section \[sec:num2\]
============================================================================================================
Here, we list the energy relation for the fully discrete schemes with boundary conditions as discussed in Section \[sec:num2\].
The results with fully implicit time discretizations are very similar to the semi-discrete case, i.e. we have that the fully implicit scheme with alternating and central fluxes satisfies $$\mathcal{E}_h^{n+1}- \mathcal{E}_h^{n}=-\frac{\Delta t}{4 \omega_p^2 \tau} \int_\Omega (J_h^{n+1} +J_h^{n})^2 dx-\frac{a\theta \Delta t}{8\omega_v^2 \tau_v} \int_\Omega (\sigma_h^{n+1} +\sigma_h^{n})^2 dx -\Delta t\Theta_{in}^{n} -\Delta t \Theta^{n}_{out}\le -\Delta t\Theta^{n}_{in},$$ and that with the upwind flux satisfies $$\begin{aligned}
\mathcal{E}_h^{n+1}- \mathcal{E}_h^{n} &=&
-\frac{\Delta t}{4 \omega_p^2 \tau} \int_\Omega (J_h^{n+1} +J_h^{n})^2 dx-\frac{a\theta \Delta t}{8\omega_v^2 \tau_v} \int_\Omega (\sigma_h^{n+1}+\sigma_h^{n})^2 dx
-\frac{\Delta t}{8\sqrt{\epsilon_\infty}} \sum_{j=1}^{N-1}[H_h^{n}+H_h^{n+1}]_{j+1/2}^2\\
&&-\frac{\Delta t\sqrt{ \epsilon_\infty}}{8}\sum_{j=1}^{N-1}[E_h^n+E_h^{n+1}]_{j+1/2}^2 -\Delta t\Theta_{in}^{n} -\Delta t \Theta^{n}_{out}\le -\Delta t\Theta^{n}_{in},\notag\end{aligned}$$ where $$\begin{aligned}
\mathcal{E}_h^n&=&\int_{\Omega} \frac{1}{2} (H_h^{n})^2 + \frac{ \epsilon_\infty}{2} (E_h^n)^2 + \frac{1}{2\omega_p^2} (J_h^n)^2 + \frac{\omega_0^2}{2 \omega_p^2} (P_h^n)^2+ \frac{a\theta}{4\omega_v^2} (\sigma_h^n)^2 + \frac{a\theta}{2} Q_h^n (E_h^n)^2 \notag\\
&& + \frac{3 a (1-\theta)}{4} (E_h^n)^4+\frac{a\theta}{4}(Q_h^n)^2 dx.\notag\end{aligned}$$ $$\begin{aligned}
0 \le\Theta^{n}_{out}&=&\left\{\begin{array}{ll}
\frac{1}{16\sqrt{\epsilon_{\infty}}} ((H^{n+1}_{h}+H^{n}_{h})^{-}_{N+1/2}
-\sqrt{\epsilon_{\infty}}(E^{n+1}_{h}+E^{n}_{h})^{-}_{N+1/2})^2, & \mbox{ for central and alternating fluxes},\\
\frac{1}{8\sqrt{\epsilon_{\infty}}}((H^{n+1}_{h}+H^{n}_{h})^{-}_{N+1/2})^2
+\frac{\sqrt{\epsilon_{\infty}}}{8}((E^{n+1}_{h}+E^{n}_{h})^{-}_{N+1/2})^2, & \mbox{ for upwind flux},\\
\end{array}
\right.\end{aligned}$$
$$\Theta^{n}_{in}=\left\{\begin{array}{ll}
\frac{1}{8} \left(E(0,t^{n+1})+E(0,t^{n})\right) \left(H^{n+1}_{h}+H^{n}_{h}\right)^{+}_{1/2}
+\frac{1}{8} \left(H(0,t^{n+1})+H(0,t^{n})\right)\\ \left(E^{n+1}_{h}+E^{n}_{h}\right)^{+}_{1/2},
& \mbox{ for central flux},\\
\frac{1}{4} \left(H(0,t^{n+1})+H(0,t^{n})\right) \left(E^{n+1}_{h}+E^{n}_{h}\right)^{+}_{1/2}, & \mbox{ for alternating flux I},\\
\frac{1}{4} \left(E(0,t^{n+1})+E(0,t^{n})\right) \left(H^{n+1}_{h}+H^{n}_{h}\right)^{+}_{1/2}, & \mbox{ for alternating flux II},\\
\frac{1}{8} \left(E(0,t^{n+1})+E(0,t^{n})\right) \left(H^{n+1}_{h}+H^{n}_{h}\right)^{+}_{1/2}
+\frac{1}{8} \left(H(0,t^{n+1})+H(0,t^{n})\right)\\ \left(E^{n+1}_{h}+E^{n}_{h}\right)^{+}_{1/2}
+\frac{1}{8\sqrt{\epsilon_\infty}}(H^{n}_{h}+H^{n+1}_{h})^{+}_{1/2}[H^{n}_{h}+H^{n+1}_{h}]_{1/2}\\
+\frac{\sqrt{\epsilon_\infty}}{8}(E^{n+1}_{h}+E^{n}_{h})^{+}_{1/2}[E^{n+1}_{h}+E^{n}_{h}]_{1/2},
& \mbox{ for upwind flux}.\\
\end{array}
\right.$$
Moreover, for the upwind flux, we have $$\begin{aligned}
\Theta_{in}=& \frac{1}{16\sqrt{\epsilon_\infty}}[H^{n+1}_{h}+H^{n}_{h}]^{2}_{1/2} +\frac{\sqrt{\epsilon_\infty}}{16}[E^{n+1}_{h}+E^{n}_{h}]^{2}_{1/2}\\
& +\frac{1}{16\sqrt{\epsilon_\infty}}\left( (H^{n+1}_{h}+H^{n}_{h})^{+}_{1/2} +\sqrt{\epsilon_\infty} \left( E(0,t^{n+1})+E(0,t^{n}) \right) \right)^2\\ &+\frac{1}{16\sqrt{\epsilon_\infty}}\left( \left( H(t^{n+1},0)+H(t^{n},0) \right) +\sqrt{\epsilon_\infty} \left( E^{n+1}_{h}+E^{n}_{h} \right) \right)^2\\
& -\frac{1}{8\sqrt{\epsilon_\infty}} \left( H(t^{n+1},0)+H(t^{n},0) \right)^2 -\frac{\sqrt{\epsilon_\infty}}{8} \left( E(t^{n+1},0)+E(t^{n},0) \right)^2.\end{aligned}$$ Thus, $$\begin{aligned}
\mathcal{E}_h^{n+1}- \mathcal{E}_h^{n}\leq
\frac{1}{8\sqrt{\epsilon_\infty}}\Delta t \left( H(t^{n+1},0)+H(t^{n},0) \right)^2 +\frac{\sqrt{\epsilon_\infty}}{8} \Delta t \left( E(t^{n+1},0)+E(t^{n},0) \right)^2.\end{aligned}$$ On the other hand, the leap-frog scheme with alternating and central fluxes satisfies $$\mathcal{E}_h^{n+1}- \mathcal{E}_h^{n}=-\frac{\Delta t}{4 \omega_p^2 \tau} \int_\Omega (J_h^{n+1} +J_h^{n})^2 dx-\frac{a\theta \Delta t}{8\omega_v^2 \tau_v} \int_\Omega (\sigma_h^{n+1} +\sigma_h^{n})^2 dx -\Delta t\Theta^{n}_{in} -\Delta t \Theta^{n}_{out}
\le -\Delta t\Theta^{n}_{in} -\Delta t \Theta^{n}_{out},$$ where $$\begin{aligned}
\mathcal{E}_h^n&=&\int_{\Omega} \frac{1}{2} H_h^{n+1/2} H_h^{n-1/2} + \frac{\epsilon_\infty}{2} (E_h^n)^2 + \frac{1}{2\omega_p^2} (J_h^n)^2 + \frac{\omega_0^2}{2 \omega_p^2} (P_h^n)^2 + \frac{a\theta}{4\omega_v^2} (\sigma_h^n)^2 + \frac{a\theta}{2} Q_h^n (E_h^n)^2, \notag \\
&& + \frac{3a (1-\theta)}{4} (E_h^n)^4+\frac{a\theta}{4}(Q_h^n)^2 dx, \notag\\
\Theta^{n}_{out}&=& \frac{\sqrt{\epsilon_\infty}}{16} \left( \left(E^{n}_{h}+E^{n+1}_{h}\right)^{-}_{N+1/2} - \frac{2}{\sqrt{\epsilon_\infty}}\left(H^{n+1/2}_{h}\right)^{-}_{N+1/2}\right)^2 +\frac{1}{16\sqrt{\epsilon_\infty}}\left(H^{n+1/2}_{h}\right)^{-}_{N+1/2}\notag\\ &&\left( H^{n-1/2}_{h}-2H^{n+1/2}_{h}+H^{n+3/2}_{h}\right)^{-}_{N+1/2}, \label{eq:nosign}\end{aligned}$$ $$\begin{aligned}
\Theta^{n}_{in}&=&\left\{\begin{array}{ll}
\frac{1}{4}\left(E(0,t^{n})+E(0,t^{n+1})\right) \left(H^{n+1/2}_{h}\right)^{+}_{1/2} + \frac{1}{4}H(0,t^{n+1/2}) \left(E^{n}_{h}+E^{n+1}_{h}\right)^{+}_{1/2}, & \mbox{ for central flux},\\
\frac{1}{2}H(0,t^{n+1/2}) \left(E^{n}_{h}+E^{n+1}_{h}\right)^{+}_{1/2},& \mbox{ for alternating flux I},\\
\frac{1}{2}\left(E(0,t^{n})+E(0,t^{n+1})\right) \left(H^{n+1/2}_{h}\right)^{+}_{1/2}, & \mbox{ for alternating flux II}.\\
\end{array}
\right.\end{aligned}$$ Unlike the previous cases, cannot be shown as non-negative, which means some energy may be injected at the right boundary in this case.
The leap-frog scheme with the upwind flux satisfies $$\begin{aligned}
\mathcal{E}_h^{n+1}- \mathcal{E}_h^{n} &=&
-\frac{\Delta t}{4 \omega_p^2 \tau} \int_\Omega (J_h^{n+1} +J_h^{n})^2 dx-\frac{a\theta \Delta t}{8\omega_v^2 \tau_v} \int_\Omega (\sigma_h^{n+1}+\sigma_h^{n})^2 dx \\
&& -\frac{\Delta t}{8\sqrt{\epsilon_\infty}} \sum_{j=1}^{N-1}[H_h^{n-1/2}+H_h^{n+1/2}]_{j+1/2}^2
-\frac{\Delta t\sqrt{ \epsilon_\infty}}{8} \sum_{j=1}^{N-1}[E_h^n+E_h^{n+1}]_{j+1/2}^2 \notag\\
&&-\Delta t\Theta_{out}^{n} -\Delta t\Theta_{in}^{n},\notag
$$ where $\Theta_{in}$, $\Theta_{out}$ and the discrete energy $\mathcal{E}_{h}^{n}$ are $$\begin{aligned}
\mathcal{E}_h^n&=&\int_{\Omega} \frac{1}{2} H_h^{n+1/2} H_h^{n-1/2} + \frac{\epsilon_\infty}{2} (E_h^n)^2 + \frac{1}{2\omega_p^2} (J_h^n)^2 + \frac{\omega_0^2}{2 \omega_p^2} (P_h^n)^2+ \frac{a\theta}{4\omega_v^2} (\sigma_h^n)^2 \notag \\
&& + \frac{a\theta}{2} Q_h^n (E_h^n)^2 + \frac{3 a (1-\theta)}{4} (E_h^n)^4+\frac{a\theta}{4}(Q_h^n)^2 dx\notag\\
&&+\frac{\Delta t}{8\sqrt{\epsilon_\infty}}\sum_{j=1}^{N-1} ([H_h^{n-1/2}] [H_h^{n-1/2}+H_h^{n+1/2}])_{j+1/2} \notag \\
&&+\frac{\Delta t}{8\sqrt{\epsilon_\infty}} (H_h^{n-1/2})^{-}_{N+1/2} (H_h^{n-1/2}+H_h^{n+1/2})|^{-}_{N+1/2}, \notag \\
\Theta_{out}^{n}&=& \frac{1}{8\sqrt{\epsilon_\infty}}\left( (H_h^{n-1/2}+H_h^{n+1/2})^{-}_{N+1/2} \right)^2
+\frac{\sqrt{ \epsilon_\infty}}{8} \left( (E_h^n+E_h^{n+1})^{-}_{N+1/2} \right)^2,\notag\\
\Theta_{in}^{n}&=& \frac{1}{4}\left(E(0,t^{n})+E(0,t^{n+1})\right) \left(H^{n+1/2}_{h}\right)^{+}_{1/2} + \frac{1}{4}H(0,t^{n+1/2}) \left(E^{n}_{h}+E^{n+1}_{h}\right)^{+}_{1/2}\notag\\
&&+\frac{1}{8\sqrt{\epsilon_\infty}}\left(H^{n+1/2}_{h}\right)^{+}_{1/2}[H^{n-1/2}_{h}+2H^{n+1/2}_{h}+H^{n+3/2}_{h}]_{1/2}
+\frac{\sqrt{\epsilon_\infty}}{8}(E^{n}_{h}+E^{n+1}_{h})^{+}_{1/2}[E^{n}_{h}+E^{n+1}_{h}]_{1/2}\notag\\
&=&\frac{1}{4\sqrt{\epsilon_\infty}}[H^{n+1/2}_{h}]^{2}_{1/2} +\frac{\sqrt{\epsilon_\infty}}{16}[E^{n+1}_{h}+E^{n}_{h}]^{2}_{1/2} +\frac{1}{4\sqrt{\epsilon_\infty}}\left( (H^{n+1/2}_{h})^{+}_{1/2} +\sqrt{\epsilon_\infty} \frac{E(0,t^{n+1})+E(0,t^{n})}{2} \right)^2\notag\\ &&+\frac{1}{4\sqrt{\epsilon_\infty}}\left( H(t^{n+1/2},0) +\sqrt{\epsilon_\infty} \frac{ E^{n+1}_{h}+E^{n}_{h}}{2} \right)^2
-\frac{1}{2\sqrt{\epsilon_\infty}} \left( H(t^{n+1/2},0) \right)^2 -\frac{\sqrt{\epsilon_\infty}}{8} \left( E(t^{n+1},0)+E(t^{n},0) \right)^2\notag\\
&&+\frac{1}{8\sqrt{\epsilon_\infty}}\left(H^{n+1/2}_{h}\right)^{+}_{1/2}[H^{n-1/2}_{h}-2H^{n+1/2}_{h}+H^{n+3/2}_{h}]_{1/2}.\label{eq:nosign2}\end{aligned}$$ Note that at fully discrete level, we can only prove energy stability for fully implicit scheme with upwind flux.
[^1]: Department of Mathematics, Oregon State University, Corvallis, OR 97331 U.S.A. [bokilv@math.oregonstate.edu]{}.
[^2]: Department of Mathematics, Michigan State University, East Lansing, MI 48824 U.S.A. [ycheng@math.msu.edu]{}. Research is supported by NSF grant DMS-1453661.
[^3]: Department of Mathematics, Michigan State University, East Lansing, MI 48824 U.S.A. [jiangyan@math.msu.edu]{}.
[^4]: Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180 U.S.A. [lif@rpi.edu]{}. Research is supported by NSF grant DMS-1318409.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Recent work has proposed various adversarial losses for training generative adversarial networks. Yet, it remains unclear what certain types of functions are valid adversarial loss functions, and how these loss functions perform against one another. In this paper, we aim to gain a deeper understanding of adversarial losses by decoupling the effects of their component functions and regularization terms. We first derive some necessary and sufficient conditions of the component functions such that the adversarial loss is a divergence-like measure between the data and the model distributions. In order to systematically compare different adversarial losses, we then propose DANTest—a new, simple framework based on discriminative adversarial networks. With this framework, we evaluate an extensive set of adversarial losses by combining different component functions and regularization approaches. This study leads to some new insights into the adversarial losses. For reproducibility, all source code is available at <https://github.com/salu133445/dan>.'
bibliography:
- 'ref.bib'
---
Introduction {#sec:intro}
============
Generative adversarial networks (GANs) [@goodfellow2014] are a class of unsupervised machine learning algorithms. In essence, a GAN learn a generative model with the guidance of another discriminative model which is trained jointly. However, the idea of adversarial losses is not limited to unsupervised learning. Adversarial losses can also be applied to supervised and semi-supervised scenarios (e.g., [@isola2017; @dossantos2017]). Over the past few years, adversarial losses have advanced the state of the art in many fields [@goodfellow2016].
Despite the success, there are several open questions that need to be addressed. On one hand, although plenty adversarial losses have been proposed, we have little theoretical understanding of what makes a loss function a valid one.
On the other hand, we note that any two adversarial losses can differ in terms of not only the *component functions* (e.g., minimax or hinge; see [Section \[sec:background\]]{}) used in the main loss function that sets up the two-player adversarial game, but also the *regularization approaches* (e.g., gradient penalties [@gulrajani2017]) used to regularize the models. However, it remains unclear how they respectively contribute to the performance of an adversarial loss. In other words, when empirically compare two adversarial losses, we need to decouple the effects of the component functions and the regularization terms, otherwise we cannot tell which one of them makes an adversarial loss better than the other.
Among existing comparative analysis of adversarial losses, to the best of our knowledge, only @lucic2018 and @kurach2018 attempted to decouple the effects of the component functions and regularization approaches. But, only few combinations of component functions and regularization approaches were tested in these two prior works, only seven and nine respectively. We attribute this to the high computational cost that may involve to conduct the experiments, and, more importantly, the lack of a framework to systematically evaluate adversarial losses.
$f$ $g$ $h$ $y^*$
--------------------------------- --------------------- ------------------------- ------------------------- ---------------
minimax [@goodfellow2014] $-\log(1 + e^{-y})$ $-y - \log(1 + e^{-y})$ $-y - \log(1 + e^{-y})$ $0$
nonsaturating [@goodfellow2014] $-\log(1 + e^{-y})$ $-y - \log(1 + e^{-y})$ $\log(1 + e^{-y})$ $0$
Wasserstein [@arjovsky2017wgan] $y$ $-y$ $-y$ $0$
least squares [@mao2017] $-(y - 1)^2$ $-y^2$ $(y - 1)^2$ $\frac{1}{2}$
hinge [@lim2017; @tran2017] $\min(0, y - 1)$ $\min(0, -y - 1)$ $-y$ $0$
These two research questions can be summarized as follows:
1. What certain types of component functions are theoretically valid adversarial loss functions?
2. How different combinations of the component functions and the regularization approaches perform empirically against one another?
We aim to tackle these two RQs in this paper to advance our understanding of the adversarial losses. Specifically, our contribution to RQ1 is based on the intuition that a favorable adversarial loss should be a divergence-like measure between the distribution of the real data and the distribution of the model output, since in this way we can use the adversarial loss as the training criterion to learn the model parameters. We derive necessary and sufficient conditions such that an adversarial loss has such a favorable property (Sections \[sec:necessary\_conditions\] and \[sec:sufficient\_conditions\]). Interestingly, our theoretical analysis leads to a new perspective to understand the underlying game dynamics of adversarial losses ([Section \[sec:psi\_function\_analysis\]]{}).
For RQ2, we need an efficient way to compare different adversarial losses. Hence, we adopt the discriminative adversarial networks (DANs) [@mirza2014], which are essentially conditional GANs with both the generator and the discriminator being discriminative models. Based on DANs, we propose *DANTest*—a new, simple framework for comparing adversarial losses ([Section \[sec:dantest\]]{}). The main idea is to first train a number of DANs for a supervised learning task (e.g., classification) using different adversarial losses, and then compare their performance using standard evaluation metrics for supervised learning (e.g., classification accuracy). With the DANTest, we systematically evaluate 168 adversarial losses featuring the combination of ten existing component functions, two new component functions we originally propose in this paper in light of our theoretical analysis, and 14 existing regularization approaches ([Section \[sec:experiments\]]{}). Moreover, we use the DANTest to empirically study the effect of the Lipschitz constant [@arjovsky2017wgan], penalty weights [@mescheder2018], momentum terms [@kingma2014], and others. We discuss the new insights that are gained, and their implications to the design of adversarial losses in future research.
Background {#sec:background}
==========
Generative Adversarial Networks {#sec:gan}
-------------------------------
A generative adversarial network [@goodfellow2014] is a generative latent variable model that aims to learn a mapping from a latent space $\mathcal{Z}$ to the data space $\mathcal{X}$, i.e., a generative model $G$, which we will refer to as the *generator*. A discriminative model $D$ (i.e., the *discriminator*) defined on $\mathcal{X}$ is trained alongside the $G$ to provide guidance for it. Let $p_d$ denote the *data distribution* and $p_g$ be the *model distribution* implicitly defined by $G({\mathbf{z}})$ when ${\mathbf{z}}\sim p_{\mathbf{z}}$. In general, most GAN loss functions proposed in the literature can be formulated as: $$\begin{aligned}
\label{eq:discriminator}
\max_{D}\;&{\mathbb{E}}_{{\mathbf{x}}\sim p_d}[f(D({\mathbf{x}}))] + {\mathbb{E}}_{{\tilde{{\mathbf{x}}}}\sim p_g}[g(D({\tilde{{\mathbf{x}}}}))]\,,\\
\label{eq:generator}
\min_{G}\;&{\mathbb{E}}_{{\tilde{{\mathbf{x}}}}\sim p_g}[h(D({\tilde{{\mathbf{x}}}}))]\,,\end{aligned}$$ where $f$, $g$ and $h$ are real functions defined on the data space (i.e., ${\mathcal{X}}\to {\mathbb{R}}$) and we will refer to them as the *compoenent functions*. We summarize in [ \[tab:loss\_functions\]]{} the component functions $f$, $g$ and $h$ used in some existing adversarial losses.
$p_{{\hat{{\mathbf{x}}}}}$ $R(x)$
--------------------------------------------- ------------------------------ -----------------------------
coupled gradient penalties [@gulrajani2017] $p_d + U[0, 1]\,(p_g - p_d)$ $(x - k)^2$ or $\max(x, k)$
local gradient penalties [@kodali2017] $p_d + c\,N(0, I)$ $(x - k)^2$ or $\max(x, k)$
R~1~ gradient penalties [@mescheder2018] $p_d$ $x$
R~2~ gradient penalties [@mescheder2018] $p_g$ $x$
-------------------------------------------------- ------------------------------------------------ --------------------------------------------- ---------------------------------------------
{width=".135\linewidth"} {width=".135\linewidth"} {width=".135\linewidth"} {width=".135\linewidth"}
\(a) coupled gradient penalties \(b) local gradient penalties \(c) R~1~ gradient penalties \(d) R~2~ gradient penalties
-------------------------------------------------- ------------------------------------------------ --------------------------------------------- ---------------------------------------------
Some prior work has also investigated the so-called IPM-based GANs, where the discriminator is trained to estimate an integral probability metric (IPM) between $p_d$ and $p_g$: $$\begin{aligned}
\label{eq:ipm_distance}
d(p_d, p_g) = -\sup_{D\in\mathcal{D}}\;{\mathbb{E}}_{{\mathbf{x}}\sim p_d}[D({\mathbf{x}})] + {\mathbb{E}}_{{\tilde{{\mathbf{x}}}}\sim p_g}[D({\tilde{{\mathbf{x}}}})]\,,\end{aligned}$$ where $\mathcal{D}$ is a set of functions from ${\mathcal{X}}$ to ${\mathbb{R}}$. For example, the Wasserstein GANs [@arjovsky2017wgan] consider $\mathcal{D}$ to be the set of all 1-Lipschitz functions. Other examples include McGAN [@mroueh2017mcgan], MMD GAN [@li2017] and Fisher GAN [@mroueh2017fishergan]. Please note that the main difference between and is that in the latter we constrain $D$ to be in some set of functions $\mathcal{D}$.
Gradient Penalties {#sec:gradient_penalties}
------------------
As the discriminator is often found to be too strong to provide reliable gradients to the generator, one regularization approach is to use some gradient penalties to constrain the modeling capability of the discriminator. Most gradient penalties proposed in the literature take the following form: $$\label{eq:gradient_penalties}
\lambda\,{\mathbb{E}}_{{\hat{{\mathbf{x}}}}\sim p_{{\hat{{\mathbf{x}}}}}}[R(||\nabla_{{\hat{{\mathbf{x}}}}} D({\hat{{\mathbf{x}}}})||)]\,,$$ where the *penality weight* $\lambda \in {\mathbb{R}}$ is a pre-defined constant, and $R(\cdot)$ is a real function. The distribution $p_{{\hat{{\mathbf{x}}}}}$ defines where the gradient penalties are enforced. [ \[tab:gradient\_penalties\]]{} shows the distribution $p_{{\hat{{\mathbf{x}}}}}$ and function $R$ used in some common gradient penalties. And, [ \[fig:gradient\_penalties\]]{} illustrates $p_{{\hat{{\mathbf{x}}}}}$.
When gradient penalties are enforced, the loss function for training the discriminator contains not only the component functions $f$ and $g$ in but also the *regularization term* .
Spectral Normalization {#sec:spectral_normalization}
----------------------
Another regularization approach we consider is the spectral normalization proposed by @miyato2018. It normalizes the spectral norm of each layer in a neural network to enforce the Lipschitz constraints. While the gradient penalties introduced in [Section \[sec:gradient\_penalties\]]{} impose local regularizations, the spectral normalization imposes a global regularization on the discriminator. Therefore, it is possible to combine the spectral normalization with the gradient penalties. We will examine this in [Section \[sec:exp\_adversarial\_losses\]]{}.
Theoretical Results {#sec:theory}
===================
In the following analysis, we follow the notations in and . Proofs can be found in [Appendix \[app:sec:proofs\]]{}.
Favorable properties for adversarial losses
-------------------------------------------
Let us first consider the minimax formulation: $$\begin{aligned}
\label{eq:minimax}
\min_{G}\;\max_{D}\;&{\mathbb{E}}_{{\mathbf{x}}\sim p_d}[f(D({\mathbf{x}}))] + {\mathbb{E}}_{{\tilde{{\mathbf{x}}}}\sim p_g}[g(D({\tilde{{\mathbf{x}}}}))]\,.\end{aligned}$$ We can see that if the discriminator is able to reach optimality, the training criterion for the generator is $$\begin{aligned}
\label{eq:g_loss}
L_G &= \max_{D}\;{\mathbb{E}}_{{\mathbf{x}}\sim p_d}[f(D({\mathbf{x}}))] + {\mathbb{E}}_{{\tilde{{\mathbf{x}}}}\sim p_g}[g(D({\tilde{{\mathbf{x}}}}))]\,.\end{aligned}$$
In general, for a valid adversarial loss, the discriminator is responsible for providing a measure of the discrepancy between the data distribution $p_d$ and the model distribution $p_g$. In principle, this will then serve as the training criterion for the generator to push $p_g$ towards $p_d$. Hence, we would like such an adversarial loss to be a divergence-like measure between $p_g$ and $p_d$. From this view, we can now define the following two favorable properties of adversarial losses.
[(Weak favorable property)]{.nodecor} For any fixed $p_d$, $L_G$ has a global minimum at $p_g = p_d$. \[prop:weak\]
[(Strong favorable property)]{.nodecor} For any fixed $p_d$, $L_G$ has a unique global minimum at $p_g = p_d$. \[prop:strong\]
We can see that [Property \[prop:strong\]]{} makes $L_G - L^*_G$ a divergence of $p_d$ and $p_g$ for any fixed $p_d$, where $L^*_G = L_G\,\big\rvert_{\,p_g = p_d}$, and [Property \[prop:weak\]]{} provides a weaker version when the identity of indiscernibles is not necessary. Note that $L_G$ is not a divergence since $L_G \geq 0$ does not always hold.
$\Psi$ and $\psi$ functions
---------------------------
In order to derive some necessary and sufficient conditions for Properties \[prop:weak\] and \[prop:strong\], we first observe from that $$\begin{aligned}
&L_G = \max_{D}\;\int_{\mathbf{x}}p_d({\mathbf{x}})\,f(D({\mathbf{x}})) + p_g({\mathbf{x}})\,g(D({\mathbf{x}}))\,d{\mathbf{x}}\\
\begin{split}
&= \int_{\mathbf{x}}(p_d({\mathbf{x}}) + p_g({\mathbf{x}}))\\
&\quad\;\;\, \max_{D} \left(\frac{p_d({\mathbf{x}})\,f(D({\mathbf{x}}))}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})} + \frac{p_g({\mathbf{x}})\,g(D({\mathbf{x}}))}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})}\right)\,d{\mathbf{x}}\,.
\end{split}\end{aligned}$$ Now, if we let $\tilde{\gamma} = \frac{p_d({\mathbf{x}})}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})}$ and $\tilde{y} = D({\mathbf{x}})$, we get $$\label{eq:g_loss_expanded}
\begin{split}
L_G &= \int_{\mathbf{x}}(p_d({\mathbf{x}}) + p_g({\mathbf{x}}))\\
&\quad\qquad\; \max_{\tilde{y}}\;\tilde{\gamma}\,f(\tilde{y}) + (1 - \tilde{\gamma})\,g(\tilde{y})\,d{\mathbf{x}}\,.
\end{split}$$ Please note that $\tilde{\gamma}({\mathbf{x}}) = \frac{1}{2}$ if and only if $p_d({\mathbf{x}}) = p_g({\mathbf{x}})$. Let us now consider the terms inside the integral and define the following two functions: $$\begin{aligned}
\label{eq:big_psi_function}
&\Psi(\gamma, y) = \gamma\,f(y) + (1 - \gamma)\,g(y)\,,\\
\label{eq:small_psi_function}
&\psi(\gamma) = \max_y\;\Psi(\gamma, y)\,,\end{aligned}$$ where $\gamma \in [0, 1]$ and $y \in {\mathbb{R}}$ are two variables independent of ${\mathbf{x}}$. We visualize in Figures \[fig:psi\_functions\](a)–(d) the $\Psi$ and $\psi$ functions for different common adversarial losses (see [Appendix \[app:sec:psi\_function\_graphs\]]{} for the graphs of the $\psi$ functions alone). These two functions actually reflect some important characteristics of the adversarial losses (see [Section \[sec:psi\_function\_analysis\]]{}) and will be used intensively in our theoretical analysis.
Necessary conditions for the favorable properties {#sec:necessary_conditions}
-------------------------------------------------
For the necessary conditions of Properties \[prop:weak\] and \[prop:strong\], we have the following two theorems.
If [Property \[prop:weak\]]{} holds, then for any $\gamma \in [0, 1]$, $\psi(\gamma) + \psi(1 - \gamma) \geq 2\,\psi(\frac{1}{2})$. \[theo:weak\_necessary\_condition\]
If [Property \[prop:strong\]]{} holds, then for any $\gamma \in [0, 1] \setminus \{\frac{1}{2}\}$, $\psi(\gamma) + \psi(1 - \gamma) > 2\,\psi(\frac{1}{2})$. \[theo:strong\_necessary\_condition\]
With Theorems \[theo:weak\_necessary\_condition\] and \[theo:strong\_necessary\_condition\], we can easily check if a pair of component functions $f$ and $g$ form a valid adversarial loss.
Sufficient conditions for the favorable properties {#sec:sufficient_conditions}
--------------------------------------------------
For sufficient conditions, we have two theorems as follows.
If $\psi(\gamma)$ has a global minimum at $\gamma = \frac{1}{2}$, then [Property \[prop:weak\]]{} holds. \[theo:weak\_sufficient\_condition\]
If $\psi(\gamma)$ has a unique global minimum at $\gamma = \frac{1}{2}$, then [Property \[prop:strong\]]{} holds. \[theo:strong\_sufficient\_condition\]
We also have the following theorem for a more specific guideline for choosing the component functions $f$ and $g$.
If $f'' + g'' \leq 0$ and there exists some $y^*$ such that $f(y^*) = g(y^*)$ and $f'(y^*) = -g'(y^*) \neq 0$, then $\psi(\gamma)$ has a unique global minimum at $\gamma = \frac{1}{2}$. \[theo:strong\_sufficient\_condition2\]
By Theorems \[theo:strong\_sufficient\_condition\] and \[theo:strong\_sufficient\_condition2\], we now see that any component function pair $f$ and $g$ that satisfies the prerequisites in [Theorem \[theo:strong\_sufficient\_condition2\]]{} makes $L_G - L^*_G$ a divergence between $p_d$ and $p_g$ for any fixed $p_d$. Interestingly, while such a theoretical analysis has not been done before, it happens that all the adversarial loss functions listed in [ \[tab:loss\_functions\]]{} have such favorable properties. We intend to examine in [Section \[sec:exp\_properties\]]{} empirically the cases when the prerequisites of [Theorem \[theo:strong\_sufficient\_condition2\]]{} do not hold.
In practice, the discriminator often cannot reach optimality at each iteration. Therefore, as discussed by @nowozin2016 [@fedus2018], the objective of the generator is similar to variational divergence minimization (i.e., to minimize a lower bound of some divergence between $p_d$ and $p_g$), where the divergence is estimated by the discriminator.
Loss functions for the generator {#sec:g_loss}
--------------------------------
Intuitively, the generator should minimize the divergence-like measure estimated by the discriminator. We have accordingly $h = g$. However, some prior works have investigated setting $h$ different from $g$. In general, most of these alternative generator losses do not change the solutions of the game and are proposed base on some heuristics. While our theoretical analysis concerns with only $f$ and $g$, we intend to empirically examine the effects of the generator loss function $h$ in [Section \[sec:exp\_g\_loss\]]{}.
Analyzing the adversarial game by the $\Psi$ functions {#sec:psi_function_analysis}
------------------------------------------------------
[ \[fig:psi\_functions\]]{} gives us some new insights regarding the adversarial behaviors of the discriminator and the generator. On one hand, if we follow and consider $\tilde{y} = D({\mathbf{x}})$ and $\tilde{\gamma}({\mathbf{x}}) = \frac{p_d({\mathbf{x}})}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})}$, then the discriminator can be viewed as maximizing $\Psi$ along the $\tilde{y}$-axis. On the other hand, since the generator is trained to push $p_g$ towards $p_d$, it can be viewed as minimizing $\Psi$ along the $\tilde{\gamma}$-axis. In this way, we can see why all these $\Psi$ functions are saddle-shaped and have saddle points at $\gamma = \frac{1}{2}$ (i.e., when $p_d({\mathbf{x}}) = p_g({\mathbf{x}})$).
Ideally, if the discriminator can be trained till optimality, then we will be on the green line, the domain of the $\psi$ function. In this case, the generator can be viewed as minimizing $\Psi$ along the green line (i.e., minizing $\psi$). Note that as $L_G$ is an integral over all possible ${\mathbf{x}}$, such adversarial game is actually being played in a (usually) high dimensional space.
By designing the landscape of $\Psi$, we propose and consider two new losses in our empirical study in [Section \[sec:exp\_adversarial\_losses\]]{}:
- The *absolute* loss, with $f(y) = -h(y) = -|1 - y|$, $g(y) = -|y|$. Its $\Psi$-landscape is similar to those of the least squares and the hinge losses (see [ \[fig:psi\_functions\]]{}(e)).
- The *asymmetric* loss, with $f(y) = -|y|$, $g(y) = h(y) = -y$. Its $\Psi$-landscape is similar to that of the Wasserstein loss, but the positive part of $y$ is blocked (see [ \[fig:psi\_functions\]]{}(f)).
DANTest {#sec:dantest}
=======
Discriminative adversarial networks (DANs) [@dossantos2017] are essentially conditional GANs [@mirza2014] where both the generator and the discriminator are discriminative models, as shown in [ \[fig:dan\]]{}. Based on DANs, we propose a new, simple framework, dubbed *DANTest*, for systematically comparing different adversarial losses. Specifically, the DANTest goes as follows:
1. Build several DANs. For each of them, the generator $G$ takes as input a real sample and outputs a fake label. The discriminator takes as input a real sample with either its true label, or a fake label made by $G$, and outputs a scalar indicating if the “sample–label” pair is real.
2. Train the DANs with different component loss functions, regularization approaches or hyperparameters.
3. Predict the labels of test data by the trained models.
4. Compare the performance of different models with standard evaluation metrics used in supervised learning.
Note that the generator is no longer a generative model in this framework, while the discriminator is still trained by the same loss function to measure the discrepancy between $p_d$ and $p_g$. This way, we can still gain insight into the performance and stability for different adversarial losses. Moreover, although we take a classification task as an example here, the proposed framework is generic and can be applied to other supervised learning tasks as well, as long as the evaluation metrics for that task are well defined.
An extension of the proposed framework is the *imbalanced dataset test*, where we examine the ability of different adversarial losses on datasts that feature class imbalance. This can serve as a measure of the *mode collapse* phenomenon [@che2017mdgan], which is a commonly-encountered failure case in GAN training. By testing on datasets with different levels of imbalance, we can examine how different adversarial losses suffer from the mode collapse problem.
Experiments and Results {#sec:experiments}
=======================
Datasets and Implementation Details {#sec:dataset}
-----------------------------------
All the experiments reported here are done based on the DANTest. If not otherwise specified, we use the MNIST handwritten digits database [@lecun1998], which we refer to as the **standard** dataset. As it is class-balanced, we create two imbalanced versions of it. The first one, referred to as the **imbalanced** dataset, is created by augmenting the training samples for digit ‘0’ by shifting them each by one pixel to the top, bottom, left and right, so that it contains *five* times more training samples of ‘0’ than the standard dataset. Moreover, we create the **very imbalanced** dataset, where we have *seven* times more training samples for digit ‘0’ than the standard dataset. For other digits, we randomly sample from the standard dataset and intentionally make the sizes of the resulting datasets identical to that of the standard dataset. We use the same test set for all the experiments.
We implement $G$ and $D$ as convolutional neural networks (see [Appendix \[app:sec:net\_architectures\]]{} for the network architectures). We use the batch normalization [@ioffe2015] in $G$. If the spectral normalization is used, we only apply it to $D$, otherwise we use the layer normalization [@ba2017] in $D$. We concatenate the label vector to each layer of $D$. For the gradient penalties, we use Euclidean norms and set $\lambda$ to $10.0$ (see ), $k$ to $1.0$ and $c$ to $0.01$ (see [ \[tab:gradient\_penalties\]]{}). We use the Adam optimizers [@kingma2014] with $\alpha = 0.001$, $\beta_1 = 0.0$ and $\beta_2 = 0.9$. We alternatively update $G$ and $D$ once in each iteration and train the model for 100,000 generator steps. The batch size is $64$. We implement the model in Python and TensorFlow [@abadi2016]. We run each experiment for ten runs and report the mean and the standard deviation of the error rates.
nonsaturating Wasserstein hinge
------------------ ------------------- --------------------- ---------------------
$\epsilon = 0.5$ 8.47$\pm$0.36 73.16$\pm$6.36 15.20$\pm$2.46
$\epsilon = 0.9$ 8.96$\pm$0.63 57.66$\pm$5.13 8.94$\pm$0.87
$\epsilon = 1.0$ **8.25$\pm$0.35** **5.89$\pm$0.26** **6.59$\pm$0.31**
$\epsilon = 1.1$ 8.62$\pm$0.45 60.30$\pm$7.61 8.02$\pm$0.35
$\epsilon = 2.0$ 9.18$\pm$0.94 69.54$\pm$5.37 11.87$\pm$0.85
: Error rates (%) for the $\epsilon$-weighted versions of the nonsaturating, the Wasserstein and the hinge losses (see ) on the standard dataset. Here, $\epsilon = 1.0$ corresponds to the original losses.[]{data-label="tab:exp_properties"}
**unregularized** **TCGP** **TLGP** **R~1~ GP** **R~2~ GP** **SN** **SN + TCGP** **SN + TLGP** **SN + R~1~ GP** **SN + R~2~ GP**
------------------------------------------------------ ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- ------------------- -------------------
**classic (M)** ([-@goodfellow2014]) 9.11$\pm$0.63 5.65$\pm$0.27 5.42$\pm$0.17 19.01$\pm$3.73 12.91$\pm$1.13 7.37$\pm$0.52 5.55$\pm$0.37 5.57$\pm$0.28 11.16$\pm$2.66 14.00$\pm$2.49
**classic (N)** ([-@goodfellow2014]) 26.83$\pm$7.17 5.64$\pm$0.23 5.56$\pm$0.31 14.67$\pm$4.86 13.80$\pm$3.20 8.25$\pm$0.35 5.52$\pm$0.16 5.61$\pm$0.50 12.98$\pm$2.71 13.50$\pm$3.78
**classic (L)** 17.38$\pm$5.16 5.66$\pm$0.36 5.55$\pm$0.16 18.49$\pm$5.51 14.92$\pm$5.20 7.98$\pm$0.36 5.70$\pm$0.36 5.48$\pm$0.29 15.45$\pm$6.54 17.61$\pm$7.60
**hinge (M)** **4.83$\pm$0.34** **4.88$\pm$0.25** 9.49$\pm$5.30 **6.22$\pm$0.23** **5.06$\pm$0.33** 10.62$\pm$2.10 12.91$\pm$4.29
**hinge (N)** 37.55$\pm$20.22 **5.00$\pm$0.24** 4.97$\pm$0.24 **7.34$\pm$1.83** 7.54$\pm$1.31 6.90$\pm$0.33 5.05$\pm$0.22 **5.06$\pm$0.39** 11.91$\pm$4.02 12.10$\pm$4.74
**hinge (L)** ([-@lim2017; -@tran2017]) 11.50$\pm$5.32 5.01$\pm$0.26 **4.89$\pm$0.18** 8.96$\pm$3.55 7.71$\pm$1.82 6.59$\pm$0.31 **4.97$\pm$0.19** 5.18$\pm$0.27 13.63$\pm$4.13 11.35$\pm$3.40
**Wasserstein** ([-@arjovsky2017wgan]) 7.69$\pm$0.33 5.04$\pm$0.19 4.92$\pm$0.23 13.89$\pm$20.64 **7.25$\pm$1.19** 5.50$\pm$0.18 5.76$\pm$0.70 13.74$\pm$5.47 13.82$\pm$4.93
**least squares** ([-@mao2017]) **7.15$\pm$0.47** 7.27$\pm$0.44 6.70$\pm$0.44 30.12$\pm$28.43 32.44$\pm$21.05 7.88$\pm$0.45 6.69$\pm$0.25 7.11$\pm$0.37 **9.91$\pm$1.55** 11.56$\pm$4.09
**relativistic** ([-@jolicoeur-martineau2018]) 90.20$\pm$0.00 5.25$\pm$0.25 5.01$\pm$0.31 **8.00$\pm$1.63** 8.75$\pm$5.83 7.14$\pm$0.39 5.35$\pm$0.29 5.25$\pm$0.26 **9.31$\pm$2.01**
**relativistic hinge** ([-@jolicoeur-martineau2018]) 52.01$\pm$9.38 8.28$\pm$10.26 8.39$\pm$1.92 7.67$\pm$1.82 6.44$\pm$0.16 **5.02$\pm$0.31** 12.56$\pm$4.42 12.40$\pm$4.55
**absolute** **6.69$\pm$0.24** 5.23$\pm$0.29 5.20$\pm$0.26 8.01$\pm$1.96 6.79$\pm$0.45 5.23$\pm$0.13 5.18$\pm$0.35 10.42$\pm$3.07 **9.93$\pm$2.28**
**asymmetric** 7.81$\pm$0.27 4.94$\pm$0.14 8.79$\pm$3.18 **7.33$\pm$1.01** **5.98$\pm$0.40** 5.60$\pm$0.29 5.82$\pm$0.44 **8.80$\pm$1.18**
Examining the necessary conditions for favorable adversarial loss functions {#sec:exp_properties}
---------------------------------------------------------------------------
As discussed in [Section \[sec:sufficient\_conditions\]]{}, we examine here the cases when the prerequisites in [Theorem \[theo:strong\_sufficient\_condition2\]]{} do not hold. We consider the classic nonsaturating, the Wasserstein and the hinge losses and change the training objective for the discriminator into $$\label{eq:imbalanced_loss}
\max_{D}\;\epsilon\,{\mathbb{E}}_{{\mathbf{x}}\sim p_d}[f(D({\mathbf{x}}))] + {\mathbb{E}}_{{\tilde{{\mathbf{x}}}}\sim p_g}[g(D({\tilde{{\mathbf{x}}}}))]\,,$$ where $\epsilon \in {\mathbb{R}}$ is a constant. The prerequisites in [Theorem \[theo:strong\_sufficient\_condition2\]]{} do not hold when $\epsilon \neq 1$. We illustrate the $\Psi$ functions of these *$\epsilon$-weighted losses* in [Appendix \[app:sec:psi\_function\_graphs\]]{}.
[ \[tab:exp\_properties\]]{} shows the results for $\epsilon = 0.5$, $0.9$, $1.0$, $1.1$, $2.0$, using the spectral normalization for regularization. We can see that all the original losses (i.e., $\epsilon = 1$) result in the lowest error rates. In general, the error rates increase as $\epsilon$ goes away from $1.0$. Notably, the Wasserstein loss turn out failing with error rates over 50% when $\epsilon \neq 1$.
On different discriminator loss functions {#sec:exp_adversarial_losses}
-----------------------------------------
In this experiment, we aim to compare different discriminator loss functions. Specifically, we evaluate an comprehensive set (in total 168) of different combinations of component functions and regularization approaches.
For the component functions, we consider the classic minimax and the classic nonsaturating losses [@goodfellow2014], the Wasserstein loss [@arjovsky2017wgan], the least squares loss [@mao2017], the hinge loss [@lim2017; @tran2017], the relativistic average and the relativistic average hinge losses [@jolicoeur-martineau2018], as well as the absolute and the asymmetric losses we propose and describe in [Section \[sec:psi\_function\_analysis\]]{}.
For the regularization approaches, we consider the coupled, the local, the R~1~ and the R~2~ gradient penalties (**GP**) and the spectral normalization (**SN**). For the coupled and the local gradient penalties, we examine both the two-side and the one-side versions (see [ \[tab:gradient\_penalties\]]{}). We will use in the captions **OCGP** and **TCGP** as the shorthands for the one-side and the two-side coupled gradient penalties, respectively, and **OLCP** and **TLCP** for the one-side and the two-side local gradient penalties, respectively. We also consider the combinations of the SN with different gradient penalties.
We report in [ \[tab:exp\_adversarial\_losses\]]{} the results for all the combinations and present in [ \[fig:training\_progress\]]{} the training progress for the nonsaturating and the hinge losses. We can see that *there is no single winning component functions and regularization approach across all different settings*. Some observations are:
With respect to the **component functions**—
- The classic minimax and nonsaturating losses never get the lowest three error rates for all different settings.
- The hinge, the asymmetric and the two relativistic losses are robust to different regularization approaches and tend to achieve lower error rates.
- The relativistic average loss outperforms both the classic minimax and nonsaturating losses across all regularization approaches. But, the relativistic average hinge loss does not always outperform the standard hinge loss.
With respect to the **regularization approaches**—
- The coupled and the local GPs outperform the R~1~ and the R~2~ GPs across nearly all different component functions, no matter whether the SN is used or not.
- The coupled and the local GPs stabilize the training (see [ \[fig:training\_progress\]]{}) and tend to have lower error rates.
- The R~2~ gradient penalties achieve lower error rates than the R~1~ gradient penalties. In some cases, they can be too strong and even stop the training early (see [ \[fig:training\_progress\]]{} (a)).[^1]
- Combining either the coupled or the local GP with the SN usually leads to higher error rates than using the coupled or the local GP only.
- Similarly, combining either the R~1~ or the R~2~ GP with the SN degrades the result. Moreover, it leads to unstable training (see Figures \[fig:training\_progress\](b) and (d)). This result implies that R~1~ and R~2~ GPs do not work well with the SN.
- Using the one-side GPs instead of their two-side counterparts increase the error rates by 0.1–9.5%. (We report the results for the one-side GPs in [Appendix \[app:sec:results\]]{} due to page limit.)
We also note that some combinations result in remarkably high error rates, e.g., “least squares loss + R~1~ GP”, “least squares loss + R~2~ GP” and “classic minimax loss + R~1~ GP”.
In sum, according to the overall performance and the robustness to different settings, for the component functions, *we recommend the hinge, the asymmetric and the two relativistic losses*. We note that these functions also feature lower computation costs as all their components functions are piecewise linear (see [ \[tab:loss\_functions\]]{} and [Section \[sec:psi\_function\_analysis\]]{}). For the regularization approaches, *we recommend the two-side coupled and the two-side local gradient penalties*.
We also conduct the imbalanced dataset test (see [Section \[sec:dantest\]]{}) on the two imbalanced datasets described in [Section \[sec:dataset\]]{} to compare the regularization approaches. We use the classic nonsaturating loss. As shown in [ \[tab:exp\_imbalanced\_dataset\]]{}, the error rates increase as the level of imbalance increases. The two-side local GP achieve the lowest error rates across all three datasets. The error rates for the R~1~ and the R~2~ GPs increase significantly when the dataset goes imbalanced.
### Effects of the Lipschitz constants {#sec:exp_lipschitz}
In this experiment, we examine the effects of the Lipschitz constant ($k$) used in the coupled and the local GPs (see [ \[tab:gradient\_penalties\]]{}). We use the classic nonsaturating loss here. We report in [ \[fig:exp\_lipschitz\]]{} the results for $k = 0.01$, $0.1$, $1$, $10$, $100$. We can see that the error rate increases as $k$ goes away from $1.0$, suggesting that $k = 1$ is indeed a good default value. Moreover, the two-side GPs are more sensitive to $k$ than their one-side counterparts.
We note that @petzka2018 suggested that the one-side coupled GP are preferable to the two-side version and showed empirically that the former has more stable behaviors. However, we observe in our experiments that the two-side penalties usually lead to faster convergence to lower error rates compared to the one-side penalties.[^2]
standard imbalanced very imbalanced
--------- --------------------- --------------------- ---------------------
TCGP 5.64$\pm$0.23 7.09$\pm$0.64 8.12$\pm$0.31
OCGP 7.20$\pm$0.39 8.86$\pm$0.65 10.23$\pm$0.75
TLGP **5.51$\pm$0.27** **6.94$\pm$0.28** **8.10$\pm$0.55**
OLGP 6.92$\pm$0.21 8.63$\pm$0.75 10.21$\pm$0.52
R~1~ GP 14.67$\pm$4.86 18.66$\pm$5.60 27.90$\pm$9.59
R~2~ GP 13.80$\pm$3.20 15.70$\pm$2.07 29.97$\pm$12.4
: Error rates (%) for different gradient penalties (using the nonsaturating loss) on datasets with different levels of imbalance.[]{data-label="tab:exp_imbalanced_dataset"}
### Effects of the penalty weights {#sec:exp_penalty_weights}
We then examine the effects of the penalty weights ($\lambda$) for the R~1~ and the R~2~ GPs (see ). We consider the classic nonsaturating, the Wasserstein and the hinge losses. We present in [ \[fig:exp\_r1r2\_penalty\_weight\]]{} the results for $\lambda = 0.01$, $0.1$, $1$, $10$, $100$. We can see that the R~1~ GP tends to outperform the R~2~ GP, while they are both sensitive to the value of $\lambda$. Hence, future research should run hyperparmeter search for $\lambda$ to find out its optimal value. When the spectral normalization is not used, the hinge loss is less sensitive to $\lambda$ than the other two losses. However, when spectral normalization is used, the error rate increases as $\lambda$ increases, which again implies that the R~1~/R~2~ GPs and the SN do not work well together.
\
(a) without the spectral normalization\
\
(b) with the spectral normalization
On different generator loss functions {#sec:exp_g_loss}
-------------------------------------
As discussed in [Section \[sec:g\_loss\]]{}, we also aim to examine the effects of the generator loss function $h(\cdot)$. We consider the classic and the hinge losses for the discriminator and the following three generator loss functions: minimax (**M**)—$h(x) = g(x)$, nonsaturating (**N**)—$h(x) = \log(1 + e^{-x})$, and linear (**L**)—$h(x) = -x$. We report the results in the first six rows of [ \[tab:exp\_adversarial\_losses\]]{}. For the classic discriminator loss, we see no single winner among the three generator loss functions across all the regularization approaches, which implies that the heuristics behind these alternative losses might not be true. For the hinge discriminator loss, the minimax generator loss is robust to different regularization approaches and achieves three lowest and four lowest-three scores. Hence, *we recommend to use hinge loss for the discriminator and minimax loss for the generator* as the overall best choice according to our experimental results.
Effects of the momentum terms of the optimizers {#sec:exp_momentum}
-----------------------------------------------
We observe a trend towards using smaller momentum [@radford2016] or even no momentum [@arjovsky2017wgan; @gulrajani2017; @miyato2018; @brock2018] in GAN training. Hence, we would also like to examine the effects of momentum terms in the optimizers with the proposed framework. As suggested by @gidel2018, we also include a negative momentum value of $-0.5$. We use the classic nonsaturating loss and the SN along with the coupled GPs for regularization. [ \[fig:exp\_momentum\]]{} shows the results for all combinations of $\beta_1 = -0.5$, $0.0$, $0.5$, $0.9$ for $G$ and $D$. We can see that for the two-side coupled GP, using larger momenta in both $G$ and $D$ leads to lower error rates, while there is no specific trend for the one-side coupled GP.
Discussions and Conclusions {#sec:discussions_and_conclusions}
===========================
In this paper, we have shown in theory what certain types of component functions form a valid adversarial loss. We have also introduced a new framework called DANTest for comparing adversarial losses. With DANTest, we systematically compared combinations of different component functions and regularization approaches to decouple their effects. Our empirical results show that there is no single winning component functions or regularization approach across all different settings. Our theoretical and empirical results can together serve as a reference for choosing or designing adversarial training objectives in future research.
As compared to the commonly used metrics for evaluating generative models, such as the Inception Score [@salimans2016] and Fréchet Inception Distance [@heusel2017] adopted in @lucic2018 and @kurach2018, the DANTest is simpler and is easier to control and extend. This allows us to easily evaluate new adversarial losses. However, we note that while the discriminator in a DAN is trained to optimize the same objectives as in a conditional GAN, the generators in the two models actually work in opposite ways ($\mathcal{X} \to \mathcal{Z}$ in a DAN versus $\mathcal{Z} \to \mathcal{X}$ in a GAN). Hence, it is unclear whether the empirical results can be generalized to conditional and unconditional GANs. Nonetheless, recent work has also adapted adversarial losses to plenty discriminative models (e.g., image-to-image translation [@isola2017] and image super-resolution [@ledig17cvpr]). Therefore, it is worth investigating the behaviors of adversarial losses in different scenarios.
In addition, our theoretical analysis provides a new perspective on adversarial losses and reveals a large class of component functions valid for adversarial losses. We note that @nowozin2016 has also shown a certain class of component functions can result in theoretically valid adversarial losses. However, in their formulations, the component functions $f$ and $g$ are not independent of each other as they considered only the *f*-divergences. A future direction is to investigate the necessary and sufficient conditions for the existence and the uniqueness of a Nash equilibrium.
Proofs of the Theorems {#app:sec:proofs}
======================
If [Property \[prop:weak\]]{} holds, then for any $\gamma \in [0, 1]$, $\psi(\gamma) + \psi(1 - \gamma) \geq 2\,\psi(\frac{1}{2})$. \[app:theo:weak\_necessary\_condition\]
Since [Property \[prop:weak\]]{} holds, we have for any fixed $p_d$, $$\label{app:eq:theo1_prerequisite}
L_G \geq L_G\,\big\rvert_{\,p_g = p_d}\,.$$
Let us consider $$\begin{aligned}
p_d({\mathbf{x}}) = \gamma\,\delta({\mathbf{x}}- \mathbf{s}) + (1 - \gamma)\,\delta({\mathbf{x}}- \mathbf{t})\,,\\
p_g({\mathbf{x}}) = (1 - \gamma)\,\delta({\mathbf{x}}- \mathbf{s}) + \gamma\,\delta({\mathbf{x}}- \mathbf{t})\,.
\end{aligned}$$ for some $\gamma \in [0, 1]$ and $\mathbf{s}, \mathbf{t} \in {\mathcal{X}}, \mathbf{s} \neq \mathbf{t}$. Then, we have $$\begin{aligned}
&L_G\,\big\rvert_{\,p_g = p_d}\\
&= \max_{D}\;\int_{\mathbf{x}}p_d({\mathbf{x}})\,f(D({\mathbf{x}})) + p_d({\mathbf{x}})\,g(D({\mathbf{x}}))\,d{\mathbf{x}}\\
&= \max_{D}\;\int_{\mathbf{x}}p_d({\mathbf{x}})\,(f(D({\mathbf{x}})) + g(D({\mathbf{x}})))\,d{\mathbf{x}}\\
\begin{split}
&= \max_{D}\;\int_{\mathbf{x}}\big(\,(\gamma\,\delta({\mathbf{x}}- \mathbf{s}) + (1 - \gamma)\,\delta({\mathbf{x}}- \mathbf{t}))\\
&\qquad\qquad\qquad\qquad\qquad (f(D({\mathbf{x}})) + g(D({\mathbf{x}})))\,\big)\,d{\mathbf{x}}\end{split}\\
\begin{split}
\label{app:eq:theo1_note1}
&= \max_{D}\;\big(\,\gamma (f(D(\mathbf{s})) + g(D(\mathbf{s})))\\
&\qquad\qquad\qquad + (1 - \gamma)\,(f(D(\mathbf{t})) + g(D(\mathbf{t})))\,\big)
\end{split}\\
\begin{split}
\label{app:eq:theo1_note2}
&= \max_{y_1, y_2}\;\big(\,\gamma (f(y_1) + g(y_1))\\
&\qquad\qquad\qquad + (1 - \gamma) (f(y_2) + g(y_2))\,\big)
\end{split}\\
\begin{split}
&= \max_{y_1}\;\gamma\,(f(y_1) + g(y_1))\\
&\qquad\qquad\qquad + \max_{y_2}\;(1 - \gamma)\,(f(y_2) + g(y_2))
\end{split}\\
&= \max_{y}\;f(y) + g(y)\\
\label{app:eq:theo1_note3}
&= 2\,\psi(\tfrac{1}{2})\,.
\end{aligned}$$
Moreover, we have $$\begin{aligned}
&L_G\nonumber\\
&= \max_{D}\;\int_{\mathbf{x}}p_d({\mathbf{x}})\,f(D({\mathbf{x}})) + p_g({\mathbf{x}})\,g(D({\mathbf{x}}))\,d{\mathbf{x}}\\
\begin{split}
&= \max_{D}\;\int_{\mathbf{x}}\big(\,(\gamma\,\delta({\mathbf{x}}- \mathbf{s}) + (1 - \gamma)\,\delta({\mathbf{x}}- \mathbf{t}))\,f(D({\mathbf{x}}))\\
&\;\qquad + ((1 - \gamma)\,\delta({\mathbf{x}}- \mathbf{s}) + \gamma\,\delta({\mathbf{x}}- \mathbf{t}))\,g(D({\mathbf{x}}))\,\big)\,d{\mathbf{x}}\end{split}\\
\begin{split}
\label{app:eq:theo1_note4}
&= \max_{D}\;\big(\,\gamma\,f(D(\mathbf{s})) + (1 - \gamma)\,f(D(\mathbf{t}))\\
&\qquad\qquad\qquad + (1 - \gamma)\,g(D(\mathbf{s})) + \gamma\,g(D(\mathbf{t}))\,\big)
\end{split}\\
\begin{split}
\label{app:eq:theo1_note5}
&= \max_{y_1, y_2}\;\big(\,\gamma\,f(y_1) + (1 - \gamma)\,g(y_1))\\
&\qquad\qquad\qquad + (1 - \gamma)\,f(y_2) + \gamma\,g(y_2)\,\big)
\end{split}\\
\begin{split}
&= \max_{y_1}\;\gamma\,f(y_1) + (1 - \gamma)\,g(y_1))\\
&\qquad\qquad\qquad + \max_{y_2} (1 - \gamma)\,f(y_2) + \gamma\,g(y_2)
\end{split}\\
\label{app:eq:theo1_note6}
&= \psi(\gamma) + \psi(1 - \gamma)\,.
\end{aligned}$$ (Note that we can obtain from and from because $D$ can be any function and thus $D(\mathbf{s})$ is independent of $D(\mathbf{t})$.)
As holds for any fixed $p_d$, by substituting and into , we get $$\psi(\gamma) + \psi(1 - \gamma) \geq 2\,\psi(\tfrac{1}{2})$$ for any $\gamma \in [0, 1]$, which concludes the proof.
If [Property \[prop:strong\]]{} holds, then for any $\gamma \in [0, 1] \setminus \{\frac{1}{2}\}$, $\psi(\gamma) + \psi(1 - \gamma) > 2\,\psi(\frac{1}{2})$. \[app:theo:strong\_necessary\_condition\]
Since [Property \[prop:strong\]]{} holds, we have for any fixed $p_d$, $$\label{app:eq:theo2_prerequisite}
L_G\big\rvert_{\,p_g \neq p_d} > L_G\,\big\rvert_{\,p_g = p_d}\,.$$
Following the proof of [Theorem \[app:theo:weak\_necessary\_condition\]]{}, consider $$\begin{aligned}
\label{app:eq:theo2_note1}
&p_d({\mathbf{x}}) = \gamma\,\delta({\mathbf{x}}- \mathbf{s}) + (1 - \gamma)\,\delta({\mathbf{x}}- \mathbf{t})\,,\\[1ex]
\label{app:eq:theo2_note2}
&p_g({\mathbf{x}}) = (1 - \gamma)\,\delta({\mathbf{x}}- \mathbf{s}) + \gamma\,\delta({\mathbf{x}}- \mathbf{t})\,,
\end{aligned}$$ for some $\gamma \in [0, 1]$ and some $\mathbf{s}, \mathbf{t} \in {\mathcal{X}}, \mathbf{s} \neq \mathbf{t}$. It can be easily shown that $p_g = p_d$ if and only if $\gamma = \tfrac{1}{2}$.
As holds for any fixed $p_d$, by substituting and into , we get $$\psi(\gamma) + \psi(1 - \gamma) > 2\,\psi(\tfrac{1}{2})\,,$$ for any $\gamma \in [0, 1] \setminus \{\tfrac{1}{2}\}$, concluding the proof.
If $\psi(\gamma)$ has a global minimum at $\gamma = \frac{1}{2}$, then [Property \[prop:weak\]]{} holds. \[app:theo:weak\_sufficient\_condition\]
First, we see that $$\begin{aligned}
&L_G\,\big\rvert_{\,p_g = p_d}\\
&= \max_{D}\;\int_{\mathbf{x}}p_d({\mathbf{x}})\,f(D({\mathbf{x}})) + p_d({\mathbf{x}})\,g(D({\mathbf{x}}))\,d{\mathbf{x}}\\
&= \max_{y}\;\int_{\mathbf{x}}p_d({\mathbf{x}})\,f(y) + p_d({\mathbf{x}})\,g(y)\,d{\mathbf{x}}\\
&= \max_{y}\;\int_{\mathbf{x}}p_d({\mathbf{x}})\,(f(y) + g(y))\,d{\mathbf{x}}\\
&= \max_{y}\;(f(y) + g(y)) \int_{\mathbf{x}}p_d({\mathbf{x}})\,d{\mathbf{x}}\\
&= \max_{y}\;f(y) + g(y)\\
\label{app:eq:theo3_note1}
&= 2\,\psi(\tfrac{1}{2})\,.
\end{aligned}$$ On the other had, we have $$\begin{aligned}
&L_G\nonumber\\
&= \max_{D}\;\int_{\mathbf{x}}p_d({\mathbf{x}})\,f(D({\mathbf{x}})) + p_g({\mathbf{x}})\,g(D({\mathbf{x}}))\,d{\mathbf{x}}\\
&= \max_{y}\;\int_{\mathbf{x}}p_d({\mathbf{x}})\,f(y) + p_g({\mathbf{x}})\,g(y)\,d{\mathbf{x}}\\
\begin{split}
&= \max_{y}\;\int_{\mathbf{x}}(p_d({\mathbf{x}}) + p_g({\mathbf{x}}))\\
&\qquad\qquad \left(\frac{p_d({\mathbf{x}})\,f(y)}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})} + \frac{p_g({\mathbf{x}})\,g(y)}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})}\right)\,d{\mathbf{x}}\end{split}\\
\begin{split}
&= \int_{\mathbf{x}}(p_d({\mathbf{x}}) + p_g({\mathbf{x}}))\\
&\qquad\qquad \max_{y} \left(\frac{p_d({\mathbf{x}})\,f(y)}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})} + \frac{p_g({\mathbf{x}})\,g(y)}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})}\right)\,d{\mathbf{x}}\,.
\end{split}
\end{aligned}$$ Since $\frac{p_d({\mathbf{x}})}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})} \in [0, 1]$, we have $$\label{app:eq:theo3_note2}
L_G = \int_{\mathbf{x}}(p_d({\mathbf{x}}) + p_g({\mathbf{x}}))\,\psi\left(\frac{p_d({\mathbf{x}})}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})}\right)\,d{\mathbf{x}}\,.$$ As $\psi(\gamma)$ has a global minimum at $\gamma = \frac{1}{2}$, now we have $$\begin{aligned}
L_G &\geq \int_{\mathbf{x}}(p_d({\mathbf{x}}) + p_g({\mathbf{x}}))\,\psi(\tfrac{1}{2})\,d{\mathbf{x}}\\
&= \psi(\tfrac{1}{2}) \int_{\mathbf{x}}(p_d({\mathbf{x}}) + p_g({\mathbf{x}}))\,d{\mathbf{x}}\\
\label{app:eq:theo3_note3}
&= 2\,\psi(\tfrac{1}{2})\,.
\end{aligned}$$ Finally, combining and yields $$L_G \geq L_G\,\big\rvert_{\,p_g = p_d}\,,$$ which holds for any $p_d$, thus concluding the proof.
If $\psi(\gamma)$ has a unique global minimum at $\gamma = \frac{1}{2}$, then [Property \[prop:strong\]]{} holds. \[app:theo:strong\_sufficient\_condition\]
Since $\psi(\gamma)$ has a unique global minimum at $\gamma = \frac{1}{2}$, we have for any $\gamma \in [0, 1] \setminus \frac{1}{2}$, $$\psi(\gamma) > \psi(\tfrac{1}{2})\,.$$ When $p_g \neq p_d$, there must be some ${\mathbf{x}}_0 \in {\mathcal{X}}$ such that $p_g({\mathbf{x}}_0) \neq p_d({\mathbf{x}}_0)$. Thus, $\frac{p_d({\mathbf{x}}_0)}{p_d({\mathbf{x}}_0) + p_g({\mathbf{x}}_0)} \neq \frac{1}{2}$, and thereby $\psi\left(\frac{p_d({\mathbf{x}}_0)}{p_d({\mathbf{x}}_0) + p_g({\mathbf{x}}_0)}\right) > \psi(\frac{1}{2})$. Now, by we have $$\begin{aligned}
&L_G\,\big\rvert_{\,p_g \neq p_d}\\
&= \int_{\mathbf{x}}(p_d({\mathbf{x}}) + p_g({\mathbf{x}}))\,\psi\left(\frac{p_d({\mathbf{x}})}{p_d({\mathbf{x}}) + p_g({\mathbf{x}})}\right)\,d{\mathbf{x}}\\
&> \int_{\mathbf{x}}(p_d({\mathbf{x}}) + p_g({\mathbf{x}}))\,\psi(\tfrac{1}{2})\,d{\mathbf{x}}\\
&= \psi(\tfrac{1}{2}) \int_{\mathbf{x}}(p_d({\mathbf{x}}) + p_g({\mathbf{x}}))\,d{\mathbf{x}}\\
\label{app:eq:theo4_note1}
&= 2\,\psi(\tfrac{1}{2})\,.
\end{aligned}$$ Finally, combining and yields $$L_G\,\big\rvert_{\,p_g \neq p_d} > L_G\,\big\rvert_{\,p_g = p_d}\,,$$ which holds for any $p_d$, thus concluding the proof.
If $f'' + g'' \leq 0$ and there exists some $y^*$ such that $f(y^*) = g(y^*)$ and $f'(y^*) = -g'(y^*) \neq 0$, then $\psi(\gamma)$ has a unique global minimum at $\gamma = \frac{1}{2}$. \[app:theo:strong\_sufficient\_condition2\]
First, we have by definition $$\Psi(\gamma, y) = \gamma\,f(y) + (1 - \gamma)\,g(y)\,.$$ By taking the partial derivatives, we get $$\begin{aligned}
\label{app:eq:theo5_partial1}
&\frac{\partial\Psi}{\partial\gamma} = f(y) - g(y)\,,\\[1ex]
\label{app:eq:theo5_partial2}
&\frac{\partial\Psi}{\partial y} = \gamma\,f'(y) + (1 - \gamma)\,g'(y)\,,\\[1ex]
\label{app:eq:theo5_partial3}
&\frac{\partial^2\Psi}{\partial y^2} = \gamma\,f''(y) + (1 - \gamma)\,g''(y)\,.
\end{aligned}$$
We know that there exists some $y^*$ such that $$\begin{aligned}
\label{app:eq:theo5_prerequisite1}
&f(y^*) = g(y^*)\,,\\[1ex]
\label{app:eq:theo5_prerequisite2}
&f'(y^*) = -g'(y^*) \neq 0\,.
\end{aligned}$$
1. By and , we see that $$\begin{aligned}
\label{app:eq:theo5_note1}
&\frac{\partial\Psi}{\partial\gamma}\,\Big\rvert_{\,y = y^*} = 0\,,\\[1ex]
\label{app:eq:theo5_note2}
&\frac{\partial\Psi}{\partial y}\,\Big\rvert_{\,(\gamma, y) = (\frac{1}{2}, y^*)} = 0\,.
\end{aligned}$$ Now, by we know that $\Psi$ is constant when $y = y^*$. That is, for any $\gamma \in [0, 1]$, $$\Psi(\gamma, y^*) = \Psi(\tfrac{1}{2}, y^*)\,.$$
2. Because $f'' + g'' \leq 0$, by we have $$\begin{aligned}
\label{app:eq:theo5_note3}
\frac{\partial^2\Psi}{\partial y^2}\,\Big\rvert_{\,\gamma = \tfrac{1}{2}} &= \tfrac{1}{2}\,f''(y) + \tfrac{1}{2}\,g''(y)\\
&\leq 0\,.
\end{aligned}$$ By and , we see that $y^*$ is a global minimum point of $\Psi\big\rvert_{\gamma = \tfrac{1}{2}}$. Thus, we now have $$\begin{aligned}
\Psi(\tfrac{1}{2}, y^*) &= \max_y\;\Psi(\tfrac{1}{2}, y)\\
\label{app:eq:theo5_note4}
&= \psi(\tfrac{1}{2})\,.
\end{aligned}$$
3. By , we see that $$\begin{aligned}
\frac{\partial\Psi}{\partial y}\,\Big\rvert_{\,y = y^*}
&= \gamma\,f'(y^*) + (1 - \gamma)\,g'(y^*)\\
&= \gamma\,f'(y^*) + (1 - \gamma)\,(-f'(y^*))\\
&= (2 \gamma - 1)\,f'(y^*)\,.
\end{aligned}$$ Since $f'(y^*) \neq 0$, we have $$\frac{\partial\Psi}{\partial y}\,\Big\rvert_{\,y = y^*} \neq 0\quad\forall\,\gamma \in [0, 1] \setminus \tfrac{1}{2}\,.$$ This shows that for any $\gamma \in [0, 1] \setminus \tfrac{1}{2}$, there must exists some $y^\circ$ such that $$\label{app:eq:theo5_note5}
\Psi(\gamma, y^\circ) > \Psi(\gamma, y^*)\,.$$ And by definition we have $$\begin{aligned}
\label{app:eq:theo5_note6}
\Psi(\gamma, y^\circ) &< \max_y\;\Psi(\gamma, y)\\
&= \psi(\gamma)\,.
\end{aligned}$$ Hence, by and we get $$\label{app:eq:theo5_note7}
\psi(\gamma) > \Psi(\gamma, y^*)\,.$$
Finally, combining , and yields $$\psi(\gamma) > \psi(\tfrac{1}{2})\quad\forall\,\gamma \in [0, 1] \setminus \tfrac{1}{2}\,,$$ which concludes the proof.
More Graphs of the $\Psi$ and $\psi$ Functions {#app:sec:psi_function_graphs}
==============================================
We show in [ \[app:fig:small\_psi\_functions\]]{} the graphs of the $\psi$ functions for different adversarial losses. Note that for the Wasserstein loss, the $\psi$ function is only defined at $\gamma = 0.5$, where it takes the value of zero, and for the asymmetric loss, the $\psi$ function is only defined when $\gamma > 0.5$, where it takes the value of zero. Hence, we do not include them in [ \[app:fig:small\_psi\_functions\]]{}.
We also present in [ \[app:fig:psi\_functions\_imbalanced\]]{} the graphs of the $\Psi$ functions for the $\epsilon$-weighted versions of the classic, the Wasserstein and the hinge losses. Moreover, Figures \[app:fig:small\_psi\_functions\](b) and (c) show the graphs of the $\psi$ functions for the $\epsilon$-weighted versions of the classic and the hinge losses, respectively.
Network Architectures {#app:sec:net_architectures}
=====================
We present in [ \[app:tab:network\_architectures\]]{} the network architectures for the generator and the discriminator used for all the experiments.
More Results {#app:sec:results}
============
We report in [ \[app:tab:exp\_loss\_functions\]]{} the results for the one-side coupled and local gradient penalties.
We also present in [ \[app:fig:exp\_momentum\]]{} the results for the experiment on the momentum terms using the hinge loss.
\
(a) common adversarial losses\
------------------------------------------------------- -----------------------------------------------------
\(b) $\epsilon$-weighted versions of the classic loss \(c) $\epsilon$-weighted versions of the hinge loss
------------------------------------------------------- -----------------------------------------------------
[lccc]{}\
*conv* &32 &3$\times$3 &3$\times$3\
*conv* &64 &3$\times$3 &3$\times$3\
*maxpool* &- &2$\times$2 &2$\times$2\
*dense* &128\
*dense* &10\
[lccc]{}\
*conv* &32 &3$\times$3 &3$\times$3\
*conv* &64 &3$\times$3 &3$\times$3\
*maxpool* &- &2$\times$2 &2$\times$2\
*dense* &128\
*dense* &1\
OCGP OLGP SN + OCGP SN + OLGP
----------------------------------------------- ------------------- ------------------- ------------------- -------------------
classic (M) [@goodfellow2014] 7.15$\pm$0.77 **6.95$\pm$0.51** 7.16$\pm$0.31 6.86$\pm$0.29
classic (N) [@goodfellow2014] 7.20$\pm$0.39 6.98$\pm$0.22 7.47$\pm$0.62 7.15$\pm$0.36
classic (L) 7.12$\pm$0.61 7.00$\pm$1.00 7.29$\pm$0.35 7.18$\pm$0.54
hinge (M) **5.82$\pm$0.31** 7.33$\pm$1.35 **5.80$\pm$0.24** 5.83$\pm$0.20
hinge (N) 7.88$\pm$1.33 5.92$\pm$0.36 **5.74$\pm$0.27**
hinge (L) [@lim2017; @tran2017] **5.77$\pm$0.29** **6.22$\pm$1.04** **5.77$\pm$0.30** **5.82$\pm$0.20**
Wasserstein [@arjovsky2017wgan] 7.60$\pm$3.02 13.34$\pm$1.49 6.35$\pm$0.43 6.06$\pm$0.45
least squares [@mao2017] 7.99$\pm$0.35 8.06$\pm$0.49 8.43$\pm$0.50 8.31$\pm$0.52
relativistic [@jolicoeur-martineau2018] 8.03$\pm$3.32 9.41$\pm$2.90 6.18$\pm$0.29 6.03$\pm$0.24
relativistic hinge [@jolicoeur-martineau2018] 10.70$\pm$2.51 14.17$\pm$1.79
absolute 5.95$\pm$0.19 6.22$\pm$0.25 6.08$\pm$0.32
asymmetric 5.85$\pm$0.35 7.57$\pm$0.98 6.21$\pm$0.34 5.92$\pm$0.37
[^1]: This is possibly because the R~1~ and the R~2~ gradient penalties encourage $D$ to have small gradients, and thus the gradients for both $D$ and $G$ might vanish when $p_g$ and $p_d$ are close enough.
[^2]: A possible reason is that as $p_g$ move towards $p_d$, the gradients for $G$ become smaller (and eventually zero when $p_d = p_g$), which can slow down the training. The two-side penalties can alleviate this by encouraging the norm of the gradients to be a fixed value.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Accelerated optimization algorithms can be generated using a double-integrator model for the search dynamics imbedded in an optimal control problem.\
address: 'Naval Postgraduate School, Monterey, CA 93943'
author:
- 'I. M. Ross'
title: An Optimal Control Theory for Accelerated Optimization
---
accelerated Newton’s method ,control Lyapunov function ,Lie derivative ,Nesterov’s accelerated gradient method ,Polyak’s heavy ball method ,Riemannian metric
Introduction
============
In [@rossJCAM-1], we proposed an optimal control theory for solving a constrained optimization problem, $$(N) \left\{\displaystyle\mathop\text{Minimize }_{{\boldsymbol x}_f \in C \subseteq {{\mathbb R}^{N_x}}} E({\boldsymbol x}_f) \right.$$ where $C$ is a constraint set in ${{\mathbb R}^{N_x}},\ N_x \in \mathbb{N}^+$ and $E: {\boldsymbol x}_f \ni {{\mathbb R}^{N_x}} \to {\mathbb R}$ is an objective function. A key concept in this framework was to view an algorithmic map $${\boldsymbol x}_0, {\boldsymbol x}_1, \ldots, {\boldsymbol x}_k, {\boldsymbol x}_{k+1}, \ldots$$ in terms of a discretization of a controllable, continuous-time trajectory, $t \mapsto {\boldsymbol x}\in {{\mathbb R}^{N_x}} $, whose dynamics is given by the single integrator model, $$\label{eq:xdot=u}
\dot{\boldsymbol x}= {{\boldsymbol u}}$$ where $t \mapsto {{\boldsymbol u}}\in {{\mathbb R}^{N_x}}$ is a control trajectory that must be designed such that at some time $t = t_f$, ${\boldsymbol x}(t_f) = {\boldsymbol x}_f$ is a solution to the given optimization problem. Starting with this simple idea, it is possible to generate a wide variety of well-known algorithms such as Newton’s method and the steepest descent method. Because continuous versions of “momentum” optimization methods involve second derivaties[@polyak64; @su], we explore the ramifications of replacing by a double-integrator model, $$\label{eq:xddot=u}
\ddot{\boldsymbol x}= {{\boldsymbol u}}$$ In essence, we show that the application of the theory presented in [@rossJCAM-1] with replaced by generates accelerated optimization techniques.
Rewriting in state-space form, $$\label{eq:dint-statespace}
\dot{\boldsymbol x}= {{\boldsymbol v}}, \quad \dot{{\boldsymbol v}}= {{\boldsymbol u}}$$ it follows from that a momentum method is essentially adding “inertia” to the “inertia-less” control of the single-integrator model.
\[rem:CG=HB\] A conjugate gradient (CG) method may also be viewed as an accelerated optimization technique in the context of . This observation follows by considering a generic CG method,
\[eq:CG\] $$\begin{aligned}
{\boldsymbol x}_{k+1} &= {\boldsymbol x}_k + \alpha_k {{\boldsymbol v}}_k \label{eq:CG-xk}\\
{{\boldsymbol v}}_{k} &= -{{\boldsymbol g}}_{k} + \beta^{CG}_k {{\boldsymbol v}}_{k-1} \label{eq:CG-pk}\end{aligned}$$
where $\alpha_k \ge 0 $ is the step length, ${{\boldsymbol v}}_k$ is the search direction, ${{\boldsymbol g}}_k := {{\boldsymbol g}}({\boldsymbol x}_k)$ is the gradient of the objective function function, and $\beta^{CG}_k \ge 0$ is the CG update parameter. Rewriting as single equation, $$\label{eq:CG=HB}
{\boldsymbol x}_{k+1} = {\boldsymbol x}_k - \alpha_k {{\boldsymbol g}}_k + \beta_k ({\boldsymbol x}_k - {\boldsymbol x}_{k-1}), \quad \beta_k := \left(\frac{\alpha_k \beta^{CG}_k}{\alpha_{k-1}}\right)$$ it follows that may be viewed as a discretization of $$\label{eq:u=HB}
\ddot{\boldsymbol x}= {{\boldsymbol u}}, \quad {{\boldsymbol u}}= -\gamma_a {{\boldsymbol g}}({\boldsymbol x}) -\gamma_b {{\boldsymbol v}}, \quad \gamma_a \in {\mathbb R}^+, \gamma_b \in {\mathbb R}^+$$ The non-control-theoretic, ordinary-differential-equation (ODE) form of , $$\label{eq:polyak}
\ddot{\boldsymbol x}+ \gamma_a {{\boldsymbol g}}({\boldsymbol x}) + \gamma_b \dot{\boldsymbol x}= {{\bf 0}}$$ is Polyak’s equation[@polyak64]. In the theory proposed in this paper, the function $({\boldsymbol x}, {{\boldsymbol v}})\mapsto -\gamma_a\, {{\boldsymbol g}}({\boldsymbol x}) - \gamma_b\, {{\boldsymbol v}}$ turns out to be a specific “optimal” feedback controller ${{\boldsymbol u}}$ for the double integrator $\ddot{\boldsymbol x}= {{\boldsymbol u}}$.
Despite their mathematical equivalence, there is a sharp change in perspective between and . Formula suggests that the search variable is velocity. According to , the search variable is acceleration.
It will be apparent shortly that the objective of the proposed theory is not to take existing algorithms and interpret them as ODEs or control systems, rather, it is to use optimal control theory as a foundational concept for optimization and as a discovery tool for algorithms[@rossJCAM-1].
Background: Optimal Control Theory for Optimization
===================================================
Consider some optimal control problem $(M)$ whose cost functional is given by a “Mayer” cost function $E : {\boldsymbol x}_f \mapsto {\mathbb R}$, where, ${\boldsymbol x}_f = {\boldsymbol x}(t_f)$ is constrained to lie in a target set $C$. A transversality condition for Problem $(M)$ is given by, $$\label{eq:tvc-cone}
{{\mbox{\boldmath $\lambda$}}}_x(t_f) \in \nu_0\,\partial E({\boldsymbol x}_f) + N_C({\boldsymbol x}_f)$$ where, ${{\mbox{\boldmath $\lambda$}}}_x(t_f)$ is the final value of an adjoint arc $t \mapsto {{\mbox{\boldmath $\lambda$}}}_x$ associated with $t \mapsto {\boldsymbol x}$, $\nu_0 \ge 0$ is a cost multiplier and $N_C({\boldsymbol x}_f)$ is the limiting normal cone[@vinter] to the set $C$ at ${\boldsymbol x}_f$. If Problem $(M)$ is designed so that ${{\mbox{\boldmath $\lambda$}}}_x(t_f)$ vanishes, then the transversality condition reduces to the necessary condition for Problem $(N)$, $${{\bf 0}}\in \nu_0\,\partial E({\boldsymbol x}_f) + N_C({\boldsymbol x}_f)$$ In [@rossJCAM-1], we showed the existence of Problem $(M)$ by direct construction for the case when $C$ is given by functional constraints, $$C = {\left\{{\boldsymbol x}\in {{\mathbb R}^{N_x}}:\ {{\boldsymbol e}}^L \le {{\boldsymbol e}}({\boldsymbol x}) \le {{\boldsymbol e}}^U \right\}}$$ where, ${{\boldsymbol e}}: {\boldsymbol x}\mapsto {{\mathbb R}^{N_e}}$ is a given function, and ${{\boldsymbol e}}^L$ and ${{\boldsymbol e}}^U$ are the specified lower and upper bounds on the values of ${{\boldsymbol e}}$. In this paper, we briefly review and revise the results obtained in [@rossJCAM-1] in the context . Furthermore, for the purposes of brevity and clarity, we limit the discussions to the unconstrained “static” optimization problem given by, $$(S) \left\{\displaystyle\mathop\text{Minimize }_{{\boldsymbol x}_f \in {{\mathbb R}^{N_x}}} E({\boldsymbol x}_f) \right.$$ In following [@rossJCAM-1], we create a vector field by “sweeping” the function $E$ backwards in time according to, $$\label{eq:idea-1}
y(t) := E({\boldsymbol x}(t))$$ Differentiating with respect to time we get, $$\label{eq:cost-evolution}
\dot y = \left[\partial_{{\boldsymbol x}} E({\boldsymbol x})\right]^T \dot{\boldsymbol x}= \left[\partial_{{\boldsymbol x}} E({\boldsymbol x})\right]^T {{\boldsymbol v}}, \quad \dot{{\boldsymbol v}}:= {{\boldsymbol u}}$$ Collecting all relevant equations, we define the following candidate optimal control problem $(R)$ that purportedly solves the optimization problem $(S)$: $$\begin{aligned}
&(R) \left\{
\begin{array}{lrl}
\textsf{Minimize } & J[y(\cdot), {\boldsymbol x}(\cdot), {{\boldsymbol v}}(\cdot), {{{{\boldsymbol u}}(\cdot)}}, t_f]
:=& y_f \\
\textsf{Subject to} & \dot{\boldsymbol x}=& {{\boldsymbol v}}\\
&\dot{{\boldsymbol v}}= & {{\boldsymbol u}}\\
&\dot y=& \left[\partial_{{\boldsymbol x}} E({\boldsymbol x})\right]^T {{\boldsymbol v}}\\
&({\boldsymbol x}(t_0), t_0) =& ({\boldsymbol x}^0, t^0) \\
& y(t_0) = & E({\boldsymbol x}^0) \\
&{{\boldsymbol v}}(t_f) = & {{\bf 0}}\end{array} \right.& \label{eq:prob-R}\end{aligned}$$ where, ${\boldsymbol x}^0$ is an initial “guess” of the solution (to Problem $(S)$). The variables $t_f, {\boldsymbol x}(t_f)$ and ${{\boldsymbol v}}(t_0)$ are free.
The main difference between and the optimal control problem for unconstrained optimization considered in [@rossJCAM-1] is the acceleration equation $\dot{{\boldsymbol v}}= {{\boldsymbol u}}$ and its associated endpoint condition ${{\boldsymbol v}}(t_f) = {{\bf 0}}$.
\[lemma:normality\] Problem $(R)$ has no abnormal extremals.
The Pontryagin Hamiltonian[@ross-book] for this problem is given by, $$\label{eq:H-unc}
H({{\mbox{\boldmath $\lambda$}}}_x, {{\mbox{\boldmath $\lambda$}}}_v, \lambda_y, {\boldsymbol x}, {{\boldsymbol v}}, y, {{\boldsymbol u}}):= {{\mbox{\boldmath $\lambda$}}}_x^T{{\boldsymbol v}}+ {{\mbox{\boldmath $\lambda$}}}_v^T{{\boldsymbol u}}+ \lambda_y \left[\partial_{{\boldsymbol x}} E({\boldsymbol x})\right]^T {{\boldsymbol v}}$$ where, ${{\mbox{\boldmath $\lambda$}}}_{x}, {{\mbox{\boldmath $\lambda$}}}_v$ and $\lambda_y$ are costates that satisfy the adjoint equations,
$$\begin{aligned}
\dot{{\mbox{\boldmath $\lambda$}}}_{x} &=-\partial_{{\boldsymbol x}}H = -\lambda_y\,\partial^2_{{\boldsymbol x}}E({\boldsymbol x})\, {{\boldsymbol v}}\label{eq:adj-x-unc}\\
\dot{{\mbox{\boldmath $\lambda$}}}_v & = -\partial_{{{\boldsymbol v}}}H = - {{\mbox{\boldmath $\lambda$}}}_x -\lambda_y\,\partial_{{\boldsymbol x}}E({\boldsymbol x})\label{eq:adj-v-unc}\\
\dot\lambda_y & =-\partial_y H = 0 \label{eq:adj-y-unc}\end{aligned}$$
The transversality conditions[@ross-book] for Problem $(R)$ are given by,
$$\begin{aligned}
{{\mbox{\boldmath $\lambda$}}}_x(t_f) &= {{\bf 0}}\label{eq:tvc-x}\\
{{\mbox{\boldmath $\lambda$}}}_v(t_0) & = {{\bf 0}}\label{eq:tvc-v}\\
{{\mbox{\boldmath $\lambda$}}}_y(t_f) & = \nu_0 \ge 0 \label{eq:tvc-y}\end{aligned}$$
where, $\nu_0$ is the cost multiplier. From and we have, $$\label{eq:lam-y-pre-tvc}
\lambda_y(t) = \nu_0$$ If $\nu_0 = 0$, then $\lambda_y(t) \equiv 0$. This implies, from and , that ${{\mbox{\boldmath $\lambda$}}}_x(t) \equiv {{\bf 0}}$. Similarly, ${{\mbox{\boldmath $\lambda$}}}_v(t) \equiv {{\bf 0}}$ from and . The vanishing of all multipliers violates the nontriviality condition. Hence $\nu_0 > 0$.
\[thm-singular\] All extremals of Problem $(R)$ are singular. Furthermore, the singular arc is of infinite order.
The Hamiltonian is linear in the control variable and the control space is unbounded; hence, if ${{\boldsymbol u}}$ is optimal, it must be singular. Furthermore, from the Hamiltonian minimization condition we have the first-order condition, $$\label{eq:hmc-order-1}
\partial_{{{\boldsymbol u}}} H = {{\mbox{\boldmath $\lambda$}}}_v(t) = {{\bf 0}}\qquad\forall t \in [t_0, t_f]$$ Differentiating with respect to time, we get, $$\label{eq:SA-deg-1}
\frac{d}{dt}\partial_{{{\boldsymbol u}}} H =\dot{{\mbox{\boldmath $\lambda$}}}_v(t) = - {{\mbox{\boldmath $\lambda$}}}_x -\nu_0\,\partial_{{\boldsymbol x}}E({\boldsymbol x}) = {{\bf 0}}$$ Equation does not generate an expression for the control function; hence, taking the second time derivative of $\partial_{{{\boldsymbol u}}}H$ we get, $$\begin{aligned}
\frac{d^2}{dt^2}\partial_{{{\boldsymbol u}}} H &= -\dot{{\mbox{\boldmath $\lambda$}}}_x - \nu_0\, \partial^2_{{\boldsymbol x}} E({\boldsymbol x})\,\dot{\boldsymbol x}\nonumber\\
&=-\dot{{\mbox{\boldmath $\lambda$}}}_x - \nu_0\, \partial^2_{{\boldsymbol x}} E({\boldsymbol x})\,{{\boldsymbol v}}\nonumber \\
&\equiv {{\bf 0}}\end{aligned}$$ where, the last equality follows from and Lemma \[lemma:normality\]. Hence, we have, $$\frac{d^k}{dt^k}\partial_{{{\boldsymbol u}}} H = {{\bf 0}}\quad\text{for\ } k = 0, 1 \ldots$$ and no $k$ yields an expression for ${{\boldsymbol u}}$.
\[thm:tmt\] The necessary condition for Problem $(S)$ is part of the transversality condition for Problem $(R)$.
From , we have $$\label{eq:lam-x-sol-generic}
{{\mbox{\boldmath $\lambda$}}}_x(t) = -\nu_0\, \partial_{{\boldsymbol x}} E({\boldsymbol x}(t))$$ From and Lemma 1, it follows that $\partial_{{\boldsymbol x}_f}E({\boldsymbol x}_f) = {{\bf 0}}$.
Minimum Principles for Accelerated Optimization
===============================================
From the results of the previous section, it follows that the primal-dual control dynamical system generated by Problem $(R)$ is given by,
\[eq:p-d-dynamics\] $$\begin{aligned}
\dot{\boldsymbol x}& = {{\boldsymbol v}}& \dot{{\mbox{\boldmath $\lambda$}}}_{x} & = -\lambda_y\,\partial^2_{{\boldsymbol x}}E({\boldsymbol x}) \, {{\boldsymbol v}}\\
\dot{{\boldsymbol v}}& = {{\boldsymbol u}}& \dot{{\mbox{\boldmath $\lambda$}}}_v & = -{{\mbox{\boldmath $\lambda$}}}_x - \lambda_y\, \partial_{{\boldsymbol x}} E({\boldsymbol x})\\
\dot y & = \big[\partial_{{\boldsymbol x}} E({\boldsymbol x})\big]^T {{\boldsymbol v}}& \dot\lambda_y & = 0\end{aligned}$$
The boundary conditions for are given by,
\[eq:BCs\] $$\begin{aligned}
{\boldsymbol x}(t^0) & = {\boldsymbol x}^0 & {{\boldsymbol v}}(t_f) & = {{\bf 0}}\\
y(t^0) &= E({\boldsymbol x}^0) & {{\mbox{\boldmath $\lambda$}}}_x(t_f) & = {{\bf 0}}\\
{{\mbox{\boldmath $\lambda$}}}_v(t^0) & = {{\bf 0}}& \lambda_y(t_f) &= \nu_0 > 0\end{aligned}$$
Because the optimal control is singular of infinite order, any control trajectory that satisfies and is optimal. Along a singular arc, ${{\mbox{\boldmath $\lambda$}}}_v(t) \equiv {{\bf 0}}$; hence, the auxiliary controllable dynamical system of interest[@rossJCAM-1] resulting from is given by, $$\label{eq:aux-dynamics}
(A) \left \{
\begin{aligned}
\dot{{\mbox{\boldmath $\lambda$}}}_{x} & = -\partial^2_{{\boldsymbol x}}E({\boldsymbol x}) \, {{\boldsymbol v}}\\
\dot{{\boldsymbol v}}& = {{\boldsymbol u}}\end{aligned}
\right.$$ where, we have scaled the adjoint covector ${{\mbox{\boldmath $\lambda$}}}_x$ by $\nu_0 > 0$ (cf. Lemma \[lemma:normality\]). The target final-time condition for $(A)$ is given by, $$\label{eq:aux-target}
(T) \left \{
\begin{aligned}
{{\mbox{\boldmath $\lambda$}}}_x(t_f) & = {{\bf 0}}\\
{{\boldsymbol v}}(t_f) & = {{\bf 0}}\end{aligned}
\right.$$ That is, any singular control that satisfies and generates a candidate “optimal” continuous-time algorithm for Problem $(S)$.
The auxiliary controllable dynamical system is equivalent to the time-derivative of the swept-back gradient function.
Application of a Minimum Principle Presented in [@rossJCAM-1]
-------------------------------------------------------------
Let ${{\mbox{\boldmath $\beta$}}}$ be a control vector field defined according to, $${{\mbox{\boldmath $\beta$}}}({\boldsymbol x}, {{\boldsymbol v}}, {{\boldsymbol u}}) := \left[
\begin{array}{c}
-\partial^2_{{\boldsymbol x}}E({\boldsymbol x}) \, {{\boldsymbol v}}\\
{{\boldsymbol u}}\\
\end{array}
\right]$$ Let $V:({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}) \mapsto {\mathbb R}$ be a control Lyapunov function for the $(A, T)$ pair. Let $\pounds_\beta V$ be the Lie derivative of $V$ along the vector field ${{\mbox{\boldmath $\beta$}}}$. Then, a sufficient condition for producing a globally convergent algorithm[@rossJCAM-1] is to design a singular control function such that $V$ is dissipative (when ${\boldsymbol x}\ne {\boldsymbol x}_f$), $$\label{eq:clf-theory-1}
\pounds_\beta V = \big[\partial V({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}})\big]^T {{\mbox{\boldmath $\beta$}}}({\boldsymbol x}, {{\boldsymbol v}}, {{\boldsymbol u}}) < 0$$ In [@rossJCAM-1], it is proposed that this objective can be achieved via the Minimum Principle, $$\begin{aligned}
&(P) \left\{
\begin{array} {lll}
\displaystyle\mathop\textsf{Minimize }_{{{\boldsymbol u}}} & \pounds_\beta V := \big[\partial V({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}})\big]^T {{\mbox{\boldmath $\beta$}}}({\boldsymbol x}, {{\boldsymbol v}}, {{\boldsymbol u}}) \\
\textsf{Subject to} & {{\boldsymbol u}}\in {\mathbb{U}}({\boldsymbol x}, {{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}, t)
\end{array} \right.& \label{prob:Min-P}\end{aligned}$$ where, ${\mathbb{U}}({\boldsymbol x}, {{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}, t)$ is an appropriate compact set that may vary with respect to the tuple $({\boldsymbol x}, {{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}, t)$. In an “unaccelerated” method, a solution to Problem ($P$) ensures the satisfaction of when ${\mathbb{U}}$ is chosen to metricize the control space[@rossJCAM-1]. Because of the presence of a drift vector field in the auxiliary dynamical system $(A)$, the Minimum Principle $(P)$ cannot guarantee $\pounds_\beta V < 0$; this follows by simply inspecting the expression for $\pounds_\beta V$, $$\pounds_\beta V = -\big[\partial_{\boldsymbol\lambda_x}V({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}})\big]^T\partial^2_{{\boldsymbol x}}E({\boldsymbol x}) \, {{\boldsymbol v}}+ \big[\partial_{{{\boldsymbol v}}}V({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}})\big]^T {{\boldsymbol u}}$$ To ensure $\pounds_\beta V < 0$, we impose the following requirement on $V$, $$\label{eq:drift-condition}
\partial_{\boldsymbol\lambda_x}V({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}})\big]^T\partial^2_{{\boldsymbol x}}E({\boldsymbol x}) \, {{\boldsymbol v}}> 0 \quad\text{if }\quad \partial_{{{\boldsymbol v}}}V({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}) = {{\bf 0}}\ \text{and } ({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}) \ne {{\bf 0}}$$ Furthermore, we set ${{\boldsymbol u}}= {{\bf 0}}$ if $\partial_{{{\boldsymbol v}}} V = {{\bf 0}}$. All of these results — in their general form — are well-known in nonlinear feedback control theory[@sontag-book]; hence, they are, technically, not new. What is new is their application to static optimization.
An Alternative Minimum Principle
--------------------------------
We can formulate an alternative Minimum Principle that essentially exchanges the cost and constraint functions in . Let $\rho:({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}, {\boldsymbol x}) \mapsto {\mathbb R}_+$ be an appropriate design function such that $-\rho$ specifies a rate of descent for $\dot V$. We propose to select a singular control ${{\boldsymbol u}}$ such that, $$\label{eq:clf-theory-strong}
\pounds_\beta V = \big[\partial V({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}})\big]^T {{\mbox{\boldmath $\beta$}}}({\boldsymbol x}, {{\boldsymbol v}}, {{\boldsymbol u}}) \le -\rho({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}, {\boldsymbol x})$$ That is, in contrast to , we seek a singular control that merely achieves a specified rate of descent given in terms of $\rho$. Let $D: ({{\boldsymbol u}}, {\boldsymbol x}, {{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}, t) \mapsto {\mathbb R}$ be an appropriate objective function. Then, a singular control ${{\boldsymbol u}}$ that solves the optimization problem, $$\begin{aligned}
&(P^*) \left\{
\begin{array} {lll}
\displaystyle\mathop\textsf{Minimize }_{{{\boldsymbol u}}} &D ({{\boldsymbol u}}, {\boldsymbol x}, {{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}, t) \\
\textsf{Subject to} & \pounds_\beta V +\rho({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}, {\boldsymbol x}) \le 0
\end{array} \right.& \label{prob:Min-P*}\end{aligned}$$ is a candidate (continuous-time) solution to the accelerated optimization problem.
Problem $(P^*)$ has been widely used in control theory for generating feedback controls[@sontag-book; @freeman-acc]. Note also that condition on $V$ specified by is implicit in .
LaSalle’s invariance principle[@sontag-book] may be used to relax the positive definite condition on $V$ and the negative definite condition on $\pounds_\beta V$.
Accelerated Optimization via Minimum Principles
===============================================
Following [@rossJCAM-1], we consider $$\label{eq:U-metric-trust}
{\mathbb{U}}({\boldsymbol x}, {{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}, t) := {\left\{{{\boldsymbol u}}:\ {{\boldsymbol u}}^T{{\boldsymbol W}}({\boldsymbol x}, {{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}, t){{\boldsymbol u}}\le \Delta({\boldsymbol x}, {{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}, t) \right\}}$$ where, $\Delta({\boldsymbol x}, {{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}, t) \ne 0$ is a positive real number and ${{\boldsymbol W}}({\boldsymbol x}, {{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}, t)$ is some appropriate positive definite matrix that metricizes the space ${\mathbb{U}}$. The quantities $\Delta$ and ${{\boldsymbol W}}$ may depend upon some or all of the variables ${\boldsymbol x}, {{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}$ and $t$. Applying the Minimum Principle given by , it is straightforward to show that if $\partial_{{{\boldsymbol v}}}V({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}) \ne {{\bf 0}}$, then ${{\boldsymbol u}}$ is given explicitly by, $$\label{eq:u-fromMinP}
{{\boldsymbol u}}= -\sigma[@t]\, {{\boldsymbol W}}^{-1}[@t] \, \partial_{{{\boldsymbol v}}} V({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}), \qquad \sigma[@t] > 0$$ where, $$\sigma^2[@t] := \frac{\Delta({\boldsymbol x}, {{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}, t)}{\big[\partial_{{{\boldsymbol v}}}V({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}})\big]^T {{\boldsymbol W}}^{-1}[@t]\big[\partial_{{{\boldsymbol v}}}V({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}})\big]}$$ and ${{\boldsymbol W}}[@t]:= {{\boldsymbol W}}({\boldsymbol x}, {{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}, t)$. That is, the $[@t]$ notation is simply a convenient shorthand for the various implicit and explicit time dependencies[@ross-book].
Switching to an application of Minimum Principle $(P^*)$ and using $$D ({{\boldsymbol u}}, {\boldsymbol x}, {{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}, t) =\frac{1}{2} \left({{\boldsymbol u}}^T{{\boldsymbol W}}({\boldsymbol x}, {{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}, t){{\boldsymbol u}}\right)$$ we get, $$\label{eq:u-fromMinP*}
{{\boldsymbol u}}= -\sigma^*[@t]\, {{\boldsymbol W}}^{-1}[@t] \, \partial_{{{\boldsymbol v}}} V({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}), \qquad \sigma^*[@t] > 0$$ where, $$\sigma^*[@t] := \frac{\rho({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}, {\boldsymbol x}) -\big[\partial_{\boldsymbol\lambda_x}V({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}})\big]^T\partial^2_{{\boldsymbol x}}E({\boldsymbol x}) \, {{\boldsymbol v}}}{\big[\partial_{{{\boldsymbol v}}}V({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}})\big]^T {{\boldsymbol W}}^{-1}[@t]\big[\partial_{{{\boldsymbol v}}}V({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}})\big]}$$ In other words, both minimum principles ($P$ and $P^*$) generate the same functional form for ${{\boldsymbol u}}$ but with different interpretations for the control “multipliers” given by $\sigma$ and $\sigma^*$.
\[thm:u-fromMPs\] Suppose we choose a quadratic positive definite Lyapunov function, $$\label{eq:V-quad-pd}
V({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}) = (a/2){{\mbox{\boldmath $\lambda$}}}_x^T{{\mbox{\boldmath $\lambda$}}}_x + (b/2){{\boldsymbol v}}^T{{\boldsymbol v}}+ c{{\mbox{\boldmath $\lambda$}}}_x^T{{\boldsymbol v}}$$ where, $$a > 0, \quad b > 0, \quad c \ne 0, \quad ab - c^2 > 0$$ are constants. Let ${{\boldsymbol W}}[@t]:= {{\boldsymbol W}}({\boldsymbol x}, {{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}, t)$ be a family of positive definite matrices that metricize the space ${\mathbb{U}}$. If $\partial^2_{{\boldsymbol x}}E({\boldsymbol x}) > 0$, then, the singular control resulting from either minimum principle ($P$ or $P^*$) is given by, $${{\boldsymbol u}}= - {{\boldsymbol W}}^{-1}[@t]\big(\gamma_a\,\partial_{{\boldsymbol x}}E({\boldsymbol x}) + \gamma_b\, {{\boldsymbol v}}\big)$$ where, $\gamma_a \in {\mathbb R}^+$ and $\gamma_b \in {\mathbb R}^+$ are (variable) controller gains.
Applying we get, $$\partial_{{{\boldsymbol v}}}V = c{{\mbox{\boldmath $\lambda$}}}_x + b{{\boldsymbol v}}= {{\bf 0}}\Rightarrow {{\mbox{\boldmath $\lambda$}}}_x = -(b/c){{\boldsymbol v}}$$ Hence, $$\begin{aligned}
\partial_{\boldsymbol\lambda_x}V({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}})\big]^T\partial^2_{{\boldsymbol x}}E({\boldsymbol x}) \, {{\boldsymbol v}}& = [a{{\mbox{\boldmath $\lambda$}}}_x + c {{\boldsymbol v}}]^T \partial^2_{{\boldsymbol x}}E({\boldsymbol x}) \, {{\boldsymbol v}}\\
&= \left(\frac{-ab + c^2}{c}\right){{\boldsymbol v}}^T \partial^2_{{\boldsymbol x}}E({\boldsymbol x}) \, {{\boldsymbol v}}\end{aligned}$$ Thus, for to hold, it follows that $c < 0$ if $\partial^2_{{\boldsymbol x}}E({\boldsymbol x}) > 0$.
The control solution resulting from the Minimum Principle $P$ or $P^*$ can be written as, $$\label{eq:uFromQuadV}
{{\boldsymbol u}}= -\sigma_q[@t] {{\boldsymbol W}}^{-1}[@t]\big(c{{\mbox{\boldmath $\lambda$}}}_x + b {{\boldsymbol v}}\big)$$ where $\sigma_q$ is given by $\sigma$ or $\sigma^*$ depending upon the choice of $P$ or $P^*$ respectively. Substituting ${{\mbox{\boldmath $\lambda$}}}_x = -\partial_{{\boldsymbol x}} E({\boldsymbol x})$ in we get, $$\label{eq:uFamilyNo1}
{{\boldsymbol u}}= - {{\boldsymbol W}}^{-1}[@t]\big(\gamma_a\,\partial_{{\boldsymbol x}}E({\boldsymbol x}) + \gamma_b\, {{\boldsymbol v}}\big)$$ where,
\[eq:gamma-def\] $$\begin{aligned}
\gamma_a &:= -c\,\sigma_q[@t] \ge 0\\
\gamma_b &:= b\,\sigma_q[@t] \ge 0\end{aligned}$$
are the (variable) controller gains.
\[corr:accel-x3\] A family of continuous accelerated optimization methods, parameterized by ${{\boldsymbol W}}$, is given by the ODE, $$\label{eq:FamilyNo1-ode}
\ddot{\boldsymbol x}= - {{\boldsymbol W}}^{-1}[@t]\big(\gamma_a\,\partial_{{\boldsymbol x}}E({\boldsymbol x}) + \gamma_b\, \dot{\boldsymbol x}\big)$$ Equation generates:
1. Polyak’s equation for the choice of a Euclidean metric (tensor) for ${{\boldsymbol W}}$ given by the identity matrix;
2. a continuous accelerated Newton’s method for a Riemannian ${{\boldsymbol W}}$ given by the Hessian, $\partial^2_{{\boldsymbol x}} E({\boldsymbol x})$; and,
3. a continuous accelerated quasi-Newton method for the choice of ${{\boldsymbol W}}$ given by ${{\boldsymbol B}}({\boldsymbol x}, {{\mbox{\boldmath $\lambda$}}}_x)$, a positive definite approximation to the Hessian.
The three special cases of are given explicitly by:
1. *Polyak’s equation:* $$\label{eq:HB-derive}
\ddot{\boldsymbol x}= - \big(\gamma_a\,\partial_{{\boldsymbol x}}E({\boldsymbol x}) + \gamma_b\, \dot{\boldsymbol x}\big)$$
2. *Continuous Accelerated Newton (${{\boldsymbol W}}({\boldsymbol x}) = \partial^2_{{\boldsymbol x}}E({\boldsymbol x})$):* $$\label{eq:CAN}
\ddot{\boldsymbol x}= - \left[\partial^2_{{\boldsymbol x}}E({\boldsymbol x})\right]^{-1}\big(\gamma_a\,\partial_{{\boldsymbol x}}E({\boldsymbol x}) + \gamma_b\, \dot{\boldsymbol x}\big)$$
3. *Continuous Accelerated Quasi-Newton (${{\boldsymbol W}}({\boldsymbol x}, {{\mbox{\boldmath $\lambda$}}}_x) = {{\boldsymbol B}}({\boldsymbol x}, {{\mbox{\boldmath $\lambda$}}}_x)$)*: $$\ddot{\boldsymbol x}= - {{\boldsymbol B}}^{-1}({\boldsymbol x}, {{\mbox{\boldmath $\lambda$}}}_x)\big(\gamma_a\,\partial_{{\boldsymbol x}}E({\boldsymbol x}) + \gamma_b\, \dot{\boldsymbol x}\big)$$
From Remark \[rem:CG=HB\], it follows that may also be viewed as a derivation of the continuous version of a conjugate gradient method.
Rewriting as, $$\partial^2_{{\boldsymbol x}}E({\boldsymbol x})\, \ddot{\boldsymbol x}+ \gamma_b\, \dot{\boldsymbol x}+ \gamma_a\,\partial_{{\boldsymbol x}}E({\boldsymbol x}) = {{\bf 0}}$$ it follows that a mechanical-system analogy for the continuous accelerated Newton’s method may be described in terms of a nonlinear mass-spring-damper system, where, the Hessian provides the variable inertia. Consequently, Polyak’s equation may be viewed as the case corresponding to the use of a constant inertia. Note also that $\gamma_a$ and $\gamma_b$ are not necessarily constants; see .
In view of Corollary \[corr:accel-x3\], and , we define a family of generalized versions of the accelerated gradient, Newton and quasi-Newton methods according to:
1. *Generalized Accelerated Gradient:* $$\ddot{\boldsymbol x}= -\sigma_q[@t]\, \partial_{{{\boldsymbol v}}} V({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}), \qquad \sigma_q[@t] > 0$$
2. *Generalized Accelerated Newton:* $$\ddot{\boldsymbol x}= -\sigma_q[@t]\, \left[\partial^2_{{\boldsymbol x}}E({\boldsymbol x})\right]^{-1} \partial_{{{\boldsymbol v}}} V({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}), \qquad \sigma_q[@t] > 0$$
3. *Generalized Accelerated Quasi-Newton*: $$\ddot{\boldsymbol x}= -\sigma_q[@t]\, {{\boldsymbol B}}^{-1}({\boldsymbol x}, {{\mbox{\boldmath $\lambda$}}}_x)\, \partial_{{{\boldsymbol v}}} V({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}), \qquad \sigma_q[@t] > 0$$
Accelerated Optimization via Direct Construction
================================================
As noted in Section 3, any singular control that satisfies and generates a candidate “optimal” continuous-time algorithm for Problem $(S)$; hence, the Minimum Principles proposed in Section 3 are not necessary conditions. They are simply a convenient systematic procedure for generating continuous accelerated optimization methods. In fact, is a sufficient condition; that is, any singular control ${{\boldsymbol u}}$ that renders $\pounds_\beta V < 0$ generates a globally convergent algorithm. In the case of the quadratic Lyapunov function given by , we have, $$\label{eq:Vdot-quadV}
\pounds_\beta V = -[a{{\mbox{\boldmath $\lambda$}}}_x + c{{\boldsymbol v}}]^T \partial^2_{{\boldsymbol x}}E({\boldsymbol x})\, {{\boldsymbol v}}+ [c {{\mbox{\boldmath $\lambda$}}}_x + b {{\boldsymbol v}}]^T {{\boldsymbol u}}$$ The optimal control resulting from either of the Minimum Principles does not directly incorporate the drift vector field for a generic metric tensor ${{\boldsymbol W}}$. Motivated by the intuition to design a control that directly incorporates the drift vector field, consider a feedback control strategy given by, $$\label{eq:uFamilyNo2}
{{\boldsymbol u}}= K_a\, {{\mbox{\boldmath $\lambda$}}}_x + K_b\, {{\boldsymbol v}}+ K_c \, \partial^2_{{\boldsymbol x}}E({\boldsymbol x})\, {{\boldsymbol v}}$$ where $K_a, K_b$ and $K_c$ are all (variable) real numbers that must be chosen so that $\pounds_\beta V$ is negative. Substituting in we get, $$\begin{aligned}
\label{eq:Vdot-wu2}
\pounds_\beta V &= \big(-a + c K_c\big){{\mbox{\boldmath $\lambda$}}}_x^T \partial^2_{{\boldsymbol x}}E({\boldsymbol x})\, {{\boldsymbol v}}+ \big(-c + b K_c\big) {{\boldsymbol v}}^T\partial^2_{{\boldsymbol x}}E({\boldsymbol x})\, {{\boldsymbol v}}\nonumber\\
& + \big(c {{\mbox{\boldmath $\lambda$}}}_x + b {{\boldsymbol v}}\big)^T \big(K_a\, {{\mbox{\boldmath $\lambda$}}}_x + K_b\, {{\boldsymbol v}}\big)\end{aligned}$$
Suppose $\partial^2_{{\boldsymbol x}}E({\boldsymbol x}) > 0$. Let $c < 0$ in . If $$K_a > 0, \quad K_b < 0, \quad b K_a = c K_b, \quad \text{and}\quad K_c = a/c < 0$$ then, $\pounds_\beta V < 0$ for all $({{\mbox{\boldmath $\lambda$}}}_x, {{\boldsymbol v}}) \ne {{\bf 0}}$ and ${{\boldsymbol u}}$ given by .
Substituting $b K_a = c K_b$ in the third term of generates, $$\label{eq:term3-result}
\big(c {{\mbox{\boldmath $\lambda$}}}_x + b {{\boldsymbol v}}\big)^T \big(K_a\, {{\mbox{\boldmath $\lambda$}}}_x + K_b\, {{\boldsymbol v}}\big) =\frac{K_b}{b}\big(c {{\mbox{\boldmath $\lambda$}}}_x + b {{\boldsymbol v}}\big)^T\big(c {{\mbox{\boldmath $\lambda$}}}_x + b {{\boldsymbol v}}\big) \le 0$$ where, the inequality in follows from the assumption that $K_b < 0$.
With $K_c = a/c$, the first term of vanishes. The second term simplifies to, $$\big(-c + b K_c\big) {{\boldsymbol v}}^T\partial^2_{{\boldsymbol x}}E({\boldsymbol x})\, {{\boldsymbol v}}= \left(\frac{-c^2 + ab}{c} \right){{\boldsymbol v}}^T\partial^2_{{\boldsymbol x}}E({\boldsymbol x})\, {{\boldsymbol v}}$$ Because $ab - c^2 >0$ and $c <0$, it follows that the second term of is negative for a positive definite Hessian; hence, $\pounds_\beta V < 0$.
Let,
$$\begin{aligned}
K_a &:= \gamma_a, &\gamma_a > 0\\
K_b &:= -\gamma_b, & \gamma_b > 0 \\
K_c &:= - \gamma_c, & \gamma_c > 0\end{aligned}$$
then, the singular control law given by generates a family of (continuous) Nesterov-type accelerated gradient methods given by, $$\label{eq:Nesterov}
\ddot{\boldsymbol x}+ \gamma_a\, \partial_{{\boldsymbol x}} E({\boldsymbol x}) + \gamma_b\, \dot{\boldsymbol x}+ \gamma_c \, \partial^2_{{\boldsymbol x}}E({\boldsymbol x})\, \dot{\boldsymbol x}= {{\bf 0}}$$
Equation follows from $\ddot{\boldsymbol x}= {{\boldsymbol u}}$, with ${{\boldsymbol u}}$ given by . The claim that the resulting ODE generates a family of Nesterov’s accelerated gradient method[@nesterov83] follows by considering a discretization of . To this end, consider first a discretization of the the last term on the left-hand-side of : $$\label{eq:term-3}
\gamma_c \, \partial^2_{{\boldsymbol x}}E({\boldsymbol x})\, \dot{\boldsymbol x}= \gamma_c \,\frac{d}{dt}\Big(\partial_{{\boldsymbol x}}E({\boldsymbol x})\Big) \longrightarrow \frac{\gamma_c}{h_k}\Big(\partial_{{\boldsymbol x}}E({\boldsymbol x}_k) - \partial_{{\boldsymbol x}}E({\boldsymbol x}_{k-1})\Big)$$ where, $h_k > 0$ is a discretization step. Next, consider the first three terms of . These are identical to Polyak’s equation (Cf. ); hence it follows from and that may be discretized as, $$\label{eq:Nesterov-derived}
{\boldsymbol x}_{k+1} = {\boldsymbol x}_k - \alpha_k \partial_{{\boldsymbol x}}E({\boldsymbol x}_k) + \beta_k ({\boldsymbol x}_k - {\boldsymbol x}_{k-1}) - \gamma_k \Big(\partial_{{\boldsymbol x}}E({\boldsymbol x}_k) - \partial_{{\boldsymbol x}}E({\boldsymbol x}_{k-1})\Big)$$ Nesterov’s method[@nesterov83] is given by,
$$\begin{aligned}
{\boldsymbol x}_k &= {{\boldsymbol y}}_k -\alpha_k\, \partial_{{{\boldsymbol y}}}E({{\boldsymbol y}}_k) \label{eq:step-1:nes}\\
{{\boldsymbol y}}_{k+1} & = {\boldsymbol x}_k + \beta_k ({\boldsymbol x}_k - {\boldsymbol x}_{k-1}) \label{eq:step-2:nes}\end{aligned}$$
Substituting in generates $$\label{eq:Nesterov-y}
{{\boldsymbol y}}_{k+1} = {{\boldsymbol y}}_k - \alpha_k \partial_{{{\boldsymbol y}}}E({{\boldsymbol y}}_k) + \beta_k ({{\boldsymbol y}}_k - {{\boldsymbol y}}_{k-1}) - \alpha_k\beta_k \Big(\partial_{{{\boldsymbol y}}}E({{\boldsymbol y}}_k) - \partial_{{{\boldsymbol y}}}E({{\boldsymbol y}}_{k-1})\Big)$$ which is the same as with $\gamma_k = \alpha_k\beta_k$.
Equation was introduced and studied by Alvarez et al[@alvarez2002] as a “dynamical inertial Newton” system. Shi et al [@shi:hi-res-Nes] generated this system as a “high-resolution” ODE that represents Nesterov’s method[@nesterov83] in continuous-time. Equation differs from by an additive “Hessian-driven damping” term[@alvarez2002] which has the effect of a “gradient correction”[@shi:hi-res-Nes] a vis-à-vis Polyak’s equation[@polyak64].
References {#references .unnumbered}
==========
[10]{}
I. M. Ross, An optimal control theory for nonlinear optimization, J. Comp. and Appl. Math., 354 (2019) 39–51. B. T. Polyak, Some methods of speeding up the convergence of iteration methods, USSR Computational Math. and Math. Phys., 4/5 (1964) 1–17 (Translated by H. F. Cleaves).
W. Su, S. Boyd, E. J. Candes, A differential equation for modeling Nesterov’s accelerated gradient method: theory and insights, J. machine learning research, 17 (2016) 1–43.
R. B. Vinter, Optimal Control, Birkhäuser, Boston, MA, 2000
I. M. Ross, A Primer on Pontryagin’s Principle in Optimal Control, second ed., Collegiate Publishers, San Francisco, CA, 2015.
E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, second ed., Springer, New York, NY, 1998.
R. A. Freeman, P. V. Kokotović, Optimal nonlinear controllers for feedback linearizable systems, Proc. ACC, Seattle, WA, June 1995.
Yu. E. Nesterov, A method of solving a convex programming problem with convergence rate $\mathcal{O}(1/k^2)$, Soviet Math. Dokl., 27/2 (1983) 371–376 (Translated by A. Rosa).
F. Alvarez, H. Attouch, J. Bolte, P. Redont, A second-order gradient-like dissipative dynamical system with Hessian-driven damping. Applications to optimization and mechanics. J. Math. Pures Appl. 81 (2002) 747–779.
B. Shi, S. S. Du, M. I. Jordan, and W. J. Su, Understanding the acceleration phenomenon via high-resolution differential equations, arXiv preprint (2018) arXiv:1810.08907.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
**Abstract.** We consider a large class of geodesic metric spaces, including Banach spaces, hyperbolic spaces and geodesic $\mathrm{CAT}(\kappa)$-spaces, and investigate the space of nonexpansive mappings on either a convex or a star-shaped subset in these settings. We prove that the strict contractions form a negligible subset of this space in the sense that they form a $\sigma$-porous subset. For certain separable and complete metric spaces we show that a generic nonexpansive mapping has Lipschitz constant one at typical points of its domain. These results contain the case of nonexpansive self-mappings and the case of nonexpansive set-valued mappings as particular cases.\
**Mathematics Subject Classification (2010).** 47H09, 47H04, 54E52\
**Keywords.** Banach space, hyperbolic space, metric space, nonexpansive mapping, porous set, set-valued mapping, star-shaped set, strict contraction
author:
- Christian Bargetz
- Michael Dymond
- Simeon Reich
title: Porosity Results for Sets of Strict Contractions on Geodesic Metric Spaces
---
Introduction
============
The question of existence of fixed points for nonexpansive mappings $$f\colon C\to C,$$ where $C$ denotes a certain nonempty closed subsets of a complete metric spaces $X$, has been well studied. Recall that a mapping $f$ is called *nonexpansive* if it satisfies, for all $x,y\in C$, the inequality $$\rho(f(x),f(y))\leq \rho(x,y),$$ where $\rho$ denotes the metric on $X$. If $X$ is a Euclidean space and $C\subset X$ is bounded, closed and convex, Brouwer’s fixed point theorem (Satz 4 in [@Bro1911Abbildungen]) states that every continuous mapping $f\colon C\to C$ has a fixed point. The example $$T\colon C \to C, \quad Tx := (1, x_1, x_2, \ldots),$$ where $C := \{x\in c_0\colon 0\leq x_n \leq 1\}$, shows that in infinite dimensions there are noncompact $C$ and nonexpansive mappings $f\colon C\to C$ without fixed points. In 1965 F. E. Browder showed in [@Bro1965HilbertSpace] that nonexpansive mappings on the closed unit ball of the Hilbert space $\ell_2$ have a fixed point. Detailed discussions of the fixed point property for nonexpansive mappings can be found, for example, in Section 1.6 of [@GR1984UniformConvexity] and in Chapter 4 of [@GK1990FixedPointTheory]. More recent results are presented, for instance, in [@Pia2015FixedPointProperty] and in the references cited therein.
Instead of characterizing the sets $C$ for which every nonexpansive self-mapping has a fixed point, F. S. De Blasi and J. Myjak took a different approach in [@DM1976Convergence; @DM1989Porosity]. They raised the question of whether the typical nonexpansive mapping has a fixed point. To be more precise, let $C$ be a bounded, closed and convex subset of a Banach space $X$, and denote by $$\mathcal{M} := \left\{f\colon C\to C\colon \|f(x)-f(y)\|\leq \|x-y\| \text{ for all }x,y\in C\right\}$$ the space of nonexpansive mappings on $C$ equipped with the metric of uniform convergence. In [@DM1976Convergence] they proved that there is a dense $G_\delta$-set $\mathcal{M}'$ in $\mathcal{M}$ such that each $f\in\mathcal{M}'$ has a unique fixed point which is the pointwise limit of the iterates of $f$. They improved this result in [@DM1989Porosity], where they showed that there is a set $\mathcal{M}_*\subset\mathcal{M}$ with a $\sigma$-porous complement such that each $f\in\mathcal{M}_*$ has a unique fixed point which is the uniform limit of the iterates of $f$. Put in different words, these results state that a generic nonexpansive mapping $f$ on a bounded, closed and convex subset of a Banach space has a unique fixed point which is the uniform limit of the iterates of $f$.
Since Banach’s fixed point theorem from 1922, see [@Ban1922EnsembesAbstraits], states that every *strict contraction*, that is, a mapping $$f\colon C\to C\quad\text{with}\quad \rho(f(x),f(y)) \leq L \rho(x,y)\text{ and } L<1,$$ has a unique fixed point which is the uniform limit of the iterates of $f$, the question arises whether a generic nonexpansive mapping on a bounded, closed and convex subset of a Banach space is, in fact, a strict contraction. Using the Kirszbraun-Valentine extension theorem, De Blasi and Myjak answered this natural question in the negative by showing in the aforementioned papers that if $X$ is a Hilbert space, then the set of strict contractions is $\sigma$-porous. In the recent article [@BD2016:Porosity] the first two authors were able to show (by employing different methods) that this also holds true for general Banach spaces $X$.
In [@Rak1962Contractive] E. Rakotch proved a generalisation of Banach’s fixed point theorem, where the Lipschitz constant can be replaced by a decreasing function. More precisely, a mapping $f\colon C\to C$ is called *contractive in the sense of Rakotch* if there exists a decreasing function $\phi^f\colon [0,\operatorname{diam}(C)]\to [0,1]$ such that $$\phi^f(t)< 1 \;\text{for}\; t>0 \quad\text{and}\quad \rho(f(x),f(y)) \leq \phi^f(\rho(x,y)) \rho(x,y)$$ for all $x,y\in C$. Theorem 2 in [@Rak1962Contractive] shows that every Rakotch contractive mapping has a unique fixed point which is the limit of the sequence of iterates of $f$. It can be shown that this fixed point is the uniform limit of the iterates of $f$.
In [@RZ2001NoncontractiveMappings] the third author together with A. J. Zaslavski showed that there is a subset $\mathcal{M}_*\subset\mathcal{M}$ such that $\mathcal{M}\setminus\mathcal{M}_*$ is $\sigma$-porous and every $f\in\mathcal{M}_*$ is Rakotch contractive. This result can be interpreted as an explanation of the results of De Blasi and Myjak.
F. Strobin showed in [@Str2012PorousAndMeager] that in the case of an *unbounded* domain $C$ this result is no longer true, but the original result of De Blasi and Myjak still holds.
In [@RZ2016TwoPorosity] the Banach space $X$ has been replaced by a hyperbolic space, that is, a complete metric space together with a family of metric lines such that the resulting triangles are thin enough. In addition, in the unbounded case, a different metric on $\mathcal{M}$ is introduced and used to show that typical nonexpansive mappings are Rakotch contractive on bounded subsets.
Corresponding results, concerning the fixed point question and the prevalence of contractive mappings, for nonexpansive *set-valued* mappings on star-shaped subsets of Banach and hyperbolic spaces have been presented in [@BMRZ2009GenericExistence] and [@PL2014ContractiveSetValued].
The aim of the present paper is to show that in all the above cases the set of strict contractions is small in the sense that it is a $\sigma$-porous subset of the space of all nonexpansive mappings. In the case where the underlying space is separable, we further distinguish the nonexpansive mappings for which the Lipschitz constant is, in a certain sense, universally equal to one. We prove that even these mappings dominate the space of all nonexpansive mappings to the extent that they form the complement of a $\sigma$-porous subset. This extends [@BD2016:Porosity Theorem 2.2] to more general settings.
The paper is structured as follows: In Section 2 we develop the necessary background, before presenting our main results in Section 3. These statements are all obtained from a construction, given in Section 4. Finally, in Section 5 we discuss an application of our main results to set-valued nonexpansive mappings. More precisely, we prove that for several important spaces of nonexpansive set-valued mappings, the subset of strict contractions is $\sigma$-porous.
Preliminaries and notations
===========================
In this section we introduce the key concepts with which we work and establish various notations which appear throughout the paper.
Nonexpansive mappings
---------------------
The central objects of study in this paper are spaces of nonexpansive mappings. Let $(X,\rho_{X})$ and $(Y,\rho_{Y})$ be complete metric spaces, and fix a point $\theta\in X$. By $${\mathcal{M}}:= \mathcal{M}(X,Y) := \{f\colon X\to Y\colon \operatorname{Lip}(f)\leq 1\}$$ we denote the space of nonexpansive mappings from $X$ to $Y$ equipped with the metric $$\label{eq:metricDTheta}
d_\theta (f,g) := \sup \left\{\frac{\rho_{Y}(f(x),g(x))}{1+\rho_{X}(x,\theta)} \colon x\in X\right\}.$$ The inequalities $$\begin{aligned}
\frac{\rho_{Y}(f(x),g(x))}{1+\rho_{X}(x,\theta)}
& \leq \frac{\rho_{Y}(f(x),f(\theta))+\rho_{Y}(f(\theta),g(\theta))+\rho_{Y}(g(\theta),g(x))}{1+\rho_{X}(x,\theta)}\\
& \leq \frac{\rho_{Y}(f(\theta),g(\theta))+2\rho_{X}(x,\theta)}{1+\rho_{X}(x,\theta)}
\leq \rho_{Y}(f(\theta),g(\theta)) + 2,\end{aligned}$$ which hold for all $x\in X$, show that $d_\theta$ is well defined. We note that the space ${\mathcal{M}}$ endowed with the metric $d_{\theta}$ is a complete metric space. Moreover, the topology of $\mathcal{M}$ does not depend on the particular choice of the point $\theta$: given $\theta_1\neq\theta$, the inequalities $$1 + \rho_{X}(x,\theta) \leq 1 + \rho_{X}(x,\theta_1) + \rho_{X}(\theta_1, \theta)
\leq (1+\rho_{X}(x,\theta_1)) (1+\rho_{X}(\theta,\theta_1)),$$ where $x\in X$, imply that the metrics $d_{\theta_1}$ and $d_\theta$ are Lipschitz equivalent. For a detailed discussion of the metric $d_\theta$, we refer the interested reader to [@RZ2016TwoPorosity].
Porosity
--------
Our main results concern a special class of exceptional sets in metric spaces, namely the class of $\sigma$-porous sets, which were introduced in [@Den1941LeconsII; @dolvzenko1967granivcnye]. We define now the notion of porosity, according to [@Zaj2005Porous]. In the context of a metric space, we write $B(x,r)$ for the open ball with centre $x$ and radius $r$, and later $\overline{B}(x,r)$ for the closed ball.
Given a metric space $(M,d)$, a subset $A\subset M$ is called *porous at a point $x\in A$* if there exist $\varepsilon_0>0$ and $\alpha>0$ such that for all $\varepsilon\in(0,\varepsilon_0)$, there exists a point $y\in B(x,\varepsilon)$ such that $B(y,\alpha\varepsilon)\cap A = \emptyset$. The set $A$ is called *porous* if it is porous at all its points and $A$ is called *$\sigma$-porous* if it is the countable union of porous sets.
Note that this definition of porosity differs from the one in some of the aforementioned literature (e.g., [@DM1989Porosity]), where $A$ is called porous if the constants $\varepsilon_0$ and $\alpha$ are independent of the point $x$. Also, there the condition on $y$ reads as $B(y,\alpha\varepsilon)\subset (M\setminus A) \cap B(x,\varepsilon)$. This condition is equivalent to the one above as can be seen by choosing the point $y$ for a smaller $\varepsilon$ and adjusting $\alpha$ appropriately. For $\sigma$-porosity it also does not matter whether we assume that $\varepsilon_0$ and $\alpha$ are independent of the point $x$: assume we have a decomposition $A = \bigcup_{i=1}^{\infty} A_i$ and every $A_i$ is porous. For every $j,k\in\mathbb{N}$, define $$A_i^{j,k} := \left\{x\in A_i\colon \varepsilon_0(x) \geq\frac{1}{j},\; \alpha(x)\geq\frac{1}{k}\right\}.$$ Then $A= \bigcup_{i,j,k=1}^{\infty} A_{i}^{j,k}$ and each set $A_i^{j,k}$ is porous in the sense of [@DM1989Porosity]. For a detailed discussion of the different concepts of porosity, we refer the reader to L. Zajíček’s survey article [@Zaj2005Porous]. For the history of porosity, we also refer to [@Bul1984DenjoyIndex; @Ren1995Porosity].
Geodesic metric spaces
----------------------
A metric space $(X,\rho_{X})$ is called *geodesic* if for every pair $x,y\in X$, there is an isometric embedding $c:[0,\rho_{X}(x,y)]\to X$ satisfying $c(0)=x$ and $c(\rho_{X}(x,y))=y$. The image of such an embedding is referred to as a *metric segment* in $X$ with endpoints $x$ and $y$, and denoted by $[x,y]$. Such metric segments may not be unique and so the notation $[x,y]$ is in general not well defined. Given $\lambda\in[0,1]$ and a choice of metric segment $[x,y]$, we denote by $(1-\lambda)x\oplus\lambda y$ the unique point $z\in[x,y]$ satisfying $\rho_X(z,x)=\lambda \rho_X(x,y)$ and $\rho_X(z,y)=(1-\lambda)\rho_X(x,y)$. In places where we wish to emphasise that this point is defined according to the geodesic structure on the metric space $X$, we will write $(1-\lambda)x\oplus_{X}\lambda y$. An image of $\mathbb{R}$ by an isometric embedding is called a *metric line*.
The most general setting in which the space of nonexpansive mappings on a convex set has so far been studied is that of a hyperbolic space; see [@RS1990Nonexpansive] and [@RZ2016TwoPorosity].
Given a metric space $(X,\rho_{X})$ and a family $\mathcal{F}$ of metric segments in $X$, we call the triple $(X,\rho_{X},{\mathcal{F}})$ *hyperbolic* if the following conditions are satisfied:
(i) For each pair $x,y\in X$, there exists a unique metric segment $[x,y]\in{\mathcal{F}}$ joining $x$ and $y$.
(ii) For all $x,y, z, w\in X$ and all $t\in[0,1]$, $$\label{eq:hyperineq}
\rho_{X}((1-t)x\oplus ty,(1-t)w\oplus tz)\leq (1-t)\rho_{X}(x,w)+t\rho_{X}(y,z).$$
(iii) The collection $\mathcal{F}$ is closed with respect to subsegements. More precisely, for all $x,y\in X$ and $u,v\in[x,y]$ we have $[u,v]\subseteq [x,y]$.
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(i) Note that our definition of hyperbolic spaces slightly differs from the one in [@RS1990Nonexpansive] since the original definition demands that every pair of points $x,y\in X$ admits a unique metric line $l\in\mathcal{F}$ such that $x,y\in l$. We note that in both variants of the definition the hyperbolic inequality can be replaced with the following inequality for midpoints: $$\rho_{X}\left(\frac{1}{2}x\oplus\frac{1}{2}y,\frac{1}{2}x\oplus\frac{1}{2}z\right)\leq \frac{1}{2}\rho_{X}(y,z).$$ A detailed discussion of different notions of hyperbolicity and convexity can be found in Remark 2.13 in [@Koh2005Metatheorems page 98].
(ii) The hyperbolic inequality was introduced by Busemann in [@Bus1948NonpositiveCurvature] and is sometimes referred to as *Busemann convexity*, cf. [@Esp1015ContinuousSelections p. 743].
Nonexpansive mappings on convex and star-shaped subsets of Banach and hyperbolic spaces have been investigated in [@BD2016:Porosity], [@Str2014Porosity], [@BMRZ2009GenericExistence] and [@RZ2016TwoPorosity]. We define below notions of convexity and star-shapedness in more general settings:
\[def:star-shapedconvex\] Let $X$ be a metric space with metric $\rho_{X}$ and let ${\mathcal{F}}$ be a family of metric segments in $X$.
1. We say that a subset $C$ of $X$ is *$\rho_{X}$-star-shaped* with respect to a point $x_{0}\in C$ if for every point $x\in C$, there is a metric segment $[x,x_{0}]\in {\mathcal{F}}$ such that $[x,x_{0}]\subseteq C$. Moreover, we write ${\operatorname{star}}(C)$ for the set of points $y\in C$ with respect to which $C$ is $\rho_{X}$-star-shaped.
2. We call a subset $C$ of $X$ *$\rho_{X}$-convex* if for each pair $x,y\in C$ there is a metric segment $[x,y]\in {\mathcal{F}}$ such that $[x,y]\subseteq C$.
Clearly, convexity is stronger than star-shapedness: A set $C\subseteq X$ is $\rho_{X}$-convex if and only if $C$ is $\rho_{X}$-star-shaped with respect to $y$ for every point $y\in C$, i.e. ${\operatorname{star}}(C)=C$. As a note of caution, we emphasise that the metric segments occurring in Definition \[def:star-shapedconvex\] need not be unique. Whenever we require that a metric segment $[x,y]$ be well defined, we will need to use condition of Definition \[def:weaklyhyperbolic\] below. Finally, let us point out that the above definitions of $\rho_{X}$-convex and $\rho_{X}$-star-shaped sets generalise the established notions in vector spaces and coincide with the notions defined for hyperbolic spaces.
Weakly hyperbolic spaces
------------------------
Whilst hyperbolic spaces form an important class of metric spaces, one can observe that even quite well-behaved metric spaces are excluded from this class. For an example, consider the unit sphere $\mathbb{S}^{2}$ in ${\mathbb{R}}^{3}$. For non-antipodal points $x,y\in \mathbb{S}^{2}$, there is a unique geodesic on the sphere with endpoints $x$ and $y$. However, antipodal points $x,-x\in \mathbb{S}^{2}$ admit infinitely many geodesics between them and there is no way to define the metric segment $[x,-x]$ so that the hyperbolic inequality is satisfied. Even if we relax the uniqueness condition on the family of metric segments, the sphere still presents problems. Taking $y=z$ in inequality , we get $$\rho_{X}((1-t)x\oplus ty,(1-t)w\oplus t y)\leq (1-t)\rho_{X}(x,w).$$ However, if we take $y\in \mathbb{S}^{2}$ to be the north pole, $x$ and $w$ to be two distinct points lying on the same line of latitude in the southern hemisphere, we observe that $$\rho_{\mathbb{S}^{2}}((1-t)x\oplus ty,(1-t)w\oplus ty)>\rho_{\mathbb{S}^{2}}(x,w).$$ for small $t\in(0,1)$. In other words, it is easy to find triangles on the sphere which become ‘fatter’ as one moves away from their base towards their peak.
Thus, we propose to weaken the hyperbolic condition, in order to capture a larger class of metric spaces, including the sphere $\mathbb{S}^{2}$ and all geodesic $\operatorname{CAT}(\kappa)$ spaces.
\[def:weaklyhyperbolic\] Given a metric space $(X,\rho_{X})$ and a family ${\mathcal{F}}$ of metric segments in $X$, we say that the triple $(X,\rho_{X},{\mathcal{F}})$ is *of temperate curvature* if the following conditions are satisfied:
(i) \[localuniqueness\] There exists a constant $D_{X}>0$ such that for any $x,y\in X$ with $\rho_{X}(x,y)<D_{X}$, there is at most one metric segment $[x,y]\in{\mathcal{F}}$ with endpoints $x$ and $y$. In the case where the metric segments in the family $\mathcal{F}$ are unique, we set $D_{X}=\infty$.
(ii) \[thinishtriangles\] For all $x,y\in X$ with $\rho_{X}(x,y)<D_{X}$ and every $\sigma>0$, there exists a positive number $\delta_{X}=\delta_{X}(x,y,\sigma)$ such that $$\label{eq:defdelta}
\rho_{X}((1-t)z\oplus ty,(1-t)w\oplus ty)\leq (1+\sigma)\rho_{X}(z,w)$$ whenever $z,w\in B(x,\delta_{X})$, $[z,y],[w,y]\in\mathcal{F}$ and $t\in [0,\delta_{X})$.
A triple $(X,\rho_{X},\mathcal{F})$ of temperate curvature is called *weakly hyperbolic* if, in addition, the following conditions are satisfied:
(iii) \[closedwrtsubseg\] $\mathcal{F}$ is closed with respect to subsegments, that is, for all metric segments $[x,y]\in\mathcal{F}$ and all points $z,w\in[x,y]$ there is a metric segment $[z,w]\in\mathcal{F}$ with $[z,w]\subseteq [x,y]$.
(iv) \[geodesic\] For all $x,y\in X$ there exists a metric segment $[x,y]\in\mathcal{F}$.
(v) \[smallballs\] For all $x\in X$ and $r\in(0,D_{X}/2)$, the ball $B(x,r)$ is a $\rho_{X}$-convex subset of $X$.
When referring to either a space of temperate curvature or to a weakly hyperbolic space $(X,\rho_{X},{\mathcal{F}})$, we often suppress the metric $\rho_{X}$ and the family of metric segments ${\mathcal{F}}$.
Condition weakens the assumption that every pair of points is connected by a unique metric segment. We note that the sphere $\mathbb{S}^{2}$ satisfies condition with $D_{\mathbb{S}^2}=\pi$. Condition is a significant weakening of the hyperbolic inequality and allows us to form ‘thin-ish’ triangles in the space $X$. Let us imagine that we wish to form a triangle $T$ with vertices $y,z,w$ in $X$. We fix first the ‘peak’ $y$ of the triangle $T$ and then consider an arbitrary location $x\in X$ with $\rho_{X}(x,y)<D_{X}$. Condition allows us to choose a small neighborhood of the point $x$ so that placing the remaining two vertices $z,w$ in this neighborhood, we produce a triangle in which the sides $[z,y]$ and $[w,y]$ do not bulge out too much as one moves a little away from the base of the triangle $[z,w]$ towards the peak $y$.
It is clear that all hyperbolic spaces are weakly hyperbolic. We now demonstrate that the class of weakly hyperbolic spaces is significantly larger than that of hyperbolic spaces. More precisely, we show that all geodesic [CAT($\kappa$) ]{}spaces are weakly hyperbolic. Let us first recall the definition of [CAT($\kappa$) ]{}spaces, from [@BH1999MetricSpaces].
1. We define a family of model spaces $(M_{\kappa})$, where $\kappa\in{\mathbb{R}}$, as follows:
1. For $\kappa>0$ we let $M_{\kappa}$ denote the metric space given by the sphere $\mathbb{S}^{2}$ with its standard path length metric, scaled by a factor of $1/\sqrt{\kappa}$.
2. We define $M_{0}$ as the Euclidean space ${\mathbb{R}}^{2}$.
3. For $\kappa<0$ we write $M_{\kappa}$ for the hyperbolic space $\mathbb{H}^{2}$ (see [@BH1999MetricSpaces Definition 2.10]) with metric scaled by a factor of $1/\sqrt{-\kappa}$.
We write $d_{\kappa}$ for the metric on $M_{\kappa}$.
2. Let $\kappa\in {\mathbb{R}}$ and $(X,\rho_{X})$ be a metric space. Given three points $x_{1},x_{2},x_{3}\in X$ and metric segments of the form $[x_{1},x_{2}],[x_{2},x_{3}],[x_{3},x_{1}]\subseteq X$ we call the union $[x_{1},x_{2}]\cup[x_{2},x_{3}]\cup[x_{3},x_{1}]$ a *geodesic triangle* with vertices $x_{1},x_{2},x_{3}$. A geodesic triangle with vertices $\overline{x}_{1},\overline{x}_{2}, \overline{x}_{3}$ in $M_{\kappa}$ is said to be a *comparison triangle* for a geodesic triangle with vertices $x_{1},x_{2},x_{3}$ in $X$ if $d_{\kappa}(\overline{x}_{i},\overline{x}_{j})=\rho_{X}(x_{i},x_{j})$. A point $\overline{x}\in [\overline{x}_{i},\overline{x}_{j}]$ is called a *comparison point* for $x\in[x_{i},x_{j}]$ if $d_{\kappa}(\overline{x},\overline{x}_{k})=\rho_{X}(x,x_{k})$ for $k=i,j$.
3. Let $(X,\rho_{X})$ be a metric space. If $\kappa\leq 0$, then $(X,\rho_{X})$ is called a [CAT($\kappa$) ]{}space if it is geodesic and every geodesic triangle $T$ in $X$ has a comparison triangle $\overline{T}$ in $M_{\kappa}$ such that $$\label{eq:comparisontriangle}
\rho_{X}(x,y)\leq d_{\kappa}(\overline{x},\overline{y})$$ whenever $\overline{x},\overline{y}\in\overline{T}$ are comparison points for $x,y\in T$. If $\kappa>0$, then we define a constant $D_{\kappa}=\operatorname{diam}M_{\kappa}=\frac{\pi}{\sqrt{\kappa}}$ and we say that $(X,\rho_{X})$ is a [CAT($\kappa$) ]{}space if for every pair of points $x,y\in X$ with $\rho_X(x,y)< D_\kappa$ there is a metric segment joining $x$ and $y$ and every geodesic triangle $T\subseteq X$ with perimeter smaller that $2D_\kappa$, that is, $\rho_X(x,y)+\rho_X(y,z)+\rho_X(z,x)< 2D_{\kappa}$, where $x,y,z$ denote the vertices of $T$, has a comparison triangle $\overline{T}$ in $M_\kappa$ such that is satisfied.
Thus, [CAT($\kappa$) ]{}spaces can be thought of as metric spaces for which every sufficiently small geodesic triangle is ‘thinner’ in all directions than a corresponding triangle in the model space $M_{\kappa}$. The classes of [CAT($\kappa$) ]{}spaces are increasing in the sense that whenever $X$ is a [CAT($\kappa$) ]{}space, it is also a $\operatorname{CAT}(\kappa')$ space for all $\kappa'\geq \kappa$; see [@BH1999MetricSpaces Theorem 1.12].
In the proof of the next proposition, the most difficult task is to establish that every [CAT($\kappa$) ]{}space satisfies condition (\[thinishtriangles\]) of Definition \[def:weaklyhyperbolic\] and, in particular, to verify inequality . A related inequality for geodesic triangles with side lengths smaller than $\pi/2$ in CAT($1$) spaces is shown in Lemma 3.3 of [@Pia2011HalpernIteration].
\[prop:CATisweaklyhyperbolic\] Every geodesic [CAT($\kappa$) ]{}space is weakly hyperbolic.
Let $(X,\rho_{X})$ be a [CAT($\kappa$) ]{}space and $\mathcal{F}$ be the collection of all geodesics in $X$. We show that the triple $(X,\rho_{X},\mathcal{F})$ is a weakly hyperbolic space. We may assume that $\kappa>0$. It is already clear that the family $\mathcal{F}$ satisfies conditions and of Defintion \[def:weaklyhyperbolic\]. For a proof that $X$ satisfies conditions and with $D_{X}=D_{\kappa}$ we refer the reader to [@BH1999MetricSpaces Proposition 1.4]. We now verify that $X$ satisfies condition . As a first step, we show that it is sufficient to verify that the sphere $\mathbb{S}^{2}$ with metric $\rho=\rho_{\mathbb{S}^{2}}$ scaled by $1/\sqrt{\kappa}$ satisfies this condition. Suppose that the model spaces satisfy conditon (\[thinishtriangles\]) of Definition \[def:weaklyhyperbolic\]. Let $(X,\rho_{X})$ be a [CAT($\kappa$) ]{}space and let $x,y\in X$ with $\rho_{X}(x,y)<D_{\kappa}$. Then we choose $\overline{x},\overline{y}\in M_{\kappa}$ with $d_{\kappa}(\overline{x},\overline{y})=\rho_{X}(x,y)$. Given $\sigma>0$, we choose $\delta=\delta_{X}(x,y,\sigma)\in(0,\delta_{\kappa}(\overline{x},\overline{y},\sigma)/4)$, where $\delta_{\kappa}(\overline{x},\overline{y},\sigma)$ is given by condition (ii) for $M_{\kappa}$, sufficiently small so that $\rho_{X}(x,y)+2\delta<D_{\kappa}$. Let $z,w\in B(x,\delta)$. Then by the triangle inequality, we have $$\rho_X(y,w)+\rho_X(w,z)+\rho_X(z,y) < 2(\rho_X(x,y)+2\delta) < 2D_\kappa.$$ Therefore we can choose a comparison triangle in $M_{\kappa}$ with vertices $\overline{y}',\overline{z},\overline{w}$ for the geodesic triangle with vertices $y,z,w$ in $X$. Since $d_{k}(\overline{u},\overline{y}')=\rho_{X}(u,y)$ for $u\in\left\{z,w\right\}$ and $z,w\in B(x,\delta)$, we have $$\left|d_{\kappa}(\overline{u},\overline{y}')-\rho_{X}(x,y)\right|< \delta$$ for $u\in\left\{z,w\right\}$. It follows that there is a great circle passing through $\overline{z}$ and $\overline{y}'$ and a point $\overline{x}'$ on this great circle with $$\label{eq:condxp}
d_{\kappa}(\overline{x}',\overline{y}')=\rho_{X}(x,y)\quad\text{and}\quad d_{\kappa}(\overline{x}',\overline{z})<\delta.$$ Since the metric $d_{\kappa}$ on $M_{\kappa}$ is invariant under isometries of the sphere, we may assume now that $\overline{y}'=\overline{y}$ and $\overline{x}'=\overline{x}$. Then we have $\overline{z},\overline{w}\in B(\overline{x},4\delta)\subset B(\overline{x},\delta_{\kappa}(\overline{x},\overline{y},\sigma))$. Therefore, by condition (ii) for $M_{\kappa}$, we get $$d_{\kappa}((1-t)\overline{z}\oplus t\overline{y},(1-t)\overline{w}\oplus t\overline{y})\leq (1+\sigma)d_{k}(\overline{z},\overline{w})=(1+\sigma)\rho_{X}(z,w)$$ for all $t\in[0,\delta_{\kappa}(\overline{x},\overline{y},\sigma))$. Consequently, by , $$\rho_{X}((1-t)z\oplus ty,(1-t)w\oplus ty)\leq d_{\kappa}((1-t)\overline{z}\oplus t\overline{y},(1-t)\overline{w}\oplus t\overline{y})\leq (1+\sigma)\rho_{X}(z,w)$$ for all $t\in[0,\delta)\subseteq(0,\delta_{\kappa}(\overline{x},\overline{y},\sigma))$.
From this point on we will assume that $\kappa=1$, since multiplying the metric $\rho$ on the sphere by a factor of $1/\sqrt{\kappa}$ does not affect any of the calculations which follow.
Let $x,y\in \mathbb{S}^{2}$ with $\rho(x,y)<D_{\kappa}=\pi$ and fix $\sigma>0$. Note that $x$ and $y$ cannot be antipodal. We consider two cases, namely $\rho(x,y)>0$ and $\rho(x,y)=0$. We start with the case $\rho(x,y)>0$ and let $\delta=\delta(x,y,\sigma)\in(0,\pi/8)$ be some positive constant to be determined later in the proof. For now we just prescribe that $\delta$ be small enough so that $$I(x,y,\delta):=[(1-\delta)(\rho(x,y)-\delta),\rho(x,y)+\delta]\subseteq (0,\pi).$$ We define constants $m_{\sin}=m_{\sin}(x,y,\delta)$ and $M_{\sin}=M_{\sin}(x,y,\delta)$ by $$\begin{aligned}
m_{\sin}&:=\min\left\{\sin \theta \colon \theta\in I(x,y,\delta)\right\},\\
M_{\sin}&:=\max\left\{\sin \theta \colon \theta\in I(x,y,\delta)\right\},
\end{aligned}$$ and define constants $m_{\cos}$, $M_{\cos}$ analogously with $\sin$ replaced by $\cos$. Note that $$m_{\sin},M_{\sin}\to\sin\rho(x,y)\quad\text{and}\quad m_{\cos},M_{\cos}\to\cos\rho(x,y)$$ as $\delta\to 0^{+}$.
For points $z\in B(x,\delta)$ we write $|z|=\rho(z,y)$. We note that $\left||z|-|x|\right|\leq \rho(z,x)\leq \delta$ and hence $|z|\in I(x,y,\delta)\subseteq(0,\pi)$ for all $z\in B(x,\delta)$. For points $z,w\in B(x,\delta)$, we let $\Theta(z,w)$ denote the angle at the vertex $y$ of the spherical triangle with vertices $z,w,y$. In what follows we use the spherical law of cosines [@Jen1994ModernGeometry Proposition 2.4.1] and the equivalent law of haversines: $$\cos c=\cos a\cos b+\sin a\sin b\cos C,\quad {\operatorname{hav}}c={\operatorname{hav}}(a-b)+\sin a\sin b{\operatorname{hav}}C,$$ where ${\operatorname{hav}}\theta:=\sin^{2}(\theta/2)$, which relate the side lengths $a$, $b$, $c$ of a spherical triangle to the angle $C$ at the vertex opposite to the side of length $c$.
Let $z,w\in B(x,\delta)$. Applying the spherical law of cosines to the spherical triangle with vertices $z$, $w$ and $y$, we deduce that $$1\geq\cos \Theta(z,w)=\frac{\cos \rho(z,w)-\cos |z|\cos |w|}{\sin |z|\sin |w|}
\geq \frac{\cos2\delta-\max\left\{M_{\cos}^{2},m_{\cos}^{2}\right\}}{M_{\sin}^{2}}$$ provided we choose $\delta$ small enough so that $\cos2\delta-\max\left\{M_{\cos}^{2},m_{\cos}^{2}\right\}\geq 0$. In the above we use the facts that $\cos $ is decreasing on the interval $(0,\pi/2)$ and $\rho(z,w)\leq 2\delta$. The last expression is independent of $z,w\in B(x,\delta)$ and converges to 1. It follows that $$\label{eq:anglecontrol}
\sup\left\{\Theta(z,w) \colon z,w\in B(x,\delta)\right\}\to0\quad\text{ as }\quad\delta\to {0^{+}}.$$ For $t\in [1-\delta,1]$ and $z,w\in B(x,\delta)$, we consider the spherical triangle with vertices $z_{t}:=tz\oplus (1-t)y$, $w_{t}:=tw\oplus (1-t)y$, and $y$. This triangle has sides of length $t\left|z\right|$, $t\left|w\right|$ and $\rho(z_{t},w_{t})$, and angle $\Theta(z,w)$ at the vertex $y$. Without loss of generality, we assume $|z|\geq |w|$ and note that the inequalities $||u|-|x||\leq \delta$ for all $u\in B(x,\delta)$ and $1-\delta\leq t \leq 1$ together with the definition of $I(x,y,\delta)$ imply that $t|z|,t|w|\in I(x,y,\delta)$. In addition note that $||z|-|w||\leq 2\delta<\pi/4$ by the triangle inequality and hence $|z|-|w|\in [0,\pi/4)$. Using the law of haversines, we obtain $$\begin{aligned}
&\frac{{\operatorname{hav}}\rho(z_{t},w_{t})}{{\operatorname{hav}}\rho(z,w)}=\frac{{\operatorname{hav}}(t(|z|-|w|))+\sin t|z|\sin t|w|
{\operatorname{hav}}\Theta(z,w)}{{\operatorname{hav}}(|z|-|w|)+\sin |z|\sin |w|{\operatorname{hav}}\Theta(z,w)}\\
&\leq 1+\frac{({\operatorname{hav}}(t(|z|-|w|))-{\operatorname{hav}}(|z|-|w|))+(\sin t|z|\sin t|w|-\sin |z|\sin |w|)
{\operatorname{hav}}\Theta(z,w)}{{\operatorname{hav}}(|z|-|w|)+\sin |z|\sin |w|{\operatorname{hav}}\Theta(z,w)}\\
&\leq 1+\frac{M_{\sin}^{2}-m_{\sin}^{2}}{m_{\sin}^{2}}.
\end{aligned}$$ To deduce the above inequalities we use the fact that ${\operatorname{hav}}$ is monotonically increasing and non-negative on the interval $[0,\pi/2)$ in combination with the constraints on $\delta$, $|z|$, $|w|$ and $t$ as discussed above. Observe that the last expression converges to $1$ as $\delta\to {0^{+}}$ and is independent of the choices of $z,w\in B(x,\delta)$ and $t\in[1-\delta,1]$. Given $\eta>0$ to be determined later in the proof, it follows that we can choose $\delta$ sufficiently small depending only on the points $x,y$ so that $$\label{eq:havct/havc}
\frac{{\operatorname{hav}}\rho(z_{t},w_{t})}{{\operatorname{hav}}\rho(z,w)}\leq 1+\eta \qquad \forall t\in[1-\delta,1],\quad \forall z,w\in B(x,\delta).$$ Next, observe that $$\begin{aligned}
{\operatorname{hav}}\rho(z_{t},w_{t})&={\operatorname{hav}}(t(|z|-|w|))+\sin t|z|\sin t|w|{\operatorname{hav}}\Theta(z,w)\\
&\leq {\operatorname{hav}}(2\delta)+M_{\sin}^{2}{\operatorname{hav}}(\sup\left\{\Theta(z,w)\colon z,w\in B(x,\delta)\right\})
\end{aligned}$$ since $t(|z|-|w|)<2\delta<\pi/2$ and ${\operatorname{hav}}$ is increasing on $[0,\pi/2)$. The last expression is independent of the choices of $z,w\in B(x,\delta)$ and $t\in[1-\delta,1]$ and, using , we see that it converges to $0$ as $\delta\to {0^{+}}$. Thus, using that ${\operatorname{hav}}\rho(z_t,w_t)\to 0$ implies $\rho(z_t,w_t)\to 0$ and the Taylor expansion of ${\operatorname{hav}}x $ at $x=0$, we can prescribe that $\delta>0$ be sufficiently small so that the following inequalities hold: $$\label{eq:havctapp}
{\operatorname{hav}}\rho(z_{t},w_{t})\geq \frac{\rho(z_{t},w_{t})^{2}}{4}-\eta \rho(z_{t},w_{t})^{2}\qquad \forall z,w\in B(x,\delta),
\quad \forall t\in[1-\delta,1],$$ $$\label{eq:havcapp}
{\operatorname{hav}}\rho(z,w)\leq \frac{\rho(z,w)^{2}}{4}+\eta \rho(z,w)^{2} \qquad \forall z,w\in B(x,\delta).$$ Combining inequalities , and , we deduce that $$\rho(z_{t},w_{t})^{2}\leq \frac{(1+\eta)(\frac{1}{4}+\eta)}{(\frac{1}{4}-\eta)}\rho(z,w)^{2} \qquad
\forall z,w\in B(x,\delta),\quad\forall t\in[1-\delta,1].$$ If we prescribe that $\eta$ be chosen sufficiently small so that the constant before $\rho(z,w)^{2}$ in the above inequality is at most $(1+\sigma)^{2}$, then we obtain the desired result.
If $\rho(x,y)=0$, we choose $\delta=\delta_{X}(x,y,\sigma)\in(0,\pi/4)$. Given $z,w\in B(x,\delta)$ and $t\in(0,1)$ we let $z_{t}:=tz\oplus (1-t)x$, $w_{t}:=tw\oplus (1-t) x$ and $\theta$ be the angle at the vertex $x$ of the spherical triangle with vertices $x,w,z$. For $u\in \mathbb{S}^{2}$ we also write $\left|u\right|$ for the distance $\rho(u,x)$. Then, for all $t\in[0,1]$, the law of haversines gives $$\begin{aligned}
{\operatorname{hav}}\rho(z_{t},w_{t})&={\operatorname{hav}}(t(\left|z\right|-\left|w\right|))+\sin(t\left|z\right|)\sin(t\left|w\right|){\operatorname{hav}}\theta\\
&\leq{\operatorname{hav}}(\left|z\right|-\left|w\right|)+\sin\left|z\right|\sin\left|w\right|{\operatorname{hav}}\theta\\
&={\operatorname{hav}}\rho(z,w).
\end{aligned}$$ In the above we used that ${\operatorname{hav}}$ is symmetric, non-negative and that ${\operatorname{hav}}$ and $\sin$ are increasing on the interval $[0,\pi/2)$. Using again that ${\operatorname{hav}}$ is increasing on the interval $[0,\pi/2)$, we conclude that $\rho(z_{t},w_{t})\leq\rho(z,w)$ for all $t\in[0,1]$. This is a stronger version of the inequality in Definition \[def:weaklyhyperbolic\], (\[thinishtriangles\]).
Given a subset $E\subset X$ of a metric space $X$ and $r>0$, we use the notations $$B(E,r) := \{x\in X\colon d(x,E) <r\}\qquad\text{and}\qquad \overline{B}(E,r) := \{x\in X\colon d(x,E) \leq r\}.$$ Note that if $X$ is a weakly hyperbolic space and $E\subseteq X$ is a nonempty subset, the set $\overline{B}(E,r)\setminus B(E,r)$ has empty interior. Indeed, any $x\in\overline{B}(E,r)\setminus B(E,r)$ satisfies $$\operatorname{dist}(x,E):=\inf\left\{\rho_{X}(x,u)\colon u\in E\right\}=r.$$ Given $0<\varepsilon<r$ we choose $x_{0}\in E$ such that $r\leq\rho_{X}(x,x_{0})<r+\varepsilon/2$. Then every point of the form $(1-\frac{\varepsilon}{\rho_{X}(x,x_{0})})x\oplus\frac{\varepsilon}{\rho_{X}(x,x_{0})}x_{0{}}$ lies in $\overline{B}(x,\varepsilon)\cap B(E,r)$. This shows that $\overline{B}(E,r)\setminus B(E,r)$ has empty interior. Note that the above argument also shows that for a $\rho_X$-star-shaped set $C\subset X$ and any $r>0$, $\overline{B}({\operatorname{star}}(C),r)\setminus B({\operatorname{star}}(C),r)$ has empty interior in $C$.
In addition, we get that in weakly hyperbolic spaces the closure of an open ball is the corresponding closed ball, that is, we have $\overline{B(x,r)} = \overline{B}(x,r)$ for all $x\in X$ and all $r>0$. The inclusion $\overline{B(x,r)}\subseteq \overline{B}(x,r)$ follows from the continuity of the metric whereas the opposite inclusion can be deduced analogously to the above argument using the fact that $[z,x]\setminus\left\{z\right\}\subseteq B(x,r)$ for any $z\in \overline{B}(x,r)$.
$\ell^{\infty}$ spaces
----------------------
We make frequent use of two special properties of $\ell_{\infty}$ spaces. Firstly, we exploit the fact that any metric space can be isometrically embedded into $\ell_{\infty}(\Omega)$ for some set $\Omega$. Thus, we often identify metric spaces with subsets of some $\ell_{\infty}$ space. Note that given two metric spaces $X$ and $Y$ which are isometrically embedded into $\ell_{\infty}(\Omega_1)$ and $\ell_{\infty}(\Omega_2)$, respectively, we can embed both $X$ and $Y$ isometrically into $\ell_{\infty}(\Omega_1\uplus\Omega_2)$, where $\Omega_1\uplus\Omega_2$ stands for the disjoint union of $\Omega_1$ and $\Omega_2$ since $\ell_{\infty}(\Omega_i)$, $i=1,2$, embeds isometrically into $\ell_{\infty}(\Omega_1\uplus\Omega_2)$. Secondly, we make use of the fact that any Lipschitz mapping defined on a subset of a metric space $M$ and taking values in some $\ell_{\infty}(\Omega)$, can be extended to a Lipschitz mapping $F\colon M\to\ell_{\infty}(\Omega)$ with the same Lipschitz constant. A detailed discussion of these special properties of $\ell_{\infty}$ spaces can be found in [@BL2000GeomtericNonlinear Chapter 1].
Main results
============
In this section we present our main results. In fact we show that all of our main results can be derived from a single theorem, Theorem \[thm:all\], which is proved in the next section. Before stating this result, we establish our general hypotheses.
\[hypotheses\] Let $(X,\rho_{X})$ be a complete, weakly hyperbolic space, $(Y,\rho_{Y})$ be a complete space of temperate curvature and $C_{X}\subseteq X$, $C_{Y}\subseteq Y$ be non-empty, closed, non-singleton and $\rho_{X}$- and $\rho_{Y}$-star-shaped subsets of $X$ and $Y$, respectively. Suppose that the set $C_{Y}$ satisfies $C_{Y}\subseteq B({\operatorname{star}}(C_{Y}),D_{Y})$. Let ${\operatorname{conv}}(C_{X})$ denote a $\rho_{X}$-convex subset of $X$ containing $C_{X}$ and choose a set $\Omega$ so that $X,Y\subset \ell_{\infty}(\Omega)$. Let $\theta\in X$ and ${\mathcal{M}}(C_{X},C_{Y})$ denote the space of nonexpansive mappings from $C_{X}$ to $C_{Y}$, equipped with the metric $d_{\theta}$. Let ${\mathcal{N}}(C_{X},C_{Y})$ denote the subset of ${\mathcal{M}}(C_{X},C_{Y})$ formed by the strict contractions. Given a mapping $f\in {\mathcal{M}}(C_{X},C_{Y})$, we let ${\mathcal{E}}(f)$ denote the set of all $1$-Lipschitz extensions $F\colon{\operatorname{conv}}(C_{X})\to \ell_{\infty}(\Omega)$ of $f$.
We note that the condition $C_{Y}\subseteq B({\operatorname{star}}(C_{Y}),D_{Y})$ is satisfied in particular in each of the following cases:
- $C_{Y}$ is $\rho_{Y}$-convex,
- $Y$ is a space of temperate curvature with $D_{Y}=\infty$. This class of spaces includes all hyperbolic spaces and [CAT($\kappa$) ]{}spaces with $\kappa\leq 0$.
In what follows, given a set $U$ and a Lipschitz mapping $f$, we write $f|_{U}$ for the restriction of $f$ to the subset of its domain contained in $U$.
\[thm:all\] Let $U$ be an open subset of $X$ with $U\cap C_{X}\neq\emptyset$ and $U\subseteq B({\operatorname{star}}(C_{X}),D_{X})$. Then the set $${\mathcal{Q}}(U)=\left\{f\in {\mathcal{M}}(C_{X},C_{Y})\colon \inf_{F\in{\mathcal{E}}(f)}\operatorname{Lip}(F|_{U})<1\right\}$$ is $\sigma$-porous in ${\mathcal{M}}(C_{X},C_{Y})$.
As a corollary of the above theorem, we obtain the $\sigma$-porosity of the set ${\mathcal{N}}(C_{X},C_{Y})$ in the space ${\mathcal{M}}(C_{X},C_{Y})$:
\[thm:Nsigporous\] The set ${\mathcal{N}}(C_{X},C_{Y})$ is a $\sigma$-porous subset of ${\mathcal{M}}(C_{X},C_{Y})$.
Any strict contraction $f\colon C_{X}\to C_{Y}$ can be extended to a strict contraction $F\colon{\operatorname{conv}}(C_{X})\to\ell_{\infty}(\Omega)$. Therefore ${\mathcal{N}}(C_{X},C_{Y})\subseteq {\mathcal{Q}}(U)$, where $U$ may be chosen arbitrarily satisfying the conditions of Theorem \[thm:all\].
Whilst Theorem \[thm:Nsigporous\] tells us that nearly all mappings in ${\mathcal{M}}(C_{X},C_{Y})$ have the maximal permitted Lipschitz constant one, we note that a large Lipschitz constant can be achieved through sporadic behavior. It is easy to find examples of mappings with a large Lipschitz constant that, when restricted to a large subset of their domain, behave like strict contractions or even constant mappings. Thus, we now consider the question of the size of the set of mappings in ${\mathcal{M}}(C_{X},C_{Y})$ for which a large set of points in $C_{X}$ in some sense witnesses the maximal Lipschitz constant. The paper [@BD2016:Porosity] proves that, for a non-empty, non-singleton, closed, convex and bounded subset $C$ of a separable Banach space $X$, there is a $\sigma$-porous subset of the space ${\mathcal{M}}(C,C)$, outside of which all mappings $f$ admit a residual subset of $C$ on which the quantity $$\operatorname{Lip}(f,x):=\limsup_{r\to 0^{+}}\left\{\frac{\rho_{Y}(f(y),f(x))}{\rho_{X}(x,y)}\colon y\in B(x,r)\setminus\left\{x\right\}\right\}$$ is uniformly one. We use the term *residual* here in the sense of the Baire Category Theorem. The proof of this result makes essential use of the fact that the Lipschitz constant of a mapping on a convex set $C$ can be expressed as the supremum of $\operatorname{Lip}(f,x)$ over all points $x\in C$. We verify this property for Lipschitz mappings on convex subsets of $X$:
\[lem:LipOnGamma\] Let $C$ be a non-empty, non-singleton, $\rho_{X}$-convex subset of $X$. Given a Lipschitz mapping $f:C\to Y$ and a number $0<L<\operatorname{Lip}(f)$, there exist points $u_{0},u_{1}\in C$ such that $$\label{eq:liminfGamma}
\liminf_{t\to 0^+}\frac{\rho_{Y}(f((1-t)u_0\oplus tu_1), f(u_0))}{t\rho_{X}(u_0,u_1)} > L.$$ In the case where $C\subseteq [w_{0},x_{0}]$ for some $w_{0},x_{0}\in X$, then such points $u_{0},u_{1}\in C$ can be found with $u_{1}=x_{0}$.
Let $L'\in(L,\operatorname{Lip}(f))$ and choose points $v,w\in C$ such that $$\frac{\rho_{Y}(f(w),f(v))}{\rho_{X}(v,w)}>L'.$$ In the case where $C\subseteq[w_{0},x_{0}]$, we identify the metric segment $[w_{0},x_{0}]$ with a closed interval in ${\mathbb{R}}$ and additionally prescribe that $v<w<x_{0}$.
Let $[v,w]$ be a metric segment in $X$ with endpoints $v$ and $w$. We identify $[v,w]$ with a closed interval in $\mathbb{R}$. Assume that $$\label{eq:LipConstContradictionGamma}
\liminf_{t\to 0^+} \frac{\rho_{Y}(f((1-t)u_0\oplus t w),f(u_0))}{t\rho_{X}(u_0,w)} < L'$$ for all $u_0\in [v,w)$, where $[u_0,w]\subseteq[v,w]$. We define a collection of metric segments $\mathcal{U}$ by $$\mathcal{U} := \left\{[\xi,\eta]\subset (v,w) \colon \frac{\rho_{Y}(f(\xi),f(\eta))}{\rho_{X}(\eta,\xi)} <L'\right\},$$ which is, by assumption , a Vitali cover of $(v,w)$. By Vitali’s covering theorem, there exist pairwise disjoint intervals $[\xi_i,\eta_i]\in\mathcal{U}$ such that $$\lambda^1 \Big( (v,w)\setminus\bigcup_{i=1}^{\infty}[\xi_i,\eta_i]\Big) = 0,$$ where $\lambda^1$ denotes the one-dimensional Lebesgue measure. We will prove that $$\frac{\rho_{Y}(f(w),f(v))}{\rho_{X}(v,w)} \leq L',$$ contradicting the choice of $v,w\in C$. From this contadiction we conclude that assumption is false. Consequently, there exists $u_{0}\in[v,w)$ such that is satisfied with $u_{1}=w$. In the case $C\subseteq [w_{0},x_{0}]$ we have $u_{0}<w<x_{0}$ and therefore is also satisfied with $u_{1}=x_{0}$.
To complete the proof, we establish the contradiction described above. For $\varepsilon>0$, choose $N$ large enough so that $$\lambda^1\Big((v,w)\setminus \bigcup_{i=1}^{N} [\xi_i,\eta_i]\Big) <\frac{\varepsilon \rho(v,w)}{\operatorname{Lip}(f)}.$$ Without loss of generality, we may assume that $\xi_1<\eta_1<\xi_2<\dots<\xi_N<\eta_N$, that is, the above intervals are in ascending order. From this, we deduce that $$\begin{aligned}
\frac{\rho_{Y}(f(w),f(v))}{\rho_{X}(v,w)}
& \leq \frac{1}{\rho_{X}(v,w)}\Big( \rho_{Y}(f(v),f(\xi_1)) +\sum_{i=1}^{N} \rho_{Y}(f(\xi_i),f(\eta_i))\\
&\hspace{4.5cm} +\sum_{i=1}^{N-1}\rho_{Y}(f(\eta_i),f(\xi_{i+1}))+\rho_{Y}(f(\eta_N),f(w))\Big) \\
& \hspace{-2.7cm} \leq \frac{1}{\rho_{X}(v,w)} \Big(L' \Big(\sum_{i=1}^{N}\rho_{X}(\xi_i,\eta_i)\Big)
+\operatorname{Lip}(f)\Big(\rho_{X}(v,\xi_1)+\sum_{i=1}^{N-1}\rho_{X}(\eta_i,\xi_{i+1})+\rho_{X}(\eta_N,w)\Big)\Big)\\
& \hspace{-2.7cm} \leq \frac{1}{\rho_{X}(v,w)} (L'\rho_{X}(v,w)+\varepsilon \rho_{X}(v,w)) = L'+\varepsilon.
\end{aligned}$$ Letting $\varepsilon\to 0^{+}$, we arrive at the desired contradiction.
For $\rho_{X}$-star-shaped domains, the conclusion of Lemma \[lem:LipOnGamma\] is, in general, not valid and so the global Lipschitz constant may not be approximated by $\operatorname{Lip}(f,x)$. We demonstrate this with an example:
Let $e=(1,0)\in\mathbb{R}^{2}$ and $u\in\mathbb{S}^{1}$ with $\|e-u\|=\frac{1}{3}$. We set $A=[0,e]$, $B=[0,u]$, $X=A\cup B$ and define $$f\colon X\to X,\quad z=(z_1,z_2)\mapsto \begin{cases}(0,0)&\text{for } z\in B\\
\frac{1}{2}\left(\max\left\{z_1-\frac{1}{3},0\right\},0\right)
& \text{for } z\in A\end{cases}.$$ Then $\operatorname{Lip}(f,x)$ is bounded above by $\frac{1}{2}$ for all $x\in X$ but $\|f(e)-f(u)\| = \frac{1}{3} = \|e-u\|$ shows that the global Lipschitz constant of $f$ is at least $1$.
Thus, for $\rho_{X}$-star-shaped domains, we consider a weaker control of the Lipschitz constant at a point. Namely, for $f\in{\mathcal{M}}(C_{X},C_{Y})$ and $x\in C_{X}$, we define the quantity $${\widehat{\operatorname{Lip}}}(f,x):=\sup\left\{\frac{\rho_{Y}(f(y),f(x))}{\rho_{X}(x,y)}\colon y\in C_{X}\setminus\left\{x\right\}\right\},$$ which satisfies $\operatorname{Lip}(f,x)\leq{\widehat{\operatorname{Lip}}}(f,x)$. Given a mapping $f\in{\mathcal{M}}(C_{X},C_{Y})$, we define sets $R(f),\widehat{R}(f)\subseteq C_X$ by $$R(f):=\left\{x\in C_{X}\colon \operatorname{Lip}(f,x)=1\right\},\qquad \widehat{R}(f):=\left\{x\in C_{X}\colon {\widehat{\operatorname{Lip}}}(f,x)=1\right\}.$$ We note that $R(f)\subseteq \widehat{R}(f)$. Under suitable additional assumptions we show that for nearly all mappings $f\in{\mathcal{M}}(C_{X},C_{Y})$, either the set $R(f)$ or the set $\widehat{R}(f)$ is a residual subset of $C_{X}$. For a given $f\in{\mathcal{M}}(C_{X},C_{Y})$, we point out that the sets $R(f)$ and $\widehat{R}(f)$ are both $G_{\delta}$ subsets of $C_{X}$. To see this, note that $$\label{eq:Rfbarint}
\widehat{R}(f)=\bigcap_{q\in\mathbb{Q}\cap(0,1)}\left\{x\in C_{X}\colon {\widehat{\operatorname{Lip}}}(f,x)>q\right\}$$ and $$\label{eq:Rfint}
R(f)=\bigcap_{q,r\in\mathbb{Q}\cap(0,1)}\left\{x\in C_{X}\colon \operatorname{Lip}(f,x,r)>q\right\},$$ where for $x\in C_{X}$ and $r>0$, we define $$\operatorname{Lip}(f,x,r):=\sup\left\{\frac{\rho_{Y}(f(y),f(x))}{\rho_{X}(x,y)}\colon y\in C_{X}\cap B(x,r)\setminus\left\{x\right\}\right\}.$$ Note that we have $\operatorname{Lip}(f,x)=\lim_{r\to0^{+}}\operatorname{Lip}(f,x,r)$. It is readily verified that each of the sets participating in the above intersections is open in $C_{X}$.
In the case where the set $C_{X}$ is separable and $\rho_{X}$-convex we obtain the following generalisation of [@BD2016:Porosity Theorem 2.2]:
\[thm:convexresidual\] Suppose $C_{X}$ is separable and $\rho_{X}$-convex. Then there exists a $\sigma$-porous set $\widetilde{{\mathcal{N}}}\subseteq {\mathcal{M}}(C_{X},C_{Y})$ such that for every $f\in{\mathcal{M}}(C_{X},C_{Y})\setminus \widetilde{{\mathcal{N}}}$, the set $$R(f)=\left\{x\in C_{X}\colon \operatorname{Lip}(f,x)=1\right\}$$ is a residual subset of $C_{X}$.
For each open set $U\subseteq X$ of diameter smaller than $D_{X}$ and non-empty intersection with $C_{X}$, we apply Theorem \[thm:all\] with ${\operatorname{conv}}(C_{X})=C_{X}$. Note that ${\operatorname{conv}}(C_X)=C_{X}$ implies in particular that $U\subset B({\operatorname{star}}(C_X),D_X)$ holds. With these settings we have ${\mathcal{E}}(f)=\left\{f\right\}$ for all $f\in{\mathcal{M}}(C_{X},C_{Y})$ and Theorem \[thm:all\] asserts that the set $${\mathcal{Q}}(U)=\left\{f\in{\mathcal{M}}(C_{X},C_{Y})\colon \operatorname{Lip}(f|_{U})<1\right\}$$ is a $\sigma$-porous subset of ${\mathcal{M}}(C_{X},C_{Y})$.
Fix a countable dense subset $\Delta$ of $C_{X}$ and define the set $\widetilde{{\mathcal{N}}}$ by $$\widetilde{{\mathcal{N}}}:=\bigcup_{i=1}^{\infty}{\mathcal{Q}}(U_{i}),$$ where $(U_{i})_{i=1}^{\infty}$ is an enumeration of all sets of the form $B(x,r)$ where $x\in \Delta$ and $r\in\mathbb{Q}\cap (0,D_{X}/2)$. It is clear that $\widetilde{{\mathcal{N}}}$ is a $\sigma$-porous subset of ${\mathcal{M}}(C_{X},C_{Y})$.
Let $f\in\mathcal{M}(C_{X},C_{Y})\setminus \widetilde{{\mathcal{N}}}$. To complete the proof, we need to verify that the set $R(f)$ is a residual subset of $C_{X}$. It suffices to show that each of the open subsets of $C_{X}$ occurring in the intersection in is a dense subset of $C_{X}$. To this end, fix an open subset $U$ of $X$ such that $U\cap C_{X}\neq\emptyset$. Given $q,r\in\mathbb{Q}\cap(0,1)$, we need to show that the set $$T_{q,r}:=\left\{x\in C_{X}\colon \operatorname{Lip}(f,x,r)>q\right\}$$ has non-empty intersection with $U$. Choose $j\geq 1$ so that $U_{j}\subset U$. Since $f\notin{\mathcal{Q}}(U_{j})$, we have $\operatorname{Lip}(f|_{U_{j}})= 1$. Using the condition of Definition \[def:weaklyhyperbolic\] on the weakly hyperbolic space $X$, we see that $C_{X}\cap U_{j}$ is $\rho_{X}$-convex, as an intersection of two $\rho_{X}$-convex sets. Therefore, we may apply Lemma \[lem:LipOnGamma\] with $C=C_{X}\cap U_{j}$ and deduce that there exists a point $u_{0}\in C_{X}\cap U_{j}$ with $\operatorname{Lip}(f,u_{0})>q$. We can do this since the set $C_{X}\cap U_{j}$ is non-singleton as open balls contain nontrivial metric segments. Hence $\operatorname{Lip}(f,u_{0},r)>q$ and $u_{0}\in U\cap T_{q,r}\neq\emptyset$.
For the remainder of this section we work towards proving a version of Theorem \[thm:convexresidual\] for $\rho_{X}$-star-shaped subsets of weakly hyperbolic spaces. Namely, we establish the following result:
\[thm:starshapedresidual\] Suppose that $C_{X}$ is separable and $C_{X}\subseteq \overline{B}({\operatorname{star}}(C_{X}),D_{X})$. Then there exists a $\sigma$-porous set $\widetilde{{\mathcal{N}}}\subseteq {\mathcal{M}}(C_{X},C_{Y})$ such that for $f\in {\mathcal{M}}(C_{X},C_{Y})\setminus\widetilde{{\mathcal{N}}}$, the set $$\widehat{R}(f)=\left\{x\in C_{X}\colon {\widehat{\operatorname{Lip}}}(f,x)=1\right\}$$ is a residual subset of $C_{X}$.
Note that for contractive mappings in the sense of Rakotch the sets $R(f)$ and $\widehat{R}(f)$ coincide. Indeed, if $f$ is contractive in the sense of Rakotch, there exists a decreasing function $\varphi\colon (0,\infty)\to [0,1)$ such that $\rho_Y(f(x),f(y))\leq \varphi(\rho_X(x,y))\,\rho_X(x,y)$ for all distinct points $x,y\in C_X$. In other words $$\frac{\rho_Y(f(x),f(y))}{\rho_X(x,y)} \leq \varphi(\rho_X(x,y))$$ for $x\neq y$, which shows that the expression on the left-hand side can only approach one when $y$ approaches $x$. With minor modifications, the proof of [@Rei2005GenericityPorosity Theorem 4] shows that, if $X$ and $Y$ are hyperbolic spaces and $C_X\subseteq X$ and $C_Y\subseteq Y$ are non-empty, non-singleton, bounded, closed and $\rho_X$- and $\rho_Y$-star-shaped subsets, respectively, there is a $\sigma$-porous subset $\bar{\mathcal{N}}\subset
\mathcal{M}(C_X,C_Y)$ such that all mappings in its complement are contractive in the sense of Rakotch.
In view of the above remark, we can get the following corollary to Theorem \[thm:starshapedresidual\], which is a strengthening of Theorems \[thm:convexresidual\] and \[thm:starshapedresidual\] restricted to the case where $X$ and $Y$ are hyperbolic spaces and $C_{X},C_{Y}$ are bounded. In particular, although we have seen that Lipschitz mappings on a star-shaped set $C$ may not satisfy $\operatorname{Lip}(f)=\sup_{x\in C}\operatorname{Lip}(f,x)$, the following corollary indicates that typical nonexpansive mappings retain this property.
Suppose $X$ and $Y$ are complete hyperbolic spaces, $C_{X}$ is separable and bounded and $C_Y$ is bounded. Then there exists a $\sigma$-porous set $\widetilde{\mathcal{N}}\subseteq \mathcal{M}(C_{X},C_{Y})$ such that for every $f\in\mathcal{M}(C_{X},C_{Y})\setminus
\widetilde{\mathcal{N}}$, the set $$R(f)=\left\{x\in C_{X}\colon \operatorname{Lip}(f,x)=1\right\}$$ is a residual subset of $C_{X}$.
For the proof of Theorem \[thm:starshapedresidual\], we require an extension lemma for Lipschitz mappings.
\[lemma:fortsetzungvermutung\] Let $(Z,d)$ and $(W,\rho)$ be metric spaces, $E\subseteq Z$ and $\Omega$ be a set such that $W\subseteq
\ell_{\infty}(\Omega)$. Let $f:E\to W$ be a $1$-Lipschitz mapping, $u_{0}\in E$, $r>0$, $q\in(0,1)$, $q'\in(q,1)$ and suppose that for every $x\in E\cap B(u_{0},r)$, we have $${\widehat{\operatorname{Lip}}}(f,x)\leq q.$$ Then there exists a $1$-Lipschitz extension $F:Z\to\ell_{\infty}(\Omega)$ of $f$ and a number $s\in(0,r)$ such that $\operatorname{Lip}(F|_{B(u_{0},s)})\leq q '$.
Using $W\subseteq \ell_{\infty}(\Omega)$, we view $f$ as a mapping from $E$ to $\ell_{\infty}(\Omega)$. Given $\omega\in\Omega$, a set $S\subseteq Z$ and a mapping $h\colon S\to \ell_{\infty}(\Omega)$ we let $h_{\omega}\colon S\to {\mathbb{R}}$ be defined by $h_{\omega}(x)=h(x)(\omega)$ for all $x\in S$. In what follows we will frequently use the identities $$\label{eq:lipcstcomp-glob}
\operatorname{Lip}(h)=\sup\left\{\operatorname{Lip}(h_{\omega})\colon \omega\in\Omega\right\},\qquad \widehat{\operatorname{Lip}}(h,x)=\sup\left\{\widehat{\operatorname{Lip}}(h_{\omega},x)\colon\omega\in\Omega\right\},$$ which are easily derived from the definitions of the Lipschitz constants and the $\ell_{\infty}$ norm. We define the mapping $F: Z\to \ell_{\infty}(\Omega)$ componentwise by $$F_{\omega}(y):=\inf\left\{f_{\omega}(z)+{\widehat{\operatorname{Lip}}}(f_{\omega},z)d(z,y)\colon z\in E\right\},\quad y\in Z,\,\omega\in\Omega.$$ This mapping is a modification of the standard Lipschitz extension of $f$, as defined in [@BL2000GeomtericNonlinear Chapter 1]. Let us verify that this mapping fulfills all the desired conditions. Firstly, we show that $F$ is an extension of $f$. Fix $\omega\in\Omega$. Letting $y\in E$ we observe from the definition that $F_{\omega}(y)\leq f_{\omega}(y)$. Moreover, given $\varepsilon>0$, we can choose $z\in E$ such that $$\label{eq:zapp}
F_{\omega}(y)\geq f_{\omega}(z)+{\widehat{\operatorname{Lip}}}(f_{\omega},z)d(z,y)-\varepsilon.$$ This leads to the observation $$\begin{aligned}
F_{\omega}(y)&\geq f_{\omega}(z)+{\widehat{\operatorname{Lip}}}(f_{\omega},z)d(z,y)-\varepsilon\\
&\geq f_{\omega}(y)-{\widehat{\operatorname{Lip}}}(f_{\omega},z)d(z,y)+{\widehat{\operatorname{Lip}}}(f_{\omega},z)d(z,y)-\varepsilon\\
&=f_{\omega}(y)-\varepsilon.
\end{aligned}$$ We conclude that $F_{\omega}(y)=f_{\omega}(y)$, as required.
We now show that $F$ is $1$-Lipschitz. Let $\omega\in\Omega$ and $y_{1},y_{2}\in Z$. Given $\varepsilon>0$, we can choose $z_{2}\in E$ so that be satisfied with $y=y_{2}$ and $z=z_{2}$. From this we deduce $$\begin{aligned}
F_{\omega}(y_{1})-F_{\omega}(y_{2})&\leq (f_{\omega}(z_{2})+{\widehat{\operatorname{Lip}}}(f_{\omega},z_{2})d(z_{2},y_{1}))-(f_{\omega}(z_{2})+{\widehat{\operatorname{Lip}}}(f_{\omega},z_{2})d(z_{2},y_{2})-\varepsilon)\\
&\leq{\widehat{\operatorname{Lip}}}(f_{\omega},z_2)d(y_{1},y_{2})+\varepsilon\leq d(y_{1},y_{2})+\varepsilon,
\end{aligned}$$ where the final inequality uses $\operatorname{Lip}(f_{\omega})\leq\operatorname{Lip}(f)\leq1$. Similarly, we can show that $F_{\omega}(y_{2})-F_{\omega}(y_{1})\leq d(y_{1},y_{2})+\varepsilon$. We have shown that $\operatorname{Lip}(F_{\omega})\leq 1$ for all $\omega\in\Omega$. Thus, by we get that $\operatorname{Lip}(F)\leq 1$. It only remains to verify that $F$ is locally a strict contraction around $u_{0}$. For this we will need the following claim.
\[lemma:dich\] There exists $N>1$ such that for every $y\in B(u_{0},r/N)$, every $z\in E$ and every $\omega\in\Omega$, at least one of the following statements holds:
(i) $f_{\omega}(z)+{\widehat{\operatorname{Lip}}}(f_{\omega},z)d(z,y)> f_{\omega}(u_{0})+{\widehat{\operatorname{Lip}}}(f_{\omega},u_{0})d(u_{0},y)$.
(ii) ${\widehat{\operatorname{Lip}}}(f_{\omega},z)\leq q'$.
We choose $N$ large enough so that $$\frac{n+1}{n-1}\leq \frac{q'}{q}$$ for all $n\geq N$. We set $s=r/N$ and fix $y\in B(u_{0},s)$ and $\omega\in\Omega$. If $z\in E\cap B(u_{0},r)$, then statement (ii) already holds, because $\widehat{\operatorname{Lip}}(f_{\omega},z)\leq \widehat{\operatorname{Lip}}(f,z)\leq q<q'$, and there is nothing to prove. Therefore, we proceed by fixing a point $z\in E\setminus B(u_{0},r)$ and supposing that $z$ fails to satisfy the inequality of (i). In other words, we have $$\label{eq:badz}
f_{\omega}(z)+{\widehat{\operatorname{Lip}}}(f_{\omega},z)d(z,y)\leq f_{\omega}(u_{0})+{\widehat{\operatorname{Lip}}}(f_{\omega},u_{0})d(u_{0},y).$$ We complete the proof by showing that statement (ii) holds for $z$. The left-hand side of can be bounded from below by the expression $$\begin{aligned}
f_{\omega}(u_{0})-q d(z,u_{0})&+{\widehat{\operatorname{Lip}}}(f_{\omega},z)(d(z,u_{0})-d(u_{0},y))\geq\\ &f_{\omega}(u_{0})-q d(z,u_{0})+{\widehat{\operatorname{Lip}}}(f_{\omega},z)(d(z,u_{0})-s).
\end{aligned}$$ Moreover, we can bound the right-hand side of from above by $f_{\omega}(u_{0})+q s$. We conclude from this that $$f_{\omega}(u_{0})-q d(z,u_{0})+{\widehat{\operatorname{Lip}}}(f_{\omega},z)(d(z,u_{0})-s)\leq f_{\omega}(u_{0})+q s.$$ Rearranging this inequality, we obtain $${\widehat{\operatorname{Lip}}}(f_{\omega},z)\leq \frac{ q(d(z,u_{0})+s)}{d(z,u_{0})-s}=q\cdot\frac{n+1}{n-1},$$ where $n:=d(z,u_{0})/s\geq r/s= N$ and $d(z,u_0)-s\geq r-s>0$ since $z\not\in B(u_0,r)$. The last expression is bounded from above by $q'$.
The proof of Lemma \[lemma:fortsetzungvermutung\] is now completed by proving the following claim:
Let $N$ be given by the statement of the previous claim. Then $$\operatorname{Lip}(F|_{B(u_{0},r/N)})\leq q'.$$
Fix $y_{1},y_{2}\in B(u_{0},r/N)$ and $\omega\in\Omega$. Given $\varepsilon>0$, we can choose $z_{2}\in E$ such that is satisfied with $y=y_{2}$, $z=z_{2}$ and $$f_{\omega}(z_{2})+{\widehat{\operatorname{Lip}}}(f_{\omega},z_{2})d(z_{2},y_{2})\leq f_{\omega}(u_{0})+{\widehat{\operatorname{Lip}}}(f_{\omega},u_{0})d(u_{0},y_{2}).$$ Then by the first claim we have ${\widehat{\operatorname{Lip}}}(f_{\omega},z_{2})\leq q'$. We conclude that $$\begin{aligned}
F_{\omega}(y_{1})-F_{\omega}(y_{2})&\leq (f_{\omega}(z_{2})+{\widehat{\operatorname{Lip}}}(f_{\omega},z_{2})d(z_{2},y_{1}))-(f_{\omega}(z_{2})+{\widehat{\operatorname{Lip}}}(f_{\omega},z_{2})d(z_{2},y_{2})-\varepsilon)\\
&\leq {\widehat{\operatorname{Lip}}}(f_{\omega},z_{2})d(y_{1},y_{2})+\varepsilon\leq q' d(y_{1},y_{2})+\varepsilon.
\end{aligned}$$ Similarly, we can show that $F_{\omega}(y_{2})-F_{\omega}(y_{1})\leq q' d(y_{1},y_{2})+\varepsilon$. The above argument establishes that $\operatorname{Lip}(f_{\omega}|_{B(u_{0},r/N)})\leq q'$ for every $\omega\in\Omega$. The conclusion of the claim follows.
This completes the proof of Lemma \[lemma:fortsetzungvermutung\].
Fix a countable dense subset $\Delta$ of $C_{X}$ and let $(U_{i})_{i=1}^{\infty}$ be an enumeration of all sets of the form $B(x,r)$, where $x\in \Delta$ and $r\in\mathbb{Q}\cap(0,1)$ with $B(x,r)\subseteq B({\operatorname{star}}(C_{X}),D_{X})$. By Theorem \[thm:all\], each set ${\mathcal{Q}}(U_{i})$ is $\sigma$-porous.
Suppose that $f\in{\mathcal{M}}(C_{X},C_{Y})$ is such that $\widehat{R}(f)$ is not residual. We complete the proof by showing that $f\in\widetilde{{\mathcal{N}}}:=\bigcup_{i=1}^{\infty}{\mathcal{Q}}(U_{i})$.
From the assumption that $\widehat{R}(f)$ is not residual, we deduce that for some $q\in\mathbb{Q}\cap (0,1)$, the open subset of $C_{X}$ $$T_{q}:=\left\{x\in C_{X}\colon {\widehat{\operatorname{Lip}}}(f,x)>q\right\},$$ which occurs in the intersection in , is not dense in $C_{X}$. Choose an open subset $U$ of $X$ such $U\cap C_{X}\neq\emptyset$ and $U\cap T_{q}=\emptyset$. Then we have ${\widehat{\operatorname{Lip}}}(f,x)\leq q$ for all $x\in C_{X}\cap U$. Using the inclusion $C_{X}\subseteq \overline{B}({\operatorname{star}}(C_{X}),D_{X})$ and the fact that the set $\overline{B}({\operatorname{star}}(C_{X}),D_{X})\setminus B({\operatorname{star}}(C_{X}),D_{X})$ has empty interior in $C_{X}$, we can find $u_{0}\in U\cap C_{X}\cap B({\operatorname{star}}(C_{X}),D_{X})$ and then choose $r>0$ such that $B(u_{0},r)\subseteq U\cap
B({\operatorname{star}}(C_{X}),D_{X})$. Applying Lemma \[lemma:fortsetzungvermutung\] with $E=C_{X}$, $Z={\operatorname{conv}}(C_{X})$ and $W=C_{Y}$, we can find an extension $F:{\operatorname{conv}}(C_{X})\to \ell_{\infty}(\Omega)$ and an open ball $B(u_{0},s)\subseteq
B(u_{0},r)$ such that $\operatorname{Lip}(F|_{B(u_{0},s)})<1$. Choosing now $i\geq 1$ such that $U_{i}\subseteq B(u_{0},s)$, we have $\operatorname{Lip}(F|_{U_{i}})<1$ and $f\in{\mathcal{Q}}(U_{i})$.
In the case where at least one of the sets $C_X$ and $C_Y$ is bounded, a more natural metric on $\mathcal{M}(C_X,C_Y)$ is the metric of uniform convergence. More generally, we consider the space $$\mathcal{M}_{B}(C_X,C_Y) := \{f\colon C_X\to C_Y \colon \operatorname{Lip}(f)\leq 1 \text{ and } f \text{ is bounded}\}$$ of bounded mappings, that is, mappings where $f(C_X)\subset C_Y$ is bounded, and equip it with the metric $$d_\infty(f,g) := \sup \left\{d(f(x),g(x))\colon x\in C_X\right\}$$ of uniform convergence.
If the set $C_X$ is bounded, then $(\mathcal{M}(C_X,C_Y),d_\theta)$ and $(\mathcal{M}_B(C_X,C_Y),d_\infty)$ coincide as topological spaces. The inequalities $$\frac{\rho_Y(f(x),g(x))}{1+\rho_X(x,\theta)} \leq \rho_Y(f(x),g(x)) \leq (1+\operatorname{diam}(C_X)) \frac{\rho_Y(f(x),g(x))}{1+\rho_X(x,\theta)}$$ show that in this case the metrics $d_\theta$ and $d_\infty$ are even Lipschitz equivalent.
With a small modification of the proof of Theorem \[thm:all\] we can also show that under the same assumptions, the set $${\mathcal{Q}}_B(U)=\left\{f\in {\mathcal{M}}_B(C_{X},C_{Y})\colon \inf_{F\in{\mathcal{E}}(f)}\operatorname{Lip}(F|_{U})<1\right\}$$ is a $\sigma$-porous subset of $\mathcal{M}_B(C_X,C_Y)$. Since Theorem \[thm:all\] is the basis for the other porosity results in this section, we may deduce that the set $\mathcal{N}_B(C_{X},C_Y)$ of bounded strict contractions is a $\sigma$-porous subset of $\mathcal{M}_B(C_X,C_Y)$ and that, in the separable setting, typical bounded nonexpansive mappings attain the maximal Lipschitz constant $1$ at typical points of their domain. In other words, all theorems in this section remain valid, if we replace ${\mathcal{M}}(C_X,C_Y)$ by ${\mathcal{M}}_B(C_X,C_Y)$ and ${\mathcal{N}}(C_X,C_Y)$ by ${\mathcal{N}}_B(C_X,C_Y)$.
Let us conclude this remark by commenting on the necessary modification of the proofs in Section 4. Since Lemma \[lemma:pert\] actually implies that the perturbed mapping is $\varepsilon$-close to the original one not only with respect to $d_\theta$ but also with respect to $d_{\infty}$, we only have to notice that starting with a bounded mapping also the perturbed mapping we obtain is bounded and that in ${\mathcal{M}}_B(C_X,C_Y)$ the inclusion $B_\infty(f,\alpha\varepsilon)\subset B_\theta (f,\alpha\varepsilon)$ holds for all $f\in{\mathcal{M}}_B(C_X,C_Y)$ and all $\alpha,\varepsilon>0$, in order to get the results for bounded mappings.
Proof of Theorem \[thm:all\]
============================
In the present section we prove Theorem \[thm:all\]. Let $X$, $Y$, $C_{X}$, $C_{Y}$, ${\operatorname{conv}}(C_{X})$, $\Omega$, $\theta$, ${\mathcal{M}}(C_{X},C_{Y})$, ${\mathcal{N}}(C_{X},C_{Y})$ and ${\mathcal{E}}(f)$ satisfy Hypotheses \[hypotheses\]. For the reader’s convenience, we repeat the statement of Theorem \[thm:all\]:
Let $U$ be an open subset of $X$ with $U\cap C_{X}\neq\emptyset$ and $U\subseteq B({\operatorname{star}}(C_{X}),D_{X})$. Then the set $${\mathcal{Q}}(U)=\left\{f\in {\mathcal{M}}(C_{X},C_{Y})\colon \inf_{F\in{\mathcal{E}}(f)}\operatorname{Lip}(F|_{U})<1\right\}$$ is $\sigma$-porous in ${\mathcal{M}}(C_{X},C_{Y})$.
Let $U\subseteq X$ satisfy the hypotheses of Theorem \[thm:all\]. From this point onwards we only work inside metric segments in the space $X$ of the form $[x,y]$, where $x,y\in X$ with $\rho_{X}(x,y)<D_{X}$. Such metric segments are well defined because $X$ satisfies condition of Definition \[def:weaklyhyperbolic\]. In particular, for $x,y\subseteq X$ with $\rho_{X}(x,y)<D_{X}$ and $\lambda\in[0,1]$, the point $(1-\lambda)x\oplus \lambda y\in X$ is well defined. We adopt a similar approach when working with metric segments in the space $Y$. In what follows we often identify a metric segment $[x,y]$ with a real interval. In particular, we endow metric segments with the natural ordering they inherit when viewed as real intervals.
Let ${\mathcal{G}}$ denote the collection of all metric segments of the form $[w_{0},w_{1}]\subseteq C_{X}\cap U$ for which there exists a point $x_{0}\in {\operatorname{star}}(C_{X})$ such that $w_{0}\in B(x_{0},D_{X})$ and $w_{1}\in[w_{0},x_{0}]$ with $w_{0}<w_{1}<x_{0}$. Since $U\subseteq B({\operatorname{star}}(C_{X}),D_{X})$ and $U\cap C_{X}\neq\emptyset$, the collection ${\mathcal{G}}$ is not empty. In the case where $C_{X}$ is convex, we note that every metric segment in $C_{X}\cap U$ contains a metric subsegment which belongs to $\mathcal{G}$. For numbers $a<b\in(0,1)$ and $p\geq 2$, we define a collection of subsets ${\mathcal{Q}}_{a,b}^{p}(U)$ of ${\mathcal{Q}}(U)$ by $${\mathcal{Q}}_{a,b}^{p}(U):=\left\{f\in{\mathcal{Q}}(U)\colon a<\sup_{\Gamma\in{\mathcal{G}}}\operatorname{Lip}(f|_{\Gamma})\leq b,\, \inf_{F\in{\mathcal{E}}(f)}\operatorname{Lip}(F|_{U})\leq 1-\frac{1}{p}\right\}.$$ The significance of the above decomposition of ${\mathcal{Q}}(U)$ is revealed in the following lemma.
\[lemma:Qabpporous\] If $a,b\in(0,1)$ and $p\geq 2$ satisfy the condition $$\label{eq:ConditionABUBStar}
b-a<\frac{a}{48(p-1)},$$ then the set ${\mathcal{Q}}_{a,b}^{p}(U)$ is porous in $\mathcal{M}(C_{X},C_{Y})$.
Let us begin working towards a proof of Lemma \[lemma:Qabpporous\]. The basic idea of the proof is to take a mapping $f\in{\mathcal{Q}}_{a,b}^{p}(U)$ and to peturb it slightly to produce a nearby mapping $g\in{\mathcal{M}}(C_{X},C_{Y})$, the distance of which from the set ${\mathcal{Q}}_{a,b}^{p}(U)$ is a relatively large proportion of its distance from $f$. In order to control the Lipschitz constant of the mapping we construct, we first extend $f$ to a mapping $F:{\operatorname{conv}}(C_{X})\to \ell_{\infty}(\Omega)$ witnessing the fact that $f\in {\mathcal{Q}}_{a,b}^{p}(U)$ and then transform $F$ to a mapping $G:{\operatorname{conv}}(C_{X})\to\ell_{\infty}(\Omega)$ satisfying $G(C_{X})\subseteq C_{Y}$. The desired mapping $g\in{\mathcal{M}}(C_{X},C_{Y})$ can then be defined as the restriction of $G$ to $C_{X}$.
The star-shaped nature of the sets $C_{X}$ and $C_{Y}$ presents two natural means of manipulating the mapping $F:{\operatorname{conv}}(C_{X})\to \ell_{\infty}(\Omega)$ in such a way that the condition $F(C_{X})\subseteq C_{Y}$ is preserved. One approach is to apply a mapping of the form $x\mapsto (1-\lambda(x))x\oplus \lambda(x)x_{0}$ with $x_{0}\in{\operatorname{star}}(C_{X})$ to the set ${\operatorname{conv}}(C_{X})$ before applying the mapping $F$. Alternatively, one can first apply the mapping $F$ and then apply a mapping of the form $y\mapsto (1-\lambda(y))y\oplus\lambda(y) y_{0}$, with $y_{0}\in {\operatorname{star}}(C_{Y})$. The latter approach is slightly more difficult than the former because the convex combination $(1-\lambda)F(x)\oplus \lambda y_{0}$ is not defined for all $x\in {\operatorname{conv}}(C_{X})$. In the present section we use both the aforementioned transformations and the next lemma captures their required properties. Given a real valued mapping $\lambda$ on $X$ we denote by $\|\lambda\|_{\infty} := \sup\{|\lambda(x)|\colon x\in X\}$ its supremum norm.
\[lemma:pert\] Let $Z\in\left\{X,Y\right\}$, $\sigma\in(0,1)$, $u_{0}\in C_{X}$, $z_{0}\in C_{Z}$ and $\pi\colon{\operatorname{conv}}(C_{X})\to \ell_{\infty}(\Omega)$ be a nonexpansive mapping such that $\pi(C_{X})\subseteq C_{Z}$ and $0<\rho_{Z}(\pi(u_{0}),z_{0})<D_{Z}$. Then there is a number $r_{0}>0$ such that the following statement holds: Let $r,\varepsilon\in(0,r_{0})$, $\lambda\colon X\to[0,1]$ be a Lipschitz mapping such that $\lambda(x)=0$ for all $x\in X\setminus B(u_{0},r)$, $$\left\|\lambda\right\|_{\infty}\leq \varepsilon/2\rho_{Z}(\pi(u_{0}),z_{0})\text{ and }\operatorname{Lip}(\lambda)\leq \sigma/\rho_{Z}(\pi(u_{0}),z_{0}),$$ and suppose that $\pi({\operatorname{conv}}(C_{X})\cap B(u_{0},r))\subseteq B(z_{0},D_{Z})$ and that every point $x\in C_{X}\cap B(u_{0},r)$ admits a unique metric segment $[\pi(x),z_{0}]\subseteq C_{Z}$. Let $\beta$ be the mapping into $\ell_{\infty}(\Omega)$ defined in the case $Z=X$ by $$\beta(x):=
(1-\lambda(x))\pi(x)\oplus \lambda(x)z_{0} \qquad \forall x\in {\operatorname{conv}}(C_{X}),$$ and in the case $Z=Y$ by $$\beta(x):=\begin{cases}
(1-\lambda(x))\pi(x)\oplus_{Y} \lambda(x)z_{0} & \text{if }x\in C_{X}\cap B(u_{0},r),
\\
\pi(x) & \text{if }x\in {\operatorname{conv}}(C_{X})\setminus B(u_{0},r).
\end{cases}$$ Then $\beta$ satisfies the following conditions:
(i) \[SxtoSy\] $\beta(C_{X})\subseteq C_{Z}$;
(ii) \[beta-p\] $\rho_{Z}(\beta(x),\pi(x))\leq\varepsilon$ for all $x\in C_{X}$;
(iii) \[lipbeta3\] $\operatorname{Lip}(\beta)\leq \max\left\{1,(1+\sigma)\operatorname{Lip}(\pi|_{B(u_{0}, r)})+2\sigma\right\}$.
We define $$r_{0}=\min\left\{\rho_{Z}(\pi(u_{0}),z_{0}),\delta_{Z}(\pi(u_{0}),z_{0},\sigma),\rho_{Z}(\pi(u_{0}),z_{0})\delta_{Z}(\pi(u_{0}),z_{0},\sigma)\right\}.$$ Let $r,\varepsilon\in(0,r_{0})$ and $\lambda\colon X\to[0,1]$ be given by the hypotheses of Lemma \[lemma:pert\]. We now verify statements -.
Statement is immediate from the definition of $\beta$, the condition that $[\pi(x),z_{0}]
\subseteq C_{Z}$ for all $x\in C_{X}\cap B(u_{0},r)$ and the fact that $\pi(C_{X})\subseteq C_{Z}$. For statement we make the following observation: If $x\in {\operatorname{conv}}(C_{X})\setminus B(u_{0},r)$, then $\beta(x)=\pi(x)$. Otherwise, we have $$\begin{aligned}
\rho_{Z}(\beta(x),\pi(x))&\leq \lambda(x)\rho_{Z}(\pi(x),z_{0})\leq\left\|\lambda\right\|_{\infty}
(\rho_{Z}(\pi(u_{0}),z_{0})+r)\leq \varepsilon,
\end{aligned}$$ using $r<\rho_{Z}(\pi(u_{0}),z_{0})$ and $\left\|\lambda\right\|_{\infty}\leq\varepsilon/2\rho_{Z}(\pi(u_{0}),z_{0})$.
To prove , we fix points $x,y$ in the intersection of the domain of $\beta$ with $B(u_{0},r)$ and observe that $$\begin{aligned}
\rho_{Z}(\beta(x),\beta(y))&\leq\rho_{Z}((1-\lambda(x))\pi(x)\oplus\lambda(x)z_{0},(1-\lambda(x))\pi(y)\oplus\lambda(x)z_{0})\nonumber\\
&\qquad\qquad\qquad+\rho_{Z}((1-\lambda(x))\pi(y)\oplus\lambda(x) z_{0},
(1-\lambda(y))\pi(y)\oplus\lambda(y)z_{0})\nonumber\\
&\leq(1+\sigma)\rho_{Z}(\pi(x),\pi(y))+\left|\lambda(y)-\lambda(x)\right|\rho_{Z}(\pi(y),z_{0})\nonumber\\
&\leq (1+\sigma)\operatorname{Lip}(\pi|_{B(u_{0},r)})\rho_{X}(x,y)\nonumber\\
&\qquad\qquad\qquad\qquad+\operatorname{Lip}(\lambda)(r+\rho_{Z}(\pi(u_{0}),z_{0}))\rho_{X}(x,y)\nonumber\\
&\leq ((1+\sigma)\operatorname{Lip}(\pi|_{B(u_{0},r)})+2\sigma)\rho_{X}(x,y). \label{eq:finallipineq}
\end{aligned}$$ In deriving the above inequalities we used the definition of $r_{0}$ and the constraints on $r,\varepsilon$ and $\lambda$ to deduce that $0\leq\lambda(x)< \delta_{Z}(\pi(u_{0}),z_{0},\sigma)$ and $\rho_{Z}(\pi(x),\pi(u_{0})),\rho_{Z}(\pi(y),\pi(u_{0}))< \delta_{Z}(\pi(u_{0}),z_{0},\sigma)$. These conditions allow us to apply condition of Definition \[def:weaklyhyperbolic\] to obtain the second inequality in the sequence above. Note that the above inequalities remain true for $x\in\partial B(u_0,r)$ when, for $z\in \ell_{\infty}(\Omega)$, we interpret the expression $(1-\lambda(x))z\oplus \lambda(x)z_{0}$ as $z$ since in that case $\lambda(x)=0$ and $\operatorname{Lip}(\pi|_{B(u_0,r)})=\operatorname{Lip}(\pi|_{\overline{B}(u_0,r)})$.
Having established and noting that $\beta$ coincides with the nonexpansive mapping $\pi$ outside of $B(u_{0},r)$, we only need to verify the Lipschitz bound for the quantity $\rho_{Z}(x,y)$ for points $x,y$ in the domain of $\beta$ with $x\in B(u_{0},r)$ and $y\notin B(u_{0},r)$. Such points admit a metric segment $[x,y]$ in ${\operatorname{conv}}(C_{X})$ and an application of the Intermediate Value Theorem provides a point $x'\in[x,y]$ with $\rho_{X}(x',u_{0})=r$, so that $x'\in\partial B(u_{0},r)$. Using the Lipschitz bound derived above for points $u,v$ in the domain of $\beta$ with $u\in\partial B(u_{0},r)$ and $v\in B(u_{0},r)$, we may now deduce that $$\begin{aligned}
\rho_{Z}(\beta(x),\beta(y)) &\leq\rho_{Z}(\beta(x),\beta(x'))+\rho_{Z}(\beta(x'),\beta(y))\\
&\leq ((1+\sigma)\operatorname{Lip}(\pi|_{B(u_{0},r)})+2\sigma) \rho_{X}(x,x')+\rho_{X}(x',y) \\
&\leq \max\{1,((1+\sigma)\operatorname{Lip}(\pi|_{B(u_{0},r)})+2\sigma)\} (\rho_{X}(x,x')+\rho_X(x',y))\\
&= \max\{1,((1+\sigma)\operatorname{Lip}(\pi|_{B(u_{0},r)})+2\sigma)\}(\rho_{X}(x,y).
\end{aligned}$$ This completes the proof of and of Lemma \[lemma:pert\] itself.
Fix a mapping $f\in {\mathcal{Q}}_{a,b}^{p}(U)$ and choose a metric segment $\Gamma=[w_{0},w_{1}]\in{\mathcal{G}}$ such that $a<\operatorname{Lip}(f|_{\Gamma})\leq b$ and an extension $F:{\operatorname{conv}}(C_{X})\to \ell_{\infty}(\Omega)$ of $f$ such that $\operatorname{Lip}(F|_{U})\leq 1-\frac{1}{p}$. Choose $x_{0}\in {\operatorname{star}}(C_{X})$ such that $w_{0}\in B(x_{0},D_{X})$ and $[w_{0},w_{1}]\subseteq [w_{0},x_{0}]$ with $w_{0}<w_{1}<x_{0}$. The mapping $F$ coincides with $f$ on the segment $\Gamma$. Therefore we have $a<\operatorname{Lip}(F|_{\Gamma})\leq b$. Applying Lemma \[lem:LipOnGamma\] with $C=(w_{0},w_{1})\subseteq [w_{0},x_{0}]$, we find a point $u_{0}\in (w_{0},w_{1})$ such that $$\liminf_{t\to 0^{+}}\frac{\rho_{Y}(F((1-t)u_0\oplus tx_0), F(u_0))}{t\rho_{X}(u_0,x_0)}>a.$$
Choose $\sigma\in(0,1)$ such that $(1-\frac{1}{p})(1+3\sigma)\leq 1$. Let $r_{0}$ be given by the conclusion of Lemma \[lemma:pert\] applied to $Z=X$, $\sigma$, $u_{0}$, $z_{0}=x_{0}$ and $\pi=\operatorname{id}_{{\operatorname{conv}}(C_{X})}\colon {\operatorname{conv}}(C_{X})\to\ell_{\infty}(\Omega)$. Let $r\in(0,r_{0})$ be small enough so that $B(u_{0},3r)\subseteq U\cap B(x_{0},D_{X})$. Using $u_{0}<w_{1}<x_{0}$, we may choose $\varepsilon_{0}\in(0,\min\left\{\sigma r/2,\rho_{X}(u_{0},x_{0})/2,1\right\})$ small enough so that $$\label{eq:liminfu0}
(1-t)u_{0}\oplus tx_{0}\in [w_{0},w_{1}]=\Gamma\quad\text{and}\quad\frac{\rho_{Y}(F((1-t)u_0\oplus t x_0),F(u_0))}{t\rho_{X}(u_0,x_0)} > a$$ for all $t\in (0,2\varepsilon_{0}/\rho_{X}(u_{0},x_{0}))$. Fixing $\varepsilon\in(0,\varepsilon_0)$, we introduce the mappings $$\psi\colon X\to [0,1], \quad x\mapsto
\begin{cases}
1-\frac{2}{r}\operatorname{dist}\left(x, B\left(u_0,\frac{r}{2}\right)\right) & x\in B(u_0,r)\\
0 & x\not\in B(u_0,r)
\end{cases}$$ and $$\varphi\colon \mathbb{R}\to\mathbb{R}, \quad t\mapsto \min\left\{|t|,\frac{\varepsilon}{\sigma}\right\}.$$ These mappings satisfy $$\operatorname{Lip}\psi = \frac{2}{r}, \quad \|\psi\|_\infty=1, \quad \operatorname{Lip}\varphi =1\quad\text{and}\quad
\|\varphi\|_\infty = \frac{\varepsilon}{\sigma}.$$
Since the metric segment $[u_0,x_0]$ is isometric to a closed real interval, it is an absolute $1$-Lipschitz retract by Proposition 1.4 in [@BL2000GeomtericNonlinear p. 13]. Let $R\colon X\to [u_0,x_0]$ be a $1$-Lipschitz retraction and $c\colon [0,\rho_{X}(u_{0},x_{0})]\to [u_0,x_0]$ be a metric embedding with $c(0)=u_0$. We define $$q \colon C_{X} \to [0,\rho_{X}(u_{0},x_{0})], \quad x \mapsto c^{-1}(R(x)).$$ Since $q$ is the composition of $1$-Lipschitz mappings, it is also a $1$-Lipschitz mapping. Finally, we also define the mapping $$\lambda\colon X\to\mathbb[0,1], \quad x\mapsto \frac{\sigma}{2\rho_{X}(u_{0},x_{0})}\psi(x)\varphi(q(x)).$$ This mapping satisfies $\lambda(x)=0$ whenever $x\in X\setminus B(u_{0},r)$, $\left\|\lambda\right\|_{\infty}\leq
\varepsilon/2\rho_{X}(u_{0},x_{0})$ and $$\begin{aligned}
\operatorname{Lip}(\lambda) &\leq \frac{\sigma}{2\rho_{X}(u_{0},x_{0})}\left(\operatorname{Lip}(\varphi)\|\psi\|_\infty+\operatorname{Lip}(\psi)\|\varphi\|_\infty\right)
= \frac{1}{2\rho_{X}(u_{0},x_{0})} \left(\sigma +\frac{2}{r} \varepsilon\right)\\
&\leq \frac{1}{2\rho_{X}(u_{0},x_{0})}\left(\sigma + \sigma\right)
\leq \frac{1}{2\rho_{X}(u_{0},x_{0})}2\sigma = \frac{\sigma}{\rho_{X}(u_{0},x_{0})}\end{aligned}$$ because $\varepsilon < \sigma \frac{r}{2}$. We observe now that the conditions of Lemma \[lemma:pert\] are satisfied for $Z=X$, $\sigma$, $u_{0}$, $z_{0}=x_{0}$ $\pi=\operatorname{id}_{{\operatorname{conv}}(C_{X})}$ $r,\varepsilon\in(0,r_{0})$ and $\lambda$. Finally also note that $$\label{eq:lambau0zero}
\lambda(u_0)=0$$ since $u_0\in[u_0,x_0]$ implies $R(u_0)=u_0$, $q(u_0)=0$ and hence $\varphi(u_0)=0$ . Applying Lemma \[lemma:pert\], we conclude that the mapping $\beta\colon {\operatorname{conv}}(C_{X})\to{\operatorname{conv}}(C_{X})$, defined by $$\beta(x):=(1-\lambda(x))x\oplus\lambda(x) x_0,$$ satisfies $\beta(C_{X})\subseteq C_{X}$, $\rho_{X}(\beta(x),x)\leq\varepsilon$ for all $x\in C_{X}$ and $\operatorname{Lip}(\beta)\leq 1+3\sigma$.
\[lem:PropertiesGStar\] The mapping $$G\colon {\operatorname{conv}}(C_{X}) \to \ell_{\infty}(\Omega), \quad x\mapsto F(\beta(x))$$ satisfies the following conditions:
(i) \[G:StoY\] $G(C_{X})\subseteq C_{Y}$;
(ii) \[eq:DistFandGStar\] $\rho_{Y}(F(x),G(x))\leq\varepsilon$ for all $x\in C_{X}$;
(iii) \[eq:LipGStar\] $\operatorname{Lip}(G)\leq 1$;
(iv) \[Gsteep\] For $s=\varepsilon/\rho_{X}(u_{0},x_{0})$, we have $(1-s)u_{0}\oplus sx_{0}\in \Gamma$ and $$\frac{\rho_{Y}(G((1-s)u_0\oplus sx_0),G(u_0))}{s\rho_{X}(u_0,x_0)} > a\left(1+\frac{\sigma}{4}\right).$$
The inclusion $\beta(C_{X})\subseteq C_{X}$ together with the fact that $F$ is an extension of the mapping $f:C_{X}\to C_{Y}$ implies condition . Condition follows immediately from the fact that $\rho_{X}(\beta(x),x)\leq\varepsilon$ for all $x\in C_{X}$. Let us now verify condition : Since $G$ coincides with $F$ outside of $B(u_{0},r)$ and is defined on a $\rho_{X}$-convex set, an argument similar to the one at the end of the proof of Lemma \[lemma:pert\] shows that it suffices to prove $\operatorname{Lip}(G|_{B(u_{0},r)})\leq 1$. If we show $\beta(B(u_0,r))\subseteq U$, this inequality follows from $\operatorname{Lip}(\beta)\leq 1+3\sigma$, $\operatorname{Lip}(F|_{U})\leq (1-\frac{1}{p})$ and $(1+3\sigma)(1-\frac{1}{p})\leq 1$. In order to show the required inclusion, we use $\rho_{X}(\beta(x),x)\leq \varepsilon$ and $\varepsilon<r$ to get that $\beta(B(u_{0},r))\subseteq \overline{B}(u_{0},r+\varepsilon)\subseteq B(u_{0},3r)\subseteq U$.
Next we turn our attention to . The choice of $\varepsilon_{0}$ and $s=\varepsilon/\rho_{X}(u_{0},x_{0})<2\varepsilon_{0}/\rho_{X}(u_{0},x_{0})$ imply that $(1-s)u_{0}\oplus sx_{0}\in\Gamma$. For $t\in(0,1)$, we define $$\gamma(t):=(1-t)[(1-s)u_{0}\oplus sx_{0}]\oplus tx_{0}.$$ Using condition of Definition \[def:weaklyhyperbolic\] in the weakly hyperbolic space $X$, we note that $\gamma(t)$ lies on the metric segment $[u_{0},x_{0}]$ in between $(1-s)u_{0}\oplus sx_{0}$ and $x_{0}$. Therefore we can compute $\rho_{X}(\gamma(t),u_{0})$ as the sum $$\begin{aligned}
\rho_{X}(\gamma(t),u_{0})&=\rho_{X}(\gamma(t),(1-s)u_{0}\oplus sx_{0})+\rho_{X}((1-s)u_{0}\oplus sx_{0},u_{0}))\nonumber\\
&=t(1-s)\rho_{X}(u_{0},x_{0})+s\rho_{X}(u_{0},x_{0})\nonumber\\
&=
(t+s(1-t))\rho_X(u_{0},x_{0}).
\end{aligned}$$ It follows that $$\label{eq:distgammat}
\gamma(t)=(1-\alpha(t))u_{0}\oplus \alpha (t)x_{0},\qquad\text{ where }\alpha(t):=t+s(1-t).$$ Using the definitions of the mappings $\varphi$, $q$ and $\psi$ together with $$\rho_{X}(u_0,(1-s)u_0\oplus sx_0)=s\rho_{X}(u_0,x_0)= \varepsilon<\varepsilon/\sigma<r/2$$ we obtain $\varphi(q((1-s)u_{0}\oplus sx_{0}))=\varepsilon$, $\psi((1-s)u_{0}\oplus sx_{0})=1$ and subsequently, $$\lambda((1-s)u_{0}\oplus sx_{0})=\sigma\varepsilon/2\rho_{X}(u_{0},x_{0})=\sigma s/2.$$ We conclude that $\beta((1-s)u_{0}\oplus sx_{0})=\gamma(\frac{\sigma s}{2})$. From we see that $\alpha(\frac{\sigma s}{2})<2s<2\varepsilon_{0}/\rho_{X}(u_{0},x_{0})$. Therefore we can apply to deduce $$\begin{aligned}
\frac{\rho_{Y}(G((1-s)u_{0}\oplus sx_{0}),G(u_{0}))}{s\rho_{X}(u_{0},x_{0})}
&=\frac{\rho_{Y}\big(F((1-\alpha(\frac{\sigma s}{2}))u_{0}\oplus\alpha(\frac{\sigma s}{2})x_{0}),F(u_{0})\big)}
{\alpha(\frac{\sigma s}{2})\rho_{X}(u_{0},x_{0})}\frac{\alpha(\frac{\sigma s}{2})}{s}\\
&>a\left(\frac{\sigma}{2}+1-\frac{\sigma s}{2}\right)>a\left(1+\frac{\sigma}{4}\right).
\end{aligned}$$ Above we used to get $G(u_0)=F(\beta(u_0))=F(u_0)$ in the first line and the condition $s<\varepsilon_{0}/\rho_{X}(u_{0},x_{0})<1/2$ to get the final inequality.
We are now ready to prove Lemma \[lemma:Qabpporous\].
Fix $f\in{\mathcal{Q}}_{a,b}^{p}(U)$ and let $\Gamma\in{\mathcal{G}}$, $F:{\operatorname{conv}}(C_{X})\to\ell_{\infty}(\Omega)$, $u_{0}\in\Gamma$, $\sigma\in(0,1)$ satisfying $(1+3\sigma)(1-\frac{1}{p})\leq 1$ and $\varepsilon_{0}>0$ be defined according to the above construction. The precise value of $\sigma$ will be determined at the end of this proof. Given $\varepsilon\in(0,\varepsilon_{0})$, let the mapping $G\colon {\operatorname{conv}}(C_{X})\to \ell_{\infty}(\Omega)$ be given by the statement of Lemma \[lem:PropertiesGStar\]. Define $g\colon C_{X}\to C_{Y}$ to be the restriction of $G$ to the set $C_{X}$. From Lemma \[lem:PropertiesGStar\] it is clear that $g\in {\mathcal{M}}(C_{X},C_{Y})$ with $d_{\theta}(g,f)\leq\varepsilon$. We complete the proof by showing that $$B_{\theta}\left(g,\frac{a\sigma}{32(1+\rho_{X}(u_{0},\theta))}\varepsilon\right)\cap {\mathcal{Q}}_{a,b}^{p}(U)=\emptyset.$$ Let $h\in B_{\theta}\left(g,\frac{a\sigma}{32(1+\rho_{X}(u_{0},\theta))}\varepsilon\right)$. Then $$\rho_{Y}(g(x),h(x)) \leq \frac{1+\rho_{X}(x,\theta)}{1+\rho_{X}(u_0,\theta)} \frac{a\sigma}{32}\varepsilon
\leq \frac{a\sigma}{16}\varepsilon$$ for $x\in C_{X}\cap B(u_0,1)$ and, in particular, $$\rho_{Y}(g((1-s)u_0\oplus s x_0),h((1-s)u_0\oplus s x_0)) \leq \frac{a\sigma}{16}\varepsilon$$ for $s=\varepsilon/\rho_{X}(u_{0},x_{0})$ because $\varepsilon<\varepsilon_{0}<1$. Therefore, using Lemma \[lem:PropertiesGStar\], part and the fact that $g$ coincides with $G$ on the segment $[u_{0},(1-s)u_{0}\oplus sx_{0}]\subseteq\Gamma\subseteq C_{X}$, we deduce that $$\begin{aligned}
\frac{\rho_{Y}(h((1-s)u_0\oplus s x_0),h(u_0))}{s\rho_{X}(u_0,x_0)}
&\geq \frac{\rho_{Y}(g((1-s)u_0\oplus s x_0),g(u_0))}{s\rho_{X}(u_0,x_0)} - 2 \frac{a\sigma}{16}\frac{\varepsilon}{s\rho_{X}(u_0,x_0)}\\
& > a \left(1+\frac{\sigma}{4}\right)-\frac{a\sigma}{8}= a\left(1+\frac{\sigma}{8}\right).
\end{aligned}$$ We conclude from the above inequalities that $\operatorname{Lip}(h|_{\Gamma})>b$, when we choose $\sigma=\frac{16(b-a)}{a}$. Condition ensures that such a choice of $\sigma$ satisfies $(1+3\sigma)(1-\frac{1}{p})\leq 1$, as required. This establishes $h\notin {\mathcal{Q}}_{a,b}^{p}(U)$ and completes the proof.
The sets ${\mathcal{Q}}_{a,b}^{p}(U)$ do not quite cover the whole of the set ${\mathcal{Q}}(U)$. In the next lemma, we verify that the elusive mappings in ${\mathcal{Q}}(U)$ form a porous subset of ${\mathcal{M}}(C_{X},C_{Y})$.
\[lemma:ConstPorousStar\] The set $${\mathcal{Q}}_{0}(U) := \left\{f\in{\mathcal{Q}}(U)\colon \sup_{\Gamma\in {\mathcal{G}}}\operatorname{Lip}(f|_{\Gamma})=0,\, \inf_{F\in{\mathcal{E}}(f)}\operatorname{Lip}(F|_{U})<1\right\}$$ is porous in $\mathcal{M}(C_{X},C_{Y})$.
Fix a mapping $f\in{\mathcal{Q}}_{0}(U)$ and choose an extension $F:{\operatorname{conv}}(C_{X})\to\ell_{\infty}(\Omega)$ of $f$ with $\operatorname{Lip}(F|_{U})<1$. Choose $x_{0}\in{\operatorname{star}}(C_{X})$ such that $U\cap B(x_{0},D_{X})\neq\emptyset$ and set $U'=U\cap B(x_{0},D_{X})\setminus\left\{x_{0}\right\}$. We make the following claim:
There exist $u_{0}\in C_{X}\cap U'$, $y_{0}\in C_{Y}\setminus \left\{f(u_{0})\right\}$ and $r>0$ such that $F({\operatorname{conv}}(C_{X})\cap B(u_{0},r))\subseteq B(y_{0},D_{Y})$ and for every $x\in C_{X}\cap B(u_{0},r)$, there is a unique metric segment $[f(x),y_{0}]\subseteq C_{Y}$ .
We distinguish between two cases. First assume that $f(x)\in{\operatorname{star}}(C_{Y})$ for all $x\in C_{X}\cap U'$. Then we choose $u_{0}\in C_{X}\cap U'$ arbitrarily and let $r>0$ be small enough so that $B(u_{0},r)\subseteq U'$ and $F({\operatorname{conv}}(C_{X})\cap B(u_{0},r))\subseteq B(f(u_{0}),D_{Y}/2)$. Let $y_{0}\in C_{Y}\cap B(f(u_{0}),D_{Y}/2)\setminus\left\{f(u_{0})\right\}$ be arbitrary. The assertion of the claim is now clear.
In the remaining case we choose $u_{0}\in C_{X}\cap U'$ such that $f(u_{0})\notin {\operatorname{star}}(C_{Y})$ and use the fact that $C_{Y}\subseteq B({\operatorname{star}}(C_{Y}),D_{Y})$ to choose $y_{0}\in {\operatorname{star}}(C_{Y})\cap B(f(u_{0}),D_{Y})$. Letting $r>0$ be sufficiently small so that $F({\operatorname{conv}}(C_{X})\cap B(u_{0},r))\subseteq B(y_{0},D_{Y})$, we verify the claim.
Let $u_{0}\in C_{X}\cap U'$, $y_{0}\in C_{Y}\setminus\left\{f(u_{0})\right\}$ and $r>0$ be given by the claim. Choose $\sigma\in(0,1)$ small enough so that $$(1+\sigma)\operatorname{Lip}(F|_{U})+2\sigma\leq 1.$$ By making $r$ smaller if necessary we may assume that $B(u_{0},r)\subseteq U'$ and $r\in(0,r_{0})$, where $r_{0}>0$ is given by the conclusion of Lemma \[lemma:pert\] with $Z=Y$, $\sigma$, $u_{0}$, $z_{0}=y_{0}$ and $\pi=F$. Set $\varepsilon_{0}=r$. Given $\varepsilon\in(0,\varepsilon_{0})$, we define a mapping $\lambda:X\to [0,1]$ by $$\lambda(x):=\frac{\sigma}{2\rho_{Y}(f(u_{0}),y_{0})}\max\left\{\varepsilon-\rho_{X}(x,u_{0}),0\right\}, \quad x\in X.$$ Then, $$\lambda(x)=0 \text{ for all } X\setminus B(u_{0},r), \;\left\|\lambda\right\|_{\infty}\leq \varepsilon/2\rho_{Y}(f(u_{0}),y_{0})\text{ and }\operatorname{Lip}(\lambda)\leq \sigma/\rho_{Y}(f(u_{0}),y_{0}).$$ Thus, the conditions of Lemma \[lemma:pert\] are satisfied for $Z=Y$, $\sigma$, $u_{0}$, $z_{0}=y_{0}$ $\pi=F$, $r,\varepsilon\in(0,r_{0})$ and $\lambda$. Therefore, Lemma \[lemma:pert\] asserts that the mapping $G$ defined by $$G(x):=\begin{cases}
(1-\lambda(x))F(x)\oplus\lambda(x)y_{0} & \text{if }x\in C_{X}\cap B(u_{0},r),\\
F(x) & \text{if }x\in{\operatorname{conv}}(C_{X})\setminus B(u_{0},r),
\end{cases}$$ satisfies $G(C_{X})\subseteq C_{Y}$, $\rho_{Y}(G(x),F(x))\leq\varepsilon$ for all $x\in C_{X}$ and $\operatorname{Lip}(G)\leq1$. Clearly, the restriction $g$ of the mapping $G$ to the set $C_{X}$ can be viewed as an element of ${\mathcal{M}}(C_{X},C_{Y})$ satisfying $d_{\theta}(g,f)\leq\varepsilon$.
Since $B(u_{0},r)\subseteq U'=U\cap B(x_{0},D_{X})\setminus\left\{x_{0}\right\}$ and $x_{0}\in {\operatorname{star}}(C_{X})$, we have that $[u_{0},x_{0}]\subseteq C_{X}$. Identifying the metric segment $[u_{0},x_{0}]$ with a real interval we have the $u_{0}<u_{0}+\varepsilon<u_{0}+r<x_{0}$. Hence $[u_{0},u_{0}+\varepsilon]\in \mathcal{G}$. Using $\lambda(u_0+\varepsilon)=0$ and the fact that $f$ is constant on the segment $[u_{0},u_{0}+\varepsilon]$, we get $$\begin{aligned}
\rho_{Y}(g(u_{0}+\varepsilon),g(u_{0}))
&=\rho_{Y}(f(u_{0}),(1-\frac{\sigma\varepsilon}{2\rho_{Y}(f(u_{0}),y_{0})})f(u_{0})\oplus
\frac{\sigma\varepsilon}{2\rho_{Y}(f(u_{0}),y_{0})}y_{0})=\frac{\sigma\varepsilon}{2}.
\end{aligned}$$ For all $h\in{\mathcal{M}}(C_{X},C_{Y})$ with $$d_{\theta}(h,g)\leq \frac{\sigma\varepsilon}{6(1+\rho_{X}(u_{0},\theta)+\varepsilon_{0})},$$ we have $\rho_{Y}(h(x),g(x))\leq\sigma\varepsilon/6$ for $x=u_{0},u_{0}+\varepsilon$ which, when combined with the above equation, implies that $h$ is non-constant on the metric segment $[u_{0},u_{0}+\varepsilon]\in{\mathcal{G}}$. Hence $$B(g,\frac{\sigma\varepsilon}{6(1+\rho_{X}(u_{0},\theta)+\varepsilon_{0})})\cap {\mathcal{Q}}_{0}(U)=\emptyset$$ and the proof is complete.
(i) The proof of Lemma 4.4 is the only place in the proof of Theorem \[thm:all\], or indeed any of the results of Section 3, where we use the hypothesis that $C_{Y}$ is $\rho_Y$-star-shaped and satisfies $C_{Y}\subseteq B({\operatorname{star}}(C_{Y}),D_{Y})$.
(ii) In the special case where $C_{X}$ is $\rho_{X}$-convex, the set $Q_{0}(U)$ becomes simply the set of all mappings $f\in {\mathcal{Q}}(U)$ which are constant on the set $C_{X}\cap U$. The conclusion of Lemma \[lemma:ConstPorousStar\] is then valid under much weaker assumptions on the set $C_{Y}$. For example, it suffices to assume that $C_{Y}$ is a metric space in which every point belongs to some non-trivial geodesic. Thus, if we restrict our attention to the case where $C_{X}$ is $\rho_{X}$-convex, the results of Section 3 can be generalised accordingly.
For each $f\in{\mathcal{Q}}(U)\setminus {\mathcal{Q}}_{0}(U)$, we have $$(\sup_{\Gamma\in {\mathcal{G}}}\operatorname{Lip}(f|_{\Gamma}),\inf_{F\in{\mathcal{E}}(f)}\operatorname{Lip}(F|_{U}))\in(0,1)^{2}.$$ The family of all rectangles of the form $(a,b)\times(0,1-\frac{1}{p})$, where $p\in\mathbb{N}$ with $p\geq 2$ and $0<a<b<1$ satisfy , is an open cover of $(0,1)^{2}$. Therefore, since $(0,1)^{2}$ is a Lindelöf space, this family admits a countable subcover $((a_{i},b_{i})\times(0,1-\frac{1}{p_{i}}))_{i=1}^{\infty}$. Hence we may write $${\mathcal{Q}}(U) = \bigcup_{i=1}^{\infty} {\mathcal{Q}}_{a_{i},b_{i}}^{p_{i}}(U) \cup {\mathcal{Q}}_0(U).$$ Applying now Lemma \[lemma:ConstPorousStar\] and Lemma \[lemma:Qabpporous\], we arrive at the asserted result.
An application to set-valued mappings
=====================================
The goal of this section is to examine properties of spaces of non-empty, closed and bounded subsets of hyperbolic spaces in order to show that these spaces can be chosen as the range of the nonexpansive mappings in the theorems which were established in the previous sections.
Let $(X,\rho)$ be a complete hyperbolic space and $C\subseteq X$ be a non-empty, non-singleton, closed and $\rho$-star-shaped set. We consider the space $$\mathcal{B}(C) := \left\{A\subseteq C\colon\; A \text{ is nonempty, closed and bounded}\right\}$$ equipped with the Pompeiu-Hausdorff metric $$h(A,B) := \max\big\{\sup\{\operatorname{dist}(a,B)\colon a\in A\},\; \sup\{\operatorname{dist}(b,A)\colon b\in B \}\big\},$$ where $\operatorname{dist}(x,A) := \inf\{\rho(x,a)\colon a\in A\}$. The space $\mathcal{B}(C)$ is a complete metric space by [@Kur1966Topology §33, IV]. In addition to the hyperspace of all bounded and closed sets, we also consider the subspaces $\mathcal{K}(C)$ of compact subsets and $\mathcal{CB}(C)$ of $\rho$-convex, bounded and closed sets.
In the case where $X$ is a Banach space, the following lemma is a consequence of Proposition 4.6 in [@Str2014Porosity].
\[lem:starShapedSetValued\] There is a family $\mathcal{F}$ of metric segments in $\mathcal{B}(C)$ such that the triple $(\mathcal{B}(C),h,\mathcal{F})$ is a space of temperate curvature with $D_{\mathcal{B}(C)}=\infty$ and $\mathcal{B}(C)$ is a $h$-star-shaped subset of this space.
For $A\in\mathcal{B}(C)$, $A\neq\{c\}$, we define $$\label{eq:SetConvex}
A^{(1-\lambda)} := \{(1-\lambda) a \oplus \lambda c\colon a\in A\} \quad\text{and}\quad
(1-\lambda) A\oplus \lambda\{c\} := \overline{A^{(1-\lambda)}},$$ and set $$\mathcal{F}:=\left\{\left\{(1-\lambda) A\oplus \lambda\{c\}\colon \lambda\in[0,1]\right\}\colon A\in\mathcal{B}(C), \; c\in{\operatorname{star}}(C)\right\}.$$ In order to show that $\mathcal{F}$ is a well-defined collection of metric segments in $\mathcal{B}(C)$, we have to show that $(1-\lambda) A\oplus \lambda\{c\}\in\mathcal{B}(C)$ for every $A\in\mathcal{B}(C)$, $A\neq\{c\}$, and that the sets $[A,\left\{c\right\}]:=\{(1-\lambda) A\oplus \lambda\{c\}\in\mathcal{B}(C)\colon \lambda\in [0,1]\}$ are metric segments. In order to show uniqueness of the metric segments in $\mathcal{F}$, note that we only have to consider the case of two singletons $\{c\}$ where $c\in{\operatorname{star}}(C)$, since for every other set $A$ the pair $(A,\{c\})$ appears only once in the definition of $\mathcal{F}$. Uniqueness of segments of the form $[\{c_1\},\{c_2\}]$, where $c_1,c_2\in{\operatorname{star}}(C)$ follows from the fact that $X$ is hyperbolic.
For $a,b\in A$, the inequality $$\rho((1-\lambda) a \oplus \lambda c, (1-\lambda) b \oplus \lambda c)\leq (1-\lambda) \rho(a,b) \leq (1-\lambda) \operatorname{diam}(A),$$ which follows from the fact that $X$ is a hyperbolic space, implies that $(1-\lambda) A\oplus \lambda\{c\}$ is a bounded set. Since it is, by definition, also non-empty and closed, we get that it is contained in $\mathcal{B}(C)$. In addition, note that for all $\mu\in [0,1]$ and all $a\in A$, the point $(1-\mu)a\oplus\mu c$ lies on the metric segment $[a,c]$, which is contained in $C$ because $C$ is $\rho$-star-shaped with respect to $c$. Therefore $(1-\mu) A \oplus \mu \{c\} \subseteq C$ for all $\mu\in[0,1]$.
Note that from $h(B,\overline{B})=0$ for arbitrary bounded sets $B\subseteq C$, we may deduce $$h((1-\lambda) A\oplus \lambda\{c\},E) = h(A^{(1-\lambda)},E)$$ for every bounded set $E\subseteq C$. Now let $A\in\mathcal{B}(C)$, $c\in{\operatorname{star}}(C)$, $\lambda,\mu\in [0,1]$ and assume without loss of generality that $\lambda > \mu$. Then $$\begin{aligned}
h\big((1-\lambda) A\oplus \lambda\{c\},\{c\}\big) & = \sup\{\rho((1-\lambda) a \oplus\lambda c,c)\colon a\in A\} \\
& = (1-\lambda) \sup\{\rho(c,a)\colon a\in A\} = (1-\lambda) h(A,\{c\}).
\end{aligned}$$ Moreover, we have $$h\big((1-\mu) A\oplus \mu\{c\}, \{c\}\big) \leq h\big((1-\mu) A\oplus \mu\{c\}, (1-\lambda) A\oplus \lambda\{c\}\big)
+ h\big((1-\lambda) A\oplus\lambda\{c\}, \{c\}\big),$$ which is equivalent to $$h\big((1-\lambda) A\oplus \lambda\{c\}, (1-\mu) A\oplus \mu\{c\}\big) \geq (\lambda - \mu) h(A, \{c\}).$$ On the other hand, we also have $$\begin{aligned}
\operatorname{dist}\big((1-\lambda)a\oplus\lambda c, (1-\mu) A\oplus \mu \{c\}\big)
& = \inf\{\rho((1-\lambda)a\oplus\lambda c,(1-\mu)b\oplus\mu c)\colon b\in A\}\\
& \leq (\lambda -\mu) \rho(a,c) \leq (\lambda -\mu) h(A, \{c\})
\end{aligned}$$ and analogously, $\operatorname{dist}\big((1-\mu)a\oplus\mu c, (1-\lambda) A \oplus \lambda \{c\}\big) \leq (\lambda-\mu) h(A,\{c\})$.\
Therefore $h\big((1-\lambda) A\oplus \lambda\{c\}, (1-\mu) A\oplus \mu\{c\}\big) = |\lambda-\mu| h(A,\{c\})$. The above facts show that for all $A\in\mathcal{B}(C)$, $A\neq\{c\}$, the mapping $$[0,h(\{c\},A)] \to \mathcal{B}(C), \quad \lambda \mapsto \left(1-\tfrac{\lambda}{h(A,\{c\})}\right) A \oplus \tfrac{\lambda}{h(A,\{c\})}\{c\}$$ is a metric embedding and therefore $[A,\{c\}]$ is a metric segment in $\mathcal{B}(C)$.
We now show that $(\mathcal{B}(C), h,\mathcal{F})$ is of temperate curvature. That this triple satisfies condition of Definition \[def:weaklyhyperbolic\] with $D_{\mathcal{B}(C)}=\infty$ is already clear. It only remains to verify condition of Definition \[def:weaklyhyperbolic\]. We will prove something stronger. Namely, that metric segments in $\mathcal{F}$ even satisfy the hyperbolic inequality , or equivalently $$\label{eq:hypineq2}
h((1-\lambda)A\oplus \lambda E,(1-\lambda)B\oplus \lambda E)\leq (1-\lambda) h(A,B)$$ for all $A,B,E\in\mathcal{B}(C)$ with $[A,E],[B,E]\in\mathcal{F}$. Note that all segments in $\mathcal{F}$ have a set of the form $\{c\}$, where $c\in{\operatorname{star}}(C)$, as one of their endpoints. Therefore we only need to verify for the case $E=\left\{c\right\}$ with $c\in {\operatorname{star}}(C)$ and the case $A=\left\{c_{1}\right\}$, $B=\left\{c_{2}\right\}$ with $c_{1},c_{2}\in {\operatorname{star}}(C)$.
Given $A, B \in\mathcal{B}(C)$ and $c\in{\operatorname{star}}(C)$, let $a\in A$ and $b\in B$. Since $X$ is a hyperbolic space, we have $$\rho((1-\lambda)b\oplus\lambda c, (1-\lambda)a\oplus\lambda c) \leq (1-\lambda) \rho(a,b)$$ and hence $$\operatorname{dist}((1-\lambda)b\oplus\lambda c, (1-\lambda) A \oplus \lambda\{c\})
\leq (1-\lambda) \inf \{\rho(a,b)\colon a\in A\} =(1-\lambda) \operatorname{dist}(b,A)$$ for all elements of $(1-\lambda) B\oplus \lambda\{c\}$. Since the situation is completely analogous if we swap the roles of $(1-\lambda) A \oplus \lambda\{c\}$ and $(1-\lambda) B \oplus \lambda \{c\}$, we may conclude that $$h((1-\lambda) A \oplus \lambda\{c\},(1-\lambda) B \oplus \lambda\{c\})\leq (1-\lambda) h(A,B).$$ This verfies inequality for the case $E=\left\{c\right\}$.
To prove the inequality in the remaining case, we take $c_1,c_2\in{\operatorname{star}}(C)$, $E\in\mathcal{B}(C)$ and observe that $$\rho((1-\lambda) c_1\oplus \lambda a, (1-\lambda) c_2\oplus \lambda a') \leq (1-\lambda) \rho(c_1,c_2) + \lambda \rho(a,a'),$$ for all $a,a'\in E$, by . From this we may deduce $$\operatorname{dist}((1-\lambda) c_1\oplus \lambda a, (1-\lambda)\{c_2\}\oplus\lambda E) \leq (1-\lambda) \rho(c_1,c_2) = (1-\lambda) h(\{c_1\},\{c_2\})$$ for all $a\in E$, and therefore, since the situation is completely symmetric with respect to $c_1$ and $c_2$, $$h((1-\lambda)\{c_1\}\oplus\lambda E, (1-\lambda) \{c_2\}\oplus\lambda E) \leq (1-\lambda) h(\{c_1\},\{c_2\}).$$ Finally, note that by the construction of $\mathcal{F}$, we get $${\operatorname{star}}(\mathcal{B}(C)) = \{\{c\}\colon c\in{\operatorname{star}}(C)\}$$ and hence $\mathcal{B}(C)$ is a $h$-star-shaped subset of $(\mathcal{B}(C),h,\mathcal{F}$).
Note that the above construction does not work if we replace the set $\{c\}$ by a non-singleton as can be seen by the following example. We consider the metric space $C:=[-1,1]^2$ equipped with the standard metric and set $A:=\{(-1,-1),(-1,1)\}$ and $B:=\{(1,-1),(1,1)\}$. We get $h(A,B)=2$ and $$\frac{1}{2}A+\frac{1}{2} B = \{(0,-1),(0,0),(0,1)\}.$$ Therefore $h(\frac{1}{2}A+\frac{1}{2} B, A)= \sqrt{2} \neq \frac{1}{2} h(A,B)$. More generally, Example 4.7 in [@Str2014Porosity] shows that even in the case of Banach spaces the hyperspace of bounded and closed subsets cannot be a hyperbolic space in the sense of Reich-Shafrir.
As a consequence of Lemma \[lem:starShapedSetValued\] and Theorems \[thm:Nsigporous\], \[thm:convexresidual\] and \[thm:starshapedresidual\], we can infer the following corollary regarding set-valued nonexpansive mappings.
\[cor:OnHyperspaces\] Let $X$ be a complete hyperbolic space and $C\subseteq X$ be a non-empty, non-singleton, closed, $\rho$-star-shaped subset. Then the following statements hold:
(i) The set $$\mathcal{N}(C,\mathcal{B}(C)) := \{f\colon C\to \mathcal{B}(C)\colon \operatorname{Lip}(f)<1\},$$ is a $\sigma$-porous subsets of the space $$\mathcal{M}(C,\mathcal{B}(C)) := \{f\colon C\to \mathcal{B}(C)\colon \operatorname{Lip}(f)\leq 1\}$$ of all nonexpansive $\mathcal{B}(C)$-valued mappings equipped with the metric $d_\theta$.
(ii) If $C$ is separable, there exists a $\sigma$-porous set $\widetilde{{\mathcal{N}}}\subseteq {\mathcal{M}}(C,\mathcal{B}(C))$ such that for all $f\in {\mathcal{M}}(C,\mathcal{B}(C))\setminus\widetilde{{\mathcal{N}}}$, the set $$\widehat{R}(f)=\left\{x\in C\colon {\widehat{\operatorname{Lip}}}(f,x)=1\right\}$$ is a residual subset of $C$.
(iii) If $C$ is separable and $\rho$-convex, there exists a $\sigma$-porous set $\widetilde{{\mathcal{N}}}\subseteq {\mathcal{M}}(C,\mathcal{B}(C))$ such that for all $f\in {\mathcal{M}}(C,\mathcal{B}(C))\setminus\widetilde{{\mathcal{N}}}$, the set $$R(f)=\left\{x\in C\colon \operatorname{Lip}(f,x)=1\right\}$$ is a residual subset of $C$.
Results analogous to Corollary \[cor:OnHyperspaces\] are valid for all hyperspaces $\mathcal{X}(C)$ with the property that $$(1-\lambda) A\oplus \lambda\{c\} \in \mathcal{X}(C),$$ where $(1-\lambda) A\oplus \lambda\{c\}$ is defined in , for all $c\in{\operatorname{star}}(C)$, $\lambda\in[0,1]$ and $A\in\mathcal{X}(C)$. In the case of $\mathcal{K}(C)$ this follows from the fact that for all $c\in{\operatorname{star}}(C)$ and all $\lambda\in[0,1]$, the mapping $$C\to C,\quad a\mapsto (1-\lambda) a\oplus \lambda c$$ is continuous. In [@PL2014ContractiveSetValued] spaces with this property are called “admissible” and, besides $\mathcal{B}(C)$ and $\mathcal{K}(C)$, the following examples are given in [@PL2014ContractiveSetValued Remark 2.5, p. 1417]: the space of singletons, the space of bounded, closed and $\rho$-convex sets, and the space of compact and $\rho$-convex sets.
In addition to the above corollary, we can also show that the set of bounded strict contractions is a $\sigma$-porous subset of the space of all bounded nonexpansive $\mathcal{B}(C)$- and $\mathcal{K}(C)$-valued mappings if we equip these spaces with the metric of uniform convergence.
Note that if $X$ is a Banach space, we do not need to take the closure in the definition of the set $(1-\lambda)A\oplus \lambda\{c\}$ in since the sum of a closed set and a compact set is closed. In addition, if we define $$\label{eq:ConvComb}
(1-\lambda)A\oplus \lambda B := \overline{\{(1-\lambda)a + \lambda b\colon a\in A,\; b\in B\}}.$$ for bounded, closed and convex sets $A$ and $B$ and $\lambda\in[0,1]$ we get analogously to above a well-defined mapping from $[0,h(A,B)]$ to the space of bounded, closed and convex sets which satisfies the hyperbolicity inequality. That the above mapping is an isometry follows from this inequality and from $$(1-\lambda)A\oplus\lambda B = \frac{1-\lambda}{1-\mu}\big((1-\mu)A\oplus\mu B\big)\oplus \frac{\lambda-\mu}{1-\mu}B.$$ for bounded, closed and convex sets $A$ and $B$ and $0\leq \mu < \lambda\leq 1$, which can be shown by interchanging the occurring convex combinations. This implies that the space of bounded, closed and convex subsets of a closed and convex subset of a Banach space is $h$-convex. In particular, the hyperspace of bounded, closed and convex subsets of a bounded and closed subset of a Banach space is a hyperbolic space. We remark in passing that convexity, in a more general sense, of hyperspaces of compact sets is studied in detail in [@Dud1970ConvexV]. For the star-shapedness and hyperbolicity properties of these hyperspaces on subsets of Banach spaces, we refer the interested reader to [@Str2014Porosity].
The authors wish to thank an anonymous referee for reading the paper very carefully and for many useful and interesting suggestions which made the article significantly more reader friendly. This research was supported in part by the Israel Science Foundation (Grant 389/12), the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund.
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Christian Bargetz\
Department of Mathematics\
The Technion—Israel Institute of Technology\
32000 Haifa, Israel\
and\
Department of Mathematics\
University of Innsbruck\
Technikerstraße 13, 6020 Innsbruck, Austria (current address)\
`christian.bargetz@uibk.ac.at`\
Michael Dymond\
Department of Mathematics\
University of Innsbruck\
Technikerstraße 13, 6020 Innsbruck, Austria\
`michael.dymond@uibk.ac.at`\
Simeon Reich\
Department of Mathematics\
The Technion—Israel Institute of Technology\
32000 Haifa, Israel\
`sreich@math.technion.ac.il`
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Nelson A. Lima $^{1}$'
- 'and Pedro G. Ferreira $^{2}$'
title: 'On the phenomenology of extended Brans-Dicke gravity'
---
=1
\[Int\]Introduction
===================
Over the next decade, we expect a step change in our understanding of gravity on cosmological scales. Surveys of large scale structure should be able to pin down the expansion of the Universe and the growth of structure with exquisite precision [@euclid; @lsst; @wfirst; @ska]. These new data sets should allow us to constrain modifications to general relativity at a level which may be comparable to those obtained on astrophysical scales.
If we are to fully take advantage of these data sets, it is essential to have a detailed and accurate understanding of how different observables depend on our assumptions about gravity. In particular, we should know how deviations from general relativity will affect our observations: whether the effects are large or small (given what we know on astrophysical scales) and how correlations between the observables themselves might be indicative of some underlying structure.
There has been a formidable campaign to develop methods for studying the effects of modified gravity on large scales (for a compendium of theories, see [@reviewall]). A different approach has been to develop a unified method of parameterizing all possible theories at the linearized level (for a selection of methods, see [@tessapar; @battye; @gubitosi; @gleyzes]). Yet, while there is an inexorable momentum that should lead to a battery of effective techniques for extracting useful information from the data, we do not have yet a firm understanding of what to expect. By this we mean that, given certain theoretical assumptions, what our observables should look like, i.e. what values should they take and how should they be interrelated as a function of whatever fundamental parameters we might consider.
In principle, the step from taking the parameters, $\alpha_i$ (with $i=1,\cdots N$), of some underlying theory and working out the resulting phenomenological parameters, $\beta_j$ (with $j=1,\cdots M$) tied to observations, should be straightforward. In practice, the process can be complicated, highly non-linear, degenerate and normally obscures the relationship between the prior assumptions on $\alpha_i$ and the resulting theoretical priors on $\beta_j$. One way around this is to develop an approximate mapping between the two sets of parameters and, wherever possible, analytic relations between the two. Furthermore, if one can find a method for restricting the range of $\alpha_i$ given some assumptions about a subset of the $\beta_i$, one can quickly surmise what correlations and covariance one should expect for the remaining phenomenological parameters. In this paper we propose an approach to do so, considering a restricted model for cosmological modifications to gravity.
Our starting point is a well known theory, the Brans-Dicke (BD) theory of gravitation [@bd1]. This theory is the simplest scalar-tensor theory one can envisage [@st1; @st2; @st3; @st4; @st5] and is considered a viable alternative to General Relativity, one which respects Mach’s Principle. Since its formulation, this theory has been exhaustively studied as a possible alternative solution for the accelerated expansion of the Universe.
It has been shown that Brans-Dicke theory can produce accelerated solutions for small, negative values of the BD parameter $\omega_{\rm{BD}}$ [@bd2; @bd6]. Given that one recovers standard GR in the limit where $\omega_{\rm{BD}} \rightarrow \infty$, such values of the $\omega_{\rm{BD}}$ clash with Solar system constraints [@will; @bertotti]; furthermore, recent constraints with the latest CMB data are also not compatible with such low values of $\omega_{\rm{BD}}$ [@planckbd; @planckbd2]. Several modifications of this theory try to include self-interacting potentials [@bd3; @bd4; @bd7] or consider a field-dependent Brans-Dicke parameter $\omega(\phi)$ [@bd5], without solving this problem successfully. Also, models with a non-minimal coupling of the scalar field have been considered in Refs. [@bd8; @bd9; @bd10; @bd11].
In this paper we construct a theory of [*designer*]{}, extended Brans-Dicke gravity and use it to characterize the form of the observables we might measure. This theory is “extended” because we include a potential for the Brans-Dicke field and we dub it “designer” (the term “designer” was first used in models of inflation that attempted to match observations by designing the density fluctuation spectra [@infdesigner]) because we reconstruct the potential (which might not have an analytic form) from a desired background evolution. While such a theory does not seem fundamental, it might be seen as an approximation to a scalar-tensor theory which has a particular, a priori, form of the background evolution. Our construction allows us to find a number of analytic approximations and, in doing so, lets us gain a firmer understanding of the phenomena we want to study.
Our designer approach for the extended Brans-Dicke gravity is novel. It allows us to retrieve the evolution of the scalar field, $\phi$, by fixing the background evolution and is robust for high values of the BD parameter, which is the regime we are interested in. This method not only works for a $\Lambda$CDM like evolution with an effective equation of state $w_{\rm{eff}} = -1$, but is also applicable for models with $w_{\rm{eff}} > -1$ as in a $w$CDM scenario. And, for both cases, we are able to retrieve analytical approximations for $\phi$ as a function of the scale factor $a$ which could prove useful for a faster and more efficient fitting of models to data.
The paper is structured as follows. In Sec. \[intro\] we introduce the Brans-Dicke theory with a constant $\omega_{\rm{BD}}$ parameter. In Sec. \[designerbd\] we describe the designer approach, motivated by an analysis of the behavior of this theory when we have a constant potential $V(\phi)$. In Sec. \[analytical\] find approximate analytic solutions to the evolution of the scalar field and use it to infer the shape of the potential. We then use these results in Sec. \[phenom\] to construct analytical approximations to the phenomenological parameter which can be constrained by data. In Sec. \[discussion\] we discuss our results.
\[intro\]Extended Brans-Dicke gravity: background equations
===========================================================
The action for extended Brans-Dicke theory in the Jordan frame, is given by $${\label{bdaction}}
S = \frac{1}{2\kappa^{2}}\int d^{4} x \sqrt{-g} {\ensuremath{\left(\phi R - \frac{\omega_{\rm{BD}}}{\phi}{\ensuremath{\left(\partial \phi\right)}}^{2} - 2 V(\phi)\right)}} + S_{\rm{m}},$$ where $S_{\rm{m}}{\ensuremath{\left[\Psi_{\rm{m}};g_{\mu \nu}\right]}}$ is the minimally coupled matter Lagrangian and $\kappa^2 = 8 \pi G$, where $G$ is Newton’s gravitational constant measured today. Varying the action with respect to the metric elements, we find the Einstein equations, $${\label{einstein}}
G_{\mu \nu} = \frac{\kappa^2}{\phi} T_{\mu \nu}^{\rm{m}} + \frac{\omega_{\rm{BD}}}{\phi^2}{\ensuremath{\left[\phi_{,\mu}\phi_{,\nu} - \frac{1}{2}g_{\mu\nu}\phi_{,\alpha}\phi^{,\alpha}\right]}} + \frac{1}{\phi}{\ensuremath{\left[\phi_{,\mu;\nu}-g_{\mu \nu} \Box \phi\right]}} - \frac{V(\phi)}{\phi} g_{\mu \nu},$$ where $T_{\mu \nu}^{\rm{m}}$ is the matter stress-energy tensor.
By varying the action (\[bdaction\]) with respect to the field, one gets the field’s equation of motion $${\label{fieldmotion}}
\Box \phi = \frac{ \kappa^2 T}{3 + 2 \omega_{\rm{BD}}} - \frac{2}{3 + 2 \omega_{\rm{BD}}}{\ensuremath{\left[2 V(\phi) - \phi V_{\phi}\right]}},$$ where $V_{\phi} \equiv dV/d\phi$. Considering a flat Friedmann-Lemaitre-Robertson-Walker (FLRW) metric, $ds^2 = -dt^2 + a^{2}(t)d\vec{x}^2$, this equation reads $${\label{fieldfrw}}
\ddot{\phi} + 3 H \dot{\phi} = \frac{\kappa^2 \rho_{\rm{m}}}{3+2 \omega_{\rm{BD}}} + \frac{4V(\phi) - 2\phi V_{\phi}}{3+2 \omega_{\rm{BD}}},$$ where $\rho_{\rm{m}}$ is the matter’s energy density and $H \equiv \dot{a}/a$ is the Hubble parameter. The latter is determined by the two Friedmann equations, which are written as $$\begin{aligned}
{\label{friedmann}}
3 H^{2} \phi &=& \kappa^2 \rho_{\rm{m}} - 3H \dot{\phi} + \frac{\omega_{\rm{BD}}}{2}\frac{\dot{\phi}^{2}}{\phi} + V(\phi) \\
2\dot{H} + 3H^{2} &=& -\kappa^2\frac{p_{\rm{m}}}{\phi} - \frac{\omega_{\rm{BD}}}{2}\frac{\dot{\phi}^{2}}{\phi^2} - 2H \frac{\dot{\phi}}{\phi} - \frac{\ddot{\phi}}{\phi} + \frac{V(\phi)}{\phi}. \nonumber\end{aligned}$$
Lastly, from the previous equations, one can define an effective equation of state for the dark energy component of our model, which is given by $${\label{weff}}
w_{\rm{eff}} = \frac{\dot{\phi}^{2} \omega(\phi) + 4 H \dot{\phi} + 2 \ddot{\phi} - 2 V(\phi)}{\dot{\phi}^{2} \omega(\phi) - 6 H \dot{\phi} + 2 V(\phi)},$$ where $\omega(\phi) = \omega_{\rm{BD}}/\phi$ and, even more straightforwardly, one can define the fractional effective dark energy density parameter, $${\label{fraceff}}
\Omega_{\phi} = \frac{\rho_{\phi}}{3 H^2 \phi},$$ where the effective energy density is given by $${\label{rhoeff}}
\rho_{\phi} = \frac{\omega_{\rm{BD}}}{\phi}\frac{\dot{\phi}^{2}}{2} - 3 H \dot{\phi} + V(\phi).$$
\[constantpot\]Constant Potential V($\phi$)
-------------------------------------------
Before proceeding to the designer approach, we can get an idea of the different effects at play in extended Brans-Dicke gravity by considering the case of a constant potential $V(\phi)$. For all our calculations in this section, we have $V(\phi) = 3 H_{0}^{2} {\ensuremath{\left(1 - \Omega_{\rm{m}}\right)}} \equiv V$, where $\Omega_{\rm{m}}$ is the fractional present-day energy density of matter. For a perfect $\Lambda$CDM scenario we should have an effective dark energy equation of state equal to $-1$ during the whole cosmological evolution, with the scalar field remaining perfectly still and showing no evolution at all. However, in the Brans-Dicke paradigm, the field should always evolve even if its dynamics are subdominant (in “slow roll") compared to the potential $V$. Hence, effectively, we will have a quasi-$\Lambda$CDM evolution.
We start by numerically solving the scalar field evolution using Eqs. (\[fieldfrw\]) and (\[friedmann\]) considering a constant potential as defined in the previous paragraph. We set the initial conditions for the scalar field deep within the matter dominated regime at a redshift around $z_{\rm{i}} \approx 1000$. For this, we consider a known solution of Brans-Dicke gravity given by [@pl1; @pl2; @pl3] $${\label{attphi}}
\phi = \phi_{0} a^{1/{\ensuremath{\left(\omega_{\rm{BD}}+1\right)}}},$$ where $\phi_{0} = {\ensuremath{\left(2\omega_{\rm{BD}}+4\right)}}/{\ensuremath{\left(2\omega_{\rm{BD}}+3\right)}}$. This solution is, in fact, an attractor solution of the system derived in the absence of a potential $V(\phi)$ and for a Universe dominated by matter alone [@pl1; @pl2; @pl3]. The scale factor, on the other hand, evolves as [@pl1; @pl2; @pl3] $${\label{atta}}
a(t) = {\ensuremath{\left(\frac{t}{t_0}\right)}}^{{\ensuremath{\left(2\omega_{\rm{BD}}+2\right)}}/{\ensuremath{\left(3\omega_{\rm{BD}}+4\right)}}},$$ and we see that, in the GR limit of $\omega_{\rm{BD}} \rightarrow \infty$, $\phi = 1$ and $a(t) \propto t^{2/3}$ throughout the matter dominated regime; $t_{0}$ is related to the inverse of the present-day value of the Hubble parameter, $H_{0}$, such that $t_{0} H_{0} = {\ensuremath{\left(2 \omega_{\rm{BD}}+2\right)}}/{\ensuremath{\left(3 \omega_{\rm{BD}}+4\right)}}$. The value of $\phi_{0}$ ensures that, in a matter dominated Universe, we would measure an effective gravitational constant today, $G_{\rm{eff}}$, equal to the actual Newton’s gravitational constant, $G$, in Cavendish-like experiments. This assumes, of course, that the Solar system value of $\phi$ is representative of the Universe as a whole, which may not be entirely accurate [@cliftonsolar].
Let us also point out that, in a matter dominated flat Universe, the matter density will not be precisely equal to the critical density due to a very small, negative, and almost negligible contribution from the scalar field dynamics. It is possible to rescale the matter density (as in Ref. [@andrew]), but we opt not to do so, since the correction is negligible in the $\omega_{\rm{BD}} >> 1$ regime we are mostly interested in this work.
In Fig. \[figure1\], we have the numerical evolution of the scalar field plotted against the power-law solution given by Eq. (\[attphi\]). We can clearly observe that, even in the presence of a constant potential $V$, the Brans-Dicke scalar field evolves according Eq. (\[attphi\]) at early-times, during the matter dominated epoch. Only at late-times, close to $a = 1$, we see a slight departure from the power-law of Eq. (\[attphi\]), when the dark energy component begins to dominate and accelerates the scalar field.
Still in Fig. \[figure1\] we can observe the numerical evolution of the dark energy effective equation of state $w_{\rm{eff}}$ as given by Eq. (\[weff\]). We observe a very sharp transition from $-0.4$ to $-1$ that we will explain later on. For now, we can conclude that, even though the scalar field is accelerated by the presence of the constant potential $V$, its dynamics remain subdominant (the aforementioned slow roll evolution) and allow for a late-time potential dominated epoch with $w_{\rm{eff}} = -1$
Having shown in Fig. [\[figure1\]]{} that we recover the power-law solution given Eq. (\[attphi\]) at early-times, we now extend its application by using it in the effective equation of state $w_{\rm{eff}}$ given by Eq. (\[weff\]) in the presence of a constant potential $V$. Hence, we approximately obtain $${\label{weffatt}}
w_{\rm{eff}} \approx \frac{4 - 4 \omega_{\rm{BD}} V a^{3}/H_{0}^{2}}{-10 + 4 \omega_{\rm{BD}}V a^{3}/H_{0}^{2}},$$ in the limit of $\omega_{\rm{BD}} >> 1$, and where we have also used Eq. (\[atta\]). Hence, in the matter dominated regime, the potential contribution is suppressed by the scale factor leading to $w_{\rm{eff}} \approx -0.4$ (unless $\omega_{\rm{BD}} \rightarrow \infty$ and $V \ne 0$). Thus, for values of $\omega_{\rm{BD}}$ which are consistent with Solar System constraints, it is impossible to get an accelerated solution without adding a potential $V(\phi)$, that may not necessarily be constant. However, with a constant potential $V(\phi)$, one gets $w_{\rm{eff}} = -1$ at late times after a sharp, non-smooth transition from $w_{\rm{eff}} \approx -0.4$, which we have seen in Fig. \[figure1\]. An effective equation of state $w_{\rm{eff}} \approx -0.4$ at early times could constitute a problem, eventually compromising the extension of the matter dominated regime and rendering the model inviable. However, calculating $\Omega_{\phi}$, given by Eq. (\[fraceff\]), explicitly during the matter dominated regime using Eqs. (\[attphi\]) and (\[atta\]), one gets $${\label{rhoeff_attractor}}
\Omega_{\phi} \approx \frac{1}{3}{\ensuremath{\left[-\frac{5}{2 \omega_{\rm{BD}}} + \frac{V(\phi)}{H_{0}^{2}}a^{3}\right]}},$$ which, for large values of $\omega_{\rm{BD}}$ is negligible at early times.
Also, we note that the discontinuity in $w_{\rm{eff}}$ happens due to a zero crossing of the denominator of Eq. (\[weff\]). If we change from physical time $t$ to the natural logarithm of the scale factor, $dt \rightarrow d\ln a$, we have that $d/dt \rightarrow H d/d\ln a$. Therefore, neglecting the $\phi^{\prime 2}$ (the prime denotes a derivative with respect to $\ln a$) term because this is proportional to $(1+\omega_{\rm{BD}})^{-2}$ before the transition and for large $\omega_{\rm{BD}}$, the denominator of $w_{\rm{eff}}$ can be approximated to just $-3\phi^{\prime} + V(\phi)/H^2$. Therefore, given that, in the matter dominated regime, $V(\phi)/H^2 \propto V(\phi) a^{3}/H_{0}^{2}$ is an increasing function of the scale factor, there will come a point at which this term will be equal to $3 \phi^{\prime}$, leading to the discontinuity in $w_{\rm{eff}}$. For the constant potential, the scale factor of the discontinuity is apprximately $${\label{adisc}}
a_{\rm{disc}} \approx {\ensuremath{\left(\frac{\Omega_{\rm{m}}} {1-\Omega_{\rm{m}}} \frac{1}{1+\omega_{\rm{BD}}}\right)}}^{1/3}.$$ The discontinuity in $w_{\rm{eff}}$ has no impact on the background expansion of the model: if we take the second Friedmann equation and $p_{\rm{m}} = 0$, we have $${\label{secondfriedmann}}
\frac{\ddot{a}}{a} = -\frac{H^{2}}{2}{\ensuremath{\left(1 + 3\frac{w_{\rm{eff}}}{\rho_{\rm{m}}/\rho_{\phi}+1}\right)}}.$$ Since the divergence in $w_{\rm{eff}}$ happens due to $\rho_{\phi}$ crossing zero, as we just discussed, no divergence is seen in the evolution of $\ddot{a}$ because the term $\rho_{\rm{m}}/\rho_{\phi}$ follows the behavior of $w_{\rm{eff}}$.
Finally, only when $\omega_{\rm{BD}} \rightarrow \infty$ (the General Relativity limit) does one get $w_{\rm{eff}} = -1$ throughout the whole evolution, as seen in Fig. \[figure1\]. Here the potential $V(\phi)$ will dominate and the scalar field dynamics is heavily suppressed. The discontinuity in $w_{\rm{eff}}$ will now happen at a much earlier time, as is clear from Eq. (\[adisc\]), leading to a smooth $w_{\rm{eff}} = -1$ in the case of a constant potential.
\[designerbd\]Designer extended Brans-Dicke gravity
===================================================
Having presented the general form for extended Brans Dicke gravity, we now proceed to construct an algorithm that will lead to a particular expansion rate or, more specifically, to an effective equation of state. Hence, effectively, we design and impose the background history we wish for our model which in turn determines the dynamical evolution of the Brans-Dicke scalar field. We note that the authors of Ref. [@acquaviva] suggested the designer method we will describe further, but did not fully explore its consequences.
$
\begin{array}{cc}
\includegraphics[scale = 0.385]{phi_designer_2.pdf} & \includegraphics[scale=0.385]{weff_designer_2.pdf}
\end{array}$
Following the previous section, we have shown that, at early-times, the scalar field will follow the matter domination attractor solution irrespective of the presence of a scalar potential $V(\phi)$. At late-times, its evolution should be dominated by $V(\phi)$, leading to a departure from the matter dominated attractor solution. Therefore, we now try fixing the background evolution to match that of a standard flat $w$CDM scenario, such that $${\label{hubbledesigner}}
H^{2}(a) = \frac{H_{0}^{2} E(a)}{\phi} \equiv \frac{H_{0}^{2}}{\phi} {\ensuremath{\left[\Omega_{\rm{m}}a^{-3} + E_{\rm{eff}}(a)\right]}},$$ where $\Omega_{\rm{m}}$ is the present-day fractional matter energy density, and the dark energy component will be fixed as $${\label{effde}}
E_{\rm{eff}}(a) = {\ensuremath{\left(1-\Omega_{\rm{m}}\right)}}e^{3\int_{a}^{1} (1+w_{\rm{eff}}) \rm{d} \ln a}.$$ We will be assuming that the effective dark energy equation of state $w_{\rm{eff}}$ is a constant such that $w_{\rm{eff}} \geq -1$. We should be clear, however, that this is not a limitation of this procedure: it can be easily extended to a varying $w_{\rm{eff}}$ by providing a $w_{\rm{eff}}$ as a function of the scale factor $a$. We merely choose to do so in hope of finding analytic expressions for some of the observables in terms of the fundamental parameters of the theory. Therefore, we can now numerically evolve the scalar field just by using Eq. (\[fieldfrw\]) without evolving the Hubble parameter using Eq. (\[friedmann\]). We are also effectively parameterizing Eq. (\[rhoeff\]) so that our dark energy component’s energy density matches a $w$CDM type and are not worried with its exact numerical evolution. We then take the approximation of considering the scalar field potential to be determined by, $${\label{designpot}}
V(\phi) = 3 H_{0}^{2} {\ensuremath{\left(1-\Omega_{\rm{m}}\right)}}e^{3\int_{a}^{1} (1+w_{\rm{eff}}) \rm{d} \ln a},$$ meaning that we are considering that the main contribution to the effective dark energy density comes from the scalar field potential, with the scalar field dynamics being sub-dominant. With this approximation we also don’t expect to affect the matter domination attractor solution at early times since, as seen before, the potential contribution to $\Omega_{\phi}$ is not relevant in the matter dominated regime.
To generate our numerical results we have fixed the initial value of the scalar field $\phi(z_i)$ and $\phi^{\prime}(z_i)$ to match the matter dominated attractor solution value at a redshift of $z_{i} = 1000$. In Fig. \[figure2\] we plot the evolution of $\phi$ and $w_{\rm{eff}}$ for different values of $\Omega_{\rm{m}}$, $w_{\rm{eff}}$, and $\omega_{\rm{BD}}$ by numerically solving Eq. (\[fieldfrw\]) and fixing the evolution of $H$ with Eq. (\[hubbledesigner\]).
We note that, similarly to what we observed in the constant potential case, the presence of the dark energy component leads to a departure of $\phi$ from the matter domination attractor solution at late times, leading to a scalar field value higher than $\phi_{0}$ at the present. And Fig. \[figure2\] makes it clear that this departure happens earlier in time and is more significant the earlier the dark energy component starts to dominate at late-times (which happens the bigger $w_{\rm{eff}}$ is or the smaller $\Omega_{\rm{m}}$ is). This means that, the higher $w_{\rm{eff}}$ is, the more relevant the scalar field dynamics becomes. Hence, our designer approach breaks down if $w_{\rm{eff}}$ is much higher than $-1$.
Looking at Eq. (\[rhoeff\]), one might be concerned about the numerical evolution of the effective dark energy density which we parameterized by Eq. (\[effde\]); we would probably not recover a flat cosmology today due to the contribution of the scalar field dynamics to the overall critical density of the Universe. If we were to compute $\rho_{\phi}$ numerically with Eq. (\[rhoeff\]), one could adjust the weight of the potential $V(\phi)$ to compensate for the dynamics of the scalar field and recover $\Omega_{\phi} = 1-\Omega_{\rm{m}}$ today. Hence, in effect, $V(\phi) = 3 H_{0}^{2} \overline{\Omega}_{\phi} a^{-3(1+w_{\rm{eff}})}$, where $\overline{\Omega}_{\phi} \ne {\ensuremath{\left(1 - \Omega_{\rm{m}}\right)}}$ could be found by performing a simple binary search, for example. We will provide an approximation for this factor using our analytical solutions for $\phi$ in Appendix \[appendix2\].
We can also study the evolution of $w_{\rm{eff}}$ in Fig. \[figure2\]. We see that we again have a sharp transition from the matter domination attractor regime $w_{\rm{eff}} = -0.4$ value at early times to the value we fix $w_{\rm{eff}}$ to at late-times. The scale factor at which this transition happens can be estimated from $${\label{adiscgen}}
a_{\rm{disc}} \approx {\ensuremath{\left(\frac{\Omega_{\rm{m}}} {1-\Omega_{\rm{m}}} \frac{1}{1+\omega_{\rm{BD}}}\right)}}^{-\frac{1}{3w_{\rm{eff}}}},$$ making it clear that, the larger $w_{\rm{eff}}$ is and the earlier our dark energy component becomes relevant, the earlier this transition happens. Also, even though we don’t show that explicitly, we recover the GR plus $w$CDM limit when we take $\omega_{\rm{BD}} \rightarrow \infty$, and $w_{\rm{eff}}$ should then be equal to the value we fix it to be throughout the whole evolution, since the discontinuity now happens earlier or may even be completely avoided.
\[analytical\]Analytical solutions for $\phi$
=============================================
With our designer approach in hand, we can now proceed to find analytical approximations to the scalar field evolution which, in turn, can be used to construct approximations to our observables. We first consider the $\Lambda$CDM-like case and then generalize to an arbitrary (but constant) effective equation of state $w_{\rm{eff}}$.
\[yes1\]$w_{\rm{eff}}=-1$
-------------------------
We start by expressing the scalar field equation of motion, given by Eq. (\[fieldfrw\]), in terms of $\ln \hspace{1 mm} a$. We then simplify it by simultaneously neglecting the $\phi^{\prime \prime}$ and $\phi^{\prime 2}$ terms, yielding $${\label{fieldfrwapprox1}}
\frac{\phi^{\prime}}{\phi}{\ensuremath{\left(1 - \frac{1}{2}\frac{\Omega_{\rm{m}}a^{-3}}{1 - \Omega_{\rm{m}} + a^{-3}\Omega_{\rm{m}}}\right)}} = \frac{4{\ensuremath{\left(1-\Omega_{\rm{m}}\right)}}+a^{-3}\Omega_{\rm{m}}}{d{\ensuremath{\left(1 - \Omega_{\rm{m}}\right)}}+a^{-3}\Omega_{\rm{m}}},$$ where $d = {\ensuremath{\left(2 \omega_{\rm{BD}} + 3\right)}}$. The solution for the scalar field will be a fully analytical expression, given by $${\label{phisolsimpleminus1}}
\phi(a) = \phi(a_i) g(a_i)^{-1} g(a),$$ where $\phi(a_i)$ is the scalar field value at a high redshift $z_i$ set by the matter dominated attractor solution, or can be fixed to be $\phi_{0}$ at $a_i = 1$. The function $g(a)$ is given by $${\label{ga}}
g(a) = a^{\frac{2}{d}}{\ensuremath{\left(2a^{3}{\ensuremath{\left(1-\Omega_{\rm{m}}\right)}}+\Omega_{\rm{m}}\right)}}^{\frac{2}{3d}}$$
We show the evolution of $\phi$ predicted by this solution in Fig. \[figure3\]. It exhibits a tendency to overestimate the deviation from the matter domination attractor solution at late-times. However, its errors are small, specially when considering the considerable simplification we have found to the full numerical analysis of our designer approach.
\[not1\]$w_{\rm{eff}}\neq-1$
----------------------------
We now extend our analytical approximation for cosmologies with $w_{\rm{eff}}\neq-1$. In these circumstances, we expect the dark energy component to become relevant earlier, and hence produce larger deviations from the matter dominated attractor prediction. We will focus mainly in the late-time evolution of $\phi$, when the dark energy component comes to dominate. For that effect, we re-express Eq. (\[fieldfrw\]) in terms of $\ln a$, and assuming $V_{\phi} = 1/\phi^{\prime}\hspace{0.5 mm}dV/d \ln a$, we approximate it as $${\label{appeqmotion1}}
\phi \frac{12 \phi^{\prime} {\ensuremath{\left(1-\Omega_{\rm{m}}\right)}} + 18 \phi {\ensuremath{\left(1-\Omega_{\rm{m}}\right)}}{\ensuremath{\left(1+w_{\rm{eff}}\right)}}}{d {\ensuremath{\left(1 - \Omega_{\rm{m}} + \Omega_{\rm{m}}a^{3w_{\rm{eff}}}\right)}}} \approx 3 \phi^{\prime 2},$$ where $d = {\ensuremath{\left(2 \omega_{\rm{BD}} + 3\right)}}$ and we have also neglected terms proportional to $\phi^{\prime \prime}$, $\phi^{\prime 3}$ and $(1+w_{\rm{eff}})(1-\Omega_{\rm{m}})$, the last two arising with the derivative of $H$. We have not included the matter driving term that dominates at early-times. Assuming that the driving term from the potential slope is much more significant than the $V(\phi)$ one- which effectively means $\phi^{\prime}$ is much smaller than unity for large $\omega_{\rm{BD}}$- we take the square root of this equation and perform a Taylor expansion of the left-hand side, obtaining:
$${\label{appeqmotion2}}
\frac{6\phi^{\prime}{\ensuremath{\left(1-\Omega_{\rm{m}}\right)}}}{\sqrt{18{\ensuremath{\left(1-\Omega_{\rm{m}}\right)}}{\ensuremath{\left(1+w_{\rm{eff}}\right)}}}} + \sqrt{18{\ensuremath{\left(1-\Omega_{\rm{m}}\right)}}{\ensuremath{\left(1+w_{\rm{eff}}\right)}}}\phi - \phi^{\prime} \sqrt{3d{\ensuremath{\left(1 - \Omega_{\rm{m}} + \Omega_{\rm{m}}a^{3w_{\rm{eff}}}\right)}}}\approx 0$$
With these approximations, the solution for this equation is given by $${\label{phi_sol_intermediate}}
\phi(a) = \phi(a_i) f(a) f(a_i)^{-1},$$ where $\phi(a_i)$ is the value of the scalar field at a desired scale factor $a_i$. This can either be set to the matter dominated attractor solution at a redshift $z_i \approx 10$ or to $\phi_0$ at $a = 1$ if one wants to fix the present-day value of the scalar field to recover $G_{\rm{eff}}/G = 1$ today. The function $f(a)$ is given approximately by $${\label{functionofa}}
f(a) \approx {\ensuremath{\left(\frac{1+x}{x-1}\right)}}^{-\frac{\sqrt{6}\sqrt{d}{\ensuremath{\left(1+w_{\rm{eff}}\right)}}^{3/2}}{w_{\rm{eff}}{\ensuremath{\left(-2+3d(1+w_{\rm{eff}})\right)}}}},$$ where $x = \sqrt{1 + \frac{\Omega_{\rm{m}}}{1-\Omega_{\rm{m}}}a^{3w_{\rm{eff}}}}$ and we have neglected similar terms whose exponents were proportional to $d^{-1}$. In Fig. \[figuresol2\] we compare the late-time evolution of $\phi$ predicted by Eq. (\[phi\_sol\_intermediate\]) with the numerical evolution found in our designer approach. We do so by fixing $\phi(a_i)$ to the matter domination attractor solution at $z_i = 10$ for all the cases. We see that this solution works better for larger $\omega_{\rm{BD}}$. Nevertheless, even if the agreement with the numerical solution is not perfect, the errors are small, and the overall form of $\phi$ is excellent for such a simple approximation.
We are now also in a position to reconstruct the effective form of the self-interaction potential $V(\phi)$ across the entire cosmological evolution. For that, we invert the solutions for $\phi$ to get the scale factor as a function of the scalar field. We use the field’s matter dominated attractor solution at early-times and our analytical approximation at late-times. Hence, the potential will be given by
$${\label{potentialphi}}
V(\phi)=\begin{cases}
3 H_0^{2} (1 - \Omega_{\rm{m}}) {\ensuremath{\left(\phi/\phi_{0}\right)}}^{-3(1+w_{\rm{eff}})(1+\omega_{\rm{BD}})}, \hspace{3 mm} \text{during matter domination} & \\
3 H_0^{2} (1 - \Omega_{\rm{m}}) {\ensuremath{\left[ \frac{2^{2/3}{\ensuremath{\left(1-\Omega_{\rm{m}}\right)}}^{1/3}{\ensuremath{\left(\phi / c\right)}}^{\frac{{\ensuremath{\left(-2+3d(1+w_{\rm{eff}})\right)}}w_{\rm{eff}}}{3\sqrt{6}\sqrt{d}{\ensuremath{\left(1+w_{\rm{eff}}\right)}}^{3/2}}}}{\Omega_{\rm{m}}^{1/3}{\ensuremath{\left( 1 - {\ensuremath{\left(\phi / c\right)}}^{\frac{{\ensuremath{\left(-2+3d(1+w_{\rm{eff}})\right)}}w_{\rm{eff}}}{\sqrt{6}\sqrt{d}{\ensuremath{\left(1+w_{\rm{eff}}\right)}}^{3/2}}}\right)}}^{2/3}} \right]}}^{\frac{-3(1+w_{\rm{eff}})}{w_{\rm{eff}}}}, \hspace{3 mm} \text{at late-times},&
\end{cases}$$
where $c = \phi(a_i) f(a_i)^{-1}$, as defined in Eq. (\[phi\_sol\_intermediate\]). We plot the late-time form of the potential $V(\phi)$ and $V_{\phi}/V$ in Fig. \[figpot\] for different values of $\omega_{\rm{BD}}$, $w_{\rm{eff}}$ and $\Omega_{\rm{m}}$. We can observe that $V(\phi)$ exhibits a simple form, as in a standard run-away potential, with the slope decreasing at higher values of $\phi$ or, equivalently, close to the present. We see as well that $V_{\phi}/V$ takes significantly high, absolute values. This justifies our assumption in considering just the effect of the slope of the potential in the evolution of $\phi$ when $w_{\rm{eff}} \ne -1$. Indeed, this is the term that will have the most effect on the scalar field dynamics, leading to a significant departure from the attractor solution at late-times for $w_{\rm{eff}}>-1$.
In Fig. \[figpot\] we can also observe that the slope of the potential becomes more significant for higher values of $\omega_{\rm{BD}}$. This seem to contradict what we have seen in Fig. \[figuresol2\], where the scalar field dynamics seem to be more relevant, the smaller $\omega_{\rm{BD}}$ is. However, the source terms for the evolution of $\phi$ are suppressed by a factor proportional to $\omega_{\rm{BD}}^{-1}$. Hence, for a larger value of $\omega_{\rm{BD}}$, the only way to have significant field dynamics at late-times, and hence induce a significant departure from the matter dominated attractor solution that produces a $w_{\rm{eff}} \ne -1$, is to have a very large source term. Finally, we can also see how, for larger $w_{\rm{eff}},$ we recover a more tilted potential: the more relevant we set our dark energy component to be, the more significant we expect the scalar field dynamics to be at late-times.
\[global\]A global solution
---------------------------
In the previous two sections, we presented solutions for the evolution of the scalar field that worked well for $w_{\rm{eff}} = -1$ and $w_{\rm{eff}}>-1$ separately. We will now propose an approximate global solution: $${\label{globalsol}}
\phi_{\rm{global}}(a) = \phi(a_i) f(a) g(a) f(a_i)^{-1} g(a_i)^{-1},$$ which is just the product of the solutions we previously found for $w_{\rm{eff}} = -1$ and for $w_{\rm{eff}} > -1$.
Note that when $w_{\rm{eff}} = -1$, we have that $f(a_i)=f(a)=1$. Hence, $\phi_{\rm{global}}(a)$ will be the exact solution for the scalar field equation of motion under the assumptions we discussed in Sec. \[yes1\]. When $w_{\rm{eff}} > -1$, we note that the main contribution will come from the $f(a)$ and $f(a_i)$ terms; we already have seen in Section \[not1\] how the scalar field dynamics are more significant when $w_{\rm{eff}} > -1$. Not only that, but we note that $d g(a) /d \ln a \propto d^{-1}$, whereas $d f(a) / d\ln a$ produces terms proportional to ${\ensuremath{\left(\sqrt{d}\right)}}^{-1}$. Hence, assuming $d g(a)/ d \ln a << d f(a)/ d \ln a$ when $w_{\rm{eff}} > -1$, $\phi_{\rm{global}}(a)$ will be a solution of Eq. (\[appeqmotion2\]). We will use this full solution in the following sections for the phenomenological parameters, and show that it is indeed a good approximation for the overall behavior of the scalar field.
\[phenom\]A model for the phenomenological parameters.
======================================================
It has been shown that, at the level of the background and linear cosmological perturbation theory, it is possible to completely characterize any modified theory of gravity in terms of a handful of time dependent functions [@tessapar]. We proceed to do so with our designer extended Brans-Dicke gravity. We have already discussed two of our time dependent functions: the time varying Newton’s constant (at the level of the background), $G_0 = G/\phi$ and the effective equation of state, $w_{\rm{eff}}$.
For linear perturbations, following the notation of Ref. [@defelice], we consider a perturbed metric about the FLRW background in the Newtonian gauge, $${\label{pertmetric}}
ds^{2} = -(1 + 2\Psi)dt^2 + a^{2}(t)(1 + 2\Phi)\delta_{ij}dx^i dx^j,$$ where $\Psi$ and $\Phi$ are the scalar perturbations that we will refer to as Newtonian potentials and are decomposed as a series of Fourier modes of scale $k$ $(h/\rm{Mpc})$.
If we are interested in the the impact of matter perturbations on galaxy and weak lensing surveys, we can focus on the modes that are well within the Hubble radius, i.e. such that the condition $k^2/a^2 \gg H^2$ is respected. In this [*quasi-static*]{} regime the evolution equation for the matter density perturbation $\delta_{\rm{m}}$ can be approximated as [@matter1] $${\label{mattpert}}
\delta_{\rm{m}}^{\prime \prime} + {\ensuremath{\left(2 + \frac{H^{\prime}}{H}\right)}} \delta_{\rm{m}}^{\prime}- \frac{3}{2} \frac{G_{\rm{eff}}}{G} \Omega_{\rm{m}}(a) \delta_{\rm{m}} \simeq 0,$$ where $\Omega_{\rm{m}}(a) = \rho_{\rm{m}}/3H^2$, and $G_{\rm{eff}}/G$ will be dependent on the model. The primes represent derivatives with respect to $\ln a$. In the extended Brans-Dicke theory, $G_{\rm{eff}}$ is given by [@defelice] $${\label{geffbd}}
\frac{G_{\rm{eff}}}{G} = \frac{1}{\phi}\frac{4 + 2\omega_{\rm{BD}} + 2\phi{\ensuremath{\left(Ma/k\right)}}^2}{3 + 2\omega_{\rm{BD}} + 2\phi{\ensuremath{\left(Ma/k\right)}}^2},$$ where the $M$ term is [@defelice] $${\label{msquared}}
M^2 = V_{\phi \phi} + \frac{\omega_{\rm{BD}}}{\phi^3}{\ensuremath{\left[\dot{\phi^2} - \phi {\ensuremath{\left(\ddot{\phi} + 3H\dot{\phi}\right)}}\right]}}.$$
At late times, when the dark energy component starts to become relevant, the mass term can be simplified and expressed in terms of the potential alone using the scalar field equation of motion, such that $M^2 \approx V_{\phi \phi} + \frac{V_{\phi}}{\phi}$ [@defelice]. where $V_{\phi \phi}$ and $V_{\phi}$ correspond to the second and first order derivatives of the potential with respect to the scalar field, respectively.
One can also define the gravitational slip $\eta$ corresponding to the ratio between the two Newtonian potentials [@defelice] $${\label{etabd}}
-\frac{\Phi}{\Psi} \equiv \eta = \frac{1 + \omega_{\rm{BD}} + \phi {\ensuremath{\left(Ma/k\right)}}^2}{2 + \omega_{\rm{BD}} + \phi {\ensuremath{\left(Ma/k\right)}}^2}$$ which, again, should depend on the specifics of the scalar-tensor model. Lastly, the sub-horizon version of the Poisson equation can be written as [@defelice] $$\frac{k^2}{a^2} \Psi \simeq -4 \pi G_{\rm{eff}} \rho_{\rm{m}} \delta_{\rm{m}}.$$
In standard GR, when we neglect matter shear, the anisotropy equation between the Newtonian potentials becomes a simple constraint equation, $\Psi = \Phi$, and $\eta$ should be $1$ throughout the cosmological evolution, as should $G_{\rm{eff}}/G$. Hence, in a modified gravity theory, a deviation in these parameters signals a departure from standard GR that can potentially be measured. From Eqs. (\[geffbd\]) and (\[etabd\]) it is clear that the GR limit is recovered when $\omega_{\rm{BD}} \rightarrow \infty$, as expected, or when the field becomes supermassive and $M^2 \rightarrow \infty$. But we now wish to understand how these functions depend on time. To do so, it is convenient to study $$\begin{aligned}
\label{subhorizonparams}
\xi_{QS} &\equiv&\lim_{k\rightarrow\infty} \frac{G_{\rm{eff}}}{G}= \frac{1}{\phi}\frac{4+2\omega_{\rm{BD}}}{3+2\omega_{\rm{BD}}}\nonumber \\
\eta_{QS}&\equiv&\lim_{k\rightarrow\infty} \eta=\frac{1+\omega_{\rm{BD}}}{2+\omega_{\rm{BD}}} \end{aligned}$$ and the inverse length scale $${\label{massscale}}
k_M\equiv\sqrt{\frac{\phi}{1+\omega_{\rm{BD}}}}Ma.$$
From Eq. (\[subhorizonparams\]), we see that $\eta_{\rm{QS}}$ is constant throughout the cosmological evolution, independently of the scalar field dynamics [@acquaviva; @defelice; @matter1]. Its GR limit is trivially recovered when we take $\omega_{\rm{BD}} \rightarrow \infty$.
On the other hand, the late-time evolution of the mass scale parameter, $k_{M}$, can be written as:
$${\label{massqs}}
k_{M}(a) \approx \frac{3} {\sqrt{2}}H_{0}\sqrt{2{\ensuremath{\left(1-\Omega_{\rm{m}}\right)}}{\ensuremath{\left(1+w_{\rm{eff}}\right)}}a^{-{\ensuremath{\left(1+3w_{\rm{eff}}\right)}}}+a^{-1}\Omega_{\rm{m}}{\ensuremath{\left(2+w_{\rm{eff}}\right)}}}$$
which is valid for $w_{\rm{eff}} > -1$ and large $\omega_{\rm{BD}}$. In the limit $w_{\rm{eff}} = -1$, this equation predicts a non-zero value for $k_{M}$, whereas it should be exactly zero throughout, as predicted by explicitly using our global solution for $\phi$ in Eq. (\[massscale\]). This should be evident as $V_{\phi} = V_{\phi \phi} = 0$ when $w_{\rm{eff}} = -1$. This approximation works fairly well for small redshifts and better for larger $\omega_{\rm{BD}}$, as can be seen in Fig. \[figure7\].
We can also observe that $k_M$ is a fairly negligible quantity, corresponding to scales which are of order or greater than the cosmological scale. To do so we compare $k_{M}$ to the comoving horizon, $aH = H_0 \sqrt{\Omega_{\rm{m}}a^{-1} + (1-\Omega_{\rm{m}})a^{-{\ensuremath{\left(1+3w_{\rm{eff}}\right)}}}}$. It is not hard to see that $k_{M}/(aH) \lesssim 1$. Hence, the scale at which $k/k_{M}$ becomes relevant is approximately the same at which the perturbations $k$-modes become sub-horizon, which is at the basis of our assumptions. Therefore, taking $k/k_{M}>>1$ is an excellent approximation on quasi-static scales.
To understand the parameter dependence of $\xi_{QS}$ we perform a Taylor expansion around $a = 1$ using our approximate global solution for the scalar field. We further simplify our functions by considering the two regimes of interest we observe in our models: one where we will have $\phi = \phi_{0}$ today if we intend to recover $G_{\rm{eff}} = G$ at the present; and another one where we do not recover $\phi_0$ at the present, meaning that, essentially, we instead recover the matter domination attractor solution for $\phi$ at early times given by Eq. (\[attphi\]). For the first case, we obtain:
$${\label{xiqsapprox1}}
\xi_{QS_1} (a) \approx 1 + {\ensuremath{\left(1-a\right)}} {\ensuremath{\left[\frac{8-6\hspace{0.1 mm}\Omega_{\rm{m}}}{d{\ensuremath{\left(2-\Omega_{\rm{m}}\right)}}} + \frac{3\sqrt{6}\sqrt{d{\ensuremath{\left(1-\Omega_{\rm{m}}\right)}}}{\ensuremath{\left(1+w_{\rm{eff}}\right)}}^{3/2}}{3d{\ensuremath{\left(1+w_{\rm{eff}}\right)}}-2}\right]}}$$
while for the second case we find $${\label{xiqsapprox2}}
\xi_{QS_2} (a) \approx {\ensuremath{\left(\frac{\Omega_{\rm{m}}}{2-\Omega_{\rm{m}}}\right)}}^{\frac{2}{3d}}{\ensuremath{\left(\frac{\sqrt{\frac{1}{1-\Omega_{\rm{m}}}}-1}{1+\sqrt{\frac{1}{1-\Omega_{\rm{m}}}}}\right)}}^{-\frac{\sqrt{6}\sqrt{d}{\ensuremath{\left(1+w_{\rm{eff}}\right)}}^{3/2}}{{\ensuremath{\left(-2 + 3d{\ensuremath{\left(1+w_{\rm{eff}}\right)}}\right)}}w_{\rm{eff}}}}\xi_{QS_1}(a)$$ The GR limit of $\xi_{QS} = 1$ is recovered in both situations when we take $\omega_{\rm{BD}} \rightarrow \infty$. For the first case, as expected, $\xi_{QS} = 1$ when $a = 1$ since we have $\phi = \phi_0$ today. In the second case, $\xi_{QS} < 1$ today since the present-day value of the scalar field in these circumstances will always be larger than $\phi_0$. This can be observed in Fig. \[figure6\], where we compare these approximations to the exact numerical solution of $\xi$ and we see they work considerably well.
We can now try and understand the dependence of $\xi_{QS}$ on the different parameters. Looking at $d = 2\omega_{\rm{BD}} + 3$, it becomes clear that increasing the Brans-Dicke parameter leads to $\xi_{QS}$ becoming closer to $1$ throughout the late-time cosmological evolution: its slope at $a = 1$, as given by Eq. (\[xiqsapprox1\]), decreases since it depends on the inverse of $d$ or $\sqrt{d}$. Then, looking at Eq. (\[xiqsapprox2\]), we see that the present-day value of $\xi_{QS}$ increases towards $1$ due to the exponents of the terms shown becoming extremely small.
Looking at the dependence of $\xi_{QS}$ on $\Omega_{\rm{m}}$, we realize it is similar to that on $\omega_{\rm{BD}}$. Increasing $\Omega_{\rm{m}}$ leads to both the slope of $\xi_{QS}$ decreasing in Eq. (\[xiqsapprox1\]) as well as the present day-value tending to $1$ in Eq. (\[xiqsapprox2\]). In Eq. (\[xiqsapprox2\]) we also see that, remarkably, our approximation recovers the matter dominated attractor solution value of $\xi_{QS} = 1$ when $\Omega_{\rm{m}} \rightarrow 1$. Lastly, we have the effective equation of state parameter, $w_{\rm{eff}}$. Looking at Eq. (\[xiqsapprox1\]), we see that the slope of $\xi_{QS}$ will increase as $w_{\rm{eff}}$ becomes less negative, making its evolution more noticeable for larger $w_{\rm{eff}}$ when all other parameters remain fixed. Also, the exponent of the second term in Eq. (\[xiqsapprox2\]) increases for large $w_{\rm{eff}}$, leading to a significant departure of $\xi_{QS}$ today, producing values of $\xi_{QS} (a=1)$ that are detectably smaller than $1$ in a clear departure from standard GR. This is a reflection of the effect of increasing $w_{\rm{eff}}$ on the evolution of the scalar field $\phi$: the higher $w_{\rm{eff}}$ is, the sooner $\phi$ departs from the matter domination attractor solution and the larger its present-day value will be.
In Fig. \[figure8\], we plot $\xi_{QS}$ as a function of $\omega_{\rm{BD}}$ at $a = 1$, using Eq. (\[xiqsapprox2\]). We see that if we don’t fix $\phi = \phi_0$ today, there is a significant, possibly detectable, deviation from the standard GR value, $\xi_{QS} = 1$, even for very large $\omega_{\rm{BD}}$. Of course, we also see that when $\omega_{\rm{BD}} \rightarrow \infty$, $\xi_{QS}$ tends to $1$. Therefore, in order to be competitive with Solar System constraints $\omega_{\rm{BD}} > 10^{4}$ [@will; @bertotti], we would have to able to measure $\xi_{QS}$ with a precision of around $10^{-4}$.
![\[figure8\]We show the evolution of $\xi_{QS}$ at a = 1 for $\Omega_{\rm{m}} = 0.308$ and $w_{\rm{eff}} = -1$ as a function of $\omega_{\rm{BD}}$. For this plot we have used Eq. (\[xiqsapprox2\]), therefore assuming that $\phi(a=1)$ may not be equal to $\phi_0$.](xi_qs_present.pdf)
\[discussion\]Discussion
========================
In this paper we have applied the designer approach to the extended Brans-Dicke theory with the explicit presence of a self-interacting potential $V(\phi)$. By fixing the expansion history to that of an effective $w$CDM dark energy model, we are able to retrieve the scalar field evolution under the assumption that the main contribution to the effective dark energy density comes from the potential $V(\phi)$.
The numerical solutions we obtain have the property of respecting the matter domination attractor solution of Brans-Dicke models at early-times. At late-times, the scalar field departs from this solution and evolves more rapidly and towards larger values, yielding a value today larger than $\phi_0$, where $\phi_0$ is the present-day value of the matter regime attractor solution that ensures that one would measure $G_{\rm{eff}}$ today equal to the actual Newton’s gravitational constant, $G$, in a matter-dominated Universe. This transition from the attractor solution happens earlier whenever we take a larger dark energy equation of state, $w_{\rm{eff}}$. However, if we constrain the present-day value of $\phi$ to be equal to $\phi_0$, our numerical solutions follow the power-law behavior of the attractor solution, shifted towards smaller values at early-times. When the evolution departs from the matter dominated behavior, we are then able to recover $\phi(a=1) = \phi_0$ as intended.
We were able to obtain separate analytical approximations for the evolution of the scalar field when $w_{\rm{eff}} = -1$ and $w_{\rm{eff}}>-1$, which we then used to construct a global solution valid for $w_{\rm{eff}} \geq -1$. These approximations work remarkably well, with errors of sub-percent for large values of $\omega_{\rm{BD}}$. These approximations also allowed us to reconstruct the late-time functional form of the potential $V(\phi)$; we found a simple run-away potential whose slope is inevitably dependent on $w_{\rm{eff}}$ and $\omega_{\rm{BD}}$. We reiterate that we have limited our analysis to constant $w_{\rm{eff}}$ so as to obtain analytical solutions which will shed light on the parameter dependence of the various observables we are considering; a non-constant $w_{\rm{eff}}$ will severely complicate any attempts at doing so. However, we stress that the numerical implementation of the designer approach presented in Sec. \[designerbd\] can be easily extended to a non-constant $w_
{\rm{eff}}$.
With these analytic approximations in hand, we then focused on the phenomenological parameters that describe the sub-horizon evolution of the linear perturbations of the theory. We showed how the effective scale of the theory, which we designated by $k_{M}$, is of order the cosmological horizon; as a result we find that there is negligible scale dependence of the phenomenological parameters on observable scales. We found that the ratio between the Newtonian potentials, $\eta = \Psi/\Phi$ is constant throughout the cosmological evolution, for large values of the Brans-Dicke parameter [@defelice]. We also found simple analytical expression for $\xi = G_{\rm{eff}}/G$ which depend explicitly on the parameters of the theory, as seen in Eqs. (\[xiqsapprox1\]) and (\[xiqsapprox2\]).
One of the main features of this model is the possibility of having $\xi_{QS} \ne 1$ today; this is due to the departure of the scalar field from the matter dominated attractor solution at late-times such that its present-day value will be larger than $\phi_0$. The present-day value of $\xi_{QS}$ at $a = 1$, given in Eq. (\[xiqsapprox2\]) tends to $1$ as $\omega_{\rm{BD}} \rightarrow \infty$ since the exponent of the terms shown tend to zero. Also, as for $w_{\rm{eff}}>-1$, the exponent of one of the terms increases, leading to smaller values of $\xi_{QS}$ today, even when $\omega_{\rm{BD}}$ is very large. If, however, we impose $\xi_{QS}$ to be $1$ today, its evolution is predicted by Eq. (\[xiqsapprox1\]). In these circumstances, the main distinguishing point between this model and standard GR will be the slope of $\xi_{QS}$ at the present: for the extended Brans-Dicke theory it can be different from zero. We note that, even when $w_{\rm{eff}} = -1$, the predicted slope is different from zero. Hence, even the simple extended Brans-Dicke+$\Lambda$CDM model could be ruled out if $\xi_{QS}$ is found to not vary close to the present.
Finally, we note that in order to attain constraints on $\omega_{\rm{BD}}$ that are competitive with those obtained in Solar-system tests [@will; @bertotti], $\xi_{QS}$ and $\eta_{QS}$ would naively need to be constrained with a precision of around $10^{-4}$. This is a formidable challenge, but one should bear in mind that $\eta_{QS}\neq1$ throughout (at least) the matter dominated era while the same is possible for $\xi_{QS}$. This means that there will be a cumulative effective (as shown in [@Baker; @Leonard]) which means that constraints on the growth rate (or weak lensing) of order $10^{-3}$ or even $10^{-2}$ might be sufficient to place competitive constraints on $w_{\rm BD}$.
We would like to thank Andrew Liddle and Tessa Baker for helpful discussions and comments on this paper. N.A.L. also acknowledges financial support from Fundação para a Ciência e a Tecnologia (FCT) through grant SFRH/BD/85164/2012. P.G.F. acknowledges support from STFC, BIPAC, a Higgs visiting fellowship and the Oxford Martin School.
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\[appendix2\]Correction factor for V($\phi$)
============================================
In this appendix we show the correction factor one can add to the potential $V(\phi)$ in order to balance the scalar field dynamics in the exact numerical evolution of $\rho_{\phi}$ in order to recover a flat Universe today. Effectively, we want to solve the equation $$-\frac{\phi^{\prime}(a_0)}{\phi(a_0)} + \frac{\omega_{\rm{BD}}}{6} {\ensuremath{\left(\frac{\phi^{\prime}(a_0)}{\phi(a_0)}\right)}}^2 + \frac{1 - D\Omega_{\rm{m}}}{\phi(a_0)} = (1 - \Omega_{\rm{m}}),$$
where $a_0 = 1$ and $D$ will be the correction factor such that $1 - D \Omega_{\rm{m}} \equiv \overline{\Omega}_{\phi}$, as discussed in Sec. \[designerbd\]. First we show that factor using our solution for $w_{\rm{eff}} = -1$ using Eq. (\[phisolsimpleminus1\]):
$${\label{corr1}}
D = \frac{1}{\Omega_{\rm{m}}} + \frac{\phi(a_i)g(a_0)}{g(a_i)}\frac{{\ensuremath{\left[3d{\ensuremath{\left(\Omega_{\rm{m}}-2\right)}}{\ensuremath{\left(8 - 6\Omega_{\rm{m}} + d{\ensuremath{\left(2-\Omega_{\rm{m}}\right)}}{\ensuremath{\left(1-\Omega_{\rm{m}}\right)}}\right)}}+2\omega_{\rm{BD}}{\ensuremath{\left(4-3\Omega_{\rm{m}}\right)}}^2\right]}}}{3\Omega_{\rm{m}}d^2{\ensuremath{\left(\Omega_{\rm{m}}-2\right)}}^2},$$
where $g(a)$ is defined in Eq. (\[ga\]) and $\phi(a_i)$ is the value of the scalar field at the starting redshift $a_i$.
Lastly, we show the correction factor for the case $w_{\rm{eff}} > -1$. For this part we have used the late-time solution for $\phi$ given by Eq. (\[phi\_sol\_intermediate\]):
$$\begin{aligned}
{\label{corr2}}
D = \frac{1}{\Omega_{\rm{m}}} - \frac{\phi(a_i)}{\Omega_{\rm{m}}} c \Bigg( 1 - \Omega_{\rm{m}} + \frac{3\sqrt{6}\sqrt{d}\sqrt{1-\Omega_{\rm{m}}}{\ensuremath{\left(1+w_{\rm{eff}}\right)}}^{3/2}}{-2 + 3d{\ensuremath{\left(1+w_{\rm{eff}}\right)}}} - \frac{9\omega_{\rm{BD}}{\ensuremath{\left(1-\Omega_{\rm{m}}\right)}}d{\ensuremath{\left(1+w_{\rm{eff}}\right)}}^{3}}{{\ensuremath{\left(2-3d{\ensuremath{\left(1+w_{\rm{eff}}\right)}}\right)}}^{2}} \Bigg),\end{aligned}$$
where $c = f(a_0)/f(a_i)$, and $f(a)$ is defined by Eq. (\[functionofa\]).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We propose a scheme to entangle multiple material qubits through interaction with single photons via non-exciting processes associated with strongly coupling systems. The basic idea is based on the material state dependent reflection and transmission for the input photons. Thus, the material qubits in several systems can be entangled when one photon interacts with each system in cascade and the photon paths are mixed by the photon detection. The character of non-exciting of the material qubits does not change the state of material qubit and thus ensures the possibility of purifying entangled states by using more photons under realistic imperfect parameters. It also guarantees directly scaling up the scheme to entangle more qubits. Detailed analysis of fidelity and success probability of the scheme in the frame of an optical Fabry-Pérot (FP) cavity based strongly coupling system is presented. It is shown that a two-qubit entangled state with fidelity above 0.99 is promised with only two photons by using currently feasible experimental parameters. Our scheme can also be directly implemented on other strongly coupled system.'
author:
- Gang Li
- Pengfei Zhang
- Tiancai Zhang
bibliography:
- 'entanglement.bib'
title: Entanglement of remote material qubits through nonexciting interaction with single photons
---
Introduction
============
Quantum entanglement is a one of the key features in quantum mechanics and has been recognized as an important resource for quantum information processing [@Nielsen00] and quantum measurement [@Roos06]. Entanglement of remote material qubits is the essential intergradient for long distance quantum communication [@Briegel98; @Duan01; @Sangouard11] and quantum networks [@Kimble08]. There have been several methods used to produce remote entanglement among qubits and all of these methods involve the excitation of the material qubits and the process of emission or absorption of photons. The first technique involves entangling a photon to the first material qubit and directly writing it into the second material qubit [@Ritter12; @Matsukevich06]. The second one is a heralded protocol [@Duan01; @Duan03], wherein two photons entangled with each of the two material qubits interfere in a 50/50 beam splitter. Upon different combination of detected photon states, the states of material qubits are projected into various entangled states [@Chou05; @Hofmann12; @Lee11; @Yuan08; @Moehring07; @Chou07]. The third way is based on the quantum interference of two separated atomic qubits [@Cabrillo99]. The detection of a single photon from two atomic emissions produces entanglement between them [@Slodi13].
In a strong coupling system, a single material particle can interact with photons without exciting the material particle. In this nonexciting regime, the path of the input photon is determined by the states of the material particle. In state where the particle strongly couples to optical cavities [@Volz11; @Aoki09; @Shomroni14; @Scheucher2016] or other structures, such as the nanoscale surface plasmons [@Chang07], the incident photon will be reflected. In the state where the particles do not couple, the photon will transmit. With such systems nondestructive detection of atoms [@Volz11] and quantum single photon circulators controlled by the state of the single atom have been experimentally demonstrated [@Scheucher2016]. A single-photon transistor based on nanoscale surface plasmons has also been theoretically proposed [@Chang07].
Here we propose a scheme to produce remote entanglement based on this non-exciting interaction between the atom and photon in a strongly coupling system. The nonexciting process does not change the state of the material qubit and thus ensures the possibility of purifying the entangled state by using more photons in a real situation where loss and other imperfections are inevitable. This also guarantees scalability for producing entanglement among additional qubits. So, our proposal can be easily scaled up to create entanglement among multiple nodes in a quantum network [@Kimble08], or to generate the Greenberger-Horne-Zeilinger (GHZ) state among remote material qubits [@Greenberger09].
We first introduce the basic ideas in section II, where the theories of state-dependent reflection and transmission in a strongly coupling system, entangling two qubits in two systems by one photon, and scaling up the scheme are presented. Next, detailed analysis of state fidelity and success probability of our scheme in the frame of an optical Fabry-Pérot (FP) cavity based strongly coupling system is given in section III, in which a set of Heisenberg equations is built to simulate the reflection and transmission of the FP cavity with pulsed input single photons. At last, we conclude our paper and discuss the outlook for realization of our scheme on possible experiment systems in section IV.
Basic idea
==========
We first take a strongly coupling CQED system between single atoms and an optical FP cavity [@Boca04; @Maunz05] as an example to describe the state-dependent reflection and transmission of the incident photons. Other systems, such as strongly coupling surface-plasmon-emitter systems and strongly coupling systems between single atom and whispering-gallery-mode optical microresonators, work in the same way. The basic concept is shown in Fig. \[fig1\](a), where the cavity is comprised of two mirrors with transmission decay rates of $\kappa_{1(2)}$, and $\kappa_\text{loss}$ represents the decay rate of unexpected cavity loss from the scattering and absorption. An atom with two ground states, $|\alpha\rangle$ and $|\beta\rangle$, and an excited state $|e\rangle$ associated with decay rate $\gamma$, resides in the cavity. The atomic transition $|\beta\rangle \leftrightarrow |e\rangle$ strongly couples to the cavity mode with coupling strength $g$. The Hamiltonian of the system is given by $$\label{eq1}
H=\hbar g \left(|\beta\rangle \langle e|a^{\dagger}+|e\rangle \langle \beta|a\right)$$ with $a$ and $a^{\dagger}$ being the annihilation and creation operators of the cavity mode. If a weak coherent light beam $a_{in1(2)}$ is incident on mirror M1(2), the dynamics of the intracavity field $a$ is then described by the Heisenberg-Langevin equation [@Walls2008; @Duan04], $$\label{eq2}
\dot{a}=-\frac{i}{\hbar}[a,H]-\kappa a + \sqrt{2 \kappa_{1(2)}} a_\text{in1(2)},$$ where $a$ is the field amplitude of cavity mode. At the same time, we have the relations between the input and output fields of the cavity $$\label{eq3}
a_\text{out1}+a_\text{in1}=\sqrt{2\kappa_1} a$$ and $$\label{eq4}
a_\text{out2}+a_\text{in2}=\sqrt{2\kappa_2} a,$$ where $a_\text{out1}$ and $a_\text{out2}$ are the cavity output fields from M1 and M2. Under weak excitation approximations, which means the excitation of the atom is negligible, Eqs. (\[eq2\]) and (\[eq3\]) can be analytically solved and the coefficients for reflection and transmission of input field $a_\text{in1(2)}$ are $$\label{eq5}
r_{1(2)}=\frac{a_\text{out1(2)}}{a_\text{in1(2)}}=1-\frac{2 \kappa_{1(2)} (i \Delta_\text{a} +\gamma)}{(i\Delta_\text{c}+\kappa)(i\Delta_\text{a}+\gamma)+g^2}$$ and $$\label{eq6}
t_{1(2)}=\frac{a_\text{out2(1)}}{a_\text{in1(2)}}=\frac{2 \sqrt{\kappa_{1} \kappa_{2}} (i \Delta_\text{a} +\gamma)}{(i\Delta_\text{c}+\kappa)(i\Delta_\text{a}+\gamma)+g^2},$$ here $\Delta_\text{a}$ and $\Delta_\text{c}$ are the frequency detunings of the incident field with respect to the atomic transition and cavity. In our case $\Delta_\text{a}=\Delta_\text{c}=0$, the reflectivity and transmission can be simplified as $$\label{eq7}
r_{1(2)}=1-2\kappa_{1(2)} \gamma/(\kappa \gamma+g^2)$$ and $$\label{eq8}
t_{1(2)}=2 \sqrt{\kappa_1 \kappa_2} \gamma/(\kappa \gamma+g^2).$$ We can see the coefficients of transmission from the both sides are the same. In the ideal case, where $\kappa_{\text{loss}}=0$ and $\kappa_1=\kappa_2$ the reflectivity and transmission are $$\label{eq9}
r=1-\kappa \gamma/(\kappa \gamma+g^2)$$ and $$\label{eq10}
t=\kappa \gamma/(\kappa \gamma+g^2).$$
![\[fig1\] (Color online) (a) Schematic of a strongly-coupled CQED system with a single atom residing in the FP cavity. (b) A system equivalent to (a) with a four-port quantum router with internal states represented by $|R\rangle$ and $|T\rangle$. The power transmission (c) and reflection (d) spectra for a cavity with atoms in states $|\alpha\rangle$ and $|\beta\rangle$ are calculated by using parameters $g=10\kappa$, $\gamma=\kappa$, $\kappa_1=\kappa_2=\kappa/2$, and $\Delta_\text{a}=\Delta_\text{c}=0$.](Fig1-2018.eps){width="8.5cm"}
If the atom is in state $|\alpha\rangle$, the coupling efficiency $g=0$, thus we get $r=0$ and $t=1$ as the behavior of an empty cavity, see Fig. \[fig1\](c) and (d). The photon will transmit through the cavity without interacting with the atom. If the atom is in state $|\beta\rangle$, where the atom-cavity coupling $g$ is switched on, due to the normal mode splitting $2g$ the incident photon becomes detuned from the coupled states and thus is reflected. We will get $r\approx 1$ and $t\approx 0$ if $g \gg \kappa, \gamma$ from Eqs. (\[eq9\]) and (\[eq10\]). In both of these cases, there is no excitation of the atom and, in principle, the atomic state will not be changed. Thus by setting the atomic state in $|\alpha\rangle$ or $|\beta\rangle$, the CQED system can route the incident photon from the input mode $|a_\text{in1}\rangle$ ($|a_\text{in2}\rangle$) to output modes $|a_\text{out2}\rangle$ ($|a_\text{out1}\rangle$) or $|a_\text{out1}\rangle$ ($|a_\text{out2}\rangle$), respectively. If the atom is in a superposition state $\cos \theta |\alpha\rangle + \sin \theta \exp{i\varphi}|\beta\rangle$ the system will route the input photons to both of the output modes with probabilities $\cos^2 \theta$ and $\sin^2 \theta$. As such, the system works exactly as a single-photon quantum router.
This system is equivalent to a four-port quantum optical beam splitter (or circulator) where the reflection and transmission of incident photons are controlled by the state of the strongly coupled atom. The corresponding schematic is shown in Fig. \[fig1\](b), where we use $|R\rangle$ and $|T\rangle$ as meaningful representations of the internal states of the strongly coupled material qubit. If the state of a router is $|R\rangle$, photons from both input modes would be reflected. In other words, the input photons in $|a_\text{in1}\rangle$ and $|a_\text{in2}\rangle$ are routed to $|a_\text{out1}\rangle$ and $|a_\text{out2}\rangle$, respectively. Otherwise, the $|T\rangle$-state router will route photons from $|a_\text{in1}\rangle$ to $|a_\text{out2}\rangle$ and photons from $|a_\text{in2}\rangle$ to $|a_\text{out1}\rangle$ via transmission.
Quantum entanglement of two material qubits can be realized using the configuration shown in Fig. \[fig2\] (a). Here two quantum routers are arranged to replace the two classical beam splitter in a Mach-Zehnder interferometer. Two single photon detectors, D1 and D2, are used to detect photons from the two output modes $|a_\text{out1}\rangle$ and $|a_\text{out2}\rangle$. Quantum states of the routers are initially prepared in its maximum coherent superposition, i.e., $[|R_1\rangle+\exp (i\varphi_1)|T_1\rangle]/\sqrt{2}$ and $[|R_2\rangle+\exp (i\varphi_2)|T_2\rangle]/\sqrt{2}$. The overall quantum state is the direct product of these two wave functions. By sending a single photon into the input mode $|a_{\text{in}1}^{(1)}\rangle $, the overall state for the whole system after the photon is transmitted through the two cascade systems is expressed as:
$$\label{eq11}
\begin{aligned}
|\Psi_{\text{out}}\rangle = &\frac{1}{2}\left[|R_1,R_2\rangle+ e^{i (\varphi_1+\varphi_2 +\Delta \varphi)}|T_1,T_2\rangle\right]|a^{(2)}_{\text{out}1}\rangle \\
&+\frac{1}{2}\left[e^{i (\varphi_2 +\Delta \varphi)}|R_1,T_2\rangle+e^{i \varphi_1}|T_1,R_2\rangle\right]|a^{(2)}_{\text{out}2}\rangle \\
= &|\Phi_2\rangle|a^{(2)}_{\text{out}1}\rangle/\sqrt{2}+ |\Psi_2\rangle|a^{(2)}_{\text{out}2}\rangle/\sqrt{2}
\end{aligned}$$
with $$\label{eq12}
|\Phi_2\rangle=\left[|R_1,R_2\rangle+e^{i (\varphi_1+\varphi_2 + \Delta \varphi)}|T_1,T_2\rangle\right]/\sqrt{2}$$ and $$\label{eq13}
|\Psi_2\rangle=\left[e^{i (\varphi_2 +\Delta \varphi)}|R_1,T_2\rangle+e^{i \varphi_1}|T_1,R_2\rangle\right]/\sqrt{2}$$ exactly representing the two maximum entangled states for the routers. Here $\Delta \varphi$ is the phase difference between two paths of the interferometer. From Eq. (\[eq11\]) we can see that upon the event of photon detection by D1 or D2 the overall state of the two routers collapses into $|{{\Phi }_{2}}\rangle $ or $|{{\Psi }_{2}}\rangle $, respectively. The process of entanglement can be understood as path 1 and path 2 (3 or 4) which can not be distinguished from each other by photon detection with D1 (D2), as shown in FIG. 2. As such, a photon click in D1 (D2) will lead to entanglement showing in Eq. (\[eq12\]) \[Eq. (\[eq13\])\]. The probability of detecting a single photon by either of the two detectors is 0.5, which implies the probability of preparing each maximum entangled state is 50%. It should be emphasized that local operations on each material qubit, such as ground state operation of a single atom either by the two-photon Raman process [@Wang2014] or microwave driving [@Xia15], can be applied separately. Thus, upon the click of D2 $|{{\Psi }_{2}}\rangle $ can be converted into $|{{\Phi }_{2}}\rangle $ by applying a local $\pi$ rotation and a phase shift on router 2 and vice versa. Thus, in theory, entangled state $|{{\Phi }_{2}}\rangle $ or $|{{\Psi }_{2}}\rangle $ can be prepared with probability of 1 on demand.
![\[fig2\] (Color online) Schematics of producing entangled states between two quantum routers (a), 3 router-entangled state (b) and GHZ state among the 3 routers (c). The inset in (b) shows one method to scale up our scheme and generate N-router entangled state. In (b) and (c) the phase difference between different paths are taken as 0 for simplicity. In all the figures, each router has been prepared in an internal quantum state $[|R_n\rangle+\exp (i\varphi_n)|T_n\rangle]/\sqrt{2}$, in which $n$ means the router number.](Fig2-2018.eps){width="8.5cm"}
Because the material qubits do not absorb the incident photon and states of the material qubits remains unchanged after the photon has been detected, our scheme can be directly scaled up to entangle more material qubits. Figs. \[fig2\] (b) and (c) show two configurations that can be used to produce different types of entangled states among three or more qubits. In Fig. \[fig2\](b), a cascade configuration of three quantum routers is displayed, where two photon detectors are used. The three involved qubits, initially prepared in their maximum superposition state $|R_n\rangle+\exp{i\phi_n |T_n\rangle}$ ($n=1,2,3$), can be entangled by sending and detecting single photon. When D1 or D2 clicks, the entangled state of $$\label{eq14}
|\Phi_3\rangle=\left(|R_3\rangle|\Phi_2\rangle+e^{i \varphi_3}|T_3\rangle|\Psi_2\rangle\right)/\sqrt{2}$$ or $$\label{eq15}
|\Psi_3\rangle=\left(|R_3\rangle|\Psi_2\rangle+e^{i \varphi_3}|T_3\rangle|\Phi_2\rangle\right)/\sqrt{2}$$ is prepared with $|\Phi_2\rangle$ and $|\Psi_2\rangle$ given by Eqs. (\[eq12\]) and (\[eq13\]). This can also be explained by the fact that a click on D1 (D2) cannot distinguish photon paths among 1–4 (5–8), as shown in FIG. \[fig2\](b), and then entangles the internal states of routers. This cascade configuration can be directly scaled up as the schematic in the inset of Fig. \[fig2\](b) to produce large scale entangled states among $N$ quantum routers.
By following a similar process, a GHZ state with three qubits can be generated if we take the polarization of photons into account. As shown in Fig. \[fig2\](c), detectors D1 and D2 (D3 and D4) are arranged to detect photons with vertical (horizontal) polarization. Two polarization beam splitters (PBSs) are used to connect router 1 to router 2 (3) by photons with vertical (horizontal) polarization. After a click of D1 or D2 and a local operation on a qubit in router 2, a maximum entangled state \[Eq. (\[eq12\])\] is produced between routers 1 and 2. Then by using a single photon with horizontal polarization as the input, a click of D3 or D4 and a corresponding local operation on qubits in router 3 will produce a maximum entangled state $[|R_1,R_3\rangle+\exp{i (\varphi_1+\varphi_3)}|T_1,T_3\rangle]/\sqrt{2}$ between router 1 and router 3. The overall state is obvious a GHZ state $|\text{GHZ}\rangle =[|{{R}_{1}},{{R}_{2}},{{R}_{3}}\rangle +\exp i({{\varphi }_{1}}+{{\varphi }_{2}}+{{\varphi }_{3}})|{{T}_{1}},{{T}_{2}},{{T}_{3}}\rangle]/\sqrt{2}$. Here we omit the phase difference between the photon paths for simplicity.
Implementation of the scheme in a strongly coupling system with single atoms and an optical FP cavity
=====================================================================================================
We now consider experimental realization of our scheme in a strongly coupled system with single atoms coupling to an optical FP cavity. We will show the achievable state fidelity and the success probability of our scheme in a situation where experimental imperfections, like limited coupling strength $g$, slow response of the cavity and extra cavity losses, are taken into account.
In order to simulate the response of the CQED system to a single photon pulse incident on one of the cavity mirrors we assume that all the input photon pulses follow a Gaussian shape $f_\text{in1(2)}(t) = C_N \exp{[-(t-T/5)^2/T^2]}$, where $C_N$ is the normalizing factor, $T$ is the total pulse length and $t$ ranges from 0 to $T$. So we have $\int_0^{T} |f_\text{in1(2)}(t)|^2=1$. Thus, the input single photons have the form $|\psi_\text{in1(2)}(t)\rangle=\int_0^{T} f_\text{in1(2)}(t) a_\text{in1(2)}^\dagger(t) dt |\text{vac}\rangle$ with $[a_\text{in1(2)} (t),a_\text{in1(2)}^\dagger (t') ]=\delta(t-t')$ and $|\text{vac}\rangle$ representing the vacuum state. Similarly, we can also define the output single photon pulse by $|\psi_\text{out1(2)}(t)\rangle=\int_0^{T} f_\text{out1(2)}(t) a_\text{out1(2)}^\dagger (t) dt |\text{vac}\rangle$ with similar commutation relation $[a_\text{out1(2)}(t),a_\text{out1(2)}^\dagger (t') ]=\delta(t-t')$ and $f_\text{out1(2)}(t)$ the output pulse shape. Since the input pulse shape is given we can get the output pulse shape through standard Heisenberg-Langevin equations.
When the atom is in state $|\beta\rangle$ the atom is strongly coupling to the cavity mode. The time evolution of cavity field $a$ with single-photon pulses incident on both sides of the cavity is then [@Walls2008] $$\label{eq16}
\dot{a}=-\frac{i}{\hbar}[a,H']-\kappa a - \sum\limits_{j=1,2}\sqrt{2 \kappa_j} a_{\text{in}j},$$ where the Hamiltonian is $$\label{eq17}
H'=H-i\gamma \sigma_{ee},$$ with $\sigma_{ee}=|e\rangle \langle e|$. The relations between input and output fields are given by Eqs. (\[eq3\]) and (\[eq4\]). The Heisenberg equation for the atomic operators is $$\label{eq18}
\dot{\sigma} = -i [\sigma, H'],$$ with $\sigma=|\beta\rangle \langle e|$. Thus by using Eqs. (\[eq16\]) and (\[eq18\]) we get two operator equations which describe the dynamics of the CQED system when the single-photon pulse is incident on one side of the cavity. They are $$\label{eq19}
\begin{split}
\dot{a} = & -i g \sigma - \kappa a - \sum\limits_{j=1,2}\sqrt{2 \kappa_j} a_{\text{in}j}, \\
\dot{\sigma} = & - i g a (\sigma_{ee}-\sigma_{\beta \beta}) - \gamma \sigma.
\end{split}$$ These equations together with Eqs. (\[eq2\]) and (\[eq3\]) describe the whole dynamics of the CQED system.
Next we will transform these equations from the Heisenberg picture to the Shrödinger picture and finally solve the reflected and transmitted pulse shapes. Since we are only considering one-photon excitation, the time-dependent wave function of the system can be defined as $$\label{eq20}
\begin{split}
|\psi(t)\rangle = & C_\beta(t)|\beta, 1, \text{vac1}, \text{vac2} \rangle + C_e(t)|e, 0, \text{vac1}, \text{vac2} \rangle \\
& + \sum\limits_{j=1,2} \int_0^T [ f_{\text{in}j}(t) a_{\text{in}j}^\dagger(t) + f_{\text{out}j}(t) a_{\text{out}j}^\dagger(t)] dt \\
& \times |\beta, 0, \text{vac1}, \text{vac2}\rangle,
\end{split}$$ where $|x, 1, \text{vac1}, \text{vac2}\rangle$ denotes the atom is in state $|x\rangle$ ($x$ can be $\beta$ or $e$), the cavity contains one photon, and both of the fields outside of cavity are in the vacuum state; $C_x(t)$ is the corresponding time-dependent coefficient. Thus we have a set of equations in the Shrödinger picture (see the Appendix) $$\label{eq21}
\begin{split}
\dot{C}_{\beta}(t) = & -i g C_{e}(t) - \kappa C_\beta(t) - \sum\limits_{j=1,2}\sqrt{2 \kappa_j} f_{\text{in}j}(t), \\
\dot{C}_{e}(t) = & - i g C_\beta(t) - \gamma C_{e}(t), \\
f_\text{out1}(t) = & f_\text{in1}(t) + \sqrt{2\kappa_\text{1}} C_\beta(t),\\
f_\text{out2}(t) = & f_\text{in2}(t) + \sqrt{2\kappa_\text{2}} C_\beta(t),
\end{split}$$ which describe the time evolution of time dependent coefficients and input and output pulse shapes.
When the atom is in state $|\alpha\rangle$ the atom does not couple to the cavity mode and the input single photon pulse encounters an empty cavity. In this case no atomic sate is involved and Eqs. (\[eq21\]) become $$\label{eq22}
\begin{split}
\dot{C}(t) = & - \kappa C(t) - \sum\limits_{j=1,2}\sqrt{2 \kappa_j} f_{\text{in}j}(t), \\
f_\text{out1}(t) = & f_\text{in1}(t) + \sqrt{2\kappa_\text{1}} C(t),\\
f_\text{out2}(t) = & f_\text{in2}(t) + \sqrt{2\kappa_\text{2}} C(t),
\end{split}$$ where $C(t)$ is the coefficient for the state $|1, \text{vac1}, \text{vac2}\rangle$ with the cavity mode having one photon, and the two outside fields are in vacuum.
![\[fig3\] (Color online) Output pulse shapes from two output ports $|a^{(2)}_\text{out1}\rangle$ (a) and $|a^{(2)}_\text{out2}\rangle$ (b) with single a Gaussian pulse incident on $|a^{(1)}_\text{in1}\rangle$ port of interferometer in FIG. \[fig2\](a) under different state combinations of two atoms. $|f_\text{in}(t)|$ is the amplitude variance of the input pulse. $|f^{xy}_\text{out1(2)}(t)|$ means the unnormalized output pulse amplitude from output port $|a^{(2)}_\text{out1(2)}\rangle$ with atoms in state $|xy\rangle$ ($xy=\alpha \alpha, \alpha \beta, \beta \alpha \text{ or } \beta \beta$). $|f^{xy,N}_\text{out1(2)}(t)|$ is the normalized pulse amplitude. The output pulse from $|a^{(2)}_\text{out2}\rangle$ has a $\pi$ phase shift with respect to the input pulse, and this is not shown in (b). In these two figures the input pulse length $T=400\kappa$ and pulse width $w=T/5$ are adopted. The CQED parameters are $g=3\kappa$, $\kappa=\gamma$, and $\kappa_1=\kappa_2=0.45\kappa$. ](Fig3-2018.eps){width="8.5cm"}
By using Eqs. (\[eq21\]) and (\[eq22\]) the output pulse shapes in two output modes $|a^{(2)}_\text{out1}\rangle$ and $|a^{(2)}_\text{out2}\rangle$ with single-photon pulses incident in $|a^{(1)}_\text{in1}\rangle$ of interferometer in FIG. \[fig2\](a) can be exactly evaluated under different state combinations of two coupled atoms. Figure \[fig3\] shows the output photon pulse shape from these two output modes with a Gaussian-shaped single-photon pulse as the input under different state combinations of the two atoms. There are two features for the output pulses:
\(1) Because of the slow response of the cavity to the input single photon pulse, the output pulse with the atom in state $|\alpha\rangle$ (empty atom) has a shape mismatch to the pulse shape with the atom in state $|\beta\rangle$, where the input pulse is directly reflected. The pulse mismatch will make the two paths to one detector distinguishable and cause deterioration of the generated entangled state. Especially for paths 1 and 2 in FIG. \[fig2\](a) the single photon is reflected or transmitted twice, thus the mismatch between the output pulses is biggest. For paths 3 and 4, both of them evenly experience transmission and reflection once thus have no mismatch between them. The slower variation of the input pulse shape $f_\text{in}(t)$, the fewer mismatches between the two output pulses. A plot of the overlap between two normalized photon pulse shapes from paths 1 and 2 with the two atoms being in states $|\alpha,\alpha\rangle$ and $|\beta,\beta\rangle$ versus input pulse length $T$ is given in Fig. \[fig4\]. We can see the overlap is greater than 0.999 when a long enough input pulse is adopted.
![\[fig4\] (Color online) Overlap between two normalized output pulse shapes from paths 1 and 2 with the two atoms being in state $|\alpha,\alpha\rangle$ and $|\beta,\beta\rangle$ versus input pulse length $T$. The input pulse width $w=T/5$ and CQED parameters with $g=3\kappa$, $\kappa=\gamma$ are adopted.](Fig4-2018.eps){width="8.5cm"}
\(2) Due to the unexpected losses ${{\kappa }_\text{loss}}$, $\gamma$, and limited coupling strength $g$, the coefficients for transmission and reflection are smaller than 1 whenever the atom is in the state $|\alpha\rangle$ or $|\beta\rangle$. Thus the generated states associated with clicks on D1 or D2 are not maximally entangled. There are other states mixed into due to the imperfect transmission and reflection.
With a single-photon pulse injected into the system and D1 clicks, the atomic state can be written as $$\label{eq23}
\begin{aligned}
|\Psi^1_{21}\rangle= \frac{1}{2\sqrt{P^1_{21}}}(A1 |\alpha \alpha\rangle+B1 |\beta \beta\rangle +C1 |\alpha \beta\rangle+D1 |\beta \alpha\rangle ),
\end{aligned}$$ with coefficients $A1=\sqrt{\int^T_0 |f^{\alpha \alpha}_\text{out1}(t)|^2dt}$, $B1=\sqrt{\int^T_0 |f^{\beta \beta}_\text{out1}(t)|^2dt}$, $C1=\sqrt{\int^T_0 |f^{\alpha \beta}_\text{out1}(t)|^2dt}$, and $D1=\sqrt{\int^T_0 |f^{\beta \alpha}_\text{out1}(t)|^2dt}$. Here we assume that $\varphi_1=\varphi_2=0$ and the phase difference between the two paths is well controlled, so that $\Delta \varphi=0$. $P^1_{21}=(A1^2+B1^2+C1^2+D1^2)/4$ is the probability of detecting the input photon by D1. The fidelity of this state to the maximum entangled state $|\Phi\rangle=(|\alpha \alpha\rangle+|\beta \beta\rangle)/\sqrt{2}$ is then $$\label{eq24}
F^1_{21}=\sqrt{\frac{(A1+B1)^2}{2(A1^2+B1^2+C1^2+D1^2)}}.$$
Through a similar process we can also get the atomic state after D2 clicks, it is $$\label{eq25}
\begin{aligned}
|\Psi^1_{22}\rangle= \frac{1}{2\sqrt{P^1_{22}}}(A2 |\alpha \alpha\rangle+B2 |\beta \beta\rangle +C2 |\alpha \beta\rangle+D2 |\beta \alpha\rangle )
\end{aligned}$$ with coefficients $A2=\sqrt{\int^T_0 |f^{\alpha \alpha}_\text{out2}(t)|^2dt}$, $B2=\sqrt{\int^T_0 |f^{\beta \beta}_\text{out2}(t)|^2dt}$, $C2=\sqrt{\int^T_0 |f^{\alpha \beta}_\text{out2}(t)|^2dt}$, and $D2=\sqrt{\int^T_0 |f^{\beta \alpha}_\text{out2}(t)|^2dt}$. $P^1_{22}=(A2^2+B2^2+C2^2+D2^2)/4$ is the probability of detecting the input photon by D2. The fidelity of Eq. (\[eq25\]) to the maximum entangled state $|\Psi\rangle=(|\alpha \beta\rangle+|\beta \alpha\rangle)/\sqrt{2}$ is $$\label{eq26}
F^1_{22}=\sqrt{\frac{(C2+D2)^2}{2(A2^2+B2^2+C2^2+D2^2)}}.$$
For a CQED system with achievable parameters such as $g=2 \kappa$ or $g=3 \kappa$, $\kappa=\gamma$, and $\kappa_\text{loss}=0.1 \kappa$, from Eqs. (\[eq24\]) and (\[eq26\]) the single-photon detection by D1 and D2 already gives a fidelity of corresponding generated states about 0.98. They are less than unity except when $A1=B1, C2=D2$, and $C1=D1=0, A2=B2=0$ in the ideal case with ${{\kappa }_\text{loss}}=0$ and $g \gg (\kappa, \gamma)$. However, as long as $C1$ and $D1$ ($A2$ and $B2$) can be controlled smaller than $A1$ and $B1$ ($C2$ and $D2$), which is easy to achieve in a low extra loss cavity, the fidelities can be further improved by simply sending more photons into the system and detecting them individually on the corresponding output mode. If we send $n$ single photons into the system one by one and after the $n$th click of D1 or D2, the wave function of the two atoms collapses into $$\label{eq27}
\begin{aligned}
|\Psi^n_{21}\rangle= & \frac{1}{2 \sqrt{P^n_{21}}}(A1^n |\alpha \alpha\rangle+B1^n |\beta \beta\rangle \\
&+C1^n |\alpha \beta\rangle+D1^n |\beta \alpha\rangle )
\end{aligned}$$ or $$\label{eq28}
\begin{aligned}
|\Psi^n_{22}\rangle= & \frac{1}{2 \sqrt{P^n_{22}}}(A2^n |\alpha \alpha\rangle+B2^n |\beta \beta\rangle \\
&+C2^n |\alpha \beta\rangle+D2^n |\beta \alpha\rangle),
\end{aligned}$$ where $$\label{eq29}
P^{n}_{21}=(A1^{2n}+B1^{2n}+C1^{2n}+D1^{2n})/4$$ and $$\label{eq30}
P^{n}_{22}=(A2^{2n}+B2^{2n}+C2^{2n}+D2^{2n})/4$$ are the probabilities of detecting the $n$th photon after $n-1$ photons have been detected by the same detector D1 or D2. As such, the generated entangled states Eqs. (\[eq27\]) and (\[eq28\]) have fidelities $$\label{eq31}
F^n_{21}=\sqrt{\frac{(A1^n+B1^n)^2}{2(A1^{2n}+B1^{2n}+C1^{2n}+D1^{2n})}}$$ and $$\label{eq32}
F^n_{22}=\sqrt{\frac{(C2^n+D2^n)^2}{2(A2^{2n}+B2^{2n}+C2^{2n}+D2^{2n})}}.$$
![\[fig5\] (Color online) Output pulse shapes from two output modes $a^{(2)}_\text{out1}$ (a) and $a^{(2)}_\text{out2}$ (b) with single Gaussian pulse incident from $a^{(1)}_\text{in1}$ port of interferometer in FIG. \[fig2\](a) under different state combinations of two atoms. All the data points in these two figures are calculated by setting the input pulse length $T=400\kappa$ and pulse width $w=T/5$. The CQED parameters are $g=3\kappa$, $\kappa=\gamma$, and $\kappa_1=\kappa_2=0.45\kappa$.](Fig5-2018.eps){width="8.5cm"}
The fidelities \[Eqs. (\[eq31\]) and (\[eq32\])\] versus detected photon numbers are plotted in Fig. \[fig5\](a). We can see that the detection of two photons does enhance the two fidelities to be greater than 0.99 in both cases of $g=2 \kappa$ and $g=3 \kappa$. However, in the case of $g=3 \kappa$ the fidelity shown in Eq. (\[eq31\]) decreases when more photons are detected. This is because that the coefficients of transmission for an empty cavity (atom in $|\alpha\rangle$) and reflection for a strongly coupling CQED (atom in $|\beta\rangle$) are not the same in a typical system, and the photon that follows path of 1 or 2 in FIG. \[fig2\](a) reflects or transmits both of the two cavity systems, which makes the coefficients $A1\neq B1$. When more photons are injected and detected, the difference between $A1^n$ and $B1^n$ becames bigger. Thus, the fidelity shown in Eq. (\[eq31\]) decreases. However, for the state given by Eq. (\[eq28\]), the photon follows path 3 or 4, where it evenly experiences both transmission and reflection once, so the resulting coefficients $C2$ and $D2$ are the same. The fidelity Eq. (\[eq32\]) will approach infinitely to unity with more photons. Anyway, by only two photons the average fidelity can already be substantially enhanced from 0.986 to 0.997 in the case of $g=3 \kappa$. In a special lower coupling example of $g=2 \kappa$, where $A1\approx B1$, both of the fidelities can be further improved. This also provides a method to improve the fidelity through tuning the coupling strength to a suitable value and making $A1\approx B1$ and $C2\approx D2$. This is feasible in a typical CQED system because the coupling $g$ can be tuned by intentionally moving the relative position of the atom with respect to the cavity mode.
Due to the decay of the atom and unexpected loss of the cavity, the input photon could be decayed out of the system and not be detected. However, once the input photons are detected by the corresponding detectors the combined atomic state collapses to the entangled states. The probability of photon detection is also the probability of successfully generating the entangled states. In the example of $g=3\kappa$ the detection of one input photon on both detectors is about 0.38, which means a total success probability of 0.76. We have already shown that higher fidelity can be achieved by more photons. Figure \[fig5\](b) shows the variation of success probability versus the detected photon number. By using more photons the fidelity of the entangled state can indeed be enhanced at a cost of low success probability. In the example of $g=3\kappa$ the probabilities of two photon detection on both detectors are around 0.28. This means that with the local operations on either one of the atoms the entangled state shown in Eq. (\[eq12\]) or (\[eq13\]) can be prepared with fidelity of 0.997 and success probability of 0.56. With suitable coupling strength and by using more photons the fidelity can be further improved, but the success probability also drops farther. For example, in the example with coupling strength $g=2\kappa$, the fidelity can be further enhanced to over 0.999 by three photons, but the success probability drops to 0.3 (0.15 for each state).
Conclusion and discussion
=========================
In conclusion, we have presented a scheme to entangle multiple remote material qubits through single photons via the nonexcitation process in strongly coupling systems. The basic idea is based on the material state dependent reflection and transmission for the input photons. If two of the strongly coupled systems are arranged as a Mach-Zehnder-interferometer-like configuration, the indistinguishability of the photon paths will finally result in the entanglement of the material qubits in the strongly coupling systems. The entangled state fidelity and success probability are analyzed in detail through strict Heisenberg equations when single photons are injected in the pulsed mode. Our analysis shows that by adopting current achievable system with $g=3\kappa$ the expected entangled state has a fidelity of about 0.986 to the maximum entangled state and success probability about 0.76 with only one photon. If two photons are used, the fidelity could be improved to over 0.99 at the cost of a lower success probability of 0.56. Using a suitable coupling strength the fidelity can be improved further with more photons and lower success probability.
Moreover, the character of no excitation of material qubits guarantees continuity and coherence of material qubits throughout the whole interaction process. Thus our scheme can be directly scaled up to entangle more qubits. Two possible configuration of entangling three or more qubits are also briefly discussed.
In our paper we discussed the fidelity and success probability in the frame of currently accessible optical FP cavity based CQED systems, but our scheme is not only executable on this system. The rapid development of fabrication of micro- or nanostructures and new materials provides more and more new strongly coupled systems [@Dayan08; @Aoki09; @Scheucher2016; @Shomroni14; @Tiecke14; @Goban15; @Kato15; @Oshea13; @Johansson06; @Wallraff04; @Englund07; @Hennessy07; @Fink08; @Hoi11; @Chang07; @Tiarks14; @Gorniaczyk14; @Manzoni14] which can also implement our scheme to entangle the remote material qubits. Especially, in the strongly coupling system between a single atom and whispering-gallery-mode optical micro-resonator fibers are used to couple the photons in and out of the system [@Dayan08; @Aoki09; @Scheucher2016; @Shomroni14]. This provides an easy way to connect the two setups and control the phase difference between different paths as shown in FIG. \[fig2\]. So the implementation of our scheme could be more direct.
This work has been supported by the National Key Research and Development Program of China (Grant No. 2017YFA0304502), the National Natural Science Foundation of China (Grants No. 11634008, No. 11674203, No. 11574187, and No. 61227902), and the Fund for Shanxi ¡°1331 Project¡± Key Subjects Construction.
Transform from Heisenberg picture to Shrödinger picture
=======================================================
The Hamiltonian which governs cavity input-output fields is [@Gardiner1985] $$\label{a1}
H_\text{cav,io}= i \int^{\infty}_{-\infty}\text{d} \omega \sum_{i=1,2} \kappa_i(\omega) [a a^\dagger_i(\omega) - a^\dagger a_i(\omega)],$$ where $a$ is the annihilation operator for the cavity field, $a_i(\omega)$ is the annihilation operator for outside continuous fields and $\kappa_i(\omega)$ is the input-output coupling efficiency. In the Markov approximation where $\kappa_i(\omega)$ is independent of the frequency $\omega$, the Eqs. (\[eq2\]) and (\[eq16\]) can be deduced from solving the Heisenberg equations [@Walls2008]. So the total Hamiltonian should be $$\label{a2}
H_\text{total}= H'+ H_\text{cav,io},$$ here $H'$ is shown in Eq. (\[eq17\]). Because state $|\beta, 0, \text{vac1},\text{vac2} \rangle$ is the ground state for $H_\text{total}$ with the eigen energy 0, thus $e^{-iH_\text{total} t /\hbar} |\beta, 0, \text{vac1},\text{vac2} \rangle = |\beta, 0, \text{vac1},\text{vac2} \rangle$.
Thus for any operator $A(t)$ we have $$\label{a3}
\begin{split}
& \langle \beta, 0, \text{vac1}, \text{vac2}|A(t)|\psi(0)\rangle \\
&= \langle \beta, 0, \text{vac1}, \text{vac2}|e^{iH_\text{total} t /\hbar}A(0) e^{-iH_\text{total} t /\hbar}|\psi(0)\rangle \\
&= \langle \beta, 0, \text{vac1}, \text{vac2}|A|\psi(t)\rangle
\end{split}$$ and $$\label{a4}
\begin{split}
& \langle \beta, 0, \text{vac1}, \text{vac2}|{\mathrm d A(t)}/{\mathrm d t}|\psi(0)\rangle \\
&=\mathrm d \left[ \langle \beta, 0, \text{vac1}, \text{vac2}|e^{iH_\text{total} t /\hbar}A(0) e^{-iH_\text{total} t /\hbar}|\psi(0)\rangle \right]/\mathrm d t\\
&= \langle \beta, 0, \text{vac1}, \text{vac2}|\mathrm d \left[A|\psi(t)\rangle\right]/\mathrm d t,
\end{split}$$ where $A=A(0)$ and $A(t)$ are the time independent and time dependent operators in Shrödinger and Heisenberg pictures. Substituting Eq. (\[eq20\]) into Eqs. (\[eq19\]), (\[eq2\]), and (\[eq3\]) and using relations shown in Eqs. (\[a3\]) and (\[a4\]) we finally have the dynamic functions for the coefficients shown in Eq. (\[eq21\]).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
There is a trend towards increased specialization of data management software for performance reasons. In this paper, we study the automatic specialization and optimization of database application programs – sequences of queries and updates, augmented with control flow constructs as they appear in database scripts, UDFs, transactional workloads and triggers in languages such as PL/SQL. We show how to build an optimizing compiler for database application programs using generative programming and state-of-the-art compiler technology.
We evaluate a hand-optimized low-level implementation of TPC-C, and identify the key optimization techniques that account for its good performance. Our compiler fully automates these optimizations and, applied to this benchmark, outperforms the manually optimized baseline by a factor of two. By selectively disabling some of the optimizations in the compiler, we derive a clinical and precise way of obtaining insight into their individual performance contributions.
author:
- |
Mohammad Dashti, Sachin Basil John, Thierry Coppey,\
Amir Shaikhha, Vojin Jovanovic, and Christoph Koch\
\
EPFL DATA Lab {firstname}.{lastname}@epfl.ch\
bibliography:
- 'refs.bib'
title: Compiling Database Application Programs
---
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this comment, we address a number of erroneous discussions and conclusions presented in a recent preprint by the HALQCD collaboration, arXiv:1703.07210 . In particular, we show that lattice QCD determinations of bound states at quark masses corresponding to a pion mass of $m_\pi=806$ MeV are robust, and that the extracted phases shifts for these systems pass all of the “sanity checks” introduced in arXiv:1703.07210 .'
author:
- 'Silas R. Beane'
- Emmanuel Chang
- Zohreh Davoudi
- William Detmold
- Kostas Orginos
- Assumpta Parreño
- 'Martin J. Savage'
- 'Brian C. Tiburzi'
- 'Phiala E. Shanahan'
- 'Michael L. Wagman'
- Frank Winter
bibliography:
- 'bibi.bib'
title: 'Comment on “Are two nucleons bound in lattice QCD for heavy quark masses? - Sanity check with Lüscher’s finite volume formula -”'
---
-1.1cm -0.5cm
In the last decade, significant progress has been made in the study of multi-hadron systems using lattice QCD, with the first calculations of multi-baryon bound states and their electroweak properties and decays having been performed [@Fukugita:1994ve; @Beane:2006mx; @Ishii:2006ec; @Aoki:2008hh; @Nemura:2008sp; @Yamazaki:2009ua; @Aoki:2009ji; @Beane:2010hg; @Inoue:2010es; @Yamazaki:2011nd; @Beane:2011iw; @Beane:2012vq; @Beane:2013br; @Inoue:2011ai; @Yamazaki:2012hi; @HALQCD:2012aa; @Beane:2013kca; @Beane:2014ora; @Beane:2015yha; @Detmold:2015daa; @Berkowitz:2015eaa; @Yamazaki:2015asa; @Yamada:2015cra; @Chang:2015qxa; @Savage:2016kon; @Shanahan:2017bgi; @Tiburzi:2017iux]. It is imperative that the methods used in these calculations be robust; investigations such as those of the HALQCD collaboration in Ref. [@Iritani:2017rlk] are vital provided they are carried out correctly. However, as we show in detail, many of the conclusions reached in Ref. [@Iritani:2017rlk] (henceforth referred to as [HAL]{}), that cast doubt on the validity of multi-baryon calculations, are incorrect. Since we have recently refined one of the analyses that is criticized in [HAL]{}, we focus our attention on the conclusions drawn regarding this case in particular, see Ref. [@Wagman:2017tmp].
The central point addressed by [HAL]{} is whether there exist bound states in the $\si$ and $\siii$ two-nucleon channels at heavy quark masses. Three independent groups have analysed lattice QCD calculations at quark masses corresponding to a heavy pion mass of $\sim800$ MeV (one set of calculations used quenched QCD) and found that there are bound states in these channels. Each of these groups has concluded this by extracting energies from two-point correlation functions (with the quantum numbers of interest) at two or more lattice volumes and demonstrating, through extrapolations based on the finite-volume formalism of Lüscher [@Luscher:1986pf; @Luscher:1990ux], that these energies correspond to an infinite-volume state that is below the two-particle threshold and is hence a bound state. Each group has used different technical approaches, and all are in reasonable agreement given the uncertainties that are reported. The HALQCD collaboration has also investigated these two-particle channels using a method (also based on the work of Lüscher [@Luscher:1986pf; @Luscher:1990ux]) that involves constructing Bethe-Salpeter wavefunctions, but do not find evidence for bound states in these channels [@Ishii:2006ec; @Aoki:2008hh; @Aoki:2009ji; @HALQCD:2012aa; @Inoue:2011ai].[^1] We note, however, that the HALQCD method introduces unquantified systematic effects as discussed in, e.g., Refs. [@Detmold:2007wk; @Beane:2010em; @Detmold:2015jda] and the nuclear physics overview talks in recent proceedings of the International Symposium on Lattice Field Theory [@Walker-Loud:2014iea; @Yamazaki:2015nka; @Savage:2016egr]). Here, we focus our criticisms of [HAL]{} on several specific points.
### Misinterpretation of energies and source independence
Figure 2 of [HAL]{} contains a compilation of results for the ground states of the $\si$ and $\siii$ two-nucleon channels. Unfortunately the figure includes a second state from Ref. [@Berkowitz:2015eaa] that the authors of Ref. [@Berkowitz:2015eaa] explicitly indicate is not the ground state, and reporting it as such is a significant error on which many of the invalid arguments of HAL are based.[^2] There is a small scatter in the remaining results that is due to statistical fluctuations, discretisation artifacts and exponentially-small residual finite-volume effects, but, taken as a whole, there is no inconsistency in these results. In addition, a further recent study of axial-current matrix elements using a different set of interpolators [@Savage:2016kon; @Shanahan:2017bgi; @Tiburzi:2017iux] (denoted in Fig. \[fig:binding\] by NPLQCD17) also finds a consistent negatively-shifted energy on the $32^3\times48$ ensemble used in this comparison. Figure 2 in [HAL]{} also fails to include the energies extracted in Ref. [@Beane:2012vq] on the largest volume, which dominate the extraction of the binding energy. Without the results from this large volume, the confidence in the binding energy in Ref. [@Beane:2012vq] would be significantly diminished. It is therefore vital that this information be included in any discussion of these results. Figure \[fig:binding\] below shows a (corrected) summary of the energy levels extracted for the ground states of the $\si$ and $\siii$ two-nucleon systems in different volumes that are published in the literature at this particular quark mass. No significant interpolator dependence is observed, as is indicated by simple fits to the reported results for each volume, with all these fits having acceptable values of $\chi^2$ per degree of freedom.
![Binding energies of the $\siii$ and $\si$ ground states at $m_\pi= 806$ MeV found in the literature: NPLQCD13 [@Beane:2012vq], Berkowitz16 [@Berkowitz:2015eaa], and NPLQCD17 [@Savage:2016kon; @Shanahan:2017bgi; @Tiburzi:2017iux] ($d=0$ and $d=2$ refer to the magnitude of the centre-of-mass momentum used in the calculations in units of $2\pi/L$). The three regions in each panel correspond to three different volumes: $L=24$, 32, and 48 from left to right. Uncertainties listed in the original references are combined in quadrature. The horizontal lines and shaded bands represent the central value and one standard deviation bands from uncorrelated fits, respectively.[]{data-label="fig:binding"}](figures/bindings.pdf){width="0.95\columnwidth"}
Figure 13 of [HAL]{} is also erroneously described as indicating that scattering state results are not source independent. The results show three energy levels where different interpolating operators are consistent within one standard deviation, and one energy level that differs at two standard deviations. This indicates broad agreement within the reported uncertainties and, contrary to statements in HAL, does not provide a sound statistical basis for a claim of inconsistency.
In summary, comparison of results from the different interpolators in Refs. [@Beane:2012vq; @Beane:2013br; @Berkowitz:2015eaa; @Tiburzi:2017iux] shows that both bound and scattering-state energy levels are source-independent within reported uncertainties. This is contrary to the claims in [HAL]{}.
### Volume scaling of energies
The authors of HAL claim that the single-exponential behaviour found in our work, Refs. [@Beane:2012vq; @Beane:2013br], and in that of Ref. [@Berkowitz:2015eaa], is a “mirage” arising from the cancellation of two or more scattering eigenstates[^3] contributing to the correlation functions with opposite signs (see Ref. [@Iritani:2016jie] for elaborations on possible “mirage” plateaus). This interpretation of the negatively-shifted states in these works is exceedingly unlikely, however, as such cancellation would need to occur in an almost identical way for multiple different volumes. For each of the different analyses of the 806 MeV ensembles in Fig. \[fig:binding\] (NPLQCD2013 $d=0$, NPLQCD2013 $d=2$ and Berkowitz2016 $d=0$), identical sources and sinks were used in each of three volumes (two volumes in the case of Berkowitz2016). Scattering-state eigenenergies necessarily change significantly with volume, having power-law dependence as dictated by the Lüscher quantisation condition. While it is possible that, in a given volume, a correlator for a particular source-sink interpolator combination could exhibit a cancellation between contributions of two scattering states that produces an energy level below threshold, it is very unlikely that the cancellation would persist in different volumes as the scattering-state eigenenergies change significantly with volume. As shown in Fig. \[fig:eff\], for example, the volume-independent interpolators used in Ref. [@Beane:2012vq; @Beane:2013br] produce energy levels in the three different volumes that are statistically indistinguishable, and even the approach to single-exponential behaviour does not depend on volume. The figure shows the effective masses of the smeared-point correlation functions, but the same features are seen in all other source-sink interpolator combinations that are studied. This rules out the possibility that the negatively-shifted signals are caused by cancellations between scattering states. The largest volume used in our works [@Beane:2012vq; @Beane:2013br] makes this an extremely robust statement as the spatial volumes from which we draw these conclusions vary by a factor of eight.
![The effective mass plots associated with the $d=0$ smeared-point correlators in the $L=24$, 32, and 48 ensembles of Ref. [@Beane:2012vq; @Beane:2013br]. The left(right) panel shows the $\siii$($\si$) channel. Quantities are expressed in lattice units. The horizontal grey line marks the infinite-volume energy of two non-interacting nucleons.[]{data-label="fig:eff"}](figures/effSP.pdf){width="0.8\columnwidth"}
### Consistency of Effective Range Expansion (“[HAL]{} Sanity Check (i)”)
If the effective range expansion (ERE) is a valid parametrization of the scattering amplitude at low energies, the analyticity of the amplitude as a function of the centre-of-mass energy implies that the ERE obtained from states with positively-shifted energies (${k^*}^2>0$, where $k^*$ is the centre-of-mass interaction momentum) must be consistent with that obtained from states with negatively-shifted energies (${k^*}^2<0$). Although [HAL]{} finds that the NPLQCD results pass this test, we demonstrate how robust the results in Refs. [@Beane:2012vq; @Beane:2013br] are in this regard through the plots presented in Fig. \[fig:ERE-n1-n2\]. This figure shows fits to the ERE using both ground states ($n=1$) and first excited states ($n=2$) (color-shaded bands). These are overlaid on ERE fits using only the ground states (hashed bands). The two sets of bands are fully consistent with each other, proving that this check is unambiguously passed. The same feature is seen for three-parameter ERE fits, with significantly larger uncertainty bands (see also Ref. [@Wagman:2017tmp]).[^4] The difference in the size of uncertainties in the phase shift between the fits with and without the $n=2$ data shows that conclusions about the behaviour and/or validity of the ERE for datasets only near the bound-state pole are likely subject to significant uncertainties. We note that scattering parameters extracted in the region near ${k^*}^2=0$ from a linear ERE will in general differ from those determined in the vicinity of a bound-state pole due to higher order terms in the ERE. Indeed, it is known that in nature, the ERE of the $\siii$ phase shift around ${k^*}^2=0$ and around the deuteron pole are different (albeit slightly) [@deSwart:1995ui].
![$k^*\cot \delta$ vs. the square of the centre-of-mass momentum of two baryons, ${k^*}^2$, along with the bands representing fits to two-parameter EREs obtained from i) only the ground states ($n=1$) and ii) from both the ground states ($n=1$) and the first excited states ($n=2$). The plots show the consistency of the ERE between negative and positive ${k^*}^2$ regions in both the $\si$ and $\siii$ channels. These results are from our recent re-analysis of these ensembles [@Wagman:2017tmp], and are consistent with the initial analysis [@Beane:2012vq; @Beane:2013br], with the mean values in agreement within one standard deviation as defined by the combined (statistical and systematic) uncertainties of each result. Quantities are expressed in lattice units (l.u.).[]{data-label="fig:ERE-n1-n2"}](figures/sc1.pdf){width="0.8\columnwidth"}
### Residue of the S-matrix at the bound-state pole (“[HAL]{} Sanity check (iii)”)
The sign of the residue of the S-matrix at the bound-state pole is fixed. This requirement leads to the following condition on $k^*\cot \delta$ : $$\begin{aligned}
\left. \frac{d}{d{k^*}^2}(k^*\cot \delta+\sqrt{-{k^*}^2}) \right |_{{k^*}^2=-{\kappa^{(\infty)}}^2} < 0,
\label{eq:slope}\end{aligned}$$ where $\kappa^{(\infty)}$ is the infinite-volume binding momentum. As is seen from Fig. \[fig:ERE-tangent\], which displays the results of The uncertainty in the tangent line to the $-\sqrt{-{k^*}^2}$ function at ${k^*}^2=-{\kappa^{(\infty)}}^2$ arises from the uncertainty in the values of $\kappa^{(\infty)}$ (see also Ref. [@Wagman:2017tmp]). A similar conclusion can be drawn from three-parameter ERE fits.
![ The two-parameter ERE is compared with the tangents to the $-\sqrt{-{k^*}^2}$ curve at values of ${k^*}^2=-{\kappa^{(\infty)}}^2$. The plots show that all the identified energy eigenstates in this work are consistent with the criterion in Eq. (\[eq:slope\]) within uncertainties. These results are from our recent re-analysis of these ensembles [@Wagman:2017tmp], and are consistent with the initial analysis [@Beane:2012vq; @Beane:2013br], with the mean values in agreement within one standard deviation as defined by the combined (statistical and systematic) uncertainties of each result. Quantities are expressed in lattice units (l.u.). []{data-label="fig:ERE-tangent"}](figures/sc3.pdf){width="0.8\columnwidth"}
![ []{data-label="fig:tangent13"}](figures/2013only.pdf){width="0.8\columnwidth"}
![ []{data-label="fig:tangent1317"}](figures/2013vs2017.pdf){width="0.8\columnwidth"}
### Discussion {#discussion .unnumbered}
Given the discussion above, the NPLQCD results presented in the “NPL2013” row of Table IV of the published version of [HAL]{} [@Iritani:2017rlk], reproduced below,
------------------- -------------- ----- -------- ------- -------------- ----- ------ -------
Data Source Source
independence (i) (ii) (iii) independence (i) (ii) (iii)
NPL2013 \[28,29\] No \* \* No No \* \* ?
------------------- -------------- ----- -------- ------- -------------- ----- ------ -------
\
should be replaced by
------ -------------- -------- -------- -------- -------------- -------- -------- --------
Data Source Source
independence (i) (ii) (iii) independence (i) (ii) (iii)
Yes Passed Passed Passed Yes Passed Passed Passed
Yes Passed Passed Passed Yes Passed Passed Passed
------ -------------- -------- -------- -------- -------------- -------- -------- --------
\
where we have taken the liberty of changing the notation (in their published version) used to indicate passing a “sanity check” in [HAL]{} from a “ \* ” entry to “Passed”. We are currently revisiting the other NPLQCD analyses discussed in [HAL]{}. Ref. [@Yamazaki:2017euu] refutes the [HAL]{} criticisms of source-dependence leveled at the works of the PACS-CS collaboration [@Yamazaki:2009ua]. Ref. [@Savage:2016egr] provides a summary of the evidence for the validity of ground-state identifications in two-nucleon systems. With the robust conclusion of the existence of bound states reached by independent groups, and argued in this Comment, the systematic uncertainties of the potential method used by the HALQCD collaboration requires further investigation to better understand the origin of its failure to identify two-nucleon bound states.
SRB was partially supported by NSF continuing grant number PHY1206498 and by the U.S. Department of Energy through grant number DE-SC001347. EC was supported in part by the USQCD SciDAC project, the U.S. Department of Energy through grant number DE-SC00-10337, and by U.S. Department of Energy grant number DE-FG02-00ER41132. ZD, WD and PES were partly supported by U.S. Department of Energy Early Career Research Award DE-SC0010495 and grant number DE-SC0011090. KO was partially supported by the U.S. Department of Energy through grant number DE- FG02-04ER41302 and through contract number DE-AC05-06OR23177 under which JSA operates the Thomas Jefferson National Accelerator Facility. A.P. is partially supported by the Spanish Ministerio de Economia y Competitividad (MINECO) under the project MDM-2014-0369 of ICCUB (Unidad de Excelencia ’María de Maeztu’), and, with additional European FEDER funds, under the contract FIS2014-54762-P, by the Generalitat de Catalunya contract 2014SGR-401, and by the Spanish Excellence Network on Hadronic Physics FIS2014-57026-REDT. MJS was supported by DOE grant number DE-FG02-00ER41132, and in part by the USQCD SciDAC project, the U.S. Department of Energy through grant number DE-SC00-10337. BCT was supported in part by the U.S. National Science Foundation, under grant number PHY15-15738. MLW was supported in part by DOE grant number DE-FG02-00ER41132. FW was partially supported through the USQCD Scientific Discovery through Advanced Computing (SciDAC) project funded by U.S. Department of Energy, Office of Science, Offices of Advanced Scientific Computing Research, Nuclear Physics and High Energy Physics and by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under contract DE-AC05-06OR23177.
[^1]: In the $\Lambda\Lambda$ channel, the HALQCD approach does indicate a bound state, but the binding energy is found to be significantly different from that determined by extrapolating finite-volume energy levels [@Beane:2012vq].
[^2]: Whether the quoted value for the second energy in Ref. [@Berkowitz:2015eaa] is a true estimate of an excited-state energy is a question for future discussion. However for the ground states, all results unambiguously agree.
[^3]: The scattering states are loosely used here to denote states in a finite volume that correspond to the continuum states of infinite volume.
[^4]: Our analysis of two-nucleon correlation functions generated from these ensembles of gauge-field configurations has been recently refined in a comprehensive re-analysis [@Wagman:2017tmp], including results at additional kinematic points. This new analysis has been used in obtaining the results shown in Figs. \[fig:ERE-n1-n2\] and \[fig:ERE-tangent\]. All of the energies extracted from the three lattice volumes, and the binding energies and ERE parameters subsequently obtained, are in agreement with our previous results; i.e., the differences in the mean values of the results from the previous and the new analyses are within one standard deviation as defined by the (statistical and systematic) uncertainties of the results combined in quadrature [@Beane:2012vq; @Beane:2013br].
| {
"pile_set_name": "ArXiv"
} |
---
address: |
$^{1}$ Center for earthquake research and information (CERI), The University of Memphis\
$^{2}$ Now at Asurion, Nashville, Tennessee\
$^{3}$ Department of Geological Sciences, Jahangirnagar University Savar, Dhaka, Bangladesh\
$^{4}$ Geological Survey of Bangladesh, Segunbagicha, Dhaka, Bangladesh
bibliography:
- 'basement\_surface\_interactions.bib'
---
Introduction {#sec:intro}
============
The Indian plate was a part of the ancient supercontinent of Gondwana. It started breaking up from Gondwana about 176 million years ago and later become a major plate [@chatterjee2013longest]. The major tectonic elements of the Indian plate started developing with the northward drift of the Indian plate since Cretaceous and its collision with the Eurasian plate by early to middle Eocene [@sikder20032]. The Bengal Basin (Figure. \[fig:regional\_tectonics\]) is one of the largest sedimentary basins, located in the north-eastern part of the Indian plate. The Indian shield makes its eastern boundary, whereas the compressional Indo-Burman folded belt makes the western boundary. The Shillong plateau marks one of the significant structural features in the northern portion of the basin.
The northwestern part of the Bengal Basin (Figure \[fig:local\_tectonics\]) has a unique geologic setup. The region has the shallowest Paleoproterozoic basement $(\sim128)$m within the Bengal Basin [@khan1991geology]. Scientists have been researching to understand the origin of the faulted basement. However, no conclusive explanations exist yet. For example, @ameen2007paleoproterozoic and @hossain2007palaeoproterozoic seperately studied the basement rock from Maddhapara, northwestern Bengal Basin to explain the tectonic evolution of the region. Uisng the SHRIMP U–Pb dating technique, both of the studies found a roughly consistent age ($1722\pm6$ Ma, and $1730\pm11$ Ma, respectively). However, their argument on the tectonic evolution of the basement differ from each other significantly. @ameen2007paleoproterozoic argued that there is no comparable age found in Indian tectonic zone on the west and Shillong plateau on the east. They proposed that the basement is a discrete trapped micro-continental block. Contrarily, @hossain2007palaeoproterozoic concluded that similar ages are found in the rocks from the Indian tectonic zone and Shillong Plateau. They concluded that the basement in the northwest Bengal basin is the continuation of the greater Indian tectonic zone. We believe that both of the hypothesizes are based on missing decisive data or observations. Therefore more data and studies are required to make a conclusive decision about the tectonic evolution of the northwestern part of the basin.
![Tectonic map of Bengal Basin and its surrounding area, modified from @reimann1993geology [@alam1972tectonic; @johnson1991sedimentation]. The blue square box is the study area, details shown in Figure \[fig:local\_tectonics\]. The black line with triangles are the thrust belts. The Shillong plateau is one of the significant structural features and located just right by our study area. The Dauki fault is a major fault along the southern boundary of the Plateau. The thick red line is the Hinge Zone which is an elongated zone separates basin in the east from the shelf zone of the west.[]{data-label="fig:regional_tectonics"}](figures/regional_tectonics.pdf)
![Major tectonic elements of northwestern Bengal Basin. The region has three distinct geological components (Dinajpur shelf, Rangpur saddle, and Bogra slope) separated by dashed lines. The black square box is the location where the shallowest Paleoproterozoic basement $(\sim128)$m is reported [@khan1991geology]. A geodynamic model is created to explore the deep basement using the E-W crustal cross-sectional profile used by @alam2003overview[]{data-label="fig:local_tectonics"}](figures/local_tectonics.pdf)
Another disagreement of this region comes on the origin of the elevated Pleistocene terrace (Figure \[fig:local\_tectonics\]). The terrace is locally known as the Barind tract. There are also two groups of opinion exists on the origin of the tract. One group believes that the tract was created by the combined effect of the basement faults and regional compressio. Using aerial photographic interpretations [@morgan1959] showed that Quaternary tectonic activities are responsible for the formation of the tract. Supporting the idea of [@morgan1959], @hussain2001geological also concluded that the tract is the product of vertical movements. Recently @Rashid2015a studied the region using borehole data and developed stratigraphy of the region. The authors found an abrupt and unusual variation in thicknesses of sedimentary covers that are correlated with the basement structures. The other hypothesis comes from @monsur1995introduction. The author argued that the tract has no connection with the regional compression; instead, it is likely to be an erosional geomorphic feature.
In this paper, we explain the interaction between basement faults and surface landforms using different types of data and approaches. We use satellite images to learn about the geomorphic process of the tract and Bouger gravity anomaly data to study the shallow and deep basement structures. Finally, we develop a geodynamic model to explain the dynamic relationship between the basement and surface of the tract.
Geology and tectonics of the study area
=======================================
The northwest part of the basin, the study area, locally known as stable-platform and have three geological components: Dinajpur shelf, Rangpur saddle, and Bogra shelf. The basement in the Dinajpur Shelf gently plunges northward with Himalayan Foredeep, which is approximately 1-3 degrees [@hossain2019synthesis] and is covered by recent sedimentary deposits [@reimann1993geology]. The Rangpur saddle, the southern block of the Dinajpur shelf, connects the Indian Shield and the Shillong Plateau [@hossain2018petrology]. The area has the shallowest basement in the Bengal Basin. The southern slope of the Rangpur Saddle, the Bogra shelf, has numerous graben, half-graben, and horsts. These faults were formed during the rifting process of the Indian plate from Gondwana in the Early Cretaceous [@alam2003overview; @reimann1993geology].
Two other important geomorphological elements in the area are elevated Pleistocene Barind tract and the Brahmaputra river (Figure. \[fig:local\_tectonics\]). The tract is the triangular wedge of landmass formed during the Pleistocene. The surface is composed of loose sediments [@Rashid2015a; @Rashid2006a]. The Brahmaputra river runs parallel to the eastern side of the tract and is believed to be linked with the lithospheric flexure of the underlying basement [@rajasekhar2008crustal].
Data and pre-processing
=======================
We use Landsat thematic mapper satellite images of four different years (1972, 1989, 2003, and 2010) to investigate the temporal surface processes in the study area (Figure \[fig:satellite\_images\]). Our visual interpretation relied on pixel values of the images and their relationship to local geologic features. For example, water bodies are generally dark in the images, and their corresponding pixel values are around zero. On the other hand, low moisture content features appear as gray to white, and their corresponding pixel values are roughly around 255. We used the Universal Transverse Mercator (UTM) projection system in all the satellite images and gravity maps.
The Bouguer anomaly data published by @rahman1990bouguer are used to study the basement of the area. The digital contour map of the gravity data was collected from the website of the Department of Interior, USA (<https://catalog.data.gov/dataset/bouguer-gravity-anomaly-map-of-bangladesh-grav8bg>). The contour map was converted to point data. Later, for convenience, we converted point data to an equally spaced grids using the kriging interpolation method. All the analysis was performed on the gridded data.
Results and discussion
======================
In this section, we discuss the surface and subsurface structures and threir relationship. We first use time-series satellite images to explore the time-dependent geomorphic processes. Then we use gravity anomaly data for the shallow and deep basement structures analysis. Finally, we develop a geodynamic model to learn about the dynamic relationship between faulted basement with surface topography.
Geomorphic process
------------------
![Time-series Landsat satellite images of the study area. Floodplains are named after @brammer1996geography. Red arrows show the location of Barind tract. This figure and associated running/plotting scripts available under @Ahamed2017.[]{data-label="fig:satellite_images"}](figures/satellite_images.pdf)
Figure. \[fig:satellite\_images\] shows the Landsat satellite images of the region in four different years. The elevated tract is visible in all of the images. The spatial color differences are mostly limited to the tract and its surrounding low lying floodplains. The southern and eastern boundaries have prominent linear sharp color contrasts. A persistent white tone is present throughout the tract, while the surrounding flood plains have a darker tone. @alam1995neotectonic relates this color variation with the amount of moisture content present in the sediments. The region with higher moisture content generally has a darker tone (e.g., floodplains), whereas the low moisture content regions have a white tone (e.g., tract).
The color contrast between the elevated tract and surrounding low-lying flood plains also provide information about the local geomorphic and tectonic processes. @rashid2018structure mapped the contrast areas as a series of lineaments. Surprisingly, the orientation of the lineaments is consistent with the direction of regional N-S and SE-SW stresses. @khandoker1987origin concluded that the tract is a horst block along with crustal weakness with compensatory subsidence of the bordering regions. We find that these sharp linear boundaries also coincide with the regional and residual gravity anomalies (Figure. \[fig:regional\_gravity\] and \[fig:svd\_gravity\]).
Deep basement structures
------------------------
We use the least-square polynomial fitting surface technique to separate deep crustal structures from shallow ones. The technique has been used for enhancing large scale long-wavelength gravity anomaly, thus regional geologic features [@beltrao1991robust; @mickus2003gravity; @telford1990applied]. In this technique, a polynomial surface with a certain degree is fitted to the Bouguer anomaly ($g(x_i, y_i)$) data. $x_i$ and $y_i$ are the locations of the anomalies. A polynomial equation with $n^th$ degree is given as: $$f(x_i, y_i) = a_0 + a_{2}x_{i} + a_{3}y_{i} + a_{4}x_{i}^2 + ....... + a_{m}y_{i}^n$$ Where n is the degree of the polynomial, m is the total number of the terms of the polynomial, $a_0, a_1..... a_m$ are the coefficients. The error between $f(x_i, y_i)$ and $g(x_i, y_i)$ depends on the several factors: quality of the original data, order used in the polynomial, and the magnitude of the area fitted [@telford1990applied]. Coefficients of the equation can be found by minimizing the least square error with trial and error iterations.
Figure \[fig:regional\_gravity\] shows the regional gravity anomalies at four different polynomial degrees (second to fifth). All the orders show a strong, negative northeast trending regional gravity anomalies with a non-uniform gradient in the northwestern region. The anomalies have high intensity and long-wavelength. Geologically the region is located on the northern slope of the *Rangpur saddle*. The saddle connects the Shillong Massif and the Mikir hills to the east. @rahman1990bouguer shows that these high magnitude anomalies are due to the combined effects of thick low-density sedimentary rocks and a north-dipping, denser substrate in the Himalayan collision zone.
![Regional Gravity anomaly map of the polynomial surface of a) second b) third c) fourth and d) fifth degree. This figure and associated running/plotting scripts available under @Ahamed2017.[]{data-label="fig:regional_gravity"}](figures/regional_gravity.pdf)
The polynomial fitted gravity data suggest that the region has a deep geologic structure that may extend up to the surface of the area, which is similar to observations from the previous studies [@Rashid2015a; @rahman1990bouguer; @reimann1993geology]. With the increasing degree of polynomial order, gravity highs disappear from the *Rangpur saddle* except some scattered and moderate magnitude regional gravity anomalies. For interpretation convenient, we denote the gravity highs as a, b, c (Figure. \[fig:regional\_gravity\]). The highs are also distinguishable on the residual gravity anomaly map (Figure. \[fig:svd\_gravity\]). Residual gravity anomalies generally represent shallow geologic structures [@telford1990applied]. Indeed, the shallowest basement of the entire Bengal basin has been found in the gravity high $b$, where the reported depth is $\sim128$ m [@ameen2007paleoproterozoic; @hossain2007palaeoproterozoic]. From the satellite images (Figure \[fig:satellite\_images\]) and regional polynomial surface fitted map, it is obvious that the Barind tract is located on the top of the gravity high a and b. The spatial correlation between the tracts and the gravity highs indicates that the barind tract is connected to the deeper basement structures.
Shallow basement structures
---------------------------
Due to the presence of the shallow basement, we analyzed the residual gravity anomalies that represent the shallow geologic features. The residual anomalies are calculated using Second Vertical Derivative(SVD). The SVD is a measure of curvature, and large curvatures enhance the high-frequency features (near-surface effects) at the expense of deeper or regional anomalies. SVD can be calculated from the second horizontal derivatives [@telford1990applied] as: $$SVD = \frac{\partial^2g}{\partial z^2} = -\left(\frac{\partial^2g}{\partial x^2} + \frac{\partial^2g}{\partial y^2}\right)$$ Where, $g$ is the Bouguer anomaly at a certain location ($x_i, y_i$). There are many numerical and Fourier transform methods available to compute $\frac{\partial^2g}{\partial z^2}$ [@telford1990applied]. In this paper, We used `Oasis montaj` software to calculate the SVD. Figure \[fig:svd\_gravity\], shows the SVD of Bouguer anomaly, where several gravity lows surround gravity highs. For interpretation purposes, we group the highs into three clusters of SVDs (SVD-1, SVD-2, SVD-3) (Figure \[fig:svd\_gravity\]). SVD-1 is located in the northernmost $\textit{Rangpur saddle}$. The location and extent of the SVD-1 are correlated with the regional gravity anomaly-a (Figure \[fig:regional\_gravity\]). Again the spatial correlation between regional and residual correlation that the surface landforms are connected to deep graben like depressed structures. Our observation is consistent with the @reimann1993geology$'$ hypothesis that the structures may be the N-S aligned grabens with Gondwana fill.
![Second Vertical Derivative (SVD) of Bouguer anomaly showing shallow crustal features. This figure and associated running/plotting scripts available under @Ahamed2017.[]{data-label="fig:svd_gravity"}](figures/svd_gravity.pdf)
Another group of scattered gravity highs (SVD-2) are present at the southern slope of the saddle, where the shallowest basement in Bengal Basin has been reported [@khan1991geology]. The third group of gravity highs (SVD-3) has a sharp boundary with gravity lows on the eastern side of the tract. The Brahmaputra river flows through the boundaries of these highs and lows. The river has been moved back and forth many times. Based on the sedimentological studies @akter2015evolution [@goodbred2003controls] linked the frequent changes of the river course with the local tectonic activities. @fergusson1863recent [@brammer1996geography; @allison1998geologic] reported that the recent river diversion happened in 1782 due to an earthquake that occurred on Dauki fault. The authors mentioned that the earthquake created an upward vertical displacement that might have been responsible for the diversion of the river. Our residual gravity anomaly map shows that river flows along with the gravity lows, which are separated by gravity highs on the eastern side of the tract. This suggests that the river flows through the fault zone or depressed graben structures.
Basement-surface relationship
-----------------------------
The satellite images and gravity data analysis shows that surface features like the Barind tract, Brahmaputra fault, and subsurface structures horsts and graben spatially correlated. However, it is still not obvious if subsurface geologic structures (horsts and grabens) combined with tectonic activities, can produce the surface features. To test the hypothesis, we created a geodynamic model. The model is initially 100 km long and 10 km thick and has a faulted granitic basement overlain by sediments (Figure \[fig:faultModelSetup\]). The model was created using the E-W crustal cross-sectional profile used by @alam2003overview. The profile is shown on Figure \[fig:local\_tectonics\]. We solve the energy balance, mass conservation, and momentum balance equations to simulate the model, which is is a Mohr-Coulomb elastoplastic layer. The energy balance equation [@Ahamed2017] is: $$\label{energy_balance_equation}
(\rho c_p+ 3P\alpha)\frac{dT}{dt}
=
\boldsymbol{\sigma} :\dot{\boldsymbol{\epsilon}}_p
+
3T\alpha\frac{dP}{dt}
-
3PT\frac{\alpha}{\rho}\frac{d \rho}{dt},$$ Where $\rho$ is the density, $c_p$ is the specific heat at constant pressure, $P$ is the pressure, $\alpha$ is the volumetric thermal expansion coefficient, $T$ is the temperature, $\boldsymbol{\sigma}$ is the Cauchy stress, $\dot{\boldsymbol{\epsilon}}_p$ is the plastic strain rate tensor and $t$ is the time. The mass conservation equation [@Ahamed2017] is given as: $$\label{mass_conservation_equation}
\frac{d \rho}{dt} =-\rho \left( \alpha \frac{dT}{dt}+\frac{1}{K}\frac{dp}{dt} \right).$$ Where $K$ is the bulk modulus. The momentum balance equation is given as: $$\label{momentum_balance_equation}
\rho\boldsymbol{\dot{u}} = \nabla\cdot \boldsymbol{\sigma} + \rho g$$ $\boldsymbol{u}$ is the velocity vector and $g$ is the acceleration due to gravity. Since the profile is at $96^{\circ}$ angle with average Indian plate velocity ($v=3.6 cm/yr$) [@mahesh2012rigid; @socquet2006india], we use the profile component $(|v\cos(96^{\circ})|$) of the velocity ($v$) to push the left boundary. The right boundary is kept as a free slip. The bottom boundary is supported by the Winkler foundation [@watts2001isostasy pp.95], and the surface is free surface. To induce the strain localization, we decrease cohesion to 4 MPa linearly as plastic strain increases to 1. We impose topographic smoothing of the diffusion type with a transport coefficient of $10^{-7} m^2/s$ [@turcotte2014geodynamics pp. 225]. Parameters used in this model are listed in Table 1:
\[tab:parametersTable\]
Parameter Symbol Sedimentary Layer Basement
---------------------------------- ---------- ------------------- ---------------
Bulk Modulus $K$ 7.24 GPa 17.89 GPa
Shear Modulus $G$ 1.4 GPa 12.5 GPa
Initial Cohesion $C$ 25 MPa 40 MPa
Friction Angle $\phi$ $30^{\circ}$ $30^{\circ}$
Dilation Angle $\Psi$ $0^{\circ}$ $0^{\circ}$
Density $\rho$ 2300 $Kg/m^3$ 2750 $Kg/m^3$
Volumetric expansion coefficient $\alpha$ 3.5 $K^{-1}$ 3.5 $K^{-1}$
\[1ex\]
: Parameters for the geodynamic model
Bulk and Shear modulus of sedimentary are calculated based on density and lower range of P wave$(V_p)$ and $(V_s)$ of porous and saturated sandstone [@bourbie1987acoustics].\
Bulk and Shear modulus have been calculated based on Young’s modulus$(37583.70)$MPa [@YounusMaddhapara2006] and Poisson’s ratio$(0.3)$.\
@bourbie1987acoustics.
![Model setup for geodynamic simulation. The model is 100 km long 10 km thick. The model was created using the E-W crustal cross-sectional profile used by @alam2003overview. Surface landforms Barind tract and the Brahmaputra river are located on the horsts and grabens. The left boundary is pushed at 0.38 cm/year while the right boundary is kept as a free slip, the bottom boundary is supported by the Winkler foundation [@watts2001isostasy pp.95], and the surface is a free surface. This figure and associated running/plotting scripts available under @Ahamed2017.[]{data-label="fig:faultModelSetup"}](figures/model_geometry.pdf)
Figure. \[fig:plastic\_strain\] shows the plains strain distribution at the different shortening of the region. Plastic strain or deformation is the permanent damage to the material. From the beginning, the plastic strain accumulation is concentrated along with the basement and sedimentary deposits interface. Conjugate thrust faults with large and thick plastic strain concentrations start to form only inside the elevated blocks(horsts) of the basement. Comparing to this wide plastic strain concentrated fault, a thin and low amount of plastic strain faults are present subsided regions (grabens) of the basement (Figure. \[fig:plastic\_strain\]a). At 1.40Km shortening, conjugate thrust faults start to extend deeper. These faults take advantage of the existing horst of the basement.
![Plastic strain distribution of four different shortenings. Platic strain scale a) 0-0.05 and b) 0-0.5. This figure and associated running/plotting scripts available under @Ahamed2017.[]{data-label="fig:plastic_strain"}](figures/plastic_strain.pdf)
The model shows that the conjugate thrust faults are responsible for the formation of the surface landforms. Faults first form inside the basement, and with time they reach the surface. The regional compression is also responsible for activating the faults and pushing the horst blocks upward (Figure. \[fig:plastic\_strain\]b). Since the deformation is accommodated mostly by the conjugate faults in the horst area, the grabens are least affected. That is why we do not see any noticeable upliftment beneath the grabens (Figure. \[fig:plastic\_strain\]b). Uplifting of the horst and subsidence of grabens are consistent with our gravity and geomorphological observations. Above the horsts, most of the gravity highs and the Barind tract are located. Whereas, the Brahmaputra river flows through the region where the gravity lows and grabens are identified.
Therefore, it is reasonable to conclude that the regional compression and the complex basement faults have a more considerable influence on the formation of surface landforms such as Barind Tract and Brahmaputra river.
Conclusion
==========
We analyze time-series satellite images, Bouguer gravity anomaly data, and construct a long-term tectonic model. Satellite images reveal significant spatial changes in the uplifted Barind tract and its surrounding low-lying subsidence floodplains. The gravity anomalies show that the basement structure may have a relationship with the surface geomorphology. We find that the uplifted Barind tract is located on top of the gravity highs, whereas low-lying flood plains and faults are on the lows. We construct a tectonic model to explore the relation between the surface and basement structures further. The model produces conjugate thrust faults beneath the gravity highs. The faults reach the surface and push the gravity highs block upward with time. However, no prominent upliftment is seen beneath the grabens. We conclude that the elevated surface tract and its surrounding low-lying floodplains are produced by the regional compression, where the existing basement has a significant role.
References
==========
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'L. Pandolfi[^1]'
title: 'The quadratic regulator problem and the Riccati equation for a process governed by a linear Volterra integrodifferential equations[^2]'
---
[**:**]{} In this paper we study the quadratic regulator problem for a process governed by a Volterra integral equation in ${{\rm I\hskip-2.1pt R}}^n$. Our main goal is the proof that it is possible to associate a Riccati differential equation to this quadratic control problem, which leads to the feedback form of the optimal control. This is in contrast with previous papers on the subject, which confine themselves to study the Fredholm integral equation which is solved by the optimal control.
[**:**]{}Quadratic regulator problem, Volterra integrodifferential equations, Riccati equation
[**:**]{} 93B22, 45D05, 49N05, 49N35
Introduction
============
The quadratic regulator problem for control processes regulated by linear differential equations both in finite and infinite dimensional spaces has been at the center of control theory at least during the last eighty years, after the proof that the synthesis of dissipative systems amounts to the study of a (singular) quadratic control problem (see [@Brune]). In this period, the theory reached a high level of maturity and the monographs [@Bittanti; @lasieckaTriggENcicl] contain the crucial ideas used in the study of the quadratic regulator problems for lumped and distributed systems (see [@BucciPANDO1; @BucciPANDO2; @PandSing1; @Pandsing2; @Pandsing3] for the singular quadratic regulator problem for distributed systems).
In recent times, the study of controllability of systems described by Voterra integrodifferential equations (in Hilbert spaces) has been stimulated by several applications (see [@Pandlibro]) while the theory of the quadratic regulator problem for these systems is still at a basic level. In essence, we can cite only the paper [@Pritch] and some applications of the results in this paper, see for example [@HUANGliWANG]. In these papers, the authors study a standard regulator problem for a system governed by a Volterra integral equation (in a Hilbert space and with bounded operators. The paper [@HUANGliWANG] and some other applications of the results in [@Pritch] studies a stochastic system) and the synthesis of the optimal control is given by relying on the usual variational approach and Fredholm integral equation for the optimal control. The authors of these papers do not develop a Riccati differential equation and this is our goal here. In order to avoid the technicalities inevitably introduce by the presence of unbounded operators which are introduced by the action of boundary controls, we confine ourselves to study Volterra integral equations in ${{\rm I\hskip-2.1pt R}}^n$.
The control problem we consider is described by $${\label}{eq:Volte}
x'={\int_0 ^t}N(t-s)x(s){\;\mbox{\rm d}}s+ Bu(t)\,,\qquad x(0)=x_0$$ where $x\in {{\rm I\hskip-2.1pt R}}^n$, $u\in {{\rm I\hskip-2.1pt R}}^m$, $B$ is a constant $n\times m$ matrix and $N(t)$ is a continuous $n\times n$ matrix (extension to $B=B(t)$ and $N=N(t,s)$ is simple). Our goal is the study of the minimization of the standard quadratic cost $${\label}{eq:costoAZERO}
{\int_0 ^T}\left [ x^*(t) Qx(t) +|u(t)|^2\right ]{\;\mbox{\rm d}}t+ x^*(T)Q_0x(T)$$ where $Q=Q^*\geq 0$, $Q_0=Q_0^*\geq 0$.
Existence of a unique optimal control in $L^2(0,T;{{\rm I\hskip-2.1pt R}}^m)$ for every fixed $x_0\in {{\rm I\hskip-2.1pt R}}^n$ is obvious.
The plan of the paper is as follows: in order to derive a Riccati differential equation, we need a suitable “state space” in which our system evolves. In fact, a Volterra integral equation is a semigroup system in a suitable infinite dimensional space (see [@Nagel Ch. 6]) and we could relay on this representation of the Volterra equation to derive a theory of the Riccati equation in a standard way but the shortcoming is that the “state space” is $ {{\rm I\hskip-2.1pt R}}^n\times L^2(0,+{\infty};{{\rm I\hskip-2.1pt R}}^n)$ and the Riccati differential equation so obtained should be solved in a space with infinite memory, even if the process is considered on a finite time interval $[0,T]$. We wish a “Riccati differential equation” in a space which has a “short memory”, say of duration at most $T$, as required by the optimization problem. So, we need the introduction of a different “state space approach” to Eq. (\[eq:Volte\]). This is done in Sect. \[sec:SateDyn\] where, using dynamic programming, we prove that the minimum of the cost is a quadratic form which satisfy a (suitable version) of the Linear Operator Inequality [**(LOI)**]{}.
Differentiability properties of the cost are studied in section \[sect:DIFFEpropri\] (using a variational approach to the optimal control related to the arguments in [@Pritch]). The regularity properties we obtain finally allows us to write explicitly a system of partial differential equations (with a quadratic nonlinearity) on $[0,T]$, which is the version of the Riccati differential equations for our system.
We believe that the introduction of the state space in Sect. \[sec:SateDyn\] is a novelty of this paper.
[sec:SateDyn]{}The state of the Volterra integral equation, and the [**(LOI)**]{}
=================================================================================
According to the general definition in [@Kalman]), the state at time ${\tau}$ is the information at time ${\tau}$ needed to uniquely solve the equation for $t>{\tau}$ (assuming the control is known for $t>{\tau}$).
It is clear that if ${\tau}=0$ then the sole vector $x_0$ is sufficient to solve equation (\[eq:Volte\]) in the future, and the state space at ${\tau}=0$ is ${{\rm I\hskip-2.1pt R}}^n$. Things are different if we solve the equation till time ${\tau}$ and we want to solve it in the future. In this case, Eq. (\[eq:Volte\]) for $t>{\tau}$ takes the form $${\label}{eq:volteat0}
x'=\int_{{\tau}}^t N(t-s)x(s){\;\mbox{\rm d}}s+B u(t)+ \int_0^{{\tau}} N(t-s)x(s){\;\mbox{\rm d}}s\,.$$ In order to solve this equation for $t>{\tau}$ we must know the pair[^3] $X_{{\tau}}=\left (x({\tau}),x_{{\tau}}(\cdot) \right )$ where $x_{{\tau}}(s) =x(s)$, $s\in (0,{\tau})$.
Note that in order to uniquely solve (\[eq:volteat0\]), $x_{{\tau}}(\cdot)$ needs not be a segment of previously computed trajectory. It can be an “arbitrary” function. This observation suggests the definition of the following state space at time ${\tau}$: $$M^2_{{\tau}}={{\rm I\hskip-2.1pt R}}^n\times L^2(0,{\tau};{{\rm I\hskip-2.1pt R}}^n)$$ (to be compare with the state space of differential equations with a fixed delay $h$ which is ${{\rm I\hskip-2.1pt R}}^n\times L^2(-h,0;{{\rm I\hskip-2.1pt R}}^n)$).
Eq. (\[eq:volteat0\]) defines, for every fixed $u$ and ${\tau}_1>{\tau}$, a solution map from $M^2_{{\tau}}$ to $M^2_{{\tau}_1}$ which is affine linear and continuous. An explicit expression of this map can be obtained easily. Let us fix an initial time ${\tau}\geq 0$. Let $t\geq {\tau}$ and let $Z(t,{\tau})$ be the $n\times n$ matrix solution of $${\label}{eq:diZgrande}
\frac{{\;\mbox{\rm d}}}{{\;\mbox{\rm d}}t}Z(t,{\tau})=\int_{{\tau}}^t Z(\xi,{\tau})N(t-\xi) {\;\mbox{\rm d}}\xi\,,\quad Z({\tau},{\tau})=I \,.$$ Then,
$${\label}{eq:evoluzione}
x(t)=Z(t,{\tau}) \hat x +\int_0 ^{{\tau}} Y(t,s;{\tau}) \tilde x(s){\;\mbox{\rm d}}s+\int _{{\tau}}^t Z(t-r+{\tau},{\tau})B u(r){\;\mbox{\rm d}}r$$
where $$Y(t,s;{\tau})=\int_{{\tau}}^t Z(t-\xi+{\tau},{\tau})N(\xi-s){\;\mbox{\rm d}}\xi\,.$$
This way, for every ${\tau}_{1}>{\tau}$ we define two linear continuous transformations: $E({\tau}_1;{\tau})$ from $M^2_{{\tau}}$ to $M^2_{{\tau}_1}$ (when $u=0$) and $\Lambda({\tau}_1;{\tau})$ from $L^2({\tau},{\tau}_1;{{\rm I\hskip-2.1pt R}}^m)$ to $M^2_{{\tau}_1}$ (when $X_{{\tau}}=0$), as follows: $$E({\tau}_1;{\tau})(\hat x,\tilde x(\cdot))=(x({\tau}_1),y)\qquad y=\left\{
\begin{array}{lll}
x(t)\ \mbox{given by~(\ref{eq:evoluzione})}&{\rm if}& {\tau}<t<t_1\\
\tilde x(t) &{\rm if} & t\in (0,{\tau})\,.
\end{array}
\right.$$ The operator $\Lambda({\tau}_1;{\tau})$ is defined by the same formula as $E({\tau}_1;{\tau})$, but when $X_{{\tau}}=0$ and $u\neq 0$.
The evolution of the system is describe by the operator $${\label}{eq:evoluSISTE}
E(t_1;{\tau})X_{{\tau}}+\Lambda(t_1;{\tau})u\,.$$
The evolutionary properties of this operator follow from the unicity of solutions of the Volterra integral equation. Let us consider Eq. (\[eq:volteat0\]) on $[{\tau},T]$ with initial condition $(\hat x,\tilde x(\cdot))$, whose solution is given by (\[eq:evoluzione\]). Let ${\tau}_1\in ({\tau},T)$ and let us consider Eq. (\[eq:volteat0\]) on $[{\tau}_1,T]$ but with initial condition $\left (x({\tau}_1),x_{{\tau}_1}\right )$. Eq. (\[eq:volteat0\]) on $[{\tau}_1,T]$ and this initial condition takes the form $$x'(t)=\int _{{\tau}_1} ^t N(t-s) x(s){\;\mbox{\rm d}}s+Bu(t)+\int_0^{{\tau}_1}N(t-s) x_{{\tau}_1}(s){\;\mbox{\rm d}}s\,,\qquad x({\tau}_1^+)=x({\tau}_1^-)$$ and so, on $[{\tau}_1,T]$ we have $$x'(t)=Z(t,{\tau}_1)x({\tau}_1^-) +\int_0^{{\tau}_1} Y(t,s;{\tau}_1) x_{{\tau}_1}(s){\;\mbox{\rm d}}s+\int _{{\tau}_1}^t Z(t-s-{\tau}_1,{\tau}_1)Bu(s){\;\mbox{\rm d}}s\,.$$ Unicity of the solutions of the Volterra integral equation shows that, for $t\in ({\tau}_1,T]$ the following equality holds $$E(t,{\tau})\left (\hat x,\tilde x\right )+\Lambda(t;{\tau})u=E(t,{\tau}_1)\left [ E({\tau}_1,{\tau})\left (\hat x,\tilde x\right )+\Lambda({\tau}_1;{\tau})u\right ]+ \Lambda(t;t_1)u\,.$$
[rema:DerivZistiniz]{}
The solution $Z(t,{\tau})$ of Eq. (\[eq:diZgrande\]) solves the following Volterra integral equation on $[{\tau},T]$: $$Z(t)=1+\int_{\tau}^t Z(\xi)M(t-\xi){\;\mbox{\rm d}}\xi\,,\qquad M(t)={\int_0 ^t}N(s){\;\mbox{\rm d}}s\,.$$ The usual Picard iteration gives $$\begin{aligned}
Z(t,{\tau})&=1+\int_{\tau}^t M(t-\xi){\;\mbox{\rm d}}\xi+\int_{\tau}^t \int_{\tau}^\xi M(\xi-\xi_1){\;\mbox{\rm d}}\xi_1 M(t-\xi){\;\mbox{\rm d}}\xi+\cdots=\\
&= 1+\int_{\tau}^t M(t-\xi){\;\mbox{\rm d}}\xi+\int_{\tau}^t \int_0^{t-s} M(t-s-r)M(r){\;\mbox{\rm d}}r{\;\mbox{\rm d}}s+\cdots \end{aligned}$$ The properties of these integrals is that, once exchanged, we have $$Z(t,{\tau})=1+\int_{\tau}^t H(t-s){\;\mbox{\rm d}}s$$ where $H(t)$ does not depend on ${\tau}$ and it is differentiable. It follows that *the function $({\tau},t)\mapsto Z(t,{\tau})$ is continuously differentiable on $0<{\tau}<t<T$ and the derivative has continuous extension to $0\leq {\tau}\leq t\leq T$.*
------------------------------------------------------------------------
Now we begin our study of the quadratic regulator problem and of the Riccati equation.
One of the possible ways to derive an expression of the optimal control and possibly a Riccati differential equation for the quadratic regulator problem is via dynamic programming. We follow this way. For every fixed ${\tau}<T$ we introduce $$J_{{\tau}}\left (X_{{\tau}},u\right )={\int_{\tau}^T}\left [ x^*(t)Qx(t) +|u(t)|^2\right ]{\;\mbox{\rm d}}t +x^*(T)Q_0x(T)$$ where $x(t)$ is the solution of (\[eq:volteat0\]) (given by (\[eq:evoluzione\])) and we define $${\label}{eq:identiLOI}
W({\tau};{X_{{\tau}}})=\min _{u\in L^2({\tau},T;{{\rm I\hskip-2.1pt R}}^m)} J_{{\tau}}\left ({X_{{\tau}}},u\right )\,.$$ Existence of the minimum is obvious and we denote $u^+(t)=u^+(t;{\tau},{X_{{\tau}}})$ the optimal control. The corresponding solution is denoted $x^+(t)=x^+(t;{\tau},{X_{{\tau}}})$ while we put $X^+_{t }=\left (x^+(t ),x^+_{t }{(\cdot)}\right )$.
Let us fix any ${{\tau}_1} \in ({\tau},T)$ and let $u(t)=u^1(t)$ if $t\in ({\tau},{{\tau}_1} )$, $u(t)=u^2(t)$ if $t\in ({{\tau}_1} ,T)$, while $$X^1_t=E(t,{\tau}){X_{{\tau}}}+\Lambda(t,{\tau})u^1\quad t\in[{\tau},{{\tau}_1} ]\,,\quad
X^2_t=E(t,{{\tau}_1} )X^1_{{{\tau}_1}} +\Lambda(t,{{\tau}_1} )u^2\quad t\in[{{\tau}_1} ,T]\,.$$ We noted that $X (t;{\tau},X_{{\tau}})$ given by (\[eq:evoluSISTE\]) on $[{\tau},T]$ is equal to $X^1_t$ on $[{\tau},{{\tau}_1} ]$ and to $X^2_t$ on $[{{\tau}_1} ,T]$.
Let $x^i$ be the ${{\rm I\hskip-2.1pt R}}^n$ component of $X^i$. Then, for every $u$ we have (we use the crochet to denote the inner product instead of the more cumberstome notation $\left (x^1(t)\right )^*Q x_1(t)$) $${\label}{eq:PREloi}
W({\tau},{X_{{\tau}}})\leq \int_{{\tau}}^{{\tau}_1} \left [{\langle}Qx^1(t),x^1(t){\rangle}+|u^1(t)|^2\right ]{\;\mbox{\rm d}}t+J_{{\tau}_1} \left ( X^1_{{\tau}_1} ,u^2\right )\,.$$ This inequality holds for every $u^1$ and $u^2$ and equality holds when $u^1$ and $u^2$ are restrictions of the optimal control $u^+$.
We keep $u^1$ fixed and we compute the minumum of the right hand side respect to $u^2$. We get the Linear Operator Inequality [**(LOI)**]{}: $${\label}{eqDiseqLOI}
W({\tau},X_{{\tau}})\leq \int _{{\tau}}^{{\tau}_1} \left [{\langle}Qx^1(t),x^1(t){\rangle}+|u^1(t)|^2\right ]{\;\mbox{\rm d}}t+W\left ({{\tau}_1} ,X^1_{{\tau}_1} \right )\,.$$ This inequality holds for every control $u\in L^2({\tau},{{\tau}_1} ;{{\rm I\hskip-2.1pt R}}^n)$. Let in particular $u^1$ be the restriction to $({\tau},{{\tau}_1} )$ of $u^+(\cdot)=u^+(\cdot;{\tau},X_{{\tau}})$. Inequality (\[eq:PREloi\]) shows that the minimum of $J_{{\tau}_1} \left ( X^1_{{\tau}_1} ,u^2\right )$ cannot be strictly less then $J_{{\tau}_1} \left ( X^1_{{\tau}_1} ,u^+\right )$, i.e. the optimal control of the cost $J_{{\tau}_1} \left ( X^1_{{\tau}_1} ,u^2\right )$ is the restriction to $({{\tau}_1} ,T)$ of $u^+(t)$, the optimal control of $J_{{\tau}}\left (X_{{\tau}},u\right )$.
Equality holds in (\[eqDiseqLOI\]) if $u^1=u^+$.
In conclusion, we divide with ${{\tau}_1-{\tau}} $ (which is positive) and we find the following inequality, *which holds with equality if $u=u^+$:* $$\frac{1}{{{\tau}_1-{\tau}} }\left [W\left ({{\tau}_1} ;X^1_{{\tau}_1} \right )-W\left ( {\tau};X_{{\tau}}\right )\right ]\geq -\frac{1}{{{\tau}_1-{\tau}} } \int _{{\tau}}^{{\tau}_1} \left [
{\langle}Qx^1(t),x^1(t){\rangle}+|u(t)|^2
\right ]{\;\mbox{\rm d}}t\,.$$ So, the following inequality holds when ${\tau}$ is a Lebesgue point of $u(t)$ (every ${\tau}$ if $u$ is continuous): $${\label}{eq:PrimaFormINEqEQ}
\lim\inf _{{{\tau}_1} \to {\tau}^+}\frac{1}{{{\tau}_1-{\tau}} }\left [W\left ({{\tau}_1} ;X^1_{{\tau}_1} \right )-W\left ( {\tau};X_{{\tau}}\right )\right ]\geq -\left [ {\langle}Q x({\tau}),x({\tau}){\rangle}+|u({\tau})|^2\right ]\,.$$ Equality holds if $u=u^+$ and ${\tau}$ is a Lebesgue point of $u^+$ and in this case we can even replace $\liminf$ with $\lim$, i.e. $W\left ({{\tau}_1} ;X^+_{{\tau}_1} \right )$ is differentiable if ${\tau}$ is a Lebesgue points of $u^+$.
The previous argument can be repeated for every ${\tau}$ so that the previous inequalities/equalities holds $a.e.$ on $[0,T]$ and we might even replace ${\tau}$ with the generic notation $t$.
If it happens that $\ker N(t)= S$, a subspace of ${{\rm I\hskip-2.1pt R}}^n$, we might also consider as the second component of the “state” $X_{{\tau}}$ the projection of $\tilde x$ on (any fixed) complement of $S$, similar to the theory developed in [@DelfourMANITIUS; @FABRIZIOPATA]. We dont’t pursue this approach here.
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[sect:DIFFEpropri]{}The regularity properties of the value function, the synthesis of the optimal control and the Riccati equation
==================================================================================================================================
We prove that $W$ is a continuous quadratic form with smooth coefficients and we prove that $u^+(t)$ is continuous (so that every time $t$ is a Lebesgue point of $u^+(t)$). We arrive at this result via the variational characterization of the optimal pair $(u^+,x^+)$ ($x^+$ is the ${{\rm I\hskip-2.1pt R}}^n$-component of $X^+$) in the style of [@Pritch]. The standard perturbation approach gives a representation of the optimal control (and a definition of the adjoint state $p(t)$): $${\label}{eq:defiOTTIMcontroVariaz}
u^+(t)=-B^*\left [\int_t^T Z^*(s-t+{\tau},{\tau})Q x^+ (r){\;\mbox{\rm d}}r +Z^*(T-t+{\tau},{\tau})Q_0x^+(T) \right ]=-B^* p(t)$$ where $p$, the function in the bracket, solves the adjoint equation $${\label}{eq:aggiunta}
p'(t)=-Qx^+(t)-\int_t^T N^*(s-t)p(s){\;\mbox{\rm d}}s\,,\qquad p(T)=Q_0 x^+(T)\,.$$
Note that $p$ depends on ${\tau}$ and that Eq. (\[eq:aggiunta\]) has to be solved (backward) on the interval $[{\tau},T]$.
The simplest way to derive the differential equation (\[eq:aggiunta\]) is to note that the function $q(t)=p(T-t)$ is given by $$\begin{aligned}
q(t)&=\int_{T-t}^T Z^*(s-T+{\tau}+t,t)Qx^+(s){\;\mbox{\rm d}}s+Z^*(t+{\tau},{\tau})Q_0x^+(T)=\\
&={\int_0 ^t}Z^*(t-r+{\tau},{\tau})Qx^+(T-r){\;\mbox{\rm d}}r+Z^*(t+{\tau},{\tau})Q_0x^+(T)\,.\end{aligned}$$ Comparison with (\[eq:evoluzione\]) shows that $q(t)$ solves $$q'(t)={\int_0 ^t}N^*(t-s)q(s){\;\mbox{\rm d}}s+Qx^+(T-t)\,,\qquad q(0)=Q_0x^+(T)$$ from which the equation of $p(t)$ is easily obtained.
We recapitulate: the equations which characterize $(x^+,u^+)$ when the initial time is ${\tau}$ and $X_{{\tau}}=\left (\hat x,\tilde x(\cdot)\right )$ is the following system of equations on the interval $[{\tau},T]$: $$\begin{array}{ll}
{\label}{eq:coppiaottima}
x'={\int_{\tau}^t}N(t-s)x(s){\;\mbox{\rm d}}s-BB^*p(t)+\int_0^{{\tau}}N(t-s) \tilde x(s){\;\mbox{\rm d}}s\,, & x({\tau})=\hat x\\[2mm]
p'(t)=-Qx(t)-\int_t^T N^*(s-t) p(s){\;\mbox{\rm d}}s\,, & p(T)=Q_0 x(T)\\[2mm]
u^+(t)=-B^*p(t)\,.
\end{array}$$
We replace $u^+(t)=u^+(t;{\tau},X_{{\tau}}) $ in (\[eq:evoluzione\]). The solution is $x^+(t)$. Then we replace the resulting expression in (\[eq:defiOTTIMcontroVariaz\]). We get the Fredholm integral equation for $u^+(t)$: $$\begin{aligned}
u^+(t)&+B^* Z^*(T-t+{\tau},{\tau}) Q_0\int _{{\tau}}^T Z(T-r+{\tau},{\tau})B u^+(r){\;\mbox{\rm d}}r+\\
&+B^*\int_t^T Z^*(s-t+{\tau},{\tau})Q
\int_{\tau}^s Z(s-r+{\tau},{\tau})Bu^+(r){\;\mbox{\rm d}}r{\;\mbox{\rm d}}s=\\
&= -B^*\left [
Z^* (T-t+{\tau},{\tau})Q_0F(T,{\tau})+
\int_t^T Z^* (s-t+{\tau},{\tau})QF(s,{\tau}){\;\mbox{\rm d}}s\right ]\end{aligned}$$ where $$F(t,{\tau})=Z(t,{\tau})\hat x+\int_0^{\tau}Y(t,s;{\tau}) \tilde x(s){\;\mbox{\rm d}}s\,.$$ This Fredholm integral equation has to be solved on $[{\tau},T]$.
By solving the Fredholm integral equation we find an expression for $u^+(t)$, of the following form: $${\label}{eq:opeloopcontrol}
u^+(t)=u^+(t;{\tau},X_{{\tau}}) = \Phi_1(t,{\tau}) \hat x+\int_0 ^{{\tau}} \Phi_2(t,s;{\tau}) \tilde x(s){\;\mbox{\rm d}}s\,,\quad t\geq {\tau}$$ and so also
$${\label}{eq:openloopSTATE}
x^+(t)=x^+(t;{\tau},X_{{\tau}}) =Z_1(t,{\tau})\hat x+\int_0 ^{{\tau}} Z_2(t,r;{\tau}) \tilde x(r){\;\mbox{\rm d}}r\,, \quad t\geq {\tau}\,.$$
The explicit form of the matrices $\Phi_1(t,{\tau})$, $\Phi_2(t,s;{\tau})$, $Z_1(t,{\tau})$, $Z_2(t,r;{\tau})$ (easily derived using the resolvent operator of the Fredholm integral equation) is not needed. The important fact is that *these matrices have continuous partial derivative respect to their arguments $t$, $s$ and ${\tau}$. In particular, $u^+(t)=u^+(t;{\tau},X_{{\tau}})$ is a continuous function of $t$ for $t\geq {\tau}$. The derivative has continuous extensions to $s={\tau}$ and to $t={\tau}$.* Differentiability respect to ${\tau}$ follows from Remark \[rema:DerivZistiniz\].
We replace (\[eq:opeloopcontrol\]) and (\[eq:openloopSTATE\]) in (\[eq:identiLOI\]) and we get $$\begin{aligned}
\nonumber W({\tau};X_{{\tau}})&=\int_{{\tau}} ^T \left |
Q^{1/2}Z_1(s,{\tau})\hat x +Q^{1/2}\int _0^{{\tau}} Z_2(s,r;{\tau}) \tilde x(r){\;\mbox{\rm d}}r
\right |^2{\;\mbox{\rm d}}s+\\
{\label}{eq:FinaPerInfoootimo}&+ \int_{{\tau}}^T \left |
\Phi_1(s;{\tau})x_0+\int_0 ^{{\tau}} \Phi_2(s,r;{\tau})\tilde x(r){\;\mbox{\rm d}}r
\right |^2{\;\mbox{\rm d}}s\,.\end{aligned}$$ This equality shows that $X_{{\tau}} \mapsto W({\tau},X_{{\tau}})$ is a continuous quadratic form of $X_{{\tau}}\in M_{{\tau}}$.
We use dynamic programming again, in particular the fact that $u^+(\cdot;{{\tau}_1} ,X^+_{{{\tau}_1} })$ is the restriction to $[{{\tau}_1} ,T]$ of $u^+(\cdot;{\tau},X_{{\tau}})$. Hence, for every ${{\tau}_1} \geq {\tau}$ we have $$\begin{aligned}
\nonumber W({{\tau}_1} ;X^+_{{{\tau}_1} })&=\int_{{{\tau}_1} } ^T \left |
Q^{1/2}Z_1(s,{{\tau}_1} ) x^+({{\tau}_1} ) +Q^{1/2}\int _0^{{{\tau}_1} } Z_2(s,r;{{\tau}_1} ) x^+(r){\;\mbox{\rm d}}r
\right |^2{\;\mbox{\rm d}}s+\\
{\label}{eq:FinaPerInfoootimo}&+ \int_{{{\tau}_1} }^T \left |
\Phi_1(s;{{\tau}_1} )x^+({{\tau}_1} )+\int_0 ^{{{\tau}_1} } \Phi_2(s,r;t{{\tau}_1} ) x^+(r){\;\mbox{\rm d}}r
\right |^2{\;\mbox{\rm d}}s\,.\end{aligned}$$
We simplify the notations: from now on we drop the ${}^+$ and we replace ${{\tau}_1} $ with $t$ but we must recall that we are computing for $t\geq {\tau}$ and, when we use equality in (\[eqDiseqLOI\]), on the optimal evolution.
By expanding the squares we see that $W({{\tau}_1} ;X_{{{\tau}_1} })$ has the following general form: $$\begin{aligned}
\nonumber
W(t ;X_{t })&= x^* (t ) P_0( t)x ( t)+ x^*( t ) {\int_0 ^t}P_1( t,s) x (s){\;\mbox{\rm d}}s+\\
&{\label}{eq:ExprreDELLAformAquaDRATI}
+\left [{\int_0 ^t}P_1(t,s) x (s){\;\mbox{\rm d}}s\right ]^* x (t)+{\int_0 ^t}{\int_0 ^t}x^* (r) K(t,\xi,r) x (\xi){\;\mbox{\rm d}}\xi{\;\mbox{\rm d}}r\,.\end{aligned}$$ For example, $$P_0(t)= \int_t^T \left [Z_1^*(s,t)QZ_1(s,t)+\Phi_1^*(s,t)\Phi_1(s,t)\right ]{\;\mbox{\rm d}}s\,.$$ Note that $P_0(t)$ is a selfadjoint differentiable matrix.
Now we consider the matrix $ K(t,\xi,r)$. We consider the contribution of the first line in (\[eq:ExprreDELLAformAquaDRATI\]) (the contribution of the second line is similar). Exchanging the order of integration and the names of the variables of integration, we see that $$\begin{aligned}
& {\int_0 ^t}x^*(r) K(t,\xi,r) x(\xi){\;\mbox{\rm d}}\xi{\;\mbox{\rm d}}r= {\int_0 ^t}{\int_0 ^t}x^*(r)\left [
\int_t^T Z_2^*(s,r,t)QZ_2(s,\xi,t){\;\mbox{\rm d}}s
\right ]x(\xi){\;\mbox{\rm d}}r{\;\mbox{\rm d}}\xi=\\
&={\int_0 ^t}{\int_0 ^t}x^*(\xi)\left [
\int_t^T Z_2^*(s,\xi,t)QZ_2(s,r,t){\;\mbox{\rm d}}s
\right ]x(r){\;\mbox{\rm d}}\xi{\;\mbox{\rm d}}r=\int_0^t\int_0^t
x^*(\xi) K^*(t,r,\xi) x(r){\;\mbox{\rm d}}\xi{\;\mbox{\rm d}}r\end{aligned}$$ so that we have $$K(t,\xi,r)=K^*(t,r,\xi)$$ and this matrix function is differentiable respect to its arguments $t$, $r$ and $\xi$.
Analogously we see differentiability of $P_1(t,s)$.
We whish a differential equations for the matrix functions $P_0(t)$, $P_1(t,s)$, $K(t,s,r)$. In order to achieve this goal, we compute the right derivative of $W(t;X _t)$ (and any continuous control) for $t>{\tau}$ and we use inequality (\[eq:PrimaFormINEqEQ\]). We use explicitly that equality holds in (\[eq:PrimaFormINEqEQ\]) when the derivative is computed along an optimal evolution.
The Riccati equation
--------------------
In order to derive a set of differential equations for the matrices $P_0(t)$, $P_1(t,s)$, $K(t,\xi,r)$ we proceed as follows: we fix (any) ${\tau}\in [0,T]$ and the initial condition $X_{{\tau}}=(\hat x,\tilde x(\cdot))$. We consider (\[eq:ExprreDELLAformAquaDRATI\]) with any continuous control $u(t)$ on $[{\tau},T]$ (the corresponding solution of the Volterra equation is $x(t)$). We consider the quadratic form $W$ with the control $u(t)$ and the corresponding solution $X_t$ given in in (\[eq:ExprreDELLAformAquaDRATI\]). In this form we separate the contribution of the functions on $(0,{\tau})$ and the contribution on $[{\tau},t]$. For example $x^*(t)P_0(t)x(t)$ remains unchanged while $x^*(t){\int_0 ^t}P_1(t,\xi)x(\xi){\;\mbox{\rm d}}\xi$ is written as $$x^*(t){\int_0 ^t}P_1(t,\xi)x(\xi){\;\mbox{\rm d}}\xi=x^*(t)\int_0^{\tau}P_1(t,s)\tilde x(s){\;\mbox{\rm d}}s+x^*(t)\int_{\tau}^t P_1(t,s) x(s){\;\mbox{\rm d}}s\,.$$ The other addenda are treated analogously.
We obtain a function of $t$ which is continuously differentiable. Its derivative at $t={\tau}$ is the left hand side of (\[eq:PrimaFormINEqEQ\]) and so it satisfies the inequality (\[eq:PrimaFormINEqEQ\]), with equality if it happens that we compute with $u=u^+$. So, the function of $u\in{{\rm I\hskip-2.1pt R}}^m$ $$u\mapsto \left [\frac{{\;\mbox{\rm d}}}{{\;\mbox{\rm d}}t}W({\tau};X _{{\tau}})+u^*({\tau})u({\tau})\right ]$$ reaches a minimum at $u=u^+_{\tau}$. Note that ${\tau}\in [0,T]$ is arbitrary and so by computing this minimum we get an expression for $u^+({\tau})$, for every ${\tau}\in[0,T]$.
It turns out that $\frac{{\;\mbox{\rm d}}}{{\;\mbox{\rm d}}t}W({\tau};X _{{\tau}})+u^*({\tau})u({\tau})$ is the sum of several terms. Some of them do not depend on $u$ and the minimization concerns solely the terms which depends on $u$. We get (we recall that $P_0({\tau})$ is selfadjoint) $$\begin{aligned}
\nonumber u^+({\tau})&={\rm arg\, min} \left \{
u^*B^*P_0({\tau})\hat x +u^*B^* \int_0^{\tau}P_1({\tau},s) \tilde x(s){\;\mbox{\rm d}}s+\right.\\
{\label}{FunzioDAMINIMperilCONTROTTIMO}&\left.+\hat x^*P_0({\tau})Bu+\left ( \int_0^{\tau}\tilde x^*(s)P_1^*({\tau},s){\;\mbox{\rm d}}s\right )Bu+u^*u
\right \}\,.\end{aligned}$$ The minimization gives $${\label}{eq:FeedbackFORMuOTTIMOpre}
u^+({\tau})=-B^*\left [P_0({\tau}) \hat x +\int_0^{\tau}P_1({\tau},s)\tilde x (s){\;\mbox{\rm d}}s\right ]\,.$$ If the system is solved up to time $t$ along an optimal evolution (so that $x^+(t)$ is equal to $\tilde x(t)$ when $t<{\tau}$ and it is the solution which corresponds to the optimal control for larger times) we have $${\label}{eq:FeedbackFORMuOTTIMO}
u^+(t)=-B^*\left [P_0({\tau}) x^+(t) +\int_0^{\tau}P_1(t,s) x^+ (s){\;\mbox{\rm d}}s\right ]$$ and this is the feedback form of the optimal control (compare [@Pritch]).
We repalce (\[eq:FeedbackFORMuOTTIMOpre\]) in the brace in (\[FunzioDAMINIMperilCONTROTTIMO\]) and we see that the minimum is $$\begin{aligned}
\nonumber&-\hat x^*P_0({\tau})BB^*P_0({\tau})\hat x-\hat x^*P_0({\tau})BB^*\int_0^{\tau}P_1({\tau},\xi)\tilde x(\xi){\;\mbox{\rm d}}\xi-\\
{\label}{eq:IlMINIMOallOOTTIMMMO}&-\left (\int_0^{\tau}\tilde x^*(r)P_1^*({\tau},r){\;\mbox{\rm d}}r\right )BB^*P_0({\tau})\hat x-\int_0^{\tau}\int_0^{\tau}\tilde x(r)P_1({\tau},r)BB^*P_1({\tau},\xi)\tilde x(\xi){\;\mbox{\rm d}}\xi{\;\mbox{\rm d}}r\,.\end{aligned}$$
Now we compute the derivative of the function $ {\tau}\mapsto W({\tau};X _{{\tau}})$ along an optimal evolution and we consider its limit for $t\to {\tau}+$. We insert this quantity in (\[eq:PrimaFormINEqEQ\]), which is an equality since we are computing the limit along an optimal evolution. We take into account that the terms which contains $u$ sum up to the expression (\[eq:IlMINIMOallOOTTIMMMO\]) and we get the following equality. In this equality, a superimposed dot denotes derivative with respect to the variable ${\tau}$: $$\dot P_0({\tau})=\frac{{\;\mbox{\rm d}}}{{\;\mbox{\rm d}}{\tau}} P_0({\tau})\,,\quad
\dot P_1({\tau},\xi)=\frac{\partial}{\partial {\tau}} P_1({\tau},\xi)\,,\qquad \dot K({\tau},\xi,r)=
\frac{\partial}{\partial {\tau}} K({\tau},\xi,r)\,.$$ The equality is: $$\begin{aligned}
&-\hat x^*P_0({\tau})BB^*P_0({\tau})\hat x-\hat x^*P_0({\tau})BB^*\int_0^{\tau}P_1({\tau},\xi)\tilde x(\xi){\;\mbox{\rm d}}\xi-\\
&-\left (\int_0^{\tau}\tilde x^*(r)P_1^*({\tau},r){\;\mbox{\rm d}}r\right )BB^*P_0({\tau})\hat x-\int_0^{\tau}\int_0^{\tau}\tilde x(r)P_1({\tau},r)BB^*P_1({\tau},\xi)\tilde x(\xi){\;\mbox{\rm d}}\xi{\;\mbox{\rm d}}r+\\
&+\left (\int_0^{\tau}\tilde x^*(r)N^*({\tau}-r){\;\mbox{\rm d}}s\right )P_0({\tau})\hat x+\hat x^*\dot P_0({\tau})\hat x+\hat x^*\int_0^{\tau}N({\tau}-\xi)\tilde x(\xi){\;\mbox{\rm d}}\xi+
\hat x^*P_1({\tau},{\tau})\hat x+\\
&+\hat x^*P_1^*({\tau},{\tau})\hat x+\left ( \int_0^{\tau}\tilde x^*(r)N^*( {\tau}-r){\;\mbox{\rm d}}r\right )\int_0^{\tau}P_1({\tau},s)\tilde x(s){\;\mbox{\rm d}}s+\hat x^*\int_0^{\tau}\dot P_1({\tau},\xi)\tilde x(\xi){\;\mbox{\rm d}}\xi+\\
&+\left (\int_0^{\tau}\tilde x^*(r)\dot P_1^*({\tau},r){\;\mbox{\rm d}}r\right )\hat x +\left ( \int_0^{\tau}\tilde x^*(r)P_1({\tau},r){\;\mbox{\rm d}}r\right )\left ( \int_0^{\tau}N({\tau}-\xi)\tilde x(\xi){\;\mbox{\rm d}}\xi\right )+\\
& +\left (\int_0^{\tau}\tilde x^*(r)K({\tau},{\tau},r){\;\mbox{\rm d}}r\right )\hat x+\hat x^*\int_0^{\tau}K({\tau},\xi,{\tau})\tilde x(\xi){\;\mbox{\rm d}}\xi+\\
&+\int_0^{\tau}\tilde x^*(r)\int_0^{\tau}\dot K({\tau},\xi,r)\tilde x(\xi){\;\mbox{\rm d}}\xi{\;\mbox{\rm d}}r+\hat x^*Q\hat x=0\end{aligned}$$
The vector $\hat x$ and the function $\tilde x(\cdot)$ are arbitrary. So, we first impose $\tilde x(\cdot)=0$ and $\hat x$ arbitrary, then the converse and finally both nonzero arbitrary. We find that the three matrix functions $P_0({\tau})$, $P_1({\tau},r)$, $K({\tau},\xi,r)$ solve the following system of differential equations in the arbitrary variable ${\tau}$. The variables $r$ and $\xi$ belong to $[0,{\tau}]$ for every ${\tau}\in [0,T]$.
$${\label}{Eq:RICCATI}\begin{array}{ll}
&\displaystyle P_0'({\tau})-P_0({\tau})B^*BP_0({\tau})+Q({\tau})+P_1({\tau},{\tau})+P_1^*({\tau},{\tau})=0\\[2mm]
&\displaystyle \frac{\partial}{\partial {\tau}}P_{1}({\tau},\xi) -P_0({\tau})BB^*P_1({\tau},\xi) +P_0({\tau})N({\tau}-\xi)
+K({\tau},\xi,{\tau})=0\\[2mm]
&\displaystyle \frac{\partial}{\partial {\tau}}K({\tau},\xi,r)-P_1^*({\tau},r)BB^*P_1({\tau},\xi)+\\
&\displaystyle{~}\quad +P^*_1({\tau},r)N({\tau}-\xi)
+N^*({\tau}-r)P_1({\tau},\xi) =0\\[2mm]
&\displaystyle P_0(T)=Q_0\,,\qquad P_1(T,\xi)=0\,,\qquad K(T,\xi,r)=0
\end{array}$$
The final conditions are obtained by noting that when ${\tau}=T$ i.e. with $X_T=(\hat x,\tilde x_T(\cdot))$ arbitrary in $M^2_T={{\rm I\hskip-2.1pt R}}^n\times L^2(0,T;{{\rm I\hskip-2.1pt R}}^n)$, the expression $W(T,X_T)$ in (\[eq:ExprreDELLAformAquaDRATI\]) is equal to $J_T(X_T;u)=\hat x^*Q_0\hat x$ for every $X_T$.
*This is the Riccati differential equation of our optimization problem.*
We note the following facts:
- We take into account the fact that $P_0$ is selfadjoint and $K^*({\tau},\xi,{\tau})=K({\tau},{\tau},\xi)$. We compute the adjoint of the second line in (\[Eq:RICCATI\]) and we find: $$\frac{\partial}{\partial {\tau}}P^*_{1}({\tau},r) -P_1^*({\tau},r)BB^*P_0({\tau}) + N^*({\tau}-r)P_0({\tau}) +K({\tau},{\tau},r) =0 \,.$$
- The form of the Riccati differential equations we derived for the Volterra integral equation (\[eq:Volte\]) has to be compared with the Riccati differential equation “ in decoupled form” which was once fashionable in the study of the quadratic regulator problem for systems with finite delays, see [@Ross].
------------------------------------------------------------------------
[99]{} [Bittanti]{} Bittanti, S., Laub, A.J., Willems, J.C. Ed.s, *The Riccati equation,* Springer-Verlag, Berlin, 1991. [Brune]{} Brune, O., Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency, *Journal of Mathematics and Physics,* [**10**]{} 191-236, 1931.
[BucciPANDO1]{} Bucci, F. Pandolfi, L., The value function of the singular quadratic regulator problem with distributed control action. *SIAM J. Control Optim.* [**36**]{} 115-136 (1998). [BucciPANDO2]{} Bucci, F. Pandolfi, L., The regulator problem with indefinite quadratic cost for boundary control systems: the finite horizon case. *Systems Control Lett.* [**39**]{} 79-86 (2000).
[DelfourMANITIUS]{} Delfour, M.C., Manitius, A., The structural operator F and its role in the theory of retarded systems. *J. Math. Analysis Appl.* Part I: [**73**]{} 466-490 (1980); Part II [**74**]{} 359-381 (1980).
[FABRIZIOPATA]{} Fabrizio M., Giorgi C., Pata V., A New Approach to Equations with Memory, *Arch. Rational Mech. Anal.* [**198**]{} 189-232 (2010).
[HUANGliWANG]{} Huang, J., Li, X., Wang, T., Mean-Field Linear-Quadratic-Gaussian (LQG) Games for Stochastic Integral Systems, *IEEE Transactions on Automatic Control* [**61**]{} 2670-2675 (2016).
[lasieckaTriggENcicl]{} Lasiecka, I., Triggiani, R., *Control theory for partial differential equations: continuous and approximation theories.* (Vol. 1 *Abstract parabolic systems* and Vol. 2 *Abstract hyperbolic-like systems over a finite time horizon.*) Cambridge University Press, Cambridge, 2000. [Kalman]{} Kalman, R. E., Falb, P. L., Arbib, M. A., *Topics in mathematical system theory.* McGraw-Hill Book Co., New York-Toronto, 1969 [Nagel]{} Engel, K.-J., Nagel, R. *One-parameter semigroups for linear evolution equations.* Springer-Verlag, New York, 2000. [PandSing1]{} Pandolfi, L. Dissipativity and the Lur’ e problem for parabolic boundary control systems. *SIAM J. Control Optim.* [**36**]{} 2061-2081 (1998) [Pandsing2]{} Pandolfi, L. The Kalman-Yakubovich-Popov theorem for stabilizable hyperbolic boundary control systems. *Integral Equations Operator Theory* [**34**]{} 478-493 (1999) [Pandsing3]{} Pandolfi, L. The Kalman-Popov-Yakubovich theorem: an overview and new results for hyperbolic control systems. *Nonlinear Anal.* [**30**]{} 735-745 (1997). [Pandlibro]{} Pandolfi, L., *Distributed systems with persistent memory. Control and moment problems.* Springer Briefs in Electrical and Computer Engineering. Control, Automation and Robotics. Springer, Cham, 2014. [Pritch]{} Pritchard, A.J., You Y., Causal feedback Optimal control for Volterra integral equations. *SIAM J. Control Optim.* [**34**]{} 1874-1890, 1996. [Ross]{} Ross, D. W., Flügge-Lotz, I., An optimal control problem for systems with differential-difference equation dynamics. *SIAM J. Control* [**7**]{} 609-623, 1969.
[^1]: Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy (luciano.pandolfi@polito.it)
[^2]: This papers fits into the research program of the GNAMPA-INDAM and has been written in the framework of the “Groupement de Recherche en Contrôle des EDP entre la France et l’Italie (CONEDP-CNRS)”.
[^3]: Remark on the notation: $x_{{\tau}}=x_{{\tau}}(s)$ is a function on $(0,{\tau})$ while $X_{{\tau}}$ (upper case letter) is the pair $(x({\tau}),x_{{\tau}})$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'It is argued that the strong coupling version of recent experiment \[Denkmayr et al., PRL 118, 010402 (2017)\] while correctly estimating the pre-selected states of the neutrons does not perform strong measurements of weak values as claimed.'
title: ' Comment on “Experimental demonstration of direct path state characterization by strongly measuring weak values in a matter-wave interferometer” '
---
Denkmayr [*et al.*]{} [@Denk] reported an experiment in which a tomographic task of “direct path state characterization” in the neutron interferometer has been performed using weak and strong coupling to neutron’s spin. I correct misleading statements in the title, abstract and conclusions regarding strong measurements of weak values.
According to the title, direct path state characterization has been achieved by “strongly measuring weak values". In the abstract: “weak measurements are not a necessary condition to determine the weak value”. In the conclusions: “we have presented a weak value determination scheme via arbitrary interaction strengths. We have applied it to experimentally determine weak values using both weak and strong interactions.” I argue that in the strong regime, the experiment does not measure weak values of the observed quantum system.
The weak value of a variable $A$ is a property of a quantum system at a particular time [@AAV]. It is specified by the forward and backward evolving quantum states at this time and it has a well defined operational meaning: any weak enough coupling to $A$ is an effective coupling to the weak value $A_w$. The pointer of a weakened von Neumann measurement is shifted in proportion to ${\rm Re} A_w$ while the shift of the conjugate pointer variable is proportional to ${\rm Im} A_w$.
Lundeen [*et al.*]{} [@Lund] pointed out that, given a particular post-selection, the weak values of local projections are proportional to the local values of the wave function and thus, measurements of these weak values provide a “direct measurement of the quantum wavefunction”. Vallone and Dequal [@Vall] showed that a modification of this procedure, in which the weak coupling is replaced by a strong coupling, provides a more efficient method for “direct measurement of the quantum wavefunction”, although one might argue that it is less “direct”, because instead of simple proportionality, we need calculations to obtain the local amplitude from a set of pointer readings.
Denkmayr [*et al.*]{} implemented these proposals in neutron interferometry, successfully accomplishing both strong and weak coupling versions of the “path state characterization”. However, the strong coupling version of their experiment is not a strong measurement of weak values as they claim.
In the experiment, polarized neutrons, $|{\uparrow}_x\rangle$, are prepared in the path state $|P_i\rangle = a |{\rm I}\rangle+b |{\rm II}\rangle$ and post-selected in $|P_f\rangle = \frac{1}{\sqrt 2} (|{\rm I}\rangle+|{\rm II}\rangle)$. The task is to determine $|P_i\rangle$. Weak values of the projections on the paths are $({\rm {\bf P}_I})_w=\frac{a}{a+b}$ and $({\rm {\bf P}_{II}})_w=\frac{b}{a+b}$. Proportionality of weak values to the complex amplitudes in the paths makes weak measurements of the projections “direct” measurements of the path state.
In a more direct version of their experiment, the polarization is rotated in one of the arms of the interferometer: $|{\uparrow_x}\rangle \rightarrow \cos \alpha |{\uparrow}_x\rangle - i\sin \alpha |{\downarrow}_x\rangle$. The spin tomography of the output beam provides the information about $|P_i\rangle$. If we choose a small coupling, say in path I, the angle of rotation in the $xy$ plane is $2\alpha{\rm Re}({\rm {\bf P}_{\rm I}})_w$ and in the $xz$ plane $2\alpha{\rm Im}({\rm {\bf P}_{\rm I}})_w$. After repeating the procedure in path II, $({\rm {\bf P}_I})_w$ and $({\rm {\bf P}_{II}})_w$ yield the pre-selected path state $|P_i\rangle$. In fact, since $({\rm {\bf P}_I})_w +({\rm {\bf P}_{II}})_w = ({\rm {\bf P}_I} +{\rm {\bf P}_{II}})_w=1$, we can calculate $({\rm {\bf P}_{\rm II}})_w $ and the second procedure is not needed.
The first procedure with strong coupling (large $\alpha$) provides $|P_i\rangle$ even more efficiently [@Vall]. But does it measure the weak values of the projection, as the authors [@Denk] claim?
When the experiment runs with large $\alpha$, the two-state vector description of the neutrons inside the interferometer is different. The weak values of projections remain constant in time, but their values are not the same as in the run with vanishing polarization rotation. At the time before the post-selection, the state of the neutron is: $
|\Psi'\rangle = a |{\rm I}\rangle |{\uparrow}_x\rangle + b |{\rm II}\rangle ( \cos \alpha |{\uparrow}_x\rangle- i\sin \alpha |{\downarrow}_x\rangle).
$ Then, the neutron is partially post-selected onto path state $|P_f\rangle$. In such a case, the weak value is given by (13.23) of [@AV2008] $$\label{psi}
({\rm {\bf P}_{I}})_w=\frac{\langle \Psi'|{\rm {\bf P}_{P_f}}{\rm {\bf P}_{I}}|\Psi'\rangle}{\langle \Psi'| {\rm {\bf P}_{P_f}}|\Psi'\rangle}=\frac{a(b^\ast \cos \alpha+a^\ast)}{b(a^\ast \cos \alpha+b^\ast)+a(b^\ast \cos \alpha+a^\ast)}.$$ The ratio of weak values of the projections, $ \frac{({\rm {\bf P}_{I}})_w}{({\rm {\bf P}_{II}})_w}=
\frac{a(b^\ast \cos \alpha+a^\star)}{b(a^\ast \cos \alpha+b^\ast)},$ yields the ratio of complex amplitudes only for vanishing interaction, $\alpha \rightarrow 0$. Direct path state characterization has not been done by strongly measuring weak values in this experiment because weak values cannot be measured strongly.
This work has been supported in part by the Israel Science Foundation Grant No. 1311/14, the German-Israeli Foundation for Scientific Research and Development Grant No. I-1275-303.14.
L. Vaidman\
Raymond and Beverly Sackler School of Physics and Astronomy\
Tel-Aviv University, Tel-Aviv 69978, Israel
[99]{}
T. Denkmayr, H. Geppert, H. Lemmel, M. Waegell, J. Dressel, Y. Hasegawa, and S. Sponar, Experimental demonstration of direct path state characterization by strongly measuring weak values in a matter-wave interferometer, Phys. Rev. Lett. [**118**]{}, 010402 (2017).
G. Vallone and D. Dequal, Strong measurements give a better direct measurement of the quantum wave function, Phys. Rev. Lett. [**116**]{}, 040502 (2016).
Y. Aharonov, D. Z. Albert, and L. Vaidman, How the result of a measurement of a component of the spin of a spin-$\frac{1}{2}$ particle can turn out to be 100, Phys. Rev. Lett. **60**, 1351 (1988).
J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, Direct measurement of the quantum wavefunction, Nature (London) [**474**]{}, 188 (2011).
Y. Aharonov and L. Vaidman, The two-state vector formalism: an updated review, Lect. Notes Phys. **734**, 399 (2008).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present the rotation velocities $V$ and velocity dispersions $\sigma$ along the principal axes of seven elliptical galaxies less luminous than $M_B= -19.5$. These kinematics extend beyond the half-light radii for all systems in this photometrically selected sample. At large radii the kinematics not only confirm that rotation and “diskiness" are important in faint ellipticals, as was previously known, but also demonstrate that in most sample galaxies the stars at large galactocentric distances have $\bigl (V/\sigma\bigr )_{max}\sim 2$, similar to the disks in [*bona-fide*]{} S0 galaxies. Comparing this high degree of ordered stellar motion in all sample galaxies with numerical simulations of dissipationless mergers argues against mergers with mass ratios $\le 3:1$ as an important mechanism in the final shaping of low-luminosity ellipticals, and favors instead the dissipative formation of a disk.'
author:
- 'Hans-Walter Rix$^{1,4}$, Marcella Carollo$^{2,5}$ and Ken Freeman$^3$'
title: Large stellar disks in small elliptical galaxies
---
Astrophysical Journal Letters, March 1999.
$^1$ Steward Observatory\
$^2$ Johns Hopkins University\
$^3$ Mount Stromlo and Sidings Springs Observatory\
$^4$ Alfred P. Sloan Fellow\
$^5$ Hubble Fellow\
[*subject headings*]{}: galaxies: formation - galaxies: evolution - galaxies: structure - galaxies: elliptical and lenticular - galaxies: kinematics and dynamics
Introduction
============
Elliptical galaxies occupy only a small volume in the stable parameter space defined by mass, luminosity, compactness, and rotational support (Kormendy and Djorgovski 1989; de Zeeuw and Franx 1991). This means that their present-day structure is not determined by stability, but by their formation history, specifically by the relative importance and the time ordering of dissipative processes [*vs.*]{} merging. Yet, the actual role of gas dissipation, resulting in gaseous and stellar disks, and of violent relaxation, leading to spheroidal systems with largely random stellar motions, is still under debate for ellipticals of different luminosity classes.
Observationally it has been established that “low-luminosity" ellipticals ($L\ltorder \frac{1}{2} L_*$) show more rotation and photometric diskiness than the brighter systems (e.g., Davies 1983; Bender et al. 1989). Yet, taken alone, these results cannot constrain uniquely their formation history, as there are many paths towards systems with modest rotation. For example, N-body simulations of dissipationless major mergers can produce rotating remnants $(V/\sigma)_{max}\ltorder 1$ with disky isophotes (e.g., Heyl, Hernquist & Spergel 1996; Weil & Hernquist 1996). On the basis of statistics, Rix & White (1990) argued that most of the so-called faint ellipticals are not just “disky” objects, but are rather the face-on counterparts of S0 galaxies, implying that they contain dynamically-cool stellar disks that make up an appreciable fraction of the total mass. If confirmed kinematically, the presence of an outer stellar disk in the majority of these systems would strongly support gas dissipation as the last significant step in their formation history. For individual objects, detailed photometric and kinematic studies have shown indeed that dynamically fragile stellar disks are present in several elliptical galaxies (Rix & White 1992; Scorza & Bender 1995).
In this paper we attempt to move beyond questions of individual misclassification of galaxies, by asking whether there is any significant fraction of “faint elliptical galaxies” that have reached their present state through largely dissipationless merging as the last formation step? To this end we present and discuss kinematic measurements (to $R > R_e$) for a photometrically selected sample of seven bona-fide ellipticals with $M_B \ge
-19.5$.
Sample and Observations
=======================
Our seven galaxies are a random sub-sample of the RC3 catalog (de Vaucouleurs et al. 1991), with Hubble types $T\le -4$ and luminosities below $M_B= -19.5$. No prior kinematic information was used, and nothing, save the luminosity cut, should have biased the selection process towards rapidly rotating objects. We added three S0 galaxies ($T =
-2,-3$) for comparison. The spectra were obtained during two different runs (February 11-14, 1997, and September 29 – October 2, 1997) at the KPNO 4-m telescope, using the RC spectrograph with the KPC-24 grating in second order. In February 1997 the detector was T2KB CCD with $2048^2$ pixels of $24\mu$m$^2$, which were binned along the slit to yield $1.38''$/pixel. In the September 1997 run we used a 3k$\times$1k F3KB CCD with $15\mu$m$^2$ pixels binned to $0.86''$/pixel along the slit. With a $2.5''$ wide slit and on-chip binning by two in the spectral direction the effective instrumental resolution was $\sigma_{instr}\approx 50$ km/s. The spectra were centered on 5150$\AA$, covering H$\beta$ $\lambda$4861$\AA$, \[OIII\] $\lambda$5007$\AA$, Mg$_b$ $\lambda$5175$\AA$, Fe $\lambda$5270$\AA$ and Fe $\lambda$5335. Table 1 lists relevant properties of the sample members and the observations, including the adopted major axis position angle and the exposure times. Spectra of several K giants were acquired with the same instrumental setup, and used as kinematic templates. The basic data reduction (bias and dark subtraction, flat-fielding, correction for slit-vignetting, wavelength calibration, airmass and Galactic extinction corrections, correction for instrumental response) was performed using [IRAF]{} and [MIDAS]{} software.
The stellar kinematics were obtained with the template fitting method, described in Rix & White (1992), after the data were binned along the slit to constant signal-to-noise. Then the best fit mean velocity $V$ and dispersion $\sigma$ were determined by $\chi^2$-minimization, along with the best fitting composite stellar template. The resulting error bars are the formal uncertainties based on the known sources of noise. External tests on this and other data sets demonstrate that they are a good approximation to the true uncertainties.
Results and Discussion
======================
Figure 1 shows the major and minor axis kinematics for the sample galaxies. Similar to S0 galaxies, the velocity dispersions in these ellipticals are found to drop from a central peak to near our instrumental resolution at large radii. Most sample members exhibit little - if any - rotation along the minor axis, in contrast to their strong rotation along the major axis. Their kinematics suggest that these systems are nearly axisymmetric and likely have a rather simple dynamical structure. Only NGC1588 shows considerable minor axis rotation over the same radial region where the photometry shows isophotal twisting; this is possibly due to the interaction with its close companion (NGC1589).
In Figure 2 we quantify the degree of rotational support in these galaxies, by plotting $V/\sigma$ as a function of the galactocentric distance along the major axis. As a diagnostic of the dynamical state beyond the effective radius, we adopt the maximum, $(V/\sigma)_{max}$, and the outermost value $(V/\sigma)_{out}$, of each $V/\sigma$ curve \[see Table 1\]. In terms of these quantities, Figure 3 re-states the main result: for four out of seven ellipticals, $(V/\sigma)_{max}\gta2$; two additional ellipticals have $(V/\sigma)_{max}\gta1.5$.
As a benchmark, we compare the observed $(V/\sigma)_{max}$ \[and $(V/\sigma)_{out}$\] values to maximally rotating, oblate “Jeans models.” We consider models where the residual velocities are isotropic in the co-rotating frame. The models are built following the method outlined in Binney, Davies, & Illingworth (1990; see also Carollo & Danziger 1994 for particulars), assuming a Jaffe (1983) law. The curves in Figure 3 represent maximum $V/\sigma$ within $2 R_e$, as a function of projected axis ratio (or inclination). We considered [*intrinsic*]{} axis-ratios of $c/a=0.6$ and $c/a=0.4$ and models with and without dark halos (which are three times more extended and more massive than the luminous component). As Figure 3 illustrates, no model can reproduce values of $(V/\sigma)_{max}\sim 2$ for any viewing angle, unless it has $c/a\ltorder 0.4$! This implies that all sample galaxies are intrinsically [*very*]{} flat. A possible exception is the one distorted galaxy, NGC1588, for which our kinematics are compatible with $c/a<0.6$, if $(V/\sigma)_{out}$ is considered instead of $(V/\sigma)_{max}$. The probability that 6 out of 7 ellipticals are that flat is vanishing ($\approx 2.4 \times 10^{-6}$) on the basis of the expected shape distribution of low-luminosity oblate spheroidals (Tremblay & Merritt 1996).
Dissipationless simulations of equal mass galaxy mergers mostly produce remnants with $(V/\sigma)_{max} \ltorder 1$ inside $\sim 2R_e$ (Heyl, Hernquist and Spergel, 1996). In addition, for most spin-orbit geometries major mergers also produce significant kinematic misalignment, which would be reflected in minor axis rotation. Neither property is consistent with our sample galaxies, ruling out formation through mergers of nearly equal mass. Unequal mass mergers ($\sim 3:1$) can produce remnants with rather disk-like shapes and kinematics (Barnes, 1996; Bekki 1998; Barnes 1998). Barnes (1998) finds remnants to be close to axisymmetric with little minor axis rotation, in this respect consistent with our sample properties. He quantifies the rotational support of the merger remnants by the parameter $\lambda'$, the total angular momentum of the most tightly bound half of the stars, normalized by the value for perfect spin alignement. For Barnes’ eight 3:1 mergers $\lambda'_{sim}=0.38$ with a scatter of $ 0.04$. As $\lambda'_{sim}$ is not observable, we attempt to construct an analogous quantity $\langle\lambda'\rangle_{obs}$ for our sample members (see Table 1), by estimating the azimuthal velocity, normalized by the local circular velocity and averaged over the inner 50% of the light. For this estimate we had to make the following assumptions: (i) the overall potential is logarithmic; (ii) the velocity dispersions are isotropic; (iii) the galaxy is axisymmetric with an intrinsic axis ratio of $0.4$ (or $0.6$; see above); (iv) the circular velocity is estimated as $v_c\approx \sqrt{v_\phi^2+2 \sigma^2}$, which is both correct for non-rotating systems in logarithmic potentials and in the “asymmetric drift" limit for $\rho_*\propto r^{-2}$; (v) the stars are on average 30$^\circ$ from the mid-plane. With this, we find $\langle\lambda'\rangle_{obs}=0.55$ with a scatter of $0.06$ (see Table 1): every observed sample galaxy has considerably more ordered motions than any of the 3:1 mergers. We checked that plausible changes in the above model assumptions will alter not $\langle\lambda'\rangle_{obs}$ by $\langle\lambda'\rangle_{obs} - \lambda'_{sim}$.
The degree of streaming motion found in our sample galaxies is nearly that of the dominant disks of S0 galaxies \[Fisher 1997 and the $(V/\sigma)_{max}$ values for the S0s of our sample\]. Without exception the sample members appear even more rotationally supported than simulated unequal mass (3:1) mergers (Barnes 1998). This strongly suggests that the dissipative formation of a massive and extended stellar disk has been the last major step in building these galaxies. This inference does not preclude, however, that for some fraction this disk has been heated considerably by subsequent gravitational interactions.
Our data, therefore allow us to push the long-known result that [*rotation is important in low-luminosity ellipticals*]{} (Davies 1982) one step further: most of these “elliptical" galaxies in an effectively luminosity-limited sample, actually contain stellar disks at large radii, comparable in mass and size to S0s. These kinematic results, combined with earlier photometric evidence ( e.g., Rix & White 1990 and references therein), now put this idea on solid observational grounds. Perhaps these results imply that at smaller mass scales the epoch of mergers ended before the epoch of star-formation.
HWR is supported by the Alfred P. Sloan Foundation. CMC is supported by NASA through the grant HF-1079.01-96a awarded by the Space Telescope Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA under contract NAS 5-26555.
Bender, R., Surma, P., Doebereiner, S., Moellenhoff, C., Madejsky, R., 1989, A&A, 217, 35 Barnes, J. 1996 in “Formation of the Galactic Halo ... Inside Out", eds. H. Morrison and A. Sarajedini, ASP Conference Series, p. 415. Barnes, J. 1998 in “Galaxies: Interactions and Induced Star Formation", Kennicutt, R., Schweizer, F and Barnes, J., Saas Fe Advanced Course 26, Springer, §7. Bekki, K. 1998, ApJ, 502, L133. Binney, J.J., Davies, R.L., Illingworth, G.D., 1990, ApJ, 361, 78 Carollo, C.M., Danziger, I.J., 1994, MNRAS, 270, 743 de Zeeuw, P. T. and Franx, M., 1991, ARA&A, 29, 239 Davies, R.L., Efstathiou, G., Fall, S.M., Illingworth, G., Schechter, P., 1983, ApJ, 266, 41 de Vaucouleurs, G., de Vaucouleurs A., Corwin H. G. Jr., Buta R. J., Paturel G., Fouqu[é]{} P., 1991, Third Reference Catalog of Bright Galaxies, Springer (RC3) Fisher, D., 1997, AJ, 113, 950 Heyl, J.S., Hernquist, L., Spergel, D.N., 1996, ApJ, 463, 69 Jaffe, W., 1983, MNRAS, 202, 995 Kormendy, J. and Djorgovski, G., 1989, ARA&A, 27, 235. Rix, H-W., White, S.D.M., 1990, ApJ, 362, 52 Rix, H-W., White, S.D.M., 1992, MNRAS, 254, 389 Scorza, C., Bender, R., 1995, A&A, 293, 20 Tremblay, B., Merritt, D., 1996, AJ, 111, 2243 Weil, M.L., Hernquist, L., 1996, ApJ, 460, 101
=
=
--------- ------ ----------- -------------- ------- ---------------- ------------ ------- --------------- ------- -------------------- -------------------- -----------------------------
Name $T$ V$_{hel}$ $D$ $M_B$ $R_e$ $\epsilon$ Run T$_{maj/min}$ $PA$ $(V/\sigma)_{out}$ $(V/\sigma)_{max}$ $ \langle{\lambda'}\rangle$
(km/s) (Mpc) (mag) (arcs) (secs) (deg)
NGC386 -5 5707 76 -19.2 11.8$\ddagger$ 0.13 Sep97 18000/8100 5 1.0 1.0 0.47 (0.43)
NGC1588 -4.6 3397 19$^\dagger$ -17.7 12.8 0.30 Sep97 10800/3400 35 1.05 1.4 0.58(0.54)
NGC1603 -5? 5038 67 -19.5 12.2 0.30 Sep97 18000/2100 50 1.5 1.5 0.50 (0.46)
NGC2592 -5 1925 26 -18.9 10.5$\ddagger$ 0.17 Feb97 5400/5400 53 3.0 3.0 0.65 (0.62)
NGC2699 -5 1660 22 -18.1 9.0$\ddagger$ 0.07 Feb97 5400/5400 45 2.1 2.1 0.63 (0.60)
NGC2778 -5 1991 27 -18.8 15.7 0.28 Feb97 5400/3600 40 2.1 2.4 0.54 (0.50)
NGC3605 -5 581 14$^\dagger$ -17.6 21.2 0.33 Feb97 4400/5400 22 1.8 1.8 0.47 (0.43)
NGC2577 -3 2068 28 -19.0 24.3$\ddagger$ 0.38 Feb97 5400/3600 105 3.0 3.8 0.71 (0.68)
NGC3156 -2 980 15$^\dagger$ -17.9 14.4 0.42 Feb97 5400/5400 47 2.4 3.0 0.60 (0.57)
NGC7617 -2 4176 56 -19.1 10.6 0.35 Sep97 18000/12560 30 1.5 1.5 0.62 (0.58)
--------- ------ ----------- -------------- ------- ---------------- ------------ ------- --------------- ------- -------------------- -------------------- -----------------------------
0.5truein
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
The web of collaborations between individuals and group of researchers continuously grows thanks to online platforms, where people can share their codes, calculations, data and results. These virtual research platforms are innovative, mostly browser-based, community-oriented and flexible. They provide a secure working environment required by modern scientific approaches. There is a wide range of open source and commercial solutions in this field and each of them emphasizes the relevant aspects of such a platform differently.
In this paper we present our open source and modular platform, [`KOOPLEX`]{}[^1], that combines such key concepts as dynamic collaboration, customizable research environment, data sharing, access to datahubs, reproducible research and reporting. It is easily deployable and scalable to serve more users or access large computational resources.
address:
- 'Department of Physics of Complex Systems, E[ö]{}tv[ö]{}s Lor[á]{}nd University, H-1117, Pázmány Péter sétány 1/a. Budapest, Hungary'
- 'Department of Information Systems, E[ö]{}tv[ö]{}s Lor[á]{}nd University, H-1117, Pázmány Péter sétány 1/c. Budapest, Hungary'
- 'Department of Computational Sciences, Wigner Research Centre for Physics of the HAS, Konkoly-Thege Miklós út 29-33., Budapest 1121, Hungary'
author:
- 'D. Visontai'
- 'J. Stéger'
- 'J. M. Szalai$-$Gindl'
- 'L. Dobos'
- 'L. Oroszl[á]{}ny'
- 'I. Csabai'
bibliography:
- 'ref.bib'
title: 'Kooplex: collaborative data analytics portal for advancing sciences'
---
kooplex ,collaboration ,platform ,jupyter ,notebook ,scalable ,open source ,data science ,reporting ,rstudio ,business intelligence ,gitea ,seafile ,kubernetes ,docker
Summary
=======
[`KOOPLEX`]{} is a platform for easy access to datahubs, for collaborative work, for developing new workflows and for creating and publishing static or interactive reports. It is clear from the user feedback regarding the platform instances mentioned in **Section \[sec:kooplex-usecase\].** that the combination of such integrated services is attractive. Accessing the various modules in the same user space speeds up analysis, code development work and sharing, not having to move data and files around between disconnected components. The platform has been designed in a way that the integration of new tools taken up by the research community is straight forward. This feature helps to keep up with the ever evolving user requirements.
Acknowledgements {#acknowledgements .unnumbered}
================
This study has received funding from the European Union’s Horizon 2020 research and innovation program under Grant Agreement No. 643476 (COMPARE) and from National Research, Development and Innovation Fund of Hungary Project (FIEK\_16-1-2016-0005 to I.C.). The authors are grateful to G. Vattay, S. Laki and students at Eötvös University for help with thorough testing of the system. This work was completed in the ELTE Excellence Program (783-3/2018/FEKUTSRAT) supported by the Hungarian Ministry of Human Capacities. The work was supported by the Hungarian National Research, Development and Innovation Office (NKFIH) through Grants No. K120660, K109577, K124351, K124152, KH129601, and the Hungarian Quantum Technology National Excellence Program (Project No. 2017-1.2.1-NKP-2017- 00001).
Conflict of interest {#conflict-of-interest .unnumbered}
====================
The authors declare that there is no conflict of interest regarding the publication of this manuscript.
[^1]: https://kooplex.github.io
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'It is demonstrated that the entropy of statistical mechanics and of information theory, $S({\bf p}) = -\sum p_i \log p_i $ may be viewed as a measure of correlation. Given a probability distribution on two discrete variables, $p_{ij}$, we define the correlation-destroying transformation $C: p_{ij} \rightarrow \pi_{ij}$, which creates a new distribution on those same variables in which no correlation exists between the variables, i.e. $\pi_{ij} = P_i Q_j$. It is then shown that the entropy obeys the relation $S({\bf p}) \leq S({\bf \pi}) = S({\bf P}) + S({\bf Q})$, i.e. the entropy is non-decreasing under these correlation-destroying transformations.'
author:
- |
John H. Van Drie\
www.johnvandrie.com\
Kalamazoo, MI 49008 USA
title: |
The Boltzmann/Shannon entropy\
as a measure of correlation
---
The concept of correlation has underlain statistical mechanics from its inception. Maxwell derived his velocity distribution law [^1] by asserting that such a distribution $\Phi(\vec{v})$ for an ideal gas should obey two properties: (1) the velocity distribution along each axis should be uncorrelated, i.e. $$\Phi(\vec{v}) d^3\vec{v} = (\phi(v_x) dv_x) (\phi(v_y) dv_y) (\phi(v_z) dv_z)$$ and (2) the velocity distribution should show no preferred orientation, $\Phi(\vec{v}) d^3\vec{v} = f(v) d^3\vec{v}$, where v is the norm of $\vec{v}$. He showed these two assumptions led to the velocity distribution $\Phi(\vec{v}) d^3\vec{v} = \exp (-\alpha v^2) d^3\vec{v}$, where $\alpha$ is a positive constant (later shown by Boltzmann to be equal to $\frac{m}{2kT}$). (This reduces to the more familiar form by writing this expression in polar coordinates, $\Phi(v_r,v_{\theta},v_{\phi}) d^3\vec{v}= \exp (-\alpha v_r^2) v_r^2 dv_r dv_{\theta} dv_{\phi}$).
Inspired by the observations of Ochs [^2], we would like to show that the Boltzmann/Shannon formula for entropy may be viewed as a measure of correlation, by showing that for a class of transformations which destroys correlations between variables in a probability distribution, the Boltzmann/Shannon formula for the entropy is non-decreasing.
Suppose that we have a set of states, $\{ X_{ij} \}$, indexed by their values along two distinct, discrete variables A and B, where i runs over the set of n discrete states of A, and j runs over the m set of discrete states of B. Consider a probability distribution over these states, $p_{ij}$. If A and B are uncorrelated, there exist some $P_i$ and $Q_j$ such that $p_{ij} = P_i Q_j \forall i,j$. In this case, it is apparent that the entropy obeys the property S(p) = S(P) + S(Q).
However, in general, this will not be true. But, given an arbitrary $p_{ij}$, we can define the following transformation, which in effect destroys the correlations between its dependence on A and on B: $$\begin{aligned}
C: & p_{ij} \rightarrow \pi_{ij} \\
\pi_{ij} & = & P_i Q_j \\
P_i & = & \sum_{j=1}^m p_{ij} \\
Q_j & = & \sum_{i=1}^n p_{ij}\end{aligned}$$
We assert that the entropy is non-decreasing under such transformations, i.e. $$S(p) \leq S(\pi) \label{eq:biggie}$$
To demonstrate this assertion, we need first to demonstrate a fundamental property of the Boltzmann/Shannon entropy formula, [*the averaging property*]{}. Given a set of states $\{Y \}_{k=1}^N$, and two probability distributions defined over these states, $G = \{g_k\}_{k=1}^N$ and $H = \{h_k\}_{k=1}^N$, one may construct a third distribution, $U = \{u_k\}, u_k = \frac{1}{\alpha + \beta}(\alpha g_k + \beta h_k)$, the weighted average of G and H, where $\alpha$ and $\beta$ are arbitrary real values. We assert that $$S(U) \geq \frac{1}{\alpha + \beta} ( \alpha S(G) + \beta S(H) )$$ This assertion can be demonstrated by observing that the similar inequality holds term-by-term in the sum. Defining $\sigma(x) = - x \ln x$, the averaging property will hold if $$\sigma(u_k) \geq \frac{1}{\alpha + \beta} ( \alpha \sigma(g_k) + \beta \sigma(h_k) )$$ This property of $\sigma(x)$ follows from it being concave everywhere over the domain of interest, $x \in [0,1]$, i.e. $\sigma''(x) < 0$.
Note, too, that a consequence of the averaging property of entropy is that, given a set of different distributions over $\{Y \}_{k=1}^K$, $Z^1, Z^2, Z^3, \dots Z^K$, and a set of weights $w_i, \sum_i w_i = 1$, that the entropy of $\bar{Z} = \sum_k w_k Z^k$, the weighted average of all these distributions, obeys the following property: $$S(\bar{Z}) \geq \sum_k w_k S(Z^k)$$ Returning to the fundamental assertion, equation \[eq:biggie\], this may be demonstrated by recalling a fundamental property of the Boltzmann/Shannon entropy formula, one which Shannon [^3] took not as a derived property but rather an axiomatic property that an entropy functional must have: If we decompose the distribution $p_{ij}$ into a two stages, where initially we distribute among the states over A by the distribution $P_i$, and next we distribute among the states of B by the distribution $\zeta^{(i)}_j$, such that $p_{ij} = P_i \zeta^{(i)}_j$, the entropy obeys the formula $$S(p) = S(P) + \sum_i P_i S(\zeta^{(i)})$$ In the general case, each of the distributions $\zeta^{(i)}$ will be different, i.e. the variables A and B are correlated. The distribution $Q_j$ above represents a weighted average of the $\zeta^{(i)}$’s, weighted by $P_i$, i.e. $$Q_j = \sum_k P_k \zeta^{(k)}_j$$ Hence, the Shannon axiom with the averaging property of the entropy leads to the desired assertion: $$\begin{aligned}
S(p) & = & S(P) + \sum_k P_k S(\zeta^{(k)}) \\
S(p) & \leq & S(P) + S(Q) \\
S(p) & \leq & S(\pi) \end{aligned}$$
Jaynes [^4] showed how Shannon’s theory could be merged with statistical mechanics, leading to the conceptualization of the thermodynamic principle of maximum entropy as a principle expressing that the distribution of energy among the microstates should be that distribution which is least-biased, given the constraint of a specified temperature.
Viewing entropy $-k \sum_k p_i \ln p_i$ as a measure of (lack of) correlation provides a new twist to Jaynes’ perspective. One may say that the equilibrium distribution is that distribution which is least-correlated given the constraint(s). The Second Law of Thermodynamics may be rephrased to state that correlations are highly unlikely to arise spontaneously, and that the natural course of evolution of a system is one in which correlations diminish.
Thinking about entropy as a measure of correlation leads to a key implicit assumption in both Boltzmann’s theory and Shannon’s theory: the individual (micro)states are assumed to be uncorrelated. This hails back to Laplace’s balls-in-urns, where the probability of finding a ball in a given urn is generally uncorrelated with the probability of finding a ball in a different urn. If the probability of occupation of the states are intrinsically correlated, the maximum entropy distribution cannot be viewed as the correct, least-biased distribution. In communication theory and statistical mechanics, this assumption may in certain circumstances be valid, but where this assumption breaks down severely is the case when we attempt to take the limit to a continuous set of states. If one constructs a discrete set of bins from an intrinsically continuous variable, the degree of bin-bin correlation grows as these bins become steadily finer. This leads to the question ’what is the correct measure of entropy for a continuous distribution?’.
Dill [^5] points out that these issues of correlation and additivity pervade our thinking about the fundamental aspects of chemical and biological phenomena. He highlights some of the pitfalls which one may encounter in settings for which an assumption of non-correlation may not be valid.
[^1]: Maxwell, J.C., [*Phil. Soc.*]{}, 1860
[^2]: Ochs, W., [*Rep. Math. Phys.*]{}, [**9**]{}, 135 (1976)
[^3]: C. Shannon and W. Weaver, [*The Mathematical Theory of Communication*]{},Urbana: Univ. of Ill. Press, 1949
[^4]: Jaynes, [*Phys. Rev.*]{}, [**106**]{}, 620 (1957)
[^5]: Dill, K.A.,[*J. Biol.Chem.*]{}, [**272**]{}, 701 (1997)
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study a real, massive Klein-Gordon field in the Poincaré fundamental domain of the $(d+1)$-dimensional anti-de Sitter (AdS) spacetime, subject to a particular choice of *dynamical* boundary conditions of generalized Wentzell type, whereby the boundary data solves a non-homogeneous, boundary Klein-Gordon equation, with the source term fixed by the normal derivative of the scalar field at the boundary. This naturally defines a field in the conformal boundary of the Poincaré fundamental domain of AdS. We completely solve the equations for the bulk and boundary fields and investigate the existence of bound state solutions, motivated by the analogous problem with Robin boundary conditions, which are recovered as a limiting case. Finally, we argue that both Robin and generalized Wentzell boundary conditions are distinguished in the sense that they are invariant under the action of the isometry group of the AdS conformal boundary, a condition which ensures in addition that the total flux of energy across the boundary vanishes.'
author:
- Claudio Dappiaggi
- 'Hugo R. C. Ferreira'
- 'Benito A. Juárez-Aubry'
title: |
Mode solutions for a Klein-Gordon field in anti-de Sitter spacetime\
with dynamical boundary conditions of Wentzell type
---
Introduction
============
Classical and quantum field theory on asymptotically anti-de Sitter (AdS) spacetimes, and generally other spacetimes with boundaries, has been the target of significant attention in the last two decades, mainly inspired by the remarkable AdS/CFT correspondence [@Maldacena:1997re; @Witten:1998qj], see [@Ammon:2015] for a modern overview. The importance of this correspondence has gone well beyond its initial connection with the quantum gravity formulation in string theory and has become relevant in many low energy physics applications, ranging from nuclear to condensed matter physics [@Hartnoll:2009sz].
From a geometric standpoint, in contrast with their asymptotically flat or asymptotically de Sitter counterparts, asymptotically AdS spacetimes are not globally hyperbolic; the conformal asymptotic boundary at infinity is timelike. As a consequence, on an asymptotically AdS background, one cannot expect to find global solutions for hyperbolic equations, such as the Klein-Gordon one, only by assigning suitable initial data. These must be supplemented with appropriate boundary conditions imposed at the conformal boundary.
In previous work [@Dappiaggi:2016fwc; @Dappiaggi:2017wvj], two of the authors analyzed the classical and quantum field theory of a massive scalar field propagating in anti-de Sitter (AdS) spacetime in $d+1$ spacetime dimensions, subject to homogeneous Robin boundary conditions, which include the familiar Dirichlet and Neumann boundary conditions as particular cases, see also [@Bussola:2017wki] for an analysis on BTZ spacetime. In that work, by means of a Fourier transform, the Klein-Gordon equation has been reduced to a Sturm-Liouville problem, which naturally provides all the admissible boundary conditions of Robin type for a specific range of the mass parameter of the field. In this context, studying all admissible Robin boundary conditions at once is a good strategy for finding the parameter space in mass and curvature coupling for which there exist bound state solutions, which decay exponentially away from the AdS boundary. These modes not only lead to instabilities in the classical linear theory but also pose an obstruction to the existence of a ground state for the underlying quantum theory.
These results call for two natural generalizations, the first by allowing the background to be a generic asymptotically AdS spacetimes, the second by fixing more general boundary conditions. In particular, in the AdS case, the second avenue implies in particular that one should treat boundary value problems outside of the realm of Sturm-Liouville theory.
From a structural standpoint, since there exist infinite choices of boundary conditions, the first step consists of identifying a natural subclass which is distinguished for its physical properties. To this end, it seems that one bit of information that could be used is the existence of a large group of isometries at the conformal boundary. Heuristically one expects that boundary conditions should be chosen in such a way to be compatible with the action of such group. The problem of translating such idea in a concrete mathematical tool can be addressed in the specific case of an AdS spacetime by adapting and reinterpreting the recent results of [@Ibort:2014sua; @Ibort2; @Pardo]. In particular one can realize that each boundary condition is nothing but an operator acting between (a suitable generalization of) the field restricted to the boundary and its normal derivative. From this perspective it is natural to require that such operator commutes with the scalar representation of the isometry group of the conformal boundary. By using this paradigm one restricts considerably the class of possible boundary conditions, while, at the same time, making clear that it is possible to go beyond those of Robin type.
In this paper, we study a massive scalar field in AdS in $d+1$ spacetime dimensions, AdS$_{d+1}$, subject to [*generalized Wentzell boundary conditions*]{} (WBCs). Specifically, we focus on the so-called Poincaré fundamental domain PAdS$_{d+1}$, which covers only a portion of the full AdS$_{d+1}$ and it has the advantage of being conformal to the half-Minkowski spacetime in $d+1$ dimensions, $\bHo^{d+1}$. As we shall see below, these boundary conditions have boundary data determined by a non-homogeneous, boundary Klein-Gordon equation, with the source term fixed by the normal derivative of the scalar field at the boundary. This naturally defines a field in the conformal boundary of the Poincaré fundamental domain of AdS, a feature which is clearly reminiscent of the AdS/CFT framework, though here we limit ourselves to considering non interacting models.
In our investigation, although we limit ourselves to considering classical features of the underlying model, the existence of bound states in particular, our ultimate goal is to provide a full-fledged quantum system. This is in particular one of the key reasons why we shall not only focus on finding smooth solutions for the underlying dynamics, but we will also be interested in the square-integrability of the relevant functions.
As we show in this paper, the WBCs are *distinguished* in the sense that they are dynamical boundary conditions which are invariant under the action of the isometry group of the AdS boundary. Furthermore, as is the case with the simpler, nondynamical Robin boundary conditions, the total fluxes of symplectic and energy currents across the AdS conformal boundary vanish, as required for a closed system. Moreover, the Robin boundary condition eigenstate solutions to the Klein-Gordon operator can be recovered as suitable limits of the WBC ones.
The treatement of WBC boundary conditions in the classical and quantum field theoretic literature has appeared in the work of one of us, together with Barbero, Margalef-Bentabol and Villaseñor, [@G.:2015yxa; @Barbero:2017kvi], where the simple $(1+1)$-dimensional mechanical model of a finite string with point masses subject to harmonic potentials in the extrema has been studied in detail. In that work, the classical system is solved and the Fock quantization is performed, with the ultimate goal of constructing [*boundary Hilbert spaces*]{} where the dynamics of the extremal masses takes place, with the aid of the PDE [*Lions trace*]{} operators. It is further shown that the quantum mechanical dynamics in the boundary Hilbert spaces is non-unitary.
These boundary conditions have also been considered in [@Zahn:2015due] in $(d+1)$-dimensional Minkowski spacetime with one or two timelike boundaries. There, it was investigated the Fock space quantization of the underlying system and, in addition it was shown that the WBC ensure that the short-distance singularities of the two-point function for the boundary field has the form expected of a field living in a $d$-dimensional spacetime, contrary to other boundary conditions, for which the two-point function inherits the short-singularity of the $(d+1)$-dimensional bulk. This seems to be a very desired feature for holographic purposes. Previous explorations of the WBC in mathematical literature appear in *e.g.* [@Ueno:1973; @Favini:2002; @Coclite:2014].
While the main inspiration of this work comes from high energy physics, namely the AdS/CFT conjecture, and the connection to holographic renormalization [@Skenderis:2002wp] is a central motivation for us, we note that the boundary conditions that we consider, as well as related dynamical boundary conditions, are suitable for studying systems in a broad sprectrum of physical problems. We point out that dynamical boundary conditions are generally interesting from the point of view of modelling open systems in condensed matter physics. They are also relevant for lower-dimensional Chern-Simons theories coupled to electrodynamics, which can model e.g. effective topological insulators [@Martin-Ruiz:2015skg]. From a gravitational perspective, dynamical boundary conditions are interesting in the study of isolated horizons, providing an avenue for associating degrees of freedom at a horizon surface. This is attractive from a quantum gravity perspective. In loop quantum gravity, for example, a procedure for counting horizon degrees of freedom yields the Bekenstein-Hawking entropy [@Ashtekar:1997yu; @Ashtekar:1999wa].
The contents of this paper are as follows. In Sec. \[sec:AdS\] we review the basic geometric properties of AdS$_{d+1}$ and the closely related PAdS$_{d+1}$, the Poincaré fundamental domain of AdS$_{d+1}$ as a spacetime in its own right. In addition we show how the Klein-Gordon equation on PAdS$_{d+1}$ can be treated as a (generally singular) Klein-Gordon equation on half-Minkowski. In Sec. \[sec:WBCs\] we introduce WBCs. We motivate them in the form of an action principle for the Klein-Gordon field with boundary dynamical contributions in half-Minkowski spacetime. We then deal with the problem in PAdS$_{d+1}$ with the aid of the aforementioned conformal techniques, by solving both the bulk and boundary field equations in full generality. We consider separately two cases: in Sec. \[sec:regular\] the regular case, corresponding to the massless, conformally coupled (conformally transformed) scalar field (in half-Minkowski), and in Sec. \[sec:singular\] the singular case, corresponding to the general, massive scalar field. Additionally, in both cases, we investigate the existence of bound state mode solutions, which decay exponentially away from the boundary. Finally, in Sec. \[sec:maths\], we explicitly show the vanishing of the fluxes of symplectic and energy currents across the AdS boundary when WBCs are imposed, and explain how these boundary conditions are invariant under the boundary isometry group, making them distinguished. Throughout the paper we employ natural units in which $c = G_{\rm N} = 1$ and a metric with signature $({-}{+}{+}\cdots)$.
Anti-de Sitter spacetime and Klein-Gordon field {#sec:AdS}
===============================================
In this section, we briefly review the basic geometric properties of anti-de Sitter spacetime AdS$_{d+1}$ and introduce the Poincaré fundamental domain PAdS$_{d+1}$, on which we consider a classical scalar field satisfying the Klein-Gordon equation.
Geometry of anti-de Sitter spacetime
------------------------------------
The maximally symmmetric solution of the Einstein field equations with negative cosmological constant, $\Lambda$, is the anti-de Sitter spacetime, which we denote by AdS$_{d+1}$ in $d+1$ Lorentzian dimensions. As a manifold it is diffeomorphic to $\mathbb{S}^1 \times \mathbb{R}^d$ and it can be realized as an embedded submanifold in the ambient space $\mathbb{R}^{d+2}$ equipped with the metric $$g_{\mathbb{R}^{d+2}} = - \dd X_0^2 - \dd X_1^2 + \sum_{i,j = 2}^{d+1} \delta^{ij} \, \dd X_i \dd X_j \ ,
\label{Rd2metric}$$ via the equation $$-X_0^2 - X_1^2 + \sum_{i = 2}^{d+1} X_i^2 = \frac{d(d-1)}{\Lambda}\,$$ where each $X_i$, $i=0,...,d+2$ is a Cartesian coordinate, while $\delta^{ij}$ stands for the Kronecker delta. As a consequence AdS$_{d+1}$ comes equipped with the induced (Lorentzian, non-degenerate) metric.
We can give an explicit representation of these geometric structures by considering the [*Poincaré fundamental domain*]{} of AdS$_{d+1}$, denoted by PAdS$_{d+1}$. This is covered by the [*Poincaré coordinate patch*]{}, with $t \in \mathbb{R}$, $z \in \mathbb{R}^+$ and $x_i \in \mathbb{R}$, defined by $$\begin{aligned}
\left\{
\begin{array}{ll}
X_0 = \cfrac{\ell}{z} t, \\
\vspace*{-0.2cm} \\
X_1 = \cfrac{z}{2} \left(1 + \cfrac{1}{z^2} \left(- t^2 + \displaystyle\sum_{i = 1}^{d-1} x_i^2 + \ell^2 \right) \right), \\
\vspace*{-0.2cm} \\
X_i = \cfrac{\ell}{z} x_{i-1}, \quad i = 2, \ldots, d,\\
\vspace*{-0.2cm} \\
X_{d+1} = \cfrac{z}{2} \left(1 + \cfrac{1}{z^2} \left(- t^2 + \displaystyle\sum_{i = 1}^{d-1} x_i^2 - \ell^2 \right) \right),
\end{array}
\right.
\label{PAdScoods}\end{aligned}$$ where $\ell^2 = -d(d-1)/ \Lambda$. Thus, PAdS$_{d+1}$ is a Lorentzian spacetime with underlying manifold $\mathbb{R}^+ \times \mathbb{R}^d$ equipped with the metric $$g_{{\rm PAdS}_{d+1}} = \frac{\ell^2}{z^2} \left( - \dd t^2 + \dd z^2 + \sum_{i, j = 1}^{d-1} \delta^{ij} \dd x_i \dd x_j \right).
\label{gPAdS}$$ Eq. shows that (PAdS$_{d+1},g_{{\rm PAdS}_{d+1}}$) is conformal to the interior of the $(d+1)$-dimensional half-Minkowski spacetime, $(\bHo^{d+1},\eta_{d+1})$, with $\eta_{d+1} = \Omega^2 g_{{\rm PAdS}_{d+1}}$, where the conformal factor is $\Omega = z/\ell$. The conformal boundary of PAdS$_{d+1}$ can be attached at $z = 0$.
The Klein-Gordon field in PAdS$_{d+1}$
--------------------------------------
In this work, we consider a classical, real Klein-Gordon field $\phi: {\rm PAdS}_{d+1} \to \mathbb{R}$. Given initial data on a hypersurface of PAdS$_{d+1}$ for the Klein-Gordon wave equation, $$P \phi = \left( \Box_g^{(d+1)} - m_0^2 - \xi R \right) \phi = 0 \ ,
\label{KGPAdS}$$ where $\Box_g^{(d+1)}$ is the d’Alembert wave operator, $m_0$ is the mass, $\xi \in \mathbb{R}$ is the coupling to the scalar curvature, while $R = -d(d+1) / \ell^2$ is the Ricci scalar of the spacetime. From now on, we set $\ell = 1$. Although, for initial data, which are smooth and compactly supported in PAdS$_{d+1}$, a unique solution exists in its domain of dependence, in order to address the problem of global existence, one needs to impose boundary conditions at timelike infinity, which, in the Poincaré patch, corresponds to the conformal boundary.
In order to control such freedom, we follow the same strategy adopted in [@Dappiaggi:2016fwc] to switch from Eq. to the associated, conformally-transformed scalar field equation in $\mathbb{\mathring{\mathbb H}}^{d+1}$. Hence, defining $\Phi = \Omega^{\frac{1-d}{2}}\phi :\bHo^{d+1} \to \mathbb{R}$, the solutions of are in one to one correspondence with those of $$P_\eta \Phi \doteq \left( \Box_\eta^{(d+1)} - \frac{m^2}{z^2} \right) \Phi = 0 \ ,
\label{KGH}$$ where we have defined $m^2 \doteq m_0^2 + (\xi -\frac{d-1}{4d}) R$. A strategy for dealing with Eq. in the case of a minimally-coupled, real scalar field in $d = 3$ that does not rely on conformal transformations has appeared in [@Ayon-Beato:2018hxz]. Here, we consider problems for which $m^2 \geq -\frac{1}{4}$, which corresponds to the Breitenlohner-Freedman (BF) bound [@Breitenlohner:1982jf].
The boundary condition imposed at the conformal boundary of PAdS$_{d+1}$ that allows one to obtain global solutions for $\phi$ corresponds in to a boundary condition at $z = 0$, where the potential term becomes singular. In [@Dappiaggi:2016fwc; @Dappiaggi:2017wvj] all possible homogeneous boundary conditions of Robin type have been analysed. It is the approach of this paper to consider more general, dynamical boundary conditions, which reduce to those of Robin type in a precise limiting sense. In particular, the boundary conditions that we consider are of [*generalized Wentzell type*]{}. As mentioned above, these have been considered in the mathematical physics literature in [@Zahn:2015due] for regular problems in the half-Minkowski spacetime, and also studied by one of the authors in [@G.:2015yxa; @Barbero:2017kvi] in the context of the quantization of a finite string coupled to point masses subject to harmonic restoring forces at the boundaries.
Wentzell boundary conditions {#sec:WBCs}
============================
In this section, we study the problem defined by Eq. in the bulk and Wentzell boundary conditions at $z = 0$. In Section \[sec:action\] we show that these boundary conditions can be obtained naturally starting from an action functional in the so-called regular case ($m^2 = 0$ in ). We study such case via a mode expansion in Section \[sec:regular\], finding the conditions under which there exists, together with the expected continuous spectrum, point spectrum contributions to the solutions. These indicate the existence of [*bound state mode solutions*]{} to the problem. Afterwards we show how to recover the solutions to the problem with Robin boundary conditions, obtaining full agreement with the results reported in [@Dappiaggi:2016fwc; @Dappiaggi:2017wvj]. In Section \[sec:singular\] we repeat the analysis for the singular problem ($m^2 \in [-\frac{1}{4}, \frac{3}{4}) \setminus \{0\}$), also characterizing the spectrum and obtaining the Robin boundary problem solutions in a suitable limit in agreement with [@Dappiaggi:2016fwc; @Dappiaggi:2017wvj].
Action {#sec:action}
-------
Let us motivate the introduction of the generalized Wentzell boundary conditions, by considering the usual action for a massless Klein-Gordon field in $\bHo^{d+1}$ together with a particular choice of boundary terms: $$\begin{aligned}
S & = - \frac{1}{2} \int_{t_1}^{t_2} \!\!\!\! \dd t \int_{\mathbb{R}^+} \!\!\!\! \dd z \int_{\mathbb{R}^{d-1}} \!\!\!\!\!\!\!\! \dd^{d-1} x \ \partial_\mu \Phi \partial^\mu \Phi \nonumber \\
&\quad +\frac{c}{2} \int_{t_1}^{t_2} \!\!\!\! \dd t \int_{\mathbb{R}^{d-1}} \!\!\!\!\!\!\!\! \dd^{d-1} x \left(-\dot{\Phi}^2 + \partial_i \Phi \partial^i \Phi + m^2_{\rm b} \Phi^2 \right) \, ,\end{aligned}$$ where $c$ and $m_{\rm b}^2$ are arbitrary real parameters at this stage and where repeated indexes are summed over with $\mu \in \{t, z, x_1, \ldots, x_{d-1}\}$ and $i \in \{x_1, \ldots, x_{d-1}\}$, and $\dd^{d-1} x = \prod_{i = 1}^{d-1} \dd x_i$. With a slight abuse of notation, we use the symbol $\Phi$ in the boundary integrand, in place of its restriction thereon. Implicitly we are also restricting our attention to kinematic configurations which are sufficiently regular at $z=0$, to make these operations meaningful.
The variation of the action yields $$\begin{aligned}
& dS(\Phi) \cdot \delta = \left. \frac{d}{d \lambda} S(\Phi + \lambda \delta) \right|_{\lambda = 0} \nonumber \\
& = - \int_{t_1}^{t_2} \!\!\!\! \dd t \int_{\mathbb{R}^+} \!\!\!\! \dd z \int_{\mathbb{R}^{d-1}} \!\!\!\!\!\!\!\! \dd^{d-1} x \left( -\dot{\Phi} \dot{\delta} + \partial_z \Phi \partial_z\delta + \partial_i \Phi \partial^i \delta \right) \nonumber \\
&\quad +c \int_{t_1}^{t_2} \!\!\!\! \dd t \int_{\mathbb{R}^{d-1}} \!\!\!\!\!\!\!\! \dd^{d-1} x \left(-\dot{\Phi} \dot{\delta} + \partial_i \Phi \partial^i \delta + m^2_{\rm b} \Phi \delta \right) \nonumber \\
& = \int_{t_1}^{t_2} \!\!\!\! \dd t \int_{\mathbb{R}^+} \!\!\!\! \dd z \int_{\mathbb{R}^{d-1}} \!\!\!\!\!\!\!\! \dd^{d-1} x \, \delta \, \Box_\eta^{(d+1)} \Phi \nonumber \\
&\quad -c \int_{t_1}^{t_2} \!\!\!\! \dd t \int_{\mathbb{R}^{d-1}} \!\!\!\!\!\!\!\! \dd^{d-1} x \, \delta \left(\Box_\eta^{(d+1)} \Phi - m^2_{\rm b} \Phi + \frac{1}{c} \partial_z \Phi \right) \, ,\end{aligned}$$ where, on the right hand side of the last equality above, the integration by parts in $z$ in the bulk action has contributed to the boundary term.
The extrema of the action, $dS(\Phi) = 0$, are $$\label{variation}
\begin{cases}
\Box_{\eta}^{(d+1)} \Phi = 0 \quad \text{in $\bHo^{d+1}$} \, , \\
\left(\Box_{\eta}^{(d)} - m^2_{\rm b} \right) F = - \dfrac{\rho}{c} \quad \text{in $\bR^d$} \, , \\
\Phi|_{z=0} = F \, , \quad \partial_z \Phi|_{z=0} = \rho \, .
\end{cases}$$ The boundary conditions of the problem are known as *generalized Wentzell boundary conditions* (WBCs), see [@Zahn:2015due] and references therein. These are dynamical boundary conditions, for which there is a boundary field $F$ required to coincide with the restriction of the bulk field at the boundary and to satisfy a Klein-Gordon equation with a source term, which is related to the derivative of the bulk field with respect to the direction orthogonal to the boundary. In the case in which the bulk field is massive, and the field equation is singular at the boundary, the explicit form of these boundary conditions need to be generalized, as the bulk field or its derivative may not be defined at the boundary. We discuss such generalization in Section \[sec:singular\].
Regular case {#sec:regular}
------------
A Klein-Gordon field, $\phi$, in PAdS$_{d+1}$, satisfying eq. with $m_0^2 = -(\xi - \frac{d-1}{4d})R$ can be mapped to the problem defined by Eq. with $m^2 = 0$. This defines, together with appropriate boundary conditions, a regular problem in $\bHo^{d+1}$. We choose the above-introduced WBCs, $$\label{eq:regsystem}
\begin{cases}
\Box_{\eta}^{(d+1)} \Phi = 0 \quad \text{in $\bHo^{d+1}$} \, , \\
\left(\Box_{\eta}^{(d)} - m^2_{\rm b} \right) F = - \dfrac{\rho}{c} \quad \text{in $\bR^d$} \, , \\
\Phi|_{z=0} = F \, , \quad \partial_z \Phi|_{z=0} = \rho \, .
\end{cases}$$ Here, the parameter $c$ is taken to be real and we restrict $m^2_{\rm b} \geq 0$, so that $m^2_{\rm b}$ is interpreted as a [*squared boundary field mass*]{} for the [*boundary field*]{} $F$. We further assume that the Fourier transforms for $\Phi$, $F$ and $\rho$ exist. It suffices for our purposes to consider these functions to identify tempered distributions.
### Bulk and boundary solutions
For the bulk field, we take the Fourier transform along the directions orthogonal to $z$, $$\label{eq:Fouriertransf}
\Phi(\underline{x},z) = \int_{\bR^d} \frac{\dd^d\underline{k}}{(2\pi)^{\frac{d}{2}}} \, e^{i\underline{k}\cdot \underline{x}} \, \widehat{\Phi}(\underline{k},z) \, ,$$ where $\underline{x} \doteq (t, x_1, \ldots, x_{d-1})$, $\underline{k} \doteq (\omega, k_1, \ldots, k_{d-1})$ and $\widehat{\Phi}$ are solutions of $$\label{eq:regmodeeq}
- \frac{\dd^2}{\dd z^2} \widehat{\Phi}(\underline{k},z) = q^2 \, \widehat{\Phi}(\underline{k},z) \, , \quad
q^2 \doteq \omega^2 - \displaystyle\sum_{i=1}^{d-1} k_i^2 \, .$$ We note that the differential operator in the LHS of is of Sturm-Liouville type [@Zettl:2005]. Therefore we will work in this framework, whose first step calls for identifying those $\widehat{\Phi}(\underline{k},z)$ which are necessary to construct the fundamental solution associated to .
For the boundary field and source term, we also take the Fourier transform along all directions,
\[eq:FourierFrho\] $$\begin{aligned}
F(\underline{x}) &= \int_{\bR^d} \frac{\dd^d\underline{k}}{(2\pi)^{\frac{d}{2}}} \, e^{i\underline{k}\cdot \underline{x}} \, \widehat{F}(\underline{k}) \, , \\
\rho(\underline{x}) &= \int_{\bR^d} \frac{\dd^d\underline{k}}{(2\pi)^{\frac{d}{2}}} \, e^{i\underline{k}\cdot \underline{x}} \, \widehat{\rho}(\underline{k}) \, .\end{aligned}$$
In view of the theory for Sturm-Liouville equations should be treated as an eigenvalue equation on $L^2((0,\infty); \dd z)$ with spectral parameter $q^2$. Being $- \frac{\dd^2}{\dd z^2}$ the standard kinetic operator, its spectrum includes a continuous part, $q^2>0$, with a basis of eigensolutions $\big\{ \widehat{\Phi}_1, \, \widehat{\Phi}_2 \big\}$, with $$\begin{aligned}
\label{eq:fundbasisreg}
\widehat{\Phi}_1(\underline{k},z) = \frac{\sin(qz)}{q} \, , \qquad
\widehat{\Phi}_2(\underline{k},z) = - \cos(qz) \, .\end{aligned}$$ Observe that both solutions are square-integrable in a neighbourhood of the origin. We call $\widehat{\Phi}_1$ the *principal solution* at $z=0$, as it is the unique solution (up to scalar multiples) such that $\lim_{z \to 0} \widehat{\Phi}_1(\underline{k},z)/\widehat{\Psi}(\underline{k},z)=0$ for every solution $\widehat{\Psi}(\underline{k},z)$ which is not a scalar multiple of $\widehat{\Phi}_1$. The solution $\widehat{\Phi}_2$ is a nonprincipal solution and is not unique.
A general solution for $q^2>0$ may then be written as $$\label{eq:reglincomb}
\widehat{\Phi}(\underline{k},z) = A(\underline{k}) \widehat{\Phi}_1(\underline{k},z) + B(\underline{k}) \widehat{\Phi}_2(\underline{k},z) \, .$$ From and , one gets $$\label{eq:ABrhoFreg}
A(\underline{k}) = \widehat{\rho}(\underline{k}) \, , \qquad
B(\underline{k}) = - \widehat{F}(\underline{k}) \, .$$ These coefficients depend explicitly in $\underline{k}$, contrarily to the more common Robin boundary conditions. However, it is possible to recover the latter, as explained below.
It remains to obtain the boundary field in terms of the source term. From , it is easy to obtain $$\label{eq:Fitorho}
\widehat{F}(\underline{k}) = - \frac{\widehat{\rho}(\underline{k})}{c \left[q(\underline{k})^2 - m^2_{\rm b} \right]} \, .$$ Hence, the solution for $q^2>0$ may be written as $$\widehat{\Phi}(\underline{k},z) = \rho(\underline{k}) \left[ \widehat{\Phi}_1(\underline{k},z) + \frac{\widehat{\Phi}_2(\underline{k},z)}{c \left[q(\underline{k})^2 - m^2_{\rm b} \right]} \right] \, .$$ Observe that the term $q(\underline{k})^2 - m^2_{\rm b}$ contributes to a singular term which corresponds to two simple poles in the Fourier transform. These can be dealt with by means of standard, complex analysis techniques.
### Existence of bound states
Above, we analyzed the continuous part of the spectrum $q^2>0$ associated with the eigenvalue problem given by . Here, we investigate if there exist also negative eigenvalues, $q^2 < 0$, in the point spectrum with eigensolutions which satisfy the WBCs. Contrary to the continuous spectrum, in these case we must look for proper eigenfunctions, that is square-integrable solutions to .
For that let $\lambda = -q^2 > 0$ and consider $$\widehat{\Phi}_{\rm bs}(\underline{k},z) = - B(\underline{k}) \, e^{-\sqrt{\lambda}z} \, .$$ This is certainly a solution of the bulk field equation. If it additionally solves the WBCs for some choice of $\lambda$, it is a *bound state mode solution*, that is, a mode which exponentially decay with $z$.
The WBCs, together with , imply that $$\label{eq:lambdaeq}
\sqrt{\lambda} = c \left(q^2 - m^2_{\rm b} \right) = c \left(- \lambda - m^2_{\rm b} \right) \, .$$ It is clear that, with $m^2_{\rm b} \geq 0$, if $c\geq 0$ there is no positive $\lambda$ which solves the equation. For $c<0$, the solutions are $$\lambda = \frac{1}{2c^2} \left(1 - 2 m^2_{\rm b} c^2 \pm \sqrt{1 - 4 m^2_{\rm b} c^2}\right) \, .$$ If $m_{\rm b}=0$ there exists *one* strictly positive value of $\lambda$, which corresponds to one negative eigenvalue and, thus, one bound state. If $m_{\rm b}>0$, we have three cases:
- If $c<-1/(2m_{\rm b})$, then there is *no* strictly positive value of $\lambda$, and thus no bound states.
- If $c=-1/(2m_{\rm b})$, then there is *one* strictly positive value of $\lambda$, which corresponds to one negative eigenvalue and, thus, one bound state.
- If $-1/(2m_{\rm b})<c<0$, there are always *two* strictly positive values of $\lambda$, corresponding to two negative eigenvalues and, thus, two bound states.
### Robin boundary conditions
It is possible to recover Robin boundary conditions at $z=0$ from the WBCs through a specific choice of the boundary field mass term $m^2_{\rm b} $ and an appropriate limit of the constant $c$.
To see that, choose the boundary field mass such that $$m^2_{\rm b} = \frac{1}{c \kappa} \, ,$$ where $\kappa$ is a real number which is positive for $c>0$ and negative for $c<0$, [*i.e.*]{} keeping the squared boundary field mass positive. Then, Robin boundary conditions are recovered in the limit $c \to 0$: $$\widehat{\Phi}(\underline{k},z) = \rho(\underline{k}) \left[ \widehat{\Phi}_1(\underline{k},z) - \kappa \, \widehat{\Phi}_2(\underline{k},z) \right] \, .$$ The usual Dirichlet boundary conditions, for which $\widehat{\Phi}(\underline{k},0)=0$, correspond to $\kappa = 0$, whereas $\kappa \to \infty$ correspond to Neumann boundary conditions, for which $\partial_z\widehat{\Phi}(\underline{k},z)|_{z=0}=0$.
In [@Dappiaggi:2016fwc] it was found that one bound state mode solution exists when $\kappa < 0$, otherwise no such solution exists. From , if we set $m^2_{\rm b} = \frac{1}{c \kappa}$ and then take $c \to 0^-$, we obtain $$\sqrt{\lambda} = - \frac{1}{\kappa} \, .$$ Hence, if $\kappa < 0$ there is a strictly positive value of $\lambda$, which furthermore agrees with the result in [@Dappiaggi:2016fwc]. If $\kappa > 0$, there are no bound states in the limit $c \to 0^+$, also in agreement with the results of [@Dappiaggi:2016fwc].
Singular case {#sec:singular}
-------------
In the case of a massive scalar field in the Poincaré patch of AdS$_{d+1}$, the corresponding field equation for the conformally related field in $\bHo^{d+1}$ is singular at $z=0$ and the previous formulation of the WBCs is no longer valid, as the bulk field or its derivative with respect to $z$ may not be defined at $z=0$. In order to bypass this hurdle, we rewrite the underlying equation of motion in the following way: $$\label{eq:singsystem}
\begin{cases}
\left(\Box_{\eta}^{(d+1)} - \frac{m^2}{z^2} \right) \Phi = 0 \quad \text{in $\bHo^{d+1}$} \, , \\
\left(\Box_{\eta}^{(d)} - m^2_{\rm b} \right) F = - \dfrac{\rho}{c} \quad \text{in $\bR^d$} \, , \\
W_z \left[\Phi, \Phi_1\right] = F \, , \quad W_z \left[\Phi, \Phi_2\right] = \rho \, ,
\end{cases}$$ where $\big\{ \Phi_1, \, \Phi_2 \big\}$ is a basis of solutions, $W_z[u,v] \doteq u\frac{\partial v}{\partial z} - \frac{\partial u}{\partial z}v$ is the Wronskian betweens, and where $\Phi_1$ is chosen so that $\widehat{\Phi}_1$ is a principal solution at $z=0$. Since both $\widehat{\Phi}_1$ and $\widehat{\Phi}_2$ turn out to be solutions of an ODE with no first order derivative, see below, their Wronskian is constant in $z$. Hence we can normalize them so that $$\label{eq:condwronskian}
W_z \left[\widehat{\Phi}_1, \, \widehat{\Phi}_2\right] = 1 \, ,$$ and $\big\{ \widehat{\Phi}_1, \, \widehat{\Phi}_2 \big\}$ reduce to in the regular case.
### Bulk and boundary solutions
Using the same Fourier expansions as in in , $\widehat{\Phi}$ is now solution of $$\label{eq:singmodeeq}
\left(- \frac{\dd^2}{\dd z^2} + \frac{m^2}{z^2} \right)\widehat{\Phi}(\underline{k},z) = q^2 \, \widehat{\Phi}(\underline{k},z) \, .$$ Here, it is useful to remind ourselves that $m^2 = m_0^2 + (\xi - \frac{d-1}{4d})R$ and to introduce the convenient notation $$\nu \doteq \frac{1}{2} \sqrt{1+4m^2} \, .$$ The BF bound implies that $\nu \geq 0$.
A basis of solutions $\big\{ \widehat{\Phi}_1, \, \widehat{\Phi}_2 \big\}$ satisfying the required properties for the continuous part of the spectrum, $q^2 > 0$, is the following:
$$\begin{aligned}
\widehat{\Phi}_1(\underline{k},z) &= \sqrt{\frac{\pi}{2}} \, q^{-\nu} \sqrt{z} \, J_{\nu}(qz) \, , \label{eq:fundamentalsolutions1} \\
\widehat{\Phi}_2(\underline{k},z) &=
\begin{cases}
- \sqrt{\dfrac{\pi}{2}} \, q^{\nu} \sqrt{z} \, J_{-\nu}(qz) \, , & \nu > 0 \, , \\
- \sqrt{\dfrac{\pi}{2}} \sqrt{z} \left[ Y_{0}(qz) - \dfrac{2}{\pi} \log(q) \right] \, , & \nu = 0 \, .
\end{cases} \label{eq:fundamentalsolutions2}\end{aligned}$$
\[eq:fundamentalsolutions\]
The solution $\widehat{\Phi}_1$ is the principal solution at $z=0$ and is square-integrable near $z=0$ for all $\nu \geq 0$. The nonprincipal solution $\widehat{\Phi}_2$ is only square-integrable near $z=0$ if $\nu \in [0,1)$, hence, we only apply the WBCs for those values of the mass, namely $m^2 \in [-\frac{1}{4}, \frac{3}{4})$.
Hence, a general solution satisfying WBCs for $q^2 > 0$ and $\nu \in [0,1)$ is $$\label{eq:fundbasissing}
\widehat{\Phi}(\underline{k},z) = A(\underline{k}) \widehat{\Phi}_1(\underline{k},z) + B(\underline{k}) \widehat{\Phi}_2(\underline{k},z) \, ,$$ with $A(\underline{k}) = \widehat{\rho}(\underline{k})$ and $B(\underline{k}) = - \widehat{F}(\underline{k})$, where we used and . For $\nu \geq 1$, the only square-integrable solution is given by $\widehat{\Phi}_1$ and no boundary conditions need to be applied at $z=0$.
The boundary field $F$ is still given by the same expression of the regular case, $$\label{eq:Fitorho2}
\widehat{F}(\underline{k}) = - \frac{\widehat{\rho}(\underline{k})}{c \left[q(\underline{k})^2 - m^2_{\rm b} \right]} \, .$$ Hence, the bulk field may be written as $$\widehat{\Phi}(\underline{k},z) = \rho(\underline{k}) \left[ \widehat{\Phi}_1(\underline{k},z) + \frac{\widehat{\Phi}_2(\underline{k},z)}{c \left[q(\underline{k})^2 - m^2_{\rm b} \right]} \right] \, .$$
### Existence of bound states
Analogously to the regular case, we investigate if there exists negative eigenvalues, $q^2 < 0$, in the point spectrum of the singular eigenvalue problem given by with proper eigenfuctions which satisfy the WBCs.
Again letting $\lambda = -q^2 > 0$, consider $$\widehat{\Phi}_{\rm bs}(\underline{k},z) = \sqrt{z} \, K_{\nu}(\sqrt{\lambda}z) \, .$$ This is a solution of , and the WBCs, together with , imply that $$\label{eq:lambdanu}
\lambda^{\nu} = c \left(q^2 - m^2_{\rm b} \right) = c \left(- \lambda - m^2_{\rm b} \right) \, .$$ For $c \geq 0$ there is no positive $\lambda$ that solves the equation.
If $c<0$, we have several possibilities. If $m_{\rm b}=0$, then there is one strictly positive root, $$\lambda = (-c)^{\frac{1}{\nu-1}} \, ,$$ corresponding to one bound state. If $m_{\rm b}>0$ and $\nu=0$, there is a positive solution, $$\lambda = - \frac{1+cm^2_{\rm b}}{c} \, ,$$ when $- 1/m^2_{\rm b} < c < 0$, otherwise there is no positive solution, and hence no bound states. If $m_{\rm b}>0$ and $\nu \in (0,1)$, we cannot find analytical solutions of , but we can still obtain the number of positive roots. Let $$f(\lambda)=\lambda^{\nu}+c\lambda+c m^2_{\rm b} \, .$$ We want to know if $f$ has any positive roots for $c<0$ and $\nu \in (0,1)$. First, note that $f(0)=c m^2_{\rm b} < 0$ and that $\lim_{\lambda\to\infty}f(\lambda)=-\infty$ for $\nu \in (0,1)$ and $m_{\rm b} > 0$. Moreover, there is only one maximum at $\lambda_{\rm max} = (-c/\nu)^{1/(\nu-1)}$ with $$f(\lambda_{\rm max}) = (1-\nu) \left(\frac{-c}{\nu}\right)^{\frac{\nu}{\nu-1}} + c m^2_{\rm b} \, .$$ The maximum is positive, and hence there are two positive roots, if $-\nu^{\nu}(m^2_{\rm b}/(1-\nu))^{\nu-1}<c<0$. Otherwise, if $c=-\nu^{\nu}(m^2_{\rm b}/(1-\nu))^{\nu-1}$, there is one positive root, and if $c<-\nu^{\nu}(m^2_{\rm b}/(1-\nu))^{\nu-1}$ then there are no positive roots.
![\[fig:plot\]Plot of $f(\lambda)=\lambda^{\nu}+c\lambda+c m^2_{\rm b}$ for $\nu=1/3$ and $m_{\rm b}=1$ for different values of $c$.](WBC-plot)
We then conclude that, if $c<0$, $m_{\rm b} > 0$ and $\nu \in (0,1)$:
- If $c<-\nu^{\nu}(m^2_{\rm b}/(1-\nu))^{\nu-1}$, then there is *no* strictly positive value of $\lambda$, and, thus, no bound states.
- If $c=-\nu^{\nu}(m^2_{\rm b}/(1-\nu))^{\nu-1}$ (or $m_{\rm b}^2 = 0$), then there is *one* strictly positive value of $\lambda$, which corresponds to one negative eigenvalue and, thus, one bound state.
- If $-\nu^{\nu}(m^2_{\rm b}/(1-\nu))^{\nu-1}<c<0$, there are always *two* strictly positive values of $\lambda$, corresponding to two negative eigenvalues and, thus, two bound states.
These results are illustrated in Fig. \[fig:plot\]. Note that they are in agreement with the regular case, $\nu=\frac{1}{2}$.
Finally, if we consider the limit in which we recover Robin boundary conditions, by setting $m^2_{\rm b} = \frac{1}{c \kappa}$ and then taking $c \to 0$, we obtain from that $$\lambda^{\nu} = - \frac{1}{\kappa} \, .$$ Hence, if $\kappa < 0$ there is a strictly positive value of $\lambda$, and no positive values of $\lambda$ for $\kappa >0$, which agrees with the result in [@Dappiaggi:2016fwc].
Distinguishing structural properties of the generalized Wentzell boundary conditions {#sec:maths}
====================================================================================
In this section, we first show that imposing the generalized Wentzell boundary conditions (WBCs) at the PAdS boundary guarantees that the total fluxes of symplectic and energy currents across the boundary vanish, thus showing that the system is closed, as is the case with Robin (hence Dirichlet and Neumann) boundary conditions. We then provide a further explanation so as to why these and the Robin boundary conditions are *distinguished* on account of their interplay the scalar represetation of the isometry group at the conformal boundary.
Vanishing symplectic and energy flux across the boundary
--------------------------------------------------------
For the bulk field $\Phi$, now assumed to be complex-valued, we may define the *bulk symplectic current* as $$J_{\mu} \doteq -i \left(\overline{\Phi} \partial_{\mu} \Phi - \Phi \partial_{\mu} \overline{\Phi} \right) \, .$$ It is covariantly conserved, $\partial_{\mu}J^{\mu} = 0$, or equivalently $\dd \ast J = 0$. Using Stokes’ theorem, $$\begin{aligned}
0 = \int_{\bHo^{d+1}} \dd \ast J = \int_{\bR^d} \ast J \, ,\end{aligned}$$ which implies that, in combination of or , $$\begin{aligned}
\int_{\bR^d} \dd^d x \, \left(\rho \overline{F} - \overline{\rho} F \right) = 0 \, .\end{aligned}$$ This is a condition that both the source term $\rho$ and the boundary field $F$ must satisfy. One possibility is that the integrand itself vanishes, which, by , implies that the ratio $B/A$ in or must be real — that is, Robin boundary conditions must be imposed at the AdS boundary. But, more generally, the integrand does not need to vanish as long as its integral over the boundary is identically zero. Using or , one has $$\begin{aligned}
{}& \int_{\bR^d} \dd^d x \int_{\bR^d} \dd^d \underline{k}_1 \int_{\bR^d} \dd^d \underline{k}_2 \,
\left[ \frac{\widehat{\rho}(\underline{k}_1) \overline{\widehat{\rho}(\underline{k}_2)}}{c \left[ q(\underline{k}_2)^2 - m_{\rm b}^2 \right]} e^{i(\underline{k}_1-\underline{k}_2)\cdot \underline{x}} \right. \\
&\quad - \left. \frac{\overline{\widehat{\rho}(\underline{k}_1)} \widehat{\rho}(\underline{k}_2)}{c \left[ q(\underline{k}_2)^2 - m_{\rm b}^2 \right]} e^{-i(\underline{k}_1-\underline{k}_2)\cdot \underline{x}} \right] \\
&= \int_{\bR^d} \dd^d \underline{k}_1 \int_{\bR^d} \frac{\dd^d \underline{k}_2}{c \left[ q(\underline{k}_2)^2 - m_{\rm b}^2 \right]} \,
\left[ \widehat{\rho}(\underline{k}_1) \overline{\widehat{\rho}(\underline{k}_2)} - \overline{\widehat{\rho}(\underline{k}_1)} \widehat{\rho}(\underline{k}_2) \right] \\
&\quad \times \delta(\underline{k}_1-\underline{k}_2) \\
&= \int_{\bR^d} \frac{\dd^d \underline{k}}{c \left[ q(\underline{k}_2)^2 - m_{\rm b}^2 \right]} \,
\left[ \widehat{\rho}(\underline{k}) \overline{\widehat{\rho}(\underline{k})} - \overline{\widehat{\rho}(\underline{k})} \widehat{\rho}(\underline{k}) \right] = 0 \, .\end{aligned}$$ This shows that WBCs guarantees that the total symplectic flux across the boundary vanishes.
Again using or , we obtain $$\begin{aligned}
\int_{\bR^d} \dd^d x \, \eta^{\alpha\beta} \partial_{\alpha} \left(\overline{F} \partial_{\beta} F - F \partial_{\beta} \overline{F}\right) = 0 \, ,\end{aligned}$$ which suggests the definition of a *boundary symplectic current* $$\label{eq:boundarysymplecticcurrent}
J^{\partial}_{\alpha} \doteq -i \, c \left(\overline{F} \partial_{\alpha} F - F \partial_{\alpha} \overline{F}\right) \, .$$ Note, however, that it is not covariantly conserved, as $$\partial^{\alpha} J^{\partial}_{\alpha} = -i \left(\rho \overline{F} - \overline{\rho} F \right) \, ,$$ except in the particular case of Robin boundary conditions.
We note that the results presented above for the the symplectic current and for its flux across the boundary apply analogously to the energy density current for a real $\Phi$, defined by $J_{\mu}^E \doteq - T_{\mu\nu} k^{\nu}$, with $k = \partial_t$ and where the *bulk stress-energy tensor* is $$T_{\mu\nu} = \partial_{\mu} \Phi \partial_{\nu} \Phi - \frac{1}{2} g_{\mu\nu} \! \Big( \partial^{\lambda} \Phi \partial_{\lambda} \Phi + \tilde{m}^2 \Phi^2 \Big) \, ,$$ where $\tilde{m}^2 \doteq m_0^2 + \xi R$. Observe that, by using the bulk equations of motion, it holds $\partial^\mu T_{\mu\nu}=0$. We may also define a *boundary stress-energy tensor* as $$T_{\alpha\beta}^{\partial} = c \Big[\partial_{\alpha} F \partial_{\beta} F - \frac{1}{2} \eta_{\alpha\beta} \! \Big( \partial^{\lambda} F \partial_{\lambda} F + \tilde{m}^2 F^2 \Big) \Big] \, ,$$ which is however not covariantly conserved, $$\partial^{\alpha} T^{\partial}_{\alpha\beta} = \left[c (m_{\rm b}^2 - \tilde{m}^2) - \rho \right] \partial_{\beta} F \, ,$$ thus indicating that energy fluxes come from the bulk towards the boundary and leave the boundary into the bulk, as expected on physical grounds.
Interplay between the boundary conditions and the boundary isometry group
-------------------------------------------------------------------------
Considering boundary conditions of Robin type, but also those as in or in might appear at a first glance a mere academic exercise. Yet, as we already discussed in the introduction, the lessons learned from the study of the AdS/CFT correspondence indicate the importance of analyzing the possible interplays between bulk and boundary theories which are both dynamical. In this section, we discuss a different, structural property which indicates that both Robin and Wentzell boundary conditions are distinguished.
To this avail it is of paramount relevance that the underlying background is static. This allows us to shift from a purely hyperbolic equation such as the one in , ruled by the wave operator $\Box_\eta^{(d+1)}-\frac{m^2}{z^2}$, to an elliptic problem, governed by $$K_{\omega,m}\doteq-\nabla^2+\omega^2-\frac{m^2}{z^2} \, ,$$ where $\nabla^2=\sum_{i=1}^d \partial_i^2$ and $\omega$ is the Fourier parameter associated to the time coordinate.
Since we are interested in theories which can be coherently quantized, it is convenient to read the operator $K_{0,m}=-\nabla^2+\frac{m^2}{z^2}$ as the Hamiltonian of the underlying system. From this viewpoint, it is natural to interpret $K_{0,m}$ as a real, symmetric operator, acting on the Hilbert space $L^2(\mathring{\mathbb H}^d)$. Hence, in order for the underlying dynamics to describe a *closed* system, one needs to pick a self-adjoint extension of $K_{0,m}$, whose selection consists in turn on fixing suitable boundary conditions at $z=0$. At the classical level this guarantees that the total flux of symplectic and energy currents across the boundary vanishes, as shown explicitly in the previous section.
To better appreciate our freedom in this choice, we divide the analysis in two cases, $d=1$ and $d>1$. In the first case, and setting without loss of generality $m=0$, the Hamiltonian reduces to the kinetic operator on the half line. By using the theory of deficiency indices [@Moretti:2013cma Ch.5], the possible self-adjoint extensions are well-known: they are in one-to-one correspondence with boundary conditions of the form $\Phi|_{z=0}+\tan\alpha\,\partial_z\Phi|_{z=0}=0$, where $\alpha\in [0,\pi)$ can at most be made to be dependent on the spectral parameter, [*i.e.*]{} $\alpha=\alpha(\omega)$. This problem has been already investigated in [@Dappiaggi:2016fwc]
If $d>1$, the scenario is more intricate since $K_{0,m}$ is either essentially self-adjoint or the associated deficiency indices are infinite. The latter instance occurs for example when $m=0$. In other words there are infinite admissible choices for the ratio between the coefficients $A(\underline{k})$ and $B(\underline{k})$ appearing in or in . A physically motivated and mathematically sound selection criterion can be implemented by considering the interplay between the isometry group of the background and the operator $K_{0,m}$. Such problem has been studied only recently in a series of papers [@Ibort:2014sua; @Ibort2; @Pardo]. Another closely related analysis can be found in [@Asorey:2015lja; @Asorey:2017euv]. We will shortly review them and apply the procedure to the case at hand.
The starting point consists of investigating whether $K_{0,m}$ is an Hermitian operator on the Sobolev space $H^2(\mathbb H^d)$, where, for all $s>0$ and for all integer $d\geq 1$, $H^s(\mathbb{R}^d)=\{\psi\in L^2(M),\;|\;(\mathbb{I}-\nabla^2)^s\psi\in L^2(M)\}$, while $H^s(\mathbb{H}^d)=\{[\Psi]\;|\Psi\in H^s(\mathbb{R}^d)\;\Psi\sim\Psi^\prime,\;\textrm{iff}\; (\Psi-\Psi^\prime)|_{\mathbb{H}^d}=0\}$ — see [@Ibort2] or [@Adams] for a survey of the theory and of the key properties of Sobolev spaces. From now on, we will not write explicitly the symbol of equivalence classes since all our statements do not depend on the representative chosen in each of these classes.
To this avail, we observe that the following Green’s formula holds true for all $\Psi,\Psi^\prime\in H^2(\mathbb H^d)$: $$\label{Green_formula}
(\Psi,K_{\omega,m}\Psi^\prime)-(K_{\omega,m}\Psi,\Psi^\prime)=\widetilde{\Sigma}(\Psi,\Psi^\prime) \, ,$$ where $(,)$ stands for the inner product in $L^2(\mathbb H^d)$ while $$\label{boundary_form}
\widetilde{\Sigma}(\Psi,\Psi^\prime)=\langle\Gamma(\Psi),\Gamma(\partial_z\Psi^\prime)\rangle-\langle\Gamma(\partial_z\Psi),\Gamma(\Psi^\prime)\rangle \, .$$ where $\langle,\rangle$ is the $L^2$ inner product on the boundary $\mathbb{R}^{d-1}$. At the same time $\Gamma:H^s(\mathbb H^{d})\to H^{s-\frac{1}{2}}(\mathbb{R}^{d-1})$ is the so called *Lions trace* [@Adams Chap. 5]. For every $s>\frac{1}{2}$ this is a continuous and surjective operator which extends at the level of Hilbert spaces the standard restriction of smooth functions, namely, for every $\Psi\in C^\infty(\mathring{\mathbb H}^{d})\cap H^s(\mathbb H^{d})$, $\Gamma(\Psi)=\Psi|_{z=0}$. $\widetilde{\Sigma}$ is also known as [*Lagrange boundary form*]{} (in the case considered in this paper, corresponds to the boundary symplectic current introduced in ). A [*dense*]{} subspace $\mathcal{D}\subseteq\mathcal{H}_{\rm b}\doteq L^2(\mathbb{R}^{d-1})$ is called [*isotropic*]{} (with respect to $\widetilde{\Sigma}$) if $\widetilde{\Sigma}(\alpha,\beta)=0$ for all $\alpha,\beta\in\mathcal{D}$.
A direct inspection of unveils that, for $K_{\omega,m}$ to be a symmetric operator, it is mandatory that $\Sigma$ vanishes on its domain. While this is automatically true if one considers smooth and compactly supported functions on $\mathring{\mathbb H}^d$, from the viewpoint of the boundary Hilbert space, this choice is not informative since $\Gamma[C^\infty_0(\mathring{\mathbb H}^d)]=\{0\}$. Hence it is useful to consider the following relevant sets:
- For any $\mathcal{W}\subseteq\mathcal{H}_b\times\mathcal{H}_b$ its [*$\Sigma$-orthogonal subspace*]{} is $$\begin{aligned}
\mathcal{W}^\perp &\doteq \big\{(\varphi,\varphi^\prime)\in\mathcal{H}_b\times\mathcal{H}_b\;|\;\Sigma((\varphi,\varphi^\prime),(\psi,\psi^\prime))=0, \notag \\
&\qquad \;\forall (\psi,\psi^\prime)\in\mathcal{W}\times\mathcal{W}\big\} \, ,
\end{aligned}$$ where $\Sigma$ is the natural generalization of , [*i.e.*]{} $$\Sigma((\varphi,\varphi^\prime),(\psi,\psi^\prime)) = \langle\varphi,\psi^\prime\rangle-\langle\varphi^\prime,\psi\rangle \, .$$
- A subspace $\mathcal{W}$ is called [*$\Sigma$-isotropic*]{} if $\mathcal{W}\subseteq\mathcal{W}^\perp$ and [*maximally $\Sigma$-isotropic*]{} if $\mathcal{W}=\mathcal{W}^\perp$.
The advantage of considering those $\mathcal{W}$ which are maximally $\Sigma$-isotropic is two-fold. On the one hand, since $\Gamma$ is surjective, $\Gamma^{-1}[\mathcal{W}]$ identifies a natural domain of $K_{0,m}$ on which the right-hand side of vanishes automatically. On the other hand, it is possible to give an explicit characterization of these spaces. As a matter of fact, as proven in [@Pardo Lemma 3.1.4 & Prop. 3.1.5], letting $\mathcal{C}:\mathcal{H}_{\rm b}\times\mathcal{H}_{\rm b}\to\mathcal{H}_{\rm b}\times\mathcal{H}_{\rm b}$ be the [*unitary Cayley transform*]{} $$\label{eq:Cayley_transform}
\mathcal{C}(\varphi,\varphi^\prime) = \frac{1}{\sqrt{2}}\left(\varphi+i\varphi^\prime,\varphi-i\varphi^\prime\right) \, ,$$ it holds that
1. $\mathcal{W}$ is maximally $\Sigma$-isotropic if and only if $\mathcal{W}_c\doteq\mathcal{C}[\mathcal{W}]$ is maximally $\Sigma_c$-isotropic, where $$\label{Sigma_c}
\Sigma_c((\varphi,\varphi^\prime),(\psi,\psi^\prime)) = -i(\langle\varphi,\psi\rangle-\langle\varphi^\prime,\psi^\prime\rangle) \, .$$
It follows from the items above that, whenever $\mathcal{W}$ is maximally $\Sigma$-isotropic and for any unitary operator $U$, we can use to write $$\label{max_isotropic}
\mathcal{W} \doteq \{(\varphi,\varphi^\prime) \in \mathcal{H}_{\rm b}\times\mathcal{H}_{\rm b} \;|\; \varphi-i\varphi^\prime = U(\varphi+i\varphi^\prime)\} \, .$$
If we recall that $\Sigma$ generalizes , we can identify $\varphi=\Gamma(\Psi)$ and $\varphi^\prime=\Gamma(\partial_z\Psi)$, $\Psi\in H^2(\mathbb H^d)$, which suggests that the choice of any $\mathcal{W}$ as in identifies a specific boundary condition. Formally this can be obtained inverting the identity in : $$\label{eq:inverse_formula}
\varphi^\prime=A_U\varphi,\quad A_U\doteq -i(\mathbb{I}-U)(\mathbb{I}+U)^{-1}.$$ As observed in [@Ibort:2014sua; @Ibort2], for to be both a well-defined mathematical expression and applicable to the case at hand, a sufficient requirement is that two conditions should be met. On the one hand, either $(\mathbb{I}+U)^{-1}$ exists or $-1$ is an element of the spectrum of $U$, which is not an accumulation point. It is noteworthy that, choosing Robin boundary conditions always falls in the first case. We stress that, if we recall the identification $\varphi^\prime=\Gamma(\partial_z\Psi)$, then we also need that $A_U$ is a continuous operator on $H^{\frac{1}{2}}(\mathbb{R}^d)$. Any unitary operator $U:\mathcal{H}_{\rm b}\to\mathcal{H}_{\rm b}$ meeting these requirements will be called [*admissible*]{}.
The next step consists of using the structures introduced above to characterize the self-adjoint extensions of the Hamiltonian operator $K_{0,m}$, whenever the deficiency indices are non-vanishing. Within this class, the prototypical case is the one in which we set $m=0$. Hence, from now on we focus our attention on $K\equiv K_{0,0}$, although all results apply also to the other scenarios.
The first step consists of translating $K$ into an Hermitian quadratic form. Following [@Ibort2], let $U$ be an admissible unitary operator so that $-1$ is not an element of its spectrum. Then we call $Q_U:D_{Q_U}\times D_{Q_U}\subset\mathcal{H}_{\rm b}\times\mathcal{H}_{\rm b}\to\mathbb{C}$, $$\label{QF}
Q_U(\Phi,\Phi^\prime)\doteq\left( d\Phi,d\Phi^\prime\right)_{\Lambda^1}+\langle\Gamma(\Phi),A_U(\Gamma(\Phi^\prime))\rangle \, ,$$ where $(,)_{\Lambda^1}$ stands for the standard $L^2$-pairing between $1$-forms on a Riemannian manifold and where $\Phi,\Phi^\prime\in D_{Q_U}\equiv H^1(\mathbb H^d)$. In [@Ibort:2014sua; @Ibort2] it has been proven that $Q_U$ enjoys several properties, the most notable being that it is closable, namely there exists a domain $D^\prime_{Q_U}\supseteq D_{Q_U}$ on which $Q_U$ is closed with respect to the norm $$\|\Phi\|_Q^2=\|d\Phi\|_{\Lambda^1}^2+(1+C_U)\|\Phi\|^2_{H^1} \, , \quad \forall\Phi\in D^\prime_{Q_U} \, .$$
Hence, we can invoke [@Ibort:2014sua Th. 2.7 & 6.7] to conclude that the quadratic form $Q_U$ identifies a unique self-adjoint operator $K_U$ such that $D(K_U)= D^\prime_{Q_U}$ and there exists $\chi\in H^2(\mathbb H^d)$ for which $Q_U(\Phi,\Phi^\prime)=(\Phi,\chi)_{H^1}$ for all $\Phi\in D^\prime_{Q_U}$. In this case we set $K_U\Phi^\prime=\chi$ and $$Q_U(\Phi,\Phi^\prime)=(\Phi,K_U\Phi^\prime) \, , \quad \forall\Phi,\Phi^\prime\in D(K_U) \, .$$
In addition, it turns out that $K_U$ is a self-adjoint extension of $K$ uniquely and unambiguously identified by an admissible unitary operator $U:\mathcal{H}_{\rm b}\to\mathcal{H}_{\rm b}$.
The above digression serves us to recall that the choice of boundary conditions for $K_{\omega,0}$ is strongly tied to the identification of a maximally $\Sigma$-isotropic $\mathcal{W}\subset\mathcal{H}_{\rm b}$, which, in turn, corresponds to selecting a self-adjoint extension for $K$ via an admissible unitary operator $U$. Yet, since the number of the latter is infinite, one might wonder whether it is at least possible to identify a distinguished subclass.
To this end we observe that the Poincaré patch of AdS$_{d+1}$ has isometry group $Iso({\rm PAdS}_{d+1})=O(d-1,1)\ltimes\mathbb{R}^d$, that is the $d$-dimensional Poincaré group. On each constant time hypersurface, the relevant subgroup is $E(d-1)\doteq O(d-1)\ltimes\mathbb{R}^{d-1}$. Let $V:E(d-1)\to\mathcal{BL}(L^2(\mathring{\mathbb H}^d))$ be such that $$(V(g)\psi)(x)=\psi(g^{-1}x) \, ,\quad\forall \psi\in L^2(\mathring{\mathbb H}^d) \, ,$$ where $g^{-1}x$ stands for the geometric action of $g^{-1}$ on the point $x\in \mathring{\mathbb H}^d$. This is a unitary, strongly continuous representation of the Euclidean group. To analyze its interplay with , we start by considering $U=\mathbb{I}$. This choice identifies the so-called [*Neumann quadratic form*]{} $Q_N$ such that $$Q_N(\Phi)=\|d\Phi\|_{\Lambda^1}^2 \, , \quad D(Q_N)=H^1(\mathbb H^d) \, .$$ Observe that yields $\varphi^\prime=0$ if $U=\mathbb{I}$, which, in the case at hand, entails that we are considering Neumann boundary conditions. In addition, a direct calculation shows that $Q_N$ is invariant under the action of $V$, namely, for every $g\in E(d-1)$, it holds $Q_N(V(g)\Phi)=Q_N(\Phi)$.
Since $E(d-1)$ can also be read as a subgroup of the isometries of the boundary of PAdS$_{d+1}$, one can infer that $V$ has a trace along the boundary (see [@Ibort:2014sua Def. 6.9]), namely for every $\Phi\in H^1(\mathbb H^d)$, it holds $$\Gamma(V(g)\Phi)=v(g)\Gamma(\Phi) \, , \quad\forall g\in E(d-1) \, ,$$ where $\Gamma:H^1(\mathbb H^d)\to H^{\frac{1}{2}}(\mathbb{R}^{d-1})$ and $v:E(d-1)\to\mathcal{BL}(L^2(\mathbb R)^{d-1})$ is the strongly continuous, unitary representation implementing the geometric action $v(g)\varphi(y)=\varphi(g^{-1}y)$, $y$ being a point of $\mathbb{R}^{d-1}$. The most notable interplay between traceable representations and self-adjoint extensions of the operator $K$ are a consequence of [@Ibort:2014sua Th. 6.10], which entails that $K_U$ is an $E(d-1)-$invariant, self-adjoint extension of $K$ if and only if $[U,v(g)]=$ for all $g\in E(d-1)$.
From a physical point of view, invariance under the action of the underlying isometry group is a desired property and hence we call [*distinguished*]{} any self-adjoint extension of $K$ which is $E(d-1)$-invariant. Two examples are certainly of interest to our analysis. In the first we choose $A_U=\cot\alpha \, \mathbb{I}$, $\alpha\in [0,\pi)$, which via Cayley transform corresponds to $U=e^{i\alpha} \, \mathbb{I}$, see [@Moretti:2013cma Th. 5.34]. We observe that, on the one hand, a multiple of the identity operator $U$ commutes with every representation of $E(d-1)$, while on the other hand, , together with the identification of $\varphi^\prime=\Gamma(\partial_z\Phi)$ and of $\varphi=\Gamma(\Phi)$, yields the standard Robin boundary condition $\varphi^\prime = \cot\alpha \, \varphi$. Hence, Robin boundary conditions identify an $E(d-1)$-invariant self-adjoint extension of $K$. We should keep in mind that our interest towards the self-adjoint extensions of $K$ arises from having transformed the wave equation on $\PAdS_{d+1}$ (in the massless case) into an eigenvalue problem for the operator $K$. Hence, although $\alpha$ is a constant one might consider to make $\alpha$ dependent on the spectral parameter $\omega$. Yet this option should be discarded since, upon inverse Fourier transform, we would obtain a boundary condition which breaks manifestly Poincaré invariance.
A second, non trivial self-adjoint operator which commutes with the unitary representation $v$ for $E(d-1)$ is certainly $-\nabla^2_{d-1}$, the (unique self-adjoint extension of the Laplace-Beltrami operator on $\mathbb{R}^{d-1}$. In this case, although the rationale behind our selection criterion is fulfilled, the unitary operator built via Cayley transform from $-\nabla^2_{d-1}$ has a spectrum whose eigenvalues admit $-1$ has an accumulation point. In this case, one is still identifying a self-adjoint extension of $K$, but a more technical analysis is required, using the so-called quasi-boundary triples, see [@boundary_triple] for a short review.
In addition, in view of our need to reinstate the time coordinate via Fourier transform, a more natural choice consists of adding to $-\nabla^2_{d-1}$ a multiple of the identity operator, dependent on the spectral parameter, namely $\nabla^2_{d-1}+(\omega^2+m^2_{\rm b})\mathbb{I}$ where $m^2_{\rm b}> 0$ is a constant. By considering once again together with the identification of $\varphi^\prime=\Gamma(\partial_z\Phi)$ and of $\varphi=\Gamma(\Phi)$, we realize that this choice consists of considering the boundary condition $\varphi^\prime=(-\nabla^2_{d-1}+\omega^2+m^2_{\rm b})\varphi$, which, after an inverse Fourier transform with respect to $\omega$ yields exactly the Wentzell boundary condition and . In other words both Robin and Wentzell boundary conditions are distinguished in view of their interplay with the action of the boundary isometry group on the underlying spaces of functions.
Conclusions {#sec:conclusions}
===========
In this paper, we have considered a real, massive scalar field in the Poincaré fundamental domain of AdS in $d+1$ dimensions, subject to dynamical boundary conditions of generalized Wentzell type at the PAdS boundary. We solved the full system, for both the bulk and boundary fields, and verified that, depending on the values of the parameters of the theory, there might exist zero, one or at most two bound state mode solutions. Although we have not dwelt into the quantization of the underlying model, this result offers a clear indication concerning those values of the mass and of the curvature coupling parameter for which we can expect or rule out the existence of a ground state. Finally, we analyzed what makes this choice of dynamical boundary conditions distinguished, as they are invariant under the action of the isometry group of the PAdS boundary, and imply zero symplectic and energy density total flux accross the boundary.
As a perspective, we outline that in order to obtain the quantization of the theory, the first step will consist of constructing the bulk propagator / fundamental solutions, relating it to the one which stems from the boundary theory. It is of particular interest to obtain a map from a Hadamard state of the boundary theory to a Hadamard state of the bulk theory, which would constitute an AdS counterpart to the result in asymptotically flat spacetimes [@Dappiaggi:2017kka]. This is work in progress.
Finally, we note that the problem that we have studied in this paper belongs to a class of systems, those with dynamical boundary conditions, that are of relevance for a broad spectrum of physical models and theories. The techniques that we have employed in this work are applicable to different problems, ranging from condensed matter to gravitational physics, as well as quantum gravity and high energy physics.
The work of C. D. was supported by the University of Pavia. The work of H. F. was supported by the INFN postdoctoral fellowship “Geometrical Methods in Quantum Field Theories and Applications”, and in part by a fellowship of the “Progetto Giovani GNFM 2017 – Wave propagation on lorentzian manifolds with boundaries and applications to algebraic QFT” fostered by the National Group of Mathematical Physics (GNFM-INdAM). H. F. also acknowledges the hospitality of the ICN-UNAM and their support through a PAPIIT-UNAM grant IG100316. The work of B. A. J.-A. was supported by a Consejo Nacional de Ciencia y Tecnología (CONACYT, México) project 101712. B. A. J.-A. also acknowledges the hospitality of the INFN – Sezione di Pavia during the realization of part of this work, as well as the support of an International Mobility Award granted by the Red Temática de Física de Altas Energías (Red FAE-CONACYT).
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We report low temperature magnetotransport measurements on a high mobility ($\mu=\unit[325\,000]{cm^{2}/V\, sec}$) 2D electron system on a H-terminated Si(111) surface. We observe the integral quantum Hall effect at all filling factors $\nu\leq6$ and find that $\nu=2$ develops in an unusually narrow temperature range. An extended, exclusively even numerator, fractional quantum Hall hierarchy occurs surrounding $\nu=3/2$, consistent with two-fold valley-degenerate composite fermions (CFs). We determine activation energies and estimate the CF mass.'
author:
- 'Tomasz M. Kott'
- Binhui Hu
- 'S. H. Brown'
- 'B. E. Kane'
bibliography:
- 'bibtex.bib'
title: Valley Degenerate 2D Electrons in the Lowest Landau Level
---
Multicomponent two dimensional (2D) systems in a single Landau level have generated interest due to the possibilities for novel correlated ground states in the integer and fractional quantum Hall (FQH) regimes when the energies of the component states become degenerate. Early experiments focused on measurements of engineered GaAs materials where the spin splitting could be reduced to zero [@leadley1997fractional; @kang1997evidence; @shukla2000largeskyrmions]. Subsequent development of AlAs quantum wells with a tunable valley degeneracy allowed the study of a spin-like degeneracy in the same limit [@shayegan2006twodimensional]. Recently, there has been great interest in the sublattice (valley) degeneracy in graphene; experiments show that the valley symmetry affects the FQH hierarchy [@dean2011multicomponent; @feldman2012unconventional2] and that valley ferromagnetism occurs when one of two degenerate valleys is occupied [@young2012spinand].
Measurements on silicon, the first multi-valley system to be considered theoretically [@rasolt1985newgapless; @rasolt1986dissipation], had been hampered by high disorder. Lately, however, Si(100)/SiGe heterostructures have shown mobilities up to $\unit[10^{6}]{cm^{2}/Vs}$ [@lu2012fractional]. Nonetheless, Si(100) is known to have an intrinsic valley splitting due to the confinement potential [@boykin2004valleysplitting; @tsui1979observation; @takashina2006valleypolarization; @goswami2007controllable]. The case of 2D electrons on (111) oriented silicon surfaces, which have three pairs of opposite momentum ($\vec{k}$) valleys, is especially interesting [@hwang2012valley]. As opposed to either AlAs or Si(100), the degeneracy of valley pairs with $\pm\vec{k}$ symmetry in Si(111) cannot be broken within the effective mass approximation or by a confinement potential [@rasolt1986dissipation], similar to the case of valleys in graphene. Additionally, both AlAs and Si(111) exhibit anisotropic mass tensors; in AlAs, this anisotropy arguably transfers to composite fermions (CFs) [@gokmen2010transference]. Novel broken symmetry states have been predicted at integer filling factor, $\nu$, when the Fermi energy, $E_{F}$, lies between two valleys with different mass tensors [@abanin2010nematic].
In this report we present transport data on a very high mobility electron system ($\mu=\unit[325\,000]{cm^{2}/Vs}$ at a temperature $T=\unit[90]{mK}$ and density $n_{s}=\unit[4.15\times10^{11}]{cm^{-2}}$) on a hydrogen-terminated Si(111) surface. We observe an extended FQH hierarchy around $\nu=3/2$. We argue that the FQH hierarchy is consistent with the SU(2) symmetry of a two-fold valley-degenerate ground state and present the first measurement of the CF mass in a multicomponent system, as well as the first in an anisotropic system. We also present preliminary evidence for many-body interactions affecting integer activation energies: the development of $\nu=2$ occurs in an unusually narrow temperature range, which may signal a transition to broken symmetry valley states.
![(Color) *(a)* $R_{xx}$, $R_{yy}$ (left), and $\rho_{xy}$ (right) vs. $B$ at $T=\unit[90]{mK}$ and $n_{s}=\unit[3.75\times10^{11}]{cm^{-2}}$ with filling factors listed on the top axis; the fractional states at $\nu=\frac{8}{5},\,\frac{10}{7},$ and $\frac{4}{3}$ have plateaus in $\rho_{xy}$. *(b)* $R_{xx}$ and $R_{yy}$ vs. $B$ for $n_{s}=\unit[3.75\times10^{11}]{cm^{-2}}$. Seven FQH states are visible at $T=\unit[110]{mK}$; the filling factors are labeled on the top axis. *Top panel*: $\rho_{xy}$ versus $B$ for $T=\unit[110]{mK}$. The derivative is also shown to accentuate the fine structure, matching the $R_{xx}$ minima. \[fig:bSweep\]](figure1){width="3.375in"}
For our samples, we replace the Si-SiO$_{2}$ interface of a typical MOSFET with an interface between H terminated Si(111) and a vacuum dielectric in order to remove the effects of strain and dangling bonds from the Si(111) surface [@eng2007integer]. An encapsulating silicon-on-insulator piece is bonded via van der Waals forces to the high resistivity H-Si(111) substrate ($<0.5^{\circ}$ miscut) and forms a gate. The sample studied in the present work is fabricated in the same way as previous ones, with further optimized cleaning and annealing steps [@mcfarland2009temperaturedependent]. Four-terminal resistance in a square van der Pauw geometry is defined as $R_{ij,lm}=V_{lm}/I_{ij}$, with $R_{xx}=R_{12,34}$ and $R_{yy}=R_{24,13}$ oriented along the [\[]{}1$\bar{1}$0[\]]{} and [\[]{}11$\bar{2}$[\]]{} directions respectively (see inset to Fig. \[fig:bSweep\]a). Hall traces, $\rho_{xy}$, are the averages of $R_{14,23}$ and $R_{23,14}$ to prevent mixing. The data were obtained at densities between $n_{s}=3.7$ and $\unit[5.7\times10^{11}]{cm^{-2}}$, adjustable via a gate voltage, in both a $^{3}$He system and $^{3}$He/$^{4}$He dilution system with base temperatures of $\unit[280]{mK}$ and $\unit[90]{mK}$, respectively. The density range is limited at high $n_{s}$ due to gate leakage and at low $n_{s}$ due to nonlinear contact resistances and a small parallel conductance channel, which are responsible for non-zero minima at some integer filling factors. Magnetotransport measurements were performed using standard lock-in techniques with a typical excitation of $1-\unit[25]{nA}$ at $\unit[5]{Hz}$ in a magnetic field $B$ up to $\unit[12]{T}$.
Figure \[fig:bSweep\]a shows $R_{xx}$, $R_{yy}$ and $\rho_{xy}$ versus $B$ taken at $T=\unit[90]{mK}$ and density $n_{s}=\unit[3.75\times10^{11}]{cm^{-2}}$. Shubnikov-de Haas oscillations (SdHO) are visible down to about $B\sim\unit[0.15]{T}$. The differences in the position of the minima at low $B$ between $R_{xx}$ and $R_{yy}$ are not an indication of density inhomogeneity; high $B$ minima of $R_{xx}$ and $R_{yy}$ are consistent to $0.01\%$ at $\nu=8/5$. While $R_{xx}\approx R_{yy}$ at $B=0$, strong anisotropy appears for $B>0$. Such anisotropy is to be expected in Si(111) when valleys with different mass tensors have an unequal density of states (DOS) at $E_{F}$. Here, we simply note that the anisotropy is essentially constant above $B=\unit[7]{T}$, and we will focus our analysis on these higher magnetic fields. By $B\approx\unit[1.3]{T}$ ($\nu=12$), only the lowest Landau level is occupied. As $B$ increases, the six-fold valley degeneracy breaks. Below $\nu=8$ all integer filling factors have minima, with the even states ($\nu=6,\,4,$ and $2$) being stronger than the odd ($\nu=3,\,5$ and $7$).
![ (Color) *(a)* Resistance data (with measurement errors) versus temperature used in Shubnikov-de Haas analysis. Open (closed) symbols are from $R_{xx}$ ($R_{yy}$). Note that only three data points are available for all $R_{yy}$ data. *(b)* Mass of composite fermions assuming that the gaps probed by Shubnikov-de Haas (SdH) oscillations have the form $\Delta_{\text{CF}}=\hbar\omega_c^{\text{CF}}$. Open symbols indicate values extracted from maxima. See text for a discussion of the statistical and systematic errors involved in this measurement. *(c)* Equivalent CF gap energy calculated from the masses. The gap value at $\nu=4/3$ is calculated from activation energy data and is approximately the same for both $R_{xx}$ and $R_{yy}$. \[fig:Mass\] ](figure2){width="3.375in"}
Figure \[fig:bSweep\]b shows the temperature dependence of $R_{xx}$ and $R_{yy}$ in the FQH regime below $\nu=2$. In this range, we observe minima at 9 fractions: $\nu=8/5$, 14/9, 20/13, 22/15, 16/11, 10/7, 4/3, (shown) as well as 6/5 and 14/11 at lower densities (not shown). Hall plateaus at $\nu=8/5$ and 4/3 are quantized to within 0.5% of their nominal values. For weaker fractions, $\nu$ is determined from $B$ evaluated at the resistance minimum relative to the value of $B$ at the sharp minimum of 8/5. Using this technique all fractions deviate less than 0.1% from their designated values.
The hierarchy of observed states is exactly that predicted by an SU(2) symmetry in which the two-fold valley degeneracy of electrons is preserved, leading to the creation of composite fermions (CF) with two and four attached vortices ($^{2}$CF and $^{4}$CF, respectively) [@park2001thespin]. For the hierarchy of fractional states around $\nu=3/2$, which are the hole-symmetric equivalents to the $\nu=1/2$ states, the filling factor $\nu^{*}$ of CFs is given by $\nu=2-\nu^{*}/(2p\nu^{*}\pm1)$ [@park2001thespin] where $p=1$ and 2 for $^{2}$CF and $^{4}$CF, respectively. With an SU(2) symmetry, only $\nu^{*}=2,4,6,\ldots$ are expected, giving rise to the hierarchy visible in Fig. \[fig:bSweep\]b. A similar, though smaller, hierarchy was observed recently in graphene with the same conclusion [@feldman2012unconventional2]. Finally, $\nu=6/5$ and 14/11 relate to $^{4}$CFs via $\nu^{*}=4/3$ and 8/5, respectively, using $2-\nu^{*}/(2\nu^{*}-1)$; the $^{4}$CF hierarchy simply reflects the FQH states of $^{2}$CFs. We note that we observe no evidence for a $\nu=5/3$ state, similar to recent experiments in both graphene [@feldman2012unconventional2] and Si/SiGe [@lu2012fractional]. However, the $5/3$ state is visible in nominally doubly-degenerate AlAs, probably due to local strains [@padmanabhan2010ferromagnetic; @shkolnikov2005observation; @abanin2010nematic].
To estimate the mass of the CFs, we use the theory of SdHO and apply it to $\nu=3/2$ by transforming to an effective magnetic field $B_{\text{eff}}=3\left(B-B_{3/2}\right)$ where $B_{3/2}=\unit[10.34]{T}$ (calculated from the density) [@du1996composite]. Although we could use either activation energy measurements or SdHO to find the gaps, analysis based on SdHO takes into account intrinsic level broadening. We first compute the amplitude $\Delta R$ of the oscillations using a linear interpolation of the minima and maxima (see [@padmanabhan2008effective]). We then use $\Delta R\propto R_{0}\exp\left\{ -\pi\Gamma/\Delta\right\} \xi/\sinh\xi$, where $\xi=2\pi^{2}T/\Delta$ and $R_{0}$ is the (temperature dependent) resistance at $B_{\text{eff}}=0$, to find the gap $\Delta$ ($\Gamma$ quantifies the intrinsic level broadening of Landau levels).
The extracted SdH data is shown in Fig. \[fig:Mass\]a; the vertical bars show the measurement errors in resistance minima. Note that due to temperature constraints, there are only three data points available for $R_{yy}$ data – this means that a straightforward comparison of masses extracted from $R_{yy}$ and $R_{xx}$ data will not have the same precision. Additionally, it is important to note that while we extract two masses ($m^{\text{CF}}_{xx}$ and $m^{\text{CF}}_{yy}$), we do not claim that these are aligned to the principal axes of the effective mass tensor (due to the van der Pauw geometry). Therefore, we cannot make any statement about the possible anisotropy of composite fermion masses as is argued by Gokmen et al[@gokmen2010transference].
We expressly assume that $\Delta_{\text{CF}}=\hbar\omega_{c}^{\text{CF}}$, $\omega_{c}^{\text{CF}}=e\left|B_{\text{eff}}\right|/m_{\text{CF}}$ and extract $m^{\text{CF}}_{xx,yy}$ directly. The results are shown in Fig. \[fig:Mass\]b. There are two distinct errors that are not shown. First is the statistical error due to the fitting process. For all cases but one, this is less than 10%, and for all $m^{\text{CF}}_{xx}$ values it is less than 3%. The second error is the error due to lack of data. While harder to quantify, it is clear that the $m^{\text{CF}}_{xx}$ values, based on five temperature values, are more robust than the $m^{\text{CF}}_{yy}$ values.
By considering the extracted SdH gap energies, however, we note that the values for both directions are consistent with activation energy data; in Fig. \[fig:Mass\]c, we plot the gap energy $\Delta_{\text{CF}}$ as a function of $B_{\text{eff}}$. The activation energy at $\nu=4/3$, measured from higher $T$ data, is consistent with a constant value for $m_{\text{CF}}$. Additionally, we find that all of the FQH data for $R_{xx}$ collapse onto a single line for $\Delta R\sinh\xi/R_{0}\xi$ vs. $1/B$ with $\Gamma^{\text{CF}}=\unit[0.83\pm0.04]{K}$, somewhat less than twice as large as the value for electrons, $\Gamma=\unit[0.49\pm0.02]{K}$, determined at low $B$ using the same approach.
![(Color) *(a)* Activation energy measurements as a function of $B$ in units of Coulomb energy. Vertical bars show the $\unit[95]{\%}$ confidence errors from fits to the linear portions of Arrhenius plots. Closed (open) symbols indicate data from $R_{yy}$ ($R_{xx}$) measurements. The $\nu=2$ data shows an enhanced energy gap greater than the Landau level spacing ($E_{LL}$) and indicates the scatter between $R_{xx}$ and $R_{yy}$ data. *(b)* Arrhenius plot of $R$ for $\nu=2$ and $6$ at $n_{s}=\unit[4.74\times10^{11}]{cm^{-2}}$. While the $\nu=6$ dependence is linear over a wide range of temperatures, the $\nu=2$ data show a very narrow linear region. Both $R_{xx}$ (open symbols) and $R_{yy}$ (closed) show this effect for $\nu=2$. *(c)* Plot of the linear portion of $\sigma_{yy}$ versus $\Delta_{\nu}/T$, showing the difference between the critical conductivities $\sigma_{yy}^{C} = \sigma_{yy}\left(1/T\rightarrow0\right)$ at different filling factors and densities. \[fig:Activation-energy-measurements\] ](figure3){width="3.375in"}
For comparison to other systems, we define a normalized CF mass $m_{\text{nor}}^{\pm}=m_{\text{CF}}\epsilon/\sqrt{B}$ (in units of $\unit[m_{e}]{T^{-1/2}}$, where the $\pm$ is the sign of $B_{\text{eff}}$) [@halperin1993theoryof]. Using $\epsilon=6.25$ ($\left(\epsilon_{\text{Si}}+\epsilon_{\text{Vac}}\right)/2$, a value appropriate at a silicon-vacuum interface for an electron gas with a perpendicular extent $z_\perp$ small compared to the interaction distance) a weighted average for $R_{xx}$ data gives $m_{\text{nor}}^{-}=2.14\pm0.04$ and $m_{\text{nor}}^{+}=2.40\pm0.05$. In other materials, $m_{\text{nor}}\approx3.2-3.5$ for GaAs at $\nu=1/2$ ([@pan2000effective] and references therein) or $m_{\text{nor}}\approx1.7$ at $\nu=3/2$ [@du1996composite], and $m_{\text{nor}}\approx1.1-1.3$ for ZnO at $\nu=1/2$ [@maryenko2012temperaturedependent]. Unlike our samples, the data for GaAs experiments appears to be symmetric around $B_{\text{eff}}=\unit[0]{T}$. Furthermore, the ZnO heterostructures show a linear dependence on the perpendicular magnetic field. Our results cannot rule out either a divergence or a linear dependence on $B$. There are several material-specific effects that may affect the CF mass, including effective-mass anisotropy [@yang2012band], the finite $z_{\perp}$ of the electron gas, and Landau level mixing [@gee1996composite; @melik-alaverdian1995composite]. The modification of short-range interactions due to the large dielectric-constant, $\epsilon$, mismatch at the surface is particularly relevant for our devices.
We turn now to gaps at integer filling factors; we use the $T$ dependence of $R_{xx}$ and $R_{yy}$ to calculate the activation energies via $R\propto\exp\left(-\Delta_{\nu}/2k_{B}T\right)$. By changing $n_{s}$, we are able to measure the gap $\Delta_{\nu}$ as a function of $B$: Figure \[fig:Activation-energy-measurements\]a shows the results of such an analysis in units of the Coulomb energy $E_{C}=e^{2}/4\pi\epsilon\epsilon_{0}l_{B}$, where $l_{B}=\sqrt{\hbar/eB}$ and we use $\epsilon=6.25$. Due to anisotropy and a lack of large resistance range in $R_{xx}$, we show only $\Delta_{\nu}^{yy}$ for most of the filling factors; for $\nu=2$ we show both transport directions as an example of scatter in the gap energy.
With the assumption that the valley splitting is smaller than the Zeeman gap, the $\nu=6$ activation energy is interpreted as $\Delta_{6}=E_{Z}=g^{*}\mu_{B}B_{\text{tot}}$, from which we estimate $g^{*}=3.63\pm0.06$, consistent with previous measurements [@eng2007integer; @mcfarland2009temperaturedependent]. The presence of minima at all filling factors $\nu\leq6$ suggests a $B$-dependent valley splitting, similar to that observed in SiGe heterostructures [@weitz1996tiltedmagnetic; @wilde2005directmeasurements]. Indeed, the appearance of odd $\nu<6$ shows that high magnetic fields increasingly break the two-fold degeneracy of opposite $\vec{k}$ valleys. The estimate at $\nu=5$ is based on a “strength” ($S$) of the state defined as the ratio of the resistance minimum to the average of the adjacent maxima [@lu2012fractional]. From the $T$ dependence, we can estimate a quasi-gap; the state is weaker than any other filling factor $\nu\leq6$. For $\nu=1$, we can only estimate an upper and lower bound while noting that the minimum remains visible at $T=\unit[1]{K}$ (see Fig. \[fig:Activation-energy-measurements\]a). The odd filling factor gaps, which are much greater than $\Gamma$, indicate that the splitting is likely due to many-body effects.
For $\nu=2$, the interpretation of the activation energy is more difficult. As shown by the solid line in Fig. \[fig:Activation-energy-measurements\]a, $\Delta_{\nu=2}$ is greater than the Landau level spacing ($E_{LL}$). To show the qualitative difference between $\nu=2$ and the much better understood $\nu=6$, we show the temperature dependence of the resistance for the two minima in Fig. \[fig:Activation-energy-measurements\]b at a density of $n_{s}=\unit[4.60\times10^{11}]{cm^{-2}}$. The data for $\nu=6$ clearly follow an Arrhenius relationship, while the resistance change for $\nu=2$ occurs over a very limited temperature change, and has a much smaller range over which it is linear. The transition for $\nu=2$ occurs in a range of $\sim\unit[200]{mK}$ near $T\approx\unit[1]{K}$. Another illustration of the peculiarity of $\nu=2$ is shown in Fig \[fig:Activation-energy-measurements\]c, where the thermally activated portion of $\sigma_{yy}$ as a function of $\Delta_{\nu}/T$ is plotted for integer filling factors. The critical conductivity $\sigma_{yy}^{C}=\sigma_{yy}\left(1/T\rightarrow0\right)$ typically observed in the IQH regime is $\sim2e^{2}/h$, with reductions possible due to short-range scattering and screening [@dtextquoterightambrumenil2011modelfor; @polyakov1995universal; @katayama1994experimental]. In our data we observe that the criticial conductivity at $\nu=2$ is 10 to 16 orders of magnitude larger than $\sigma_{yy}^{C}$ at higher filling factors. We therefore consider other interpretations for the large activation energy.
At $\nu=2$ one pair of valleys is filled, so added electrons must occupy a new valley with a different mass tensor. In this regime, Abanin et al. [@abanin2010nematic] predict a novel nematic phase with the electron gas broken up into domains of differing valley polarization. Indeed, it is characterized by a very narrow $T$ range where $R$ is thermally activated, leading to a large extrapolated $R$ for $1/T\rightarrow0$ in an Arrhenius plot of the data. Abanin et al. argue that in an anisotropic valley-degenerate system, the transport mechanism in the limit of valley-polarized domains is variable-range hopping due to the low $T$ localization of edge currents along domain walls separating areas of different valley polarization. Fits of the temperature dependent resistance to different models, including $\sigma=\left(\sigma^{C*}/T\right)\exp\left\{ \Delta_{\nu}/2k_{B}T\right\} $, reproduce the large energy gap and do not reduce the discrepancy in $\sigma_{yy}^{C}$ [@katayama1994experimental]. To the best of our knowledge, the thermal behavior reported here for $\nu=2$ has not been seen in any other QHE state.
In summary, we have shown evidence of electron-electron interactions in a high mobility Si(111) system. The six-fold valley degeneracy breaks at high magnetic fields into an apparent SU(2) symmetry reflected by the fractional quantum Hall state hierarchy. While the SU(2) symmetry is not unexpected due to the underlying valley structure of Si(111), the extended hierarchy reiterates the need to fully understand the valley degeneracy breaking mechanism in this system. We estimated the mass of composite fermions near $\nu=3/2$ by assuming that the Shubnikov-de Haas oscillation gaps can be interpreted as a cyclotron energy. Further experiments are necessary. First, hexagonal samples would allow unambiguous measurements of $\rho$ on Si(111) surfaces. Second, higher magnetic fields are necessary to probe the $\nu<1$ regime at similar densities, which would shed further light on the valley degeneracy of composite fermions. Finally, tilted magnetic fields would introduce controlled valley splitting of inequivalent valleys [@eng2007integer; @gokmen2008parallel] and in particular would help shed light on the behavior at $\nu=2$.
This work was funded by the Laboratory for Physical Sciences. The authors thank Jainendra Jain for useful discussions.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In earlier work, a decentralized optimal control framework was established for coordinating online connected and automated vehicles (CAVs) at urban intersections. The policy designating the sequence that each CAV crosses the intersection, however, was based on a first-in-first-out queue, imposing limitations on the optimal solution. Moreover, no lane changing, or left and right turns were considered. In this paper, we formulate an upper-level optimization problem, the solution of which yields, for each CAV, the optimal sequence and lane to cross the intersection. The effectiveness of the proposed approach is illustrated through simulation.'
author:
- 'Andreas A. Malikopoulos, [*[Senior Member, IEEE]{}*]{}, and Liuhui Zhao, [*[Member, IEEE]{}*]{} [^1][^2]'
bibliography:
- 'TCST\_references.bib'
title: '**Optimal Path Planning for Connected and Automated Vehicles at Urban Intersections** '
---
Introduction {#sec:1}
============
We are currently witnessing an increasing integration of our energy, transportation, and cyber networks, which, coupled with the human interactions, is giving rise to a new level of complexity in the transportation network. As we move to increasingly complex emerging mobility systems, new control approaches are needed to optimize the impact on system behavior of the interplay between vehicles at different transportation scenarios, e.g., intersections, merging roadways, roundabouts, speed reduction zones. These scenarios along with the driver responses to various disturbances [@Malikopoulos2013] are the primary sources of bottlenecks that contribute to traffic congestion [@Margiotta2011].
An automated transportation system [@Zhao2019] can alleviate congestion, reduce energy use and emissions, and improve safety by increasing significantly traffic flow as a result of closer packing of automatically controlled vehicles in platoons. One of the very early efforts in this direction was proposed in 1969 by Athans [@Athans1969] for safe and efficient coordination of merging maneuvers with the intention to avoid congestion. Varaiya [@Varaiya1993] has discussed extensively the key features of an automated intelligent vehicle-highway system and proposed a related control system architecture.
Connected and automated vehicles (CAVs) provide the most intriguing opportunity for enabling decision makers to better monitor transportation network conditions and make better operating decisions to improve safety and reduce pollution, energy consumption, and travel delays. Several research efforts have been reported in the literature on coordinating CAVs at at different transportation scenarios, e.g., intersections, merging roadways, roundabouts, speed reduction zones. In 2004, Dresner and Stone [@Dresner2004] proposed the use of the reservation scheme to control a single intersection of two roads with vehicles traveling with similar speed on a single direction on each road. Since then, several approaches have been proposed [@Dresner2008; @DeLaFortelle2010] to maximize the throughput of signalized-free intersections including extensions of the reservation scheme in [@Dresner2004]. Some approaches have focused on coordinating vehicles at intersections to improve travel time [@Yan2009]. Other approaches have considered minimizing the overlap in the position of vehicles inside the intersection, rather than arrival time [@Lee2012]. Kim and Kumar [@Kim2014] proposed an approach based on model predictive control that allows each vehicle to optimize its movement locally in a distributed manner with respect to any objective of interest. A detailed discussion of the research efforts in this area that have been reported in the literature to date can be found in [@Malikopoulos2016a].
In earlier work, a decentralized optimal control framework was established for coordinating online CAVs in different transportation scenarios, e.g., merging roadways, urban intersections, speed reduction zones, and roundabouts. The analytical solution without considering state and control constraints was presented in [@Rios-Torres2015], [@Rios-Torres2], and [@Ntousakis:2016aa] for coordinating online CAVs at highway on-ramps, in [@Zhang2016a] at two adjacent intersections, and in [@Malikopoulos2018a] at roundabouts. The solution of the unconstrained problem was also validated experimentally at the University of Delaware’s Scaled Smart City using 10 CAV robotic cars [@Malikopoulos2018b] in a merging roadway scenario. The solution of the optimal control problem considering state and control constraints was presented in [@Malikopoulos2017] at an urban intersection.
However, the policy designating the sequence that each CAV crosses the intersection in the aforementioned approaches, was based on a first-in-first-out queue, imposing limitations on the optimal solution. Moreover, no lane changing, or left and right turns were considered. In this paper, we formulate an upper-level optimization problem, the solution of which yields, for each CAV, the optimal sequence and lane to cross the intersection. The effectiveness of the solution is illustrated through simulation.
The structure of the paper is organized as follows. In Section II, we formulate the problem of vehicle coordination at an urban intersection and provide the modeling framework. In Section III, we briefly present the analytical, closed form solution for the low-level optimization problem. In Section IV, we present the upper-level optimization problem the solution of yields, for each CAV, the optimal sequence and lane to cross the intersection. Finally in Section V, we validate the effectiveness of the solution through simulation. We offer concluding remarks in Section VI.
Problem Formulation {#sec:2}
===================
Modeling Framework {#sec:2a}
------------------
We consider CAVs at a 100% penetration rate crossing a signalized-free intersection (Fig. \[fig:1\]). The region at the center of the intersection, called *merging zone*, is the area of potential lateral collision of the vehicles. The intersection has a *control zone* inside of which the CAVs can communicate with each other and with the intersection’s *crossing protocol*. The *crossing protocol*, defined formally in the next subsection, stores the vehicles’ path trajectories from the time they enter until the time they exit the control zone. The distance from the entry of the control zone until the entry of the merging zone is $S_c$ and, although it is not restrictive, we consider to be the same for all entry points of the control zone. We also consider the merging zone to be a square of side $S_m$ (Fig. \[fig:1\]). Note that the length $S_c$ could be in the order of hundreds of $m$ depending on the crossing protocol’s communication range capability, while $S_m$ is the length of a typical intersection. The CAVs crossing the intersection can also make a right turn of radius $R_r$, or a left turn of radius $R_l$ (Fig. \[fig:1\]). The intersection’s geometry is not restrictive in our modeling framework, and is used only to determine the total distance travelled by each CAV inside the control zone.
![A signalized-free intersection.[]{data-label="fig:1"}](figures/fig1.pdf){width="3.4"}
Let $\mathcal{N}(t)=\{1,\ldots,N(t)\}$, $N(t)\in\mathbb{N}$, be the set of CAVs inside the control zone at time $t\in\mathbb{R}^{+}$. Let $t_{i}^{f}$ be the assigned time for vehicle $i$ to exit the control zone. There is a number of ways to assign $t_{i}^{f}$ for each vehicle $i$. For example, we may impose a strict first-in-first-out queuing structure [@Malikopoulos2017], where each CAV must exit the control zone in the same order it entered the control zone. The policy, which determines the time $t_{i}^{f}$ that each vehicle $i$ exits the control zone, is the result of an upper-level optimization problem and can aim at maximizing the throughput of the intersection. On the other hand, deriving the optimal control input (minimum acceleration/deceleration) for each vehicle $i$ from the time $t_{i}^{0}$ it enters the control zone to achieve the target $t_{i}^{f}$ can aim at minimizing its energy [@Malikopoulos2010a].
In what follows, we present a two-level, joint optimization framework: (1) an upper level optimization that yields for each CAV $i\in\mathcal{N}(t)$ with a given origin (entry of the control zone) and desired destination (exit of the control zone) the sequence that will be exiting the control zone, namely, (a) minimum time $t_{i}^{f}$ to exit the control zone and (b) optimal path including the lanes that each CAV should be occupying while traveling inside the control zone; and (2) a low-level optimization that yields, for CAV $i\in\mathcal{N}(t),$ its optimal control input (acceleration/deceleration) to achieve the optimal path and $t_{i}^{f}$ derived in (1) subject to the state, control, and safety constraints.
The two-level optimization framework is used by each CAV $i\in\mathcal{N}(t)$ as follows. When vehicle $i$ enters the control zone at $t_{i}^{0}$, it accesses the intersection’s *crossing protocol* that includes the path trajectories, defined formally in the next subsection, of all CAVs inside the control zone. Then, vehicle $i$ solves the upper-level optimization problem and derives the minimum time $t_{i}^{f}$ to exit the control zone along with its optimal path including the appropriate lanes that it should occupy. The outcome of the upper-level optimization problem becomes the input of the low-level optimization problem. In particular, once the CAV derives the minimum time $t_{i}^{f}$, it derives its minimum acceleration/deceleration profile, in terms of energy, to achieve the exit time $t_{i}^{f}$.
The implications of the proposed optimization framework are that CAVs do not have to come to a full stop at the intersection, thereby conserving momentum and energy while also improving travel time. Moreover, by optimizing each vehicle’s acceleration/deceleration, we minimize transient engine operation [@Malikopoulos2008b], and thus we have additional benefits in fuel consumption.
Vehicle Model, Constraints, and Assumptions {#sec:2b}
-------------------------------------------
In our analysis, we consider that each CAV $i\in\mathcal{N}(t)$ is governed by the following dynamics $$%
\begin{split}
\dot{p}_{i} & =v_{i}(t)\\
\dot{v}_{i} & =u_{i}(t)\\
\dot{s}_{i} & = \xi_i \cdot (v_{k}(t)-v_{i}(t))
\label{eq:model2}
\end{split}$$ where $p_{i}(t)\in\mathcal{P}_{i}$, $v_{i}(t)\in\mathcal{V}_{i}$, and $u_{i}(t)\in\mathcal{U}_{i}$ denote the position, speed and acceleration/deceleration (control input) of each vehicle $i$ inside the control zone at time $t\in[t_{i}^{0}, t_{i}^{f}]$, where $t_i^0$ and $t_i^f$ are the times that vehicle $i$ enters and exits the control zone respectively; $s_{i}(t)\in\mathcal{S}_{i}$, with $s_{i}(t)=p_{k}(t)-p_{i}(t),$ denotes the distance of vehicle $i$ from the CAV $k\in\mathcal{N}(t)$ which is physically immediately ahead of $i$ in the same lane, and $\xi_{i}$ is a reaction constant of vehicle $i$. The sets $\mathcal{P}_{i}$,$\mathcal{V}_{i}$, $\mathcal{U}_{i}$, and $\mathcal{S}_{i}$, $i\in\mathcal{N}(t),$ are complete and totally bounded subsets of $\mathbb{R}$.
Let $x_{i}(t)=\left[p_{i}(t) ~ v_{i}(t) ~ s_{i}(t)\right] ^{T}$ denote the state of each vehicle $i$ taking values in $\mathcal{X}_{i}%
=\mathcal{P}_{i}\times\mathcal{V}_{i}\times\mathcal{S}_{i}$, with initial value $x_{i}(t_{i}^{0})=x_{i}^{0}=\left[p_{i}^{0} ~ v_{i}^{0} ~s_{i}^{0}\right] ^{T},$ where $p_{i}^{0}= p_{i}(t_{i}^{0})=0$, $v_{i}^{0}= v_{i}(t_{i}^{0})$, and $s_{i}^{0}= s_{i}(t_{i}^{0})$ at the entry of the control zone. The state space $\mathcal{X}_{i}$ for each vehicle $i$ is closed with respect to the induced topology on $\mathcal{P}_{i}\times
\mathcal{V}_{i}\times\mathcal{S}_{i}$ and thus, it is compact. We need to ensure that for any initial state $(t_i^0, x_i^0)$ and every admissible control $u(t)$, the system has a unique solution $x(t)$ on some interval $[t_i^0, t_i^f]$. The following observations from satisfy some regularity conditions required both on the state equations and admissible controls $u(t)$ to guarantee local existence and uniqueness of solutions for : a) the state equations are continuous in $u$ and continuously differentiable in the state $x$, b) the first derivative of the state equations in $x$, is continuous in $u$, and c) the admissible control $u(t)$ is continuous with respect to $t$.
To ensure that the control input and vehicle speed are within a given admissible range, the following constraints are imposed. $$\begin{gathered}
%
u_{i,min} \leq u_{i}(t)\leq u_{i,max}, \label{speed_accel constraints} \quad\text{and}\\
0 < v_{min}\leq v_{i}(t)\leq v_{max},\label{speed}\quad\forall t\in\lbrack t_{i}%
^{0},t_{i}^{f}],\end{gathered}$$ where $u_{i,min}$, $u_{i,max}$ are the minimum deceleration and maximum acceleration for each vehicle $i\in\mathcal{N}(t)$, and $v_{min}$, $v_{max}$ are the minimum and maximum speed limits respectively.
To ensure the absence of rear-end collision of two consecutive vehicles traveling on the same lane, the position of the preceding vehicle should be greater than or equal to the position of the following vehicle plus a predefined safe distance $\delta_i(t)$. Thus we impose the rear-end safety constraint $$\begin{split}
s_{i}(t)=\xi_i \cdot (p_{k}(t)-p_{i}(t)) \ge \delta_i(t),~ \forall t\in [t_i^0, t_i^f].
\label{eq:rearend}
\end{split}$$ We consider constant time headway instead of constant distance that each vehicle should keep when following the other vehicles, thus, the minimum safe distance $\delta_i(t)$ is expressed as a function of speed $v_i(t)$ and minimum time headway between vehicle $i$ and its preceding vehicle $k$, denoted as $\rho_i$. $$\begin{split}
\delta_i(t)=\gamma_i + \rho_i \cdot v_i(t),~ \forall t\in [t_i^0, t_i^f],
\label{eq:safedist}
\end{split}$$ where $\gamma_i$ is the standstill distance (i.e., the distance between two vehicles when they both stop).
A lateral collision can occur if a vehicle $j\in\mathcal{N}(t)$ cruising on a different road from $i$ inside the merging zone. In this case, the lateral safety constraint between $i$ and $j$ is $$\begin{split}
s_{i}(t)=\xi_i \cdot (p_{j,i}(t)-p_{i}(t)) \ge \delta_i(t),~ \forall t\in [t_i^0, t_i^f],
\label{eq:lateral}
\end{split}$$ where $p_{j,i}(t)$ is the distance of vehicle $j$ from the entry point that vehicle $i$ entered the control zone.
\[def:lanes\] The set of all lanes at the roads of the intersection is denoted by $\mathcal{L}:=\{1,\dots,M\}, M\in\mathbb{N}.$
\[def:lanesfunction\] For each vehicle $i\in\mathcal{N}(t)$, the function $l_i(t): [t_i^0, t_i^f]\to \mathcal{L}$ yields the lane the vehicle $i$ occupies inside the control zone at time $t$.
\[def:cardinal\] For each vehicle $i\in\mathcal{N}(t)$, the pair of the cardinal point that the vehicle enters the control zone and the cardinal point that the vehicle exits the control zone is denoted by $o_i$.
For example, based on Definition \[def:cardinal\], for a vehicle $i$ that enters the control zone from the West entry (Fig. \[fig:1\]) and exits the control zone from the South exit, $o_i=(W,S)$.
\[def:path\] For each vehicle $i\in\mathcal{N}(t)$, the function $t_{p_i,l_i}\big(p_i(t),l_i(t)\big): \mathcal{P}_i\times \mathcal{L}\to[t_i^0, t_i^f],$ is called the *path trajectory* of vehicle $i$, and it yields the time when vehicle $i$ is at the position $p_i(t)$ inside the control zone and occupies lane $l_i(t)$.
\[def:protocol\] The intersection’s *crossing protocol* is denoted by $\Pi(t)$ and includes the following information $$\begin{gathered}
\label{eq:protocol}
\Pi(t):=\{t_{p_i,l_i}\big(p_i(t),l_i(t)\big), l_i(t), o_i, t_i^0, t_i^f\}, \\ \nonumber
\forall i\in\mathcal{N}(t), t\in\mathbb{R}^+.
\end{gathered}$$
\[ass:feas\] The vehicles traveling inside the control zone can change lanes either (1) in the lateral direction (e.g., move to a neighbor lane), or (2) when making a right (or a left) turn inside the merging zone. In the former case, when the vehicle changes lane it travels along the hypotenuse $dy$ of the triangle created by the width of the lane and the longitudinal displacement $dp$ if it had not changed lane. Thus, in this case, the vehicle travels an additional distance which is equal to the difference between the hypotenuse $dy$ and the longitudinal displacement $dp$, i.e., $dy-dp$.
\[ass:feas\] When a vehicle is about to make a right turn it must occupy the right lane of the road before it enters the merging zone. Similarly, when a vehicle is about to make a left turn it must occupy the left lane before it enters the merging zone.
In the modeling framework presented above, we impose the following assumptions:
\[ass:lane\] The vehicle’s additional distance $dy-dp$ traveled when it changes lanes in the lateral direction can be neglected.
\[ass:noise\] Each CAV $i\in\mathcal{N}(t)$ has proximity sensors and can communicate with other CAVs and the *crossing protocol* without any errors or delays.
The first assumption can be justified since we consider an intersection and the speed limit inside the control zone is relatively low, hence $dy\approx dp$. The second assumption may be strong, but it is relatively straightforward to relax it as long as the noise in the communication, measurements and delays are bounded. In this case, we can determine upper bounds on the state uncertainties as a result of sensing or communication errors and delays, and incorporate these into more conservative safety constraints.
When each vehicle $i$ with a given $o_i$ enters the control zone, it accesses the intersection’s *crossing protocol* and solves two optimization problems: (1) an upper-level optimization problem, the solution of which yields its path trajectory $t_{p_i,l_i}\big(p_i(t),l_i(t)\big)$ and the minimum time $t_{i}^{f}$ to exit the control zone; and (2) a low-level optimization problem, the solution of which yields its optimal control input (acceleration/deceleration) to achieve the optimal path and $t_{i}^{f}$ derived in (1) subject to the state, control, and safety constraints.
We start our exposition with the low-level optimization problem, and then we discuss the upper-level problem.
Low-level optimization {#sec:3}
======================
In this section, we consider that the solution of the upper-level optimization problem is given, and thus, the minimum time $t_{i}^{f}$ for each vehicle $i\in\mathcal{N}(t)$ is known, and we focus on a low-level optimization problem that yields for each vehicle $i$ the optimal control input (acceleration/deceleration) to achieve the assigned $t_{i}^{f}$ subject to the state, control, and safety constraints.
\[problem1\] Once $t_{i}^{f}$ is determined, the low-level problem for each vehicle $i\in\mathcal{N}(t)$ is to minimize the cost functional $J_{i}(u(t))$, which is the $L^2$-norm of the control input in $[t_i^0, t_i^f]$ $$\begin{gathered}
\label{eq:decentral}
\min_{u(t)\in U_i} J_{i}(u(t))= \frac{1}{2} \int_{t^0_i}^{t^f_i} u^2_i(t)~dt,\\
\text{subject to}%
:\eqref{eq:model2},\eqref{speed_accel constraints},\eqref{speed}, \eqref{eq:rearend},\nonumber\\
\text{and given }t_{i}^{0}\text{, }v_{i}^{0}\text{, }t_{i}^{f}\text{,
}p_{i}(t_{i}^{0})\text{, }p_{i}(t_{i}^{f}),\nonumber\end{gathered}$$ where $p_{i}(t_{i}^{0})=0$, while the value of $p_{i}(t_{i}^{f})$ for each $i\in\mathcal{N}(t)$ depends on $o_i$ and, based on Assumption \[ass:lane\], can take the following values (Fig. \[fig:1\]): (1) $p_{i}(t_{i}^{f})=2 S_c + S_m$, if the CAV crosses the merging zone, (2) $p_{i}(t_{i}^{f})=2 S_c + \frac{\pi R_r}{2}$, if the CAV makes a right turn at the merging zone, and (3) $p_{i}(t_{i}^{f})=2 S_c + \frac{\pi R_l}{2}$, if the CAV makes a left turn at the merging zone.
For the analytical solution of , we formulate the Hamiltonian $$\begin{gathered}
H_{i}\big(t, p_{i}(t), v_{i}(t), s_{i}(t), u_{i}(t)\big) \nonumber \\
=\frac{1}{2} u_i(t)^{2}_{i} + \lambda^{p}_{i} \cdot v_{i}(t) + \lambda^{v}_{i} \cdot u_{i}(t) +\lambda^{s}_{i} \cdot \xi_i \cdot (v_{k}(t) - v_{i}(t)) \nonumber\\
+ \mu^{a}_{i} \cdot(u_{i}(t) - u_{max})
+ \mu^{b}_{i} \cdot(u_{min} - u_{i}(t)) \nonumber\\
+ \mu^{c}_{i} \cdot u_{i}(t) - \mu^{d}_{i} \cdot u_{i}(t) \nonumber\\
+ \mu^{s}_{i} \cdot (\rho_i \cdot u_i(t) - \xi_i\big(v_{k}(t) - v_i(t)\big)) ,\label{eq:16b}\end{gathered}$$ where $\lambda^{p}_{i}$, $\lambda^{v}_{i}$, and $\lambda^{s}_{i}$ are the influence functions [@Bryson:1963], and $\mu^{T}$ is the vector of the Lagrange multipliers. To address this problem, the constrained and unconstrained arcs will be pieced together to satisfy the Euler-Lagrange equations and necessary condition of optimality.
For the case that none of the state and control constraints become active, the optimal control is [@Malikopoulos2019ACC] $$u^{*}_{i}(t) = (a_{i} - b_{i} \cdot \xi_i) \cdot t + c_{i}, ~ t \in[t^{0}_{i}, t_i^f]. \label{eq:20}$$ Substituting the last equation into we find the optimal speed and position for each vehicle, namely $$\begin{gathered}
v^{*}_{i}(t) = \frac{1}{2} (a_{i} - b_{i} \cdot \xi_i) \cdot t^2 + c_{i} \cdot t +d_{i}, ~ t \in[t^{0}_{i}, t_i^f], \label{eq:21}\\
p^{*}_{i}(t) = \frac{1}{6} (a_{i} - b_{i} \cdot \xi_i) \cdot t^3 +\frac{1}{2} c_{i} \cdot t^2 + d_{i}\cdot t +e_{i}, \label{eq:22} \\~ t \in[t^{0}_{i}, t_i^f], \nonumber \end{gathered}$$ where $a_{i}$, $b_{i}$, $c_{i}$, $d_{i}$ and $e_{i}$ are constants of integration that can be computed by the initial, final, and transversality conditions [@Malikopoulos2019ACC].
Upper-level optimization {#sec:4}
========================
When a vehicle $i\in\mathcal{N}(t),$ with a given $o_i$, enters the control zone, it accesses the intersection’s *crossing protocol* and solves an upper-level optimization problem. The solution of this problem yields for $i$ the path trajectory $t_{p_i,l_i}\big(p_i(t),l_i(t)\big)$ and the minimum time $t_{i}^{f}$ to exit the control zone. In our exposition, we seek to derive the minimum $t_{i}^{f}$ without activating any of the state and control constraints of the low-level optimization Problem \[problem1\]. Therefore, the upper-level optimization problem should yield a $t_{i}^{f}$ such that the solution of the low-level optimization problem will result in the unconstrained case - .
There is an apparent trade off between the two problems. The lower the value of $t_{i}^{f}$ in the upper-level problem, the higher the value of the control input in $[t_{i}^{0}, t_{i}^{f}]$ in the low-level problem. The low-level problem is directly related to minimizing energy for each vehicle (individually optimal solution). On the other hand, the upper-level problem is related to maximizing the throughput of the intersection, thus eliminating stop-and-go driving (social optimal solution). Therefore, by seeking a solution for the upper-level problem which guarantees that none of the state and control constraints become active may be considered an appropriate compromise between the two.
For simplicity of notation, for each vehicle $i\in\mathcal{N}(t)$ we write the optimal position of the unconstrained case in the following form
$$\begin{gathered}
p^{*}_{i}(t) = \phi_{i,3} \cdot t^3 +\phi_{i,2} \cdot t^2 + \phi_{i,1} \cdot t +\phi_{i,0} , ~ t\in [t_{i}^{0}, t_{i}^{f}], \label{eq:upper_p}%\end{gathered}$$
where $\phi_{i,3}, \phi_{i,2}, \phi_{i,1}, \phi_{i,0}\in\mathbb{R}$ are the constants of integration derived in the Hamiltonian analysis, in Section \[sec:3\], for the unconstrained case.
\[rem:3\] For each $i\in\mathcal{N}(t),$ the optimal position is a continuous and differentiable function. Based on , it is also an increasing function with respect to $t\in\mathbb{R}^+$.
Next, we investigate some properties of .
\[lem:1\] For each $i\in\mathcal{N}(t)$, the optimal position $p_i^*$ given by is an one-one function.
Since, for each $i\in\mathcal{N}(t),$ $p_i^*(t)$ is an increasing function with respect to $t\in\mathbb{R}^+$ and from , for any $t_1, t_2\in[t_{i}^{0}, t_{i}^{f}]$, $p_i^*(t_{1})\neq p_i^*(t_{2}).$
\[cor:1\] Since, for each $i\in\mathcal{N}(t),$ is an one-one function, there exist an inverse function $p_i^*(t)^{-1}$ such that $$\begin{gathered}
\label{eq:upper_inversep}
p^{*}_{i}(t)^{-1} = \omega_{i,3} \cdot p^3 +\omega_{i,2} \cdot p^2 + \omega_{i,1} \cdot p +\omega_{i,0} , \end{gathered}$$ where $\omega_{i,3}, \omega_{i,2}, \omega_{i,1}, \omega_{i,0}\in\mathbb{R}$ are constants that are a function of $\phi_{i,3}, \phi_{i,2}, \phi_{i,1}, \phi_{i,0}$.
\[rem:4\] For each $i\in\mathcal{N}(t),$ $t\in [t_{i}^{0}, t_{i}^{f}]$, we rewrite as follows $$\begin{gathered}
p^{*}_{i}(t) = \phi_{i,3} \cdot t_i^3 +\phi_{i,2} \cdot t_i^2 + \phi_{i,1} \cdot t_i +\phi_{i,0}. \label{eq:upper_pi}%
\end{gathered}$$
\[lem:2\] Let $p^{*}_{i}(t)^{-1}$ be the inverse function of for each vehicle $i\in\mathcal{N}(t).$ Then the constants $\phi_{i,3}, \phi_{i,2}, \phi_{i,1}, \phi_{i,0}\in\mathbb{R}$ can be derived by $\omega_{i,3}, \omega_{i,2}, \omega_{i,1}, \omega_{i,0}\in\mathbb{R}.$
Due to space limitation the proof is omitted. However, the result is trivial.
\[rem:5\] The inverse function $p_i^*(t)^{-1}=t_i(p^*(t)),$ where $t_i(p^*(t))\in[t_{i}^{0}, t_{i}^{f}]$, yields the time that vehicle $i\in\mathcal{N}(t)$ is at the position $p^{*}_{i}(t)$ inside the control zone.
\[lem3\] For each $i\in\mathcal{N}(t),$ the domain of $t_i(p^*(t))$ is the closed interval $[p_i(t_{i}^{0}),p_i(t_{i}^{f})]$.
Since, for each $i\in\mathcal{N}(t),$ $p_i^*(t)$ is an increasing function in $[t_{i}^{0}, t_{i}^{f}]$, then by the Intermediate Value Theorem, $p_i^*(t)$ takes values on the closed interval $[p_i(t_{i}^{0}),p_i(t_{i}^{f})]$.
\[cor:2\] Since $p_i^*(t)$ is a continuous and one-one function in $[t_{i}^{0}, t_{i}^{f}]$ for each $i\in\mathcal{N}(t),$ then $t_i(p^*(t))$ is also continuous.
\[cor:3\] For each $i\in\mathcal{N}(t),$ $p'\big(t_i(p(t)) \big)\neq 0$ for all $p\in [p_i(t_{i}^{0}),p_i(t_{i}^{f})]$. Hence, $t_i(p^*(t))$ is differentiable in $[p_i(t_{i}^{0}),p_i(t_{i}^{f})]$.
\[lem4\] For each $i\in\mathcal{N}(t),$ $t_i(p^*(t))$ is an increasing function in $[p_i(t_{i}^{0}),p_i(t_{i}^{f})]$.
From Lemma \[lem3\], for each $i\in\mathcal{N}(t)$ the domain of $t_i(p^*(t))$ is $[p_i(t_{i}^{0}), p_i(t_{i}^{f})]$. Let $p_i(t_{i}^{0})<\alpha_1<\alpha_2< p_i(t_{i}^{f})$ with $t_i(p_i(t_{i}^{0}) < t_i(\alpha_1)$. If we had $t_i(p_i(t_{i}^{0})) > t_i(\alpha_2),$ then by applying the Intermediate Value Theorem to the interval $[\alpha_2, p_i(t_{i}^{f})]$ would give an $\alpha_3$ with $\alpha_2<\alpha_3< p_i(t_{i}^{f})$ and $t_i(p_i(t_{i}^{0})) = t_i(\alpha_3)$ contradicting the fact that $t_i(p^*(t))$ is one-one on $[p_i(t_{i}^{0}),p_i(t_{i}^{f})]$.
Since each vehicle $i\in\mathcal{N}(t)$ can change lanes inside the control zone, its position should be associated with the function $l_i(t)$ (Definition \[def:lanesfunction\]) that yields the lane vehicle $i$ occupies inside the control zone at $t$.
\[def:pos\_lane\] The position of each vehicle $i\in\mathcal{N}(t)$ using lane $l_i(t)$, $m\in\mathcal{L},$ is denoted by $p_{i,l}(t,l)$.
Based on Definition \[def:pos\_lane\], we augment the optimal position of $i\in\mathcal{N}(t)$ given by to capture the lane that vehicle $i$ as follows $$\begin{gathered}
p^*_{i}(t,l) = p^*(t)\cdot {I}_{1}(l) + p^*(t)\cdot {I}_{2}(l) + \dots + p^*(t)\cdot {I}_{M}(l), \nonumber \\
t\in [t_{i}^{0}, t_{i}^{f}], \label{eq:upper_pl}%\end{gathered}$$ where ${I}_{m}(l)$, $m\in\mathcal{L}$, is the indicator function with ${I}_{m}(l=m)=1$, if $i$ occupies lane $m\in\mathcal{L}$ and ${I}_{m}(l\neq m)=0$ otherwise. For each vehicle $i\in\mathcal{N}(t)$, the inverse function of enhanced with the lane that vehicle $i$ occupies is the path trajectory (Definition \[def:path\]) and can be written as follows $$\begin{gathered}
t_{p_i,l_i}(p_i(t),l_i(t))) = \omega_{i,3} \cdot p^3_{i}(t,l) +\omega_{i,2} \cdot p^2_{i}(t,l) \nonumber \\
+\omega_{i,1} \cdot p_{i}(t,l) +\omega_{i,0} . \label{eq:path_traj}\end{gathered}$$
The path trajectory $t_{p_i,l_i}(p_i(t),l_i(t))) $ yields the time that vehicle $i$ is at the position $p_i(t)$ inside the control zone and occupies lane $l_i(t)$ and is used as the cost function for the upper-level optimization problem.
In the upper-level optimization problem, each vehicle $i\in\mathcal{N}(t)$ derives its optimal path trajectory which yields the minimum time $t_i^f$ that vehicle $i$ exits the control zone along with the lane $l^*\in\mathcal{L}$ that should occupy at each $p_i^*$. To formulate this problem, we need to minimize , evaluated at $p_i(t_i^f),$ with respect to $\omega_{i,3}, \omega_{i,2}, \omega_{i,1}, \omega_{i,0}$ that determine the shape of the path trajectory of the vehicle in $[p_i^0, p_i^f]$. Note that the value of $p_{i}(t_{i}^{f})$ for each $i\in\mathcal{N}(t)$ depends on $o_i$ and, based on the Assumption \[ass:lane\], it can be equal to (see Fig. \[fig:1\]): (1) $p_{i}(t_{i}^{f})=2 S_c + S_m$, if the vehicle crosses the merging zone, (2) $p_{i}(t_{i}^{f})=2 S_c + \frac{\pi R_r}{2}$, if the vehicle makes a right turn at the merging zone, and (3) $p_{i}(t_{i}^{f})=2 S_c + \frac{\pi R_l}{2}$, if the vehicle makes a left turn at the merging zone. For simplicity of notation, we denote the total distance travelled by the vehicle $i\in\mathcal{N}(t)$ in $[t_{i}^{0}, t_{i}^{f}]$ with $S_{i,total}$, thus $p_{i}(t_{i}^{f})=S_{i,total}$. Hence, the upper-level optimization problem is formulated as follows.
\[problem2\]
$$\begin{gathered}
\label{eq:decentral2}
\min_{\omega_{i,3}, \omega_{i,2}, \omega_{i,1}, \omega_{i,0}}t_{p_i,l_i}\big(S_{i,total},l_i(t)\big)\\
% =\min_{\omega_{i,3}, \omega_{i,2}, \omega_{i,1}, \omega_{i,0}\in\mathbb{R}} \Big(\omega_{i,3} \cdot S_{i,total}^3 +\omega_{i,2} \cdot S_{i,total}^2 \nonumber\\
% + \omega_{i,1} \cdot S_{i,total} +\omega_{i,0} \Big)\nonumber\\
\text{subject to}%
: \eqref{speed_accel constraints},\eqref{speed}, \eqref{eq:rearend}, \text{and given }t_{i}^{0}\text{, }v_{i}^{0}\text{, }t_{i}^{f}\text{,
}p_{i}(t_{i}^{0})\text{, }p_{i}(t_{i}^{f}).\nonumber
\end{gathered}$$
From Lemma \[lem:2\], the constants $\phi_{i,3}, \phi_{i,2}, \phi_{i,1}, \phi_{i,0}\in\mathbb{R}$ corresponding to the constraints imposed through can be derived by $\omega_{i,3}, \omega_{i,2}, \omega_{i,1}, \omega_{i,0}\in\mathbb{R}$. This is a nonlinear programming problem that each vehicle can solve using Lagrange multiplier theory.
Simulation Results {#sec:5}
==================
Validation of Upper-Level Optimization {#sec:5a}
--------------------------------------
To evaluate the effectiveness of the solution of the proposed upper-level optimization problem, we conduct a simulation in MATLAB. The simulation setting is as follows. The intersection contains two roads, each of which has one lane per direction. The length of each direction is 300 $m$, the merging zone of the intersection is 25 $m$ by 25 $m$, and the entry of merging zone is located at 125 $m$ from the entry point for both directions. The maximum and minimum speed are 18 $m/s$ and 2 $m/s$, respectively. The maximum and minimum acceleration are 3.0 $m/s^2$ and -3.0 $m/s^2$. The safety (minimum allowed) headway is 1.0 $s$, and the standstill distance is 1.5 $m$. Six vehicles are entering into the intersection from three directions at different time steps. Since vehicle 1 is the first vehicle in the network, it cruises through the intersection without any constraints imposed. The trajectories of all vehicles along with the safety distance are shown in Fig. \[fig:distance\]. Negative values of the safety distance means violation of the rear-end constraint. We see from Fig. \[fig:distance\] that both rear-end and lateral collision constraints are satisfied. The control input (acceleration) and speed profiles for the vehicles in the network is shown in Fig. \[fig:speed\]. We note that for all vehicles driving through the intersection, none of the acceleration and speed constraints are activated.
![Trajectories and safety distances of vehicles.[]{data-label="fig:distance"}](figures/distance_v2.jpg){width=".48\textwidth"}
![Speed and control profiles of vehicles.[]{data-label="fig:speed"}](figures/speed_v2.jpg){width=".48\textwidth"}
Concluding Remarks
==================
In this paper, we formulated an upper-level optimization problem, the solution of which yields, for each CAV, the optimal sequence and lane to cross the intersection. The effectiveness of the solution was illustrated through simulation. We showed, through numerical results, that vehicles are successfully crossing an intersection without any rear-end or lateral collision. In addition, the state and control constraints did not become active for the entire trajectory for each vehicle. While the potential benefits of full penetration of CAVs to alleviate traffic congestion and reduce energy have become apparent, different penetrations of CAVs can alter significantly the efficiency of the entire system. Therefore, future research should look into this direction.
[^1]: This research was supported in part by ARPAE’s NEXTCAR program under the award number DE-AR0000796 and by the Delaware Energy Institute (DEI).
[^2]: The authors are with the Department of Mechanical Engineering, University of Delaware, Newark, DE 19716 USA (email: ;
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The theoretical approach proposed recently for description of redistribution of electronic charge in multilayered selectively doped systems is modified for a system with finite number of layers. A special attention is payed to the case of a finite heterostructure made of copper-oxide layers which are all non-superconducting (including non-conducting) because of doping levels being beyond the well-known characteristic interval for superconductivity. Specific finite structures and doping configurations are proposed to obtain atomically thin superconducting heterojunctions of different compositions.'
author:
- 'V. M. Loktev$^1$ and Yu. G. Pogorelov$^2$'
title: 'Superconducting junctions from non-superconducting doped CuO$_2$ layers'
---
An interesting area in nanoengineering of materials was opened in a series of experiments by Bozovič *et al* [@Boz1] on atomically perfect stacks of selectively doped perovskite layers. These and some other papers [@Goz1; @Boz2] mainly used periodic multilayered structures where essential new electronic effects, as interface SC between nominally non-SC layers [@Boz2], appeared due to charge redistribution between layers and related shifts of in-plane energy bands. The basic condition for SC to appear within few perovskite layers or even in a single layer is that the local density of hole charge carriers occurs within a definite, rather narrow, interval: $p_{min} \geq p \geq p_{max}$ with $p_{min} \approx 0.07$ and $p_{max} \approx 0.2$ (carriers per site). The required density distribution results from the corresponding shifts of in-plane energy bands by local Coulomb potentials. A simple theoretical model for such processes was proposed [@lok], combining a discrete version of Poisson equation for potential with a band-structure modified self-consistent Thomas-Fermi charge density. This approach gives exact solutions for infinite periodical and some other unbounded systems. However recent studies [@Goz2; @Smad; @Log] showed that pronounced modification of electronic ground state and related SC transitions can be obtained either in stacks of finite (and small) number of layers which is quite promising for practical applications in nanoengineered composite devices. The following consideration aims on an extension of the previous model on an arbitrary layered system and establishing criteria for its optimum SC performance. This line of study can be seen as a practical realization of long envisaged Ginzburg’s program for ultrathin superconducting states [@ginz].
Following the same assumptions as in Ref. [@lok], we express local charge density in *j*-th layer: $\r_j = e\left(p_j - x_j\right)$, through the densities $p_j$ of mobile holes and $x_j$ of ionized dopants ($e$ is the elementary charge) and then present the potential difference $\varphi_{j+1} - \varphi_j$ between neighbor layers as: \_[j+1]{} - \_j = (\_[l=j+1]{}\^N \_l - \_[l=1]{}\^j \_l). This value is obtained considering the electric field $E_{j,j+1}$ in the $j,j+1$ spacer as the geometric sum of fields $E_l$ emitted by each *l*-th charged layer: $E_l = 2\pi
\r_l/\left(\e_{\rm eff}a^2\right)$, on both sides of this spacer. Eq. \[eq1\] involves the in-plane and *c*-axis lattice constants $a$ and $c$, and $\e_{\rm eff}$ is the (static) dielectric constant that effectively reduces the Coulomb field in the *c*-direction. Eq. \[eq1\] would be exact for a stack of mathematical planes, with uniform in-plane charge densities $\r_j$ and separation $c$, and it should be a reasonable model for real ${\rm La}_{2-x}{\rm Sr}_x{\rm CuO}_4$ layers where $p_j$ delocalized holes and $x_j$ localized dopants are distributed in different atomic planes within the period $c$ of *j*th layer. The adopted form of purely dielectric screening is justified in neglect of *c*-hopping processes, accordingly to their above mentioned weakness, also this model neglects polarization effects from the insulating substrate. We note that the charge densities $\r_j$ naturally vanish in uniformly doped ($p_j = x_j$), including undoped ($p_j = x_j = 0$), systems.
![Schematic of nanostructured system with selectively introduced dopants (white circles) into each of $N$ layers of ${\rm La}_2{\rm CuO}_4$.[]{data-label="fig1"}](fig1.eps "fig:"){width="8cm"}\
Otherwise, the hole carrier density $p_j$ can be related to the local potential $\varphi_j$ using the respective density of states (DOS) $g_j(\e)$: p\_j = 2\_[\_[F]{}]{}\^[W/2 - e\_j]{}g\_j()d with the spin factor 2 (this zero-temperature formula is justified for all the considered temperatures $T \lesssim T_c$). Thus the role of *c*-hopping in this model is reduced to establishing the common Fermi level $\e_{\rm F}$ for all the layers. Using the simplest approximation of rectangular DOS: $g_j(\e) = 1/W$ within the bandwidth $W$, we arrive at the linear relation between $p_j$ and $\varphi_j$: e\_j = 2 W - \_[F]{}. Then, inserting Eq. \[eq3\] into Eq. \[eq1\], we obtain the linear relation between carrier and dopant densities, referred to the $j,j+1$ spacer: p\_[j+1]{} - p\_j = 2, where the dimensionless quantity: = is the single material parameter of the model. Finally, subtracting the relations, Eq. \[eq4\], for $j,j+1$ and $j - 1,j$ spacers leads to simple linear equations for the hole carrier densities in neighbor layers only: p\_[j+1]{} + p\_[j-1]{} - (2 + )p\_j = -x\_j. The advantage of Eq. \[eq5\] against possible analogous relations between the potentials $\varphi_j$ is in avoiding the need to know the Fermi level position (doping dependent). For an infinite stack of layers, summing these equations in all $j$ would automatically assure the total electroneutrality condition $\sum_j \r_j = 0$, and this was just the way used in Ref. [@lok] to obtain a more detailed alternative to the phenomenological Thomas-Fermi treatment [@Smad]. However, for a finite stack of $j = 1,\dots,N$ layers, this condition should be additionally imposed besides the $N - 1$ relations, Eq. \[eq4\], in order to completely define all the $N$ densities $p_j$. Since Eq. \[eq5\] in this case only applies for the *internal* layers $j = 2,\dots,N-1$, one needs two more equations which can be obtained from Eq. \[eq4\] for terminal layers, $j = 1$ and $j = N -1$, under the electroneutrality condition: (1 + )p\_1 - p\_2 & = & x\_1\
- p\_[N -1]{} + (1 + )p\_N & = & x\_N. Thus the non-uniform linear system of $N$ Eqs. \[eq5\], \[eq6\] can be presented in the matrix form as: (1 + \^[-1]{}L) = , where the finite $N$-stack of layers generates the “Laplacian” matrix: L = (
[cccccc]{} 1 & -1 & 0 & 0 & …& 0\
-1 & 2 & -1 & 0 & …& 0\
0 & -1 & 2 & -1 & …& 0\
…& …& …& …& …& …\
0 & …& 0 & -1 & 2 & -1\
0 & …& 0 & 0 & -1 & 1
) and the $N$-vectors: $$\overrightarrow{p} = \left(\begin{array}{c}
p_1 \\
\dots \\
p_N
\end{array}\right),\qquad{\rm and}\qquad \overrightarrow{x}=\left(\begin{array}{c}
x_1 \\
\dots \\
x_N
\end{array}\right).$$
![Modulated electronic configurations by shifted energy bands for the samples of selectively doped layered systems. a) For a finite stack of $N = 6$ layers with doping levels $x_1 = x_2 = x_3 = 0.45$, $x_4 = x_5 = x_6 = 0$ (light columns) and the localization parameter $\a = 1$, the carrier densities (dark columns) are calculated from Eqs. \[eq7\], \[eq8\]. b) For an unlimited system with periodic repetition of the same stack, those are calculated from Eqs. \[eq7\], \[eq9\]. The dashed lines mark the values $p_{min}$ and $p_{max}$, delimiting the interval of carrier densities where superconductivity should exist.[]{data-label="fig2"}](fig2.eps "fig:"){width="7cm"}\
It is seen from Eq. \[eq7\] that $\a$ plays the role of localization parameter: the carrier density strictly coincides with the doping distribution in the limit $\a \to \infty$, otherwise it is spread beyond this distribution, the stronger the smaller $\a$ is. Generally, the standard solution $\overrightarrow{p} = \hat R(\a)\overrightarrow{x}$ with the resolvent $\hat R =
\left(1 + \a^{-1}\hat L\right)^{-1}$ gives the densities $p_j$ in terms of the doping levels $x_j$ and of the localization parameter $\a$ as for the instance in Fig. \[fig2\] with $N =
6$, $\a = 1$ (a reliable estimate for real La$_2$CuO$_4$ [@pog; @kast; @chen]) and $x_1 = x_2
= x_3 = 0.45,\, x_4 = x_5 = x_6 = 0$. It is of interest to compare this solution to that for an unlimited system with the same distribution of dopants $x_j$ but periodically repeated, so that the $\hat L$ matrix is replaced by its periodic version: L’ = (
[cccccc]{} 2 & -1 & 0 & …& 0 & -1\
-1 & 2 & -1 & 0 & …& 0\
0 & -1 & 2 & -1 & …& 0\
…& …& …& …& …& …\
0 & …& 0 & -1 & 2 & -1\
-1 & 0 & …& 0 & -1 & 2
). A notable difference in the resulting distributions of densities $p_j$ is seen in Fig. \[fig2\]a,b. An important feature of superconducting layers formed in each of these structures (4th layer in Fig. \[fig2\]a and 4th and 6th layers in Fig. \[fig2\]b) is that they are realized in nominally undoped La$_2$CuO$_4$ and thus can be expected almost free of undesirable disorder effects (as scattering by defects and consequent fluctuations of the SC order parameter [@lang]).
![Comparison of calculated (dark columns) and experimentally determined (white circles) carrier densities for the real system with $N = 12$, the doping levels $x_1 = x_2
= x_4 = x_5 = x_6 = 0.45,\, x_7 = 0.17,\, x_8 = 0.06, x_9 = 0.02,\,x_{10} = x_{11} = x_{12}
= 0$ (light columns) and the localization parameter $\a = 1$.[]{data-label="fig3"}](fig3.eps "fig:"){width="8cm"}\
Also a comparison of such calculation with the experiment data [@Log] is presented in Fig. \[fig3\], using the real values of structure parameters: $N = 12$ and $x_1 = x_2 =
x_4 = x_5 = x_6 = 0.45,\, x_7 = 0.17,\, x_8 = 0.06, x_9 = 0.02$ (the three last values are due to the interdiffusion of Sr dopants into nominally undoped La$_2$CuO$_4$ layers), $x_{10}
= x_{11} = x_{12} = 0$ and the same localization parameter $\a = 1$ as before. The resulting distribution of carrier densities (black columns) displays an excellent coincidence with its measured values. In particular, the carrier density occurs within the SC interval just in the $N = 8$ layer with the value $p_8 \approx 0.106$ that just corresponds to the observed transition temperature $T_c \approx 32$ K when used in the phenomenological formula [@lok] $T_c(p) =
\left(p - p_{min}\right)\left(p_{max} - p\right)T^\ast$ with $T^\ast \approx 9000$ K.
From such a good agreement, one expects that this approach can be also used for design of new configurations with tailored superconducting performance. In particular, an interesting possibility is to build ultra-quantum heterojunctions of two types: superconductor- insulator-superconductor (SIS) and superconductor-normal metal-superconductor (SNS), each component being restricted to a single cuprate layer. Such junctions could realize a single layer limit of already discussed thicker SIS and SNS nanostructures with giant proximity effect [@Boz4]. It should be noted that since the localization parameter $\a$ value (see Eq. \[eq5\]) is rather fixed by the choice of the building material (with $\a \approx 1$ for La$_2$CuO$_4$), the practical control parameters in this process must be the total number $N$ of layers in the stack and particular doping levels $x_j$ in each layer.
Thus, one possible simple structure to produce a SNS junction can consist of $N = 5$ cuprate layers with the doping levels defined for instance as: $x_1 = x_5 = 0.45,\, x_2 = x_3 = x_4
= 0$ (that is, nominally all non-superconducting). From Eq. \[eq7\], the resulting carrier density distribution: $p_1 = p_5 \approx 0.29$, correspond to nomal (overdoped) metal layers, $p_2 = p_4 \approx 0.12$ to superconducting layers with a high enough transition temperature $T_c^{high} \approx 37$ K, separated by the layer with $p_3 \approx 0.08$ and low transition temperature $T_c^{low} \approx 11$ K. Then, in the temperature range $T_c^{low} < T <
T_c^{high}$ one should obtain a SNS heterostructure. If so, the quasiparticle spectrum of this junction will present a peculiar combination of gapped (a kind of size quantization) and gapless branches with interesting IR absorption and electric transport properties.
As to the SIS heterostructure, it is rather difficult to be obtained in the suggested $N =
5$ stack, but it can be achieved, e.g., by adding one more undoped layer (i.e., from 3 to 4) to the above structure, or introducing *electronic* doping in the central layer (making $x_3 < 0$). Unlike the above SNS case, the resulting SIS junction should display a quasiparticle spectrum with gapped branches only.
In conclusion, an extension of the recent electrostatic model for charge redistribution in non-uniformly doped multilayered systems is proposed for finite (mostly small) number of layers. Formal solutions of this model are mainly analyzed in the parameter range actual for the experimentally investigated ${\rm La}_{2-x}{\rm Sr}_x{\rm CuO}_4$ multilayers. The distinctions of finite stacks from previously studied unlimited or cyclic systems are indicated. Some new specific arrangements of doped and undoped layers are suggested for realization of artificial atomically thin heterostructures with unusual electronic excitation spectrum, potentially interesting for applications in nanocomputing devices. Such artificial structures may present also an interest for their behavior under applied magnetic field.
This work was partially supported by the Special Program of Fundamental Research of the Department of Physics and Astronomy of NAS of Ukraine. The authors are grateful to I. Bozovič for reading the manuscript and valuable discussion.
[1]{}
I. Bozovič, IEEE Trans. Appl. Superconduct. **11**, 2686 (2001).
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Topological insulators are solid state systems of independent electrons for which the Fermi level lies in a mobility gap, but the Fermi projection is nevertheless topologically non-trivial, namely it cannot be deformed into that of a normal insulator. This non-trivial topology is encoded in adequately defined invariants and implies the existence of surface states that are not susceptible to Anderson localization. This non-technical review reports on recent progress in the understanding of the underlying mathematical structures, with a particular focus on index theory.'
author:
- |
Hermann Schulz-Baldes\
Department Mathematik, FAU Erlangen-Nürnberg, Germany
date:
title: 'Topological insulators from the perspective of non-commutative geometry and index theory '
---
In solid state physics, one distinguishes between conductors and insulators. In conductors the electrons at the Fermi level can move through the solid and lead to electric currents, while in insulators there are either no such electrons due to a spectral gap or they are localized due to destructive interferences and thus cannot move freely, and one then speaks of a mobility gap due to Anderson localization. Whether a given solid is a conductor or an insulator can be determined from a quantum mechanical one-particle Hamiltonian modeling the electrons near the Fermi level. The parameters of this so-called tight-binding model are obtained as the exchange integrals between the orbitals of the chemical compounds of the solid.
In the last decade, a multitude of both theoretical and experimental physics contributions showed that insulators can be topological in the sense that the Fermi projections have non-trivial topological invariants (given in terms of winding numbers, Chern numbers, etc.). These topological insulators or topological materials are distinct from conventional normal insulators, and this difference manifests itself typically by non-trivial boundary physics. More precisely, on the boundaries of a topological material there are delocalized surface modes that make the insulator conducting after all. This makes such materials also interesting for technological applications. The connection between non-trivial topology and these surface modes is called the bulk-boundary correspondence. It is a very robust principle and also the main reason that the field more recently attracted the interest of a several mathematical physicists. Indeed, methods from $K$-theory, index theory and non-commutative geometry can be put to work in the models describing topological insulators. In some situations, the analysis even allows to explain experimental facts that can be observed in a lab. This review starts out with a short description of the physics and the underlying quantum mechanical models, and then offers an (admittedly personal) overview of recent mathematical results on them. Hopefully, this allows the interested reader to get a glimpse of the intrinsic beauty due to the materialization of deep mathematical concepts in real world solid state physics systems.
What is a topological insulator? {#sec-WhatIs}
================================
The title of this section and to some extent also its content may seem better fitted for a physics than a mathematics review. Indeed, what now follows are a few pages outlining a mathematicians simplistic view of the modern solid state physics background needed to describe topological insulators. The focus will here be on systems of electrons in a solid. These are microscopic particles, so there is no way of escaping quantum mechanics. On the other hand, in a first approximation, the electrons are often described as independent non-interacting particles only submitted to the Pauli exclusion principle, even though there are actually many of them in a solid and they seem to interact strongly via the Coulomb interaction. What is usually put forward as a justification is that screening plays an important role (from a distance, the charge of an electron close to a charged nucleus doesn’t look so large any more) and that, anyhow, a quasiparticle description of the excitations of an interacting electron system should hold. On rigorous grounds, very little is known on how to derive the effective quantum theories used below from first principle. A further standard simplification is the so-called tight-binding approximation. This means that each atom in the solid offers the itinerant electron only a finite number of bound states, say $L$ of them. The single-electron quantum Hamiltonian $H$ in the $d$-dimensional tight-binding approximation is then a linear, self-adjoint and discrete Schrödinger operator on the Hilbert space $\ell^2({{\mathbb Z}}^d)\otimes{{\mathbb C}}^L$ which is typically of the form $$\label{eq-Hbasic}
H\;=\;
\Delta^B\,+\,W
\;.$$ Here the operator $\Delta^B$ is the kinetic part given by a discrete magnetic Laplacian with magnetic field $B$ and $W$ is matrix-valued potential, namely an operator which commutes the position operator $X$ on $\ell^2({{\mathbb Z}}^d)$, trivially extended to $\ell^2({{\mathbb Z}}^d)\otimes{{\mathbb C}}^L$ (for more details, see below). This potential is, unlike in scattering theory, homogeneous in space. The easiest example of such a homogeneous potential is a periodic potential, other classes are quasi-periodic and random potentials. The latter are of particular relevance for the description of solid state systems, which are rarely perfectly clean and may even be doped on purpose with impurities on purpose. There is an interesting and important physical phenomenon linked to such random potentials, and random quantum Hamiltonians in general: Anderson localization. On an intuitive level it can be described as follows. The electron wave undergoes multiple scattering processes in the random environment; all the scattering phases along the path have to be added up; this can lead to destructive interferences, namely the electron can be trapped in the random environment. This is by now known to be a very effective mechanism in dimensions $1$ and $2$ as well as for states energetically close to a band edge, and in all these cases one speaks of dynamical strong [*Anderson localization*]{}, and an energy interval within this regime is called a [*mobility gap*]{}. Many mathematical physicists have contributed to the mathematical understanding of this regime [@FS; @AM], but a proof for all energies in dimension $d=2$ remains a very challenging issue. On the other hand, in dimension $d=3$ and higher, physicists predict the existence of a regime of so-called weak localization in which the destructive interferences merely lead to a diffusive motion, surely slower than a ballistic spreading, but also not localized in a strict sense. To provide a rigorous proof of this is another challenging open question in mathematical physics. Now, what does all this have to do with topological insulators?
As the electrons are Fermions, the zero temperature ground state for Fermi energy $\mu\in{{\mathbb R}}$ is characterized by the Fermi projection obtained via functional calculus from the Hamiltonian: $$P\;=\;\chi(H\leq\mu)
\;,$$ where $\chi$ is the indicator function. The system is then said to be an insulator if $\mu$ lies in a mobility gap of the spectrum of $H$. A special case of this is that $\mu$ lies in a true gap with no spectrum at all. Now there is something that makes the insulator topological. Let us attempt to explain this in an example, namely that of a periodic quantum Hall system. Then the spatial dimension is $d=2$ and only one state per lattice site is needed (that is, $L=1$). Furthermore the uniform magnetic flux through a unit is supposed to be rational (in units of $2\pi$). Then the Fermi projection $P$ below a gap of the spectrum can be diagonalized by a Bloch-Floquet transform ${{\cal F}}$: $${{\cal F}}P{{\cal F}}^*\;=\;\int^\oplus_{{{\mathbb T}}^2} dk\;P(k)
\;,$$ where ${{\mathbb T}}^2={{\mathbb R}}^2\slash 2\pi{{\mathbb Z}}^2$ is the two-dimensional Brillouin torus, $P(k)$ is a projection depending smoothly on $k\in{{\mathbb T}}^2$ and the symbol $\oplus$ indicates that we have a direct integral representation. Such a projection is, via its range, naturally identified with a vector bundle over the torus. The topological invariant is now the Chern number associated to this vector bundle. It is defined as $$\label{eq-Chern2D}
{{\rm Ch}}_{\{1,2\}}(P)
\;=\;
\int_{{{\mathbb T}}^2} \frac{dk}{2\pi\imath}\;{\mbox{\rm Tr}}\big(P(k)[\partial_{k_1}P(k),\partial_{k_2}P(k)]\big)
\;.$$ This index $\{1,2\}$ indicates that this invariant involves derivatives in directions $1$ and $2$. Even though this is not obvious on first sight, it is not too difficult to check that ${{\rm Ch}}_{\{1,2\}}(P)$ is an integer which is homotopy invariant under smooth changes of $P(k)$, and, moreover, has the additivity property ${{\rm Ch}}_{\{1,2\}}(P+Q)={{\rm Ch}}_{\{1,2\}}(P)+{{\rm Ch}}_{\{1,2\}}(Q)$ for two orthogonal projections $P$ and $Q$, [*e.g.*]{} [@BES]. When this integer is non-vanishing, one says that the insulator is topological. This non-trivial topology is then responsible for physical effects, in this case the quantum Hall effect, namely the Hall conductance of the system takes an integer value in units of the inverse Klitzing constant $e^2\slash h$ . The reader may have noticed that up to now, ${{\rm Ch}}_{\{1,2\}}(P)$ was defined merely for a periodic system for which the Bloch-Floquet transform could be applied. Hence the system is topological, but it is not a disordered system. However, the quantum Hall effect results only from the interplay between topology and Anderson localization. One of the challenges for mathematical physicists in the 1980s was to develop a theory which could define the Chern number also for disordered systems. This was achieved in the work of Bellissard [@Bel; @Bel2] and Avron, Seiler and Simon [@ASS], a situation that will be covered by the theory below. The Chern numbers are called the [*bulk invariants*]{} of the quantum Hall system as they are calculated merely from a model on the entire two-dimensional plane. It was also clear to physicists [@Hat] that these bulk systems go along with the existence of so-called chiral edge states when boundaries are introduced, [*e.g.*]{} by restricting the system to a half-plane and imposing Dirichlet boundary conditions. From these edge states it is possible to calculate a [*boundary invariant*]{} as an adequately defined winding number. Moreover, this winding number is equal to the Chern number, a fact that is now commonly referred to as the [*bulk-boundary correspondence*]{} (BBC). All this was put into a $K$-theoretic mathematical description in [@KRS] which also applies to disordered systems and will be described in some more detail below.
Quantum Hall systems are historically the first topological insulators, even though this terminology was only used much latter. The breakthrough came on the theoretical side with the work of Kane and Mele [@KM1; @KM2]. They realized that systems of independent Fermions with half-integer spin may have non-trivial topology even in presence of time-reversal symmetry (TRS). Such a symmetry means that the Fermi projection satisfies $$\label{eq-TRS}
{S_{\mbox{\rm\tiny tr}}}^*\, \overline{P}\,{S_{\mbox{\rm\tiny tr}}}\;=\;
P
\;,
\qquad
{S_{\mbox{\rm\tiny tr}}}\;=\;e^{-\imath \pi \sigma_2}
\;.$$ Here $\overline{P}={{\cal C}}P{{\cal C}}$ denotes the complex conjugate of the Fermi projection w.r.t. a given complex conjugation on Hilbert space (an anti-linear involution), and $\sigma_2=\binom{0\,-\imath}{\imath\;\;\;0}$ is the second Pauli matrix acting on the spin degrees of freedom (recall that TRS is given by complex conjugation and a rotation in spin space by $180^{\circ}$) so that ${S_{\mbox{\rm\tiny tr}}}$ is a real unitary squaring to minus the identity for half-integer spin, which is often called an odd TRS. Now for such Fermi projections one readily checks that the Chern number vanishes. Hence the discovery [@KM1] that there are nevertheless two distinguishable classes of projections came as a surprise, and made solid states richer by adding the new so-called [*quantum spin Hall phase*]{}. It is not described here how Kane and Mele [@KM2] constructed for periodic systems a ${{\mathbb Z}}_2$-invariant distinguishing the new phase from a conventional normal insulator. However, Section \[sec-symmetries\] defines such an invariant as a ${{\mathbb Z}}_2$-index of a Fredholm operator with a symmetry. The new ${{\mathbb Z}}_2$-invariant is not as directly related to an observable quantity (as the Chern number to the Hall conductance), it is nevertheless related to a physical phenomenon, namely via the BBC to the existence of quantum mechanical boundary states. These states were, moreover, predicted to be remarkably stable under perturbations [@KM2]. The field then got a further boost when systems with such edge states were discovered experimentally (see the references in [@QZ]). In the last several years, several groups [@KRY; @Ma] observed experimentally that these delocalized edge states are also very stable under the perturbation by very strong magnetic fields. This leads to serious doubts about the theoretical interpretation – which after all is rooted in TRS – and indicates that theory may not yet have reached its definite state, see Section \[sec-SpinCh\]. From a mathematical perspective, the aim is first to distinguish and classify Fermi projections with TRS . This is achieved using $K$-theory with symmetries which following Atiyah is called $KR$-theory. The second aim is then to calculate these invariants and to analyze the physical effects that go along with these invariants.
The TRS invokes complex conjugation. It can be odd as in , or even if ${S_{\mbox{\rm\tiny tr}}}^2={{\bf 1}}$. There is another class of Fermionic systems for which the Fermi projection satisfies the similar, but distinct invariance property: $$\label{eq-PHS}
{S_{\mbox{\rm\tiny ph}}}^*\, \overline{P}\,{S_{\mbox{\rm\tiny ph}}}\;=\;
{{\bf 1}}-P
\;.$$ Here the Fermi level is $\mu=0$ and ${S_{\mbox{\rm\tiny ph}}}$ is a real unitary which can either be even (${S_{\mbox{\rm\tiny ph}}}^2={{\bf 1}}$) or odd (${S_{\mbox{\rm\tiny ph}}}^2=-{{\bf 1}}$). Such a system is then said to have a particle-hole symmetry (PHS) because the symmetry exchanges particles above the Fermi level with holes below the Fermi level. A PHS naturally appears when a quadratic many-body Hamiltonian describing a superconductor is written using the Bogoliubov-de Gennes (BdG) Hamiltonian $H$ [@AZ]. Again such systems can be topological and may have non-trivial boundary modes, and it is interesting to distinguish the possible phases by $K$-theoretic means and index theorems. Furthermore, by combining TRS and PHS one obtains a so-called chiral symmetry (CHS) of the form $$\label{eq-CHS}
{S_{\mbox{\rm\tiny ch}}}^*\, {P}\,{S_{\mbox{\rm\tiny ch}}}\;=\;
{{\bf 1}}-P
\;.$$ Here ${S_{\mbox{\rm\tiny ch}}}={S_{\mbox{\rm\tiny ph}}}{S_{\mbox{\rm\tiny tr}}}$, but a phase may be added to make this symmetry always even. All combinations of TRS, PHS and CHS lead to $10$ classes, the so-called Altland-Zirnbauer classes [@AZ] which also correspond nicely to the $10$ classical groups of Weyl if they are classified by symmetries (see [*e.g.*]{} [@AZ] or the appendix in [@GS]). Within each class and for every given dimension $d$ of physical space, there may or may not be different distinguishable topological phases [@SRFL]. A $K$-theoretic classification scheme for such phases was put forward by Kitaev [@Kit], and this will be elaborated on here. One further important element making the connection to operator algebraic $K$-theory (which considers homotopy classes of both projections and unitaries) is the following. Every projection having a CHS is encoded by a unitary operator $U$ if one goes into the eigenbasis of the (even) symmetry operator ${S_{\mbox{\rm\tiny ch}}}$: $$\label{eq-CHS2}
P
\;=\;
\frac{1}{2}
\begin{pmatrix}
{{\bf 1}}& U^* \\ U & {{\bf 1}}\end{pmatrix}
\;,
\qquad
{S_{\mbox{\rm\tiny ch}}}\;=\;
\begin{pmatrix}
{{\bf 1}}& 0 \\ 0 & -{{\bf 1}}\end{pmatrix}
\;.$$ This $U$ is also called the Fermi unitary of a chiral system and again there may be non-trivial topology encoded in it. Let us briefly discuss the Su-Schrieffer-Heeger model [@SSH] as a simple one-dimensional example of such a situation (a detailed discussion of this model can be found in [@PSB]). Its Hamiltonian acting on $\ell^2({{\mathbb Z}})\otimes{{\mathbb C}}^2$ is of the form $$H
\; =\;
\tfrac{1}{2}\,S\otimes (\sigma_1+\imath \sigma_2)\; + \;\tfrac{1}{2}\,S^*\otimes (\sigma_1-\imath \sigma_2)\; +\;m \,{{\bf 1}}\otimes\sigma_2
\;,$$ where $S$ is the bilateral shift on $\ell^2({{\mathbb Z}})$, $m\in{{\mathbb R}}$ is a mass and $\sigma_1=\binom{0\;1}{1\;0}$ and $\sigma_2=\binom{0\,-\imath}{\imath\;\;\;0}$ as above. In the grading of the Pauli matrices, $$H\;=\;\begin{pmatrix} 0 & S-\imath m \\ S^*+\imath m & 0
\end{pmatrix}
\;.$$ This Hamiltonian is off-diagonal which reflects the CHS ${S_{\mbox{\rm\tiny ch}}}H {S_{\mbox{\rm\tiny ch}}}=-H$ with ${S_{\mbox{\rm\tiny ch}}}=\sigma_3=\binom{1\;\;\;0}{0\;-1}$. The Fermi projection $P=\chi(H\leq 0)$ hence satisfies and is of the form as long as $m\not\in\{-1,1\}$. One finds $U=(S^*+\imath m)|S^*+\imath m|^{-1}$. Upon discrete Fourier transform, it becomes obvious that this unitary has a winding number ${{\rm Ch}}_{\{1\}}(U)$ equal to $1$ for $m\in(-1,1)$ and no winding for $|m|>1$. This is the robust topology of this model. If it is non-trivial it goes along with the existence of chiral zero modes at the edges, notably bound states of the half-line restriction of $H$ with energy $0$ which are also eigenvectors of ${S_{\mbox{\rm\tiny ch}}}$.
Before going on, let us briefly resume what makes out a topological insulator:
$\bullet$ $d$-dimensional disordered system of independent Fermions
$\bullet$ Fermi level in a gap or a mobility gap
$\bullet$ System is submitted to a combination of basic symmetries (TRS, PHS, CHS)
$\bullet$ non-trivial topology of bulk states (winding numbers, Chern numbers, etc.)
$\bullet$ Delocalized edge modes resulting from a bulk-boundary correspondence (BBC)
Let us note that there are other physical effects linked to the non-trivial topology in these systems, [*e.g.*]{} topological bound states at point defects. Furthermore, topological states of matter have been found in the physics literature in a great variety of other physical system: Fermions with spacial symmetries (such as reflection, rotation), interacting Fermions, bosonic systems (topological photonic crystals, topological magnons), and also in spin systems. Here we focus only on non-interacting Fermions.
Short overview of the mathematical physics literature
=====================================================
Following the first two papers of Kane and Mele [@KM1; @KM2] appeared several particularly influential theoretical physics papers [@RSFL; @QHZ; @Kit; @SRFL; @SCR]. No attempt will be made to list further references from the abundant theoretical and experimental physics literature. There are by now several longer review papers available, a particularly nice one being [@QZ]. Here an overview of rigorous mathematical analysis of topological insulator systems is offered with, obviously, a focus on the results of the author and his collaborators, in particular [@Sch; @SB], the joint recent monograph with Emil Prodan [@PSB], as well as the papers with Giuseppe De Nittis [@DS] and Julian Grossmann [@GS]. The book only covers the so-called complex classes of topological insulators, namely those Fermionic systems either have no symmetry or a chiral symmetry, but not any of the symmetries requiring a real structure (TRS and PHS). The other papers then concern the latter cases, mainly by implementing these symmetries in the complex classes. Before going into some detail in the remainder of this review, this is a good spot to at least point the reader to interesting recent contributions of several other mathematical physicists.
Several authors followed up on the physics literature by studying periodic systems. Upon Bloch-Floquet transform, one is then naturally led to the question of classification of vector bundles. De Nittis and Gomi achieved that for several symmetry classes ([*e.g.*]{} in [@DG; @DG2]), others with an approach close to homotopy theory [@KZ], $K$-theory [@LKK] or obstruction theory [@FMP]. Building on [@Kit; @SCR], a unifying approach based on Bott periodicity was put forward by Kennedy and Zirnbauer [@KZ]. A bit earlier, it was shown by Freed and Moore how twisted equivariant $K$-theory could be used for a classification of insulators [@FM]. This was still restricted to vector bundles, but extensions to an operator algebraic framework were given by Thiang [@Thi], Kellendonk [@Kel] and Kubota [@Kub]. Focussing on quantum spin Hall systems, Graf and Porta [@GP] proved a bulk-boundary correspondence by functional analytic techniques, extending results from [@ASV]. The bulk-boundary correspondence was also approached using $T$-duality [@MT2] and very successfully by means of Kasparov’s $KK$-theory [@BCR; @BKR]. Katsura and Koma [@KK] studied the ${{\mathbb Z}}_2$-index of a pair of projections, which is closely tied to [@Sch]. Other rigorous work was done by Loring and Hastings who developed local signatures to detect non-trivial invariants [@LH; @Lor]. This has been applied to several systems now, and the connection to the more conventional index approach is currently under investigation.
Bulk models and associated observable algebras
==============================================
Let us now be a bit more precise about the Fermionic one-particle bulk Hamiltonians to be studied. Here the word “bulk" indicates that the system is extended in all directions of $d$-dimensional space and that there are no boundaries present. These operators will be of the form acting on a tight-binding Hilbert space $\ell^2({{\mathbb Z}}^d)\otimes{{\mathbb C}}^L$, but, moreover, will be indexed by a variable $\omega$ from a compact topological space $\Omega$ modeling the disordered configurations of the solid: $$\label{eq-HamDis}
H_\omega\;=\;
\Delta^B\,+\,W_\omega
\;.$$ The discrete magnetic Laplacian for a magnetic field given by an anti-symmetric real matrix $B=(B_{i,j})_{i,j=1,\ldots,d}$ is typically of the form $$\Delta^B
\;=\;
\sum_{i=1}^{d} \;(t_i^*U_i+t_iU_i^*)
\;,$$ where the $t_i$ are $L\times L$-matrices allowing to describe, [*e.g.*]{}, spin-orbit coupling, and the $U_i$ are the magnetic translations on the lattice satisfying the commutation relations $$\label{eq-ComRel}
U_jU_i
\;=\;
e^{\imath B_{i,j}}U_iU_j
\;.$$ There are many gauges encoding the magnetic translations, one being the Landau gauge presented next. Let $S_j:\ell^2({{\mathbb Z}}^d)\to \ell^2({{\mathbb Z}}^d)$ be the shift operator in the $j$th direction defined by $S_j|n\rangle=|n+e_j\rangle$ where the Dirac notation $|n\rangle=\delta_n$ for localized states on the lattice site $n\in{{\mathbb Z}}^d$ was used and $e_j$ is the standard basis of ${{\mathbb Z}}^d$. If then $X_j$ are the unbounded self-adjoint position operators defined by $X_j|n\rangle=n_j|n\rangle$, then the Landau gauge for $d=3$ is $$\label{eq-Landau}
U_1\,=\,e^{\imath B_{1,2} X_2+ \imath B_{1,3} X_3}S_1\;,
\qquad
U_2\,=\,e^{\imath B_{2,3} X_3}S_2\;,
\qquad
U_3\,=\,S_3
\;.$$ This can readily be extended to higher dimensions, always such that $U_d=S_d$. The matrix potential in is typically of the form $$W_\omega
\;=\;
\sum_{n\in{{\mathbb Z}}^d}\, |n\rangle\omega_n\langle n|
\;,$$ with self-adjoint $L\times L$ matrices $\omega_n$. Together these matrices form a configuration $\omega=(\omega_n)_{n\in{{\mathbb Z}}^d}$ of the solid and the set of all such configurations is denoted by $\Omega$. All $\omega_n$ are supposed to be drawn independently and identically from a compact subset of the self-adjoint $L\times L$ matrices. Then the set $\Omega$ equipped with the Tychonov topology is also a compact space on which the product probability measure is denoted by ${{\mathbb P}}$. For averages over ${{\mathbb P}}$ we simple write ${{\bf E}}_{{\mathbb P}}$. Finally, there is a natural shift action $T:{{\mathbb Z}}^d\times\Omega \to\Omega$ on $\Omega$ and ${{\mathbb P}}$ is invariant and ergodic w.r.t. this action. It will furthermore be assumed that the support of the distribution of $\omega_n$ is a contractible set. Then also $\Omega$ is contractible.
Summing up, we have a random family of Hamiltonians $H=(H_\omega)_{\omega\in\Omega}$ of the form , indexed by points $\omega$ from a compact and ergodic dynamical system $(\Omega,T,{{\mathbb Z}}^d,{{\mathbb P}})$. Actually, the detailed from of the kinetic part $\Delta_B$ and the potential part $W_\omega$ are not so important and may be generalized to include, say, random matrix elements between lattice points farther apart. What is crucial, however, is the covariance properties of the family $(H_\omega)_{\omega\in\Omega}$, namely there are so-called dual magnetic translations $a\in{{\mathbb Z}}^d\mapsto V_a$ commuting with the $U_j$ (and actually constructed similarly as in ) so that the following covariance relation holds: $$\label{eq-CovRel}
V_a H_\omega V_a^*\;=\;H_{T_a\omega}
\;,
\qquad
a\in{{\mathbb Z}}^d
\;.$$ Now the set of all covariant operator families of finite range (no matrix elements from $|n\rangle$ to $|m\rangle$ for $|n-m|$ arbitrarily large) forms a $*$-algebra because linear combinations and products of two such covariant families are again covariant and of finite range, and so is the adjoint. Hence one can introduce $C^*$-algebra of bulk-observables as $${{\cal A}}_d
\; =\;
\mbox{\rm C}^*\,\big\{A=(A_\omega)_{\omega\in\Omega} \mbox{ finite range covariant operators}\big\}
\;.$$ The $C^*$-closure is concretely given w.r.t. the norm $\|A\|=\sup_{\omega\in\Omega}\|A_\omega\|$ where $\|A_\omega\|$ is the operator norm on $\ell^2({{\mathbb Z}}^d)\otimes{{\mathbb C}}^L$. Let us point out that if $\Omega$ consists of just one point and the magnetic field vanishes, then the covariance relation reduces to translation invariance and, upon discrete Fourier transform, the algebra ${{\cal A}}_d$ is simply given by $C({{\mathbb T}}^d)\otimes{{\rm Mat}}(L\times L,{{\mathbb C}})$, the matrix valued continuous functions on the $d$-dimensional torus. In general, the algebra ${{\cal A}}_d$ can be seen to be isomorphic to a twisted crossed product algebra $C(\Omega)\rtimes_B {{\mathbb Z}}^d$ as well as an $d$-fold iterated crossed product $C(\Omega)\rtimes {{\mathbb Z}}\ldots \rtimes {{\mathbb Z}}$ [@Bel; @PSB]. Even though this is not spelled out in detail, these are important facts for the sequel. Further reasons to put the $C^*$-algebra ${{\cal A}}_d$ into the spotlight are the following:
$\bullet$ ${{\cal A}}_d$ contains most physically relevant operators constructed from the Hamiltonian.
$\bullet$ ${{\cal A}}_d$ is not too large, so it still contains interesting topological information.
$\bullet$ The $C^*$-algebraic framework allows to use the associated algebraic $K$-theory,
$\bullet$ as well as Connes’ non-commutative geometry
$\bullet$ by defining a quantized calculus (non-commutative differentiation and integration).
$\bullet$ The Toeplitz extension of ${{\cal A}}_d$ connects bulk and boundary topology.
$\bullet$ Physical symmetries (TRS, PHS, CHS) can be implemented by involutive automorphisms.
The aim of the remainder of this review is to support all these claims.
Classification of bulk phases by $K$-theory {#sec-KClass}
===========================================
Topological $K$-theory classifies vector bundles over a manifold, and algebraic $K$-theory analyzes homotopy classes of projections in a C$^*$-algebra. This latter will be used to distinguish topological phases. Let us begin by explaining why, and then give a more formal definition of the $K$-groups. The bulk phase of a system of independent Fermions described by $(H_\omega)_{\omega\in\Omega}$ at zero temperature is specified by the associated Fermi projections $P_\omega=\chi(H_\omega\leq \mu)$. If $\mu$ lies in a (bulk) gap of the system, namely the Fermi level $\mu\in{{\mathbb R}}$ is not in the spectrum $H_\omega$, then the characteristic function $\chi$ can be replaced by a continuous function, so that $P_\omega$ is obtained by continuous functional calculus from the Hamiltonian. Moreover, the family $P=(P_\omega)_{\omega\in\Omega}$ inherits the covariance property and therefore is a projection in the C$^*$-algebra ${{\cal A}}_d$. Now a continuous deformation of the Hamiltonian leads to a continuous deformation of the projection (as long as the gap remains open) and all covariant projections obtained in this manner belong to the same bulk phase of the system. Hence one is naturally led to using homotopy equivalence classes of projections in ${{\cal A}}_d$ as phase labels for the system. These classes almost define the $K_0$-group of ${{\cal A}}_d$ which also considers classes of projections in matrix algebras of ${{\cal A}}_d$: $$K_0({{\cal A}}_d)
\;=\;
\{[P]_0-[Q]_0\,:\,P=P^2=P^*,\,Q=Q^2=Q^*\in{{\rm Mat}}(N\times N,{{\cal A}}_d) \mbox{ with }N<\infty\}
\;.$$ Here $[\,.\,]_0$ denotes the homotopy equivalence classes, and the minus sign in the difference results from the Grothendieck construction of a group from a semigroup which is defined by $[P]_0+[P']_0=[{{\rm diag}}(P,P')]_0$ (see [@Bla0; @WO; @RLL] for mathematical details). From a physical point of view, the additional matrix degrees of freedom in $K$-theory reflect that there are high energy bands present which are discarded when modelling in a tight-binding framework (effectively a high energy cut-off). There seem to be no known physical effects linked to a fixed finite number of degrees of freedom per cell (the $L$ above), so that it is reasonable to classify phases by $K$-groups rather than just homotopy classes with fixed dimension of the fiber.
In complex $K$-theory there is a second group $K_1({{\cal A}}_d)$ made up of homotopy equivalence classes of unitaries, again without fixing the matrix degree of freedom: $$K_1({{\cal A}}_d)
\;=\;
\{[U]_1\,:\,U=(U^*)^{-1}\in{{\rm Mat}}(N\times N,{{\cal A}}_d) \mbox{ with }N<\infty\}
\;.$$ The group structure is given $[U]_1\cdot[U']_1=[{{\rm diag}}(U,U')]_1$, or equivalently by $[U]_1\cdot [U']_1=[UU']_1$ in case the matrices are of same size. The group $K_1({{\cal A}}_d)$ is used to classify $d$-dimensional Fermionic systems with a chiral symmetry. Indeed, the Fermi projection is then determined by a Fermi unitary as in . Actually, if the chiral symmetry only holds approximately in the sense that $HJ+JH$ is sufficiently small in norm, then the block representation of $P$ as in still has an invertible off-diagonal entry and its phase still defines a Fermi unitary and thus a class in $K_1({{\cal A}}_d)$ [@PSB]. In conclusion, the two complex $K$-groups $K_0({{\cal A}}_d)$ and $K_1({{\cal A}}_d)$ allow to classify the Fermi projections of gapped covariant systems without further symmetry and an approximate chiral symmetry respectively. Hence it is of great interest to calculate these groups and understand their structure. The first simplifying step is elementary [@PSB]:
If $\Omega$ is contractible, then $K_j({{\cal A}}_d)=K_j({{\cal A}}_B)$ where ${{\cal A}}_B=C^*(U_1,\ldots,U_d)$ is the rotation algebra generated by $d$ unitaries satisfying the commutation relations associated to a given anti-symmetric real $d\times d$ matrix $B$.
As indicated above, the hypothesis of contractibility on $\Omega$ is adequate for disordered systems, however, for other models like quasicrystals it is not satisfied. The $K$-theory of the rotation algebra was first determined in a seminal paper by Pimsner and Voiculescu. Let us first state the result and then provide some further explanations.
$K_0({{\cal A}}_d)={{\mathbb Z}}^{2^{d-1}}$ and $K_1({{\cal A}}_d)={{\mathbb Z}}^{2^{d-1}}$.
Let us stress that these $K$-groups are, in particular, independent of $B$ which can be viewed as a deformation parameter of the commutation relations . For $B=0$, the rotation algebra is isomorphic to $C({{\mathbb T}}^d)$ as already stressed above, and its algebraic $K$-groups coincide with the topological $K$-groups classifying the vector bundles over the $d$-torus ${{\mathbb T}}^d$. The paper by Pimsner and Voiculescu [@PV] also provides an algorithm to construct the generators of the $K$-groups, and it is helpful in the context of topological insulators to understand what these generators actually are in low dimensions. For $d=1$, the group $K_0({{\cal A}}_1)={{\mathbb Z}}$ is generated by the identity $1$, and $K_1({{\cal A}}_1)={{\mathbb Z}}$ is generated by the unitary $U_1$. For $d=2$, the group $K_0({{\cal A}}_2)={{\mathbb Z}}^2$ has again $1$ as one generator, and then the Powers-Rieffel projection as the second (for $B\not=0$) or the Bott projection (for $B=0$), while $K_1({{\cal A}}_2)={{\mathbb Z}}^2$ is simply generated by $U_1$ and $U_2$. For $d=3$, the group $K_0({{\cal A}}_3)={{\mathbb Z}}^4$ is generated by $1$ and three Powers-Rieffel (or Bott) projection in the three coordinate planes, and $K_1({{\cal A}}_3)={{\mathbb Z}}^4$ by $U_1$, $U_2$ and $U_3$, as well as one essentially new and intrinsically three-dimensional generator, which is given by the pre-image of the Powers-Rieffel projection under $K$-theoretic index map (for details on this and the following, see Chapter 4 of [@PSB]). This procedure can then be iterated. The generators $G_I$ of $K({{\cal A}}_d)$ can then be labelled by subsets $I\subset\{1,\ldots,d\}$, for even cardinality of $I$ giving an element in $K_0({{\cal A}}_d)$, and for odd in $K_1({{\cal A}}_d)$. The set $I$ indicates which spacial directions are involved. For example, $G_{\{1,2\}}$ is the Powers-Rieffel projection and $G_{\{1,2,3\}}$ the intrinsically three-dimensional generator described above. The class of a projection $P\in {{\rm Mat}}(N\times N,{{\cal A}}_d)$ can be decomposed as $$\label{eq-Pdecomp}
[P]_0
\;=\!
\sum_{I\subset\{1,\ldots,d\},\,|I|\,{\rm even}}
\!n_I\;[G_I]_0\;,
\qquad
n_I\in{{\mathbb Z}}\;.$$ The coefficients $n_I$ are called the topological invariants of $P$ (up to a constant) and the aim below will be to calculate them by non-commutative geometry (actually rather non-commutative differential topology). For $I={\{1,\ldots,d\}}$, the integer $n_I$ is called the strong topological invariant (or also top invariant), all other invariants are called weak. The reason for this terminology is that the top invariants are known to remain stable in the Anderson localization regime [@PLB; @PSB]. It is believed that the weak invariants are less robust, but this issue is still under investigation. This concludes the discussion of the so-called complex classes of topological insulators, namely those not invoking a real structure (needed to implement TRS and PHS). The same approach can also be used in these cases, and it will be briefly explained in Section \[sec-symmetries\] how to proceed.
Topological invariants and their main properties {#sec-BulkInv}
================================================
Let us begin by introducing the so-called non-commutative analysis tools on the algebra ${{\cal A}}_d$. Integration is given by tracial normalized state on ${{\cal A}}_d$ specified by the trace per unit volume: $${{\cal T}}(A)
\;=\;
\lim_{N\to\infty}\frac{1}{(2N+1)^d}\,\sum_{n\in [-N,N]^d}\,\langle n|A_\omega|n\rangle
\;=\;
{{\bf E}}_{\mathbb{P}}\;
\langle 0| A_\omega|0\rangle
\;,
\qquad
A=(A_\omega)_{\omega\in\Omega}\in{{\cal A}}_d
\;.$$ The convergence for $N\to\infty$ holds ${{\mathbb P}}$-almost surely by Birkhoff’s ergodic theorem (for the ergodic action of ${{\mathbb Z}}$ on $\Omega$), which, due to the covariance relation, also shows that the almost sure value is given by the average over ${{\mathbb P}}$ on the r.h.s.. Non-commutative derivations $\nabla_j$ are densely defined by $$\nabla_{j}A_\omega
\;=\;
\imath[X_j,A_\omega]
\;,
\qquad
j=1,\ldots,d$$ where the position operators are given by $X_j|n\rangle=n_j|n\rangle$. Operators in ${{\cal A}}_d$ which are infinitely many times differentiable are called smooth, and these smooth operators form a Fréchet sub-algebra of ${{\cal A}}_d$. Both ${{\cal T}}$ and $\nabla_j$ naturally extend to matrix algebras ${{\rm Mat}}(N\times N,{{\cal A}}_d)$ over ${{\cal A}}_d$. One has the invariance property ${{\cal T}}(\nabla_j A)=0$ so that partial integration holds. Furthermore, if one deals with a periodic system so that ${{\cal A}}_d=C({{\mathbb T}}^d)$, the trace ${{\cal T}}$ reduces to the integration of the Brillouin torus ${{\mathbb T}}_d$ and $\nabla_j$ to the partial derivative w.r.t. the $j$-th component of the quasimomentum. Hence the non-commuative analysis tools naturally extend the well-known semiclassical analysis approach to solid state physics systems. Equipped with these tools, there is now a standard procedure to produce Connes-Chern characters on ${{\cal A}}_d$ [@Con] spelled out next. The somewhat lengthy formulas cannot be avoided, but are very natural when the full structure behind them is uncovered (as in [@Con]), and there are even good reasons for the choice of the normalization constants, as will become apparent further below.
\[def-CC\] For even cardinality $|I|$ and a smooth projection $P\in {{\rm Mat}}(N\times N,{{\cal A}}_d)$, the $I$-th Chern number of $P$ is $$\mbox{\rm Ch}_{I} (P)
\;=\;
\frac{(2\imath \pi)^{\frac{|I|}{2}}}{\frac{|I|}{2}!}\; \sum_{\rho\in S_I} (-1)^\rho \,{{\cal T}}\left(P\prod_{j=1}^{|I|} \nabla_{\rho_j} P \right)
\;.$$ Here the sum runs over bijections $\rho:\{1,\ldots,|I|\}\to I$ for which the definition of the signature $(-1)^\rho$ is extended using the natural order on $I$. For odd $|I|$ and an invertible smooth $A\in{{\rm Mat}}(N\times N,{{\cal A}}_d)$, the $I$-th Chern number is $$\mbox{\rm Ch}_I (A)
\;=\;
\frac{\imath(\imath \pi)^\frac{|I|-1}{2}}{|I|!!}\; \sum_{\rho\in S_{I}} (-1)^\rho \;{{\cal T}}\left(\prod_{j=1}^{|I|}A^{-1} \nabla_{\rho_j}A \right)
\;,$$ where $(2n+1)!!=\prod^n_{k=1}(2k+1)$.
Before going on to describe the mathematical properties of these objects, let us stress that they often linked to physical quantities. It is by now a classic fact that $\mbox{\rm Ch}_{\{1,2\}} (P)$ is the zero-temperature Hall conductance of two-dimensional electron system described by the Fermi projection $P$ [@BES]. Actually, for a periodic system $\mbox{\rm Ch}_{\{1,2\}} (P)$ reduces to after Fourier transform. Furthermore, if one of the $d$ direction is interpreted as time, it is essentially the orbital polarization of a periodically driven system [@ST; @DL] and then $\mbox{\rm Ch}_{\{1,2,3,4\}} (P)$ describes the non-linear magneto-electric response [@PLB; @PSB]. For chiral systems with Fermi unitary $U$, $\mbox{\rm Ch}_{\{j\}}(U)$ is the so-called chiral polarization [@PSB]. Further examples are described in [@PSB]. All these quantities are of topological nature due to the following classic result.
$\mbox{\rm Ch}_{I} (P)$ and $\mbox{\rm Ch}_I (A)$ are homotopy invariants under smooth deformations of $P\in{{\cal A}}_d$ and $A\in{{\cal A}}_d$ and thus establish pairings with $K_*({{\cal A}}_d)$.
The rather complicated choice of the normalization constants in Definition \[def-CC\] guarantees, beneath other things, that the top pairings with $I=\{1,\ldots,d\}$ are integer valued, and that the generalized Streda formulas described below hold. Furthermore, by a suspension argument [@PSB] a connection to the invariants introduced by Essin and Gurarie [@EG1] (based on earlier work by Volovik) can be established. They express the invariants in terms of the resolvents and hence fairly directly in terms of the Hamiltonian.
Let $|I|$ be even. Consider a path $z:[0,1]\to {{\mathbb C}}\setminus\sigma(H)$ encircling $(-\infty,\mu]\cap\sigma(H)$ and set $G(t)=(H-z(t))^{-1}$. Then for the Fermi projection $P=\chi(H\leq \mu)$ of the gapped Hamiltonian $H$, $$\mbox{\rm Ch}_{I}(P_\mu)
\;=\;
\frac{(\imath \pi)^\frac{|I|}{2}}{\imath (|I|-1)!!} \sum_{\rho\in S_{I\cup \{0\}}}\!\! (-1)^\rho\! \int^{1}_0 \!dt\,{{\cal T}}\!
\left(\prod_{j=1}^{|I|}G(t)^{-1} \nabla_{\rho_j}G(t) \right)
\;,$$ where $\nabla_0=\partial_t$ and $\rho:\{1,\ldots,|I|+1\}\to \{0\}\cup I$. A similar formula holds for odd $|I|$.
The next result concerns the dependence of the invariants on the magnetic field. Interestingly, the magnetic derivatives of weak invariants are given by other invariants. The first such relation was found by Streda [@Str] who considered two-dimensional quantum Hall systems and showed that the integrated density of states grows linearly in the magnetic field with derivative essentially given by the Chern number. The integrated density of states is actually ${{\bf E}}\, \langle 0|P|0\rangle=\mbox{\rm Ch}_\emptyset(P)$, and Streda’s formula then states $$\partial_{B_{1,2}}\, \mbox{\rm Ch}_\emptyset(P)
\;=\;
\frac{1}{2\pi}\;\mbox{\rm Ch}_{\{1,2\}}(P)
\;.$$ In this generality, the Streda formula was first proved by Rammal and Bellissard [@RB], and their techniques can be extended to obtain the following.
\[theo-Streda\] For sufficiently smooth projections $P$ and invertibles $A$ and $|I|$ even and odd respectively, one has for $i,j\not\in I$ $$\partial_{B_{i,j}}\,\mbox{\rm Ch}_{I}(P)\;=\;\frac{1}{2\pi}\;
\mbox{\rm Ch}_{I\cup \{i,j\}}(P)
\;,
\qquad
\partial_{B_{i,j}}\,\mbox{\rm Ch}_{I}(A)
\;=\;\frac{1}{2\pi}\;
\mbox{\rm Ch}_{I\cup \{i,j\}}(A)
\;.$$
Apart from the quantum Hall effect, this result is relevant for magneto-electric effects in dimension $d=3$. Here time is added as a $4$th direction needed for calculation of polarization, and the non-linear response is the derivative w.r.t. the magnetic field of the polarization, which is thus connected to an integer $\mbox{\rm Ch}_{\{1,2,3,4\}}(P)$. Another example in $d=3$ is the chiral magneto-electric response [@PSB]. Furthermore, combining Theorem \[theo-Streda\] with results by Elliott [@Ell] leads to the following statement on the ranges of parings.
\[theo-PairingRange\] Let $I,J\subset\{1,\ldots,d\}$ be increasingly ordered sets with a cardinality of equal parity. Then, with the generators $G_J$ of $K({{\cal A}}_{d})$ described above, $$\begin{aligned}
\label{eq-paringrangeeven1}
& {{\rm Ch}}_{I}(G_J)
\;=\;
0\;,
& I\setminus J\not=\emptyset\;,
\\
\label{eq-paringrangeeven2}
& {{\rm Ch}}_{I}(G_J)
\;=\;
1 \;, & I=J\;,
\\
\label{eq-paringrangeeven3}
& {{\rm Ch}}_{I}(G_J)
\;=\;
(2\pi)^{-\frac{1}{2}| J\setminus I|} \; {\rm Pf}(B_{J\setminus I})\;,
& I\subset J\;,\end{aligned}$$ where $ {\rm Pf}$ denotes the Pfaffian and $B_{J\setminus I}$ is the antisymmetric matrix retaining just the indices of $J$ not contained in $I$.
When these identities are combined with and the additivity property $${{\rm Ch}}_I(P)
\;=\!
\sum_{I\subset\{1,\ldots,d\},\,|I|\,{\rm even}}
\!n_I\;{{\rm Ch}}_I(G_I)
\;,
\qquad
n_I\in{{\mathbb Z}}\;,$$ the whole range of ${{\rm Ch}}_I(P)\in{{\mathbb R}}$ is indeed determined with its full dependence on the magnetic field. Let us note that, in particular, the strong topological invariants for $I=\{1,\ldots,d\}$ are integer valued, as are the next lower weak ones with $|I|=d-1$. Actually, for the strong topological invariants (also of a chiral system), an index theorem is known. This is described next. Let $\gamma_1,\ldots,\gamma_d$ be an irreducible representation of the complex Clifford algebra ${{\mathbb C}}_d$ on ${{\mathbb C}}^{2^{(d-1)/2}}$. Then the Dirac operator on $\ell^2({{\mathbb Z}}^d)\otimes{{\mathbb C}}^L\otimes {{\mathbb C}}^{2^{(d-1)/2}}$ is defined as $$\label{eq-Dirac}
D\;=\;\sum_{j=1}^d X_j\otimes {{\bf 1}}\otimes\gamma_j
\;.$$ This may not look like a Dirac operator at first sight, but after Fourier transform it does become a first order differential operator on the Brillouin torus. The Dirac phase is then defined by $F=D|D|^{-1}$ (after having lifted the non-relevant kernel of $D$). It can be checked that $F$ defines a so-called odd Fredholm module over ${{\cal A}}_d$, namely (by definition) $F^2={{\bf 1}}$ and the commutators $[F,A_\omega]$ are compact for all $A=(A_\omega)_{\omega\in\Omega}\in{{\cal A}}_d$. Actually for smooth $A$ the commutators $[F,A_\omega]$ even lie in the Schatten ideal ${{\cal L}}^{d+\epsilon}$. For even $d$, one furthermore has a grading induced by $\Gamma=-i^{-d/2}\gamma_1\cdots \gamma_d$ which anti-commutes with $D$. After an adequate basis change on the representation space ${{\mathbb C}}^{2^{(d-1)/2}}$, $\Gamma={{\rm diag}}({{\bf 1}},-{{\bf 1}})$ and then $F=\binom{0\;\;G}{G^*\;0}$. An odd Fredholm module over ${{\cal A}}_d$ together with such a grading defines an even Fredholm module, so that is what one has for even $d$. By a standard procedure [@Con0; @Con] one now obtains Fredholm operators associated to these Fredholm modules. Non-standard is, however, the connection to the invariants defined in Definition \[def-CC\]. Such a connection is called a local index formula. Possibly it follows, similar as in [@Andr], from the general Connes-Moscovici local index formula [@CGX], but the proofs in [@PLB; @PSB] are based on interesting new geometric identities for the volume of higher-dimensional simplices, which generalize Connes’ two-dimensional triangle identity [@Con0; @ASS; @BES].
\[Local index formula, [@PLB; @PSB]\] \[theo-IndexPair\] For even $d$ and a smooth projection $P\in{{\cal A}}_d$, the operator $P_\omega GP_\omega$ is ${{\mathbb P}}$-almost surely Fredholm with almost sure index given by $${{\rm Ind}}(P_\omega GP_\omega)
\;=\;
\mbox{\rm Ch}_{\{1,\ldots,d\}} (P)
\;.$$ For odd $d$, let $E=\frac{1}{2}(F+{{\bf 1}})$ be the Hardy projection for $F$. For invertible smooth $A\in{{\cal A}}_d$, the operator $EA_\omega E$ is ${{\mathbb P}}$-almost surely Fredholm with almost sure index given by $$\mbox{\rm Ind}(E\,A_\omega E)
\;=\;
\mbox{\rm Ch}_{\{1,\ldots,d\}} (A)
\;.$$
For $d=2$, $G=\frac{X_1+\imath X_2}{|X_1+\imath X_2|}$ and the theorem provides the well-known connection between the Hall conductance given by the Kubo formula and a Fredholm index [@BES; @ASS]. One substantial advantage of the index theorem is that it can be extended to the regime of strong Anderson localization [@PLB; @PSB], for which the Fermi level lies in a mobility gap. Here the Fermi projection is [*not* ]{} in the C$^*$-algebra, but only in the subspace of sufficiently smooth elements of the enveloping von Neumann algebra with ${{\cal T}}(|\nabla P|^p)<\infty$ for $p>d$ so that ${{\rm Ch}}_{\{1,\ldots,d\}} (P)$ is well-defined (in the terminology of [@BES], this is the defining property of a non-commutative Sobolev space which has, however, [*not*]{} the property of lying in the C$^*$-algebra as in the commutative case). Hence one can use the strong invariants and the associated indices as a phase label also in this localization regime. This is of crucial importance, for example, for the explanation of the quantum Hall effect [@BES].
As a final comment on bulk topological invariants let us mention the approach by Loring and Hastings [@LH; @Lor] which introduced local signature indices for topological insulators which have the advantage that they can be very easily calculated numerically for a given model and even be used to define local changes in the ground states. While this is far from obvious, it is now becoming evident that these invariants coincide with the above (work in preparation with Terry Loring).
Bulk-boundary correspondence {#sec-BBC}
============================
Up to now, only bulk topological invariants were introduced and used to distinguish bulk ground states of systems of independent Fermions. One of the key features of topological insulators is that these very invariants are responsible for a large number of effects linked to defects. Most prominent is the existence of surface, edge or boundary states. These states appear in the bulk gap for the models restricted to half-spaces. That wave equations on half-spaces have such modes was already known to Rayleigh and Sommerfeld, however, the boundary states in topological insulators, moreover, come with a remarkable stability. They are not susceptible to Anderson localization if perturbed by a weak random potential (which is particularly surprising if the boundary is one-dimensional) and appear for any type of boundary condition, as long as it is local (for example, the spectral boundary conditions of Atiyah-Patodi-Singer are non-local). Less prominent, but just as important and based on similar mathematical principles, are bound states attached to particular types of point defects in topological insulators and this will be briefly described in Section \[sec-SF\].
Let us begin by describing the half-space Hamiltonian, for sake of concreteness by simply imposing Dirichlet boundary conditions. We choose ${{\mathbb Z}}^{d-1}\times{{\mathbb N}}\subset{{\mathbb Z}}^d$ as the half-space, with the restriction $x_d\geq 0$ on the last coordinate. Let $\Pi$ be the projection from $\ell^2({{\mathbb Z}}^d)$ to $\ell^2({{\mathbb Z}}^{d-1}\times{{\mathbb N}})$, naturally extended to $\ell^2({{\mathbb Z}}^d)\otimes{{\mathbb C}}^L$. Then the half-space Hamiltonians acting on $\ell^2({{\mathbb Z}}^{d-1}\times{{\mathbb N}})\otimes{{\mathbb C}}^L$ are simply given by $\widehat{H}_\omega=\Pi H_\omega \Pi$. This operator as well as all other half-space operators will carry a hat. It lies in the C$^*$-algebra $\widehat{{{\cal A}}}_d$ of half-space operators which is generated by $\widehat{U}_j=\Pi U_j\Pi$ and the functions $f\in C(\Omega)$ acting as a potential $f_\omega|n\rangle=f(T_n\omega)|n\rangle$, $n\in{{\mathbb Z}}^{d-1}\times{{\mathbb N}}$. Note that the $\widehat{U}_j$ are still unitaries for $j=1,\ldots,d-1$, but $\widehat{U}_d$ is now only a partial isometry. In the Landau gauge it is actually simply the unilateral shift so that $\widehat{U}_d\widehat{U}_d^*={{\bf 1}}-\Pi_d$ where $\Pi_d$ is the projection on the boundary surface. The C$^*$-algebra generated by $\Pi_d$, $\widehat{U}_1,\ldots,\widehat{U}_{d-1}$ and the $f_\omega$ is called the boundary or edge algebra and denoted by ${{\cal E}}_d$. It is naturally embedded in $\widehat{{{\cal A}}}_d$, which in turn is naturally mapped onto ${{\cal A}}_d$ by discarding the hats and all summands containing $\Pi_d$. Hence
The boundary, half-space and bulk algebra from a short exact sequence of C$^*$-algebras $$\label{eq-ExactSeq}
\begin{array}{ccccccccc}
0 & \to & {{\cal E}}_d & \to & \widehat{{{\cal A}}}_d & \to &{{\cal A}}_d & \to & 0\;.
\end{array}$$ Moreover, there is an isomorphism ${{\cal E}}_d\cong{{\cal A}}_{d-1}\otimes{{\cal K}}(\ell^2({{\mathbb N}}))$.
Actually, is isomorphic to the well-known Pimsner-Voiculescu short exact sequence for the crossed product ${{\cal A}}_d={{\cal A}}_{d-1}\rtimes{{\mathbb Z}}$. Associated to any short exact sequence of C$^*$-algebras there is the exact sequence of $K$-theory: $$\begin{array}{cccccccccc}
& K_0({{\cal E}}_d) & {\longrightarrow} & & K_0(\widehat{{{\cal A}}}_d) & & {\longrightarrow}& & K_0({{\cal A}}_d)&
\\
& & & & & & & & &
\\
& {{\rm Ind}}\;\;\uparrow & & & & & & & \downarrow\;\;\mbox{\rm Exp}&
\\
& & & & & & & & &
\\
& K_1({{\cal A}}_d) & {\longleftarrow} & & K_1(\widehat{{{\cal A}}}_d) & & {\longleftarrow} & & K_1({{\cal E}}_d) &
\end{array}$$ By stability of $K$-theory and the above proposition, one has $K_j({{\cal E}}_d)\cong K_j({{\cal A}}_{d-1})$. Pimsner and Voiculescu, moreover, proved $K_j(\widehat{{{\cal A}}}_d)\cong K_j({{{\cal A}}}_{d-1})$ and that the maps from $K_j({{\cal E}}_d) $ to $K_j(\widehat{{{\cal A}}}_d)$ are trivial. The definition of the interesting connecting maps (the exponential map Exp and the index map Ind) will not be described in detail here (see [@Bla0; @WO]), but rather only the outcome for topological insulators. This is the $K$-theoretic content of the BBC.
\[theo-KBBC\] Let the open set $\Delta\subset{{\mathbb R}}$ be a gap of $H\in{{\cal A}}_d$ and $P=\chi(H\leq\mu)$ be the Fermi projection to $\mu\in\Delta$. Then $$\label{ExpMapFormula}
\mbox{\rm Exp}([P]_0)
\;=\;
\left[\exp(2\pi\imath\, {f_{\mbox{\rm\tiny Exp}}}(\widehat{H}))\right]_1
\;,$$ where ${f_{\mbox{\rm\tiny Exp}}}:{{\mathbb R}}\to [0,1]$ is a non-decreasing continuous function equal to $0$ below $\Delta$ and to $1$ above $\Delta$. If $H$ has a chiral symmetry and $U$ is the Fermi unitary given by , then $$\label{IndMapFormula}
\mbox{\rm Ind} \big ([U]_1 \big )
\;=\;
\left[
e^{-\imath\frac{\pi}{2} {f_{\mbox{\rm\tiny Ind}}}(\widehat{H})}
\,
{\rm diag}({{\bf 1}}_L,0_L)
\,e^{\imath\frac{\pi}{2} {f_{\mbox{\rm\tiny Ind}}}(\widehat{H})}
\right]_0
\;-\;
\left[{\rm diag}({{\bf 1}}_L, 0_L)
\right]_0
\;,$$ where ${f_{\mbox{\rm\tiny Ind}}}:{{\mathbb R}}\to [-1,1]$ is a non-decreasing smooth odd function, equal to $\pm 1$ above/below $\Delta$. If, moreover, there is a gap in the surface spectrum and $\widehat{P}=\widehat{P}_++\widehat{P}_-$ is the decomposition of the projection $\widehat{P}$ on the central surface band into sectors of positive and negative chirality, then $$\mbox{\rm Ind} \big ([U]_1 \big )\;=\;[\widehat{P}_+]_0-[\widehat{P}_-]_0
\;.$$
It is remarkable that the r.h.s. in and indeed define classes in ${{\cal E}}_d$, even though they are obtained by functional calculus of $\widehat{H}$ which lies in $\widehat{{{\cal A}}}_d$ and not in ${{\cal E}}_d$. Hence Theorem \[theo-KBBC\] connects $K$-theoretic invariants of the bulk and the boundary.
We have seen in Section \[sec-BulkInv\] how to extract numerical Chern-Connes invariants from the bulk classes. The same can be done for the boundary invariants. Indeed, Definition \[def-CC\] verbatim transposes to define pairings with projections and unitaries in ${{\cal E}}_d\cong{{\cal A}}_{d-1}\otimes{{\cal K}}(\ell^2({{\mathbb N}}))$. One merely has to include a trace over ${{\cal K}}(\ell^2({{\mathbb N}}))$ and consequently verify a supplementary trace class condition, which is satisfied, for example, for the r.h.s. of and . Again the Connes-Chern invariants over ${{\cal E}}_d$ are denoted by ${{\rm Ch}}_I$, but $I$ now cannot contain the index $d$. The connection to the bulk invariants is as follows.
\[theo-BBC\] Provided the traceclass conditions hold, one has for $I\subset\{1,\ldots,d-1\}$ of even and odd cardinality respectively, $$\label{eq-BBC}
\mbox{\rm Ch}_{I\cup \{d\}}(U)\;=\;\mbox{\rm Ch}_I({{\rm Ind}}(U))
\;,
\qquad
\mbox{\rm Ch}_{I\cup \{d\}}(P)\;=\;\mbox{\rm Ch}_I(\mbox{\rm Exp}(P))
\;.$$ Here ${{\rm Ind}}(U)$ and $\mbox{\rm Exp}(P)$ are representatives of ${{\rm Ind}}([U]_1)\in K_0({{\cal A}}_d)$ and $\mbox{\rm Exp}([P]_0)\in K_1({{\cal A}}_d)$.
The proof in [@KRS; @PSB] is an explicit calculation based on a formula for the index map and the suspension. Recently, a $KK$-theoretic proof has been put forward [@BCR; @BKR] which, however, uses the full arsenal of Kasparov’s $KK$-theory. This is a far reaching generalization of $K$-theory. The $K$-groups are described by generalized Fredholm operators on Hilbert modules, similar in spirit to the Atiyah-Jänich theorem. Furthermore, every exact sequence of C$^*$-algebras, such as , also leads to a $KK$-group element. The key element of the theory is then the Kasparov product of two generalized Fredholm operators describing $KK$-group elements. We will not attempt to describe any details on all of this (see [@Bla0]), but rather sketch for the reader familiar with $KK$-theory how it is applied in the present context. Let us consider the second identity in more precisely. The key fact is that the exact sequence defines an extension class and thus an element $[\xi]\in KK^1({{\cal A}}_d,{{\cal E}}_d)$. Its Kasparov product $[P]_0\times [\xi]\in KK^0({{\mathbb C}},{{\cal A}}_d)$ with $[P]_0\in K_0({{\cal A}}_d)\cong KK^1({{\mathbb C}},{{\cal E}}_d)$ is $[U]_1$, if $[U]_1={{\rm Exp}}([P]_0)$. On the other hand, using $K$-homology $[{{\rm Ch}}_{I}]\in KK^0({{\cal E}}_d,{{\mathbb C}})$ and $[{{\rm Ch}}_{I\cup \{d\}}]\in KK^0({{\cal A}}_d,{{\mathbb C}})$, and the Kasparov product can be shown to yield $[\xi]\times [{{\rm Ch}}_{I}]=[{{\rm Ch}}_{I\cup \{d\}}]$. Hence by the associativity of the Kasparov product, $[U]_1\times [{{\rm Ch}}_{I}]=[P]_0\times [\xi]\times [{{\rm Ch}}_{I}]=[P]_0\times [{{\rm Ch}}_{I\cup \{d\}}]$, but this is just claim. Yet another novel approach to the BBC is based on $T$-duality [@MT2]. Let us exhibit two concrete instances to which the general Theorem \[theo-BBC\] about the duality of pairings associated to the exact sequence applies.
\[theo-BoundCur\] Let $d=2$ and $\mu$ lie in a bulk gap. Then the Hall conductance $\mbox{\rm Ch}_{\{1,2\}}(P)$ as given by the Kubo formula dictates the quantization of chiral boundary currents: $$\mbox{\rm Ch}_{\{1,2\}}(P)
\;=\;
\mbox{\rm Ch}_{\{1\}}(\mbox{\rm Exp}(P))
\;=\;
{{\bf E}}_{{\mathbb P}}\sum_{n_2\geq 0} \langle 0,n_2| {f_{\mbox{\rm\tiny Exp}}}'(\widehat{H})\imath[X_1,\widehat{H}]|0,n_2\rangle
\;.$$
The first equality here is just restating Theorem \[theo-BBC\], and the second one results from a calculation. The r.h.s. is physically interpreted as the quantum mechanical expectation value of the current flowing along the boundary, measured by the current operator $\imath[X_1,\widehat{H}]$, of all states in the density matrix ${f_{\mbox{\rm\tiny Exp}}}'(\widehat{H})$ of the half-space operator. This density matrix can be used to approximate $|\Delta|^{-1}\chi_\Delta(\widehat{H})$ and the expression appearing in Theorem \[theo-BoundCur\] can be interpreted as the net contribution of boundary currents (left edge minus right edge due to different chemical potentials). The fact that this difference is quantized with the same integer as the bulk conductivity is crucial for the existence of the integer quantum Hall effect. The second application concerns a surface quantum Hall effect in chiral systems, which is termed [*anomalous*]{} because no external magnetic field is needed. The surface Hall conductance ${{\rm Ch}}_{\{1,2\}}({{\rm Ind}}(U))$ is rather dictated by the bulk topological invariant. No chiral insulator where this can be measured seems to be known.
\[theo-AnoSur\] Let $d=3$ and $\mu$ lie in a bulk gap. Suppose the system is chiral and described by a Fermi unitary $U$, and that the surface spectrum has a gap, opened [*e.g.*]{} by a weak magnetic field perpendicular to surface. Then the last claim of Theorem \[theo-KBBC\] applies and Theorem \[theo-BBC\] shows $$\mbox{\rm Ch}_{\{1,2,3\}}(U)
\;=\;
\mbox{\rm Ch}_{\{1,2\}}(\widehat{P}_+)-\mbox{\rm Ch}_{\{1,2\}}(\widehat{P}_-)
\;.$$
It is conjectured that generically either $\widehat{P}_+=0$ or $\widehat{P}_-=0$. In this situation, a non-trivial bulk-invariant $ \mbox{\rm Ch}_{\{1,2,3\}}(A)\not =0$ leads by Theorem \[theo-AnoSur\] to a surface quantum Hall effect. As a last result let us state what can indeed be proved about the delocalized nature of the surface spectrum.
Suppose that $d$ is even and the Fermi projection below the bulk gap has a non-trivial strong invariant $\mbox{\rm Ch}_{\{1,\ldots,d\}} (P)\not=0$. Then for a disordered potential in an arbitrary finite strip along the boundary, the surface spectrum is not Anderson localized in the sense the Aizenman-Molcanov estimate [@AM; @DDS] on low moments of the Green matrix does not hold for any energy in the bulk gap. For odd $d\geq 3$ and a Fermi unitary with $\mbox{\rm Ch}_{\{1,\ldots,d\}} (U)\not=0$, there is no Anderson localization of the surface states at $\mu=0$.
Discrete real symmetries and periodic table {#sec-symmetries}
===========================================
Except for the second part of Section \[sec-WhatIs\], only systems without symmetry or with a chiral symmetry have been considered so far. The ground states of such systems were classified by complex $K$-theory. Now time reversal symmetry (TRS) and particle hole symmetry (PHS) will be considered, both of which require a real structure allowing to define the complex conjugate of an operator. The ground state (Fermi projection $P$) of systems having these symmetries satisfy respectively $$\label{eq-STRPHS}
{S_{\mbox{\rm\tiny tr}}}^*\,\overline{P}\,{S_{\mbox{\rm\tiny tr}}}\;=\;P\;,
\qquad
{S_{\mbox{\rm\tiny ph}}}^*\,\overline{P}\,{S_{\mbox{\rm\tiny ph}}}\;=\;{{\bf 1}}\,-\,P
\;.$$ Both symmetry operators ${S_{\mbox{\rm\tiny tr}}}$ and ${S_{\mbox{\rm\tiny ph}}}$ are real and square to either ${{\bf 1}}$ or $-{{\bf 1}}$, so that one can either have an even or and odd TRS/PHS. Furthermore, these two operators are strictly local and act in a translation invariant manner only on the fiber ${{\mathbb C}}^L$ of the Hilbert space. It ought to be stressed again that the PHS is actually not a physical symmetry of the system, but rather a property of the BdG Hamiltonian describing a quadratic Fermionic Hamiltonian on Fock space, see [*e.g.*]{} [@KZ]. As already pointed out in Section \[sec-WhatIs\], whenever both TRS and PHS are present, they combine to a chiral symmetry. Hence there are $8$ real symmetry classes, listed in the lower part of Table \[table2\]. Within each so-called Cartan-Altland-Zirnbauer (CAZ) symmetry class and depending on the dimension $d$ of physical space, one can now distinguish possible ground states. Again this can be done using $K$-theory by studying homotopy classes of projections satisfying the corresponding symmetry relations for the given CAZ class. This is done by means of so-called $KR$-theory [@Kar; @BL]. If there is no magnetic field and the system is periodic, one is then lead to use the group $KR_{j}(C({{\mathbb T}}^d_\tau))$ where $j$ is as indicated in Table \[table2\] and $\tau$ denotes the involution $\tau(k)=-k$ on ${{\mathbb T}}^d$. This group can be calculated [@Kar; @Kit] and contains all the strong and weak invariants. It is rather large (like its complex counterpart described in Section \[sec-KClass\]), and also contains so-called torsion invariants which are ${{\mathbb Z}}_2$-valued. To get a more clear representation of the possible phases, in a first step one restricts the attention to the strong invariants, which are known to persist in the Anderson localization regime, as will be described below. The strong invariants are those which are intrinsically $d$-dimensional and thus stem from the groups $KR_{j}(C_0({{\mathbb R}}^d_\tau))$. Here $\tau$ still is the $k$-space inversion. These groups are well-known [@Kar] to coincide with the homotopy groups of the stable orthogonal group $O$: $$\label{eq-invarform}
KR_{j}(C_0({{\mathbb R}}^d_\tau))
\;=\;
\pi_{j-1-d}(O)
\;,$$ given by $$\label{tab-HomO}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|}
\hline
$j$ & $0$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ \\
\hline
$\pi_{j}(O)$ &${{\mathbb Z}}_2$ & ${{\mathbb Z}}_2$ & $0$ & $2\,{{\mathbb Z}}$ & $0$ & $0$ & $0$ & ${{\mathbb Z}}$
\\
\hline
\end{tabular}
\;.$$
$\!\!j\backslash d\!\!$ $\!$TRS$\!$ $\!$PHS$\!$ $\!$CHS$\!$ $\!$CAZ$\!$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$
------------------------- ------------- ------------- ------------- ------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- -------------------- --------------------
$0$ $0$ $0$ $0$ A ${{\mathbb Z}}$ ${{\mathbb Z}}$ ${{\mathbb Z}}$ ${{\mathbb Z}}$
$1$ $0$ $0$ $ 1$ AIII ${{\mathbb Z}}$ ${{\mathbb Z}}$ ${{\mathbb Z}}$ ${{\mathbb Z}}$
$0$ $+1$ $0$ $0$ AI $2\,{{\mathbb Z}}$ ${{\mathbb Z}}_2$ ${{\mathbb Z}}_2$ ${{\mathbb Z}}$
$1$ $+1$ $+1$ $1$ BDI ${{\mathbb Z}}$ $2\,{{\mathbb Z}}$ ${{\mathbb Z}}_2$ ${{\mathbb Z}}_2$
$2$ $0$ $+1$ $0$ D ${{\mathbb Z}}_2$ ${{\mathbb Z}}$ $2\,{{\mathbb Z}}$ ${{\mathbb Z}}_2$
$3$ $-1$ $+1$ $1$ DIII ${{\mathbb Z}}_2$ ${{\mathbb Z}}_2$ ${{\mathbb Z}}$ $2\,{{\mathbb Z}}$
$4$ $-1$ $0$ $0$ AII ${{\mathbb Z}}_2$ ${{\mathbb Z}}_2$ ${{\mathbb Z}}$ $2\,{{\mathbb Z}}$
$5$ $-1$ $-1$ $1$ CII $2\,{{\mathbb Z}}$ ${{\mathbb Z}}_2$ ${{\mathbb Z}}_2$ ${{\mathbb Z}}$
$6$ $0$ $-1$ $0$ C $2\,{{\mathbb Z}}$ ${{\mathbb Z}}_2$ ${{\mathbb Z}}_2$ ${{\mathbb Z}}$
$7$ $+1$ $-1$ $1$ CI $2\,{{\mathbb Z}}$ ${{\mathbb Z}}_2$ ${{\mathbb Z}}_2$ ${{\mathbb Z}}$
: *So-called periodic table of topological insulators, listing possible values of the strong invariants for the CAZ classes in dependence of the dimension $d$.* []{data-label="table2"}
Bott periodicity states that $\pi_j(O)=\pi_{j+8}(O)$. Reporting this data into a table now produces the celebrated periodic table for topological insulators [@Kit; @RSFL]. Why there appears a $2\,{{\mathbb Z}}$ instead of the isomorphic group ${{\mathbb Z}}$ will be explained below. The above discussion did not explain why the CAZ classes are placed in the particular order labelled by the index $j$. Indeed, it requires quite a lengthy series of arguments of successively implementing symmetries to show that the CAZ classes correspond to the index $j$ as stated, see [@SCR; @FM; @Thi; @KZ; @GS; @Kel].
An alternative approach [@Sch; @GS] is to directly implement the symmetries in the index pairings described in Theorem \[theo-IndexPair\]. This has the advantage that disordered systems can be dealt with even when the Fermi energy lies in a mobility gap. Let first explain how this works by considering the example of a two-dimensional system having an odd TRS (just as in the first novel topological insulator studied in [@KM2]). Then ${S_{\mbox{\rm\tiny tr}}}^* \overline{P}{S_{\mbox{\rm\tiny tr}}}=P$ with ${S_{\mbox{\rm\tiny tr}}}^2=-{{\bf 1}}$. As the Dirac phase $G=\frac{X_1+\imath X_2}{|X_1+\imath X_2|}$ commutes with ${S_{\mbox{\rm\tiny tr}}}$ and satisfies $G^t=G$ where $A^t=(\overline{A})^*$ denotes the transpose of an operator $A$. The Fredholm operator $T_\omega=P_\omega G P_\omega$ in Theorem \[theo-IndexPair\] thus satisfies ${S_{\mbox{\rm\tiny tr}}}^* (T_\omega)^t{S_{\mbox{\rm\tiny tr}}}=T_\omega$, or equivalently the Fredholm operator $T_\omega{S_{\mbox{\rm\tiny tr}}}$ is anti-symmetric. It can then be proved [@AS; @Sch] that the set of anti-symmetric Fredholm operators has exactly $2$ connected components labelled by the compactly stable and homotopy invariant ${{\mathbb Z}}_2$-valued index $$\label{eq-Ind2}
\mbox{\rm Ind}_2(T)
\;=\;
\dim(\mbox{\rm Ker}(T))\;\mbox{\rm mod}\;2
\;\in\;{{\mathbb Z}}_2
\;.$$ As already stressed above, this allows to define a ${{\mathbb Z}}_2$ phase label for Kane-Mele model in the regime of a mobility gap. As a second example, let us consider a system with odd TRS in dimension $d=8$. Then one can check that the Dirac phase satisfies $\overline{G}=G$. Consequently, the Fredholm operator $T_\omega=P_\omega G P_\omega$ satisfies ${S_{\mbox{\rm\tiny tr}}}^* \overline{T_\omega}{S_{\mbox{\rm\tiny tr}}}=T_\omega$. It is hence quaternionic and has an even dimensional kernel and cokernel so that the Fredholm index is even (thus the entry $2\,{{\mathbb Z}}$ for $(j,d)=(4,8)$ in Table \[table2\]). This parity can actually be of some physical significance, see Theorem \[theo-PHSbound\] below.
Let us now sketch how this approach can be extended to treat all cases appearing in the periodic table. The symmetry of the Fermi projection (even/odd TRS/PHS) is one ingredient entering into the symmetry of the index pairings of Theorem \[theo-IndexPair\], a second one comes from the symmetry of the homological part given by the Dirac operator . In the above, this was $G^t=G$ and $\overline{G}=G$, and the general situation is described next. The irreducible representation $\gamma_1,\ldots,\gamma_d$ of the Clifford algebra ${{\mathbb C}}_d$ is chosen such that $\gamma_{2j}=-\overline{\gamma_{2j}}$ and $\gamma_{2j+1}=\overline{\gamma_{2j+1}}$. Recall that for $d$ even there is a grading $\Gamma=-i^{-d/2}\gamma_1\cdots \gamma_d$. Now one can show that there exists a real unitary $\Sigma$ (essentially unique) such that
$d=8-i$ $8$ $7$ $6$ $5$ $4$ $3$ $2$ $1$
-------------------------------------- ------------- ------------- -------------- -------------- -------------- -------------- ------------- -------------
${\Sigma}^2$ ${{\bf 1}}$ ${{\bf 1}}$ $-{{\bf 1}}$ $-{{\bf 1}}$ $-{{\bf 1}}$ $-{{\bf 1}}$ ${{\bf 1}}$ ${{\bf 1}}$
${\Sigma}^*\,\overline{D}\,{\Sigma}$ $D$ $-D$ $D$ $D$ $D$ $-D$ $D$ $D$
$\Gamma\,{\Sigma}\,\Gamma$ ${\Sigma}$ $-{\Sigma}$ ${\Sigma}$ $-{\Sigma}$
.
The construction of this representation is explicit [@GS] and is related to irreducible representations of the real Clifford algebra. The data of the table sates that $(D,\Gamma,\Sigma)$ defines a $KR^i$-cycle, also called a spectral triple with real structure (this is particularly clear by comparing with [@GVF]). Now one can do a careful symmetry analysis of the index pairings in Theorem \[theo-IndexPair\]. Even though this can be achieved with basic functional analytic tools, there are actually a number of intricate spectral degeneracy arguments needed (one of them being the classical Kramers’ degeneracy).
The strong index paring from Theorem \[theo-IndexPair\] leads to a ${{\mathbb Z}}$, ${{\mathbb Z}}_2$ or $2\,{{\mathbb Z}}$ index as stated in Table \[table2\], provided the Fermi level lies in a region of Anderson localization.
As there are index theorems for the boundary invariants in complete analogy with Theorem \[theo-IndexPair\], the very same techniques also allows to define ${{\mathbb Z}}$, ${{\mathbb Z}}_2$ or $2\,{{\mathbb Z}}$ indices for the boundary invariants and then establish a BBC for these invariants based on the implementation of symmetries in Theorem \[theo-KBBC\], but there is no published report on this. As there is a very intimate relation between Bott periodicity and real Clifford algebras, it is also possible to prove Clifford-module-valued index pairings [@LM] for the strong bulk invariants in the realm of $KK$-theory [@BCR2] (this is still restricted to gapped systems for now). This also allows to connect bulk and boundary invariants in a natural manner, by an extension of the argument sketched after Theorem \[theo-BBC\], see [@BKR]. Once all these invariants are defined, an important question is about their physical significance. Some further insight is presented in the remaining two sections.
Spectral flow in topological insulators {#sec-SF}
=======================================
The spectral flow of a given family $t\in[0,1]\mapsto T_t$ of self-adjoint Fredholm operators is the net number of eigenvalues moving through $0$, namely those moving up count for $1$ and those moving down with $-1$ (if simple, otherwise the multiplicity appears). For analytic paths, one can readily refer to analytic perturbation theory in order to make a rigorous definition out of this basic idea, but for merely continuous paths this is somewhat delicate and was only made precise by Phillips [@Phi]. The basic properties of the spectral flow are its homotopy invariance (with fixed endpoints) and a natural concatenation property. Furthermore, there is a close link to index theory for paths of with unitarily equivalent end points $T_1=U^*T_0U$, namely their spectral flow is equal to the index of the Fredholm operator $PUP$ where $P=\chi(T_0)$ [@Phi]. In topological insulators, the parameter $t$ stems from a local perturbation of the Hamiltonian denoted by $H_t$, and then $T_t=H_t-\mu$ for $\mu$ in a gap of $H=H_0$ is a family of self-adjoint Fredholm operators. The main example of the type is the insertion of a flux tube in a two-dimensional system, that is the addition of a supplementary magnetic field $2\pi t$ through one cell of the lattice ${{\mathbb Z}}^2$. This situation and actually also the result below are usually referred to as a Laughlin argument, see for example [@BES; @ASS]. While the gauges used to realized the (local) flux tube are not compact, one can check that $T_t=H_t-\mu$ is nevertheless a path of Fredholm operators [@Mac; @DS].
Let $d=2$ and $\mu$ lie in a gap of $H=H_0$. Set $P=\chi(H\leq\mu)$. The spectral flow associated to the insertion of a flux tube satisfies $$\mbox{\rm SF}(t\in[0,1]\mapsto H_t-\mu)
\;=\;
{{\rm Ch}}_{\{1,2\}}(P)
\;.$$
It is now possible to analyze the fate of this spectral flow in a two-dimensional system with odd TRS, namely in a quantum spin Hall system. As the Chern number ${{\rm Ch}}_{\{1,2\}}(P)$ vanishes, there is actually no spectral flow. However, the path has the symmetry property $\overline{H_t}=-{S_{\mbox{\rm\tiny tr}}}^* G^* H_{1-t} G{S_{\mbox{\rm\tiny tr}}}$ with $G$ as in Theorem \[theo-IndexPair\]. Therefore also the spectral curves have symmetry property $\sigma(H_t)=-\sigma(H_{1-t})$. Let us further point out that the insertion of the flux breaks the odd TRS, except for a half-flux $t=\frac{1}{2}$ which can be interpreted as a local defect respecting the symmetry. Thus one has Kramers’ degeneracy for eigenvalues of $H_0$, $H_{\frac{1}{2}}$ and $H_1$. It can be shown [@ASV] that there are two topologically distinct classes of spectral curves having these Kramers’ degeneracies and the symmetry described above. The non-trivial one appears exactly when the bulk ${{\mathbb Z}}_2$ invariant is non-trivial. Thus one can conclude that the half-flux defect necessarily leads to a Kramers’ degenerate bound state, if the system is topologically non-trivial:
Let $d=2$ and suppose that $\mu$ is in a gap of $H=H_0$ having odd TRS. If the ${{\mathbb Z}}_2$-index ${{\rm Ind}}_2(P_\omega GP_\omega)$, defined as in , is non-trivial, then $H_{\frac{1}{2}}$ has Kramers degenerate bound state in the gap of $\mu$.
It is also possible to consider other symmetry classes. Particularly interesting are the BdG Hamiltonians with PHS. Insertion of a flux breaks the PHS, but again a half-flux is local defect respecting PHS. Now the attached bound state is a so-called Majorana zero mode (namely in second quantization, it induces a self-adjoint creation operator).
\[theo-PHSbound\] Let $d=2$ and suppose that $0$ is in a gap of $H=H_0$ having an even or odd PHS. Then $$\dim ({{\rm Ker}}(H_{\frac{1}{2}}))
\;\mbox{\rm mod }2
\;=\;
{{\rm Ch}}_{\{1,2\}}(P)
\;\mbox{\rm mod }2
\;.$$ Here the Chern number ${{\rm Ch}}_{\{1,2\}}(P)$ of $P=\chi(H\leq 0)$ is the strong invariant of the system. If it is odd, $H_{\frac{1}{2}}$ must have a zero mode. For a system with odd PHS, ${{\rm Ch}}_{\{1,2\}}(P)$ is even and thus Majorana zero modes are unstable in the sense that they can be lifted by a generic perturbation.
Let us briefly describe another type of spectral flow which is of interest in the context of topological insulators, as recently shown in [@CPS]. Consider again a BdG Hamiltonian $H$ with an even PHS ${S_{\mbox{\rm\tiny ph}}}^*H{S_{\mbox{\rm\tiny ph}}}=-H$. Suppose ${S_{\mbox{\rm\tiny ph}}}=\binom{0 \;1}{1\;0}$ and let $C=2^{-\frac{1}{2}}\binom{1\,-\imath}{1\;\;\imath}$ be the Cayley transform in the same grading. Then the so-called Majorana representation $H_{{\mbox{\rm\tiny Maj}}}=C^* HC$ is purely imaginary and can thus be written as $H_{{\mbox{\rm\tiny Maj}}}=\imath T$ with a skew-adjoint real operator $T$, which is Fredholm provided that $0$ lies in a gap of $H$. Therefore paths of gapped BdG operators lead to paths of skew-adjoint Fredholm operators. For such paths one can define a ${{\mathbb Z}}_2$-valued spectral flow ${{\rm SF}}_2$ by counting the number orientation changes of the eigenfunctions at eigenvalue crossings through $0$ [@CPS]. A particular example of this is obtained by inserting a flux into the one-dimensional Kitaev chain with even PHS and even TRS given by $$\label{eq-KitaevDef}
H
\;=\;
\frac{1}{2}\,
\begin{pmatrix}
S+S^*+2\mu & \imath(S-S^*) \\
\imath(S-S^*) & -(S+S^*+2\mu)
\end{pmatrix}
\;+\;
W
\;,$$ where $W$ a random matrix potential respecting the even PHS and the even TRS ${S_{\mbox{\rm\tiny tr}}}^*\overline{W}{S_{\mbox{\rm\tiny tr}}}=W$ with ${S_{\mbox{\rm\tiny tr}}}=\binom{1 \;\,0}{0\;-1}$. The operator acts on $\ell^2({{\mathbb Z}})\otimes{{\mathbb C}}^2=\ell^2({{\mathbb Z}}\times\{0,1\})$ and $S$ is the bilateral shift as before. Now one can again insert a flux tube trough one cell of the lattice strip ${{\mathbb Z}}\times\{0,1\}$ which now leads to a path $H_t$ of BdG operators and thus also a path of skew-adjoint Fredholm operators. If $|\mu|<1-\|W\|$, this path can be shown to have non-trivial ${{\mathbb Z}}_2$-valued spectral flow and this non-triviality can again be used to show the existence of bound states to the defect given by a half-flux.
\[theo-defect\] For $|\mu|<1-\|W\|$, the Kitaev Hamiltonian ${H}_{\frac{1}{2}}$ with a half-flux defect has an odd number of evenly degenerate zero eigenvalues: $$\tfrac{1}{2}\,\dim({{\rm Ker}}({H}_{\frac{1}{2}}))\;\mbox{\rm mod}\,2
\;=\;1
\;.$$
There are several other situations where the spectral flow is interesting in topological insulators [@PSB; @CPS]. What is, however, missing is a spectral flow in chiral three-dimensional systems which detects their strong topological invariant.
Spin Chern numbers {#sec-SpinCh}
==================
The last sections documented that the ${{\mathbb Z}}_2$ invariants are well-established on a theoretical side and can be used to distinguish phases of systems having exact symmetries (like TRS and PHS). On the other hand, their importance for experimental observations is still disputed. In several recent experiments on (two-dimensional) quantum spin Hall systems [@KRY; @Ma], it became apparent that the effect of having delocalized surface modes is remarkably stable under perturbation by magnetic fields (up to $9$ Tesla!). As these magnetic fields break TRS, there is no notion of ${{\mathbb Z}}_2$-invariant in this situation and the question is whether there is another mechanism leading to these stable surface states. Here it will be argued that the so-called spin Chern numbers introduced by Prodan [@Pro] are topological invariants that may be at the root of the phenomena. Their definition does not require TRS, but rather an approximate conservation of one component of the spin, say $s^z$. In mathematical terms, this is expressed by requiring that $$\label{eq-SCon}
\|[H,s^z]\|\;\leq\;C
\;,$$ for some sufficiently small constant $C$. If then $P$ is the Fermi projection below the gap, one can show that the spectrum of the self-adjoint operator $Ps^zP$ has a gap at $0$ (the origin itself lies in the spectrum, but is irrelevant here). Therefore there are two Riesz projections $P_\pm$ associated to the positive and negative spectrum and they provide an orthogonal decomposition $P=P_++P_-$. Furthermore, the existence of the gap readily implies that both $P_\pm$ are smooth so that the Chern numbers ${{\rm Ch}}_{\{1,2\}}(P_\pm)$ are well-defined. One has the sum rule ${{\rm Ch}}_{\{1,2\}}(P_+)+{{\rm Ch}}_{\{1,2\}}(P_-)={{\rm Ch}}_{\{1,2\}}(P)$ and, as typically ${{\rm Ch}}_{\{1,2\}}(P)=0$, it is sufficient to consider one of these invariants. The spin Chen number is then defined as $\mbox{\rm SCh}(P)={{\rm Ch}}_{\{1,2\}}(P_+)$. It clearly also exists for systems without TRS. Moreover, it is connected to the ${{\mathbb Z}}_2$-invariants discussed above.
Let $H$ have odd TRS and approximate spin conservation . Then $$\mbox{\rm Ind}_2(P_\omega G P_\omega)
\;=\;
\mbox{\rm SCh}(P)\;\mbox{\rm mod}\;2
\;.$$
While this result was stated for $d=2$ and odd TRS, it can readily be extended to other situations where there is an approximately conserved observable with spectral gap. Hence in many situations, there may be non-trivial invariants resulting from complex pairings present in systems with non-trivial ${{\mathbb Z}}_2$ invariants. This is similar to the situation of invariants in chiral systems, which remain stable also when the system is only approximately chiral. It still remains to discuss the importance of spin Chern number for physical effects. A partial answer is provided by the following result which should be compared with Theorem \[theo-BoundCur\].
If $\mbox{\rm SCh}(P)\not =0$, then spin filtered edge currents in of states in the bulk gap are stable w.r.t. perturbations by magnetic field and disorder: $${{\bf E}}_{{\mathbb P}}\sum_{n_2\geq 0} \langle 0,n_2| {f_{\mbox{\rm\tiny Exp}}}'(\widehat{H})\,\tfrac{1}{2}\big\{\imath[X_1,\widehat{H}],s^z\big\}|0,n_2\rangle
\;=\;
\mbox{\rm SCh}(P) \;+\; \mbox{corrections}
\;,$$ where the corrections depend linearly on the size of the commutator and the $C^6$-norm of ${f_{\mbox{\rm\tiny Exp}}}$. Here $\{\,,\,\}$ denotes the anti-commutator used to produce a selfadjoint observable.
The presence of corrections here implies that the l.h.s. is not quantized, but it is non-vanishing when the spin Chern number is non-vanishing. What is still somewhat unsatisfactory about this result is that the spin current itself is ill-defined when $[H,s^z]\not=0$. Nevertheless, the result clearly hints at the existence of surface states with continuous spectrum, even when TRS is broken.
Future directions
=================
Topological materials still remain a very active field of research in physics. Those currently investigated include topological photonic crystals, topological bosonic systems, topological mechanical and optomechanical systems, spin systems, as well as the role of interactions in Fermionic systems. As the long list of references suggests, there is also a growing interest in the mathematical physics community. Let us list a few open questions within the framework of topological insulators made up of independent Fermions:
$\bullet$ Index theory for weak topological invariants and definite answer on their stability;
$\bullet$ more detailed description of the bulk-edge correspondence in cases with real symmetries;
$\bullet$ topological invariants linked to spacial symmetries (like rotation, inversion, reflection);
$\bullet$ an analysis of the stability of the invariants in the previous item;
$\bullet$ further investigation of physical implications of the invariants, [*e.g.*]{} for heat transport.
Some of these issues should be settled in the near future. Also applications to the other physical systems listed above should be within reach for rigorous analysis. A major challenge, however, remains the definition and analysis of topological invariants in interacting systems and spin systems - even though there are already numerous contributions from theoretical physicists.
[**Acknowledgements:**]{} The author thanks his main collaborators Jean Bellissard, Johannes Kellendonk and Emil Prodan as well as further coauthors Giuseppe De Nittis, Stefan Teufel, Julian Grossmann, Carlos Villegas, Julio Cesar Avila, Alan Carey and John Phillips for inspiring intellectual input and persistence. This work is in part supported by the DFG.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The extender based Magidor-Radin forcing is being generalized to supercompact type extenders.'
address: |
School of Computer Science\
Tel-Aviv Academic College\
Rabenu Yeroham St.\
Tel-Aviv 68182\
Israel
author:
- Carmi Merimovich
date: 'August 1, 2016'
title: 'Supercompact Extender Based Magidor-Radin Forcing'
---
Introduction
============
This work[^1] continues the project of of generalizing the extender based Prikry forcing [@GitikMagidor1992] to larger and larger cardinals. In [@Merimovich2003; @Merimovich2011b] the methods introduced in [@GitikMagidor1992] (which generalized Prikry forcing [@Prikry1968] from using a measure to using an extender), were used to generalize the Magidor [@Magidor1978] and Radin [@Radin1982] forcing notions to use a sequence of extenders. In a different direction [@Merimovich2011c] used the methods of [@GitikMagidor1992] to define the extender based Prikry forcing over extenders which have higher directedness properties than their critical point. Such extenders give rise to supercompact type embeddings. Generalization of Prikry forcing to fine ultrafilters yielding supercompact type embeddings appeared in [@Magidor1977I]. Extending this forcing notion to Magidor-Radin type forcing notions were done in [@ForemanWoodin1991] and [@Krueger2007]. In the current paper we use extenders with higher directedness properties to define the extender based Magidor-Radin forcing notion. All of the forcing notions mentioned above are of course of Prikry type. For more information on Prikry type forcing notions one should consult [@Gitik2010].
Before stating the theorem of this paper we need to make some notions precise. Assume $E$ is an extender. We let $j_E{\mathrel{:}}V \to M \simeq \operatorname{Ult}(V,E)$ be the natural embedding of $V$ into the transitive collpase of the ultrapower $\operatorname{Ult}(V,E)$. We denote by $\operatorname{crit}E$ the critical point of the embedding $j_E$. In principle, an extender is a directed family of ultrafilters and projections. We denote by ${\lambda}(E)$ a degree of directedness holding for the extender $E$. We do not require ${\lambda}(E)$ to be optimal, i.e., ${\lambda}(E)$ is not necessarily the minimum cardinal for which $E$ is not ${\lambda}(E)^+$-directed. Note $M \supseteq {{\vphantom{M}}^{<{\lambda}(E)}{M}}$.
A sequence of extenders $\Vec{E} = {\ensuremath{{\ensuremath{\langle E_{\xi}\mid {\xi}< \operatorname{o}(\Vec{E}) \rangle}}}}$, all with the same critical point $\operatorname{crit}E_{\xi}$ and the same directedness size ${\lambda}(E_{\xi})$, is said be Mitchell increasing if for each ${\xi}< \operatorname{o}(\Vec{E})$ we have ${\ensuremath{{\ensuremath{\langle E_{{\xi}'} \mid {\xi}' < {\xi}\rangle}}}} \in M_{\xi}\simeq \operatorname{Ult}(V,E_{\xi})$. We will denote by $\operatorname{crit}(\Vec{E})$ and ${\lambda}(\Vec{E})$ the common values of $\operatorname{crit}E_{\xi}$ and ${\lambda}(E_{\xi})$, respectively.
If $\Vec{E} = {\ensuremath{{\ensuremath{\langle E_{\xi}\mid {\xi}< \operatorname{o}(\Vec{E}) \rangle}}}}$ is a Mithcell increasing sequence of extenders and ${\alpha}\in [\operatorname{crit}{\Vec{E}}, j_{E_0}({\kappa}))$ then ${\Bar{E}}= {\ensuremath{\langle {\alpha}, \Vec{E} \rangle}}$ is said to be an extender sequence. Hence an extender sequence is an ordered pair with the first coordinate being an ordinal and the second coordinate being a Mitchell increasing sequence of extenders. Note that an empty sequence of extenders is legal in an extender sequence, e.g., ${\ensuremath{\langle {\alpha}, {\ensuremath{\langle \rangle}} \rangle}}$ is an extender sequence. Let $\operatorname{ES}$ be the collection of extender sequences. If ${\Bar{E}}$ is an extender sequence then we denote the projections to the first and second coordinates by $\mathring{{\Bar{E}}}$ and $\accentset{\mid}{{\Bar{E}}}$, respectively. The ordinals at the first coordinate of an extender sequence induce an order $<$ on $\operatorname{ES}$ by setting ${\Bar{{\nu}}}< {\Bar{{\mu}}}$ if $\mathring{{\Bar{{\nu}}}} < \mathring{{\Bar{{\mu}}}}$. We lift the functions defined on the Mitchell increasing sequence of extenders to extender sequences in the obvious way, i.e., $\operatorname{o}({\Bar{E}}) = \operatorname{o}(\accentset{\mid}{{\Bar{E}}})$ and ${\lambda}({\Bar{E}})={\lambda}(\accentset{\mid}{{\Bar{E}}})$. We will also abuse notation by writing ${\Bar{E}}_{\xi}$ for the extender $E_{\xi}$. There are two restrictions we have on ${\lambda}(\Vec{E})$. The first one seems a bit technical. We demand ${\lambda}({\Bar{E}})^{<\operatorname{crit}{\Bar{E}}} = {\lambda}({\Bar{E}})$ due to limitations we encountered in \[GetGoodPair\]. The second one is more substantial. We demand ${\lambda}({\Bar{E}}) \leq j_{E_0}(\operatorname{crit}(E_0))$. (It seems this last demand can be removed for the special case $\operatorname{o}(\Vec{E}) = 1$.) With all these preliminaries at hand we can write the theorem proved in this paper.
Assume the GCH. Let $\Vec{E}$ be a Mitchell increasing sequence such that ${\lambda}({\Vec{E}}) < j_{E_0}(\operatorname{crit}(\Vec{E}))$ and ${\mu}^{<\operatorname{crit}{\Vec{E}}} < {\lambda}({\Vec{E}})$ for each ${\mu}< {\lambda}({\Vec{E}})$. Furthermore, assume ${\epsilon}\leq j_{E_0}({\kappa})$. Then there is a forcing notion ${\mathbb{P}}(\Vec{E}, {\epsilon})$ such that the following hold in $V[G]$, where $G \subseteq {\mathbb{P}}(\Vec{E}, {\epsilon})$ is generic. There is a set $G^{\kappa}\subseteq \operatorname{ES}$ such that $G^{\kappa}{\cup}{\ensuremath{\{ {\ensuremath{\langle \operatorname{crit}\Vec{E}, \Vec{E} \rangle}} \}}}$ is increasing and for each ${\Bar{{\nu}}}\in G^{\kappa}{\cup}{\ensuremath{\{ {\ensuremath{\langle \operatorname{crit}\Vec{E}, \Vec{E} \rangle}} \}}}$ such that $\operatorname{o}({\Bar{{\nu}}}) > 0$ the following hold:
1. ${\ensuremath{\{ \operatorname{crit}{\Bar{{\mu}}}\mid {\Bar{{\mu}}}\in G^{\kappa}, {\Bar{{\mu}}}< {\Bar{{\nu}}}\}}} \subseteq \mathring{{\Bar{{\nu}}}}$ is a club.
2. $\operatorname{crit}{\Bar{{\nu}}}$ and ${\lambda}({\Bar{{\nu}}})$ are preserved in $V[G]$, and $(\operatorname{crit}{\Bar{{\nu}}}^+ = {\lambda}({\Bar{{\nu}}}))^{V[G]}$.
3. If $\operatorname{o}({\Bar{{\nu}}})<\operatorname{crit}{\Bar{{\nu}}}$ is $V$-regular then $\operatorname{cf}\operatorname{crit}{\Bar{{\nu}}}= \operatorname{cf}\operatorname{o}({\Bar{{\nu}}})$ in $V[G]$.
4. (Gitik) If $\operatorname{o}({\Bar{{\nu}}}) \in [\operatorname{crit}{\Bar{{\nu}}}, {\lambda}({\Bar{{\nu}}}))$ and $\operatorname{cf}(\operatorname{o}({\Bar{{\nu}}})) \geq \operatorname{crit}({\Bar{{\nu}}})$ then $\operatorname{cf}\operatorname{crit}{\Bar{{\nu}}}= {\omega}$ in $V[G]$.
5. If $\operatorname{o}({\Bar{{\nu}}}) \in [\operatorname{crit}{\Bar{{\nu}}}, {\lambda}({\Bar{{\nu}}}))$ and $\operatorname{cf}(\operatorname{o}({\Bar{{\nu}}})) < \operatorname{crit}({\Bar{{\nu}}})$ then $\operatorname{cf}\operatorname{crit}{\Bar{{\nu}}}= \operatorname{cf}\operatorname{o}({\Bar{{\nu}}})$ in $V[G]$.
6. If $\operatorname{o}({\Bar{{\nu}}})= \operatorname{crit}({\Bar{{\nu}}})$ then $\operatorname{cf}\operatorname{crit}{\Bar{{\nu}}}= {\omega}$ in $V[G]$.
7. If $\operatorname{o}({\Bar{{\nu}}})= {\lambda}({\Bar{{\nu}}})$ then $\operatorname{crit}{\Bar{{\nu}}}$ is regular in $V[G]$.
8. If $\operatorname{o}({\Bar{{\nu}}})= {\lambda}({\Bar{{\nu}}})^{++}$ then $\operatorname{crit}{\Bar{{\nu}}}$ is measurable in $V[G]$.
9. $2^{\operatorname{crit}{\Bar{{\nu}}}} = \max {\ensuremath{\{ {\lambda}({\Bar{{\nu}}}), {\lvert{\epsilon}\rvert} \}}}$.
Thus for example, if we assume ${\ensuremath{{\ensuremath{\langle E_{\xi}\mid {\xi}< {\omega}_1 \rangle}}}}$ is a Mitchell increasing sequence of extenders on ${\kappa}$ giving rise to a $<{\kappa}^{++}$-closed elementary embeddings (and no more), then in the generic extension ${\kappa}$ will change its cofinality to ${\omega}_1$, and ${\kappa}^+$ would be collapsed. Moreover, there is a club of ordertype ${\omega}_1$ cofinal in ${\kappa}$, and for each limit point ${\tau}$ in this club ${\tau}^+$ of the ground model is collapsed. The GCH would be preserved, and no other cardinals are collapsed.
As another example, assume ${\ensuremath{{\ensuremath{\langle E_{\xi}\mid {\xi}< {\omega}_1 \rangle}}}}$ is a Mitchell increasing sequence of extenders on ${\kappa}$ giving rise to a $<{\kappa}^{++}$-close elementary embeddings which are also ${\kappa}^{+3}-strong$ (and no more), then in the generic extension ${\kappa}$ will change its cofinality to ${\omega}_1$, and ${\kappa}^+$ would be collapsed. Moreover, there is a club of ordertype ${\omega}_1$ cofinal in ${\kappa}$, and for each limit point ${\tau}$ in this club ${\tau}^+$ of the ground model is collapsed. In this case we get $2^{\kappa}= {\kappa}^{++}$ and $2^{\tau}= {\tau}^{++}$ for the limit points of the club. In fact we have $2^{\kappa}= ({\kappa}^{+3})_V$ and $2^{\tau}= ({\tau}^{+3})_V$, and we see only gap-2 in the generic extension since ${\kappa}^+$ of the ground mode gets collapsed as do all the ${\tau}^+$ of the ground model. No other cardinal get collapsed. The structure of the work is as follows. In section \[sec:ExtendersAndNormality\] a formulation of extenders useful for ${\lambda}$-directed extenders is presented, and an appropriate diagonal intersection operation is introduced. In section \[sec:Forcing\] the forcing notion is defined and the properties of it which do not rely on understanding the dense subsets of the forcing are presented. In section \[sec:Dense\] claims regarding the dense subsets of the forcing notion are presented. This section is highly combinatorial in nature. In section \[sec:KappaProperties\] the influence of $\operatorname{o}({\Vec{E}})$ on the properties of ${\kappa}$ in the generic extension is shown. The claims here rely on the structure of the dense subsets as analyzed in section \[sec:Dense\].
This work is self contained assuming large cardinals and forcing are known.
${\lambda}$-Directed Extenders and Normality
============================================
Assume the GCH. Let ${\Vec{E}}= {\ensuremath{{\ensuremath{\langle E_{\xi}\mid {\xi}< \operatorname{o}(\Vec{E}) \rangle}}}}$ be a Mitchell increasing sequence of ${\lambda}$-directed extenders such that ${\lambda}\leq j_{E_0}({\kappa})$ is regular and ${\lambda}^{<{\kappa}} = {\lambda}$, where ${\kappa}= \operatorname{crit}\Vec{E}$. For each ${\xi}< \operatorname{o}(\Vec{E})$ let $j_{E_{\xi}} {\mathrel{:}}V \to M_{\xi}\simeq \operatorname{Ult}(V, E_{\xi})$ be the natural embedding. Assume $d \in [{\epsilon}]^{<{\lambda}}$ and ${\lvertd\rvert}+1 \subseteq d$. We let $\operatorname{OB}(d)$ be the set of functions ${\nu}{\mathrel{:}}\operatorname{dom}{\nu}\to \operatorname{ES}$ such that ${\kappa}\in \operatorname{dom}{\nu}\subseteq d$, and if ${\alpha}, {\beta}\in \operatorname{dom}{\nu}$ and ${\alpha}< {\beta}$ then ${\mathring{{\nu}}}({\alpha}) < {\mathring{{\nu}}}({\beta})$. Define an order on $\operatorname{OB}(d)$ by saying for each pair ${\nu},{\mu}\in \operatorname{OB}(d)$ that ${\nu}< {\mu}$ if $\operatorname{dom}{\nu}\subseteq \operatorname{dom}{\mu}$, ${\lvert{\nu}\rvert} < {\mathring{{\mu}}}({\kappa})$, and for each ${\alpha}\in \operatorname{dom}{\nu}$, ${\nu}({\alpha}) < {\mathring{{\mu}}}({\kappa})$.
For ${\xi}< \operatorname{o}({\Vec{E}})$ and a set $d \in [{\epsilon}]^{<{\lambda}}$ define the measure $E_{\xi}(d)$ on $\operatorname{OB}(d)$ as follows: $$\begin{aligned}
X \in E_{\xi}(d) \iff {\ensuremath{\{ {\ensuremath{\langle j_{E_{\xi}}({\alpha}), {\ensuremath{\langle {\alpha}, {\ensuremath{{\ensuremath{\langle E_{{\xi}'} \mid {\xi}' < {\xi}\rangle}}}} \rangle}} \rangle}} \mid {\alpha}\in d \}}} \in j_{E_{\xi}}(X).
\end{aligned}$$ For a set $d \in [{\epsilon}]^{<{\lambda}}$ let $\Vec{E}(d) = {\bigcap}{\ensuremath{\{ E_{\xi}(d \mid {\xi}< \operatorname{o}({\Bar{E}}) \}}}$. It is clear $E_{\xi}(d)$ is a ${\kappa}$-complete ultrafilter over $\operatorname{OB}(d)$ and ${\Vec{E}}(d)$ is a ${\kappa}$-complete filter over $\operatorname{OB}(d)$. In addition to this, the filter ${\Vec{E}}(d)$ has a useful normality property with a matching diagonal intersection soon to be introduced.
If $S \subseteq \operatorname{OB}(d)$, ${\nu}^* \in j_{E_{\xi}}(S)$, and ${\nu}^* < \operatorname{mc}_{\xi}(d)$, then there is ${\nu}\in S$ such that ${\nu}^* = j_{E_{\xi}}({\nu})$.
Assume $S \subseteq \operatorname{OB}(d)$ and for each ${\nu}\in S$ there is a set $X({\nu}) \subseteq \operatorname{OB}(d)$. Define the diagonal intersection of the family ${\ensuremath{\{ X({\nu}) \mid {\nu}\in S \}}}$ as follows: $$\begin{aligned}
\operatorname*{\triangle}_{{\nu}\in S}X({\nu}) = {\ensuremath{\{ {\nu}\in \operatorname{OB}(d) \mid \forall {\mu}\in S\ ({\mu}< {\nu}\implies {\nu}\in X({\mu})) \}}}.\end{aligned}$$
Assume $S \subseteq \operatorname{OB}(d)$, and for each ${\nu}\in S$, $X({\nu}) \in {\Vec{E}}(d)$. Then $X^* = \operatorname*{\triangle}_{{\nu}\in S}X({\nu}) \in {\Vec{E}}(d)$.
We need to show for each ${\xi}< \operatorname{o}({\Vec{E}})$, $\operatorname{mc}_{\xi}(d) \in j_{E_{\xi}}(X^*)$. I.e., we need to show $\operatorname{mc}_{\xi}(d) \in j_{E_{\xi}}(X)({\nu}^*)$ for each ${\nu}^* \in j_{E_{\xi}}(S)$ such that ${\nu}^* < \operatorname{mc}_{\xi}(d)$. Fix ${\nu}^* \in j_{E_{\xi}}(S)$ such that ${\nu}^* < \operatorname{mc}_{\xi}(d)$. There is ${\nu}\in S$ such that ${\nu}^* = j_{E_{\xi}}({\nu})$. Hence $j_{E_{\xi}}(X)({\nu}^*) = j_{E_{\xi}}(X({\nu}))$. Since $X({\nu}) \in E_{\xi}(d)$ we get $\operatorname{mc}_{\zeta}(d) \in j_{E_{\zeta}}(X({\nu}))$, by which we are done.
The diagonal intersection above can be generalized to work with more than one measure in the following way. A set $T \subseteq {{\vphantom{\operatorname{OB}(d)}}^{n}{\operatorname{OB}(d)}}$, where $n<{\omega}$, is said to be a tree if the following hold:
1. Each ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}} \in T$ is increasing.
2. For each $k<n$ and ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}} \in T$ we have ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_k \rangle}} \in T$.
Assume $T \subseteq {{\vphantom{\operatorname{OB}(d)}}^{n}{\operatorname{OB}(d)}}$ is a tree and ${\ensuremath{\langle {\nu}\rangle}} \in T$. Set $T_{{\ensuremath{\langle {\nu}\rangle}}}= {\ensuremath{\{ {\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{n-2} \rangle}} \mid {\ensuremath{\langle {\nu}, {\mu}_0, \dotsc, {\mu}_{n-2} \rangle}} \in T \}}}$. Denote the $k$-level of the tree $T$ by $\operatorname{Lev}_k(T)$, i.e., $\operatorname{Lev}_k(T) = T {\cap}{{\vphantom{\operatorname{OB}(d)}}^{k+1}{\operatorname{OB}(d)}}$. We will use ${{\Vec{{\nu}}}}$ as a shorthand for ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}}$. For each ${{\Vec{{\nu}}}}\in T$ we define the successor level of ${{\Vec{{\nu}}}}$ in $T$ by setting $\operatorname{Suc}_T({{\Vec{{\nu}}}}) = {\ensuremath{\{ {\mu}\mid {{\Vec{{\nu}}}}{\mathop{{}^\frown}}{\mu}\in T \}}}$. A tree $S\subseteq {{\vphantom{\operatorname{OB}(d)}}^{n}{\operatorname{OB}(d)}}$, with all maximal branches having the same finite height $n<{\omega}$, is said to be an ${\Vec{E}}(d)$-tree if the following hold:
1. There is ${\xi}< \operatorname{o}({\Vec{E}})$ such that $\operatorname{Lev}_0(S) \in E_{\xi}(d)$.
2. For each ${{\Vec{{\nu}}}}{\mathop{{}^\frown}}{\mu}\in S$ there is ${\xi}< \operatorname{o}({\Vec{E}})$ such that $\operatorname{Suc}_S({{\Vec{{\nu}}}}) \in E_{\xi}(d)$.
If $S$ is a tree of finite height $n<{\omega}$ then we write $\operatorname{Lev}_{\max} S$ for $\operatorname{Lev}_{n-1}S$.
Assume $S$ is an ${\Vec{E}}(d)$-tree, and for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}(S)$ there is a set $X({{\Vec{{\nu}}}}) \subseteq \operatorname{OB}(d)$. By recursion define the diagonal intersection of the family ${\ensuremath{\{ X({{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S \}}}$ by setting $\operatorname*{\triangle}{\ensuremath{\{ X({{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S \}}} =
\operatorname*{\triangle}{\ensuremath{\{ X^*({{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^* \}}}$, where $S^* = S {\cap}{{\vphantom{\operatorname{OB}(d)}}^{n-1}{\operatorname{OB}(d)}}$ and $X^*({{\Vec{{\mu}}}}) = \operatorname*{\triangle}{\ensuremath{\{ X({{\Vec{{\mu}}}}{\mathop{{}^\frown}}{\ensuremath{\langle {\nu}\rangle}}) \mid {\nu}\in S_{{\ensuremath{\langle {{\Vec{{\mu}}}}\rangle}}} \}}}$. The following is immediate.
Assume $S$ is an ${\Vec{E}}(d)$-tree, and for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}(S)$ there is a set $X({{\Vec{{\nu}}}}) \in {\Vec{E}}(d)$. Then $\operatorname*{\triangle}{\ensuremath{\{ X({{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S \}}} \in {\Vec{E}}(d)$.
The Forcing Notion {#sec:Forcing}
==================
A finite sequence ${\ensuremath{\langle {\Bar{{\nu}}}_0, \dotsc, {\Bar{{\nu}}}_k \rangle}} \in {{\vphantom{\operatorname{ES}}}^{<{\omega}}{\operatorname{ES}}}$ is said to be $\operatorname{o}$-decreasing if it is increasing and ${\ensuremath{\langle \operatorname{o}({\Bar{{\nu}}}_0), \dotsc, \operatorname{o}({\Bar{{\nu}}}_k \rangle}})$ is non-increasing.
A condition $f$ is in the forcing notion ${\mathbb{P}}^*_f({\Vec{E}}, {\epsilon})$ if $f$ is a function $f {\mathrel{:}}d \to {{\vphantom{\operatorname{ES}}}^{<{\omega}}{\operatorname{ES}}}$ such that:
1. $d \in [{\epsilon}]^{<{\lambda}}$.
2. $d \supseteq ({\lvertd\rvert}+1)$.
3. For each ${\alpha}\in d$, $f({\alpha})$ is $\operatorname{o}$-decreasing.
Assume $f, g \in {\mathbb{P}}_f^*({\Vec{E}}, {\epsilon})$ are conditions. We say $f$ is an extension of $g$ ($f \leq^*_{{\mathbb{P}}_f^*({\Vec{E}},{\epsilon})} g$) if $f \supseteq g$.
For a condition $f \in {\mathbb{P}}_f^*(\Vec{E})$ we will write $E_{\xi}(f)$ and ${\Vec{E}}(f)$ instead of $E_{\xi}(\operatorname{dom}f)$ and $\Vec{E}(\operatorname{dom}f)$, respectively. If $T \subseteq \operatorname{OB}(e)$ and $d \subseteq e$ then $T {\mathrel{\restriction}}d = {\ensuremath{\{ {\nu}{\mathrel{\restriction}}d \mid {\nu}\in T \}}}$.
A condition $p$ is in the forcing notion ${\mathbb{P}}^*(\Vec{E}, {\epsilon})$ if $p$ is of the form ${\ensuremath{\langle f^p, T^p \rangle}}$, where $f^p \in {\mathbb{P}}^*_f(\Vec{E}, {\epsilon})$, $T^p \in {\Vec{E}}(f^p)$, and for each ${\nu}\in T^p$ and each ${\alpha}\in \operatorname{dom}{\nu}$, $\max \mathring{f}^p({\alpha}) < {\mathring{{\nu}}}({\kappa})$.
Assume $p, q \in {\mathbb{P}}^*(\Vec{E}, {\epsilon})$ are conditions. We say $p$ is a direct extension of $q$ ($p \leq^*_{{\mathbb{P}}^*(\Vec{E}, {\epsilon})} q$) if $f^p \supseteq f^q$ and $T^p {\mathrel{\restriction}}\operatorname{dom}f^q \subseteq T^q$. We say $p$ is a strong direct extension of $q$ ($p \leq^{**}_{{\mathbb{P}}^*(\Vec{E}, {\epsilon})} q$) if $p$ is a direct extension of $q$ and $f^p =f^q$.
Since ${\epsilon}$ and the sequence ${\Vec{E}}$ are fixed througout this work we designate ${\mathbb{P}}^*(\Vec{E}, {\epsilon})$ by ${\mathbb{P}}^*$.
A condition $p$ is in the forcing $\Bar{{\mathbb{P}}}$ if $p = {\ensuremath{\langle p_0, \dotsc, p_{n^p-1} \rangle}}$, where $n^p < {\omega}$, there is a sequence ${\ensuremath{{\ensuremath{\langle \Vec{E}^p_i \mid i < n^p \rangle}}}}$ such that each $\Vec{E}^p_i$ is a Mitchell increasing sequence of extenders, ${\ensuremath{\langle \operatorname{crit}(\Vec{E}^p_0), \dotsc, \operatorname{crit}(\Vec{E}^p_{n^p-1}) \rangle}}$ is strictly increasing, $\sup {\ensuremath{\{ j_{E^p_{i,{\xi}}}(\operatorname{crit}\Vec{E}^p_i) \mid {\xi}< \operatorname{o}(\Vec{E}^p_i) \}}} < \operatorname{crit}\Vec{E}^p_{i+1}$, ${\lambda}(\Vec{E}^p_i) < \operatorname{crit}(\Vec{E}^p_{i+1})$, and for each $i < n^p$, $p_i \in {\mathbb{P}}^*(\Vec{E}^p_i, {\epsilon}^p_i)$.
Assume $p, q \in \Bar{{\mathbb{P}}}$ are conditions. We say $p$ is a direct extension of $q$ ($p \leq^*_{\Bar{{\mathbb{P}}}} q$) if $n^p = n^q$ and for each $i < n^p$, $p^i \leq^* q^i$. We say $p$ is a strong direct extension of $q$ ($p \leq^{**}_{\Bar{{\mathbb{P}}}} q$) if $n^p = n^q$ and for each $i < n^p$, $p^i \leq^{**} q^i$.
The following sequence of definitions leads to the definition of the order $\leq_{\Bar{{\mathbb{P}}}}$ (which is somewhat involved, hence the breakup to several steps). If ${\nu}\in \operatorname{OB}(d)$ we let $\operatorname{o}({\nu}) = \operatorname{o}({\nu}({\kappa}))$.
Assume $f {\mathrel{:}}d \to {{\vphantom{\operatorname{ES}}}^{<{\omega}}{\operatorname{ES}}}$ is a function, ${\nu}\in \operatorname{OB}(d)$, and for each ${\alpha}\in \operatorname{dom}{\nu}$, $\max \mathring{f}({\alpha}) < {\mathring{{\nu}}}({\kappa})$. Define $f_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}$ and $f_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}$ as follows.
1. If $\operatorname{o}({\nu})=0$ then $f_{{\ensuremath{\langle {\nu}\rangle}}\downarrow} = \emptyset$. If $\operatorname{o}({\nu}) > 0$ then $f_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}$ is the function $g$, where:
1. $\operatorname{dom}g = \operatorname{ran}{\mathring{{\nu}}}$.
2. For each ${\alpha}\in \operatorname{dom}{\nu}$, $g({\mathring{{\nu}}}({\alpha})) = {\ensuremath{\langle {\Bar{{\tau}}}_{k+1}, \dotsc, {\Bar{{\tau}}}_{n-1} \rangle}}$, where $f({\alpha}) = {\ensuremath{\langle {\Bar{{\tau}}}_0, \dotsc, {\Bar{{\tau}}}_{n-1} \rangle}}$ and $k<n$ is maximal such that $\operatorname{o}({\Bar{{\tau}}}_k) \geq \operatorname{o}({\nu}({\alpha}))$. Set $k = -1$ if there is no $k < n$ such that $\operatorname{o}({\Bar{{\tau}}}_k) \geq \operatorname{o}({\nu}({\alpha}))$.
2. Define $f_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}$ to be the function $g$ where:
1. $\operatorname{dom}g = \operatorname{dom}f$.
2. For each ${\alpha}\in \operatorname{dom}{\nu}$, $g({\alpha})= {\ensuremath{\langle {\Bar{{\tau}}}_0, \dotsc, {\Bar{{\tau}}}_{k} \rangle}}$, where $f({\alpha}) = {\ensuremath{\langle {\Bar{{\tau}}}_0, \dotsc, {\Bar{{\tau}}}_{n-1} \rangle}}$ and $k<n$ is maximal such that $\operatorname{o}({\Bar{{\tau}}}_k) \geq \operatorname{o}({\nu}({\alpha}))$. Set $k = -1$ if there is no $k < n$ such that $\operatorname{o}({\Bar{{\tau}}}_k) \geq \operatorname{o}({\nu}({\alpha}))$.
The following definitions show how to reflect down a function ${\mu}\in \operatorname{OB}(d)$ using a larger function ${\nu}\in \operatorname{OB}(d)$.
1. Assume ${\mu}, {\nu}\in \operatorname{OB}(d)$, ${\mu}< {\nu}$, and $\operatorname{o}({\mu}) < \min(\operatorname{o}({\nu}), {\mathring{{\nu}}}({\kappa}))$. Define the function ${\tau}= {\mu}\downarrow {\nu}\in \operatorname{OB}(\operatorname{ran}{\mathring{{\nu}}})$ by:
1. $\operatorname{dom}{\tau}= {\ensuremath{\{ {\mathring{{\nu}}}({\alpha}) \mid {\alpha}\in \operatorname{dom}{\mu}{\cap}\operatorname{dom}{\nu}\}}}$.
2. For each ${\xi}\in \operatorname{dom}{\tau}$, ${\tau}({\xi}) = {\mu}({\alpha})$, were ${\xi}= {\mathring{{\nu}}}({\alpha})$.
2. Assume $T \subseteq \operatorname{OB}(d)$ and ${\nu}\in \operatorname{OB}(d)$. If $\operatorname{o}({\nu}) = 0$ then set $T_{{\ensuremath{\langle {\nu}\rangle}}\downarrow} = \emptyset$. If $\operatorname{o}({\nu}) > 0$ then $T_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}= {\ensuremath{\{ {\mu}\downarrow {\nu}\mid {\mu}\in T,\ {\mu}< {\nu},\ \operatorname{o}({\mu}) < \min(\operatorname{o}({\nu}), {\mathring{{\nu}}}({\kappa})) \}}}$.
Assume $p \in {\mathbb{P}}^*({\Vec{E}})$ and ${\nu}\in T^p$. We define $p_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}$ as follows. If $\operatorname{o}({\nu}) = 0$ then $p_{{\ensuremath{\langle {\nu}\rangle}}\downarrow} = \emptyset$. If $\operatorname{o}({\nu}) > 0$ then $p_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}$ is the condition $q \in {\mathbb{P}}^*({\accentset{\mid}{{\nu}}})$ defined by setting $f^q = f^p_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}$ and $T^q = T^p_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}$. Define $p_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}$ to be the condition $q \in {\mathbb{P}}^*({\Vec{E}})$, where $f^q = f^p_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}$ and $T^q = T^p_{{\ensuremath{\langle {\nu}\rangle}}}$. Finally set $p_{{\ensuremath{\langle {\nu}\rangle}}} = {\ensuremath{\langle p_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}, p_{{\ensuremath{\langle {\nu}\rangle}}\uparrow} \rangle}}$.
Of course for the above definition to make sense $T^p_{{\ensuremath{\langle {\nu}\rangle}}\downarrow} \in {\accentset{\mid}{{\nu}}}(\operatorname{ran}{\nu})$ should hold, which we prove in \[MakesSense\]. If $T \subseteq \operatorname{OB}(d)$ hen we let ${{{\vphantom{T}}^{<{\omega}}{T}}}= {\ensuremath{\{ {\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_n \rangle}} \mid n<{\omega},\ {\nu}_0, \dotsc, {\nu}_n \in T,\ {\nu}_0 < \dotsb < {\nu}_n \}}}$.
Assume $p, q \in \Bar{{\mathbb{P}}}$. We say $p$ is an extension of $q$ ($p \leq_{\Bar{{\mathbb{P}}}} q$) if the following hold:
1. $n^p \geq n^q$.
2. ${\ensuremath{\{ \Vec{E}^q_j \mid j < n^q \}}} \subseteq {\ensuremath{\{ \Vec{E}^p_i \mid i < n^p \}}}$ and ${\Vec{E}}^q_{n^q-1} = {\Vec{E}}^p_{n^p-1}$.
3. For each $i < n^q$ there is ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}} \in {{{\vphantom{T}}^{<{\omega}}{T}}}^{q_i}$ such that ${\ensuremath{\langle p_{j_0+1}, \dotsc, p_{j_1} \rangle}} \leq^* q_{i{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}}}$, where $i$, $j_0$ and $j_1$, are being set as follows. Let $j_1 < n^p$ satisfy $\Vec{E}^p_{j_1} = \Vec{E}^q_{i}$. If $i = 0$ then set $j_0 = -1$. If $i > 0$ then let $j_0 < j_1$ satisfy $\Vec{E}^p_{j_0} = \Vec{E}^q_{i-1}$.
Finally we give the definition of the forcing notion we are going to work with:
${\mathbb{P}}(\Vec{E}, {\epsilon}) = {\ensuremath{\{ q \leq_{\Bar{{\mathbb{P}}}} p \mid p \in {\mathbb{P}}^*(\Vec{E}, {\epsilon}) \}}}$. The partial orders $\leq_{{\mathbb{P}}(\Vec{E}, {\epsilon})}$ and $\leq^*_{{\mathbb{P}}(\Vec{E}, {\epsilon})} $ are inherited from $\leq_{\Bar{{\mathbb{P}}}}$ and $\leq^*_{\Bar{{\mathbb{P}}}}$.
Since ${\epsilon}$ and the sequence ${\Vec{E}}$ are fixed throughout this work we will write ${\mathbb{P}}$ instead of ${\mathbb{P}}(\Vec{E}, {\epsilon})$ throughout this paper. is needed in order to show the forcing notion defined above makes sense.
If $T \in {\Vec{E}}(d)$ then $X = {\ensuremath{\{ {\nu}\in T \mid T_{{\ensuremath{\langle {\nu}\rangle}}\downarrow} \in {\accentset{\mid}{{\nu}}}(\operatorname{ran}{\mathring{{\nu}}}) \}}} \in {\Vec{E}}(d)$.
We need to show $X \in {\Vec{E}}(d)$. I.e., we need to show for each ${\xi}< \operatorname{o}(\Vec{E})$, $X \in E_{\xi}(d)$. Fix ${\xi}< \operatorname{o}(\Vec{E})$. We need to show $\operatorname{mc}_{\xi}(d_{\xi}) \in
j_{E_{\xi}}(X)$. Hence it is enough showing $\operatorname{mc}_{\xi}(d) \in j_{E_{\xi}}(T)$ and $ j_{E_{\xi}}(T)_{{\ensuremath{\langle \operatorname{mc}_{\xi}(d) \rangle}}\downarrow} \in {\Vec{E}}{\mathrel{\restriction}}{\xi}(d)$. Since $T \in \Vec{E}(d)$ we have $\operatorname{mc}_{\xi}(d) \in j_{E_{\xi}}(T)$. So we are left with showing $j_{E_{\xi}}(T)_{{\ensuremath{\langle \operatorname{mc}_{\xi}(d) \rangle}}\downarrow} \in {\Vec{E}}{\mathrel{\restriction}}{\xi}(d)$. From the definition of the operation $\downarrow$ we get $$\begin{aligned}
j_{E_{\xi}}(T)_{{\ensuremath{\langle \operatorname{mc}_{\xi}(d) \rangle}}\downarrow}= \begin{aligned}[t]
{\ensuremath{\{ {\mu}\downarrow \operatorname{mc}_{\xi}(d) \mid {\mu}\in j_{E_{\xi}}(T), \ {\mu}< \operatorname{mc}_{\xi}(d),\
\operatorname{o}({\mu}) < \min({\kappa},{\xi}) \}}}.
\end{aligned}\end{aligned}$$ Consider ${\mu}\in j_{E_{\xi}}(T)$ such that ${\mu}< \operatorname{mc}_{\xi}(d)$. There is ${\mu}^* \in T$ such that ${\mu}= j_{E_{\xi}}({\mu}^*)$. Since for each ${\mu}^* \in T$ such that $\operatorname{o}({\mu}^*) < {\xi}$ we have $j_{E_{\xi}}({\mu}^*) \downarrow \operatorname{mc}_{\xi}(d) = {\mu}^*$, we get $j_{E_{\xi}}(T)_{{\ensuremath{\langle \operatorname{mc}_{\xi}(d) \rangle}}\downarrow} =
{\ensuremath{\{ {\mu}\in T \mid \operatorname{o}({\mu}) < {\xi}\}}} \in {\Vec{E}}{\mathrel{\restriction}}{\xi}(d)$.
For each condition $p \in {\mathbb{P}}$ let ${\mathbb{P}}/p = {\ensuremath{\{ q \in {\mathbb{P}}\mid q \leq p \}}}$. It is immediate from the definitions above that for each $0<i<n^p-1$ the forcing notion ${\mathbb{P}}/p$ factors to $P_0 \times P_1$, where $P_0 = {\ensuremath{\{ q^0 \leq p^0 \mid q^0 {\mathop{{}^\frown}}p^1 \in {\mathbb{P}}\}}}$, $P_1 = {\ensuremath{\{ q^1 \leq p^1 \mid p^0 {\mathop{{}^\frown}}q^1 \in {\mathbb{P}}\}}}$, $p^0 = {\ensuremath{\langle p_0, \dotsc, p_{i-1} \rangle}}$, and $p^1 = {\ensuremath{\langle p_i, \dotsc, p_{n^p-1} \rangle}}$. Together with the Prikry property (\[PrikryProperty\]) and the closure of the direct order, one can analyze the cardinal structure in $V^{{\mathbb{P}}}$ straightforwardly.
If $e\supseteq d$ we define ${\pi}^{-1}_{e,d}$ to be the inverse of the operation ${\mathrel{\restriction}}d$, i.e., for each $X \subseteq \operatorname{OB}(d)$ we let ${\pi}_{e,d}^{-1}(X) = {\ensuremath{\{ {\nu}\in \operatorname{OB}(e) \mid {\nu}{\mathrel{\restriction}}d \in X \}}}$. If $f, g \in {\mathbb{P}}^*_f$ are conditions then we write ${\pi}_{f,g}^{-1}$ for ${\pi}_{\operatorname{dom}f, \operatorname{dom}g}^{-1}$. We end this section with the analysis of the cardinal structure above ${\kappa}$ in the generic extension: The cardinals between ${\kappa}$ and ${\lambda}$ are collapsed, and ${\lambda}$ and the cardinals above it are preserved. The properties of cardinals up to ${\kappa}$ will be dealt with in later sections.
${\mathbb{P}}$ satisfies the ${\lambda}^{+}$-cc.
Begin with a family of conditions ${\ensuremath{{\ensuremath{\langle p^{\xi}\mid {\xi}< {\lambda}^{+} \rangle}}}}$. Without loss of generality we can assume $n^{p^{{\xi}_0}} = n^{p^{{\xi}_1}}$ for each ${\xi}_0, {\xi}_1 < {\lambda}^{+}$. Without loss of generality we can assume ${\ensuremath{\langle p^{{\xi}_0}_0, \dotsc, p^{{\xi}_0}_{n^{p^{{\xi}_0}}-2} \rangle}} = {\ensuremath{\langle p^{{\xi}_1}_0, \dotsc, p^{{\xi}_1}_{n^{p^{{\xi}_1}}-2} \rangle}}$ for each ${\xi}_0, {\xi}_1 < {\lambda}^{+}$. Thus, without loss of generality, we can assume $n^{p^{{\xi}}} = 1$ for each ${\xi}< {\lambda}^{+}$. By the ${\Delta}$-system lemma we can assume ${\ensuremath{\{ \operatorname{dom}f^{p^{{\xi}}} \mid {\xi}< {\lambda}^+ \}}}$ is a ${\Delta}$-system with kernel $d$. Since ${\lvertd\rvert} < {\lambda}$ we can assume that for each ${\xi}_0, {\xi}_1 < {\lambda}^+$ and ${\alpha}\in d$, $f^{p^{{\xi}_0}}({\alpha}) = f^{p^{{\xi}_1}}({\alpha})$. Fix ${\xi}_0 < {\xi}_1 < {\lambda}^+$. Set $f = f^{p^{{\xi}_0}}{\cup}f^{p^{{\xi}_1}}$, $T = {\pi}^{-1}_{f,f^{p^{{\xi}_0}}} T^{p^{{\xi}_0}}{\cap}{\pi}^{-1}_{f,f^{p^{{\xi}_1}}}T^{p^{{\xi}_1}}$, and let $p = {\ensuremath{\langle f, T \rangle}}$. Then $p \leq p^{{\xi}_0}, p^{{\xi}_1}$.
${\mathrel\Vdash}{{}\text{``} \text{There are no cardinals between ${\kappa}$ and ${\lambda}$} {}\text{''}}$.
Fix a $V$-regular cardinal ${\tau}\in ({\kappa}, {\lambda})$. Fix a condition $p \in {\mathbb{P}}$ such that $\operatorname{dom}f^{p_{n^p-1}} \supseteq {\tau}\setminus {\kappa}$ will hold. Let $G \subseteq {\mathbb{P}}$ be generic such that $p \in G$. Set $C = {\ensuremath{\{ {{\Vec{{\nu}}}}\in {{\vphantom{T^{p_{n^p-1}}}}^{<{\omega}}{T^{p_{n^p-1}}}} \mid p_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}} \in G \}}}$. Then $\sup {\ensuremath{\{ \sup({\tau}{\cap}{\bigcup}\operatorname{dom}{{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in C \}}} = {\tau}$. Since ${\mathrel\Vdash}{{}\text{``} {\lvertC\rvert} \leq {\kappa}{}\text{''}}$ we get $p {\mathrel\Vdash}{{}\text{``} \operatorname{cf}{\tau}\leq {\kappa}{}\text{''}}$.
Preservation of ${\lambda}$ will be proved by a properness type argument (\[WeAreProper\]) for which we need some preparation.
We say the elementary substructure $N {\prec}H_{\chi}$, where ${\chi}$ is large enough, is ${\kappa}$-internally approachable if there is an increasing continuous sequence of elementary substructures ${\ensuremath{{\ensuremath{\langle N_{\xi}\mid {\xi}<{\kappa}\rangle}}}}$ such that $N = {\bigcup}{\ensuremath{\{ N_{\xi}\mid {\xi}<{\kappa}\}}}$, for each ${\xi}< {\kappa}$, $N_{\xi}{\prec}H_{\chi}$, ${\lvertN_{\xi}\rvert} < {\lambda}$, $N_{{\xi}+1} \supseteq \operatorname{{\mathcal{P}}}_{{\kappa}}({\lvertN_{{\xi}}\rvert})$, $N_{\xi}{\cap}{\lambda}\in {\ensuremath{\text{On}}}$, ${\mathbb{P}}^*_f \in N_{\xi}$, $N_{{\xi}+1} \supseteq {{\vphantom{N_{{\xi}+1}}}^{<{\kappa}}{N_{{\xi}+1}}}$, and ${\ensuremath{{\ensuremath{\langle N_{{\xi}'} \mid {\xi}'<{\xi}\rangle}}}} \in N_{{\xi}+1}$.
We say the pair ${\ensuremath{\langle N, f \rangle}}$ is a good pair if $N {\prec}H_{\chi}$ is a ${\kappa}$-internally approachable elementary substructure and there is a sequence ${\ensuremath{{\ensuremath{\langle {\ensuremath{\langle N_{\xi}, f_{\xi}\rangle}} \mid {\xi}<{\kappa}\rangle}}}}$ such that ${\ensuremath{{\ensuremath{\langle N_{\xi}\mid {\xi}< {\kappa}\rangle}}}}$ witnesses the ${\kappa}$-internal approachablity of $N$, $f = {\bigcup}{\ensuremath{\{ f_{\xi}\mid {\xi}< {\kappa}\}}}$, ${\ensuremath{{\ensuremath{\langle f_{\xi}\mid {\xi}<{\kappa}\rangle}}}}$ is a $\leq^*$-decreasing continuous sequence in ${\mathbb{P}}^*_f$, and for each ${\xi}< {\kappa}$, $f_{{\xi}} \in {\bigcap}{\ensuremath{\{ D \in N_{\xi}\mid D \text{ is a dense open subset of }
{\mathbb{P}}^*_f \}}}$, $f_{\xi}\subseteq N_{{\xi}+1}$, and $f_{\xi}\in N_{{\xi}+1}$. Note that if $N {\prec}H_{\chi}$ is an elementary substructure such that ${\lvertN\rvert} < {\lambda}$, $N \supseteq {{\vphantom{N}}^{<{\kappa}}{N}}$, ${\mathbb{P}}_f^* \in N$, $f \in {\bigcap}{\ensuremath{\{ D \in N \mid D \text{ is a dense open subset of }{\mathbb{P}}_f^* \}}}$, $f \subseteq N$, and ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}} \in N{\cap}\operatorname{OB}(\operatorname{dom}f)$, then $f_{{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}} } \in
{\bigcap}{\ensuremath{\{ D \in N \mid D \text{ is a dense open subset of }{\mathbb{P}}_f^* \}}}$.
Hence if ${\ensuremath{\langle N, f \rangle}}$ is a good pair and ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}} \in N {\cap}\operatorname{OB}(\operatorname{dom}f)$, then ${\ensuremath{\langle N, f_{{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}}} \rangle}}$ is a good pair also. The following is immediate.
For each set $X$ and $f \in {\mathbb{P}}^*_f$ there is a good pair${\ensuremath{\langle N, f^* \rangle}}$ such that $ f^* \leq^* f$ and $X,f \in N$.
Assume ${\chi}$ is large enough and $N {\prec}H_{\chi}$ is an elementary substructure such that ${\mathbb{P}}\in N$. We say the condition $p \in N$ is $N$-generic if for each dense open subset $D \in N$ of ${\mathbb{P}}$ we have $p {\mathrel\Vdash}{{}\text{``} {\Check{{\mathbb{P}}}} {\cap}{\underset{\widetilde{}}{G}} {\cap}{\Check{N}} \neq \emptyset {}\text{''}}$.
We say the forcing notion ${\mathbb{P}}$ is ${\lambda}$-proper if for an unbounded set of structures $N {\prec}H_{\chi}$ such that ${\mathbb{P}}\in N$ and ${\lvertN\rvert} < {\lambda}$, and for each condition $p \in {\mathbb{P}}{\cap}N$ there is a stronger $N$-generic condition. The followig lemma shows a property stronger than properness.
Let $N{\prec}H_{\chi}$ be a ${\kappa}$-internally approachable structure, ${\mathbb{P}}\in N$, and $p \in N {\cap}{\mathbb{P}}$ a condition. Then there is a direct extension $p^* \leq^* p$ such that for each dense open subset $D \in N$ of ${\mathbb{P}}$ the set ${\ensuremath{\{ s {\mathop{{}^\frown}}p^*_{{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}}\uparrow} \in D \mid {\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}} \in {{\vphantom{T}}^{<{\omega}}{T}}^{p^*},\
s \leq^* p^*_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}\downarrow} \}}}$ is predense below $p^*$. Moreover, if $s \leq^* p^*_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}\downarrow}$ and $s {\mathop{{}^\frown}}p^*_{{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}}\uparrow}
\in D$ then there is a weaker condition $q \geq^* p^*_{{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}}\uparrow}$ such that $s {\mathop{{}^\frown}}q \in D {\cap}N$.
Let ${\ensuremath{\langle N, f^* \rangle}}$ be a good pair such that $f^* \leq^* f^{p_{n^p-1}}$. Choose a set $T \in {\Vec{E}}(f^*)$ such that ${\ensuremath{\langle f^*, T \rangle}} \leq^* p_{n^p-1}$. Let ${\ensuremath{{\ensuremath{\langle D_{\alpha}\mid {\alpha}< {\lvertN\rvert} \rangle}}}}$ be an enumeration of the dense open subsets of ${\mathbb{P}}$ appearing in $N$. Let ${\ensuremath{{\ensuremath{\langle {\ensuremath{\langle N_{\iota},f_{\iota}\rangle}} \mid {\iota}< {\kappa}\rangle}}}}$ be a sequence witnessing ${\ensuremath{\langle N, f^* \rangle}}$ is a good pair. For each ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}} \in {{\vphantom{T}}^{<{\omega}}{T}}$ construct the set $T^{{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}}}$ as follows.
Fix ${{\Vec{{\nu}}}}= {\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}} \in {{\vphantom{T}}^{<{\omega}}{T}}$.
Let ${\mathcal{D}}= {\ensuremath{\{ D_{\alpha}\mid {\alpha}\in \operatorname{dom}{\nu}_{k-1} \}}}$. Note ${\mathcal{D}}\in N$ since ${\lvert{\nu}_{k-1}\rvert} < {\kappa}$ and $N \supseteq {{\vphantom{N}}^{<{\kappa}}{N}}$. For each $s \in {\mathbb{P}}({\accentset{\mid}{{\nu}}}_{k-1})$ and $D \in {\mathcal{D}}$ define the sets $D^\in_{{{\Vec{{\nu}}}},s,D}$, $D^{\perp}_{{{\Vec{{\nu}}}},s,D}$, and $D^*_{{{\Vec{{\nu}}}},s,D}$, as follows: Let $g \in D^\in_{{{\Vec{{\nu}}}},s,D}$ if $g \leq f^{p_{n^p-1}}$, $\operatorname{dom}g \supseteq \operatorname{dom}{\nu}_{k-1}$, and $s {\mathop{{}^\frown}}{\ensuremath{\langle g_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}},T' \rangle}} \in D$ for some $T' \in {\Vec{E}}(g)$. Let $h \in D^{\perp}_{{{\Vec{{\nu}}}},s,D}$ if $h {\perp}g$ for each $g \in D^\in_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}},s,D}$. Set $D^*_{{{\Vec{{\nu}}}},s,D} =
D^\in_{{{\Vec{{\nu}}}},s,D} {\cup}D^{\perp}_{{{\Vec{{\nu}}}},s,D}$. It is immediate $D^\in_{{{\Vec{{\nu}}}},s,D}$ and $D^{\perp}_{{{\Vec{{\nu}}}},s,D}$ are open subsets of ${\mathbb{P}}^*_f$ below$f^{p_{n^p-1}}$. Thus $D^*_{{{\Vec{{\nu}}}},s,D}$ is a dense open subset of ${\mathbb{P}}^*_f$ below $f^{p_{n^p-1}}$. Set $D^*_{{{\Vec{{\nu}}}}} =
{\bigcap}{\ensuremath{\{ D^*_{{{\Vec{{\nu}}}},s,D} \mid s \in {\mathbb{P}}({\accentset{\mid}{{\nu}}}_{k-1}),\ D \in {\mathcal{D}}\}}}$. Note $D^*_{{{\Vec{{\nu}}}}} \in N$ is a dense open subset of ${\mathbb{P}}^*_f$ below $f^{p_{n^p-1}}$. Let ${\iota}< {\kappa}$ be minimal such that ${{\Vec{{\nu}}}},{\mathcal{D}}, D^*_{{{\Vec{{\nu}}}}} \in N_{{\iota}}$. Then $f_{{\iota}} \in D^*_{{{\Vec{{\nu}}}}}{\cap}N_{{\iota}+1}$. Thus for each $s \in {\mathbb{P}}({\accentset{\mid}{{\nu}}}_{k-1})$ and $D \in {\mathcal{D}}$ either there is a set $T^{{{\Vec{{\nu}}}},s,D} \in {\Vec{E}}(f_{{\iota}}) {\cap}N_{{\iota}+1}$ such that $s {\mathop{{}^\frown}}{\ensuremath{\langle f_{{\iota}{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}},
T^{{{\Vec{{\nu}}}},s,D} \rangle}} \in D$ or $s {\mathop{{}^\frown}}{\ensuremath{\langle h,T'' \rangle}}\notin D$ for each $h \leq^* f_{{\iota}{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}$ and $T''\in {\Vec{E}}(h)$. Set $T^{\nu}= {\bigcap}{\ensuremath{\{ T^{{{\Vec{{\nu}}}},s,D} \mid s \in {\mathbb{P}}({\accentset{\mid}{{\nu}}}_{k-1}),\ D \in {\mathcal{D}},\
s {\mathop{{}^\frown}}{\ensuremath{\langle f_{{\iota}{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}},T^{{\nu},s,D} \rangle}}\in D \}}}$.
Set $T^* = \operatorname*{\triangle}{\ensuremath{\{ {\pi}^{-1}_{f^*,f_{{\iota}({{\Vec{{\nu}}}})}}T^{{\Vec{{\nu}}}}\mid {{\Vec{{\nu}}}}\in {{\vphantom{T}}^{<{\omega}}{T}} \}}}$. Set $p^* = p{\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}{\ensuremath{\langle f^*, T^* \rangle}}$. We claim $p^*$ satisfies the lemma. To show this fix a dense open subset $D \in N$ and a condition $q \leq p^*$.
Let ${\alpha}< {\lvertN\rvert}$ be such that $D = D_{\alpha}$. Without loss of generality assume $q \in D$, $q_{n^q-1} \leq p^*_{{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}}\uparrow}$, ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}} \in {{\vphantom{T}}^{<{\omega}}{T}}^*$, and ${\alpha}\in \operatorname{dom}{\nu}_{k-1}$. Set $s = q {\mathrel{\restriction}}n^q - 1$. Thus $q = s {\mathop{{}^\frown}}{\ensuremath{\langle f^{q_{n^q-1}}, T^{q_{n^q-1}} \rangle}} \in D$. Let ${\iota}<{\kappa}$ be minimal such that ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}}, {\ensuremath{\{ D_{\alpha}\mid {\alpha}\in \operatorname{dom}{\nu}_{k-1} \}}}\in N_{\iota}$. Since $f^{q_{n^q-1}} \leq^* f_{{\iota}{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{k-1} \rangle}}}$ we must have $s {\mathop{{}^\frown}}{\ensuremath{\langle f_{{\iota}{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}, T^{{{\Vec{{\nu}}}},s,D} \rangle}} \in D$, hence $s {\mathop{{}^\frown}}p^*_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}\uparrow} \in D$. It is clear $q$ and $s {\mathop{{}^\frown}}p^*_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}\uparrow}$ are compatible. In addition $s {\mathop{{}^\frown}}{\ensuremath{\langle f_{{\iota}{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}, T^{{{\Vec{{\nu}}}},s,D} \rangle}} \in N$, thus we are done.
${\mathbb{P}}$ is ${\lambda}$-proper.
${\mathrel\Vdash}{{}\text{``} {\lambda}\text{ is a cardinal} {}\text{''}}$.
Dense open sets and measure one sets
====================================
In order to reduce clutter later on, given a condition $p \in {\mathbb{P}}^*$, we will say a tree is a $p$-tree instead of saying it is an ${\Vec{E}}(f^p)$-tree. If $S$ is a $p$-tree and $r$ is a function with domain $S$ then we define the function $\Vec{r}$ by setting for each ${{\Vec{{\nu}}}}= {\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_n \rangle}} \in S$, $\Vec{r}({{\Vec{{\nu}}}}) = r({\nu}_0) {\mathop{{}^\frown}}\dotsb {\mathop{{}^\frown}}r({\nu}_0, \dotsc, {\nu}_{n})$. A function $r$ is said to be a ${\ensuremath{\langle p, S \rangle}}$-function if $S$ is a $p$-tree, for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{<\max}S$, $\Vec{r}({{\Vec{{\nu}}}}) \leq^{**} p_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}\downarrow}$, and for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S$, $\Vec{r}({{\Vec{{\nu}}}}) \leq^{**} p_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}$.
One of the measures suffices.
-----------------------------
The aim of this subsection is to prove \[GetPreDense\], which together with \[DenseHomogen\] will allow the investigation of the cardinal structure below ${\kappa}$. Note the proof of \[DenseHomogen\] depends on \[GetPreDense\]. The following lemma, which is quite technical, takes its core argument from the proof of the Prikry property for Radin forcing.
Assume $p \in {\mathbb{P}}^*$ is a condition, $S$ is a $p$-tree of height one, and $r$ is a ${\ensuremath{\langle p,S \rangle}}$-function. Then there is a strong direct extension $p^* \leq^{**} p$ such that ${\ensuremath{\{ r({\nu}) \mid {\ensuremath{\langle {\nu}\rangle}} \in S \}}}$ is predense below $p^*$.
Define the functions $r_0$ and $r_1$, both with domain $S$, so that $r({\nu}) = r_0({\nu}) {\mathop{{}^\frown}}r_1({\nu})$ will hold for each ${\ensuremath{\langle {\nu}\rangle}} \in S$. Fix ${\xi}< \operatorname{o}(\Vec{E})$ so that $S \in E_{{\xi}}(f^p)$ will hold. We need to collect the information from the sets $T^{r_0({\nu})}$ and $T^{r_1({\nu})}$ into one set $T^*$. The information from the sets $T^{r_1({\nu})}$’s is collected by setting $R = \operatorname*{\triangle}_{{\ensuremath{\langle {\nu}\rangle}} \in S} T^{r_1({\nu})}$. By \[DiagonalIsBig\] $R \in {\Vec{E}}(f^p)$.
The information from the sets $T^{(r_0({\nu}))}$’s is collected into the set $T^*$ as follows. The set $T^*$ will be the union of the three sets $T^0, T^1$, and $T^2$, which we construct now. The construction of $T^0$ is easy. Set $T^{0} = T^{j_{E_{\xi}}(r_0)(\operatorname{mc}_{\xi}(f^p))}$. It is obvious $T^0 \in {\Vec{E}}{\mathrel{\restriction}}{\xi}(f^p)$.
The constructin of $T^1$ is slightly more involved than the construction of $T^0$. Set $T^{1\prime} = {\ensuremath{\{ {\ensuremath{\langle {\nu}\rangle}} \in S \mid T^0_{{\ensuremath{\langle {\nu}\rangle}} \downarrow}= T^{r_0({\nu})} \}}}$. From the construction of $T^0$ it is clear $T^{1\prime} \in E_{{\xi}}(f^p)$. For each ${\mu}\in T^0$ set $X({\mu}) = {\ensuremath{\{ {\ensuremath{\langle {\nu}\rangle}} \in S \mid {\mu}< {\nu},\ {\mu}\downarrow {\nu}\in T^{r_0({\nu})} \}}}$. From the construction of $T^0$ we get $X({\mu}) \in E_{\xi}(f^p)$. Set $T^1 = {\ensuremath{\{ {\nu}\in T^{1\prime} \mid \forall {\mu}\in T^0\ ({\mu}< {\nu}\implies {\nu}\in X({\mu})) \}}}$. We show $T^1 \in E_{\xi}(f^p)$. Thus we need to show $\operatorname{mc}_{\xi}(f^p) \in j_{E_{\xi}}(T^1)$. Since $\operatorname{mc}_{\xi}(f^p) \in j_{E_{\xi}}(T^{1\prime})$ it is enough to show that if ${\mu}\in j_{E_{\xi}}(T^0)$ and ${\mu}< \operatorname{mc}_{\xi}(f^p)$ then $\operatorname{mc}_{\xi}(f^p) \in j_{E_{\xi}}(X)({\mu})$. So fix ${\mu}\in j_{E_{\xi}}(T^0)$ such that ${\mu}< \operatorname{mc}_{\xi}(f^p)$. Then ${\lvert{\mu}\rvert} < {\kappa}$, $\operatorname{dom}{\mu}\subseteq j''_{E_{\xi}}(\operatorname{dom}f^p)$, and $\sup \operatorname{ran}{\mathring{{\mu}}}< {\kappa}$. Necessarily there is ${\mu}^* \in T^0$ such that ${\mu}= j_{E_{\xi}}({\mu}^*)$. Hence $j_{E_{\xi}}(X)({\mu}) = j_{E_{\xi}}(X({\mu}^*)) \ni \operatorname{mc}_{\xi}(f^p)$, by which we are done.
We construct now the set $T^2$. For each ${\mu}\in R$ set $Y({\mu}) = {\ensuremath{\{ {\nu}\downarrow{\mu}\in R_{{\ensuremath{\langle {\mu}\rangle}}\downarrow} \mid {\nu}\in T^1,\
R _{
{\ensuremath{\langle {\mu}\rangle}}\downarrow
{\ensuremath{\langle {\nu}\downarrow{\mu}\rangle}}\downarrow}\in {\accentset{\mid}{{\nu}}}(\operatorname{dom}{\nu})
\}}}$. Now let $T^2 = {\ensuremath{\{ {\mu}\in R \mid \exists {\tau}< \operatorname{o}({\mu})\ Y({\mu}) \in {\accentset{\mid}{{\mu}}}_{\tau}(\operatorname{dom}{\mu}) \}}}$. We show $T^2 \in E_{\zeta}(f^p)$ for each ${\zeta}> {\xi}$. We need to show for each ${\zeta}> {\xi}$, $\operatorname{mc}_{\zeta}(f^p) \in j_{E_{\zeta}}(T^2)$. Fix ${\zeta}> {\xi}$. We show $\operatorname{mc}_{\zeta}(f^p) \in j_{E_{\zeta}}(T^2)$. It is enough to show there is ${\tau}< {\zeta}$ such that $j_{E_{\zeta}}(Y)(\operatorname{mc}_{\zeta}(f^p)) \in E_{\tau}(f^p)$. We claim ${\xi}$ can serve as the needed ${\tau}< {\zeta}$. Thus it is enough to show $j_{E_{\zeta}}(Y)(\operatorname{mc}_{\zeta}(f^p)) \in E_{\xi}(f^p)$. Hence we need to show $$\begin{gathered}
{\ensuremath{\{
{\nu}\downarrow \operatorname{mc}_{\zeta}(f^p) \in j_{E_{\zeta}}(R)_
{{\ensuremath{\langle \operatorname{mc}_{\zeta}(f^p) \rangle}}\downarrow}
\mid
{\nu}\in j_{E_{\zeta}}(T^1),\\
\ j_{E_{\zeta}}(R)_{
{\ensuremath{\langle \operatorname{mc}_{\zeta}(f^p) \rangle}}\downarrow
{\ensuremath{\langle {\nu}\downarrow\operatorname{mc}_{\zeta}(f^p) \rangle}}\downarrow
}
\in {\accentset{\mid}{{\nu}}}(\operatorname{dom}{\nu})
\}}} \in E_{\xi}(f^p).\end{gathered}$$ Note $R^* = j_{E_{\zeta}}(R)_{{\ensuremath{\langle \operatorname{mc}_{\zeta}(f^p) \rangle}}\downarrow}
\in {\Vec{E}}{\mathrel{\restriction}}{\zeta}(f^p)$, and if ${\nu}\in j_{E_{\zeta}}(T^1)$ and ${\nu}< \operatorname{mc}_{\zeta}(f^p)$, then there is ${\nu}^* \in T^1$ such that ${\nu}= j_{E_{\zeta}}({\nu}^*)$. Moreover, ${\nu}^* = {\nu}\downarrow \operatorname{mc}_{\zeta}(f^p)$. Hence it is enough to show $$\begin{aligned}
{\ensuremath{\{
{\nu}^* \in R^*
\mid
{\nu}^* \in T^1,
\ R^*_{
{\ensuremath{\langle {\nu}^* \rangle}}\downarrow
}
\in {\accentset{\mid}{{\nu}}}^*(\operatorname{dom}{\nu}^*)
\}}} \in E_{\xi}(f^p).\end{aligned}$$ We are done since the last formula holds.
Having constructed $T^0$, $T^1$, and $T^2$ we set $p^* = {\ensuremath{\langle f^p, T^* {\cap}R \rangle}}$. We will be done by showing ${\ensuremath{\{ r({\nu}) \mid {\nu}\in S \}}}$ is predense below $p^*$. Assume $q \leq p^*$. We need to exhibit ${\nu}\in S$ so that $q {\parallel}r({\nu})$. We work as follows. Fix ${\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{n-1} \rangle}} \in {{{\vphantom{T}}^{<{\omega}}{T}}}^{p^*}$ such that $q \leq^* p^*_{{\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{n-1} \rangle}}}$. There are three cases to handle:
1. Assume there is $i<n$ such that ${\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{i-1} \rangle}} \in {{{\vphantom{T}}^{<{\omega}}{T}}}^0$ and ${\mu}_i \in T^1$. The construction of $T^1$ yields ${\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{i-1} \rangle}} \in T^{r_0({\mu}_i)}$ and the construction of $R$ yields ${\ensuremath{\langle {\mu}_{i+1}, \dotsc, {\mu}_{n-1} \rangle}} \in {{{\vphantom{T}}^{<{\omega}}{T}}}^{r_1({\mu}_i)}$. Hence $r_{0}({\mu}_i)_{{\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{i-1} \rangle}}} {\mathop{{}^\frown}}r_1({\mu}_i)_{{\ensuremath{\langle {\mu}_{i+1}, \dotsc, {\mu}_{n-1} \rangle}}}$ and $q$ are $\leq^*$-compatible, by which this case is done.
2. Assume ${\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{n-1} \rangle}} \in {{{\vphantom{T}}^{<{\omega}}{T}}}^0$. By the construction of $T^1$ the set $X = {\ensuremath{\{ {\nu}\in T^1 \mid {\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{n-1} \rangle}}\downarrow {\nu}\in {{{\vphantom{T}}^{<{\omega}}{T}}}^0_{{\ensuremath{\langle {\nu}\rangle}}\downarrow} \}}} \in E_{\xi}(f^p)$. Choose ${\nu}^* \in T^{q_{n^{n^q}-1}}$ such that ${\nu}= {\nu}^* {\mathrel{\restriction}}f^p \in X$. Then $q_{{\ensuremath{\langle {\nu}^* \rangle}}} \leq^*
p_{{\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{n-1}, {\nu}\rangle}}}$. Now we can procced as in the first case above.
3. The last case is when there is $i<n$ such that ${\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{i-1} \rangle}} \in {{{\vphantom{T}}^{<{\omega}}{T}}}^0$ and ${\mu}_i \notin T^0 {\cup}T^1$. By the construction of $T^2$ there is ${\tau}< \operatorname{o}({\mu}_i)$ such that $Y = Y({\mu}_i) \in {\accentset{\mid}{{\mu}}}_{i{\tau}}(\operatorname{dom}{\mu}_i)$. Hence there are ${\mu}_{i{\tau}}(\operatorname{dom}{\mu}_i)$-many ${\nu}\downarrow {\mu}_i$ such that ${\nu}\in T^1$, ${\nu}\downarrow{\mu}_i \in T^*_{{\ensuremath{\langle {\mu}_i \rangle}}\downarrow}$ and $T^*_{{\ensuremath{\langle {\mu}_i \rangle}}\downarrow {\ensuremath{\langle {\nu}\rangle}}\downarrow {\mu}_i} \in
{\nu}(\operatorname{dom}{\nu})$.
Thus there is ${\sigma}^* \in T^{q_i}$ such that ${\sigma}= {\sigma}^*{\mathrel{\restriction}}\operatorname{dom}f^{p_i} \in Y$, where ${\sigma}= {\nu}\downarrow {\mu}_i$ and ${\nu}\in T^1$. Thus $q_{{\ensuremath{\langle {\sigma}^* \rangle}}} \leq^*
p_{{\ensuremath{\langle {\mu}_0, \dotsc, {\mu}_{i-1}, {\nu}, {\mu}_i, \dotsc, {\mu}_{n-1} \rangle}}}$ and we can proceed as in the first case above.
Assume $p \in {\mathbb{P}}$ is a condition, $S$ is a $p_{n^p-1}$-tree of height one, and $r$ is a ${\ensuremath{\langle p_{n^p-1},S \rangle}}$-function. Then there is a strong direct extension $p^* \leq^{**} p$ such that $p^* {\mathrel{\restriction}}n^p-1 = p {\mathrel{\restriction}}n^p-1$ and ${\ensuremath{\{ p {\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}r({\nu}) \mid {\ensuremath{\langle {\nu}\rangle}} \in S \}}}$ is predense open below $p^*$.
Generalize the notions of $p$-tree and ${\ensuremath{\langle p, S \rangle}}$-function to arbitrary condition $p \in {\mathbb{P}}$ as follows. By recursion we say the tree $S$ is a $p$-tree if there is $n<{\omega}$ for which following hold:
1. $\operatorname{Lev}_{< n}(S)$ is a $p{\mathrel{\restriction}}n^p-1$-tree.
2. For each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{n-1}(S)$, $S_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}$ is a $p_{n^p-1}$-tree.
Let $p \in {\mathbb{P}}$ be an arbitrary condition. By recursion we say the function $r$ is a ${\ensuremath{\langle p,S \rangle}}$-function if there is $n < {\omega}$ such that:
1. $S$ is a $p$-tree.
2. $\operatorname{Lev}_{< n}(S)$ is a $p {\mathrel{\restriction}}n^p-1$-tree.
3. $ r{\mathrel{\restriction}}\operatorname{Lev}_{< n}S$ is a ${\ensuremath{\langle \operatorname{Lev}_{< n}S, p{\mathrel{\restriction}}n^p-1 \rangle}}$-function.
4. For each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{n-1}(S)$ the function $s$ with domain $S_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}$, define by setting $s({{\Vec{{\mu}}}}) = r({{\Vec{{\nu}}}}{\mathop{{}^\frown}}{{\Vec{{\mu}}}})$, is a ${\ensuremath{\langle p_{n^p-1}, S_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}} \rangle}}$-function.
Assume $p \in {\mathbb{P}}$ is a condition, $S$ is a $p$-tree, and $r$ is a ${\ensuremath{\langle p,S \rangle}}$-function. Then there is a strong direct extension $p^* \leq^{**} p$ such that ${\ensuremath{\{ \Vec{r}({{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in S \}}}$ is predense below $p^*$.
If $S$ is a $p_{n^p-1}$-tree then we are done by \[GetPreDense1\]. Thus assume there is $n<{\omega}$ such that $\operatorname{Lev}_{<n}S$ is a $p{\mathrel{\restriction}}n^p-1$-tree. Construct the strong direct extension $q({{\Vec{{\nu}}}}) \leq^{**} p_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}\uparrow}$ and the ${\ensuremath{\langle p_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}\uparrow}, S_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}} \rangle}}$-function $s_{{{\Vec{{\nu}}}}}$ for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{n-1}S$ as follows. For each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{n-1}S$ let $s_{{\Vec{{\nu}}}}$ be the function with domain $S_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}$ defined by setting $s_{{\Vec{{\nu}}}}({{\Vec{{\mu}}}}) = r({{\Vec{{\nu}}}}{\mathop{{}^\frown}}{{\Vec{{\mu}}}})$ for each ${{\Vec{{\mu}}}}\in S_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}$. By \[GetPreDense1\] there is a strong direct extension $q({{\Vec{{\nu}}}}) \leq^{**} p_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}\uparrow}$ such that ${\ensuremath{\{ s_{{\Vec{{\nu}}}}({{\Vec{{\mu}}}}) \mid {{\Vec{{\mu}}}}\in \operatorname{Lev}_{\max}(S({{\Vec{{\nu}}}})) \}}}$ is predense below $q({{\Vec{{\nu}}}})$. Let $q \leq^{**} p_{n^p-1}$ be a strong direct extension satisfying $q \leq^{**} q({{\Vec{{\nu}}}})$ for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{n-1}(S)$. Hence ${\ensuremath{\{ s_{{\Vec{{\nu}}}}({{\Vec{{\mu}}}}) \mid {{\Vec{{\mu}}}}\in \operatorname{Lev}_{\max}(S({{\Vec{{\nu}}}})) \}}}$ is predense below $q$ for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{n-1}S$. Hence ${\ensuremath{\{ \Vec{r}({{\Vec{{\nu}}}}) {\mathop{{}^\frown}}s_{{\Vec{{\nu}}}}({{\Vec{{\mu}}}}) \mid {{\Vec{{\mu}}}}\in \operatorname{Lev}_{\max}S({{\Vec{{\nu}}}}) \}}}$ is predense below $\Vec{r}({{\Vec{{\nu}}}}) {\mathop{{}^\frown}}q$. Let $p^* \leq^{**} p$ be a strong direct extension such that $p^*_{n^p-1} = q$ and $p^* {\mathrel{\restriction}}n^p-1 \leq^{**} p{\mathrel{\restriction}}n^p-1$ is a strong direct extension constructed by recursion so as to satisfy ${\ensuremath{\{ \Vec{r}({{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{n-1}S \}}}$ is predense below $p^*{\mathrel{\restriction}}n^p-1$. Necessarily ${\ensuremath{\{ \Vec{r}({{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S \}}}$ is predense below $p^*$
Dense open sets and direct extensions
-------------------------------------
In this subsection we prove \[DenseHomogen\], which is the basic tool to be used in the next section to analyse the properties of the cardinal ${\kappa}$ and the cardinal structure below it.
An essential obstacle in the extender based Radin forcing in comparison to the plain extender forcing is that while in the later forcig notion if we have two direct extensions $q,r\leq^* p$ then $q$ and $r$ are compatible, in the former forcing notion this does not hold. This usually entails some inductions, taking place inside elementary substructures, which construct long increasing seqeunce of conditions from ${\mathbb{P}}^*_f$, which at the end will be combined into one conditions. This method breaks if the elementary substructures in question are not closed enough (which is our case if we want to handle ${\lambda}$ successor of singular). The point of \[DenseHomogenOneBlock\] is to show how we can construction a condition $p$ such that if a direct extension $q \leq^* p$ has some favorable circumstances then the condition $p$ will suffice for this circumstances. This will enable us to work more like in a plain Radin forcing. So as we just pointed out, we aim to prove \[DenseHomogenOneBlock\]. This lemma is proved by recursion with the non-recursive case being \[DenseHomogenOneBlockCase0\]. Since the notation in \[DenseHomogenOneBlock\] is kind of hairy we present the cases $k=1$ and $k = 2$ in \[DenseHomogenOneBlockCase1\] and \[DenseHomogenOneBlockCase2\], respectively.
Assume ${\ensuremath{\langle N, f^* \rangle}}$ is a good pair and $D \in N$ is a dense open set. Let $p \in {\mathbb{P}}$ be a condition such that $f^{p_{n^p-1}} = f^*$. If there is an extension $s \leq p {\mathrel{\restriction}}n^p-1$ and a direct extension $q \leq^* p_{n^p-1}$ such that $s {\mathop{{}^\frown}}q \in D$ then there is a set $T^* \in {\Vec{E}}(f^*)$ such that ${\ensuremath{\langle f^*, T^* \rangle}} \leq^{**} p_{n^p-1}$ and $s {\mathop{{}^\frown}}{\ensuremath{\langle f^*, T^* \rangle}} \in D$.
Assume $s \leq p {\mathrel{\restriction}}n^p-1$, $q \leq^* p_{n^p-1}$, and $s {\mathop{{}^\frown}}q \in D$. Set $D^\in = {\ensuremath{\{ g \mid \exists T\in {\Vec{E}}(g)\ \
s {\mathop{{}^\frown}}{\ensuremath{\langle g, T \rangle}} \in D \}}}$ and $D^{\perp}= {\ensuremath{\{ g \mid \forall h \in D^\in\ g {\perp}h \}}}$. Then $D^{\perp}\in N$ is open by its definiton and $D^\in \in N$ is open since $D$ is open. The set $D^* = D^\in {\cup}D^{\perp}\in N$ is dense open, hence $f^* \in D^*$. Since $f^* \geq f^{q} \in D^\in$ we get $f^* \notin D^{\perp}$, thus $f^* \in D^\in$.
Assume ${\ensuremath{\langle N, f^* \rangle}}$ is a good pair, $D \in N$ is a dense open set, and $p \in {\mathbb{P}}$ is a condition such that $f^{p_{n^p-1}} = f^*$. If there is an extension $s \leq p{\mathrel{\restriction}}n^p-1$ and ${\xi}< \operatorname{o}({\Vec{E}})$ such that ${\ensuremath{\{ {\nu}\in T^{p_{n^p-1}} \mid \exists q \leq^* p_{n^p-1{\ensuremath{\langle {\nu}\rangle}}}\ s {\mathop{{}^\frown}}q \in D \}}} \in E_{\xi}(f^*)$, then there is a $p_{n^p-1}$-tree $S$ of height one, and a ${\ensuremath{\langle p_{n^p-1},S \rangle}}$-function r, such that for each ${\ensuremath{\langle {\nu}\rangle}} \in S$, $s {\mathop{{}^\frown}}r({\nu}) \in D$.
Assume $X = {\ensuremath{\{ {\nu}\in T^{p_{n^p-1}} \mid \exists q \leq^* p_{n^p-1{\ensuremath{\langle {\nu}\rangle}}}\ s {\mathop{{}^\frown}}q \in D \}}}
\in E_{\xi}(f^*)$. Set $$\begin{gathered}
D^\in = {\ensuremath{\{ g \mid \exists T\in {\Vec{E}}(g),\
\text{there is a ${\ensuremath{\langle g,T \rangle}}$-tree $S$ of height one and} \\
\text{a ${\ensuremath{\langle {\ensuremath{\langle g,T \rangle}},S \rangle}}$-function $r$ such that }
\forall {\ensuremath{\langle {\nu}\rangle}} \in S\
s {\mathop{{}^\frown}}r({\nu})\in D \}}}
\end{gathered}$$ and $D^{\perp}= {\ensuremath{\{ g \mid \forall h \in D^\in\ g {\perp}h \}}}$. Then $D^{\perp}\in N$ is open by its definiton and $D^\in \in N$ is open since $D$ is open. The set $D^* = D^\in {\cup}D^{\perp}\in N$ is dense open, hence $f^* \in D^*$. For each ${\nu}\in X$ fix a direct extension $t({\nu}) \leq^* p_{n^p-1{\ensuremath{\langle {\nu}\rangle}}\downarrow}$, and a direct extension $q({\nu}) \leq^* p_{n^p-1{\ensuremath{\langle {\nu}\rangle}}\uparrow}$ such that $s {\mathop{{}^\frown}}t({\nu}) {\mathop{{}^\frown}}q({\nu}) \in D$. Since ${\ensuremath{\langle N, f^*_{{\ensuremath{\langle {\nu}\rangle}}\uparrow} \rangle}}$ is a good pair we get by the previous lemma a set $T({\nu}) \in {\Vec{E}}(f^*_{{\ensuremath{\langle {\nu}\rangle}}\uparrow})$ satisfying ${\ensuremath{\langle f^*_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}, T({\nu}) \rangle}} \leq^{**} p_{n^p-1{\ensuremath{\langle {\nu}\rangle}}\uparrow}$ and $s {\mathop{{}^\frown}}t({\nu}) {\mathop{{}^\frown}}{\ensuremath{\langle f^*_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}, T({\nu}) \rangle}} \in D$.
Set $g = f^* {\cup}f^{j_{E_{\xi}}(t)(\operatorname{mc}_{\xi}(f^*))}$. Set $X^* = {\pi}^{-1}_{g,f^*}(X)$. By removing a measure zero set from $X^*$ we can assume for each ${\nu}\in X^*$, $g_{{\ensuremath{\langle {\nu}\rangle}}\downarrow} = f^{t({\nu}{\mathrel{\restriction}}\operatorname{dom}f^*)}$. Choose a set $T \in {\Vec{E}}(g)$ such that ${\ensuremath{\langle g,T \rangle}} \leq^* p_{n^p-1}$. Define the funcrion $r$ with domain $X^*$ by setting for each ${\nu}\in X^*$, $r({\nu}) = {\ensuremath{\langle g_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}, T^{(t({\nu}{\mathrel{\restriction}}\operatorname{dom}f^*))} {\cap}T_{{\ensuremath{\langle {\nu}\rangle}}\downarrow} \rangle}} {\mathop{{}^\frown}}{\ensuremath{\langle g_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}, {\pi}^{-1}_{g,f^*}T({\nu}) {\cap}T \rangle}}$. Note $r({\nu}) \leq^{**} {\ensuremath{\langle g, T \rangle}}_{{\ensuremath{\langle {\nu}\rangle}}}$, thus $r$ is a ${\ensuremath{\langle g,X^* \rangle}}$-function. Since $D$ is open we get for each ${\nu}\in X^*$, $s {\mathop{{}^\frown}}r({\nu}) \in D$. Thus $g \in D^\in$. Since $g \leq f^* \in D^*$ we get $f^* \in D^\in$.
Assume ${\ensuremath{\langle N, f^* \rangle}}$ is a good pair, $D \in N$ is a dense open set, and $p \in {\mathbb{P}}$ is a condition such that $f^{p_{n^p-1}} = f^*$. If there is $s \leq p{\mathrel{\restriction}}n^p-1$ such that ${\ensuremath{\{ {\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}} \in {{\vphantom{T}}^{2}{T}}^{p_{n^p-1}} \mid \exists q \leq^* p_{n^p-1{\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}}}\ s {\mathop{{}^\frown}}q \in D \}}}$ is an ${\Vec{E}}(f^*)$-tree, then there is a $p_{n-1}$-tree $S$ of height two, and a ${\ensuremath{\langle p_{n^p-1},S \rangle}}$-function r such that for each ${\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}} \in S$, $s {\mathop{{}^\frown}}\Vec{r}({\nu}_0, {\nu}_1) \in D$.
Assume $X = {\ensuremath{\{ {\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}} \in {{\vphantom{T}}^{2}{T}}^{p^*_{n^p-1}} \mid \exists s \leq p{\mathrel{\restriction}}n^p-1\ \exists q \leq^* p_{n^p-1{\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}}}\ q \in D \}}}$ is an ${\Vec{E}}(f^*)$-tree. Set $$\begin{gathered}
D^\in = {\ensuremath{\{ g \mid \exists T\in {\Vec{E}}(g),\
\text{there is a ${\ensuremath{\langle g,T \rangle}}$-tree $S$ of height two and} \\
\text{a ${\ensuremath{\langle {\ensuremath{\langle g,T \rangle}},S \rangle}}$-function $r$ such that }\\
\forall {\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}} \in S\
s {\mathop{{}^\frown}}\Vec{r}({\nu}_0, {\nu}_1)\in D \}}}
\end{gathered}$$ and $D^{\perp}= {\ensuremath{\{ g \mid \forall h \in D^\in\ g {\perp}h \}}}$. Then $D^{\perp}\in N$ is open by its definiton and $D^\in \in N$ is open since $D$ is open. The set $D^* = D^\in {\cup}D^{\perp}\in N$ is dense open, hence $f^* \in D^*$. For each ${\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}} \in X$ fix a direct extension $t({\nu}_0, {\nu}_1) \leq^* p_{n^p-1{\ensuremath{\langle {\nu}_0 \rangle}}\downarrow }$, a direct extension $t_0({\nu}_0, {\nu}_1) \leq^* p_{n^p-1{\ensuremath{\langle {\nu}_0 \rangle}}\uparrow{\ensuremath{\langle {\nu}_1 \rangle}}\downarrow}$, and a direct extension $q({\nu}_0, {\nu}_1) \leq^* p_{n^p-1{\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}}\uparrow}$ such that $s {\mathop{{}^\frown}}t({\nu}_0, {\nu}_1) {\mathop{{}^\frown}}t_0({\nu}_0, {\nu}_1) {\mathop{{}^\frown}}q({\nu}_0, {\nu}_1) \in D$. For each ${\nu}_0 \in \operatorname{Lev}_0(X)$ we can remove a measure zero set from $\operatorname{Suc}_X({\nu}_0)$ so that we can assume there is a direct extension $t({\nu}_0) \leq^* p^*_{n^p-1{\ensuremath{\langle {\nu}_0 \rangle}} \downarrow}$ such that $t({\nu}_0) = t({\nu}_0, {\nu}_1)$ for each ${\nu}_1 \in \operatorname{Suc}_X({\nu}_0)$. By the previous lemma there is a $p_{n^p-1{\ensuremath{\langle {\nu}_0 \rangle}}\uparrow}$-tree $S({\nu}_0)$ of height one, and a ${\ensuremath{\langle p_{n^p-1{\ensuremath{\langle {\nu}_0 \rangle}}\uparrow}, S({\nu}_0) \rangle}}$-function $r_{{\nu}_0}$ satisfying for each ${\nu}_1 \in S({\nu}_0)$, $s {\mathop{{}^\frown}}t({\nu}_0) {\mathop{{}^\frown}}r_{{\nu}_0}({\nu}_1) \in D$.
Set $g = f^* {\cup}f^{j_{E_{\xi}}(t)(\operatorname{mc}_{\xi}(f^*))}$, where $\operatorname{Lev}_0(X) \in E_{\xi}(f^*)$. Set $X^* = {\ensuremath{\{ {\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}} \mid {\nu}_0 \in {\pi}^{-1}_{g,f^*}\operatorname{Lev}_0(X), \
{\nu}_1 \in {\pi}^{-1}_{g,f^*}S({\nu}_0{\mathrel{\restriction}}\operatorname{dom}f^*) \}}}$. By removing a measure zero set from $\operatorname{Lev}_0(X^*)$ we can assume for each ${\nu}_0 \in X^*$, $g_{{\ensuremath{\langle {\nu}_0 \rangle}}\downarrow} = f^{t({\nu}_0{\mathrel{\restriction}}\operatorname{dom}f^*)}$. Choose a set $T \in {\Vec{E}}(g)$ such that ${\ensuremath{\langle g,T \rangle}} \leq^* p^*_{n^p-1}$. For each ${\nu}_0 \in \operatorname{Lev}_0(X^*)$ let $r'_{{\nu}_0}$ be the function with domain $\operatorname{Suc}_{X^*}({\nu}_0)$ defined by shrinking the trees in $r_{{\nu}_0}$ so that both $r'_{{\nu}_0}({\nu}_1) \leq^{**}
r_{{\nu}_0{\mathrel{\restriction}}\operatorname{dom}f^*}({\nu}_1 {\mathrel{\restriction}}\operatorname{dom}f^*)$ and $r'_{{\nu}_0}({\nu}_1) \leq^{**} {\ensuremath{\langle g,T \rangle}}_{{\ensuremath{\langle {\nu}_0 \rangle}}\uparrow{\ensuremath{\langle {\nu}_1 \rangle}}}$ will hold for each ${\nu}_1 \in \operatorname{Suc}_{X^*}({\nu}_0)$. Define the function $r$ with domain $X^*$ by setting for each ${\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}} \in X^*$, $r({\nu}_0) = {\ensuremath{\langle g_{{\ensuremath{\langle {\nu}_0 \rangle}}\downarrow},
T_{{\ensuremath{\langle {\nu}_0 \rangle}}\downarrow} {\cap}{\pi}^{-1}_{g_{{\ensuremath{\langle {\nu}_0 \rangle}}\downarrow},f^{t({\nu}_0{\mathrel{\restriction}}\operatorname{dom}f^*)}}T^{(t({\nu}_0 {\mathrel{\restriction}}\operatorname{dom}f^*))}
\rangle}}$ and $r({\nu}_0, {\nu}_1) =
r'_{{\nu}_0}
({\nu}_1)$.
Note $\Vec{r}({\nu}_0, {\nu}_1) \leq^{**} {\ensuremath{\langle g, T \rangle}}_{{\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}}}$, thus $r$ is a ${\ensuremath{\langle g,X^* \rangle}}$-function. Since $D$ is open we get $s {\mathop{{}^\frown}}\Vec{r}({\nu}_0, {\nu}_1) \in D$ for each ${\ensuremath{\langle {\nu}_0, {\nu}_1 \rangle}} \in X^*$. Thus $g \in D^\in$. Since $g \leq f^* \in D^*$ we get $f^* \in D^\in$.
As discussed earlier, the following lemma is the intended one, with the previous ones serving as an introduction to the technique used in the proof.
Assume ${\ensuremath{\langle N, f^* \rangle}}$ is a good pair, $k<{\omega}$, $D \in N$ is a dense open set, and $p \in {\mathbb{P}}$ is a condition such that $f^{p_{n^p-1}} = f^*$. If there is $s \leq p{\mathrel{\restriction}}n^p-1$ such that ${\ensuremath{\{ {{\Vec{{\nu}}}}\in {{\vphantom{T}}^{k}{T}}^{p_{n^p-1}} \mid \exists q \leq^* p_{n^p-1{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}\ s {\mathop{{}^\frown}}q \in D \}}}$ is an ${\Vec{E}}(f^*)$-tree, then there is a $p_{n^p-1}$-tree $S$ of height $k$, and a ${\ensuremath{\langle p_{n^p-1},S \rangle}}$-function r such that for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S$, $s {\mathop{{}^\frown}}\Vec{r}({{\Vec{{\nu}}}}) \in D$.
Assume $X = {\ensuremath{\{ {\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}\in {{\vphantom{T}}^{k}{T}}{p_{n^p-1}} \mid \exists q \leq^* p_{n^p-1{\ensuremath{\langle {\ensuremath{\langle {\mu}\rangle}}{\mathop{{}^\frown}}{{\Vec{{\nu}}}}\rangle}}}\ s {\mathop{{}^\frown}}q \in D \}}}$ is an ${\Vec{E}}(f^*)$-tree. For each ${\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}\in X$ fix a direct extension $t({\mu}{\mathop{{}^\frown}}{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}) \leq^* p_{n^p-1{\ensuremath{\langle {\mu}\rangle}}\downarrow}$ and a direct extension $q({\mu}{\mathop{{}^\frown}}{{\Vec{{\nu}}}}) \leq^* p_{n^p-1{\ensuremath{\langle {\mu}\rangle}}\uparrow{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}$ such that $s {\mathop{{}^\frown}}t({\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}){\mathop{{}^\frown}}q({\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}) \in D$. For each ${\mu}\in \operatorname{Lev}_0(X)$ we can remove a measure zero set from $X_{{\ensuremath{\langle {\mu}\rangle}}}$ so that we will have a direct extension $t({\mu}) \leq^* p_{n^p-1{\ensuremath{\langle {\mu}\rangle}}\downarrow}$ such that $t({\mu}) = t({\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}})$ for each ${{\Vec{{\nu}}}}\in X_{{\ensuremath{\langle {\mu}\rangle}}}$. By recursion there is a $p_{n^p-1{\ensuremath{\langle {\mu}\rangle}}\uparrow}$-tree $S({\mu})$ of height $k-1$, and a ${\ensuremath{\langle p_{n^p-1{\ensuremath{\langle {\mu}\rangle}}\uparrow}, S({\mu}) \rangle}}$-function $r_{{\mu}}$ satisfying for each ${{\Vec{{\nu}}}}\in S({\mu})$, $s {\mathop{{}^\frown}}t({\mu}) {\mathop{{}^\frown}}r_{{\mu}}({{\Vec{{\nu}}}}) \in D$.
Set $g = f^* {\cup}f^{j_{E_{\xi}}(t)(\operatorname{mc}_{\xi}(f^*))}$, where $\operatorname{Lev}_0(X) \in E_{\xi}(f^*)$. Set $X^* = {\ensuremath{\{ {\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}\mid {\mu}\in {\pi}^{-1}_{g,f^*}\operatorname{Lev}_0(X), \
{{\Vec{{\nu}}}}\in {\pi}^{-1}_{g,f^*}(S({\mu}{\mathrel{\restriction}}\operatorname{dom}f^*)) \}}}$. By removing a measure zero set from $\operatorname{Lev}_0(X^*)$ we can assume for each ${\mu}\in X^*$, $g_{{\ensuremath{\langle {\mu}\rangle}}\downarrow} = f^{t({\mu}{\mathrel{\restriction}}\operatorname{dom}f^*)}$. Choose a set $T \in {\Vec{E}}(g)$ such that ${\ensuremath{\langle g,T \rangle}} \leq^* p_{n^p-1}$. For each ${\mu}\in \operatorname{Lev}_0(X^*)$ let $r'_{{\nu}_0}$ be the function with domain $X^*_{{\ensuremath{\langle {\mu}\rangle}}}$ defined by shrinking the trees in $r_{{\mu}}$ so that both $\Vec{r}'_{{\mu}}({{\Vec{{\nu}}}}) \leq^{**}
\Vec{r}_{{\mu}{\mathrel{\restriction}}\operatorname{dom}f^*}({{\Vec{{\nu}}}}{\mathrel{\restriction}}X_{{\ensuremath{\langle {\mu}\rangle}} {\mathrel{\restriction}}\operatorname{dom}f^*})$ and $r'_{{\mu}}({{\Vec{{\nu}}}}) \leq^{**} {\ensuremath{\langle g,T \rangle}}_{{\ensuremath{\langle {\mu}\rangle}}\uparrow{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}$ will hold for each ${{\Vec{{\nu}}}}\in X^*_{{\ensuremath{\langle {\mu}\rangle}}}$. Define the function $r$ with domain $X^*$ by setting for each ${\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}\in X^*$, $r({\mu}) = {\ensuremath{\langle g_{{\ensuremath{\langle {\mu}\rangle}}\downarrow},
T_{{\ensuremath{\langle {\mu}\rangle}}\downarrow} {\cap}{\pi}^{-1}_
{g_{{\ensuremath{\langle {\mu}\rangle}}\downarrow},f^{t({\mu}{\mathrel{\restriction}}\operatorname{dom}f^*)}
}T^{(t({\mu}{\mathrel{\restriction}}\operatorname{dom}f^*))}
\rangle}}$ and $r({\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}) =
r'_{{\mu}}
({{\Vec{{\nu}}}})$.
Note $\Vec{r}({\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}) \leq^{**}
{\ensuremath{\langle g, T \rangle}}_{{\ensuremath{\langle {\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}\rangle}}}$, thus $r$ is a ${\ensuremath{\langle g, X^* \rangle}}$-function. Since $D$ is open we get $s {\mathop{{}^\frown}}\Vec{r}({\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}) \in D$ for each ${\ensuremath{\langle {\ensuremath{\langle {\mu}\rangle}} {\mathop{{}^\frown}}{{\Vec{{\nu}}}}\rangle}} \in X^*$. Thus $g \in D^\in$. Since $g \leq f^* \in D^*$ we get $f^* \in D^\in$.
Assume ${\ensuremath{\langle f, T \rangle}} \in {\mathbb{P}}$ is a condition, $k < {\omega}$, and $S \subseteq {{\vphantom{T}}^{k}{T}}$ is not an ${\Vec{E}}(f)$-tree. Then there is a set $T^* \in {\Vec{E}}(f)$ such that ${\ensuremath{\langle f, T^* \rangle}} \leq^* {\ensuremath{\langle f, T \rangle}}$ and ${{\vphantom{T}}^{k}{T}}^{*} {\cap}S = \emptyset$.
By removing measure zero sets from the levels of $S$ we can find $n < k$ so that the following will hold:
1. For each $l < n$ and ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{l-1} \rangle}} \in S$, $\operatorname{Suc}_S({\nu}_0, \dotsc, {\nu}_l) \in E_{\xi}(f)$ for some ${\xi}< \operatorname{o}({\Vec{E}})$.
2. For each ${\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}} \in S$, $\operatorname{Suc}_S({\nu}_0, \dotsc, {\nu}_{n-1}) \notin E_{\xi}(f)$ for each ${\xi}< \operatorname{o}({\Vec{E}})$.
Shrink $T$ so that ${\ensuremath{\{ {\ensuremath{\langle f,T \rangle}}_{{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}}} \mid {\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}} \in \operatorname{Lev}_n(S) \}}}$ is predense below ${\ensuremath{\langle f,T \rangle}}$. We are done by setting $A = \operatorname*{\triangle}{\ensuremath{\{ T \setminus \operatorname{Suc}_{S_n}({\nu}_0, \dotsc, {\nu}_{n-1}) \mid {\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{n-1} \rangle}} \in S_n \}}}$ and $T^* = T {\cap}A$.
Assume ${\ensuremath{\langle N, f^* \rangle}}$ is a good pair, $D \in N$ is a dense open set, and $p \in {\mathbb{P}}$ is a condition such that $f^{p_{n^p-1}} = f^*$. Assume $s \leq p {\mathrel{\restriction}}n^p-1$. Then one and only one of the following holds:
1. There is a $p_{n^p-1}$-tree S, and a ${\ensuremath{\langle p_{n^p-1}, S \rangle}}$-function $r$, such that for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max} S$, $s {\mathop{{}^\frown}}\Vec{r}({{\Vec{{\nu}}}}) \in D$.
2. There is a set $T^* \in {\Vec{E}}(f^*)$ such that ${\ensuremath{\langle f^*, T^* \rangle}} \leq^{**} p_{n^p-1}$ and for each ${{\Vec{{\nu}}}}\in {{\vphantom{T}}^{<{\omega}}{T}}^*$ and $q \leq^* {\ensuremath{\langle f^*, T^* \rangle}}_{{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}$, $s {\mathop{{}^\frown}}q \notin D$.
It is about time we get rid of the conditional appearing in the former statements show we have densely many times $p_{n^p-1}$-trees and functions.
Assume $D \in N$ is a dense open set, and $p \in {\mathbb{P}}$ is a condition. Then there is an extension $s \leq p {\mathrel{\restriction}}n^p-1$ and $k < {\omega}$ so that ${\ensuremath{\{ {{\Vec{{\nu}}}}\in {{\vphantom{T}}^{k}{T}}^{p_{n^p-1}} \mid \exists q\leq^* p_{n^p-1{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}\ s {\mathop{{}^\frown}}q \in D \}}}$ is an ${\Vec{E}}(f^p)$-tree.
Towards contradiction assume the claim fails. Then for each $s \leq p{\mathrel{\restriction}}n^p - 1$ and $k<{\omega}$ the set $S(s,k) = {\ensuremath{\{ {{\Vec{{\nu}}}}\in {{\vphantom{T}}^{k}{T}}^{p_{n^p-1}} \mid \exists q\leq^* p_{n^p-1{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}\ s {\mathop{{}^\frown}}q \in D \}}}$ is not an ${\Vec{E}}(f^p)$-tree. Thus there is a set $T(s, {\kappa}) \in {\Vec{E}}(f^{p_{n^p-1}})$ satisfying ${{\vphantom{T}}^{k}{T}}(s, k) {\cap}S(s,k) = \emptyset$. Set $T^* = {\bigcap}{\ensuremath{\{ T(s,k) \mid k<{\omega},\ s\leq p{\mathrel{\restriction}}n^p-1 \}}}$. Consider the condition $p^* = p{\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}{\ensuremath{\langle f^{p_{n^p-1}}, T^* \rangle}}$. By the density of the set $D$ there is an extension $s \leq p {\mathrel{\restriction}}n^p-1$, ${{\Vec{{\nu}}}}\in {{\vphantom{T}}^{k}{T}}^*$, and $q \leq^* p^*_{n^p-1{\ensuremath{\langle {{\Vec{{\nu}}}}\rangle}}}$, such that $s {\mathop{{}^\frown}}q \in D$. Hence ${{\Vec{{\nu}}}}\in S(s, k)$. However ${{\vphantom{T^*}}^{k}{T^*}} {\cap}S(n,k) = \emptyset$, contradiction.
Assume ${\ensuremath{\langle N, f^* \rangle}}$ is a good pair, $D \in N$ is a dense open set, and $p \in {\mathbb{P}}$ is a condition such that $f^{p_{n^p-1}} = f^*$. Then there is a maximal antichain $A$ below $p {\mathrel{\restriction}}n^p-1$ such that for each $s \in A$ there is a $p_{n^p-1}$-tree S and a ${\ensuremath{\langle p_{n^p-1}, S \rangle}}$-function $r$, such that for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max} S$, $s {\mathop{{}^\frown}}\Vec{r}({{\Vec{{\nu}}}}) \in D$.
Assume $D \in N$ is a dense open set and $p \in {\mathbb{P}}$ is a condition. Then there is a direct extension $p^* \leq^* p$, a $p^*$-tree $S$, and a ${\ensuremath{\langle p^*, S \rangle}}$-function $r$ such that for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}(S)$, $\Vec{r}({{\Vec{{\nu}}}}) \in D$.
${\kappa}$ Properties in the Genric Extension
=============================================
The forcing notion ${\mathbb{P}}$ is of Prikry type.
Assume $p \in {\mathbb{P}}$ is a condition and ${\sigma}$ is a formula in the ${\mathbb{P}}$-forcing language.We will be done by exhibiting a direct extension $p^* \leq^* p$ such that $p^* {\mathrel\Vert}{\sigma}$. Set $D = {\ensuremath{\{ q \leq p \mid q {\mathrel\Vert}{\sigma}\}}}$. The set $D$ is dense open, hence by \[DenseHomogen\] there is a direct extension $p^* \leq^* p$, a $p^*$-tree $S$, and a ${\ensuremath{\langle p^*, S \rangle}}$-function $r$, such that for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S$, $\Vec{r}({{\Vec{{\nu}}}}) \in D$. Set $X_0 = {\ensuremath{\{ {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S \mid \Vec{r}({{\Vec{{\nu}}}}) {\mathrel\Vdash}\lnot {\sigma}\}}}$ and $X_1 = {\ensuremath{\{ {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S \mid \Vec{r}({{\Vec{{\nu}}}}) {\mathrel\Vdash}{\sigma}\}}}$. Since the sets $X_0$ and $X_1$ are a disjoint partition of $\operatorname{Lev}_{\max}S$, only one of them is a measure one set. Fix $i < 2$ such that $X_i$ is a measure one set. Set $S_i = {\ensuremath{\{ {\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_k \rangle}} \mid {\ensuremath{\langle {\nu}_0, \dotsc {\nu}_n \rangle}}\in X_i,\ k \leq n \}}}$. Using \[GetPreDense\] shrink the trees appearing in the condition $p^*$ so that ${\ensuremath{\{ \Vec{r}({{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S_i \}}}$ is predense below $p^*$. Thus $p^* {\mathrel\Vdash}{\sigma}_i$, where ${\sigma}_0 = {{}\text{``} \lnot{\sigma}{}\text{''}}$ and ${\sigma}_1 = {{}\text{``} {\sigma}{}\text{''}}$.
${\mathrel\Vdash}{{}\text{``} {\kappa}\text{ is a cardinal} {}\text{''}}$.
If $\operatorname{o}(\Vec{E}) = 1$ then there are no new bounded subset of ${\kappa}$ in $V^{{\mathbb{P}}}$, hence no cardinal below ${\kappa}$ is collapsed, hence ${\kappa}$ is preserved. If $\operatorname{o}(\Vec{E}) > 1$ then an unbounded number of cardinals below ${\kappa}$ is preserved, hence ${\kappa}$ is preserved.
\[BecomesSingular\] If $\operatorname{o}(\Vec{E}) < {\kappa}$ is regular then ${\mathrel\Vdash}{{}\text{``} \operatorname{cf}{\kappa}=\operatorname{cf}\operatorname{o}(\Vec{E}) {}\text{''}}$.
It is immediate ${\mathrel\Vdash}{{}\text{``} \operatorname{cf}{\kappa}\leq \operatorname{cf}\operatorname{o}(\Vec{E}) {}\text{''}}$. Hence we need to show ${\mathrel\Vdash}{{}\text{``} \operatorname{cf}{\kappa}\not< \operatorname{cf}\operatorname{o}(\Vec{E}) {}\text{''}}$. Assume ${\sigma}< {\kappa}$ and $p {\mathrel\Vdash}{{}\text{``} {\sigma}< \operatorname{cf}\operatorname{o}(\Vec{E}) \text{ and }
{\utilde{f}}{\mathrel{:}}{\sigma}\to {\kappa}{}\text{''}}$. We will be done by exhibiting a direct extension $p^* \leq^* p$ such that $p^* {\mathrel\Vdash}{{}\text{``} {\utilde{f}} \text{ is bounded} {}\text{''}}$. Let ${\ensuremath{\langle N, f^* \rangle}}$ be a good pair such that $p, {\utilde{f}}, {\sigma}\in N$ and $f^* \leq^* f^{p_{n^p-1}}$. Shrink $T^{p_{n^p-1}}$ so as to satisfy for each ${\nu}\in T^{p_{n^p-1}}$, ${\mathring{{\nu}}}({\kappa}) > {\sigma}$.
Factor ${\mathbb{P}}$ as follows. Set $P_0 = {\ensuremath{\{ s \leq p{\mathrel{\restriction}}n^p-1 \mid \exists q \leq p_{n^p-1}\ s {\mathop{{}^\frown}}q \in {\mathbb{P}}\}}}$ and $P_1 = {\ensuremath{\{ q \leq p_{n^p-1} \mid \exists s \leq p {\mathrel{\restriction}}n^p-1\ s {\mathop{{}^\frown}}q \in {\mathbb{P}}\}}}$. For each ${\xi}< {\sigma}$ work as follows. Set $D_{\xi}= {\ensuremath{\{ q \leq p_{ n^{p}-1} \mid \text{There exists a } P_0\text{-name
${\utilde{{\rho}}}$ such that } q {\mathrel\Vdash}_{P_1} {{}\text{``} {\utilde{f}}({\xi}) = {\utilde{{\rho}}} {}\text{''}} \}}}$. Since $D_{\xi}\in N$ is a dense open subset of ${\mathbb{P}}$ below $p_{n^p-1}$ there is a a direct extension $p^{\xi}= {\ensuremath{\langle f^*,T^{\xi}\rangle}} \leq^* p_{n^p-1}$, a $p^{\xi}$-tree $S^{\xi}$, and a ${\ensuremath{\langle p^{\xi}, S^{\xi}\rangle}}$-function $r_{\xi}$ satisfying for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^{\xi}$, $\Vec{r}_{\xi}({{\Vec{{\nu}}}}) \in D_{\xi}$. Thus for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max} S^{\xi}$ there is a $P_0$-name ${\utilde{{\rho}}}^{{\xi},{{\Vec{{\nu}}}}}$ so that $\Vec{r}_{\xi}({{\Vec{{\nu}}}}){\mathrel\Vdash}_{P_1} {{}\text{``} {\utilde{f}}({\xi}) = {\utilde{{\rho}}}^{{\xi},{{\Vec{{\nu}}}}} {}\text{''}}$. Since ${\lvertP_0\rvert}<{\kappa}$ there is ${\zeta}^{{\xi},{{\Vec{{\nu}}}}} < {\kappa}$ such that $p{\mathrel{\restriction}}n^p-1 {\mathrel\Vdash}_{P_0} {{}\text{``} {\utilde{{\rho}}}^{{\xi},{{\Vec{{\nu}}}}}<{\zeta}^{{\xi},{{\Vec{{\nu}}}}} {}\text{''}}$.
Let $m_{\xi}$ be a function witnessing $S^{\xi}$ is a $p_{n^p-1}$-tree, i.e., $m_{\xi}{\mathrel{:}}{\ensuremath{\{ \emptyset \}}} {\cup}\operatorname{Lev}_{<\max}S \to \operatorname{o}({\Vec{E}})$ is a function satisfying for each ${{\Vec{{\nu}}}}\in \operatorname{dom}m_{\xi}$, $\operatorname{Suc}_S({{\Vec{{\nu}}}}) \in E_{m_{\xi}({{\Vec{{\nu}}}})}(f^{p_{n^p-1}})$. (We use the convention $\operatorname{Suc}_S({\ensuremath{\langle \rangle}}) = \operatorname{Lev}_0(S)$.) Since $\operatorname{o}(\Vec{E}) < {\kappa}$ we can remove a measure zero set from $S^{\xi}$ (and $\operatorname{dom}m_{\xi}$) and get for each ${{\Vec{{\nu}}}}_0, {{\Vec{{\nu}}}}_1 \in \operatorname{dom}m^{\xi}$, if ${\lvert{{\Vec{{\nu}}}}_0\rvert} = {\lvert{{\Vec{{\nu}}}}_1\rvert}$ then $m_{\xi}({{\Vec{{\nu}}}}_0) = m_{\xi}({{\Vec{{\nu}}}}_1)$. Thus ${\lvert\operatorname{ran}m_{\xi}\rvert} < {\omega}$. Set ${\tau}_{\xi}= \sup \operatorname{ran}m_{\xi}$. Shrink $T^{\xi}$ so that ${\ensuremath{\{ \Vec{r}_{\xi}({{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^{\xi}\}}}$ is predense below $p^{\xi}$. Note, if ${\mu}\in \operatorname{Lev}_0 T^{{\xi}}$, ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^{\xi}$, $\operatorname{o}({\mu}) > {\tau}_{\xi}$ and ${{\Vec{{\nu}}}}\not< {\mu}$, then $\Vec{r}({{\Vec{{\nu}}}}) {\perp}p^{{\xi}}_{{\ensuremath{\langle {\mu}\rangle}}}$. Hence $p{\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}p^{{\xi}}_{{\ensuremath{\langle {\mu}\rangle}}} {\mathrel\Vdash}{{}\text{``} {\utilde{f}}({\xi}) < \sup {\ensuremath{\{ {\zeta}^{{\xi},{{\Vec{{\nu}}}}} \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^{\xi}, {{\Vec{{\nu}}}}< {\mu}\}}} {}\text{''}}$.
Set $T^* = {\bigcap}_{{\xi}< {\sigma}} T^{\xi}$ and $p^* = p{\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}{\ensuremath{\langle f^*, T^* \rangle}}$. We claim $p^* {\mathrel\Vdash}{{}\text{``} {\utilde{f}} \text{ is bounded} {}\text{''}}$. To show this set ${\tau}= \sup {\ensuremath{\{ {\tau}_{\xi}\mid {\xi}< {\sigma}\}}}$. Note ${\tau}< \operatorname{o}(\Vec{E})$. Since ${\ensuremath{\{ p^*_{{\ensuremath{\langle {\mu}\rangle}}} \mid {\mu}\in T^{*},\
\operatorname{o}({\mu}) = {\tau}\}}}$ is predense below $p^*$ it is enough to show that $p^*_{{\ensuremath{\langle {\mu}\rangle}}} {\mathrel\Vdash}{{}\text{``} {\utilde{f}} \text{ is bounded} {}\text{''}}$ for each ${\mu}\in T^{*}$ such that $\operatorname{o}({\mu}) = {\tau}$. So fix ${\mu}\in T^{*}$ such that $\operatorname{o}({\mu}) = {\tau}$. Set ${\zeta}= \sup {\ensuremath{\{ {\zeta}^{{\xi},{{\Vec{{\nu}}}}} \mid {\xi}< {\sigma}, {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max} S^{\xi}, {{\Vec{{\nu}}}}<{\mu}\}}}$. Note ${\zeta}< {\kappa}$. We get for each ${\xi}< {\sigma}$, $p^*_{{\ensuremath{\langle {\mu}\rangle}}} \leq^* p{\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}p^{{\xi}}_{{\ensuremath{\langle {\mu}\rangle}}} {\mathrel\Vdash}{{}\text{``} {\utilde{f}}({\xi}) < \sup {\ensuremath{\{ {\zeta}^{{\xi},{{\Vec{{\nu}}}}} \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^{\xi}, {{\Vec{{\nu}}}}< {\mu}\}}}<{\zeta}<{\kappa}{}\text{''}}$.
\[Gitik\] If $\operatorname{o}(\Vec{E}) \in [{\kappa}, {\lambda})$ and $\operatorname{cf}(\operatorname{o}({\Vec{E}})) \geq {\kappa}$ then ${\mathrel\Vdash}{{}\text{``} \operatorname{cf}{\kappa}= {\omega}{}\text{''}}$.
Fix a condition $p \in {\mathbb{P}}$ such that $\operatorname{o}({\Vec{E}})+1 \subseteq \in \operatorname{dom}f^{p_{n^p-1}}$. Partition $T^{p_{n^p-1}}$ into $\operatorname{o}({\Vec{E}})$ disjoint subsets ${\ensuremath{\{ A_{\xi}\mid {\xi}< \operatorname{o}({\Vec{E}}) \}}}$ by setting for each ${\xi}< \operatorname{o}({\Vec{E}})$, $$\begin{aligned}
A_{{\xi}} = {\ensuremath{\{ {\nu}\in T^{p_{n^p-1}} \mid {\xi}\in \operatorname{dom}{\nu},\
\operatorname{o}({\nu}({\kappa})) = \operatorname{otp}((\operatorname{dom}{\nu}) {\cap}{\xi}) \}}}.\end{aligned}$$ Let $G$ be generic. Choose a condition $p \in G$. Let ${\ensuremath{{\ensuremath{\langle {\nu}_{\xi}\mid {\xi}<{\kappa}\rangle}}}}$ be the increasing enumeration of the set ${\ensuremath{\{ {\nu}_0, \dotsc, {\nu}_k \mid p{\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}p_{n^p-1{\ensuremath{\langle {\nu}_0, \dotsc, {\nu}_{{\kappa}} \rangle}}} \in G \}}}$. Set ${\zeta}_0 = 0$. For each $n < {\omega}$ set ${\zeta}_{n+1} = \min {\ensuremath{\{ {\xi}> {\zeta}_n \mid
{\nu}_{\xi}\in A_{sup ((\operatorname{dom}{\nu}_{{\zeta}_n}) {\cap}\operatorname{o}({\Vec{E}})) } \}}}$ . We are done since ${\kappa}= \sup_{n<{\omega}} {\zeta}_n$.
Using the same method as above we get the following claim.
If $\operatorname{o}(\Vec{E}) \in [{\kappa}, {\lambda})$ and $\operatorname{cf}(\operatorname{o}({\Vec{E}})) < {\kappa}$ then ${\mathrel\Vdash}{{}\text{``} \operatorname{cf}{\kappa}= \operatorname{cf}(\operatorname{o}({\Vec{E}})) {}\text{''}}$.
\[BecomesRegular\] If $\operatorname{o}(\Vec{E}) = {\lambda}$ then ${\mathrel\Vdash}{{}\text{``} {\kappa}\text{ is regular} {}\text{''}}$.
Assume ${\sigma}< {\kappa}$ and $p {\mathrel\Vdash}{{}\text{``} {\utilde{f}}{\mathrel{:}}{\sigma}\to {\kappa}{}\text{''}}$. We will be done by exhibiting a direct extension $p^* \leq^* p$ such that $p^* {\mathrel\Vdash}{{}\text{``} {\utilde{f}} \text{ is bounded} {}\text{''}}$. Let ${\ensuremath{\langle N, f^* \rangle}}$ be a good pair such that $p, {\utilde{f}}, {\sigma}\in N$ and $f^* \leq^* f^{p_{n^p-1}}$. Shrink $T^{p_{n^p-1}}$ so as to satisfy for each ${\nu}\in \operatorname{Lev}_0(T^{p_{n^p-1}})$, ${\mathring{{\nu}}}({\kappa}) > {\sigma}$.
Factor ${\mathbb{P}}({\Vec{E}})$ as follows. Set $P_0 = {\ensuremath{\{ s \leq p{\mathrel{\restriction}}n^p-1 \mid \exists q \leq p_{n^p-1}\ s {\mathop{{}^\frown}}q \in {\mathbb{P}}({\Vec{E}}) \}}}$ and $P_1 = {\ensuremath{\{ q \leq p_{n^p-1} \mid \exists s \leq p\ {\mathrel{\restriction}}n^p-1 s {\mathop{{}^\frown}}q \in {\mathbb{P}}({\Vec{E}}) \}}}$. For each ${\xi}< {\sigma}$ work as follows. Set $D_{\xi}= {\ensuremath{\{ q \leq p_{ n^{p}-1} \mid \text{There exists a } P_0\text{-name
${\utilde{{\rho}}}$ such that } q {\mathrel\Vdash}_{P_1} {{}\text{``} {\utilde{f}}({\xi}) = {\utilde{{\rho}}} {}\text{''}} \}}}$. Since $D_{\xi}\in N$ is a dense open subset of ${\mathbb{P}}$ below $p_{n^p-1}$ there is a a direct extension $p^{\xi}= {\ensuremath{\langle f^*,T^{\xi}\rangle}} \leq^* p_{n^p-1}$, a $p^{\xi}$-tree $S^{\xi}$, and a ${\ensuremath{\langle p^{\xi}, S^{\xi}\rangle}}$-function $r_{\xi}$ satisfying for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^{\xi}$, $\Vec{r}_{\xi}({{\Vec{{\nu}}}}) \in D_{\xi}$. Thus for each ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max} S^{\xi}$ there is a $P_0$-name ${\utilde{{\rho}}}^{{\xi},{{\Vec{{\nu}}}}}$ so that $\Vec{r}_{\xi}({{\Vec{{\nu}}}}){\mathrel\Vdash}{{}\text{``} {\utilde{f}}({\xi}) = {\utilde{{\rho}}}^{{\xi},{{\Vec{{\nu}}}}} {}\text{''}}$. Since ${\lvertP_0\rvert}<{\kappa}$ there is ${\zeta}^{{\xi},{{\Vec{{\nu}}}}} < {\kappa}$ such that $p{\mathrel{\restriction}}n^p-1 {\mathrel\Vdash}_{P_0} {{}\text{``} {\utilde{{\rho}}}^{{\xi},{{\Vec{{\nu}}}}}<{\zeta}^{{\xi},{{\Vec{{\nu}}}}} {}\text{''}}$.
Let $m_{\xi}$ be a function witnessing $S^{\xi}$ is a $p_{n^p-1}$-tree, i.e., $m_{\xi}{\mathrel{:}}{\ensuremath{\{ \emptyset \}}} {\cup}\operatorname{Lev}_{<\max}S \to \operatorname{o}({\Vec{E}})$ is a function satisfying for each ${{\Vec{{\nu}}}}\in \operatorname{dom}m_{\xi}$, $\operatorname{Suc}_S({{\Vec{{\nu}}}}) \in E_{m_{\xi}({{\Vec{{\nu}}}})}(f^{p_{n^p-1}})$. (We use the convention $\operatorname{Suc}_S({\ensuremath{\langle \rangle}}) = \operatorname{Lev}_0(S)$.) Since ${\lambda}=\operatorname{o}(\Vec{E})$ is regular and ${\lvertS^{\xi}\rvert} < {\lambda}$ we get ${\tau}_{\xi}= \sup \operatorname{ran}m_{\xi}< {\lambda}$. Shrink $T^{\xi}$ so that ${\ensuremath{\{ \Vec{r}_{\xi}({{\Vec{{\nu}}}}) \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^{\xi}\}}}$ is predense below $p^{\xi}$. Note, if ${\mu}\in T^{{\xi}}$, ${{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^{\xi}$, $\operatorname{o}({\mu}) > {\mathring{{\mu}}}({\tau}_{\xi})$ and ${{\Vec{{\nu}}}}\not< {\mu}$, then $\Vec{r}({{\Vec{{\nu}}}}) {\perp}p^{{\xi}}_{{\ensuremath{\langle {\mu}\rangle}}}$. Hence $p{\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}p^{{\xi}}_{{\ensuremath{\langle {\mu}\rangle}}} {\mathrel\Vdash}{{}\text{``} {\utilde{f}}({\xi}) < \sup {\ensuremath{\{ {\zeta}^{{\xi},{{\Vec{{\nu}}}}} \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^{\xi}, {{\Vec{{\nu}}}}< {\mu}\}}} {}\text{''}}$.
Set $T^* = {\bigcap}_{{\xi}< {\sigma}} T^{\xi}$ and $p^* = p{\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}{\ensuremath{\langle f^*, T^* \rangle}}$. We claim $p^* {\mathrel\Vdash}{{}\text{``} {\utilde{f}} \text{ is bounded} {}\text{''}}$. To show this set ${\tau}= \sup {\ensuremath{\{ {\tau}_{\xi}\mid {\xi}< {\sigma}\}}}$. Note ${\tau}< \operatorname{o}(\Vec{E})={\lambda}$. Since ${\ensuremath{\{ p^*_{{\ensuremath{\langle {\mu}\rangle}}} \mid {\mu}\in T^{*},\
\operatorname{o}({\mu}) = {\mathring{{\mu}}}({\tau}) \}}}$ is predense below $p^*$ it is enough to show that $p^*_{{\ensuremath{\langle {\mu}\rangle}}} {\mathrel\Vdash}{{}\text{``} {\utilde{f}} \text{ is bounded} {}\text{''}}$ for each ${\mu}\in \operatorname{Lev}_0 T^{*}$ such that $\operatorname{o}({\mu}) = {\mathring{{\mu}}}({\tau})$. So fix ${\mu}\in \operatorname{Lev}_0 T^{*}$ such that $\operatorname{o}({\mu}) = {\mathring{{\mu}}}({\tau})$. Set ${\zeta}= \sup {\ensuremath{\{ {\zeta}^{{\xi},{{\Vec{{\nu}}}}} \mid {\xi}< {\sigma}, {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max}S^{\xi}, {{\Vec{{\nu}}}}<{\mu}\}}}$. Note ${\zeta}< {\kappa}$. We get for each ${\xi}< {\sigma}$, $p^*_{{\ensuremath{\langle {\mu}\rangle}}} \leq^* p{\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}p^{{\xi}}_{{\ensuremath{\langle {\mu}\rangle}}} {\mathrel\Vdash}{{}\text{``} {\utilde{f}}({\xi}) < \sup {\ensuremath{\{ {\zeta}^{{\xi},{{\Vec{{\nu}}}}} \mid {{\Vec{{\nu}}}}\in \operatorname{Lev}_{\max} S^{\xi}, {{\Vec{{\nu}}}}< {\mu}\}}}<{\zeta}<{\kappa}{}\text{''}}$.
An ordinal ${\rho}< \operatorname{o}(\Vec{E})$ is a repeat point of $\Vec{E}$ if for each $d \in [{\epsilon}]^{<{\lambda}}$, ${\bigcap}_{{\xi}< {\rho}}E_{\xi}(d) = {\bigcap}_{{\xi}< \operatorname{o}(\Vec{E})} E_{\xi}(d)$.
Assume ${\rho}< \operatorname{o}(\Vec{E})$ is a repeat point of $\Vec{E}$.
1. If $p,q \in {\mathbb{P}}$ are compatible then $j_{E_{\rho}}(p)_{{\ensuremath{\langle \operatorname{mc}_{\rho}(p) \rangle}}}$ and $j_{E_{\rho}}(q)_{{\ensuremath{\langle \operatorname{mc}_{\rho}(q) \rangle}}}$ are compatible.
2. For each $p \in {\mathbb{P}}$ there is a direct extension $p^* \leq^* p$ such that $j_{E_{\rho}}(p^*)_{{\ensuremath{\langle \operatorname{mc}_{\rho}(p^*) \rangle}}} {\mathrel\Vert}{{}\text{``} \operatorname{mc}_{\rho}(p^*) \in j_{E_{\rho}}({\utilde{A}}) {}\text{''}}$.
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1. Let $r \leq p,q$. By definition of the order there are extensions $p' \leq p$ and $q' \leq q$ such that $r \leq^* p',q'$. By elementarity $j_{E_{\rho}}(p')_{{\ensuremath{\langle \operatorname{mc}_{\rho}(p') \rangle}}} \leq j_{E_{\rho}}(p)_{{\ensuremath{\langle \operatorname{mc}_{\rho}(p) \rangle}}}$ and $j_{E_{\rho}}(q')_{{\ensuremath{\langle \operatorname{mc}_{\rho}(q') \rangle}}} \leq j_{E_{\rho}}(q)_{{\ensuremath{\langle \operatorname{mc}_{\rho}(q) \rangle}}}$. Thus we will be done by showing $j_{E_{\rho}}(p')_{{\ensuremath{\langle \operatorname{mc}_{\rho}(p') \rangle}}}$ and $j_{E_{\rho}}(q')_{{\ensuremath{\langle \operatorname{mc}_{\rho}(q') \rangle}}}$ are compatible.
So, without loss of generality assume $p$ and $q$ are $\leq^*$ compatible. By elementarity $j_{E_{\rho}}(p)$ and $j_{E_{\rho}}(q)$ are compatible. Note $$\begin{aligned}
& p_{n^p-1} = j_{E_{\rho}}(p_{n^p-1})_{
{\ensuremath{\langle \operatorname{mc}_{\rho}(p_{n^p-1}) \rangle}}\downarrow}
\intertext{and}
& q_{n^q-1} = j_{E_{\rho}}(q_{n^q-1})_{
{\ensuremath{\langle \operatorname{mc}_{\rho}(q_{n^q-1}) \rangle}}\downarrow}.
\end{aligned}$$ Then $$\begin{aligned}
& j_{E_{\rho}}(p)_{{\ensuremath{\langle \operatorname{mc}_{\rho}(p_{n^p-1}) \rangle}}} =
p {\mathop{{}^\frown}}{\ensuremath{\langle
j_{E_{\rho}}(p_{n^p-1})_{{\ensuremath{\langle \operatorname{mc}_{\rho}(p_{n^p-1}) \rangle}}\uparrow} \rangle}}
\intertext{and}
& j_{E_{\rho}}(q)_{{\ensuremath{\langle \operatorname{mc}_{\rho}(q_{n^q-1}) \rangle}}} =
q {\mathop{{}^\frown}}{\ensuremath{\langle
j_{E_{\rho}}(q_{n^q-1})_{{\ensuremath{\langle \operatorname{mc}_{\rho}(q_{n^q-1}) \rangle}}\uparrow} \rangle}}.
\end{aligned}$$ We are done.
2. Let ${\ensuremath{\langle N, f^* \rangle}}$ be a good pair such that $f^* \leq^* f^{p_{n^p-1}}$ and $p, {\utilde{A}} \in N$. Set $T = {\pi}^{-1}_{f^*, f^{p_{n^p-1}}}T^p$. Fix ${\nu}\in T$ and consider the condition $p {\mathrel{\restriction}}n^p-1 {\mathop{{}^\frown}}{\ensuremath{\langle f^*,T \rangle}}_{{\ensuremath{\langle {\nu}\rangle}}}$. By the Prikry property there is $s \leq^* p {\mathrel{\restriction}}n^p-1$, $r \leq^* {\ensuremath{\langle f^*, T \rangle}}_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}$, and $q \leq^* {\ensuremath{\langle f^*, T \rangle}}_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}$ such that $s {\mathop{{}^\frown}}r {\mathop{{}^\frown}}q {\mathrel\Vert}{{}\text{``} {\nu}\in {\utilde{A}} {}\text{''}}$. Since the set ${\ensuremath{\{ t \leq p \mid t {\mathrel\Vert}{{}\text{``} {\nu}\in {\utilde{A}} {}\text{''}} \}}}$ is dense open below $p$ and belongs to $N$, we get by \[BasicReflection\] that there is a set $T_1({\nu}) \in \Vec{E}(f^*)$ such that $s {\mathop{{}^\frown}}r {\mathop{{}^\frown}}{\ensuremath{\langle f^*_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}, T_1({\nu}) \rangle}} {\mathrel\Vert}{{}\text{``} {\nu}\in {\utilde{A}} {}\text{''}}$.
Thus for each ${\nu}\in T$ there is $s({\nu}) \leq^* p {\mathrel{\restriction}}n^p-1$, $r({\nu}) \leq^* {\ensuremath{\langle f^*, T \rangle}}_{{\ensuremath{\langle {\nu}\rangle}}\downarrow}$, and $T_1({\nu}) \in \Vec{E}(f^*)$ such that $s({\nu}) {\mathop{{}^\frown}}r({\nu}) {\mathop{{}^\frown}}{\ensuremath{\langle f^*_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}, T_1({\nu}) \rangle}} {\mathrel\Vert}{{}\text{``} {\nu}\in {\utilde{A}} {}\text{''}}$. We can find a set $T_{={\rho}} \in E_{\rho}(f^*)$ and $s \leq^* p {\mathrel{\restriction}}n^p-1$ such that for each ${\nu}\in T_{={\rho}}$, $s({\nu}) = s$. Thus for each ${\nu}\in T_{={\rho}}$, $s {\mathop{{}^\frown}}r({\nu}) {\mathop{{}^\frown}}{\ensuremath{\langle f^*_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}, T_1({\nu}) \rangle}} {\mathrel\Vert}{{}\text{``} {\nu}\in {\utilde{A}} {}\text{''}}$. Then by removing a measure set from $T_{={\rho}}$ we can have either $$\begin{aligned}
& \forall {\nu}\in T_{={\rho}}\
s {\mathop{{}^\frown}}r({\nu}) {\mathop{{}^\frown}}{\ensuremath{\langle f^*_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}, T_1({\nu}) \rangle}} {\mathrel\Vdash}{{}\text{``} {\nu}\in {\utilde{A}} {}\text{''}}
\intertext{or}
& \forall {\nu}\in T_{={\rho}}\
s {\mathop{{}^\frown}}r({\nu}) {\mathop{{}^\frown}}{\ensuremath{\langle f^*_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}, T_1({\nu}) \rangle}} {\mathrel\Vdash}{{}\text{``} {\nu}\notin {\utilde{A}} {}\text{''}}.\end{aligned}$$ Let $g = f^* {\cup}f^{j(r)(\operatorname{mc}_{\rho}(f^*))}$. Set $T^* = {\pi}^{-1}_{g,f^*}T^{j_{E_{\rho}(r)}(\operatorname{mc}_{\rho}(f^*))} {\cap}{\pi}^{-1}_{g,f^*}\operatorname*{\triangle}_{{\nu}\in T_{={\rho}}} T_1({\nu})$. Setting $p^* = s {\mathop{{}^\frown}}{\ensuremath{\langle g,T^* \rangle}}$ we get for each ${\nu}\in T^*_{={\rho}}$, $p^*_{{\ensuremath{\langle {\nu}\rangle}}} \leq^*
s {\mathop{{}^\frown}}r({\nu}) {\mathop{{}^\frown}}{\ensuremath{\langle f^*_{{\ensuremath{\langle {\nu}\rangle}}\uparrow}, T_1({\nu}) \rangle}}$. Thus by removing a measure zero set from $T^*_{={\rho}}$ we get either $$\begin{aligned}
& \forall {\nu}\in T^*_{={\rho}}\
p^*_{{\ensuremath{\langle {\nu}\rangle}}} {\mathrel\Vdash}{{}\text{``} {\nu}\in {\utilde{A}} {}\text{''}}
\intertext{or}
& \forall {\nu}\in T^*_{={\rho}}\
p^*_{{\ensuremath{\langle {\nu}\rangle}}} {\mathrel\Vdash}{{}\text{``} {\nu}\notin {\utilde{A}} {}\text{''}}.\end{aligned}$$ Going to the ultrapower we get $j_{E_{\rho}}(p^*)_{{\ensuremath{\langle \operatorname{mc}_{\rho}(g) \rangle}}} {\mathrel\Vert}{{}\text{``} \operatorname{mc}_{\rho}(g) \in j_{E_{\rho}}({\utilde{A}}) {}\text{''}}$.
Assume ${\rho}< \operatorname{o}(\Vec{E})$ is a repeat point of $\Vec{E}$. Then $ {\mathrel\Vdash}{{}\text{``} {\kappa}\text{ is measurable} {}\text{''}}$.
If $G \subseteq {\mathbb{P}}$ is generic then it is a simple matter to check that $$\begin{aligned}
U = {\ensuremath{\{ {\utilde{A}}[G] \mid p \in G,\
j_{E_{\rho}}(p)_{{\ensuremath{\langle \operatorname{mc}_{\rho}(p_{n^p-1}) \rangle}}} {\mathrel\Vdash}{{}\text{``} \operatorname{mc}_{{\rho}}(p_{n^p-1})\in j_{E_{\rho}}({\utilde{A}}) {}\text{''}} \}}}\end{aligned}$$ is the witnessing ultrafilter.
\[BecomesMeasurable\] If $\operatorname{o}(\Vec{E}) = {\lambda}^{++}$ then ${\mathrel\Vdash}{{}\text{``} {\kappa}\text{ is measurable} {}\text{''}}$.
By the previous corollary it is enough to exhibit ${\rho}< {\lambda}^{++}$ which is a repeat point of $\Vec{E}$. Fix $d \in [{\epsilon}]^{<{\lambda}}$ and consider the sequence ${\ensuremath{{\ensuremath{\langle {\bigcap}_{{\xi}' < {\xi}} E_{{\xi}'}(d) \mid {\xi}< {\lambda}^{++} \rangle}}}}$. This is a $\subseteq$-decreasing sequence of filters on $\operatorname{OB}(d)$. Since there are ${\lambda}^+$ filters on $\operatorname{OB}(d)$ there is ${\rho}_d < {\lambda}^{++}$ such that ${\bigcap}_{{\xi}< {\rho}_d} E_{{\xi}}(d) =
{\bigcap}_{{\xi}< {\lambda}^{++}} E_{{\xi}}(d)$. Set ${\rho}= \sup {\ensuremath{\{ {\rho}_d \mid d \in [{\epsilon}]^{<{\lambda}} \}}}$. Then ${\rho}$ is a repeat point of ${\Vec{E}}$.
[10]{}
Matthew Foreman and W. Hugh Woodin. . , 133(1):1–35, 1991. <http://www.jstor.org/stable/2944324>.
Moti Gitik. Prikry type forcings. In [Matthew Foreman and Akihiro Kanamoril]{}, editor, [*[Handbook of Set Theory]{}*]{}, pages 1351–1447. Springer, 2010. .
Moti Gitik and Menachem Magidor. . In Haim Judah, Winfried Just, and W. Hugh Woodin, editors, [*Set theory of the continuum*]{}, volume 26 of [*Mathematical Sciences Research Institute publications*]{}, pages 243–279. Springer, 1992.
Moti Gitik and Carmi Merimovich. . Preprint.
John Krueger. . , 46(3-4):223–252, 2007. .
Menachem Magidor. . , 28(1):1–31. .
Menachem Magidor. . , 99(1):61–71, 1978.
Carmi Merimovich. Extender-based [R]{}adin forcing. , 355:1729–1772, 2003. .
Carmi Merimovich. Extender based [M]{}agidor-[R]{}adin forcing. , 182(1):439–480, April 2011. .
Carmi Merimovich. . , 50(5-6):592—601, June 2011. .
Karel Prikry. . PhD thesis, [Department of Mathematics, UC Berkeley]{}, 1968.
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[^1]: Most of this work was done somewhat after [@Merimovich2011c] was completed. Lacking an application, which admittedly was lacking also in [@Merimovich2011c], it was mainly distributed among interested parties. Gitik, observing the utility of this forcing to some HOD constructions (see [@GitikMerimovichPreprint]), has urged us to bring the work into publishable state.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We rewrite the Klein-Gordon (KG) equation in an arbitrary space-time transforming it into a generalized Schrödinger equation. Then, we take the weak field limit and show that this equation has certain differences with the traditional Schrödinger equation plus a gravitational field. Thus, this procedure shows that the Schrödinger equation derived in a covariant manner is different from the traditional one. We study the KG equation in a Newtonian space-time to describe the behavior of a scalar particle in an inertial system. This particle is immersed in a gravitational field with the new Schrödinger equation. We study particular physical systems given examples for which we find their energy levels, effective potential and the wave function of the systems. The results contain the gravitational effects due to the curvature of the space-time. Finally, we discuss the possibility of the experimental verification of these effects in a laboratory using non-inertial reference frames.'
author:
- 'Omar Gallegos[^1]'
- 'Tonatiuh Matos[^2]'
bibliography:
- 'ref.bib'
date: February 2019
title: Weak gravitational quantum effects in boson particles
---
Introduction
============
In the last century, General Relativity (GR) and Quantum Mechanics (QM), the two pillars of modern physics, have been developed and verified independently with great precision, while quantum physics describes successfully the behavior of tiny particles, GR is very accurate for forces at cosmic scales. However, in some cases, the two theories produce incompatible results which give rise to different definitions for the same concept. We think the inconsistency between GR and QM owes to the concept of interaction between particles. In QM two particles interact when they exchange a virtual particle, while in GR the interaction is just due to the geometry of space-time. In this work we adopt the geometrical GR concept, instead of the exchange of virtual particles to test if a boson gas follows the Klein- Gordon equation in a curved space-time, and to test GR in quantum regime. These results could be useful either for laboratory particles as well as for the study of the quantum character of boson particles proposed as dark matter (see for example [@Matos:1998vk], [@Hui:2016ltb]).
An important problem in theoretical and fundamental physics is to have a Theory of Everything where the principal theories in physics, GR and QM, can be compatible. In the last decades, some theories have been proposed [@Loop],[@String] to this unification. Nevertheless, the experimental verification of these candidates and their theoretical problems are so far too complex. Several physicists have tried to test if gravity has a quantum nature with different proposals for experiments and observations[@Casimir][@entaglement], but some proposals are not feasible with the technology available to this day. We do not pretend to propose a new Theory of Everything, instead we want to give a new different way to measure the gravitational effect due to the curvature of space-time on quantum systems, specially on scalar particles. These results do not definitely prove if gravity is or not a quantum interaction but they present a closer path to answer this fundamental question.
Einstein’s Equivalence Principle (EEP), one of the foremost ideas for developing General Relativity, where the concept of inertia takes a different role from the one used in Newtonian mechanics. EEP states that experiments in a sufficiently small freely falling laboratory, over a sufficiently short time, give results that are indistinguishable from those of the same experiments in an inertial frame in the presence of a gravitational field. Hence, we use this principle to develop this work, studying some different examples of QM on an inertial frame immersed in a gravitational field. To measure these effects in a laboratory, we will place a quantum system on a non-inertial frame, hoping to get the same results both in the theoretical part and in the experimental one. Furthermore, we expect to obtain results on quantization as in QM. So, one of the objectives of this paper is to test the EEP in quantum scales, thus a proof of GR in a quantum scale.\
We would expect that there exists a regime where the quantum aspects of gravity are detectable. With a simple dimensional analysis we may have an idea where the gravitational effects on a quantum system are important, i.e. the physical scale where these effects can be observed. It is possible to compare two well-known quantities in QM and GR, which are the Bohr radius $r_{bohr}$ and the Schwarzschild radius $r_{sch}$. Both radii are defined as $r_{bohr}=\hbar/(\mu c \alpha)$ and $r_{sch}=2GM/c^2$ where $c$, $G$, $\alpha$ are the speed of light, the gravitational constant and the fine-structure constant respectively. Also $\hbar=h/2\pi$ is the Planck’s constant, while $M$ is the mass that produces the gravitational field, $\mu$ is the reduced mass of a couple of particles. In a case where the comparison between masses is such that $M>>m$, it leads us to the approximation $\mu\approx m$, where $m$ is the mass of a boson particle, which is inside a gravitational field generated by a source of mass $M$. Thereby, the following expression is obtained when both radii $r_{bohr}\sim r_{sch}$ are comparable $$\label{dimensional analysis}
M\sim\dfrac{\hbar c}{2G\alpha m}=\dfrac{m_{pl}^2}{2\alpha m}\approx\dfrac{3.25\times10^{-14}}{m}\text{kg}^2,$$ where $m_{pl}=\sqrt{\hbar c/G}$ is the Planck’s mass. This result was inspired by the micro black holes described in[@microBH] and it enables us to find the limit for measuring quantum gravitational effects. If we assume $m=m_e$ is the mass of an electron, we obtain that $M\sim 3.57\times10^{16}$kg is the mass for which an electron should feel a gravitational effect. On the other hand, the mass of Earth is $\sim 5\times 10^{24}$kg, namely, we should measure in Earth the quantum gravitational effects on an electron. If we consider now the mass of an ultralight scalar particle $\sim 10^{-22}eV/c^2$, the mass proposed in models of Scalar Field Dark Matter (SFDM)[@Matos:1998vk], [@Hui:2016ltb], we calculate from Eq.(\[dimensional analysis\]) that $M\sim 10^{12}M_\odot$ (the galaxies have a mass of this order). Thus, we can say, if dark matter is a scalar field with mass $m\sim 10^{-22}eV/c^2$ [@Matos:1998vk], [@Hui:2016ltb], the galaxy should present a gravitational quantum behavior. In this work, we study well-known examples of QM for scalar particles with corrections due to the curvature of space-time.
Generalized Schrödinger Equation
================================
In order to analyze the gravitational effect due to curvature of space-time in a quantum system, we focus on a scalar field following reference [@Matos:2016uxo], in which the KG covariant equation with an external potential is described $$\label{KG covariant}
\Box\Phi-\dfrac{d\mathscr{V}}{d\Phi^*}=0,$$ here $\Box=g^{\mu\nu}\nabla_\mu\nabla_\nu=\dfrac{1}{\sqrt{-g}}\partial_\mu(\sqrt{-g}g^{\mu\nu}\partial_\nu)$ is the D’Alembertian operator associated to an arbitrary metric $g_{\mu\nu}$, $\Phi=\Phi(t,\vec{x})$ is the scalar field, $\Phi^*=\Phi^*(t,\vec{x})$ is its conjugated complex and the scalar field potential is $\mathscr{V}=\mathscr{V}(\Phi,\Phi^*)$ endowed with an external potential $V$ just as it is shown in the following equation $$\label{potential}
\mathscr{V}=\left(\dfrac{m^2c^2}{\hbar^2}+\dfrac{\lambda n_0}{2}+\dfrac{2m}{\hbar^2}V\right)\Phi\Phi^*,$$ where $m$ is the mass of the scalar field and $n_0=n_0(t,\vec{x})$ is defined as the scalar field density, namely $n_0=|\Phi|^2$. The space-time is expanded in a 3+1 slices, such that the coordinate $t$ here is the parameter of evolution, the 3+1 metric then reads $$\label{metric}
\mathrm{d}s^2 = - N^2 c^2
\mathrm{d}t^2 + \gamma_{ij} \left(\mathrm{d}x^i + N^i c \, \mathrm{d}t \right) \left(
\mathrm{d}x^j + N^j c \, \mathrm{d}t \right) ,$$ $N$ represents the lapse function which measures the proper time of the observers traveling along the world line, $N^i$ is the shift vector that measures the displacement of the observers between the spatial slices and $\gamma_{ij}$ is the spatial metric.
The KG equation is a covariant equation whose origin lies in quantum field theory. We can obtain a Schrödinger equation, starting from equation (\[KG covariant\]) with the potential in Eq.(\[potential\]), using the metric (\[metric\]) and transforming the scalar field by $\Phi(t,\vec{x})=\Psi(t,\vec{x}) e^{-i\omega t}$. Following [@Matos:2016uxo] we obtain $$\begin{aligned}
i \nabla^0 \Psi - \frac{1}{2\omega} \Box_G \Psi
+ \frac{1}{2\omega} \left( \widetilde{m}^2 + \lambda n_0+V \right) \Psi &&
\nonumber\\
+\quad \frac{1}{2} \left( - \frac{\omega}{N^2}
+ i \, \Box_G \, t \right) \Psi &=& 0 ,
\label{eq:GP}\end{aligned}$$ being $\lambda$ the coupling parameter. Here $\omega=mc^2/\hbar$ is the characteristic frequency of the scalar field. The D’Alembertian operator $\Box_G=\nabla^\mu\nabla_\mu$ is associated to the metric (\[metric\]), and $\widetilde{m}$ stands for the mass in units where $c=\hbar=1$, i.e, $\widetilde{m}=m^2c^2/\hbar^
2$ (see [@Matos:2016uxo]). We can interpret the function $\Psi$ as a wave function analogously as in QM.
Beginning with Eq.(\[KG covariant\]) but now using the $\Psi$ variable, we obtain Eq.(\[eq:GP\]) using the 3+1 metric (\[metric\]) and the potential (\[potential\]). Equation (\[eq:GP\]) can be interpreted as the covariant generalization of the Schrödinger equation for any curved space-time, (see[@Chavanis:2016shp] and [@Matos:2016uxo]) where this equation in the weak field limit reduces to the standard Schrödinger one. Hereafter, we use the Newtonian geometry given by the Newtonian metric $$\label{Newtonian metric}
ds^2=-\left(1-\dfrac{2GM}{rc^2}\right)c^2dt^2+\left(1+\dfrac{2GM}{rc^2}\right)dx_idx^i,$$ Newtonian gravity is known to be valid when the gravitational fields are weak, that is $GM/rc^2<<1$. Using the Newtonian metric in Eq.(\[eq:GP\]) yields $$\label{KG-N equation}
\dfrac{\hbar^2}{2m}\Box_N\Psi+\left(1+\dfrac{2U}{c^2}\right)^{-1}\left(i\hbar\dfrac{\partial \Psi}{\partial t}+\dfrac{mc^2}{2}\Psi\right)-\dfrac{\hbar^2}{2m}V\Psi=0,$$ where $U=-GM/r$ is the gravitational potential, hither the D’Alembertian operator $\Box_N$ is related to the metric given by Eq.(\[Newtonian metric\]). Furthermore, we do not consider self-interaction, here the contribution of the term $\lambda$ is negligible. We take into account that the evolution of this function is small, thus the contribution of $\partial_0^2\Phi$ can be taken as null because the Newtonian potential fulfills $U/c^2<<1$. Additionally, one can ignore terms of equal or greater order than $\left(\dfrac{2U}{c^2}\right)^2$. With all this in mind, the linearization of the generalized Schrödinger equation (\[KG-N equation\]) gives rise to the following equation $$\label{KG-N motion equation}
-\dfrac{\hbar^2}{2m}\nabla^2\Psi+V\Psi+mU\Psi +\left(\dfrac{2U}{c^2}V-\dfrac{2\hbar^2U}{mc^2}\nabla^2\right)\Psi=i\hbar\dfrac{\partial\Psi}{\partial t}.$$ Henceforth, we are going to work with the Laplacian operator $\nabla^2=\nabla\cdot\nabla$ in flat space, specially for the examples in the following chapters where we use spherical symmetry. In other words, equation (8) comes from the KG equation in a curved space-time immersed in a weak gravitational field, which is described by a Newtonian geometry.
Focusing on the comparison of the traditional Schrödinger equation plus a gravitational potential, we note that now there are two additional terms inside the parenthesis in Eq.(\[KG-N motion equation\]). These terms were not simply added in the Schrödinger equation, they appear from the covariant meaning of the equation. From a qualitative point of view, this means that the QM version of the interactions between particles fulfills the Schrödinger equation, while the GR version of these interactions fulfills the generalized Schrödinger Eq.(\[KG-N motion equation\]). This difference is the main idea of this work. In what follows, we calculate the quantum quantities for different external potentials of well known problems in QM and compare them with those corresponding to GR. For the subsequent chapters our equation of motion will be given by Eq.(\[KG-N motion equation\]).
With the goal of testing EEP in a quantum region, we are going to show in the subsequent chapters distinct calculations about well known examples in QM. In them, the correction terms due to the curvature of space-time which were introduced in the KG covariant equation derived in Eq.(\[KG-N motion equation\]) are going to appear. To compare the cases with and without a gravitational source, the extra terms in Eq.(\[KG-N motion equation\]) will be taken as perturbations in the Schrödinger equation, for this reason we decide to opt for the QM formalism, using perturbation theory to give a clearer comparison between the cases with and without gravitational field. The results of the gravitational field contribution have been well studied in the standard literature.
Free Particle
=============
Firstly, we consider a free scalar particle with mass $m$ under the influence of a gravitational field generated by a source of mass $M$. Using an external potential $V_{free}=0$ in the equation of motion (\[KG-N motion equation\]), we have that $$-\dfrac{\hbar^2}{2m}\nabla^2\Psi+mU\Psi-\dfrac{2\hbar^2U}{mc^2}\nabla^2\Psi=i\hbar\dfrac{\partial\Psi}{\partial t},$$ we recall that the gravitational potential is defined by $U=-GM/r$ and that an extra term $-\dfrac{2\hbar^2U}{mc^2}\nabla^2\Psi$ appears in the Schrödinger equation, although the treatment on these equations will be in the QM formalism. Using perturbation theory, from the previous equation we can write a principal Hamiltonian operator $\hat{H}_0$ and a perturbed Hamiltonian operator $\hat{H}_p$ in the following way $$\begin{aligned}
\label{H0 free particle}
&\hat{H}_0\Psi=-\dfrac{\hbar^2}{2m}\nabla^2\Psi+mU\Psi=-\dfrac{\hbar^2}{2m}\nabla^2\Psi-\dfrac{GMm}{r}\Psi,\\
\label{Hp free particle}
&\hat{H}_p\Psi=-\dfrac{2\hbar^2U}{mc^2}\nabla^2\Psi=\dfrac{4U}{c^2}(E_n^{(0)}-mU)\Psi.\end{aligned}$$ In general, to find the corrections to eigenvalues of energy due to a perturbation in QM, the following expression is used $$\label{energy correction}
E_n=E_n^{(0)}+\bra{\psi^{(0)}_n}\hat{H}_p\ket{\psi^{(0)}_n}+\sum_{m\neq n}\dfrac{|\bra{\psi^{(0)}_m}\hat{H}_p\ket{\psi^{(0)}_n}|^2}{E_n^{(0)}-E_m^{(0)}}+...,$$ we can associate the Eq.(\[energy correction\]) as a series of contributions of higher order correction of energies $E_n=E_n^{(0)}+E_n^{(1)}+E_n^{(2)}+...$, where $E_n^{(j)}$ is the $j$th-order correction of the eigenvalues in the $n$th-state of energy, hence we can say that the first-order correction is given by $E_n^{(1)}=\bra{\psi^{(0)}_n}\hat{H}_p\ket{\psi^{(0)}_n}$. The zero-order correction for eigenvalues of energy $E_n^{(0)}$ and the eigenfunctions are the well-known exact solutions of QM (without gravitational field). Equation (\[energy correction\]) gives the correction for a non-degenerated quantum system, though first-order correction is valid for both cases (degenerate and non degenerate systems). We do not present an expression for the second order correction in a degenerate system because the corrections we make are at most of first-order. Nevertheless, such a case can be found in standard QM textbooks.\
Going back to our example of a free scalar particle inside a gravitational field, the first order correction of energy using the perturbed Hamiltonian from the hydrogen atom-like problem in QM is $$\begin{aligned}
E_n&\approx E^{(0)}_n\left(1-\dfrac{4GM}{\rho_0c^2n^2}+...\right),
\\&=E^{(0)}_n\left(1+\dfrac{8E^{(0)}_n}{mc^2}+...\right)\notag,\end{aligned}$$ being $E^{(0)}_n$ the well known energy for the hydrogen atom, that is $$\label{energy atom gravitation}
E^{(0)}_n=-\dfrac{(GM)^2m^3}{2\hbar^2n^2}=-\dfrac{GMm}{2\rho_0n^2},$$ where $\rho_0=\hbar^2/(GMm^2)$ is the Bohr radius for the gravitational case. Thus, we expect that the gravitational field modifies the energy of a free particle in a quadratic level, suppressed by the rest energy of the boson particle. This result is surprising for an ultra-light dark matter boson [@Matos:1998vk][@Matos:2000ss], because this model postulates a boson particle with a mass of the order of $10^{-22}$eV$/c^2$. With such a mass the self-gravitation effects of the boson field are important in a system of particles, a feature that a heavy boson system does not present.
Isotropic Harmonic Oscillator
=============================
Another important example, not only in QM but in physics as a whole, is the study of the harmonic oscillator potential. With this in mind, we analyze the case of potential $V_{osc}$ for an isotropic harmonic oscillator. This analysis can be done in two ways. The first one is to start from Eq.(\[KG-N motion equation\]) with the principal potential given by the gravitational type $U=-GM/r$ and taking the isotropic harmonic oscillator $V_{osc}=\dfrac{1}{2}m\omega_0^2r^2$ as perturbation. The other way is to regard the principal Hamiltonian like as that of an isotropic harmonic oscillator, while the perturbation is taken from the gravitational potential. Therefore, equation (\[KG-N motion equation\]) with the potential $V_{osc}$, transforms into $$-\dfrac{\hbar^2}{2m}\nabla^2\Psi+mU\Psi +\dfrac{1}{2}m\omega_0^2r^2\Psi+\left[\dfrac{2U}{c^2}\left(\dfrac{1}{2}m\omega_0^2r^2\right)-\dfrac{2\hbar^2U}{mc^2}\nabla^2\right]\Psi=i\hbar\dfrac{\partial\Psi}{\partial t}.$$
Harmonic Oscillator inside a Gravitational Field
------------------------------------------------
We start with the first cases exposed previously, we suppose that one has an isotropic harmonic oscillator immersed in a gravitational field, that means, $V_{osc}<<U$. To solve this problem we can take the principal Hamiltonian operator $\hat{H}_0$ with the gravitational part and the perturbed Hamiltonian $\hat{H}_p$ in the following way $$\begin{aligned}
\label{principal H IHO in curved space}
\hat{H}_0\Psi&=-\dfrac{\hbar^2}{2m}\nabla^2\Psi-\dfrac{GMm}{r}\Psi,\\
\label{Hp IHO in curved space}
\hat{H}_p&=\dfrac{1}{2}m\omega_0^2r^2-\dfrac{GMm\omega_0^2r}{c^2}+\dfrac{4GME^{(0)}_n}{rc^2}.\end{aligned}$$ We are interested in the wave function with spherical symmetry $\Psi=\Psi(t,r,\theta,\phi)$. Thus, we apply the separation of variables method for $\Psi=R_{nl}(r)Y_{lj}(\theta,\phi)\exp(-\dfrac{iE_n}{\hbar}t)$, where $Y_{lj}$ are the spherical harmonics and $R_{nl}(r)$ is the radial function for the hydrogen atom problem. Here $n,l$, and $j$ play the role of quantum numbers as in QM. The $\hat{H}_0$ Hamiltonian contains the well-known solutions of the eigenvalues from the hydrogen atom problem in terms of the recurrence relation for powers of $r$. It can be shown that the correction for first order of energy from Eq.(\[energy correction\]) is given by $$\begin{aligned}
E_n=&E^{(0)}_n\left[1-\dfrac{8E^{(0)}_n}{mc^2}-\dfrac{\omega_0^2\rho_0^2}{c^2}(3n^2-l(l+1))\right]
\\ &+\dfrac{1}{4}m\omega_0^2\rho_0^2\left[n^2(5n^2+1-3l(l+1))\right],\notag\end{aligned}$$ where $E^{(0)}_n$ is given by Eq.(\[energy atom gravitation\]). Note that if $\omega_0=0$, we return to the case of a free particle on a gravitational field. Observe that as in the previous case, the modifications due to the gravitational field are of second order, but now, an additional term is added which is proportional to the mass and the frequency $\omega_0$. In this case, the second term of the contributions of the perturbations of the gravitational potential due to the harmonic oscillator are negligible for ultra-light masses. Nevertheless, if it is a massive boson then the quadratic contributions are not important. In any case, for any boson mass there is a contribution of the harmonic oscillator that must be taken into account.
Gravitational Field inside Harmonic Oscillator
----------------------------------------------
On the other hand, assuming the gravitational field as a perturbation in an isotropic harmonic oscillator with spherical symmetry, the principal Hamiltonian operator $\hat{H}_0$ is given by $$\begin{aligned}
\hat{H}_0\Psi&=-\dfrac{\hbar^2}{2m}\nabla^2\Psi+\dfrac{1}{2}m\omega_0^2\left[\left(r-\dfrac{GM}{c^2}\right)^2-\left(\dfrac{GM}{c^2}\right)^2\right]\Psi,\notag
\\&\approx-\dfrac{\hbar^2}{2m}\nabla^2\Psi+\dfrac{1}{2}m\omega_0^2r^2-\Psi-\dfrac{1}{2}m\omega_0^2\left(\dfrac{GM}{c^2}\right)^2\Psi.\end{aligned}$$ Since the quantum system is far from the source, it is possible to do the previous approximation. For the perturbed Hamiltonian $\hat{H}_p$ we have $$\hat{H}_p=-\dfrac{GMm}{r}+\dfrac{4GME^{(0)}_n}{rc^2}.$$ The principal Hamiltonian is that of an isotropic harmonic oscillator with spherical symmetry whose eigenfunctions are well known from QM. These solutions are obtained after applying the separation of variables method, just as in the previous case $\Psi_{nklj}(t,r,\theta,\phi)=R_{kl}(r)Y_{lj}(\theta,\phi)\exp(-iE_nt)$. Here $R_{nl}(r)$ is the radial function for an isotropic harmonic oscillator $R_{kl}(r)=r^le^{-\gamma r^2}L^{(l+1/2)}_k(2\gamma r^2)$, with $\gamma=\dfrac{m\omega}{2\hbar}$ and $L^{p}_q(x)$ being the generalized associated Laguerre polynomials. The set $(n,k,l,j)$ become the quantum numbers for the isotropic harmonic oscillator with spherical symmetry from QM.Thus, the first order correction for the energy from Eq.(\[energy correction\]) is given by $$E_n=\hbar\omega_0\left(n+\dfrac{3}{2}\right)-\dfrac{1}{2}m\omega_0^2\left(\dfrac{GM}{c^2}\right)^2-GMm\left[1-\dfrac{4E^{(0)}_n}{mc^2}\right]\left\langle \dfrac{1}{r}\right\rangle,$$ where $E^{(0)}_n=\hbar\omega_0\left(n+\dfrac{3}{2}\right)$ is a well known result from QM. Also note that there is a degeneration of these numbers since $n=2k+l$. A more general solution of $\left\langle \dfrac{1}{r}\right\rangle$ was found in [@Amrmstrong], that for the case we are discussing reduces to the expression $$\begin{aligned}
\label{iho correction}
\left\langle \dfrac{1}{r}\right\rangle=&\sqrt{\dfrac{m\omega_0}{\hbar}}\Gamma(l+1)[\dfrac{1}{2}(n-l-1)]!\times
\\&\sum_t\dfrac{(-1)^t}{[1/2(n-l-1)-t]!\Gamma(1/2(2l+1)+t+1)}\binom{-1/2}{t}^2\notag,\end{aligned}$$ where $\Gamma(z)$ is the Gamma function $\Gamma(z)=\int^\infty_0t^{z-1}\exp(-t)dt$. When only the first term is taken from the sum of Eq.(\[iho correction\]), we obtain the energy value $$\begin{aligned}
\label{correction oscillator}
E_n=&\hbar\omega\left(n+\dfrac{3}{2}\right)-\dfrac{1}{2}m\omega_0^2\left(\dfrac{GM}{c^2}\right)^2\notag\\
&-GMm\sqrt{\dfrac{m\omega_0}{\hbar}}\left[1-\dfrac{4E^{(0)}_n}{mc^2}\right]\Gamma(l+1)[\dfrac{1}{2}(n-l-1)]!.\end{aligned}$$
Note that if $G=0$, we return to the solution for an isotropic harmonic oscillator with spherical symmetry without gravitational field. It is peculiar that the Schwarzschild radius $r_{sch}$ is obtained in a very natural way.
The two cases we have analyzed in this section are extreme, and correspond to those where the gravitational field is much more intense than the harmonic oscillator and vice-versa. The case where the two potentials are comparable is much more complex and we leave it for a future work.
Infinite Spherical Well {#Infinite Spherical Potential G}
=======================
In this section we continue discussing well known examples from QM with the novelty of the presence of a gravitational field. We now deal with the problem of an infinite spherical well barrier $V_{inf}(r)$ that has two regions, this problem is commonly used in experimental verification or applications of QM. The barrier potential is given by $$V_{inf}(r)= \left\{
\begin{array}{ll}
0\ \ \ \text{if} \ \ \ 0<r<a, \\
\infty \ \ \ \text{otherwise}. \\
\end{array}
\right.$$ We are only interested in studying the region where $0<r<a$, since outside of this region the wave function is zero, thus the probability to find a particle in the $r>a$ is null. The equation of motion (\[KG-N motion equation\]) in the region of interest is $$-\dfrac{\hbar^2}{2m}\nabla^2\Psi+mU\Psi -\dfrac{2\hbar^2U}{mc^2}\nabla^2\Psi=i\hbar\dfrac{\partial\Psi}{\partial t}.$$ We can choose a principal Hamiltonian operator $\hat{H}_0$ from the motion equation, where $$\hat{H}_0\Psi=-\dfrac{\hbar^2}{2m}\nabla^2\Psi=E^{(0)}\Psi.$$ The zero-order correction of the energy is then $E^{(0)}_{ln}=\dfrac{\hbar^2}{2m}\dfrac{q_{ln}^2}{a^2}$. We can define the perturbed Hamiltonian $\hat{H}_p$ from the KG equation $$\begin{aligned}
\hat{H}_p=-\dfrac{GMm}{r}\left(1+\dfrac{4E^{(0)}}{mc^2}\right).\end{aligned}$$ The first-order correction of the energy $E^{(1)}$ can be calculated using the zero-order correction of the eigenfunction $\Psi_{nlk}^{(0)}(r,\theta,\phi)=A_{ln}j_l\left(\dfrac{q_{ln}}{a}r\right)Y_{lj}(\theta,\phi)$. Here $q_{ln}$ is n-th root of the Bessel spherical functions $j_l(x)$ and $A_{ln}$ is the constant of normalization $$A_{ln}^2=\dfrac{2}{a^3[j_{l+1}(q_{ln})]^2}.$$ Therefore $$\label{first order infinity sph energy}
E_{ln}^{(1)}=-GMm\left(1+\dfrac{4E^{(0)}}{mc^2}\right)\left\langle \dfrac{1}{r}\right\rangle,
%$$ where, from perturbation theory we have $$\left\langle \dfrac{1}{r}\right\rangle=A_{ln}^2\int_0^{a}\abs{j_l\left(\dfrac{q_{ln}}{a}r\right)}^2rdr.$$ Unfortunately, it is not possible to solve the last integral with analytic methods, thus we will integrate it numerically. In general, there exists for each state $n=2l+1$ degeneration, thus for $l=0$, we have that $j_0(x)=\sin(x)/x$, whose n-th root is $q_{0n}=n\pi$ and the normalization’s constant is $A_{01}=\sqrt{2/a}$. Hence $$\left\langle \dfrac{1}{r}\right\rangle\approx\dfrac{2a}{\pi^2}(1.218),$$ Now, for $l=1$ the first excited state is three-fold degenerate, which means we need the first three roots of $j_1(x)=\dfrac{\sin(x)}{x^2}-\dfrac{\cos(x)}{x}$. The roots are $q_{11}\approx 4.49340$, $q_{12}\approx 7.72525$ and $q_{13}\approx 10.90412$. Therefore for $n=1$ $$\left\langle \dfrac{1}{r}\right\rangle\approx A_{11}^2\left( \dfrac{a}{q_{11}}\right)^2 (0.4124),$$ for $n=2$ $$\left\langle \dfrac{1}{r}\right\rangle\approx A_{12}^2\left( \dfrac{a}{q_{12}}\right)^2 (0.4590),$$ and for $n=3$ $$\left\langle \dfrac{1}{r}\right\rangle\approx A_{13}^2\left( \dfrac{a}{q_{13}}\right)^2 (0.4778).$$ We can continue this process for the next excited states, as in the previous cases. If we make $G=0$, we return to solutions for QM without gravitational field.
Spherical Potential Barrier
===========================
The problem of square well potential with certain symmetry (spherical, cartesian or cylindrical) is important for some experiments of quantum systems. In this section, we study a square well barrier with spherical symmetry. Similarly as in the previous sections, we analyze the case of QM for a square well potential with space-time curvature. We use the Newtonian metric from (\[Newtonian metric\]), so the equation of motion (\[KG-N motion equation\]) transforms into $$\label{spherical KG motion eq}
-\dfrac{\hbar^2}{2m}\nabla^2\Psi+mU\Psi +V_B\left(1+\dfrac{2U}{c^2}\right)\Psi-\dfrac{4U}{c^2}\dfrac{\hbar^2}{2m}\nabla^2\Psi=i\hbar\dfrac{\partial\Psi}{\partial t}$$ where again $U=-GM/r$ is the gravitational potential. We take the Laplacian operator $\nabla^2$ in spherical coordinates and the potential $V_B$ as $$V_B(r)= \left\{
\begin{array}{ll}
-U_0\ \ \ \text{if} \ \ \ r<a, \\
0 \ \ \ \ \ \ \ \text{if} \ \ \ r>a,
\end{array}
\right.$$ Thus, from Eq.(\[spherical KG motion eq\]) we can identify a principal Hamiltonian operator $\hat{H}_0$ $$\hat{H}_0\Psi=-\dfrac{\hbar^2}{2m}\nabla^2\Psi+V_B\Psi=E^{(0)}\Psi.
\label{perturbed sph barrier}$$ The perturbed Hamiltonian $\hat{H}_p$ can be defined from Eq.(\[spherical KG motion eq\]) $$\hat{H}_p=-\dfrac{GMm}{r}\left(1+\dfrac{4E^{(0)}}{mc^2}-\dfrac{2V_B}{mc^2}\right)$$ In general, we can find the fist-order correction of energy from Eq.(\[energy correction\]) with the perturbed Hamiltonian operator in Eq.(\[perturbed sph barrier\]). This yields $$E^{(1)}=-GMm\left(1+\dfrac{4E^{(0)}}{mc^2}-\dfrac{2V_B}{mc^2}\right)\left\langle \dfrac{1}{r}\right\rangle.$$ The case where $E<0$ shows the quantum nature of the system due to the fact that the energy spectrum is discrete. Potential $V_B$ defines naturally two regions, $Region\ I$ ($r<a$) and $Region\ II$ ($r>a$). Both regions without gravitational field are well-known. For $Region\ I$ the first-order correction of energy $E^{(1)}_{I}$ is given by $$E^{(1)}_I=-GMm\left(1+\dfrac{4E_{I}^{(0)}}{mc^2}+\dfrac{2U_0}{mc^2}\right)\left\langle \dfrac{1}{r}\right\rangle_I,$$ where the zero-order correction of the eigenvalue is $E_{I}^{(0)}=\dfrac{\hbar^2}{2m}(\gamma_{ln})^2$ and the zero-order correction eigenfunction reads $\Psi_{lnj}(r,\theta,\phi)=R_{ln}(r)Y_{lp}(\theta,\phi)$. As usual, $Y_{lp}(\theta,\phi)$ are the spherical harmonic functions and $R_{ln}(r)=A_{ln}j_l(\gamma_{ln}r)$ is the radial solution, such that $j_l(x)$ are the spherical Bessel functions. Also $k_1=\gamma_{ln}$ is the n-th solution of the transcendental equation due to the boundary and continuity condition given by $$\label{trascendental eq}
\dfrac{1}{h_l^{(1)}(ik_2r)}\dfrac{dh_l^{(1)}(ik_2r)}{dr}|_{r=a}=\dfrac{1}{j_l(ik_1r)}\dfrac{dj_l(ik_1r)}{dr}|_{r=a}.$$ For $l=0$, it reduces to the transcendental equation $$-k_2=k_1\cot(k_1a).$$ In the limit $|E|<<U_0$, we return to the solution for the first root. When $l=0$ we recover the same result as in the last section, namely $k_1a\approx \pi/2$. In general for Region I, we need to integrate $$\left\langle \dfrac{1}{r}\right\rangle_I=A_{ln}^2\int^a_0\abs{j_l\left(\dfrac{\sigma_{ln}}{a}r\right)}^2rdr,$$ where $\sigma_{ln}=a\gamma_{ln}$. If we define the parameter $\eta=\sigma_{ln}/q_{ln}$ that compares the n-th solution of the transcendental equation with the n-th root, we see that when $\eta=1$ we return to case of Section \[Infinite Spherical Potential G\]. However, the solution of this integral should be calculated using numerical methods.
In $Region\ II$, the zero-order correction is given by $$E_{II}^{(0)}=\dfrac{\hbar^2}{2m}(\gamma^*_{ln})^2=\dfrac{\hbar^2}{2M}\left(\dfrac{\sigma^*_{ln}}{a}\right)^2,$$ where we have introduced the new parameters $\sigma^*_{ln}=a\gamma^*_{ln}$, and $\eta^*=\gamma^*_{ln}/q_{ln}$. The first-order correction is similar to that of $Region\ I$ $$E^{(1)}_{II}=-GMm\left(1+\dfrac{4E_{II}^{(0)}}{mc^2}\right)\left\langle \dfrac{1}{r}\right\rangle_{II},$$ with the eigenfunction $\Psi_{lnj}(r,\theta,\phi)=R_{ln}(r)Y_{lj}(\theta,\phi)$. Nevertheless, the radial function has a new form $R_{ln}(r)=Bh_l^{(1)}(i\gamma_{ln}^*r)=Bj_l(i\gamma_{ln}^*r)+i Bn_l(i\gamma_{ln}^*r)$, where $B$ is a constant of normalization, $h_l^{(1)}(x)$ are the spherical Hankel functions of the first kind, $n_l(x)$ are the spherical Neumann function and $k_2=\gamma_{ln}^*$ is the solution of the transcendental equation (\[trascendental eq\]). Therefore $$\begin{aligned}
\left\langle \dfrac{1}{r}\right\rangle_{II}=&B^2\int_a^\infty\abs{h_l^{(1)}\left(i\gamma^*_{ln}\right)}^2rdr\notag\\
=&B^2\int_a^\infty\left(\abs{j_l(i\gamma_{ln}^*r)}^2- \abs{n_l(i\gamma_{ln}^*r)}^2\right) rdr,\end{aligned}$$ This is a general solution of this perturbation, and if we want to solve this integral, we need to set every parameter for a specific case. As expected, just like in the previous cases, if $G=0$ we return to the solutions without a gravitational field.
Conclusions
===========
Throughout the course of this work we studied the KG equation in a weak gravitational field for different external potentials with the objective of understanding the quantum effects of a boson gas on a gravitational field. Typical examples of QM were analyzed featuring the addition of space-time curvature (or gravitational effects). Starting from the most general equation for bosons in Quantum Field Theory in curved space-times, we found a generalized Schrödinger equation that is simply the KG covariant one. To solve the differential equation, we identified the principal and perturbed Hamiltonian operator in each case, and compared them with the well-known results in QM. Each example was worked out on an inertial frame. This is an important aspect to highlight due to the main idea of this work. We expect to obtain the same results when gravitational effects are measured on a quantum system in the laboratory. This quantum system (with scalar particles) will be on a non-inertial frame, where we predict that the Einstein’s equivalence principle gives us a correspondence between experimental and theoretical results.\
These results let us find the limit, where we could measure quantum gravitational effects, namely $2\alpha\, mM/m_{pl}^2\sim 1$. For this analysis we could be able to apply our results in micro black holes scale[@microBH]. For example, if we consider a mass of a scalar particle $m$ as the mass of the electron, we obtain that the gravitational mass $M$ that affects the electron by quantum gravitational effects is $\sim 10^{16}$kg. In the same way, a $M\sim 10^{13}$kg could affect a particle with a proton mass. If we think on the easiest non-inertial frame, we could think on a rotating system, for which there would not be a real mass $M$, but rather an effective mass given by angular frequency $\Omega$, where $\Omega^2=GM/r^3$. Using a detector to $r=1$ meter, the corresponding gravity can be reached with a rotational wheel spinning with an angular frequency of $\Omega\approx 1543$ rad/s=245 rev/s for $M\sim 10^{16}$kg, and $\Omega\approx 25$ rad/s=3.9 rev/s for $M\sim 10^{13}$kg. However, for a mass $M\sim M_\odot$, we would need an angular frequency of $\Omega\sim 10^{9}$rev/s, to obtain relativistic effects.\
Analyzing the typical experiments on a harmonic oscillator [@oscillator] and on an infinite well barrier [@pozo] for a particle with an electron mass, the correction due to the gravitational effects has, in both cases, a dominant term which is given by $GMm\left\langle \dfrac{1}{r}\right\rangle$. For the case of a harmonic oscillator with frequency $\omega_0\sim G$Hz and using the value for the mass $M\sim 10^{16}$kg. The dominant term of the first-order correction of the energy in this case is $\sim 10^{-5}E_{osc}^{(0)}$ from Eq.(\[correction oscillator\]), while other contributions in this equation are $<10^{-9}E^{(0)}_{osc}$. On the other hand, for a particle with an electron mass confined in an infinite well barrier of width $a\sim 10$nm, it is possible to obtain that $E_{inf}^{(0)}\sim 10$eV. The dominant term in Eq.(\[first order infinity sph energy\]) is $\sim 10^{-5}E_{inf}^{(0)}$ for the first-order correction of energy, and the other term in this correction is $ <10^{-20}E_{inf}^{(0)}$. Note that in either cases, the first-order corrections of energy have the same order if we compare them with their zero-order counterparts. With this analysis it is possible to conclude that we can measure in a laboratory the effects of the quantization of a weak gravitational field directly or using non-inertial systems.
Acknowledgments
===============
This work was partially supported by CONACyT México under grants CB-2011 No. 166212, CB-2014-01 No. 240512, Project No. 269652 and Fronteras Project 281; Xiuhcoatl and Abacus clusters at Cinvestav, IPN; I0101/131/07 C-234/07 of the Instituto Avanzado de Cosmología (IAC) collaboration (http:// www.iac.edu.mx). O.G. acknowledge financial support from CONACyT doctoral fellowship. Works of T.M. are partially supported by Conacyt through the Fondo Sectorial de Investigación para la Educación, grant CB-2014-1, No. 240512
[^1]: ogallegos@fis.cinvestav.mx
[^2]: tmatos@fis.cinvestav.mx
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Autonomous aerial robots provide new possibilities to study the habitats and behaviors of endangered species through the efficient gathering of location information at temporal and spatial granularities not possible with traditional manual survey methods. We present a novel autonomous aerial vehicle system—*TrackerBots*—to track and localize multiple radio-tagged animals. The simplicity of measuring the received signal strength indicator (RSSI) values of very high frequency (VHF) radio-collars commonly used in the field is exploited to realize a low cost and lightweight tracking platform suitable for integration with unmanned aerial vehicles (UAVs). Due to uncertainty and the nonlinearity of the system based on RSSI measurements, our tracking and planning approaches integrate a particle filter for tracking and localizing; a partially observable Markov decision process (POMDP) for dynamic path planning. This approach allows autonomous navigation of a UAV in a direction of maximum information gain to locate multiple mobile animals and reduce exploration time; and, consequently, conserve on-board battery power. We also employ the concept of a search termination criteria to maximize the number of located animals within power constraints of the aerial system. We validated our real-time and online approach through both extensive simulations and field experiments with two mobile VHF radio-tags.'
author:
- |
Hoa Van Nguyen\
School of Computer Science\
The University of Adelaide\
SA 5005, Australia\
`hoavan.nguyen@adelaide.edu.au`\
Michael Chesser\
School of Computer Science\
The University of Adelaide\
SA 5005, Australia\
`michael.chess@adelaide.edu.au`\
Lian Pin Koh\
School of Ecology and Environmental Science\
The University of Adelaide\
SA 5005, Australia\
`lianpin.koh@adelaide.edu.au`\
S. Hamid Rezatofighi\
School of Computer Science\
The University of Adelaide\
SA 5005, Australia\
`hamid.rezatofighi@adelaide.edu.au`\
Damith C. Ranasinghe\
School of Computer Science\
The University of Adelaide\
SA 5005, Australia\
`damith.ranasinghe@adelaide.edu.au`\
bibliography:
- 'references.bib'
title: 'TrackerBots: Autonomous UAV for Real-Time Localization and Tracking of Multiple Radio-Tagged Animals'
---
Introduction
============
Understanding basic questions of ecology such as how animals use their habitat, their movements and activities are necessary for addressing numerous environmental challenges ranging from invasive species to diseases spread by animals and saving endangered species from extinction. Conservation biologists, ecologists as well as natural resource management agencies around the world rely on numerous methods to monitor animals. Today, capturing and collaring concerned species with Very High Frequency (VHF) radio tags and the subsequent use of VHF telemetry or radio tracking is the most important and cost effective tool employed to study the movement of a wide range of animal sizes [@wikelski2007going] in their natural environments [@kays2011tracking; @thomas2012wildlife]. However, the traditional method of radio tracking typically requires researchers to trek long distances in the field, armed with cumbersome VHF radio receivers with hand-held antennas and battery packs to manually home in on radio signals emitted from radio-tagged or collared animals. Consequently, the precious spatial data acquired through radio tracking comes at a significant cost to researchers in terms of manpower, time and funding. The problem is often compounded by other challenges, such as low animal recapture rates, equipment failures, and the inability to track animals that move into inaccessible terrain. Furthermore, many of our most endangered species also happen to be the most difficult to track due to their small size, inconspicuousness, and location in remote habitats.
Automated tracking and location of wildlife with autonomous unmanned aerial vehicles (UAVs) can provide *new* possibilities to better understand ecology and our native wildlife to safeguard biodiversity and manage our natural resources cost-effectively. We present a low-cost approach capable of realization in a lightweight payload for transforming existing commodity drone platforms into autonomous aerial vehicle systems as shown in Fig. \[fig\_UAV\_System\_Overview\] to empower conservation biologists to track and localize multiple radio-tagged animals.
![*TrackerBots*: An overview of the UAV tracking platform with its sensor system.[]{data-label="fig_UAV_System_Overview"}](Figures/UAV_System.png){width="14cm"}
The main contribution of our work is a new autonomous aerial vehicle system for simultaneously tracking and localizing multiple mobile radio-tagged animals using VHF radio-collars, commonly used in the field. In particular:
- Our system is realized in a 260 g payload suitable for a multitude of low-cost, versatile, easy to operate multi-rotor UAVs without a remote pilot license. Our lightweight realization—of less than 2 kg system mass—is achieved through a new sensor design that exploits the simplicity of a software defined radio architecture for capturing received signal strength indicator (RSSI) value from multiple VHF radio tags and a compact, lightweight VHF antenna geometry.
- We formulate a *joint* tracking and path planning problem to realize a real-time and online autonomous system. Due to the noisy, complex and nonlinear characteristics of RSSI data we integrate a sequential Monte Carlo implementation of a Bayesian filter, also known as particle filter (PF), for real-time tracking and localization *jointly* with a partially observable Markov decision process (POMDP) with Réyni divergence between prior and posterior estimates of animal locations for autonomy and dynamic online path planning to minimize flight time while maximizing number of located animals. Further, our formulation considers the trade-off between location accuracy and resource constraints of the UAV, its maneuverability, and power constraints to develop a practical solution.
- We validate our method through extensive simulations and field experiments with mobile VHF radio-tags.
- In order to support researchers in the field and facilitate adoption of new technologies in the field, we provide a complete design description of *TrackerBots*, including a repository of source code to develop our fully autonomous system [^1]
Related Work
============
Since this application is related to locating VHF collared animals, we will focus on progress made towards the autonomous localization and tracking of multiple VHF radio-tagged animals here.
Off-line estimations of a radio beacon from data logged from a UAV have been demonstrated in [@jensen2014monte; @wagle2011spatio]. Pioneering achievements in autonomous wildlife tracking have been made through simulation studies [@Posch2009] and experimentally demonstrated systems [@cliff2015online; @Korner2010; @tokekar2010robotic; @vander2014cautious] in recent years. In particular, the first demonstration of a UAV was presented in [@cliff2015online].
The recent approaches [@cliff2015online; @vander2014cautious] for real-time localization of a static target (assuming stationary wildlife) used wireless signal characteristics captured by a narrowband receiver to estimate location; in particular, the angle-of-arrival (AoA) of a radio beacon was determined using an array of antennas with the information related to a ground-based receiver for location estimations. Although the approach can conveniently manage topological variations in terrain, AoA systems require a large bulky receiver system and multiple antenna elements as well as long observation times; 45 seconds per observation as reported in [@cliff2015online]. Moreover, the antenna systems being mounted on top of the UAV [@cliff2015online] is likely to lead to difficulty in tracking terrestrial animals although being suitable for locating avian species dwelling in trees.
We can see that there are few investigations that have studied the problem of locating radio-collared animals using autonomous robots. Although a system based on angle-of-arrival was recently evaluated to locate a stationary animal, the development of a low-cost and lightweight autonomous system capable of long-range flights and localization of multiple *mobile* radio-collared animals still remains. This is especially significant in the realization of a system that is widely accessible to conservation biologists in the field where a very small UAV—of less than 2 kg—can be flown without a formal pilot license and with fewer restrictions given the exclusion of this category of UAVs from regulatory regimes [@casaac101]
We present an alternative approach exploiting RSSI based range only measurements because of the ability to use a simpler sensing system on board commodity UAVs to realize lower cost and longer flight range UAVs for tracking and localizing multiple animals. Together with a strongly principled approach for joint tracking and planning, our lightweight autonomous aerial robot platform provides a cost-effective method for wildlife conservation and management.
Tracking and Planning Problem Formulation
=========================================
Real-time tracking requires an online estimator and a dynamic planning method. This section presents our tracking and localizing formulation under a principled framework of a Bayesian filter for tracking and POMDP for planning strategy.
Tracking and localizing
-----------------------
For tracking, we use a Bayesian filter. It is an online estimation technique which deals with the problem of inferring knowledge about the unobserved state of a dynamic system—in our problem, wildlife—which changes over time, from a sequence of noisy measurements. Suppose $\mathbf{x} \in \mathcal{X}$ and $\mathbf{z} \in \mathcal{Z}$ are respectively the system (kinematic) state vector in the state space $\mathcal{X}$ and the measurement (observation) vector in the observation space $\mathcal{Z}$. The problem is estimating the state $ \mathbf{x} \in \mathcal{X} $ from the measurement $ \mathbf{z} \in \mathcal{Z} $ or calculating the marginal posterior distribution $\mathit{p(\mathbf{x}_\mathit{k} | \mathbf{z}_{1:k})}$ sequentially through ***prediction*** and ***update*** steps.
$$\begin{aligned}
\mathit{p(\mathbf{x}_\mathit{k} | \mathbf{z}_\mathit{1:k-1})} &= \mathit{\int{p(\mathbf{x}_\mathit{k} | \mathbf{x}_\mathit{k-1}) p(\mathbf{x}_\mathit{k-1} | \mathbf{z}_\mathit{1:k-1}) d\mathbf{x}_\mathit{k-1 } }} \label{eq:chapman} \\
\mathit{p(\mathbf{x}_\mathit{k} | \mathbf{z}_{1:k})} &= \mathit{\dfrac{p(\mathbf{z}_\mathit{k} | \mathbf{x}_\mathit{k})p(\mathbf{x}_\mathit{k} | \mathbf{z}_\mathit{1:k-1})}{\int{p(\mathbf{z}_\mathit{k} |\mathbf{x}_\mathit{k})p(\mathbf{x}_\mathit{k}|\mathbf{z}_\mathit{1:k-1})d\mathbf{x}_k}}} \label{eq:bayesfilter}
\end{aligned}$$
In the case of a nonlinear system or non-Gaussian noise, there is no general closed-form solution for the Bayesian recursion and $\mathit{p(\mathbf{x}_\mathit{k} | \mathbf{z}_{1:k})}$ generally has a non-parametric form. Therefore, in our problem, we use a particle filter implementation as an approximate solution for the Bayesian filtering problem due to our highly nonlinear measurement model.
**Particle Filter (PF):** \[sec\_PF\] A particle filter uses a sampling approach to represent the non-parametric form of the posterior density $\mathit{p(\mathbf{x}_\mathit{k} | \mathbf{z}_{1:k})}$. The samples from the distribution are represented by a set of particles; each particle has a weight assigned to represent the probability of that particle being sampled from the probability density function. Then, these particles representing the nonparametric form of $\mathit{p(\mathbf{x}_\mathit{k} | \mathbf{z}_{1:k})}$ are propagated over time. In the simplest version of the particle filter, known as the bootstrap filter first introduced by Gordon in [@Gordon1993], the samples are directly generated from the transitional dynamic model. Then, to reduce the particle degeneracy, resampling and injection techniques are implemented; a detailed algorithm can be found in [@Ristic2004].
**Measurement model:** The update process of a PF requires the derivation of a likelihood of measurements. In our problem, based on estimating a target’s—VHF radio tag’s—range from the receiver, we require a realistic signal propagation model to obtain the likelihood of receiving a given measurement. We employ two VHF signal propagation models suitable for describing RSSI measurements in non-urban outdoor environments [@patwari2005locating; @wc1974microwave]. Denoting $\mathbf{h(x,u)}$ as the RSSI measurement function between target $\mathbf{x}$ and observer (UAV) state $\mathbf{u}$, we have:
***i)*** ***Log Distance Path Loss Model (LogPath)*:** The received power is the only line of sight power component transmitted from a transmitter subjected to signal attenuation such as through absorption and propagation loss [@patwari2005locating]: $$\label{eq_LogPath}
\begin{aligned}
\mathbf{h(x,u)} = P_r^{d_0}- 10n\log_{10}(d(\mathbf{x},\mathbf{u}_p)/d_0) + G_r(\mathbf{x,u})
\end{aligned}$$ where
- $\mathbf{x} = [p_x^{t}, p_y^{t}, p_z^{t}]^{T}$ is the target’s position; $\mathbf{u}_p = [p_x^{u}, p_y^{u}, p_z^{u}]^T$ is the observer’s (UAV) position in Cartesian coordinates; $\mathbf{u} = [\mathbf{u}_p; \theta^{u}]$ is the UAV’s state which includes its heading angle $\theta^{u}$.
- $d(\mathbf{x},\mathbf{u}_p)$ is the euclidean distance between the target’s position and UAV’s position.
- $G_r(\mathbf{x,u})$ is the UAV receiver antenna gain which depends on its heading, its position, and target’s position (details explained in Sec. \[sec\_emperical\_meas\]).
- $P_r^{d_0}$ is received power at a reference distance $d_0$.
- $n$ is the path-loss exponent that characterizes the signal losses such as absorption and propagation losses and this parameter depends on the environment with typical values ranging from 2 to 4 [@patwari2005locating].
***ii)*** ***Log Distance Path Loss Model with Multi-Path Fading (MultiPath)*:** The received power is composed of both line of sight power component transmitted from a transmitter and the multi-path power component reflected from the ground plane subjected to signal attenuation such as through absorption and propagation loss: [@wc1974microwave p. 81]:
$$\label{eq_MultiPath}
\begin{aligned}
\mathbf{h(x,u)} &= P_r^{d_0}- 10n\log_{10}(d(\mathbf{x},\mathbf{u}_p)/d_0) \\
&+ G_r(\mathbf{x,u}) + 10n\log_{10}(|1+\Gamma(\psi) e^{-j\triangle \varphi}|)
\end{aligned}$$
where, in addition to terms in \[eq\_LogPath\]
- $\psi$ is the angle of incidence between the reflected path and the ground plane.
- $\Gamma(\psi) = [\sin(\psi) - \sqrt{\varepsilon_g - \cos^2(\psi)}]/[\sin(\psi) + \sqrt{\varepsilon_g - \cos^2(\psi)}]$ is the ground reflection coefficient with $\varepsilon_g$ is the relative permittivity of the ground.
- $\triangle \varphi = 2\pi \triangle d / \lambda$ is the phase difference between two waves where $\lambda$ is the wave length and $\triangle d = ((p_x^{t} - p_x^{u})^2 + (p_y^{t} - p_y^{u})^2 + (p_z^{t} + p_z^{u})^2)^{1/2} -d(\mathbf{x}, \mathbf{u}). $
In non-urban environments, received power is usually corrupted by environmental noise, with the assumption that the noise is white, the total received power $\mathbf{z} = P_r(\mathbf{x,u})$ \[dBm\] is: $$\begin{aligned}
\label{eq_pathloss}
\mathbf{z} &= \mathbf{h(x,u)} + \eta_{P}
\end{aligned}$$ where $\eta_{P} \sim \mathcal{N}(0,\sigma_P^2)$ is Gaussian white noise with covariance $\sigma_P^2$. Notably, even if RSSI noise is not completely characterized by a white noise model, we found it practical to characterize the received noise with a white Gaussian model as shown in Fig. \[fig\_Meas\_Model\].
We use data captured in experiments using our sensor system to validate the physical sensor characteristics $G_r(\mathbf{x,u})$ (see Sec. \[sec\_Antenna\_Gain\]) and $n$ defined by environmental characteristics, as well as estimate the propagation model reference power parameter $ P_r^{d0}$ and noise $\sigma_P$ (see Sec. \[sec\_emperical\_meas\]).
***Measurement likelihood*:** Based on with Gaussian noise $\eta_P$, the likelihood of measurement $\mathbf{z}_k$, given target and sensor position are $\mathbf{x}_{k}$ and $\mathbf{u}_{k}$, respectively, at time $k$ is $$\begin{aligned}
p(\mathbf{z}_{k}|\mathbf{x}_k, \mathbf{u}_{k}) \sim \mathcal{N}(\mathbf{z}_{k};\mathbf{h}(\mathbf{x}_k,\mathbf{u}_k) , \sigma^2_P )
\end{aligned}$$ where $\mathcal{N}(\cdot;\mu,\textcolor{black}{\sigma^2} )$ is a normal distribution with mean $\mu$ and .
Path Planning
-------------
The UAV planning problem is similar to the problem of an agent computing optimal actions under partially observable Markov decision process (POMDP) to maximize its reward. [@KAELBLING199899] have shown that a POMDP framework implements an efficient and *optimal* approach based on previous actions and observations to determine the true world states. POMDP in conjunction with a particle filter provides a principled approach for evaluating planning decision to realize an autonomous system for tracking.
In general, a POMDP is described by the 6-tuple $ (\mathcal{S,A,T,R,O,Z}) $ where $\mathcal{S} $ is set of both UAV and target states ($\mathbf{s} = \{\mathbf{x,u}\} \in S$), $\mathcal{A} $ is , $ \mathcal{T} $ is state-transition function $\mathcal{T}(\mathbf{s,a,s'}) = p(\mathbf{s'|s,a})$ , $\mathcal{R}(\mathbf{a})$ is reward function, $\mathcal{O} $ is set of observations and $\mathcal{Z} $ is observation likelihood with $\mathbf{s,s'} \in \mathcal{S}$ is current state and next state respectively, $\mathbf{a} \in \mathcal{A} $ is taken action and $\mathbf{o} \in \mathcal{O} $ is the observation—i.e measurement. The goal of a POMDP is to find an optimal policy to maximize the total expected reward where $H$ is look-ahead horizon steps, $\gamma $ is the discount factor which serves as the value difference between the current reward versus the future reward; and $\mathbb{E}[\cdot]$ is the expectation operator [@Hsu2008].
The reward function can be calculated using different methods such as task-driven or information-driven strategies. When uncertainty is high, the information gain approach is preferable to reduce a target’s location uncertainty [@beard2017void]; hence, we used this method to calculate our reward function. There are several approaches to evaluate information gain in robotic path planning such as Shannon entropy [@cliff2015online], Kullback-Leibler (KL) divergence or Rényi divergence [@Hero2008]. We adapted the approach in [@Ristic2013; @Ristic2010] to implement Rényi divergence as our reward function since it fits naturally with our Monte-Carlo sampling method. Here, Rényi divergence measures the information gain between prior and posterior densities [@beard2017void; @Ristic2010]:
$$\label{ReyniDivergence}
\begin{aligned}
\mathcal{R}^{(i)}_{k+H}(\mathbf{a}_k) &= \dfrac{1}{\alpha-1} \log \int
\begin{bmatrix}
p(\mathbf{x}_{k+H}|\mathbf{z}_{1:k})
\end{bmatrix}^{\alpha} \\
& \times
\begin{bmatrix}
p(\mathbf{x}_{k+H}|\mathbf{z}_{1:k},\mathbf{z}^{(i)}_{k+1:k+H}(\mathbf{a}_k))
\end{bmatrix}^{1-\alpha}
d\mathbf{x}_{k+H},
\end{aligned}$$
where $\alpha \geq 0$ is a scale factor to decide the effect from the tails of two distributions. The prior density $p(\mathbf{x}_{k+H}|\mathbf{z}_{1:k})$ is calculated by propagating current posterior particles sampled from $p(\mathbf{x}_{k}|\mathbf{z}_{1:k})$ to time $k+H$ using the prediction step . The posterior density $p(\mathbf{x}_{k+H}|\mathbf{z}_{1:k},\mathbf{z}^{(m)}_{k+1:k+H}(\mathbf{a}_k))$ where $\mathbf{z}^{(m)}_{k+1:k+H}(\mathbf{a}_k)$ is the *future* measurement set that will be observed if action $\mathbf{a}_k \in \mathcal{A}_{k}$ is taken; this is calculated by applying both prediction and update steps up to time $k+H$. However, using Bayes update procedure is computationally expensive and prohibitive in a real-time setting. Instead, we implement a computationally efficient approach using a black box simulator proposed in [@silver2010monte] along with the Monte Carlo sampling approach. Hence, the problem transforms to find an optimal action $\mathbf{a}^*_k\in \mathcal{A}_{k} $ to maximize total expected reward: $$\begin{aligned}
\mathbf{a}^*_k \approx \arg \max\limits_{\mathbf{a}_k \in \mathcal{A}_{k}} \dfrac{1}{M_s} \sum_{t=k}^{k+H} \sum_{m=1}^{M_s} \gamma^{t-k}\mathcal{R}^{(m)}_{t}(\mathbf{a}_k),
\end{aligned}$$ where $M_s$ is the number of future measurements.
Multi-targets Tracking
----------------------
The particle filter proposed in Sec. \[sec\_PF\] can be extended to multi-target tracking (MTT). However, MTT normally deals with the complex data association problem where it is difficult to determine which measurement belongs to which target. In contrast, for our system, each target can be estimated from the measurement based on the signal frequency and tracked independently. Thus, we do not need to solve the data association problem. Notably, not all of the targets are detected due to, for example, the UAV movements, the measurement range limits imposed by propagations losses and receiver sensitivity. Therefore, if the target is not detected, the solver does not update its estimated position.
Besides maximizing the number of targets localized and tracked, we formulated a termination condition for each target to conserve UAV battery power; a target is considered localized if its location uncertainty, determined by a determinant of its particles covariance, is sufficiently small (<$N_{Th}$). Then, those found targets are *forgotten* to aid the solver to prioritize its computing resources on those targets with high uncertainty.
System Implementation
=====================
We implemented an experimental aerial robot system based on our tracking and planning formulation. An overview of the complete system is described in Fig. \[fig\_Combine\_PC\_UAV\_Communication\_And\_Antenna\_Design\]a. Our experimental system used a 3DR IRIS+ UAV platform and a new sensor system built with: i) a compact directional VHF antenna design, and ii) a software-defined signal processing module capable of simultaneously processing signals from multiple targets and remotely communicate with a ground control system for tracking and planning.
![a) The full communication channels between UAV and the ground control system with its main software components and protocols. b) The folded 2-element Yagi antenna design used in our sensor system for observations.[]{data-label="fig_Combine_PC_UAV_Communication_And_Antenna_Design"}](Figures/Combine_PC_UAV_Communication_And_Antenna_Design.png){width="17cm"}
In our system, the ArduPilotMega (APM) on the IRIS+ UAV transmits back its global positioning system (GPS) location to the Telemetry Host tool developed by our group to communicate with the APM module using the MAVLink protocol over a 915 MHz full duplex radio channel. The sensor system together with the Antenna, SDR receiver, and the embedded compute module delivers targets’ RSSI data through a 2.40 GHz radio channel to the ground control system.
GPS location of the UAV platform and targets’ RSSI data are delivered to our tracking and planning algorithm—*solver*—through the Telemetry Host using a RESTful web service. The solver estimates target locations and calculates new control actions per each POMDP cycle to command the UAV through MAVLink to fly to a new location. In order to ensure safety and meet University regulatory requirements, we also employ QGroundControl—a popular cross-platform flight control and mission planning software—to monitor and abort autonomous navigation. We detail our Sensor System below.
**Signal Processing Module:** Fig. \[fig\_dsp\_chains\] illustrates the components of the proposed signal processing module. We propose using a *software defined radio* (SDR) receiver to implement the signal processing components. The key advantages of our choice are the ability to: i) reduce the weight of the receiver; ii) rapidly scan a large frequency spectrum to track multiple animals beaconing on different VHF frequency channels; and iii) because the signal processing chain is defined in software, we have the ability to reconfigure the system on the fly.
In this work, we use the *HackRF One* SDR—an open source platform developed by [@Ossmann2015] capable of directly converting radio frequency (RF) signals to digital signals using an analog-to-digital converter (ADC)—together with an Intel Edison board as our embedded compute module. We implemented a Discrete Fourier Transform (DFT) filter to isolate, from multiple signals, each unique VHF frequency channel associated with an animal radio collar and measure the signal strength of the received signal.
![A signal processing module. (a) *Software defined radio:* raw input RF signals gone through the HackRF SDR device with different configurable amplifiers - Low Noise Amplifier (LNA) and Variable Gain Amplifier (VGA), and an ADC to convert to digital signals. (b) *Embedded compute module:* digital signals processed on an Edison board using a DFT-based frequency filter with configurable input frequencies, edge filter and peak detector algorithms to derive pulse RSSI measurements. []{data-label="fig_dsp_chains"}](Figures/dsp-chain-v2.png){width="17cm"}
**Antenna:** \[sec\_Antenna\_Gain\] A lightweight folded 2-element Yagi antenna was specially designed for our sensor system. Our design achieves a low profile antenna capable of being within the form factor of low-cost commodity UAVs suitable for easy operation in the field. Similar to a standard 2-element Yagi antenna, the folded design has one reflector and one driven element as shown in Fig. \[fig\_Combine\_PC\_UAV\_Communication\_And\_Antenna\_Design\]b.
The antenna operates in the frequency range from 145 to 155 MHz (a typical range for wildlife radio tags), and a center frequency of $f=150$ MHz. The length of driven and reflector elements are $D_d= 0.3975 \lambda$ and $\ D_r = 0.402\lambda$, respectively, while $d_1=0.1\lambda$, $d_2 = 0.03\lambda$ and the inductive loading ring diameter is $d_3 = 0.015\lambda$. Here, the wavelength $\lambda = c/f = 2$ (m) with $c=3\cdot10^8$ (m/s). The antenna gain model calculated for the the design is shown in Fig. \[Combine\_RotorNoise\_AntennaGain\_MeasureModel\]b.
Planning implementation for a real-time system
----------------------------------------------
Implementing planning algorithms on any real-time systems is always challenging because of its high computational demand. Thus, in this following, we present the approaches to minimize the planning computational time while not scarifying the overall localization performance:
1. Notably, for RSSI data, the uncertainty in the estimation of a target’s location is reduced when the maximum gain of the directional antenna mounted on the UAV points toward the target position. Hence, to increase the localization accuracy, the UAV heading angle $\theta^{u}_{k}$ must be controlled during the path planning process, although the multi-rotor UAV can be maneuvered without changing its heading. We select a set of discrete UAV rotation angles for the control actions $\mathcal{A}_{k} $ based on a simulation based study to reduce the computational complexity of the POMDP planning process by limiting the number of possible actions to evaluate.
2. The solver performs planning in every $N_{p}$ observation cycles with $N_{p} > 1$ instead of every observation. This approach helps to ensure that the solver prioritizes its limited computational resource on tracking targets instead of only performing planning steps.
3. A coarse planning interval $t_p$ in the planning procedure is implemented to minimize the computational time by reducing the number of look-ahead steps while still having the same look-ahead horizon. For example, we want to estimate the target’s state in a 10 second horizon, we can use the normal interval $t_{p}= 1 (s)$ and estimate the target’s state 10 times or use the coarser interval $t_{p} = 5 (s)$ and perform the estimation twice; the latter approach is computationally less expensive.
4. Instead of selecting the best action from the possible action space $\mathcal{A}_k$, the domain knowledge of the receiver antenna gain is used to select a subset of actions that give the highest received gain using **Alg.** \[Algo\_UAV\_SubSet\_Possible\_Location\].
\[!htbp\]
Get $\mathbf{u}^{l}_{k+H} \in \mathcal{A}_{k}(l)$ Calculate $G_r^{l} = G_r(\mathbf{x}_{k+H}, \mathbf{u}^{l}_{k+H})$ $\mathcal{A}_{k}^{s} = \mathcal{A}_{k}(G_r^{l} \geq \textbf{ Top } N_{\mathcal{A},s} \text{ of } G_r)$
Following the above implementation approach, UAV motion includes two modes: i) changing its heading angle while hovering; and ii) moving forward to its direct location. In one planning procedure with $N_p$ cycles, the UAV needs $\lfloor |\triangle\theta|/\theta_{max} \rfloor$ cycles to rotate, and spends the remaining cycles $N_p -\lfloor |\triangle\theta|/\theta_{max} \rfloor$ to move forward without changing its heading. Here $\lfloor\cdot\rfloor$ and $|\cdot|$ are the floor and absolute operator respectively, and $\theta_{max}$ is the UAV maximum rotation angle in one cycle . The sign of $\triangle\theta$ decides the rotation direction ($+$ for the clockwise, and $-$ for the counter-clockwise).
Simulation Experiments
======================
Implementing on a real system is time-consuming and difficult. Hence, we want to validate our systems first through several simulation experiments to: ***i)*** verify our tracking and planning algorithms; ***ii)*** investigate how our planning parameters such as different $\alpha$ values of the Rényi divergence or number of discrete actions $N_{\mathcal{A},s} = |\mathcal{A}^{s}_k|$ created in Alg. \[Algo\_UAV\_SubSet\_Possible\_Location\] contribute to the overall algorithm performance; and ***iii)*** compare our proposed Rényi divergence based planning technique with other well-known methods, and the impact of the look-ahead horizon parameters on computational time and localization accuracy. All of the simulation experiments were processed on a PC with an Intel(R) Core(TM) i7-6700 CPU @ 3.40GHz, 32GB RAM and MATLAB-2016b.
Tracking and Planning Simulation {#subsec_SIM_Target_Tracking}
--------------------------------
This simulation was implemented to validate our approach under a synthetic scenario where all parameters ([*e.g.* ]{}velocity of the UAV $v_u$) are set to those expected in practice. In this experiment, the UAV attempted to search and localize 10 moving targets randomly located in an area of $ 500~\text{m} \times 500~\text{m}$. The following are list of parameters used in this simulation: the sampling time step is $1$ second since the tag emits pulse signals every $1$ second. The solver performed a planning procedure every $N_p = 5$ s, and the look-ahead horizon parameters: $H =N_H t_{p} = 5$ s with number of horizon $\ N_H = 1$ and the planning interval $\ t_{p} = 5$ s. The UAV started from its home location at $u_{1} = [0,0,20,0]^{T}$, moved under the constant velocity $v_u = 5$ m/s with its maximum heading rotation angle $\theta_{max} = \pi/6$ rad/s. Number of particles for each target were capped at $N_s = 10,000$, with the future sample measurement $M_s = 50$, Rényi divergence parameter $\alpha = 0.5$, number of actions $N_{\mathcal{A},s} = 5$. In addition, a target is considered localized if its location uncertainty, determined by the determinant of its particles covariance, is small enough—$N_{Th} = 10,000~\text{m}^{2N_s}$ was chosen as the limit. The *LogPath* measurement model with $P^{d0}_r = 7.7~ \text{dBm}, n = 3.1, \sigma_P = 4.22~\text{dB}$ was used to verify our proposed algorithm. To demonstrate that our algorithm was able to localize mobile targets, a *wombat*—an animal that usually wanders around its area was considered. Hence, a *random walk* model was used to describe its behavior with a single target’s transitional density: $$\begin{aligned}
p(\mathbf{x}_k|\mathbf{x}_{k-1}) = \mathcal{N}(\mathbf{x}_k;\mathbf{Fx}_{k-1},\mathcal{Q})
\end{aligned}$$ where $F = \mathbf{I}_{3}$ with $\mathbf{I}_n$ is $n \times n$ identity matrix , $\mathcal{Q} = \sigma^2_{Q}\operatorname{diag}([1,1,0]^T)$, $\sigma_Q = 2$ m/s.
Fig. \[fig\_SIM\_Estimate\_Postion\] shows localization results for 10 mobile targets where the estimation details are annotated next to the target’s position with two indicators: *RMS* and *flight time*—see Sec. \[sec\_5c\_MC\] for definitions. In summary, for this scenario, it took the UAV **587** seconds to localize all ten moving targets a the maximum error distance less than 15 m, except for the outlier, target \#2 (RMS = 26.3 m). At flight time 587 s, after finishing localizing the last target (target \#7), the UAV was sent a command to fly back to its original station. In this case, the total UAV travel distance was 1.93 km. The results demonstrate that our algorithm can search and accurately localize multiple numbers of targets in real time (about 10 minutes) at the travel distance 1.93 km well within the capacity of commercial off the shelf drones under the 2 kg mass category.
![Simulation results with 10 mobile targets localized using a single UAV by Particle Filter and POMDP.[]{data-label="fig_SIM_Estimate_Postion"}](Figures/SIM_Target_Estimated_Pos.png){width="11cm"}
Monte Carlo simulations {#sec_5c_MC}
-----------------------
For this experiment, all of Monte Carlo setup parameters were kept the same as in Sec. \[subsec\_SIM\_Target\_Tracking\], except for those under investigation. In addition, to ensure that the results were not random, all of the conducted experiments were performed over 100 Monte Carlo runs. The tracking algorithm was evaluated based on the following criterion:
- *Estimation Error* is the absolute distance between ground truth and estimated target location $\mathcal{D}_{rms} = \sum_{j=1}^{N_{tg}} d^j_{rms}/N_{tg} $ with $d^j_{rms} = [(x^j_{truth} - x^j_{est})^2 + (y^j_{truth} - y^j_{est})^2]^{1/2} $.
- *Flight time* (s) for UAV to localize **all** of the targets which includes hovering time when the UAV waits for commands from the solver to take an action.
- *UAV travel distance:* the total distance traveled by the UAV to track and locate all of the targets to the required location uncertainty bound; i.e the determinant of covariance being adequately small—$N_{Th} \leq 10,000~\text{m}^{2N_s}$ .
- *Computational cost*: We evaluate the computational cost in terms of two components: i) execution time for the solver to execute the tracking algorithm only (called *non-planning time*), and ii) the execution time for the solver to select the best action—planning step—as well as complete the tracking task (called *planning time*).
First, our search and localization algorithms were evaluated using different $\alpha$ values for Réyni reward function in . Table \[table\_MC\_alpha\] presents the Monte Carlo results for $\alpha = \{0.1, 0.5, 0.9999\}$. In general, the $\alpha$ values do not significantly impact the overall performance. However, applying $\alpha = 0.1$ provides the best localization results in terms of estimation error and search duration. Applying $\alpha = 0.5$ proposed in [@Ristic2010; @ristic2010information] results in the worst performance, it increases flight time and travel distance necessary to complete the localization task. Using $\alpha = 0.9999$ (considered as using KL divergence which is a popular information gain measure) helps to save UAV travel distance while sacrificing location accuracy. One explanation for this scenario is that our noisy measurement causes the predicted posterior $p(\mathbf{x}_{k+H}|\mathbf{z}_{1:k},\mathbf{z}^{(m)}_{k+1:k+H}(\mathbf{a}))$ in to be less informative due to high uncertainty. Therefore, the reward function should place more emphasis on the current posterior instead by using a small $\alpha$ value or setting $\alpha \rightarrow 1$ to completely ignore the future posterior. This also explains the reason for the worst localization performance observed when $\alpha = 0.5$ (equally weighting the current and the future posterior).
$\alpha = 0.1$ $\alpha = 0.5$ $\alpha = 0.9999$
-------------------------- ---------------- ---------------- -------------------
RMS (m) **12.35** 12.77 12.96
Flight time (s) **724** 741 727
UAV travel distance (km) 2.38 2.41 **2.34**
: Localization performance for different alpha values
\[table\_MC\_alpha\]
Second, we conducted experiments to understand how the action space set $N_{\mathcal{A},s}$ created by Alg. \[Algo\_UAV\_SubSet\_Possible\_Location\] affects our tracking performance in term of planning time and localization error. Table \[table\_MC\_action\] shows Monte Carlo results for $N_{\mathcal{A},s} = \{2,3,4,5,6,7\}$. Increasing the number of actions beyond four does not necessarily lead to better planning decisions because of the directionality of the antenna gain. Since the antenna gain is not omnidirectional, some actions result in changing the heading where antenna gain along the propagation path between the UAV and the target is lower; when the number of actions evaluated are increased, we encounter instances when an action leading to such a lower antenna gain in fact results in a higher reward. This result is a consequence of the inherent uncertainties in the models used in tracking and planning. Thus, we can see that $N_{\mathcal{A},s}=4$ provides an adequate pool of actions to yield the best localization performance in terms of estimation error, flight duration and travel distance; a desirable result for realizing real time planning with limited computational resources.
[l | \*[6]{}[r]{}]{} Number of actions $N_{\mathcal{A},s}$ & 2 & 3 & 4 & 5 & 6 & 7\
RMS (m) & 14.18 & 12.64 &**12.17** & 12.27 & 12.83 & 12.63\
Flight time (s) & 840 & 781 & **693** & 723 & 756 & 799\
UAV travel distance (km) & 2.62 & 2.53 & **2.39** & 2.50 & 2.52 & 2.70\
Planning time (s) & **1.16** & 1.19 & 1.23 & 1.27 & 1.36 & 1.47\
\[table\_MC\_action\]
Third, we want to examine the performance of our proposed algorithm under increasing number of targets; in this study we increase the maximum number of targets $N_{tg}$ from 1 to 10. As depicted in the Fig. \[fig\_MC\_Target\_Change\_Comparison\], our algorithm’s estimation error was stable and invariant to the number of targets. Moreover, it is reasonable that the flight time and the travel distance increased linearly with target numbers because it took more time and power to track more targets.
![Localization performance for different number of targets $N_{tg}$ increase from 1 to 10.[]{data-label="fig_MC_Target_Change_Comparison"}](Figures/MC_Target_Change_Comparison.png){width="10.0cm"}
Fourth, we examined the performance of the information gain measure, Rényi divergence, under different look-ahead horizons $H = N_{H} t_p$ compared to: i) Shannon entropy [@cliff2015online]; ii) a naive approach that moves UAV to the closest estimated target location; and iii) a uniform search with predefined path used in [@ristic2010information]. Table \[table\_MC\_planning\] shows the Monte Carlo comparison results among various planning algorithms. All the parameters were reused from the Sec. \[subsec\_SIM\_Target\_Tracking\], except for $\alpha = 0.1$ and $N_{\mathcal{A},s} = 4$ were updated based on the previous experimental results. The result has demonstrated that Rényi divergence reward function is superior to other planning strategies in term of localization accuracy, including the Shannon entropy with the same horizon settings. For Rényi reward function itself, the large look ahead horizon number $N_H > 1$ helps to improve the localization accuracy; however, it requires higher computational power (planning) and causes the UAV to travel further. Using $N_H = 1;t_p = 5$ provides the best trade-off between computational time and accuracy.
[L[3cm]{} | \*[2]{}[R[2cm]{}]{} R[2.4cm]{} | \*[4]{}[R[1cm]{}]{} ]{} & **Uniform** & **Closest Target** & **Shannon** [@cliff2015online] &\
$N_{H}$ & N/A & N/A & 1 & 1 & 3 & 5 & 10\
$t_p$ (s) & N/A & N/A & 5 & 5 & 1 & 1 & 1\
RMS (m) & 18.8 & 13.4 & 12.6 & 12.5 & 12.4 & 12.0 & **11.6**\
Flight time (s) & 921 & 799 & 774 & **699**& 889 & 811 & 822\
UAV travel distance (km) & 3.72 & 2.29 & 2.54 & **2.27** & 2.99 & 2.82 & 2.42\
Planning Time (s) & 1.58 & **1.11** & 1.38 & 1.28 & 1.53 & 1.65 & 2.71\
Non-planning Time (s) & 1.58 & 1.03 & 0.99 & 0.97& **0.96** & 0.97 & **0.96**\
\[table\_MC\_planning\]
**Summary:** According to the above simulation results, we select $\alpha = 0.1$, $N_{\mathcal{A},s} = 4$, and $N_H = 1, t_p = 5 ~s$ as the planning parameters for the field experiment since these parameters provides the lowest computational cost, best performance in term of location estimation error, travel distance and flight time.
Field Experiments
=================
We describe here our extensive experiments regime to validate our approach and evaluate the performance of our aerial robot system in the field. Our aim is to: ***i)*** investigate the possibility for signal interference from spinning motors of a UAV on RSSI measurements; ***ii)*** estimate the model parameters in the sensor model and validate the proposed model; and ***iii)*** conduct field trials to demonstrate and evaluate our system capabilities.
Rotors noise
------------
We investigated the rotor noise to confirm that our system is not affected by the electromagnetic interference from the UAV’s motors. It also helps to clear the concern raised in [@cliff2015online] that the rotor noise may affect the RSSI measurements. Four motors of the 3DR IRIS+ quad-copter shown in Fig.\[fig\_UAV\_System\_Overview\] were used in this experiment. The RSSI data of a radio collar were measured across 149 MHz to 151 MHz frequency spectrum when four motors were operating at $20\%$, $ 50\%$, $100\%$ of its maximum speed of $10,212$ rounds per minute. Fig \[Combine\_RotorNoise\_AntennaGain\_MeasureModel\] (a) shows the frequency spectrum of the received signal. We can see that there was no difference in the frequency characteristics when the rotors were in ON and OFF states. This result confirms that the rotors do not spin fast enough to generate high-frequency interference to impact our RSSI measurements.
{width="14cm"}
Sensor model validation and parameter estimation {#sec_emperical_meas}
------------------------------------------------
**Antenna Gain:** The antenna gain pattern was measured to verify its directivity compared to the antenna gain model $G_r(\mathbf{x,u}) = G_r(\phi)$ calculated—following [@orfanidis2002electromagnetic p.1252]—based on the physical design as discussed in Sec. \[sec\_Antenna\_Gain\]. Fig. \[Combine\_RotorNoise\_AntennaGain\_MeasureModel\]b shows the measured and modeled radiation patterns $G_r(\phi)$ in the E-plane. In the measurement process, $\phi$ is evaluated as the angle between the UAV heading, changed through to , and the direction from its position to a fixed location of a VHF radio tag. The result shows that the front-to-back ratio is smaller (2 dB) than expected and this is an artifact of folding the reflector on our design.
**Signal propagation model parameter validation:** We collected RSSI data points over a range from 10 m to 320 m between the UAV and a VHF radio tag. The tag and the UAV were kept at a height of 5 m above ground during this experiment. The tag was stationary at all times, while the UAV was directed to move away in a straight line from the tag at 10 m intervals whilst hovering at each location to allow the collection of approximately 30 measurements. The UAV heading was maintained to ensure consistent antenna gain during the experiment. Since we operated in an open terrain over a grassland, we selected the path loss exponent $n=2$ suitable for modeling free space path loss. Fig. \[fig\_Meas\_Model\] shows the measured RSSI and the propagation models obtained using a nonlinear regression algorithm to estimate model parameters; we have the following results for reference power $P_r^{d0}$ in , at the reference distance $d_0 = 1$ m, and measurement noise variance $\sigma_{P}$ in :
- **LogPath model :** $P_r^{d0} = -15.69 \text{ (dBm)}$ ; $\sigma_{P} = 4.21$ (dB).
- **MultiPath model :** $P_r^{d0} = -15.28 \text{ (dBm)}$ ; $\sigma_{P} = 2.31$ (dB).
The results show that both models, as expected, derived a similar reference power $P_r^{d0}$ whilst providing a reasonable fit to measurement data and this affirms the validity of our propagation model. Although *LogPath* model is reasonable, *MultiPath* model is more accurate and yields a smaller measurement noise variance. The results confirm the impact of ground reflections, especially close to the signal source.
![Plot of measured RSSI data points and its model estimates over a distance from 10 m to 320 m at 10 m intervals. []{data-label="fig_Meas_Model"}](Figures/Field_Experiment_Measurement_Model.png){width="10cm"}
Field Trials
------------
We present two sets of field experiments to validate the two measurement models and conducted a total of 16 autonomous flights to demonstrate our system capabilities. Our experiments were designed around University of Adelaide regulations governing the conduct of experimental UAV research. Given the need to operate in an autonomous mode, our flight zone, as well as the scope of the experiment, was restricted to University-owned property designated for UAV flight tests. Therefore: i) the UAV task was set to search and localize two mobile targets in a search area $75~m \times 300~m$ (2.25 Hectares); and ii) instead of wildlife, we relied on two people, each wore a VHF radio tag on their forearm, and a mobile phone-based GPS data logger on their hands to obtain ground truth; with two extra personnel stationed to maintain constant sight of the UAV as well a pilot in the field and abort the autonomous mode to transfer control to manual operations. The volunteers with the radio tags were asked to walk randomly and no other instructions were given.
Fig. \[fig\_Field\_Experiment\_Result\] shows the tracking and localization results along with UAV trajectories based on the two different measurement models As expected, we observe the UAV planning has a tendency to approach the target’s position since when the distance between the UAV and targets reduces, the RSSI measurement uncertainty is reduced. thus it helps to reduce the uncertainty and increase the information gain. We can observe clear difference in the *LogPath* model and *MultiPath* model where UAV pursues the second target after completing the tracking task for target 1. The more accurate *MultiPath* model is able to track and localize the second target without needing a close approach. We can also observe that using *LogPath* model, where multipath propagation is not modeled but is clearly dominant close to the target, leads to a poorer localization accuracy despite the path planning algorithm leading the UAV close to the target.
Fig. \[fig\_Field\_Experiment\_Particle\] shows the particle distribution after the first observation is updated and when the targets are tracked and localized using the two measurement models We can see that the solver is able to estimate the two tag positions quiet accurately even after the update using the initial observation; however, the uncertainty (as noted by the particle distribution) is still very high. Interestingly, *MultiPath* model location uncertainty is significantly less where target 1 is placed in the bottom half of the field while target 2 is placed in the top half of the field. Target 1, being closer to the UAV, is localized first, with under $55$ measurements for both measurement models. At the time when target 1 is localized, the uncertainty of target 2 is relatively higher for the *LogPath* model. The *MultiPath* model required significantly less measurements to track and localize target 2. As expected, both measurement models required significantly more measurements to localize the second target given the high measurement uncertainty associated with being much further than the first target from the UAV during its flight. Furthermore, random walk of the second target provided a challenging scenario since target 2 typically moved a larger distance around the field compared to the random walk performed by target 1.
Although the solver guides the UAV to move toward a target’s position in both measurement models, as expected, the standard *LogPath* model is less accurate compared to the *MultiPath* model shown in Fig. \[fig\_Meas\_Model\]; thus the uncertainty when using the *LogPath* model is higher and leads to longer time durations to localize the two tags. The consequence of model uncertainty resulting from the simple *LogPath* model, albeit still capable of locating both moving targets within the flight time capability of the UAV, is more apparent when the UAV makes an approach to the target and the distance to the target is less than $50$ m depicted in Fig. \[fig\_Field\_Experiment\_Particle\]c in comparison to Fig. \[fig\_Field\_Experiment\_Particle\]f. We can see that the target location uncertainty increase for the *LogPath* model in the vicinity of 50 m and as a result the UAV requires an increasing number of maneuvers to in its attempt to track and locate the target; this is clearly evident in the path followed for tracking and locating the second target. Table \[table\_Field\_Experiment\_Result\] presents the summary comparison results of location estimates between the two measurement models. Smaller RMS (root mean square) estimation error values suggest a higher accuracy, while shorter flight times and travel distance to localize all targets are highly desirable for a practicable system given the power constrained nature of commodity UAVs. The results confirm that the *MultiPath* model is superior to the standard *LogPath* model since it has been able to account for ground reflections and the UAV is not required to approach the target as closely when using the *LogPath* model to reduce its measurement uncertainty.
![Field experiment results to search, track and localize two mobile tags for the two different measurement models. a) Standard **LogPath**. b) **MultiPath**.[]{data-label="fig_Field_Experiment_Result"}](Figures/Field_Experiment.png){width="16cm"}
![Field experiment results with the distribution of particles to search, track and localize two mobile tags. (a), (b) and (c) demonstrate the convergence of particles using the standard **LogPath** measurement model after the first observation is updated, the tag 1 is localized, and the tag 2 is localized, respectively. Similarly, (d), (e) and(f) demonstrate the convergence of particles using the **MultiPath** measurement model after the first observation is updated, the tag 1 is localized, and the tag 2 is localized, respectively. The blue and orange dots represent the start positions of the tag 1 and tag 2 respectively; the square symbols denote the ground truths of the localized tags; the star symbols denote the estimated positions of the tags; the solid yellow lines represent the UAV trajectories. []{data-label="fig_Field_Experiment_Particle"}](Figures/LogPath_ParticleDistribution.png "fig:"){width="16cm"} ![Field experiment results with the distribution of particles to search, track and localize two mobile tags. (a), (b) and (c) demonstrate the convergence of particles using the standard **LogPath** measurement model after the first observation is updated, the tag 1 is localized, and the tag 2 is localized, respectively. Similarly, (d), (e) and(f) demonstrate the convergence of particles using the **MultiPath** measurement model after the first observation is updated, the tag 1 is localized, and the tag 2 is localized, respectively. The blue and orange dots represent the start positions of the tag 1 and tag 2 respectively; the square symbols denote the ground truths of the localized tags; the star symbols denote the estimated positions of the tags; the solid yellow lines represent the UAV trajectories. []{data-label="fig_Field_Experiment_Particle"}](Figures/MultiPath_ParticleDistribution.png "fig:"){width="16cm"}
[L[3cm]{} | C[2cm]{} C[1cm]{} C[2cm]{} \*[2]{}[C[3cm]{}]{}]{} **Model** & Target Type & Trials & RMS (m) & Total Flight Time (s) & Travel Distance (m)\
LogPath & Mobile & 8 & $30.1\pm 12.8$ & $255\pm104$ & $549\pm167$\
MultiPath & Mobile & 8 & $\mathbf{22.7}\pm13.9$ & $\mathbf{138}\pm53$ & $\mathbf{286}\pm121$\
[@cliff2015online] & Stationary & 6 & $23.8\pm14.0$ & 838[^2] & N/A\
\[table\_Field\_Experiment\_Result\]
[L[3cm]{} | L[6cm]{} | L[6cm]{}]{} & Ours & [@cliff2015online]\
*Payload* (g) & **260** & 750\
*Total mass (g)* & **1,280** & 2,200\
*Drone type* & Quadcopters (**smaller** drone)& Octocopters (relatively larger drone)\
*Receiver Architecture* & Software defined radio (digital-based,rapidly scan multiple frequencies to support multiple frequencies) & Analog filtering circuit and a fixed frequency narrowband receiver (analog-based, difficult to re-configure for a new frequency)\
*Antenna elements* & Compact, lightweight, folded 2-element Yagi antenna (designed for small drone form factor) & Antenna array structure requiring a large spatial separation of two antenna elements and wire ground plane\
*Detection range (m)* & 320 & **500**\
*Measurement model* & Range-only (exploiting the simplicity of a range-only measurement system)& Bearing-only (antenna array, and UAV rotation at grid points with a phase difference measurement system)\
*Filtering method* & Particle filter ($\mathrm{O}(N)$ operations per iteration)& Grid-based filter ($\mathrm{O}(N^2)$ operations per iteration)\
*Planning algorithm* & Rényi divergence & Shannon entropy\
*Nature of targets* & **Multiple mobile** target tracking & A single stationary target localization\
\[table\_Field\_Experiment\_System\]
Discussion
----------
In this section, we summarize results from our approach as well as compare and discuss our results in the context of the recent study by [@cliff2015online]. Table \[table\_Field\_Experiment\_Result\] presents a summary of localization field study results while Table \[table\_Field\_Experiment\_System\] presents a complete comparison between our proposed system and [@cliff2015online] system. Notably, our search area is smaller compared to [@cliff2015online] ($75~m \times 300~m$ v.s $1000~m \times 1000~m$) due to our test flight zone restrictions, however, we have set up our initial distance from the UAV home position to its farthest target’s position (target \#2 in this case) to be equivalent to the distance of the stationary target in [@cliff2015online]; approximately 300 m.
The results in Table \[table\_Field\_Experiment\_Result\] demonstrate that our proposed method can localize two mobile targets in a shorter flight time (the flight time of *MultiPath* model is one-sixth of that in [@cliff2015online]) with better accuracy. Moreover, we search and locate two mobile targets; in contrast, [@cliff2015online] method was implemented to locate a single and stationary target.
In general, as shown in Table \[table\_Field\_Experiment\_System\], our system is more compact, lighter, and has a payload that is one-third of that in [@cliff2015online] and consequently capable of longer flight times on any given UAV. Although our reliance on an SDR without a pre-amplifier has resulted in a shorter detection range, our total system mass being under $2$ kg is significant since it enables ecologists to operate our system without a remote pilot licenses (RePL) [@casaac101]. Moreover, the ability to instantly collect range-only measurements also helps reduce flight time significantly compared to the bearing-only method, requiring full rotations of a UAV at each observation point, as shown in the Table \[table\_Field\_Experiment\_Result\]. Furthermore, as discussed in [@arulampalam2002tutorial], the computational cost for grid-based methods used in [@cliff2015online] increases dramatically with the number of cells whilst the grid must be dense enough to achieve accurate estimations; [*e.g.* ]{}, a grid-based filter with $N$ cells conducts $\mathrm{O}(N^2)$ operations per iteration, while a similar particle filter with $N$ particles only requires $\mathrm{O}(N)$ operations. Hence, the grid-based filter method only works in case of stationary targets as in [@cliff2015online] where the most expensive computational step, the prediction step, is skipped. Moreover, as shown in Table. \[table\_MC\_planning\], our planning algorithm based on Rényi divergence is superior to the Shannon entropy approach in [@cliff2015online] in terms of two important metrics: accuracy and UAV flight time.
Conclusion {#sec:conclusion}
==========
We have developed and demonstrated an autonomous aerial vehicle system for range only tracking and localization of VHF radio-tagged animals under RSSI based measurement uncertainty and mobility of targets during their discovery in the field. The joint particle filter and POMDP with Rényi divergence based reward function provided an accurate method to explore, track and locate multiple animals while considering the resource constraints of the underlying UAV platform. In addition, we have realized a UAV system under 2 kg to ensure both the practicability and the accessibility of the technology to conservation biologists.
While we have demonstrated a successful system, we have only formulated our approach as a two-dimensional tracking problem that is ideally suitable for tracking endangered species in largely flat terrains and grasslands. Consequently, the current approach is not suitable for tracking wildlife in hills or mountainous areas and it would require: i) a UAV capability to maintain a fixed relative altitude above the ground; or ii) formulating a 3D tracking problem to extend our method to all topographical conditions. We leave the latter for future work.
### Acknowledgments {#acknowledgments .unnumbered}
This work was jointly supported by the Western Australia Parks and Wildlife (WA Parks), the Australian Research Council (LP160101177), the Defense Science and Technology Group (DSTG), and the University of Adelaide’s Unmanned Research Aircraft Facility. We would like to thank the support and guidance provided by Mr. Adam Kilpatrick, Chief Remote Pilot and Maintenance Controller at the University of Adelaide, for making the field trials possible and and Remote Pilot, Mr. Fei Chen, Auto-ID Lab, The University of Adelaide for support provided in conducting all of the field experiments in the study.
[^1]: see: <https://github.com/AdelaideAuto-IDLab/TrackerBots> (please note that project material will be fully uploaded upon the acceptance of the article for publication)
[^2]: Information regarding the total flight is not reported in [@cliff2015online], however, as shown in Fig. 9 in [@cliff2015online], one observation took 76.21s and one trial needed 11 observations, hence total flight time is $11 \times 76.21 = 838.31 s$
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present a fully dynamic algorithm for maintaining approximate maximum weight matching in general weighted graphs. The algorithm maintains a matching ${\cal M}$ whose weight is at least $\frac{1}{8} M^{*}$ where $M^{*}$ is the weight of the maximum weight matching. The algorithm achieves an expected amortized $O(\log n \log \mathcal C)$ time per edge insertion or deletion, where $\mathcal C$ is the ratio of the weights of the highest weight edge to the smallest weight edge in the given graph. Using a simple randomized scaling technique, we are able to obtain a matching whith expected approximation ratio 4.9108.'
author:
- |
Abhash Anand\
Department of CSE,\
I.I.T. Kanpur, India\
[`abhash@cse.iitk.ac.in`]{}
- |
Surender Baswana\
Department of CSE,\
I.I.T. Kanpur, India\
[`sbaswana@cse.iitk.ac.in`]{}[^1]
- |
Manoj Gupta\
Department of CSE,\
I.I.T. Delhi, India\
[`gmanoj@cse.iitd.ernet.in`]{}
- |
Sandeep Sen\
Department of CSE,\
I.I.T. Delhi, India\
[`ssen@cse.iitd.ernet.in`]{}
bibliography:
- 'references.bib'
title: Maintaining Approximate Maximum Weighted Matching in Fully Dynamic Graphs
---
[^1]: Research supported by the Indo-German Max Planck Center for Computer Science (IMPECS).
| {
"pile_set_name": "ArXiv"
} |
---
author:
- Grzegorz Kopacki and Andrzej Pigulski
title: Variable stars in the globular cluster M79
---
Introduction, Observations and Results
======================================
In the last two decades we observed a rapid increase in the number of variable stars detected in Galactic globular clusters. The main reason for this was the invention of the image subtraction method (ISM, [@ala98]) and its application to CCD data obtained with small telescopes (e.g. [@kop09]). The ISM enables making a complete inventory of bright variable stars, such as RR Lyrae variables, because it works well in crowded stellar fields like the cluster core. However, there are still many globular clusters poorly searched for variable stars, especially pulsating stars of the RR Lyrae and SX Phoenicis types.
Here we present results of a variability analysis for the southern globular cluster M79 (NGC1904) of intermediate metallicity (\[Fe/H\]${}=-1.57$). The Catalogue of Variable Stars in Globular Clusters (CVSGC, [@cle01]) listed 13 objects in the field of this cluster including nine RR Lyrae stars, one semiregular variable and three stars suspected for variability.
We used two sets of observations. The first one consisted of 690 $V$-filter and 230 $I_{\rm C}$-filter CCD frames obtained during one-month observing run in Feb/Apr, 2008 using 40-inch telescope at SSO, Australia. The other one included 80 $V$-filter CCD frames acquired during one week of observation in Apr, 2008 using 40-inch telescope at SAAO, South Africa.
We have detected two new RR Lyrae stars, both of the RRc type. The mean period of RRab stars in M79 is equal to $\langle P\rangle_{\rm ab}=0.69$ d, and relative percentage of RRc stars amounts to $N_{\rm c}/(N_{\rm ab}+N_{\rm c})=44$ %. With these values we conclude that M79 belongs to the Oosterhoff’s II group of globular clusters.
We show that v7 is a W Virginis-type star. Moreover, three SX Phoenicis stars were found among cluster’s blue stragglers. One of them, n18, and RRc star v9 are multiperiodic pulsators. The period ratio for n18 indicates that this SX Phoenicis star is a double-mode radial pulsator. Positions and periods of all observed periodic stars are given in Table \[tab:1\]. Irregular light variations were also found in a dozen of red giants located at the tip of the cluster’s red giant branch.
[p[0.8cm]{}p[2cm]{}p[2cm]{}p[2cm]{}p[0.8cm]{}]{} Var& $\alpha_{2000}$ \[$^{\rm h}$ $^{\rm m}$ $^{\rm s}$\]& $\delta_{2000}$ \[$^\circ$ $^\prime$ $^{\prime\prime}$\]& P \[d\]& Type\
v3& 5 24 13.54& $-$24 32 29.1& 0.73602& RRab\
v4& 5 24 17.77& $-$24 32 16.2& 0.63339& RRab\
v5& 5 24 10.23& $-$24 31 03.6& 0.66894& RRab\
v6& 5 24 06.03& $-$24 29 32.9& 0.339065& RRc\
v7& 5 24 12.68& $-$24 31 41.9& 13.946& WVir\
v9& 5 24 12.58& $-$24 31 52.6& 0.37905& RRc\
& & & 0.36099&\
& & & 0.37049&\
v10& 5 24 12.13& $-$24 31 34.5& 0.72894& RRab\
v11& 5 24 11.93& $-$24 31 34.6& 0.8232& RRab\
v12& 5 24 11.35& $-$24 31 28.3& 0.32394& RRc\
v13& 5 24 10.59& $-$24 31 11.5& 0.68931& RRab\
n14& 5 23 23.74& $-$24 27 46.3& 0.309058& RRc\
n15& 5 24 07.77& $-$24 31 00.3& 0.323758& RRc\
n16& 5 24 09.97& $-$24 31 07.3& 0.038763& SXPhe\
n17& 5 24 14.15& $-$24 33 20.7& 0.044803& SXPhe\
n18& 5 24 10.86& $-$24 31 11.8& 0.050276& SXPhe\
& & & 0.039169&\
This work was supported by Polish Ministry of Science grant N203 014 31/2650.
[99.]{} Alard, C., Lupton R.H.: A Method for Optimal Image Subtraction. ApJ **503**, 325 (1998) Clement, C.M., et al.: Variable Stars in Galactic Globular Clusters. AJ **122**, 2587 (2001) Kopacki, G.: Search for Pulsating Stars in the Globular Cluster M 80 from Ground- and Space-based Observations. AIPC **1170**, 194 (2009)
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The off-shell and the on-shell Sudakov form factors in theories with broken gauge symmetry are calculated in the double-logarithmic approximation. We have used different infrared cut-offs, i.e. different mass scales, for virtual photons and weak gauge bosons.'
---
[Mass scale effects for the Sudakov form factors in theories with the broken gauge symmetry]{}
[A. Barroso and B.I. Ermolaev[^1]\
Centro de Física Teórica e Computacional, Faculdade de Ciências, Universidade de Lisboa, Av. Prof. Gama Pinto 2, P-1649-003 Lisboa, Portugal\
]{}
Introduction
============
In QED the electro-magnetic vertex function, $\Gamma_{\mu}$, can be written as: $$\label{gammaborn}
\Gamma_{\mu} = \bar{u}(p_2)\Big[ \gamma_{\mu} f(p_1,~p_2) -
(1/2m)\sigma_{\mu \nu} q_{\nu} g(p_1, p_2) \Big] u(p_1) ~,$$ where $f$ and $g$ are the form factors. In the fifties V.V. Sudakov showed[@sud] that in the limit of large momentum transfer, i.e., $$\label{kin}
q^2 = (p_2 - p_1)^2 \gg p^2_1, p^2_2 ~,$$ the most important radiative corrections to the form factor $f(p_1, ~p_2)$ are the double- logarithmic ones (DL). The summing of these corrections to all orders in $\alpha$ leads to, $$\label{foffshell}
f = e^{-(\alpha/2 \pi) \ln (q^2/p^2_1)\ln(q^2/p^2_2)},$$ for off-shell momenta $p_1$ and $p_2$ and to the formula $$\label{fonshell}
f = e^{-(\alpha/4 \pi) \ln^2 (q^2/m^2)},$$ if $p^2_1 = p^2_2 = m^2$.
After this pioneer work, the Sudakov form factor $f$ was calculated in QCD [@qcd] and recently it has also been considered in the electroweak (EW) theory [@flmm],[@ciaf],[@k]. In non-Abelian theories the direct graph-by-graph calculation to all orders in the couplings is a very complicated procedure, even when the double- logarithmic approximation (DLA) is used. Technically, it is more convenient to use some evolution equation. In particular, the infrared evolution equation (IREE) approach was used in ref. [@flmm] to calculate the “inclusive” EW Sudakov form factor, i.e., where summation over the left handed lepton flavour was assumed. The IREE method is based on the Gribov bremsstrahlung theorem [@g]. It was applied earlier, in ref. [@efl], to calculate the radiative form factors for $e^+e^-$ annihilation into quarks and gluons. It has been proved to be a very efficient and simple method. This theorem was formulated and proved first for QED and then generalised in refs.[@kl],[@efl],[@e],[@ce] to QCD. Besides the calculation of the Sudakov form factor $f(q^2)$, the IREE turned out to be also useful [@et] in order to calculate the form factor $g(q^2)$ of electrons and quarks in the kinematic region specified by eq.(\[kin\]).
The IREE exploits the evolution of the scattering amplitudes with respect to the infrared cut-off $\mu$ introduced in the space of the transverse momenta of virtual particles. This cut-off plays the role of a mass scale and with DL accuracy all other masses can be safely neglected. So, in theories with unbroken gauge symmetry it is enough to have one mass scale. On the other hand, for the electroweak theory, the $SU(2)\times U(1)$ symmetry is broken down to the $U_{EM}(1)$ symmetry. This introduces a second mass parameter, $M$, associated with the symmetry breaking scale. Therefore, besides the conventional DL contributions of the order of $\ln^{2n}(q^2/M^2)$ in $n$-th order of the perturbative expansion, there appear other corrections of the type $$\label{c}
\sim a_k \ln^k (M^2/ \mu^2) \ln^{(n - k)}(q^2/M^2) ,$$ where $a_k$ are numerical coefficients and $k$ runs from 1 to $n$. Since $\mu$ and $M$ can be widely different (we assume the value of $\mu$ to be equal or greater than masses of the involved fermions whereas $M$ is comparable with masses of the weak bosons), the impact of these two mass scales on the value of the inclusive Sudakov form factor can be important. To calculate these contributions to all orders in the electroweak couplings is the aim of the present work. In Sect. 2 we obtain the Sudakov form factor for a $U(1)\times U(1)$ model with two “photons”, using the graph-by-graph calculation. In Sect. 3 we derive and solve the IREE for the Sudakov form factor for the same model. Then, in Sect. 4 a similar derivation is made for the Sudakov form factor in the electroweak theory. Expressions for the off-shell electroweak Sudakov form factor are obtained in Sect. 5. Finally, in Sect. 6 we summarise and discuss our results.
The Sudakov form factor in the Abelian model
============================================
As a toy model it is instructive to consider a $U(1)\times U(1)$ gauge theory. The first group is the normal electro-magnetic gauge group and the second $U(1)$ is broken. The corresponding gauge boson, called $B$ meson has a mass $M$. Besides the photon and the $B$ meson the model has only one charged fermion, and a neutral scalar Higgs particle arising from the spontaneous breaking of the second $U(1)$.
At one loop order the Sudakov form factor is obtained from the diagrams in figure \[fig1\] where the dashed line represents the photon and the wavy line represents the $B$ meson.
Summing both contributions in $DL$ approximation the result is $$\label{f1}
f^{(1)} = -(g_1^2/16 \pi^2) \ln^2(q^2/ \mu^2) -
(g_2^2/16 \pi^2) \ln^2(q^2/ M^2)$$ where $g_1, g_2$ are the gauge couplings corresponding to the unbroken and broken groups respectively. It is interesting to point out that there is also a similar diagram with the higgs particle in loop. However this contribution vanishes as $(m/M)^2$ when the fermion mass, $m$, goes to zero. This is obviously true in all orders of perturbation theory.
At two loop order the DL contributions stems from the diagrams of figure Fig. \[fig2\].
\
The calculation of the first four diagrams in Fig.\[fig2\] is similar to the QED calculation. Hence, we simply quote the result, $$\label{ab}
f^{(2)}_{a+b+c+d} = \frac{1}{2}[(g_1^2/16 \pi^2) \ln^2(q^2/ \mu^2)]^2
+\frac{1}{2}[(g_2^2/16 \pi^2)\ln^2(q^2/ M^2)]^2.$$ The calculation of the remaining four diagrams is less trivial. Let us consider diagram $e)$, for instance. It is well known that in DLA one can perform the integration over the momenta $k_i$, $i=1,2$, replacing the boson propagators by $-2\pi i \delta(k_i^2)$. Then, using the Sudakov parametrisation, $$\label{sud}
k_i = \alpha_i p_2 + \beta_i p_1 + k_{i\perp},$$ it is easy to integrate over $k_{i\perp}$ and to obtain $$\label{e}
f_e^{(2)} = \frac{g^2_1 g^2_2}{(8 \pi^2)^2} \int_{D_e}
\frac{d \alpha_1 d \beta_1 d \alpha_2 d \beta_2}
{\alpha_1 \beta_1 \alpha_2 \beta_2}
\Theta (\alpha_1\beta_1 - \lambda^2_1)~
\Theta(\alpha_2\beta_2 - \lambda^2_2),$$ where $\lambda^2_1=M^2/s$, $\lambda^2_2=\mu^2/s$, and $s=|q^2|$. Notice that the $DL$ arises if one uses the approximation $\alpha_1 +
\alpha_2 \approx \alpha_2$ and $\beta_1+ \beta_2 \approx \beta_2$. This implies $\alpha_2 \gg \alpha_1$ and $\beta_2 \gg \beta_1$. These conditions plus the boundaries stemming from the arguments of the $\Theta$ functions define the integration region, $D_e$. In Fig. \[fig3\] we show this region.
The full curve corresponds to the equation $\alpha_2\beta_2=\lambda_2^2$ and the dashed curve represents the condition $\alpha_1\beta_1=\lambda_1^2$. Because the lower limits of the $\alpha_2$ and $\beta_2$ integrals are $\alpha_1$ and $\beta_1$, respectively, it is easy to see that we are integrating over the rectangle $E$. So we obtain $$\label{e}
f_e^{(2)} = \frac{g^2_1}{8 \pi^2}\frac{g^2_2}{8 \pi^2}
\int_{\lambda_1^2}^1\frac{d\alpha_1}{\alpha_1}
\int_{\lambda_1^2/\alpha_1}^1\frac{d\beta_1}{\beta_1}
\int_{\alpha_1}^1\frac{d\alpha_2}{\alpha_2}
\int_{\beta_1}^1\frac{d\beta_2}{\beta_2}.$$ The integrals are now trivial. However, rather than doing this it is better to realize that the remaining diagrams give rise to similar integrals but with the integration regions $F$, $G$ and $H$ of Fig. \[fig3\], respectively. Then, the sum of the diagrams $e)$, $f)$, $g)$ and $h$ corresponds to $$f_{e+f+g+h}^{(2)} = \frac{g^2_1}{8 \pi^2}\frac{g^2_2}{8 \pi^2}
\int_{\lambda_1^2}^1\frac{d\alpha_1}{\alpha_1}
\int_{\lambda_1^2/\alpha_1}^1\frac{d\beta_1}{\beta_1}
\int_{\lambda_2^2}^1\frac{d\alpha_2}{\alpha_2}
\int_{\lambda_2^2/\alpha_2}^1\frac{d\beta_2}{\beta_2}.$$ which immediately leads to $$\label{eh}
f_{e+f+g+h}^{(2)} = \frac{g_1^2}{16 \pi^2} \ln^2(q^2/ \mu^2)
\frac{g_2^2}{16 \pi^2} \ln^2(q^2/ M^2)~.$$
Adding these results to eq.(\[ab\]) one obtains the total two-loop contribution to the form factor, namely: $$\label{ah}
f^{(2)} = (1/2)\big[(g_1^2/16 \pi^2) \ln^2(q^2/ \mu^2) +
(g_2^2/16 \pi^2) \ln^2(q^2/ M^2) \big]^2~.$$ Repeating the same analyses in higher orders in $g_1$ and $g_2$, we would arrive at the simple exponentiation of the double-logarithmic corrections to $\Gamma_{\mu}$, i.e., $$\label{fmum}
\Gamma_{\mu}^{DL} = \Gamma_{\mu}^{Born} f(q^2, \mu^2, m^2)~,$$ with $$\label{exp}
f(q^2, \mu^2, M^2) =
e^{- \big[(g_1^2/16 \pi^2) \ln^2(q^2/ \mu^2) +
(g_2^2/16 \pi^2) \ln^2(q^2/ M^2) \big]} .$$ This equation accounts for all DL contributions described by Eq. (\[c\]).
Infrared evolution equations with two mass scales
=================================================
In order to avoid the direct graph-by-graph summation of the DL contributions, it is possible to obtain the Sudakov form factor $f(q^2, \mu^2, m^2)$ as a solution of some integral equation. The method of obtaining this infrared evolution equation (IREE) can be extended in order to include two mass scales, $\mu$ and $M$.
Let us first notice that the boson mass $M$ in virtual propagators can be regarded in DLA as the infrared cut-off for integrating over transverse momentum space for $B$-bosons. In fact, with logarithmic accuracy we have $$\label{ircutoff}
\int_0^s \frac{d k^2_{\perp}}{k^2_{\perp} + M^2} =
\int_{M^2}^s\frac{d k^2_{\perp}}{k^2_{\perp}} ~.$$ In DLA the integrals over the longitudinal momenta have the transverse momenta as the lowest limit. So, after introducing the cut-off shown in the previous equation one can neglect the mass $M$ in the $B$-propagators and still be free of infra-red singularities. On the other hand, $\mu$ is the IR cut-off in the transverse momentum space for photons. Therefore, with the DL accuracy, we have QED with two kinds of photons, each one has an independent infrared cut-off. We remind the reader that we have assumed that $\mu \ll M$. According to the generalisation[@efl; @ce] of the Gribov bremsstrahlung theorem[@g], the boson with the minimal $k_{\perp}$ can be factorized, i.e. the main contribution from integrating over its momentum comes from the graphs where its propagator is attached, in all possible ways, to the external charged lines whereas its $k_{\perp}$ acts as a new IR cut-off for the integrations over the remaining loop momenta and becomes, in DLA, a new effective mass scale. Therefore the blob in Fig.\[fig4\] is on-shell.
It depends on the new infrared cut-off $k_{\perp}$ and does not depend on the longitudinal components of momentum $k$. However, in contrast to the usual QED situation, now we have two options: the factorized particle with minimal $k_{\perp}$ can be either a photon or a $B$-boson. Applying the Feynman rules to the graph in Fig.\[fig4\], we obtain for the first possibility $$\begin{aligned}
\label{photon}
M^{\gamma} = - \frac{g_1^2}{8 \pi^2}
\Big[ \int_{\mu^2}^{M^2} \frac{d k^2_{\perp}}{k^2_{\perp}}
\ln(s / k^2_{\perp}) f(s/ k^2_{\perp},~s/M^2) + \\ \nonumber
\int_{M^2}^{s} \frac{d k^2_{\perp}}{k^2_{\perp}}
\ln(s / k^2_{\perp}) f(s/ k^2_{\perp},~s/k^2_{\perp}) \Big]~, \end{aligned}$$ where $\ln(s/ k^2_{\perp})$ appears as a result of integrating over the longitudinal momentum of the factorized photon quite similarly to the way it appears in the first loop calculation of $f$ in QED. Note that the form factors $f$ in the right hand side (rhs) of eq. (\[photon\]) have the same first argument, $s/k^2_{\perp}$, but different second arguments, $s/M^2$, in the first integral and $s/k^2_{\perp}$ in the second one. The reason for this is that the propagators of the $B$-bosons depend on $M$ through $(k^2_{\perp} +
M^2)$ and in DLA they are approximated by $max(k^2_{\perp}, ~M^2)$ . The second possibility is that the factorized particle is the $B$-boson. Now, we can regard $M$ as the infra-red cut-off for the integration over the $k_{\perp}$ of the $B$-meson. Hence the result is $$\label{boson}
M^{B} = - \frac{g_2^2}{8 \pi^2}
\int_{M^2}^{s} \frac{d k^2_{\perp}}{k^2_{\perp}}
\ln(s / k^2_{\perp}) f(s/ k^2_{\perp},~s/k^2_{\perp}) .$$
Adding eqs. (\[photon\]) and (\[boson\]) and including the Born contribution, $f_{Born} = 1$, one obtains the integral IREE for the form factor $f(s/ \mu^2,~ s/m^2)$, i.e. $$\label{eqf}
f(s/ \mu^2,~ s/M^2) = 1 + M^{\gamma} + M^{B} ~.$$ The solution of Eq. (\[eqf\]) can be obtained, for example, by iterations: substituting $f_{Born} = 1$ into the integrand of the rhs of eq. (\[eqf\]) we obtain $f(s/ \mu^2,~ s/m^2)$ in the one loop approximation and so on. It is easy to see that, after summing up contributions to all orders, the solution to eq. (\[eqf\]) coincides with Eq. (\[exp\]) obtained by the direct graph-by-graph summation.\
Alternatively, the integral IREE given in eq. (\[eqf\]) can be rewritten in the differential form (cf [@flmm]). Indeed, differentiating Eq. (\[eqf\]) with respect to $\mu$ yields $$\label{dmu}
\frac{\partial f(\rho_1,~\rho_2)}{\partial \rho_1} =
- \frac{g_1^2}{8 \pi^2} \rho_1 f(\rho_1,~\rho_2)$$ where $\rho_1 = \ln(s/\mu^2)$ and $\rho_2 = \ln(s/M^2)$. Obviously the solution is $$\label{fmu}
f(s/ \mu^2,~ s/M^2) = G(\rho_2) e^{ -g_1^2/ 16 \pi^2 \rho_1^2}.$$ Substituting it into Eq. (\[dmu\]) and differentiating with respect to $\rho_2$ gives $$\label{dm}
\frac{d G(\rho_2)}{d \rho_2} =
- \frac{g_2^2}{8 \pi^2} \rho_2 G(\rho_2)~.$$ With the boundary condition $G(0) = 1$ it is easy to solve this equation. Again one obtains eq.(\[exp\])
The Sudakov form factor in the electroweak theory
=================================================
The fermions in the electroweak theory are such that the left-handed fields are doublets of the weak isospin group and the right handed fields are singlets of $SU(2)\times U_y(1)$. So one has a left, $F_L$, and a right, $F_R$, Sudakov form factors. In line with the approximation of massless particles there is no chirality flip amplitude. Because the right-handed fermions couple to the $U_Y(1)$ boson, $F_R$ only gets contributions from $Z$ and photon exchange. Hence, borrowing directly from the results of the previous section, one can easily obtain: $$\label{fr}
F_R(s/ \mu^2, ~s/M^2) =\exp(- \psi_R)~,$$ with $$\begin{aligned}
\label{psir}
\psi_R =
\frac{\alpha Q^2}{4 \pi}
[\ln^2(s/\mu^2) + \tan^2 \theta \ln^2(s/M^2)]~, \end{aligned}$$ where $M=M_Z$ and $\theta_W$ is the Weinberg angle.
Now, lets us derive the IREE for $F_L(s/\mu^2,s/M^2)$, neglecting the mass difference between the $Z$ and the $W$ boson, i.e., $M_W=M_Z=M$. To do this, applying the Gribov bremsstrahlung theorem, one should factorize the boson with the minimal $k_\perp$. If the integration over the $k_\perp$ of the factorized boson is done in the region $M<k_\perp<\sqrt{s}$ this particle can be any of the four bosons present in the theory, the $W^\pm$, the $Z$ and the photon. This yields the following contribution to the IREE: $$\label{wz}
\tilde{M}^{WZ} = - \frac{e^2Q^2 + C_{WZ}}{8 \pi^2}
\int_{M^2}^{s} \frac{d k^2_{\perp}}{k^2_{\perp}}
\ln(s / k^2_{\perp}) F(s/ k^2_{\perp},~s/k^2_{\perp}) ,$$ where we have explicitly separated the photon from the $WZ$ contributions. The latter are proportional to $$\label{cwz}
C_{WZ} = g^2 [(t^2_1 + t^2_2) +
(1/ \cos^2 \theta_W)(t_3 - \sin^2 \theta_W Q)^2].$$ The second DL region is when $\mu<k_\perp<M$, but, now, the integration over the $k_\perp$ of the factorized boson only gives a DL contribution if this boson is a photon. So we obtain $$\label{phot}
\tilde{M}^{\gamma} = - \frac{e^2 Q^2}{8 \pi^2}
\int_{\mu^2}^{M^2} \frac{d k^2_{\perp}}{k^2_{\perp}}
\ln(s / k^2_{\perp}) F(s/ k^2_{\perp},~s/M^2)~.$$
The sum of the factorized contributions given by eqs.(\[wz\]) and (\[phot\]) together with the Born value $F_{Born} = 1$ leads to the IREE for the Sudakov form factor $F_L(s/ \mu^2,~s/M^2)$ in the integral form,
$$\begin{aligned}
\label{eqF}
F(s/ \mu^2,~s/M^2) = 1
- \frac{e^2Q^2}{8 \pi^2}
\int_{\mu^2}^{M^2} \frac{d k^2_{\perp}}{k^2_{\perp}}
\ln(s / k^2_{\perp}) F(s/ k^2_{\perp},~s/M^2) - \nonumber \\
- \frac{(C_{WZ} + e^2 Q^2)}{8 \pi^2}
\int_{M^2}^{s} \frac{d k^2_{\perp}}{k^2_{\perp}}
\ln(s / k^2_{\perp}) F(s/ k^2_{\perp},~s/k^2_{\perp}). \end{aligned}$$
This equation is similar to Eq. (\[eqf\]) where $g_1$ is replaced by $e Q$ and $g_2$ is replaced by $C_{WZ}$ . Differentiating Eq. (\[eqF\]) first with respect to $\mu$ and then with respect to $M$ we obtain the IREE in the differential form (cf eqs. (\[dmu\]),(\[dm\])):
$$\label{dmuF}
\frac{\partial F_L(\rho_1,~\rho_2)}{\partial \rho_1} =
- \frac{e^2 Q^2}{8 \pi^2} \rho_1 F_L(\rho_1,~\rho_2) ~,$$
$$\label{dmF}
\frac{d \tilde{G}(\rho_2)}{d \rho_2} =
- \frac{C_{WZ}}{8 \pi^2} \rho_2 \tilde{G}(\rho_2)~,$$
where $\tilde{G}$ is the general solution of eq. (\[dmuF\]). Finally, solving this equation in the same way that we have solved eq. (\[dm\]) we obtain $$\label{F}
F_L(s/ \mu^2, ~s/M^2) = \exp(-\Psi_L) ~,$$ with $$\begin{aligned}
\label{psi}
\Psi_L &=& \frac{e^2 Q^2}{16 \pi^2} \ln^2(s/ \mu^2) \nonumber \\
& & + \frac{g^2}{16 \pi^2}
[(t^2_1 + t^2_2) + \frac{1}{\cos^2 \theta_W}(t_3 - \sin^2 \theta_W Q)^2]
\ln^2(s/ M^2) ~.\end{aligned}$$
The off-shell Sudakov electroweak form factor
==============================================
Eqs. (\[foffshell\]) and (\[fonshell\]) show that even in the simplest QED case the on-shell Sudakov form factor cannot be obtained from the expression for the off-shell form factor with the simple replacement of the electron $p^2_1$ and $p^2_2$ by the electron mass or by the infrared cut-off. Both form factors have to be calculated independently. Obviously, the same is true for the off-shell electroweak Sudakov form factor $\tilde{F}_L$. Before we calculate $\tilde{F}_L$, it is instructive to demonstrate how the IREE for the off-shell QED form factor can be obtained. This form factor $\tilde{f}$ depends on $q^2,~p^2_1,~p^2_2$ and also depend on the infrared cut-off $\mu$, i.e., $\tilde{f} =
\tilde{f}((q^2/ \mu^2,~p^2_1/\mu^2, ~p^2_2/\mu^2 )$.
Similarly to the on-shell case, factorizing the contribution of the virtual photon with the minimal $k_{\perp}$ leads to the following IREE: $$\label{offint}
\tilde{f}((q^2/ \mu^2,~p^2_1/\mu^2, ~p^2_2/\mu^2 ) = 1 -
\frac{\alpha}{2 \pi}
\int_D \frac{d k^2_{\perp}}{k^2_{\perp}}\frac{d \beta}{\beta}
\tilde{f}((q^2/ k^2_{\perp},~p^2_1/k^2_{\perp}, ~p^2_2/k^2_{\perp} )~.$$ However, in contrast to the IREE for the on-shell form factor, the region $D$ in the previous equation now depends on $p^2_1$ and $p^2_2$. The region $D$ is different for the particular case when the virtualities $p^2_{1,2}$ are small enough such that $$\label{d1}
p^2_1 p^2_2 < q^2 \mu^2$$ or large enough so that $$\label{d2}
p^2_1p^2_2 > q^2 \mu^2 ~.$$ In the former case we denote the integrating region $D_1$ and call it $D_2$ in the latter case. In Fig. \[fig5\] we show $D_1$.
In the plane $k_\perp^2=\alpha\beta$, $\beta$, $D_1$ is bounded by the “on-shell” curves $s\beta=k_\perp^2$ and $s\beta=s$ and also by the new curves $s\beta=p_2^2$ and $p_1^2p_2^2=q^2\mu^2$. From now on we call $\tilde{f}_1$ the contribution to $\tilde{f}$ from the region $D_1$. Performing the $\beta$ integration in eq.(\[offint\]) and after that differentiating with respect to $\mu^2$ we obtain $$\label{offdif}
\frac{\partial \tilde{f}_1}{\partial x} +
\frac{\partial \tilde{f}_1}{z_1} + \frac{\partial \tilde{f}_1}{ z_2} =
- \frac{\alpha}{2 \pi}\big[ x - z_1 - z_2 \big] \tilde{f}_1 ,$$ where $x = \ln(q^2/ \mu^2)$, $~z_1 = \ln(p^2_1/\mu^2)$ and $~z_2 = \ln(~p^2_2/\mu^2)$. Obviously, the solution to this equation is, $$\label{offsol1}
\tilde{f}_1(q^2,p^2_1,p^2_2, \mu^2) =
e^{-\big[(\alpha/4 \pi)(x^2 - z^2_1 - z^2_2) \big]}~.$$
In the kinematical region specified by eq.(\[d2\]), the integration of eq.(\[offint\]) is over the domain $D_2$. This region does not involve $\mu$ so we have $$\label{eqd2}
-\mu^2\frac{\partial \tilde{f}_2}{\partial \mu^2} =
\frac{\partial \tilde{f}_2}{\partial x} +
\frac{\partial \tilde{f}_2}{z_1} + \frac{\partial \tilde{f}_2}{ z_2} = 0$$ and its general solution is $$\label{d2sol}
\tilde{f}_2 = \Phi(x - z_2, x - z_2)$$ where $\Phi$ is an arbitrary function. The matching condition $\tilde{f}_1 = \tilde{f}_2$ when $p^2_1p^2_2 = q^2 \mu^2 $, which is equivalent to $x=z_1+z_2$, leads to $$\tilde{f}_2 = \exp \big[
-(\alpha/2\pi)\ln(q^2/p^2_1)\ln(q^2/p^2_2)\big].$$ This is exactly the expression that we had anticipated in the introduction (cf. eq.(\[foffshell\])).
Now, it should be clear that following a similar prescription one obtains the IREE for $\tilde{F}_L$, namely $$\begin{aligned}
\label{eqFo}
\tilde{F}(s/ \mu^2,~s/M^2) = 1
- \frac{e^2}{8 \pi^2}
\int_{D} \frac{d k^2_{\perp}}{k^2_{\perp}}
\ln(s / k^2_{\perp}) \tilde{F}(s/ k^2_{\perp},~s/M^2) - \nonumber\\
- \frac{(C_{WZ} + e^2 Q^2)}{8 \pi^2}
\int_{D'} \frac{d k^2_{\perp}}{k^2_{\perp}}
\ln(s / k^2_{\perp}) \tilde{F}(s/ k^2_{\perp},~s/k^2_{\perp}). \end{aligned}$$ The regions $D$ and $D'$ are bounded, in addition to the “on-shell” requirements, by the relation between $q^2 \mu^2$, $q^2 M^2 $ and $p^2_1p^2_2$. We specify the following basic off-shell kinematic regions:\
\
$R_1$: $\mu^2<p^2_1,p^2_2 < M^2,~~~~~p^2_1p^2_2 <q^2 \mu^2 $,\
\
$R_2$: $\mu^2<p^2_1,p^2_2 < M^2,~~~~~q^2 \mu^2 < p^2_1p^2_2 < q^2 M^2
$\
\
$R_3$: $p^2_1p^2_2 > M^2,~~~~~~p^2_1p^2_2 <q^2 M^2 $.\
\
$R_3$: $p^2_1p^2_2 > M^2,~~~~~~p^2_1p^2_2 >q^2 M^2 $.\
\
\
For each region, $R_i$ ($i=1\dots 4$), we obtain $$\label{offF}
\tilde{F_L} = \exp(-\psi_i),$$ with $$\label{psi1}
\psi_1 = \frac{e^2 Q^2}{16 \pi^2}
[\ln^2(q^2/ \mu^2) - \ln^2(p^2_1/\mu^2) - \ln^2(p^2_2/\mu^2)]
+ \frac{g^2}{16 \pi^2}
%[(t^2_1 + t^2_2) + \frac{1}{\cos^2 \theta_W}(t_3 - \sin^2 \theta_W Q)^2]
C_{WZ}\ln^2(s/ M^2) ~$$ for the region $R_1$, $$\label{psi2}
\psi_2 = \frac{e^2 Q^2}{8 \pi^2} \ln(q^2/ p^2_1)\ln(q^2/ p^2_2)
+ \frac{g^2}{16 \pi^2}
C_{WZ}\ln^2(s/ M^2) ~$$ for the region $R_2$, $$\label{psi3}
\psi_3 = \frac{e^2 Q^2}{8 \pi^2} \ln(q^2/ p^2_1)\ln(q^2/ p^2_2)
+ \frac{g^2}{16 \pi^2}
C_{WZ}[\ln^2(q^2/ M^2) - \ln^2(p^2_1/M^2) - \ln^2(p^2_2/M^2)] ~$$ for the region $R_3$, and finally $$\label{psi3}
\psi_4 = \frac{e^2 Q^2 + C_{WZ}}{8 \pi^2} \ln(q^2/ p^2_1)\ln(q^2/ p^2_2)$$ for the region $R_4$.
Discussion
==========
Expression (\[F\]) for the electroweak Sudakov form factor, $F_L$, accounts for the mass difference between the photon and the weak bosons. On the other hand, it neglects the difference between the masses of the $W$ and the $Z$. It has been obtained introducing different infrared cut-offs for the integration over transverse momenta of different gauge bosons: cut-off $\mu$ for photons and cut-off $M$ for the $W$ and the $Z$. Expanding Eq. (\[F\]) into serias, one can easily extract the first-loop and the second-loop DL contributions. The first-loop contribution (save the minus sign) is given by Eq. (\[psi\]). The DL contributions to $F_L$ in two loops were also calculated in [@bw]. Before comparing our results with results of [@bw], let us notice that besides the DL contributions we account for, the DL contributions in [@bw] account also for the double logarithms of $m^2/\mu^2$, where $m$ stands for fermion masses. Such contributions are absent if the photon cut-off is equal or greater than masses of involved fermions as we assume in this paper. Having dropped them, we arrive at agreement with the two-loop results of [@bw]. Usually, DL calculations involve only one mass scale. Using one mass scale, for instance $M$, for all DL terms in $\Psi_L$ of Eq. (\[psi\]) allows us to rewrite it as $$\begin{aligned}
\label{onescale}
\Psi_L = \frac{g^2}{16 \pi^2}
(t^2_i + g'^2(Y/2)^2)\ln^2(s/M^2) +
\frac{e^2 Q^2}{16 \pi^2} [2\ln(s/ M^2) \ln(M^2/\mu^2) \nonumber \\
+ \ln^2(M^2/\mu^2)].~~~\end{aligned}$$ The first term in the rhs of this equation is the DL contribution with the same scale $M$ for both the electro-magnetic and the weak interactions. The second term is formally single-logarithmic and therefore it is beyond control of the IREE with one mass scale. Finally, the third term, $\ln^2(M^2/\mu^2)$, does not depend on $s$. It is usually dropped in the IREE with one mass scale.
An expression for $\Psi_L$ similar to this one was obtained earlier in ref.[@flmm] using a similar approach of writing an IREE with two infrared cut-offs. However, there are certain differences between our result and the one given in eq. (28) of ref. [@flmm]. Besides the cut-offs $\mu$ and $M$, Fadin [*et al.*]{} [@flmm] have another mass scale $m$ defined such that $m<M$. Setting $m=M$ the result derived by Fadin [*et al.*]{} agrees with ours except for an over all factor $1/2$ which we don’t have. We believe that the origin of this disagreement could be traced back to the use of Fadin [*et al.*]{} of the axial gauge. In fact, in this gauge, the DL contributions arise from fermion self-energy diagrams. Then, it could be that the authors of ref. [@flmm] give the one electron self-energy contribution rather than the Sudakov form factor, which is the double of it. There is another difference between our IREE for $F_L$ (see eq.\[eqF\]) and the corresponding equation in ref [@flmm]. In the work of Fadin [*et al.*]{} the rhs of the evolution equation contains an extra logarithmic dependence on the fermion mass $m$. The $m$ dependence comes from the fermion propagators. But, with DL accuracy, one can write the propagators in terms of $\alpha$ and $\beta$ as $(p_2 - k)^2 - m^2 = k^2 - 2p_2 k \approx -s\beta - k^2_{\perp}
$. To obtain a log term from the $\beta$ integration one has to require that $s\beta \gg k_\perp^2$. Then, this condition fixes the lower limit of integration as $k_\perp^2$ and the remaining integral is $$\int_{\mu^2}^s\frac{dk_\perp^2}{k_\perp^2}\ln{(s/k_\perp^2)}$$ with no logarithmic contribution depending on $m$.
As a final remark, we would like to stress that the exponentiation of the DL contributions to the EW Sudakov form factor, $F_L$, takes place when both the initial and the final state are not specified and summation over their weak isospin is done. This means that, in contrast to QCD, such form factors should be regarded as a theoretical object with rather limited applications. In principle the off-shell version of $F_L$ could be considered as an ingredient in the calculation of some more complicated physical processes.
Acknowledgement
===============
The work is supported by grants CERN/2000/FIS/40131/ and INTAS-97-30494.
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[^1]: Permanent address: A.F. Ioffe Physico-Technical Institute, St.Petersburg 194021, Russia
| {
"pile_set_name": "ArXiv"
} |
---
address: 'Instituto de Matemática, UFF, Rua Mário Santos Braga S/N Valonguinho, Niterói, Rio de Janeiro, Brasil 24020-140'
author:
- 'Javier Rib[ó]{}n'
bibliography:
- 'rendu.bib'
title: Topological rigidity of unfoldings of resonant diffeomorphisms
---
[^1] [^2] [^3]
Abstract {#abstract .unnumbered}
========
We prove that a topological homeomorphism conjugating two generic $1$-parameter unfoldings of $1$-variable complex analytic resonant diffeomorphisms is holomorphic or anti-holomorphic by restriction to the unperturbed parameter. We provide examples that show that the genericity hypothesis is necessary. Moreover we characterize the possible behavior of conjugacies for the unperturbed parameter in the general case. In particular they are always real analytic outside of the origin.
We describe the structure of the limits of orbits when we approach the unperturbed parameter. The proof of the rigidity results is based on the study of the action of a topological conjugation on the limits of orbits.
Introduction
============
We are interested in the study of the topological properties of unfoldings of tangent to the identity diffeomorphisms. We define ${\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{n+1},0)$}}}$ as the set of $n$-parameter unfoldings of local complex analytic $1$-variable tangent to the identity diffeomorphisms. An element $\varphi$ of ${\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{n+1},0)$}}}$ is of the form $$\varphi(x_{1},\hdots,x_{n},y)=(x_{1},\hdots,x_{n},f(x_{1},\hdots,x_{n},y))$$ where $f(0)=0$ and $(\partial f/\partial y)(0)=0$. We define ${\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ as the set of diffeomorphisms whose fixed points set does not contain $x=0$. We denote by $N(\varphi)$ or $N$ if there is no confusion the number of points in $\{x=x_{0}\} \cap \mathrm{Fix} (\varphi)$ for any $x_{0} \neq 0$. Clearly it is a topological invariant. We prove the following rigidity theorem:
\[teo:main\] Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$ such that there exists a homeomorphism $\sigma$ satisfying $\sigma \circ \varphi = \eta \circ \sigma$. Suppose that either $\varphi_{|x=0}$ or $\eta_{|x=0}$ is non-analytically trivial. Then $\sigma_{|x=0}$ is holomorphic or anti-holomorphic.
We denote by ${\mbox{{\rm Diff}{${\,}_{1}({\mathbb C}^{},0)$}}}$ the group of local complex analytic $1$-dimensional diffeomorphisms whose linear part is the identity. An element $\phi \in {\mbox{{\rm Diff}{${\,}_{1}({\mathbb C}^{},0)$}}}$ is analytically trivial if it is embedded in an analytic flow, i.e. $\phi$ is the exponential of an analytic singular local vector field $X=g(y) \partial / \partial y$. A consequence of the Ecalle-Voronin analytic classification of tangent to the identity diffeomorphisms [@Ecalle] [@V] [@mal:ast] is that elements of ${\mbox{{\rm Diff}{${\,}_{1}({\mathbb C}^{},0)$}}}$ are generically non-analytically trivial.
In particular if a generic element $\varphi$ of ${\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ is topologically conjugated to $\eta \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ then $\varphi_{|x=0}$ and $\eta_{|x=0}$ are either holomorphically or anti-holomorphically conjugated. The result is far from trivial since a topological class of conjugacy of a tangent to the identity diffeomorphism in one variable contains a continuous infinitely dimensional moduli of analytic classes of conjugacy.
Let us point out that all the topological conjugations in this paper between elements of ${\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ preserve the fibration $dx =0$. In other words they are of the form $\sigma(x,y) =(\sigma_{0}(x), \sigma_{1}(x,y))$. This is a natural hypothesis since we are interested in the topological classification of unfoldings.
A natural problem is determining the classes of conjugacy of unfoldings up to topological, formal or analytic equivalence.
The study of the analytic properties of unfoldings is an active field of research. A natural idea to study an unfolding $\varphi$ in ${\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ is comparing the dynamics of $\varphi$ and ${\rm exp}(X)$ where $X = g(x,y) \partial / \partial y$ is a vector field with $\mathrm{Fix} (\varphi) = \mathrm{Sing} (X)$ whose time $1$ flow “approximates" $\varphi$. This point of view has been developed by Glutsyuk [@Gluglu]. In this way extensions of the Ecalle-Voronin invariants [@Ecalle] [@V] to some sectors in the parameter space are obtained. The extensions are uniquely defined. The sectors of definition have to avoid a finite set of directions of instability, typically associated (but not exclusively) to small divisors phenomena. The rich dynamics of $\varphi$ around the directions of instability prevents the extension of the Ecalle-Voronin invariants to be defined in the neighborhood of the instability directions. Interestingly the study of the dynamics around instability directions is one of the key elements of the proof of the Main Theorem.
A different point of view was introduced by Shishikura for codimension $1$ unfoldings [@Shishi]. The idea is constructing appropriate fundamental domains bounded by two curves with common ends at singular points: one curve is the image of the other one. Pasting the boundary curves by the dynamics yields (by quasiconformal surgery) a Riemann surface that is conformally equivalent to the Riemann sphere. The logarithm of an appropriate affine complex coordinate on the sphere induces a Fatou coordinate for $\varphi$. These ideas were generalized to higher codimension unfoldings by Oudkerk [@Oudkerk]. In this approach the first curve is a phase curve of an appropriate vector field transversal to the real flow of $X$. In both cases the Fatou coordinates provide Lavaurs vector fields $X^{\varphi}$ such that $\varphi= {\rm exp}(X^{\varphi})$ [@Lavaurs]. The Shishikura’s approach was used by Mardesic, Roussarie and Rousseau to provide a complete system of invariants for unfoldings of codimension $1$ tangent to the identity diffeomorphisms [@MRR]. Rousseau and Christopher classified the generic unfoldings of codimension $1$ resonant diffeomorphisms [@Rou-Chris:mod]. The analytic classification for the unfoldings of finite codimension resonant diffeomorphisms was completed in [@JR:mod] by using the Oudkerk’s point of view.
We described the formal invariants of elements $\varphi$ of ${\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{n+1},0)$}}}$ for any $n \in {\mathbb N}$ in [@UPD]. The invariants are divided in two sets, namely those that are analogous to the $1$-dimensional formal invariants and invariants that are associated to the position of $\mathrm{Fix} (\varphi)$ with respect to the fibration $dx_{1}= \hdots = dx_{n} = 0$.
Topological classification
--------------------------
In contrast with the analytic and formal cases there is no topological classification of unfoldings of tangent to the identity diffeomorphisms. One of the obstacles is the absence of a complete system of analytic invariants for elements of ${\mbox{{\rm Diff}{${\,}_{}({\mathbb C}^{},0)$}}}$. More precisely the problem is associated with small divisors; it is not known the topological classification of elements $\phi(z) = \lambda z + O(z^{2}) \in {\mbox{{\rm Diff}{${\,}_{}({\mathbb C}^{},0)$}}}$ such that $\lambda \in {\mathbb S}^{1}$ is not a root of the unit and $\phi$ is not analytically linearizable.
Let $\varphi = (x,f(x,y)) \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$. We denote by $m(\varphi)$ the vanishing order of $f-y$ at the line $x=0$. We study unfoldings $\varphi$ such that $(N,m)(\varphi) \neq (1,0)$. The remaining case is trivial and hence uninteresting since the only topological invariant is the vanishing order of $f(0,y)-y$ at $0$. From now on we assume $(N,m) \neq (1,0)$.
The situation in absence of small divisors (multi-parabolic case) has been studied in [@rib-mams]. An element $\varphi(x,y)=(x,f(x,y))$ of ${\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ is multi-parabolic if $(\partial f/\partial y)_{|\mathrm{Fix} (\varphi)} \equiv 1$. A complete system of topological invariants is presented in [@rib-mams] for the classification of multi-parabolic diffeomorphisms under the assumption that a conjugation $\sigma$ such that $\sigma \circ \varphi = \eta \circ \sigma$ is of the form $\sigma(x,y)=(x, f(x,y))$ and satisfies $\sigma_{|\mathrm{Fix} (\varphi)} \equiv Id$. One of the topological invariants is the analytic class of the unperturbed diffeomorphism of the unfolding. Moreover $\sigma_{|x=0}$ is always a local biholomorphism.
A key point of the classification is a shadowing property for multi-parabolic diffeomorphisms. Roughly speaking, given a multi-parabolic $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ there exists a vector field $X = g(x,y) \partial / \partial y$ with $\mathrm{Fix} (\varphi) = \mathrm{Sing} (X)$ such that every orbit of $\varphi$ can be approximated by an orbit of ${\rm exp}(X)$ (Theorem 7.1 [@rib-mams]). As a consequence the continuous dynamical system defined by the real flow $\Re (X)$ of $X$ is a good model of the topological behavior of $\varphi$. In spite of this, generically there is no shadowing for unfoldings of tangent to the identity diffeomorphisms. Indeed the existence of a shadowing property for a non-multi-parabolic element $\varphi$ of ${\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ implies that $\varphi$ is embedded in an analytic flow [@rib-ast]. Our strategy in this paper includes, as in the multi-parabolic case, approximating $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ with ${\rm exp}(X)$ for some local vector field $X = g(x,y) \partial / \partial y$ and then studying the real flow of $X$ to try to obtain interesting dynamical phenomena associated to $\varphi$. Since there is no shadowing property for all orbits of $\varphi$ we have to show that the dynamics of $\Re (X)$ that we are trying to replicate for $\varphi$ takes place in regions in which the orbits of ${\rm exp}(X)$ and $\varphi$ remain close.
The main tool in this paper is the study of Long Trajectories and Long Orbits. These concepts were introduced in [@rib-mams]. They are analogous to the concept of homoclinic trajectories for polynomial vector fields introduced by Douady, Estrada and Sentenac [@DES]. Let us focus on vector fields since the concepts are analogous and the presentation is a little simpler. Consider a local vector field $X=g(x,y) \partial / \partial y$ with $g(0)=0$, $(\partial g/\partial y)(0)=0$ and $g(0,y) \not \equiv 0$, i.e. an unfolding of a non-trivial vector field of vanishing order higher than $1$. Roughly speaking a Long Trajectory is given by the choice of a point $y_{+} \neq 0$, a curve $\beta$ in the parameter space and a continuous function $T:{\beta} \to {\mathbb R}^{+}$ such that $$(0,y_{-}) \stackrel{\mathrm{def}}{=} \lim_{x \in \beta, x \to 0} \mathrm{exp}(T(x)X)(x,y_{+})$$ exists and $\lim_{x \in \beta, x \to 0} T(x) = \infty$. In general $(0,y_{-})$ does not belong to the trajectory through $(0,y_{+})$. We go from $(0,y_{+})$ to $(0,y_{-})$ by following the real flow of $X$ an infinite time. We say that $(X,y_{+},\beta,T)$ generates a [*Long Trajectory*]{} of $X$ containing $(0,y_{-})$. Denote $\varphi = {\rm exp}(X)$. The point $(0,y_{-})$ is in the limit of the orbits of $\varphi$ passing through points $(x,y_{+})$ with $x \in T^{-1}({\mathbb N})$ when $x \to 0$. We say that $(\varphi,y_{+},\beta,T)$ generates a [*Long Orbit*]{} containing $(0,y_{-})$. By replacing $T$ with $T+s$ for $s \in {\mathbb R}$ we obtain that $\mathrm{exp}(s X)(0,y_{-})$ is in the Long Orbit generated by $(\varphi,y_{+},\beta,T+s)$. The rest of the points in a neighborhood of $(0,y_{-})$ in $x=0$ are also in Long Orbits of $\varphi$ through $(0,y_{+})$. They are obtained by varying the curve $\beta$. In particular the complex flow of the infinitesimal generator of $\varphi_{|x=0}$ in the repelling petal containing $(0,y_{-})$ can be retrieved from Long Orbits through $(0,y_{+})$. In other words such complex flow is in the topological closure of the pseudogroup generated by $\varphi$.
The Long Orbits phenomenon reminds Shcherbakov and Nakai’s results [@Shcherbakov-topan] [@Nakai-nonsolvable] for non-solvable pseudogroups of holomorphic diffeomorphisms of open neighborhoods of $0$ in ${\mathbb C}$. A pseudogroup is non-solvable if its associated group of local diffeomorphisms is non-solvable. More precisely Nakai proves that there exists a real semianalytic subset $\Sigma$ such that any orbit of the pseudogroup is dense or empty in every connected component of the complementary of $\Sigma$ (see [@Nakai-nonsolvable] for further details). Moreover the proof of Scherbakov’s theorem (a homeomorphism conjugating non-solvable pseudogroups is holomorphic or anti-holomorphic) by Nakai is based on finding real flows of holomorphic vector fields that are in the topological closure of a non-solvable pseudogroup.
Long Orbits are interesting in themselves. Long Trajectories and Long Orbits are phenomena related to instable behavior in the unfolding. Given $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ and a curve $\beta$ in the parameter space supporting a Long Orbit then $\beta$ is tangent at $0$ to a unique semi-line $\lambda {\mathbb R}^{+}$ for some $\lambda \in {\mathbb S}^{1}$. Moreover $\lambda$ belongs to a finite set that only depends on $\varphi$. Generically in the parameter space there are no Long Orbits. Notice that the absence of Long Orbits is a necessary condition in the Glutsyuk point of view described above. In spite of being scarce Long Orbits somehow vary continuously. For instance the function $T$ in the definition can be calculated by applying conveniently the residue theorem. Indeed $T$ is (up to a bounded additive function) a sum of meromorphic functions that are formal invariants of the unfolding. The residue formula allows to describe the evolution of the Long Orbits when we replace $\beta$ with nearby curves. On the one hand Long Orbits appear in the regions of instability of the unfolding and generically together with small divisors phenomena. On the other hand they have a (rich) regular structure. The main technical difficulty regarding Long Orbits is proving their existence and properties. Once the setup is established the Main Theorem is obtained by a relatively simple description of the action of topological conjugations on Long Orbits.
The analytic classification of elements of ${\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ depends on studying transversal structures to the dynamics of the unfolding. The point of view behind the Main Theorem is closer to Glutsyuk’s point of view. Anyway the focus on the parameter space is of complementary type. The extension of the Ecalle-Voronin invariants à la Glutsyuk is obtained for regions of stability of the unfolding. Nevertheless the topological dynamics in stability regions is uninteresting. The significant topological information is located in the neighborhood of the instability directions.
Rigidity of unfoldings
----------------------
The rigidity result of the Main Theorem extends to the general case.
\[def:simb\] Let $\phi, \rho \in {\mbox{{\rm Diff}{${\,}_{1}({\mathbb C}^{},0)$}}} \setminus \{Id\}$. We say that $\phi$ and $\rho$ have the same topological bifurcation type and we denote $\phi \sim_{b} \rho$ if there exist topologically conjugated unfoldings $\varphi$, $\varrho$ such that $\varphi_{|x=0} \equiv \phi$, $\varrho_{|x=0} \equiv \rho$ and $N(\varphi) \neq 1$. If the restriction $\sigma_{|x=0}$ of the topological conjugation $\sigma$ to the unperturbed line is orientation-preserving (resp. orientation-reversing) we denote $\phi {\sim}_{b}^{+} \rho$ (resp. $\phi {\sim}_{b}^{-} \rho$).
We could naively think that this equivalence relation is the same induced by the topological classification. The Main Theorem implies that a topological class of conjugacy contains a continuous moduli of classes of ${\sim}_{b}$. This is even true if we restrict ourselves to diffeomorphisms that are embedded in analytic flows (Lemma \[lem:res\]) since residues are not topological invariants. Somehow surprisingly the analytic nature of a generic $\phi \in {\mbox{{\rm Diff}{${\,}_{1}({\mathbb C}^{},0)$}}}$ is encoded in the topological dynamics of any of its non-trivial unfoldings.
This kind of rigidity properties are typical in theory of complex analytic foliations. We already mentioned the results on non-solvable groups by Scherbakov and Nakai. Other instances of the rigidity of the moduli topological/analytic can be found in Ilyashenko [@Ilya-toppor], Cerveau and Sad [@Cerveau-Sad:modules], Lins Neto, Sad and Scardua [@NSS:rigidity], Marín [@Marin-rigidity], Rebelo [@Rebelo-rigidity]... Moreover Cerveau and Moussu proved that in the context of non-solvable non-exceptional groups, formal conjugacy implies analytic conjugacy [@CM:bsmf].
A natural question is what happens in the setup of the Main Theorem if $\varphi_{|x=0}$ is analytically trivial. It turns out that the situation is still rigid.
\[teo:general\] (General Theorem) Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ with $(N,m)(\varphi) \neq (1,0)$ such that there exists a homeomorphism $\sigma$ satisfying $\sigma \circ \varphi = \eta \circ \sigma$. Then $\sigma_{|x=0}$ is affine in Fatou coordinates. Moreover $\sigma_{|x=0}$ is orientation-preserving if and only if the action of $\sigma$ on the parameter space is orientation-preserving.
Topological conjugations are of the form $\sigma(x,y) =(\sigma_{0}(x), \sigma_{1}(x,y))$. We say that the action of $\sigma$ in the parameter space is orientation-preserving if $\sigma_{0}$ is. Analogously we define the concept of holomorphic action on the parameter space.
The definition of affine in Fatou coordinates is provided in Definitions \[def:afffc\] and \[def:afffc2\]. Affine in Fatou coordinates implies real analytic outside the origin. In order to compare the Main Theorem and Theorem \[teo:general\] let us point out that holomorphic conjugations between elements of ${\mbox{{\rm Diff}{${\,}_{1}({\mathbb C}^{},0)$}}} \setminus \{Id\}$ are translations in Fatou coordinates. The Main Theorem is a consequence of Theorem \[teo:general\]. Indeed we show that affine in Fatou coordinates implies holomorphic or anti-holomorphic in the non-analytically trivial case.
How to strengthen the General Theorem? A first approach is provided by the Main Theorem by considering generic classes of analytic conjugacy. Another possibility is trying to impose conditions on the action of conjugations on the parameter space. Finally we notice that for analytically trivial elements of ${\mbox{{\rm Diff}{${\,}_{1}({\mathbb C}^{},0)$}}}$ the formal and analytic conjugacy classes coincide. So it is interesting to study the action of $\sigma_{|x=0}$ on formal invariants. The next propositions establish a relation between the topological, formal and analytic classifications.
\[pro:holpar\] Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ with $(N,m)(\varphi) \neq (1,0)$ such that there exists a homeomorphism $\sigma$ satisfying $\sigma \circ \varphi = \eta \circ \sigma$. Suppose that the action of $\sigma$ on the parameter space is holomorphic (resp. anti-holomorphic). Then $\sigma_{|x=0}$ is holomorphic (resp. anti-holomorphic).
Let $\phi(y) = y + c y^{\nu +1} + h.o.t. \in {\mbox{{\rm Diff}{${\,}_{1}({\mathbb C}^{},0)$}}}$ with $\nu \in {\mathbb N}$ and $c \in {\mathbb C}^{*}$. The number $\nu$ determines the class of topological conjugacy of $\phi$. The diffeomorphism $\phi$ is formally conjugated to a unique diffeomorphism $y + y^{\nu +1} + ((\nu+1)/2 - \lambda) y^{2 \nu +1}$ for some $\lambda \in {\mathbb C}$. The pair $(\nu,\lambda)$ provides a complete system of formal invariants. We define $Res_{\phi}(0)=\lambda$ and $Res_{\varphi}(0,0) = Res_{\varphi_{|x=0}}(0)$ for $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$.
\[pro:rigi\] Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$ such that there exists a homeomorphism $\sigma$ satisfying $\sigma \circ \varphi = \eta \circ \sigma$. Suppose that either $\varphi_{|x=0}$ or $\eta_{|x=0}$ is analytically trivial. Suppose that either $Res_{\varphi}(0,0) \not \in i {\mathbb R}$ or $Res_{\eta}(0,0) \not \in i {\mathbb R}$. Then
- If $\sigma_{|x=0}$ is orientation-preserving then $\sigma_{|x=0}$ is holomorphic if and only if $Res_{\varphi}(0,0) =Res_{\eta}(0,0)$.
- If $\sigma_{|x=0}$ is orientation-reversing then $\sigma_{|x=0}$ is anti-holomorphic if and only if $Res_{\varphi}(0,0) =\overline{Res_{\eta}(0,0)}$.
On the one hand it is possible to construct examples of diffeomorphisms $\varphi, \eta$ satisfying the hypotheses of the previous proposition such that $\varphi_{|x=0}$ and $\eta_{|x=0}$ are neither holomorphically nor anti-holomorphically conjugated (Section \[sec:build\]). On the other hand if they are holomorphically conjugated (in the orientation-preserving case) then $\sigma_{|x=0}$ is also holomorphic. In other words given $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ as in Proposition \[pro:rigi\] the analytic class of $\eta_{|x=0}$ is not determined for $\eta$ in the class of topological conjugacy of $\varphi$ but the conjugation $\sigma_{|x=0}$ is determined up to composition with a holomorphic diffeomorphism (see Proposition \[pro:atu\]). The condition $Res_{\varphi}(0,0) \not \in i {\mathbb R}$ on formal invariants implies flexibility in the analytic classes of $\eta_{|x=0}$ but once they are fixed there is rigidity of the conjugating mappings.
Next we consider the case of purely imaginary formal invariants.
\[pro:atui\] Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$ such that there exists a homeomorphism $\sigma$ satisfying $\sigma \circ \varphi = \eta \circ \sigma$. Suppose that either $\varphi_{|x=0}$ or $\eta_{|x=0}$ is analytically trivial. Suppose that either $Res_{\varphi}(0,0) \in i {\mathbb R}$ or $Res_{\eta}(0,0) \in i {\mathbb R}$. Then $\varphi_{|x=0}$ and $\eta_{|x=0}$ are analytically conjugated (resp. anti-analytically conjugated) if $\sigma$ is orientation-preserving (resp. orientation-reversing) on the parameter space.
The roles of analytic classes and conjugacies are reversed with respect to Proposition \[pro:rigi\]. Indeed there are at most $2$ classes of analytic conjugacy of $\eta_{|x=0}$ in the set composed of the diffeomorphisms $\eta \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ in the topological class of $\varphi$. In spite of the rigidity of analytic classes, conjugations are not rigid. Even if $\varphi_{|x=0}$ and $\eta_{|x=0}$ are analytically conjugated the mapping $\sigma_{|x=0}$ is not necessarily holomorphic. Examples of this behavior are presented in Section \[sec:build\].
The following result is an immediate consequence of Proposition \[pro:atui\], the Main and the General Theorems.
\[cor:topiana\] Let $\phi, \rho \in {\mbox{{\rm Diff}{${\,}_{1}({\mathbb C}^{},0)$}}} \setminus \{Id\}$ with $Res_{\phi}(0) \in i {\mathbb R}$. Then $\phi$ and $\rho$ have the same topological bifurcation type if and only if $\phi$ and $\rho$ are holomorphically or anti-holomorphically conjugated. Moreover $\phi \sim_{b}^{+} \rho$ (resp. $\phi \sim_{b}^{-} \rho$) if and only if $\phi$ and $\rho$ are holomorphically (resp. anti-holomorphically) conjugated.
Generalizations and consequences
--------------------------------
The results have a straightforward generalization to unfoldings of resonant diffeomorphisms. A diffeomorphism $\phi \in {\mbox{{\rm Diff}{${\,}_{}({\mathbb C}^{},0)$}}}$ is resonant if $\phi'(0)$ is a root of the unit of order $q \in {\mathbb N}$. An unfolding $\varphi(x,y) =(x,f(x,y))$ of $\phi$ satisfies that the iterate $\varphi^{q}$ belongs to ${\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$.
Consider unfoldings $\varphi, \varrho$ of resonant diffeomorphisms $\phi, \rho \in {\mbox{{\rm Diff}{${\,}_{}({\mathbb C}^{},0)$}}}$ and a local homeomorphism $\sigma$ such that $\sigma \circ \varphi = \varrho \circ \sigma$. We have $\phi'(0) = \rho'(0)$ if $\sigma_{|x=0}$ is orientation-preserving and $\phi'(0) = \overline{\rho'(0)}$ if $\sigma_{|x=0}$ is orientation-reversing by Naishul’s theorem [@Naishul]. Since $\sigma$ conjugates iterates of $\varphi$ and $\varrho$ then all theorems in the introduction have obvious generalizations. Moreover all results (except Proposition \[pro:atui\] and Corollary \[cor:topiana\]) describe properties of $\sigma$ so the generalizations are trivial consequences.
The generalizations of Proposition \[pro:atui\] and Corollary \[cor:topiana\] are also simple. We apply our results to the iterates. Then it suffices to prove that given resonant $\phi, \rho \in {\mbox{{\rm Diff}{${\,}_{}({\mathbb C}^{},0)$}}}$ such that $\phi'(0) = \rho'(0)$, $\phi^{q} \in {\mbox{{\rm Diff}{${\,}_{1}({\mathbb C}^{},0)$}}}$ and $\phi^{q}$ is analytically conjugated to $\rho^{q}$ then $\phi$ and $\rho$ are analytically conjugated. This is a trivial consequence of the description of the formal centralizer of $\phi^{q}$ (see Corollary 6.17, p. 88 [@Ilya-Yako]).
A very simple consequence of our results is that a homeomorphism conjugating two generic unfolding of saddle-nodes is either transversaly conformal or transversaly anti-conformal by restriction to the unperturbed parameter.
Outline of the paper
--------------------
The properties of Long Trajectories and Long Orbits are studied by dividing a neighborhood of the origin in two kind of sets: exterior sets in which the unfolding behaves as a trivial one ($N=1$) and compact-like sets in which the dynamics of the unfolding is described in terms of the dynamics of a polynomial vector field. This decomposition is called [*dynamical splitting*]{} and it is explained in Section \[sec:dynspl\].
The existence of Long Trajectories and Long Orbits in the multi-parabolic case was proved in [@rib-mams]. We introduce a simpler proof that is valid in a more general setting. The idea is taking profit of the polynomial vector fields that are canonically associated to the unfolding. The dynamics of the real flow of polynomial vector fields is treated in Section \[sec:dpvf\]. At this point it is good to point out that we need to compare the dynamics of elements of ${\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ with exponentials of vector fields. In Section \[sec:dynext\] we develop the tools required for such a task in the exterior sets of the dynamical splitting. We complete the proof of the existence of Long Trajectories in Section \[sec:lt\].
It is easy to see that the existence of Long trajectories implies the existence of Long Orbits for elements of ${\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$ (Proposition \[pro:ltlo\]). Indeed the Long Orbits are constructed in the neighborhood of Long Trajectories of the real flow of a holomorphic vector field $X=g(x,y) \partial / \partial y$ such that ${\rm exp}(X)$ approximates $\varphi$. A topological homeomorphism $\sigma$ conjugating $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$ does not conjugate the real flows of $X$ and $Y$ if ${\rm exp}(Y)$ approximates $\eta$. Then it is not clear a priori that the image by $\sigma$ of a Long Orbit is in the neighborhood of a Long Trajectory of the real flow of $Y$. This shadowing property is important since it is the base for the residue formula that provides the quantitative estimates of Long Orbits. The tracking (or shadowing) property is proved in Section \[sec:tracking\] by showing that trajectories of the real flow of $Y$ in the neighborhood of Long Orbits of $\eta$ satisfy a Rolle property.
The rigidity results in the introduction are proved in Section \[sec:infinite\] for unfoldings of the identity map and in Section \[sec:finite\] for the remaining cases. Examples showing that the hypotheses in the results are optimal are presented in Section \[sec:build\].
Notations
=========
We denote by ${\mbox{{\rm Diff}{${\,}_{}({\mathbb C}^{n},0)$}}}$ the group of local complex analytic diffeomorphisms defined in a neighborhood of $0$ in ${\mathbb C}^{n}$. We denote by ${\mbox{{\rm Diff}{${\,}_{1}({\mathbb C}^{},0)$}}}$ the group of local complex analytic one-dimensional diffeomorphisms whose linear part is the identity.
We define ${\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{n+1},0)$}}}$ as the set of $n$-parameter unfoldings of local complex analytic tangent to the identity diffeomorphisms. In other words $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{n+1},0)$}}}$ is of the form $\varphi(x_{1},\hdots,x_{n},y)=(x_{1},\hdots,x_{n},f(x_{1},\hdots,x_{n},y))$ where $f \in {\mathbb C}\{x_{1},\hdots,x_{n},y\}$ and the unperturbed diffeomorphism $f(0,\hdots,0,y)$ is tangent to the identity, i.e. $f(0)=0$ and $(\partial f/\partial y)(0,\hdots,0)=1$. We denote by ${\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{n+1},0)$}}}$ the subset of elements $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{n+1},0)$}}}$ such that $\varphi_{|x_{1}=\hdots=x_{n}=0} \neq Id$. Indeed ${\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{n+1},0)$}}} \setminus {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{n+1},0)$}}}$ is the set of unfoldings of the identity map.
We relate the topological properties of unfoldings of tangent to the identity diffeomorphisms and unfoldings of vector fields with a multiple singular point.
We denote by ${{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$ the set of local complex analytic vector fields of the form $X=g(x,y) \partial / \partial y$ where $g \in {\mathbb C}\{x,y\}$ satisfies $g(0,0)=0$ and $(\partial g/\partial y)(0,0)=0$. In other words $X$ is an unfolding of the vector field $g(0,y) \partial /\partial y$ that has a multiple zero at the origin. We denote ${{\mathcal X}_{p1}^{*} {\mbox{(${\mathbb C}^{2},0$)}}}=\{ X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}: X_{|x=0} \not \equiv 0\}$.
We denote by ${{\mathcal X}_{tp1}{\mbox{(${\mathbb C}^{2},0$)}}}$ the subset of ${{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$ of local complex analytic vector fields $X$ such that any irreducible component of $\mathrm{Sing}(X)$ different than $x=0$ is of the form $y=\gamma(x)$ for some $\gamma \in {\mathbb C}\{x\}$. In other words the irreducible components of $\mathrm{Sing}(X)$ are transversal to the fibration $dx=0$. Let us remark that given $g(x,y) \partial /\partial y \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$ there exists $k \in {\mathbb N}$ such that $g(x^{k},y) \partial /\partial y$ belongs to ${{\mathcal X}_{tp1}{\mbox{(${\mathbb C}^{2},0$)}}}$. We denote ${{\mathcal X}_{tp1}^{*} {\mbox{(${\mathbb C}^{2},0$)}}}= {{\mathcal X}_{tp1}{\mbox{(${\mathbb C}^{2},0$)}}}\cap {{\mathcal X}_{p1}^{*} {\mbox{(${\mathbb C}^{2},0$)}}}$.
Given a vector field $X$ defined in a domain $U \subset {\mathbb C}^{n}$ we denote by $\Re (X)$ the real flow of $X$, namely the flow defined in ${\mathbb R}^{2n} = {\mathbb C}^{n}$ by considering real times. For instance if $X$ is of the form $a(x,y) \partial/\partial x + b(x,y) \partial / \partial y$ we have $$\Re (X) = Re (a) \frac{\partial}{\partial x_{1}} + Im (a) \frac{\partial}{\partial x_{2}} +
Re(b) \frac{\partial}{\partial y_{1}} + Im(b) \frac{\partial}{\partial y_{2}}$$ where $x=x_{1} + i x_{2}$ and $y = y_{1} + i y_{2}$.
\[def:traj\] Let $\gamma_{P}(s)$ be the trajectory of $\Re (Z)$ such that $\gamma_{P}(0)=P$. We define ${\mathcal I}(Z, P,F)$ the maximal interval where $\gamma_{P}(s)$ is well-defined and belongs to $F$ for any $s \in {\mathcal I}(Z, P,F)$ whereas $\gamma_{P}(s)$ belongs to the interior $\accentset{\circ}{F}$ of $F$ for any $s \neq 0$ in the interior of ${\mathcal I}(Z, P,F)$. We denote $\Gamma(Z,P,F)=\gamma_{P}({\mathcal I}(Z, P,F))$. We define $$\partial {\mathcal I}(Z, P,F) = \{ \inf({\mathcal I}(Z, P,F)), \sup({\mathcal I}(Z, P,F)) \}
\subset {\mathbb R} \cup \{-\infty, \infty\}.$$ We denote $\Gamma(Z,P,F)(s) = \gamma_{P}(s)$.
\[def:normal\] Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{1}({\mathbb C}^{},0)$}}}$ (resp. ${\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$). Consider a vector field $X=g \partial / \partial y$ for $g \in {\mathbb C}\{y\}$ (resp. ${\mathbb C}\{x,y\}$) such that $y \circ \varphi - y \circ \mathrm{exp}(X) \in (y \circ \varphi -y)^{3}$. We say that $X$ and ${\mathfrak F}_{\varphi}=\mathrm{exp}(X)$ are [*convergent normal forms*]{} of $\varphi$. There exist convergent normal forms (Proposition 1.1 of [@UPD]).
The idea is that the dynamics of $\mathrm{exp}(X)$ is much simpler than the dynamics of $\varphi$. In particular the orbits of $\mathrm{exp}(X)$ are contained in the trajectories of $\Re (X)$. Generically the orbits of $\mathrm{exp}(X)$ and $\varphi$ are very different. In spite of this $\Re (X)$ provides valuable information of the dynamics of $\varphi$ (Section \[sec:tracking\]).
\[def:delta\] Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{}({\mathbb C}^{},0)$}}} \cup {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$. Fix a convergent normal form $X$ of $\varphi$ and ${\mathfrak F}_{\varphi}=\mathrm{exp}(X)$. We define $$\Delta_{\varphi} = \psi_{X} \circ \varphi - \psi_{X} \circ {\mathfrak F}_{\varphi} =
\psi_{X} \circ \varphi - (\psi_{X} +1) .$$ Indeed we have $$\Delta_{\varphi} = \psi_{X} \circ \varphi - \psi_{X} \circ {\mathfrak F}_{\varphi} \sim
\frac{\psi_{X}}{\partial y} (y \circ \varphi - y \circ \mathrm{exp}(X)) =
O \left( \frac{(y \circ \varphi -y)^{3}}{X(y)} \right).$$ The function $\Delta_{\varphi}$ belongs to the ideal $(y \circ \varphi -y)^{2} = (X(y))^{2}$ of ${\mathbb C}\{x,y\}$ (see Lemma 7.2.1 of [@rib-mams]).
The function $\Delta_{\varphi}$ measures how good is the approximation of $\varphi$ provided by $\mathrm{exp}(X)$.
Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$. We define $N(X)$ as the number of points in $\mathrm{Sing} (X) \cap \{x=x_{0}\}$ for $x_{0} \neq 0$. Analogously we define $N(\varphi)$ for $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ by replacing $\mathrm{Sing} (X)$ with $\mathrm{Fix} (\varphi)$. We have $N(\varphi)=N(X)$ if $X$ is a convergent normal form of $\varphi$.
Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$. We define $m(X) \in {\mathbb N} \cup \{0\}$ as the multiplicity of $x=0$ in $\mathrm{Sing} (X)$. More precisely $X$ is of the form $x^{m(X)} X_{0}$ for some holomorphic vector field such that $\{x=0\} \not \subset \mathrm{Sing} (X)$. We define $m(\varphi)$ as the multiplicity of $x=0$ in $\mathrm{Fix} (\varphi)$.
Let $X=g(y) \partial / \partial y$ with $g \in {\mathbb C}\{y\}$. We define $\nu (X)=a-1$ where $a$ is the vanishing order of of $g(y)$ at $0$.
Let $\varphi(y) \in {\mbox{{\rm Diff}{${\,}_{}({\mathbb C}^{},0)$}}}$. We define $\nu (\varphi)=a-1$ where $a$ is the vanishing order of of $\varphi(y)-y$ at $0$.
Let $X =x^{m(X)} g(x,y) \partial / \partial y \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$. We define $\nu (X)$ as $\nu (g(0,y) \partial / \partial y)$. Let $\varphi=(x,y + x^{m(\varphi)} f(x,y)) \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$. We define $\nu (\varphi)$ as $a-1$ where $a$ is the vanishing order of $f(0,y)$ at $0$.
\[def:atpetvf\] Let $X = g(y) \partial / \partial y$ with $X \neq 0$ and $\nu(X)>0$. We say that ${\mathcal P}$ is an [*attracting petal*]{} of $\Re (X)_{|B(0,\epsilon)}$ if it is a connected component of $$\{ y \in B(0,\epsilon): [0,\infty) \subset {\mathcal I}(y,X,B(0,\epsilon)) \
\mathrm{and} \ \lim_{s \to \infty} \Gamma(y,X,B(0,\epsilon))(s) = 0\}$$ where $B(0,\epsilon)$ is the open ball of center at the origin and radius $\epsilon$. Analogously a repelling petal of $\Re (X)_{|B(0,\epsilon)}$ is an attracting petal of $\Re (-X)_{|B(0,\epsilon)}$. We consider the petals ${\{ {\mathcal P}_{j}\}}_{j \in {\mathbb Z}/(2 \nu(X){\mathbb Z})}$ ordered in counter clock wise sense (see [@Loray5]).
A vector field $X=(a_{\nu +1} y^{\nu +1} + h.o.t.) \partial /\partial y$, $a_{\nu+1} \neq 0$, has very similar petals as $a_{\nu +1} y^{\nu +1} \partial /\partial y$. Consider a half line $e^{i \theta_{0}} {\mathbb R}^{+}$ with $a_{\nu+1} e^{i \theta_{0} \nu} \in {\mathbb R}^{*}$. The set of half lines in $\{ a_{\nu+1} y^{\nu} \in {\mathbb R}\}$ is ${\{ e^{i \theta_{j}} {\mathbb R}^{+} \}}_{j \in {\mathbb Z}/(2 \nu {\mathbb Z})}$ where $\theta_{j} = \theta_{0} + \pi j /\nu$. Given $j \in {\mathbb Z}/(2 \nu {\mathbb Z})$ there exists a petal ${\mathcal P}_{j}$ that is bisected by $e^{i \theta_{j}} {\mathbb R}^{+}$. More precisely given $\eta>0$ the sector $(0,\delta) e^{i (\theta_{j} - \pi /\nu + \eta, \theta_{j} + \pi / \nu - \eta)}$ is contained in ${\mathcal P}_{j}$ for $\delta >0$ small enough. Moreover ${\mathcal P}_{j}$ is attracting if and only if $a_{\nu+1} e^{i \theta_{j} \nu} \in {\mathbb R}^{-}$. Two petals have non-empty intersection if and only if they are consecutive. These properties can be easily proved by using the change of coordinates $z = -1/(\nu a_{\nu} y^{\nu})$. The vector field $X$ is of the form $(1 + o(1)) \partial / \partial z$ where $z$ is defined in a neighborhood of $\infty$.
Let $X$ be a holomorphic vector field defined in an open set $U$ of ${\mathbb C}^{n}$. We say that a holomorphic $\psi: U \to {\mathbb C}$ is a [*Fatou coordinate*]{} of $X$ if $X(\psi) \equiv 1$ in $U$.
\[def:atpetd\] Let $\phi \in {\mbox{{\rm Diff}{${\,}_{1}({\mathbb C}^{},0)$}}}$ with $\phi \neq Id$. We say that ${\mathcal P}'$ is an [*attracting petal*]{} of $\phi_{|B(0,\epsilon)}$ if it is a connected component of $$\{ y \in B(0,\epsilon): \phi^{j}(y) \in B(0,\epsilon) \ \forall j \in {\mathbb N}
\ \mathrm{and} \ \lim_{j \to \infty} \phi^{j}(y)=0 \} .$$ Analogously a [*repelling petal*]{} of $\phi_{|B(0,\epsilon)}$ is an attracting petal of $\phi_{|B(0,\epsilon)}^{-1}$. We consider the petals ${\{ {\mathcal P}_{j}'\}}_{j \in {\mathbb Z}/(2 \nu(\phi){\mathbb Z})}$ ordered in counter clock wise sense (see [@Loray5]).
The petals of $y + a_{\nu +1} y^{\nu +1} + h.o.t.$, $a_{\nu +1} \neq 0$, satisfy the properties described below Definition \[def:atpetvf\] for the petals of $a_{\nu +1} y^{\nu +1} \partial / \partial y$.
Let $\phi \in {\mbox{{\rm Diff}{${\,}_{1}({\mathbb C}^{},0)$}}}$ with $\phi \neq Id$. Consider a petal ${\mathcal P}'$ of $\phi$. Consider a convergent normal form $X$ of $\phi$ and a Fatou coordinate $\psi$ of $X$ in ${\mathcal P}'$. We say that $\psi_{{\mathcal P}'}^{\phi}$ is a [*Fatou coordinate*]{} of $\phi$ in ${\mathcal P}_{+}'$ if $\psi_{{\mathcal P}_{+}'}^{\phi} \circ \varphi \equiv \psi_{{\mathcal P}_{+}'}^{\phi} +1$ and there exists $c \in {\mathbb C}$ such that $$(\psi_{{\mathcal P}'}^{\phi} - (\psi + c))(y) = o(\max_{j \geq 0} |\phi^{j}(y)|) \ \ \mathrm{or}
\ \ (\psi_{{\mathcal P}'}^{\phi} - (\psi + c))(y) = o(\max_{j \geq 0} |\phi^{-j}(y)|)$$ depending on wether ${\mathcal P}'$ is attracting or repelling. The definition depends only on $\phi$ and ${\mathcal P}'$. The Fatou coordinate is unique up to an additive constant. Indeed $$\psi_{{\mathcal P}'}^{\phi}(y) = \psi(y) + \sum_{j=0}^{\infty} \Delta_{\phi}(\phi^{j}(y)) \ \ \mathrm{or} \ \
\psi_{{\mathcal P}'}^{\phi}(y) = \psi(y) - \sum_{j=1}^{\infty} \Delta_{\phi}(\phi^{-j}(y))$$ is a Fatou coordinate of $\phi$ in ${\mathcal P}'$ (see Definition \[def:delta\]) depending on wether ${\mathcal P}'$ is attracting or repelling (see [@Loray5]).
\[def:infgen\] Let $\phi \in {\mbox{{\rm Diff}{${\,}_{1}({\mathbb C}^{},0)$}}}$. Consider a petal ${\mathcal P}'$ of $\phi$. There exists a unique vector field $X_{{\mathcal P}'}^{\phi}$ defined in ${\mathcal P}'$ such that $X_{{\mathcal P}'}^{\phi}$ is the $1/\nu(\phi)$ Gevrey sum of the infinitesimal generator of $\phi$ in ${\mathcal P}'$ and $\phi = \mathrm{exp}(X_{{\mathcal P}'}^{\phi})$ [@Ecalle]. Equivalently $X_{{\mathcal P}'}^{\phi}$ is the unique holomorphic vector field defined in ${\mathcal P}'$ such that $X_{{\mathcal P}'}^{\phi}(\psi_{{\mathcal P}'}^{\phi}) \equiv 1$ for some (and then every) Fatou coordinate $\psi_{{\mathcal P}'}^{\phi}$ of $\phi$ in ${\mathcal P}'$. If ${\mathcal P}'$ is a petal of $\varphi_{|x=0}$ for $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ we denote $X_{{\mathcal P}'}^{\varphi} = X_{{\mathcal P}'}^{\varphi_{|x=0}}$.
\[def:afffc\] Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$ and a homeomorphism $\sigma$ conjugating $\varphi$ and $\sigma$. We say that $\sigma_{|x=0}$ is [*affine in Fatou coordinates*]{} if there exists a ${\mathbb R}$-linear isomorphism ${\mathfrak h}:{\mathbb C} \to {\mathbb C}$ such that $${\mathfrak h} (z) = (\psi_{\sigma({\mathcal P}')}^{\eta} \circ \sigma \circ
\mathrm{exp}(z X_{{\mathcal P}'}^{\varphi}) -
\psi_{\sigma({\mathcal P}')}^{\eta} \circ \sigma)(0,y)$$ for any petal ${\mathcal P}'$ of $\varphi_{|x=0}$. The previous property implies that $\psi_{\sigma({\mathcal P}')}^{\eta} \circ \sigma \circ (\psi_{{\mathcal P}'}^{\varphi})^{-1}$ is affine for any choice $\psi_{{\mathcal P}'}^{\varphi}$, $\psi_{\sigma({\mathcal P}')}^{\eta}$ of Fatou coordinates.
\[def:afffc2\] Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ with $m(\varphi)>0$ such that there exists a homeomorphism $\sigma$ satisfying $\sigma \circ \varphi = \eta \circ \sigma$. Consider normal forms $X=x^{m(\varphi)} X_{0}$ and $Y=x^{m(\eta)} Y_{0}$ for $\varphi$ and $\eta$ respectively. Let $\psi_{0}^{\varphi}$, $\psi_{0}^{\eta}$ Fatou coordinates of $(X_{0})_{|x=0}$ and $(Y_{0})_{|y=0}$ respectively. We say that $\sigma_{|x=0}$ is [*affine in Fatou coordinates*]{} if $\psi_{0}^{\eta} \circ \sigma \circ (\psi_{0}^{\varphi})^{-1}$ is an affine isomorphism.
The vector fields $(X_{0})_{|x=0}$ and $(Y_{0})_{|y=0}$ are well-defined up to multiplication by a non-zero complex number. They do not depend on the choices of convergent normal forms. Hence the previous property is well-defined.
\[def:contsets\] We consider coordinates $(x,y) \in {\mathbb C} \times {\mathbb C}$ or $(r,\lambda,y) \in
{\mathbb R}_{\geq 0} \times {\mathbb S}^{1} \times {\mathbb C}$ in ${\mathbb C}^{2}$. Given a set $F \subset {\mathbb C}^{2}$ we denote by $F(x_{0})$ the set $F \cap \{ x=x_{0} \}$ and by $F(r_{0},\lambda_{0})$ the set $F \cap \{ (r,\lambda)=(r_{0},\lambda_{0}) \}$.
\[def:ue\] We define $U_{\epsilon} = {\mathbb C} \times B(0,\epsilon)$. In practice we always work with domains of the form $B(0,\delta) \times B(0,\epsilon)$ for some small $\delta \in {\mathbb R}^{+}$.
Dynamical splitting {#sec:dynspl}
===================
We define a dynamical splitting $\digamma_{X}$ associated to an element $X$ of ${{\mathcal X}_{tp1}{\mbox{(${\mathbb C}^{2},0$)}}}$ such that $N(X) \geq 1$. Most of the concepts were introduced in [@JR:mod]. The idea is dividing a neighborhood of the origin ${\mathcal T}_{0}^{\epsilon}=\{(x,y) \in B(0,\delta) \times \overline{B(0,\epsilon)} \}$, where $B(0,\epsilon)$ is the open ball of center at the origin and radius $\epsilon$, in sets in which the dynamics of $\Re (X)$ is simpler to describe. The sets of the division are obtained through a process of desingularization of $\mathrm{Sing} (X)$.
We say that ${\mathcal T}_{0}^{\epsilon}=\{(x,y) \in B(0,\delta) \times \overline{B(0,\epsilon)} \}$ is a [*seed*]{}. Let us explain the terminology. The set ${\mathcal T}_{0}={\mathcal T}_{0}^{\epsilon}$ is the starting point of the division. Throughout the process we obtain sets of the form ${\mathcal T}_{\beta} = \{ (x,t) \in B(0,\delta) \times \overline{B(0,\eta)} \}$ for some new coordinate $t$. Since these sets share analogous properties as ${\mathcal T}_{0}$ we can define a recursive process of division. The sets of the form ${\mathcal T}_{\beta}$ are called [*seeds*]{} and the coordinate $t$ is canonically associated to ${\mathcal T}_{\beta}$ along the process.
We provide a recursive method to divide ${\mathcal T}_{0}$. At each step we have a vector $\beta=(0, \beta_{1}, \hdots, \beta_{k}) \in \{0\} \times {\mathbb C}^{k}$ with $k \geq 0$ and a seed ${\mathcal T}_{\beta} = \{ (x,t) \in B(0,\delta) \times \overline{B(0,\eta)} \}$ in coordinates $(x,t)$ canonically associated to ${\mathcal T}_{\beta}$. We say that the coordinates $(x,t)$ are [*adapted*]{} to ${\mathcal T}_{\beta}$ and ${\mathcal E}_{\beta}$ (it is defined below). In the first step we have $k=0$, $\beta=0$ and $t=y$. Suppose also that $$\label{for:forx}
X = x^{e({\mathcal E}_{\beta})}
v(x,t) (t-\gamma_{1}(x))^{s_{1}} \hdots (t-\gamma_{p}(x))^{s_{p}}
\partial / \partial{t}$$ in ${\mathcal T}_{\beta}$ where $\gamma_{1}(0)=\hdots=\gamma_{p}(0)=0$ and $\{ v=0 \} \cap {\mathcal T}_{\beta}=\emptyset$.
For $p=1$ we define the [*terminal exterior basic set*]{} ${\mathcal E}_{\beta}={\mathcal T}_{\beta}$, we do not split the terminal seed ${\mathcal T}_{\beta}$. The singular set of $X$ in ${\mathcal T}_{\beta}$ is already desingularized. Suppose $p>1$. We define $t=xw$ and $S_{\beta} = \{ (\partial \gamma_{1}/\partial x)(0), \hdots,
(\partial \gamma_{p}/\partial x)(0) \}$. Consider the blow-up of the point $(x,t)=(0,0)$; the set $S_{\beta}$ can be interpreted as the intersection of the strict transform of $\prod_{j=1}^{p} (t-\gamma_{j}(x))^{s_{j}}=0$ and the divisor. We define the [*exterior basic set*]{} ${\mathcal E}_{\beta} = {\mathcal T}_{\beta} \cap \{ |t| \geq |x|\rho \}$ and $M_{\beta}= \{ (x,w) \in B(0,\delta) \times \overline{B(0,\rho)} \}$ for some $\rho>>0$. The set $M_{\beta}$ contains $\prod_{j=1}^{p} (t-\gamma_{j}(x))^{s_{j}}=0$. One of the ideas of the construction is that since ${\mathcal E}_{\beta}$ is far away of the singular points then the dynamics of the vector fields $\Re (X)_{|{\mathcal E}_{\beta}}$ and $\Re (x^{e({\mathcal E}_{\beta})}
v(x,t) (t-\gamma_{1}(x))^{s_{1}+\hdots+s_{p}} \partial / \partial{t})_{|{\mathcal E}_{\beta}}$ are very similar. We denote $$\partial_{e} {\mathcal E}_{\beta} = \{ (x,t) \in B(0,\delta) \times \partial B(0,\eta) \}
\ {\rm and} \
\nu({\mathcal E}_{\beta}) = s_{1}+\hdots+s_{p}-1.$$ We define $$\partial_{I} {\mathcal E}_{\beta} = \{ (x,t) \in B(0,\delta) \times \overline{B(0,\eta)} :
|t| = |x| \rho \}$$ if ${\mathcal E}_{\beta}$ is not terminal. We say that the sets $\partial_{e} {\mathcal E}_{\beta}$ and $\partial_{I} {\mathcal E}_{\beta}$ are the [*exterior*]{} and [*interior boundaries*]{} of ${\mathcal E}_{\beta}$ respectively and $e({\mathcal E}_{\beta})$ is the [*exterior exponent*]{} of ${\mathcal E}_{\beta}$.
We say that an exterior set ${\mathcal E}_{\beta}$ is [*parabolic*]{} if $\nu({\mathcal E}_{\beta})>0$. Every non-parabolic exterior set is terminal but a terminal exterior set can be parabolic.
\[def:rvfe\] Given an exterior set ${\mathcal E}_{\beta}$ we define $X_{{\mathcal E}_{\beta}}$ as the vector field defined in ${\mathcal T}_{\beta}$ such that $X= x^{e({\mathcal E}_{\beta})} X_{{\mathcal E}_{\beta}}$.
We have $$X =
x^{e({\mathcal E}_{\beta})+s_{1}+\hdots+s_{p}-1}v(x,xw)
{\left({ w - \gamma_{1}(x) / x }\right)}^{s_{1}} \hdots
{\left({ w - \gamma_{p}(x) / x }\right)}^{s_{p}}
\partial / \partial{w}.$$
\[def:rvfc\] We define $\nu({\mathcal C}_{\beta}) = \nu({\mathcal E}_{\beta})$ and $e({\mathcal C}_{\beta}) = e({\mathcal E}_{\beta})+ \nu({\mathcal E}_{\beta})$. We define $X_{{\mathcal C}_{\beta}}$ as the vector field defined in $M_{\beta}$ such that $X= x^{e({\mathcal C}_{\beta})} X_{{\mathcal C}_{\beta}}$.
\[def:pol\] We define the polynomial vector field $$X_{\beta}(\lambda) = \lambda^{e({\mathcal C}_{\beta})} v(0,0)
{\left({ w - (\partial \gamma_{1} / \partial x)(0) }\right)}^{s_{1}} \hdots
{\left({ w - (\partial \gamma_{p} / \partial x)(0) }\right)}^{s_{p}}
\partial / \partial{w}$$ for $\lambda \in {\mathbb S}^{1}$ (see Eq. (\[for:forx\]), note that $t=xw$) associated to $X$, ${\mathcal T}_{\beta}$ and ${\mathcal C}_{\beta}$.
The dynamics of $\Re(X)_{|M_{\beta}}$ and $\Re(X_{\beta}(\lambda))_{|M_{\beta}}$ are similar (up to reparametrization of the trajectories) outside of a neighborhood of the singular set. If $X_{\beta}(\lambda)$ has a multiple zero at $\zeta \in S_{\beta}$ we just choose $w_{\zeta}=w-\zeta$. Suppose now that $X_{\beta}(\lambda)$ has a simple zero at $\zeta \in S_{\beta}$. Assume that $(\partial \gamma_{1} / \partial x)(0)=\zeta$. As a consequence there exist coordinates $(r,\lambda,w_{\zeta})$ defined in the neighborhood of $\{ (r,\lambda,w) \in \{0\} \times {\mathbb S}^{1} \times \{\zeta\}\}$ such that $\lambda^{e({\mathcal C}_{\beta})} X_{{\mathcal C}_{\beta}}$ is of the form $\lambda^{e({\mathcal C}_{\beta})} h(r \lambda) w_{\zeta} \partial / \partial w_{\zeta}$ for some function $h$ (see [@Loray5]). Indeed $w_{\zeta}$ is a linearizing coordinate. Hence $|w_{\zeta}| =\eta'$ for $\eta'>0$ small is transversal to $\Re (X)$ if $(x,w)=(x_{0},\gamma_{1} (x_{0})/x_{0})$ is an attractor or a repeller. Moreover $|w_{\zeta}|=\eta'$ is invariant by $\Re (X)$ if $(x,w)=(x_{0},\gamma_{1} (x_{0})/x_{0})$ is an indifferent singular point.
We define the [*compact-like basic set*]{} $${\mathcal C}_{\beta}=\{
(x,w) \in B(0,\delta) \times \overline{B(0,\rho)} \} \setminus
(\cup_{\zeta \in S_{\beta}}
\{ (x,w_{\zeta}) \in B(0,\delta) \times B(0,\eta_{\beta,\zeta}) \})$$ where $\eta_{\beta,\zeta}>0$ is small enough for any $\zeta \in S_{\beta}$. We denote $$\partial_{e} {\mathcal C}_{\beta} = \{ (x,w) \in B(0,\delta) \times \partial B(0,\rho) \}, \
\partial_{I} {\mathcal C}_{\beta} = \cup_{\zeta \in S_{\beta}} \{ (x,w_{\zeta}) \in B(0,\delta) \times
\partial B(0, \eta_{\beta,\zeta}) \} .$$ We say that $e({\mathcal C}_{\beta})$ (see Definition \[def:rvfc\]) is the [*exponent*]{} of ${\mathcal C}_{\beta}$. Notice that the vector field $X_{\beta}(\lambda)$ determines the dynamics of $\Re (X)$ in ${\mathcal C}_{\beta}$ since $X /|x|^{e({\mathcal C}_{\beta})} \to X_{\beta}(\lambda_{0})$ in ${\mathcal C}_{\beta}$ if $x \to 0$ and $x/|x| \to \lambda_{0}$.
Fix $\zeta \in S_{\beta}$. We define the seed ${\mathcal T}_{\beta, \zeta}= \{ (x,t') \in B(0,\delta) \times \overline{B(0,\eta_{\beta,\zeta})} \}$ where $t'$ is the coordinate $w_{\zeta}$. By definition $(x,t')$ is the set of adapted coordinates associated to ${\mathcal T}_{\beta, \zeta}$. We denote $e({\mathcal E}_{\beta, \zeta}) = e({\mathcal C}_{\beta})$. We have $$X = x^{e({\mathcal E}_{\beta, \zeta})} h(x,t')
\prod_{(\partial \gamma_{j}/\partial x)(0) = \zeta}
{\left({ t' - \left({
\frac{\gamma_{j}(x)}{x} - \zeta }\right)
}\right)}^{s_{j}} \frac{\partial}{\partial{t'}}, \
X = x^{e({\mathcal E}_{\beta, \zeta})} h(x)
t' \frac{\partial}{\partial{t'}}$$ depending on wether or not $X_{\beta}(\lambda)$ has a multiple zero at $\zeta$.
Resuming, in each step of the process we either decide not to split ${\mathcal T}_{\beta}$ or we divide it in sets ${\mathcal E}_{\beta}$, ${\mathcal C}_{\beta}$, ${\{{\mathcal T}_{\beta, \zeta}\}}_{\zeta \in S_{\beta}}$ where $S_{\beta}$ is a finite subset of ${\mathbb C}$. The seeds ${\mathcal T}_{\beta, \zeta}$ for $(\beta,\zeta) \in \{0\} \times {\mathbb C}^{k+1}$ with $\zeta \in S_{\beta}$ are divided in ulterior steps of the process. The sets ${\mathcal T}_{\beta}$, ${\mathcal E}_{\beta}$ and ${\mathcal C}_{\beta}$ are defined by induction on $k$. The sets ${\mathcal E}_{\beta}$ are called [*exterior basic sets*]{} whereas the sets ${\mathcal C}_{\beta}$ are called [*compact-like basic sets*]{} (see Example \[exa:dynsplit\]).
\[def:dynfx\] A dynamical splitting $\digamma_{X}$ associated to $X \in {{\mathcal X}_{tp1}{\mbox{(${\mathbb C}^{2},0$)}}}$ is a division of a neighborhood of the origin ${\mathcal T}_{0}^{\epsilon}=B(0,\delta) \times \overline{B(0,\epsilon)}$ in exterior and compact-like basic sets. The choice of dynamical splitting is not unique.
![Splitting for $X = y(y-{x}^{2})(y-x) \partial/\partial{y}$ in a line $x=x_{0}$[]{data-label="fig5"}](torfig3.eps){height="6cm" width="6cm"}
\[exa:dynsplit\] Consider $X=y (y-x^{2}) (y-x) \partial / \partial y$. Denote $w=y/x$. The vector field $X$ has the form $x^{2} w (w-x) (w-1) \partial / \partial w$ in coordinates $(x,w)$. The polynomial vector field $X_{0}(\lambda)$ associated to the seed ${\mathcal T}_{0}=B(0,\delta) \times \overline{B(0,\epsilon)}$ is equal to $\lambda^{2} w^{2} (w-1) \partial / \partial w$.
The exterior and compact-like sets associated to ${\mathcal T}_{0}$ are ${\mathcal E}_{0}={\mathcal T}_{0} \cap \{ |y| \geq \rho_{0} |x| \}$ and ${\mathcal C}_{0}=\{ |y| \leq \rho_{0} |x| \} \setminus (\{ |w|<\eta_{00} \} \cup \{ |w_{1}|<\eta_{01} \} )$ respectively. The set ${\mathcal C}_{0}$ encloses the seeds ${\mathcal T}_{00}=\{ |w| \leq \eta_{00} \}$ and ${\mathcal E}_{01} = {\mathcal T}_{01}= \{ |w_{1}| \leq \eta_{01} \}$. The seed ${\mathcal T}_{01}$ is terminal since it only contains one irreducible component of $\mathrm{Sing} X$.
Denote $w'=w/x$. We have $X=x^{3} w' (w'-1) (xw'-1) \partial / \partial w'$ in coordinates $(x,w')$. Thus $- \lambda^{3} w' (w'-1) \partial / \partial w'$ is the polynomial vector field $X_{00}(\lambda)$ associated to ${\mathcal T}_{00}$. The seed ${\mathcal T}_{00}$ contains an exterior set ${\mathcal E}_{00} = {\mathcal T}_{00} \cap \{ |w| \geq \rho_{00} |x| \}$ for $\rho_{00}>>1$, a compact-like set ${\mathcal C}_{00} =
\{ |w'| \leq \rho_{00} \} \setminus ( \{ |w_{0}'|<\eta_{000} \} \cup \{ |w_{1}'|<\eta_{001} \} )$ and two terminal seeds ${\mathcal E}_{000}={\mathcal T}_{000}=\{ |w_{0}'| \leq \eta_{000} \}$ and ${\mathcal E}_{001}={\mathcal T}_{001}= \{ |w_{1}'| \leq \eta_{001} \}$ for some $0 < \eta_{000},\eta_{001} <<1$. We have $e({\mathcal E}_{0})=0$, $e({\mathcal C}_{0})=e({\mathcal E}_{00})=e({\mathcal E}_{01})=2$ and $e({\mathcal C}_{00})=e({\mathcal E}_{000})=e({\mathcal E}_{001})=3$.
This construction reminds the Fulton-MacPherson compactification of the configuration space of $n$ distinct labeled points in a nonsingular algebraic variety $X$ [@Fulton-MacPherson]. Indeed the analogue of the [*seeds*]{} are the Fulton-MacPherson’s [*screens*]{}.
Dynamics of polynomial vector fields {#sec:dpvf}
====================================
Given a vector field $X \in {{\mathcal X}_{tp1}{\mbox{(${\mathbb C}^{2},0$)}}}$ we divide a neighborhood of the origin in exterior and compact-like basic sets. The dynamics of $\Re (X)_{|x=x_{0}}$ in an exterior set is simple, namely an attractor, a repeller or a Fatou flower (see Section \[sec:dynext\]). The dynamics of $\Re (X)$ in compact-like sets determines the dynamics of $\Re (X)$ in a neighborhood of the origin. It turns out that the behavior of $\Re (X)_{|{\mathcal C}}$ for a compact-like basic set ${\mathcal C}$ can be described in terms of the polynomial vector field $X_{\mathcal C}(\lambda)$ associated to ${\mathcal C}$ (see Definition \[def:pol\]). In this section we study polynomial vector fields and their stability properties.
Polynomial vector fields have been studied by Douady, Estrada and Sentenac [@DES]. We include here the main properties and their proofs for the sake of completeness.
Let $Y = P(w) \partial / \partial w \in {\mathbb C}[w] \partial / \partial w$ be a polynomial vector field. We define $\nu (Y) = \deg (P) -1$. We consider polynomial vector fields of degree greater than $1$.
We define the set $Tr_{\to \infty}(Y)$ of trajectories $\gamma:(c,d) \to {\mathbb C}$ of $\Re(Y)$ such that $c \in {\mathbb R} \cup \{-\infty\}$, $d \in {\mathbb R}$ and $\lim_{\zeta \to d} \gamma(\zeta) = \infty$. In an analogous way we define $Tr_{\leftarrow \infty}(Y)=Tr_{\to \infty}(-Y)$. We define $Tr_{\infty}(Y) = Tr_{\leftarrow \infty}(Y) \cup Tr_{\to \infty}(Y)$.
\[rem:frinfty\] Let $Y = P(w) \partial / \partial w \in {\mathbb C}[w] \partial / \partial w$ with $\nu (Y) \geq 1$. The vector field $Y$ is analytically conjugated to $\nu(Y)^{-1} z^{1-\nu(Y)} \partial / \partial z$ in the neighborhood of $\infty$ by a change of coordinates of the form $w = c/z +O(1)$ for some $c \in {\mathbb C}$ (see [@JR:mod], p. 348). We have $Tr_{\to \infty}(\partial/\partial{z})= {\mathbb R}^{+}$ and $Tr_{\leftarrow \infty}(\partial/\partial{z})= {\mathbb R}^{-}$. Since $\nu(Y)^{-1} z^{1-\nu(Y)} \partial / \partial z$ is equal to $(z^{\nu(Y)})^{*} (\partial / \partial z)$ each of the sets $Tr_{\to \infty}(Y)$, $Tr_{\leftarrow \infty}(Y)$ contains $\nu (Y)$ trajectories of $\Re (Y)$ in a neighborhood of $\infty$.
\[def:angle\] The complementary of the set $Tr_{\infty}(Y) \cup \{ \infty \}$ has $2 \nu(Y)$ connected components in the neighborhood of $w=\infty$. Each of these components is called an [*angle*]{}, the boundary of an angle contains exactly one $\to \infty$-trajectory and one $\leftarrow \infty$-trajectory.
\[def:homtr\] We say that $\Re(Y)$ has $\infty$-connections or homoclinic trajectories if $Tr_{\to \infty}(Y) \cap Tr_{\leftarrow \infty}(Y) \neq \emptyset$, i.e. there exists a trajectory $\gamma:(c_{-1},c_{1}) \to {\mathbb C}$ of $\Re(Y)$ such that $c_{-1},c_{1} \in {\mathbb R}$ and $\lim_{\zeta \to c_{s}} \gamma(\zeta)=\infty$ for any $s \in \{-1,1\}$. The notion of homoclinic trajectory has been introduced in [@DES] for the study of deformations of elements of ${\mbox{{\rm Diff}{${\,}_{1}({\mathbb C}^{},0)$}}}$.
We define the $\alpha$ and $\omega$-limits $\alpha^{Y}(P)$ and $\omega^{Y}(P)$ respectively of a point $P \in {\mathbb C}$ by the vector field $\Re(Y)$. If $P \in Tr_{\to \infty}(Y)$ we denote $\omega^{Y}(P) = \{\infty\}$ whereas if $P \in Tr_{\leftarrow \infty}(Y)$ we denote $\alpha^{Y}(P)=\{\infty\}$.
\[lem:infom\] Let $Y = P(w) \partial / \partial w$ be a polynomial vector field such that $\nu(Y) \geq 1$. Then $\omega^{Y}(w_{0})=\{ \infty \}$ is equivalent to $w_{0} \in Tr_{\to \infty}(Y)$. Analogously $\alpha^{Y}(w_{0})=\{ \infty \}$ is equivalent to $w_{0} \in Tr_{\leftarrow \infty}(Y)$
The vector field $Y$ is a ramification of a regular vector field in a neighborhood of $\infty$. Thus there exists an open neighborhood $V$ of $\infty$ and $c \in {\mathbb R}^{+}$ such that $$\mathrm{exp}(c Y)(V \setminus Tr_{\to \infty}(Y)) \cap V = \emptyset
\ \ \mathrm{and} \ \
\mathrm{exp}(-c Y)(V \setminus Tr_{\leftarrow \infty}(Y)) \cap V = \emptyset.$$ We are done since $w_{0} \not \in Tr_{\to \infty}(Y)$ implies $\omega^{Y}(w_{0}) \cap ({{\mathbb P}^{1}({\mathbb C})} \setminus V) \neq \emptyset$.
\[lem:doml\] Let $Y = P(w) \partial / \partial w$ be a polynomial vector field such that $\nu(Y) \geq 1$. Let $w_{1} \in {\mathbb C}$ be a point such that $w_{1} \in \omega^{Y} (w_{0})$ for some $w_{0} \in {\mathbb C}$. Then either $\omega^{Y}(w_{1}) = \{ \infty \}$ or $w_{1} \in \mathrm{Sing} (Y)$ and $\omega^{Y}(w_{0}) = \{w_{1}\}$ or $w_{1}$ belongs to a closed trajectory of $\Re (Y)$. Moreover, in the latter case $w_{0}$ and $w_{1}$ belong to the same trajectory of $\Re (Y)$.
If $\omega^{Y} (w_{1})$ contains a singular point $\tilde{w}$ then $\omega^{Y} (w_{0}) = \{\tilde{w}\}$ and $w_{1}=\tilde{w}$ since basins of attractions of attractors and parabolic points are open sets.
Suppose that $\omega^{Y} (w_{1}) \neq \{\infty\}$ and $\omega^{Y} (w_{1}) \cap \mathrm{Sing} (Y) = \emptyset$. Then $\omega^{Y} (w_{1})$ contains a regular point $w_{2}$ of $Y$ by Lemma \[lem:infom\]. Consider a transversal $T$ to $\Re (Y)$ passing through $w_{2}$. Trajectories through points in $\alpha$ or $\omega$-limits intersect connected transversals at most once (Proposition 2, p. 246 [@Hirsch-Smale]). Thus $w_{2}$ is in the same trajectory $\gamma$ of $\Re (Y)$ as $w_{1}$ and $\gamma$ is a closed trajectory. The neighborhood of a closed trajectory of $\Re (Y)$ is composed by closed trajectories of the same period by the isolated zeros principle. We deduce that $w_{0}$ and $w_{1}$ both belong to $\gamma$.
\[cor:trninf\] Let $Y = P(w) \partial / \partial w$ be a polynomial vector field with $\nu(Y) \geq 1$. Assume that $\infty \not \in \omega^{Y} (w_{0})$. Then either $\omega^{Y} (w_{0})$ is a singleton contained in $\mathrm{Sing} (Y)$ or $w_{0}$ belongs to a closed trajectory.
Either $\omega^{Y} (w_{0})$ is a singleton or there exists $w_{1} \in \omega^{Y} (w_{0}) \cap ({\mathbb C} \setminus \mathrm{Sing} (Y))$. We have $\infty \not \in \omega^{Y} (w_{1})$, otherwise we obtain $\infty \in \omega^{Y} (w_{0})$. Lemma \[lem:doml\] implies that either $w_{1}$ is singular and $\omega^{Y} (w_{0}) = \{w_{1}\}$ or $w_{0}$ belongs to a closed trajectory.
The previous corollary has an analogous version for elements of ${{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$.
\[def:alom\] Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$. We define $\alpha^{X}(P)$ as the $\alpha$-limit of $\Gamma(X, P, {\mathcal T}_{0})$ for any $P \in B(0,\delta) \times \overline{B(0,\epsilon)}$ such that ${\mathcal I}(X,P,{\mathcal T}_{0})$ contains $(- \infty,0)$. Otherwise we define $\alpha^{X}(P)=\{ \infty \}$. We define $\omega^{X}(P)$ in an analogous way.
\[pro:lxp1\] Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$. Consider a point $P \in B(0,\delta) \times B(0,\epsilon)$ such that $\omega^{X}(P) \neq \{\infty\}$. Then either $\omega^{X}(P)$ is a singleton contained in $\mathrm{Sing} (X)$ or the trajectory of $\Re (X)$ through $P$ is closed.
The proof is analogous to the proof of Corollary \[cor:trninf\]. The analogue for $P$ of the condition $\infty \not \in \omega^{Y} (w_{0})$ is $\omega^{X}(P) \neq \{\infty\}$.
Our next goal is showing that the dynamics of the real flow of a polynomial vector field of degree greater than $1$ is simple if there are no homoclinic trajectories.
\[lem:aolnht\] Let $Y = P(w) \partial / \partial w$ be a polynomial vector field such that $\nu(Y) \geq 1$. Suppose that $\Re (Y)$ has no homoclinic trajectories. Then either $\omega^{Y}(w_{0})=\{\infty\}$ or $\omega^{Y}(w_{0})$ is a singleton contained in $\mathrm{Sing} (Y)$. In particular $\Re (Y)$ does not have periodic trajectories.
We claim that $\omega^{Y}(w_{0})$ can not contain a point $w_{1} \in {\mathbb C} \setminus \mathrm{Sing} (Y)$ such that $\omega^{Y}(w_{1})=\{\infty\}$. Otherwise there exists an angle $A$ containing points of $\Gamma(Y,w_{0},{\mathbb C})[0,\infty)$ in every neighborhood of $\infty$. The angle $A$ is limited by a trajectory in $Tr_{\to \infty}(Y)$ and a trajectory $\gamma$ in $Tr_{\leftarrow \infty}(Y)$. Moreover $\gamma$ is contained in $\omega^{Y}(w_{0})$. It satisfies $\gamma \cap Tr_{\to \infty}(Y) = \emptyset$ since there are no homoclinic trajectories. We obtain a contradiction since Lemma \[lem:doml\] implies that $\gamma$ is a closed orbit.
We obtain that $w_{0} \in Tr_{\to \infty}(Y)$ or $\omega^{Y}(w_{0}) \subset \mathrm{Sing} (Y)$ or $w_{0}$ belongs to a closed orbit of $\Re (Y)$ for any $w_{0} \in {\mathbb C}$ by Lemma \[lem:doml\].
Let us prove that if $\Re (Y)$ has a closed trajectory $\gamma: [0,a] \to {\mathbb C}$ ($\gamma(0)=\gamma(a)$) we obtain a contradiction. Let $D$ be the union of all the closed trajectories of $\Re (Y)$. We denote by $D_{0}$ the connected component of $D$ containing $\gamma[0,a]$. Since $Tr_{\infty}(Y) \cap D = \emptyset$ we can choose $w_{0} \in \partial D_{0} \setminus \mathrm{Sing} (Y)$. If $\omega^{Y}(w_{0}) \subset \mathrm{Sing} (Y)$ or $w_{0}$ belongs to a closed trajectory then the analogous property also holds true for the points in the neighborhood of $w_{0}$ and $w_{0} \not \in \partial D_{0}$. We deduce that $w_{0}$ belongs to $Tr_{\to \infty}(Y)$. Analogously $w_{0}$ is contained in $Tr_{\leftarrow \infty}(Y)$. Thus $w_{0}$ belongs to a homoclinic trajectory.
The next result is of technical type, it will be used in the proof of Lemma \[lem:inst\].
\[lem:sptcl\] Let $Y = P(w) \partial / \partial w$ be a polynomial vector field such that $\nu(Y) \geq 1$. Suppose that $\Re (Y)$ has no homoclinic trajectories. Let $w_{0} \in {\mathbb C} \setminus \mathrm{Sing} (Y)$ be a point such that $\omega^{Y}(w_{0}) =\{w_{1}\}$ for some $w_{1} \in \mathrm{Sing} (Y)$. Then there exists a trajectory $\gamma$ in $Tr_{\leftarrow \infty}(Y)$ such that $\omega^{Y}(\gamma) = \{w_{1}\}$.
Let $D = \{ w \in {\mathbb C} \setminus \mathrm{Sing} (Y) : \omega^{Y}(w) =\{w_{1}\}\}$. We denote by $D_{0}$ the connected component of $D$ containing $w_{0}$. Since $Tr_{\to \infty}(Y) \cap D = \emptyset$ we can choose $\tilde{w} \in \partial D_{0} \setminus \mathrm{Sing} (Y)$. Lemma \[lem:aolnht\] implies that $\tilde{w} \in Tr_{\to \infty}(Y)$ by the openness of basins of attraction of singular points. Thus we have $\infty \in \overline{D_{0}}$. The set $D_{0}$ contains an angle $A$. The angle $A$ is limited by a trajectory in $Tr_{\to \infty}(Y)$ and a trajectory $\gamma$ in $Tr_{\leftarrow \infty}(Y)$. Lemma \[lem:aolnht\] implies $\omega^{Y}(\gamma)=\{w_{1}\}$.
Next we study the notion of stability of polynomial vector fields as introduced in [@DES]. It is crucial in the paper since the rigidity results for conjugacies between elemens of ${\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ are obtained by analyzing the directions of instability in the parameter space for unfoldings in ${{\mathcal X}_{p1}^{*} {\mbox{(${\mathbb C}^{2},0$)}}}$ and ${\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$.
\[def:noinfcon\] We denote by ${\mathcal X}_{stable}{\mbox{(${\mathbb C}^{},0$)}}$ the set of polynomial vector fields such that $\nu(Y) \geq 1$ and $\Re (Y)$ is orbitally equivalent to $\Re (\mu Y)$ for any $\mu \in {\mathbb S}^{1}$ in a neighborhood of $1$.
\[def:res1\] Let $X$ be a holomorphic vector field defined in a connected domain $U \subset {\mathbb C}$ such that $X \neq 0$. Consider $P \in \mathrm{Sing} (X)$. There exists a unique meromorphic differential form $\omega$ in $U$ such that $\omega(X)=1$. We denote by $Res(X,P)$ the residue of $\omega$ at the point $P$.
\[def:res12\] Given $\phi \in {\mbox{{\rm Diff}{${\,}_{1}({\mathbb C}^{},0)$}}} \setminus \{Id\}$ we consider a convergent normal form $X=g(y) \partial /\partial y$. We define $Res_{\phi}(0) =Res (X,0)$. The definition does not depend on the choice of $X$. Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$; we define $Res_{\varphi}(0,0) = Res_{\varphi_{|x=0}}(0)$.
\[def:res2\] Let $X=f(x,y) \partial/\partial y \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$. Given $(x_{0},y_{0}) \in \mathrm{Sing} (X)$ such that $\{x=x_{0}\} \not \subset \mathrm{Sing} (X)$ we define $Res(X,(x_{0},y_{0}))=Res(f(x_{0},y) \partial/\partial y,y_{0})$.
We introduce the main result in this section.
\[pro:DES\] (See [@DES]) Let $Y = P(w) \partial / \partial w$ be a polynomial vector field such that $\nu(Y) \geq 1$. Then $Y$ belongs to ${\mathcal X}_{stable}{\mbox{(${\mathbb C}^{},0$)}}$ if and only if $\Re (Y)$ has no homoclinic trajectories.
Let $\Omega$ be the meromorphic $1$-form such that $\Omega (Y)=1$. Let $\gamma:(c_{-1},c_{1}) \to {\mathbb C}$ be a homoclinic trajectory of $\Re (\mu Y)$ (see Definition \[def:homtr\]). We obtain $${\mathbb R}^{+} \ni c_{1}-c_{-1} = \int_{\gamma(c_{-1},c_{1})} \frac{\Omega}{\mu} =
\frac{1}{\mu} 2 \pi i \sum_{\omega \in E} Res(Y, w)$$ where $E$ is the set of singular points of $Y$ enclosed by $\gamma$. The set of directions $\mu \in {\mathbb S}^{1}$ such that $2 \pi i \sum_{\omega \in E} Res(Y, w) \in {\mathbb R}^{*} \mu$ for some subset $E$ of $\mathrm{Sing} (Y)$ is finite. Hence $Y$ is not stable if it has a homoclinic trajectory.
The trajectories $\eta_{1,\mu}$, $\hdots$, $\eta_{2 \nu(Y),\mu}$ of $Tr_{\infty}(\mu Y)$ depend continuously on $\mu$. Suppose that $\Re (Y)$ has no homoclinic trajectories. Given a trajectory $\eta_{j,\mu}$ in $Tr_{\to \infty}(Y)$ (resp. $Tr_{\leftarrow \infty}(Y)$) we have that $\alpha^{Y}(\eta_{j,\mu})$ (resp. $\omega^{Y}(\eta_{j,\mu})$) is a singleton contained in $\mathrm{Sing} (Y)$ by Lemma \[lem:aolnht\]. Since basins of attractions are open sets then the previous limits do not depend on $\mu$ for $\mu \in {\mathbb S}^{1}$ in a neighborhood of $1$. By extending an analytical conjugacy defined in a neighborhood of $\infty$ we obtain that there exists a topological orbital equivalence between $\Re (Y)$ and $\Re (\mu Y)$ defined in $F$ where $F(\mu) = Tr_{\to \infty}(\mu Y) \cup Tr_{\leftarrow \infty}(\mu Y) \cup \{\infty\} \cup \mathrm{Sing} (Y)$. The set $F(\mu)$ depends continuously on $\mu$. The functions $\alpha^{\mu Y}$ and $\omega^{\mu Y}$ are constant in each component of ${{\mathbb P}^{1}({\mathbb C})} \setminus F(\mu)$. Moreover $$\begin{array}{cccc}
(\alpha, \omega) : &
({{\mathbb P}^{1}({\mathbb C})} \times V) \setminus \cup_{\mu \in V} (F(\mu) \times \{\mu\}) & \to & \mathrm{Sing} (Y) \times \mathrm{Sing} (Y) \\
& (w,\mu) & \mapsto & (\alpha^{\mu Y}(w), \omega^{\mu Y}(w))
\end{array}$$ is constant in the connected components of $({{\mathbb P}^{1}({\mathbb C})} \times V) \setminus \cup_{\mu \in V} (F(\mu) \times \{\mu\})$ for some connected open neighborhood $V$ of $1$ in ${\mathbb S}^{1}$. As a consequence the topological orbital equivalence can be extended to ${{\mathbb P}^{1}({\mathbb C})}$.
\[def:plevels\] Let $Y = P(w) \partial / \partial w$ be a polynomial vector field with $\nu(Y) \geq 1$. We define ${\mathcal U}_{Y}^{1} = \left\{{ \lambda \in {\mathbb S}^{1} : \lambda Y \not \in
{\mathcal X}_{stable} {\mbox{(${\mathbb C}^{},0$)}} }\right\}$.
\[def:levels\] Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$. Consider the compact-like sets ${\mathcal C}_{1}$, $\hdots$, ${\mathcal C}_{q}$ associated to $X$. Let $X_{j}(\lambda) = \lambda^{e({\mathcal C}_{j})} P_{j}(w) \frac{\partial}{\partial w}$ be the polynomial vector field associated to ${\mathcal C}_{j}$ and $X$ for $1 \leq j \leq q$. We define $${\mathcal U}_{X}^{j,1} = \left\{{ \lambda \in {\mathbb S}^{1} : X_{j}(\lambda) \not \in
{\mathcal X}_{stable} {\mbox{(${\mathbb C}^{},0$)}} }\right\}, \ \
{\mathcal U}_{X}^{1} = \cup_{1 \leq j \leq q} {\mathcal U}_{X}^{j,1} .$$
On the one hand the dynamics of $\Re (X)$ in an exterior set is trivial. On the other hand the behavior of $\Re (X)$ in ${\mathcal C}_{j}$ is controlled by the vector field $X_{j}(\lambda)$. Hence the dynamics of $\Re (X)$ is stable in the neighborhood of the directions in ${\mathbb S}^{1} \setminus {\mathcal U}_{X}^{1}$.
\[def:dir\] Let $A$ be a subset of ${\mathbb C} \setminus \{0\}$. Consider the blow-up mapping $\pi : ({\mathbb R}^{+} \cup \{0\}) \times {\mathbb S}^{1} \to {\mathbb C}$ defined by $\pi (r,\lambda)=r \lambda$. We denote $A_{\pi}$ the subset $\overline{\pi^{-1}(A)}$ of $({\mathbb R}^{+} \cup \{0\}) \times {\mathbb S}^{1}$. We say that $A$ adheres the directions in $A_{\pi} \cap (\{0\} \times {\mathbb S}^{1})$.
\[lem:inst\] Let $Y = P(w) \partial / \partial w$ be a polynomial vector field with $\nu(Y) \geq 1$. Then ${\mathcal U}_{Y}^{1} = \emptyset$ if and only if $\sharp \mathrm{Sing} (Y) = 1$.
If $\sharp \mathrm{Sing} (Y) = 1$ then we have $Y = w^{\nu(Y) +1} \partial / \partial w$ for some $\nu \geq 1$ up to an affine change of coordinates. The vector fields $Y$ and $\mu Y$ are analytically conjugated by a linear change of coordinates for any $\mu \in {\mathbb S}^{1}$.
Suppose ${\mathcal U}_{Y}^{1} = \emptyset$. The dynamics of $\Re (\mu Y)$ depends continuously on $\mu$. Consider the trajectories $\eta_{1,\mu}$, $\hdots$, $\eta_{2 \nu(Y),\mu}$ of $Tr_{\infty}(\mu Y)$ ordered in counter clock wise sense. We can suppose that $\eta_{j,\mu} \in Tr_{\leftarrow \infty}(\mu Y)$ if $j$ is odd whereas $\eta_{j,\mu} \in Tr_{\to \infty}(\mu Y)$ if $j$ is even. The trajectory $\eta_{j,e^{i \theta}}$ adheres to a direction $\lambda_{0} e^{-i \theta / \nu} e^{j \pi i/\nu}$ for some $\lambda_{0} \in {\mathbb S}^{1}$ and all $\theta \in {\mathbb R}$ and $1 \leq j \leq 2 \nu$. When we follow the path $e^{i \theta}$ for $\theta \in [0,\pi]$ the direction $\lambda_{0} e^{j \pi i/\nu}$ becomes $\lambda_{0} e^{-i \pi / \nu} e^{j \pi i/\nu}$ and $Y$ becomes $-Y$. Hence the trajectories $\eta_{j,1}$ and $\eta_{j+1,-1}$ are the same as sets. We obtain $$\omega^{Y}(\eta_{j,1}) = \alpha^{Y}(\eta_{j+1,1}) \
\mathrm{for} \ j \ \mathrm{odd} \ \mathrm{and} \
\alpha^{Y}(\eta_{j,1}) = \omega^{Y}(\eta_{j+1,1}) \
\mathrm{for} \ j \ \mathrm{even} .$$ This implies that there exists $w_{1} \in \mathrm{Sing} (Y)$ such that $\omega^{Y}(\eta_{j,1})=\{w_{1}\}$ if $j$ is odd and $\alpha^{Y}(\eta_{j,1})=\{w_{1}\}$ if $j$ is even.
We claim that $w_{1}$ is the unique point of $\mathrm{Sing} (Y)$. Consider $w_{2} \in \mathrm{Sing} (Y)$. The point $w_{2}$ is a parabolic point, i.e $P' (w_{2})=0$. Otherwise there exists $\mu_{0}$ such that $w_{2}$ is a center of $\Re (\mu_{0} Y)$ and $\mu_{0}$ belongs to ${\mathcal U}_{Y}^{1}$ by Lemma \[lem:aolnht\]. The set $G = \{ w \in {\mathbb C} \setminus \mathrm{Sing} (Y) : \omega^{Y}(w) = \{w_{2}\} \} \cap Tr_{\leftarrow \infty}(Y)$ is non-empty by Lemma \[lem:sptcl\]. The discussion in the previous paragraph implies $w_{2}=w_{1}$.
\[cor:exir\] Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$. Then ${\mathcal U}_{X}^{j,1} = \emptyset$ iff $\sharp \mathrm{Sing} (X_{j}(1)) = 1$. Moreover we have ${\mathcal U}_{X}^{1} = \emptyset$ iff $N(X)=1$.
Corollary \[cor:exir\] implies that there are instability phenomena for any $X =g(x,y) \partial /\partial y$ in ${{\mathcal X}_{p1}^{*} {\mbox{(${\mathbb C}^{2},0$)}}}$ with $N(X) \neq 1$. The remaining case $N(X) = 1$ is a topological product. More precisely $\Re (X)$ is topologically conjugated to $\Re (g(0,y) \partial /\partial y)$ by a mapping of the form $\sigma(x,y) = (x,\tilde{\sigma}(x,y))$.
\[def:unstext\] Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$. Consider a non-parabolic exterior set ${\mathcal E}$. We define ${\mathcal U}_{X}^{{\mathcal E},1}=
\{ \lambda \in {\mathbb S}^{1} : \lambda^{e({\mathcal E})} h(0) \in i {\mathbb R}\}$ where $X = x^{e({\mathcal E})}
h(x) t \partial / \partial{t}$ in adapted coordinates in ${\mathcal E}$. We define ${\mathcal U}_{X}^{{\mathcal E},1} = \emptyset$ if ${\mathcal E}$ is a parabolic exterior set.
Let us stress that if ${\mathcal C}_{j}$ is the compact-like set enclosing the non-parabolic exterior set ${\mathcal E}$ then ${\mathcal U}_{X}^{{\mathcal E},1} \subset {\mathcal U}_{X}^{j,1}$. The point $t=0$ is an attractor in $\{ x \in B(0,\delta) \setminus \{0\} : Re(\lambda^{e({\mathcal E})} h(x)) <0 \}$ where $x=|x| \lambda$.
Let us analyze the dynamics of $\Re (X)$ in the stable directions. The next result is Lemma 6.13 of [@JR:mod].
\[lem:goins\] Let $X \in {{\mathcal X}_{p1}^{*} {\mbox{(${\mathbb C}^{2},0$)}}}$. Let $K$ be a compact subset of ${\mathbb S}^{1} \setminus {\mathcal U}_{X}^{1}$. Then $[0,\infty)$ is contained in ${\mathcal I}(X, P_{0},{\mathcal T}_{0}^{\epsilon})$ and $\lim_{\zeta \to \infty} \mathrm{exp}(\zeta X)(P_{0}) \in \mathrm{Sing} (X)$ for any $P_{0} \in [0,\delta(\epsilon,K)) K \times \partial{B(0,\epsilon)}$ such that $\Re (X)$ does not point towards ${\mathbb C}^{2} \setminus {\mathcal T}_{0}^{\epsilon}$ at $P_{0}$. Moreover, there exists a dynamical splitting $\digamma_{K}$ such that the intersection of $\mathrm{exp}((0,\infty) X)(P_{0})$ with every compact-like or exterior set is connected for any $P_{0} \in [0,\delta(\epsilon,K)) K \times \partial{B(0,\epsilon)}$.
Let us remark that the dynamical splitting $\digamma_{K}$ depends on $K$ but it does not depend on $\epsilon$. Indeed stable behavior degrades as we approach the directions in ${\mathcal U}_{X}^{1}$ in the parameter space. Therefore a unique dynamical splitting does not satisfy the result in the theorem for any compact set $K \subset {\mathbb S}^{1} \setminus {\mathcal U}_{X}^{1}$. On the other hand instability of the dynamics of $\Re (X)$ is related to a finite set of data, namely the polynomial vector fields restricted to the finitely many directions in ${\mathcal U}_{X}^{1}$. Hence we can choose a unique dynamical splitting to describe instability phenomena.
\[cor:stdir\] Let $X \in {{\mathcal X}_{p1}^{*} {\mbox{(${\mathbb C}^{2},0$)}}}$. Let $K$ be a compact subset of ${\mathbb S}^{1} \setminus {\mathcal U}_{X}^{1}$. Given any $\epsilon>0$ small there exists $\delta_{0}=\delta_{0}(\epsilon,K) \in {\mathbb R}^{+}$ such that
- There is no $P \in [0,\delta_{0}) K \times B(0,\epsilon)$ such that $\alpha^{X}(P)=\omega^{X}(P) = \{\infty\}$.
- $\alpha^{X}(P) \subset \mathrm{Sing} (X)$ or $\omega^{X}(P) \subset \mathrm{Sing} (X)$ for any $P \in [0,\delta_{0}) K \times B(0,\epsilon)$.
- There are no closed trajectories or centers of $\Re (X)$ in $[0,\delta_{0}) K \times B(0,\epsilon)$.
Let $\delta_{0} \in {\mathbb R}^{+}$ be the constant provided by Lemma \[lem:goins\]. Suppose there exists $P \in [0,\delta_{0}) K \times B(0,\epsilon)$ such that $\alpha^{X}(P) = \{\infty\}$. Consider $(a,b) = {\mathcal I}(X, P, U_{\epsilon})$ (see Definition \[def:ue\]). Let $P_{0} = \Gamma (X, P, {\mathcal T}_{0})(a)$. Lemma \[lem:goins\] implies $b = \infty$ and $\omega^{X}(P_{0}) \subset \mathrm{Sing} (X)$. Analogously $\omega^{X}(P) = \{\infty\}$ implies $\alpha^{X}(P) \subset \mathrm{Sing} (X)$.
It remains to consider the case $\alpha^{X}(P) \neq \{\infty\}$, $\omega^{X}(P) \neq \{\infty\}$. We have that either both $\alpha^{X}(P)$, $\omega^{X}(P)$ are contained in $\mathrm{Sing} (X)$ or $P$ belongs to a closed trajectory $\gamma$ by Proposition \[pro:lxp1\]. Suppose that $\gamma$ is contained in $x=x_{0}$. Consider the union $D$ of closed trajectories of $\Re (X)$ in $\{x_{0}\} \times B(0,\epsilon)$. Let $(x_{0},y_{0})$ be a point in $(\partial D_{0} \setminus \mathrm{Sing} (X)) \cap (\{x_{0}\} \times B(0,\epsilon))$ where $D_{0}$ is the connected component of $D$ containing $\gamma$. The point $(x_{0},y_{0})$ satisfies $\alpha^{X}(x_{0},y_{0})=\omega^{X}(x_{0},y_{0}) = \{\infty\}$. This contradicts the first part of the corollary.
Dynamics in exterior basic sets {#sec:dynext}
===============================
We study the dynamics of diffeomorphisms $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$. The idea is comparing the dynamics of $\varphi$ with the dynamics of a convergent normal form ${\rm exp}(X)$ (see Definition \[def:normal\]). This section is intended to describe the behavior of $\Re (X)$ in exterior sets.
The main technique to prove the results in the paper is the study of special orbits for $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$. Part of the proof is showing that such orbits are close to trajectories of $\Re (X)$ where $X$ is a normal form of $\varphi$. In order to compare the orbits of $\varphi$ and ${\mathfrak F}_{\varphi}=\mathrm{exp}(X)$ we need some estimates that are introduced below.
Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$ and $P \in U_{\epsilon}$. Given $M \in {\mathbb R}^{+}$ we define $B_{X}(P,M) = \cup_{z \in B(0,M)} \{ \mathrm{exp}(z X)(P) \}$.
Exterior sets
-------------
\[def:psiext\] Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$. Let ${\mathcal E} = \{ \eta \geq |t| \geq \rho|x| \}$ be an exterior set associated to a seed ${\mathcal T}$. The vector field $X_{\mathcal E}$ is of the form $$X_{\mathcal E} =v(x,t) (t-\gamma_{1}(x))^{s_{1}} \hdots (t-\gamma_{p}(x))^{s_{p}}
\partial / \partial{t}$$ where $v$ is a function never vanishing in ${\mathcal T}$. Denote $\gamma_{\mathcal E}=\gamma_{1}$. Denote $\psi_{\mathcal E}^{0}$ a Fatou coordinate of $v(0,t-\gamma_{\mathcal E}(x)) (t-\gamma_{\mathcal E}(x))^{\nu({\mathcal E})+1}
\partial / \partial{t}$ defined in the neighborhood of ${\mathcal E} \setminus \mathrm{Sing} X$.
The idea behind the definitions in this section is that the dynamics of the vector fields $X_{\mathcal E}$ and $v(0,t-\gamma_{\mathcal E}(x)) (t-\gamma_{\mathcal E}(x))^{\nu({\mathcal E})+1}
\partial / \partial{t}$ in ${\mathcal E}$ are analogous. But the latter vector field is much simpler since it is conjugated to $v(0,t) t^{\nu({\mathcal E})+1} \partial / \partial t$, that does not depend on $x$, by a diffeomorphism $(x,t+\gamma_{\mathcal E}(x))$.
\[rem:psiext0\] Suppose $\nu({\mathcal E})=0$. We have $X_{\mathcal E} = h(x) t \partial / \partial t$. The function $\psi_{\mathcal E}^{0}$ is of the form $$\psi_{\mathcal E}^{0} =
\frac{1}{h(0)} \ln t + b(x)$$ where $b(x)$ is a holomorphic function in the neighborhood of $0$. The Fatou coordinates are useful to study trajectories of $\Re (X)$ intersecting the boundaries of the basic sets. We will use determinations of $\psi_{\mathcal E}^{0}$ that are bounded by above in the exterior boundary of ${\mathcal E}$.
\[rem:psiext\] Suppose $\nu({\mathcal E})>0$. The function $\psi_{\mathcal E}^{0}$ is of the form $$\psi_{\mathcal E}^{0} =
\frac{-1}{\nu({\mathcal E}) v(0,0)} \frac{1}{(t-\gamma_{\mathcal E}(x))^{\nu({\mathcal E})}}
+ Res(X_{\mathcal E},(0,0)) \ln (t-\gamma_{\mathcal E}(x)) + h(t-\gamma_{\mathcal E}(x)) + b(x)$$ where $h(z)$ is a $O(1/z^{\nu({\mathcal E})-1})$ meromorphic function and $b(x)$ is a holomorphic function in the neighborhood of $0$. We make analogous choices of determinations of $\psi_{\mathcal E}^{0}$ as in the case $\nu({\mathcal E})=0$. We obtain that given $\zeta>0$ there exists $C_{\zeta} \in {\mathbb R}^{+}$ such that $$\frac{1}{C_{\zeta}} \frac{1}{|t-\gamma_{\mathcal E}(x)|^{\nu({\mathcal E})}} \leq
|\psi_{\mathcal E}^{0}|(x,t) \leq C_{\zeta} \frac{1}{|t-\gamma_{\mathcal E}(x)|^{\nu({\mathcal E})}}$$ in ${\mathcal E} \cap \{t - \gamma_{\mathcal E}(x) \in {\mathbb R}^{+} e^{i[-\zeta, \zeta]}\}
\cap \{x \in B(0,\delta(\zeta)) \}$.
\[def:psiE\] Let ${\mathcal E} = \{ \eta \geq |t| \geq \rho|x| \}$ be a parabolic exterior set associated to $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$. Denote by $\psi_{\mathcal E}$ a Fatou coordinate of $X_{\mathcal E}$ defined in the neighborhood of ${\mathcal E} \setminus \mathrm{Sing} X$ such that $\psi_{\mathcal E}(0,\lambda,t) \equiv \psi_{\mathcal E}^{0}(0,\lambda,t)$. The function $\psi_{\mathcal E}$ is multi-valued.
### Non-parabolic exterior sets
The next lemma is used in Proposition \[pro:boufespre\] to show that far away from ${\mathcal U}_{X}^{{\mathcal E},1}$ (see Definition \[def:unstext\]) the orbits of $\varphi$ track, i.e. are very close to, orbits of the normal form ${\mathfrak F}_{\varphi}$. The lemma will be applied to the function $\Delta_{\varphi}$ (see Definition \[def:delta\]) that measures how good is the approximation of $\varphi$ provided by ${\mathfrak F}_{\varphi}$.
\[lem:cansumnp\] Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$ with $N \geq 1$. Fix a non-parabolic exterior set ${\mathcal E}$. Consider a function $\Delta=O({x}^{a} t)$ where $X = x^{e({\mathcal E})}
h(x) t \partial / \partial{t}$ in adapted coordinates. Fix a closed subset $S$ of $\{ \lambda \in {\mathbb S}^{1} : Re(\lambda^{e({\mathcal E})} h(0)) <0 \}$ and $b \in {\mathbb N}$. Then there exists $C \in {\mathbb R}^{+}$ such that $|\Delta| \leq C {|x|}^{a}/ |\psi_{{\mathcal E}}|^{b}$ in a neighborhood of any trajectory $\gamma$ of $\Re (X)$ in ${\mathcal E} \cap \{ x \in (0,\delta) S\}$.
In Lemma \[lem:cansumnp\] we consider neighborhoods of the form $B_{X}(\gamma,M) = \cup_{P \in \gamma} B_{X}(P,M)$ for some a priori fixed $M \in {\mathbb R}^{+}$, for instance $M=1$. In such neighborhoods the function $\psi_{\mathcal E}$ is uni-valuated.
We have $$X_{\mathcal E} =
h(x) t \frac{\partial}{\partial{t}}
\implies \psi_{\mathcal E} = \frac{\ln t}{h(x)} .$$ We obtain $\Delta = O({x}^{a} e^{h(x) \psi_{\mathcal E}})$. We have ($x = r \lambda$ with $r \in {\mathbb R}^{+} \cup \{0\}$ and $\lambda \in {\mathbb S}^{1}$) $$(h(x) \psi_{\mathcal E}) \circ \mathrm{exp}(z X)(x,y) =
(h(x) \psi_{\mathcal E})(x,y) +
r^{e({\mathcal E})} z h(x) {\lambda}^{e({\mathcal E})}$$ for any $z \in {\mathbb C}$. The trajectories of $\Re (X)$ are obtained by considering $z \in {\mathbb R}$. We obtain $$| e^{h(x) \psi_{\mathcal E}} | =
e^{Re(h(x) \psi_{\mathcal E})} \leq
C_{1} e^{- C_{2} |\psi_{\mathcal E}|}$$ along any trajectory of $\Re (X)$. The constants $C_{1},C_{2} \in {\mathbb R}^{+}$ do not depend on the trajectory because of the choice of $\psi_{\mathcal E}$ (see Remark \[rem:psiext0\] and Definition \[def:psiE\]). It is clear that $e^{-C_{2} |\psi_{\mathcal E}|} = O(1/\psi_{\mathcal E}^{b})$.
Parabolic exterior sets
-----------------------
Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$. Consider a parabolic exterior set ${\mathcal E}$. The approximation of $X_{\mathcal E}$ with $v(0,t-\gamma_{\mathcal E}(x)) (t-\gamma_{\mathcal E}(x))^{\nu({\mathcal E})+1}
\partial / \partial{t}$ is accurate.
(Lemma 6.5 [@JR:mod]) \[lem:itf\] Let ${\mathcal E} = \{(x,t) \in B(0,\delta) \times {\mathbb C} : \eta \geq |t| \geq \rho|x| \}$ be a parabolic exterior set associated to $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$. Let $\upsilon>0$, $\zeta>0$. Suppose ${\mathcal E}$ is terminal. Then $|\psi_{\mathcal E}/\psi_{\mathcal E}^{0} -1| \leq \upsilon$ in ${\mathcal E} \cap \{t - \gamma_{\mathcal E}(x) \in {\mathbb R}^{+} e^{i[-\zeta, \zeta]}\}
\cap \{x \in B(0,\delta(\upsilon,\zeta)) \}$ for some $\delta(\upsilon,\zeta) \in {\mathbb R}^{+}$. The same inequality is true for a non-terminal ${\mathcal E}$ if $\rho >0$ is big enough.
The behavior of a multi-transversal flow in a parabolic exterior set is also analogous to a Fatou flower from a quantitative point of view. In particular we prove that the spiraling behavior is bounded in exterior basic sets.
\[pro:estext\] Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$ and let ${\mathcal E} = \{ \eta \geq |t| \geq \rho |x| \}$ be a parabolic exterior set associated to $X$. Consider a trajectory $\Gamma =
\Gamma( \lambda^{e({\mathcal E})} X_{\mathcal E},(r, \lambda,t),{\mathcal E})$ for $r \lambda$ in a neighborhood of $0$. Then $\Gamma$ is contained in a sector centered at $t=\gamma_{\mathcal E}(r \lambda)$ (see Definition \[def:psiext\]) of angle less than $\zeta$ for some $\zeta>0$ independent of $r, \lambda$ and $\Gamma$.
Let us explain the statement. Consider the universal covering $$(r,\lambda,\gamma_{\mathcal E}(r \lambda) + e^{z}):
{\mathcal E}^{\flat} \to {\mathcal E} \setminus \mathrm{Sing} X .$$ Let ${\Gamma}^{\flat}$ the lifting of $\Gamma$ by $(r,\lambda,e^{z})$. We claim that the set $(Im(z))({\Gamma}^{\flat})$ is contained in an interval of length $\zeta$.
We have $$X = x^{e({\mathcal E})} v(x,t) (t-\gamma_{1}(x))^{s_{1}} \hdots (t-\gamma_{p}(x))^{s_{p}}
\partial / \partial{t}$$ where we consider $\gamma_{\mathcal E} \equiv \gamma_{1}$. We denote $$\psi_{\mathcal E}^{00} =
\frac{-1}{\nu({\mathcal E}) v(0,0)} \frac{1}{(t-\gamma_{\mathcal E}(x))^{\nu({\mathcal E})}} =
\frac{-1}{\nu({\mathcal E}) v(0,0)} \frac{1}{(t-\gamma_{1}(x))^{\nu({\mathcal E})}}.$$ Given $\upsilon>0$ and $\zeta_{0}>0$ we can consider $\eta>0$ small to obtain that $|\psi_{\mathcal E}^{0}/\psi_{\mathcal E}^{00} -1| < \upsilon$ in the set $\{ (x,t) \in {\mathcal E} : |\arg(t-\gamma_{\mathcal E}(x))| \leq \zeta_{0} \}$ (see Remark \[rem:psiext\]). Therefore we obtain $|\psi_{\mathcal E}/\psi_{\mathcal E}^{00} -1| < \upsilon$ in $\{ (x,t) \in {\mathcal E} : |\arg(t-\gamma_{1}(x))| \leq \zeta_{0} \}$ by considering $\eta >0$ small enough and $\rho >0$ big enough if ${\mathcal E}$ is not terminal (Lemma \[lem:itf\]).
We have that either $$(\psi_{\mathcal E}/\lambda^{e({\mathcal E})})(\Gamma) \cap i ({\mathbb R}^{+} \cup \{0\}) = \emptyset \
\mathrm{or} \ (\psi_{\mathcal E}/\lambda^{e({\mathcal E})})(\Gamma) \cap i ({\mathbb R}^{-} \cup \{0\}) = \emptyset .$$ Thus $(\psi_{\mathcal E}/\lambda^{e({\mathcal E})})(\Gamma)$ lies in a sector of angle of angle $2 \pi$. Since $\psi_{\mathcal E} / \psi_{\mathcal E}^{00} \sim 1$ then $\Gamma$ lies in a sector of center $t=\gamma_{\mathcal E}(r, \lambda)$ and angle close to $2 \pi/\nu({\mathcal E})$.
The next result plays an analogous role as Lemma \[lem:cansumnp\] for parabolic exterior sets.
\[lem:cansum\] Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$ with $N \geq 1$. Fix a parabolic exterior set ${\mathcal E}$. Consider a function $\Delta=O({x}^{a} {f'}^{b})$ where $X = x^{e({\mathcal E})} f' \partial / \partial{t}$ in adapted coordinates. Then $\Delta$ is of the form $O({x}^{a}/ \psi_{{\mathcal E}}^{b})$ in ${\mathcal E}$.
Let $$X = x^{e({\mathcal E})}
v(x,t) (t-\gamma_{1}(x))^{s_{1}} \hdots (t-\gamma_{p}(x))^{s_{p}}
\frac{\partial}{\partial{t}}$$ be the expression of $X$ in adapted coordinates in the exterior set ${\mathcal E}$. Let $\nu = \nu_{{\mathcal E}} (X)$. We obtain $\Delta = O({x}^{a} {(t-\gamma_{1}(x))}^{b (\nu+1)})$ in ${\mathcal E}$. We have $$\psi_{{\mathcal E}} \sim \psi_{{\mathcal E}}^{0} \sim 1 /(t-\gamma_{1}(x))^{\nu}$$ by Lemma \[lem:itf\]. Let us remark that the previous property holds true at a sector centered at $t=\gamma_{1}(r \lambda)$ and angle uniformly bounded. We obtain $\Delta = O({x}^{a}/\psi_{{\mathcal E}}^{be})$ for $e = (\nu+1)/\nu$ by Proposition \[pro:estext\].
Long Trajectories {#sec:lt}
=================
Consider a subset $\beta$ of ${\mathbb C}$. We say that $\upsilon: \beta \to {\mathbb C}^{2}$ is a section if there exists a continuous function $\tilde{\upsilon}: \beta \to {\mathbb C}$ such that $\upsilon(x) \equiv (x, \tilde{\upsilon}(x))$.
The proof of Theorem \[teo:main\] depends on the instability properties of elements $\varphi$ of ${\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$. Roughly speaking, we consider sections $\upsilon: \beta \cup \{0\} \to U_{\epsilon}$, where $\beta$ is a connected set with $0 \in \overline{\beta}$, such that the limit of the orbits of $\varphi$ through $\upsilon(x)$ splits in two orbits in the limit. One of the orbits is obviously the orbit through $\upsilon(0)$ whereas the other orbit is composed of points $(0,y_{-})$ such that to go from $\upsilon(x)$ to a neighborhood of $(0,y_{-})$ we have to iterate $\varphi$ a number of times $T(x)$ that tends to $\infty$ when $x \to 0$. These so called Long Orbits appear when $N(\varphi) >1$ even if for simplicity we only consider the case $N(\varphi) >1$, $m(\varphi)=0$. Anyway, this is a non-generic phenomenon since the parameters containing Long Orbits adhere directions of ${\mathcal U}_{X}^{1}$ (Proposition \[pro:udlo\]). Next we introduce the rigorous definition of Long Orbits and its analogue (Long Trajectories) for vector fields.
Let $X \in {{\mathcal X}_{p1}^{*} {\mbox{(${\mathbb C}^{2},0$)}}}$ be a vector field defined in $B(0,\delta) \times B(0,\epsilon)$. Consider a set $\beta \subset {\mathbb C}^{*}$ adhering $0 \in {\mathbb C}$ and a point $y_{+} \in B(0,\epsilon) \setminus \{0\}$ such that $\omega^{X} (0,y_{+}) = \{(0,0)\}$. Assume that $T: \beta \to {\mathbb R}^{+}$ is a continuous function such that $\lim_{x \in \beta, \ x \to 0} T(x) = \infty$. We are interested in describing the limit of the trajectory $\Gamma (X, (x,y_{+}), U_{\epsilon})$ when $x \in \beta$ and $x \to 0$.
\[def:wlt\] We say that ${\mathcal O}=(X, y_{+},\beta,T)$ generates a [*weak Long Trajectory*]{} if there exist a submersion $\vartheta_{\mathcal O}: \beta \to {\mathcal S}_{\mathcal O}$ where ${\mathcal S}_{\mathcal O}$ is a connected subset of ${\mathbb R}$ containing $0$ and a section $\upsilon_{\mathcal O}: \beta \cup \{0\} \to U_{\epsilon}$ with $\upsilon_{\mathcal O}(0)=(0,y_{+})$ such that
- $\vartheta_{\mathcal O}^{-1}(s)$ is a germ of connected curve for any $s \in {\mathcal S}_{\mathcal O}$.
- $[0,T(x)] \subset {\mathcal I}(\Gamma (X, \upsilon_{\mathcal O}(x), U_{\epsilon}))$ for any $x \in \beta$.
- Given a compact subset $K$ of ${\mathcal S}_{\mathcal O}$ there exists $\epsilon_{K}>0$ such that $$\Gamma (X, \upsilon_{\mathcal O}(x), U_{\epsilon})(T(x)) \not \in
U_{\epsilon_{K}}$$ for any $x \in \vartheta_{\mathcal O}^{-1}(K)$ close to $0$.
- Given any $\epsilon''>0$ there exists $M \in {\mathbb N}$ such that $$\Gamma (X, \upsilon_{\mathcal O}(x), U_{\epsilon})[M,T(x)-M] \subset
U_{\epsilon''}$$ for any $x \in \beta$ in a neighborhood of $0$.
\[def:lt\] We say that ${\mathcal O}=(X, y_{+},\beta,T)$ generates a [*Long Trajectory*]{} if ${\mathcal O}$ generates a weak Long Trajectory and there exists a continuous $\chi_{\mathcal O}:i {\mathcal S}_{\mathcal O} \to U_{\epsilon}(0)$ (see Definition \[def:ue\]) such that $$\chi_{\mathcal O}(i s) =
\lim_{x \in \beta, \ \vartheta_{\mathcal O}(x) \to s, \ \ x \to 0}
\Gamma (X, (x,y_{+}), U_{\epsilon})(T(x))$$ for any $s \in {\mathcal S}_{\mathcal O}$ and $\chi_{\mathcal O}(i s) = \mathrm{exp}(i s X)(\chi_{\mathcal O}(0))$ for any $s \in {\mathcal S}_{\mathcal O}$.
Fix $s \in {\mathcal S}_{\mathcal O}$ for a Long Trajectory ${\mathcal O}$. The trajectory $\Gamma (X, (x,y_{+}), U_{\epsilon})[0,T(x)]$ converges to $\Gamma (X, (0,y_{+}), U_{\epsilon})[0,\infty) \cup \{(0,0) \} \cup
\Gamma (X, \chi_{\mathcal O}(i s), U_{\epsilon})(-\infty,0]$ when $x \in \vartheta_{\mathcal O}^{-1}(s)$ tends to $0$. We consider the Hausdorff topology for compact sets. Moreover $\Gamma (X, \chi_{\mathcal O}(i s), U_{\epsilon})$ describes all trajectories in a petal of $\Re (X)_{|U_{\epsilon}(0)}$ if ${\mathcal S}_{\mathcal O}={\mathbb R}$.
Consider a weak Long Trajectory. An accumulation point of $\Gamma (X, (x,y_{+}), U_{\epsilon})[0,T(x)]$ when $x \in \vartheta_{\mathcal O}^{-1}(s)$ tends to $0$ is a union of two trajectories of $\Re (X)_{|U_{\epsilon}(0)}$ and the origin by the last two properties of Definition \[def:wlt\]. Anyway the accumulation set is not necessarily unique. The definition of Long Trajectory was introduced in [@rib-mams]. There the section $\upsilon_{\mathcal O}$ is defined in a curve denoted by $\beta \cup \{0\}$ in [@rib-mams] and whose analogue in this paper is $\vartheta_{\mathcal O}^{-1}(0) \cup \{0\}$. Then the evolution of the Long Trajectories is studied when that curve varies in a family as ${\{\vartheta_{\mathcal O}^{-1}(s)\}}_{s \in {\mathcal S}_{\mathcal O}}$. In this paper we can deal with all the curves simultaneously since the proofs have been improved by using the properties of polynomial vector fields.
\[rem:udll\] Corollary \[cor:stdir\] implies the non-existence of weak Long Trajectories ${\mathcal O}$ such that ${\mathcal S}_{\mathcal O}$ is compact and $\beta$ adheres directions in ${\mathbb S}^{1} \setminus {\mathcal U}_{X}^{1}$. More precisely we have $\lim_{n \to \infty} \Gamma (X, \upsilon_{\mathcal O}(x_{n}), U_{\epsilon})(T(x_{n}))=(0,0)$ for any sequence $x_{n}$ in $\beta$ such that $x_{n} \to 0$ and $x_{n}/|x_{n}|$ converges to a point in ${\mathbb S}^{1} \setminus {\mathcal U}_{X}^{1}$. Thus any set $\beta$ supporting a weak Long Trajectory with ${\mathcal S}_{\mathcal O}$ compact adheres to a unique direction in the finite set ${\mathcal U}_{X}^{1}$.
Next we introduce the analogue of Long Trajectories for diffeomorphisms. Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$. Let $y_{+} \neq 0$ be a point such that $\varphi^{j}(0,y_{+})$ is well-defined and belongs to $U_{\epsilon}$ for any $j \in {\mathbb N}$ and $\lim_{j \to \infty} \varphi^{j}(0,y_{+}) = (0,0)$. Consider a germ of set $\beta \subset {\mathbb C}^{*}$ at $0$ and a continuous function $T: \beta \to {\mathbb R}^{+}$ with $\lim_{x \in \beta, \ x \to 0} T(x) = \infty$. We denote by $[s]$ and $\lceil s \rceil$ the integer part and the ceiling of $s \in {\mathbb R}$ respectively. Let us remind that $\lceil s \rceil$ is the smallest integer not less than $s$.
\[def:lo\] We say that ${\mathcal O}=(\varphi, y_{+},\beta,T)$ generates a Long Orbit if there exist a submersion $\vartheta_{\mathcal O}: \beta \to {\mathcal S}_{\mathcal O}$ where ${\mathcal S}_{\mathcal O}$ is a connected subset of ${\mathbb R}$ containing $0$ and continuous $\upsilon_{\mathcal O}: \beta \cup \{0\} \to U_{\epsilon}$ and $\chi_{\mathcal O}: [0,1]+ i {\mathcal S}_{\mathcal O} \to U_{\epsilon}(0) \setminus \{(0,0)\}$ such that
- $\vartheta_{\mathcal O}^{-1}(s)$ is a germ of connected curve for any $s \in {\mathcal S}_{\mathcal O}$.
- $\upsilon_{\mathcal O}$ is a section, $\upsilon_{\mathcal O}(0)=(0,y_{+})$ and $\chi_{\mathcal O} (1+is) = \varphi(\chi_{\mathcal O}(is))$ for any $s \in {\mathcal S}_{\mathcal O}$.
- $\varphi^{j}(\upsilon_{\mathcal O}(x))$ is well-defined and belongs to $U_{\epsilon}$ for any $0 \leq j \leq [T(x)]+1$ and any $x \in \beta$.
- Given any $z=s +i u \in [0,1] + i {\mathcal S}_{\mathcal O}$ and a sequence $\{x_{n}\}$ in $\beta$ with $x_{n} \to 0$ and $$s = \lim_{n \to \infty} ( \lceil T(x_{n}) \rceil - T(x_{n})), \ \
\lim_{n \to \infty} \vartheta_{\mathcal O}(x_{n}) = u$$ we obtain $\lim_{n \to \infty} \varphi^{\lceil T(x_{n}) \rceil} (\upsilon_{\mathcal O}(x_{n})) = \chi_{\mathcal O}(z)$.
- Given any $\epsilon'>0$ there exists $M \in {\mathbb N}$ such that $$\{\varphi^{M}(\upsilon(x)), \hdots, \varphi^{\lceil T(x) \rceil -M}(\upsilon(x))\}$$ is contained in $U_{\epsilon'}$ for any $x \in \beta$ in a neighborhood of $0$.
Notice that if $(X, y_{+},\beta,T)$ generates a Long Trajectory then $({\rm exp}(X), y_{+},\beta,T)$ generates a Long Orbit. The definitions of Long Trajectories and Orbits are analogous. Obviously the definition for flows is a bit simpler since for diffeomorphisms we can only iterate an integer number of times.
Long Orbits are topological invariants.
\[rem:idgnc\] In the previous definitions $\beta$ is a germ of set at $0$ if ${\mathcal S}_{\mathcal O}$ is compact. Otherwise we identify $\beta$ and $\tilde{\beta}$ if the germs of $\vartheta_{\mathcal O}^{-1}[-n,n] \cap \beta$ and $\vartheta_{\mathcal O}^{-1}[-n,n] \cap \tilde{\beta}$ coincide for any $n \in {\mathbb N}$.
\[def:trim\] Suppose that $(X, y_{+},\beta,T)$ generates a weak Long Trajectory. Let $(0,y_{+}') = \mathrm{exp}(MX) (0,y_{+})$ for some $M \in {\mathbb N}$. Then $(X,y_{+}',\beta,T-2M)$ generates a weak Long Trajectory. We say that the latter weak Long Trajectory is obtained by [*trimming*]{} the former one. Given $\epsilon'>0$ any weak Long Trajectory is contained in $U_{\epsilon'}$ up to trimming by the last condition in Definition \[def:wlt\].
Analogously suppose that $(\varphi, y_{+},\beta,T)$ generates a Long Orbit. Let $(0,y_{+}') = \varphi^{M} (0,y_{+})$ for some $M \in {\mathbb N}$. Then $(\varphi,y_{+}',\beta,T-2M)$ generates a Long Orbit. We say that the latter Long Orbit is obtained by trimming the former one.
Trimming does not change the fundamental properties of a Long Trajectory. Moreover it is easy to define germs of Long Trajectory. Trimming maps a Long Trajectory to another one in the same equivalence class.
The residue formula {#subsec:res}
-------------------
The quantitative properties of the Long Trajectories are obtained by applying the residue formula. It allows to calculate the “length" of the Long Trajectories or more precisely the function $T$ (see Definition \[def:lt\]).
Consider a vector field $Z=a(y) \partial /\partial y$ defined in a neighborhood of $\overline{B(0,\epsilon')}$ such that $\mathrm{Sing} (Z) \cap \partial B(0,\epsilon') = \emptyset$. Let $\gamma:[0,c] \to {\mathbb C}$ a trajectory of $\Re (Z)$ such that $\gamma(0) \in \partial B(0,\epsilon') \ni \gamma(c)$ and $\gamma (0,c) \subset B(0,\epsilon')$. Let $\kappa$ be a path in $\partial B(0,\epsilon')$ going from $\gamma(0)$ to $\gamma (c)$ in counter clock wise sense. Consider the bounded connected component $C_{-}$ of ${\mathbb C} \setminus \gamma \kappa^{-1}$. We denote $E_{-} = C_{-} \cap \mathrm{Sing} (Z)$.
Consider a Fatou coordinate $\psi_{+}$ of $Z$ defined in a neighborhood of $\gamma(0)$. We define $\psi_{-}$ and $\psi_{-}'$ Fatou coordinates of $Z$ defined in the neighborhood of $\gamma(c)$. More precisely, $\psi_{-}$ and $\psi_{-}'$ are obtained by analytic continuation of $\psi_{+}$ along the paths $\kappa$ and $\gamma$ respectively. We have $${\mathbb R}^{+} \ni c = \psi_{-}'(\gamma(c)) - \psi_{+}(\gamma(0)) =
\psi_{-}(\gamma(c)) - 2 \pi i \sum_{y \in E_{-}} Res (Z,y) - \psi_{+}(\gamma(0)) .$$ This is the residue formula. Of course it can be extended to other setups. For instance if $\gamma:[0,c] \to {\mathbb C}$ is a trajectory of $\Re (Z)$ such that there exists $c_{1},c_{2} \in [0,c]$ with $\gamma(0,c_{1}) \subset {\mathbb C} \setminus \overline{B(0,\epsilon')}$, $\gamma(c_{1},c_{2}) \subset B(0,\epsilon')$ and $\gamma(c_{2},c) \subset {\mathbb C} \setminus \overline{B(0,\epsilon')}$ the we obtain $$\label{equ:res}
\psi_{-}(\gamma(c)) - 2 \pi i \sum_{y \in E_{-}} Res (Z,y) - \psi_{+}(\gamma(0)) = c \in {\mathbb R}^{+}$$ where $\psi_{+}$ is a Fatou coordinate of $Z$ defined in a neighborhood of $\gamma(0)$ and $\psi_{-}$ is the analytic continuation of $\psi_{+}$ along $\gamma[0,c_{1}] \kappa \gamma[c_{2},c]$.
Eq. (\[equ:res\]) is interesting to study weak Long Trajectories. Given a Long Trajectory ${\mathcal O}$ (see Definition \[def:lt\]) we can define a holomorphic Fatou coordinate $\psi_{+}$ of $X$ in the neighborhood of $(0,y_{+})$. Then we can consider the Fatou coordinates $\psi_{-}$ and $\psi_{-}'$ defined in the neighborhood of $\chi_{\mathcal O}(0)$. The Fatou coordinate $\psi_{-}$ is holomorphic in the neighborhood of $\chi_{\mathcal O}(0)$ whereas $\psi_{-}'$ is equal to $\infty$ in $x=0$. Thus the length and properties of Long Trajectories are intimately related to the properties of the meromorphic residue functions.
Suppose that ${\mathcal O}=(X, y_{+},\beta,T)$ generates a Long Trajectory. Then the trajectory $\Gamma(X,\upsilon_{\mathcal O}(x),U_{\epsilon})[0,T(x)]$ establishes a division of $\mathrm{Sing} (X)$ in sets $E_{-}(x)$ and $E_{+}(x) = (\mathrm{Sing} (X))(x) \setminus E_{-}(x)$ as described above. Moreover the sets $E_{-}(x)$ and $E_{+}(x)$ depend continuously on $x$. We say that $(E_{-},E_{+})$ is the division of $\mathrm{Sing} (X)$ induced by ${\mathcal O}$.
Behavior of trajectories in adapted coordinates
-----------------------------------------------
The Long Trajectories of an element in ${{\mathcal X}_{p1}^{*} {\mbox{(${\mathbb C}^{2},0$)}}}$ with $N>1$ are obtained by analyzing the dynamics in the most exterior compact-like set ${\mathcal C}_{j_{0}}$ such that ${\mathcal U}_{X,j_{0}}^{1} \neq \emptyset$. This section is devoted to describe the dynamics of $\Re (X)$ in the basic sets enclosing ${\mathcal C}_{j_{0}}$.
We study the properties of the sets of tangencies between $\Re (X)$ and the boundaries of the basic sets in the next results. This is useful to understand the topological behavior of $\Re (X)$. Moreover the set of tangencies determines the dynamics of $\Re (X)$ for some simple basic sets and in particular for the basic sets that are the subject of this section.
Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$. Consider an exterior set $${\mathcal E} = \{(x,t) \in B(0,\delta) \times {\mathbb C} : \eta \geq |t| \geq \rho|x| \}$$ associated to $X$ with $\eta >0$ and $\rho \geq 0$. We define $T{\mathcal E}_{X}^{\eta}(r, \lambda)$ the set of tangent points between $|t|=\eta$ and $\Re(\lambda^{e({\mathcal E})} X_{\mathcal E})_{|x=r \lambda}$ for $(r,\lambda) \in {\mathbb R}_{\geq 0} \times {\mathbb S}^{1}$. We denote $T_{X}^{\epsilon}(r \lambda)= T {\mathcal E}_{X}^{\epsilon}(r, \lambda)$ for the particular case ${\mathcal E}= {\mathcal E}_{0}$.
We have $X = r^{e({\mathcal E})} \lambda^{e({\mathcal E})} X_{\mathcal E}$. Thus $T{\mathcal E}_{X}^{\eta}(r, \lambda)$ is the set of tangent points between $\Re (X)_{|x=r \lambda}$ and $|t|=\eta$ for $r \neq 0$. The definition allows to extend the concept to $r=0$ in adapted coordinates.
\[def:taintpt0\] Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$. Consider a compact-like set $${\mathcal C} =\{
(x,w) \in B(0,\delta) \times \overline{B(0,\rho)} \} \setminus
(\cup_{\zeta \in S_{\mathcal C}}
\{ (x,w_{\zeta}) \in B(0,\delta) \times B(0,\eta_{{\mathcal C}, \zeta}) \})$$ associated to $X$. We denote $T{\mathcal C}_{X}^{\rho}(r, \lambda)$ the set of tangent points between the exterior boundary $|w|=\rho$ of ${\mathcal C}$ and $\Re(\lambda^{e({\mathcal C})} X_{\mathcal C})_{|x=r \lambda}$.
Let ${\mathcal B}$ a basic set. We say that a point $P \in T{\mathcal B}_{X}(r, \lambda)$ is [*convex*]{} if the germ of trajectory of $\Re(\lambda^{e({\mathcal B})} X_{\mathcal B})_{|x=r \lambda}$ through $P$ is contained in ${\mathcal B}$.
We describe the tangent sets $T{\mathcal B}_{X}(r, \lambda)$ for parabolic exterior sets and compact-like sets.
(See [@JR:mod]) \[lem:tgpt20\] Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$. Let ${\mathcal E}=\{ \eta \geq |t| \geq \rho|x| \}$ be a parabolic exterior set associated to $X$ with $0 < \eta <<1$ and $\rho \geq 0$. Then the set $T{\mathcal E}_{\mu X}^{\eta}(r, \lambda)$ is composed of $2 \nu({\mathcal E})$ convex points for all $(\lambda,\mu) \in {\mathbb S}^{1} \times {\mathbb S}^{1}$ and $r$ close to $0$. Each connected component of $(\partial_{e} {\mathcal E})(r, \lambda) \setminus T{\mathcal E}_{\mu X}^{\eta}(r, \lambda)$ contains a unique point of $T{\mathcal E}_{\mu' X}^{\eta}(r, \lambda)$ for all $r \in [0,\delta)$, $\lambda$, $\mu$, $\mu' \in {\mathbb S}^{1}$ such that $\mu' \in {\mathbb S}^{1} \setminus \{ -\mu, \mu \}$.
(See [@JR:mod]) \[lem:tgpt30\] Let $X \in {{\mathcal X}_{tp1}{\mbox{(${\mathbb C}^{2},0$)}}}$ and a compact-like set $${\mathcal C} =\{
(x,w) \in B(0,\delta) \times \overline{B(0,\rho)} \} \setminus
(\cup_{\zeta \in S_{\mathcal C}}
\{ (x,w_{\zeta}) \in B(0,\delta) \times B(0,\eta_{{\mathcal C}, \zeta}) \})$$ associated to $X$ with $\rho >>0$. Then $T{\mathcal C}_{\mu X}^{\rho}(r, \lambda)$ is composed of $2 \nu({\mathcal C})$ convex points for all $(\lambda,\mu) \in {\mathbb S}^{1} \times {\mathbb S}^{1}$ and $r$ close to $0$. Moreover each connected component of $(\partial_{e} {\mathcal C})(r, \lambda) \setminus T{\mathcal C}_{\mu X}^{\rho}(r, \lambda)$ contains a unique point of $T {\mathcal C}_{\mu' X}^{\rho}(r, \lambda)$ for all $r \in [0,\delta)$, $\lambda$, $\mu$, $\mu' \in {\mathbb S}^{1}$ such that $\mu' \in {\mathbb S}^{1} \setminus \{ -\mu, \mu \}$.
\[def:bdtr\] Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$ with $N>1$. Let ${\mathcal B}$ be a basic set. The set $(\partial_{e} {\mathcal B} \setminus T_{X} {\mathcal B})(x)$ has $2 \nu ({\mathcal B})>0$ connected components if $\nu ({\mathcal B})>0$ (Lemmas \[lem:tgpt20\] and \[lem:tgpt30\]). Otherwise ${\mathcal B}$ is a non-parabolic exterior set and $(\partial_{e} {\mathcal B} \setminus T_{X} {\mathcal B})(x)$ is either empty or coincides with $(\partial_{e} {\mathcal B})(x)$. We say that a set is a [*boundary transversal*]{} if it is the closure of a connected component of $(\partial_{e} {\mathcal B} \setminus T_{X} {\mathcal B})$. for some basic set ${\mathcal B}$.
Suppose $\nu ({\mathcal B})>0$. We define the set $\partial_{\downarrow} {\mathcal B}$ of points in $\partial_{e} {\mathcal B}$ where $\Re (X)$ does not point towards the exterior of ${\mathcal B}$. It is the union of $\nu ({\mathcal B})$ boundary transversals.
Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$ with $N(X)>1$. Corollary \[cor:exir\] implies that there exists a sequence of basic sets ${\mathcal E}_{0}$, ${\mathcal C}_{1}$, ${\mathcal E}_{1}$, ${\mathcal C}_{2}$, $\hdots$, ${\mathcal E}_{j_{0}-1}$, ${\mathcal C}_{j_{0}}$ such that $$\partial_{I} {\mathcal E}_{0} = \partial_{e} {\mathcal C}_{1}, \
\partial_{I} {\mathcal C}_{1} = \partial_{e} {\mathcal E}_{1}, \hdots, \
\partial_{I} {\mathcal C}_{j_{0}-1} = \partial_{e} {\mathcal E}_{j_{0}-1}, \
\partial_{I} {\mathcal E}_{j_{0}-1} = \partial_{e} {\mathcal C}_{j_{0}},$$ ${\mathcal U}_{X,j}^{1} = \emptyset$ for any $1 \leq j < j_{0}$ and ${\mathcal U}_{X,j_{0}}^{1} \neq \emptyset$. We deduce $\nu({\mathcal E}_{0}) = \hdots = \nu({\mathcal C}_{j_{0}})$. We say that ${\mathcal E}_{0}$, ${\mathcal C}_{1}$, ${\mathcal E}_{1}$, ${\mathcal C}_{2}$, $\hdots$, ${\mathcal E}_{j_{0}-1}$, ${\mathcal C}_{j_{0}}$ is a [*simple sequence*]{} associated to $X$.
Let ${\mathcal B}$ be a basic set in the sequence with ${\mathcal B} \neq {\mathcal C}_{j_{0}}$. The set ${\mathcal B}(r_{0},\lambda_{0})$ is an annulus that does not contain singular points of $X$. Moreover the number of tangent points between $\Re (X)$ and $\partial_{e} {\mathcal B}$ coincides with the number of tangent points between $\Re (X)$ and $\partial_{I} {\mathcal B}$ and it is equal to $2 \nu (X)$ in any line $(r,\lambda)=(r_{0},\lambda_{0})$ with $r_{0} \in [0,\delta)$ and $\lambda_{0} \in {\mathbb S}^{1}$. Both sets of tangent points are composed of convex points. It is easy to show that in this setting the dynamics of $\Re (\lambda^{e({\mathcal B})} X_{\mathcal B})_{|x= r \lambda}$ is as described in Figure (\[EVfig6\]) (see Proposition 6.1 and Corollary 6.1 of [@JR:mod]). The dynamics of $\Re (X)$ in ${\mathcal B}$ is a truncated Fatou flower.
![Dynamics in basic set ${\mathcal B} \neq {\mathcal C}_{j_{0}}$ of a simple sequence[]{data-label="EVfig6"}](EVfig6.eps){height="6cm" width="7cm"}
\[def:hit\] Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$ with $N>1$. Let ${\mathcal E}_{0}$, ${\mathcal C}_{1}$, $\hdots$, ${\mathcal C}_{j_{0}}$ be a simple sequence associated to $X$. Consider a continuous section $\tau: [0,\delta) \times {\mathbb S}^{1} \to \partial_{\downarrow} {\mathcal E}_{0}$. Denote $\Gamma_{r,\lambda} =
\Gamma (X, \tau(r,\lambda), {\mathcal E}_{0} \cup {\mathcal C}_{1} \cup \hdots \cup {\mathcal E}_{j_{0}-1})$. The dynamics of $\Re (X)$ implies $\sup {\mathcal I}(\Gamma_{r,\lambda}) < \infty$ and $\Gamma_{r,\lambda}(\sup {\mathcal I}(\Gamma_{r,\lambda})) \in \partial_{\downarrow} {\mathcal C}_{j_{0}}$ for any $(r,\lambda) \in (0,\delta) \times {\mathbb S}^{1}$. The formula $$\partial \tau (r,\lambda) = \Gamma_{r,\lambda}(\sup {\mathcal I}(\Gamma_{r,\lambda}))$$ defines a continuous section $\partial \tau: (0,\delta) \times {\mathbb S}^{1} \to \partial_{\downarrow} {\mathcal C}_{j_{0}}$.
In other words $\partial \tau (r,\lambda)$ is the first point of the positive trajectory of $\Re (X)$ through $\tau (r,\lambda)$ that belongs to ${\mathcal C}_{j_{0}}$.
A section $\tau$ as introduced in Definition \[def:hit\] satisfies that $\tau ([0,\delta) \times {\mathbb S}^{1})$ is contained in a component $\kappa$ of $\partial_{\downarrow} {\mathcal E}_{0}$. The set $(\partial \tau) (B(0,\delta) \setminus \{0\})$ is contained in a connected component $\tilde{\kappa}$ of $\partial_{\downarrow} {\mathcal C}_{j_{0}}$. Moreover $\tilde{\kappa}$ depends only on $\kappa$ and the mapping $\kappa \to \tilde{\kappa}$ is a bijection from the boundary transversals of $\partial_{\downarrow} {\mathcal E}_{0}$ onto the boundary transversals of $\partial_{\downarrow} {\mathcal C}_{j_{0}}$.
The next proposition shows that $\partial \tau$ is continuous in adapted coordinates.
\[pro:hit\] Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$ with $N>1$. Let ${\mathcal E}_{0}$, $\hdots$, ${\mathcal C}_{j_{0}}$ be a simple sequence. Consider a continuous section $\tau: [0,\delta) \times {\mathbb S}^{1} \to \partial_{\downarrow} {\mathcal E}_{0}$. Then $\partial \tau$ admits a continuous extension to $[0,\delta) \times {\mathbb S}^{1}$ in the adapted coordinates of ${\mathcal C}_{j_{0}}$. Moreover we have $(\partial \tau)(0,\lambda) \in Tr_{\leftarrow \infty}(X_{j_{0}}(\lambda))$ for any $\lambda \in {\mathbb S}^{1}$. The mapping $(\partial \tau)_{|r=0}$ depends only on the connected component $\kappa$ of $\partial_{\downarrow} {\mathcal E}_{0}$ containing $\tau([0,\delta) \times {\mathbb S}^{1})$. The mapping $\kappa \to (\partial \tau)_{|r=0}$ is a bijection from $\partial_{\downarrow} {\mathcal E}_{0}$ onto the continuous sections of $Tr_{\leftarrow \infty}(X_{j_{0}}(\lambda))$.
We denote $${\mathcal E}_{j_{0}-1}^{\rho'} =
\{(x,t) \in B(0,\delta) \times {\mathbb C} : \eta \geq |t| \geq \rho' |x| \}$$ in adapted coordinates associated to ${\mathcal E}_{j_{0}-1}$. We have ${\mathcal E}_{j_{0}-1}={\mathcal E}_{j_{0}-1}^{\rho}$. Analogously we denote $${\mathcal C}_{j_{0}}^{\rho'} =\{
(x,w) \in B(0,\delta) \times \overline{B(0,\rho')} \} \setminus
(\cup_{\zeta \in S_{\mathcal C}}
\{ (x,w_{\zeta}) \in B(0,\delta) \times B(0,\eta_{{\mathcal C}, \zeta}) \})$$ in adapted coordinates associated to ${\mathcal C}_{j_{0}}$ ($t=xw$). We obtain ${\mathcal C}_{j_{0}}={\mathcal C}_{j_{0}}^{\rho}$. Consider the section $(\partial \tau)^{\rho'}$ associated to $\tau$ and ${\mathcal E}_{0}$, ${\mathcal C}_{1}$, $\hdots$, ${\mathcal C}_{j_{0}-1}$, ${\mathcal E}_{j_{0}-1}^{\rho'}$, ${\mathcal C}_{j_{0}}^{\rho'}$ for $\rho' \geq \rho$. The image of $(\partial \tau)^{\rho'}$ is contained in a connected component $\kappa_{\rho'}$ of $\partial_{\downarrow} {\mathcal C}_{j_{0}}^{\rho'}$. Moreover $\kappa_{\rho'}$ depends continuously on $\rho'$ for $\rho' \geq \rho$.
Consider $\rho' > \rho$ and $\lambda \in {\mathbb S}^{1}$. We define $\iota_{\lambda}^{\rho'}: \kappa_{\rho'}(0,\lambda) \to \kappa_{\rho}(0,\lambda)$ as the mapping given by the formula $$\iota_{\lambda}^{\rho'}(P) = \Gamma_{P}^{\rho'}
(\sup {\mathcal I}(\Gamma_{P}^{\rho'})) \
\mathrm{where} \ \Gamma_{P}^{\rho'} = \Gamma ( X_{j_{0}}(\lambda), P,
\overline{B(0,\rho')} \setminus B(0,\rho) ).$$ Notice that all the accumulation points of sequences of the form $(\partial \tau)(r_{n},\lambda_{n})$ with $(r_{n},\lambda_{n}) \to (0,\lambda)$ are contained in $\iota_{\lambda}^{\rho'}( \kappa_{\rho'}(0,\lambda))$ for any $\rho' \geq \rho$. Hence any accumulation point belongs to $E=\{ P \in \kappa_{\rho}(0,\lambda) : \Gamma_{P}^{\rho'}
(\inf {\mathcal I}(\Gamma_{P}^{\rho'})) \in \partial B(0,\rho') \ \forall \rho' > \rho \}$. A neighborhood of $\infty$ in ${\mathbb C}$ is a union of angles of $\Re (X_{j_{0}}(\lambda))$ (see Remark \[rem:frinfty\] and Definition \[def:angle\]) limited by trajectories in $Tr_{\infty}(X_{j_{0}}(\lambda))$. As a consequence the set $E$ is the singleton $\kappa_{\rho}(0,\lambda) \cap Tr_{\leftarrow \infty}(X_{j_{0}}(\lambda))$ for any $\lambda \in {\mathbb S}^{1}$.
Let $X \in {{\mathcal X}_{p1}^{*} {\mbox{(${\mathbb C}^{2},0$)}}}$ with $N>1$. Let ${\mathcal E}_{0}$, $\hdots$, ${\mathcal C}_{j_{0}}$ be a simple sequence associated to $X$. By applying Proposition \[pro:hit\] to $X$ and $-X$ we obtain the existence of a bijection between the connected components of $(B(0,\delta) \times \partial B(0,\epsilon)) \setminus T_{X}^{\epsilon}$ and the continuous sections of $Tr_{\infty}(X_{j_{0}}(\lambda))$.
\[def:parj\] Let ${\mathcal P}_{1}$, $\hdots$, ${\mathcal P}_{2 \nu(X)}$ be the petals of $\Re (X)_{|U_{\epsilon}(0)}$ (see Definition \[def:atpetvf\]). Let $a_{1}$, $\hdots$, $a_{2 \nu(X)}$ be the $2 \nu (X)$ connected components of $(B(0,\delta) \times \partial B(0,\epsilon)) \setminus T_{X}^{\epsilon}$. We can enumerate them so we have $a_{j}(0) \subset \overline{{\mathcal P}_{j}}$ for any $1 \leq j \leq 2 \nu (X)$. Moreover if ${\mathcal P}_{j}$ is an attracting petal we have $a_{j} \subset \partial_{\downarrow} {\mathcal E}_{0}$. We denote by $\partial_{j}:{\mathbb S}^{1} \to \partial_{e} {\mathcal C}_{j_{0}}$ the mapping $(\partial \tau)_{|r=0}$ for $\tau (B(0,\delta)) \subset a_{j}$.
There exists a bijection between attracting (resp. repelling) petals of $\Re (X)_{|x=0}$ and continuous sections of $Tr_{\leftarrow \infty}(X_{j_{0}}(\lambda))$ (resp. $Tr_{\to \infty}(X_{j_{0}}(\lambda))$).
Let us explain the remark. A section $\tau_{1}: B(0,\delta) \to B(0,\delta) \times B(0,\epsilon)$ with $\tau_{1}(0)$ in an attracting petal ${\mathcal P}_{j}$ induces a continuous section $\partial \tau_{1}: [0,\delta) \times {\mathbb S}^{1} \to \partial_{e} {\mathcal C}_{j_{0}}$ in adapted coordinates such that $\partial_{j}(\lambda) = (\partial \tau_{1})(0,\lambda)$ for any $\lambda \in {\mathbb S}^{1}$. More precisely, consider $\epsilon'<0$ such that $\epsilon' < |\tau_{1}(0)|$. Then $\tau_{1}$ induces a section $\tau_{1}':B(0,\delta) \to B(0,\delta) \times \partial B(0,\epsilon')$ where $\tau_{1}'(x)$ is the first point in $B(0,\delta) \times \partial B(0,\epsilon')$ of the positive trajectory of $\Re (X)$ through $\tau_{1}(x)$. In the same way we can define $\tau'$ for $\tau (B(0,\delta)) \subset a_{j}$. We define $\partial \tau_{1} = \partial \tau_{1}'$ and we have $\partial \tau = \partial \tau'$. Since $\tau'(x)$ and $\tau_{1}'(x)$ are contained in the same component of $(B(0,\delta) \times \partial B(0,\epsilon')) \setminus T_{X}^{\epsilon'}$ for any $x \in B(0,\delta)$ we obtain $\partial_{j} \equiv (\partial \tau_{1})_{|r=0}$ by Proposition \[pro:hit\].
Existence of Long Trajectories
------------------------------
Next we prove the existence of Long Trajectories for $N>1$. The main idea is that Long Trajectories appear naturally in the neighborhood of some homoclinic trajectories of polynomial vector fields associated to the unfolding.
\[pro:exll\] Let $X \in {{\mathcal X}_{p1}^{*} {\mbox{(${\mathbb C}^{2},0$)}}}$ with $N>1$. There exist $y_{+} \in B(0,\epsilon) \setminus \{0\}$, a germ of set $\beta$ at $0$ and a continuous function $T: \beta \to {\mathbb R}^{+}$ such that $(X,y_{+},\beta,T)$ generates a Long Trajectory with ${\mathcal S}_{\mathcal O}={\mathbb R}$.
The first part of the proof is intended to introduce the objects that define the Long Trajectory of $\Re (X)$. Let ${\mathcal E}_{0}$, $\hdots$, ${\mathcal C}_{j_{0}}$ be a simple sequence associated to $X$. Consider $\lambda_{0} \in {\mathcal U}_{X,j_{0}}^{1}$ and a homoclinic trajectory $\Gamma = \Gamma(X_{j_{0}}(\lambda_{0}), w_{0}, {\mathbb C})$ (Proposition \[pro:DES\]). We have ${\mathcal I}(\Gamma) = (s_{0},s_{1})$. The choice of the compact-like set $${\mathcal C}_{j_{0}} =\{
(x,w) \in B(0,\delta) \times \overline{B(0,\rho)} \} \setminus
(\cup_{\zeta \in S_{j_{0}}}
\{ (x,w_{\zeta}) \in B(0,\delta) \times B(0,\eta_{\zeta}) \})$$ implies $\sharp (\Gamma \cap \partial B(0,\rho)) =2$ because of the local dynamics of $\Re (X_{j_{0}}(\lambda_{0}))$ in the neighborhood of $\infty$. Indeed we have $\Gamma \cap \partial B(0,\rho) = \{ \Gamma(t_{0}), \Gamma(t_{1}) \}$ for some $s_{0} < t_{0} < t_{1} < s_{1}$. There exists a unique attracting petal ${\mathcal P}_{+}={\mathcal P}_{j}$ of $\Re (X)_{|U_{\epsilon}(0)}$ such that $\partial_{j}(\lambda_{0}) = \Gamma (t_{0})$ (see Definition \[def:parj\]). Denote $\partial_{+}=\partial_{j}$. Analogously, there exists a unique repelling petal ${\mathcal P}_{-}$ of $\Re (X)_{|x=0}$ such that $\partial_{-}(\lambda_{0}) = \Gamma (t_{1})$. Let $\psi$ be a Fatou coordinate of $X_{j_{0}}(\lambda_{0})$ defined in the neighborhood of $\infty$ and such that $\psi(\infty)=0$. There exists a unique connected $C_{-}$ component of ${\mathbb C} \setminus \Gamma(s_{0},s_{1})$ such that $\Gamma$ parametrizes $\partial C_{-}$ in clock wise sense. Denote $E_{-} = C_{-} \cap \mathrm{Sing} (X_{j_{0}}(1))$. We obtain $$\label{equ:resp}
{\mathbb R}^{+} \ni s_{1}-s_{0} =
\psi(\infty) - \psi(\infty) - 2 \pi i \sum_{P \in E_{-}} Res(X_{j_{0}}(\lambda_{0}),P)$$ by the residue formula. Consider the set $E_{-}(r,\lambda) \subset (\mathrm{Sing} (X))(r,\lambda)$ that varies continuously with respect to $(r,\lambda)$ and satisfies $E_{-}(0,1)=E_{-}$. We have $$\label{equ:resa}
\sum_{P \in E_{-}(x)} Res(X,P) = \frac{1}{|x|^{e({\mathcal C}_{j_{0}})}}
\frac{\lambda_{0}^{e({\mathcal C}_{j_{0}})}}{\mu^{e({\mathcal C}_{j_{0}})}}
\left( \sum_{P \in E_{-}} Res(X_{j_{0}}(\lambda_{0}),P) + o(1) \right)$$ where $\mu = x/|x|$. The function $\sum_{P \in E_{-}(x)} Res(X,P)$ is meromorphic (Proposition 5.2 of [@UPD]) and it has a pole of order greater than $0$.
Fix $(0,y_{+}) \in {\mathcal P}_{+}$. Consider $(0,y_{-}) \in {\mathcal P}_{-}$ such that $\mathrm{exp}(z X)(0,y_{-})$ is well-defined and belongs to $U_{\epsilon}$ for any $z \in i {\mathbb R}$. Given any $(0,y_{-}') \in {\mathcal P}_{-}$ the point $\mathrm{exp}(-j X)(0,y_{-}')$ satisfies the previous property for some $j \in {\mathbb N}$ big enough. Let $\psi_{+}$ be a Fatou coordinate of $X$ defined in the neighborhood of ${\mathcal P}_{+}$ in ${\mathbb C}^{2}$. We define a Fatou coordinate $\psi_{-}$ of $X$ in a neighborhood of ${\mathcal P}_{-}$ as in Subsection \[subsec:res\]. We define $$\label{equ:deftf}
T_{0}(x) =
\psi_{-}(0,y_{-}) - \psi_{+}(0,y_{+}) - 2 \pi i \sum_{P \in E_{-}(x)} Res(X,P), \ \
T_{s}(x) =T_{0}(x) + is$$ for $s \in {\mathbb R}$. Eqs. (\[equ:resp\]) and (\[equ:resa\]) imply that there exists a curve $\beta_{s}$ adhering $\lambda_{0}$ at $0$ (see Definition \[def:dir\]) and contained in $T_{s}^{-1}({\mathbb R}^{+})$ for any $s \in {\mathbb R}$. We define $\beta$ as a connected set such that $\beta \subset \cup_{s \in {\mathbb R}} \beta_{s}$, $\beta_{\pi} \cap (\{0\} \times {\mathbb S}^{1}) =\{(0,\lambda_{0})\}$ (see Definition \[def:dir\]) and contains the germ of $\beta_{s}$ for any $s \in {\mathbb R}$. Let us clarify that we do not define $\beta = \cup_{s \in {\mathbb R}} \beta_{s}$ straight up because then $\beta_{\pi} \cap (\{0\} \times {\mathbb S}^{1}) =\{(0,\lambda_{0})\}$ does not hold true. We define $T = Re (T_{0})$, $\upsilon_{\mathcal O}(x)=(x,y_{+})$, $(\vartheta_{\mathcal O})_{|\beta_{s}} \equiv s$ and $\chi_{\mathcal O}(z) = \mathrm{exp}(z X)(0,y_{-})$. Our goal is proving that ${\mathcal O}=(X,y_{+},\beta,T)$ generates a Long Trajectory.
There exists a continuous section $\upsilon^{1}: \beta \to U_{\epsilon}$ such that $$\psi_{-}(\upsilon^{1}(x)) - \psi_{+}(x,y_{+}) \equiv
\psi_{-}(0,y_{-}) - \psi_{+}(0,y_{+}) + i \vartheta_{\mathcal O}(x)$$ and $\lim_{x \in \beta, \ \vartheta_{\mathcal O}(x) \to s, \ \ x \to 0}
\upsilon^{1}(x) = \chi_{\mathcal O}(i s)$ for any $s \in {\mathbb R}$. Notice that $$\psi_{-}(\upsilon^{1}(x)) - \psi_{+}(x,y_{+}) - 2 \pi i \sum_{Q \in E_{-}(x)} Res(X,Q) = T(x) \in {\mathbb R}^{+}$$ for any $x \in \beta$. We define $$\Gamma_{0}=\Gamma(X,(x,y_{+}),U_{\epsilon}), \ \
\Gamma_{1}=\Gamma(X,\upsilon^{1}(x),U_{\epsilon}) .$$ We claim that $\upsilon^{1}(x) = \Gamma_{0}(T(x))$ for $x \in \beta$. Let $u_{j}(x)$ be the smallest positive real number such that $\Gamma_{j}((-1)^{j} u_{j}(x)) \in \partial_{e} {\mathcal C}_{j_{0}}$ for $j \in \{0,1\}$. Denote $\kappa_{+}(x) = \Gamma_{0}(u_{0}(x))$, $\kappa_{-}(x) = \Gamma_{1}(-u_{1}(x))$. Moreover we get $\lim_{x \in \beta, x \to 0} \kappa_{+}(x) = \partial_{+}(\lambda_{0})$ and $\lim_{x \in \beta, x \to 0} \kappa_{-}(x) = \partial_{-}(\lambda_{0})$. Since $\partial_{+}(\lambda_{0})$ and $\partial_{-}(\lambda_{0})$ belong to the same trajectory of $\Re (X_{j_{0}}(\lambda_{0}))$ there exists $u_{2}(x) \in {\mathbb R}^{+}$ such that $\Gamma(X,\kappa_{+}(x), U_{\epsilon})(0,u_{2}(x)) \in \accentset{\circ}{{\mathcal C}_{j_{0}}}$ and $$\tilde{\kappa}_{+}(x) \stackrel{def}{=} \Gamma(X,\kappa_{+}(x),U_{\epsilon})(u_{2}(x)) \in
\partial_{e} {\mathcal C}_{j_{0}}.$$ We have $\lim_{x \in \beta, x \to 0} \tilde{\kappa}_{+}(x) = \partial_{-}(\lambda_{0})$. Since $\psi_{-} - 2 \pi i \sum_{P \in E_{-}(x)} Res(X,P)$ is an analytic continuation of $\psi_{+}$ along $\Gamma_{0}$ then there exists a Fatou coordinate $\tilde{\psi}_{+}$ defined in a neighborhood of $(r,\lambda,w)=(0,\lambda_{0},\partial_{-}(\lambda_{0}))$ such that $\tilde{\psi}_{+}(\kappa_{-}(x)) - \tilde{\psi}_{+}(\tilde{\kappa}_{+}(x)) \in {\mathbb R}$ for any $x \in \beta$. The point $\partial_{-}(\lambda_{0})$ does not belong to $T ({\mathcal C}_{j_{0}})_{X}^{\rho}(0,\lambda_{0})$. Hence $\tilde{\kappa}_{+}(x)$ and $\kappa_{-}(x)$ belong to a common connected transversal to $\Re (X)$ for any $x \in \beta$. We deduce that $\tilde{\kappa}_{+}(x)=\kappa_{-}(x)$ for any $x \in \beta$.
We have that $\upsilon^{1}(x) = \Gamma_{0}(T(x))$ for any $x \in \beta$. Given $\epsilon'>0$ small there exists a continuous function $v_{0}: \beta \to {\mathbb R}^{+} \cup \{0\}$ such that $\Gamma_{0}[0,v_{0}(x)) \cap \overline{U_{\epsilon'}} = \emptyset$ and $\Gamma_{0}(v_{0}(x)) \in \overline{U_{\epsilon'}}$. The function $v_{0}$ is bounded by above. Moreover there exists $v_{1} \in {\mathbb R}^{+}$ such that ${\mathcal I}(\Gamma(X,{\rm exp}(-v_{1}X)(\chi_{\mathcal O}(i s)),U_{\epsilon'}))$ contains $(-\infty,0]$ for any $s \in {\mathbb R}$. Proposition \[pro:hit\] applied to $\Gamma_{0}(v_{0})$, $\Gamma_{1}(-v_{1})$ and the construction of $\Gamma_{0}$ imply that $\Gamma_{0}(v_{0}(x),T(x)-v_{1})$ is contained in $U_{\epsilon'}$. Thus ${\mathcal O}=(X,y_{+},\beta,T)$ generates a Long Trajectory.
\[rem:fhevol\] Consider the setting in Proposition \[pro:exll\]. All the Long Trajectories of points of ${\mathcal P}_{+}$ with respect to $\beta$ have an analogous behavior. Let $y_{+}' \in {\mathcal P}_{+}$ and $y_{-}' \in {\mathcal P}_{-}$ such that $$\psi_{-}(0,y_{-}') - \psi_{+}(0,y_{+}') =
\psi_{-}(0,y_{-}) - \psi_{+}(0,y_{+}) + j$$ for some $j \in {\mathbb Z}$. If $\mathrm{exp}(i s X)(0,y_{-}') \in U_{\epsilon}$ for any $s \in {\mathbb R}$ then $\tilde{\mathcal O}=(X,y_{+}',\beta,T+j)$ generates a Long Orbit and $\chi_{\tilde{\mathcal O}}(0)=(0,y_{-}')$ by the proof of Proposition \[pro:exll\], see Eq. (\[equ:deftf\]). In general there exists $j_{0} \in {\mathbb N}$ such that $\breve{\mathcal O}=(X,y_{+}',\beta,T+j-j_{0})$ generates a Long Orbit and $\chi_{\breve{\mathcal O}}(0)=\mathrm{exp}(-j_{0} X)(0,y_{-}')$. Hence, up to replace $T$ with $T-j_{1}$ for some $j_{1} \in {\mathbb N} \cup \{0\}$, the Long Trajectory of a point of ${\mathcal P}_{+}$ with respect to $\beta$ is always non-empty.
It is natural to ask if we can choose $y_{+}$ in any attracting petal ${\mathcal P}_{+}$ of $\Re(X)_{|U_{\epsilon}(0)}$. The answer is positive and the proof exploits the symmetries of the polynomial vector fields associated to $X$.
\[pro:exll2\] Let $X \in {{\mathcal X}_{p1}^{*} {\mbox{(${\mathbb C}^{2},0$)}}}$ with $N>1$. Let ${\mathcal P}_{+}$ be an attracting petal of $Re(X)_{|x=0}$. Then there exists a germ of set $\beta$ at $0$ such that the Long Trajectory associated to $X$, $y_{+}$ with respect to $\beta$ is not empty for any $y_{+} \in {\mathcal P}_{+}$. Moreover given a repelling petal ${\mathcal P}_{-}$ and a point $(0,y_{-}) \in {\mathcal P}_{-}$ there exist $d \in {\mathbb N} \cup \{0\}$ and a Long Trajectory ${\mathcal O} = (X,y_{+}',\beta,T)$ such that $\chi_{\mathcal O}(0) = {\rm exp}(-d X)(0,y_{-})$.
Up to a ramification $(x,y) \mapsto (x^{l},y)$ we can suppose $X \in {{\mathcal X}_{tp1}^{*} {\mbox{(${\mathbb C}^{2},0$)}}}$. We have $$X = u(x,y) (y - \gamma_{1}(x))^{n_{1}} \hdots (y - \gamma_{p}(x))^{n_{p}} \frac{\partial}{\partial y}$$ where $u \in {\mathbb C}\{x,y\}$ is a unit and $\nu(X) + 1 = n_{1} + \hdots + n_{p}$. Denote $\nu = \nu (X)$.
Consider the notations in Proposition \[pro:exll\]. We have ${\mathcal P}_{j}={\mathcal P}_{+}$ and ${\mathcal P}_{k}={\mathcal P}_{-}$ for some $j,k \in {\mathbb Z}/(2 \nu {\mathbb Z})$. We have $X_{j_{0}}(\lambda) = \lambda^{j_{0} \nu} P_{j_{0}}(w) \partial / \partial w$. Notice that $u(0,0)$ is the the coefficient of highest degree in $P_{j_{0}}$. The trajectories in $Tr_{\infty} (X_{j_{0}}(\lambda))$ adhere to the directions in $\lambda^{j_{0} \nu} u(0,0) w^{\nu} \in {\mathbb R}$ at $\infty$. These directions rotate at a speed of $-j_{0}$ (in other words if $\lambda$ rotates an angle of $\theta$ then the directions rotate an angle of $-j_{0} \theta$). In particular the tangent directions to $Tr_{\infty} (X_{j_{0}}(\lambda e^{2 \pi i s/(j_{0} \nu)}))$ are obtained by rotating an angle of $-2 \pi s/\nu$ the tangent directions to $Tr_{\infty} (X_{j_{0}}(\lambda))$ for $s \in [0,1]$. Since $X_{j_{0}}(\lambda_{0}) = X_{j_{0}}(\lambda_{0} e^{2 \pi i /(j_{0} \nu)})$ every trajectory in $Tr_{\to \infty} (X_{j_{0}}(\lambda))$ (resp. $Tr_{\leftarrow \infty} (X_{j_{0}}(\lambda))$) is transformed into the previous one when $s$ goes from $0$ to $1$. The previous discussion implies (see Definition \[def:parj\]) $$\partial_{j+2} (\lambda_{0} e^{2 \pi i /(j_{0} \nu)}) = \Gamma(t_{0}), \ \mathrm{and} \
\partial_{k+2} (\lambda_{0} e^{2 \pi i /(j_{0} \nu)}) = \Gamma(t_{1})$$ where $\Gamma$ is the homoclinic trajectory such that $\partial_{j}(0,\lambda_{0}) \in \Gamma \ni \partial_{k}(0,\lambda_{0})$. Consider any points $(0,y_{+}') \in {\mathcal P}_{j+2}$ and $(0,y_{-}') \in {\mathcal P}_{k+2}$. Up to replace $(0,y_{-}')$ with $\mathrm{exp}(-d X)(0,y_{-}')$ if necessary there exists a set $\beta'$ tangent to $\lambda_{0} e^{2 \pi i /(j_{0} \nu)}$ such that $\tilde{\mathcal O}=(X,y_{+}',\beta',T')$ generates a Long Trajectory for some $T':\beta' \to {\mathbb R}^{+}$ with $\chi_{\tilde{\mathcal O}}(0)=(0,y_{-}')$ by Proposition \[pro:exll\]. Long Trajectories of points of ${\mathcal P}_{j+2}$ with respect to $\beta'$ are not empty by Remark \[rem:fhevol\].
We proved that if the result is true for ${\mathcal P}_{l}$ then it is also true for ${\mathcal P}_{l+2}$. Hence it is satisfied for any petal of $\Re (X)_{|x=0}$.
Tracking Long Orbits {#sec:tracking}
====================
Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ and a convergent normal form $X$ of $\varphi$. Let ${\mathcal O}=(X,y_{+},\beta,T)$ be a Long Trajectory. It is natural to ask whether $(\varphi,y_{+},\beta,T)$ generates a Long Orbit. A priori this is not clear since orbits of $\varphi$ could be (and are!) very different than orbits of ${\mathfrak F}_{\varphi}$. Anyway the orbits ${\{ \varphi^{j}(\upsilon_{\mathcal O}(x))\}}_{0 \leq j \leq \lceil T(x) \rceil}$ and ${\{ {\mathfrak F}_{\varphi}^{j}(\upsilon_{\mathcal O}(x))\}}_{0 \leq j \leq \lceil T(x) \rceil}$ remain close for $x \in \beta$. The dynamics of $\varphi$ “tracks" the dynamics of ${\mathfrak F}_{\varphi}$ along Long Trajectories of $\Re (X)$ (Proposition \[pro:tracking\]). The idea is that Long Trajectories change of basic set a number of times that is bounded by above uniformly and that in basic sets the tracking property is simple to prove.
We study topological conjugacies $\sigma$ between elements $\varphi$, $\eta$ of ${\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$. Long Orbits are topological invariants but $\sigma$ does not conjugate ${\mathfrak F}_{\varphi}$ and ${\mathfrak F}_{\eta}$ and does not preserve the dynamical splitting in general. Hence it is not clear that the image of a Long Orbit of $\varphi$ by $\sigma$ is close to a Long Trajectory of $Y$ where $Y$ is a convergent normal form of $\eta$. We prove in Section \[sec:Rolle\] that Long Orbits are always in the neighborhood of Long Trajectories of the normal form since the latter one satisfies a sort of Rolle property. The tracking phenomenon allows to generalize the residue formula for diffeomorphisms (Propositions \[pro:ltlo\] and \[pro:lores\]).
Let $X \in {{\mathcal X}_{p1}^{*} {\mbox{(${\mathbb C}^{2},0$)}}}$ with $N \geq 1$. Let $\beta$ be a germ of connected set at $0 \in {\mathbb C}$. Consider a family of sub-trajectories $\Gamma_{x}:[0,T'(x)] \to U_{\epsilon}(x)$ of $\Re (X)$ defined for $x \in \beta$. We say that ${\{ \Gamma_{x} \}}_{x \in \beta}$ is [*stable*]{} if $\{ x \in \beta: \Gamma_{x} \cap {\mathcal E} \neq \emptyset \}$ does not adhere the directions in ${\mathcal U}_{X}^{{\mathcal E},1}$ (see Definition \[def:unstext\]) for any exterior set ${\mathcal E}$.
The orbits of $\varphi$ and ${\mathfrak F}_{\varphi}$ are very different in the neighborhood of the indifferent fixed points of $\varphi$. Roughly speaking a stable family is a family far away from indifferent fixed points.
Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ with $N \geq 1$. Fix a convergent normal form $X$ of $\varphi$ and a basic set ${\mathcal B}$. We define $\nu_{{\mathcal B}}(\Delta_{\varphi}) \in {\mathbb N} \cup \{0\}$ as the integer such that $\Delta_{\varphi}$ (see Definition \[def:delta\]) is of the form $x^{\nu_{{\mathcal B}}(\Delta_{\varphi})} g(x,t)$ in the adapted coordinates $(x,t)$ associated to ${\mathcal B}$ where $g(0,t) \not \equiv 0$.
The next propositions provide the tracking properties for basic sets.
\[pro:boufespre\] Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ with $N \geq 1$. Fix a convergent normal form $X$ of $\varphi$. Let $\beta$ be a germ of connected set at $0 \in {\mathbb C}$. Consider an exterior set ${\mathcal E} = \{(x,t) \in B(0,\delta) \times {\mathbb C} : \eta \geq |t| \geq \rho|x| \}$. Let $\nu =\nu_{{\mathcal E}}(\Delta_{\varphi}) - e({\mathcal E})$ (see Definitions \[def:rvfe\], \[def:rvfc\]). Fix a closed set $S \subset {\mathbb S}^{1} \setminus {\mathcal U}_{X}^{{\mathcal E},1}$ (see Definition \[def:unstext\]). There exists $\xi > 0$ such that the properties $\mathrm{exp}([0,j]X)(x,t_{1}) \subset \cup_{x \in (0,\delta) S} {\mathcal E}(x)$ for some $j \in {\mathbb N} \cup \{ 0 \}$ and $(x,t_{2}) \in B_{X}((x,t_{1}),1)$ imply $$\label{equ:trackes}
|\psi_{X} \circ {\varphi}^{j+1}(x,t_{2}) - \psi_{X} \circ {{\mathfrak F}}_{\varphi}^{j+1}(x,t_{1})| \leq
|\psi_{X}(x,t_{2}) - \psi_{X}(x,t_{1})| + \xi {|x|}^{\nu}.$$ Moreover we can choose $\xi>0$ as small as desired by considering a small $\eta>0$.
We denote $$G = (\psi_{X} \circ {\varphi}^{j+1}(x,t_{2}) - \psi_{X} \circ {{\mathfrak F}}_{\varphi}^{j+1}(x,t_{1})) -
(\psi_{X}(x,t_{2}) - \psi_{X}(x,t_{1})) .$$ We have $G=\sum_{k=0}^{j} \Delta_{\varphi}(\varphi^{j}(x,t_{2}))$. We obtain $$|G| \leq \sum_{k=0}^{\infty}
\frac{C |x|^{\nu_{\mathcal E}(\Delta_{\varphi})}}{(\min_{0 \leq l \leq j}
|\psi_{\mathcal E}({\mathfrak F}_{\varphi}^{l}(x,t_{1}))|
+k |x|^{e({\mathcal E})})^{2}} =
O \left( \frac{|x|^{\nu_{\mathcal E}(\Delta_{\varphi})-e({\mathcal E})}}{\min_{0 \leq l \leq j}
|\psi_{\mathcal E}({\mathfrak F}_{\varphi}^{l}(x,t_{1}))|} \right)$$ for some $C \in {\mathbb R}^{+}$ by Lemmas \[lem:cansumnp\] and \[lem:cansum\]. Since $\psi_{\mathcal E} (x,t) \to \infty$ uniformly in ${\mathcal E}$ when $\eta \to 0$ we get Eq. (\[equ:trackes\]).
\[pro:bouis\] Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ with $N > 1$. Fix a convergent normal form $X$ of $\varphi$. Fix a compact-like basic set ${\mathcal C}$. Let $\nu =\nu_{{\mathcal C}}(\Delta_{\varphi}) - e({\mathcal C})$. Fix $B \in {\mathbb R}^{+}$. There exists a constant $C'>0$ such that $\mathrm{exp}([0,j]X)(P) \subset U_{\epsilon}(x) \cap {\mathcal C}$ for some $x \in B(0,\delta)$, $0 \leq j \leq B/|x|^{e({\mathcal C})}$ and $Q \in B_{X}(P,1)$ imply $$\label{equ:trackcs}
|\psi_{X} \circ {\varphi}^{j+1}(Q) - \psi_{X} \circ {{\mathfrak F}}_{\varphi}^{j+1}(P)| \leq
|\psi_{X}(Q) - \psi_{X}(P)| + C' {|x|}^{\nu}.$$
We denote $$G = (\psi_{X} \circ {\varphi}^{j+1}(Q) - \psi_{X} \circ {{\mathfrak F}}_{\varphi}^{j+1}(P)) -
(\psi_{X}(Q) - \psi_{X}(P)) .$$ We have $G=\sum_{k=0}^{j} \Delta_{\varphi}(\varphi^{j}(Q))$. The inequalities $$|G| \leq |x|^{\nu_{\mathcal C}(\Delta_{\varphi})}
C \sum_{k=0}^{j} 1 \leq B C |x|^{\nu_{\mathcal C}(\Delta_{\varphi})-e({\mathcal C})}$$ lead us to Eq. (\[equ:trackcs\]).
\[def:ab\] Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$ with $N \geq 1$. Let $\beta$ be a germ of connected set at $0 \in {\mathbb C}$. Consider a family of sub-trajectories $\Gamma_{x}:[0,T'(x)] \to U_{\epsilon}(x)$ of $\Re (X)$ defined for $x \in \beta$. We say that the family is $(A,B)$ bounded if
- $\Gamma_{x}$ changes at most $A$ times of basic set and
- $\Gamma_{x}[j,j'] \subset {\mathcal C}$ and $0 \leq j \leq j' \leq T'(x) \implies j'-j \leq B/|x|^{e({\mathcal C})}$
for any compact-like set ${\mathcal C}$ and any $x \in \beta$.
We compare orbits of $\varphi$ and ${\mathfrak F}_{\varphi}=\mathrm{exp}(X)$ by analyzing the sub-orbits contained in the basic sets of the dynamical splitting. The first condition in Definition \[def:ab\] is a natural finiteness property. The second property allows to apply Proposition \[pro:bouis\] to ${\{\Gamma_{x}[0,T'(x)]\}}_{x \in \beta}$. They assure that the dynamics of $\varphi$ and $\mathrm{exp}(X)$ are similar in a neighborhood of ${\{\Gamma_{x}[0,T'(x)]\}}_{x \in \beta}$.
It is clear that the trajectories of $\Re (X)$ ($N >1$) provided by Propositions \[pro:exll\] and \[pro:exll2\] are $(A,B)$ bounded for some values $A,B \in {\mathbb R}^{+}$. Indeed all Long Trajectories are $(A,B)$ bounded (Proposition \[pro:uniform\]).
Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$. A posteriori the Long Orbits of $\varphi$ are $(A,B)$ bounded for some values $A,B \in {\mathbb R}^{+}$ that do not depend on the Long Orbit. We introduce next these a priori bounds.
\[def:B\] Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$. Consider the dynamical splitting associated to $X$ in Section \[sec:dynspl\]. The number of boundary transversals (see Definition \[def:bdtr\]) is bounded by a number $A_{X} \in {\mathbb N}$ depending only on $X$. Let ${\mathcal C}_{j}$ be a compact-like set and $\lambda_{0} \in {\mathbb S}^{1}$. We define $B_{j,\lambda_{0}}^{0}$ as the maximum of the periods of the closed trajectories of $X_{j}(\lambda_{0})$ (Definition \[def:levels\]). We define $B_{j,\lambda_{0}}^{1}$ as the maximum of the values $s \in {\mathbb R}^{+}$ such that exists a trajectory $\Gamma[0,s]$ of $\Re (X_{j}(\lambda_{0}))$ contained in ${\mathcal C}_{j}(0,\lambda_{0})$ and not contained in a closed trajectory of $\Re (X_{j}(\lambda_{0}))$ in ${\mathcal C}_{j}(0,\lambda_{0})$. We define $B_{\digamma}^{k}= 1+\max_{1 \leq j \leq q, \ \lambda \in {\mathcal U}_{X}^{1}} B_{j,\lambda}^{k}$ for $k \in \{0,1\}$ and $B_{\digamma} = \max (B_{\digamma}^{0},B_{\digamma}^{1})$.
The polynomial vector fields associated to a convergent normal form $X$ of $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ only depend on $\varphi$. Thus the constant $B_{\digamma}$ depends only on $\varphi$ and the dynamical splitting $\digamma$.
\[def:A\] Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ with convergent normal form $X$. The number $A_{X}$ depends on $\varphi$ but not on $X$. We denote $A_{\varphi}=A_{X}$. We denote $B_{\varphi} = B_{\digamma_{X}}$. Of course $B_{\varphi}$ depends on the choice of $\digamma_{X}$ (see Definition \[def:dynfx\]).
\[lem:abist\] Let $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$ with $N \geq 1$. Let $\beta$ be a germ of connected set at $0 \in {\mathbb C}$. Consider a family of sub-trajectories ${\{\Gamma_{x}[0,T'(x)]\}}_{x \in \beta}$ of $\Re (X)$ such that $\lim_{x \to \beta} \Gamma_{x}(0)$ exists and it is not $(0,0)$. Suppose that the family is $(A,B)$ bounded. Then ${\{\Gamma_{x}[0,T'(x)]\}}_{x \in \beta}$ is stable.
A family ${\{\Gamma_{x}[0,T'(x)]\}}_{x \in \beta}$ that is $(A,B)$ bounded and stable satisfies the hypotheses in Propositions \[pro:boufespre\] and \[pro:bouis\] that guarantee tracking in basic sets. The lemma shows that the stability condition is superfluous.
Suppose that it is not stable. There exists a sequence $x_{n} \in \beta$, $x_{n} \to 0$ such that $\Gamma_{x_{n}}[0,T'(x_{n})] \cap {\mathcal E} \neq \emptyset$, and $x_{n}/|x_{n}|$ tends to $\lambda_{0} \in {\mathcal U}_{X}^{{\mathcal E},1}$ for some non-parabolic exterior set ${\mathcal E}$. The exterior set ${\mathcal E}$ is enclosed by a compact-like set ${\mathcal C}$. Let $X_{\mathcal C}(\lambda)$ be the polynomial vector field associated to ${\mathcal C}$. Since the point in $(\mathrm{Sing} X \cap {\mathcal E})(0,\lambda_{0})$ is indifferent for $X_{\mathcal C}(\lambda_{0})$ then $\Gamma_{x_{n}}[0,T'(x_{n})]$ adheres in (adapted coordinates) to all periodic trajectories in ${\mathcal C}(0,\lambda_{0})$ enclosing $(\mathrm{Sing} (X) \cap {\mathcal E})(0,\lambda_{0})$. Periodic trajectories in ${\mathcal C}$ never quit ${\mathcal C}$. Thus the family does not satisfy the last condition in Definition \[def:ab\].
\[def:fam\] Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ with $N > 1$. Fix a convergent normal form $X$ of $\varphi$. Suppose that ${\mathcal O}$ is either a weak Long Trajectory $(X, y_{+},\beta,T)$ or a Long Orbit $(\varphi, y_{+},\beta,T)$. Consider $\Gamma_{x}=\Gamma(X, \upsilon_{\mathcal O}(x), U_{\epsilon})$ for $x \in \beta$. A sub-family associated to ${\mathcal O}$ is a family of the form ${\{ \Gamma_{x}[0,T_{1}(x)]\}}_{x \in \beta}$ for some function $0 \leq T_{1} \leq T$. If $[0,T(x)] \subset {\mathcal I}(\Gamma_{x})$ for any $x \in \beta$ we say that ${\{ \Gamma_{x}[0,T(x)]\}}_{x \in \beta}$ is the family associated to ${\mathcal O}$
In order to prove that a Long Orbit tracks its associated family it suffices to show that it is $(A,B)$ bounded by Lemma \[lem:abist\]. We briefly outline the proof. First we see that there is tracking for $(A,B)$ bounded sub-families (Proposition \[pro:tracking\]). Then we prove that $(A,B)$ boundness plus tracking implies that the families associated to Long Orbits satisfy a Rolle property (Proposition \[pro:rolle\]). Finally if the families associated to Long Orbits are not $(A,B)$ bounded we construct $(A,B)$-bounded subfamilies that do not satisfy the Rolle property, obtaining a contradiction. Along the way we obtain a formula for the length of Long Orbits (Propositions \[pro:ltlo\] and \[pro:lores\]).
The residue formula for diffeomorphisms {#subsec:resdif}
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In this section we show that Long Trajectories of a convergent normal form induce Long Orbits of a diffeomorphism. We also obtain a generalization of the residue formula.
\[pro:tracking\] Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N > 1$. Fix a convergent normal form $X$ of $\varphi$. Suppose that ${\mathcal O}$ is either a weak Long Trajectory $(X, y_{+},\beta,T)$ or a Long Orbit $(\varphi, y_{+},\beta,T)$. Then, up to trimming ${\mathcal O}$, any $(A,B)$ bounded sub-family ${\{ \Gamma_{x}[0,T_{1}(x)]\}}_{x \in \beta}$ of ${\mathcal O}$ satisfies that (see Definition \[def:normal\]) $$\varphi^{j}(\upsilon_{\mathcal O}(x)) \in
B_{X}({\mathfrak F}_{\varphi}^{j}(\upsilon_{\mathcal O}(x)),1)$$ for all $0 \leq j \leq \lceil T_{1}(x) \rceil$ and $x \in \beta$. Moreover, if ${\{ \Gamma_{x}[0,T(x)]\}}_{x \in \beta}$ is $(A,B)$ bounded and $\lim_{n \to \infty} \varphi^{\lceil T(x_{n} \rceil}(\upsilon_{\mathcal O}(x_{n}))$ converges to $(0,y_{-})$ for some sequence $x_{n} \in \beta$ then $$\label{equ:noise}
\sum_{j=0}^{\lceil T(x_{n} \rceil -1} \Delta_{\varphi}(\varphi^{j}(\upsilon_{\mathcal O}(x_{n}))) -
\sum_{j=0}^{\infty} \Delta_{\varphi}(\varphi^{j}(0,y_{+}))
-
\sum_{j=1}^{\infty} \Delta_{\varphi}(\varphi^{-j}(0,y_{-}))$$ converges to $0$ when $n \to \infty$ (see Definition \[def:delta\]).
The idea is that Long Orbits of elements of ${\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ have good tracking properties if their associated families are $(A,B)$ bounded. The convergence to $0$ of the expression in Eq. (\[equ:noise\]) is key to generalize the residue formula for diffeomorphisms.
Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$. Consider a convergent normal form $X$ of $\varphi$. There exists $\epsilon'>0$ such that $$\left| \sum_{j \geq 0} \Delta_{\varphi} \circ \varphi^{j}(0,y_{+}) \right| < \frac{1}{4}
\ \ \mathrm{and} \ \
\left| \sum_{j \geq 1} \Delta_{\varphi} \circ \varphi^{-j}(0,y_{-}) \right| < \frac{1}{4} .$$ for all $(0,y_{+})$ in an attracting petal of $\Re (X)_{|U_{\epsilon'}(0)}$ and $(0,y_{-})$ in a repelling petal. From now on and up to trimming we suppose that Long Orbits are contained in $U_{\epsilon'}$.
We have $\nu_{\mathcal B} (\Delta_{\varphi}) - e({\mathcal B})>0$ for any basic set ${\mathcal B}$ different than the first exterior set ${\mathcal E}_{0}$. On the other hand we have $\nu_{{\mathcal E}_{0}} (\Delta_{\varphi}) - e({\mathcal E}_{0})=0$ if $m=0$. Let us use the notations for families and sub-families in Definition \[def:fam\].
Fix $0 < \xi <1/(4(A+1))$. The first exterior set is parabolic, so we can choose $\epsilon'' >0$ such that Eq. (\[equ:trackes\]) in Proposition \[pro:boufespre\] holds for trajectories contained in $${\mathcal E} = \{(x,y) \in B(0,\delta) \times {\mathbb C} : \epsilon'' \geq |y| \geq \rho|x| \}.$$ We claim that there exists $M'>0$ such that $\Gamma_{x}[M',\min(T_{1}(x),T(x)-M')]$ is contained in $U_{\epsilon''}$ for any $x \in \beta$. This is obvious if ${\mathcal O}$ is a weak Long Trajectory. Let us prove it for Long Orbits. We choose $M' \in {\mathbb N}$ satisfying that $$\label{equ:auxu}
\{\varphi^{M'}(\upsilon(x)), \hdots, \varphi^{\lceil T(x) \rceil -M'}(\upsilon(x))\}
\subset U_{\tilde{\epsilon}}$$ for some $\tilde{\epsilon} >0$ such that $\cup_{z \in B(0,2)} \mathrm{exp}(z X)(\overline{U_{\tilde{\epsilon}}}) \subset U_{\epsilon''}$. If the property does not hold true we define $$T_{2}(x) = \min \{ s \in [M',\min(T_{1}(x),T(x)-M')]: \Gamma_{x}(s) \not \in U_{\epsilon''} \};$$ it is well-defined for a sequence $x_{n} \in \beta$, $x_{n} \to 0$. The family ${\{\Gamma_{x}[0,T_{2}(x)]\}}_{x \in \beta}$ is stable by Lemma \[lem:abist\]. Propositions \[pro:boufespre\] and \[pro:bouis\] imply that $$|\psi_{X} (\varphi^{[T_{2}(x_{n})]}(\upsilon_{\mathcal O}(x_{n}))) -
\psi_{X}(\mathrm{exp}([T_{2}(x_{n})] X)(\upsilon_{\mathcal O}(x_{n})))| \leq \frac{1}{4} + A \xi < 1 \
\mathrm{for} \ n >>0.$$ The left hand side of the previous equation is greater than $2-1=1$ by Eq. (\[equ:auxu\]) and the choice of $T_{2}$. We obtain a contradiction.
The family ${\{\Gamma_{x}[0,T_{1}(x)]\}}_{x \in \beta}$ is stable by Lemma \[lem:abist\]. Propositions \[pro:boufespre\] and \[pro:bouis\] imply $$|\psi_{X} (\varphi^{j}(\upsilon_{\mathcal O}(x))) -
\psi_{X}({\mathfrak F}_{\varphi}^{j}(\upsilon_{\mathcal O}(x)))| < \frac{1}{4} + \frac{1}{4} + A \xi < 1$$ for all $0 \leq j \leq \lceil T_{1}(x) \rceil$ and $x \in \beta$ in a neighborhood of $0$.
Suppose that ${\{ \Gamma_{x}[0,T(x)]\}}_{x \in \beta}$ is $(A,B)$ bounded and $\lim_{n \to \infty} \varphi^{\lceil T(x_{n} \rceil}(\upsilon_{\mathcal O}(x_{n})) =(0,y_{-})$. We denote by $G(x_{n})$ the expression in Eq. (\[equ:noise\]). We define $$G_{0} = \sum_{j=0}^{\infty} \Delta_{\varphi}(\varphi^{j}(0,y_{+})) \ \mathrm{and} \
G_{1} = \sum_{j=1}^{\infty} \Delta_{\varphi}(\varphi^{-j}(0,y_{-})).$$ We have $$G(x_{n})=\psi_{X} (\varphi^{\lceil T(x_{n}) \rceil}(\upsilon(x_{n}))) -
\psi_{X}(\mathrm{exp}(\lceil T(x_{n}) \rceil X)(\upsilon(x_{n})))
- G_{0} - G_{1}$$ Given $0 < \xi <1/(4(A+1))$ Propositions \[pro:boufespre\] and \[pro:bouis\] imply $|G(x_{n})| \leq o(1) + A \xi$ for $n>>1$. We deduce that $\lim_{n \to \infty} G(x_{n}) =0$.
The next proposition is the analogue of Remark \[rem:udll\] for Long Orbits. The non-existence of Long Orbits is a generic phenomenon in the parameter space.
\[pro:udlo\] Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N > 1$. Fix a convergent normal form $X$ of $\varphi$. Consider a Long Orbit ${\mathcal O}=(\varphi, y_{+},\beta,T)$ such that ${\mathcal S}_{\mathcal O}$ is compact. Then $\beta$ adheres a unique direction in ${\mathcal U}_{X}^{1}$.
Fix $\lambda_{0} \in {\mathbb S}^{1} \setminus {\mathcal U}_{X}^{1}$. Consider a compact connected small neighborhood $K$ of $\lambda_{0}$ in ${\mathbb S}^{1} \setminus {\mathcal U}_{X}^{1}$ and $\tilde{\beta} = (0,\delta) K$. Up to trimming the Long Orbit we can suppose that $(0,y_{+})$ is in an attracting petal of $\Re (X)_{|U_{\epsilon}(0)}$. Fix the dynamical splitting $\digamma_{K}$ provided by Lemma \[lem:goins\]. Thus given $\epsilon''>0$ there exists $M \in {\mathbb N}$ such that ${\mathfrak F}_{\varphi}^{j}(x,y_{+}) \in U_{\epsilon''}$ for all $j \geq M$ and $x \in \tilde{\beta}$ close to $0$. Corollary \[cor:stdir\] implies $\lim_{n \to \infty} {\mathfrak F}_{\varphi}^{n}(x,y_{+}) \in \mathrm{Fix} (\varphi)$ for any $x \in \tilde{\beta}$ close to $0$. Consider any family of sub-trajectories $\Gamma_{x}:[0,T'(x)] \to U_{\epsilon}(x)$ of $\Re (X)$ defined for $x \in \tilde{\beta}$ and such that $\lim_{x \in \tilde{\beta}, \ x \to 0} \Gamma_{x}(0) =(0,y_{+})$. Lemma \[lem:goins\] implies that ${\{ \Gamma_{x}\}}_{x \in \tilde{\beta}}$ is $(A,1+\max_{1 \leq j \leq q} B_{j,\lambda_{0}}^{1})$ bounded for some $A \in {\mathbb R}^{+}$ that depends only on $X$ (see Definition \[def:B\]). We can proceed as in the proof of Proposition \[pro:tracking\] to show that $\varphi^{j}(\upsilon_{\mathcal O}(x)) \in
B_{X}({\mathfrak F}_{\varphi}^{j}(\upsilon_{\mathcal O}(x)),1)$ for all $j \geq 0$ and $x \in \tilde{\beta}$. We deduce that $\tilde{\beta} \cap \beta$ does not contain any point in the neighborhood of $0$. Hence $\beta_{\pi} \cap (\{0\} \times {\mathbb S}^{1})$ is a singleton since it is a connected set contained in $\{0\} \times {\mathcal U}_{X}^{1}$.
The residue formula (\[equ:deftf\]) for Long Trajectories of $\Re (X)$ with $X \in {{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$ involves Fatou coordinates $\psi_{+}$ and $\psi_{-}$ of $X$. In order to obtain a generalization for Long Orbits of $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ it is natural to replace the previous functions with Fatou coordinates of $\varphi_{|x=0}$.
\[def:fatl\] Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N > 1$. Fix a convergent normal form $X$ of $\varphi$. Consider an attracting petal ${\mathcal P}_{+}'$ and a repelling petal ${\mathcal P}_{-}'$ of $\varphi_{|x=0}$. We define $$\psi_{{\mathcal P}_{+}'}^{\varphi}(0,y) = \psi_{+}(0,y) + \sum_{j=0}^{\infty} \Delta_{\varphi}(\varphi^{j}(0,y)), \ \
\psi_{{\mathcal P}_{-}'}^{\varphi}(0,y) = \psi_{-}(0,y) - \sum_{j=1}^{\infty} \Delta_{\varphi}(\varphi^{-j}(0,y))$$ in ${\mathcal P}_{+}'$ and ${\mathcal P}_{-}'$ respectively where $\psi_{+}$, $\psi_{-}$ are Fatou coordinates of $X$. The function $\psi_{{\mathcal P}_{j}'}^{\varphi}$ is a Fatou coordinates of $\varphi_{|{\mathcal P}_{j}'}$, i.e $\psi_{{\mathcal P}_{j}'}^{\varphi} \circ \varphi \equiv \psi_{{\mathcal P}_{j}'}^{\varphi} +1$ for $j \in \{+,-\}$.
We introduce the main result of this section.
\[pro:ltlo\] Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N > 1$. Fix a convergent normal form $X$ of $\varphi$. Suppose that ${\mathcal O}=(X, y_{+},\beta,T)$ is a weak Long Trajectory and that ${\{ \Gamma_{x}[0,T(x)]\}}_{x \in \beta}$ is $(A,B)$ bounded. Suppose that, up to trimming ${\mathcal O}$, $\varphi^{\lceil T(x_{n}) \rceil}(\upsilon_{\mathcal O}(x_{n}))$ converges to $(0,y_{-}) \neq (0,0)$ for some sequence $x_{n} \in \beta$, $x_{n} \to 0$. Then we obtain $$\label{equ:ltlo1}
\psi_{{\mathcal P}_{-}'}^{\varphi}(0,y_{-}) - \psi_{{\mathcal P}_{+}'}^{\varphi}(0,y_{+}) =
\lim_{n \to \infty} \left( \lceil T(x_{n}) \rceil + 2 \pi i \sum_{Q \in E_{-}(x_{n})} Res(X,Q) \right)$$ where $(E_{-},E_{+})$ is the division of $\mathrm{Sing}(X)$ induced by ${\mathcal O}$. Suppose now that ${\mathcal O}$ is a Long Trajectory. Then ${\mathcal O}'=(\varphi, y_{+},\beta,T)$ is a Long Orbit. Moreover, ${\mathcal O}'$ satisfies ${\mathcal S}_{{\mathcal O}'} = {\mathcal S}_{\mathcal O}$ and $$\psi_{{\mathcal P}_{-}'}^{\varphi}(\chi_{{\mathcal O}'}(s+i u))
- \psi_{{\mathcal P}_{+}'}^{\varphi}(0,y_{+}) =
\lim_{\vartheta_{\mathcal O}(x) \to u, \ x \to 0}^{\lceil T(x) \rceil - T(x) \to s}
\left(
\lceil T(x) \rceil + 2 \pi i \sum_{Q \in E_{-}(x)} Res(X,Q) \right)$$ for any $s+i u \in [0,1] + i {\mathcal S}_{{\mathcal O}'}$. In particular we get $\psi_{{\mathcal P}_{-}'}^{\varphi}(\chi_{{\mathcal O}'}(z)) =
\psi_{{\mathcal P}_{-}'}^{\varphi}(\chi_{{\mathcal O}'}(0)) +z$ for any $z \in [0,1] + i {\mathcal S}_{{\mathcal O}'}$.
Let us remark that $(0,y_{+})$ belongs to an attractive petal ${\mathcal P}_{+}'$ and all possible limits of sequences of the form $\varphi^{\lceil T(x_{n}) \rceil}(\upsilon_{\mathcal O}(x_{n}))$ are contained in a repelling petal ${\mathcal P}_{-}'$ of $\varphi_{|U_{\epsilon}(0)}$.
The family associated to a weak Long Trajectory is always $(A_{\varphi},B_{\varphi})$ bounded. A direct proof is not difficult and it is also a consequence of Proposition \[pro:uniform\]. The corresponding hypothesis in Proposition \[pro:ltlo\] is a posteriori unnecessary. Long Trajectories of $\Re (X)$ always induce Long Orbits of $\varphi$.
Denote $\upsilon^{1}(x)=\mathrm{exp}(T(x)X)(\upsilon_{\mathcal O}(x))$. By defining $\psi_{+}$ and $\psi_{-}$ as in Subsection \[subsec:res\] we obtain $$\label{equ:ltlo3}
T(x) = \psi_{-}(\upsilon^{1}(x)) - \psi_{+}(\upsilon_{\mathcal O}(x)) -
2 \pi i \sum_{Q \in E_{-}(x)} Res(X,Q) \ \ \forall x \in \beta$$ for some division $(E_{-},E_{+})$ of $\mathrm{Sing} (X)$. Denote $G(x)=- 2 \pi i \sum_{Q \in E_{-}(x)} Res(X,Q)$. We want to express $T$ as a function of data depending on $\varphi$. Since $$\psi_{-} \circ \varphi^{\lceil T(x) \rceil}(\upsilon_{\mathcal O}(x)) -
\psi_{-} \circ \mathrm{exp}(\lceil T(x) \rceil X) (\upsilon_{\mathcal O}(x)) =
\sum_{j=0}^{\lceil T(x) \rceil -1} \Delta_{\varphi}(\varphi^{j}(\upsilon_{\mathcal O}(x)))$$ (see Definition \[def:delta\]) we obtain $$\lceil T(x) \rceil= \psi_{-} \circ \varphi^{\lceil T(x) \rceil}(\upsilon_{\mathcal O}(x)) -
\sum_{j=0}^{\lceil T(x) \rceil-1} \Delta_{\varphi}(\varphi^{j}(\upsilon_{\mathcal O}(x)))
- \psi_{+}(\upsilon_{\mathcal O}(x)) + G(x)$$ for any $x \in \beta$. We obtain $\varphi^{\lceil T(x) \rceil}(\upsilon_{\mathcal O}(x)) \in
B_{X}({\mathfrak F}_{\varphi}^{\lceil T(x) \rceil}(\upsilon_{\mathcal O}(x)),1)$ by the tracking phenomenon. Thus any $\varphi^{\lceil T(x_{n}) \rceil}(\upsilon_{\mathcal O}(x_{n}))$ has a convergent subsequence. Suppose that $\varphi^{\lceil T(x_{n}) \rceil}(\upsilon_{\mathcal O}(x_{n}))$ converges to $(0,y_{-}) \neq (0,0)$. Proposition \[pro:tracking\] implies Eq. (\[equ:ltlo1\]), see Definition \[def:fatl\].
Suppose that ${\mathcal O}$ is a Long Trajectory. We have $$\lim_{\vartheta_{\mathcal O}(x) \to u, \ x \to 0}^{\lceil T(x) \rceil - T(x) \to s}
\left(
\lceil T(x) \rceil + 2 \pi i \sum_{Q \in E_{-}(x)} Res(X,Q) \right)=
\psi_{-}(\chi_{\mathcal O}(iu)) -\psi_{+}(0,y_{+}) + s$$ for all $s \in [0,1]$ and $u \in {\mathcal S}_{\mathcal O}$. We obtain $$\label{equ:ltlo2}
\psi_{{\mathcal P}_{-}'}^{\varphi}(\chi_{{\mathcal O}'}(s+i u)) - \psi_{{\mathcal P}_{+}'}^{\varphi}(0,y_{+}) =
\psi_{-}(\chi_{\mathcal O}(iu)) -\psi_{+}(0,y_{+}) + s .$$ by Eq. (\[equ:ltlo1\]) since $\psi_{{\mathcal P}_{-}'}^{\varphi}$ is injective ${\mathcal P}_{-}'$. Moreover $\chi_{{\mathcal O}'}$ satisfies $$\psi_{{\mathcal P}_{-}'}^{\varphi}(\chi_{{\mathcal O}'}(z)) -
\psi_{{\mathcal P}_{-}'}^{\varphi}(\chi_{{\mathcal O}'}(0)) = z$$ for any $z \in [0,1] + i {\mathcal S}_{{\mathcal O}'}$. We deduce $\chi_{{\mathcal O}'}(1+iu) = \varphi(\chi_{{\mathcal O}'}(iu))$ for any $u \in {\mathcal S}_{{\mathcal O}'}$.
The Long Trajectories provided in Propositions \[pro:exll\] and \[pro:exll2\] are $(A,B)$ bounded. Thus given $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$ and a point $(0,y_{+})$ contained in an attracting petal of $\varphi_{|x=0}$ there exists a Long Orbit $(\varphi,y_{+},\beta,T)$.
The Rolle property {#sec:Rolle}
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Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$ and let $X$ be a convergent normal form. Long Trajectories of $\Re (X)$ induce Long Orbits of $\varphi$ but the reciprocal is not clear. If $\varphi(x,y)=(x,f(x,y))$ is multi-parabolic, i.e. if $(\partial f/\partial y)_{|\mathrm{Fix} (\varphi)} \equiv 1$ the situation is much simpler. Indeed trajectories of $\Re (X)$ satisfy the Rolle property, i.e. they do not intersect twice connected transversals (Proposition 2.1.1 of [@rib-mams]). In particular $\Re (X)$ has no closed trajectories. We deduce that any family of trajectories of $\Re (X)$ is $(A_{\varphi},B_{\varphi})$ bounded. This situation is quite special and corresponds to the case when orbits of $\varphi$ [*always*]{} track orbits of ${\mathfrak F}_{\varphi}$. In the general case the Rolle property still holds true for families associated to Long Orbits.
\[def:rolle\] Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N > 1$. Fix a convergent normal form $X$ of $\varphi$. Suppose that ${\mathcal O}=(\varphi, y_{+},\beta,T)$ generates a Long Orbit. We say that a sub-family ${\{ \Gamma_{x}[0,T_{1}(x)]\}}_{x \in \beta}$ of ${\mathcal O}$ satisfies the Rolle property if there is no choice of a sequence $x_{n} \in \beta$, $x_{n} \to 0$ such that
- There exist $0 \leq T_{2}(x_{n}) < T_{3}(x_{n}) \leq T_{1}(x_{n})$ for any $n \in {\mathbb N}$ such that $\lim_{n \to \infty} \Gamma_{x_{n}}(T_{j}(x_{n})) = (0,0)$ for $j \in \{2,3\}$.
- There exists a trajectory $\gamma_{n}:[0,T_{4}(x_{n})] \to U_{\epsilon}(x_{n})$ of $\Re (iX)$ or $\Re (-iX)$ such that $\gamma_{n}(0)=\Gamma_{x_{n}}(T_{2}(x_{n}))$ and $\gamma_{n}(T_{4}(x_{n}))=\Gamma_{x_{n}}(T_{3}(x_{n}))$ for any $n \in {\mathbb N}$.
- Given any $\epsilon''>0$ there exists $n_{0} \in {\mathbb N}$ such that $\gamma_{n}[0,T_{4}(x_{n})] \subset U_{\epsilon''}$ for any $n \geq n_{0}$.
We can always suppose that $\Gamma_{x_{n}}[T_{2}(x_{n}), T_{3}(x_{n})] \cup \gamma_{n}[0,T_{4}(x_{n})]$ is a closed simple curve by changing slightly the trajectories. We denote by $D_{n}$ the bounded component of $(\{x_{n}\} \times {\mathbb C})
\setminus (\Gamma_{x_{n}}[T_{2}(x_{n}), T_{3}(x_{n})] \cup \gamma_{n}[0,T_{4}(x_{n})])$. We define $\mathrm{gap}_{n}=T_{4}(x_{n})$.
We prove that families associated to Long Orbits are $(A,B)$ bounded by reductio ad absurdum. More precisely we construct sub-families that are $(A,B)$ bounded and fail to satisfy the Rolle property. This contradicts the next proposition.
\[pro:rolle\] Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N > 1$. Fix a convergent normal form $X$ of $\varphi$. Suppose that ${\mathcal O}=(\varphi, y_{+},\beta,T)$ generates a Long Orbit. Suppose that ${\mathcal S}_{\mathcal O}$ is a compact set. Then, up to trimming ${\mathcal O}$, any $(A,B)$ bounded sub-family ${\{ \Gamma_{x}[0,T_{1}(x)]\}}_{x \in \beta}$ of ${\mathcal O}$ satisfies the Rolle property.
Suppose that the Rolle property is not satisfied. The set $\chi_{\mathcal O} ([0,1] + i {\mathcal S}_{\mathcal O})$ is compact and it does not contain $(0,0)$. The first condition in Definition \[def:rolle\] and Proposition \[pro:tracking\] imply that $\lim_{n \to \infty} T_{2}(x_{n}) = \lim_{n \to \infty} (T-T_{3})(x_{n}) = \infty$ and that $\Gamma_{x_{n}}[T_{2}(x_{n}), T_{3}(x_{n})]$ converges to $\{(0,0)\}$ in the Hausdorff topology for compact sets. Hence $\overline{D_{n}}$ converges to $\{(0,0)\}$ by the last condition in Definition \[def:rolle\].
If $\Re (X)$ points towards $D_{n}$ at $\gamma_{n}(0,T_{4}(x_{n}))$ then $D_{n}$ is invariant by the positive flow of $\Re (X)$. Otherwise $D_{n}$ is invariant by the negative flow of $\Re (X)$. We claim that we are always in the former situation for $n>>0$. Otherwise $\Gamma_{x_{n}}(0) \in \overline{D_{n}}$ for a subsequence and we obtain a contradiction since $$(0,y_{+}) = \lim_{n \to \infty} \upsilon_{\mathcal O}(x_{n}) =
\lim_{n \to \infty} \Gamma_{x_{n}}(0) = (0,0)$$ and $y_{+} \neq 0$. The last equality is a consequence of $\lim_{n \to \infty} \overline{D_{n}} = \{(0,0)\}$.
Consider a subsequence such that $\mathrm{gap}_{n} \leq K'$ for some $K' \in {\mathbb R}^{+}$. Let us prove that $\lim_{n \to \infty} (T_{3}-T_{2})(x_{n}) = \infty$. The vector field $X_{|x=0}$ has a multiple singular point at $(0,0)$. Thus the diffeomorphism $(z,x,y) \mapsto (z, \mathrm{exp}(zX)(x,y))$ defined in a neighborhood of ${\mathbb C} \times \{(0,0)\}$ is of the form $(z,x,y+z u(z,x,y) X(y))$ where $u(z,0,0) \equiv 1$. Hence given $C \in {\mathbb R}^{+}$ there exists a neighborhood $W$ of $(0,0)$ in ${\mathbb C}^{2}$ such that $z \mapsto \mathrm{exp}(zX)(x,y)$ is injective in $B(0,C)$ for any $(x,y) \in W \setminus \mathrm{Sing} (X)$. If a subsequence satisfies $\mathrm{gap}_{n} \leq K'$ and $(T_{3}-T_{2})(x_{n}) < K''$ then $\partial D_{n}$ and $D_{n}$ are contained in $B_{X}(\Gamma_{x_{n}}(T_{2}(x_{n})),K'+K''+1)$ where clearly trajectories of $\Re (X)$ can not intersect twice trajectories of $\Re (iX)$. We obtain $\lim_{n \to \infty} (T_{3}-T_{2})(x_{n}) = \infty$. We have $${\mathfrak F}_{\varphi}^{[T_{3}(x_{n})-T_{2}(x_{n})]}(\upsilon_{\mathcal O}(x_{n}))=
\mathrm{exp}([T_{3}(x_{n})-T_{2}(x_{n})]X)(\upsilon_{\mathcal O}(x_{n}))
\in B_{X}(\upsilon_{\mathcal O}(x_{n}),K'+1) .$$ The tracking phenomenon (Proposition \[pro:tracking\]) implies that $$|\psi_{X} (\varphi^{[T_{3}(x_{n})-T_{2}(x_{n})]}(\upsilon_{\mathcal O}(x_{n}))) -
\psi_{X} ({\mathfrak F}_{\varphi}^{[T_{3}(x_{n})-T_{2}(x_{n})]}(\upsilon_{\mathcal O}(x_{n})))| < 1 .$$ Since $\lim_{n \to \infty} (T_{3}-T_{2})(x_{n}) = \lim_{n \to \infty} T(x_{n})- (T_{3}-T_{2})(x_{n}) =\infty$ and ${\mathcal O}$ is a Long Orbit we have $\lim_{n \to \infty} \varphi^{[T_{3}(x_{n})-T_{2}(x_{n})]}(\upsilon_{\mathcal O}(x_{n})) = (0,0)$. This leads us to $$\lim_{n \to \infty} {\mathfrak F}_{\varphi}^{[T_{3}(x_{n})-T_{2}(x_{n})]}(\upsilon_{\mathcal O}(x_{n})) = (0,0) \
\mathrm{and} \
(0,y_{+}) = \lim_{n \to \infty} \upsilon_{\mathcal O}(x_{n}) =(0,0).$$ The last property contradicts $y_{+} \neq 0$.
Resuming $D_{n}$ is invariant by the positive flow of $\Re (X)$ and $\lim_{n \to \infty} \mathrm{gap}_{n} = \infty$. Let us suppose that $\mathrm{gap}_{n} \geq 4$ for any $n \in {\mathbb N}$. Our goal is proving that there exists $s_{n} \in {\mathbb N}$ such that $\varphi^{j}(B_{X}(\overline{D_{n}},1)) \subset B_{X}(\overline{D_{n}},2)$ and $\varphi^{s_{n}}(B_{X}(\overline{D_{n}},1)) \subset D_{n}$ for $0 \leq j <s_{n}$ and $n>>0$. Consider the segment $$\eta_{n} = \Gamma(iX, \Gamma_{x_{n}}(T_{3}(x_{n})), U_{\epsilon})(-\mathrm{gap}_{n}+1,\mathrm{gap}_{n}-1) .$$ It satisfies $\mathrm{exp}(s X)(\eta_{n}) \subset D_{n}$ for all $s \in {\mathbb R}^{+}$ and $n \in {\mathbb N}$. We define $$\tilde{D}_{n} = \cup_{s \in {\mathbb R}^{+}} \mathrm{exp}(s X)(\eta_{n}), \ \
\iota_{n}^{c} = B_{X}(\Gamma_{x_{n}}[T_{2}(x_{n}), T_{3}(x_{n})],c) .$$
If $Q \in \iota_{n}^{2}$ there exists $s \in [T_{2}(x_{n}), T_{3}(x_{n})]$ such that $Q \in B_{X}(\Gamma_{x_{n}}(s),2)$. The tracking phenomenon implies that $\varphi^{[T_{3}(x_{n})-s] +4}(Q) \in \tilde{D}_{n}$.
If $Q \in B_{X}(\overline{D_{n}},1) \setminus (\tilde{D}_{n} \cup \iota_{n}^{2} )$ then $Q$ is of the form $\mathrm{exp}(s X)(Q_{0})$ where $s \in [-1,0]$ and $Q_{0}$ belongs to $\gamma_{n}[\sqrt{3},T_{4}(x_{n})-\sqrt{3}]$. We obtain that $\varphi^{2}(Q)$ belongs to $\tilde{D}_{n}$. It suffices to prove that $\varphi^{j}$ is well-defined in $\tilde{D}_{n}$ and $\varphi^{j}(\tilde{D}_{n}) \subset D_{n}$ for all $j \geq 0$ and $n \in {\mathbb N}$. We can define $s_{n}=[T_{3}(x_{n})-T_{2}(x_{n})] +4$.
Suppose that $\varphi^{j}(Q) \not \in D_{n}$ for some $Q \in \tilde{D}_{n}$ and $j \in {\mathbb N}$. We can assume that $\varphi^{k}(Q) \in D_{n}$ for $0 <k<j$. We obtain that $\varphi^{j}(Q) \in B_{X}(\overline{D_{n}},\epsilon_{n})$ where $\lim_{n \to \infty} \epsilon_{n}=0$. We claim that $\varphi^{j}(Q) \in \iota_{n}^{1/2}$; otherwise we obtain $\varphi^{j-1}(Q) \not \in D_{n}$. The point $\varphi^{j}(Q)$ belongs to $B_{X}(\Gamma_{x_{n}}(s),1/2)$ for some $s \in [T_{2}(x_{n}), T_{3}(x_{n})]$. The tracking phenomenon implies that there exists $0 \leq j' \leq [s-T_{2}(x)]+2$ such that $\varphi^{-k}(\varphi^{j}(Q)) \in \iota_{n}^{3/4}$ for $0 \leq k <j'$ and $\varphi^{-j'}(\varphi^{j}(Q)) \not \in D_{n}$. This is impossible since $\varphi^{-1}(\iota_{n}^{3/4}) \cap \tilde{D}_{n} =\emptyset$.
It is clear that $\varphi^{s_{n}}$ contracts the Poincaré metric in $D_{n}$. Thus $\varphi^{j s_{n}}$ converges uniformly to a point $P_{n}$ in $B_{X}(\overline{D_{n}},1)$. The point $P_{n}$ is an attractor for $\varphi^{s_{n}}$ and then for $\varphi$. The orbit ${\{ \varphi^{j}(Q)\}}_{j \in {\mathbb N}}$ is contained in $B_{X}(\overline{D_{n}},2)$ for any $Q \in B_{X}(\overline{D_{n}},1)$ and $\lim_{j \to \infty} \varphi^{j}(Q) = P_{n}$. The point $\varphi^{\lceil T_{2}(x_{n}) \rceil}(\upsilon_{\mathcal O}(x_{n}))$ belongs to $B_{X}(\overline{D_{n}},1)$. We deduce that $\varphi^{\lceil T(x_{n}) \rceil}(\upsilon_{\mathcal O}(x_{n}))$ belongs to $B_{X}(\overline{D_{n}},2)$ for $n>>0$. This implies that there exists $z \in [0,1]+i {\mathcal S}_{\mathcal O}$ such that $\chi_{\mathcal O} (z)=(0,0)$. This property contradicts Definition \[def:lo\].
Let us remark that the analogue for Long Trajectories admits a much simpler proof. It is a version of the proof of the case $\mathrm{gap}_{n} \not \to 0$. Indeed this condition guarantees that the attracting nature of $D_{n}$ is stable under small deformations.
\[pro:uniform\] Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N > 1$. Fix a convergent normal form $X$ of $\varphi$. Suppose that ${\mathcal O}=(\varphi, y_{+},\beta,T)$ generates a Long Orbit. Then, up to trimming ${\mathcal O}$, the family ${\{ \Gamma_{x}[0,T(x)]\}}_{x \in \tilde{\beta}}$ of ${\mathcal O}$ is $(A_{\varphi},B_{\varphi})$ bounded (see Definition \[def:A\]) where $\tilde{\beta}$ is a subset of $\beta$ such that $\tilde{\mathcal O}=(\varphi, y_{+},\tilde{\beta},T)$ satisfies ${\mathcal S}_{\tilde{\mathcal O}} = {\mathcal S}_{\mathcal O}$.
The proposition implies that the $(A,B)$ boundness condition is automatic for Long Orbits. Let us explain the setup of the proof.
A priori all the trajectories in the family ${\{ \Gamma_{x} \}}_{x \in \beta}$ are labeled [**(a)**]{}. We want to have $$\label{equ:pba}
\varphi^{j}(\upsilon_{\mathcal O}(x)) \in B_{X}({\mathfrak F}_{\varphi}^{j}(\upsilon_{\mathcal O}(x)),1)$$ for any $0 \leq j \leq \lceil T(x) \rceil$. Suppose that the property does not hold true for some $x \in \beta$. Then we replace $T(x)$ with $T_{1}(x) \leq T(x)$ such that Eq. (\[equ:pba\]) holds true for $0 \leq j < \lceil T_{1}(x) \rceil$ but it is false for $j= \lceil T_{1}(x) \rceil$. In such a case the label [**(a)**]{} for $\Gamma_{x}$ is replaced with [**(b)**]{}. Otherwise we define $T_{1}(x)=T(x)$.
Given a trajectory $\Gamma_{x}[0,T_{1}(x)]$ we keep the label if $\Gamma_{x}[0,T_{1}(x)]$ does not intersect twice a boundary transversal (see Definition \[def:bdtr\]). Otherwise we replace the previous label with [**(c)**]{}. We also replace $T_{1}(x)$ with a smaller or equal value such that $\Gamma_{x}[0,T_{1}(x)]$ intersects twice a boundary transversal $\gamma$ but $\Gamma_{x}[0,T_{1}(x))$ does not. The curve $\gamma$ is contained in the exterior boundary of a basic set ${\mathcal B} \neq {\mathcal E}_{0}$. It is easy to consider $T_{1}'(x)$ such that $\Gamma_{x}[0,T_{1}'(x)]$ intersects twice a sub-trajectory of $\Re (iX)$ contained in ${\mathcal B}$ (passing through a point in $T{\mathcal B}_{i X}(x)$) and either $T_{1}'(x) \leq T_{1}(x)$ or $T_{1}'(x) > T_{1}(x)$ and $\Gamma_{x}[T_{1}(x),T_{1}'(x)]$ is contained in ${\mathcal B}$ (see Figure (\[dr14\])).
![The trajectories of $\Re (X)$ and $\Re (iX)$ are solid and dashed thin curves respectively; the boundary of ${\mathcal B}$ is a thick curve[]{data-label="dr14"}](dr14.eps){height="5cm" width="10cm"}
The trajectories $\Gamma_{x}[0,T_{1}(x)]$ and $\Gamma_{x}[0,T_{1}'(x)]$ change of basic set a number of times that is bounded a priori.
We replace the label if there exists a compact-like set ${\mathcal C}'$, a sequence $x_{n} \in \beta$, $x_{n} \to 0$ and sequences $0 \leq j_{n} < j_{n}' \leq T_{1}(x_{n})$ such that $$\label{equ:auxlb}
\Gamma_{x_{n}}[j_{n},j_{n}'] \subset {\mathcal C}', \ \mathrm{and} \
|x_{n}|^{e({\mathcal C}')} (j_{n}' - j_{n}) > B_{\varphi} \ \forall n \in {\mathbb N}.$$ We can refine the sequence and consider a smaller value of $T_{1}$, $T_{1}< T_{2} \leq T$ and a compact-like basic set ${\mathcal C}$ such that $$\Gamma_{x_{n}}(T_{1}(x_{n})) \in \partial {\mathcal C}, \
\Gamma_{x_{n}}[T_{1}(x_{n})), T_{2}(x_{n}))] \subset {\mathcal C}, \
|x_{n}|^{e({\mathcal C})} (T_{2} - T_{1})(x_{n}) > B_{\varphi}$$ for any $n \in {\mathbb N}$ and there is no choice of subsequences $j_{n}$, $j_{n}'$ and a compact-like set ${\mathcal C}'$ such that Eq. (\[equ:auxlb\]) holds true. We replace the previous label with [**(d)**]{} for the parameters in the sequence $x_{n}$.
Let us prove that the labels [**(b)**]{}, [**(c)**]{} and [**(d)**]{} never happen if ${\mathcal S}_{\mathcal O}$ is compact. The set $\beta$ adheres a unique direction $\lambda_{0}$ in ${\mathcal U}_{X}^{1}$ (Proposition \[pro:udlo\]). Let us remark that $T_{1}(x) \leq T(x)$ and $T_{2}(x) \leq T(x)$ for any $x \in \beta$.
Suppose that we are in the situation [**(d)**]{}. The family ${\{ \Gamma_{x_{n}}[0,T_{1}(x_{n})]\}}_{n \in {\mathbb N}}$ is $(A_{\varphi},B_{\varphi})$ bounded by the previous construction. The sequence $x_{n}/|x_{n}|$ tends to $\lambda_{0}$. We can also suppose that $\Gamma_{x_{n}}(T_{1}(x_{n}))$ converges to a point $(0,w_{0})$ in the adapted coordinates $(x,w)$ of ${\mathcal C}$ up to consider a subsequence. Let $X_{\mathcal C}(\lambda)$ be the polynomial vector field associated to ${\mathcal C}$. Since $X /|x|^{e({\mathcal C})} \to X_{\mathcal C}(\lambda_{0})$ in ${\mathcal C}$ if $x \to 0$ and $x/|x| \to \lambda_{0}$ the condition $|x_{n}|^{e({\mathcal C})} (T_{2} - T_{1})(x_{n}) > B_{\varphi}$ for any $n \in {\mathbb N}$ implies that the trajectory $\Gamma$ of $\Re (X_{\mathcal C}(\lambda_{0}))$ through $(r,\lambda,w)=(0,\lambda_{0},w_{0})$ in ${\mathcal C}$ is closed. The period of $\Gamma$ is less or equal than $B_{\varphi}-1$. We can replace $T_{2}(x_{n})$ with $T_{1}(x_{n}) + B_{\varphi} / |x_{n}|^{e({\mathcal C})}$. It is clear that $\Gamma_{x_{n}}[0,T_{2}(x_{n})]$ is $(A_{\varphi},B_{\varphi})$ bounded and does not satisfy the Rolle property for $n >>0$ since it adheres a periodic trajectory. This contradicts Proposition \[pro:rolle\].
Suppose that we are in the situation [**(c)**]{}, i.e. the set $E$ of parameters in $\beta$ having the label [**(c)**]{} adheres $0$. The families ${\{ \Gamma_{x}[0,T_{1}(x)]\}}_{x \in E}$ and ${\{ \Gamma_{x}[0,T_{1}'(x)]\}}_{x \in E}$ are $(A_{\varphi}+1,B_{\varphi})$ bounded by construction. If the set $\{x \in E : T_{1}'(x) < T(x) \}$ adheres $0$ then we obtain a violation of the Rolle property (Proposition \[pro:rolle\]). Otherwise we have $T_{1}(x) \leq T(x) \leq T_{1}'(x)$ for any $x \in \beta$. The tracking phenomenon (Proposition \[pro:tracking\]) implies that $\varphi^{\lceil T(x_{n}) \rceil}(\upsilon_{\mathcal O}(x_{n}))$ tends to $(0,0)$. This contradicts the definition of Long Orbit.
Suppose that we are in the situation [**(b)**]{}, i.e. the set $E$ of parameters in $\beta$ having the label [**(b)**]{} adheres $0$. The family ${\{ \Gamma_{x}[0,T_{1}(x)]\}}_{x \in E}$ is $(A_{\varphi},B_{\varphi})$ bounded by construction. This setup is incompatible with the tracking phenomenon (Proposition \[pro:tracking\]).
We can suppose that every point $x \in \beta$ has the label [**(a)**]{}. As a consequence the family associated to ${\mathcal O}$ is $(A_{\varphi},B_{\varphi})$ bounded.
Consider the general case, i.e. ${\mathcal S}_{\mathcal O}$ is not necessarily compact. Let $\beta_{n}$ be a neighborhood of $0$ in $\vartheta_{\mathcal O}^{-1}[-n,n]$ such that ${\{ \Gamma_{x}[0,T(x)]\}}_{x \in \beta_{n}}$ is $(A_{\varphi},B_{\varphi})$ bounded for $n \in {\mathbb N}$. The family ${\{ \Gamma_{x}[0,T(x)]\}}_{x \in \tilde{\beta}}$ is $(A_{\varphi},B_{\varphi})$ bounded for $\tilde{\beta} = \cup_{n \in {\mathbb N}} \beta_{n}$.
Next we provide the generalization of the residue formula in the discrete case.
\[pro:lores\] Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N > 1$. Fix a convergent normal form $X$ of $\varphi$. Suppose that ${\mathcal O}=(\varphi, y_{+},\beta,T)$ is a Long Orbit. Then, up to trimming ${\mathcal O}$, $\chi_{{\mathcal O}}$ satisfies $$\psi_{{\mathcal P}_{-}'}^{\varphi}(\chi_{{\mathcal O}}(s+iu)) - \psi_{{\mathcal P}_{+}'}^{\varphi}(0,y_{+}) =
\lim_{\vartheta_{\mathcal O}(x) \to u, \ x \to 0}^{\lceil T(x) \rceil - T(x) \to s}
\left(
\lceil T(x) \rceil + 2 \pi i \sum_{Q \in E_{-}(x)} Res(X,Q) \right)$$ for any $s +iu \in [0,1]+i{\mathcal S}_{\mathcal O}$ where $(E_{-},E_{+})$ is the division of $\mathrm{Fix} (\varphi)$ induced by ${\mathcal O}$.
The concept of division of the fixed points induced by a Long Orbit is introduced during the proof.
The family ${\{\Gamma_{x}[0,T(x)]\}}_{x \in \beta}$ associated to ${\mathcal O}$ is $(A_{\varphi},B_{\varphi})$ bounded by Proposition \[pro:uniform\]. The tracking phenomenon (Proposition \[pro:tracking\]) implies that $(X,y_{+},\beta,T)$ is a weak Long Trajectory. Hence it induces a division $(E_{-},E_{+})$ of $\mathrm{Fix} (\varphi) = \mathrm{Sing} (X)$; it is the division induced by ${\mathcal O}$. We complete the proof by applying Eq. (\[equ:ltlo1\]) in Proposition \[pro:ltlo\].
Long Orbits are invariant under translations in Fatou coordinates.
\[pro:evol\] Let $\varphi \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N > 1$. Fix a convergent normal form $X$ of $\varphi$. Suppose that ${\mathcal O}=(\varphi, y_{+},\beta,T)$ is a Long Orbit. Let ${\mathcal P}_{+}'$ and ${\mathcal P}_{-}'$ be the petals of $\varphi_{|x=0}$ containing $y_{+}$ and $\chi_{\mathcal O}(0)$ respectively. Consider $y_{+}' \in {\mathcal P}_{+}'$ such that there exists $\chi':[0,1] + i {\mathcal S}_{\mathcal O} \to {\mathcal P}_{-}'$ satisfying $$\psi_{{\mathcal P}_{-}'}^{\varphi}(\chi'(z)) -\psi_{{\mathcal P}_{+}'}^{\varphi}(0,y_{+}') =
\psi_{{\mathcal P}_{-}'}^{\varphi}(\chi_{\mathcal O}(z))
- \psi_{{\mathcal P}_{+}'}^{\varphi}(0,y_{+})$$ for any $z \in [0,1] + i {\mathcal S}_{\mathcal O}$. Then ${\mathcal O}'=(\varphi, y_{+}',\beta,T)$ generates a Long Orbit such that $\chi_{{\mathcal O}'} \equiv \chi'$.
A consequence of the proposition is that a topological conjugacy $\sigma$ between $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$ conjugates translations in Fatou coordinates of $\varphi_{|x=0}$ and $\eta_{|x=0}$. We will see that $\sigma_{|x=0}$ is affine in Fatou coordinates (Corollary \[cor:orp\]).
Let $(E_{-},E_{+})$ be the division of $\mathrm{Sing}(X)$ induced by ${\mathcal O}$. The family associated to ${\mathcal O}$ is $(A_{\varphi},B_{\varphi})$ bounded by Proposition \[pro:uniform\]. Consider $\Gamma_{x} =\Gamma(X,(x,y_{+}'),U_{\epsilon})$ for $x \in \beta$. We deduce that the family ${\{\Gamma_{x}[0,T(x)]\}}_{x \in \beta}$ is also $(A,B)$ bounded. It is a weak Long Trajectory by the tracking phenomenon. The trajectories in the family ${\{\Gamma_{x}[0,T(x)]\}}_{x \in \beta}$ induce the division $(E_{-},E_{+})$. We can apply the residue formula.
Consider ${\mathcal O}'=(\varphi, y_{+}',\beta,T)$. We have $$\psi_{{\mathcal P}_{-}'}^{\varphi}(\chi_{{\mathcal O}}(s+iu)) - \psi_{{\mathcal P}_{+}'}^{\varphi}(0,y_{+}) =
\lim_{\vartheta_{\mathcal O}(x) \to u, \ x \to 0}^{\lceil T(x) \rceil - T(x) \to s}
\left(
\lceil T(x) \rceil + 2 \pi i \sum_{Q \in E_{-}(x)} Res(X,Q) \right)$$ for any $s+iu \in [0,1] + i {\mathcal S}_{\mathcal O}$ (Proposition \[pro:lores\]). Equation (\[equ:ltlo1\]) in Proposition \[pro:ltlo\] and the definition of $\chi'$ imply $$\psi_{{\mathcal P}_{-}'}^{\varphi}(\chi_{{\mathcal O}'}(s+iu)) - \psi_{{\mathcal P}_{+}'}^{\varphi}(0,y_{+}') =
\lim_{\vartheta_{\mathcal O}(x) \to u, \ x \to 0}^{\lceil T(x) \rceil - T(x) \to s}
\left(
\lceil T(x) \rceil + 2 \pi i \sum_{Q \in E_{-}(x)} Res(X,Q) \right)$$ for any $s+iu \in [0,1] + i {\mathcal S}_{\mathcal O}$ and $\chi_{{\mathcal O}'} \equiv \chi'$.
Topological conjugacies in the generic case {#sec:finite}
===========================================
We present some consequences of the existence of Long Orbits and their properties for elements of ${\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N >1$. We are interested in the study of the rigidity properties of topological conjugacies at the unperturbed line $x=0$. More precisely we want to describe the behavior of $\sigma_{|x=0}$ where $\sigma$ is a homeomorphism conjugating $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$. Let us remark that we always consider that the topological conjugation $\sigma$ preserves the fibration $dx=0$. In other words $\sigma$ is of the form $\sigma(x,y) = (\sigma_{0}(x), \sigma_{1}(x,y))$. The dynamics of $\varphi$ can be very rich, containing for instance small divisors phenomena. It is then natural to think that conjugating all the dynamics of $\varphi$, $\eta$ for the lines $x=cte$ should impose heavy restrictions on the conjugacy at $x=0$. We prove that $\sigma_{|x=0}$ is affine in Fatou coordinates and generically holomorphic or anti-holomorphic.
Affine conjugacies
------------------
In the next proposition we analyze the behavior of topological conjugacies in a petal of the unperturbed line $x=0$.
\[pro:defh\] Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$ such that there exists a homeomorphism $\sigma$ satisfying $\sigma \circ \varphi = \eta \circ \sigma$. Consider a petal ${\mathcal P}'$ of $\varphi_{|x=0}$. Let $\psi_{\sigma({\mathcal P}')}^{\eta}$ be a Fatou coordinate of $\eta$ in the petal $\sigma ({\mathcal P}')$. Then the function $${\mathfrak h}_{{\mathcal P}'}(y, z) = (\psi_{\sigma({\mathcal P}')}^{\eta} \circ \sigma \circ
\mathrm{exp}(z X_{{\mathcal P}'}^{\varphi}) -
\psi_{\sigma({\mathcal P}')}^{\eta} \circ \sigma)(0,y)$$ (see Definition \[def:infgen\]) does not depend on $y \in {\mathcal P}'$.
The mapping $\sigma$ conjugates the translations $\psi + z$ and $\psi + {\mathfrak h}_{{\mathcal P}'}(z)$ in Fatou coordinates of $\varphi_{|{\mathcal P}'}$ and $\eta_{|\sigma({\mathcal P}')}$ respectively. Moreover the proof of the proposition provides an expression for ${\mathfrak h}_{{\mathcal P}'}(z)$.
The function ${\mathfrak h}_{{\mathcal P}'}$ satisfies $${\mathfrak h}_{{\mathcal P}'}(\varphi(0,y), z) \equiv {\mathfrak h}_{{\mathcal P}_{-}'}(y, z) \ \
\mathrm{and} \ \
{\mathfrak h}_{{\mathcal P}'}(y, z+1) \equiv {\mathfrak h}_{{\mathcal P}'}(y, z) +1.$$ In particular it suffices to prove the proposition for $z \in [0,1) + i {\mathbb R}$ and $y^{1} \in {\mathcal P}'$ such that $\mathrm{exp}(z X_{{\mathcal P}'}^{\varphi})(0,y^{1}) \in {\mathcal P}'$ for any $z \in [0,1] + i {\mathbb R}$.
It suffices to prove the result when ${\mathcal P}'={\mathcal P}_{-}'$ is a repelling petal. Otherwise ${\mathcal P}'$ is a repelling petal of $\varphi^{-1}$ and $\sigma$ conjugates $\varphi^{-1}$ and $\eta^{-1}$. Consider $\epsilon>0$ small enough. In particular $\sigma(U_{\epsilon})$ is contained in $U_{\tilde{\epsilon}}$ for a small $\tilde{\epsilon}>0$. Let $X$, $Y$ be convergent normal forms of $\varphi$ and $\eta$ respectively.
By Proposition \[pro:exll2\] there exists an attracting petal ${\mathcal P}_{+}$ of $\Re(X)_{|U_{\epsilon}(0)}$ and a Long Trajectory ${\mathcal O}=(X,y_{+},\beta,T)$ such that ${\mathcal S}_{\mathcal O}= {\mathbb R}$ and $\chi_{\mathcal O} (i {\mathbb R}) \subset {\mathcal P}_{-}$. Consider $y_{+}' \in {\mathcal P}_{+}'$ such that $$\psi_{{\mathcal P}_{-}'}^{\varphi}(0,y^{1}) - \psi_{{\mathcal P}_{+}'}^{\varphi}(0,y_{+}')=
\psi_{-}(\chi_{\mathcal O} (0)) - \psi_{+}(0,y_{+}).$$ Such a choice is possible by replacing $T$ with $T-j$ and $(0,y_{+})$ with ${\mathfrak F}_{\varphi}^{j}(0,y_{+})$ for some $j \in {\mathbb N}$ if necessary. Consider the Long Orbit ${\mathcal O}' = (\varphi, y_{+}, \beta, T)$. We have $$\psi_{{\mathcal P}_{-}'}^{\varphi}(\chi_{{\mathcal O}'}(z)) - \psi_{{\mathcal P}_{+}'}^{\varphi}(0,y_{+})
\equiv \psi_{-}(\chi_{\mathcal O} (0)) - \psi_{+}(0,y_{+}) + z$$ by Proposition \[pro:ltlo\] (see Eq. (\[equ:ltlo2\])). Proposition \[pro:evol\] implies that ${\mathcal O}'' = (\varphi, y_{+}', \beta, T)$ is a Long Orbit such that $\chi_{{\mathcal O}''}(z) = \mathrm{exp}(z X_{{\mathcal P}'}^{\varphi})(0,y^{1})$ for any $z \in [0,1] + i {\mathbb R}$.
We define $(0,\tilde{y}_{+}) = \sigma(0,y_{+}')$ and $\tilde{\mathcal O} = (\eta, \tilde{y}_{+}, \sigma(\beta), T \circ \sigma^{-1})$. It is clear that $\tilde{\mathcal O}$ is a Long Orbit such that $\chi_{\tilde{\mathcal O}} \equiv \sigma \circ \chi_{{\mathcal O}''}$. Up to trimming $(Y, \tilde{y}_{+}, \sigma(\beta), T \circ \sigma^{-1})$ generates a weak Long Trajectory. Hence $(Y, \tilde{y}_{+}, \sigma(\beta), T \circ \sigma^{-1})$ induces a division $(\tilde{E}_{-},\tilde{E}_{+})$ of $\mathrm{Sing} (Y)$.
Fix $z = s+iu \in [0,1) + i {\mathbb R}$. We consider the sequence ${\{x_{n}^{z}\}}$ contained in $\vartheta_{\mathcal O}^{-1}(u)$ such that $T(x_{n}^{z})=n-s$. It is defined for $n>>0$. We have $$\label{equ:con1}
\lim_{n \to \infty} \left(
\psi_{{\mathcal P}_{-}'}^{\varphi}(\chi_{{\mathcal O}''}(z)) - \psi_{{\mathcal P}_{+}'}^{\varphi}(0,y_{+}') - n
- 2 \pi i \sum_{Q \in E_{-}(x_{n}^{z})} Res(X,Q) \right) =0$$ and $$\label{equ:con2}
\lim_{n \to \infty} (
\psi_{\sigma({\mathcal P}_{-}')}^{\eta}( \sigma(\chi_{{\mathcal O}''}(z)))
- \psi_{\sigma({\mathcal P}_{+}')}^{\eta} (0,\tilde{y}_{+}) - n
- 2 \pi i \sum_{Q \in \tilde{E}_{-}(\sigma(x_{n}^{z}))} Res(Y,Q) ) =0$$ by Proposition \[pro:lores\]. Denote $L=\psi_{{\mathcal P}_{-}'}^{\varphi}(0,y^{1}) + \psi_{\sigma({\mathcal P}_{+}')}^{\eta}(0,\tilde{y}_{+})
-\psi_{{\mathcal P}_{+}'}^{\varphi}(0,y_{+}')$ and $$G_{n}^{z} = 2 \pi i \left( \sum_{Q \in \tilde{E}_{-}(\sigma(x_{n}^{z}))} Res(Y,Q) -
\sum_{Q \in E_{-}(x_{n}^{z})} Res(X,Q) \right)$$ By subtracting Eqs. (\[equ:con2\]) and (\[equ:con1\]) we get $$\label{equ:inv}
\psi_{\sigma({\mathcal P}_{-}')}^{\eta}(\sigma(\mathrm{exp}(z X_{{\mathcal P}_{-}'}^{\varphi})(0,y^{1}))) = z+ L +
\lim_{n \to \infty} G_{n}^{z}$$ and then $$\psi_{\sigma({\mathcal P}_{-}')}^{\eta}(\sigma(\mathrm{exp}(z X_{{\mathcal P}_{-}'}^{\varphi})(0,y^{1}))) -
\psi_{\sigma({\mathcal P}_{-}')}^{\eta}(\sigma(0,y^{1})) = z+
\lim_{n \to \infty} G_{n}^{z} - \lim_{n \to \infty} G_{n}^{0} .$$ Let us remark that we replace $T$ with $T-j$ for some $j \in {\mathbb N}$ during the proof. The sequence $\textmd{x}_{n}^{z}$ that we associate to $T-j$ satisfies $\textmd{x}_{n-j}^{z}=x_{n}^{z}$. Thus the right hand side of Eq. (\[equ:inv\]) does not change through the process. Clearly it does not depend on $y^{1}$.
\[pro:rlinear\] Consider the setting of Proposition \[pro:defh\]. Then ${\mathfrak h}_{{\mathcal P}_{-}}:{\mathbb C} \to {\mathbb C}$ is a ${\mathbb R}$-linear isomorphism such that ${\mathfrak h}_{{\mathcal P}'}(1)=1$.
We have $${\mathfrak h}_{{\mathcal P}'}(z_{1}+z_{2}) =
{\mathfrak h}_{{\mathcal P}'}(\mathrm{exp}(z_{2} X_{{\mathcal P}'}^{\varphi})(0,y), z_{1})
+ {\mathfrak h}_{{\mathcal P}'}(y, z_{2})=
{\mathfrak h}_{{\mathcal P}'}(z_{1}) + {\mathfrak h}_{{\mathcal P}'}(z_{2})$$ for any $z_{1},z_{2} \in {\mathbb C}$. Since ${\mathfrak h}_{{\mathcal P}'}$ is continuous by definition then ${\mathfrak h}_{{\mathcal P}'}$ is ${\mathbb R}$-linear. The function ${\mathfrak h}_{{\mathcal P}'}$ is injective in a neighborhood of $0$, hence ${\mathfrak h}_{{\mathcal P}'}$ is an isomorphism. The condition ${\mathfrak h}_{{\mathcal P}'}(1)=1$ is obvious.
Consider the setting of Proposition \[pro:defh\]. Proposition \[pro:rlinear\] implies that ${\mathfrak h}_{{\mathcal P}'}(s) =s$ for all petal ${\mathcal P}'$ of $\varphi_{|x=0}$ and $s \in {\mathbb R}$. As a consequence $\sigma$ conjugates the real flows of $X_{{\mathcal P}'}^{\varphi}$ and $X_{\sigma({\mathcal P}')}^{\eta}$.
So far we proved that $\psi_{\sigma({\mathcal P}')}^{\eta} \circ \sigma \circ (\psi_{{\mathcal P}'}^{\varphi})^{-1}$ is affine for any petal ${\mathcal P}'$ of $\varphi_{|x=0}$. Next we show that these mappings do not depend on the petal.
\[lem:eqcon\] Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$ such that there exists a homeomorphism $\sigma$ satisfying $\sigma \circ \varphi = \eta \circ \sigma$. Then ${\mathfrak h}_{{\mathcal P}'} \equiv {\mathfrak h}_{{\mathcal Q}'}$ for all petals ${\mathcal P}'$, ${\mathcal Q}'$ of $\varphi_{|x=0}$.
It suffices to prove the lemma for consecutive petals ${\mathcal P}_{+}'$ and ${\mathcal P}_{-}'$ of $\varphi_{|x=0}$. Let $X$, $Y$ be convergent normal forms of $\varphi$ and $\eta$ respectively. Consider Fatou coordinates $\psi$, $\tilde{\psi}$ of $X$, $Y$ defined in the neighborhood of ${\mathcal P}_{+}' \cup {\mathcal P}_{-}'$ and $\sigma({\mathcal P}_{+}' \cup {\mathcal P}_{-}')$ respectively. We have $$\lim_{ |Im(\psi(0,y))| \to \infty}
\sum_{j \in {\mathbb Z}} |\Delta_{\varphi}(\varphi^{j}(0,y))| =0, \ \
\lim_{|Im(\tilde{\psi}(0,y))| \to \infty}
\sum_{j \in {\mathbb Z}} |\Delta_{\eta}(\eta^{j}(0,y))|=0,$$ see Definition \[def:delta\]. The first limit is calculated for $y \in {\mathcal P}_{+}' \cup {\mathcal P}_{-}'$ whereas the second limit is calculated for $y \in \sigma({\mathcal P}_{+}' \cup {\mathcal P}_{-}')$. The condition $|Im(\psi(0,y))| \to \infty$ is equivalent to the orbit ${\{ \varphi^{j}(0,y)\}}_{j \in {\mathbb Z}}$ tending uniformly to the origin. The proof of the previous equations are analogous to the proof of Proposition \[pro:boufespre\]. We obtain $$\label{equ:auxpr}
\lim_{|Im(\psi(0,y))| \to \infty}
(\psi_{{\mathcal P}_{k}'}^{\varphi} - \psi)(0,y) =0,
\lim_{|Im(\tilde{\psi}(0,y))| \to \infty}
(\psi_{\sigma({\mathcal P}_{k}')}^{\eta} - \tilde{\psi})(0,y) =0$$ for $k \in \{+,-\}$. We also get $$\label{equ:auxpr2}
\lim_{|Im(\psi(0,y))| \to \infty}
(\psi_{{\mathcal P}_{+}'}^{\varphi} - \psi_{{\mathcal P}_{-}'}^{\varphi})(0,y) =
\lim_{|Im(\tilde{\psi}(0,y))| \to \infty}
(\psi_{\sigma({\mathcal P}_{+}')}^{\eta} - \psi_{\sigma({\mathcal P}_{-}')}^{\eta})(0,y) =0.$$
Consider a sequence $y_{n}$ in ${\mathcal P}_{+}' \cap {\mathcal P}_{-}'$ such that $|Im(\psi(0,y_{n}))| \to \infty$. Fix $z \in {\mathbb C}$. We have $|Im(\psi(\mathrm{exp}(z X_{{\mathcal P}'}^{\varphi})(0,y_{n})))| \to \infty$ by Eq. (\[equ:auxpr\]). Let $z_{n}$ be the complex number such that $\mathrm{exp}(z X_{{\mathcal P}_{-}'}^{\varphi})(0,y_{n})=
\mathrm{exp}(z_{n} X_{{\mathcal P}_{+}'}^{\varphi})(0,y_{n})$. The sequence $z_{n}$ satisfies $\lim_{n \to \infty} z_{n}=z$ by Eq. (\[equ:auxpr2\]). We have $${\mathfrak h}_{{\mathcal P}_{-}'}(z) =
(\psi_{\sigma({\mathcal P}_{-}')}^{\eta} \circ \sigma \circ
\mathrm{exp}(z X_{{\mathcal P}_{-}'}^{\varphi}) -
\psi_{\sigma({\mathcal P}_{-}')}^{\eta} \circ \sigma)(0,y_{n})$$ and $${\mathfrak h}_{{\mathcal P}_{+}'}(z_{n}) =
(\psi_{\sigma({\mathcal P}_{+}')}^{\eta} \circ \sigma \circ
\mathrm{exp}(z_{n} X_{{\mathcal P}_{+}'}^{\varphi}) -
\psi_{\sigma({\mathcal P}_{+}')}^{\eta} \circ \sigma)(0,y_{n}).$$ Equation (\[equ:auxpr2\]) implies $\lim_{n \to \infty}({\mathfrak h}_{{\mathcal P}_{-}'}(z)-{\mathfrak h}_{{\mathcal P}_{+}'}(z_{n})) =0$. Since $\lim_{n \to \infty} z_{n}=z$ and ${\mathfrak h}_{{\mathcal P}_{+}'}$ is continuous we obtain ${\mathfrak h}_{{\mathcal P}_{-}'}(z)={\mathfrak h}_{{\mathcal P}_{+}'}(z)$ for any $z \in {\mathbb C}$.
Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$ such that there exists a homeomorphism $\sigma$ satisfying $\sigma \circ \varphi = \eta \circ \sigma$. We denote by ${\mathfrak h}_{\varphi,\eta,\sigma}$ any of the functions ${\mathfrak h}_{{\mathcal P}'}$ defined in Proposition \[pro:defh\]. We denote ${\mathfrak h}={\mathfrak h}_{\varphi,\eta,\sigma}$ if the data are implicit.
Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ such that a homeomorphism $\sigma$ conjugates $\varphi$ and $\eta$. The mapping $\sigma$ is of the form $\sigma(x,y) = (\sigma_{0}(x), \sigma_{1}(x,y))$. We say that the action of $\sigma$ on the parameter space is holomorphic (resp. anti-holomorphic, orientation-preserving) if $\sigma_{0}$ is holomorphic (resp. anti-holomorphic, orientation-preserving).
The orientation properties of the restriction of the conjugation to $x=0$ and of its action on the parameter space are the same.
\[lem:orip\] Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$ such that there exists a homeomorphism $\sigma$ satisfying $\sigma \circ \varphi = \eta \circ \sigma$. Then the isomorphism ${\mathfrak h}_{\varphi,\eta,\sigma}$ is orientation-preserving if and only if the action of $\sigma$ on the parameter space is orientation-preserving.
Suppose that $\sigma$ does not preserve orientation in the parameter space. We denote $$\zeta(x,y) = (\overline{x},\overline{y}), \
\tilde{\eta} = \zeta \circ \eta \circ \zeta \ \mathrm{and} \
\tilde{\sigma} = \zeta \circ \sigma$$ where $\overline{x}$ is the complex conjugation. The diffeomorphism $\tilde{\eta}$ belongs to ${\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ and $\tilde{\sigma} \circ \varphi = \tilde{\eta} \circ \tilde{\sigma}$. The action of $\tilde{\sigma}$ on the parameter space is orientation-preserving. We have $${\mathfrak h}_{\varphi, \tilde{\eta}, \tilde{\sigma}}(z) = \overline{{\mathfrak h}_{\varphi, \eta, \sigma}(z)}$$ for any $z \in {\mathbb C}$. Therefore it suffices to prove that if the action of $\sigma$ in the parameter space is orientation-preserving so is ${\mathfrak h}={\mathfrak h}_{\varphi,\eta,\sigma}$.
Fix a repelling petal ${\mathcal P}'={\mathcal P}_{-}'$. Consider the notations in Proposition \[pro:defh\]. We define $$G_{0}(x) = - 2 \pi i \sum_{Q \in {E}_{-}(x)} Res(X,Q) \ \mathrm{and} \
G(x) = - 2 \pi i \sum_{Q \in \tilde{E}_{-}(x)} Res(Y,Q) .$$ We have $$\psi_{\sigma({\mathcal P}_{-}')}^{\eta}(\chi_{\tilde{\mathcal O}}(z)) -
\psi_{\sigma({\mathcal P}_{+}')}^{\eta}(0,\tilde{y}_{+}) =
\lim_{(\lceil T \rceil - T)(x) \to s, \ x \in \vartheta_{\tilde{\mathcal O}}^{-1}(u), \ x \to 0} (
\lceil T(x) \rceil - G(x) )$$ for any $z = s+iu \in [0,1] +i {\mathbb R}$ by Proposition \[pro:lores\]. Since ${\mathfrak h}_{|{\mathbb R}} \equiv Id$, $\chi_{\tilde{\mathcal O}} \equiv \sigma \circ \chi_{{\mathcal O}''}$ and $\psi_{{\mathcal P}_{-}'}^{\varphi}(\chi_{{\mathcal O}''}(z)) -
\psi_{{\mathcal P}_{-}'}^{\varphi}(\chi_{{\mathcal O}''}(0)) \equiv z$ we deduce that $$\label{equ:auxor}
\lim_{x \in \vartheta_{\tilde{\mathcal O}}^{-1}(u), \ x \to 0} Im (G(x)) = c_{u} \ \mathrm{and} \
\lim_{x \in \vartheta_{\tilde{\mathcal O}}^{-1}(u), \ x \to 0} Re (G(x)) = \infty.$$ where $c_{u}=
- Im(\psi_{\sigma({\mathcal P}_{-}')}^{\eta}( \sigma(\chi_{{\mathcal O}''}(i u))) -
\psi_{\sigma({\mathcal P}_{+}')}^{\eta}(0,\tilde{y}_{+}))$. The function $G$ is meromorphic (Proposition 5.2 of [@UPD]) and has a pole of order greater than $0$. Every curve $\vartheta_{\tilde{\mathcal O}}^{-1}(u)$ adheres to the same direction in the parameter space (Proposition \[pro:udlo\]). We obtain $$\lim_{x \in \vartheta_{\tilde{\mathcal O}}^{-1}(u), \ x \to 0} Im (G(x)) -
\lim_{x \in \vartheta_{\tilde{\mathcal O}}^{-1}(0), \ x \to 0} Im (G(x)) =
- Im ({\mathfrak h}(iu))$$ for any $u \in {\mathbb R}$.
Consider the connected curve $\iota_{u}$ such that $\iota_{u}$ adheres to the same direction as $\vartheta_{\tilde{\mathcal O}}^{-1}(0)$ and $$\iota_{u} \subset \{ x \in B(0,\delta) \setminus \{0\} :
Im (G(x)) = c_{u}. \}$$ We have $\lim_{x \in \iota_{u}, \ x \to 0} Re (G(x)) = \infty$ for any $u \in {\mathbb R}$. The curves (see Eq. (\[equ:deftf\])) $$\varsigma_{u} = \{ x \in {\mathbb C}: \psi_{-}(0,y_{-}) - \psi_{+}(0,y_{+})
+ x +iu \in {\mathbb R}^{+} \}$$ move in clock wise sense when we increase $u$. The function $G_{0}$ is meromorphic, it satisfies $G_{0}^{-1}(\varsigma_{u}) = \vartheta_{{\mathcal O}''}^{-1}(u)$ for $u \in {\mathbb R}$ and has a pole of order greater than $0$. Thus the curves $\vartheta_{{\mathcal O}''}^{-1}(u)$ move in counter clock wise sense when $u$ increases. Since the action of $\sigma$ on the parameter space is orientation-preserving the same property holds true for $\vartheta_{\tilde{\mathcal O}}^{-1}(u)$. Equation (\[equ:auxor\]) implies that the curves $\iota_{u}$ move in counter clockwise sense when $u$ increases. The function $G$ is meromorphic and has a pole of order greater than $0$. Thus the curves $\{x \in {\mathbb C}: Re(x) \in {\mathbb R}^{+} \ \mathrm{and} \ Im(x)=c_{u} \}$ move in clock wise sense when we increase $u$. Hence $c_{u}$ is a decreasing function of $u$. We obtain $$Im ({\mathfrak h}(i)) = c_{0} - c_{1} >0 .$$ As a consequence ${\mathfrak h}$ is orientation-preserving.
\[cor:holact\] Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$ such that there exists a homeomorphism $\sigma$ satisfying $\sigma \circ \varphi = \eta \circ \sigma$. Suppose that the action of $\sigma$ on the parameter space is holomorphic (resp. anti-holomorphic). Then $\sigma_{|x=0}$ is holomorphic (resp. anti-holomorphic).
The corollary implies Proposition \[pro:holpar\] in the case $m(\varphi) = 0$. The case $m(\varphi) > 0$ is treated in Corollary \[cor:holpar\].
Suppose that the action of $\sigma$ on the parameter space is holomorphic. Consider the notations in Proposition \[pro:defh\]. Let ${\mathcal P}'$ be a petal of $\varphi_{|x=0}$. The functions $$\sum_{Q \in E_{-}(x)} Res(X,Q) \ \ \mathrm{and} \
\sum_{Q \in \tilde{E}_{-}(\sigma(x))} Res(Y,Q)$$ are meromorphic (Proposition 5.2 of [@UPD]). The function $$2 \pi i \left(
\sum_{Q \in \tilde{E}_{-}(\sigma(x))} Res(Y,Q) - \sum_{Q \in E_{-}(x)} Res(X,Q)
\right)$$ is meromorphic. Hence all the limits of the previous function in sequences tending to $0$ are equal. We deduce that $\lim_{n \to \infty} G_{n}^{z} = \lim_{n \to \infty} G_{n}^{0}$ for any $z \in {\mathbb C}$. We obtain $$\psi_{\sigma({\mathcal P}')}^{\eta}(\sigma(\mathrm{exp}(z X_{{\mathcal P}'}^{\varphi})(0,y))) -
\psi_{\sigma({\mathcal P}')}^{\eta}(\sigma(0,y)) = z$$ for $y \in {\mathcal P}'$. The mappings $\psi_{\sigma({\mathcal P}')}^{\eta}$ and $z \to \mathrm{exp}(z X_{{\mathcal P}'}^{\varphi})(0,y)$ are biholomorphic. As a consequence $\sigma_{|x=0}$ is holomorphic in ${\mathcal P}'$. Since the union of the petals is a pointed neighborhood of the origin we obtain that $\sigma_{|x=0}$ is holomorphic by Riemann’s removable singularity theorem.
Suppose that the action of $\sigma$ on the parameter space is anti-holomorphic. Denote $\zeta(x,y) = (\overline{x},\overline{y})$, $\tilde{\eta} = \zeta \circ \eta \circ \zeta$ and $\tilde{\sigma} = \zeta \circ \sigma$. We have $\tilde{\sigma} \circ \varphi = \tilde{\eta} \circ \tilde{\sigma}$. The action of $\tilde{\sigma}$ on the parameter space is holomorphic. Therefore $\tilde{\sigma}_{|x=0}$ is holomorphic and $\sigma_{|x=0}$ is anti-holomorphic.
The next result is the General Theorem for the case $m(\varphi)=0$.
\[cor:orp\] Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$ such that there exists a homeomorphism $\sigma$ satisfying $\sigma \circ \varphi = \eta \circ \sigma$. Then $\sigma_{|x=0}$ is affine in Fatou coordinates (see Definition \[def:afffc\]). Moreover $\sigma_{|x=0}$ is orientation-preserving if and only if the action of $\sigma$ on the parameter space is orientation-preserving.
Let us remark that if $\sigma_{|x=0}$ is affine in Fatou coordinates then it is real analytic in $\{x=0\} \setminus \{(0,0)\}$. The corollary is a consequence of Propositions \[pro:defh\], \[pro:rlinear\] and Lemmas \[lem:eqcon\], \[lem:orip\].
Let $\phi \in {\mbox{{\rm Diff}{${\,}_{1}({\mathbb C}^{},0)$}}}$. We say that $\phi$ is analytically trivial if there exists a local holomorphic singular vector field $Z=a(z) \partial / \partial z$ such that $\phi = \mathrm{exp}(Z)$. This condition is equivalent to $X_{{\mathcal P}'}^{\phi} \equiv X_{{\mathcal Q}'}^{\phi}$ for all petals ${\mathcal P}'$, ${\mathcal Q}'$ of $\phi$ such that ${\mathcal P}' \cap {\mathcal Q}' \neq \emptyset$. It is also equivalent to $\psi_{{\mathcal P}'}^{\phi} - \psi_{{\mathcal Q}'}^{\phi}$ being constant for all petals ${\mathcal P}'$, ${\mathcal Q}'$ of $\phi$ such that ${\mathcal P}' \cap {\mathcal Q}' \neq \emptyset$ ($\psi_{{\mathcal P}'}^{\phi}$ and $\psi_{{\mathcal Q}'}^{\phi}$ are Fatou coordinates of $\phi$ in ${\mathcal P}'$ and ${\mathcal Q}'$ respectively).
The condition of being non-analytically trivial is generic among the tangent to the identity local diffeomorphisms in one variable. More precisely, every formal class of conjugacy (i.e. a class of equivalence for the relation given by the formal conjugation) contains a continuous moduli of analytic classes of conjugacy and a unique analytically trivial class. These properties are a consequence of the analytic classification of tangent to the identity diffeomorphisms (see [@Loray5]).
It is possible to construct affine conjugacies in Fatou coordinates that are not holomorphic or anti-holomorphic by restriction to $x=0$ if both $\varphi$ and $\eta$ are embedded in analytic flows (see Section \[sec:build\]). They are essentially the only examples.
\[pro:natc\] Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$ such that there exists a homeomorphism $\sigma$ satisfying $\sigma \circ \varphi = \eta \circ \sigma$. Suppose that either $\varphi_{|x=0}$ or $\eta_{|x=0}$ is non-analytically trivial. Then either ${\mathfrak h}_{\varphi,\eta,\sigma} \equiv z$ and $\sigma_{|x=0}$ is holomorphic or ${\mathfrak h}_{\varphi,\eta,\sigma} \equiv \overline{z}$ and $\sigma_{|x=0}$ is anti-holomorphic.
The proposition implies the Main Theorem.
The isomorphism ${\mathfrak h}_{\eta,\varphi,\sigma^{-1}}$ is the inverse of ${\mathfrak h}_{\varphi,\eta,\sigma}$. Thus we can suppose that $\varphi_{|x=0}$ is non-analytically trivial.
Let ${\mathcal P}'$ be a petal of $\varphi_{|x=0}$. Fix $(0,y_{+}) \in {\mathcal P}'$. We have $$\psi_{\sigma({\mathcal P}')}^{\eta} (\sigma(0,y)) =
\psi_{\sigma({\mathcal P}')}^{\eta} (\sigma(0,y_{+})) +
{\mathfrak h}( \psi_{{\mathcal P}'}^{\varphi}(0,y) - \psi_{{\mathcal P}'}^{\varphi}(0,y_{+}))$$ and then $$(\psi_{\sigma({\mathcal P}')}^{\eta} \circ \sigma)(0,y) =
((z+ c_{{\mathcal P}'}) \circ {\mathfrak h} \circ \psi_{{\mathcal P}'}^{\varphi})(0,y)$$ for some $c_{{\mathcal P}'} \in {\mathbb C}$ and any $y \in {\mathcal P}'$. Consider two consecutive petals ${\mathcal P}'$ and ${\mathcal Q}'$ of $\varphi_{|x=0}$. We consider the changes of charts $$\psi_{\sigma({\mathcal Q}')}^{\eta} \circ (\psi_{\sigma({\mathcal P}')}^{\eta})^{-1}=
(\psi_{\sigma({\mathcal Q}')}^{\eta} \circ \sigma) \circ
(\psi_{\sigma({\mathcal P}')}^{\eta} \circ \sigma)^{-1} =$$ $$(z+ c_{{\mathcal Q}'}) \circ {\mathfrak h} \circ \psi_{{\mathcal Q}'}^{\varphi} \circ
(\psi_{{\mathcal P}'}^{\varphi})^{-1} \circ {\mathfrak h}^{-1} \circ (z- c_{{\mathcal P}'}).$$ The left hand side is holomorphic, thus ${\mathfrak h} \circ \psi_{{\mathcal Q}'}^{\varphi} \circ
(\psi_{{\mathcal P}'}^{\varphi})^{-1} \circ {\mathfrak h}^{-1}$ is also holomorphic. We denote $H = \psi_{{\mathcal Q}'}^{\varphi} \circ
(\psi_{{\mathcal P}'}^{\varphi})^{-1}$, it is holomorphic. The isomorphisms ${\mathfrak h}$ and ${\mathfrak h}^{-1}$ are of the form ${\mathfrak h}(z) = \varsigma_{0} z + \varsigma_{1} \overline{z}$ and ${\mathfrak h}^{-1}(z) = \varrho_{0} z + \varrho_{1} \overline{z}$ where $\varrho_{0} = \overline{\varsigma_{0}}/(|\varsigma_{0}|^{2}- |\varsigma_{1}|^{2})$ and $\varrho_{1} = -\varsigma_{1}/(|\varsigma_{0}|^{2}- |\varsigma_{1}|^{2})$. We have $$\frac{\partial ({\mathfrak h} \circ H \circ {\mathfrak h}^{-1})}{\partial \overline{z}} =
\frac{\partial {\mathfrak h}}{\partial z} \frac{\partial (H \circ {\mathfrak h}^{-1})}{\partial \overline{z}} +
\frac{\partial {\mathfrak h}}{\partial \overline{z}}
\frac{\partial (\overline{H \circ {\mathfrak h}^{-1}})}{\partial \overline{z}}=
\varsigma_{0} \frac{\partial H}{\partial z} \varrho_{1} +
\varsigma_{1} \overline{ \frac{\partial H}{\partial z}} \overline{\varrho_{0}} =$$ $$\frac{\varsigma_{0} \varsigma_{1}}{|\varsigma_{0}|^{2}- |\varsigma_{1}|^{2}}
\left( \overline{ \frac{\partial H}{\partial z}} - \frac{\partial H}{\partial z} \right).$$ Suppose $\varsigma_{0}=0$. Since ${\mathfrak h}(1)=1$ we deduce ${\mathfrak h} \equiv \overline{z}$. Hence $\sigma_{|x=0}$ is anti-holomorphic. Suppose $\varsigma_{1}=0$. Since ${\mathfrak h}(1)=1$ we deduce ${\mathfrak h} \equiv z$. Hence $\sigma_{|x=0}$ is holomorphic. Since ${\mathfrak h} \circ H \circ {\mathfrak h}^{-1}$ is holomorphic we can suppose that $\overline{\partial H/\partial z} \equiv \partial H/\partial z$. The function $\partial H / \partial z$ is real and then constant by the open mapping theorem. Therefore $H$ is of the form $az + b$ for some constants $a,b \in {\mathbb C}$. The constant $a$ is equal to $1$ since $H(z+1) \equiv H(z)+1$. We obtain $\psi_{{\mathcal Q}'}^{\varphi} \equiv \psi_{{\mathcal P}'}^{\varphi} + b$. We deduce that $X_{{\mathcal Q}'}^{\varphi} \equiv X_{{\mathcal P}'}^{\varphi}$. The last property does not hold true for every pair of consecutive petals of $\varphi_{|x=0}$ by hypothesis. Thus $\sigma_{|x=0}$ is either holomorphic or anti-holomorphic.
Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$ such that there exists a homeomorphism $\sigma$ satisfying $\sigma \circ \varphi = \eta \circ \sigma$. Then $\varphi_{|x=0}$ is analytically trivial if and only if $\eta_{|x=0}$ is analytically trivial.
Proposition \[pro:natc\] describes the invariance of the analytic classes of the unperturbed diffeomorphisms by topological conjugation. Next lemma describes the action on formal invariants.
\[lem:res\] Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$ such that there exists a homeomorphism $\sigma$ satisfying $\sigma \circ \varphi = \eta \circ \sigma$. Then we have $${\mathfrak h}_{\varphi,\eta,\sigma}(2 \pi i Res_{\varphi}(0,0)) = 2 \pi i Res_{\eta}(0,0) \ \mathrm{or} \
{\mathfrak h}_{\varphi,\eta,\sigma}(2 \pi i Res_{\varphi}(0,0)) = -2 \pi i Res_{\eta}(0,0)$$ (see Definition \[def:res12\]) depending on whether or not the action of $\sigma$ on the parameter space is orientation-preserving. In particular $Re (Res_{\varphi}(0,0))$ and $Re (Res_{\eta}(0,0))$ have the same sign.
The sign can be positive, negative or $0$. We denote $R_{\varphi}=Res_{\varphi}(0,0)$ and $R_{\eta}=Res_{\eta}(0,0)$. Suppose that either $\varphi_{|x=0}$ or $\eta_{|x=0}$ is non-analytically trivial. If $\sigma$ is orientation-preserving on the parameter space then ${\mathfrak h} \equiv z$ and $\sigma_{|x=0}$ is holomorphic by Corollary \[cor:orp\] and Proposition \[pro:natc\]. The result is a consequence of the residues being analytic invariants. If $\sigma$ is orientation-reversing on the parameter space then $\sigma_{|x=0}$ is anti-holomorphic and ${\mathfrak h} \equiv \overline{z}$ by Corollary \[cor:orp\] and Proposition \[pro:natc\]. The equation $R_{\eta} = \overline{R_{\varphi}}$ implies ${\mathfrak h}(2 \pi i R_{\varphi}) = -2 \pi i R_{\eta}$.
Suppose that both $\varphi_{|x=0}$ and $\eta_{|x=0}$ are analytically trivial. Let $X^{\varphi}$ and $X^{\eta}$ the local vector fields defined in $x=0$ such that $\varphi_{|x=0} =\mathrm{exp} (X^{\varphi})$ and $\eta_{|x=0} =\mathrm{exp} (X^{\eta})$. Consider Fatou coordinates $\psi_{X}^{\varphi}$ and $\psi_{X}^{\eta}$ of $X^{\varphi}$ and $X^{\eta}$ respectively. The complex number $2 \pi i R_{\varphi}$ is the additive monodromy of $\psi_{X}^{\varphi}$ along a path turning once around the origin in counter clock wise sense. Since we have $$(\psi_{X}^{\eta} \circ \sigma)(0,y) \equiv ({\mathfrak h} \circ \psi_{X}^{\varphi})(0,y) + c$$ for some $c \in {\mathbb C}$ then ${\mathfrak h}(2 \pi i R_{\varphi}) = \pm 2 \pi i R_{\eta}$ depending on whether ${\mathfrak h}$ is orientation-preserving or orientation-reversing.
Suppose that ${\mathfrak h}$ is orientation-preserving. We obtain $$\mathrm{sign}(Re(R_{\eta})) =\mathrm{sign}(Im({\mathfrak h}(2 \pi i R_{\varphi})))=
\mathrm{sign}(Im(2 \pi i R_{\varphi})) = \mathrm{sign}(Re(R_{\varphi})) .$$ We have $$\mathrm{sign}(Re(R_{\eta})) =-\mathrm{sign}(Im({\mathfrak h}(2 \pi i R_{\varphi})))=
\mathrm{sign}(Im(2 \pi i R_{\varphi})) = \mathrm{sign}(Re(R_{\varphi}))$$ when ${\mathfrak h}$ is orientation-reversing.
Analytically trivial case
-------------------------
In this section we study the properties of affine conjugacies (in Fatou coordinates) in the non-analytically trivial case. Examples of the specific kind of behavior described in next propositions are presented in Section \[sec:build\].
Consider $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$ and a topological conjugation $\sigma$. Suppose that $\varphi_{|x=0}$ is analytically trivial. There are two fundamentally different cases. On the one hand if $Res_{\varphi}(0,0) \in i {\mathbb R}$ there are plenty of (Fatou) affine mappings conjugating $\varphi_{|x=0}$ and $\eta_{|x=0}$ but $\varphi_{|x=0}$ and $\eta_{|x=0}$ are always analytically or anti-analytically conjugated. On the other hand if $Res_{\varphi}(0,0) \not \in i {\mathbb R}$ there is rigidity of conjugations. In fact there are at most two conjugations (up to precomposition with elements of the center of $\varphi_{|x=0}$ in ${\mbox{{\rm Diff}{${\,}_{}({\mathbb C}^{},0)$}}}$) and in general none of them is holomorphic or anti-holomorphic.
Both residues $Res_{\varphi}(0,0)$ and $Res_{\eta}(0,0)$ belong to $i {\mathbb R}$ by Lemma \[lem:res\]. Suppose that $\sigma$ is orientation-preserving on the parameter space. The map ${\mathfrak h}$ satisfies ${\mathfrak h}_{|{\mathbb R}} \equiv Id$ (Proposition \[pro:rlinear\]). Then we have $$2 \pi i Res_{\varphi}(0,0) = {\mathfrak h}(2 \pi i Res_{\varphi}(0,0)) = 2 \pi i Res_{\eta}(0,0)$$ by Lemma \[lem:res\]. Since $\nu(\varphi_{|x=0}) = \nu(\eta_{|x=0})$, $Res_{\varphi}(0,0) = Res_{\eta}(0,0)$ and $\varphi_{|x=0}$, $\eta_{|x=0}$ are analytically trivial then $\varphi_{|x=0}$ and $\eta_{|x=0}$ are analytically conjugated.
Suppose that $\sigma$ is orientation-reversing on the parameter space. We have $$2 \pi i Res_{\varphi}(0,0) = {\mathfrak h}(2 \pi i Res_{\varphi}(0,0)) = -2 \pi i Res_{\eta}(0,0)=
2 \pi i \overline{Res_{\eta}(0,0)}$$ by Lemma \[lem:res\]. Since $\nu(\varphi_{|x=0}) = \nu(\eta_{|x=0})$, $Res_{\varphi}(0,0) = \overline{Res_{\eta}(0,0)}$ and $\varphi_{|x=0}$, $\eta_{|x=0}$ are analytically trivial then $\varphi_{|x=0}$, $\eta_{|x=0}$ are anti-holomorphically conjugated.
\[pro:atu\] Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$ with $N>1$ such that there exists a homeomorphism $\sigma$ satisfying $\sigma \circ \varphi = \eta \circ \sigma$. Suppose that either $\varphi_{|x=0}$ or $\eta_{|x=0}$ is analytically trivial. Suppose that either $Res_{\varphi}(0,0) \not \in i {\mathbb R}$ or $Res_{\eta}(0,0) \not \in i {\mathbb R}$. Then
- If $\sigma_{|x=0}$ is orientation-preserving then $\sigma_{|x=0}$ is holomorphic if and only if $Res_{\varphi}(0,0) =Res_{\eta}(0,0)$.
- If $\sigma_{|x=0}$ is orientation-reversing then $\sigma_{|x=0}$ is anti-holomorphic if and only if $Res_{\varphi}(0,0) =\overline{Res_{\eta}(0,0)}$.
- If $Res_{\varphi}(0,0) =Res_{\eta}(0,0) \in {\mathbb R}^{*}$ then $\sigma_{|x=0}$ is holomorphic or anti-holomorphic.
- If $Res_{\varphi}(0,0) \not \in
\{Res_{\eta}(0,0), \overline{Res_{\eta}(0,0)}\}$ then $\varphi_{|x=0}$ and $\eta_{|x=0}$ are neither holomorphically nor anti-holomorphically conjugated. In particular $\sigma_{|x=0}$ is neither holomorphic nor anti-holomorphic.
Consider a pair of homeomorphisms $\sigma$, $\tilde{\sigma}$ conjugating $\varphi$, $\eta$ and such that both are orientation-preserving or orientation-reversing. Then we obtain $\tilde{\sigma}_{|x=0} = \sigma_{|x=0} \circ \phi$ for some holomorphic $\phi \in {\mbox{{\rm Diff}{${\,}_{}({\mathbb C}^{},0)$}}}$ commuting with $\varphi_{|x=0}$.
Proposition \[pro:atu\] implies Proposition \[pro:rigi\].
The isomorphism ${\mathfrak h}$ satisfies ${\mathfrak h}(2 \pi i Res_{\varphi}(0,0)) =\pm 2 \pi i Res_{\eta}(0,0)$ (Lemma \[lem:res\]) and ${\mathfrak h}_{|{\mathbb R}} \equiv Id$ (Proposition \[pro:rlinear\]). Thus ${\mathfrak h}$ depends only on wether or not $\sigma_{|x=0}$ is orientation-preserving. We deduce $${\mathfrak h}_{\varphi,\varphi, \sigma^{-1} \circ \tilde{\sigma}} =
{\mathfrak h}_{\varphi,\eta, \sigma}^{-1}\circ
{\mathfrak h}_{\varphi,\eta, \tilde{\sigma}} = Id$$ if $\sigma$, $\tilde{\sigma}$ have the same orientation. The mapping $(\sigma^{-1} \circ \tilde{\sigma})_{|x=0}$ is holomorphic and commutes with $\varphi_{|x=0}$.
We denote $R_{\varphi}=Res_{\varphi}(0,0)$ and $R_{\eta}=Res_{\eta}(0,0)$. Suppose that $\sigma_{|x=0}$ is orientation-preserving. The equation ${\mathfrak h}(2 \pi i R_{\varphi})=2 \pi i R_{\eta}$ in Lemma \[lem:res\] implies that ${\mathfrak h} \equiv z$ is equivalent to $R_{\varphi} = R_{\eta}$. Hence $\sigma_{|x=0}$ is holomorphic if and only if $R_{\varphi} = R_{\eta}$.
Suppose that $\sigma_{|x=0}$ is orientation-reversing. The equation ${\mathfrak h}(2 \pi i R_{\varphi})=-2 \pi i R_{\eta}$ in Lemma \[lem:res\] implies that ${\mathfrak h} \equiv \overline{z}$ is equivalent to $R_{\varphi} = \overline{R_{\eta}}$. Hence $\sigma_{|x=0}$ is anti-holomorphic if and only if $R_{\varphi} = \overline{R_{\eta}}$.
The third item is a consequence of the previous ones. Suppose $R_{\varphi} \not \in \{R_{\eta}, \overline{R_{\eta}}\}$. Then $\varphi_{|x=0}$ and $\eta_{|x=0}$ are neither holomorphically nor anti-holomorphically conjugated.
The results in this section (except the statements involving orientation-preserving actions on the parameter space) hold true for higher dimensional unfoldings $\varphi(x_{1},\hdots,x_{n},y) =(x_{1},\hdots,x_{n}, F(x_{1},\hdots,x_{n},y))$ of tangent to the identity diffeomorphisms with $N>1$. The idea is that Long Orbits ${\mathcal O}= (\varphi,y_{+},\beta,T)$ satisfy Propositions \[pro:tracking\], \[pro:lores\] and \[pro:evol\]. Otherwise there exists a sequence $a_{n} \in \beta$ such that no subsequence satisfies the tracking properties in Proposition \[pro:tracking\]. The construction of the dynamical splitting amounts to desingularize the fixed points set by a finite sequence of blow-ups. Then up to take a subsequence of $a_{n}$ we can use the machinery in Section \[sec:tracking\] to obtain a contradiction.
Topological conjugacies for unfoldings of the identity map {#sec:infinite}
==========================================================
Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ such that there exists a homeomorphism $\sigma$ satisfying $\sigma \circ \varphi = \eta \circ \sigma$. Suppose that $m \stackrel{def}{=} m(\varphi) >0$. The unfoldings of the identity map are easier to study than the elements of ${\mbox{{\rm Diff}{${\,}_{p1}^{*}({\mathbb C}^{2},0)$}}}$. We can construct analogues of the Long Orbits by iterating $O(1/|x|^{m})$ times a diffeomorphism starting at a point $(x,y_{0})$. In this way we obtain analogous results to the ones in Section \[sec:finite\].
The number $\tilde{m} \stackrel{def}{=} m(\eta)$ is positive too. Consider convergent normal forms $X$, $Y$ of $\varphi$ and $\eta$ respectively. The vector fields $X$, $Y$ can be written in the form $X=x^{m} X_{0}$ and $Y=x^{\tilde{m}} Y_{0}$ where $\mathrm{Sing} (X_{0})$ and $\mathrm{Sing} (Y_{0})$ do not contain $x=0$. We obtain $\Delta_{\varphi} \in (x^{2m})$ and $\Delta_{\eta} \in (x^{2 \tilde{m}})$ (see Definition \[def:delta\]).
Fix $\mu \in {\mathbb S}^{1}$. Let $x_{n}$ be a sequence such that $x_{n} \to 0$ and $x_{n}/|x_{n}| \to \mu$. Consider $s \in {\mathbb R}$ and $T_{n} = [s/|x_{n}|^{m}]$. We want to study the behavior of the sequence $\varphi^{T_{n}}(x_{n},y_{0})$. Let $\psi$ be a Fatou coordinate of $X$. We have $$(\psi \circ \varphi^{T_{n}} - (\psi +T_{n}))(x_{n},y_{0}) =
\sum_{j=0}^{T_{n}-1} (\Delta_{\varphi} \circ \varphi^{j})(x_{n},y_{0}) = O(x_{n}^{m}) .$$ It implies $$\label{equ:short}
\lim_{n \to \infty} \varphi^{T_{n}}(x_{n},y_{0}) = \lim_{n \to \infty}
\mathrm{exp}(T_{n} X) (x_{n},y_{0}) = \mathrm{exp}(s \mu^{m} X_{0})(0,y_{0})$$ for any $y_{0} \in B(0,\epsilon)$.
The next two lemmas are intended to show that the conjugation $\sigma$ is well-behaved even if we consider a real blow-up in the parameter space.
There is no subsequence of $|\sigma(x_{n})|^{\tilde{m}}/|x_{n}|^{m}$ converging to $0$ or $\infty$.
The conjugating mapping $\sigma$ is of the form $\sigma(x,y) =(\sigma_{0}(x), \sigma_{1}(x,y))$. We define $\sigma(x) = \sigma_{0}(x)$.
We denote $a_{n}=|\sigma(x_{n})|^{\tilde{m}}/|x_{n}|^{m}$. Suppose $\lim_{n \to \infty} a_{n}=\infty$. Up to consider a subsequence we can suppose that $\sigma(x_{n})/|\sigma(x_{n})|$ converges to $\tilde{\mu} \in {\mathbb S}^{1}$. Since $|\sigma(x_{n})|^{\tilde{m}} = a_{n}|x_{n}|^{m}$ then $\lim_{n \to \infty} a_{n}|x_{n}|^{m}=0$. We define $$T_{n}'= \left[ \frac{s}{a_{n}|x_{n}|^{m}} \right]= \left[ \frac{s}{|\sigma(x_{n})|^{\tilde{m}}} \right].$$ We deduce $\lim_{n \to \infty} \varphi^{T_{n}'}(x_{n},y_{0}) = (0,y_{0})$ for any $y_{0} \in B(0,\epsilon)$. The equation $$\lim_{n \to \infty} \eta^{T_{n}'}(\sigma(x_{n},y_{0})) = \lim_{n \to \infty}
\mathrm{exp}(T_{n}' Y)(\sigma(x_{n},y_{0})) = \mathrm{exp}(s \tilde{\mu}^{m} Y_{0})(\sigma(0,y_{0}))$$ is the analogue of Eq. (\[equ:short\]) for $\eta$. The mappings $\varphi$ and $\eta$ are conjugated, hence we obtain $\sigma(0,y) \equiv \mathrm{exp}(s \tilde{\mu}^{\tilde{m}} Y_{0})(\sigma(0,y))$ for any $s \in {\mathbb R}$. This is impossible since $x=0$ is not contained in $\mathrm{Sing} (Y_{0})$.
The case $\lim_{n \to \infty} a_{n}=0$ is impossible too. The proof is analogous by defining $T_{n}' = [s/|x_{n}|^{m}]$.
The limits $${\sigma}_{\sharp}(\mu) \stackrel{def}{=}
\lim_{x/|x| \to \mu, \ x \to 0} \frac{|\sigma(x)|^{\tilde{m}/m}}{|x|} \ \ \mathrm{and} \ \
\breve{\sigma}(\mu) \stackrel{def}{=} \lim_{x/|x| \to \mu, \ x \to 0} \frac{\sigma(x)}{|\sigma(x)|}$$ exist. In particular ${\sigma}_{\sharp}:{\mathbb S}^{1} \to {\mathbb R}^{+}$ and $\breve{\sigma}:{\mathbb S}^{1} \to {\mathbb S}^{1}$ are continuous.
Consider sequences $x_{n}$, $\tilde{x}_{n}$ such that $\lim_{n \to \infty} x_{n} = \lim_{n \to \infty} \tilde{x}_{n} =0$, both sequences $x_{n}/|x_{n}|$, $\tilde{x}_{n}/|\tilde{x}_{n}|$ converge to $\mu$ and the limits $$a_{0} = \lim_{n \to \infty} \frac{|\sigma(x_{n})|^{\tilde{m}}}{|x_{n}|^{m}}, \
\mu_{0} = \lim_{n \to \infty} \frac{\sigma(x_{n})}{|\sigma(x_{n})|}, \
a_{1} = \lim_{n \to \infty} \frac{|\sigma(\tilde{x}_{n})|^{\tilde{m}}}{|\tilde{x}_{n}|^{m}}, \
\mu_{1} = \lim_{n \to \infty} \frac{\sigma(\tilde{x}_{n})}{|\sigma(\tilde{x}_{n})|}$$ are well-defined. It suffices to prove that $a_{0}=a_{1}$ and $\mu_{0}=\mu_{1}$.
We define $T_{n}=[s/|x_{n}|^{m}]$ and $\tilde{T}_{n}=[s/|\tilde{x}_{n}|^{m}]$. We obtain $$\lim_{n \to \infty} \varphi^{T_{n}}(x_{n},y_{0}) =
\lim_{n \to \infty} \mathrm{exp}(s \mu^{m} X_{0})(0,y_{0})=
\lim_{n \to \infty} \varphi^{\tilde{T}_{n}}(\tilde{x}_{n},y_{0})$$ for any $y_{0} \in B(0,\epsilon)$ whereas we have $$\lim_{n \to \infty} \eta^{T_{n}}(\sigma(x_{n},y_{0})) =
\mathrm{exp}(s a_{0} \mu_{0}^{\tilde{m}} Y_{0})(\sigma(0,y_{0}))$$ and $$\lim_{n \to \infty} \eta^{\tilde{T}_{n}}(\sigma(\tilde{x}_{n},y_{0})) =
\mathrm{exp}(s a_{1} \mu_{1}^{\tilde{m}} Y_{0})(\sigma(0,y_{0}))$$ for $y_{0} \in B(0,\epsilon)$ and $s \in {\mathbb R}$. We deduce $a_{0} \mu_{0}^{\tilde{m}} =a_{1} \mu_{1}^{\tilde{m}}$ and then $a_{0}=a_{1}$ and $\mu_{0}^{\tilde{m}} =\mu_{1}^{\tilde{m}}$. Consider a connected set $E$ such that $E_{\pi} \cap (\{0\} \times {\mathbb S}^{1}) = \{(0,\mu)\}$ and containing the sequences $x_{n}$ and $\tilde{x}_{n}$. Hence $\sigma(E)$ adheres to a connected set of directions contained in the finite set $\{ \lambda \in {\mathbb S}^{1}: \lambda^{\tilde{m}} = \mu_{0}^{\tilde{m}} \}$. As a consequence $\sigma(E)$ adheres to a unique direction and $\mu_{0}=\mu_{1}$.
\[lem:defhmp\] Let $\tilde{\psi}$ be a Fatou coordinate of $Y_{0}$. There exists a ${\mathbb R}$-linear isomorphism ${\mathfrak h}:{\mathbb C} \to {\mathbb C}$ such that $$(\tilde{\psi} \circ \sigma \circ \mathrm{exp}(z X_{0}))(0,y) -
(\tilde{\psi} \circ \sigma)(0,y) \equiv {\mathfrak h}(z) .$$ In particular ${\mathfrak h}$ does not depend on $y$. The mapping $\sigma$ is orientation-preserving on the parameter space if and only if the mapping ${\mathfrak h}$ is orientation-preserving. Moreover $\breve{\sigma}: {\mathbb S}^{1} \to {\mathbb S}^{1}$ is a homeomorphism and $m=\tilde{m}$.
Fix $z = \lambda |z|$. Consider $\mu \in {\mathbb S}^{1}$ with $\lambda = \mu^{m}$. We define $T'(x) = [|z|/|x|^{m}]$. We have $$\lim_{x \to 0}^{x \in \mu {\mathbb R}^{+}} \varphi^{T'(x)}(x,y_{0}) =
\lim_{x \to 0}^{x \in \mu {\mathbb R}^{+}} \mathrm{exp}
\left( \left[\frac{|z|}{|x|^{m}} \right] x^{m} X_{0} \right)(x,y_{0}) =
\mathrm{exp}(z X_{0})(0,y_{0})$$ for $y_{0} \in B(0,\epsilon)$. We obtain $$\lim_{x \to 0}^{x \in \mu {\mathbb R}^{+}} \eta^{T'(x)}(\sigma(x,y_{0})) =
\lim_{x \to 0}^{x \in \mu {\mathbb R}^{+}} \mathrm{exp}
\left( \left[ \frac{|z|}{|x|^{m}} \right] \sigma(x)^{\tilde{m}} Y_{0} \right)(\sigma(x,y_{0}))$$ and then $$\lim_{x \to 0}^{x \in \mu {\mathbb R}^{+}} \eta^{T'(x)}(\sigma(x,y_{0})) =
\mathrm{exp}(|z| \sigma_{\sharp}(\mu)^{m} \breve{\sigma}(\mu)^{\tilde{m}} Y_{0})(\sigma(0,y_{0}))$$ for $y_{0} \in B(0,\epsilon)$. This implies ${\mathfrak h}(z) = |z| \sigma_{\sharp}(\mu)^{m} \breve{\sigma}(\mu)^{\tilde{m}}$. Clearly the value of ${\mathfrak h}$ does not depend on $y_{0}$.
Analogously as in Proposition \[pro:rlinear\] we can show that ${\mathfrak h}$ is a linear isomorphism. The mapping $\breve{\sigma}$ is a homeomorphism that satisfies ${\mathfrak h}(\mu^{m}) = \sigma_{\sharp}(\mu)^{m} \breve{\sigma}(\mu)^{\tilde{m}}$ for any $\mu \in {\mathbb S}^{1}$. Thus ${\mathfrak h}$ is orientation-preserving if and only if the action of $\sigma$ on the parameter space is orientation-preserving. Notice that when $\mu$ turns once around $0$ the path $\mu \to {\mathfrak h}(\mu^{m})$ turns $m$ times and $\mu \to \breve{\sigma}(\mu)^{\tilde{m}}$ turns $\tilde{m}$ times. We obtain $m=\tilde{m}$.
\[cor:affm\] Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ with $m(\varphi)>0$ such that there exists a homeomorphism $\sigma$ satisfying $\sigma \circ \varphi = \eta \circ \sigma$. Then we obtain $m(\varphi)=m(\eta)$. Moreover $\sigma_{|x=0}$ is affine in Fatou coordinates. The mapping $\sigma_{|x=0}$ is orientation-preserving if and only if the action of $\sigma$ on the parameter space is orientation-preserving.
Corollary \[cor:affm\] is the General Theorem for the case $m(\varphi)>0$. Let us remark that $\sigma_{|x=0}$ is real analytic in $x=0$ if $N=0$ and it is real analytic in $\{x=0\} \setminus \{(0,0)\}$ if $N \geq 1$. The proof of Corollary \[cor:affm\] is analogous to the proof of Corollary \[cor:orp\].
The next result completes the proof of Proposition \[pro:holpar\].
\[cor:holpar\] Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{2},0)$}}}$ with $m(\varphi)>0$ such that there exists a homeomorphism $\sigma$ satisfying $\sigma \circ \varphi = \eta \circ \sigma$. Suppose that the action of $\sigma$ on the parameter space is holomorphic (resp. anti-holomorphic). Then $\sigma_{|x=0}$ is holomorphic (resp. anti-holomorphic).
Suppose that the action of $\sigma$ on the parameter space is holomorphic. Hence ${\sigma}_{\sharp}$ is constant and $\breve{\sigma}$ is of the form $\lambda \mapsto \mu \lambda$ for some $\mu \in {\mathbb S}^{1}$. We obtain that ${\mathfrak h}$ is of the form $\varsigma z$ for some $\varsigma \in {\mathbb C}^{*}$. Thus $\sigma_{|x=0}$ is holomorphic. If the action of $\sigma$ is anti-holomorphic we argue as in Corollary \[cor:holact\] to reduce the situation to the previous one.
We present two examples of conjugating mappings $\sigma$ such that $\sigma_{|x=0}$ is not necessarily holomorphic or anti-holomorphic. They correspond to the cases $N=0$ and $N=1$ respectively.
Consider $X=x^{m} \partial /\partial y$. Fix a ${\mathbb R}$-linear mapping ${\mathfrak h}:{\mathbb C} \to {\mathbb C}$. We define $\sigma_{\sharp}:{\mathbb S}^{1} \to {\mathbb R}^{+}$ and $\breve{\sigma}:{\mathbb S}^{1} \to {\mathbb S}^{1}$ by using the formula ${\mathfrak h}(\lambda^{m}) = \sigma_{\sharp}(\lambda)^{m} \breve{\sigma}(\lambda)^{m}$ for $\lambda \in {\mathbb S}^{1}$. We define the homeomorphism $$\sigma (r \lambda, y) = (r \sigma_{\sharp}(\lambda) \breve{\sigma}(\lambda), {\mathfrak h}(y)) .$$ The vector field $X_{|x=\sigma(r \lambda)}$ is $r^{m} \sigma_{\sharp}(\lambda)^{m} \breve{\sigma}(\lambda)^{m} \partial / \partial y=
r^{m} {\mathfrak h}(\lambda^{m}) \partial / \partial y$. The real flows of $X_{|x=r \lambda}$ and $X_{|x=\sigma(r \lambda)}$ are $$\theta_{s}(y) = y + s r^{m} \lambda^{m} \ \ \mathrm{and} \ \
\tilde{\theta}_{s}(y) = y + s r^{m} {\mathfrak h}(\lambda^{m})$$ respectively for $s \in {\mathbb R}$. Clearly the ${\mathbb R}$-linear mapping ${\mathfrak h}$ conjugates them. Thus $\sigma$ commutes with ${\rm exp}(X)$. It is ${\mathbb R}$-linear when $m=1$. Moreover $\sigma_{|x=0}$ is holomorphic (resp. anti-holomorphic) if and only if ${\mathfrak h}$ is holomorphic (resp. anti-holomorphic).
Consider $X = x^{m} y \partial / \partial y$ for some $m \in {\mathbb N}$. We define ${\mathfrak h}(z) = 3z/2 + \overline{z}/2$ and the homeomorphism $$\sigma (r \lambda, y) = (r \sigma_{\sharp}(\lambda) \breve{\sigma}(\lambda), y |y|)$$ by using the formula ${\mathfrak h}(\lambda^{m}) = \sigma_{\sharp}(\lambda)^{m} \breve{\sigma}(\lambda)^{m}$. The vector fields $X_{|x=r \lambda}$ and $X_{|x=\sigma(r \lambda)}$ are equal to $r^{m} \lambda^{m} y \partial / \partial y$ and ${\mathfrak h}(r^{m} \lambda^{m}) y \partial / \partial y$ respectively. The real flows of $X_{|x=r \lambda}$ and $X_{|x=\sigma(r \lambda)}$ are $$\theta_{s}(y) = e^{s r^{m} \lambda^{m}} y \ \ \mathrm{and} \ \
\tilde{\theta}_{s}(y) = e^{s r^{m} {\mathfrak h}(\lambda^{m})} y$$ respectively for $s \in {\mathbb R}$. We have $$(y |y|) \circ \theta_{s}(y) = e^{s r^{m} (\lambda^{m} + Re(\lambda^{m}))} y |y| =
e^{s r^{m} {\mathfrak h}(\lambda^{m})} y |y| =\tilde{\theta}_{s}(y |y|)$$ for $y \in {\mathbb C}$ and $s \in {\mathbb R}$. Thus $\sigma$ commutes with the real flow of $X$ and then with $\mathrm{exp}(X)$. The mapping $\sigma_{|x=0}$ is neither holomorphic nor anti-holomorphic. Moreover $\sigma_{|x=0}$ is real analytic in $\{x=0\} \setminus \{(0,0)\}$ and it is not a $C^{1}$ diffeomorphism in the neighborhood of $0$.
Let $\varphi, \eta \in {\mbox{{\rm Diff}{${\,}_{p1}({\mathbb C}^{n+1},0)$}}}$ with ${\mathfrak l} \stackrel{def}{=} \{x_{1}=\hdots=x_{n}=0\} \subset \mathrm{Fix} (\varphi)$ such that there exists a homeomorphism $\sigma$ satisfying $\sigma \circ \varphi = \eta \circ \sigma$. It is easy to prove analogous results to the ones in this section if every irreducible component of $\mathrm{Fix} (\varphi)$ containing ${\mathfrak l}$ is a union of fibers of the fibration $dx_{1}=\hdots=d x_{n}=0$. In general we can reduce the situation to the previous one by blowing up centers that are union of lines of the form $\{x_{1}=c_{1}\} \cap \hdots \cap \{x_{n}=c_{n}\}$.
Building examples {#sec:build}
=================
Let us construct topological conjugations between real flows of vector fields in ${{\mathcal X}_{p1} {\mbox{(${\mathbb C}^{2},0$)}}}$ whose restrictions to $x=0$ are neither holomorphic nor anti-holomorphic in the case $N>1$, $m=0$. We choose the case $N=2$ for the sake of simplicity. This section provides examples for the exceptional cases covered by Propositions \[pro:atui\] and \[pro:atu\].
Description of the construction
-------------------------------
Consider complex numbers $a,b \in {\mathbb C}$ and a ${\mathbb R}$-linear orientation-preserving isomorphism ${\mathfrak h}:{\mathbb C} \to {\mathbb C}$ such that ${\mathfrak h}(1)=1$ and ${\mathfrak h}(2 \pi i a) = 2 \pi i b$. In particular $Re (a)$ and $Re(b)$ have the same sign. We define $$X = \frac{y^{2}-x}{1+ay} \frac{\partial}{\partial y} \ \ \mathrm{and} \ \
Y = \frac{y^{2}-x}{1+by} \frac{\partial}{\partial y} .$$ The residue functions of $X$ satisfy $$Res_{X}(x,\sqrt{x}) = \frac{1}{2} \left( \frac{1}{\sqrt{x}} + a \right), \ \
Res_{X}(x,-\sqrt{x}) = \frac{1}{2} \left( \frac{-1}{\sqrt{x}} + a \right).$$ Consider a germ of homeomorphism $\tau$ defined in a neighborhood of $0$ in ${\mathbb C}$ such that ${\mathfrak h}(\pi i /\sqrt{x}) = \pi i / \sqrt{\tau(x)}$ for any $x$ defined in a neighborhood of $0$. In this way we obtain $$\label{equ:hpres}
{\mathfrak h}(2 \pi i Res_{X}(x,\pm \sqrt{x})) = 2 \pi i Res_{Y}(\tau(x),\pm \sqrt{\tau(x)})$$ for any $x$ in a neighborhood of $0$. The isomorphism ${\mathfrak h}$ is of the form ${\mathfrak h}(z) = \varsigma_{0} z + \varsigma_{1} \overline{z}$ with $|\varsigma_{0}|>|\varsigma_{1}|$. We obtain $$\label{equ:dtq}
\frac{\varsigma_{0}}{\sqrt{x}} - \frac{\varsigma_{1}}{\overline{\sqrt{x}}} \equiv
\frac{1}{\sqrt{\tau(x)}} \implies
\frac{\sqrt{x}}{\sqrt{\tau(x)}} \equiv
\varsigma_{0} - \varsigma_{1} \frac{\sqrt{x}}{\overline{\sqrt{x}}}.$$ Fix a point $y_{0} \in B(0,\epsilon) \setminus \{0\}$ close to $0$. Let $\psi_{X}$, $\psi_{Y}$ be Fatou coordinates of $X$, $Y$ such that $\psi_{X}(x,y_{0}) \equiv \psi_{Y}(x,y_{0}) \equiv 0$. We want to define a homeomorphism $\sigma$ conjugating $\Re (X)$ and $\Re (Y)$ such that
- $\sigma$ is of the form $\sigma(x,y) =(\tau(x), \sigma_{\natural}(x,y))$.
- $\sigma(x,y_{0})= (\tau(x),y_{0})$ for any $x$ in a neighborhood of $0$.
- $\psi_{Y} \circ \sigma \circ \mathrm{exp}(z X) - \psi_{Y} \circ \sigma \equiv {\mathfrak h}(z)$ for any $z \in {\mathbb C}$.
Let us remark that the monodromies of $({\mathfrak h} \circ\psi_{X})(x_{0},y)$ and $\psi_{Y}(\tau(x_{0}), y)$ around $(x_{0}, \pm \sqrt{x_{0}})$ and $(\tau(x_{0}), \pm \sqrt{\tau(x_{0})})$ respectively coincide by Eq. (\[equ:hpres\]). The natural way of defining $\sigma$ is by using the equation ${\mathfrak h} \circ\psi_{X} \equiv \psi_{Y} \circ \sigma$.
The method of the path
----------------------
In order to prove the existence of $\sigma$ satisfying ${\mathfrak h} \circ\psi_{X} \equiv \psi_{Y} \circ \sigma$ we apply the method of the path (see [@Rou:ast], [@Mar:ast]). First we relocate the points in $\mathrm{Sing}(Y)$ by considering the change of coordinates $y = y \sqrt{\tau(x)/x}$ for the parameter $\tau(x)$. This corresponds to the change of coordinates $\tilde{\sigma}(x,y) = (x, y \sqrt{x/\tau^{-1}(x)})$. We obtain $$Y(\tau(x), y) =
\frac{{ \left( \frac{\sqrt{\tau(x)}}{\sqrt{x}} y \right) }^{2} -
\tau(x)}{1 + b y \sqrt{\tau(x)} / \sqrt{x}}
\frac{\sqrt{x}}{\sqrt{\tau(x)}} \frac{\partial}{\partial y} =
\frac{\sqrt{\tau(x)}}{\sqrt{x}} \frac{y^{2}-x}{1 + b y \sqrt{\tau(x)} / \sqrt{x}}
\frac{\partial}{\partial y}$$ in the new coordinates. Let us point out that the change of coordinates $\tilde{\sigma}$ is not well-defined at $x=0$. It is not very pathological either since $\sqrt{x/\tau^{-1}(x)}$ is bounded away from $0$ and $\infty$ for $x$ in a neighborhood of $0$ (see Eq. (\[equ:dtq\])).
Let us consider the function $$\Psi = (1-s) ({\mathfrak h} \circ \psi_{X})(x,y) + s \psi_{Y}
\left( \tau(x), \frac{\sqrt{\tau(x)}}{\sqrt{x}} y \right) = \Psi_{1} + i \Psi_{2}.$$ In general $\Psi$ is neither holomorphic nor anti-holomorphic. Roughly speaking all the functions $\Psi(s_{0},x,y)$ have the same poles and the monodromy around those poles does not depend on $s_{0}$. We are trying to conjugate $\Re (X)$ and $\tilde{\sigma}^{*} (\Re (Y))$. In order to do this we want to find a continuous vector field $Z$ defined in coordinates $(x,y,s)$ in a neighborhood of $\{(0,0)\} \times [0,1]$ in $(({\mathbb C}^{*} \times {\mathbb C}) \cup \{(0,0)\}) \times {\mathbb C}$ of the form $$Z = \frac{\partial}{\partial s} + c_{1}(x,y,s) \frac{\partial}{\partial y_{1}} +
c_{2}(x,y,s) \frac{\partial}{\partial y_{2}}$$ where $y=y_{1}+iy_{2}$. Moreover we ask $Z$ to satisfy
- $Z(\Psi) \equiv 0$.
- $c_{j}$ is a continuous function defined in a neighborhood of $\{(0,0)\} \times [0,1]$ in $(({\mathbb C}^{*} \times {\mathbb C}) \cup \{(0,0)\}) \times {\mathbb C}$ and vanishing at $y^{2}-x=0$ for any $j \in \{1,2\}$.
The idea is that for such $Z$ the mapping $\mathrm{exp}(Z)(x,y,0)$ conjugates $\Re(X)$ and $\tilde{\sigma}^{*}(\Re (Y))$ in a neighborhood of $(0,0)$ in $({\mathbb C}^{*} \times {\mathbb C}) \cup \{(0,0)\}$. We define $$\rho = ({\mathfrak h} \circ \psi_{X})(x,y) - \psi_{Y}
\left( \tau(x), \frac{\sqrt{\tau(x)}}{\sqrt{x}} y \right) .$$ The equation $Z(\Psi) \equiv 0$ is equivalent to $$\left\{
\begin{array}{ccccl}
c_{1} \frac{\partial \Psi_{1}}{\partial y_{1}} & + & c_{2} \frac{\partial \Psi_{1}}{\partial y_{2}} &
= & Re (\rho) \\
c_{1} \frac{\partial \Psi_{2}}{\partial y_{1}} & + & c_{2} \frac{\partial \Psi_{2}}{\partial y_{2}} &
= & Im (\rho).
\end{array}
\right.$$ The solutions are $$\label{equ:c1c2}
c_{1} =
\frac{\left|
\begin{array}{cc}
Re(\rho) & \frac{\partial \Psi_{1}}{\partial y_{2}} \\
Im(\rho) & \frac{\partial \Psi_{2}}{\partial y_{2}}
\end{array}
\right|}{\left|
\begin{array}{cc}
\frac{\partial \Psi_{1}}{\partial y_{1}} & \frac{\partial \Psi_{1}}{\partial y_{2}} \\
\frac{\partial \Psi_{2}}{\partial y_{1}} & \frac{\partial \Psi_{2}}{\partial y_{2}}
\end{array}
\right|}, \ \ \
c_{2} =
\frac{\left|
\begin{array}{cc}
\frac{\partial \Psi_{1}}{\partial y_{1}} & Re(\rho) \\
\frac{\partial \Psi_{2}}{\partial y_{1}} & Im(\rho)
\end{array}
\right|}{\left|
\begin{array}{cc}
\frac{\partial \Psi_{1}}{\partial y_{1}} & \frac{\partial \Psi_{1}}{\partial y_{2}} \\
\frac{\partial \Psi_{2}}{\partial y_{1}} & \frac{\partial \Psi_{2}}{\partial y_{2}}
\end{array}
\right|}.$$ The denominator $D$ of the previous expressions satisfies $D= |\partial \Psi/\partial y|^{2} - |\partial \Psi/\partial \overline{y}|^{2}$. We have $$\left\{
\begin{array}{ccl}
\frac{\partial \Psi}{\partial y} & = &
(1-s) \varsigma_{0} \frac{1+ay}{y^{2}-x} + s \frac{\sqrt{x}}{\sqrt{\tau(x)}}
\frac{1 + b y \sqrt{\tau(x)} / \sqrt{x}}{y^{2}-x} \\
\frac{\partial \Psi}{\partial \overline{y}} & = &
(1-s) \varsigma_{1} \overline{ \left( \frac{1+ay}{y^{2}-x} \right) } .
\end{array}
\right.$$
\[lem:den\] The function $D |y^{2}-x|^{2}$ is bounded away from $0$ and $\infty$.
Consider the function $\kappa = (1-s) \varsigma_{0} + s \sqrt{x} / \sqrt{\tau(x)}$. It suffices to prove that $\kappa$ is bounded by below for $x$ in a neighborhood of $0$ and $s$ in a neighborhood of $[0,1]$ and that we have $|\kappa(x,s)|/|(1-s) \varsigma_{1}| \geq C >1$ for some constant $C \in {\mathbb R}^{+}$. Notice that $\kappa$ is bounded by above. The property $\sqrt{x}/\sqrt{\tau(x)} \equiv
\varsigma_{0} - \varsigma_{1} \sqrt{x} / \overline{\sqrt{x}}$ implies $$|\kappa(x,s)| =
\left| \varsigma_{0} - s \varsigma_{1} \frac{\sqrt{x}}{\overline{\sqrt{x}}} \right| \geq
| \varsigma_{0}| - | \varsigma_{1}| >0$$ for $s \in [0,1]$. We have $$|\kappa(x,s)|^{2} = (1-s)^{2} | \varsigma_{0}|^{2} +
s^{2} {\left| \frac{\sqrt{x}}{\sqrt{\tau(x)}} \right|}^{2} +
2 s (1-s) Re \left(
\varsigma_{0} \overline{\frac{\sqrt{x}}{\sqrt{\tau(x)}}}
\right)$$ and $$\varsigma_{0} \overline{\frac{\sqrt{x}}{\sqrt{\tau(x)}}} =
\varsigma_{0}
\left( \overline{\varsigma_{0}} - \overline{\varsigma_{1}} \frac{\overline{\sqrt{x}}}{\sqrt{x}} \right)
\implies
Re \left(
\varsigma_{0} \overline{\frac{\sqrt{x}}{\sqrt{\tau(x)}}}
\right) \geq |\varsigma_{0}|^{2}-|\varsigma_{0}||\varsigma_{1}| >0.$$ We deduce that $|\kappa| \geq |(1-s) \varsigma_{0}| \geq C |(1-s) \varsigma_{1}|$ for some constant $C >1$.
The next step of the proof is showing that $(y^{2}-x) \rho$ can be extended as a continuous function vanishing at $y^{2}-x=0$. We define the auxiliary functions $$R_{\pm} = \frac{\pm 1}{2 \sqrt{x}}
\left( \varsigma_{0} - \frac{\sqrt{x}}{\sqrt{\tau(x)}} \right) + \frac{1}{2} (\varsigma_{0} a-b)$$ and $$\tilde{\rho} = R_{+}(x) \ln |y - \sqrt{x}|^{2} + R_{-}(x) \ln |y + \sqrt{x}|^{2} .$$ Notice that ${\mathfrak h}(2 \pi i a) = 2 \pi i b$ implies $b = \varsigma_{0} a - \varsigma_{1} \overline{a}$. We have $$\frac{\partial \tilde{\rho}}{\partial y} =
\frac{R_{+}(x) (y+\sqrt{x}) + R_{-}(x)(y-\sqrt{x}) }{y^{2}-x} =\frac{\partial \rho}{\partial y}$$ and $$\frac{\partial \tilde{\rho}}{\partial \overline{y}} =
\frac{(\varsigma_{0} a -b) \overline{y} +\varsigma_{1} }{\overline{y^{2}-x}} =
\varsigma_{1} \overline{\left( \frac{1+ay}{y^{2}-x} \right)}=
\frac{\partial \rho}{\partial y} .$$
\[lem:tilrho\] The function $\rho - \tilde{\rho}$ is a bounded function of $x$.
It is obvious that $\rho - \tilde{\rho}$ is constant in each line $x=x_{0}$. Since $\rho(x,y_{0})$ is bounded by construction it suffices to show that $\tilde{\rho}(x,y_{0})$ is bounded in the neighborhood of $0$. We have $$\tilde{\rho}(x,y_{0}) = \frac{1}{2 \sqrt{x}}
\left( \varsigma_{0} - \frac{\sqrt{x}}{\sqrt{\tau(x)}} \right)
\ln {\left| \frac{y_{0} - \sqrt{x}}{y_{0} + \sqrt{x}} \right|}^{2}
+(\varsigma_{0} a -b) \ln |y_{0}^{2}-x| .$$ It suffices to prove that $$\hat{\rho}(x,y_{0}) \stackrel{\mathrm{def}}{=} \frac{1}{2 \sqrt{x}}
\left( \varsigma_{0} - \frac{\sqrt{x}}{\sqrt{\tau(x)}} \right)
\ln {\left| \frac{y_{0} - \sqrt{x}}{y_{0} + \sqrt{x}} \right|}^{2}$$ is bounded. We have $$\hat{\rho}(x,y_{0})= \frac{1}{2 \sqrt{x}}
\left( \varsigma_{0} - \frac{\sqrt{x}}{\sqrt{\tau(x)}} \right)
\ln \left(
1 - 2 \frac{\sqrt{x} \overline{y_{0}} + \overline{\sqrt{x}} y_{0} }{
|y_{0}|^{2} +\sqrt{x} \overline{y_{0}} + \overline{\sqrt{x}} y_{0} + |x|}
\right)$$ and then $$\hat{\rho}(x,y_{0}) \sim -
\left( \varsigma_{0} - \frac{\sqrt{x}}{\sqrt{\tau(x)}} \right)
\left( \frac{1}{y_{0}} +\frac{1}{\overline{y_{0}}} \frac{\overline{\sqrt{x}}}{\sqrt{x}} \right)$$ is bounded.
\[lem:limrho\] We have $\lim_{(x,y) \to (x_{0},y_{0})} \rho(x,y) (y^{2}-x)=0$ for any $(x_{0},y_{0})$ in the curve $y^{2}-x=0$.
It suffices to show the result for $\hat{\rho}$, see the proof of Lemma \[lem:tilrho\]. Suppose $(x_{0},y_{0}) = (0,0)$, otherwise the proof is straightforward.
Suppose that $|y/\sqrt{x}| \leq 8$. We have $$|\hat{\rho}(x,y) (y^{2}-x)| \leq
\frac{9}{2} \left| \varsigma_{0} - \frac{\sqrt{x}}{\sqrt{\tau(x)}} \right|
\left(
|y - \sqrt{x}| |\ln |y - \sqrt{x}|^{2}| +
|y + \sqrt{x}| |\ln |y + \sqrt{x}|^{2}| \right) .$$ We deduce $\lim_{|y/\sqrt{x}| \leq 8, \ (x,y) \to (0,0)} \rho(x,y) (y^{2}-x)=0$.
Suppose that $|y/\sqrt{x}| \geq 8$. We have $$|\hat{\rho}(x,y) (y^{2}-x)| \leq \frac{1}{2 \sqrt{|x|}}
\left| \varsigma_{0} - \frac{\sqrt{x}}{\sqrt{\tau(x)}} \right|
|y^{2}-x|
\left| \ln {\left| \frac{y - \sqrt{x}}{y + \sqrt{x}} \right|}^{2} \right|$$ and then $$|\hat{\rho}(x,y) (y^{2}-x)| \leq \frac{1}{2 \sqrt{|x|}}
\left| \varsigma_{0} - \frac{\sqrt{x}}{\sqrt{\tau(x)}} \right|
|y^{2}-x| C \frac{\sqrt{|x|}}{|y|}$$ for some $C \in {\mathbb R}^{+}$. We obtain $$|\hat{\rho}(x,y) (y^{2}-x)| \leq \frac{C}{2}
\left| \varsigma_{0} - \frac{\sqrt{x}}{\sqrt{\tau(x)}} \right|
|y| \left| 1 - \frac{x}{y^{2}} \right| \leq
\frac{65 C}{128} |y| \left| \varsigma_{0} - \frac{\sqrt{x}}{\sqrt{\tau(x)}} \right|.$$ We deduce $\lim_{|y/\sqrt{x}| \geq 8, \ (x,y) \to (0,0)} \rho(x,y) (y^{2}-x)=0$.
The functions $c_{1}$ and $c_{2}$ are continuous. They are defined in a neighborhood of $\{ (0,0) \} \times [0,1]$ in $(({\mathbb C}^{*} \times {\mathbb C}) \cup \{(0,0)\}) \times {\mathbb C}$ and vanish at $y^{2}-x=0$.
Let us prove the result for $c_{1}$ without lack of generality. Equation (\[equ:c1c2\]) and Lemma \[lem:den\] imply that it suffices to show that $$|y^{2}-x|^{2}
\left|
\begin{array}{cc}
Re(\rho) & \frac{\partial \Psi_{1}}{\partial y_{2}} \\
Im(\rho) & \frac{\partial \Psi_{2}}{\partial y_{2}}
\end{array}
\right|$$ tends to $0$ when we approach a point of $y^{2}-x=0$. Since $\partial \Psi_{j} / \partial y_{k} = O(1/(y^{2}-x))$ for all $j, k \in \{1,2\}$ the property is a consequence of Lemma \[lem:limrho\].
Consequences of the construction
--------------------------------
We have all the ingredients required to provide examples of exceptional conjugating mappings that are not holomorphic or anti-holomorphic. Hence the condition requiring the unperturbed diffeomorphisms to be non-analytically trivial in the Main Theorem can not be removed.
\[pro:sigma\] The mapping $\sigma_{\flat}(x,y) = {\rm exp}(Z)(x,y,0)$ conjugates $\Re (X)$ and $\tilde{\sigma}^{*} (\Re (Y))$ in a neighborhood of $(0,0)$ in $({\mathbb C}^{*} \times {\mathbb C}) \cup \{(0,0)\}$. Moreover the mapping $\sigma = \tilde{\sigma} \circ \sigma_{\flat}$ is a homeomorphism conjugating $\Re (X)$ and $\Re (Y)$ defined in a neighborhood of $(0,0)$. Moreover $\sigma$ and $\sigma^{-1}$ are real analytic outside $y^{2}-x=0$.
It is clear that $\sigma_{\flat}$ and its inverse ${\rm exp}(-Z)(x,y,1)$ satisfy the properties by construction. The mapping $\sigma$ conjugates $\Re (X)$ and $\Re (Y)$. The remaining issue is the study of the properties of $\sigma$ in the neighborhood of the line $x=0$.
Suppose $a \in i {\mathbb R}$. Then we have $a=b$ and we define ${\mathfrak h}_{u}(z)=(1-u) z + u {\mathfrak h}(z)$ and $a_{u}=a$ for $u \in [0,1]$. If $a \not \in i {\mathbb R}$ the real parts of $a$ and $b$ have the same sign since ${\mathfrak h}$ is orientation preserving. We define $a_{u}=(1-u)a +ub$ and ${\mathfrak h}_{u}:{\mathbb C} \to {\mathbb C}$ as the ${\mathbb R}$-linear mapping such that ${\mathfrak h}_{u}(1)=1$ and ${\mathfrak h}_{u}(2 \pi i a)=2 \pi i a_{u}$ for $u \in [0,1]$. We define $X_{u}=[(y^{2}-x)/(1+a_{u} y)] \partial / \partial y$ and a Fatou coordinate $\psi_{u}$ of $X_{u}$ such that $\psi_{u}(x,y_{0}) \equiv 0$ for $u \in [0,1]$. We denote by $\sigma_{u}$ the diffeomorphism obtained by applying the previous method to $X$, $X_{u}$ and ${\mathfrak h}_{u}$. We obtain $\sigma_{0}=Id$ and $\sigma_{1} = \sigma$. Since $\sigma_{u}$ depends continuously on $u$ and $\sigma_{u}$ conjugates the functions ${\mathfrak h}_{u} \circ \psi_{X}$ and $\psi_{u}$ we obtain $\sigma_{u}(x,y_{0}) = (\tau_{u}(x), y_{0})$ for all $u \in [0,1]$ and $x$ in a neighborhood of $0$. In particular we deduce $\sigma(x,y_{0}) = (\tau(x), y_{0})$ for any $x$ in a neighborhood of $0$. The equation ${\mathfrak h} \circ\psi_{X} \equiv \psi_{Y} \circ \sigma$ implies that $\sigma$ and $\sigma^{-1}$ are real analytic outside $y^{2}-x=0$.
Consider $b= a \in i {\mathbb R}$ and a ${\mathbb R}$-linear orientation-preserving isomorphism such that ${\mathfrak h}(1)=1$. Then the mapping $\sigma$ provided by Proposition \[pro:sigma\] satisfies $\sigma^{*}(\Re (X)) = \Re(X)$ and $\sigma \circ \mathrm{exp}(X) = \mathrm{exp}(X) \circ \sigma$. Moreover $\sigma$ is holomorphic if and only if ${\mathfrak h} \equiv z$.
If ${\mathfrak h}$ is orientation-reversing consider the conjugation $\sigma$ associated to $(X,X,\overline{z} \circ {\mathfrak h})$. Now $\zeta \circ \sigma$ conjugates $\Re (X)$ and $\Re (Y)$ where $Y= [(y^{2}-x)/(1+\overline{a} y)] \partial / \partial y$ and $\zeta(x,y) = (\overline{x},\overline{y})$. The mapping ${\mathfrak h}_{{\rm exp}(X), {\rm exp}(Y), \zeta \circ \sigma}$ is equal to ${\mathfrak h}$. We described both situations in Proposition \[pro:atui\] where the diffeomorphisms by restriction to $x=0$ are holomorphically (resp. anti-holomorphically) conjugated but the conjugation is not holomorphic (resp. anti-holomorphic) in general.
Consider $a,b \not \in i {\mathbb R}$ such that $Re (a) Re (b) >0$. Let ${\mathfrak h}$ be the ${\mathbb R}$-linear mapping such that ${\mathfrak h}(1)=1$ and ${\mathfrak h}(2 \pi i a) = 2 \pi i b$. Then the mapping $\sigma$ provided by Proposition \[pro:sigma\] satisfies $\sigma^{*}(\Re (Y)) = \Re(X)$ and $\sigma \circ \mathrm{exp}(X) = \mathrm{exp}(Y) \circ \sigma$. Moreover $\sigma$ is holomorphic if and only if $a=b$.
Let ${\mathfrak h}$ be the ${\mathbb R}$-linear mapping such that ${\mathfrak h}(1)=1$ and ${\mathfrak h}(2 \pi i a) = -2 \pi i b$. We obtain $(\overline{z} \circ {\mathfrak h})(2 \pi i a) = 2 \pi i \overline{b}$. Then the mapping $\sigma$ provided by Proposition \[pro:sigma\] and associated to $X$, $\tilde{Y}=[(y^{2}-x)/(1+\overline{b} y)] \partial / \partial y$ and $\overline{z} \circ {\mathfrak h}$ satisfies $\sigma^{*}(\Re (\tilde{Y})) = \Re(X)$ and $\sigma \circ \mathrm{exp}(X) = \mathrm{exp}(\tilde{Y}) \circ \sigma$. The homeomorphism $\zeta \circ \sigma$ conjugates $\Re (X)$ and $\Re (Y)$. Moreover $\zeta \circ \sigma$ is anti-holomorphic if and only if $a=\overline{b}$. We described these situations in Proposition \[pro:atu\].
[^1]: e-mail address: javier@mat.uff.br
[^2]: MSC-class. Primary: 37F45, 37F75; Secondary: 37G10, 34E10
[^3]: Keywords: resonant diffeomorphism, bifurcation theory, topological classification, normal form
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We investigate the $L^p$ boundedness of the lacunary maximal function $A_rf$ associated to the spherical means on the Heisenberg group. By suitable adaptation of an approach of M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions, which lead to new weighted estimates. In order to prove the result, several properties of the spherical means have to be accomplished, namely, the $L^p$ improving property of the operator $A_rf$ and a continuity property of the difference $A_rf-\tau_y A_rf$, where $\tau_yf(x)=f(xy^{-1})$ is the right translation operator.'
address:
- 'Stat-Math Unit, Indian Statistical Institute, Kolkata, India.'
- |
Department of Mathematics\
Indian Institute of Science\
560 012 Bangalore, India
- |
BCAM - Basque Center for Applied Mathematics\
48009 Bilbao, Spain and Ikerbasque, Basque Foundation for Science, 48011 Bilbao, Spain
- |
Department of Mathematics\
Indian Institute of Science\
560 012 Bangalore, India\
and BCAM - Basque Center for Applied Mathematics\
48009 Bilbao, Spain
author:
- Sayan Bagchi
- Sourav Hait
- Luz Roncal
- Sundaram Thangavelu
title: |
On the maximal function associated to the\
lacunary spherical means on the Heisenberg group
---
Introduction and main results
=============================
A celebrated theorem of Stein [@Stein] proved in 1976 says that the spherical maximal function $ M $ defined by $$Mf(x) = \sup_{r>0} |f\ast \sigma_r(x)| = \sup_{r>0}\Big| \int_{|y|=r} f(x-y) d\sigma_r(y)\Big|$$ is bounded on $ L^p({\mathbb R}^n)$, $n \ge 2$, if and only if $ p > n/(n-1).$ Here $ \sigma_r $ stands for the normalised surface measure on the sphere $ S_r = \{ x\in {\mathbb R}^n: |x|=r\} $ in $ {\mathbb R}^n.$ The case $ n =2 $ was proved later by Bourgain [@Bourgain]. As opposed to this, in 1979, C. P. Calderón [@C] proved that the lacunary maximal function $$M_{\text{lac}}f(x) = \sup_{ j \in {\mathbb Z}} \Big|\int_{|y|=2^j} f(x-y) d\sigma_{2^j}(y)\Big|$$ is bounded on $ L^p({\mathbb R}^n) $ for all $ 1 < p < \infty $ for any $ n \ge 1.$ In a recent article, Lacey [@Lacey] has revisited the spherical maximal function. Using a new approach he has managed to prove certain sparse bounds for these maximal functions which has led him to obtain new weighted norm inequalities. Our main goal in this paper is to adapt the method of Lacey to prove an analogue of Calderón’s theorem in the context of certain spherical means on the Heisenberg group, and deduce weighted inequalities as immediate consequences.
Let ${\mathbb H}^n={{\mathbb C}}^n\times {\mathbb R}$ be the $(2n+1)$-dimensional Heisenberg group with the group law $$(z,t)(w,s)=\Big(z+w,t+s+\frac12{\operatorname{Im}}z\cdot \overline{w}\Big).$$ Given a function $f$ on ${\mathbb H}^n$, consider the spherical means $$\label{eq:defin}
A_rf(z,t):=f\ast \mu_r(z,t)=\int_{|w|=r}f\Big(z-w,t-\frac12{\operatorname{Im}}z\cdot \overline{w}\Big)\,d\mu_r(w)$$ where $\mu_r$ is the normalised surface measure on the sphere $S_r=\{(z,0):|z|=r\}$ in ${\mathbb H}^n$. The maximal function associated to these spherical means was first studied by Nevo and Thangavelu in [@NeT]. Later, improving the results in [@NeT], Narayanan and Thangavelu [@NaT] and Müller and Seeger [@MS], independently, proved the following sharp maximal theorem: the full maximal function $ Mf(z,t) = \sup_{r>0} |A_rf(z,t)| $ is bounded on $ L^p({\mathbb H}^n), n \geq 2 $ if and only if $ p > (2n)/(2n-1).$ In this work we consider the lacunary maximal function $$M_{\operatorname{lac}}f(z,t)=\sup_{j\in {\mathbb Z}}|A_{\delta^j}f(z,t)|, \quad \delta>0,$$ associated to the spherical means and prove the following result.
\[thm:spherical\] Assume that $ n \geq 2.$ Then for any $ 0 < \delta <\frac{1}{96}$ the associated lacunary maximal funcion $ M_{\operatorname{lac}}$ is bounded on $ L^p({\mathbb H}^n) $ for any $ 1 < p < \infty.$
Actually we can take any $\delta$ in Theorem \[thm:spherical\]. For example, we can take $\delta=2$. In our result we are taking $\delta< \frac{1}{96}$ not because the proof requires the restriction, but because we want to keep the proof simple, see more explanation after the statement of Lemma \[lem:Hn\].
We remark that another kind of spherical maximal function on the Heisenberg group has been considered by Cowling. In [@Cowling] he has studied the maximal function associated to the spherical means taken over genuine Heisenberg spheres, i.e., averages taken over spheres defined in terms of a homogeneous norm on $ {\mathbb H}^n.$ It would be interesting to see if lacunary maximal functions asociated these spherical means also have better mapping properties. We remark in passing that the spherical means studied in [@NeT; @NaT; @MS] (and hence in this paper) are more singular than the one studied in [@Cowling] as these means are supported on codimension two submanifolds.
Theorem \[thm:spherical\], as well as certain weighted versions, are easy consequences of the sparse bound in Theorem \[thm:sparse\], which is the main result of this paper. Before stating the result let us set up the notation. As in the case of $ {\mathbb R}^n $, there is a notion of dyadic grids on $ {\mathbb H}^n $, the members of which are called (dyadic) cubes. A collection of cubes $\mathcal{S}$ in ${\mathbb H}^n $ is said to be $ \eta$-sparse if there are sets $\{E_S \subset S:S\in \mathcal{S}\}$ which are pairwise disjoint and satisfy $|E_S|>\eta|S|$ for all $S\in \mathcal{S}$. For any cube $Q$ and $1<p<\infty$, we define $$\langle f\rangle_{Q,p}:=\bigg(\frac{1}{|Q|}\int_Q|f(x)|^pdx\bigg)^{1/p}, \qquad \langle f\rangle_{Q}:=\frac{1}{|Q|}\int_Q|f(x)|dx.$$ In the above $ x =(z,t) \in {\mathbb H}^n $ and $ dx = dz dt $ is the Lebesgue measure on $ {{\mathbb C}}^n \times {\mathbb R}$ which incidentally is the Haar measure on the Heisenberg group. Following Lacey [@Lacey], by the term $(p,q)$-sparse form we mean the following: $$\Lambda_{\mathcal{S},p,q}(f,g)=\sum_{S\in\mathcal{S}}|S|\langle f\rangle_{S,p}\langle g\rangle_{S,q}.$$
\[thm:sparse\] Assume $ n \geq 2$ and fix $ 0 < \delta < \frac{1}{96}.$ Let $ 1 < p, q < \infty $ be such that $ (\frac{1}{p},\frac{1}{q}) $ belongs to the interior of the triangle joining the points $ (0,1), (1,0) $ and $ (\frac{n}{n+1},\frac{n}{n+1}).$ Then for any pair of compactly supported bounded functions $ (f,g) $ there exists a $ (p,q)$-sparse form such that $ \langle M_{\operatorname{lac}}f, g\rangle \leq C \Lambda_{\mathcal{S},p,q}(f,g).$
In proving the corresponding result for the spherical means on $ {\mathbb R}^n $, Lacey [@Lacey] has made use of two important properties of the spherical means, namely, the $ L^p $ improving property of the operator $ S_r f = f \ast \sigma_r $ for a fixed $ r $, and the continuity property of the difference $ S_r f- \tau_y S_r f $ where $ \tau_y f(x) = f(x-y) $ is the translation operator.
A remark is in order. In order to keep the shape of our main results analogous to the ones related to the lacunary maximal function in ${\mathbb R}^n$, we decided to restrict the range of $(p,q)$ to the same regions as in the Euclidean case. Nevertheless, enhanced results for the lacunary maximal function in ${\mathbb H}^n$ are obtained (although we cannot say anything about the sharpness of such results) and we will also state them throughout the paper, see Subsections \[rem:sharp\], \[sub:sharpenc\], \[sub:sharpens\] and \[sub:sharpenw\]. In particular, a sharpened version of Theorem \[thm:sparse\] is given in Theorem \[thm:mainSH\].
In the next section we establish $ L^p-L^q $ estimates for our spherical means $ A_r f $ on the Heisenberg group. In Section \[sec:continuity\] we prove the continuity property of $ A_r f- A_r\tau_y f$, where now $ \tau_yf(x) =f(xy^{-1}) $ is the right translation operator. In Section \[sec:sparse\] we establish the sparse bound and finally in the last section we deduce weighted boundedness properties of the lacunary maximal function.
The $L^p$ improving property of the spherical mean value operator {#sec:Lp}
=================================================================
The observation that the spherical mean value operator $ S_r f := f\ast \sigma_r $ on $ {\mathbb R}^n $ is a Fourier multiplier plays an important role in every work dealing with the spherical maximal function. In fact, we know that $$\label{eq:ArEuc}
f \ast \sigma_r(x) = (2\pi)^{-n/2} \int_{{\mathbb R}^n} e^{i x \cdot \xi} \widehat{f}(\xi) \frac{ J_{n/2-1}(r|\xi|)} {(r|\xi|)^{n/2-1}} d\xi$$ where $ J_{n/2-1} $ is the Bessel function of order $ n/2-1.$ As Bessel functions $ J_\alpha $ are defined even for complex values of $ \alpha $ the above allows one to embed $ S_r f$ in an analytic family of operators and Stein’s analytic interpolation theorem comes in handy in studying the spherical maximal function. The same technique was employed by Strichartz [@S] who studied the $ L^p $ improving properties of $ S_r .$ For the spherical means on the Heisenberg group we do have such a representation if we replace the Euclidean Fourier transform by the group Fourier transform on $ {\mathbb H}^n.$
For the group $ {\mathbb H}^n $ we have a family of irreducible unitary representations $ \pi_\lambda $ indexed by non-zero reals $ \lambda $ and realised on $ L^2({\mathbb R}^n)$. The action of $\pi_{\lambda}(z,t)$ on $L^2({\mathbb R}^n)$ is explicitly given by $$\label{eq:pilambda}
\pi_{\lambda}(z,t)\varphi(\xi)=e^{i\lambda t}e^{i\lambda(x\cdot \xi+\frac12 x\cdot y)}\varphi(\xi+y)$$ where $\varphi\in L^2({\mathbb R}^n)$ and $z=x+iy$. By the theorem of Stone and von Neumann, which classifies all the irreducible unitary representations of $ {\mathbb H}^n $, combined with the fact that the Plancherel measure for $ {\mathbb H}^n $ is supported only on the infinite dimensional representations, it is enough to consider the following operator valued function known as the group Fourier transform of a given function $ f $ on $ {\mathbb H}^n$: $$\widehat{f}(\lambda) = \int_{{\mathbb H}^n} f(z,t) \pi_\lambda(z,t) \,dz \,dt.$$ The above is well defined, e.g., when $ f \in L^1({\mathbb H}^n) $ and for each $ \lambda \neq 0,$ $ \widehat{f}(\lambda) $ is a bounded linear operator on $ L^2({\mathbb R}^n).$ Observe that the above definition makes sense even if we replace $ f $ by a finite Borel measure $ \mu.$ In particular, $ \widehat{\mu}_r(\lambda) $ are well defined bounded operators on $ L^2({\mathbb R}^n)$ which can be described explicitly. Combined with the fact that $ \widehat{ f\ast g}(\lambda) = \widehat{f}(\lambda) \widehat{g}(\lambda) $ we obtain $ \widehat{A_rf}(\lambda) = \widehat{f}(\lambda)\widehat{\mu}_r(\lambda).$
The operators $ \widehat{\mu}_r(\lambda) $ turn out to be functions of the Hermite operator $H(\lambda) = -\Delta+\lambda^2 |x|^2.$ Indeed, if the spectral decomposition of $ H(\lambda) $ is written as $$H(\lambda) = \sum_{k=0}^\infty (2k+n)|\lambda| P_k(\lambda)$$ where $ P_k(\lambda) $ are the Hermite projection operators, then (see [@TRMI Proposition 4.1]) $$\widehat{\mu}_r(\lambda) = \sum_{k=0}^\infty \psi_k^{n-1}(\sqrt{|\lambda|}r) P_k(\lambda),$$ where for any $ \delta > -1 $ the normalised Laguerre functions are defined by $$\label{eq:LagF}
\psi_k^{\delta}(r)=\frac{\Gamma(k+1)\Gamma(\delta+1)}{\Gamma(k+\delta+1)}L_k^{\delta}\Big(\frac12r^2\Big)e^{-\frac14r^2}.$$ In the above definition $ L_k^\delta(r) $ stands for the Laguerre polynomials of type $ \delta.$ Thus we have the relation $$\label{ArHe}
\widehat{A_rf}(\lambda) = \widehat{f}(\lambda) \sum_{k=0}^\infty \psi_k^{n-1}(\sqrt{|\lambda|}r) P_k(\lambda),$$ which is the analogue of in our situation. Thus, as in the Euclidean case, the spherical mean value operators $ A_r $ are (right) Fourier multipliers on the Heisenberg group. We now proceed to rewrite in terms of Laguerre expansions, which is more suitable for defining an analytic family of operators containing the spherical means.
The irreducible unitary representations $ \pi_\lambda $ admit the factorisation $ \pi_\lambda(z,t) = e^{i\lambda t} \pi_\lambda(z,0) $ and hence we can write the Fourier transform as $$\widehat{f}(\lambda) = \int_{{{\mathbb C}}^n} f^\lambda(z) \pi_\lambda(z,0)\, dz,$$ where for a function $f$ on ${\mathbb H}^n$, $f^{\lambda}(z)$ stands for the partial inverse Fourier transform $$f^{\lambda}(z)=\int_{-\infty}^{\infty}e^{i\lambda t}f(z,t)\,dt.$$ We now make use of the special Hermite expansion of the function $ f^\lambda $, which can be put in a compact form as follows. Let $ \varphi_k^{\lambda}(z)=L_k^{n-1}\big(\frac12|\lambda||z|^2\big)e^{-\frac14|\lambda||z|^2}$ stand for the Laguerre functions of type $ (n-1) $ on $ {{\mathbb C}}^n.$ The $\lambda$-twisted convolution $f^{\lambda}\ast_{\lambda}\varphi_k^{\lambda}(z)$ is then defined by $$f^{\lambda}\ast_{\lambda}\varphi_k^{\lambda}(z)=\int_{{{\mathbb C}}^n}f^{\lambda}(z-w)\varphi_k^{\lambda}(w)e^{i\frac{\lambda}{2}{\operatorname{Im}}z\cdot \overline{w}}\,dw.$$ It is well known that one has the expansion (see [@STH Chapter 3, proof of Theorem 3.5.6]) $$f^\lambda(z) = (2\pi)^{-n} |\lambda|^n \sum_{k=0}^\infty f^{\lambda}\ast_{\lambda}\varphi_k^{\lambda}(z),$$ which leads to the formula (see [@STH Theorem 2.1.1]) $$f(z,t) = (2\pi)^{-n-1} \int_{-\infty}^\infty e^{-i\lambda t} \Big( \sum_{k=0}^\infty f^{\lambda}\ast_{\lambda}\varphi_k^{\lambda}(z)\Big) |\lambda|^n d\lambda$$ Applying this to $f\ast \mu_r$ we have the formula $$f\ast \mu_r(z,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-i\lambda t}f^{\lambda}\ast_{\lambda}\mu_r(z) \,d\lambda$$ where we used the fact that $(f\ast \mu_r)^{\lambda}(z)=f^{\lambda}\ast_{\lambda}\mu_r(z).$ It can be shown that [@TRMI Theorem 4.1], [@NeT Proof of Proposition 6.1], $$f^{\lambda}\ast_{\lambda}\mu_r(z)= (2\pi)^{-n} |\lambda|^n \sum_{k=0}^{\infty}\frac{k!(n-1)!}{(k+n-1)!}\varphi_k^{\lambda}(r)f^{\lambda}\ast_{\lambda}\varphi_k^{\lambda}(z),$$ leading to the expansion (see [@NeT; @NaT]) $$\label{eq:expression}
A_rf(z,t)=(2\pi)^{-n-1}\int_{-\infty}^{\infty}e^{-i\lambda t}\Big(\sum_{k=0}^{\infty}\psi_k^{n-1}(\sqrt{|\lambda|}r)f^{\lambda}\ast_{\lambda}\varphi_k^{\lambda}(z)\Big)|\lambda|^n\,d\lambda.$$ By replacing $ \psi_k^{n-1} $ by $ \psi_k^\delta $ we get the family of operators taking $ f $ into $$(2\pi)^{-n-1}\sum_{k=0}^{\infty}\int_{-\infty}^{\infty}e^{-i\lambda t}\psi_k^{\delta}(\sqrt{|\lambda|}r)f^{\lambda}\ast_{\lambda}\varphi_k^{\lambda}(z)|\lambda|^n\, d\lambda.$$ We make use of these operators in studying the $ L^p $ improving properties of the spherical mean value operator.
In what follows we require sharp estimates on the normalised Laguerre functions given in . It is convenient to express $\psi_k^{\delta}(r)$ in terms of the standard Laguerre functions $$\mathcal{L}_k^{\delta}(r)=\Big(\frac{\Gamma(k+1)\Gamma(\delta+1)}{\Gamma(k+\delta+1)}\Big)^{\frac12}L_k^{\delta}(r)e^{-\frac12r}r^{\delta/2}$$ which form an orthonormal system in $L^2((0,\infty),dr)$. In terms of $\mathcal{L}_k^{\delta}(r),$ we have $$\psi_k^{\delta}(r)=2^{\delta}\Big(\frac{\Gamma(k+1)\Gamma(\delta+1)}{\Gamma(k+\delta+1)}\Big)^{\frac12}r^{-\delta}\mathcal{L}_k^{\delta}\Big(\frac12r^2\Big).$$ Asymptotic properties of $\mathcal{L}_k^{\delta}(r)$ are well known in the literature and we have the following result, see [@T Lemma 1.5.3] (actually, the estimates in Lemma \[lem:T\] below are sharp, see [@M Section 2] and [@Mu Section 7]).
\[lem:T\] For $\delta>-1$, we have the following: $$|\mathcal{L}_k^{\delta}(r)|\le C\begin{cases}(kr)^{\delta/2}, &0\le r\le \frac{1}{k}\\
(kr)^{-\frac14}, &\frac{1}{k}\le r\le \frac{k}{2}\\
k^{-\frac14}(k^{\frac13}+|k-r|)^{-\frac14}, &\frac{k}{2}\le r\le \frac{3k}{2}\\
e^{-\gamma r}, & r\ge \frac{3k}{2},
\end{cases}$$ where $\gamma>0$ is a fixed constant.
We can now rewrite the above estimates of $\mathcal{L}_k^{\delta}$ in terms of estimates for the normalised Laguerre functions $\psi_k^{\delta}$.
\[lem:uniform\] For any $\delta\ge-\frac13,$ we have the uniform estimates $$\sup_k|\psi_k^{\delta}(\sqrt{|\lambda|})|\le C\begin{cases}
1, &\text{ if } |\lambda|\le 1\\
|\lambda|^{-\delta-\frac13}, &\text{ if } |\lambda|> 1.
\end{cases}$$
Since $\frac{\Gamma(k+1)\Gamma(\delta+1)}{\Gamma(k+\delta+1)}\le Ck^{-\delta}$ we need to bound $(k|\lambda|)^{-\delta/2}\mathcal{L}_k^{\delta}\big(\frac12|\lambda|\big)$ for $|\lambda|\ge 1$. When $1\le \frac12|\lambda|\le \frac{k}{2}$ we have the estimate $$|\psi_k^{\delta}(\sqrt{|\lambda|})|\le C(k|\lambda|)^{-\delta/2-1/4}.$$ From here, since $\delta+\frac12\ge 0$, $\lambda^2\le k|\lambda|$, we get $$|\psi_k^{\delta}(\sqrt{|\lambda|})|\le C|\lambda|^{-\delta-1/2}.$$ When $\frac{k}{2}\le \frac12|\lambda|\le \frac{3k}{2}$, $|\lambda|$ is comparable to $k$ and hence we have $$|\psi_k^{\delta}(\sqrt{|\lambda|})|\le C(k|\lambda|)^{-\delta/2}k^{-\frac14}k^{-\frac{1}{12}}\le C|\lambda|^{-\delta-\frac13}.$$ On the region $|\lambda|\ge \frac{3k}{2}$ we have exponential decay. Finally, the estimate $\sup_k|\psi_k^{\delta}(\sqrt{|\lambda|})|\le C$ for $0\le |\lambda|\le1$ is immediate, in view of Lemma \[lem:T\]. With this we prove the lemma.
\[lem:uniform2\] For any $\delta\ge \frac12$ and $|\lambda|\ge1$ we have $$\sup_k(k|\lambda|)^{\frac12}|\psi_k^{\delta}(\sqrt{|\lambda|})|\le C|\lambda|^{-\delta+\frac23}.$$
As in the proof of Lemma \[lem:uniform\], in the range $1\le \frac12|\lambda|\le \frac{k}{2}$, $$(k|\lambda|)^{\frac12}|\psi_k^{\delta}(\sqrt{|\lambda|})|\le C(k|\lambda|)^{-\delta/2+1/4}\le C|\lambda|^{-\delta+\frac12}$$ as $\delta\ge\frac12$. When $\frac{k}{2}\le \frac12|\lambda|\le \frac{3k}{2}$, as before $$(k|\lambda|)^{\frac12}|\psi_k^{\delta}(\sqrt{|\lambda|})|\le C|\lambda|^{-\delta+1-\frac13}=C|\lambda|^{-\delta+\frac23}.$$
The Laguerre functions $\psi_k^{\delta}$ can be defined for all values of $\delta>-1$, even for complex $\delta$ with ${\operatorname{Re}}\delta>-1$ and we would like to use this fact to embed $A_1$ into an analytic family of operators. With the analytic interpolation in mind we define $$\label{eq:Abeta}
\mathcal{A}^{\beta}f(z,t)=(2\pi)^{-n-1}\int_{-\infty}^{\infty}e^{-i\lambda t}\Big(\sum_{k=0}^{\infty}\psi_k^{\beta+n-1}(\sqrt{|\lambda|})f^{\lambda}\ast_{\lambda}\varphi_k^{\lambda}(z)\Big)|\lambda|^n\,d\lambda,$$ for ${\operatorname{Re}}(\beta+n-1)>-1$. Note that for $\beta=0$ we recover $A_1$, thus $A_1=\mathcal{A}^0$. We will use the following relation between Laguerre polynomials of different types in order to express $\mathcal{A}^{\beta}$ in terms of $A_1$ (see [@PBM (2.19.2.12)]) $$\label{eq:connection}
L_k^{\mu+\nu}(r)=\frac{\Gamma(k+\mu+\nu+1)}{\Gamma(\nu)\Gamma(k+\mu+1)}\int_0^1t^{\mu}(1-t)^{\nu-1}L_k^{\mu}(rt)\,dt,$$ valid for ${\operatorname{Re}}\mu>-1$ and ${\operatorname{Re}}\nu>0$. We define $$\label{eq:Poisson}
P_rf(z,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-i\lambda t}e^{-\frac14|\lambda|r}f^{\lambda}(z)\,d\lambda$$ to be the Poisson integral of $f$ in the $t$-variable. We see that for ${\operatorname{Re}}\beta>0$, $\mathcal{A}^{\beta}$ is given by the following representation.
\[lem:family\] Let ${\operatorname{Re}}\beta>0$. The operator $\mathcal{A}^{\beta}$ is given by the formula $$\mathcal{A}^{\beta}f(z,t)=2\frac{\Gamma(\beta+n)}{\Gamma(\beta)\Gamma(n)}\int_0^1r^{2n-1}(1-r^2)^{\beta-1}P_{1-r^2}f\ast \mu_r(z,t)\,dr.$$
In view of , it is enough to verify $$\begin{gathered}
2\frac{\Gamma(\beta+n)}{\Gamma(\beta)\Gamma(n)}\int_0^1r^{2n-1}(1-r^2)^{\beta-1}P_{1-r^2}f\ast \mu_r(z,t)\,dr\\
=(2\pi)^{-n-1}\int_{-\infty}^{\infty}e^{-i\lambda t}\Big(\sum_{k=0}^{\infty}\psi_k^{\beta+n-1}(\sqrt{|\lambda|})f^{\lambda}\ast_{\lambda}\varphi_k^{\lambda}(z)\Big)|\lambda|^n\,d\lambda.\end{gathered}$$ Note that the left hand side of the above equation is well defined only for ${\operatorname{Re}}\beta>0$ whereas the right hand side makes sense for all ${\operatorname{Re}}\beta>-n$. We can thus think of the right hand side as an analytic continuation of the left hand side. In view of , the Poisson integral $P_rf$ of $f$ in the $t$-variable can be written as $$(P_rf)^{\lambda}(z)=e^{-\frac14|\lambda|r}f^{\lambda}(z).$$ Then, by we consider the equation $$P_{1-r^2}f\ast \mu_r(z,t)=(2\pi)^{-n-1}\sum_{k=0}^{\infty}\int_{-\infty}^{\infty}e^{-i\lambda t}\psi_k^{n-1}(\sqrt{|\lambda|}r)e^{-\frac14|\lambda|(1-r^2)}f^{\lambda}\ast_{\lambda}\varphi_k^{\lambda}(z)|\lambda|^n\,d\lambda.$$ Integrating the above equation against $r^{2n-1}(1-r^2)^{\beta-1}\,dr$, we obtain $$\int_0^1r^{2n-1}(1-r^2)^{\beta-1}P_{1-r^2}f\ast \mu_r(z,t)\,dr=(2\pi)^{-n-1}\sum_{k=0}^{\infty}\int_{-\infty}^{\infty}e^{-i\lambda t}\rho_k(\sqrt{|\lambda|})f^{\lambda}\ast_{\lambda}\varphi_k^{\lambda}(z)|\lambda|^n\,d\lambda,$$ where $$\label{eq:rho}
\rho_k(\sqrt{|\lambda|})=\int_0^1r^{2n-1}(1-r^2)^{\beta-1}\psi_k^{n-1}(\sqrt{|\lambda|}r)e^{-\frac14|\lambda|(1-r^2)}\,dr.$$ Recalling the definition of $\psi_k^{n-1}$ given in we have $$\rho_k(\sqrt{|\lambda|})=\frac{\Gamma(k+1)\Gamma(n)}{\Gamma(k+n)}\int_0^1r^{2n-1}(1-r^2)^{\beta-1}L_k^{n-1}\Big(\frac12r^2|\lambda|\Big)e^{-\frac14|\lambda|}\,dr.$$ We now use the formula . First we make a change of variables $t\to s^2$ and then choose $\mu=n-1$ and $\nu=\beta$, so that $$\label{eq:rho2}
\rho_k(\sqrt{|\lambda|})=\frac{\Gamma(k+1)\Gamma(n)}{\Gamma(k+n)}\frac12\frac{\Gamma(\beta)\Gamma(k+n)}{\Gamma(k+n+\beta)}e^{-\frac14|\lambda|}L_k^{n+\beta-1}\Big(\frac{|\lambda|}{2}\Big)=\frac12\frac{\Gamma(\beta)\Gamma(n)}{\Gamma(\beta+n)}\psi_k^{n+\beta-1}(\sqrt{|\lambda|}).$$ The proof is complete.
In particular, from the computations in the proof of Lemma \[lem:family\], we infer the following identity.
\[cor:ident\] Let ${\operatorname{Re}}\beta>0$ and $\alpha>-1$. Then, for $t>0$, $$\psi_k^{\alpha+\beta}(t)=2\frac{\Gamma(\beta+\alpha+1)}{\Gamma(\beta)\Gamma(\alpha+1)}\int_0^1s^{\alpha}(1-s)^{\beta-1}\psi_k^{\alpha}(t\sqrt{s})e^{-\frac14t^2(1-s)}\,ds.$$
The identity follows from and , after a change of variable.
We slightly modify the family in Lemma \[lem:family\] and define a new family $T_{\beta}$. The modification becomes necessary since we want our family to have some $ L^p $ improving property for large values of $ \beta.$ The original operator $ \mathcal{A}^\beta $ remains as convolution with a distribution supported on $ {{\mathbb C}}^n \times \{0\} $ however large $ \beta$ is. This is in sharp contrast with the Euclidean case, see [@S]. As we will see below the modified family of operators $ T_\beta$ has a better behaviour for $ \beta \ge 1.$ Let $k_\beta(t)=\frac{1}{\Gamma(\beta)}t_+^{\beta-1}e^{-t}$, ${\operatorname{Re}}\beta>0$, which defines a family of distributions on ${\mathbb R}$ and $\lim_{\beta\to 0} k_\beta(t)=\delta_0$, the Dirac distribution at $0$. Given a function $f$ on ${\mathbb H}^n$ and $\varphi$ on ${\mathbb R}$ we use the notation $f\ast_3 \varphi$ to stand for the convolution in the central variable: $$f\ast_3 \varphi(z,t)=\int_{-\infty}^{\infty}f(z,t-s)\varphi(s)\,ds.$$ Thus we note that $P_{1-r^2}f(z,t)=f\ast_3 p_{1-r^2}(z,t)$ where $p_{1-r^2}$ is the usual Poisson kernel in the one dimensional variable $t$, associated to $P_{1-r^2}$. In fact, $p_r(t) $ is defined by the relation $ \int_{-\infty}^\infty e^{i \lambda t} p_r(t) dt = e^{-\frac{1}{4}r|\lambda|} $ and it is explicitly known: $ p_r(t) = c r(r^2+16 t^2)^{-1}$ for some constant $ c>0,$ see for example [@SW]. With the above notation we define the new family by $$T_{\beta}f(z,t)=\frac{\Gamma(\beta+n)}{\Gamma(\beta)\Gamma(n)}\int_0^1r^{2n-1}(1-r^2)^{\beta-1}P_{1-r^2}(f\ast_3 k_\beta)\ast \mu_r(z,t)\,dr.$$ In other words $$T_{\beta}f=\mathcal{A}^{\beta}(f\ast_3 k_\beta).$$
The operator $T_{\beta}f$ has the explicit expansion $$T_{\beta}f(z,t)=(2\pi)^{-n-1}\int_{-\infty}^{\infty}e^{-i\lambda t}(1-i\lambda)^{-\beta}\Big(\sum_{k=0}^{\infty}\psi_k^{\beta+n-1}(\sqrt{|\lambda|})f^{\lambda}\ast_{\lambda}\varphi_k^{\lambda}(z)\Big)|\lambda|^n\,d\lambda.$$
The statement follows from Lemma \[lem:family\], , and from the fact $$\int_{-\infty}^{\infty}e^{i\lambda t}k_\beta(t)\,dt=\frac{1}{\Gamma(\beta)}\int_0^{\infty}e^{i\lambda t}t^{\beta-1}e^{-t}\,dt=(1-i\lambda)^{-\beta}.$$ This can be verified by looking at the function $$F(\beta,z)=\frac{1}{\Gamma(\beta)}\int_0^{\infty}t^{\beta-1}e^{-tz}\,dt$$ defined and holomorphic for ${\operatorname{Re}}\beta>0$, ${\operatorname{Re}}z>0$. Indeed, when $z$, with ${\operatorname{Re}}z>0$, is fixed, we have the relation $F(\beta,z)=zF(\beta+1,z)$ which allows us to holomorphically extend $F(\beta,z)$ in the $\beta$ variable. It is clear that when $z>0$, $F(\beta,z)=z^{-\beta}$, which allows us to conclude that the Fourier transform of $ k_\beta$ at $\lambda$ is given by $(1-i\lambda)^{-\beta}$, as claimed.
We will show that when $\beta=1+i\gamma$, $T_{\beta}$ is bounded from $L^p({\mathbb H}^n)$ into $L^{\infty}({\mathbb H}^n)$ for any $p>1$, and for certain negative values of $\beta$, $T_{\beta}$ is bounded on $L^2({\mathbb H}^n)$. We can then use analytic interpolation to obtain a result for $T_0=\mathcal{A}^0=A_1$.
\[prop:Linfty\] For any $\delta>0$, $\gamma\in {\mathbb R}$ $$\|T_{1+i\gamma}f\|_{\infty}\le C_1(\gamma)\|f\|_{1+\delta},$$ where $C_1(\gamma)$ is of admissible growth.
Without loss of generality we can assume that $f\ge0$. For $\beta=1+i\gamma$ it follows that $$|T_{1+i\gamma}f(z,t)|\le \frac{|\Gamma(1+i\gamma+n)|}{|\Gamma(1+i\gamma)|^2\Gamma(n)}\int_0^1r^{2n-1}P_{1-r^2}(f\ast_3 \varphi)\ast \mu_r(z,t)\,dr$$ where $\varphi(t)=e^{-t}\chi_{(0,\infty)}(t)$. Since $\varphi\ge0$ it follows that $$P_{1-r^2}(f\ast_3 \varphi)=\varphi\ast_3p_{1-r^2}\ast_3f\le \varphi\ast_3\Lambda f$$ where $\Lambda f$ is the Hardy–Littlewood maximal function in the $t$-variable. In proving the above we have used the well known fact that $\sup_{r>0}|P_rg(t)|\le C\Lambda g(t)$ for any $g$ on ${\mathbb R}$. Thus we have the estimate $$|T_{1+i\gamma}f(z,t)|\le C_1(\gamma)\int_0^1 (\Lambda f\ast_3 \varphi)\ast \mu_r(z,t)r^{2n-1}\,dr.$$ Now we make the following observation: Suppose $K(z,t)=k(|z|)\varphi(t)$. Then $$f\ast K(z,t)=\int_0^{\infty}(f\ast_3 \varphi)\ast \mu_r(z,t)k(r)r^{2n-1}\,dr,$$ which can be verified by recalling the definition of the spherical means $f\ast\mu_r(z,t)$ in and integrating in polar coordinates. This gives us $$|T_{1+i\gamma}f(z,t)|\le C_1(\gamma)\Lambda f\ast K(z,t)$$ where $K(z,t)=\chi_{|z|\le 1}(z)\varphi(t)$. As $\Lambda f\in L^{1+\delta}({\mathbb H}^n)$ and $K\in L^q({\mathbb H}^n)$ for any $q\ge 1$, by Hölder we get $$\|T_{1+i\gamma}f\|_{\infty}\le C_1(\gamma)\|\Lambda f\|_{1+\delta}\le C_1(\gamma)\|f\|_{1+\delta}.$$
In the next proposition we show that $T_{\beta}$ is bounded on $L^2({\mathbb H}^n)$ for some $\beta<0$. It is possible to sharpen the following result, see Subsection \[rem:sharp\], but for the sake of simplicity (and to mimic the corresponding Euclidean result), we consider only the case ${\operatorname{Re}}\beta \ge -\frac{(n-1)}{2}$.
\[prop:L2\] Assume that $n\ge1$ and $\beta\ge -\frac{(n-1)}{2}$. Then for any $\gamma\in {\mathbb R}$ $$\|T_{\beta+i\gamma}f\|_{2}\le C_2(\gamma)\|f\|_{2}.$$
We only have to check that the functions $$(1+\lambda^2)^{-\beta/2}|\psi_k^{\beta+i\gamma+n-1}(\sqrt{|\lambda|})|\le C_2(\gamma)$$ where $C_2(\gamma)$ is independent of $K$ and $\lambda$. When $\gamma=0$, it follows from the estimates of Lemma \[lem:uniform\] that $$(1+\lambda^2)^{-\beta/2}|\psi_k^{\beta+n-1}(\sqrt{|\lambda|})|\le C|\lambda|^{-\beta}|\lambda|^{-\beta-(n-1)-\frac13}$$ for $|\lambda|\ge 1$, which is clearly bounded for $\beta\ge -\frac{n-1}{2}$ (actually, it is bounded for $\beta\ge -\frac{n}{2}+\frac13$, so it is for $\beta\ge -\frac{n}{2}+\frac12$). For $\gamma \neq 0$ we can express $\psi_k^{\beta+i\gamma+n-1}(\sqrt{|\lambda|})$ in terms of $\psi_k^{\beta-\varepsilon+n-1}(\sqrt{|\lambda|})$ for a small enough $\varepsilon>0$ and obtain the same estimate. Indeed, by Corollary \[cor:ident\] and using the asymptotic formula $|\Gamma(\mu+iv)|\sim \sqrt{2\pi}|v|^{\mu-1/2}e^{-\pi|v|/2}$, as $v\to \infty$ (see for instance [@SW p. 281 bottom note]) $$\begin{aligned}
|\psi_k^{\beta+i\gamma+n-1}(\sqrt{|\lambda|})|&=\Big|2\frac{\Gamma(\beta+i\gamma+n)}{\Gamma(\varepsilon+i\gamma)\Gamma(\beta-\varepsilon+n)}\\
&\quad \times \int_0^1s^{\beta-\varepsilon+n-1}(1-s)^{\varepsilon+i\gamma-1}\psi_k^{\beta-\varepsilon+n-1}(\sqrt{|\lambda|s})e^{-\frac14|\lambda|(1-s)}\,ds\Big|\\
&\lesssim \frac{|\gamma|^{\beta+n-1/2}}{|\gamma|^{\varepsilon-1/2}}\Big|\int_0^1s^{\beta-\varepsilon+n-1}(1-s)^{\varepsilon+i\gamma-1}\psi_k^{\beta-\varepsilon+n-1}(\sqrt{|\lambda|s})e^{-\frac14|\lambda|(1-s)}\,ds\Big|,\end{aligned}$$ where the constant depends on $\beta$. Now, by the estimate for $\psi_k^{\delta}$ in Lemma \[lem:uniform\] and the integrability of the function $s^{\beta-\varepsilon+n-1}(1-s)^{\varepsilon+i\gamma-1}$ we have $$(1+\lambda^2)^{-\beta/2}|\psi_k^{\beta+i\gamma+n-1}(\sqrt{|\lambda|})|\le C|\lambda|^{-\beta}|\gamma|^{\beta+n-1-\varepsilon}|\lambda|^{-(\beta+n-1-\varepsilon)-1/3}.$$ For $|\lambda|\ge 1$, the above is bounded for $\beta\ge -\frac{n-1}{2}$ (actually, it is bounded for $\beta-\varepsilon\ge -\frac{n}{2}+\frac13$ with $\varepsilon$ small enough, so it is for $\beta\ge -\frac{n}{2}+\frac12$). The proof is complete.
\[thm:Lp\] Assume that $n\ge 2$. Then $A_1:L^p({\mathbb H}^n)\to L^{n+1}({\mathbb H}^n)$ for any $\frac{n+1}{n} < p < (n+1)$.
For $ \frac{n+1}{n} < p < (n+1)$ choose $\delta>0$ such that $p=\frac{(n+1)(1+\delta)}{n+\delta} $, which is possible as $ \frac{1}{n} < \frac{1+\delta}{n+\delta} < 1.$ By considering the analytic family $T_{\alpha(z)}$ where $\alpha(z)=\frac{n-1}{2}(z-1)+z$ with $z=u+iv$, in view of Propositions \[prop:Linfty\] and \[prop:L2\], and interpolation between the endpoints ${\operatorname{Re}}z=0$ and ${\operatorname{Re}}z=1$ we obtain $$T_{\alpha(u)}:L^{p_u}({\mathbb H}^n)\to L^{q_u}({\mathbb H}^n)$$ where $\frac{1}{p_u}=\frac{1-u}{2}+\frac{u}{1+\delta}$ and $\frac{1}{q_u}=\frac{1-u}{2}$. The choice $u=\frac{n-1}{n+1}$ gives $q_u=n+1$ and $p_u=\frac{(n+1)(1+\delta)}{n+\delta}=p$. Since $\alpha\big(\frac{n-1}{n+1}\big)=0$ we obtain the result.
Observe the restriction on the dimension in Theorem \[thm:Lp\], that comes into play due to the restriction (that we imposed, a bit artificially, for cosmetic reasons) on the parameter $\beta$ in Proposition \[prop:L2\]. This is the only place in the $L^p$-improving estimates where the dimensional restriction arises, but we insist that we imposed that. Actually, the results we can obtain concerning $L^p$-improving estimates are sharp and valid for all the dimensions, see Subsection \[rem:sharp\].
\[cor:LpLq\] Assume that $n\ge 2$. Then $$A_1:L^{p}({\mathbb H}^n)\to L^{q}({\mathbb H}^n)$$ whenever $\big(\frac{1}{p},\frac{1}{q}\big)$ lies in the interior of the triangle joining the points $(0,0), (1,1)$ and $\big(\frac{n}{n+1},\frac{1}{n+1}\big)$, as well as the straight line segment joining the points $(0,0), (1,1)$, see $\mathbf{L}_n'$ in Figure \[figurea\].
The result follows from Theorem \[thm:Lp\] after applying Marcinkiewicz interpolation theorem with the obvious estimates $\|A_1f\|_1\le \|f\|_1$ and $\|A_1f\|_{\infty}\le \|f\|_{\infty}$.
![Triangle $\mathbf{L}_n'$ shows the region for $L^p-L^q$ estimates for $A_1$. The dual triangle $\mathbf{L}_n$ is on the left.[]{data-label="figurea"}](FiguresaDual.pdf "fig:") ![Triangle $\mathbf{L}_n'$ shows the region for $L^p-L^q$ estimates for $A_1$. The dual triangle $\mathbf{L}_n$ is on the left.[]{data-label="figurea"}](Figuresa.pdf "fig:")
A sharpened result {#rem:sharp}
------------------
As indicated in the proof of Proposition \[prop:L2\], we could state an enhanced result as follows.
\[prop:L2s\] Assume that $n\ge1$ and $\beta> -\frac{n}{2}+\frac13$. Then for any $\gamma\in {\mathbb R}$ $$\|T_{\beta+i\gamma}f\|_{2}\le C_2(\gamma)\|f\|_{2}.$$
On the other hand, let us consider the following holomorphic function $\alpha(z)$ on the strip $\{z:0\le {\operatorname{Re}}z\le 1\}$, given by $\alpha(z)=\big(\frac{n}{2}-\frac13\big)(z-1+\varepsilon)+z$. We have $\alpha(0)=\big(-\frac{n}{2}+\frac13\big)(1-\varepsilon)$ and $\alpha(1)=1$. Then, $T_{\alpha(z)}$ is an analytic family of linear operators and it was already shown that $T_{1+i\gamma}$ is bounded from $L^{1+\delta}({\mathbb H}^n)$ to $L^{\infty}({\mathbb H}^n)$. Therefore, we can apply Stein’s interpolation theorem. Letting $z=u+iv$, we have $$\alpha(z)=0 \Longleftrightarrow \Big(\frac{n}{2}-\frac13\Big)(u-1+\varepsilon)+u=0 \Longleftrightarrow u=\frac{3n-2}{3n+4}(1-\varepsilon).$$ Since $\varepsilon>0$ is arbitrary, we obtain $$T_{\alpha(u)}:L^{p_u}({\mathbb H}^n)\to L^{q_u}({\mathbb H}^n)$$ where $$\frac{1}{p_u}=\frac{3n+1}{3n+4}-\varepsilon\frac{3n-2}{2(3n+4)},\quad\frac{1}{q_u}=\frac{3+\frac12(3n-2)\varepsilon}{3n+4}.$$ This leads to the following result, the enhanced version of Theorem \[thm:Lp\].
\[prop:L2en\] Assume that $n\ge 1$ and $\varepsilon>0$. Then $A_1:L^{p}({\mathbb H}^n)\to L^{q}({\mathbb H}^n)$ for any $p,q$ such that $$\frac{1}{p}=\frac{3n+1}{3n+4}-\varepsilon\frac{3n-2}{2(3n+4)},\quad\frac{1}{q}=\frac{3+\frac12(3n-2)\varepsilon}{3n+4}.$$
And we deduce the following corollary.
\[cor:LPs\] Assume that $n\ge 1$. Then $$A_1:L^{p}({\mathbb H}^n)\to L^{q}({\mathbb H}^n)$$ whenever $\big(\frac{1}{p},\frac{1}{q}\big)$ lies in the interior of the triangle joining the points $(0,0), (1,1)$ and $\big(\frac{3n+1}{3n+4},\frac{3}{3n+4}\big)$, as well as the straight line segment joining the points $(0,0), (1,1)$, see $\mathbf{S}_n'$ in Figure \[figureaSH\].
![Triangle $\mathbf{S}_n'$ shows the region for sharpened $L^p-L^q$ estimates for $A_1$. The dual triangle $\mathbf{S}_n$ is on the left.[]{data-label="figureaSH"}](FiguresaSHDual.pdf "fig:") ![Triangle $\mathbf{S}_n'$ shows the region for sharpened $L^p-L^q$ estimates for $A_1$. The dual triangle $\mathbf{S}_n$ is on the left.[]{data-label="figureaSH"}](FiguresaSH.pdf "fig:")
The continuity property of the spherical mean value operator {#sec:continuity}
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In the work of Lacey [@Lacey] dealing with the lacunary spherical maximal function on $ {\mathbb R}^n $, the continuity property of the spherical mean value operator plays a crucial role. In the case of the Heisenberg group we require the following continuity property.
\[prop:continuity\] Assume that $ n \geq 2.$ Then for $ y \in {\mathbb H}^n, |y| \leq 1, $ we have $$\| A_1 - A_1 \tau_y \|_{L^2\to L^2} \leq C |y|$$ where $ \tau_y f(x) = f(xy^{-1}) $ is the right translation operator.
For $ f \in L^2({\mathbb H}^n) $ we estimate the $ L^2 $ norm of $ A_1f -A_1(\tau_yf) $ using Plancherel theorem for the Fourier transform on $ {\mathbb H}^n.$ Recall that $ A_1f(x) = f \ast \mu_1(x) $ so that $ \widehat{A_1f}(\lambda) = \widehat{f}(\lambda)\widehat{\mu_1}(\lambda)$, where $ \widehat{\mu_1}(\lambda) $ is explicitly given by $$\widehat{\mu_1}(\lambda) = \sum_{k=0}^\infty \psi_k^{n-1}(\sqrt{|\lambda|}) P_k(\lambda) .$$ We also have $$\widehat{\tau_yf}(\lambda) = \int_{{\mathbb H}^n} f(xy^{-1}) \pi_\lambda(x) dx = \widehat{f}(\lambda)\pi_\lambda(y).$$ Thus by the Plancherel theorem for the Fourier transform we have $$\| A_1f -A_1(\tau_yf)\|_2^2 = c_n \int_{-\infty}^\infty \| \widehat{f}(\lambda)(I-\pi_\lambda(y))\widehat{\mu_1}(\lambda)\|_{\operatorname{HS}}^2 |\lambda |^n d\lambda.$$ Since the space of all Hilbert-Schmidt operators is a two sided ideal in the space of all bounded linear operators, it is enough to estimate the operator norm of $ (I-\pi_\lambda(y))\widehat{\mu_1}(\lambda).$ (For more about Hilbert-Schmidt operators see V. S. Sunder [@VSS].) Again, $ \widehat{\mu_1}(\lambda) $ is self adjoint and $ \pi_\lambda(y)^\ast = \pi_\lambda(y^{-1})$ and so we will estimate $ \widehat{\mu_1}(\lambda) (I-\pi_\lambda(y)).$
We make use of the fact that for every $ \sigma \in U(n) $ there is a unitary operator $ \mu_\lambda(\sigma) $ acting on $ L^2({\mathbb R}^n) $ such that $ \pi_\lambda(\sigma z, t) = \mu_\lambda(\sigma)^\ast \pi_\lambda(z,t) \mu_\lambda(\sigma) $ for all $ (z,t) \in {\mathbb H}^n.$ This follows from the well known Stone–von Neumann theorem which says that any irreducible unitary representation of the Heisenberg group which acts like $ e^{i\lambda t}I $ when restricted to the center is unitarily equivalent to $ \pi_\lambda,$ see [@F]. Actually, $ \mu_\lambda $ has an extension to a double cover of the symplectic group as a unitary representation and is called the metaplectic representation. Given $ y = (z, t) \in {\mathbb H}^n $ we can choose $ \sigma \in U(n) $ such that $ y =(|z|\sigma e_1,t) $ where $ e_1 = (1,0,....,0).$ Thus $$\pi_\lambda(y) = \mu_\lambda(\sigma)^\ast \pi_\lambda(|z|e_1,t)\mu_\lambda(\sigma).$$ Also, it is well known that $ \mu_\lambda(\sigma) $ commutes with functions of the Hermite operator $H(\lambda)$ given by $H(\lambda)=-\Delta+\lambda^2|x|^2$. Since $ \widehat{\mu_1}(\lambda) $ is a function of $ H(\lambda) $ it follows that $$\widehat{\mu_1}(\lambda) (I-\pi_\lambda(z,t)) =\mu_\lambda(\sigma)^\ast \widehat{\mu_1}(\lambda) (I-\pi_\lambda(|z|e_1,t))\mu_\lambda(\sigma).$$ Thus it is enough to estimate the operator norm of $ \widehat{\mu_1}(\lambda) (I-\pi_\lambda(|z|e_1,t)).$ In view of the factorisation $ \pi_\lambda(|z|e_1,t) = \pi_\lambda(|z|e_1,0) \pi_\lambda(0,t) $ we have that $$I-\pi_\lambda(|z|e_1,t)=I- \pi_\lambda(|z|e_1,0) \pi_\lambda(0,t)=(I- \pi_\lambda(0,t))+(I-\pi_\lambda(|z|e_1,0)) \pi_\lambda(0,t)$$ so it suffices to estimate the norms of $ \widehat{\mu_1}(\lambda) (I-\pi_\lambda(0,t)) $ and $\widehat{\mu_1}(\lambda) (I-\pi_\lambda(|z|e_1,0)) \pi_\lambda(0,t) $ separately. Moreover, we only have to estimate them for $ |\lambda| \geq 1$ as they are uniformly bounded for $ |\lambda| \leq 1.$
Assuming $ |\lambda| \geq 1 $ we have, in view of , $$\widehat{\mu_1}(\lambda) (I-\pi_\lambda(0,t)) \varphi(\xi) = (1-e^{i\lambda t}) \widehat{\mu_1}(\lambda)\varphi(\xi), \quad \varphi \in L^2({\mathbb R}^n).$$ By mean value theorem, the operator norm of $ (1-e^{i\lambda t}) \widehat{\mu_1}(\lambda) $ is bounded by $$C |t| |\lambda| \sup_{k} |\psi_k^{n-1}(\sqrt{|\lambda|})| \leq C |t| |\lambda|^{-(n-1)+2/3}$$ where we have used the estimate in Lemma \[lem:uniform\]. Thus for $ n \geq 2 ,$ $$\|\widehat{\mu_1}(\lambda) (I-\pi_\lambda(0,t)) \|_{L^2\to L^2} \leq C |t| \leq C |(z,t)|^2,$$ where $ |x| = |(z,t)| = (|z|^4+t^2)^{1/4}$ is the Koranyi norm on $ {\mathbb H}^n $. In order to estimate $\widehat{\mu_1}(\lambda) (I-\pi_\lambda(|z|e_1,0))$ we recall that $$\pi_\lambda(|z|e_1,0)\varphi(\xi) = e^{i\lambda |z| \xi_1} \varphi(\xi), \quad \varphi \in L^2({\mathbb R}^n).$$ Since we can write $$(1-e^{i\lambda |z| \xi_1}) = -i\lambda |z| \xi_1 \int_0^1 e^{it\lambda |z| \xi_1} dt = \lambda |z| \xi_1 m_\lambda(|z|, \xi)$$ with a bounded function $ m_\lambda(|z|,\xi), $ it is enough to estimate the norm of the operator $ |z| \widehat{\mu_1}(\lambda) M_\lambda $ where $ M_\lambda \varphi (\xi) = \lambda \xi_1 \varphi(\xi).$
Let $ A(\lambda) = \frac{\partial}{\partial \xi_1}+|\lambda| \xi_1 $ and $ A(\lambda)^\ast = -\frac{\partial}{\partial \xi_1}+|\lambda| \xi_1 $ be the annihilation and creation operators, so that we can express $ M_\lambda $ as $ M_\lambda = \frac{1}{2} ( A(\lambda) + A(\lambda)^\ast) .$ Thus it is enough to consider $ |z| \widehat{\mu_1}(\lambda) A(\lambda) $ and $ |z| \widehat{\mu_1}(\lambda) A(\lambda)^\ast. $ Moreover as the Riesz transforms $ H(\lambda)^{-1/2}A(\lambda) $ and $ H(\lambda)^{-1/2}A(\lambda)^\ast $ are bounded on $ L^2({\mathbb R}^n) $ we only need to consider $ |z|\widehat{\mu_1}(\lambda) H(\lambda)^{1/2}.$ But the operator norm of $
\widehat{\mu_1}(\lambda) H(\lambda)^{1/2} $ is given by $\sup_{k} ((2k+n)|\lambda|)^{1/2} |\psi_k^{n-1}(\sqrt{|\lambda|})|$ which, in view of Lemma \[lem:uniform2\], is bounded by $ C |\lambda|^{-(n-1)+2/3}.$ Thus for $ n \geq 2 $ we obtain $$\|\widehat{\mu_1}(\lambda) (I-\pi_\lambda(|z|e_1,0))\|_{L^2\to L^2} \leq C |z| \leq C |(z,t)|.$$ This completes the proof of the proposition.
Observe that the result above is restricted to the case $n\ge2$, and this due to the restriction on the available (and sharp!) estimates for the Laguerre functions in Lemmas \[lem:T\], \[lem:uniform\] and \[lem:uniform2\]. We do not know whether there is a way to reach $n=1$ with our approach.
\[cor:A1\] Assume that $n\ge 2$. Then for $ y \in {\mathbb H}^n$, $|y| \leq 1$, and for $\big(\frac1p,\frac1q\big)$ in the interior of the triangle joining the points $(0,0), (1,1)$ and $\big(\frac{n}{n+1},\frac{1}{n+1}\big)$, we have the inequalities $$\| A_1 - A_1 \tau_y \|_{L^p\to L^q} \leq C |y|^{\eta}$$ for some $0<\eta<1$, where $ \tau_y f(x) = f(xy^{-1}) $ is the right translation operator.
The result follows by Riesz-Thorin interpolation theorem, taking into account Corollary \[cor:LpLq\] and Proposition \[prop:continuity\].
We need a version of the inequality in Corollary \[cor:A1\] when $ A_1 $ is replaced by $A_r.$ This can be easily achieved by making use of the following lemma which expresses $ A_r $ in terms of $ A_1.$ Let $\delta_r\varphi(w,t)=\varphi(rw,r^2t)$ stand for the non-isotropic dilation on $ {\mathbb H}^n.$
\[lem:dilat\] For any $r> 0$ we have $A_rf=\delta_r^{-1}A_1\delta_rf.$
This is just an easy verification. Starting from the expression in we have $$\begin{aligned}
A_rf(z,t)&=(2\pi)^{-n-1}\int_{-\infty}^{\infty}e^{-i\lambda t}\Big(\sum_{k=0}^{\infty}\psi_k^{n-1}(\sqrt{|\lambda|}r)f^{\lambda}\ast_{\lambda}\varphi_k^{\lambda}(z)\Big)|\lambda|^n\,d\lambda\\
&=(2\pi)^{-n-1}r^{-2n-2}\int_{-\infty}^{\infty}e^{-i\frac{\lambda}{r^2} t}\Big(\sum_{k=0}^{\infty}\psi_k^{n-1}(\sqrt{|\lambda|})f^{\lambda/r^2}\ast_{\lambda/r^2}\varphi_k^{\lambda/r^2}(z)\Big) |\lambda|^n\,d\lambda.\end{aligned}$$ In view of the relation $$f^{\lambda/r^2}(rw)=\int_{-\infty}^{\infty}f(rw,t)e^{i\lambda/r^2t}\,dt=r^2\int_{-\infty}^{\infty}f(rw,r^2t)e^{i\lambda t}\,dt$$ we make the following simple computation: $$\begin{aligned}
f^{\lambda/r^2}\ast_{\lambda/r^2}\varphi_k^{\lambda/r^2}(z)&=\int_{{{\mathbb C}}^n}f^{\lambda/r^2}(w)\varphi_k^{\lambda/r^2}(z-w)e^{-i\frac{\lambda}{r^2} {\operatorname{Im}}z\cdot \bar{w}}\,dw\\
&=\int_{{{\mathbb C}}^n}f^{\lambda/r^2}(rw)\varphi_k^{\lambda}(z/r-w)e^{-i\frac{\lambda}{r^2} {\operatorname{Im}}\frac{z}{r}\cdot \bar{w}}r^{2n}\,dw\\
&=r^{2+2n}\int_{{{\mathbb C}}^n}(\delta_rf)^{\lambda}(w)\varphi_k^{\lambda}(z/r-w)e^{-i\frac{\lambda}{r^2} {\operatorname{Im}}\frac{z}{r}\cdot \bar{w}}r^{2n}\,dw\\
&=r^{2+2n}(\delta_rf\ast_{\lambda}\varphi_k^{\lambda})(z/r).\end{aligned}$$ Therefore, we have $$\begin{aligned}
A_rf(z,t)&=(2\pi)^{-n-1}\int_{-\infty}^{\infty}e^{-i\frac{\lambda}{r^2} t}\Big(\sum_{k=0}^{\infty}\psi_k^{n-1}(\sqrt{|\lambda|})(\delta_rf\ast_{\lambda}\varphi_k^{\lambda})(z/r)\Big)|\lambda|^nd\lambda=A_1(\delta_rf)\Big(\frac{z}{r},\frac{t}{r^2}\Big),\end{aligned}$$ which proves the stated result.
\[cor:dilat2\] Assume that $n\ge 2$. Then for $ y \in {\mathbb H}^n$, $|y| \leq 1$, and for $\big(\frac1p,\frac1q\big)$ in the interior of the triangle joining the points $(0,0), (1,1)$ and $\big(\frac{n}{n+1},\frac{1}{n+1}\big),$ we have the inequality $$\| A_r - A_r \tau_y \|_{L^p\to L^q} \leq C r^{-\eta} |y|^{\eta} r^{(2n+2)(\frac{1}{q}-\frac{1}{p})}$$ for some $ \eta >0$.
Observe that $ \delta_r( \tau_y f) = \tau_{\delta_r^{-1}y}(\delta_r f )$, which follows from the fact that $ \delta_r : {\mathbb H}^n \rightarrow {\mathbb H}^n $ is an automorphism. The corollary follows from Corollary \[cor:A1\], Lemma \[lem:dilat\] and the fact that $ \|\delta_r f\|_p = r^{-\frac{(2n+2)}{p}}$ for any $ 1 \leq p < \infty.$
A sharpened continuity property {#sub:sharpenc}
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By using Corollary \[cor:LPs\] instead of Corollary \[cor:LpLq\], we could obtain a sharpened version of Corollary \[cor:A1\], so that we indeed can obtain the following.
\[cor:dilat2s\] Assume that $n\ge 2$. Then for $ y \in {\mathbb H}^n, |y| \leq 1$, and for $\big(\frac1p,\frac1q\big)$ in the interior of the triangle joining the points $(0,0), (1,1)$ and $\big(\frac{3n+1}{3n+4},\frac{3}{3n+4}\big),$ we have the inequality $$\| A_r - A_r \tau_y \|_{L^p\to L^q} \leq C r^{-\eta} |y|^{\eta} r^{(2n+2)(\frac{1}{q}-\frac{1}{p})}$$ for some $ \eta >0$.
Sparse bounds {#sec:sparse}
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Our aim in this section is to prove the sparse bounds for the lacunary spherical maximal function stated in Theorem \[thm:sparse\]. In doing so we closely follow [@Lacey] with suitable modifications that are necessary since we are dealing with a non-commutative set up. We can equip $ {\mathbb H}^n $ with a metric induced by the Koranyi norm which makes it a homogeneous space. On such spaces there is a well defined notion of dyadic cubes and grids with properties similar to their counter parts in the Euclidean setting. However, we need to be careful with the metric we choose since the group is non-commutative.
Recall that the Koranyi norm on $ {\mathbb H}^n $ is defined by $ |x| = |(z,t)| = (|z|^4+t^2)^{1/4}$ which is homogeneous of degree one with respect to the non-isotropic dilations. Since we are considering $ f \ast \mu_r $ it is necessary to work with the left invariant metric $ d_L(x,y) = |x^{-1}y| = d_L(0, x^{-1}y)$ instead of the standard metric $ d(x,y) = |xy^{-1}| = d(0, xy^{-1})$, which is right invariant. The balls and cubes are then defined using $ d_L $. Thus $ B(a,r) = \{ x \in {\mathbb H}^n: |a^{-1}x| < r \}.$ With this definition we note that $ B(a,r) =a\cdot B(0,r) $, a fact which is crucial. This allows us to conclude that when $ f $ is supported in $ B(a,r) $ then $ f \ast \mu_s $ is supported in $ B(a,r+s).$ Indeed, as support of $ \mu_s $ is contained in $ B(0,s) $ we see that $ f \ast \mu_s $ is supported in $ B(a,r)\cdot B(0,s) \subset a\cdot B(0,r)\cdot B(0,s) \subset B(a,r+s).$
Let $\delta\in (0,1)$ with $\delta\le 1/96$. Then there exists a countable set of points $\{z_{\nu}^{k,\alpha} : \nu\in \mathscr{A}_k\}$, $k\in {\mathbb Z}$, $\alpha=1,2,\ldots,N$ and a finite number of dyadic systems $\mathcal{D}^{\alpha}:=\cup_{k\in {\mathbb Z}}\mathcal{D}_k^{\alpha}$, $\mathcal{D}_k^{\alpha}:=\{Q_{\nu}^{k,\alpha}:\nu\in \mathscr{A}_k\}$ such that
1. For every $\alpha\in \{1,2,\ldots, N\}$ and $k\in {\mathbb Z}$ we have
- ${\mathbb H}^n=\cup_{Q\in \mathcal{D}_k^{\alpha}}Q$ (disjoint union).
- $Q,P\in \mathcal{D}^{\alpha}\Rightarrow Q\cap P\in \{\emptyset, P, Q\}$.
- $Q_{\nu}^{k,\alpha}\in \mathcal{D}^{\alpha}\Rightarrow B\big(z_{\nu}^{k,\alpha}, \frac{1}{12}\delta^k\big)\subseteq Q_{\nu}^{k,\alpha}\subseteq B\big(z_{\nu}^{k,\alpha}, 4\delta^k\big)$. In this situation $z_\nu^{k,\alpha} $ is called the center of the cube and the side length $\ell{ (Q_\nu^{k,\alpha})}$ is defined to be $ \delta^k.$
2. For every ball $B=B(x,r)$, there exists a cube $Q_B\in \cup_{\alpha}\mathcal{D}^{\alpha}$ such that $B\subseteq Q_B$ and $\ell(Q_B)=\delta^{k-1}$, where $k$ is the unique integer such that $\delta^{k+1}<r\le \delta^k.$
It follows from Theorem 4.1, the proof of Lemma 4.12, Remark 4.13 and Theorem 2.2 in [@HK], where the choices $c_0=1/4$ and $C_0=2$ in [@HK Theorem 2.2] are made so that the property $(2)$ holds (see [@HK Lemma 4.10]).
We will first prove a lemma that is the analogue of [@Lacey Lemma 2.3].
\[lem:anal23\] Let $f$ and $g$ be supported on a cube $Q$ and let $\ell(Q)=r$. For $\big(\frac1p,\frac1q\big)$ in the interior of the triangle joining the points $(0,1), (1,0)$ and $\big(\frac{n}{n+1},\frac{n}{n+1}\big)$, there holds $$|\langle A_rf-A_r\tau_yf,g\rangle|\lesssim |y/r|^{\eta}|Q|\langle f\rangle_{Q,p}\langle g\rangle_{Q,q}, \quad |y|<r.$$
Observe that continuity property holds for the pair $\big(\frac1p,\frac1{q'}\big)$. By Hölder’s inequality and Corollary \[cor:dilat2\] we have, for $|y|<r$, $$\begin{aligned}
|\langle A_rf-A_r\tau_yf,g\rangle|&\le \|A_rf-A_r\tau_yf\|_{q'}\|g\|_{q}\\
&\le Cr^{(2n+2)(\frac{1}{q'}-\frac{1}{p})}r^{-\eta} |y|^{\eta}\|f\|_p\|g\|_{q}\\
&=Cr^{(2n+2)(\frac{1}{q'}-\frac{1}{p})}r^{-\eta} |y|^{\eta}|Q|^{\frac{1}{p}+\frac{1}{q}}\langle f\rangle_{Q,p}\langle g\rangle_{Q,q}\\
&\lesssim |Q|^{\frac{1}{q'}-\frac{1}{p}}|Q|^{\frac{1}{p}+\frac{1}{q}}r^{-\eta} |y|^{\eta}\langle f\rangle_{Q,p}\langle g\rangle_{Q,q}\\
&\lesssim |Q|r^{-\eta} |y|^{\eta}\langle g\rangle_{Q,p}\langle g\rangle_{Q,q},\end{aligned}$$ as $ |Q|$ is comparable to $ r^{2n+2}$.
\[lem:Hn\] For $Q$ with $\ell(Q)=\delta^k$ we consider $$\mathbb{V}_{Q}=\{P\in \mathcal{D}^1_{k+3}: B(z_{P},\delta^{k+1})\subseteq Q\}.$$ and define $$A_{Q}f=A_{\delta^{k+2}}(f{\bf{1}}_{V_Q})$$ where $ V_{Q}=\cup_{P\in \mathbb{V}_{Q}}P.$ Then for any $ f $ supported in $ Q $ the support of $A_{Q}f $ is also contained in $ Q.$ Moreover, $$A_{\delta^{k+2}}f\le \sum_{\alpha=1}^N\sum_{Q\in \mathcal{D}_k^{\alpha}}A_Q(f).$$
We emphasize that we can take any $\delta$ in Lemma \[lem:Hn\] (and in the rest of the paper), in particular we could take $\delta=2$. In that case we have to do some modifications in defining $A_Q f$, where one has to use the fact that if $\delta<\frac{1}{96}$ then the number of points of the form $2^m$, $m\in {\mathbb Z}$, liying between $\delta^j$ and $\delta^{j+1}$, $j\in {\mathbb Z}$, does not depend on $j$.
Observe that for any $ x \in {\mathbb H}^n$ there exists $P\in \mathcal{D}^1_{k+3}$ such that $ x \in P \subseteq B(z_P, 4\delta^{k+3}).$ Then $ P \subseteq B(z_P,\delta^{k+1})\subseteq Q$ for some $Q$ in $\mathcal{D}_k^{\alpha}$, for some $\alpha$. Therefore $P\in V_{Q}$ and hence $ x \in V_{Q}$. This proves that $ {\mathbb H}^n=\bigcup_{\alpha=1}^N\bigcup_{Q\in \mathcal{D}_k^{\alpha}}\mathbb{V}_Q $ and hence we have $f\le \sum_{\alpha=1}^N\sum_{Q\in \mathcal{D}_k^{\alpha}}f{\bf{1}}_{V_Q}$ and consequently, $A_{\delta^{k+2}}f\le \sum_{\alpha=1}^N\sum_{Q\in \mathcal{D}_k^{\alpha}}A_Qf$. It remains to be proved that $ A_Qf $ is supported in $ Q.$ Now assume that ${\operatorname{supp}}f\subseteq Q$ and recall $A_{\delta^{k+2}}f(x)=f\ast \mu_{\delta^{k+2}}(x)$. Then it is enough to show that ${\operatorname{supp}}A_{\delta^{k+2}}(f{\bf{1}}_P) \subseteq B(z_P, \delta^{k+1})$ for every $P \in \mathbb{V}_Q$. Indeed, $${\operatorname{supp}}(f {\bf{1}}_P) \ast \mu_{\delta^{k+2}}\subseteq ({\operatorname{supp}}(f {\bf{1}}_P))\cdot({\operatorname{supp}}\mu_{\delta^{k+2}})\subseteq z_P\cdot B(0, \delta^{k+2})\cdot B(0, \delta^{k+2})$$ which is contained in $ B(z_P, \delta^{k+1}) \subseteq Q $ by the definition of $ V_Q$. Observe that the above argument fails if we use balls defined by the standard right invariant metric. The lemma is proved.
In view of Lemma \[lem:Hn\] it suffices to prove the sparse bound for each $M_{\mathcal{D}^{\alpha}}f=\sup_{Q\in \mathcal{D}^{\alpha}}A_Qf$ for $\alpha=1,2,\ldots,N$. Let us fix then $\mathcal{D}=\mathcal{D}^{\alpha}$. We will linearise the supremum. Let $\mathcal{Q}$ be the collection of all dyadic subcubes of $Q_0\in \mathcal{D}$. Let us define $$E_Q:=\big\{x\in Q : A_Qf(x)\ge \frac{1}{2} \sup_{P\in \mathcal{Q}} A_Pf(x)\big\}$$ for $Q\in \mathcal{Q}$. Note that for any $ x \in {\mathbb H}^n $ there exists a $Q \in \mathcal{Q}$ such that $$A_Qf(x)\ge \frac{1}{2} \sup_{P\in \mathcal{Q}} A_Pf(x)$$ and hence $ x \in E_Q.$ If we define $B_Q=E_Q\setminus \cup_{Q'\supseteq Q}E_{Q'}$, then $\{B_Q: Q\in \mathcal{Q}\}$ are disjoint and also, $\cup_{Q\in \mathcal{Q}}B_Q=\cup_{Q\in \mathcal{Q}}E_Q$. Let $f,g>0$. Then $$\begin{aligned}
\notag\langle\sup_{Q\in \mathcal{Q}}A_Qf,g\rangle&=\sum_{Q\in \mathcal{Q}}\int_{B_Q}\sup_{P\in \mathcal{Q}}A_Pf(x)g(x)\,dx\\
\notag&\le 2 \sum_{Q\in \mathcal{Q}}\int_{B_Q}A_Qf(x)g(x)\,dx\\
\notag&\le 2 \sum_{Q\in \mathcal{Q}}\int_{{\mathbb H}^n}A_Qf(x)g(x)\mathbf{1}_{B_Q}(x)\,dx\\
&\le 2 \sum_{Q\in \mathcal{Q}}\langle A_Qf,g\mathbf{1}_{B_Q}\rangle.\end{aligned}$$ Defining $g_Q=g\mathbf{1}_{B_Q}$ we will deal with $\sum_{Q\in \mathcal{Q}}\langle A_Qf,g_Q\rangle$.
\[lem:key\] Let $1<p,q<\infty$ be such that $\big(\frac1p,\frac1q\big)$ in the interior of the triangle joining the points $(0,1), (1,0)$ and $\big(\frac{n}{n+1},\frac{n}{n+1}\big)$. Let $f=\mathbf{1}_{F}$ and let $ g $ be any bounded function supported in $ Q_0$. Let $C_0>1$ be a constant and let $\mathcal{Q}$ be a collection of dyadic subcubes of $Q_0\in \mathcal{D}$ for which the following holds $$\label{eq:keyCondf1}
\sup_{Q'\in \mathcal{Q}}\sup_{Q: Q'\subset Q\subset Q_0}\frac{\langle f\rangle_{Q,p}}{\langle f\rangle_{Q_0,p}}<C_0.$$ Then there holds $$\sum_{Q\in \mathcal{Q}}\langle A_Qf,g_Q\rangle\lesssim |Q_0|\langle f\rangle_{Q_0,p}\langle g\rangle_{Q_0,q}.$$
We perform a Calderón–Zygmund decomposition of $f$ at height $2C_0\langle f \rangle_{Q_0,p}$. Let us denote by $\mathcal{B}$ the resulting collection of (maximal) dyadic subcubes of $Q_0$ so that $$\label{eq:stopp}
\langle f\rangle_{Q,p}>2C_0\langle f \rangle_{Q_0,p}.$$ Set $f=g_1+b_1$, where $$b_1=\sum_{P\in \mathcal{B}}(f-\langle f\rangle_{P})\mathbf{1}_P=\sum_{k=s_0+1}^\infty \sum_{P\in \mathcal{B}(k)}(f-\langle f\rangle_{P})\mathbf{1}_P=:\sum_{k=s_0+1}^{\infty}B_{1,k},$$ where $\ell(Q_0)=\delta^{s_0}$ and $\mathcal{B}(k)=\{P\in \mathcal{B}: \ell(P)=\delta^k\}$. Now $$\big|\sum_{Q\in \mathcal{Q}}\langle A_Qf,g_Q\rangle\big|\le \sum_{Q\in \mathcal{Q}}|\langle A_Qg_1,g_Q\rangle|+\sum_{Q\in \mathcal{Q}}|\langle A_Qb_1,g_Q\rangle|.$$ Since $g_1$ is a bounded function then $\|A_Qg_1\|_{\infty}<\infty$. Hence $$\sum_{Q\in \mathcal{Q}}|\langle A_Qg_1,g_Q\rangle|\lesssim \sum_{Q\in \mathcal{Q}}\|g\mathbf{1}_{B_Q}\|_1\lesssim |Q_0|.$$ We now make the following useful observation. For all $Q\in \mathcal{Q}$ and $P\in \mathcal{B}$, if $P\cap Q\neq \emptyset$ then $P$ is properly contained in $ Q$. For otherwise, $Q\subseteq P$ and by the assumption on $ \mathcal{Q}$, we get $\langle f\rangle_{P,p}<C_0\langle f \rangle_{Q_0,p}$. But this contradicts the Calderón–Zygmund decomposition since $ \langle f\rangle_{P,p}>2C_0\langle f \rangle_{Q_0,p}$. Therefore, for any $Q\in \mathcal{Q}$ with $\ell(Q)=\delta^s$ we have $$\langle A_Qb_1,g_Q\rangle=\sum_{k>s}\langle A_QB_{1,k},g_Q\rangle=\sum_{k=1}^{\infty}\langle A_QB_{1,s+k},g_Q\rangle$$ and so $$\big|\sum_{Q\in \mathcal{Q}}\langle A_Qb_1,g_Q\rangle\big|\le \sum_{k=1}^{\infty}\sum_{Q\in \mathcal{Q}}|\langle A_QB_{1,s+k},g_Q\rangle|.$$ By making use of the mean zero property of $b_1$, we see that $$\begin{aligned}
|\langle A_QB_{1,s+k},g_Q\rangle|&=|\langle B_{1,s+k},A_Q^*g_Q\rangle|\\
&= \sum_{P\in B(s+k)}\big|\int_P A_Q^*g_Q(x)B_{1,s+k}(x)\,dx\big|\\
&\le \sum_{P\in B(s+k)}\frac{1}{|P|}\Big|\int_P\int_P\big[A_Q^*g_Q(x)-A_Q^*g_Q(x')\big]B_{1,s+k}(x)\,dx\,dx'\Big|.\end{aligned}$$ In the integral with respect to $ x'$ we make the change of variables $x'=xy^{-1}$ and note that $ P^{-1}x \subset P^{-1}P.$ Since $ P \subset B(z_P, 4\delta^{s+k}) = z_P\cdot B(0,4\delta^{s+k})$ it follows that $ P^{-1} \subset B(0, 4\delta^{s+k})z_P^{-1} $ and hence $ P^{-1}P \subset P_0 = B(0, 8\delta^{s+k}) \subset B(0,\delta^{s+k-1})$ (observe that for the above argument it is important that the balls are defined using the left invariant metric). Thus we have $$\begin{aligned}
|\langle A_QB_{1,s+k},g_Q\rangle|&\le \sum_{P\in B(s+k)}\frac{1}{|P|}\Big|\int_{P^{-1}P}\int_P\big[A_Q^*g_Q(x)-\tau_yA_Q^*g_Q(x)\big]B_{1,s+k}(x)\,dx\,dy\Big|\\
&\lesssim \frac{1}{|P_0|}\int_{P_0}\Big|\int_Qg_Q(x)(A_Q-A_Q\tau_{y^{-1}})B_{1,s+k}(x)\,dx\Big|\,dy\\
&\lesssim \frac{1}{|P_0|}\int_{P_0}\Big|\frac{y}{\ell(Q)}\Big|^{\eta}|Q|\langle B_{1,s+k}\mathbf{1}_Q\rangle_{Q,p}\langle g_Q\rangle_{Q,q}\,dy\\
&\lesssim \frac{\delta^{(q+k-1)\eta}}{\delta^{q\eta}}|Q|\langle B_{1,s+k}\mathbf{1}_Q\rangle_{Q,p}\langle g_Q\rangle_{Q,q}\\
&\lesssim \delta^{k\eta}|Q|\langle B_{1,s+k}\mathbf{1}_Q\rangle_{Q,p}\langle g_Q\rangle_{Q,q},\end{aligned}$$ where we used Lemma \[lem:anal23\] in the third inequality.
Now we will prove $$\label{eq:claimII}
\sum_{Q\in \mathcal{Q}}|Q|\langle B_{1,s+k}\mathbf{1}_Q\rangle_{Q,p}\langle g\mathbf{1}_{B_Q}\rangle_{Q,q}\lesssim |Q_0|\langle f\rangle_{Q_0,p}\langle g\rangle_{Q_0,q},$$ for all $k\ge1$ and for all $1<p,q<\infty$ such that $\big(\frac1p,\frac1q\big)$ are in the interior of the triangle joining the points $(0,1), (1,0)$ and $(1,1)$ (including the segment joining $(0,1)$ and $(1,0)$, excluding the endpoints).
Let us fix the integer $k$. From the definition and it follows that we can dominate $$|B_{1,s+k}|\lesssim \langle f\rangle_{Q_0,p}\mathbf{1}_{E_s}+\mathbf{1}_{F_{1,s}},$$ where $E_s=E_{s,k}$ are pairwise disjoint sets in $Q_0$ as $s$ varies, and $F_{1,s}=F_{1,s,k}$ are pairwise disjoint sets in $F_1$. This produces two terms to control. For the first one, we will show that $$\label{eq:first}
\langle f\rangle_{Q_0,p}\sum_{Q\in \mathcal{Q}}|Q|\langle \mathbf{1}_{E_s}\rangle_{Q,p}\langle g\mathbf{1}_{B_Q}\rangle_{Q,q}\lesssim |Q_0|\langle f\rangle_{Q_0,p}\langle g\rangle_{Q_0,q}.$$
First we consider the case when $1/p+1/q=1,$ i.e. $ p =q'$, for $1<p<\infty$. $$\begin{aligned}
\sum_{Q\in \mathcal{Q}}|Q|\langle \mathbf{1}_{E_s}\rangle_{Q,p}\langle g\mathbf{1}_{B_Q}\rangle_{Q,p'}&=\sum_{Q\in \mathcal{Q}}\Big(\int_Q\mathbf{1}_{E_s} \,dx\Big)^{1/p}\Big(\int_{Q}|g(x)|^{p'}\mathbf{1}_{B_Q}\,dx\Big)^{1/p'}\\
& \le \Big(\sum_{Q\in \mathcal{Q}}\int_Q\mathbf{1}_{E_s} \,dx\Big)^{1/p}\Big(\sum_{Q\in \mathcal{Q}}\int_{Q}|g(x)|^{p'}\mathbf{1}_{B_Q}\,dx\Big)^{1/p'}.\end{aligned}$$ On one hand, from the disjointness of $B_Q$, $$\sum_{Q\in \mathcal{Q}}\int_{Q}|g(x)|^{p'}\mathbf{1}_{B_Q}\,dx = \int_{\cup B_Q}|g(x)|^{p'}\,dx\leq \Big( \frac{1}{|Q_0|}\int_{Q_0}|g(x)|^{p'}\,dx \Big) |Q_0| = |Q_0|\langle g\rangle_{Q_0,p'}^{p'}.$$ On the other hand, as $E_s \cap Q $ are disjoing subsets of $ Q_0 $ we finally obtain $$\sum_{Q\in \mathcal{Q}}\int_Q\mathbf{1}_{E_s} \,dx= \sum_{Q\in \mathcal{Q}}|E_s \cap Q| \leq |Q_0|.$$ Thus the required inequality is proved in the case $ 1/p +1/q =1$. In the case $1/p +1/q =1+\tau>1$, set $1/\widetilde{p}=1/p-\tau$. Then, $1/\widetilde{p}+1/q=1$, and $p<\widetilde{p}$, so that $$\langle \mathbf{1}_{E_s}\rangle_{Q,p}\langle g\mathbf{1}_{B_Q}\rangle_{Q,q}\lesssim \langle \mathbf{1}_{E_s}\rangle_{Q,\widetilde{p}}\langle g\mathbf{1}_{B_Q}\rangle_{Q,q}.$$ Then, follows from the previous case since $1/\widetilde{p}+1/q'=1$.
Concerning the second term, we will show that $$\label{eq:second}
\sum_{Q\in \mathcal{Q}}|Q|\langle \mathbf{1}_{F_{1,s}}\rangle_{Q,p}\langle g\mathbf{1}_{B_Q}\rangle_{Q,q}\lesssim |Q_0|\langle f\rangle_{Q_0,p}\langle g\rangle_{Q_0,q}.$$ Again, the inequality holds in the case of $1/p+1/q=1$. For $1/p+1/q=1+\tau>1$, we define $\widetilde{p}$ as above. By using the stopping condition we have then $$\langle \mathbf{1}_{F_{1,s}}\rangle_{Q,p}\langle g\mathbf{1}_{B_Q}\rangle_{Q,q}\lesssim\langle \mathbf{1}_{F_{1}}\rangle_{Q_0}^{\tau}\langle \mathbf{1}_{F_{1,s}}\rangle_{Q,\widetilde{p}}\langle g\mathbf{1}_{B_Q}\rangle_{Q,q}.$$ From this and by using the previous case, since $1/\widetilde{p}+1/q=1$, we can conclude , and therefore . The proof is complete.
Let us proceed to prove Theorem \[thm:sparse\]. We will state it also here, for the sake of the reading.
\[thm:main\] Assume $ n \geq 2$ and fix $ 0 < \delta < \frac{1}{96}.$ Let $ 1 < p, q < \infty $ be such that $ (\frac{1}{p},\frac{1}{q}) $ belongs to the interior of the triangle joining the points $ (0,1), (1,0) $ and $ (\frac{n}{n+1},\frac{n}{n+1}).$ Then for any pair of compactly supported bounded functions $ (f,g) $ there exists a $ (p,q)$-sparse form such that $ \langle M_{\operatorname{lac}}f, g\rangle \leq C \Lambda_{\mathcal{S},p,q}(f,g).$
Fix a dyadic grid $\mathcal{D}$ and consider the maximal function $$M_{\mathcal{D}}f(x)=\sup_{Q\in \mathcal{D}}|A_{Q}f(x)|.$$ We can assume that $f\ge0$ and supported in $Q_0$ so that $A_Qf=0$ for all large enough cubes. According to this, we will therefore prove the sparse bound for the maximal function $$M_{\mathcal{D}\cap Q_0}f(x)=\sup_{Q\in \mathcal{D}}|A_{Q}f(x)|.$$ From this, it follows that $M_{\operatorname{lac}}$ is bounded by the sum of a finite number of sparse forms. But it is known that there exists one universal dominating sparse form (see for instance [@LM Lemma 4.7] and [@CaR Proposition 2.1]). Namely, given $f,g$, there is a constant $C>1$ and sparse family of dyadic cubes $\mathcal{S}_0$ so that $\sup_{\mathcal{S}}\Lambda_{\mathcal{S},p,q}(f,g)\le C \Lambda_{\mathcal{S}_0,p,q}(f,g)$. This fact, proved in the Euclidean setting, is also valid in our case and we will not enter into details. Therefore, the claimed sparse bound holds.
As explained above, by linearising the supremum it is enough to prove the sparse bound for the sum $$\label{eq:linear}
\sum_{Q\in\mathcal{D}\cap Q_0}\langle A_Qf,g\mathbf{1}_{B_Q}\rangle$$ for the collection of pairwise disjoint $B_{Q}\subset Q$ described just before Lemma \[lem:key\].
Given $1<p,q<\infty$ so that the $L^p$ improving and continuity properties of the spherical means hold for $\big(\frac1p,\frac1q\big)$ (i.e., Corollaries \[cor:LpLq\] and \[cor:dilat2\] hold), we have to produce a sparse family $\mathcal{S}$ of subcubes of $Q_0$ such that $$\langle M_{\mathcal{D}\cap Q_0}f, g\rangle \le 2 \sum_{Q\in\mathcal{D}\cap Q_0}\langle A_Qf,g\mathbf{1}_{B_Q}\rangle \le C\sum_{S\in \mathcal{S}}|S|\langle f\rangle_{S,p}\langle g \rangle_{S,q}$$ where for each $S\in \mathcal{S}$, there exists $F_S\subset S$ with $|F_S|\ge \frac12 |S|$.
We first prove when $f$ is the characteristic function of a set $F\subset Q_0$. Consider the collection $\mathcal{E}_{Q_0}$ of maximal children $P\subset Q_0$ for which $$\langle f\rangle_{P,p}>2\langle f\rangle_{Q_0,p}.$$ Let $E_{Q_0}=\cup_{P\in \mathcal{E}_{Q_0}}$. For a suitable choice of $c_n>1$ we can arrange $|E_{Q_0}|<\frac12|Q_0|$. We let $F_{Q_0}=Q_0\setminus E_{Q_0}$ so that $|F_{Q_0}|\ge \frac12|Q_0|$. We define $$\label{eq:zero}
\mathcal{Q}_0=\{Q\in \mathcal{D}\cap Q_0: Q\cap E_{Q_0}=\emptyset\}.$$ Note that when $Q\in \mathcal{Q}_0$ then $\langle f\rangle_{Q,p}\le 2\langle f\rangle_{Q_0,p}$. For otherwise, if $\langle f\rangle_{Q,p}>2\langle f\rangle_{Q_0,p}$ then there exists $P\in \mathcal{E}_{Q_0}$ such that $P\supset Q$, which is a contradiction. For the same reason, if $Q'\in \mathcal{Q}_0$ and $Q'\subset Q\subset Q_0$ then $\langle f\rangle_{Q,p}\le 2\langle f\rangle_{Q_0,p}$. Thus $$\sup_{Q'\in \mathcal{Q}_0}\sup_{Q:Q'\subset Q\subset Q_0}\langle f\rangle_{Q,p}\le 2\langle f\rangle_{Q_0,p}.$$ Note that for any $Q\in \mathcal{D}\cap Q_0$, either $Q\in \mathcal{Q}_0$ or $Q\subset P$ for some $P\in \mathcal{E}_{Q_0}$. Thus $$\sum_{Q\in \mathcal{D}\cap Q_0}\langle A_Q f,g\mathbf{1}_{B_Q}\rangle=\sum_{Q\in Q_0}\langle A_Q f,g\mathbf{1}_{B_Q}\rangle+\sum_{P\in \mathcal{E}_{Q_0}}\sum_{Q\subset P}\langle A_Q f,g\mathbf{1}_{B_Q}\rangle$$ for any $Q\in \mathcal{Q}_0$, $Q\subset F_{Q_0}$ and hence $$\sum_{Q\in \mathcal{Q}_0}\langle A_Q f,g\mathbf{1}_{B_Q}\rangle=\sum_{Q\in \mathcal{Q}_0}\langle A_Q f,g\mathbf{1}_{F_{Q_0}}\mathbf{1}_{B_Q}\rangle.$$ Applying Lemma \[lem:key\] we obtain $$\sum_{Q\in \mathcal{Q}_0}\langle A_Q f,g\mathbf{1}_{B_Q}\rangle\le C|Q_0|\langle f\rangle_{Q_0,p} \langle g\mathbf{1}_{F_{Q_0}}\rangle_{Q_0,q}.$$
Let $\{P_j\}$ be an enumeration of the cubes in $\mathcal{E}_{Q_0}$. Then the second sum above is given by $$\sum_{j=1}^{\infty}\sum_{Q\in P_j\cap \mathcal{D}}\langle A_Q f,g\mathbf{1}_{B_Q}\rangle.$$ For each $j$ we can repeat the above argument recursively. Putting everything together we get a sparse collection $\mathcal{S}$ for which $$\label{eq:toget}
\sum_{Q\in \mathcal{D}\cap Q_0}\langle A_Q f,g\mathbf{1}_{B_Q}\rangle\le C\sum_{S\in \mathcal{S}}|S| |\langle f\rangle_{S,p} \langle g\mathbf{1}_{F_{S}}\rangle_{S,q}.$$ This proves the result when $f=\mathbf{1}_F$. We pause for a moment to remark that we have actually proved a sparse domination stronger than the one stated in the theorem. However, we are not able to prove such a result for general $ f.$
Now we prove the theorem for any bounded $f\ge0$ supported in $Q_0$. We start as in the case of $f=\mathbf{1}_F$ but now we define $\mathcal{Q}_0$ using stopping conditions on both $f$ and $g$. Thus we let $\mathcal{E}_{Q_0}$ stand for the collection of maximal subcubes $P$ of $Q_0$ for which either $\langle f\rangle_{P,p}>2\langle f\rangle_{Q_0,p}$ or $\langle g\rangle_{P,q}>2\langle g\rangle_{Q_0,q}$. As before, we define $E_{Q_0}=\cup_{P\in \mathcal{E}_{Q_0}}$ and $F_{Q_0}=Q_0\setminus E_{Q_0}$ so that $|F_{Q_0}|\ge \frac12|Q_0|$. We let $$\mathcal{Q}_0=\{Q\in \mathcal{D}\cap Q_0: Q\cap E_{Q_0}=\emptyset\}.$$ Then it follows that $$\sup_{Q'\in \mathcal{Q}_0}\sup_{Q:Q'\subset Q\subset Q_0}\langle f\rangle_{Q,p}\le 2\langle f\rangle_{Q_0,p}$$ and $$\sup_{Q'\in \mathcal{Q}_0}\sup_{Q:Q'\subset Q\subset Q_0}\langle g\rangle_{Q,q}\le 2\langle g\rangle_{Q_0,q}.$$ If we can show that $$\label{eq:star}
\sum_{Q\in \mathcal{Q}_0}\langle A_Q f,g\mathbf{1}_{B_Q}\rangle\le C|Q_0|\langle f\rangle_{Q_0,\rho} \langle g\rangle_{Q_0,q}$$ for some $\rho>p$, then we can proceed as in the case of $f=\mathbf{1}_F$ to get the sparse domination $$\langle M_{\mathcal{D}}f,g\rangle\le C\sum_{S\in \mathcal{S}}|S| |\langle f\rangle_{S,\rho} \langle g\rangle_{S,q}.$$
In order to prove $\eqref{eq:star}$ we make use of the sparse domination already proved for $f=\mathbf{1}_F$. Defining $E_m=\{x\in Q_0: 2^m\le f(x)\le 2^{m+1}\}$ and $f_m=f\mathbf{1}_{E_m}$ we have the decomposition $f=\sum_mf_m$ (since $f$ is bounded it follows that $E_m=\emptyset$ for all $m\ge m_0$ for some $m_0\in {\mathbb Z}$). By applying the sparse domination to $\mathbf{1}_{E_m}$ we obtain the following: $$\begin{aligned}
\sum_{Q\in \mathcal{Q}_0}\langle A_Q f_m,g\mathbf{1}_{B_Q}\rangle&\le 2^{m+1}\sum_{Q\in \mathcal{Q}_0}\langle A_Q\mathbf{1}_{E_m},g\mathbf{1}_{B_Q}\rangle\\
&=2^{m+1}\sum_{Q\in \mathcal{Q}_0}\langle A_Q\mathbf{1}_{E_m},g\mathbf{1}_{F_{Q_0}}\mathbf{1}_{B_Q}\rangle\\
&\le2^{m+1}\sum_{Q\in Q_0\cap \mathcal{D}}\langle A_Q\mathbf{1}_{E_m},g\mathbf{1}_{F_{Q_0}}\mathbf{1}_{B_Q}\rangle\\
&\le C2^{m+1}\sum_{S\in \mathcal{S}_m}|S|\langle \mathbf{1}_{E_m}\rangle_{S,p}\langle g\mathbf{1}_{F_{Q_0}}\rangle_{S,q},\end{aligned}$$ where in the last three lines we used that for any $Q\in \mathcal{Q}_0$, $Q\subset F_{Q_0}$, and . In the above sum, $\langle g\mathbf{1}_{F_{Q_0}}\rangle_{S,q}=0$ unless $S\cap F_{Q_0}\neq \emptyset$. If $S\subset F_{Q_0}$ then by the definition of $\mathcal{Q}_0$ in it follows that $S\in \mathcal{Q}_0$ and $$\langle g\mathbf{1}_{F_{Q_0}}\rangle_{S,q}\le \langle g\rangle_{S,q}\le c_n \langle g\rangle_{Q_0,q}.$$ If $S\cap F_{Q_0}\neq \emptyset$ as well as $S\cap E_{Q_0}\neq \emptyset$ then for some $P\in \mathcal{E}_{{Q}_0}$, $P\subset S$. But then by the maximality of $P$ we have $$\langle g\mathbf{1}_{F_{Q_0}}\rangle_{S,q}\le \langle g\rangle_{S,q}\le 2 \langle g\rangle_{Q_0,q}.$$ Using this we obtain $$\sum_{Q\in \mathcal{Q}_0}\langle A_Q f_m,g\mathbf{1}_{B_Q}\rangle\le C2^{m+1}\langle g\rangle_{Q_0,q}\sum_{S\in \mathcal{S}_m}|S|\langle \mathbf{1}_{E_m}\rangle_{S,p}.$$ By Lemma \[lem:carleson\] we get $$\sum_{Q\in \mathcal{Q}_0}\langle A_Q f_m,g\mathbf{1}_{B_Q}\rangle\le C2^{m+1}\langle g\rangle_{Q_0,q}\langle \mathbf{1}_{E_m}\rangle_{Q_0,\rho_1}|Q_0|$$ for some $\rho_1>p$. As $f=\sum_m f_m$ it follows that $$\sum_{Q\in \mathcal{Q}_0}\langle A_Q f,g\mathbf{1}_{B_Q}\rangle\le C\langle g\rangle_{Q_0,q}|Q_0|\sum_m2^m\langle \mathbf{1}_{E_m}\rangle_{Q_0,\rho_1}.$$ We now claim that (see Lemma \[lem:lorentz\] below) $$\label{eq:twostar}
\sum_m2^m\langle \mathbf{1}_{E_m}\rangle_{Q_0,\rho_1}\le C\|f\|_{L^{\rho_1,1}(Q_0,d\mu)}$$ where $L^{\rho_1,1}(Q_0,d\mu)$ stands for the Lorentz space defined on the measure space $(Q_0,d\mu)$, $d\mu=\frac{1}{|Q_0|}dx.$ We also know that on a probability space, the $L^{\rho_1,1}(Q_0,d\mu)$ norm is dominated by the $L^{\rho}(Q_0,d\mu)$ norm for any $\rho>\rho_1$ (Lemma \[lem:proba\]). Using these two results we see that $$\sum_{Q\in \mathcal{Q}_0}\langle A_Q f,g\mathbf{1}_{B_Q}\rangle\le C\langle g\rangle_{Q_0,q}|Q_0|\langle f\rangle_{Q_0,\rho}.$$ Hence is proved and thus completes the proof of Theorem \[thm:main\].
It remains to prove Lemma \[lem:proba\] and the claim . The first one is a well known fact which we include here for the sake of completeness.
\[lem:proba\] On a probability space $ (X,d\mu)$, $L^p(X,d\mu)\subset L^{r,1}(X,d\mu) \text{ for } p>r.$
Recall that the Lorentz spaces $ L^{p,q}(X,d\mu) $ are defined in terms of the Lorentz norms (see [@GM1]) $$\| f\|_{p,q} =
\begin{cases}
\Big(\int_0^{\infty}\big(t^{\frac{1}{p}}f^*(t)\big)^q\frac{dt}{t}\Big)^{\frac{1}{q}} \quad &\text{ if } q<\infty,\\
\sup_{t>0}t^{\frac{1}{p}}f^*(t)\quad &\text{ if } q=\infty,
\end{cases}$$ where $ f^*(t) $ stands for the non-decreasing rearrangement of $ f.$ When $f\in L^p(X,d\mu)$, as $d\mu$ is a probability measure, we know that the distribution function $df(s)$ of $f$ is bounded by $1$ and hence $f^*(t)=0$ for $t\ge 1$. Now $$\|f\|_{L^{r,1}(X,d\mu)}=\int_0^{\infty}t^{\frac{1}{r}-1}f^*(t)\,dt=\int_0^{1}t^{-\frac{1}{r'}}f^*(t)\,dt.$$ By Hölder’s inequality $$\|f\|_{L^{r,1}(X,d\mu)}\le \Big(\int_0^1 t^{-\frac{p'}{r'}} dt\Big)^{1/p'} \Big(\int_0^1f^*(t)^p\,dt\Big)^{1/p} =C_{r,p}\Big(\int_0^1f^*(t)^p\,dt\Big)^{1/p}$$ where $C_{r,p}<\infty$ since $p'<r'$. This proves the claim since $$\Big(\int_0^1f^*(t)^p\,dt\Big)^{\frac1p}=\|f\|_{L^p(X,d\mu)}.$$
The claim is the content of the next lemma.
\[lem:lorentz\] Let $f=\sum_mf_m$, $f_m=f\mathbf{1}_{E_m}$ where $E_m=\{x\in Q_0: 2^m\le |f(x)|\le 2^{m+1}\}.$ We consider the probability measure $ d\mu = |Q_0|^{-1} dx $ on $ X = Q_0.$ Then for any $ r >1 $ we have $$\sum_m2^m\langle \mathbf{1}_{E_m}\rangle_{Q_0,r}\le C\|f\|_{L^{r,1}(Q_0,d\mu)}.$$
We make use the following definition of the Lorentz norm in terms of $df(s)$: $$\|f\|_{L^{r,1}(X,d\mu)}=\int_0^{\infty}df(s)^{\frac{1}{r}}\,ds.$$ As $df(s)$ is a decreasing function of $s$ we have $$\begin{aligned}
\|f\|_{L^{r,1}(X,d\mu)}&= \sum_m\int_{2^m}^{2^{m+1}}df(s)^{\frac{1}{r}}\,ds\\
&\ge \sum_mdf(2^{m})^{\frac1r}(2^{m+1}-2^m)\\
&=\frac12\sum_mdf(2^m)^{\frac1r}2^m.\end{aligned}$$ As $f_m=f\mathbf{1}_{E_m}$, it follows that $
\mu(E_m) =df(2^m)-df(2^{m+1})\le df(2^m)$ and consequently, $$\sum_m \mu(E_m)^{\frac1r}2^m\le \sum_mdf(2^m)^{\frac1r}2^m\le 2\|f\|_{L^{r,1}(X,d\mu)}.$$ This proves the lemma.
In proving Theorem \[thm:main\] we have made use of the following lemma, which is proved in [@Lacey Proposition 2.19]. We include a proof here for the convenience of the reader.
\[lem:carleson\] Let $\mathcal{S}$ be a collection of sparse subcubes of a fixed dyadic cube $Q_0$ and let $1\le s<t<\infty$. Then, for a bounded function $\phi$, $$\sum_{Q\in \mathcal{S}}\langle \phi\rangle_{Q,s}|Q|\lesssim \langle \phi\rangle_{Q_0,t}|Q_0|.$$
By sparsity, $$\begin{aligned}
\sum_{Q\in \mathcal{S}}\langle \phi\rangle_{Q,s}|Q|&=\sum_{Q\in \mathcal{S}}\langle \phi\rangle_{Q,s}|Q|^{1/t+1/t'}\\
&\le \Big(\sum_{Q\in \mathcal{S}}\langle \phi\rangle_{Q,s}^t|Q|\Big)^{1/t} \Big(\sum_{Q\in \mathcal{S}}|Q|\Big)^{1/t'}\\
&\lesssim \Big(\sum_{Q\in \mathcal{S}}\langle |\phi|^s\rangle_{Q}^{t/s}|Q|\Big)^{1/t}|Q_0|^{1/t'}\\
&\lesssim \|\phi\mathbf{1}_{Q_0}\|_t|Q_0|^{1/t'}.\end{aligned}$$
A sharpened sparse domination {#sub:sharpens}
-----------------------------
Although we have stated Theorem \[thm:main\] for a slightly more restricted region $\mathbf{L}_n$, indeed the sparse domination holds for $\big(\frac1p,\frac1q\big)$ in the interior of the triangle $\mathbf{S}_n$ (of course Lemma \[lem:key\] holds also for the enlarged triangle since Lemma \[lem:anal23\] does and so on). This means, in particular, that Theorem \[thm:main\] is true for the closed triangle $\mathbf{L}_n$.
\[thm:mainSH\] Assume $ n \geq 2$ and fix $ 0 < \delta < \frac{1}{96}.$ Let $ 1 < p, q < \infty $ be such that $ (\frac{1}{p},\frac{1}{q}) $ belongs to the interior of the triangle joining the points $ (0,1), (1,0) $ and $ (\frac{3n+1}{3n+4},\frac{3n+1}{3n+4}).$ Then for any pair of compactly supported bounded functions $ (f,g) $ there exists a $ (p,q)$-sparse form such that $ \langle M_{\operatorname{lac}}f, g\rangle \leq C \Lambda_{\mathcal{S},p,q}(f,g).$
Boundedness properties
======================
Consequences inferred from sparse domination are well-known and have been studied in the literature. We refer to [@BC Section 4] for an account of the same. In particular, sparse domination provides unweighted and weighted inequalities for the operators under consideration.
The strong boundedness is a result by now standard, see [@CUMP], also [@Lacey Proposition 6.1]. Our Theorem \[thm:spherical\] follows from Theorem \[thm:mainSH\] (or just Theorem \[thm:sparse\]) and Proposition \[prop:un\].
\[prop:un\] Let $1\le r<s'\le\infty$. Then, $$\Lambda_{r,s}(f,g)\lesssim \|f\|_{L^p}\|g\|_{L^{p'}}, \quad r<p<s'.$$
Once again for the sake of completeness we reproduce the proof which is quite simple: as the collection $ \mathcal{S} $ is sparse, we have $$\Lambda_{r,s}(f,g) \leq C \sum_{S \in \mathcal{S}} \int_{E_S} \langle f\rangle_{S,r} \langle g \rangle_{S,s} \mathbf{1}_{E_S} dx$$ where $ E_S \subset S $ are disjoint with the property that $ |E_S| \geq \eta |S|.$ The above leads to the estimate $$\Lambda_{r,s}(f,g) \leq C \int_{{\mathbb H}^n} \big(\Lambda |f|^r(x)\big)^{1/r} \big(\Lambda |g|^s(x)\big)^{1/s} dx$$ where $ \Lambda h $ stands for the Hardy-Littlewood maximal function of $ h.$ In view of the boundedness of $\Lambda$, an application of Hölder’s inequality completes the proof of the proposition.
A weight $w$ is a nonnegative locally integrable function defined on ${\mathbb H}^n$. Given $1<p<\infty$, the Muckhenhoupt class of weights $A_p$ consists of all $w$ satisfying $$[w]_{A_p}:=\sup_Q \langle w\rangle_{Q}\langle \sigma \rangle_{Q}^{p-1}<\infty,\quad \sigma:= w^{1-p'}$$ where the supremum is taken over all cubes $Q$ in ${\mathbb H}^n$. On the other hand, a weight $w$ is in the reverse Hölder class $\operatorname{RH}_p$, $1\le p<\infty$, if $$[w]_{\operatorname{RH}_p}=\sup_Q\langle w\rangle_Q^{-1}\langle w\rangle_{Q,p}<\infty,$$ again the supremum taken over all cubes in ${\mathbb H}^n$.
The following theorem was shown in [@BFP Section 6].
\[thm:BFP\] Let $1\le p_0<q_0'\le\infty$. Then, $$\Lambda_{p_0,q_0}(f,g)\le \{[w]_{A_{p/p_0}}\cdot[w]_{\operatorname{RH}_{(q_0'/p)'}}\}^{\alpha}\|f\|_{L^p(w)}\|g\|_{L^{p'}(\sigma)}, \quad p_0<p<q_0',$$ with $\alpha=\max\Big\{\frac{1}{p-1},\frac{q_0'-1}{q_0'-p}\Big\}$.
In view of Theorem \[thm:BFP\] and with the sharpened sparse domination in Theorem \[thm:mainSH\] at hand, but restricting ourselves to values of $(1/p,1/q)$ on $\mathbf{L}_n$, we can obtain the following corollary: it provides unprecedented weighted estimates for the lacunary maximal spherical means in ${\mathbb H}^n$.
\[cor:weight\] Let $n\ge2$ and define $$\frac{1}{\phi(1/p_0)}=\begin{cases}1-\frac{1}{np_0}, \quad 0<\frac1p_0\le \frac{n}{n+1},\\
n\Big(1-\frac1p_0\Big), \quad \frac{n}{n+1}<\frac1p_0<1.
\end{cases}$$ Then $M_{\operatorname{lac}}$ is bounded on $L^p(w)$ for $w\in A_{p/p_0}\cap \operatorname{RH}_{(\phi(1/p_0)'/p)'}$ and all $1<p_0<p<(\phi(1/p_0))'$.
Quantitative weighted estimates could have been stated in Corollary \[cor:weight\], because by Theorem \[thm:mainSH\] we have the sparse domination in the closed triangle $\textbf{L}_n$.
A sharpened weighted inequality {#sub:sharpenw}
-------------------------------
Finally, we remark that an enhanced version of Corollary \[cor:weight\], with the range of $(1/p,1/q)$ in the interior of $\mathbf{S}_n$, might be also stated (see [@Lacey Corollary 6.3] and [@BC Corollary 4.2] for similar discussions).
\[cor:weightSH\] Let $n\ge2$ and define $$\frac{1}{\phi(1/p_0)}=\begin{cases}1-\frac{1}{p_0}\frac{3}{3n+1}, \quad 0<\frac1p_0\le \frac{3n+1}{3n+4},\\
\frac{3n+1}{3}\Big(1-\frac1p_0\Big), \quad \frac{3n+1}{3n+4}<\frac1p_0<1.
\end{cases}$$ Then $M_{\operatorname{lac}}$ is bounded on $L^p(w)$ for $w\in A_{p/p_0}\cap \operatorname{RH}_{(\phi(1/p_0)'/p)'}$ and all $1<p_0<p<(\phi(1/p_0))'$.
[**Acknowledgments**]{}
This work was mainly carried out when the first, second and the fourth author were visiting the third author in Bilbao. They wish to thank BCAM in general and Luz Roncal in particular for the warm hospitality they enjoyed during their visit. The last author fondly remembers the daily shots of cortado as well as the changing colours of The Puppy!
All the four authors are thankful to Michael Lacey for answering several queries and offering clarifications regarding his work [@Lacey]. They also acknowledge helpful discussions with David Beltran and Kangwei Li.
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| {
"pile_set_name": "ArXiv"
} |
---
bibliography:
- 'ChiralFN.bib'
---
CPHT-RR051.082019\
DESY 19-154
[Chiral Froggatt-Nielsen models, gauge anomalies and flavourful axions]{}\
\
[${}^a$*DESY, Notkestrasse 85, 22607 Hamburg, Germany\
*]{}
[${}^b$*Centre de Physique Théorique, CNRS, École Polytechnique, IP Paris,\
F-91128 Palaiseau, France\
*]{}
[${}^c$*Institute of Theoretical Physics, Faculty of Physics, University of Warsaw,\
ul. Pasteura 5, PL-02-093 Warsaw, Poland\
*]{}
We study UV-complete Froggatt-Nielsen-like models for the generation of mass and mixing hierarchies, assuming that the integrated heavy fields are chiral with respect to an abelian Froggatt-Nielsen symmetry. It modifies the mixed anomalies with respect to the Standard Model gauge group, which opens up the possibility to gauge the Froggatt-Nielsen symmetry without the need to introduce additional spectator fermions, while keeping mass matrices usually associated to anomalous flavour symmetries. We give specific examples where this happens, and we study the flavourful axion which arises from an accidental Peccei-Quinn symmetry in some of those models. Such an axion is typically more coupled to matter than in models with spectator fermions.
Introduction
============
As efficient as the Standard Model (SM) may be to describe particle physics phenomenology, it still has unsatisfactory features. Among those, the unexplained hierarchies in masses and mixings between elementary particles has motivated intense theoretical work, leading to precise BSM scenarii. The latter deal with the flavour hierarchies, as well as with the several discrepancies with the SM predictions in magnetic dipole moments or heavy meson decays, while abiding by the conclusions of precision tests of the SM.
Many flavour models for the mass hierarchies involve additional symmetries, whose nature and origin are diverse: they can be global abelian or non-abelian, local abelian or non-abelian as well as discrete. In particular, Froggatt-Nielsen (FN) models [@Froggatt:1978nt; @Dimopoulos:1983rz; @Leurer:1992wg; @Leurer:1993gy] are leading candidates to account for the flavour hierarchies. They rely on an extended scalar and fermionic heavy sector and on an additional spontaneously broken symmetry. Their study has recently been revived by the focus on flavourful axions which arise in FN-like setups [@Wilczek:1982rv; @Ema:2016ops; @Calibbi:2016hwq; @Ema:2018abj] and whose EFT is very much constrained by flavour physics [@Choi:2017gpf; @Bjorkeroth:2018dzu; @Gavela:2019wzg]. Such flavourful axions can also be linked with dark matter studies [@Jaeckel:2013uva].
The nature of the FN symmetry is debatable, and the question of whether it can be gauged is raised, in particular in order to evade quantum gravity corrections which explicitly break global symmetries [@Hawking:1987mz; @Giddings:1988cx; @Banks:2010zn; @Harlow:2018jwu; @Harlow:2018tng; @Fichet:2019ugl]. For instance, it has been shown [@Ibanez:1994ig; @Jain:1994hd; @Binetruy:1994ru; @Dudas:1995yu] that in minimal supersymmetric models, the MSSM spectrum induces gauge anomalies when charged under a FN symmetry, such that one must design an extra fermionic spectrum or a Green-Schwarz (GS) mechanism in order to make the model consistent (see e.g. [@Dudas:1996fe; @Binetruy:1996xk; @Irges:1998ax; @Froggatt:1998he; @King:1999mb; @King:1999cm; @Berger:2000sc; @Dreiner:2003yr; @Chen:2006hn; @Dreiner:2007vp; @DelleRose:2017xil; @Bonnefoy:2018hdo]). One way to do this is to add chiral spectator fermions at the scale where the FN symmetry is broken.
In this paper, we explore the possibility of gauging the FN symmetry without adding any other extra field than the ones required to implement the FN mechanism. In particular, we do not need to introduce both heavy vector-like fields which generate the flavour hierarchies [à la]{} Froggatt-Nielsen and chiral ones which take care of anomalies. This is indeed possible if the fields participating in the FN mechanism are chiral with respect to the FN symmetry, which we choose to be abelian in what follows. We show it by presenting specific examples of two kinds, without (as already shown in [@Alonso:2018bcg; @Smolkovic:2019jow]) and with an accidental global symmetry, focusing for concreteness on supersymmetric models (we briefly comment on non-SUSY models at the end of the discussion). In particular, we explicitly display a model with a physical flavourful axion, which we analyze and compare to flavourful axions arising from global FN symmetries [@Ema:2016ops; @Calibbi:2016hwq; @Ema:2018abj]. In our example, although the qualitative axion phenomenology is similar to the one of global flavourful axions, meaning that the axion couplings are mainly dictated by low-energy physics, there are slight changes in the axion couplings to gauge fields since the latter are already generated by the integrating-out of the heavy FN sector. An other obvious but significant difference between the global and the gauged FN models with an axion is that the shift symmetry of the latter can easily be protected in the second kind of models.
We also establish constraints coming from the perturbativity of the (MS)SM gauge couplings, which imposes that the scale of spontaneous FN symmetry breaking is at least intermediate ($10^{12-13}$ GeV). This allows us to compare chiral FN models with gauged vector-like FN models which use spectator fields to cancel the anomalies. An immediate consequence of using a chiral heavy sector instead of a vector-like one is that there are in general less SM-charged heavy particles, such that constraints from the running of gauge couplings are weaker. In particular, axions can be more coupled to matter in chiral models. An other feature of the latter is that they reduce the number of necessary input scales, since they do not need to introduce the mass scale of the vector-like FN fields.
The plan of the paper is as follows: in section \[motivSection\], we review minimal supersymmetric abelian FN models and their naive gauging to motivate the present work. In section \[sectionMain\], we study in details how the conclusions of section \[motivSection\] are evaded if the fields which generate the mass and mixing hierarchies are chiral with respect to the FN symmetry. We discuss our general framework in section \[chiralGeneral\], illustrate it with specific examples in section \[chiralExamples\], while section \[gaugeRunningSection\] presents the constraints coming from the running of the MSSM gauge couplings. At this point, we also compare chiral and vector-like gauged FN models, in the spirit of what was sketched above. In section \[accidentalAxion\], we elaborate on the accidental flavourful Peccei-Quinn symmetry [@Peccei:1977hh; @Weinberg:1977ma; @Wilczek:1977pj] and its associated axion which arise in some of the models we scrutinize, and we briefly discuss constraints on the model parameters derived from the consistency of the model. We also discuss constraints arising from the axion phenomenology in section \[axionPhenoSession\]. We mention non-SUSY models in section \[nonSUSY\], by discussing again some examples. After a conclusive summary, appendix \[fermionCouplingsAppendix\] covers a discussion of gauge-invariant superpotential terms which were postponed in section \[chiralGeneral\], appendix \[anomaliesUnificationAppendix\] discusses the link between the GS conditions for anomaly cancellation and the unification of gauge couplings and appendix \[vectorlikeAppendix\] details the construction of some of the gauged vector-like FN models discussed in section \[gaugeRunningSection\]. The preliminary results of this work have been presented in [@DudasPlanck2019].
Motivation {#motivSection}
==========
Yukawa matrices and Froggatt-Nielsen models
-------------------------------------------
The flavour structure of the SM is a consequence of the number of families and of the structure of the Yukawa sector: -(Y\^u\_[ji]{}HQ\_[L,j]{}+Y\^d\_[ji]{}H\^cQ\_[L,j]{}+Y\^e\_[ji]{}H\^cL\_[L,j]{})+h.c. . \[SMYukawas\] Its phenomenological predictions are fully characterized by fermion masses $m_{i=1..3}^X$ (with $X=u,d,e$, a notation we use throughout this paper) and by the CKM matrix [@Cabibbo:1963yz; @Kobayashi:1973fv]. The latter reads, in the Wolfenstein parametrization [@Wolfenstein:1983yz]: V\_=1-&&A\^3(-i)\
-&1-&A\^2\
A\^3(1--i)&-A\^2&1+(\^4) , \[hierarchiesMixings\] where $\lambda$ is linked to the Cabibbo angle $\theta_C$: $\lambda=\sin(\theta_C)\approx 0.22$, and $A,\rho,\eta=\cO(1)$. Orders of magnitude for the quark and lepton masses can also be expressed in terms of $\lambda$: &\~\^8 , \~\^4 , \~\^4 , \~\^2 , \~\^2 , \~\^4 , \~\^2 , \~\^2 \[hierarchiesMasses\] at the GUT scale $M_\text{GUT}\sim 10^{16}$ GeV. The strong hierarchies between the particle masses as well as the milder ones appearing in the CKM matrix are unexplained input parameters in the SM. They can be traced back to hierarchies which must be present in the Yukawa matrices $Y^{u,d,e}$ of . In the minimal supersymmetric standard model (MSSM), to which we stick except in section \[nonSUSY\], the flavour structure as well as the mass and mixing hierarchies are found in the superpotential[^1]: W Y\^u\_[ij]{}Q\_[i]{}H\_uU\_[j]{}+Y\^d\_[ij]{}Q\_[i]{}H\_dD\_[j]{}+Y\^e\_[ij]{}L\_[i]{}H\_dE\_[j]{} . \[MSSMYukawas\]
Froggatt-Nielsen models [@Froggatt:1978nt] address the origin of flavour hierarchies by means of a symmetry explanation: the masses and mixings arise after spontaneous breaking of a chiral symmetry, which forbids their existence when it is exact in the UV (except for the top quark Yukawa term, as well as the bottom quark one if $\tan\beta$ is large). For instance, one can postulate a global horizontal/family symmetry $U(1)_\text{FN}$ acting on the different MSSM fields and on a standard model singlet superfield $\phi$, the flavon. Then, $U(1)_\text{FN}$ invariance of the Yukawa sector of the MSSM requires a dressing of the Yukawa matrices by powers of $\phi$: W h\^u\_[ij]{} ()\^[n\^u\_[ij]{}]{}Q\_[i]{}H\_uU\_[j]{}+h\^d\_[ij]{} ()\^[n\^d\_[ij]{}]{}Q\_[i]{}H\_dD\_[j]{}+ h\^e\_[ij]{} ()\^[n\^e\_[ij]{}]{}L\_[i]{}H\_dE\_[j]{} , \[minimalFN\] where the $h^X_{ij}$ are order one numbers, $M$ is a high scale of new physics, for instance the mass scale of heavy fields which mix with the standard model ones (see section \[chiralGeneral\] for explicit examples) or the Planck mass if those higher-dimensional operators are generated by supergravity, and the $n^X_{ij}$ are the $U(1)_\text{FN}$ charges of the MSSM Yukawa couplings in units of the charge of $\overline\phi$. Indeed, $U(1)_\text{FN}$ invariance imposes that the $n^X$’s are n\^u\_[ij]{}=- , n\^d\_[ij]{}=- , n\^e\_[ij]{}=- , \[linkNQ\] where the $q$’s denote with transparent subscripts the $U(1)_\text{FN}$ charges of the different superfields. In particular, they are such that n\^X\_[11]{}-n\^X\_[i1]{}=n\^X\_[1j]{}-n\^X\_[ij]{} , n\^u\_[11]{}-n\^u\_[i1]{}=n\^d\_[11]{}-n\^d\_[i1]{} . Once $U(1)_\text{FN}$ is spontaneously broken by a vacuum expectation value (vev) of $\phi$, the hierarchies in the fermion mass matrices are naturally explained in terms of a small parameter $\epsilon=\abs{\frac{\langle\phi\rangle}{M}}$, assumed to be $\sim\lambda$, and larger charges for the light generations (see section \[chiralExamples\] for explicit examples). Indeed, the low-energy Yukawa couplings are given by $$Y_{ij}^X = h_{ij}^X \epsilon^{n_{ij}^X}$$ and have the required hierarchies for $h_{ij}^X \sim \cO(1)$.
Gauged $U(1)_\text{FN}$ and anomaly cancellation {#anomalousFN}
------------------------------------------------
Since global symmetries are threatened by quantum gravity [@Hawking:1987mz; @Giddings:1988cx; @Banks:2010zn; @Harlow:2018jwu; @Harlow:2018tng; @Fichet:2019ugl], one could be tempted to gauge $U(1)_\text{FN}$ to protect it against explicit breaking, which could in principle generate uncontrolled $U(1)_\text{FN}$-breaking Yukawa terms and spoil the symmetry-based hierarchies -. However, the possible $U(1)_\text{FN}$ charges are constrained by the flavour structure of the SM and the question of whether they can be chosen such that all gauge anomalies vanish is raised [@Ibanez:1994ig; @Jain:1994hd; @Binetruy:1994ru; @Dudas:1995yu]. In particular, defining anomaly coefficients such that \_[U(1)\_]{}=&-\^G\_\^aG\_\^a-\^W\_\^iW\_\^i-\^F\_F\_-...\
&+() , where $G_\mu^a$ is the gluon field of field strength $G_{\mu\nu}^a$, $W_{\mu(\nu)}^i$ the $SU(2)_W$ gauge boson field (strength) and $F_{\mu(\nu)}$ the $U(1)_Y$ gauge boson field (strength), it has been shown that[^2] \~\^[(A\_1+A\_2-2A\_3)]{} . The determinant of the left hand side is clearly $\epsilon$-suppressed when we insert phenomenologically relevant Yukawa matrices. For instance, assuming $\tan\beta=1$, \~\~\^[30]{} , \[YuYdfirst\] where $v$ is the Higgs vev. Thus, we understand that introduces a $U(1)_\text{FN}$ which has mixed anomalies with the SM gauge group $G_\text{SM}$. This enables one to interpret the phase of $\phi$ as a flavourful QCD axion when $U(1)_\text{FN}$ is global [@Wilczek:1982rv; @Ema:2016ops; @Calibbi:2016hwq; @Ema:2018abj], with couplings to gauge fields and SM fermions which are fully determined by the mass matrices. On the other hand, it also means that $U(1)_\text{FN}$ cannot be naively gauged. Ways out would either introduce additional chiral fermions, extend the scalar sector or rely on a Green-Schwarz-inspired mechanism [@Green:1984sg]. In what follows, we will explore the first and second options. Let us point out in passing that, due to the phenomenological interest in additional abelian factors to the SM gauge group, there are recent works about anomaly cancellation in such models with a general focus, see e.g. [@Ellis:2017nrp; @Allanach:2018vjg; @Correia:2019pnn; @Costa:2019zzy].
Chiral Froggatt-Nielsen models {#sectionMain}
==============================
A key assumption in , which we now relax, is that the only low-energy contribution of the heavy sector at scale $M$ is the generation of the Yukawa terms. This is true if the heavy sector is vector-like with respect to the SM gauge group, but if it is chiral there could also be in the EFT anomalous couplings between (the longitudinal component of) the $U(1)_\text{FN}$ gauge field and the SM gauge bosons [@Anastasopoulos:2006cz]. In this section, we explore this possibility.
We focus on models with two singlet superfields $\phi_1$ and $\phi_2$, which respectively replace the flavon $\phi$ and the mass $M$ in , such that the Yukawa sector is as follows: W h\^u\_[ij]{} ()\^[n\^u\_[ij]{}]{}Q\_[i]{}H\_uU\_[j]{}+ h\^d\_[ij]{} ()\^[n\^d\_[ij]{}]{}Q\_[i]{}H\_dD\_[j]{}+ h\^e\_[ij]{} ()\^[n\^e\_[ij]{}]{}L\_[i]{}H\_dE\_[j]{} , \[chiralFN\] and we allow in particular $\phi_2$ to be charged under $U(1)_\text{FN}$. Generalizing , we now have n\^u\_[ij]{}=- , n\^d\_[ij]{}=- , n\^e\_[ij]{}=- , \[linkNQChiral\] and we define $x_{1,2}\equiv -q_{\phi_{1,2}},h_{u,d}\equiv q_{H_{u,d}}$ for the sake of reducing the subscripts in what follows. In order to trace back the role of the hierarchies in the Yukawa matrices, we also trade most of the charges for the integers $n^X_{ij}$ using , such that for instance q\_[U\_1]{}=-q\_[Q\_1]{}-h\_u+(x\_1-x\_2)n\^u\_[11]{} . Working out other relations leads to the charges of the superfields which appear in Table \[MSSMcharges\].
[|c|c|c|c|c|]{} &SU(3)\_C&SU(2)\_W&U(1)\_Y&U(1)\_\
\_1&**1**&**1**&0&-x\_1\
\_2&**1**&**1**&0&-x\_2\
H\_u&**1**&**2**&1/2&h\_u\
H\_d&**1**&**2**&-1/2&h\_d\
Q\_[i]{}&**3**&**2**&1/6&X\_Q-(x\_1-x\_2)(n\^u\_[11]{}-n\^u\_[i1]{})\
U\_[j]{}&&**1**&-2/3&-X\_Q-h\_u+(x\_1-x\_2)n\^u\_[1j]{}\
D\_[j]{}&&**1**&1/3&-X\_Q-h\_d+(x\_1-x\_2)n\^d\_[1j]{}\
L\_[i]{}&**1**&**2**&-1/2&X\_L-(x\_1-x\_2)(n\^e\_[11]{}-n\^e\_[i1]{})\
E\_[j]{}&**1**&**1**&1&-X\_L-h\_d+(x\_1-x\_2)n\^e\_[1j]{}\
We again assume $\langle\phi_1\rangle=\epsilon\langle\phi_2\rangle$, with $\epsilon\approx\lambda$, and formulas to follow will encompass cases where $\phi_1$ or $\phi_2$ is uncharged and equivalent to a mass $M$. However, we always impose $x_1\neq x_2$ such that $U(1)_\text{FN}$ acts non-trivially on the MSSM fields.
The contribution of the MSSM fields to the mixed anomaly coefficients are as follows: SU(3)\_C\^2U(1)\_: A\_[3,]{} =& \_i(2q\_[Q\_i]{}+q\_[U\_i]{}+q\_[D\_i]{})= -3(h\_u+h\_d)+(x\_1-x\_2)\_i(n\^u\_[ii]{}+n\^d\_[ii]{})\
SU(2)\_W\^2U(1)\_: A\_[2,]{}=& \_i(3q\_[Q\_i]{}+q\_[L\_i]{})+q\_[H\_u]{}+q\_[H\_d]{}\
=& 3(3X\_Q+X\_L)+h\_u+h\_d\
&-(x\_1-x\_2)(3(2n\^u\_[11]{}-n\^u\_[21]{}-n\^u\_[31]{})+2n\^e\_[11]{}-n\^e\_[21]{}-n\^e\_[31]{})\
U(1)\_Y\^2U(1)\_: A\_[1,]{}=& \_i(+++q\_[L\_i]{}+2q\_[E\_i]{})+q\_[H\_u]{}+q\_[H\_d]{}\
=& -3(3X\_Q+X\_L)-7(h\_u+h\_d)\
&+(x\_1 - x\_2)(+\
&++2n\^e\_[12]{}+2n\^e\_[13]{}+n\^e\_[21]{}+n\^e\_[31]{}) . \[newSManomalies\] The vanishing of the mixed $U(1)_Y\times U(1)_\text{FN}^2$ anomaly is also imposed, but for brevity we do not display it explicitly. On the other hand, we ignore the $U(1)_\text{FN}^3$ or $U(1)_\text{FN}\times$gravity anomalies. Those could for instance be modified if we added to this setup some sterile neutrino superfields charged under $U(1)_\text{FN}$.
The anomalies in are non-vanishing, since the discussion of section \[anomalousFN\] still applies. Nonetheless, they can be cancelled by taking into account the gauge anomalies induced by the heavy FN sector, as we now discuss.
Heavy FN sector and anomalies {#chiralGeneral}
-----------------------------
We now design a UV theory which generates in the IR. We understand as being perturbatively generated[^3], closely following the original FN picture. The setup, together with our notations, can be understood by looking at Figure \[treeDiags\]: ignoring for the time being the second and third SM generations, we introduce the heavy fermions shown in Table \[Heavycharges\], vector-like under the SM gauge group but chiral with respect to $U(1)_\text{FN}$.
[|c|c|c|c|c|]{} &SU(3)\_C&SU(2)\_W&U(1)\_Y&U(1)\_\
\
\^Q\_[i=1,...,n\_[Q,1]{}]{}&&**2**&-1/6&-X\_Q+i(x\_1-x\_2)+x\_2\
\^Q\_[i=1,...,n\_[Q,1]{}]{}&**3**&**2**&1/6&X\_Q-i(x\_1-x\_2)\
\
\^u\_[i=n\_[Q,1]{}+1,...,n\^u\_[11]{}]{}&&**1**&-2/3&-X\_Q+(i-1)(x\_1-x\_2)-h\_u\
\^u\_[i=n\_[Q,1]{}+1,...,,n\^u\_[11]{}]{}&**3**&**1**&2/3&X\_Q-(i-1)(x\_1-x\_2)+x\_2+h\_u\
\^d\_[i=n\_[Q,1]{}+1,...,,n\^d\_[11]{}]{}&&**1**&1/3&-X\_Q+(i-1)(x\_1-x\_2)-h\_d\
\^d\_[i=n\_[Q,1]{}+1,...,,n\^d\_[11]{}]{}&**3**&**1**&-1/3&X\_Q-(i-1)(x\_1-x\_2)+x\_2+h\_d\
\
\^L\_[i=1,...,n\_[L,1]{},n\^e\_[11]{}]{}&**1**&**2**&1/2&-X\_L+i(x\_1-x\_2)+x\_2\
\^L\_[i=1,...,n\_[L,1]{},n\^e\_[11]{}]{}&**1**&**2**&-1/2&X\_L- i(x\_1-x\_2)\
\^e\_[i=n\_[L,1]{}+1,...,,n\^e\_[11]{}]{}&**1**&**1**&1&-X\_L+(i-1)(x\_1-x\_2)-h\_d\
\^e\_[i=n\_[L,1]{}+1,...,,n\^e\_[11]{}]{}&**1**&**1**&-1&X\_L-(i-1)(x\_1-x\_2)+ x\_2+h\_d\
We define $n_{Q/L,1}$ to be the numbers of $SU(2)_W$ doublets pairs in the heavy sector associated to the quark and lepton mass matrices respectively, i.e. which mix with the quark or lepton $SU(2)_W$ doublets of the MSSM (see Figure \[treeDiags\] to understand how those contribute to the FN mechanism). The subscript $1$ anticipates that there will be equivalent numbers of doublets for each generation. Analogously, there could be heavy pairs of $SU(2)_W$ singlets mixing with the up-type and the down-type quarks or with the electron-like fields, and we denote their numbers by $n_{U,i}$, $n_{D,i}$ and $n_{E,i}$ respectively. In this notation, the total number of heavy pairs needed for generating the Yukawa coupling $h_{ii}^u$ (for example) is $n_{Q,i} + n_{U,i}$. Said differently, the number of heavy pairs of doublets and singlets are related by relations of the type $$n_{U,i} = (n_{ii}^u-n_{Q,i}) \theta (n_{ii}^u-n_{Q,i}) \ , \ n_{D,i} = (n_{ii}^d-n_{Q,i}) \theta (n_{ii}^d-n_{Q,i}) \ ,
\label{defNsNonDoublet}$$ as clearly depicted on Figure 1.
Those fields together with the MSSM fields form a renormalizable UV theory, with a superpotential formed of (here only for the first generation) W& \_1\^X\_[i]{}\^X\_[i+1]{} ( \_1\^Q\_[i]{}\^Q\_[i+1n\_[Q,1]{}]{}) , \_2\^X\_[i]{}\^X\_[i]{} ,\
& H\_u\^Q\_[n\_[Q,1]{}]{}\^u\_[n\_[Q,1]{}+1]{} , H\_d\^Q\_[n\_[Q,1]{}]{}\^d\_[n\_[Q,1]{}+1]{} , H\_d\^L\_[n\_[L,1]{}]{}\^e\_[n\_[L,1]{}+1]{} , \[goodcouplings\] where the $\Psi^X$ and $\tilde\Psi^X$ can also be MSSM fields according to the following replacement rules: Q\_[1]{} \^Q\_[0]{} , U\_[1]{} \^u\_[n\^u\_[11]{}+1]{} , D\_[1]{} \^d\_[n\^d\_[11]{}+1]{} , L\_[1]{} \^L\_[0]{} , E\_[1]{} \^e\_[n\^e\_[11]{}+1]{} . \[replace\] Those couplings are (generically) the only ones one can write at renormalizable order (see appendix \[fermionCouplingsAppendix\]) and they are precisely the ones needed to generate , via diagrams such as the one of Figure \[treeDiags\].
![Tree diagram generating the $d$-quark mass, when $n^d_{11}=6$ and $n_{Q,1}=4$\
The gray line indicates how it should be modified to generate a mixing to $Q_{2}$ when $n^d_{21}=4$[]{data-label="treeDiags"}](diagramExtended.pdf)
Mixings to other generations can be similarly implemented via couplings between e.g. $Q_{i>1}$ and one of the $(\phi_1)\Psi^Q$ (again, see Figure \[treeDiags\] for an example of a diagram which results). However, in order to have mass matrices of rank 3 each, we need to supplement the FN fields of Table \[Heavycharges\] by their equivalent for the second and third families (see e.g. [@Leurer:1992wg; @Calibbi:2012yj]), in which case the indices $i$ in Table \[Heavycharges\] range between $1$ and $n^u_{22},n^d_{22},n^e_{22}$ for the second family, and between $1$ and $n^u_{33}=0,n^d_{33},n^e_{33}$ for the third one. The charges $X_Q$ and $X_L$ in Table \[Heavycharges\] should also be replaced by $X_Q-(x_1-x_2)(n^u_{11}-n^u_{21})$ and $X_L-(x_1-x_2)(n^e_{11}-n^e_{21})$ for the second family, or by $X_Q-(x_1-x_2)(n^u_{11}-n^u_{31})$ and $X_L-(x_1-x_2)(n^e_{11}-n^e_{31})$ for the third one.
The contribution of the FN fields to the mixed anomaly coefficients are as follows: A\_[3,]{} =& x\_2 \_i (2 n\_[Q,i]{} + n\_[U,i]{} +n\_[D,i]{}) = x\_2(2(n\_[Q,1]{}+n\_[Q,2]{}+n\_[Q,3]{})+(n\^u\_[11]{}-n\_[Q,1]{}) (n\^u\_[11]{}-n\_[Q,1]{})\
& +(n\^d\_[11]{}-n\_[Q,1]{}) (n\^d\_[11]{}-n\_[Q,1]{})+ (n\^u\_[22]{}-n\_[Q,2]{}) (n\^u\_[22]{}-n\_[Q,2]{})\
& + (n\^d\_[22]{}-n\_[Q,2]{}) (n\^d\_[22]{}-n\_[Q,2]{})+(n\^d\_[33]{}-n\_[Q,3]{}) (n\^d\_[33]{}-n\_[Q,3]{}))\
A\_[2,]{}=& x\_2(3(n\_[Q,1]{}+n\_[Q,2]{}+n\_[Q,3]{})+n\_[L,1]{}+n\_[L,2]{}+n\_[L,3]{})\
A\_[1,]{}=& x\_2((n\_[Q,1]{}+n\_[Q,2]{}+n\_[Q,3]{})+n\_[L,1]{}+n\_[L,2]{}+n\_[L,3]{}\
& +\[ (n\^u\_[11]{}-n\_[Q,1]{}) (n\^u\_[11]{}-n\_[Q,1]{})+ (n\^u\_[22]{}-n\_[Q,2]{}) (n\^u\_[22]{}-n\_[Q,2]{})\]\
& +\[ (n\^d\_[11]{}-n\_[Q,1]{}) (n\^d\_[11]{}-n\_[Q,1]{})+ (n\^d\_[22]{}-n\_[Q,2]{}) (n\^d\_[22]{}-n\_[Q,2]{})+ (n\^d\_[33]{}-n\_[Q,3]{}) (n\^d\_[33]{}-n\_[Q,3]{})\]\
& + 2\[(n\^e\_[11]{}-n\_[L,1]{}) (n\^e\_[11]{}-n\_[L,1]{})+ (n\^e\_[22]{}-n\_[L,2]{}) (n\^e\_[22]{}-n\_[L,2]{})+(n\^e\_[33]{}-n\_[L,3]{})(n\^e\_[33]{}-n\_[L,3]{})\]) ,\
where $\theta(x)$ is the Heaviside step function. Hence, we understand that the integrating out of those FN fields generate in addition to the following anomalous axionic term in the lagrangian[^4] Wd\^2(-(\_2)(W\^a)\^2+...) , \[FNanomalousW\] where we only displayed the consequence of the QCD anomaly. This would not happen for a vector-like FN sector. Note that only $\phi_2$ appears in since it is the field which gives its mass to the heavy sector in our construction.
Anomaly-free models {#chiralExamples}
-------------------
The presence of allows one to build “minimal” models where the fermions which participate in the FN mechanism, meaning those which are necessary to generate the hierarchies in masses and mixings, are sufficient to make the model anomaly-free, providing what could be called a minimal anomaly-free gauged FN model. We will not study thoroughly all possible models which achieve this, but, as proofs of principle, we restrict to two specific models.
The first one, which we call Model A in what follows, has only one singlet field $\phi_2$ (and corresponds to a case where $x_1=0$, hence $\phi_1=M$). It reproduces the following Yukawa matrices Y\^u=(
[ccc]{} \^[8]{}&\^5&\^3\
\^[7]{}&\^4&\^2\
\^[5]{}&\^2&1\
) , Y\^d=(
[ccc]{} \^4&\^3&\^3\
\^3&\^2&\^2\
&1&1\
) , Y\^e=(
[ccc]{} \^4&\^3&\^3\
\^3&\^2&\^2\
&1&1\
) , \[YukawasChosen\] which fit well the phenomenological values for quarks and mixings when $\tan\beta$ is large. When the FN superfields do not feature any doublet (i.e. $n_{Q,i}=n_{L,i}=0$, leading to a number of heavy fields derived from Table \[Heavycharges\] and reminded in Table \[fermionsTwoModels\]),
[max width=]{} $\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
&n_{Q,1}&n_{Q,2}&n_{Q,3}&n_{U,1}&n_{U,2}&n_{U,3}&n_{D,1}&n_{D,2}&n_{D,3}&n_{L,1}&n_{L,2}&n_{L,3}&n_{E,1}&n_{E,2}&n_{E,3}\\
\hline
\text{Model A}&0&0&0&8&4&0&4&2&0&0&0&0&4&2&0\\
\hline
\text{Model B}&4&2&0&4&2&0&0&0&0&0&0&0&4&2&0\\
\hline
\end{array}$
choosing $h_u=h_d=0$ and $x_2=-\frac{3(3X_Q+X_L)}{16}$ makes all anomalies vanish (and the $\mu$-term $\mu H_uH_d$ is allowed in the superpotential). This amounts to the usual FN model, with the exception that $\frac{\phi}{M}$ is replaced by $\frac{M}{\phi}$. This model is discussed in [@Alonso:2018bcg; @Smolkovic:2019jow]. It is interesting to note that this anomaly-free and supersymmetric model has the same field content as the ones which were doomed to be saved by a GS mechanism [@Ibanez:1994ig; @Jain:1994hd; @Binetruy:1994ru; @Dudas:1995yu]. If one insists on using $\phi_1$ as a dynamical scalar, it is a pure singlet and there will be terms such as $\phi_1^n$ in the superpotential. There is no light degree of freedom in the FN sector in this scenario, which can be constrained by the running of gauge couplings, as discussed in section \[gaugeRunningSection\].
On our second model, called Model B, we impose the condition that the heavy FN fields should respect the qualitatively satisfying gauge coupling unification obtained in the MSSM, which can be obtained if the FN fields contribute to the running of the MSSM gauge couplings as $SU(5)$ multiplets (albeit with different $U(1)_\text{FN}$ charges within a same “$SU(5)$ multiplet”). We thus demand that[^5] n\_[Q,1]{}+n\_[Q,2]{}+n\_[Q,3]{}=& n\_[U,1]{}+n\_[U,2]{}= (n\^u\_[11]{}-n\_[Q,1]{}) (n\^u\_[11]{}-n\_[Q,1]{})+(n\^u\_[22]{}-n\_[Q,2]{}) (n\^u\_[22]{}-n\_[Q,2]{})\
=& n\_[E,1]{}+n\_[E,2]{}+n\_[E,3]{}=\_i (n\^e\_[ii]{}-n\_[L,i]{}) (n\^e\_[ii]{}-n\_[L,i]{}) ,\
n\_[L,1]{}+n\_[L,2]{}+n\_[L,3]{}=& n\_[D,1]{}+n\_[D,2]{}+n\_[D,3]{} =\_i (n\^d\_[ii]{}-n\_[Q,i]{}) (n\^d\_[ii]{}-n\_[Q,i]{}) . \[unifConditions\] One can check that we need this time two singlets $\phi_1$ and $\phi_2$, if we insist on not using additional spectator fields beyond the ones which enter the FN mechanism. Choosing $x_1=1,x_2=10$, $h_u=h_d=\frac{9}{2},X_Q=-\frac{67}{2},X_L=-\frac{39}{2}$ and the number of heavy fields again displayed in Table \[fermionsTwoModels\], all the anomalies vanish and we obtain the following mass matrices (which reproduce the correct masses and mixings up to two $\cO(\lambda)$ deviations [@Dudas:1995yu]) Y\^u=(
[ccc]{} \^[8]{}&\^5&\^4\
\^[7]{}&\^4&\^3\
\^[4]{}&&1\
) , Y\^d=(
[ccc]{} \^3&\^3&\^4\
\^2&\^2&\^3\
()&()&1\
) , Y\^e=(
[ccc]{} \^4&\^3&\^3\
\^3&\^2&\^2\
&1&1\
) , \[massMatDeviations\] where by the the parenthesis in the last row of $Y^d$, we mean that those entries are forbidden by holomorphy. However, they might be generated after field redefinitions to take care of corrections to the Kähler potential [@Dudas:1995yu]. We nevertheless leave them in , since they indicate what we choose for the charges of the different fields. Furthermore, notice that, in order to generate the $(1,3)$ and $(2,3)$ entries of $Y^d$, the heavy sector in Table \[Heavycharges\] should be modified such that, for instance, the index $i$ for the $d$-quark-like heavy fields of the first generation in Table \[Heavycharges\] is bounded by $n^{d}_{13}$ instead of $n^{d}_{11}$.
In this model, the $\mu$-term is forbidden and should be generated from the Kähler potential via the Giudice-Masiero mechanism [@Giudice:1988yz], by writing KH\_uH\_d\_2 , \[GMasiero\] assuming $\phi_1$ has a non-vanishing $F$-term.
An interesting aspect of this model is that it has a light mode, since out of the two phases of $\phi_1$ and $\phi_2$ only one is absorbed by the $U(1)_\text{FN}$ gauge boson and the last one is left as a physical Nambu-Goldstone boson (GB). This feature is generic of the models with two singlets, so we generally comment on it in section \[accidentalAxion\].
Constraints from the running of gauge couplings {#gaugeRunningSection}
-----------------------------------------------
The presence of the heavy FN sector adds to the theory many new particles charged under the SM gauge group, so that the running of the MSSM gauge couplings is strongly modified above their mass. In particular, demanding that the model remains perturbative up to some fundamental scale sets strong constraints on the possible masses for the heavy modes (see e.g. [@Calibbi:2012yj]). For concreteness, we look at the specific cases of the two models discussed in section \[chiralExamples\].
Assuming that all the superpartners kick in at a TeV and all the heavy superfields at a high scale $v_2\equiv\langle\phi_2\rangle$, Figure \[perturbativityRunning\] shows the MSSM gauge couplings running at 1-loop in the model A of section \[chiralExamples\], for $v_2=10^{14}$ and $10^{16}$ GeV respectively. We see there that the hypercharge Landau pole, if it is to be above the Planck mass, imposes $v_2\geq10^{16}$ GeV.
![Running coupling constants of the MSSM in model A, assuming $m_\text{soft}=$ TeV[]{data-label="perturbativityRunning"}](11014GeV.pdf "fig:")![Running coupling constants of the MSSM in model A, assuming $m_\text{soft}=$ TeV[]{data-label="perturbativityRunning"}](11016GeV.pdf "fig:")![Running coupling constants of the MSSM in model A, assuming $m_\text{soft}=$ TeV[]{data-label="perturbativityRunning"}](legendRunning.png "fig:")
With the same assumptions about the supersymmetric spectrum and at 1-loop, Figure \[perturbativityRunningP\] shows the MSSM gauge couplings running in model B, for $v_2=3\times 10^{12}$ and $10^{15}$ GeV respectively. Here, we see that the hypercharge Landau pole being above the Planck mass imposes $v_2\geq 10^{15}$ GeV. If we instead only impose that the unification happens before any Landau pole, we find that $v_2\geq 3\times10^{12}$ GeV.
![Running coupling constants of the MSSM in model B, assuming $m_\text{soft}=$ TeV[]{data-label="perturbativityRunningP"}](Unif31012GeV.pdf "fig:")![Running coupling constants of the MSSM in model B, assuming $m_\text{soft}=$ TeV[]{data-label="perturbativityRunningP"}](Unif11015GeV.pdf "fig:")![Running coupling constants of the MSSM in model B, assuming $m_\text{soft}=$ TeV[]{data-label="perturbativityRunningP"}](legendRunning.png "fig:")
Discussing the running of gauge couplings is a good opportunity to emphasize one interesting aspect of chiral models: since their FN sector takes care of both the flavour hierarchies and the anomalies, their heavy field content is expected to be somehow minimal. Consequently, they should be least constrained by the running of gauge couplings. Indeed, the bounds coming from the latter running become stronger when additional charged particles are added to the model, which is necessary if anomalies remain after one integrates the heavy sector which generates the Yukawa couplings. This holds of course for vector-like FN models.
The minimality of chiral models can be understood as follows: given a mass matrix, one can read off how many heavy fields will be necessary in the FN sector for each generation (for instance, one will need at least $n^u_{11}$ coloured particles to generate the entry $Y^u_{11}$ in a renormalizable model). Quark-like heavy doublets do not overload the model since they are used in both the $U$-like heavy sector and the $D$-like one (unless they are too many such that some of them are only used to generate one Yukawa entry, which happens when $n_{Q,i}>\min(n^u_{ii},n^d_{ii})$). Lepton-like doublets are not minimal in this respect, therefore we can already conclude that if all $n_{L,i}=0$ (and $n_{Q,i}\leq\min(n^u_{ii},n^d_{ii})$), the chiral models we discuss here realize the minimal number of necessary heavy fields. This is the case of models A and B of section \[chiralExamples\]. Any model using spectator fields to cancel anomalies will have more (or at least as many) heavy SM-charged particles and will be more constrained (maybe marginally) by the running of gauge couplings. If such a model has a physical axion of the kind we discuss later in section \[accidentalAxion\], this axion will be less coupled to matter, thus less detectable, than axions originating from chiral models.
For one-singlet models, one can give a clear estimate of how many additional particles would be needed. For instance, in a vector-like counterpart to model A, meaning a model which has one SM singlet and the matrices , the anomalies from the MSSM(+FN) sector as well as the holomorphy of the supersymmetric couplings impose that one needs at least six pairs of $SU(3)_C$-triplet spectators which contribute to the running of the colour gauge coupling (details can be found in appendix \[vectorlikeAppendix\]). Such additional particles already have a significant impact on the bounds implied by the $SU(3)_C$ gauge coupling (as illustrated in Figure \[perturbativityRunningVectorLike\] in appendix \[vectorlikeAppendix\]). However, in this model they can be singlets under $SU(2)_W$ and without any hypercharge, such that the hypercharge running is unchanged with respect to the chiral case, while it gave the strongest constraint in Figure \[perturbativityRunning\]. Thus, the chiral model is as much constrained as (or only marginally less constrained than) its vector-like counterpart, although it contains less heavy particles.
A net strengthening of the bounds arises if we instead try to find a vector-like counterpart to model B. We again leave details to appendix \[vectorlikeAppendix\], and we only report here the following result: vector-like counterparts to model B (i.e. vector-like models with spectator fields which preserve the unification of the MSSM gauge couplings) are more constrained than model B itself. For instance, demanding that unification happens within the perturbative regime imposes on those vector-like models that $v_2\geq 4.5\times 10^{13}$ GeV at least, meaning an increase of more than an order of magnitude with respect to model B.
Of course, making such considerations general depend a lot on the $U(1)_\text{FN}$ charges of the Yukawa couplings, as well as on the field content of the theory. For instance, one can find in [@Dudas:1995yu] a (two-singlets) model such that the $U(1)_\text{FN}$ charges of the Yukawa couplings are anomaly-free. Hence, a vector-like heavy sector generating them is enough and chiral models do not perform better with respect to the running of gauge couplings. On the other hand, two-singlets chiral models such as the ones we presented come with only two input scales, the vevs of the two singlet scalars, whereas the model aforementioned comes with three: the vevs of the two scalars and the mass scale of the heavy sector, all constrained to reproduce the correct mass hierarchies. In this respect, chiral models have the advantage of minimality.
An accidental flavourful Peccei-Quinn symmetry {#accidentalAxion}
----------------------------------------------
We now turn to the systematic discussion of the physical GB which arises in models with two singlets $\phi_1$ and $\phi_2$. We stick to the kind of models discussed in sections \[chiralGeneral\] and \[chiralExamples\], namely those where the heavy sector (only or mostly fields participating in the FN mechanism) gets its mass via couplings to $\phi_2$.
In this section, we assume that the light GB is only made of the phases of $\phi_1$ and $\phi_2$, and that the physical pseudoscalar originating from $H_u$ and $H_d$ gets a large mass. This is for instance a valid assumption if the “$b_\mu$” soft term $b_\mu H_uH_d$ is present (i.e. gauge-invariant). Nonetheless, it turns out that the formulas written below are still valid for model B, sometimes thanks to the large values of $\frac{\langle\phi_{1,2}\rangle}{M_W}$ imposed by the running of the gauge couplings, which make the leading order correct. For the same reason, the pseudoscalar $a_\text{FN}$ which gives the longitudinal component of the $U(1)_\text{FN}$ gauge boson is also given at leading order by the contribution of $\phi_1$ and $\phi_2$. Its expression is thus: a\_x\_1v\_1\_1+x\_2v\_2\_2 , where we wrote $\phi_{1,2}=\frac{r_{1,2}+v_{1,2}}{\sqrt{2}}e^{i\frac{\theta_{1,2}}{v_{1,2}}}$. Then, the physical leftover GB $a$ is given by ax\_2v\_2\_1-x\_1v\_1\_2 . Depending on the $U(1)_\text{FN}$ charges of the different scalar fields, the first gauge-invariant operator one could write which violates the shift symmetry of $a$ may be of very high dimension, thus rendering this shift symmetry accidentally protected (more on this below).
We now show that the mode $a$ has couplings similar to the one of flavourful axions [@Wilczek:1982rv; @Ema:2016ops; @Calibbi:2016hwq; @Ema:2018abj], albeit slightly different numerically, meaning that the family symmetry $U(1)_\text{FN}$ imposes that it has anomalous couplings to gauge fields (and in particular to QCD, making it a Peccei-Quinn axion) and direct couplings to SM fermions.
In the kind of models we consider, the couplings to gauge fields are completely specified by the mass matrices. Indeed, as already mentioned in , the heavy sector contributes to the (axionic) anomalous couplings as Wd\^2(-(\_2)(W\_A\^2)) , \[heavyFNAxionContribution\] where $A$ refers to either $SU(3)_C,SU(2)_W$ or $U(1)_Y$, $\cC=1,2$ respectively for a $SU(N)$ or an abelian factor of the gauge group, and we used our assumption that all mass terms come from couplings to $\phi_2$. This contribution should be such that its gauge variation precisely cancels that of the contribution from the MSSM fields (here only focusing on QCD): Wd\^2(-(()\^[\_i(n\^u\_[ii]{}+n\^d\_[ii]{})]{}(H\_uH\_d)\^3)(W\_[SU(3)\_C]{}\^2)+...) , Neglecting $H_{u,d}$ as we assumed, $\sum_i(n^u_{ii}+n^d_{ii})=\frac{A_{3,\text{SM}}}{x_1-x_2}$, and since anomaly cancellation imposes $A_{3,\text{heavy}}=-A_{3,\text{SM}}$, we end up with a total contribution Wd\^2(-(\_1\_2\^[-]{})(W\_[SU(3)\_C]{}\^2)+...) , \[aQCDcoupling\] which is obviously gauge-invariant ($a_\text{FN}$ exactly disappears from the $\log$), as it should.
On the other hand, induces a coupling between $a$ and the gluons, since -i(\_1\_2\^[-]{})=a , \[aFromLog\] where we used the canonical normalization for $a$ so that the axion decay constant[^6] can be read off from and : f\_a= . Besides the coupling to gluons, the heavy chiral fields also feed in the axion-photons coupling. A same line of reasoning gives us the latter: Wd\^2(-(\_1\_2\^[-]{})W\_[U(1)\_]{}\^2+...) , \[aPhotonCoupling\] where $A_\text{em,SM}=\frac{A_{1,\text{SM}}+A_{2,\text{SM}}}{2}$ is the MSSM electromagnetic anomaly, so that we understand that =-= , with the conventions of [@Marsh:2015xka]. For instance, model B has $E/N=8/3$, which is the same as in the DFSZ model [@Dine:1981rt; @Zhitnitsky:1980tq]. Thus, in this respect, our models’ predictions do not deviate qualitatively from those of usual flavourful axions.
Dominant couplings between the axion and the SM fermions arise at tree-level from , such that the (schematic) coupling between the axion and the SM fermions is as follows: h\_[ij]{} ()\^[n\_[ij]{}]{}\_[L,i]{}H\^[(c)]{}h\_[ij]{} e\^[in\_[ij]{} (-)]{}\_[L,i]{}H\^[(c)]{}h\_[ij]{} e\^[i]{}\_[L,i]{}H\^[(c)]{} , where we neglected radial degrees of freedom in the first step, and projected the scalar phase onto the physical axion in the second. We also identified the scale of axion-fermions coupling: f\_[ij]{}= , where we see that the axion couples more strongly to lighter generations, since those have larger charges, i.e. larger $n_{ij}$’s. The ratio between the axion coupling to gauge fields $C_a$ and the coupling to fermions $C_{ij}$ is \~=\~ . \[CoverCchiral\] As a comparison, flavourful axions models [@Ema:2016ops; @Calibbi:2016hwq] find \~ \[CoverCglobal\] (where $x_1-x_2$ should be understood as the $U(1)_\text{FN}$ charge of the flavon field). features qualitative differences with , for instance it is not only sensitive to $\abs{x_1-x_2}$, which sets the magnitude of the $U(1)_\text{FN}$ charges of the MSSM fields, but also to the absolute value of e.g. $x_2$, such that it contains non-trivial information about the UV physics. Nonetheless, provided the $x_i$’s take reasonable values, the magnitude of and are comparable (they are actually equal at order zero in $\epsilon$) and the phenomenological predictions of either kinds of models are qualitatively robust.
An upper bound can actually be imposed on $\langle\phi_{1,2}\rangle$ by requiring that the shift symmetry of the axion $a$ is of high enough quality [@PhysRevD.46.539; @Kamionkowski:1992mf; @Holman:1992us; @Fukuda:2017ylt; @Bonnefoy:2018ibr] to actually solve the strong CP problem once quantum gravity corrections [@Hawking:1987mz; @Giddings:1988cx; @Banks:2010zn; @Harlow:2018jwu; @Harlow:2018tng; @Fichet:2019ugl] are taken into account. Indeed, we started with gauge symmetries considerations and did not impose any global symmetry on the model. Consequently, we expect to be able to write some gauge invariant operator which would break the shift symmetry of the physical axion. On the other hand, the presence of the $U(1)_\text{FN}$($\times G_\text{SM})$ gauge symmetry may force such an operator to be of very high dimension such that it has no relevant impact on the axion dynamics.
For instance, in the model B discussed in section \[chiralExamples\], the first gauge-invariant operator one could write (beyond those such as which respect the axion shift symmetry) is[^7] c\_1\^[10]{} , \[axionMassOperator\] with $c$ a coupling constant. In the latter case for instance, to be consistent with the measured value of the $\theta$-angle of QCD, $\theta<10^{-10}$ [@Baker:2006ts] , we must ensure that: &>10\^5\
&\
v\_2(10\^[-5]{}& \^\^[-]{}\^[-5]{}m\_f\_)\^\~2\^[-]{}10\^[11]{} , \[protectQCD\] where $M_P$ is the reduced Planck mass. We immediately see that this is in tension with the perturbativity bound of section \[gaugeRunningSection\], even though not in strict contradiction since there are lots of undetermined order one numbers (e.g. the precise heavy fermion mass or the coefficient $c$). For instance, in our supersymmetric framework, would be present in the scalar potential if it is also present in the Kähler potential and SUSY is broken. Then c\~ , with $m_{3/2}$ the gravitino mass, such that the upper bound in can for instance increase by a factor $\sim5\times10^5$ if $m_{3/2}=10^{-4}$ eV, compatible with the gauge mediation of SUSY breaking. , or its analog in an other model, could also be generated from interference terms between superpotential terms, in which case similar increases of the upper bound may happen.
It is then presumably possible to satisfy both bounds if $v_2\sim 10^{11-13}$ GeV, also implying that explicit breaking of the Peccei-Quinn symmetry could be observable in future experiments aiming at better measuring the neutron (or proton) EDM [@Kirch:2013jsa; @Anastassopoulos:2015ura]. Furthermore, this value for $v_2$ implies a value for $f_a$ which is compatible with the fact that the flavourful axion makes up part or all of dark matter [@Preskill:1982cy; @Abbott:1982af; @Dine:1982ah] (see also [@Marsh:2015xka] for a review), and which is close to the values probed by precision flavour measurements, as discussed now.
Flavourful axion phenomenology {#axionPhenoSession}
------------------------------
The low-energy phenomenology of our flavourful axion is similar to the one of the flaxion/axiflavon of [@Ema:2016ops; @Calibbi:2016hwq; @Ema:2018abj]. The axion-induced flavour changing transitions of the type $d_i \to d_j + a$, where $d_i$ are $d$-type quarks, and $e_i \to e_j + a$, where $e_i$ are charged leptons, generate decays with the axion in the final state. Experimental limits on such processes set lower limits on the axion decay constant, for a fixed axion-induced flavour changing vertex. Flavour transitions in the quark sector lead to meson decays, the most constraining ones being $K^+ \to \pi^+ + a$ , $B^+ \to \pi^+ + a$. The first decay, for example, is bound experimentally [@Adler:2008zza] to be $\text{Br}(K^+ \to \pi^+ + a) < 7.3 \times 10^{-11}$, which leads to the limit $$f_a \geq 2 \times 10^{10} \text{ GeV} \times \frac{26}{N_\text{DW}} \left| \frac{(k_v^d)_{12}}{m_s} \right| \ ,
\label{bound1}$$ where $k_v^d$ are vector-like fermionic couplings to the axion[^8] $\frac{ia}{\sqrt{2} f_a} ( k_a^{\psi} {\bar \psi}_i \gamma_5 \psi_i + k_v^{\psi} {\bar \psi}_i \psi_i )$ and $N_\text{DW} = \frac{\sum_i (2 q_{Q_i} + q_{U_i} + q_{D_i})}{x_2\abs{x_1-x_2}}$ is the domain wall number.
Flavour transitions in the charged lepton sector lead to lepton number per species non-conserving processes , most constraining one being $\mu \to e + a + \gamma$, constrained experimentally to be $\text{Br}(\mu \to e + a + \gamma) < 1.1 \times 10^{-9}$, which leads to the bound [@Goldman:1987hy] $$f_a \geq 1 \times 10^{8} \text{ GeV}\times \frac{26}{N_{DW}} \left| \frac{(k_v^l)_{12}}{m_{\mu}} \right| \ .
\label{bound2}$$ Our perturbativity bounds due to the running effects of heavy fields $v_2 \gtrsim 10^{12}$ GeV are compatible with all these bounds. However, one-two orders of magnitude improvement of experimental data in the near future is expected and will start probing our models[^9].
Another source of flavour violation is the coupling of quarks and leptons to the $U(1)_\text{FN}$ gauge boson $Z'$. Baryon number cannot be violated in this way, otherwise the flavon would carry baryon number. Lepton number per species could be violated, but due to the high scale of $U(1)_\text{FN}$ symmetry breaking $v_2 \gtrsim 10^{12}$ GeV, $Z'$-induced lepton number non-conserving processes are currently unobservable.
Non-supersymmetric models {#nonSUSY}
-------------------------
We now briefly comment on non-SUSY chiral FN models, by again explicitly displaying such models for definiteness. We focus for simplicity on holomorphic models with two Higgs doublets, meaning that , once complemented by its hermitian conjugate, now defines the lagrangian of the theory, with $H_{u,d}$ referring to scalar fields and $Q_i,U_j,D_j,E_j$ to left-handed Weyl fermions. The heavy FN fields are similarly all fermionic, except $\phi_{1,2}$ which are scalars.
It is straightforward to check that the model A of section \[chiralExamples\] is also valid as a non-SUSY model, consistently with the findings of [@Alonso:2018bcg; @Smolkovic:2019jow] (indeed, higgsinos carry no charge under $U(1)_\text{FN}$, so they can be removed at no cost, and all the heavy fermions remain)[^10]. For the Yukawa matrices in , there are again models with two charged scalars, for instance if $x_1=1,x_2=\frac{1}{2}$, $h_u=h_d=3,X_Q=0,X_L=-\frac{1}{6}$ and $n_{Q,1}=5, n_{Q,2}=1, n_{Q,3}=0, n_{L,1}=0, n_{L,2}=2, n_{L,3}=0$ (so that $n_{U,1}=3, n_{U,2}=3, n_{U,3}=0,n_{D,1}=0, n_{D,2}=1, n_{D,3}=0,n_{E,1}=4, n_{E,2}=0, n_{E,3}=0$). With such charges the physical pseudoscalar contained in $\phi_{1,2}$, which has a QCD axion-like coupling to gluons, is too heavy to be a proper QCD axion since there can be operators such as \_1\^2 in the lagrangian. On the other hand, the “$\mu$-term” has the following form H\_uH\_d\_1\^2()\^4 , with $\Lambda$ a high scale, so that the pseudoscalar in $H_{u,d}$ is heavier than e.g. a TeV if v\_1\^2()\^4>\^2 , i.e. v\_1>(\^2)\^\_[=M\_P]{}210\^[13]{} . This may give a stronger bound on $v_{1,2}$ than the running of the SM gauge couplings: the latter is indeed slightly less severe than the ones seen in section \[gaugeRunningSection\] for SUSY models. Figure \[perturbativityRunningNonSUSY\] shows the running for the two models aforementioned in this section, and we see there that the heavy sector masses for which the hypercharge gauge coupling blows up at the Planck scale are reduced by a few orders of magnitude with respect to what appears on Figures \[perturbativityRunning\] and \[perturbativityRunningP\].
![Running coupling constants of the SM\
Left panel: first model discussed in section \[nonSUSY\], right panel: second model[]{data-label="perturbativityRunningNonSUSY"}](phiOverMNonSUSY.pdf "fig:")![Running coupling constants of the SM\
Left panel: first model discussed in section \[nonSUSY\], right panel: second model[]{data-label="perturbativityRunningNonSUSY"}](twoFieldsNonSUSY.pdf "fig:")![Running coupling constants of the SM\
Left panel: first model discussed in section \[nonSUSY\], right panel: second model[]{data-label="perturbativityRunningNonSUSY"}](legendRunning.png "fig:")
Conclusion
==========
We studied the gauging of a horizontal abelian symmetry generating the Froggatt-Nielsen mechanism, when the heavy fields in the UV completion of the mechanism are chiral with respect to this family symmetry. This for instance happens when the small parameter which explains the flavour hierarchies is composed of the vevs of two charged scalar fields which respectively mix and give masses to the heavy sector. The mixed anomalies between the Standard Model gauge group and the new symmetry are modified in this setup, such that the anomaly-free completions of the model are not the same as in the usual case when the heavy sector is vector-like.
We mostly focused on supersymmetric models, since their holomorphy properties usually do not leave much freedom for the anomalies to cancel. Unlike the vector-like heavy sector case, for which it has been shown that the minimal embedding of the FN symmetry is always anomalous at the level of the MSSM, with our chiral heavy sectors the mixed anomalies are enough disentangled from the mass matrices so that they sometimes vanish without adding a Green-Schwarz mechanism or any other spectator field than the ones which are necessary for the FN mechanism to take place. We gave specific examples where this “minimal” UV content is realized, and compared them to gauged vector-like models, with spectator fields cancelling the anomalies, to precisely illustrate what we mean by “minimal”: chiral models push Landau poles to the highest possible values, so that bounds on the input scales are the loosest possible, and they minimize the number of input scales in the problem. We also presented non-supersymmetric examples with the same behaviour.
Moreover, we emphasized the fact that chiral models often come with a physical axion mode, which has couplings typical of a flavourful QCD axion. In such models, the gauging of the FN symmetry makes it easy to protect the axion mass, which is a significant difference with respect to flavourful axions originating from a global FN mechanism. The qualitative axion phenomenology is similar to the one of global flavourful axion models, meaning that the axion couplings are mainly dictated by low-energy physics, which remains as a robust prediction. However, there are slight changes in the axion couplings to gauge fields, since the latter are already generated by the integrating-out of heavy FN fermions, not only by the SM ones. In addition, irrespective of the model, strong bounds on the input scales describing the heavy sector can be derived from the running of the (MS)SM gauge couplings, which imposes that the scale of spontaneous FN symmetry breaking is at least intermediate ($10^{12-13}$ GeV). Those lower bounds are stricter in gauged vector-like FN models, so that the axions arising in chiral models are maximally coupled/detectable. Nonetheless, such bounds are enough for the models to be automatically compatible with experimental results on flavour-changing processes, albeit not too high so that one will start scanning the couplings of the flavourful axions after the experimental sensitivity increases slightly. Finally, since the SM mass matrices make anomaly cancellation compatible with the MSSM gauge coupling unification, one can define anomaly-free models which preserve gauge coupling unification such that the couplings of the axion, anomaly cancellation and the gauge couplings running are all entangled.
Acknowledgments {#acknowledgments .unnumbered}
===============
Q.B. would like to thank the Institute of Theoretical Physics at the University of Warsaw for their financial support and their kind hospitality. E.D. was supported in part by the Agence Nationale de la Recherche project ANR Black-dS-String. S.P. thanks the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support via the Centro de Excelencia Severo Ochoa Program under Grant SEV-2016-0597 and Belen Gavela and Pablo Quilez for very useful discussions. The S.P. research was partially supported by the Munich Institute for Astro- and Particle Physics (MIAPP) of the DFG Excellence Cluster Origins (www.origins-cluster.de).
All possible superpotential terms {#fermionCouplingsAppendix}
=================================
In this appendix, we study the most general renormalizable couplings which can appear in the superpotential for the kind of models discussed in section \[sectionMain\]. We assume that $x_1-x_2\neq 0$, so that the Froggatt-Nielsen mechanism is operating, as well as R-parity.
With the matter content of Table \[Heavycharges\] (together with the replacement rules ), Table \[bilinears\] displays the renormalizable invariant-under-$G_\text{SM}$ terms which one may write down in the superpotential if they are also $U(1)_\text{FN}$-invariant (i.e. if they are invariant under the full gauge group).
[|c|c|c|]{} G\_&U(1)\_&\
&&\
.
\^Q\_[i]{}\^Q\_[j]{}\
\^u\_[i]{}\^u\_[j]{}\
\^d\_[i]{}\^d\_[j]{}\
\^L\_[i]{}\^L\_[j]{}
}&(i-j)x\_1-(i-j-1)x\_2&
i=jx\_2=0\
\
ijx\_1=
\
&&\
.
\^Q\_[i]{}\^Q\_[j]{}\
\^u\_[i]{}\^u\_[j]{}\
\^d\_[i]{}\^d\_[j]{}\
\^L\_[i]{}\^L\_[j]{}
}\_1&(i-j-1)(x\_1-x\_2)&i=j+1\
&&\
.
\^Q\_[i]{}\^Q\_[j]{}\
\^u\_[i]{}\^u\_[j]{}\
\^d\_[i]{}\^d\_[j]{}\
\^L\_[i]{}\^L\_[j]{}
}\_2&(i-j)(x\_1-x\_2)&i=j\
&&\
.
\^Q\_[i]{}\^u\_[j]{}H\_u\
\^Q\_[i]{}\^d\_[j]{}H\_d\
\^L\_[i]{}\^e\_[j]{}H\_d
}&(j-i-1)(x\_1-x\_2)&i=j-1\
&&\
.
\^Q\_[i]{}\^u\_[j]{}H\_d\
\^Q\_[i]{}\^d\_[j]{}H\_u\
\^L\_[i]{}\^e\_[j]{}H\_u
} &(i-j+1)x\_1-(i-j-1)x\_2+h\_u+h\_d&
i=j-1x\_2=\
\
ij-1x\_1=
\
&&\
There, the three sets of possibilities in the middle of the Table are those which are required to implement the FN mechanism as emphasized in . They can always be written down provided the values of the subscripts $i$ and $j$ are chosen appropriately. The other two can appear when $x_1,x_2,h_u$ and $h_d$ verify specific arithmetical relations. For instance, they are not allowed in the two models discussed in section \[chiralExamples\].
Anomalies and unification {#anomaliesUnificationAppendix}
=========================
In this appendix, we show that the “unification” relations can be reexpressed in terms of the mixed anomalies in the MSSM sector. To see this, let us recall (see the discussion around in section \[accidentalAxion\]) that the contribution to the axion couplings generated by the integrating out of the heavy FN sector is: Wd\^2(-(\_2)(W\_A\^2)) , where again $\cC=1,2$ respectively for a $SU(N)$ or an abelian factor of the gauge group and, with our conventions, $\frac{A_{A,\text{heavy}}}{x_2}$ counts the heavy chiral fields which are charged under the gauge factor $A$ (with multiplicity and charge squared for an abelian gauge factor). Thanks to the holomorphy in our SUSY model, the same anomaly coefficients appear in the $\beta$-function for the gauge couplings: =+()-() . Respecting the gauge unification of the MSSM thus demands A\_[SU(3)\_C,]{}=A\_[SU(2)\_W,]{}=A\_[U(1)\_Y,]{} , which, for anomaly-free models, is extended to A\_[SU(3)\_C,]{}=A\_[SU(2)\_W,]{}=A\_[U(1)\_Y,]{} . This relation agrees well with phenomenological mass matrices, and is the one required to implement the Green-Schwarz mechanism [@Ibanez:1994ig; @Jain:1994hd; @Binetruy:1994ru; @Dudas:1995yu], consistently with the fact that the phase $\theta_2$ of $\phi_2$ does generate a GS mechanism here once the heavy fields are integrated out.
As a consistency check, it is straightforward to check that model B of section \[chiralExamples\] indeed verifies A\_[SU(3)\_C,]{}&=A\_[SU(2)\_W,]{}=A\_[U(1)\_Y,]{}\
&=-A\_[SU(3)\_C,]{}=-A\_[SU(2)\_W,]{}=-A\_[U(1)\_Y,]{}=-180 .
Minimal vector-like models {#vectorlikeAppendix}
==========================
We detail in this appendix the construction of and the bounds on the vector-like counterparts to model A and B.
A vector-like counterpart to model A is a model which has one SM singlet (which amounts to the choice $x_1=1$ and $x_2=0$) and the matrices . Thus, the anomalies from the MSSM(+FN) sector are A\_[3,]{} =& 18 - 3 (h\_u + h\_d)\
A\_[2,]{}=& -16+3(3X\_Q+X\_L)+h\_u + h\_d\
A\_[1,]{}=& 64-3(3X\_Q+X\_L) -7(h\_u + h\_d)\
A’\_[1,]{}=& -128 +X\_Q(6h\_d-12h\_u+36)+X\_L(6h\_d-12)+64h\_u-40h\_d+5(h\_d\^2-h\_u\^2) , \[SManomaliesVectorModelA\] the last coefficient referring to the $U(1)_Y\times U(1)_\text{FN}^2$ anomaly. Restricting ourselves to spectators $(\chi^i,\tilde\chi^i)$ which get their mass from couplings to the singlet $\phi_1$ as follows W c\_[i]{}\_1\^i\^i , and which live in the singlet or fundamental representations of the SM gauge groups (as shown in Table \[spectatorFiedsArray\]),
$\begin{array}{|c|c|c|c|c|}
\hline
&SU(3)_C&SU(2)_W&U(1)_Y&U(1)_\text{FN}\\
\hline
\chi^i&\textbf{3} \text{ or } \textbf{1}&\textbf{2} \text{ or } \textbf{1}&y_{\chi^i}&q_{\chi^i}\\
\tilde\chi^i&\overline{\textbf{3}} \text{ or } \textbf{1}&\textbf{2} \text{ or } \textbf{1}&-y_{\chi^i}&-q_{\chi^i}+1\\
\hline
\end{array}$
their contribution to the anomalies are A\_[3,]{} =& \_i(1+\^i\_2)\^i\_3\
A\_[2,]{}=& \_i(1+2\^i\_3)\^i\_2\
A\_[1,]{}=& \_i(1+\^i\_2)(1+2\^i\_3)y\_[\^i]{}\^2\
A’\_[1,]{}=& \_i(1+\^i\_2)(1+2\^i\_3)y\_[\^i]{}\[q\_[\^i]{}\^2-(q\_[\^i]{}-1)\^2\] , \[SpectatorAnomaliesVectorModelA\] where $\delta^i_{2/3}$ is equal to one if the corresponding spectator field is in the fundamental representation of $SU(3)_C/SU(2)_W$, and zero if it is a singlet. In particular, $A_{1-2-3,\text{spect.}}$ are positive, and $A_{2-3,\text{spect.}}$ are integer, such that $A_{1-2-3,\text{SM}}$ should be negative and $A_{2-3,\text{SM}}$ should be integer for the anomalies to cancel. One can also write A\_[1,]{}+A\_[2,]{}=12+2A\_[3,]{} , such that $A_{1,\text{SM}}+A_{2,\text{SM}}\leq0\implies A_{3,\text{SM}}\leq-6$, which in turn implies that at least six pairs of spectator triplets will contribute to the running of the colour gauge coupling. Such additional particles already have a significant impact on the bounds implied by the $SU(3)_C$ gauge coupling, as illustrated in Figure \[perturbativityRunningVectorLike\].
![Running coupling constant of $SU(3)_C$ in model A when $v_2=10^{14}$ GeV,\
and in its gauged vector-like counterpart with six triplets when $v_2=1.5\times10^{15}$ GeV, assuming $m_\text{soft}=1$ TeV[]{data-label="perturbativityRunningVectorLike"}](runningVectorLike.pdf "fig:")![Running coupling constant of $SU(3)_C$ in model A when $v_2=10^{14}$ GeV,\
and in its gauged vector-like counterpart with six triplets when $v_2=1.5\times10^{15}$ GeV, assuming $m_\text{soft}=1$ TeV[]{data-label="perturbativityRunningVectorLike"}](legendVectorLike.png "fig:")
All the anomalies can be cancelled here by choosing[^11] $h_u=h_d=4$, $X_Q=0$, $X_L=\frac{8}{3}$ and by adding exactly six pairs of spectator coloured triplets, singlets under $SU(2)_W$ and without any hypercharge. In this case, the hypercharge running is unchanged with respect to the chiral case, while it gave the strongest constraint in Figure \[perturbativityRunning\], so that the chiral model is as much constrained as (or only marginally less constrained than) its vector-like counterpart, although it contains less heavy particles.
Let us now turn to the derivation of the bounds on vector-like counterparts to model B, meaning models which have two SM singlets of $U(1)_\text{FN}$ charges $-1$ and $-10$, the matrices and $SU(5)$-like unification as in . Assuming that the charges of the MSSM particles are the same as in model B and that the Yukawa couplings are all expressed in terms of the same singlet, then normalizing all the $U(1)_\text{FN}$ charges such that the Yukawa couplings have natural integer charges (i.e. using with $x_1-x_2=1$), the MSSM anomalies are A\_[3,]{} =& 17 - 3 (h\_u + h\_d)\
A\_[2,]{}=& -19+3(3X\_Q+X\_L)+h\_u + h\_d\
A\_[1,]{}=& -3(3X\_Q+X\_L) -7(h\_u + h\_d)\
A’\_[1,]{}=& -135 +X\_Q(6h\_d-12h\_u+38)+X\_L(6h\_d-12)+68h\_u-40h\_d+5(h\_d\^2-h\_u\^2) . \[SManomaliesVectorModelA\] With our assumptions and normalizations, model B contains either a SM singlet of $U(1)_\text{FN}$ charge $-1$ and an other one of charge $-10$, or a SM singlet of $U(1)_\text{FN}$ charge $-1$ and an other one of charge $-\frac{1}{10}$. $A_{1-2-3,\text{SM}}$ should be negative in both cases, and $A_{2-3,\text{SM}}$ should be either integer or multiples of $\frac{1}{10}$, for the same reasons as presented previously. Since the running of the hypercharge coupling gives the strongest constraint in Figure \[perturbativityRunningP\], the bound it induces will strengthen if at least a spectator field has a non-vanishing hypercharge[^12]. Assuming the opposite would imply $A_{1,\text{spect.}}=A_{1,\text{SM}}=0$, which gives (since $A_{1,\text{SM}}+A_{2,\text{SM}}-2A_{3,\text{SM}}=\frac{40}{3}$) A\_[2,]{}-2A\_[3,]{}= , whereas this quantity should be a multiple of $1$ or $\frac{1}{10}$. We thus conclude that one needs at least one spectator with a non-zero hypercharge, which would in turn make the running of the hypercharge coupling a bit steeper than it was in the chiral model[^13].
In order to make this quantitative, we scanned over all possible realizations compatible with the assumptions listed at the beginning of this appendix, and we found that the least constrained vector-like models must have $v_2\geq4.5\times 10^{13}$ GeV for the unification to happen within the perturbative regime, more than one order of magnitude above the chiral model. A realization of this is obtained as follows: choose $h_u+h_d=17$, $3X_Q+X_L=-\frac{32}{3}$ and $(\phi_1,\phi_2)$ of charges $(-1,-10)$, add a set of $8$ $Q$-like and $1$ $L$-like $SU(5)$ heavy multiplets, use $6$ $Q$-like multiplets among those to generate the mass matrices and couple the rest in a chiral way to $\phi_1$ and $\phi_2$, distributing the remaining available superfields as in Table \[realizationVectorLikeMinimal\].
$\begin{array}{|c|c|c|c|}
\hline
\text{Heavy superfield type}&\text{Number of (chiral) spectators}&\text{coupled to }\phi_1&\text{coupled to }\phi_2\\
\hline
Q&2&1&1\\
U&2&2&0\\
D&1&0&1\\
E&2&2&0\\
L&1&1&0\\
\hline
\end{array}$
This way, $A_1=A_{1,\text{SM}}+A_{1,\text{spect.}},A_2$ and $A_3$ all vanish, and one can choose $X_Q$ so that $A'_1$ vanishes as well.
[^1]: Our definitions and conventions for the MSSM superfields can be found in Table \[MSSMcharges\].
[^2]: Many order one coefficients have been and will be dropped, which may change slightly the estimated orders of magnitude. However, they do not change qualitatively the anomaly discussion.
[^3]: All the operators we consider in this paper are either perturbatively generated or present in the UV.
[^4]: For an explicit derivation, see e.g. [@Anastasopoulos:2006cz] or the appendix D of [@Bonnefoy:2018ibr].
[^5]: This condition can be rewritten in terms of the standard model anomalies, see appendix \[anomaliesUnificationAppendix\].
[^6]: The domain of $a$ is given by $a=a+2\pi f$. In the model defined around , $f\equiv\frac{v_1v_2}{\sqrt{v_1^2 +100v_2^2}}\times\min\{\abs{10m-n},(m,n)\in\mathbb{Z}^2\}=\frac{v_1v_2}{\sqrt{v_1^2 +100v_2^2}}$. Thus, $N_\text{DW}=\frac{A_{3,\text{SM}}}{x_2\abs{x_1-x_2}}=2$ in this model.
[^7]: An other dimension 11 option would be $H_uH_d\overline\phi_1^{9}$ but this is more suppressed since the weak scale is much below $\langle\phi_2\rangle$.
[^8]: In the notations of section \[accidentalAxion\], we have $k^\psi\sim\frac{f_a}{f_{ij}}$, estimated in .
[^9]: As we emphasized in section \[gaugeRunningSection\], for a given vector-like gauged FN model with an axion, the bound on $v_2$ may increase (perhaps weakly) with respect to those derived in chiral models, so that the axion is in principle less detectable than ones from chiral models.
[^10]: Since the Higgs fields carry no $U(1)_\text{FN}$ charge, one of them can be discarded by defining e.g. $H_u=H_d^c$.
[^11]: We also see in passing that the $\mu$-term cannot be gauge-invariant in such a one-singlet vector-like model, since $h_u+h_d=0\implies A_{3,\text{SM}}>0$.
[^12]: The bound could also increase due to contributions of the other gauge couplings.
[^13]: This conclusion might be evaded, and the running brought back to the one of the chiral model, by relaxing some of the assumptions we made, for instance if one starts using both singlets to generate the mass matrices, contrary to what we assumed here. On the other hand, this means that the $U(1)_\text{FN}$ charges of some fields should be modified as well. Indeed, if the two singlets have charges $-1$ and $-10$, for certain charges of the Yukawa couplings such as the ones we choose in this appendix, only the singlet of charge $-1$ can enter the Yukawas.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Recently, graph convolutional network (GCN) has been widely used for semi-supervised classification and deep feature representation on graph-structured data. However, existing GCN generally fails to consider the local invariance constraint in learning and representation process. That is, if two data points $X_i$ and $X_j$ are close in the intrinsic geometry of the data distribution, then their labels/representations should also be close to each other. This is known as *local invariance assumption* which plays an essential role in the development of various kinds of traditional algorithms, such as dimensionality reduction and semi-supervised learning, in machine learning area. To overcome this limitation, we introduce a graph Laplacian GCN (gLGCN) approach for graph data representation and semi-supervised classification. The proposed gLGCN model is capable of encoding both graph structure and node features together while maintains the local invariance constraint naturally for robust data representation and semi-supervised classification. Experiments show the benefit of the benefits the proposed gLGCN network.'
author:
- |
Bo Jiang, Doudou Lin\
School of Computer Science and Technology\
Anhui University\
Hefei, China\
`jiangbo@ahu.edu.cn`\
bibliography:
- 'nmfgm.bib'
title: 'Graph Laplacian Regularized Graph Convolutional Networks for Semi-supervised Learning'
---
Introduction
============
Given a graph $G(V, E)$ with $V$ denoting the $n$ nodes and $E$ representing the edges. Let $A \in \mathbb{R}^{n\times n}$ be the corresponding adjacency matrix, and $X=(X_1,X_2,\cdots X_n)\in \mathbb{R}^{p\times n}$ be the collection of node features where $X_i$ denotes the feature descriptor for node $v_i\in V$. For semi-supervised learning tasks, let ${L}$ indicates the set of labelled nodes and $Y_{{L}}$ be the corresponding labels for labelled nodes. The aim of semi-supervised learning is to predict the labels for the unlabelled nodes.
Graph Laplacian regularization
------------------------------
One kind of popular method for semi-supervised learning problem is to use graph-based semi-supervised learning, where the label information is smoothed over the graph via graph Laplacian regularization [@belkin2006manifold; @zhu2003semi] i.e., $$\label{EQ:GLR}
\mathcal{L} = \mathcal{L}_{\mathrm{label}} + \lambda \mathcal{L}_{\mathrm{reg}}$$ Here $ \mathcal{L}_{\mathrm{label}}$ and $\mathcal{L}_{\mathrm{reg}}$ are defined as, $$\label{EQ:fit}
\mathcal{L}_{\mathrm{label}} = \sum\nolimits_{i\in L} l(Y_i, f(X_i)) \ \ \ \ \ \mathrm{and} \ \ \ \ \mathcal{L}_{\mathrm{reg}} = \sum^n\nolimits_{i,j=1}S_{ij} \|f(X_i) - f(X_j)\|^2 $$ where $l(\cdot)$ denotes some standard supervised loss function and $\mathcal{L}_{\mathrm{reg}}$ is called as graph Laplacian regularization. Function $f(X_i)$ denotes the label prediction of node $v_i$ and $S_{ij}$ denotes some kind of relationship (e.g., affinity and similarity) between graph node $v_i$ and $v_i$. We can set $S$ as adjacency matrix $A$ or some other graph construction. One traditional graph is to use a k nearest neighborhood graph with edge weighted by some kernel metric (e.g., Gaussian kernel) $K(X_i, X_j)$ between feature $X_i$ and $X_j$. Parameter $\lambda >0$ balances two terms. The objective function Eq.(1) encourages that if two data points $X_i$ and $X_j$ are close in data distribution, then their corresponding labels should also be close with each other.
Graph convolutional network
---------------------------
Recently, graph convolutional network (GCN) [@defferrard2016convolutional; @kipf2016semi] has been proposed for semi-supervised tasks. It aims to seek a nonlinear function $f(X,A)$ to predict labels for unlabelled nodes. It contains several propagation layers and one final perceptron layer together. Given any input feature $X$ and graph structure (adjacency matrix) $A$, GCN conducts the following layer-wise propagation rule [@kipf2016semi], $$\begin{aligned}
& X^{(1)} = \mathrm{ReLu}(\widetilde{A}XW^{(0)}) \nonumber \\
& \,\,\,\,\,\,\,\, \cdots \nonumber \\
& X^{(K)} = \mathrm{ReLu}(\widetilde{A}X^{(K-1)}W^{(K-1)}) \\
&Z = \mathrm{softmax} (\widetilde{A}X^{(K)} W^{(K)} ) \nonumber\end{aligned}$$ Here, $\widetilde{A}=\bar{D}^{-1/2}\bar{A}\bar{D}^{-1/2}$ and $\bar{A}=A +I$, where $I$ is the identity matrix and $\bar{D}$ is a diagonal matrix with $\bar{D}_{ii}=\sum_j\bar{A}_{ij}$. $\{X^{(1)}, X^{(2)},\cdots X^{(K)}\}$ denotes the feature output of the different layers and $Z$ is the label output of the final layer where $Z_i$ is the label indication vector of the node $v_i$. For semi-supervised learning, the optimal weights $\{W^{(0)}, W^{(1)},\cdots W^{(K)}\}$ can be trained by minimizing the following cross-entropy loss function over all labeled nodes $L$. $$\mathcal{L}_{\mathrm{GCN}} = -\sum\nolimits_{i\in L} \sum^d\nolimits_{j=1} Y_{ij}\mathrm{ln} Z_{ij}$$ **Remark.** When $W^{(k)}\in \mathbb{R}^{d_{k-1}\times d_k}$ and $d_{k-1}<d_k$, the above GCN provides a series of low-dimensional embedding $X^{(k)}$ for the original input feature $X$.
Graph Laplacian GCN
===================
In this section, we present two types of graph Laplacian GCN. Inspired by traditional graph based semi-supervised learning model, we first propose a graph Laplacian GCN for robust semi-supervised learning. In addition, motivated by manifold assumption, we propose to incorporate manifold regularization in GCN feature representation.
Graph Laplacian label prediction
--------------------------------
GCN [@kipf2016semi] predicts the labels for unlabelled nodes by using label propagation on graph. One limitation of GCN is that it fails to consider the local consistency of nodes with similar features in label propagation, i.e., if the features of neighboring node $v_i$ and $v_j$ are similar, then their corresponding labels should also be close. This point has commonly used in traditional graph based semi-supervised learning model [@belkin2006manifold; @weston2012deep]. This motivate us to propose an improved graph Laplacian GCN (gLGCN), which aims to conduct local label propagation via GCN while maintains the local consistency via graph Laplacian regularization. This can be obtained by optimizing the following loss function, $$\begin{aligned}
&\mathcal{L}_{\mathrm{gLGCN}}(Z) = \mathcal{L}_{\mathrm{GCN}}(Z) + \lambda \mathcal{L}_{\mathrm{reg}}(Z)\nonumber\\
&= -\sum\nolimits_{i\in L} \sum^d\nolimits_{k=1} Y_{ik}\mathrm{ln} Z_{ik} + \lambda \sum^n\nolimits_{i,j=1}S_{ij}\|Z_i - Z_j\|^2
\end{aligned}$$ where $Z_i$ denotes the $i$-th row of matrix $Z$ and $S_{ij}$ denotes the similarity between node $i$ and $j$, as mentioned in Eq.(2).
![ Architecture of the proposed Graph Laplacian GCN for robust data representation. []{data-label="fig::lambda"}](demo_LapGCN.pdf){width="99.00000%"}
Graph Laplacian feature representation
--------------------------------------
On the other hand, to boost the effectiveness of the learned deep representation, similar to [@weston2012deep; @te2018rgcnn], we also incorporate the graph Laplacian regularization into the feature generation layer and introduce a regularization loss, as shown in Figure 1.
The regularization subnetwork can be regarded as a kind of pair-wise siamese network, which first takes two low-dimensional features of the final layer $X^{(K)}_i$ and $X^{(K)}_j$ as input, and then calculates the distance between them. If the similarity $S_{ij}$ between node $v_i$ and $v_j$ is larger, then the generated low-dimensional feature representation $X^{(K)}_i$ and $X^{(K)}_j$ should be close with each other. This is known as manifold assumption which has been widely used in dimensionality reduction in machine learning area. We adopt a loss function as $$\begin{aligned}
\mathcal{L}_{\mathrm{reg}}(X^{(K)}) = \sum^n\nolimits_{i,j=1}S_{ij}\|X^{(K)}_i - X^{(K)}_j\|^2
\end{aligned}$$ The above semi-supervised learning and manifold regularized feature representation are optimized at the same time in a unified network. Thus, we can write the total objective function as $$\begin{aligned}
\mathcal{L} = \mathcal{L}_{\mathrm{GCN}}(Z) + \lambda \mathcal{L}_{\mathrm{reg}}(X^{(l)})\end{aligned}$$ where $\lambda >0$ is the balanced parameter.
**Comparison with related works.** Our model is different from previous works [@te2018rgcnn; @weston2012deep] in several aspects. First, we focus on semi-supervised learning problem. The proposed regularization on the final layer provides a label propagation for semi-supervised learning problem. Second, in our model, the feature $X^{(K)}_i$ provides a low-dimensional representation for graph node $v_i$ and thus our model provides a kind of local preserving low-dimensional embedding for semi-supervised learning. Third, the proposed model conducts feature propagation on graph $\widetilde{A}$ and linear projection via $W^{(k)}$ together in each layer of the network. In contrast, in previous work [@weston2012deep], it only conducts linear projection in each layer. Overall, it integrates the benefits of work [@weston2012deep] and GCN [@kipf2016semi] simultaneously for semi-supervised learning.
In addition, for semi-supervised learning, the labels of some nodes are known. We can define the label correlation $C_{ij}$ as follows, $$C_{ij} =
\begin{cases}
1 & \text{if} \ \ v_i, v_j \in L \ \ \text{and} \ \ Y_i=Y_j\\
-\alpha & \text{if} \ \ v_i, v_j \in L \ \ \text{and} \ \ Y_i\neq Y_j\\
0 & \text{otherwise}
\end{cases}$$ Based on $C$, we can incorporate the label information via the regularization loss as $$\begin{aligned}
\mathcal{L}_{\mathrm{reg}}(X^{(K)}) = \sum^n\nolimits_{i,j=1}C_{ij}\|X^{(K)}_i - X^{(K)}_j\|^2
\end{aligned}$$ which is similar to the widely used triplet loss function used in deep networks.
Evaluation
==========
To evaluate the effectiveness of the proposed gLGCN network. We follow the experimental setup in work [@Yang:2016] and test our model on the citation network datasets including Citeseer, Cora and Pubmed [@sen2008collective] The detail introduction of datasets used in our experiments are summarized in Table 1. The optimal regularization parameter $\lambda$ is chosen based on validation.
Dataset Type Nodes Edges Classes Features Label rate
---------- ------------------ ------- ------- --------- ---------- ------------
Citeseer Citation network 3327 4732 6 3703 0.036
Cora Citation network 2708 5429 7 1433 0.052
Pubmed Citation network 19717 44338 3 500 0.003
: Dataset description in experiments
We compare against the same baseline methods including traditional label propagation (LP) [@zhu2003semi], semi-supervised embedding (SemiEmb) [@weston2012deep], manifold regularization (ManiReg) [@belkin2006manifold], Planetoid [@Yang:2016] and graph convolutional network (GCN) [@kipf2016semi]. For GCN [@kipf2016semi], we implement it using the pythorch code provided by the authors. For fair comparison, we also implement our gLGCN by using pythorch. Results for the other baseline methods are taken from work [@Yang:2016; @kipf2016semi] For component analysis, we implement it with three versions, i.e., 1) gLGCN-F that incorporates Laplacian regularization in feature representation. 2) gLGCN-L that incorporates Laplacian regularization in label prediction. 3) gLGCN-F-L that incorporates Laplacian regularization in both feature representation and label prediction. Table 2 summarizes the comparison results. Here we can note that, our gLGCN performs better than traditional LP, ManiReg and GCN, which clearly indicates the benefit of the proposed gLGCN network method.
Methond Citeseer Cora Pubmed
------------------------------ ---------- ------ --------
ManiReg[@belkin2006manifold] 60.1 59.5 70.7
SemiEmb[@weston2012deep] 59.6 59.0 71.1
LP[@zhu2003semi] 45.3 68.0 63.0
Planetoid[@Yang:2016] 64.7 75.7 77.2
GCN[@kipf2016semi] 70.4 81.4 78.6
gLGCN-F 70.8 82.2 79.2
gLGCN-L 71.3 82.7 79.2
gLGCN-F-L 71.4 83.3 79.3
: Comparison results on different datasets
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Gas-phase NMR spectra demonstrating the effect of weak intermolecular forces on the NMR shielding constants of the interacting species are reported. We analyse the interaction of the molecular hydrogen isotopomers with He, Ne, and Ar, and the interaction in the He–CO$_2$ dimer. The same effects are studied for all these systems in the [*ab initio*]{} calculations. The comparison of the experimental and computed shielding constants is shown to depend strongly on the treatment of the bulk susceptibility effects, which determine in practice the pressure dependence of the experimental values. Best agreement of the results is obtained when the bulk susceptibility correction in rare gas solvents is evaluated from the analysis of the He-rare gas interactions, and when the shielding of deuterium in D$_2$–rare gas systems is considered.'
author:
- 'Piotr Garbacz, Konrad Piszczatowski, Karol Jackowski, Robert Moszynski'
- 'Micha[ł]{} Jaszuński'
bibliography:
- 'artikler.bib'
- 'propert.bib'
- 'mppubl.bib'
- 'mjpubl.bib'
- 'spinrev.bib'
title: 'Weak intermolecular interactions in gas-phase NMR'
---
Introduction {#sec1}
============
The importance of intermolecular interactions in physics, chemistry, and biology does not need to be stressed. Intermolecular potentials determine the properties of non-ideal gases, (pure) liquids, solutions, molecular solids, and the behavior of complex molecular ensembles encountered in biological systems. They describe the so-called non-bonded contributions, as well as the special hydrogen bonding terms, that are part of the force fields used in simulations of processes such as enzyme-substrate binding, drug-receptor interactions, etc. A few examples showing important applications of intermolecular potentials include the Monte Carlo and molecular dynamics simulations of biological systems, studies of processes in the earth’s atmosphere, or interstellar chemistry.
Also the NMR spectra, in particular the observed chemical shifts, depend not only on the molecular structure but also on the intermolecular forces. The changes due to the environment are difficult to interpret theoretically and make the comparison of the computed and observed spectra unreliable. The role of the intermolecular forces is undoubtedly the largest in the condensed phase, and much smaller in dilute gas-phase solutions. Moreover, it is particularly small if we analyse a system where only weak van der Waals intermolecular forces play a significant role. In this work, we describe gas-phase NMR spectra for such systems, analyse the dependence of the observed shielding constants on the intermolecular forces, and present [*ab initio*]{} calculations which describe this dependence.
Early NMR studies in the gas phase were reviewed by Rummens [@fhar-rev], another review has been written in 1991 by Jameson [@cjjcr91]. However, the role of the intermolecular interactions in the gas phase was almost exclusively interpreted on the basis of binary collision gas model introduced by Raynes, Buckingham, and Bernstein [@wtradbhjbjcp36]. In this RBB model the change in the shielding constant is qualitatively described as a sum of contributions due to the bulk susceptibility, neighbor-molecule magnetic anisotropy, polar effects, and van der Waals effects. At present, by applying state-of-the-art methods of quantum chemistry we should be able to predict accurately the small changes of the shielding constants due to weak intermolecular forces. For the first time this should be possible within an [*ab initio*]{} approach, which is in principle more reliable than the standard methods used to describe for instance the solvent effects in liquids, such as various polarizable continuum models based on classical approximations. Therefore, a study of gas-phase model systems has a specific advantage for the comparison between experiment and theory.
Theoretical studies of the interaction-induced changes in the NMR parameters are scarce, and mostly restricted to supermolecule calculations of the interaction-induced shielding constants and spin-spin coupling constants; see, e.g. Refs. [@abmjthkrcpl250; @mpjscp234; @mpjscp248; @kjmwmpjsjpca104; @mparmp100; @mbmkolmvgmdrsjcc20; @mhplnrjjjvjcp121; @mhplmihjajjvjcp127] for typical applications. To our knowledge only one paper [@abmjthkrcpl250] analysed (comparing the theory with the numerical results) the asymptotic long-range behavior of the shielding constant and its anisotropy in a dimer. Most of the papers reporting [*ab initio*]{} calculations of the NMR parameters that could directly be compared with the gas phase NMR experiment were devoted to studies of atom-atom interactions [@aamjarjcp126; @mparmp100; @mbmkolmvgmdrsjcc20; @mhplnrjjjvjcp121; @mhplmihjajjvjcp127] (this is in sharp contrast with the electric properties of molecular complexes for which a general long-range theory and applications to the optical and dielectric properties of gases are available [@tgahrmpeswavdamp89; @rmtgahavdacpl247; @arscdmjlcbfchmp104]). There are very few [*ab initio*]{} studies of the NMR effects of weak interactions between a molecule and an atom or two molecules. The whole property surface has been computed for the interactions in the C$_{2}$H$_{2}$–He and C$_{2}$H$_{2}$–H$^+$ complexes [@mpjscp248], but no Boltzmann averaging has been performed; NMR properties were also examined [@mpjscp234] for the optimized geometries of other binary complexes of acetylene.
Also on the experimental side not too much has been done. The effects of weak intermolecular interactions on NMR shielding of $^1$H, $^2$H and $^3$He in gases are small and buried in the much larger bulk susceptibility correction, therefore a detailed analysis of such systems is practically impossible and the RBB model has mostly been used. We recall here that for $^3$He the effect of the weak interactions is particularly small and difficult to observe (see for instance the study of gas-to-liquid shifts [@rspdrkmjjjmra101]). For $^{21}$Ne the precision of the NMR measurements is limited because the magnetically active isotope has a large nuclear quadrupole moment; for argon the only magnetically active $^{39}$Ar isotope is radioactive. On the other hand, for $^{129}$Xe the effects are very large. They have been observed in the xenon dimer and nearly quantitative agreement of theory with experiment was reached in state-of-the-art [*ab initio*]{} calculations [@mhplnrjjjvjcp121; @mhplmihjajjvjcp127]. Also density functional theory (DFT) calculations for Xe-rare gas dimers yield satisfactory agreement with experimental data, see Refs. [@cjjakjsmcjcp62; @cjjdnsacdjcp118]. Most recently, the chemical shift of Xe dissolved in liquid benzene was studied in the calculations combining the DFT methods with the classical molecular dynamics [@sspkrmjhpbmstca129]. However, there are no similar studies of atom-molecule systems applying well established state-of-the art wavefunction methods and comparing the results with known experimental data.
In this paper we fill this gap and report a joint experimental and theoretical study of the gas phase shielding constants in the mixtures of atomic and molecular gases. We study the effects resulting from the weak interactions between a molecule and an atom in series of model systems: H$_2$–He, H$_2$–Ne and H$_2$–Ar dimers and their deuterium-substituted isotopomers and in He–CO$_2$. For the selected magnetically active nuclei—$^1$H, $^2$H, $^3$He and $^{13}$C— we observe the dependence of the NMR spectrum on the density of the solvent gas, which enables next a comparison of the [*ab initio*]{} and experimental results. In the analysis of the NMR spectra we take into account the bulk susceptibility correction, dependent on the magnetizability of the medium and on the shape of the NMR sample [@rspdrkmjjjmra101; @japwgshjbbook]. In the case of the weakly interacting systems which we study this correction dominates in the density dependence of the spectrum and its proper description is essential when we extract the information on the role of the intermolecular interactions from the experimental data and compare the experimental and computed quantities.
The plan of this paper is as follows. We start with the virial expansion of the shielding constant in terms of the gas density and discuss all quantities needed on the route from the theory to a direct comparison with the experiment. This is thoroughly discussed in sec. \[sec2\]. The details of the computational procedures adopted in the [*ab initio*]{} calculations, fitting of the interaction potential and shielding surfaces, and some numerical integration procedures will be discussed in sec. \[sec3\]. The experiment is described in detail in sec. \[sec4\]. The results of the measurements and calculations are reported and compared in sec. \[sec5\]. Finally, sec. \[sec6\] concludes our paper.
Shielding constants in the gas-phase solutions {#sec2}
==============================================
For a binary mixture of a gas $A$, containing the nucleus $X$ whose shielding $\sigma^A(X)$ is observed, and gas $B$ as the solvent, $\sigma^A(X)$ can be expressed as [@wtradbhjbjcp36]: $$\sigma^A(X) = \sigma_0^A(X) + \sigma_{1}^{AA}(X) \rho_A + \sigma_{1}^{AB}(X) \rho_B
\label{eq:AB}$$ where $\rho_A$ and $\rho_B$ are the densities of $A$ and $B$, respectively, and $\sigma_0^A(X)$ is the shielding in the zero-density limit. All higher terms in Eq. (\[eq:AB\]), which represents a truncated virial expansion, can safely be neglected if the experimental dependence of the shielding on the density is linear. The coefficients $\sigma_{1}^{AA}(X)$ and $\sigma_{1}^{AB}(X)$ are then the only terms responsible for the medium effects. They contain the bulk susceptibility corrections, $\sigma_{{\rm 1bulk}}^A$ and $\sigma_{{\rm 1bulk}}^B$, and the terms directly taking account of the intermolecular interactions during the binary collisions of the $A-A$ and $A-B$ molecules: $\sigma_1^{A-A}(X)$ and $\sigma_1^{A-B}(X)$, respectively. The shielding parameters in Eq. (\[eq:AB\]) are temperature dependent and for this reason all the present measurements are performed at the constant temperature of 300 K. Moreover, in the experiments the density of $A$, $\rho_A$, is always kept very low in order to eliminate the solute-solute molecular interactions and Eq. (\[eq:AB\]) can be simplified to: $$\sigma^A(X) = \sigma_0^A(X) + \sigma_{1}^{AB}(X) \rho_B
\label{eq:B}$$ where $$\sigma_{1}^{AB}(X) = \sigma_{{\rm 1bulk}}^B + \sigma_1^{A-B}(X) .
\label{eq:ABA-B}$$
Fig. \[figd2\] displays, as an example, the dependence of the helium and deuteron magnetic shielding (given with respect to the isolated systems) on the density of the rare gas solvent in gaseous solutions. The plots in Fig. \[figd2\] are linear, which proves that Eq. (\[eq:B\]) is a valid approximation and allows the determination of the $\sigma_0^A(X)$ and $\sigma_1^{AB}(X)$ shielding parameters. The part of the shielding constant $\sigma^A(X)$ which is exclusively due to pair intermolecular interactions between the solute and solvent molecules is given by $\sigma_1^{A-B}(X)$. An inspection of Eqs. (\[eq:B\]) and (\[eq:ABA-B\]) shows that $\sigma_1^{A-B}(X)$ can be extracted from the experimental results once the measured shielding constant becomes linear in the gas density $\rho_B$, and if the bulk susceptibility correction, $\sigma_{{\rm 1bulk}}^B$, is known.
![The observed density-dependent $^3$He shielding of atomic helium and $^2$H shielding of deuterium in gaseous solutions (for comparison, the best estimate of the $\sigma_{{\rm 1bulk}}^{{\rm Ar}}\, \rho_{{\rm Ar}}$ contribution is shown).[]{data-label="figd2"}](3he_2h.ps)
In the experiment it is not easy to measure the gas number density $\rho$, but rather the pressure $p$. Therefore, the following form of Eq. (\[eq:B\]) was used: $$\sigma^{A}(X)=\sigma_0^{A}(X)+\sigma_{1p}^{AB}(X)p.
\label{eq:Bp}$$ For an ideal gas Eqs. (\[eq:B\]) and (\[eq:Bp\]) are equivalent, and the coefficients $\sigma_{1p}^{AB}(X)$ and $\sigma_{1}^{AB}(X)$ are inter-related by the following simple expression: $$\sigma_{1p}^{AB}(X)=\sigma_{1}^{AB}(X)/k_BT.
\label{rel}$$ In general Eq. (\[rel\]) is not valid since the pressure depends on the gas number density in a more complicated way: $$p = k_BT\rho + B_2(T)\rho^2 + B_3(T)\rho^3 + \cdots,
\label{pvir}$$ where $B_2(T)$ and $B_3(T)$ are the second and third thermodynamic virial coefficients, respectively. Assuming that the NMR active molecules are infinitely diluted in the bath, $B_2(T)$ exclusively depends on the pair interactions between the molecules in the bath. The third virial coefficient $B_3(T)$ also depends on the non-additive three-body interactions in the bath. We assume that we are dealing with an infinitely diluted solutions. In such a case we can assume that the concentration of the solute is very small and that the contribution of the partial pressure of the solute to the total pressure is negligible. This means that the thermodynamics of the system is described by Eq. (\[pvir\]) with the characteristic coefficients of the solvent used in the experiment, while Eq. (\[eq:B\]) describes the change of the shielding constant due to binary collisions of the NMR active molecule with the bath molecules. It is worth noting at this point that the virial expansions (\[pvir\]) and (\[eq:B\]) follow from the theory, and that in the virial expansions the gas number density appears as the variable of the power series.
A precise evaluation of $\sigma_{{\rm 1bulk}}^B$, the bulk susceptibility correction, is particularly important in the present work, because for all the nuclei the total change of the shielding due to intermolecular interactions in the gas phase is very small. We consider first the determination of $\sigma_{{\rm 1bulk}}^B$ terms from the available $\chi_{\rm M}$ —molar magnetic susceptibilities of gases— and applying the standard formula for a infinitely long cylindrical tube parallel to the external magnetic field [@rspdrkmjjjmra101; @japwgshjbbook; @beckerbook]: $$\sigma_{{\rm 1bulk}} = - \frac{4 \pi}{3} \; \chi_{\rm M} \;
\label{eq:4pi}$$ where $\chi_{\rm M}$ is given in ppm cgs and $\sigma_{{\rm 1bulk}}$ in ppm mL/mol. The macroscopic molar magnetic susceptibility $\chi_{\rm M}$ is for closed shell systems proportional to the microscopic molecular magnetizability (1 ppm cgs corresponding to 16.60529 $\times 10^{-30}$ JT$^{-2}$). Equation (\[eq:4pi\]) may be applied for the cylindrical geometry of the sample, and assuming that the molecules of the solvent do not interact. However, this assumption is not always true, the cylinder is not infinite, and in such an approach various additional corrections are undoubtedly necessary to get realistic values of $\sigma_{{\rm 1bulk}}^B$ (see for instance Seydoux [*et al.*]{} [@rspdrkmjjjmra101]). We use a different approach to determine $\sigma_{{\rm 1bulk}}^B$, which will be discussed in detail in section \[sub5\].
Theoretical determination of $\sigma_1^{A-B}(X)$ requires two steps: [*ab initio*]{} calculations of the interaction potential and interaction-induced shielding constant for the binary complex , and the average of the latter quantity with the Boltzmann factor depending on the interaction potential. The interaction potential $V$ is given by the standard expression: $$V=E_{AB}-E_A-E_B,
\label{vint}$$ where $E_{AB}$, $E_A$, and $E_B$ are the energies of the collisional dimer $A-B$, and of the solvent ($A$) and solute ($B$) molecules, respectively. The interaction-induced shielding constant $\sigma_{\rm int}^{A-B}(X)$ is given by: $$\sigma_{\rm int}^{A-B}(X)=\sigma_0^{A-B}(X)-\sigma_0^A(X),
\label{sint}$$ where $\sigma_0^{A-B}(X)$ and $\sigma_0^A(X)$ are the shielding constants of the nucleus $X$ in the dimer $A-B$ and in the solute molecule $A$, respectively. Finally, $\sigma_{1}^{A-B}(X)$ appearing in Eq. (\[eq:ABA-B\]) is defined as: $$\begin{split}
&\sigma_{1}^{A-B}(X)=\\&\int\int\int\sigma_{\rm int}^{A-B}(X)
\exp\left(-\beta V(\omega_A,\omega_B,R)\right)R^2{\rm d}R{\rm d}\omega_A
{\rm d}\omega_B,\
\end{split}
\label{boltz}$$ where $\omega_A$ and $\omega_B$ denote the two sets of the angles specifying the orientations of the monomers $A$ and $B$, $R$ is the distance between the centers of mass of the monomers, $\beta=(k_BT)^{-1}$, $k_B$ is the Boltzmann constant, and $T$ is the temperature in Kelvin. We note that in general the calculation of $\sigma_{1}^{A-B}(X)$ is not an easy task. For rigid molecules $A$ and $B$ it requires a six-dimensional integration over five angles and one distance. For systems considered in the present paper the integral of Eq. (\[boltz\]) reduces to a two-dimensional integral that can easily be evaluated. The details of the computational procedures adopted in [*ab initio*]{} calculations, fitting, and numerical integration will be discussed in the next section.
Computational approach {#sec3}
======================
Ab initio calculations {#sub1}
----------------------
In all calculations the bond lengths of the interacting subsystems were kept fixed at their experimental geometries. We report below the results for H$_2$ obtained with the H–H distance fixed at $r$(HH) = 1.449 a$_0$ [@wklwjcp41]. In test calculations we have verified that practically the same results are obtained using noticeably smaller values of $r$(HH). Thus, we can compare the same set of [*ab initio*]{} results with the experimental data for different isotopomers of the hydrogen molecule. For He–CO$_2$, following the previous studies of the potential energy surface [@tkrmftjmlbbhjbpeswjcp115], we have used $r$(CO)= 2.1944 a$_0$, an experimental value deduced from the microwave spectra.
All calculations of the energies and of the shielding constants have been performed with the coupled cluster method restricted to single, double, and noniterative triple excitations, CCSD(T). The NMR shielding constants and the magnetizabilities were obtained by applying the coupled cluster linear response theory [@jgjfsjcp102; @jgjfsjcp104]. Gauge-including atomic orbitals, GIAO’s [@fljpr8; @kwjfhppjacs112], were used in all calculations of the magnetic properties, and we have systematically corrected all the interaction-induced changes in the energies and in the shielding constants by eliminating the basis set superposition error, i.e. all calculations for the monomers were done in the full basis of the dimer. We have used the d-aug-cc-pVXZ basis sets [@rakthdrjhjcp96]; d-aug-cc-pVQZ for the smallest H$_2$–He system, and the d-aug-cc-pVTZ basis set for the larger H$_2$–Ne, H$_2$–Ar, and He–CO$_2$ dimers. The calculations of the energies and shielding constants were performed using the ACES II [@aces2:2006] program, while the magnetizabilities were computed using the more recent CFOUR program [@cfour:09].
Interaction potentials {#sub2}
----------------------
For two systems we used the available fitted interaction potential energy surfaces: for H$_2$–Ar taken from Ref. [@hlwksbjrmsrjcp98] and for He–CO$_2$ taken from Ref. [@tkrmftjmlbbhjbpeswjcp115]. These potentials were obtained from the symmetry-adapted perturbation theory (SAPT) calculations (see Refs. [@bjrmkscr94; @rm:rev] for a review of the SAPT methodology and of the accuracy of the SAPT potentials). These potentials were shown to reproduce the high-resolution infrared spectra of the H$_2$–Ar [@rmbjpeswavdacpl221; @fmrmjcp109] and the He–CO$_2$ [@tkrmftjmlbbhjbpeswjcp115] van der Waals complexes. More importantly, they also reproduce very accurately the thermodynamic (pressure) virial coefficients [@rmtktgahpeswavdabspjch72]. For other systems the interaction potential $V(R,\theta)$ was calculated by the supermolecular method according to Eq. (\[vint\]).
We use spherical coordinates defined with respect to the center of mass of the molecule. Calculations were performed for several angles $\theta$ ranging from 0 to 180$^\circ$ and for several radial distances $R$. For each angle $\theta$ radial dependence of the interaction potential $V$ was fitted with the function: $$\begin{split}
V_\theta(R)&=e^{-\alpha(\theta) R}(A_0(\theta)+A_1(\theta) R+A_2(\theta) R^2)\\
&-\frac{C_6(\theta)}{R^6}-\frac{C_8(\theta)}{R^8}\,,
\end{split}
\label{vfit}$$ where $\alpha$, $A_0$, $A_1$, $A_2$, $C_6$, and $C_8$ were adjusted to fit the computed points at a given angle $\theta$. We note parenthetically that odd powers of $R^{-1}$ do not appear in the long-range asymptotics of Eq. (\[vfit\]) because the H$_2$ and CO$_2$ molecules are centrosymmetric. Next, interpolation was used to obtain the full interaction energy surface. The points calculated for a given radial distance $R$ from each $V_\theta$ fit were interpolated with a third-order polynomial in $\theta$. This procedure leads to a fitted/interpolated interaction energy surface $V(R,\theta)$, which was used in further calculations.
Shielding constants {#sub3}
-------------------
The same technique was applied to obtain the $\sigma_{\rm int}^{A-B}(R,\theta)$ surface. For each angle the radial dependence of $\sigma$ was fitted to the following function: $$S_\theta(R)=e^{-\alpha(\theta) R}\sum_{k=0}^NA_k(\theta) R^k-\sum_{m\in M}\frac{C_m(\theta)}{R^m},$$ where all the parameters appearing on the r.h.s. of the expression above were adjusted to fit the computed values.
For the hydrogen atom in H$_2$–He, H$_2$–Ne, and H$_2$–Ar a modification to the procedure described above was introduced. Since the interaction energy surface for these systems is symmetric we were allowed to use symmetrized $\sigma$-surface $\bar\sigma(R,\theta)$ calculated as an arithmetical average of interaction-induced shifts for both H nuclei. This improved the accuracy of the further integration of the $\sigma_{\rm int}^{A-B}$ function with the Boltzmann factor.
Final integration {#sub4}
-----------------
To obtain the final result one has to calculate for the temperature of interest the Boltzmann average of $\sigma_{\rm int}^{A-B}(R,\theta)$: $$\label{eq:Boltz}
\begin{split}
&\sigma_1^{A-B} =\\
&\int_0^\infty \!\!\!{\rm d}R \int_0^\pi\!\!\! {\rm d}\theta
\int_0^{2\pi}\!\!\!{\rm d}\phi R^2\sin\theta \; \exp\left(-\beta V(R,\theta)\right) \sigma_{\rm int}^{A-B}(R,\theta).
\end{split}$$ Due to the axial symmetry of the considered systems the integration over $\phi$ gives 2$\pi$. Integrations over $R$ nad $\theta$ were done numerically with the [Mathematica]{} [@Mathematica7] package. First, for each angle the radial integration was performed. The integration range \[0,$\infty$\[ was substituted by \[$R_{\rm min}$, $R_{\max}$\] with properly defined $R_{\rm min}$ and $R_{\rm max}$. The $R_{\rm min}$ value was chosen to ensure that $V(R_{\rm min},\theta)$ was positive and large enough to make the Boltzmann factor close to zero. The value of $R_{\rm max}$ was chosen in such a way that $\sigma(R_{\rm max},\theta)$ was almost zero at $R_{\rm max}$, independent of the angle $\theta$. This choice leads to $R_{\rm min}$ = 2 a$_0$ and $R_{\rm max}$ = 300 a$_0$ for H$_2$–He, for other systems the required integration range is smaller and within the same limits. The results obtained from the radial integration for each angle were interpolated with third-order function and this function was integrated over $\theta$.
Bulk susceptibilities {#sub5}
---------------------
To estimate the bulk susceptibility correction (BSC) we first used new values of magnetizabilities obtained from CCSD(T) calculations. Using the d-aug-cc-pVQZ basis set we obtain for CO$_2$ at the experimental geometry –22.254 ppm cgs, with the basis set error estimated to be smaller than 0.2 ppm cgs. This value is consistent with the results of Ref. [@krprtmjjpca104] and confirmed by new CCSD(T)/d-aug-cc-pCVQZ-unc calculations for the Ne and Ar atoms, which give –7.601 and –20.610 ppm cgs, also in agreement with Ref. [@krprtmjjpca104]. The values of $\chi_{\rm M}$ and the corresponding bulk susceptibility effects derived from Eq. (\[eq:4pi\]) are given in Table \[tab:magn\]. These values may only be considered as a crude approximation to the real $\sigma_{{\rm 1bulk}}^B$ quantities. First, because the geometric factor is unable to reproduce accurately the susceptibility corrections in nuclear shielding; this problem was frequently discussed in the literature from the early days of NMR [@aabregjcp26; @aabjmsp5]. Secondly, we have used a special high-pressure tube, cf. sec. \[sec4\], which was not spinning and this may induce nonnegligible unknown effects.
[l d d d d r]{} & & & &\
$\chi_{\rm M}$ & -1.8915 & -7.601 & -20.610 & -22.254\
$\sigma_{{\rm 1bulk}}^B$&7.923 & 31.839 &86.333 & 93.217\
$\sigma^{HeB}_{\rm 1}\, $&8.29(12) & 28.79(6) & 77.36(30) & &\
$\sigma_{{\rm 1bulk}}^B$&8.62(12) & 29.56(6) & 80.26(30) & &\
Since a precise determination of the BSC value according to Eq. (\[eq:4pi\]) is impossible, we have applied our own experimental approach to estimate the bulk susceptibility corrections. It is well known that molecular interactions between the atoms of rare gases disturb the $^3$He shielding only to very small extent [@rspdrkmjjjmra101]. Moreover, a description of such interactions is available from the theoretical studies of the shielding in these gas mixtures [@aamjarjcp126]. In the present work we have measured the density dependent $^3$He shielding in helium, neon, and argon gases. It gave us the $\sigma_{1}^{HeHe}$, $\sigma_{1}^{HeNe}$ and $\sigma_{1}^{HeAr}$ coefficients of Eq. (\[eq:B\]), which were used next in Eq. (\[eq:ABA-B\]) together with the interatomic interaction coefficients, to obtain $ \sigma_{{\rm 1bulk}}^B$ as $\sigma_{1}^{AB}(X) - \sigma_1^{A-B}(X)$. We have used the values of $\sigma_1^{He-He}$, $\sigma_1^{He-Ne}$ and $\sigma_1^{He-Ar}$, based on the theoretical results of Ref. [@aamjarjcp126]: –0.328, –0.776 and –2.901 ppm mL/mol, respectively (another available value of $\sigma_1^{He-He}$, derived from the full configuration interaction calculations, but with a smaller basis set, is equal to –0.353 ppm mL/mol [@mparmp100]). In this way we determined the final values of the bulk susceptibility effects in the present NMR experiments, shown in Table \[tab:magn\]. Finally, we note that the problems related to precise determination of the bulk susceptibility effects are known, they have been recently analysed [@mdpbjctc6; @rehjmr178] and discrepancies of the order of $\approx$10% between the computed and experimental data have been observed [@mdpbjctc6].
Experiment {#sec4}
==========
The $^1$H, $^2$H, $^3$He and $^{13}$C NMR chemical shifts were measured on a Varian INOVA 500 spectrometer at 300 K operated at 500.61, 76.85, 381.36 and 125.88 MHz, respectively. $^2$H and $^{13}$C spectra were acquired with a standard two channel Varian switchable 5 mm probe, while $^3$He and $^1$H spectra in the self reconstructed helium probe [@kjmjbkmwjmr193]. Nitromethane-d$_3$ was used for a lock system when the $^1$H, $^3$He, and $^{13}$C NMR measurements were carried out. The $^2$H experiments required a high-band lock operating on the proton signal of liquid tetramethylsilane (TMS). For this purpose a special set of coaxial glass capillaries was prepared and the same set was also used for the external referencing of all the chemical shifts. The set of capillaries contained nitromethane-d$_3$ in the outer chamber and pure liquid TMS in the inner container. The capillaries were placed in a special non-spinning NMR tube which was used for all our measurements. The tube was made of zirconia and equipped with a metal valve for gas filling at high pressure (Daedalus Innovations, USA).
The described sample setup was complex, the zirconia tube with the capillaries affects the external magnetic field, and therefore we could not apply Eq. (\[eq:4pi\]) in our experimental work (see also the discussion in Ref. [@rspdrkmjjjmra101]). We have bypassed the problem of bulk susceptibility corrections performing analogous measurements of $^3$He shielding in $^4$He, Ne and Ar as gaseous solvents using exactly the same setup of the sample tube with the same set of capillaries. This series of measurements was specially designed for precise determination of the bulk susceptibility effects in our experiments, according to the approach discussed in section \[sub5\].
![A high pressure system for filling the zirconia NMR tube with a gas up to the pressure of 300 bar.[]{data-label="fig:spec"}](HPG.ps)
An efficient high-pressure system built in our laboratory permitted the NMR investigations of the hydrogen and helium gases for a wide range of densities. As indicated in Fig. \[fig:spec\] the measurements in this system can be carried out continuously up to the total pressure of 300 bar. All compartments were degassed when they were connected to the vacuum line, then a small amount of the solute gas was supplied from the same vacuum line and finally gas solvent was added and mechanically compressed. The pressure of gaseous solution was read by the calibrated gauge and converted into the number of moles following the van der Waals equation and appropriate coefficients for real gases [@handbook77]. Gases: H$_2$ (Air Product, 99.9999%), HD (Isotec, 98% D), D$_2$ (Isotec, 99.96%), $^3$He (Isotec, 99.96%), $^4$He (Air Product, 99.9%), Ne (Air Product, 99.999%), Ar (Air Product, 99.9999%) and CO$_2$ (Aldrich, 99.8%) from lecture bottles were used for the preparation of samples without further purification.
We have performed all the measurements of the $^1$H and $^2$H shielding for the hydrogen isotopomers, H$_2$, HD and D$_2$, as a function of the solvent density where helium, neon and argon were used as the solvents. Comparing the $^1$H and $^2$H NMR signals from the H$_2$ and D$_2$ molecules we found that the width at the half maximum of the deuterium signal is over an order of magnitude smaller than the same parameter of protons in H$_2$, e.g. $\Delta\nu_{1/2}$ = 86.9 Hz for H$_2$ in helium at 60 bar while $\Delta\nu_{1/2}$ = 6.0 Hz for D$_2$ in helium at the same pressure. In practice, this means that the deuterium experiments deliver much more precise data for the analysis than can be obtained from the $^1$H NMR observations of the H$_2$ molecule. Consequently, we use next the $^2$H NMR experimental data for comparison of the theoretical and experimental results.
For each discussed system, measurements have been performed for more than 20 different solvent gas densities. In each case, the linear fit represents well the density dependence of the results, with the adjusted coefficient of determination larger than 0.995. The experimental shielding constants were corrected for the gas imperfection, and not only derived from the relations (\[eq:Bp\]) and (\[rel\]).
Results and discussion {#sec5}
======================
We begin the discussion of the effects of intermolecular interactions on the shielding constants with a brief summary of the [*ab initio*]{} results. Three-dimensional plots of $\sigma_{\rm int}^{A-B}(X)$ for the H$_2$–He, $^3$He–CO$_2$, and $^{13}$CO$_2$–He complexes are presented in Fig. \[shieldrt\]. Similar plots for H$_2$–Ne and H$_2$–Ar are not reported since their $R$ and $\theta$ dependence is nearly the same as for H$_2$–He. An inspection of Fig. \[shieldrt\] shows that $\sigma_{\rm int}^{H_2-He}$(H) does not show any strong variations on $R$ and $\theta$. Only at very small intermolecular distances a stronger dependence shows up, but at these geometries the interaction potential is strongly repulsive, so the exponential Boltzmann factor is almost zero and these large variations of $\sigma_{\rm int}^{H_2-He}$(H) do not contribute to $\sigma_{1}^{H_2-He}$(H). Slightly more pronounced is the geometry dependence of the interaction-induced shielding for both $^3$He–CO$_2$ and $^{13}$CO$_2$–He.
$T$ (K) $\sigma_1^{H_2-He}$(H) $\sigma_1^{H_2-Ne}$(H) $\sigma_1^{H_2-Ar}$(H)
--------- -------------------------- ------------------------- ------------------------
150 –0.324 –0.230 –4.071
200 –0.352 –0.281 –4.025
250 –0.381 –0.330 –4.085
280 –0.398 –0.358 –4.144
300 –0.410 –0.377 –4.189
320 –0.422 –0.396 –4.237
350 –0.440 –0.424 –4.314
$\sigma_1^{He-CO_2}$(He) $\sigma_1^{CO_2-He}$(C)
150 –6.579 1.3050
200 –6.514 1.2894
250 –6.548 1.2871
280 –6.590 1.2876
300 –6.622 1.2883
320 –6.658 1.2889
350 –6.714 1.2898
: Calculated [*ab initio*]{} values of $\sigma_1^{A-B}$ (ppm mL/mol)[]{data-label="tab:temp"}
Let us now analyse the temperature dependence of the $\sigma_1^{A-B}$ coefficients calculated from Eq. (\[eq:Boltz\]). The results for all the systems are shown in Table \[tab:temp\]. An inspection of the Table shows that the temperature effects are too small to be reliably determined from the experimental data (thus, in what follows we shall only compare theoretical results with the experiment for $T$=300 K). The dependence of the computed $\sigma_1^{A-B}$ on the temperature $T$ is almost linear. It is interesting to note that for H$_2$–He, H$_2$–Ne, H$_2$–Ar, and $^3$He–CO$_2$ systems $\sigma_1^{A-B}$ decreases with $T$, while for $^{13}$CO$_2$–He the opposite is found. The values at the lowest temperature, 150 K, do not differ considerably from the room temperature data, suggesting that the quantum effects will start to play a noticeable role at still lower temperatures.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\(a) ![Geometry dependence of the interaction-induced shielding constant for the (a) $^3$He–CO$_2$, (b) $^{13}$CO$_2$–He and (c) H$_2$–He complexes.[]{data-label="shieldrt"}](S-He-CO2.eps "fig:")
\(b) ![Geometry dependence of the interaction-induced shielding constant for the (a) $^3$He–CO$_2$, (b) $^{13}$CO$_2$–He and (c) H$_2$–He complexes.[]{data-label="shieldrt"}](S-CO2-He.eps "fig:")
[(c) ![Geometry dependence of the interaction-induced shielding constant for the (a) $^3$He–CO$_2$, (b) $^{13}$CO$_2$–He and (c) H$_2$–He complexes.[]{data-label="shieldrt"}](S-He-H2.eps "fig:")]{}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The temperature dependence of $\sigma_1^{A-B}$ was studied experimentally in glass samples as described earlier [@kjmjbkmwjmr193], but only for the $^3$He–CO$_2$ system (rare gases like He, Ne and Ar could not in practice be used as solvents in such experiments). Unfortunately, it was not possible to achieve sufficient precision to perform a quantitative analysis of the results. In addition, a most important factor required to determine $\sigma$($^3$He) in $^3$He–CO$_2$ as a function of the temperature—the temperature dependence of CO$_2$ bulk susceptibility—is not known.
Before we consider the comparison of the [*ab initio*]{} and experimental results, we recall that for all the systems the bulk susceptibility corrections are dominant. The observed density dependence of $^2$H and $^3$He shielding in gaseous solutions is shown in Fig. \[figd2\] (the depicted range of densities corresponds to the pressure of the solvent rare gas increasing up to 300 bar). For comparison, we have shown the effect of $\sigma_{{\rm 1bulk}}^B$ for Ar. It is clear that for all the nuclei the total change of the shielding in the gas phase is very small, determined largely by $\sigma_{{\rm 1bulk}}^B$, and the precise evaluation of the bulk susceptibility corrections is crucial for the present study of weak molecular interactions. The BSC effect can be neglected in the analysis of the experimental data in two cases—when a spherical NMR sample is prepared or when the sample is fast spinning at the magic angle. Unfortunately, neither of these methods can provide accurate results for compressed gas at high density.
Our final results, obtained for 300 K, are shown in Table \[tab:sum\]. For each system, the BSC constitutes the essential part of the measured effect, thus a minor error in the evaluation of $\sigma_{{\rm 1bulk}}$ clearly leads to a very significant error in $\sigma_1^{A-B}$. In particular, the standard approximation, $\sigma_{{\rm 1bulk}} = -({4\pi}/{3})\; \chi_{\rm M}$, is not sufficiently accurate. The error bars shown in Table \[tab:magn\] and Table \[tab:sum\] do not account for any systematic errors in the experiment, they represent only the errors of the linear fits to the observed density dependence of the results. We have considered systematic errors arising from a limited precision of the nominal reading of the absolute frequency, and errors in the control of the stability of the external magnetic field; they limit the precision of the measured shielding constants to $\pm$0.015 ppm. However, let us recall that in many cases the determination of $\sigma_1^{A-B}(X)$ required two NMR experiments, one for the observed $A-B$ binary system and one for the $^3$He$-B$ solvent. Consequently, the error bars are at least doubled, to $\pm$0.030 ppm for the $A-B$ system. Moreover, our experimental setup could slightly disturb the magnetic field as the sample was not spinning during the measurements; observing repeatedly the same samples we noticed deviations of up to ±2 Hz in the measured frequencies. A complete analysis of these systematic errors, following the discussed precision of frequency measurements and possible deviations in the gas density inside the NMR tube, gives $\pm$0.50 ppm mL/mol as an estimate of their contribution to the error bars in the $\sigma_1^{AB}$ values. This estimate does not include the tabulated errors of the linear fitting of the results, and does not take into account the left-over errors in the analysis of the bulk susceptibility effects.
The errors in the [*ab initio*]{} calculations are also difficult to estimate. The point-wise determined shielding surface is presumably accurate for the smallest H$_2$–He system, the correlation and basis set errors becoming larger for the other systems. Although the following stages—fitting the potential and the property surfaces, followed by the Boltzmann average—appear to be straightforward, approximations made in this part of the calculation contribute significantly to the final error bars. As shown in Fig. \[shieldrt\], there are regions of the shielding surface of opposite contributions to the induced shielding constant, and therefore the final result depends heavily on a significant cancellation of positive and negative contributions, which in turn depends on the potential surface. Following various test calculations we estimate that the errors of the computed $\sigma_1^{A-B}$ should not exceed 15-20% of the discussed above final [*ab initio*]{} values.
[l d d d d]{} & & & &\
& & & &\
& 8.07(8) & 8.62(12) & 0.55(20) &0.41\
& 29.83(3)& 29.56(6) & 0.27(9) &0.38\
& 76.44(32) & 80.26(30) & 3.78(62) & 4.19\
& 84.7(24) & 93.22& 8.5(24) & 6.62\
& 11.09(9) & 8.62(12)& 2.47(21) & 1.29\
Conclusions {#sec6}
===========
In this paper, we reported the first measurement of the changes of the NMR shielding constants due to weak intermolecular interactions. It became possible due to a new approach for the determination of bulk susceptibility effects, which are dominant in the studied systems. The interpretation of the results is related to the corresponding [*ab initio*]{} calculations, and we observe qualitative agreement of the [*ab initio*]{} values with those derived from the experimental data. There is a series of approximations that should be analysed to improve this agreement. In particular, it is obvious that one cannot expect quantitative agreement without a better description of the bulk susceptibility effects. We have bypassed this problem transferring the necessary information from one set of the experimental data—for pairs of rare gas atoms systems— to another, that is to the studied molecule–atom systems. Such an approach appears to yield satisfactory results in our case, but in general a better theory, describing accurately the bulk susceptibility corrections, is needed. For larger systems, for instance involving molecule-molecule interactions, the experiment may be easier, but without a proper description of these effects the interpretation of the results is almost impossible. Last but not least, we note that the corresponding theoretical calculations are also demanding, high level of the [*ab initio*]{} theory is required to obtain a reliable description of the small changes of the shielding constants due to weak intermolecular forces. For larger systems it may be difficult to achieve satisfactory accuracy of the results, in particular when the effects due to different parts of the shielding surface partially cancel out.
Acknowledgments {#acknowledgments .unnumbered}
===============
We acknowledge support of the Polish Ministry of Science and Higher Education research grant N N204 244134 (2008-2011). This project was partly co-operated within the Foundation for Polish Science MPD Programme co-financed by the EU European Regional Development Fund.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Person Re-identification (re-id) faces two major challenges: the lack of cross-view paired training data and learning discriminative identity-sensitive and view-invariant features in the presence of large pose variations. In this work, we address both problems by proposing a novel deep person image generation model for synthesizing realistic person images conditional on pose. The model is based on a generative adversarial network (GAN) designed specifically for pose normalization in re-id, thus termed pose-normalization GAN (PN-GAN). With the synthesized images, we can learn a new type of deep re-id feature free of the influence of pose variations. We show that this feature is strong on its own and complementary to features learned with the original images. Importantly, under the transfer learning setting, we show that our model generalizes well to any new re-id dataset without the need for collecting any training data for model fine-tuning. The model thus has the potential to make re-id model truly scalable.'
author:
- |
Xuelin Qian$^{1}$, Yanwei Fu$^{1}$, Tao Xiang$^{2}$, Wenxuan Wang$^{1}$\
Jie Qiu$^{3}$, Yang Wu$^{3}$, Yu-Gang Jiang$^{1}$, Xiangyang Xue$^{1}$\
$^{1}$Fudan University; $^{2}$Queen Mary University of London;\
$^{3}$Nara Institute of Science and Technology;\
bibliography:
- 'egbib.bib'
title: 'Pose-Normalized Image Generation for Person Re-identification'
---
Introduction
============
Person Re-identification (re-id) aims to match a person across multiple non-overlapping camera views [@gong2011person]. It is a very challenging problem because a person’s appearance can change drastically across views, due to the changes in various covariate factors independent of the person’s identity. These factors include viewpoint, body configuration, lighting, and occlusion (see Fig. \[fig:The-different\_pose\]). Among these factors, pose plays the most important role in causing a person’s appearance changes. Here pose is defined as a combination of viewpoint and body configuration. It is thus also a cause of self-occlusion. For instance, in the bottom row examples in Fig. \[fig:The-different\_pose\], the big backpacks carried by the three persons are in full display from the back, but reduced to mostly the straps from the front.
![The same person’s appearance can be very different across camera views, due to the presence of large pose variations.[]{data-label="fig:The-different_pose"}](figure/intro)
Most existing re-id approaches [@deepreid; @Ejaz_cvpr2015; @de_cheng_2016; @yu2017cross; @qian2017multi; @SVDNet; @reid_in_wild; @PersonSearch] are based on learning identity-sensitive and view-insensitive features using deep neural networks (DNNs). To learn the features, a large number of persons’ images need to be collected in each camera view with variable poses. With the collected images, the model can have a chance to learn what features are discriminative and invariant to the camera view and pose changes. These approaches thus have a number of limitations. The first limitation is **lack of scalability** to large camera networks. Existing models require sufficient identities and sufficient images per identity to be collected from each camera view. However, manually annotating persons across views in the camera networks is tedious and difficult even for humans. Importantly, in a real-world application, a camera network can easily consist of hundreds of cameras (i.e. those in an airport or shopping mall); annotating enough training identities from all camera views are infeasible. The second limitation is **lack of generalizability** to new camera networks. Specifically, when an existing deep re-id model is deployed to a new camera network, view points and body poses are often different across the networks; additional data thus need to be collected for model fine-tuning, which severely limits its generalization ability. As a result of both limitations, although deep re-id models are far superior for large re-id benchmarks such as Market-1501 [@market1501] and CUHK03 [@deepreid], they still struggle to beat hand-crafted feature based models on smaller datasets such as CUHK01 [@cuhk01], even when they are pre-trained on the larger re-id datasets.
Even with sufficient labeled training data, existing deep re-id models face the challenge of learning identity-sensitive and view-insensitive features in the presence of large pose variations. This is because a person’s appearance is determined by a combination of identity-sensitive but view-insensitive factors and identity-insensitive but view-sensitive ones, which are inter-connected. The former correspond to semantic related identity properties, such as gender, carrying, clothing style, color, and texture. The latter are the covariates mentioned earlier including pose. Existing models aim to keep the former and remove the latter in the learned feature representations. However, these two aspects of the appearance are not independent, e.g., the appearance of the carrying depends on the pose. Making the learned features pose-insensitive means that the features supposed to represent the backpacks in the bottom row examples in Fig. \[fig:The-different\_pose\] are reduced to those representing only the straps a much harder type of features to learn.
In this paper, we argue that the key to learning an effective, scalable and generalizable re-id model is to remove the influence of pose on the person’s appearance. Without the pose variation, we can learn a model with much less data thus making the model scalable to large camera networks. Furthermore, without the need to worry about the pose variation, the model can concentrate on learning identity-sensitive features and coping with other covariates such as different lighting conditions and backgrounds. The model is thus far more likely to generalize to a new dataset from a new camera network. Moreover, with the different focus, the features learned without the presence of pose variation would be different and complementary to those learned with pose variation.
To this end, a novel deep re-id framework is proposed. Key to the framework is a deep person image generation model. The model is based on a generative adversarial network (GAN) designed specifically for pose normalization in re-id. It is thus termed pose-normalization GAN (PN-GAN). Given any person’s image and a desirable pose as input, the model will output a synthesized image of the same identity with the original pose replaced with the new one. In practice, we define a set of eight canonical poses, and synthesize eight new images for any given image, resulting in a 8-fold increase in the training data size. The pose-normalized images are used to train a pose-normalized re-id model which produces a set of features that are complementary to the feature learned with the original images. The two sets of feature are thus fused as the final feature representation. Critically, once trained, the model can be applied to a new dataset without any model fine-tuning as long as the test image’s pose is also normalized.
**Contributions**. Our contributions are as follows. (1) We identify pose as the chief culprit for preventing a deep re-id model from learning effective identity-sensitive and view-insensitive features, and propose a novel solution based on generating pose-normalized images. This also addresses the scalability and generalizability issues of existing models. (2) A novel person image generation model PN-GAN is proposed to generate pose-normalized images, which are realistic, identity-preserving and pose controllable. With the synthesized images of canonical poses, strong and complementary features are learned to be combined with features learned with the original images. Extensively experiments on several benchmarks show that the efficacy of our proposed model. (3) A more realistic unsupervised transfer learning setting is considered in this paper. Under this setting, no data from the target dataset is used for model updating: the model trained from labeled source datasets/domains is applied to the target domain without any modification.
Related Work
============
**Deep re-id models**Most recently proposed re-id models employ a DNN to learn discriminative view-invariant features [@deepreid; @Ejaz_cvpr2015; @de_cheng_2016; @yu2017cross; @qian2017multi; @SVDNet; @reid_in_wild; @PersonSearch]. They differ in the DNN architectures some adopt a standard DNN developed for other tasks, whilst others have architectures tailor-made. They differ also in the training objectives. Different models use different training losses including identity classification, pairwise verification, and triplet ranking losses. A comprehensive study on the effectiveness of different losses and their combinations on re-id can be found in [@deeptransfer2016]. The focus of this paper is not on designing new re-id deep model architecture or loss we use an off-the-shelf ResNet architecture [@resnet] and the standard identity classification loss. We show that once the pose variation problem is solved, such a general-purpose model can achieve the state-of-the-art re-id performance, beating many existing models with more elaborative architectures and losses.
**Pose-guided deep re-id** The negative effects of pose variation on deep re-id models have been recognised recently. A number of models [@su2017pose; @zheng2017pose; @zhao2017deeply; @zhao2017spindle; @li2017learning; @yao2017deep; @wei2017glad] are proposed to address this problem. Most of them are pose-guided based on body part detection. For example, [@su2017pose; @zhao2017spindle] utilize detect normalized part regions from a person image, and then fuse the features extracted from the original images and the part region images. These body part regions are predefined and the region detectors are trained beforehand. Differently, [@zhao2017deeply] combine region selection and detection with deep re-id in one model. Our model differs significantly from these models in that we synthesize realistic whole-body images using the proposed PN-GAN, rather than only focusing on body parts for pose normalization. Note that body parts are related to semantic attributes which are often specific to different body parts. A number of attributes based re-id models [@wang2017attribute; @sarfraz2017deep; @yu2016weakly; @deng2015learning] have been proposed. They use attributes to provide additional supervision for learning identity-sensitive features. In contrast, without using the additional attribute information, our PN-GAN is learned as a conditional image generation model for the re-id problem.
**Deep image generation** Generating realistic images of objects using DNNs has received much interest recently, thanks largely to the development of GAN [@goodfellow2014generative]. GAN is designed to find the optimal discriminator network $D$ between training data and generated samples using a min-max game and simultaneously enhance the performance of an image generator network $G$. It is formulated to optimize the following objective functions: $$\begin{aligned}
\underset{G}{\mathrm{min}}\underset{D}{\mathrm{max}}\mathcal{L}_{GAN} & =\mathbb{E}_{x\sim p_{data}\left(x\right)}\left[\mathrm{log}D\left(x\right)\right]+\label{eq:gan}\\
& \mathbb{E}_{z\sim p_{prior}\left(z\right)}\left[\mathrm{log}\left(1-D\left(G\left(z\right)\right)\right)\right]\nonumber \end{aligned}$$ where $p_{data}\left(x\right)$ and $p_{prior}\left(z\right)$ are the distributions of real data $x$ and Gaussian prior $z\sim\mathcal{N}\left(\mathbf{0},\mathbf{1}\right)$. The training process iteratively updates the parameters of $G$ and $D$ with the loss functions $\mathcal{L}_{D}=-\mathcal{L}_{GAN}$ and $\mathcal{L}_{G}=\mathcal{L}_{GAN}$ for the generator and discriminator respectively. The generator can draw a sample $z\sim p_{prior}\left(z\right)=\mathcal{N}\left(\mathbf{0},\mathbf{1}\right)$ and utilize the generator network $G$, i.e., $G(z)$ to generate an image.
Among all the variants of GAN, our pose normalization GAN is built upon deep convolutional generative adversarial networks (DCGANs) [@radford2015unsupervised]. Based on a standard convolutional decoder, DCGAN scales up GAN using Convolutional Neural Networks (CNNs) and it results in stable training across various datasets. Many other variants of GAN, such as VAEGAN [@larsen2015autoencoding], Conditional GAN [@isola2016image], stackGAN [@zhang2016stackgan] also exist. However, most of them are designed for training with high-quality images of objects such as celebrity faces, instead of low-quality surveillance video frames of pedestrians. This problem is tackled in a very recent work [@poseguid2017nips], which also aims to synthesize person images in different poses.
Nonetheless, our model differs significant from the existing variants of GAN. In particular, built upon the residual blocks, our PN-GAN is learned to change the poses and yet keeps the identity of input person. Note that the only work so far that uses deep image generator for re-id is [@zheng2017unlabeled]. However, their model is not a conditional GAN and thus cannot control either identity or pose in the generated person images. As a result, the generated images can only be used as unlabeled or weakly labeled data. In contrast, our model generate strongly labeled data with its ability to preserve the identity and remove the influence of pose variation.
![\[fig:Overview\] Overview of our framework.](figure/overview)
Methodology
===========
Problem Definition and Overview\[subsec:Problem-Setup-and\]
-----------------------------------------------------------
**Problem definition**. Assume we have a training dataset of $N$ persons $\mathcal{D}_{Tr}=\left\{ \mathbf{I}_{k},y_{k}\right\} _{k=1}^{N}$, where $\mathbf{I}_{k}$ and $y_{k}$ are the person image and person id of the $k$-th person. In the training stage we learn a feature extraction function $\phi$ so that a given image $\mathbf{I}$ can be represented by a feature vector $\mathbf{f}_{\mathbf{I}}=\phi(\mathbf{I})$. In the testing stage, given a pair of person images $\left\{ \mathbf{I}_{i},\mathbf{I}_{j}\right\} $ in the testing dataset $\mathcal{D}_{Te}$, we need to judge whether $y_{i}=y_{j}$ or $y_{i}\neq y_{j}$. This is done by simply computing the Euclidean distance between $\mathbf{f}_{\mathbf{I}_{i}}$ and $\mathbf{f}_{\mathbf{I}_{j}}$ as the identity-similarity measure.
**Framework Overview**. As shown in Fig. \[fig:Overview\], our framework has two key components, *i.e.*, a GAN based person image generation model (Sec. \[subsec:GAN-based-person\]) and a person re-id feature learning model (Sec. \[subsec:person-re-id-classification\]).
Deep Image Generator\[subsec:GAN-based-person\]
-----------------------------------------------
Our image generator aims at producing the same person’s images under different poses. Particularly, given an input person image $\mathbf{I}_{i}$ and a desired pose image $\mathbf{I}_{\mathcal{P}_{j}}$, our image generator aims to synthesize a new person image $\hat{\mathbf{I}}_{j}$, which contains the same person but with a different pose defined by $\mathbf{I}_{\mathcal{P}_{j}}$. As in any GAN model, the image generator has two components, a Generator $G_{P}$ and a Discriminator $D_{P}$. The generator is learned to edit the person image conditional on a given pose; the discriminator discriminates real data samples from the generated samples and help to improve the quality of generated images.
![\[fig:gan\] Schematic of our PN-GAN model](figure/GAN)
**Pose estimation.** The image generation process is conditional on the input image and one factor: the desired pose represented by a skeleton pose image. Pose estimation is obtained by a pretrained off-the-shelf model. More concretely, the off-the-shelf pose detection toolkit OpenPose [@cao2017realtime] is deployed, which is trained without using any re-id benchmark data. Given an input person image $\mathbf{I}_{i}$, the pose estimator can produce a pose image $\mathbf{I}_{\mathcal{P}_{i}}$, which localizes and detects 18 anatomical key-points as well as their connections. In the pose images, the orientation of limbs is encoded by color (see Fig. \[fig:Overview\], target pose). In theory, any pose from any person image can be used as a condition to control the pose of another person’s generated image. In this work, we focus on pose normalization so we stick to eight canonical poses as shown in Fig. \[fig:The-eight-poses\](a), to be detailed later.
**Generator.** As shown in Fig. \[fig:gan\], given an input person image $\mathbf{I}_{i}$, and a target person image $\mathbf{I}_{j}$ which contains the same person as $\mathbf{I}_{i}$ but a different pose $\mathbf{I}_{\mathcal{P}_{j}}$, our generator will learn to replace pose information in $\mathbf{I}_{i}$ with the target pose $\mathbf{I}_{\mathcal{P}_{j}}$ and generate the new pose $\hat{\mathbf{I}}_{j}$. The input to the generator is the concatenation of the input person image $\mathbf{I}_{i}$ and target pose image $\mathbf{I}_{\mathcal{P}_{j}}$. Specifically, we treat the target body pose image $\mathbf{I}_{\mathcal{P}_{j}}$ as a three-channel image and directly concatenate it with the three-channel source person image as the input of the generator. The generator $G_{P}$ is designed based on the “ResNet” architecture and is an encoder-decoder network [@hinton2006science]. The encoder-decoder network progressively down-samples $\mathbf{I}_{i}$ to a bottleneck layer, and then reverse the process to generate $\hat{\mathbf{I}}_{j}$. The encoder contains $9$ ResNet basic blocks[^1].
The motivation of designing such a generator is to take advantage of learning residual information in generating new images. The general shape of “ResNet” is learning $y=f(x)+x$ which can be used to pass invariable information from the bottom layers of the encoder to the decoder, and change the variable information of pose. To this end, the other features (e.g., clothing, and the background) will also be reserved and passed to the decoder in order to generate $\hat{\mathbf{I}}_{j}$. With this architecture (see Fig. \[fig:gan\]), we have the best of both worlds: the encoder-decoder network can help learn to extract the semantic information, stored in the bottleneck layer, while the ResNet blocks can pass rich invariable information of person identity to help synthesize more realistic images, and change variable information of poses to realize pose normalization at the same time.
Formally, let $G_{P}\left(\cdot\right)$ be the generator network which is composed of an encoder subnet $G_{Enc}\left(\cdot\right)$ and a decoder subnet $G_{Dec}\left(\cdot\right)$, the objective of the generator network can be expressed as $$\mathcal{L}_{_{G_{P}}=}\mathcal{L}_{GAN}+\lambda_{1}\cdot\mathcal{L}_{L_{1}},\label{eq:generator}$$ where $\mathcal{L}_{GAN}$ is the loss of the generator in Eq (\[eq:gan\]) with the generator $G_{P}\left(\cdot\right)$ and discriminator $D_{P}\left(\cdot\right)$ respectively, $$\begin{aligned}
\mathcal{L}_{GAN} & =\mathbb{E}_{\mathbf{I}_{j}\sim p_{data}\left(\mathbf{I}_{j}\right)}\left\{ \mathrm{log}D_{P}\left(\mathbf{I}_{j}\right)\right.\label{eq:updated_GAN}\\
+ & \left.\mathrm{log}\left(1-D_{P}\left(G_{P}\left(\mathbf{I}_{i},\mathbf{I}_{\mathcal{P}_{j}}\right)\right)\right)\right\} \nonumber \end{aligned}$$
and $\mathcal{L}_{L_{1}}=\mathbb{E}_{\mathbf{I}_{j}\sim p_{data}\left(\mathbf{I}_{j}\right)}\left[\left\Vert \mathbf{I}_{j}-\hat{\mathbf{I}}_{j}\right\Vert _{1}\right]$, and $\hat{\mathbf{I}}_{j}=G_{Dec}\left(G_{Enc}\left(\mathbf{I}_{i},\mathbf{I}_{\mathcal{P}_{j}}\right)\right)$ is the reconstructed image for $\mathbf{I}_{j}$ from the input image $\mathbf{I}_{i}$ with the body pose $\mathbf{I}_{\mathcal{P}_{j}}$. Here the $L_{1}-$norm is used to yield sharper and cleaner images. $\lambda_{1}$ is the weighting coefficient to balance the importance of each term.
**Discriminator**. The discriminator $D_{P}\left(\cdot\right)$ aims at learning to differentiate the input images is real or fake (i.e., a binary classification task). Given the input image $\mathbf{I}_{i}$ and target output image $\mathbf{I}_{j}$, the objective of the discriminator network can be formulated as $$\mathcal{L}_{D_{P}}=-\mathcal{L}_{GAN},\label{eq:discriminator}$$ Since our final goal is to obtain the best generator $G_{P}$, the optimization step would be to iteratively minimize the loss function $\mathcal{L}_{G_{P}}$ and $\mathcal{L}_{D_{P}}$ until convergence. Please refer to the Supplementary Material for the detailed structures and parameters of the generator and discriminator.
Person re-id with Pose Normalization \[subsec:person-re-id-classification\]
---------------------------------------------------------------------------
------------------------------------------- ----------------------------------------------
 
\(a) Eight canonical poses on Market-1501 \(b) t-SNE visualization of different poses.
------------------------------------------- ----------------------------------------------
As shown in Fig. \[fig:Overview\], we train two re-id models. One model is trained using the original images in a training set to extract identity-invariant features in the presence of pose variation. The other is trained using the synthesized images with normalized poses using our PN-GAN to compute re-id features free of pose variation. They are then fused as the final feature representation.
**Pose Normalization.** We need to obtain a set of canonical poses, which are representative of the typical viewpoint and body-configurations exhibited by people in public captured by surveillance cameras. To this end, we predict the poses of all training images in a dataset and then group the poses into eight clusters $\left\{ \mathbf{I}_{\mathcal{P}_{C}}\right\} _{c=1}^{8}$. We use VGG-19 [@returnDevil2014BMVC] pre-trained on the ImageNet ILSVRC-2012 dataset to extract the features of each pose images, and K-means algorithm is used to cluster the training pose images into canonical poses. The mean pose images of these clusters are then used as the canonical poses. The eight poses obtained on Market-1501 [@market1501] is shown in Fig. \[fig:The-eight-poses\](a). With these poses, given each image $\mathbf{I}_{i}$, our generator will synthesize eight images $\left\{ \hat{\mathbf{I}}_{i,\mathcal{P}_{C}}\right\} _{C=1}^{8}$ by replacing the original pose with these poses.
**Re-id Feature with pose variation.** We train one re-id model with the original training images to extract re-id features with pose variation. The ResNet-50 model [@resnet] is used as the base network. It is pre-trained on the ILSVRC-2012 dataset, and fine-tuned on the training set of a given re-id dataset to classify the training identities. We name this network ResNet-50-A (Base Network A), as shown in Fig. (\[fig:Overview\]). Given an input image $\mathbf{I}_{i}$, ResNet-50-A produces a feature set $\left\{ \mathbf{f}_{\mathbf{I}_{i},layer}\right\} $, where $layer$ indicates from which layer of the network, the re-id features are extracted. Note that, in most existing deep re-id models, features are computed from the final convolutional layer. Inspired by [@liu2017hydraplus] which shows that layers before the final layer in a DNN often contain useful mid-level identity-sensitive information. We thus merge the $5a$, $5b$ and $5c$ convolutional layers of ResNet-50 structures into a $1024\textendash{}d$ feature vector after an FC layer.
**Re-id Feature without pose variation.** The second model called ResNet-50-B has the same architecture as ResNet-50-A, but performs feature learning using the pose-normalized synthetic images. We thus obtain eight sets of features for the eight poses $\mathbf{f}_{\hat{\mathbf{I}}{}_{i,\mathcal{P}_{C}}}=\left\{ \mathbf{f}_{\hat{\mathbf{I}}_{i,\mathcal{P}_{C}}}\right\} _{C=1}^{8}$.
**Testing stage.** Once ResNet-50-A and ResNet-50-B are trained, during testing, for each gallery image, we feed it into ResNet-50-A to obtain one feature vector; and synthesize eight images of the canonical poses, feed them into ResNet-50-B to obtain 8 pose-free features. This can be done offline. Then given a query image $\mathbf{I}_{q}$, we do the same to obtain nine feature vectors $\left\{ \mathbf{f}_{\mathbf{I}_{q}},\mathbf{f}_{\hat{\mathbf{I}}_{q,\mathcal{P}_{C}}}\right\} $. Since Maxout and Max-pooling have been widely used in multi-query video re-id, we thus obtain one final feature vector by fusing the nine feature vectors by element-wise maximum operation. We then calculate the Euclidean distance between the final feature vectors of the query and gallery images and use the distance to rank the gallery images.
Experiments
===========
Datasets and Settings
---------------------
Experiments are carried out on four benchmark datasets:
**Market-1501** [@market1501] is collected from 6 different camera views. It has 32,668 bounding boxes of 1,501 identities obtained using a Deformable Part Model (DPM) person detector. Following the standards split [@market1501], we use 751 identities with 12,936 images as training and the rest 750 identities with 19,732 images for testing. The training set is used to train our PN-GAN model. ****
**CUHK03** [@deepreid] contains 14,096 images of 1,467 identities, captured by six camera views with 4.8 images for each identity in each camera on average. We utilize the more realistic yet harder detected person images setting. The training, validation and testing sets consist of 1,367 identities, 100 identities and 100 identities respectively. The testing process is repeated with 20 random splits following [@deepreid]. ****
**DukeMTMC-reID** [@Duke_ori_data] is constructed from the multi-camera tracking dataset DukeMTMC. It contains 1,812 identities. Following the evaluation protocol [@zheng2017unlabeled], 702 identities are used as the training set and the remaining 1,110 identities as the testing set. During testing, one query image for each identity in each camera is used for query and the remaining as the gallery set.
**CUHK01** [@cuhk01] has 971 identities with 2 images per person captured in two disjoint camera views respectively. As in [@cuhk01], we use as probe the images of camera A and utilize those from camera B as gallery. 486 identities are randomly selected for testing and the remaining are used for training. The experiments are repeated for 10 times with the average results reported.
---------------------------------------------------- ---------------- ---------------- ----------- ------------
[R-1]{} [mAP]{} [R-1]{} [mAP ]{}
[TMA [@martinel2016eccv]]{} [47.90]{} [22.3]{} [ ]{}
[SCSP [@chen2016cvpr]]{} [51.90]{} [26.40]{} [ ]{}
[DNS [@null_space_cvpr2016]]{} [61.02]{} [35.68]{} [71.56]{} [46.03]{}
[LSTM Siamese [@lstm2016eccv]]{} [61.60]{} [35.31]{}
[Gated\_Sia [@gated_siamese_eccv2016]]{} [65.88]{} [39.55]{} [76.50]{} [48.50 ]{}
[HP-net [@liu2017hydraplus]]{} [76.90]{}
[Spindle [@zhao2017spindle]]{} [76.90]{}
[Basel.+LSRO [@zheng2017unlabeled][\*]{}]{} [78.06]{} [56.23]{} [85.12]{} [68.52]{}
[PIE [@zheng2017pose]]{} [79.33]{} [55.95]{} [ ]{}
[Verif.-Identif. [@verif]]{} [79.51]{} [59.87]{} [85.84]{} [70.33 ]{}
[DLPAR[@zhao2017deeply]]{} [81.00]{} [63.40]{} [ ]{}
[DeepTransfer [@deeptransfer2016]]{} [83.70]{} [65.50]{} [89.60]{} [73.80 ]{}
[Verif-Identif.+LSRO[@zheng2017unlabeled][\*]{}]{} [83.97]{} [66.07]{} [88.42]{} [76.10 ]{}
[PDC [@su2017pose]]{} [84.14]{} [63.41]{} [ ]{}
[DML [@zhang2017deep]]{} [87.7]{} [68.8]{} [ ]{}
[SSM [@bai2017scalable]]{} [82.2]{} [68.8]{} [88.2]{} [76.2]{}
[JLML [@li2017person]]{} [85.10]{} [65.50]{} [89.70]{} [74.50]{}
[ResNet-50-A]{} [87.26]{} [69.32]{} 91.81 77.85
[Ours (SL)]{} **89.43**[ ]{} **72.58** **** **92.93** **80.19**
---------------------------------------------------- ---------------- ---------------- ----------- ------------
: \[tab:Results-of-market\]Results on Market-1501. ‘-’ indicates not reported. Note that [\*]{}: [on [@zheng2017unlabeled], we report the results of using both ]{}Basel.+LSRO and Verif-Identif.+LSRO. Our model only uses the identification loss, so should be compared with Basel. + LSRO which uses the same ResNet-50 base network and the same loss.
**Evaluation metrics**. Two evaluation metrics are used to quantitatively measure the re-id performance. The first one is Rank-1, Rank-5 and Rank-10 accuracy. For Market-1501 and DukeMTMC-reID datasets, the mean Average Precision (mAP) is also used.
**Implementation details.** Our model is implemented on Tensorflow [@tensorflow] (PN-GAN part) and Caffe [@caffe] (re-id feature learning part) framework. The $\lambda_{1}$ in Eq (\[eq:generator\]) is empirically set as 10 in all experiments. We utilize the two-stepped fine-tuning strategy in [@geng2016deep] to fine-tune ResNet-50-A and ResNet-50-B. The input images are resized into $256\times128$. Adam [@adam] is used to train both the PN-GAN model and re-id networks with a learning rate of 0.0002, $\beta_{1}=0.5$, a batch size of 32, and a learning rate of 0.00035, $\beta_{1}=0.9$, a batch size of 16, respectively. The dropout ratio is set as 0.5. Our PN-GAN models and re-id networks are converged in 19 hours and 8 hours individually on Market-1501 with one NVIDIA 1080Ti GPU card. Codes and trained models will be made available on the first author’s webpage.
[cc]{}
[Method]{} [R-1]{} [R-5]{} [R-10]{}
------------------------------------------ ------------ ---------------- ------------
[DeepReid [@deepreid]]{} [19.89]{} [50.00]{} [64.00]{}
[Imp-Deep [@Ejaz_cvpr2015]]{} [44.96]{} [76.01]{} [83.47]{}
[EMD [@hailin_shi]]{} [52.09]{} [82.87]{} [91.78]{}
[SI-CI [@joint_learning_cvpr16]]{} [52.17]{} [84.30]{} [92.30]{}
[LSTM Siamese [@lstm2016eccv]]{} [57.30]{} [80.10]{} [88.30]{}
[PIE [@zheng2017pose]]{} [67.10]{} [92.20]{} [96.60]{}
[Gated\_Sia [@gated_siamese_eccv2016]]{} [68.10]{} [88.10]{} [94.60]{}
[Basel. + LSRO [@zheng2017unlabeled]]{} [73.10]{} [92.70]{} [96.70 ]{}
[DGD [@xiao2016learning]]{} [75.30]{}
[OIM [@xiao2017joint]]{} [77.50]{}
[PDC [@su2017pose]]{} [78.92]{} [94.83]{} [97.15]{}
[DLPAR[@zhao2017deeply]]{} **81.60** **97.30** **** [98.40]{}
[ResNet-50-A]{} (SL) [ 76.83]{} [ 93.79]{} [97.27]{}
[Ours (SL)]{} [79.76 ]{} [96.24 ]{} **98.56**
[ResNet-50-A (TL)]{} 16.50 38.60 52.84
Ours (TL) 16.85 39.05 53.32
&
[Method]{} [R-1]{} [R-5]{} [R-10]{}
--------------------------------------- ---------------- ---------------- -----------
[eSDC [@unsupervised_per_reid]]{} [19.76]{} [32.72]{} [40.29]{}
[kLFDA [@kLFDA]]{} [32.76]{} [59.01]{} [69.63]{}
[mFilter [@mFilter]]{} [34.30]{} [55.00]{} [65.30]{}
[Imp-Deep [@Ejaz_cvpr2015]]{} [47.53]{} [71.50]{} [80.00]{}
[DeepRanking [@deepranking2016TIP]]{} [50.41]{} [75.93]{} [84.07]{}
[Ensembles [@Ensembles]]{} [53.40]{} [76.30]{} [84.40]{}
[ImpTrpLoss [@ImpTrpLoss]]{} [53.70]{} [84.30]{} [91.00]{}
[GOG [@GOG]]{} [57.80]{} [79.10]{} [86.20]{}
[Quadruplet [@quadruplet]]{} [62.55]{} [83.44]{} [89.71]{}
[NullReid [@NullReid]]{} [64.98]{} [84.96]{} [89.92]{}
[ResNet-50-A]{} (SL) [64.56]{} [83.66]{} [89.74]{}
[Ours (SL)]{} **67.65**[ ]{} **86.64**[ ]{} **91.82**
[ResNet-50-A]{} (TL) 27.20 48.60 59.20
Ours (TL) 27.58 49.17 59.57
[\
]{}(a) Results on CUHK03 & (b) Results on CUHK01[\
]{}
**Experimental Settings.** Experiments are conducted under two settings. The first is the standard **Supervised Learning** (SL) setting on all datasets: the models are trained on the training set of the dataset, and evaluated on the testing set. The other one is the **Transfer Learning** (TL) setting only for the datasets, CUHK03, CUHK01, and DukeMTMC-reID. Specifically, the re-id model is trained on Market-1501 dataset. We then directly utilize the trained single model to do the testing (*i.e*., to synthesize images with canonical poses and to extract the nine feature vectors) on the test set of CUHK03, CUHK01, and DukeMTMC-reID. That is, no model updating is done using any data from these three datasets. The TL setting is especially useful in real-world scenarios, where a pre-trained model needs to be deployed to a new camera network without any model fine-tuning. This setting thus tests how generalizable a re-id model is.
Supervised Learning Results
---------------------------
[Methods ]{} [R-1 ]{} [R-10 ]{} [mAP ]{}
----------------------------------------- ------------ ----------- -----------
[LOMO+XQDA[@XQDA] ]{} [30.80 ]{} [ ]{} [17.00]{}
[ResNet50 [@resnet] ]{} [65.20 ]{} [ ]{} [45.00]{}
[Basel. +LSRO [@zheng2017unlabeled] ]{} [67.70 ]{} [ ]{} [47.10]{}
[AttIDNet [@lin2017improving] ]{} [70.69 ]{} [ ]{} [51.88]{}
[ResNet-50-A (SL)]{} 72.80 87.90 52.48
[Ours (SL)]{} **73.58** **88.75** **53.20**
[ResNet-50-A (TL)]{} 27.872 51.122 13.942
[Ours (TL)]{} 29.937 51.615 15.768
: \[tab:Results-on-the-Duke\]Results on DukeMTMC-reID.
**Results on large-scale datasets**. Tables \[tab:Results-of-market\], \[tab:Results-on-the-Duke\] and \[tab:Results-of-CUHK03.\] (a) compare our model with the best performing alternative models. We can make the following observations:
\(1) On all three datasets, the results clearly show that, in the supervised learning settings, our results are improved over those of ResNet-50-A baselines by a clear margin. This validates that the synthetic person images generated by PN-GAN can indeed help the person re-id tasks. (2) Compared with the existing pose-guided re-id models [@zhao2017spindle; @zheng2017pose; @su2017pose], our model is clearly better, indicating that synthesizing multiple normalized poses is a more effective way to deal with the large pose variation problem.
\(3) Compared with the only other re-id model that uses synthesized images for re-id model training [@zheng2017unlabeled], our model yields better performance for all datasets, the gap on Market-1501 and DukeMTCM-reID being particularly clear. This is because our model can synthesize images with different poses, which can thus be used for supervised training. In contrast, the synthesized images in [@zheng2017unlabeled] do not correspond to any particular person identities or poses, so can only be used as unlabeled or weakly-labeled data.
**Results on small-scale dataset**. On the smaller dataset CUHK01, Table \[tab:Results-of-CUHK03.\](b) shows that, again our ResNet-50-A is a pretty strong baseline which can beat almost all the other methods. And by using the normalized pose images generated by PN-GAN, our framework further boosts the performance of ResNet-50-A by more than $3\%$ in the supervised setting. This demonstrates the efficacy of our framework. Note that on the small dataset CUHK01, the handcrafted feature + metric learning based models (e.g., NullReid [@NullReid]) are still quite competitive, often beating the more recent deep models. This reveals the limitations of the existing deep models on scalability and generalizability. In particular, previous deep re-id models are pre-trained on some large-scale training datasets, such as CUHK03 and Market-1501. But the models still struggle to fine-tune on the small datasets such as CUHK01 due to the covariate condition differences between them. With the pose normalization, our model is more adaptive to the small datasets and the model pre-trained on only Market-1501 can be easily fine-tuned on the small datasets, achieving much better result than existing models.
Transfer Learning Results
-------------------------
[Dataset]{}
----------------- ----------- ---------------- ---------------- ---------------- ----------- ----------- ----------- -----------
[Methods]{} [R-1]{} [mAP]{} [R-1]{} [mAP]{} [R-1]{} [R-5]{} [R-1]{} [R-5]{}
[ResNet-50-A]{} [87.26]{} [69.32]{} [72.80]{} [52.48]{} [76.83]{} [93.79]{} [64.56]{} [83.66]{}
[ResNet-50-B]{} [63.75]{} [41.29]{} [26.62]{} [14.30]{} [32.54]{} [55.12]{} [36.18]{} [51.17]{}
[Ours]{} **89.43** **72.58**[ ]{} **73.58**[ ]{} **53.20**[ ]{} **79.76** **96.24** **67.65** **86.64**
[Feature(s) ]{}
------------------ ----------- ----------- ----------- -----------
[Methods ]{} [R-1 ]{} [mAP ]{} [R-1 ]{} [mAP]{}
[ResNet-50-A ]{} [87.26]{} [69.34]{} [87.26]{} [69.34]{}
[ResNet-50-B ]{} 58.70 36.69 [63.75]{} [41.67]{}
[Ours (SL) ]{} 87.65 69.60 [89.40]{} [72.58]{}
: \[tab:The-Ablation-Study-2\]The Ablation Study of Market-1501 on 1 pose feature and 8 pose features.
We report our results obtained under the TL settings on the three datasets CUHK03, CUHK01, and DukeMTMC-reID in Table \[tab:Results-of-CUHK03.\](b), and Table \[tab:Results-on-the-Duke\] respectively. On **** CUHK01 dataset, we can achieve $27.58\%$ Rank-1 accuracy in Table \[tab:Results-of-CUHK03.\](b) which is comparable to some models trained under the supervised learning setting, such as eSDC [[@unsupervised_per_reid]]{}. These results thus show that our model has the potential to be truly generalizable to a new re-id data from new camera networks when operating in a ‘plug-and-play’ mode. Our results are also compared against those of ResNet-50-A (TL) baseline. On all three datasets, we can observe that our model gets improved over those of ResNet-50-A (TL) baseline. Again, this demonstrates that our pose normalized person images can also help the person re-id in the transfer learning settings. Note that due to the intrinsic difficulty of transfer setting, the results are still much lower than those in supervised setting.
Further Evaluations
-------------------
**Ablation Studies**. We first evaluate the contributions from the two types of features computed using ResNet-50-A and ResNet-50-B respectively towards the final performance. Table \[tab:The-Ablation-Study-2\] shows that: (1) Each model on its own is quite strong - better than many existing models compared earlier. (2) When the two types of features are combined, there is an improvement in the final results on all four datasets. This clearly indicates that the two types of features are complementary to each other. In a second study, we compare the result obtained when features are merged with 8 poses and that obtained with only one pose, in Table \[tab:The-Ablation-Study-2\]. The result drops from $72.58$ to $69.60$ on Market-1501 on mAP. This suggests that having eight canonical poses is beneficial the quality of generated image under one particular pose may be poor; using all eight poses thus reduces the sensitivity to the quality of the generated images for specific poses.
**Examples of the synthesized images**. Figure \[fig:Visualization\] gives some examples of the synthesized image poses. Given one input image, our image generator can produce realistic images under different poses, while keeping the similar visual appearance as the input person image. We find that, (1) Even though we did not explicitly use the attributes to guide the PN-GAN, the generated images of different poses have roughly the same visual attributes as the original images. (2) Our model can help alleviate the problems caused by occlusion as shown in the last row of Fig. \[fig:Visualization\]: a man with yellow shirt and grey trousers is blocked by a bicycle, while our image generator can generate synthesized images to keep his key attributes whilst removing the occlusion.
Conclusion
==========
We have proposed a novel deep person image generation model by synthesizing pose-normalized person images for re-id. In contrast to previous re-id approaches that try to extract discriminator features which are identity-sensitive but view-insensitive, the proposed method learns complementary features from both original images and pose-normalized synthetic images. Extensive experiments on four benchmarks showed that our model achieves state-of-the-art performance. More importantly, we demonstrated that our model can be generalized to new re-id datasets collected from new camera networks without any additional data collection and model fine-tuning.
[^1]: Details of structure are in the Supplementary.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We introduce and study a new type of integral equations called anticipating backward stochastic Volterra integral equations (anticipating BSVIEs). In these equations the generator involves not only the present values but also the future values of the solutions. We obtain the existence and uniqueness theorem and a comparison theorem for the solutions to these anticipating BSVIEs.'
author:
- Jiaqiang Wenand Yufeng Shi
title: '**Anticipating backward stochastic Volterra integral equations** '
---
**keywords**: Anticipating backward stochastic Volterra integral equation, Backward stochastic Volterra integral equation, Comparison theorem.
: 60H10, 60H20.
Introduction
============
Stochastic Volterra integral equations (SVIEs, for short) were introduced by Berger and Mizel [@Berger], and developed to the anticipating SVIEs by Pardoux and Protter [@Protter], Alòs and Nualart [@Nualart]. General backward stochastic Volterra integral equations (BSVIEs, for short) were introduced by Yong [@Yong2; @Yong3]. In more details, let $(\Omega,\mathcal{F},P,\mathcal{F}_{t},t\geq 0)$ be a complete stochastic basis such that $\mathcal{F}_{0}$ contains all $P$-null elements of $\mathcal{F}$ and suppose that the filtration is generated by a $d$-dimensional standard Brownian motion $W=\{W(t); t\geq 0\}$. Let $(Y(\cdot),Z(\cdot,\cdot))$ be the solution of the following backward stochastic Volterra integral equation: $$\label{3}
Y(t)=\psi(t) + \int_t^T g(t,s,Y(s),Z(t,s),Z(s,t)) ds - \int_t^T Z(t,s) dW(s), \ \ t\in [0,T],$$ where $g:\Omega\times \Delta\times \mathbb{R}^{m}\times \mathbb{R}^{m\times d}\times \mathbb{R}^{m\times d}\rightarrow \mathbb{R}^{m}$ and $\psi:\Omega\times[0,T]\rightarrow \mathbb{R}^{m}$ are given maps with $\Delta=\{ (t,s)\in[0,T]^{2}| \ t\leq s\}$. Such an equation was introduced by Yong [@Yong2; @Yong3]. A special case of (\[3\]) with $g(\cdot)$ independent of $Z(s,t)$ and $\psi(t)\equiv \xi$ was studied by Lin [@Lin] a little earlier. Some recent developments of BSVIEs can be found in Eduard and Ludger [@Kromer], Shi, Wen and Yong [@Wen], Shi and Wang [@Wang], Shi, Wang and Yong [@Shi2; @Shi4], Wang and Yong [@Yong], Yong [@Yong4], Zhang [@Zhangx], etc., among theories and applications. The same as anticipating SVIEs, it is a natural question if there are the corresponding “anticipating” BSVIEs.
Recently, Peng and Yang [@Yang] introduced the anticipating (or anticipated) backward stochastic differential equation (BSDE, for short) as follows, $$\label{21}
\begin{cases}
-dY_{t}=f(t,Y_{t},Z_{t},Y_{t+\delta_{t}},Z_{t+\zeta_{t}}) dt - Z_{t} dW_{t}, \ \ \ 0\leq t\leq T; \\
Y_{t}=\xi_{t}, \ \ Z_{t}=\eta_{t}, \ \ \ T\leq t\leq T+K,
\end{cases}$$ where $\xi_{\cdot}, \eta_{\cdot}$ are given adapted stochastic processes, and $\delta_{\cdot}, \zeta_{\cdot}$ are given nonnegative deterministic functions. See Pardoux and Peng [@Peng], El Karoui, Peng, and Quenez [@Peng2], Ma and Yong [@Ma], Chen and Wu [@Wu], Yang and Elliott [@Yang2], etc., for systematic discussions about BSDEs and anticipating BSDEs.
These tempt us to introduce the following new type of BSVIEs: $$\label{18}
\begin{cases}
Y(t)=\psi(t) + \int_t^T g(t,s,Y(s),Z(t,s),Z(s,t),Y(s+\delta_{s}),Z(t,s+\zeta_{s}),Z(s+\zeta_{s},t)) ds \\
\ \ \ \ \ \ \ \ \ \ - \int_t^T Z(t,s) dW(s), \ \ t\in [0,T]; \\
Y(t)=\psi(t), \ \ t\in [T,T+K];\\
Z(t,s)=\eta(t,s), \ \ (t,s)\in [0,T+K]^{2}\setminus [0,T]^{2}.
\end{cases}$$ See Section 3 for detailed discussions. We call equation (\[18\]) the anticipating backward stochastic Volterra integral equation (ABSVIE, for short). One can note that, comparing with BSVIE (\[3\]), the distinct development of ABSVIE (\[18\]) is that the generator of (\[18\]) involves not only the present values of solutions but also the future ones of solutions.
In this paper, we establish the existence and uniqueness of solutions of ABSVIE (\[18\]) under Lipschitz condition. The method used to prove the existence and uniqueness theorem (Theorem \[0\] below) is convenient than the four steps method in Yong [@Yong3]. Since the comparison theorem is a fundamental tool, which plays an important role in the theory and applications of BSVIEs, we also prove a comparison theorem for ABSVIEs, which generalises one of the main results in Wang and Yong [@Yong]. Similar to BSVIE (\[3\]) and the anticipating BSDE (\[21\]), ABSVIE (\[18\]) can also be applied in mathematical finance, risk management, especially in the field of stochastic optimal controls. About this topic, we will give some further studies in the coming future researches.
The rest of the paper is organized as follows. In Section 2, we introduce some preliminaries. Section 3 is devoted to the proof of the existence and uniqueness theorem for ABSVIEs. A comparison theorem for ABSVIEs is also established in Section 4.
Preliminaries
=============
Let $(\Omega,\mathcal{F},P,\mathcal{F}_{t},t\geq 0)$ be a complete stochastic basis such that $\mathcal{F}_{0}$ contains all $P$-null elements of $\mathcal{F}$ and suppose that the filtration is generated by a $d$-dimensional standard Brownian motion $W=\{W(t),t\geq 0\}$. The Euclidean norm of a vector $x\in\mathbb{R}^{m}$ will be denoted by $|x|$, and for a $m\times d$ matrix $A$, we define $\| A \|=\sqrt{Tr AA^{\ast}}$. Given $T>0$, and let $K\geq 0$ be a constant, denote $$\Delta=\{ (t,s)\in[0,T]^{2}| \ 0\leq t\leq s\leq T \}; \ \ \ \widetilde{\Delta}=\{ (t,s)\in[0,T+K]^{2}| \ 0\leq t\leq s\leq T+K \}.$$ Also, for $H=\mathbb{R}^{m},\mathbb{R}^{m\times d}$ and $t\in[0,T],$ denote\
$\bullet$ $L^{2}(\mathcal{F}_{t};H) =\{\xi:\Omega\rightarrow H \mid \xi$ is $\mathcal{F}_{t}$-measurable, $E[|\xi|^{2}]< \infty\}$;\
$\bullet$ $L_{\mathcal{F}_{T}}^2(0,T;H)
=\big \{\psi:\Omega\times [0,T]\rightarrow H \mid \psi(t)$ is $\mathcal{F}_{T\vee t}$-measurable, $E\int_0^{T} |\psi(t)|^{2} dt < \infty\big \}$;\
$\bullet$ $L_{\mathcal{F}}^2(0,T;H)
=\big \{X:\Omega\times [0,T]\rightarrow H \mid X(t)$ is $\mathcal{F}_{t}$-measurable, $E\int_0^T |X(t)|^{2} dt$ $< \infty\big \}$;\
$\bullet$ $L_{\mathcal{F}}^2(\Delta;H)
=\big\{Z:\Omega\times \Delta \rightarrow H \mid Z(t,s)$ is $\mathcal{F}_{s}$-measurable, $E\int_0^{T} \int_t^{T} |Z(t,s)|^{2} dsdt$ $< \infty\big\}$;\
$\bullet$ $L_{\mathcal{F}}^2([0,T]^{2};H)
=\big\{Z:\Omega\times [0,T]^{2}\rightarrow H \mid Z(t,s)$ is $\mathcal{F}_{s}$-measurable, $E\int_0^{T} \int_0^{T} |Z(t,s)|^{2} dsdt$ $< \infty\big\}$.\
For any $\beta\geq 0$, let $\mathcal{H}^{2}_{\Delta}$ be the space of all pairs $(Y,Z)\in L_{\mathcal{F}}^2(0,T;\mathbb{R}^{m})\times L_{\mathcal{F}}^2(\Delta;\mathbb{R}^{m\times d})$ under the following norm $$\| (Y(\cdot),Z(\cdot,\cdot)) \|_{\mathcal{H}_{\Delta}^{2}}
\equiv \left[ E\int_0^{T} \left(e^{\beta t}|Y(t)|^{2} + \int_t^{T} e^{\beta s}\|Z(t,s)\|^{2} ds\right) dt \right]^{\frac{1}{2}}< \infty.$$ Clearly, $\mathcal{H}_{\Delta}^{2}$ is a Hilbert space. Similarly, we can define $L_{\mathcal{F}_{T}}^2(0,T+K;H)$, $L_{\mathcal{F}}^2(0,T+K;H)$, $L_{\mathcal{F}}^2(\widetilde{\Delta};H)$, $L_{\mathcal{F}}^2([0,T+K]^{2};H)$ and $\mathcal{H}_{\widetilde{\Delta}}^{2}$. From the definition, we note that the space $L_{\mathcal{F}_{T}}^2(T,T+K;H)$ is equivalent to the space $L_{\mathcal{F}}^2(T,T+K;H)$.
Let’s consider the following BSVIE, which was introduced by Yong [@Yong2; @Yong3], $$\label{22}
Y(t)=\psi(t) + \int_t^T g(t,s,Y(s),Z(t,s),Z(s,t)) ds - \int_t^T Z(t,s) dW(s), \ \ t\in [0,T],$$ where $\psi(\cdot)\in L_{\mathcal{F}_{T}}^2(0,T;\mathbb{R}^{m})$, and $g:\Delta\times \mathbb{R}^{m}\times \mathbb{R}^{m\times d}\times \mathbb{R}^{m\times d}\times\Omega\longrightarrow \mathbb{R}^{m} $ is $\mathcal{B}(\Delta\times \mathbb{R}^{m}\times \mathbb{R}^{m\times d}\times \mathbb{R}^{m\times d})\otimes\mathcal{F}_{T}$-measurable such that $s\mapsto g(t,s,y,z,\vartheta)$ is $\mathcal{F}$-progressively measurable for all $(t,y,z,\vartheta)\in [0,s]\times \mathbb{R}^{m}\times \mathbb{R}^{m\times d}\times \mathbb{R}^{m\times d}$, $s\in[0,T]$.
- Suppose there exists a constant $L>0$ such that, $P$-a.s., for all $(t,s) \in \Delta, \ y,y' \in \mathbb{R}^{m}, \ z,z',\vartheta,$ $\vartheta' \in \mathbb{R}^{m\times d}$, $$\begin{split}
& |g(t,s,y,z,\vartheta)-g(t,s,y',z',\vartheta')|\leq L\left(|y-y'| + \|z-z'\| + \|\vartheta-\vartheta'\|\right);\\
& and \ E\int_0^T\int_t^T |g_{0}(t,s)|^{2} ds dt< \infty, \ where \ g_{0}(t,s)=g(t,s,0,0,0).
\end{split}$$
An adapted solution $(Y(\cdot),Z(\cdot,\cdot))$ of BSVIE (\[22\]) is called an adapted M-solution if the following holds: $$\label{54}
Y(t) =E[Y(t)] + \int_0^t Z(t,s)dW(s), \ \ 0\leq t\leq T.$$
The following propositions can be found in [@Wang; @Yong; @Yong3].
\[19\] Under the assumption (H1), for any $\psi(\cdot)\in L_{\mathcal{F}_{T}}^2(0,T;\mathbb{R}^{m})$, BSVIE (\[22\]) admits a unique adapted M-solution.
\[2\] Consider the following simple BSVIE $$\label{8}
Y(t)=\psi(t) + \int_t^T g(t,s) ds - \int_t^T Z(t,s) dW(s), \ t\in [0,T],$$ where $\psi(\cdot)\in L_{\mathcal{F}_{T}}^{2}(0,T;\mathbb{R}^{m})$ and $g\in L_{\mathcal{F}}^2(\Delta;\mathbb{R}^{m})$. Then the above equation has a unique adapted solution $(Y,Z)\in \mathcal{H}_{\Delta}^{2}$, and the following estimate holds: $$\label{10}
\begin{split}
&E \int_0^T \left(e^{\beta t}|Y(t)|^{2} + \int_t^T e^{\beta s}\|Z(t,s)\|^{2} ds\right) dt\\
\leq& Ce^{\beta T}E \int_0^T |\psi(t)|^{2} dt + \frac{C}{\beta}E \int_0^T \int_t^T e^{\beta s}|g(t,s)|^{2} dsdt.
\end{split}$$ Hereafter C is a positive constant which may be different from line to line.
\[16\] For $i=0,1$, assume $g^{i}=g^{i}(t,s,y,z)$ satisfies (H1). Let $(Y^{i},Z^{i})\in \mathcal{H}_{\Delta}^{2}$ be respectively the solutions of the following BSVIEs, $$Y^{i}(t)=\psi^{i}(t) + \int_t^T g^{i}(t,s,Y^{i}(s),Z^{i}(t,s)) ds - \int_t^T Z^{i}(t,s) dW(s), \ \ t\in [0,T].$$ Suppose $\overline{g}(t,s,y,z)$ satisfies (H1) such that $y\mapsto \overline{g}(t,s,y,z)$ is nondecreasing with $$g^{0}(t,s,y,z)\leq \overline{g}(t,s,y,z)\leq g^{1}(t,s,y,z),
\ \ \forall (t,y,z)\in [0,s]\times \mathbb{R}^{m}\times \mathbb{R}^{m\times d}, \ a.s., \ a.e. \ s\in[0,T].$$ Moreover, $\overline{g}_{z}(t,s,y,z)$ exists and $$\overline{g}_{z_{1}}(t,s,y,z),...,\overline{g}_{z_{d}}(t,s,y,z) \in \mathbb{R}_{d}^{m\times m},
\ \ \forall (t,y,z)\in [0,s]\times \mathbb{R}^{m}\times \mathbb{R}^{m\times d}, \ a.s., \ a.e. \ s\in[0,T].$$ Then for any $\psi^{i}(\cdot)\in L_{\mathcal{F}_{T}}^2(0,T;\mathbb{R}^{m})$ satisfying $ \psi^{0}(t)\leq \psi^{1}(t), \ \ a.s., \ t\in[0,T],$ the corresponding unique adapted solution $(Y^{i},Z^{i})\in \mathcal{H}_{\Delta}^{2}$ satisfies $$Y^{0}(t)\leq Y^{1}(t), \ \ a.s., \ t\in[0,T].$$
Existence and uniqueness theorem
================================
We now consider a new form of BSVIEs as follows: $$\label{4}
\begin{cases}
Y(t)=\psi(t) + \int_t^T g(t,s,Y(s),Z(t,s),Z(s,t),Y(s+\delta_{s}),Z(t,s+\zeta_{s}),Z(s+\zeta_{s},t)) ds \\
\ \ \ \ \ \ \ \ \ \ - \int_t^T Z(t,s) dW(s), \ \ t\in [0,T]; \\
Y(t)=\psi(t), \ \ t\in [T,T+K];\\
Z(t,s)=\eta(t,s), \ \ (t,s)\in [0,T+K]^{2}\setminus [0,T]^{2}. \\
\end{cases}$$ where $\delta_{\cdot}$ and $\zeta_{\cdot}$ are two $\mathbb{R}^{+}$-valued continuous functions defined on $[0,T]$ such that:
- There exists a constant $K\geq 0$ such that, for all $ s\in [0,T]$, $$s+\delta_{s}\leq T+K; \ \ s+\zeta_{s}\leq T+K.$$
- There exists a constant $M\geq 0$ such that, for all non-negative and integrable $g_{1}(\cdot), g_{2}(\cdot,\cdot)$, $t\in [0,T]$, $$\begin{cases}
\int_t^T g_{1}(s+\delta_{s}) ds\leq M\int_t^{T+K} g_{1}(s) ds; \\
\int_t^T g_{2}(t,s+\zeta_{s}) ds\leq M \int_t^{T+K} g_{2}(t,s) ds; \\
\int_t^T g_{2}(s+\zeta_{s},t) ds\leq M \int_t^{T+K} g_{2}(s,t) ds.
\end{cases}$$
We call equation (\[4\]) the anticipating BSVIE.
Assume that for all $(t,s)\in \Delta,$ $g(t,s,y,z,x,\xi,\eta,\varsigma,\omega): \mathbb{R}^{m}\times \mathbb{R}^{m\times d} \times \mathbb{R}^{m\times d}
\times L^{2}(\mathcal{F}_{r_{1}};\mathbb{R}^{m}) \times L^{2}(\mathcal{F}_{r_{2}};\mathbb{R}^{m\times d})\times L^{2}(\mathcal{F}_{r_{3}};\mathbb{R}^{m
\times d})\times\Omega
\longrightarrow L^{2}(\mathcal{F}_{s};\mathbb{R}^{m})$, where $r_{1},r_{2},r_{3} \in[s,T+K],$ and $g$ satisfies the following conditions:
- There exists a constant $L>0$, such that, $P$-a.s., for all $(t,s)\in \Delta, y,y'\in \mathbb{R}^{m},z,z',x,x'\in {R}^{m\times d},
\xi(\cdot),\xi'(\cdot)\in L_{\mathcal{F}}^2(s,T+K;\mathbb{R}^{m}),\eta(t,\cdot),\eta'(t,\cdot),
\varsigma(\cdot,t),$ $\varsigma'(\cdot,t)\in L_{\mathcal{F}}^2(s,T+K;\mathbb{R}^{m\times d}), r,r'\in [s,T+K],$ we have $$\begin{split}
&|g(t,s,y,z,x,\xi(r),\eta(t,r'),\varsigma(r',t))-g(t,s,y',z',x',\xi'(r),\eta'(t,r'),\varsigma'(r',t)) | \\
\leq& L\bigg(|y-y'|+\|z-z'\|+\|x-x'\| \\
& +E^{\mathcal{F}_{s}}\left[|\xi(r)-\xi'(r)|+\|\eta(t,r')-\eta'(t,r')\|+\|\varsigma(r',t)-\varsigma'(r',t)\|\right]\bigg);\\
&and \ E\int_0^T \int_t^T |g_{0}(t,s)|^{2} ds dt < \infty, \ where \ g_{0}(t,s)=g(t,s,0,0,0,0,0,0). \\
\end{split}$$
Note that for all $(t,s)\in \Delta,$ $g(t,s,\cdot,\cdot,\cdot,\cdot,\cdot,\cdot)$ is $\mathcal{F}_{s}$-measurable ensures the solution to the anticipating BSVIE is $\mathcal{F}_{s}$-adapted.
In order to establish the well-posedness of anticipating BSVIE (\[4\]), we introduce the following space. For any $\beta\geq 0$, we let $\mathcal{H}^{2}[0,T+K]$ be the space of all pairs $$(Y,Z)\in L_{\mathcal{F}}^2(0,T+K;\mathbb{R}^{m})\times L_{\mathcal{F}}^2([0,T+K]^{2};\mathbb{R}^{m\times d})$$ under the following norm $$\| (Y(\cdot),Z(\cdot,\cdot)) \|_{\mathcal{H}^{2}[0,T+K]}
\equiv \left[ E\int_0^{T+K} \left(e^{\beta t}|Y(t)|^{2} + \int_0^{T+K} e^{\beta s}\|Z(t,s)\|^{2} ds\right) dt \right]^{\frac{1}{2}}< \infty.$$ Different from $\mathcal{H}_{\Delta}^{2}$ defined in the previous section, we see that $Z(\cdot,\cdot)$ is defined on $[0,T+K]^{2}$. Similar to $\mathcal{H}_{\Delta}^{2}$, we know $\mathcal{H}^{2}[0,T+K]$ is also a Hilbert space.
Next, let $\mathcal{M}^{2}[0,T+K]$ be the set of all pairs $(Y,Z)\in \mathcal{H}^{2}[0,T+K]$ such that Eq. (\[54\]) holds in $[0,T+K]$, i.e., $$\label{24}
Y(t) =E[Y(t)] + \int_0^t Z(t,s)dW(s), \ \ 0\leq t\leq T+K.$$ Then for any $(Y,Z)\in \mathcal{M}^{2}[0,T+K]$, one can show that $$\label{7}
\begin{split}
&E\int_0^{T+K} \left(e^{\beta t}|Y(t)|^{2} + \int_0^{T+K} e^{\beta s}\|Z(t,s)\|^{2} ds \right) dt\\
\leq& 2E\int_0^{T+K} \left(e^{\beta t}|Y(t)|^{2} + \int_t^{T+K} e^{\beta s}\|Z(t,s)\|^{2} ds \right) dt,
\end{split}$$ since from (\[24\]) one has $$\label{23}
E\int_0^{T+K} \int_0^t e^{\beta s}\|Z(t,s)\|^{2} ds dt \leq E\int_0^{T+K} e^{\beta t}|Y(t)|^{2} dt.$$ This means that we can use the following as an equivalent norm in $\mathcal{M}^{2}[0,T+K]$: $$\| (Y(\cdot),Z(\cdot,\cdot)) \|_{\mathcal{M}^{2}[0,T+K]}
\equiv \bigg[ E\int_0^{T+K} \bigg(e^{\beta t}|Y(t)|^{2} + \int_t^{T+K} e^{\beta s}\|Z(t,s)\|^{2} ds\bigg) dt \bigg]^{\frac{1}{2}}.$$ We also let $\overline{\mathcal{M}}^{2}[0,T+K]$ be the set of all pairs $(\psi,\eta)\in L_{\mathcal{F}_{T}}^2(0,T+K;\mathbb{R}^{m})\times L_{\mathcal{F}}^2([0,T+K]^{2};\mathbb{R}^{m\times d})$ such that $$\psi(t) =E[\psi(t)] + \int_0^t \eta(t,s)dW(s), \ \ 0\leq t\leq T+K.$$ Since the space $L_{\mathcal{F}_{T}}^2(T,T+K;\mathbb{R}^{m})$ is equivalent to the space $L_{\mathcal{F}}^2(T,T+K;\mathbb{R}^{m})$, the space $\overline{\mathcal{M}}^{2}[T,T+K]$ is equivalent to the space $\mathcal{M}^{2}[T,T+K]$ too.
We now state and proof the well-posedness theorem for anticipating BSVIE (\[4\]).
\[0\] Suppose that $g$ satisfies (H2), and $\delta,\zeta$ satisfy (i) and (ii). Then for any $(\psi(\cdot),\eta(\cdot,\cdot))\in \overline{\mathcal{M}}^{2}[0,T+K]$, the anticipating BSVIE (\[4\]) admits a unique adapted M-solution $(Y(\cdot),Z(\cdot,\cdot))\in \mathcal{H}^{2}[0,T+K]$.
For any $(y(\cdot),z(\cdot,\cdot))\in \mathcal{M}^{2}[0,T+K]$, consider the following BSVIE: $$\label{6}
\begin{cases}
Y(t)=\psi(t) + \int_t^T \overline{g}(t,s) ds - \int_t^T Z(t,s) dW(s), \ \ t\in [0,T]; \\
Y(t)=\psi(t), \ \ t\in [T,T+K];\\
Z(t,s)=\eta(t,s), \ \ (t,s)\in [0,T+K]^{2}\setminus [0,T]^{2}, \\
\end{cases}$$ where $$\overline{g}(t,s)=g(t,s,y(s),z(t,s),z(s,t),y(s+\delta_{s}),z(t,s+\zeta_{s}),z(s+\zeta_{s},t)).$$ From Proposition \[19\] and \[2\], Eq. (\[6\]) admits a unique adapted solution $(Y(\cdot),Z(\cdot,\cdot))\in \mathcal{H}_{\widetilde{\Delta}}^{2}$. Now we define $Z(\cdot,\cdot)$ on $\widetilde{\Delta}^{c}$ from the following: $$Y(t)=EY(t) + \int_0^t Z(t,s) dW(s), \ \ t\in [0,T+K].$$ Then $(Y(\cdot),Z(\cdot,\cdot))\in \mathcal{M}^{2}[0,T+K]$ is an adapted M-solution to Eq. (\[6\]). Thus, the maping $(y(\cdot),z(\cdot,\cdot))\mapsto (Y(\cdot),Z(\cdot,\cdot))$ is well-defined. By the estimate (\[10\]) in Proposition \[2\], one has $$\begin{split}
&E\int_0^T \left(e^{\beta t}|Y(t)|^{2} + \int_t^T e^{\beta s}\|Z(t,s)\|^{2} ds\right) dt \\
\leq& Ce^{\beta T}E \int_0^T |\psi(t)|^{2} dt
+ \frac{C}{\beta}E\int_0^T \int_t^T e^{\beta s}|\overline{g}(t,s)|^{2} dsdt.\\
\end{split}$$ From (H2) and (\[23\]), and note that $\delta,\zeta$ satisfy (i) and (ii), we have $$\label{25}
\begin{split}
&E\int_0^T \int_t^T e^{\beta s}|\overline{g}(t,s)|^{2} dsdt\\
\leq& 5L^{2}E \int_0^T \int_t^T e^{\beta s}\bigg(|g_{0}(t,s)|^{2} + |y(s)|^{2} + \|z(t,s)\|^{2} + \|z(s,t)\|^{2} \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +3\big[|y(s+\delta_{s})|^{2} + \|z(t,s+\zeta_{s})\|^{2} + \|z(s+\zeta_{s},t)\|^{2}\big]\bigg) dsdt \\
\leq& 5L^{2}E \int_0^T \int_t^T e^{\beta s}|g_{0}(t,s)|^{2} dsdt + 10L^{2}(T+1)E \int_0^T \left(e^{\beta t}|y(t)|^{2}
+ \int_0^T e^{\beta s}\|z(t,s)\|^{2} ds\right)dt \\
&+30L^{2}M(T+K+1)E \int_0^{T+K}\left(e^{\beta t}|y(t)|^{2} + \int_0^{T+K} e^{\beta s}\|z(t,s)\|^{2} ds\right)dt\\
\leq& 5L^{2}E \int_0^T \int_t^T e^{\beta s}|g_{0}(t,s)|^{2} dsdt \\
& +60L^{2}(M+1)(T+K+1)E \int_0^{T+K}\left(e^{\beta t}|y(t)|^{2} + \int_0^{T+K} e^{\beta s}\|z(t,s)\|^{2} ds\right)dt.\\
\end{split}$$ Hence $$\label{26}
\begin{split}
&E \int_0^T \left(e^{\beta t}|Y(t)|^{2} + \int_t^T e^{\beta s}\|Z(t,s)\|^{2} ds\right) dt\\
\leq& Ce^{\beta T}E \int_0^T |\psi(t)|^{2} dt + \frac{C}{\beta}E \int_0^T \int_t^T e^{\beta s}|g_{0}(t,s)|^{2} dsdt \\
&+ \frac{C}{\beta}E \int_0^{T+K}\left(e^{\beta t}|y(t)|^{2} + \int_0^{T+K} e^{\beta s}\|z(t,s)\|^{2} ds\right)dt.\\
\end{split}$$ Now if $(Y_{i}(\cdot),Z_{i}(\cdot,\cdot))$ is the corresponding adapted M-solution of $(y_{i}(\cdot),z_{i}(\cdot,\cdot))$ to BSVIE (\[6\]), $i=1,2$, note (\[7\]), then $$\begin{split}
&E \int_0^{T+K} \left(e^{\beta t}|Y_{1}(t)-Y_{2}(t)|^{2} + \int_0^{T+K} e^{\beta s}\|Z_{1}(t,s)-Z_{2}(t,s)\|^{2} ds\right) dt\\
\leq& \frac{C}{\beta}E\int_0^{T+K} \left(e^{\beta t}|y_{1}(t)-y_{2}(t)|^{2} + \int_0^{T+K} e^{\beta s}\|z_{1}(t,s)-z_{2}(t,s)\|^{2} ds\right) dt.
\end{split}$$ Let $\beta=2C+1$, then $(y(\cdot),z(\cdot,\cdot))\mapsto (Y(\cdot),Z(\cdot,\cdot))$ is a contraction on $\mathcal{M}^{2}[0,T+K]$. This completes the proof.
Under the assumptions of Theorem \[0\], the solution of anticipating BSVIE (\[4\]) satisfies $$\label{28}
\begin{split}
& E\bigg[\int_0^T e^{\beta t} |Y(t)|^{2} dt + \int_0^T \int_t^T e^{\beta s} \|Z(t,s)\|^{2} dsdt\bigg]\\
\leq& CE\bigg[\int_0^{T+K} |\psi(t)|^{2} dt +\int_0^T \int_t^T e^{\beta s}|g_{0}(t,s)|^{2} dsdt
+ \int_T^{T+K} \int_T^{T+K} e^{\beta s}\|\eta(t,s)\|^{2} dsdt\\
& + \int_0^T \int_T^{T+K} \big(e^{\beta s}\|\eta(t,s)\|^{2}+ e^{\beta t}\|\eta(s,t)\|^{2}\big) dsdt \bigg].
\end{split}$$
By the estimate (\[10\]), similar to (\[25\]) and (\[26\]), and note (\[7\]), we have $$\begin{split}
&E \left(\int_0^T e^{\beta t}|Y(t)|^{2}dt + \int_0^T\int_0^T e^{\beta s}\|Z(t,s)\|^{2} ds dt\right)\\
\leq& Ce^{\beta T}E \int_0^T |\psi(t)|^{2} dt + \frac{C}{\beta}E \int_0^T \int_t^T e^{\beta s}|g_{0}(t,s)|^{2} dsdt \\
&+ \frac{C}{\beta}E \int_0^{T+K}e^{\beta t}|Y(t)|^{2}dt + \frac{C}{\beta}E \int_0^{T+K}\int_0^{T+K} e^{\beta s}\|Z(t,s)\|^{2} dsdt.
\end{split}$$ Since $$\begin{split}
& \frac{C}{\beta}E \int_0^{T+K}e^{\beta t}|Y(t)|^{2}dt + \frac{C}{\beta}E \int_0^{T+K}\int_0^{T+K} e^{\beta s}\|Z(t,s)\|^{2} dsdt\\
=& \frac{C}{\beta}E \bigg(\int_0^{T}e^{\beta t}|Y(t)|^{2}dt + \int_0^{T}\int_0^{T} e^{\beta s}\|Z(t,s)\|^{2} dsdt\bigg)\\
&+ \frac{C}{\beta}E\int_T^{T+K} e^{\beta t}|\psi(t)|^{2}dt + \frac{C}{\beta}E\int_T^{T+K} \int_T^{T+K} e^{\beta s}\|\eta(t,s)\|^{2} dsdt\\
&+ \frac{C}{\beta}E \bigg(\int_0^{T}\int_T^{T+K} e^{\beta s}\|\eta(t,s)\|^{2} dsdt + \int_T^{T+K}\int_0^{T} e^{\beta s}\|\eta(t,s)\|^{2} dsdt \bigg).
\end{split}$$ Now let $\beta=2C$, then we obtain the estimate (\[28\]).
Comparison theorem
==================
In this section we prove a comparison theorem for ABSVIEs of the following type: For $i=0,1,$ $$\label{9}
\begin{cases}
Y^{i}(t)=\psi^{i}(t) + \int_t^T g^{i}(t,s,Y^{i}(s),Z^{i}(t,s),Y^{i}(s+\delta_{s})) ds - \int_t^T Z^{i}(t,s) dW(s), \ \ t\in [0,T]; \\
Y^{i}(t)=\psi^{i}(t), \ \ t\in [T,T+K].
\end{cases}$$ For ABSVIEs of the above form, we need only the values $Z^{i}(t,s)$ of $Z^{i}(\cdot,\cdot)$ for $(t,s)\in \Delta$ and the notation of $M$-solution is not necessary. It is easy to see that under the assumption of Theorem \[0\], for any $\psi^{i}(\cdot)\in L_{\mathcal{F}_{T}}^2(0,T+K;\mathbb{R}^{m})$, ABSVIE (\[9\]) admits a unique adapted solution $(Y^{i}(\cdot),Z^{i}(\cdot,\cdot))\in L_{\mathcal{F}}^2(0,T+K;\mathbb{R}^{m})\times L_{\mathcal{F}}^2(\Delta;\mathbb{R}^{m\times d})$.
\[15\] Let $\delta$ and $\zeta$ satisfy (i)-(ii), and $g^{i}$ satisfies (H2), $i=1,2$. Suppose $\overline{g}=\overline{g}(t,s,y,z,\xi)$ satisfies (H2) and for all $(t,s,y,z)\in \Delta\times\mathbb{R}^{m}\times \mathbb{R}^{m\times d}$, $\overline{g}(t,s,y,z,\cdot)$ is increasing, i.e., $\overline{g}(t,s,y,z,\xi_{1}(r))\leq \overline{g}(t,s,y,z,\xi_{2}(r))$, if $\xi_{1}(r)\leq \xi_{2}(r)$, $\xi_{1}(\cdot),\xi_{2}(\cdot)\in L^{2}_{\mathcal{F}}(s,T+K;\mathbb{R})$, $r\in [s,T+K]$. Moreover $$\begin{split}
g^{0}&(t,s,y,z,\xi)\leq \overline{g}(t,s,y,z,\xi)\leq g^{1}(t,s,y,z,\xi), \\
\ \ &\forall (t,s,y,z,\xi)\in \Delta\times \mathbb{R}^{m}\times \mathbb{R}^{m\times d}\times L^{2}(\mathcal{F}_{r};\mathbb{R}^{m}), \ a.s., \ a.e.,
\end{split}$$ and $\overline{g}_{z}(t,s,y,z,\xi)$ exists with $$\begin{split}
\overline{g}_{z_{1}}&(t,s,y,z,\xi),...,\overline{g}_{z_{d}}(t,s,y,z,\xi) \in \mathbb{R}_{d}^{m\times m},\\
\ \ &\forall (t,s,y,z,\xi)\in \Delta\times \mathbb{R}^{m}\times \mathbb{R}^{m\times d}\times L^{2}(\mathcal{F}_{r};\mathbb{R}^{m}), \ a.s., \ a.e.
\end{split}$$ Then for any $\psi^{i}(\cdot)\in L_{\mathcal{F}_{T}}^2(0,T+K;\mathbb{R}^{m})$ satisfying $\psi^{0}(t)\leq \psi^{1}(t), \ a.s., \ t\in[0,T+K],$ we have $$Y^{0}(t)\leq Y^{1}(t), \ a.s., \ t\in[0,T+K].$$
Let $\overline{\psi}(\cdot)\in L_{\mathcal{F}_{T}}^2(0,T+K;\mathbb{R}^{m})$ and $$\psi^{0}(t)\leq \overline{\psi}(t)\leq \psi^{1}(t), \ a.s. \ t\in[0,T+K].$$ Let $(\overline{Y}(\cdot),\overline{Z}(\cdot,\cdot))\in L_{\mathcal{F}}^2(0,T+K;\mathbb{R}^{m})\times L_{\mathcal{F}}^2(\Delta;\mathbb{R}^{m\times d})$ be the unique adapted solution to the following ABSVIE: $$\label{11}
\begin{cases}
\overline{Y}(t)=\overline{\psi}(t) + \int_t^T \overline{g}(t,s,\overline{Y}(s),\overline{Z}(t,s),
\overline{Y}(s+\delta_{s}) ds - \int_t^T \overline{Z}(t,s) dW(s), \ \ t\in [0,T]; \\
\overline{Y}(t)=\overline{\psi}(t), \ \ t\in [T,T+K].\\
\end{cases}$$ Now we set $\widetilde{Y}_{0}(\cdot)=Y^{1}(\cdot)$ and consider the following BSVIE: $$\begin{cases}
\widetilde{Y}_{1}(t)=\overline{\psi}(t) + \int_t^T \overline{g}(t,s,\widetilde{Y}_{1}(s),\widetilde{Z}_{1}(t,s),
\widetilde{Y}_{0}(s+\delta_{s})) ds - \int_t^T \widetilde{Z}_{1}(t,s) dW(s), \ \ t\in [0,T]; \\
\widetilde{Y}_{1}(t)=\overline{\psi}(t), \ \ t\in [T,T+K].\\
\end{cases}$$ Let $(\widetilde{Y}_{1}(\cdot),\widetilde{Z}_{1}(\cdot,\cdot))\in L_{\mathcal{F}}^2(0,T+K;\mathbb{R}^{m})\times L_{\mathcal{F}}^2(\Delta;\mathbb{R}^{m\times d})$ be the unique adapted solution to the above equation. Since $$\begin{cases}
\overline{g}(t,s,y,z,\widetilde{Y}_{0}(s+\delta_{s}))
\leq g^{1}(t,s,y,z,\widetilde{Y}_{0}(s+\delta_{s})), \ \
(t,s,y,z)\in \Delta \times \mathbb{R}^{m}\times \mathbb{R}^{m\times d}, \ a.s., \ a.e.;\\
\overline{\psi}(t) \leq \psi^{1}(t),\ \ \ a.s. \ \ t\in[0,T+K].
\end{cases}$$ By Proposition \[16\], we obtain that $$\widetilde{Y}_{1}(t)\leq \widetilde{Y}_{0}(t), \ \ a.s. \ t\in[0,T+K].$$ Next, we consider the following BSVIE: $$\begin{cases}
\widetilde{Y}_{2}(t)=\overline{\psi}(t) + \int_t^T \overline{g}(t,s,\widetilde{Y}_{2}(s),\widetilde{Z}_{2}(t,s),\widetilde{Y}_{1}(s+\delta_{s})) ds
- \int_t^T \widetilde{Z}_{2}(t,s) dW(s), \ \ t\in [0,T]; \\
\widetilde{Y}_{2}(t)=\overline{\psi}(t), \ \ t\in [T,T+K].\\
\end{cases}$$ Let $(\widetilde{Y}_{2}(\cdot),\widetilde{Z}_{2}(\cdot,\cdot))
\in L_{\mathcal{F}}^2(0,T+K;\mathbb{R}^{m})\times L_{\mathcal{F}}^2(\Delta;\mathbb{R}^{m\times d})$ be the adapted solution to the above equation. Now, since $\xi\mapsto \overline{g}(t,s,y,z,\xi)$ is increasing, we have $$\overline{g}(t,s,y,z,\widetilde{Y}_{1}(s+\delta_{s})) \leq \overline{g}(t,s,y,z,\widetilde{Y}_{0}(s+\delta_{s})), \ \
(t,s,y,z)\in \Delta\times \mathbb{R}^{m}\times \mathbb{R}^{m\times d}, \ a.s., \ a.e.$$ Hence, similar to the above, we obtain $$\widetilde{Y}_{2}(t)\leq \widetilde{Y}_{1}(t), \ \ \ a.s. \ t\in[0,T+K].$$ By induction, we can construct a sequence $\{(\widetilde{Y}_{k}(\cdot),\widetilde{Z}_{k}(\cdot,\cdot))\}_{k\geq 1}
\in L_{\mathcal{F}}^2(0,T+K;\mathbb{R}^{m})\times L_{\mathcal{F}}^2(\Delta;\mathbb{R}^{m\times d})$ such that $$\begin{cases}
\widetilde{Y}_{k}(t)=\overline{\psi}(t) + \int_t^T \overline{g}(t,s,\widetilde{Y}_{k}(s),\widetilde{Z}_{k}(t,s),
\widetilde{Y}_{k-1}(s+\delta_{s})) ds - \int_t^T \widetilde{Z}_{k}(t,s) dW(s), \ t\in [0,T]; \\
\widetilde{Y}_{k}(t)=\overline{\psi}(t), \ t\in [T,T+K].\\
\end{cases}$$ Similarly, we deduce $$Y^{1}(t)=\widetilde{Y}_{0}(t)\geq \widetilde{Y}_{1}(t)\geq \widetilde{Y}_{2}(t)\cdots, \ \ \ a.s. \ t\in[0,T+K].$$ Next we will show that the sequence $\{(\widetilde{Y}_{k}(\cdot),\widetilde{Z}_{k}(\cdot,\cdot))\}_{k\geq 2}$ is Cauchy sequence. By utilizing the estimate (\[10\]), we have $$\begin{split}
&E\int_0^T \left(e^{\beta t}|\widetilde{Y}_{k}(t)-\widetilde{Y}_{k-1}(t)|^{2}
+ \int_t^T e^{\beta s}\|\widetilde{Z}_{k}(t,s)-\widetilde{Z}_{k-1}(t,s)\|^{2} ds \right) dt\\
\leq& \frac{C}{\beta}E\int_0^T \int_t^T e^{\beta s}
\bigg(\overline{g}(t,s,\widetilde{Y}_{k}(s),\widetilde{Z}_{k}(t,s),\widetilde{Y}_{k-1}(s+\delta_{s}))\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
-\overline{g}(t,s,\widetilde{Y}_{k-1}(s),\widetilde{Z}_{k-1}(t,s),\widetilde{Y}_{k-2}(s+\delta_{s}))\bigg)^{2} dsdt\\
\leq& \frac{C}{\beta}E\int_0^T \int_t^T e^{\beta s}\bigg(|\widetilde{Y}_{k}(s)-\widetilde{Y}_{k-1}(s)|^{2} +\|\widetilde{Z}_{k}(t,s)-\widetilde{Z}_{k-1}(t,s)\|^{2}\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +|\widetilde{Y}_{k-1}(s+\delta_{s})-\widetilde{Y}_{k-2}(s+\delta_{s})|^{2}\bigg) dsdt\\
\leq& \frac{C}{\beta}E\int_0^T\left( e^{\beta t}|\widetilde{Y}_{k}(t)-\widetilde{Y}_{k-1}(t)|^{2}
+\int_t^T e^{\beta s}\|\widetilde{Z}_{k}(t,s)-\widetilde{Z}_{k-1}(t,s)\|^{2} ds\right) dt \\
& \ \ \ +\frac{C}{\beta}E\int_0^{T+K} e^{\beta t}|\widetilde{Y}_{k-1}(t)-\widetilde{Y}_{k-2}(t)|^{2} dt.
\end{split}$$ Hence $$\begin{split}\label{13}
&(1-\frac{C}{\beta})E\int_0^T \left(e^{\beta t}|\widetilde{Y}_{k}(t)-\widetilde{Y}_{k-1}(t)|^{2}
+ \int_t^T e^{\beta s}\|\widetilde{Z}_{k}(t,s)-\widetilde{Z}_{k-1}(t,s)\|^{2} ds\right) dt \\
\leq& \frac{C}{\beta} E\int_0^{T+K} e^{\beta t}|\widetilde{Y}_{k-1}(t)-\widetilde{Y}_{k-2}(t)|^{2} dt.
\end{split}$$ Note that the constant $C>0$ in the above can be chosen independent of $\beta\geq 0$. Thus by choosing $\beta=3C$, we obtain $$\begin{split}
& E\int_0^{T+K} \left(e^{\beta t}|\widetilde{Y}_{k}(t)-\widetilde{Y}_{k-1}(t)|^{2}
+ \int_t^{T+K} e^{\beta s}\|\widetilde{Z}_{k}(t,s)-\widetilde{Z}_{k-1}(t,s)\|^{2} ds\right) dt\\
\leq& \frac{1}{2} E\int_0^{T+K} e^{\beta t}|\widetilde{Y}_{k-1}(t)-\widetilde{Y}_{k-2}(t)|^{2} dt\\
\leq& \frac{1}{2} E\int_0^{T+K} \left(e^{\beta t}|\widetilde{Y}_{k-1}(t)-\widetilde{Y}_{k-2}(t)|^{2}
+ \int_t^{T+K} e^{\beta s}\|\widetilde{Z}_{k-1}(t,s)-\widetilde{Z}_{k-2}(t,s)\|^{2} ds\right) dt\\
\leq& (\frac{1}{2})^{k-2} E\int_0^{T+K} \left(e^{\beta t}|\widetilde{Y}_{2}(t)-\widetilde{Y}_{1}(t)|^{2}
+ \int_t^{T+K} e^{\beta s}\|\widetilde{Z}_{2}(t,s)-\widetilde{Z}_{1}(t,s)\|^{2} ds \right) dt.
\end{split}$$ It follows that $\{(\widetilde{Y}_{k}(\cdot),\widetilde{Z}_{k}(\cdot,\cdot))\}_{k\geq 1}$ is a Cauchy sequence in the Banach space $ L_{\mathcal{F}}^2(0,T+K;\mathbb{R}^{m})\times L_{\mathcal{F}}^2(\Delta;\mathbb{R}^{m\times d})$. Denote their limits by $\widetilde{Y}(\cdot)$ and $\widetilde{Z}(\cdot,\cdot)$, respectively. Then $(\widetilde{Y}(\cdot),\widetilde{Z}(\cdot,\cdot))$ $\in L_{\mathcal{F}}^2(0,T+K;\mathbb{R}^{m})\times L_{\mathcal{F}}^2(\Delta;\mathbb{R}^{m\times d})$ and $$\lim_{k\rightarrow \infty}\left(E\int_0^{T+K} e^{\beta t}|\widetilde{Y}_{k}(t)-\widetilde{Y}(t)|^{2} dt
+ E\int_0^{T+K}\int_t^{T+K} e^{\beta s}\|\widetilde{Z}_{k}(t,s)-\widetilde{Z}(t,s)\|^{2} ds dt\right)=0,$$ also we have $$\begin{cases}
\widetilde{Y}(t)=\overline{\psi}(t) + \int_t^T \overline{g}(t,s,\widetilde{Y}(s),\widetilde{Z}(t,s),
\widetilde{Y}(s+\delta_{s})) ds
- \int_t^T \widetilde{Z}(t,s) dW(s), \ t\in [0,T]; \\
\widetilde{Y}(t)=\overline{\psi}(t), \ t\in [T,T+K].
\end{cases}$$ Connecting the above equation with the equation (\[11\]), by Theorem \[0\], we have $$\overline{Y}(t)= \widetilde{Y}(t), \ \ \ a.s. \ t\in[0,T+K].$$ Hence we obtain $$\overline{Y}(t)\leq Y^{1}(t), \ \ \ a.s. \ t\in[0,T+K].$$ Similarly, we can prove that $$Y^{0}(t)\leq \overline{Y}(t), \ \ \ a.s. \ t\in[0,T+K].$$ Therefore, our conclusion follows.
Let $g^{0}(t,s,\xi(r))=-E^{\mathcal{F}_{s}}[|\xi(r)|]-ln2$, $g^{1}(t,s,\xi(r))=E^{\mathcal{F}_{s}}[|\xi(r)|]+\pi$. We choose $\overline{g}(t,s,\xi(r))=E^{\mathcal{F}_{s}}[\xi(r)]+1$. It’s easy to check that $g^{0},g^{1}$ and $\overline{g}$ satisfy the assumptions of Theorem \[15\]. If the terminal condition satisfies $\psi^{0}(t)\leq \psi^{1}(t), \ a.s., \ t\in[0,T+K],$ we derive $$Y^{0}(t)\leq Y^{1}(t), \ \ a.s. \ t\in[0,T+K].$$
Acknowledgements {#acknowledgements .unnumbered}
================
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| {
"pile_set_name": "ArXiv"
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---
abstract: 'We investigate the phase structure of pure $SU(2)$ LGT at finite temperature in the mixed fundamental and adjoint representation modified with a $\mathbb{Z}_2$ monopole chemical potential. The decoupling of the finite temperature phase transition from unphysical zero temperature bulk phase transitions is analyzed with special emphasis on the continuum limit. The possible relation of the adjoint Polyakov loop to an order parameter for the finite temperature phase transition and to the topological structure of the theory is discussed.'
author:
- |
Andrea Barresi, Giuseppe Burgio[^1], Michael Müller-Preussker\
$\;$\
Humboldt-Universität zu Berlin, Institut für Physik, 10115, Germany
title: 'Finite temperature phase transition, adjoint Polyakov loop and topology in $SU(2)$ LGT[^2]'
---
INTRODUCTION
============
Pure $SU(N)$ lattice gauge theories within the fundamental representation of the gauge group show a finite temperature deconfinement phase transition together with the breaking of a global $\mathbb{Z}_N$ center symmetry. But if confinement is a feature of the Yang-Mills continuum degrees of freedom it should be independent of the group representation for the lattice action. As Polyakov’s center symmetry breaking mechanism is available only to half-integer representations of the group, a finite temperature investigation of Wilson’s action for $SU(2)$ in the adjoint representation, i.e. $SO(3)$, might offer interesting insight to the present understanding of confinement.
The $SU(2)$ mixed fundamental-adjoint action was originally studied by Bhanot and Creutz [@1BC81]: $$\label{eq1}
S\!=\!\sum_{P}\Bigg[\!\beta_{A}\Bigg(1-\frac{\mathrm{Tr}_{A}U_{P}}{3}\Bigg)+\beta_{F}\Bigg(1-\frac{\mathrm{Tr}_{F}U_{P}}{2}\Bigg)\!\Bigg]$$ They found the well known non-trivial phase diagram characterized by first order $T=0$ bulk phase transition lines. A similar phase diagram is shared by $SU(N)$ theories with $N\ge 3$ [@2BC81].
Halliday and Schwimmer [@1HS81] found a similar phase diagram using a Villain discretization for the center blind part of action (\[eq1\]) $$S=\!\!\sum_{P}\!\!\Bigg[\!\beta_{V}\Bigg(\!1-\frac{\sigma_{P}\mathrm{Tr}_{F}U_{P}}{2}\!\Bigg)\!+\beta_{F}\Bigg(\!1-\frac{\mathrm{Tr}_{F}U_{P}}{2}\!\Bigg)\!\Bigg]$$ $\sigma_{P}$ being an auxiliary $\mathbb{Z}_2$ plaquette variable. By defining $\mathbb{Z}_2$ magnetic monopole and electric vortex densities $M=1-\langle\frac{1}{N_{c}}\sum_{c}\sigma_{c}\rangle$, $E=1-\langle\frac{1}{N_{l}}\sum_{l}\sigma_{l}\rangle$ with $\sigma_{c}=\prod_{P\epsilon\partial c}\sigma_{P}$ and $\sigma_{l}=\prod_{P\epsilon\hat{\partial} l}\sigma_{P}$ they argued that the bulk phase transitions were caused by condensation of these lattice artifacts. They also suggested [@2HS81] a possible suppression mechanism via the introduction of chemical potentials of the form $\lambda\sum_{c}(1-\sigma_{c})$ and $\gamma\sum_{l}(1-\sigma_{l})$.
Recently Gavai and Datta [@1G99] explicitely realized this suggestion, studying the $\beta_{V}-\beta_{F}$ phase diagram as a function of $\lambda$ and $\gamma$. They found lines of second order finite temperature phase transitions crossing the $\beta_V$ and $\beta_F$ axes for $\lambda\ge 1$ and $\gamma\ge 5 $. In the limiting case $\beta_{F}=0$ and $\gamma=0$, i.e. $SO(3)$ theory with a $\mathbb{Z}_2$ monopole chemical potential, a quantitative study is difficult because of the lack of an order parameter. The $\mathbb{Z}_2$ global symmetry remains trivially unbroken. A thermodynamical approach [@2G99] shows a steep rise in the energy density for asymmetric lattices with $N_{\tau}=2,4$ and a peak in the specific heat at least for $N_{\tau}=2$, supporting the idea of a second order deconfinement phase transition. The authors have seen the adjoint Polyakov loop to fluctuate around zero below the phase transition and to take the values $1$ and $-\frac{1}{3}$ above the phase transition as $\beta_V\to \infty$.
ADJOINT ACTION WITH CHEMICAL POTENTIAL
======================================
We study an adjoint representation Wilson action modified by a chemical potential suppressing the $\mathbb{Z}_2$ magnetic monopoles $$S=\frac{4}{3}\beta_{A}\sum_{P}\Bigg(1-\frac{\mathrm{Tr}_{F}^{2}U_{P}}{4}\Bigg)+\lambda\sum_{c}(1-\sigma_{c})$$
The link variables are taken in the fundamental representation only to improve the speed of our simulations, after checking that with links represented by $SO(3)$ matrices nothing changes. A standard Metropolis algorithm is used to update the links. The term $\sigma_{c}=\prod_{P\epsilon\partial c}\mathrm{sign}(\mathrm{Tr}_{F}U_{P})$ is completely center blind, i.e. $U_{\mu}(x)\rightarrow -U_{\mu}(x)\Rightarrow\sigma_{c}\rightarrow\sigma_{c}\;\;\;\forall \mu,x,c$.
\[fig:btlam\] {width="45.00000%"}
Fig. 1 shows the phase diagram in the $\beta_A-\lambda$ plane at finite temperature. The two phases (I-II) are separated by a bulk first order line at which $\mathbb{Z}_2$ monopoles condense, phase I being continously connected with the physical $SU(2)$ phase as $\beta_F$ is turned on. Finite temperature lines, at which $\langle L_A \rangle$ shows a jump, cross the plane more or less horizontally. Putting aside the order parameter problem, the scaling behaviour at the critical temperature $T_c\equiv\frac{1}{aN_\tau}$ as a function of $\beta_A$ and $\lambda$ turns out difficult in phase I, whereas in phase II it shows a nice scaling behaviour in $\beta_A$ at fixed $\lambda \gtrsim 1$.
SYMMETRY AND ORDER PARAMETER
============================
A quantitative study of the observed finite temperature transition is viable either relying on pure thermodynamical quantities [@2G99] or defining a reasonable order parameter, i.e. by understanding the underlying symmetry breaking mechanism, if any. The only hints we have are the change in the distribution of the adjoint Polyakov line operator $\frac{1}{3}\mathrm{Tr} L_A(\vec{x})$ and the values it takes in the continuum limit. After maximal abelian gauge (MAG) [@MAG] and abelian projection it is indeed possible to establish an exact global symmetry which can be broken at the phase transition and a related order parameter. Taking $$O_{\mu}(x)=I+ \sin 2 \theta_{\mu}(x) T_3 + (1-\cos 2 \theta_{\mu}(x))T_3^2$$ as the projected link in the adjoint theory, with $\vec{T}$ the adjoint representation generators of the Lie algebra, it is easy to see that the “parity” operator $P=I+2 T_3^2$ acting on all links living at fixed time as $$P O_{\mu}(x)=I- \sin 2 \theta_{\mu}(x)T_3 + (1+\cos 2 \theta_{\mu}(x))T_3^2$$ leaves all the plaquettes (and thus the action) invariant, while changing the Polyakov line. If $\Theta_L (\vec{x}) =\sum_{n=0}^{N_{\tau}-1} \theta_4(\vec{x}+\,n\,a\,\hat{4})$ is the Polyakov line global abelian phase, then for the spatial average $\langle \mathrm{Tr}L_A\rangle=1+2\langle \mathrm{cos} 2 \Theta_L (\vec{x})\rangle$ and $\langle \mathrm{Tr}PL_A\rangle=1-2\langle \mathrm{cos} 2 \Theta_L (\vec{x})\rangle$. If this symmetry is broken at the phase transition, then $\langle \mathrm{Tr}L_A\rangle=1$ below and $\langle \mathrm{Tr}L_A\rangle=1\pm2\Delta$ above, with $\Delta=\langle \mathrm{cos} 2 \Theta_L\rangle$. Thus, a reasonable order parameter should be $|\Delta|=\frac{1}{2}|\langle\mathrm{Tr}L_A\rangle-1|$.
\[fig:dist\]
 
Fig. 2 shows the volume distribution of the Polyakov line angle at the phase transition for some typical configurations. Although such a sharp change can be observed also for the full ${\rm Tr} \mathrm{L}_A(\vec{x})$ distribution, in the latter case a quantitative analysis is made difficult by the asymmetry of the values at which it peaks. In the abelian projected case, after MAG, $\Theta_L(\vec{x})$ is clearly flat below the phase transition, peaking around $0 (\pi)$ and $\frac{\pi}{2}$ above. In Fig. 3 the proposed order parameter is plotted as a function of $\beta_A$ for $\lambda=1$ and $N_\tau=4$. A singular behaviour around $\beta_A\simeq 1$ is starting to show at $V=16^3$. At $N_\tau=6$ the critical $\beta_A$ increases by roughly $35\%$.
\[fig:magn\] 
The results show that the proposed symmetry breaking mechanism is plausible and that the order parameter behaves as one expects for a $2^{\rm nd}$ order transition, although more data at higher volumes and a study of the susceptibility would be necessary to asses such statements. The analysis of Binder cumulants is also feasible with our definitions. A study of the critical exponents and of the cluster properties of $\langle \mathrm{Tr}L_A \rangle$ would be as well interesting in order to establish whether the features of such a system are similar to those of the usual $SU(2)$ phase transition. All these questions will be addressed in a forthcoming paper.
CONCLUSIONS
===========
We have studied the phase diagram of the adjoint Wilson action with a chemical potential $\lambda$ for the $\mathbb{Z}_2$ magnetic monopoles. The finite temperature phase transition can be decoupled from the bulk phase transition both for positive and negative $\lambda$. The scaling behaviour of $\beta_A$ with $N_\tau$ is established in both cases, although the type I phase presents some difficulties in taking the continuum limit. In the context of abelian dominance we propose a symmetry breaking mechanism and an order parameter for the phase transition, giving promising results for numerical simulations. A deeper numerical analysis and possible extensions of the definitions will be the subject of a forthcoming paper. This work was funded by a EU-TMR network under the contract FMRX-CT97-0122 and by the DFG-GK 271.
[9]{} G. Bhanot and M. Creutz, Phys. Rev. D24 (1981) 3212. M. Creutz and K.J.M. Moriarty, Nucl. Phys. B210 (1982) 59. I.G. Halliday and A. Schwimmer, Phys. Lett. B101 (1981) 327. I.G. Halliday and A. Schwimmer, Phys. Lett. B102 (1981) 337. S. Datta and R.V. Gavai, Nucl. Phys. Proc. Suppl. 83 (2000) 366-368. S. Datta and R.V. Gavai, Phys. Rev. D60 (1999) 34505. A.S. Kronfeld, M.L. Laursen, G. Schierholz and U.J. Wiese, Phys. Lett. B198 (1987) 516.
[^1]: Address from Nov. $1^{\rm st}$ 2001: School of Mathematics, Trinity College, Dublin 2, Ireland.
[^2]: Talk given by A. Barresi at Lattice2001, Berlin. HU-EP-01/41
| {
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bibliography:
- 'microstates.bib'
---
=10000
IPHT-T16/004\
[<span style="font-variant:small-caps;">Momentum Fractionation on Superstrata</span>]{}
[<span style="font-variant:small-caps;">Iosif Bena$^1$, Emil Martinec$^{2}$, David Turton$^{1}$, Nicholas P. Warner$^{3}$</span>]{}
$^1$Institut de Physique Théorique,\
Université Paris Saclay,\
CEA, CNRS, F-91191 Gif sur Yvette, France\
$^2$Enrico Fermi Inst. and Dept. of Physics,
University of Chicago, 5640 S. Ellis Ave.,
Chicago, IL 60637-1433, USA
$^3$Department of Physics and Astronomy
and Department of Mathematics,
University of Southern California,
Los Angeles, CA 90089, USA
[iosif.bena @ cea.fr, ejmartin @ uchicago.edu,]{}\
[david.turton @ cea.fr, warner @ usc.edu]{}\
<span style="font-variant:small-caps;">Abstract</span>
[13.5mm]{}[13.5mm]{} Superstrata are bound states in string theory that carry D1, D5, and momentum charges, and whose supergravity descriptions are parameterized by arbitrary functions of (at least) two variables. In the D1-D5 CFT, typical three-charge states reside in high-degree twisted sectors, and their momentum charge is carried by modes that individually have fractional momentum. Understanding this momentum fractionation holographically is crucial for understanding typical black-hole microstates in this system. We use solution-generating techniques to add momentum to a multi-wound supertube and thereby construct the first examples of asymptotically-flat superstrata. The resulting supergravity solutions are horizonless and smooth up to well-understood orbifold singularities. Upon taking the AdS$_3$ decoupling limit, our solutions are dual to CFT states with momentum fractionation. We give a precise proposal for these dual CFT states. Our construction establishes the very nontrivial fact that large classes of CFT states with momentum fractionation can be realized in the bulk as smooth horizonless supergravity solutions.
=14.5pt
Introduction {#Sect:introduction}
============
String theory has been successful in counting the microstates of black holes in the regime of parameters where stringy effects overwhelm gravitational effects at the horizon scale. When supersymmetry is present, this counting carries over to the regime of parameters where gravitational effects are dominant at the horizon scale, and the entropy of these microstates reproduces the Bekenstein-Hawking entropy of the black hole [@Strominger:1996sh; @Sen:1995in]. However, the exploration of the implications of this achievement for resolving the information paradox [@Hawking:1976ra] and for understanding the physics of an infalling observer [@Almheiri:2012rt; @*Almheiri:2013hfa; @Mathur:2012jk; @*Mathur:2013gua] is still in its infancy. Indeed, very little is known about the fate of the individual stringy microstates, counted in the zero-gravity regime, as one increases the gravitational coupling and goes to the regime in which the configuration corresponds to a classical black hole with a large event horizon.
There are several possibilities as to what this fate might be. One is that, as gravity becomes stronger, all these microstates develop a horizon and end up looking identical to the black hole [@Horowitz:1996nw; @Damour:1999aw; @Papadodimas:2012aq]. Another is that some of the microstates that one constructs at zero gravitational coupling will develop a horizon, and others will remain horizonless. A third possibility is that none of these microstates develop a horizon, and they all grow into horizon-sized bound states that have the same mass, charges and angular momentum as the black hole, but have no horizon [@Lunin:2001jy; @Mathur:2005zp; @Bena:2007kg; @Skenderis:2008qn; @Balasubramanian:2008da; @Chowdhury:2010ct; @Mathur:2012zp; @Bena:2013dka]. There are then a range of “sub-possibilities”: At one extreme, typical black-hole microstates would not be describable in supergravity, but will be intrinsically quantum or non-geometrical; at the other extreme, in the sector dual to the typical microstates, one could find a [*basis*]{} of Hilbert space vectors that correspond to coherent states that have a supergravity description, or at least a stringy limit thereof.
In the context of the AdS-CFT correspondence [@Maldacena:1997re], one can similarly ask whether a typical CFT microstate corresponds to a classical black hole with an event horizon, or to some horizonless configuration. The latter might either be impossible to describe in supergravity because of large quantum fluctuations or stringy corrections, or might be described using a Hilbert state basis given by smooth low-curvature solutions, or might correspond to some hybrid configuration (such as an intrinsically quantum configuration lying in a smooth, horizonless supergravity solution).
There exist pieces of evidence that can be taken as bringing support to any of these possible outcomes, some founded on calculations, and some based more on intuition and conjecture. Perhaps the strongest evidence that at least some microstates become smooth horizonless supergravity solutions at strong gravitational coupling comes from the explicit construction of numerous families of smooth horizonless solutions that have the same charges as black holes [@Bena:2006kb]. The largest family of solutions are parametrized by arbitrary continuous functions of two variables [@Bena:2015bea], and come from the back-reaction of certain families of [*superstrata*]{} [@Bena:2011uw]. Superstrata are string theory bound states whose counting has been argued to reproduce a finite fraction of the entropy of three-charge supersymmetric black holes [@Bena:2014qxa].
However, even if the existence of these large families of solutions rules out the possibility that all the microstates one counts at zero gravity develop a horizon, it does not prove that all microstates remain horizonless, nor does it establish whether typical horizonless configurations are smooth and describable in supergravity, or are instead non-geometric or strongly-curved.
For example, it has been argued [@Chen:2014loa] that for the two-charge D1-D5 black hole, the typical states of the dual symmetric product orbifold CFT [@Vafa:1995bm; @Douglas:1995bn; @Vafa:1995zh; @Bershadsky:1995qy; @Maldacena:1997re] are not well-described by the microstate geometries of [@Lunin:2001jy; @Lunin:2002iz; @Kanitscheider:2007wq] when the average harmonic of the two-charge profile function becomes larger than $\sqrt{N_1 N_5}$. The harmonics of the profile function correspond to the winding of strands in the D1-D5 orbifold CFT; since typical three-charge microstates come from adding momentum to CFT strands whose length is of order $N_1 N_5$, one might naively conclude that all typical three-charge microstate geometries would be strongly-curved and hence not describable by supergravity.
There are also arguments that the bulk configurations dual to typical CFT states will be non-geometrical. One such argument comes from an analysis of the possible supertube transitions that can occur in three-charge systems, which indicate that the configurations resulting from these transitions will be generically non-geometric [@deBoer:2010ud; @*deBoer:2012ma]. It has also been suggested that the states that carry fractionated momentum modes, which are the typical states that contribute in the entropy counting, will involve multi-valued wavefunctions [@Bena:2015bea].
Furthermore, there are also conjectures that when tracking microstates of the D1-D5-P system to the regime of parameters where gravity becomes important, only very few states will give rise to horizonless geometries, while most states will correspond to a black hole with a horizon [@Sen:2009bm]. According to this perspective, the more typical the state, the larger the likelihood that its bulk dual will not be a horizonless solution, but will be a solution with a horizon.
The purpose of this paper is to provide evidence that these alternative scenarios are not realized, by showing that highly-nontrivial CFT states whose momentum is carried by fractionated carriers are dual to smooth horizonless supergravity solutions (with localized orbifold singularities). We construct these solutions using a combination of two solution-generating techniques: [*Spectral interchange*]{} (also known as [*spectral inversion*]{}) and adding charge density oscillations to a supertube. Spectral interchange is a transformation of the D1-D5(-P) BPS solutions that interchanges the null coordinate along the D1 and D5 branes, $v=t+y$, with the Gibbons-Hawking fiber of the transverse space [@Bena:2008wt; @Niehoff:2013kia]. Modifying the charge density distribution along the supertube source profile has been studied, for example, in [@Lunin:2002iz; @Bena:2008dw; @Bena:2010gg].
In this paper we show that by combining these techniques one can add $y$-momentum to a seed solution with D1 and D5 charges, as follows: First perform a spectral inversion, then use a charge density oscillation to introduce $\psi$-dependence and associated angular momentum, then spectrally invert back to the original frame to obtain a new solution carrying $v$-dependence and momentum. The $\psi$-dependent solutions in the spectrally inverted frame can be generated by integrating the Green functions against the modified charge and angular momentum densities along the supertube.
For our explicit construction, we apply this combination of techniques to a simple seed solution – a multiwound circular D1-D5 supertube. The multiwinding of the seed solution is what will allow us to study the physics of momentum fractionation. While in principle the Green function/spectral interchange method can be used to construct new general classes of superstrata, a particular class of examples is amenable to a direct analysis of the equations governing all supersymmetric solutions of six-dimensional supergravity [@Gutowski:2003rg; @Cariglia:2004kk; @Bena:2011dd; @Giusto:2013rxa]. These equations determine the various potentials that enter in the supergravity solution, and are arranged in stages or layers, where the potentials to be solved for one layer satisfy linear equations sourced by the potentials determined in previous layers [@Bena:2011dd].
Our solutions are regular up to the usual orbifold singularity at the location of the multiwound supertube. We arrange the regularity of our supergravity solutions by imposing constraints on Fourier modes and coefficients; this procedure is known as [*coiffuring*]{} [@Mathur:2013nja; @Bena:2013ora; @Bena:2014rea]. We find two classes of regular solutions, corresponding to two “Styles” of coiffuring. We analyze the conserved charges and other properties of the solutions.
Our construction also yields the first examples of asymptotically-flat superstrata; in a particular limit our solutions contain the generalization to asymptotically-flat space of one class of the asymptotically-AdS superstrata constructed in [@Bena:2015bea].
Upon taking the decoupling limit, we obtain solutions that are asymptotically AdS$_3\times$S$^3\times \cM$ (where $\cM$ is either $\bbT^4$ or $K3$) and we investigate the corresponding dual CFT description. We do this by assembling a variety of clues. We observe that the relation between the $y$-momentum and the angular momenta of the solutions suggest that the dual CFT states involve repeated applications of fractionally moded $SU(2)$ $\cR$-symmetry generators, and also that they can be generated by [*fractional spectral flow*]{} [@Martinec:2001cf; @*Martinec:2002xq] applied to a [*subset*]{} of strands of certain two-charge seed states.
We then study the vevs of operators and find that they have the right properties to reproduce the vevs of the supergravity fields at the linearized level, using the technology of [@Skenderis:2006ah; @*Kanitscheider:2006zf; @Kanitscheider:2007wq; @Taylor:2007hs]. We find however that the supergravity regularity constraints are not visible at this order. Finally, by analyzing the possible two-charge seed solutions, we determine the precise proposal for the CFT states dual to both styles of coiffuring in supergravity.
Prior to the present work, there were only two classes of supergravity solutions, one BPS and one non-BPS [@Jejjala:2005yu; @Bena:2005va; @*Berglund:2005vb], which had been shown to be dual to CFT states involving momentum fractionation [@Giusto:2012yz; @Chakrabarty:2015foa].[^1] These states came from fractional spectral flow applied to [*all*]{} strands of certain two-charge states, and hence are very special. One way to see this is that the $AdS$ region of their dual bulk solutions can be obtained from global $AdS_3 \times S^3$ by a coordinate transformation.[^2] In contrast, our technology produces supergravity solutions that are much more general, and cannot be written in this way.
The remainder of this paper is structured as follows. In Section \[Sect:sixD\], we review the class of five- and six-dimensional supergravity solutions of interest, the BPS equations they satisfy, and the multiwound circular D1-D5 supertube. In Section \[Sect:SpecInter\], we apply the sequence of solution-generating techniques to add momentum to the seed solution. We perform a direct analysis of the BPS equations in Section \[Sect:MomST\], and find two classes of regular solutions via coiffuring. In Section \[Sect:CFT\], we first review the states in the CFT, the operators that are dual to linearized supergravity field modes, and spectral flow. We then develop the precise proposal for the CFT states dual to our supergravity solutions. Section \[Sect:Discussion\] summarizes our results and discusses open questions.
BPS solutions in supergravity {#Sect:sixD}
=============================
We work in type IIB string theory on $\mathbb{R}^{4,1}\times \bbS^1\times \cM$ where $\cM$ is $\bbT^4$ or $K3$. We take the size of $\cM$ to be microscopic and the $\bbS^1$ to be macroscopic. The $\bbS^1$ is parameterized by the coordinate $y$ which we take to have radius $R_y$, $$y ~\sim~ y \,+\, 2 \pi R_y \,.
\label{yperiod}$$ We reduce on $\cM$ and work in the supergravity limit. The six-dimensional truncation of interest is an $\Neql1$ supergravity coupled to two (anti-self-dual) tensor multiplets. This is the theory in which the first superstrata were constructed [@Bena:2015bea]; the theory contains all the fields expected from D1-D5-P string emission calculations [@Giusto:2011fy]. The BPS system of equations describing all 1/8-BPS D1-D5-P solutions of this theory has been found in [@Giusto:2013rxa], and is a generalization of the system discussed in [@Gutowski:2003rg; @Cariglia:2004kk] and greatly simplified in [@Bena:2011dd].
The BPS equations in six dimensions {#ss:BPSeqns}
-----------------------------------
To exploit the structure of the six-dimensional BPS equations, we work with null coordinates $u$ and $v$, defined by: $$u ~\equiv~ \frac{1}{\sqrt{2}}\, (t-y) \,, \qquad v ~\equiv~ \frac{1}{\sqrt{2}}\, (t+y) \,. \label{uvdefn}$$ The periodicity of the $y$ circle induces an identification on $u$ and $v$. It will be convenient to parameterize this as follows: $$(u,v) ~\sim~ (u,v) + (-4\pi R, 4\pi R)\,, \qquad R ~\equiv~ \frac{R_y}{2 \sqrt{2}} \,.
\label{uviden}$$
For supersymmetric solutions, the metric is required to have the local form: $$ds_6^2 ~=~ -\frac{2}{\sqrt{\cP}} \, (dv+\beta) \big(du + \omega + \tfrac{1}{2}\, \mathcal{F} \, (dv+\beta)\big)
~+~ \sqrt{\cP} \, ds_4^2(\cB)\,, \label{sixmet}$$ Note that we can always shift $\mathcal{F}$ by a constant, $c$, by sending $u \to u- \frac{1}{2} c v$ and $\omega \to \omega - \frac{1}{2} c \beta$. Given our choice of $t$ and $y$ coordinates in , to obtain our desired asymptotics we require that $\mathcal{F}$ vanishes at infinity throughout this paper.
Introducing the quantities $Z_3$ and $\mathbf{k}$ via[^3] \[eq:Z3k\] Z\_3 = 1- , = , one can write the metric in the form $$ds_6^2 ~= -\frac{1}{Z_3\sqrt{\cP} } \, (dt + \mathbf{k})^2 \,+\,
\frac{Z_3}{\sqrt{\cP}}\, \left[dy +\left(1- Z_3^{-1}\right) (dt + \mathbf{k}) +\frac{\beta-\omega}{\sqrt{2}} \right]^2 + \sqrt{\cP} \, ds_4^2(\cB)\,.
\label{sixmet-sqty}$$ This form of the metric is useful in the analysis of closed time-like curves (CTC’s). In particular, if there are closed curves whose length in the metric $ds_4^2(\cB)$ vanishes, then it is essential that the remaining part of the metric does not make these curves time-like. The relevant condition is manifest from (\[sixmet-sqty\]): The danger arises if one chooses a curve along which $dy$ is related to the other angles such that the second square vanishes.[^4] We thus require that for any such curve, in the limit where the length of the curve in $ds_4^2(\cB)$ tends to zero, the one-form $\mathbf{k}$ acting on the tangent vector to the curve must also tend to zero (appropriately quickly).
The four-dimensional base, $\cB$, has a metric, $ds_4^2$, and is required to be an “almost hyper-Kähler” manifold [@Gutowski:2003rg]. However we are going to simplify things by assuming that the base has a Gibbons-Hawking metric: $$ds_4^2 ~=~ V^{-1} \, \big( d\psi + A)^2 ~+~ V\, d \vec y \cdot d
\vec y \,, \label{GHmetric}$$ where the periodicity of $\psi$ will be given below in and where, on the flat $\IR^3$ defined by the coordinates $\vec y$, one has: $$\nabla^2 V ~=~ 0\,, \qquad \vec \nabla \times \vec A ~=~ \vec \nabla V\,.
\label{AVreln}$$ We take $V$ to have the form $$V ~=~ h ~+~ \sum_{j=1}^N \, \frac{q_j}{|\vec y - \vec y^{(j)}| } \,,
\label{Vform}$$ for some fixed points, $\vec y^{(j)} \in \IR^3$, some charges, $q_j \in \ZZ$, and some constant $h$.
We will also require that the one-form, $\beta$, is $v$-independent and then the BPS equations require that $\beta$ has self-dual field strength: $$\label{eqbeta}
\Theta_3 ~\equiv~ d \beta = *_4 d\beta\,,$$ where $*_4$ denotes the four-dimensional Hodge dual in the Gibbons-Hawking metric. We will also assume that $\beta$ is $\psi$-independent and solve the self-duality by taking $$\beta ~=~ \frac{K^3}{V} \, (d\psi +A) ~+~ \vec \sigma^{(3)} \cdot \vec{dy} \,,
\label{betaform1}$$ where $K^3$ is harmonic on $\IR^3$ and $$\vec \nabla \times \vec \sigma^{(3)}~=~ -\vec \nabla K^3 \,.
\label{Asigform}$$ The supergravity theory has three tensor gauge fields (one is in the graviton multiplet) and two scalars (one in each tensor multiplet). The scalars may be thought of as the dilaton, $\Phi$, and axion, $C_0$, of the IIB theory. The tensor fields of BPS solutions may be described in terms of three potential functions, $Z_1$, $Z_2$, $Z_4$ and three sets of two-forms, $\Theta_1$, $\Theta_2$, $\Theta_4$, on the base $\cB$.
The BPS condition then requires a suitable generalization of the “floating brane Ansatz” [@Bena:2009fi] in which the metric warp factor and scalars are expressed in terms of the potentials: $$\cP ~=~ Z_1 \, Z_2 - Z_4^2 \,, \qquad e^{2\Phi}~=~ \frac{Z_1^2}{\cP}\,, \qquad C_0~=~\frac{Z_4}{Z_1} \,.
\label{Psimp}$$ Since we are allowing the scalars and tensor gauge fields (but not $\beta$ or $ds_4^2$) to depend upon $v$, the BPS equations impose the following linear differential equations on the potentials and the two-forms $(Z_I, \Theta_I)$:[^5] $$\label{BPSlayer1a}
*_4 \mathcal{D} \dot{Z}_1 = \mathcal{D} \Theta_2\,,\quad \mathcal{D}*_4\mathcal{D}Z_1 = -\Theta_2\wedge d\beta\,,\quad \Theta_2=*_4 \Theta_2\,,$$ $$\label{BPSlayer1b}
*_4 \mathcal{D} \dot{Z}_2 = \mathcal{D} \Theta_1\,,\quad \mathcal{D}*_4\mathcal{D}Z_2 = -\Theta_1\wedge d\beta\,,\quad \Theta_1=*_4 \Theta_1\,,$$ $$\label{BPSlayer1c}
*_4 \mathcal{D} \dot{Z}_4 = \mathcal{D} \Theta_4\,,\quad \mathcal{D}*_4\mathcal{D}Z_4 = -\Theta_4\wedge d\beta\,,\quad \Theta_4=*_4 \Theta_4\,.$$ where the dot denotes $ \frac{\partial}{\partial v}$, $\mathcal{D}$ is defined by $$\mathcal{D} \equiv \tilde d - \beta\wedge \frac{\partial}{\partial v}\,,$$ and $\tilde d$ denotes the exterior differential on the spatial base $\cB$.
In (\[BPSlayer1a\])–(\[BPSlayer1c\]), the first equation in each set involves four component equations, while the second equation in each set is essentially an integrability condition for the first equation. The self-duality condition reduces each $ \Theta_I$ to three independent components and adding in the corresponding $Z_J$ yields four independent functional components upon which there are four constraints.
If we separate the $Z_I$ into their $v$-independent (zero-mode) and $v$-dependent parts, $Z_I = Z_I^{(0)}+ Z_I^{(v)}$, then the $v$-dependent parts $Z_I^{(v)}$ satisfy simpler equations, as follows. It is convenient to define two-forms $\xi_I$ via: $$\Theta_I ~\equiv~ \partial_v \xi_I \,, \qquad I =1,2,4\,.
\label{ThetaIdefn}$$ Then for the $v$-dependent parts, one can simplify the BPS equations (\[BPSlayer1a\])–(\[BPSlayer1c\]) by integrating, as follows: $$*_4 \, \mathcal{D} Z_1^{(v)} ~=~ \mathcal{D} \xi_2 \,, \qquad *_4 \, \mathcal{D} Z_2^{(v)} ~=~ \mathcal{D} \xi_1 \,, \qquad*_4 \, \mathcal{D} Z_4^{(v)} ~=~ \mathcal{D} \xi_4 \,. \label{xieqn1}$$ The final set of BPS equations are linear differential equations for $\omega$ and $\mathcal{F}$: $$\label{BPSlayer2a}
\mathcal{D} \omega + *_4 \mathcal{D}\omega = Z_1 \Theta_1+ Z_2 \Theta_2 - \mathcal{F}\Theta_3 -2\,Z_4 \Theta_4 \,,$$ and a second-order constraint that follows from the $vv$ component of Einstein’s equations[^6], $$\label{BPSlayer2b}
\begin{aligned}
*_4\mathcal{D} *_4\!\big(\dot{\omega} + \coeff{1}{2}\,\mathcal{D} \mathcal{F} \big)&~=~\dot{Z}_1\dot{Z}_2+Z_1 \ddot{Z}_2 + Z_2 \ddot{Z}_1 -(\dot{Z}_4)^2 -2 Z_4 \ddot{Z}_4-\coeff{1}{2} *_4\!\big(\Theta_1\wedge \Theta_2 - \Theta_4 \wedge \Theta_4\big) \\
&~=~\partial_v^2 (Z_1 Z_2 - {Z}_4^2) -(\dot{Z}_1\dot{Z}_2 -(\dot{Z}_4)^2 )-\coeff{1}{2} *_4\!\big(\Theta_1\wedge \Theta_2 - \Theta_4 \wedge \Theta_4\big)\,.
\end{aligned}$$
BPS solutions in five dimensions {#ss:5dim}
--------------------------------
We now recall how $v$-independent solutions reduce to five-dimensions and our discussion will closely follow that of [@Bena:2010gg]. We will assume that the magnetic $2$-forms, $\Theta^{(I)}$, are independent of the GH fiber coordinate, $\psi$. This means that one may use the same class of solutions as in (\[betaform1\]) by introducing more harmonic functions, $K^{I}$, on $\IR^3$ and taking $$\Theta^{(I)} ~=~ d B^{(I)} \,, \label{GHdipoleforms}$$ with $$B^{(I)}=V^{-1} K^{I} \, (d \psi + A) ~+~ \vec{\sigma}^{(I)}\cdot d\vec{y} \,,
\qquad \vec \nabla \times \vec \sigma^{(I)} ~\equiv~ - \vec \nabla K^I \,.
\label{vecpotdefns}$$ The sources in BPS equations for $Z_I$ ($I=1,2,3,4$) are independent of $v$ and $\psi$ and so the inhomogeneous solutions for the functions $Z_I$ follow the standard form: $$Z_I ~=~ \coeff{1}{2}\, C_{IJK} V^{-1} K^{J}K^{K} ~+~ L_I \,,
\label{ZIform}$$ where $C_{IJK}$ are the usual (completely symmetric) structure constants for supergravity coupled to vector multiplets. The particular theory that we use can be written in this form if one sends $Z_4 \to -Z_4$ and takes $$C_{123} ~=~ 1 \,, \qquad C_{344} ~=~ -2 \,,
\label{RecC}$$ with other (non-cyclically related) components equal to zero.
The functions $L_I$ in (\[ZIform\]) are required to be harmonic on the GH base, $\cB$, and can be allowed to depend upon all the coordinates, including $\psi$. Thus we have $$\nabla^2_{(4)} L_I ~=~0 \,.
\label{Lharmcond}$$ One can then make a simple Ansatz for the angular momentum, one-form $\omega$: $$\omega ~=~ \mu (d\psi+ A) + \vec{\varpi}\cdot d\vec{y} \,.
\label{omansatz}$$ If one introduces the covariant derivative $$\vec { D} ~\equiv~ \vec \nabla ~-~ \vec A \,\partial_\psi \,,
\label{covD}$$ then the last BPS equation can be written as: $$( \mu \vec {D} V - V\vec { D} \mu ) ~+~ \vec {D} \times \vec \varpi ~+~ V \partial_\psi \vec \varpi ~=~ - V\, \sum_{I=1}^3 \, Z_I \, \vec \nabla \big(V^{-1} K^I \big) \,.
\label{covomeqn}$$
The BPS equations have a gauge invariance: $\omega \to \omega + df$ and this reduces to: $$\mu \to \mu ~+~ \partial_\psi f \,, \qquad \vec \varpi \to \vec \varpi ~+~ \vec {D} f \,,
\label{fgaugetrf}$$ The Lorentz gauge-fixing condition, $d\star_4\omega =0$, reduces to $$V^2 \, \partial_\psi \mu ~+~\vec {D} \cdot \vec \varpi ~=~ 0 \,,
\label{Lorgauge}$$ and we will impose this gauge choice.
Taking the covariant divergence, using $\vec { D}$, of (\[covomeqn\]) and using the Lorentz gauge choice, one obtains: $$V^{^2}\, \nabla^2_{(4)} \mu ~=~ \vec { D} \cdot \Big( V\, \sum_{I=1}^3 \, Z_I \, \vec { D} \big(V^{-1} K^I \big) \Big) \,.
\label{mueqn}$$ It is useful to note that the four-dimensional Laplacian may be written as: $$\nabla^2_{(4)} F ~=~ V^{-1} \big[ V^2 \, \partial_\psi^2 F ~+~ \vec { D} \cdot \vec {D} F \big] \,.
\label{Lapl}$$ The equation for $\mu$ is solved by taking: $$\mu ~=~ \coeff{1}{6}\, V^{-2}C_{IJK} K^{I}K^{J}K^{K} ~+~ \coeff{1}{2}\, V^{-1} K^{I}L_{I} ~+~ M\,,
\label{muform}$$ where, once again, $M$ is a harmonic function on $\cB$.
Finally, we can use this solution back in (\[covomeqn\]) to simplify the right-hand side and obtain: $$\vec {D} \times \vec \varpi ~+~ V \partial_\psi \vec \varpi ~=~ V \vec {D} M - M\vec {D} V
+\frac{1}{2} \big( K^{I} \vec {D} L_{I} - L_{I} \vec { D} K^{I} \big).
\label{omegaeqn}$$ Once again one sees the emergence of the familiar symplectic form on the right-hand side of this equation. One can also verify that the covariant divergence (using $\vec { D}$) generates an identity that is trivially satisfied as a consequence of $ \vec\nabla V = \vec \nabla \times \vec A$, (\[Lorgauge\]), (\[muform\]) and $$\nabla^2_{(4)} L_I ~=~\nabla^2_{(4)} M ~=~ 0 \,.
\label{harmonicLM}$$ An explicit, closed form for all the components of $\vec \varpi$ was not given in [@Bena:2010gg], but for our solutions we will be able to construct them.
A round supertube in flat space {#ss:rndstube}
-------------------------------
The simplest supertube Ansatz is to take the base, $\cB$, to be flat $\IR^4$ and set $\Theta_3$ and $\beta$ to be that of a simple magnetic monopole. There are two convenient ways to formulate this. First, one can take $\beta$ given by (\[betaform1\]) and write $\IR^4$ in Gibbons-Hawking form using spherical polar coordinates $(\rho_{-},\vartheta_{-},\phi)$: $$ds_4^2 ~=~ V^{-1} \, (d\psi +A)^2 ~+~ V\,(d\rho_{-}^2 + \rho_{-}^2 \, d\vartheta_{-}^2 + \rho_{-}^2 \, \sin^2 \vartheta_{-}\, d\phi^2 ) \,,
\label{GHmet}$$ where in terms of the three-dimensional Cartesian coordinates $y_1, y_2, y_3$ we have $$V ~=~ \frac{1}{\rminus} \,, \qquad K^3 ~=~ \frac{k R}{\rplus} \,, \qquad \rpm ~\equiv~ \sqrt{y_1^2 + y_2^2 +({y_3 \mp \coeff{1}{2}\ell})^2} \,,
\label{GHVdefn}$$ where the dipole moment $k$ is an integer. One then has: $$A ~=~\frac{(y_3 + \coeff{1}{2}\ell)} { \rminus} \, d\phi \,, \qquad \sigma ~=~- k R\, \frac{(y_3 - \coeff{1}{2}\ell)} {\rplus} \, d\phi \,.
\label{Asigres1}$$ The periodicity identifications on $\psi$ and $\phi$ are as usual $$\psi ~\sim~ \psi + 4\pi \,,
\qquad (\psi,\phi) ~\sim~ (\psi,\phi) + (2\pi,-2\pi) \,.
\label{eq:psi-phi-periods}$$ One can then follow through with the construction outlined in Section \[ss:5dim\]. However, we subsequently want to make heavy use of the results and formalism employed in [@Bena:2015bea] and so we will use this as an opportunity to introduce the geometry and flux components that make up the second convenient description of supertubes.
One starts by describing the base manifold in terms of spherical bipolar coordinates, defined by[^7] $$\begin{aligned}
4\, \rplus &~=~ \Sigma ~\equiv~(r^2 + a^2 \cos^2 \theta) \,, \qquad\quad~ 4\, \rminus =~\Lambda ~\equiv~(r^2 + a^2 \sin^2 \theta) \,, \\
\cos\frac{\vartheta_-}{2} &~=~ \frac{(r^2 + a^2)^{1/2}}{ \Lambda^{1/2}} \sin \theta \,, \qquad\qquad
\sin\frac{\vartheta_-}{2} ~=~ \frac{r \cos \theta}{ \Lambda^{1/2}} \,, \\
\psi &~=~ \varphi_1 + \varphi_2 \,, \qquad~~ \phi ~=~ \varphi_1 - \varphi_2 \,, \qquad~~ \ell ~\equiv~ \coeff{1}{4}\, a^2 \,.
\label{polarchng}\end{aligned}$$ The metric becomes: $$ds_4^2 ~=~ \Sigma\, \bigg( \frac{dr^2}{(r^2 + a^2)} ~+~ d \theta^2 \bigg) ~+~ (r^2 + a^2) \sin^2 \theta \, d\varphi_1^2 ~+~ r^2 \cos^2 \theta \, d\varphi_2^2 \,,
\label{bipolmet}$$ and we choose the natural system of frames $$e_1 ~=~ \frac{\Sigma^{1/2} }{(r^2 + a^2)^{1/2}} \, dr\,, \quad e_2 ~=~\Sigma^{1/2} \, d\theta\,, \quad e_3 ~=~(r^2 + a^2)^{1/2} \sin \theta \, d\varphi_1\,, \quad e_4 ~=~ r \cos \theta \, d\varphi_2 \,.
\label{frames1}$$ Following [@Bena:2015bea], it is convenient to introduce the self-dual two-forms $\Omega^{(1)}$, $\Omega^{(2)}$ and $\Omega^{(3)}$: $$\label{selfdualbasis}
\begin{aligned}
\Omega^{(1)} &\equiv \frac{dr\wedge d\theta}{(r^2+a^2)\cos\theta} + \frac{r\sin\theta}{\Sigma} d\varphi_1\wedge d\varphi_2 ~=~ \frac{1}{\Sigma \, (r^2+a^2)^\frac{1}{2} \cos\theta} \,(e_1 \wedge e_2 + e_3 \wedge e_4)\,,\\
\Omega^{(2)} &\equiv \frac{r}{r^2+a^2} dr\wedge d\varphi_2 + \tan\theta\, d\theta\wedge d\varphi_1 ~=~ \frac{1}{\Sigma^\frac{1}{2}\, (r^2+a^2)^\frac{1}{2} \cos\theta} \,(e_1 \wedge e_4 + e_2 \wedge e_3) \,,\\
\Omega^{(3)} &\equiv \frac{dr\wedge d\varphi_1}{r} - \cot\theta\, d\theta\wedge d\varphi_2~=~ \frac{1}{\Sigma^\frac{1}{2}\, r \sin\theta} \,(e_1 \wedge e_3 - e_2 \wedge e_4) \,,
\end{aligned}$$ and note that $$\begin{aligned}
*_4(\Omega^{(1)}\wedge \Omega^{(1)})
&={2\over (r^2+a^2)\Sigma^2\cos^2\theta},\quad&
*_4(\Omega^{(2)}\wedge \Omega^{(2)})
&={2\over (r^2+a^2)\Sigma\cos^2\theta},\\
*_4(\Omega^{(3)}\wedge \Omega^{(3)})
&={2\over r^2\Sigma\sin^2\theta},&
\Omega^{(i)}\wedge \Omega^{(j)} &=0,\quad i\neq j.
\end{aligned}$$
The vector field $\beta$ corresponding to the harmonic functions in (\[GHVdefn\]) is $$\hat\beta ~=~\frac{2\,k R a^2 }{\Sigma} \, ( \sin^2 \theta \, d\varphi_1- \cos^2 \theta \, d\varphi_2 )+ k R\,( d\varphi_1+ d\varphi_2) \,.
\label{betaform2}$$ To obtain flat asymptotics, we see from that $\beta$ and $\omega$ must vanish at infinity. We thus make a coordinate transformation to gauge away the constant part of $\hat\beta$, obtaining $$\beta ~\equiv~\beta_1 \, d\varphi_1+ \beta_2\, d\varphi_2 ~=~\frac{2\,k R a^2 }{\Sigma} \, ( \sin^2 \theta \, d\varphi_1- \cos^2 \theta \, d\varphi_2 ) \,.
\label{betaform3}$$ The two-form $\Theta_3=d\beta$ is given by $$\Theta_3 ~=~ d\beta ~=~\frac{4\,k R a^2 }{\Sigma^2} \, ( (r^2 + a^2)\cos^2 \theta \, \Omega^{(2)} - r^2 \sin^2 \theta \, \Omega^{(3)} ) \,.
\label{Theta3form1}$$
The basic, round $v$-independent asymptotically-flat supertube solution is then given by: $$\begin{aligned}
Z_1 &~=~ 1 ~+~ \frac{Q_1}{\Sigma} \,, \qquad Z_2 ~=~ 1 ~+~ \frac{Q_2}{\Sigma} \,, \qquad \mathcal{F} ~=~ 0 \,, \qquad Z_4 ~=~ 0 \,,\nonumber \\
\Theta_3 &~=~ d\beta \,, \qquad \Theta_I ~=~ 0\,, \ \ I=1,2,4 \nonumber \\
\omega &~\equiv~\omega_1 \, d\varphi_1+ \omega_2\, d\varphi_2 ~=~ \Big(c_1 + \frac{J\,(r^2 + a^2) }{a^2 \, \Sigma} \Big)\, d \varphi_1 ~+~ \Big(c_2 - \frac{J\,r^2}{a^2 \, \Sigma} \Big)\, d \varphi_2 \,,
\label{STsol1}\end{aligned}$$ where $c_1$ and $c_2$ are constants to be determined via regularity and asymptotics. The constants $Q_1$, $Q_2$ and $J$ are harmonic sources that encode charges and angular momentum.
At the center of space ($r=0, \theta =0$) the size of the $\varphi_1$-circle and of the $\varphi_2$-circle collapse to zero size as measured in the spatial base metric, $ds_4^2$, in (\[sixmet\]). Moreover, ${\cal P}$ goes to a constant at the center of space. It is evident from this and the discussion around (\[sixmet-sqty\]) that to avoid closed time-like curves at the center of space one must have $\omega + \beta =0$ at $r=0, \theta =0$. This implies: $$c_1 ~=~ - \frac{J}{a^2} \,, \qquad c_2 ~=~ 2 k R \,.
\label{cres1}$$ In addition, $\omega$ must also fall off when $r \to \infty$ and hence we require $$J ~=~2 \, k R \, a^2 \,.
\label{STreg1}$$ Thus $\omega$ is given by $$\omega ~=~\frac{2\,k R a^2 }{\Sigma} \, ( \sin^2 \theta \, d\varphi_1 + \cos^2 \theta \, d\varphi_2 ) \,.
\label{omegaring}$$
Finally there is the regularity of the metric near the supertube, which means that as one approaches $\Sigma =0$, or $r=0, \theta =\frac{\pi}{2}$, the metric must remain smooth. One can easily check that the only potentially singular parts of the metric are the $d\varphi_1^2$ terms and these are proportional to: $$- \frac{2}{\sqrt{\cP}}\,\beta_1 \, \omega_1 ~+~ \sqrt{\cP} \,a^2 \, d\varphi_1^2 \,
\label{JSTdiv1}$$ The vanishing of the singularity at $\Sigma=0$ requires $$J ~=~ \frac{Q_1Q_2}{4kR} \, \qquad \Rightarrow \qquad a^2 ~=~ \frac{Q_1Q_2}{k^2R_y^2} \,.
\label{STreg0}$$ Thus supertube regularity determines the radius, $a$, and the angular momentum, $J$, in terms of the charges $Q_1$, $Q_2$ and the dipole charge $k$. We thus recover the supertube metric of[@Balasubramanian:2000rt; @*Maldacena:2000dr] and its Gibbons-Hawking description [@Giusto:2004kj].
Having made these choices, the $\psi$-fiber limits to a fixed size as one approaches the supertube while the remaining part of the spatial metric limits to (in spherical polar coordinates ($\rplus,\vartheta_+,\phi$) centered around the supertube): $$\widetilde{ds}_4^2 ~=~ \frac{ \sqrt{Q_1Q_2}}{4\,\ell} \, \bigg[ \frac{16 \, \ell}{Q_1Q_2} \, \rplus \big(dy+\coeff{1}{\sqrt{2}}\, ( \sigma-\varpi)\big)^2~+~ \frac{1}{ \rplus} \,(d\rplus^2 + \rplus^2 \, d\vartheta_+^2 + \rplus^2 \, \sin^2 \vartheta_+\, d\phi^2 ) \bigg]\,.
\label{nearST1}$$ Setting $\rplus = \coeff{1}{4} r_+^2$ and using (\[polarchng\]) and (\[STreg0\]) one obtains: $$\widetilde{ds}_4^2 ~=~ \frac{ \sqrt{Q_1Q_2}}{4\,\ell} \, \Big[ dr_+^2 +\coeff{1}{4}\, r_+^2 \, \big(d\vartheta_+^2 + \sin^2 \vartheta_+\, d\phi^2 + \coeff{1}{k^2}\, \big[ \coeff{1}{\sqrt{2} \,R} \, \big(dy+\coeff{1}{\sqrt{2}}\, ( \sigma-\varpi)\big)\big]^2 \big)\Big]\,.
\label{nearST2}$$ Since $R_y = 2 \sqrt{2} R$, one has $y \sim y + 4 \pi \sqrt{2} R$ and so the coordinate $\coeff{y}{\sqrt{2} \,R} $ has period $4 \pi$, which means that the metric in (\[nearST2\]) represents the standard $\ZZ_k$ orbifold of $\IR^4$.
Supertubes with momentum via spectral interchange {#Sect:SpecInter}
=================================================
The original D1-D5 supertube solution [@Lunin:2001jy; @Lunin:2002iz] was defined in terms of an arbitrary profile function, $\vec F(\v)$, in $\IR^4$. While this manifestly describes the shape of the supertube, the supertube solution is [*not*]{} invariant under reparameterizations of $\v$, indeed, reparameterizations encode the choice of the charge-density functions. Put differently, the supertube can be given two charge densities, $\rhoone$ and $\rhotwo$, and an angular momentum density, $\rhohat$. However, supertube regularity and the absence of closed time-like curves (CTC’s) places two functional constraints (local analogues of (\[STreg0\])) on these densities leaving a free choice of one function. This function encodes the degrees of freedom represented by the choice of reparameterization in the original formulation.
Spectral interchange can then be combined with the addition of such a charge-density fluctuation so as to generate a third (momentum) charge.
Spectral interchange in general {#ss:SIflip}
-------------------------------
The idea behind spectral interchange is extremely simple. When the base space, $\cB$, has a Gibbons-Hawking form then the entire solution can be written as a torus fibration over a flat $\IR^3$. The torus is, of course, described by $(v,\psi)$ and one can act on this torus with elements of $GL(2,\ZZ)$[^8]. Since these transformations are generated by simple changes of coordinate, they must map BPS solutions to BPS solutions. Some elements of this transformation group generate what are known as gauge transformations [@Bena:2005ni] and generalized spectral flows [@Bena:2008wt], that mix $K^3$ and $V$. Of relevance later will be the gauge transformations: $$\begin{aligned}
K^I &\to& K^I + \alpha^I V \cr
L_I &\to& L_I ~-~ C_{IJK}\,\alpha^J K^K ~-~ \coeff{1}{2} \, C_{IJK}\, \alpha^J \alpha^K V \cr
M &\to& M ~-~ \coeff{1}{2} \,\alpha^I L_I ~+~ \coeff{1}{12} \, C_{IJK}\,
\left( V \alpha^I \alpha^J \alpha^K +3 \alpha^I \alpha^J K^K \right) \,. \label{BPSgaugetrf}\end{aligned}$$ Such transformations are pure gauge in that, while they reshuffle the potentials, while leaving the physical properties of the solution invariant.
Spectral interchange is a subset of the generalized spectral flow transformations [@Bena:2008wt], and is simply the modular inversion that interchanges $v$ and $\psi$ on the torus [@Niehoff:2013kia]. It corresponds to a global diffeomorphism on the fibers: $$v \to - \psi \,, \quad \psi \to - v \,; \qquad \Leftrightarrow \qquad V ~\leftrightarrow~ K_3\,, \quad A \to - \xi \,, \quad \xi \to- A \,.
\label{SpecInv}$$ This mapping also interchanges all the harmonic functions that make up the BPS solutions outlined in the previous section, as we now describe.
To make the mapping more precise, we must normalize the torus angles that we interchange. The periodicity of the $y$ circle induces an identification on $u$ and $v$. As described in above, we parameterize this as $$(u,v) ~\sim~ (u,v) + (-4\pi R, 4\pi R) \,.
\label{uviden-2}$$ Recalling the periodicity identifications on $\psi$ and $\phi$ given in , we see that the relevant lengths are $4\pi R$ for $v$ and $4\pi$ for $\psi$. Thus the spectral interchange is more precisely written as: $$\begin{aligned}
\frac{v}{R} &\to& -\psi \,, \qquad \psi \,~\to~\, -\frac{v}{R} \,.
\label{eq:specinvdef-2}\end{aligned}$$ Setting $Z_4=0$ and $\Theta^{(4)}=0$, spectral interchange implies that the following must also give a BPS solution: $$\begin{aligned}
\widetilde V &=& \frac{K^3}{R} \,, \quad \widetilde K^3 ~=~ R\, V \,, \quad
\widetilde K^1 ~=~ \frac{L_2}{R} \,, \quad \widetilde K^2 ~=~ \frac{L_1}{R} \,, \cr
\widetilde L_1 &=& R K_2 \,, \quad \widetilde L_2 ~=~ R\, K_1 \,, \quad
\widetilde L_3 ~=~ -\frac{2\, M}{R} \,, \quad \widetilde M ~=~ -\coeff{1}{2}\, R\, L_3
\label{eq:specinv5d}\end{aligned}$$ where any $\psi$-dependence is converted to $v$-dependence in accordance with (\[eq:specinvdef-2\]). Observe, in particular, that if the $L_I$ have some non-trivial $\psi$-dependence, then $\widetilde K^1$, $\widetilde K^2$ and $\widetilde L_3$ and hence $\widetilde{\mathcal{F}}$ inherit a non-trivial $v$-dependence. Thus the new solution involves a momentum wave and carries a momentum charge. We now implement this general idea in a specific explicit construction.
Spectral interchange: An example {#ss:SIexample}
--------------------------------
Our goal it to obtain a supertube with a magnetic dipole, $k$, and generic momentum densities and we will do this via spectral interchange.
Performing spectral interchange on the round $k$-wound supertube , combined with a gauge transformation with parameters $$\alpha^1 \,=\, -\frac{\bar{Q}_2}{k R} \,, \qquad \alpha^2 \,=\, -\frac{\bar{Q}_1}{k R} \,, \qquad \alpha^3 ~=~ 0\,,
\qquad\quad \bar{Q}_i ~\equiv~ \frac{Q_i}{4} \,, \quad i=1,2 \,,
\label{eq:gauge-trans}$$ results in a solution specified by the harmonic functions $$\begin{aligned}
V &~=~ \frac{k}{\rplus} \,, \qquad
K^1 ~=~ K^2 ~=~ \frac{1}{R}\,, \qquad
K^3 ~=~ \frac{R}{\rminus} \,,\label{ConstSeedSol1} \\
L_1 &~=~ \frac{\bar{Q}_1}{k} \frac{1}{\rminus} \,, \qquad
L_2 ~=~ \frac{\bar{Q}_2}{k} \frac{1}{\rminus} \,, \qquad
L_3 ~=~ \left( k + \frac{\bar{Q}_1+\bar{Q}_2}{k R^2} \right) \,,\label{ConstSeedSol2} \\
M &~=~ - \frac{1}{2} R
~+~ \frac12 \frac{\bar{Q}_1\bar{Q}_2}{k^2R} \frac{1}{\rminus} \,.
\label{ConstSeedSol3}\end{aligned}$$ This solution describes a supertube that is singly-wound, in a base space which is $\mR^4/\mZ_k$. The spectral interchange has thus had the effect of exchanging the original $\mZ_k$ orbifold at the location of the supertube for a $\mZ_k$ orbifold at the center of space, and the original smooth center of space has become the location of a singly-wound supertube.
On this supertube in the spectrally-inverted frame, we introduce charge densities as studied in [@Bena:2010gg], $$\begin{aligned}
V &~=~ \frac{k}{\rplus} \,, \qquad
K^1 ~=~ K^2 ~=~ \frac{1}{R}\,, \qquad
K^3 ~=~ \frac{R}{\rminus} \,,\label{SeedSol1} \\
L_1 &~=~ \frac{\bar{Q}_1}{k}\, \lambda_1(\psi,\vec{y}) \,, \qquad
L_2 ~=~ \frac{\bar{Q}_2}{k} \, \lambda_2(\psi,\vec{y}) \,, \qquad
L_3 ~=~ \left( k + \frac{\bar{Q}_1+\bar{Q}_2}{k R^2} \right) \,,\label{SeedSol2} \\
M &~=~ - \frac{1}{2} R
~+~ \frac12 \frac{\bar{Q}_1\bar{Q}_2}{k^2R} \, j(\psi,\vec{y}) \,,
\label{SeedSol3}\end{aligned}$$ where the $\lambda_A$ and $j$ are harmonic functions on $\IR^4$ written as a Gibbons-Hawking space, and are sourced by normalized densities $\rhoone$, $\rhotwo$, and $\rhohat$ localized at the supertube location $\vec{y}=\vec{y}_{-}$, that is $\rho_-=0$ or $(y_1=0, y_2=0, y_3=-\frac{\ell}{2})$: $$\begin{aligned}
\lambda_A(\psi,\vec{y}) ~=~ 4\pi \int\!d^3 y' \int_0^{4\pi} \!d \psi' \;
\widehat G(\psi,\vec{y}; \, \psi',\vec{y}\;\!')\, \rhoA (\psi' - k \phi') \delta^3(\vec{y}\,'-\vec{y}_{-}) \,, \nonumber \\
j(\psi,\vec{y}) ~=~ 4\pi \int\!d^3 y' \int_0^{4\pi} \!d \psi' \;
\widehat G(\psi,\vec{y}; \, \psi',\vec{y}\;\!')\, \rhohat (\psi' - k \phi') \delta^3(\vec{y}\,'-\vec{y}_{-}) \,.
\label{wigglybits-2}\end{aligned}$$ The dependence of the densities on the combination of angles $\psi - k \phi$ will become clear when we use the Green function on $\IR^4/\mZ_k$ in the next subsection to construct explicit solutions. For now, we keep the discussion general to explain our overall strategy.
We now transform back to the original supertube frame, first performing the inverse gauge transformation to and then performing spectral inversion. This results in the new BPS solution: $$\begin{aligned}
V &~=~ \frac{1}{\rminus} \,, \qquad
K_1 ~=~ \frac{\bar{Q}_2}{k R}\, \Big( \lambda_2(-\tfrac{v}{R},\vec{y}) ~-~ \frac{1}{\rminus} \Big) \,, \qquad
K_2 ~=~ \frac{\bar{Q}_1}{k R}\, \Big( \lambda_1(-\tfrac{v}{R},\vec{y}) ~-~ \frac{1}{\rminus}\Big) \,, \label{SIresult1}\\
K_3 &~=~ \frac{ k R}{\rplus} \,, \qquad
L_1 ~=~ 1~+~ \frac{\bar{Q}_1}{\rplus} \,, \qquad
L_2 ~=~ 1~+~ \frac{\bar{Q}_2}{\rplus} \,, \label{SIresult2}\\
L_3 & ~=~ 1 ~-~ \frac{\bar{Q}_1\bar{Q}_2}{(k R)^2} \,
\left( j(-\tfrac{v}{R},\vec{y}) ~-~ \lambda_1(-\tfrac{v}{R},\vec{y}) - \lambda_2(-\tfrac{v}{R},\vec{y}) + \frac{1}{\rminus}\right) \,, \label{SIresult3}\\
M &~=~ -\frac{k R}{2} ~+~ \frac{1}{2} \frac{\bar{Q}_1\bar{Q}_2}{k R}\, \frac{1}{\rplus} \,.
\label{SIresult4}\end{aligned}$$ The form of $V$ means that the base, $\cB$, has returned to flat $\IR^4$. There is a supertube with a dipole charge $k$ (corresponding to a pole in $K^3$), and charges $\bar{Q}_A$ located at $\rplus =0$. In addition, the harmonic functions $\lambda_A$ and $j$ describe a momentum wave along the $v$ direction that is sourced at $\rminus =0$. We have therefore succeeded in adding momentum to a standard two charge supertube solution.
Spectral interchange is simply a global diffeomorphism and so regularity conditions can be imposed on the supertube in the spectral-inverted frame. Before we do this, one should note that in the original seed solution (\[SeedSol1\])–(\[SeedSol3\]), the parameters, $\bar{Q}_A$, could be absorbed into the normalization of the charge densities, $\rhoA$ and $\rhohat$. We are therefore free to adjust them in some convenient manner and we choose to impose the constraint: $$\ell ~=~ \frac{\bar{Q}_1\bar{Q}_2}{(k R)^2} \,.
\label{basicreg}$$ As we will see, this choice will mean that one of the supertube regularity conditions is automatically satisfied for $\rhoA = \rhohat =1$.
Supertube regularity with varying charge density was extensively studied in [@Bena:2010gg] (following [@Bena:2008dw]) where it was shown that the supertube (\[SeedSol1\])–(\[SeedSol3\]) is regular if one imposes the following functional constraints at each point of the GH fiber: $$\lim_{\rminus \to 0}\, \rminus \, \left[ V \mu ~-~ Z_3 K^3 \right] ~=~ 0 \,, \label{regcondb}$$ $$\lim_{\rminus \to 0}\, \rminus^2 \, \left[ V Z_1 Z_2 ~-~ Z_3 (K^3)^2 \right] ~=~ 0 \,.
\label{regcondc}$$ Using (\[basicreg\]), the first equation can be reduced to $${k R} \left( \rhohat - 1 \right) ~+~ \frac{1}{k R} \left[ \bar Q_1 \left( \rhoone - 1 \right) + \bar Q_2 \left( \rhotwo - 1 \right) \right] ~=~ 0 \,,
\label{regcondd}$$ where $\rhoA$ and $\rhohat$ are defined in (\[wigglybits-2\]). The second regularity condition (\[regcondc\]), when combined with (\[regcondd\]) reduces to a simple, local constraint on the charge densities [@Bena:2010gg], $$\label{regconde}
\rhohat ~=~ \rhoone \;\! \rhotwo \,.$$ The regularity conditions (\[regcondd\]) and (\[regconde\]) can be thought of as “coiffuring” the charge densities so as to achieve regularity. One should note that while one can certainly satisfy (\[regcondd\]) using finite sets of Fourier modes, the charge density condition, (\[regconde\]), generically requires one of the Fourier series to be infinite. As we will see below, coiffuring and the holographic interpretation of the modes is somewhat simpler if one switches on $(Z_4, \Theta_4)$. One could repeat the foregoing analysis by introducing an additional charge density $\rhofour$, however for ease of presentation we will continue without introducing $\rhofour$ explicitly, and introduce $(Z_4, \Theta_4)$ in Section \[Sect:MomST\].
The Green function and mode expansions on an $\mR^4/\mZ_k$ base {#ss:GreenModes}
---------------------------------------------------------------
To construct explicit solutions of the form (\[SeedSol1\])–(\[SeedSol3\]), we need the scalar Green function for a GH base space with $V = \frac{k}{\rplus}$ and with a source located at $\rminus =0$. It is straightforward to obtain this via a coordinate transformation of the standard flat $\IR^4$ Green function, or one can use the general result of Page [@Page:1979ga]. One finds that the Green function for the response at the point $(\psi, \vec{y})$ caused by a source at the point $(\psi',\vec{y}\,'=\vec{y}_{-})$ defined by $\rminus =0$ is: $$\widehat G(\psi,\vec{y};\psi',\vec{y}\;\!')~=~{1\over 16\pi^2 \rminus} {\sinh\Bigl[ {k \over 2} \log
{ \rplus +\ell + \rminus \over \rplus +\ell - \rminus}\Bigr]\over
\cosh \Bigl[{k \over 2} \log
{ \rplus +\ell + \rminus \over \rplus +\ell - \rminus}\Bigr]
-\cos {\left[ \tfrac12(\psi-\psi') - \tfrac{k}{2} (\phi-\phi') \right]}}\,.
\label{PageGF}$$ Note that this function depends upon the combination of angular coordinates: $$\psi - k \phi\,.$$ This should not be surprising because the GH fiber is defined by $(d \psi +A)$ and, at $\rminus =0$, this becomes $(d \psi - k d \phi)$. Thus the charge density functions and solutions will depend upon precisely this mixture of angles, explaining the form of Eq.. If one expands the charge densities into Fourier modes, $$\rhoA(\psi-k \phi) ~=~ \sum_n {b_{A,n} \, e^{i \frac{n}{2}(\psi-k\phi) }} \,,$$ then the solutions are elementary to obtain from the Green function using contour integration (see for example [@Bena:2013ora]): $$\lambda_A(\psi, \vec{y}) ~=~ \sum_n {\frac{b_{A,n}}{\rminus}
\left[
\left( \frac{\rplus - \rminus +\ell}{\rplus + \rminus +\ell}
\right)^{\frac{k}{2}}
e^{\frac{i}{2}(\psi-k \phi) }
\right]^{n} }
~\equiv~ \sum_n \, {\frac{b_{A,n}}{\rminus} \, \hat{F}^n} \,
\label{eq:lambda-sol}$$ where $\hat{F}^n$ is defined through the above equation. Similarly, for $j$ we have $$\rhohat(\psi-k\phi) ~=~ \sum_n \, {\hat{b}_{n} \, e^{i \frac{n}{2}(\psi-k\phi) }} \,, \qquad j(\psi, \vec{\rminus}) ~=~ \sum_n \, {\frac{\hat{b}_{n}}{\rminus} \, \hat{F}^n} \,.
\label{eq:jsol}$$ Note that in the limit $\rminus \to 0$ these reduce to the following simple forms: $$\lambda_A(\psi, \vec{y}) ~\to~\sum_n {\frac{b_{A,n}}{\rminus} \,
e^{i\frac{n}{2}(\psi-k\phi) }}
~=~ \frac{\rhoA(\psi-k\phi)}{\rminus} \,, \qquad
j(\psi, \vec{y}) ~\to~ \frac{\rhohat(\psi-k\phi)}{\rminus} \,. \label{eq:r0limit}$$ Introducing spherical polar coordinates $(\rho,\vartheta,\phi)$ centered at the origin (halfway between the supertube and the GH center), we observe that for $\rho \gg \ell$, $$\rpm ~\simeq ~ \rho \left(1 ~\mp~ \frac{\ell}{2\, \rho}\cos\vartheta \right) \,.$$ This means that $\hat F^n$ falls off as $\rho^{-{\frac{k n}{2}}}$ at large $\rho$: $$\hat F^n ~\sim~ \left( \frac{\ell(1-\cos\vartheta)}{2\, \rho} \right)^{\frac{k n}{2}} e^{\frac{in}{2}(\psi-k\phi)}$$ and so higher orbifolds lead to more rapid fall-off at infinity.
When we come to imposing regularity constraints, we will find it useful to introduce non-zero $(Z_4, \Theta_4)$. In principle one could repeat the above analysis with an additional density profile function $\rhofour$ and analyze the modified supertube regularity conditions in the spectral inverted frame. Rather than pursue this route, we will find it more convenient to perform a direct analysis of the BPS equations using the techniques of [@Bena:2015bea] to construct our explicit solutions. This will lead to the complete solution in a manner that is well-adapted to coiffuring and holography.
Adding momentum to the supertube {#Sect:MomST}
================================
As we have seen, adding momentum to a supertube naturally leads us to consider $v$-dependent fluctuations. We now do this by generalizing the circular supertube seed solution described in Section \[ss:rndstube\]. In this way we will also obtain the complete solution including all components of the angular-momentum vector.
A natural way to construct $v$-dependent solutions is to introduce fluctuating charge-density sources along the $v$-fiber above the center of space, $r=0, \theta =0$ or $\rminus =0$, as described in [@Niehoff:2013kia]. Indeed, the $\psi$-fiber pinches off at the center of space while the $v$-fiber remains finite: $$( dv + \beta) ~\to~ ( dv - 2 k R \, d\varphi_2) \,.
\label{vfiber1}$$ This means that a single-valued source introduced along the $v$-fiber must have a Fourier expansion with the following dependence: $$e^{-i p(\frac{v}{2 R} - k \varphi_2)}\,, \qquad p \in \ZZ \,.
\label{Fourier1}$$ We will therefore seek solutions based upon these Fourier modes. Thus we define the phase: $$\zeta ~=~ \frac{v}{2 R} - k \varphi_2 \,.
\label{zetadefn}$$
The first layer of equations {#ss:Layer1}
----------------------------
Based upon the form of Eqs., and and the results of [@Niehoff:2013kia; @Shigemori:2013lta; @Giusto:2013bda; @Bena:2015bea] it is not hard to infer a solution to the first layer of BPS equations. Define $$\Delta ~\equiv~ \frac{a \, \cos \theta}{(r^2 + a^2)^\frac{1}{2}} \,,
\label{Deltadefn}$$ then a somewhat lengthy computation shows that the following fields satisfy the first layer of equations (\[BPSlayer1a\])–(\[BPSlayer1c\]) for some complex Fourier coefficients, $b_1$ and $b_2$: $$\begin{aligned}
Z_A &~=~ 1 ~+~ \frac{Q_A}{\Sigma} \, \big(1 + \Delta^{k n} \,( b_A \, e^{-i n \zeta} + \bar b_A \, e^{i n \zeta}) \big) \,, \quad A= 1,2 \,, \label{Z12osc1} \\
\Theta_1 &~=~ -\frac{n\, Q_2 }{2 \, R} \, \Delta^{k n} \,\Big[ \, b_2\, e^{-i n \zeta} \,(\Omega^{(2)} + i r \sin \theta \, \Omega^{(1)}) + \bar b_2\, e^{i n \zeta} \,(\Omega^{(2)} - i r \sin \theta\, \Omega^{(1)}) \Big] \,,\label{T1osc1}\\
\Theta_2 &~=~ -\frac{n\, Q_1 }{2 \, R} \, \Delta^{k n} \,\Big[ \, b_1\, e^{-i n \zeta} \,(\Omega^{(2)} + i r \sin \theta \, \Omega^{(1)}) + \bar b_1\, e^{i n \zeta} \,(\Omega^{(2)} - i r \sin \theta\, \Omega^{(1)}) \Big] \,,\label{T2osc1}\end{aligned}$$ To these fields one can add a completely independent, purely oscillating set of modes for $(Z_4, \Theta_4)$: $$\begin{aligned}
Z_4 &~= \frac{\Delta^{k p}}{\Sigma} \, ( b_4\, e^{-i p \zeta} + \bar b_4 \, e^{i p \zeta}) \,, \label{Z4osc1} \\
\Theta_4 &~=~ -\frac{p}{2 \, R} \, \Delta^{k p} \,\Big[ \, b_4\, e^{-i p \zeta} \,(\Omega^{(2)} + i r \sin \theta \, \Omega^{(1)}) + \bar b_4\, e^{i p \zeta} \,(\Omega^{(2)} - i r \sin \theta\, \Omega^{(1)}) \Big] \,. \label{T4osc1} \end{aligned}$$ Regularity of the metric and dilaton factors mean that one should have $Z_A >0$ for $A=1,2$. This means that the terms in the parentheses in (\[Z12osc1\]) must be strictly positive and since $|\Delta| < 1$ away from the source, one can certainly guarantee $Z_A >0$ by taking: $$|b_A| ~\le~\frac{1}{2}\,, \qquad A=1,2 \,. \label{babound}$$ One may be able to improve this bound slightly, but the important point is that $|b_A|$ will always be bounded by a number of order $1$.
The second layer of equations {#ss:Layer2}
-----------------------------
Consider a single mode of $\omega$ and $\mathcal{F}$: $$\omega ~=~ e^{- i q \zeta}(\hat \omega_r dr + \hat \omega_\theta d \theta + \hat \omega_1 d \varphi_1 +\hat \omega_2 d \varphi_2) \,, \qquad \mathcal{F} =~ - W \;\! e^{- i q \zeta}$$ then the differential operators in (\[BPSlayer2a\]) and (\[BPSlayer2b\]) may be written as: $$\begin{aligned}
& \mathcal{D}\omega +*_4 \mathcal{D}\omega + {\mathcal{F}}\, \Theta_3 \nonumber \\
& \qquad \qquad~\equiv~ e^{- i q \zeta} \Big[ (r^2 + a^2) \cos \theta \,\Omega^{(1)}\,\mathcal{L}^{(q)}_1 +r \sin \theta \,\Omega^{(3)}\,\mathcal{L}^{(q)}_3 + \frac{(r^2 + a^2)}{r} \cos \theta \, \Omega^{(2)}\,\mathcal{L}^{(q)}_2 \Big] \,, \\
&*_4\mathcal{D} *_4 \Big(\mathcal{D} \mathcal{F} - 2\,\dot{\omega} \Big) ~\equiv~ e^{- i q \zeta} \Big[ \, \widehat \cL^{(q)} \, W - \frac{i \, q}{R}\,\mathcal{L}^{(q)}_0 \Big] \,, \end{aligned}$$ where we define $$\begin{aligned}
\mathcal{L}^{(q)}_0 & \equiv \frac{1}{\Sigma}\,\Big[\frac{1}{r}\,\partial_r (r \,(r^2+a^2)\, \hat \omega_r) + \frac{1}{\sin\theta\cos\theta}\,\partial_\theta(\sin\theta\cos\theta\,\hat \omega_\theta)+ \frac{i \, k q \, a^2 }{(r^2 + a^2) }\, \hat \omega_1 + \frac{i \, k q }{\cos^2\theta}\, \hat \omega_2\Big] \,, \\
\mathcal{L}^{(q)}_1 &\equiv (\partial_r\hat \omega_\theta-\partial_\theta\hat \omega_r)-
\frac{i\, k q}{r(r^2+a^2) \sin\theta\, \cos\theta}
\,(r^2 \hat \omega_1 - a^2 \sin^2\theta \,\hat \omega_2) \,,\\
\mathcal{L}^{(q)}_2 &\equiv \frac{1}{\cos \theta}\, \,\partial_r\hat \omega_2 + \frac{r}{(r^2 +a^2) \sin\theta} \,\partial_\theta\hat \omega_1 - \frac{i \, k q \, r^2 }{\Sigma \cos \theta}\,\hat \omega_r - \frac{i \, k q a^2 r\, \sin \theta }{\Sigma \,(r^2 +a^2)}\,\hat \omega_\theta
- \frac{4 k R a^2 r \cos \theta}{\Sigma^2} \, W \,,\\
\mathcal{L}^{(q)}_3 &\equiv \frac{1}{\sin \theta}\,\partial_r\hat \omega_1-\frac{1}{r \cos \theta}\,\partial_\theta\hat \omega_2 - \frac{i\, k q}{\Sigma \cos\theta\, } \, (a^2 \sin\theta \cos\theta\, \hat \omega_r - r\,\hat \omega_\theta)
+ \frac{4 k R a^2 r \sin \theta}{\Sigma^2} \, W \,,\\
\widehat \cL^{(q)} W & \equiv \frac{1}{\Sigma}\,\Big[\frac{1}{r}\,\partial_r (r \,(r^2+a^2)\, \partial_r W) + \frac{1}{\sin\theta\cos\theta}\,\partial_\theta(\sin\theta\cos\theta\, \partial_\theta W ) - \frac{k^2 q^2 (r^2 + a^2 \sin^2 \theta)}{(r^2 + a^2)\cos^2 \theta }\, W\Big] \,.\end{aligned}$$ Using the solutions in (\[Z12osc1\])–(\[T4osc1\]), the source terms in (\[BPSlayer2a\]) and (\[BPSlayer2b\]) give rise, [*a priori*]{}, to four non-trivial kinds of source terms:
- Terms arising from products of modes with same phase. These depend upon $e^{\pm 2 i n \zeta}$ and have singularities involving $\Sigma^{-2}$.
- Terms arising from the product of a mode and a $\frac{Q_A}{\Sigma}$ term. These depend upon $e^{\pm i n \zeta}$ and have singularities involving $\Sigma^{-2}$.
- Terms arising from the product of a mode and the constant ($1$) in $Z_A$. These depend upon $e^{\pm i n \zeta}$ and have singularities involving $\Sigma^{-1}$.
- Terms arising from product of modes with the opposite phase. These are independent of $\zeta$ and have singularities involving $\Sigma^{-2}$.
However, the sources of types 1(a) and 1(b) are not really distinct in that the solution is the same but simply with a different mode number. We therefore break down the sources into types 1,2 and 3 and write the particular equations that need to be solved and find the particular solutions.
These systems of equations are:
$$\begin{aligned}
\mathcal{L}^{(q)}_1 ~=~& - \frac{i\, q}{2\,R}\,\frac{\Delta^{k q} \, r \sin\theta}{\Sigma (r^2 + a^2) \cos\theta} \,, \qquad
{
\mathcal{L}^{(q)}_3 ~=~0 \,,
} \\
{
\mathcal{L}^{(q)}_2 ~=~ } & - \frac{q}{2\,R}\,\frac{\Delta^{k q} \, r }{\Sigma (r^2 + a^2) \cos\theta} \,,
\qquad
\widehat \cL^{(q)} \, W - \frac{i \, q}{R}\,\mathcal{L}^{(q)}_0 ~=~
\frac{q^2}{2\,R^2 }\,\frac{\Delta^{k q}}{\Sigma^2} \,,
\label{source1}
\end{aligned}$$
$$\begin{aligned}
\mathcal{L}^{(q)}_1 ~=~& - \frac{i\, q}{2\,R}\,\frac{\Delta^{k q} \, r \sin\theta}{ (r^2 + a^2) \cos\theta} \,, \qquad
{
\mathcal{L}^{(q)}_3 ~=~0 \,, } \\
{
\mathcal{L}^{(q)}_2 ~=~ } &
- \frac{q}{2\,R}\,\frac{\Delta^{k q} \, r }{ (r^2 + a^2) \cos\theta} \,, \qquad
\widehat \cL^{(q)} \, W - \frac{i \, q}{R}\,\mathcal{L}^{(q)}_0 ~=~
\frac{q^2}{2\,R^2 }\,\frac{\Delta^{k q}}{\Sigma} \,,
\label{source2}
\end{aligned}$$ $$\begin{aligned}
\mathcal{L}^{(q=0)}_1 ~=~& 0 \,, \qquad\qquad\qquad\qquad\qquad\quad
{
\mathcal{L}^{(q=0)}_3 ~=~0 \,, } \\
{
\mathcal{L}^{(q=0)}_2 ~=~ } &
- \frac{m}{R}\,\frac{\Delta^{2mk} \,r}{\Sigma (r^2 + a^2) \cos\theta} \,, \qquad
\widehat \cL^{(q=0)} \, W ~=~
{
\frac{m^2}{R^2 }\,\frac{\Delta^{2km}}{\Sigma (r^2 + a^2) \cos^2\theta} \,,
}
\label{source3}
\end{aligned}$$
These equations have a gauge invariance associated with changing the $u$-coordinate: $$\label{gaugeu}
u\to u + f(x^i,v)\,,\quad \omega \to \omega - df + \beta \, \partial_v{f} \,,\quad {\mathcal{F}} \to {\mathcal{F}} - 2\, \partial_v{f}\,.$$ In terms of the $q^{\rm th}$ mode this becomes $$(\hat \omega_r, \hat \omega_\theta, \hat \omega_1, \hat \omega_2 ; W) \to (\hat \omega_r, \hat \omega_\theta, \hat \omega_1, \hat \omega_2 ; W) + (\partial_r h, \partial_\theta h, \frac{i\,k q\, a^2 \sin^2 \theta}{\Sigma}\, h , \frac{i\,k q\, r^2 }{\Sigma} \, h ; \frac{i\, q\,}{R} \, h) \,,
\label{gaugeh}$$ for an arbitrary function $h(x^i)$ on the base, $\cB$. In particular, for $q\ne 0$ one can choose a gauge with $W=0$.
It is relatively easy to find the explicit solutions for each of these sources:
$$(\hat \omega_r, \hat \omega_\theta, \hat \omega_1, \hat \omega_2 ; W) ~=~ \frac{\Delta^{k q}}{4 \,k R}\, \Big(- \frac{i }{r (r^2 + a^2) }, 0, \frac{\sin^2 \theta}{\Sigma} ,\frac{\cos^2 \theta}{\Sigma} ; 0 \Big) \,,
\label{solution1}$$ $$(\hat \omega_r, \hat \omega_\theta, \hat \omega_1, \hat \omega_2 ; W) ~=~ \frac{\Delta^{k q}}{4 \,k R}\, \Big(- \frac{i }{r}, i \tan \theta, 0 ,1 ; 0 \Big) \,,
\label{solution2}$$ $$\begin{aligned}
\hat \omega_r ~=~ & \hat \omega_\theta ~=~ 0 \,, \qquad W~=~ - \frac{1}{4 \,k^2 R^2}\,\frac{1}{(r^2 + a^2 \sin^2 \theta)} \, \big(1 - \Delta^{2 k m}\big)\,, \label{solution3a} \\
\hat \omega_1 ~=~ &\frac{1}{2 \,k R }\,\frac{(r^2 + a^2)}{\Sigma} \, \bigg( \frac{\big( \Delta^{2 k m}-1\big)\sin^2 \theta}{(r^2 + a^2 \sin^2 \theta)} ~+~ \frac{1}{a^2} \bigg) + \frac{\widehat J}{a^2 }\,\frac{(r^2 + a^2)}{\Sigma} + \hat c_1 \,, \label{solution3b} \\
\hat \omega_2 ~=~ &\frac{1}{2 \,k R }\,\frac{r^2}{\Sigma} \, \bigg( \frac{\big( \Delta^{2 k m}-1\big)\cos^2 \theta}{(r^2 + a^2 \sin^2 \theta)} ~-~ \frac{1}{a^2} \bigg) - \frac{\widehat J}{a^2 }\,\frac{r^2}{\Sigma} + \hat c_2 \,, \label{solution3c}\end{aligned}$$ where $\widehat J$, $\hat c_1$ and $\hat c_2$ are constants to be determined.
The complete angular momentum vector {#ss:compomega}
------------------------------------
Writing the components of $\omega$ so as to include all the phases: $$\omega ~=~ ( \omega_r dr + \omega_\theta d \theta + \omega_1 d \varphi_1 + \omega_2 d \varphi_2) \,,$$ putting together all the source terms and, for the moment setting $b_4=0$, we find: $$\begin{aligned}
\omega_r ~=~ & - \frac{i \, Q_1 Q_2}{4 \,k R}\, \frac{\Delta^{2 k n}}{r (r^2 + a^2) } \, \big( b_1 b_2 \, e^{- 2 i n \zeta} - \bar b_1 \bar b_2 \, e^{2 i n \zeta} \big) \nonumber \\
& - \frac{i}{4 \,k R}\, \frac{\Delta^{ k n}}{r (r^2 + a^2) } \, \big[ \big((b_1 + b_2) Q_1 Q_2 + (r^2 + a^2)(b_1 Q_1+ b_2 Q_2) \big) e^{- i n \zeta} \nonumber\\
&- \big((\bar b_1 + \bar b_2) Q_1 Q_2 + (r^2 + a^2)(\bar b_1 Q_1+ \bar b_2 Q_2) \big) e^{ i n \zeta} \big] \qquad \quad \label{omrsol1} \\
\omega_\theta ~=~ & \frac{i \, \Delta^{ k n} }{4 \,k R}\,\tan \theta \, \big( (b_1 Q_1+ b_2 Q_2) e^{- i n \zeta} - (\bar b_1 Q_1+ \bar b_2 Q_2) e^{ i n \zeta} \big) \,, \label{omthsol1} \\
\omega_1 ~=~ & \frac{ Q_1 Q_2}{4 \,k R}\, \frac{\Delta^{2 k n} \sin^2\theta}{\Sigma } \, \big( b_1 b_2 \, e^{- 2 i n \zeta} + \bar b_1 \bar b_2 \, e^{2 i n \zeta} \big) \nonumber \\
& + \frac{ Q_1 Q_2}{4 \,k R}\, \frac{\Delta^{k n} \sin^2\theta}{\Sigma } \, \big( (b_1 + b_2) \, e^{- i n \zeta} + (\bar b_1 + \bar b_2)\, e^{ i n \zeta} \big) \nonumber \\
& + \frac{Q_1 Q_2}{2 \,k R }( b_1 \bar b_2 + b_2 \bar b_1 ) \,\frac{(r^2 + a^2)}{\Sigma} \, \bigg( \frac{\big( \Delta^{2 k n}-1\big)\sin^2 \theta}{(r^2 + a^2 \sin^2 \theta)} + \frac{1}{a^2} \bigg) ~+~ \frac{J}{a^2 }\,\frac{(r^2 + a^2)}{\Sigma} + c_1 \,,\label{om1sol1} \end{aligned}$$ $$\begin{aligned}
\omega_2 ~=~ & \frac{ Q_1 Q_2}{4 \,k R}\, \frac{\Delta^{2 k n} \cos^2\theta}{\Sigma } \, \big( b_1 b_2 \, e^{- 2 i n \zeta} + \bar b_1 \bar b_2 \, e^{2 i n \zeta} \big) \nonumber \\
& + \frac{ Q_1 Q_2}{4 \,k R}\, \frac{\Delta^{k n} \cos^2\theta}{\Sigma } \, \big( (b_1 + b_2) \, e^{- i n \zeta} + (\bar b_1 + \bar b_2)\, e^{ i n \zeta} \big) \nonumber \\
& + \frac{ \Delta^{k n} }{4 \,k R}\, \big( (b_1 Q_1+ b_2 Q_2) \, e^{- i n \zeta} + (\bar b_1 Q_1 + \bar b_2 Q_2)\, e^{ i n \zeta} \big) \nonumber \\
& + \frac{Q_1 Q_2}{2 \,k R }( b_1 \bar b_2 + b_2 \bar b_1 ) \,\frac{r^2 }{\Sigma} \, \bigg( \frac{\big( \Delta^{2 k n}-1\big)\cos^2 \theta}{(r^2 + a^2 \sin^2 \theta)} - \frac{1}{a^2} \bigg) ~-~ \frac{J}{a^2 }\,\frac{r^2 }{\Sigma} + c_2 \,,\label{om2sol1} \\
\mathcal{F}~=~ & \frac{Q_1 Q_2}{4 \,k^2 R^2}\,( b_1 \bar b_2 + b_2 \bar b_1 ) \, \frac{1}{(r^2 + a^2 \sin^2 \theta)} \, \big(1 - \Delta^{2 k n}\big)\,, \label{omZ3sol1}\end{aligned}$$ where $J$, $c_1$ and $c_2$ are constants to be determined. Note that $\mathcal{F}$ vanishes at infinity.
The solutions for $(Z_4,\Theta_4)$ will be allowed to have different moding from $(Z_A,\Theta_B)$, where $\{A,B\} = \{1,2\}$. Using (\[Z4osc1\]) and (\[T4osc1\]) and the solutions for “Source 3,” we find $$\begin{aligned}
\omega_r ~=~ & \frac{i }{4 \,k R}\, \frac{\Delta^{2 k p}}{r (r^2 + a^2) } \, \big( b_4^2 \, e^{- 2 i p \zeta} - \bar b_4^2 \, e^{2 i p \zeta} \big) \,, \qquad \omega_\theta ~=~ 0 \,, \label{omrthsol2} \\
\omega_1 ~=~ &- \frac{1}{4 \,k R}\, \frac{\Delta^{2 k p} \sin^2\theta}{\Sigma } \, \big( b_4^2 \, e^{- 2 i n \zeta} + \bar b_4^2 \, e^{2 i n \zeta} \big) - \frac{|b_4|^2 }{k R }\,\frac{(r^2 + a^2)}{\Sigma} \, \bigg( \frac{\big( \Delta^{2 k n}-1\big)\sin^2 \theta}{(r^2 + a^2 \sin^2 \theta)} + \frac{1}{a^2} \bigg) \,,\label{om1sol2} \\
\omega_2 ~=~ & - \frac{ 1}{4 \,k R}\, \frac{\Delta^{2 k p} \cos^2\theta}{\Sigma } \, \big( b_4^2 \, e^{- 2 i p \zeta} + \bar b_4^2 \, e^{2 i p \zeta} \big) - \frac{|b_4|^2}{k R } \,\frac{r^2 }{\Sigma} \, \bigg( \frac{\big( \Delta^{2 k p}-1\big)\cos^2 \theta}{(r^2 + a^2 \sin^2 \theta)} - \frac{1}{a^2} \bigg) \,,\label{om2sol2} \\
\mathcal{F}~=~ & {}- \frac{|b_4|^2}{2 \,k^2 R^2}\, \frac{1}{(r^2 + a^2 \sin^2 \theta)} \, \big(1 - \Delta^{2 k p}\big)\,. \label{omZ3sol2}\end{aligned}$$ One should note that these solutions are singular: $\omega_r$ diverges at $r=0$. We therefore need to smooth these solutions out by adjusting the Fourier coefficients appropriately.
Coiffuring and regularity {#ss:STReg}
-------------------------
As we have discussed in Section \[ss:SIflip\], the core of this solution can be obtained via spectral interchange [@Niehoff:2013kia]. Moreover, supertube regularity requires that the charge density functions satisfy (\[regcondd\]) and (\[regconde\]). The important point here this the (\[regconde\]) may be viewed as the continuum analog of (\[STreg0\]) and, as such, determines $\rhohat$ in terms of $\rhoone$ and $\rhotwo$ This can easily be implemented explicitly in a finite Fourier expansion. On the other hand (\[regcondd\]) and (\[regconde\]) together mean that one cannot have a regular solution that involves [*finite*]{} Fourier series for both $\rhoone$ and $\rhotwo$: Regularity with only two fluctuating charge densities means (at least) one of the two Fourier series must be infinite. Since the solution in this paper is the spectral interchange of such a charge density fluctuation, the conclusion will be exactly the same.
In [@Bena:2015bea], regularity was achieved in a different manner: If one introduces one more charge species one cancels the singular terms between the species in a process that is known as “coiffuring” [@Mathur:2013nja; @Bena:2014rea; @Bena:2013ora]. In the discussion above, the addition of the additional species is represented by a new charge density, $\rhofour$, and replaces the $\rhoone \rhotwo$ terms by $\rhoone \rhotwo- \rhofour^2$; one can cancel the problematic quadratic terms and achieve regularity by simple linear constraints on Fourier coefficients and this can be implemented in a finite Fourier series. Thus coiffuring simply represents a very convenient way to solve the standard supertube regularity conditions using finite Fourier expansions.
At a practical level, our problem is simply to cancel all the $\frac{1}{r}$ singularities in $\omega_r$ and there are two natural ways to achieve this.
### Coiffuring: Style 1 {#ss:Coiffure1 .unnumbered}
The first is to give $(Z_4,\Theta_4)$ the same mode-dependence as the $(Z_A,\Theta_B)$. That is, to take $p=n$ in Eqs.(\[Z12osc1\])–(\[T4osc1\]) and then combine the corresponding contributions to $\omega$ and $\mathcal{F}$.
The singular terms that depend on $e^{\pm 2 i n \zeta}$ are then cancelled by setting $$b_4^2 ~=~ Q_1 Q_2 \, b_1 b_2 \,.
\label{breln1}$$ There are still singular terms that depend upon $e^{\pm i n \zeta}$ and these can be cancelled (at $r=0$) by setting $$(b_1 + b_2) \, Q_1 Q_2 ~+~ a^2 \, (b_1 Q_1+ b_2 Q_2) ~=~ 0 \,.
\label{breln2}$$ Eliminating $b_1$ and $b_2$ in terms of $b_4$ gives $$b_1 ~=~ \frac{i\,b_4}{Q_1} \sqrt{ \frac{Q_1 + a^2}{Q_2 + a^2}} \,, \qquad b_2 ~=~ - \frac{i\,b_4}{Q_2} \sqrt{ \frac{Q_2 + a^2}{Q_1 + a^2}} \,.
\label{breln0}$$ The solution for $\mathcal{F}$ and $\omega$ then reduces to: $$\begin{aligned}
\omega_r ~=~ & - \frac{i}{4 \,k R}\, \frac{r \, \Delta^{ k n}}{(r^2 + a^2) } \, \big[(b_1 Q_1+ b_2 Q_2) e^{- i n \zeta} - (\bar b_1 Q_1+ \bar b_2 Q_2)e^{ i n \zeta} \big] \qquad \quad \label{omrcoiff1} \\
\omega_\theta ~=~ & \frac{i \, \Delta^{ k n} }{4 \,k R}\, \tan \theta \, \big[ (b_1 Q_1+ b_2 Q_2) e^{- i n \zeta} - (\bar b_1 Q_1+ \bar b_2 Q_2) e^{ i n \zeta} \big] \,, \label{omthcoiff1} \\
\omega_1 ~=~ & - \frac{a^2}{4 \,k R}\, \frac{\Delta^{k n} \sin^2\theta}{\Sigma } \, \big[ (b_1 Q_1+ b_2 Q_2) \, e^{- i n \zeta} + (\bar b_1 Q_1+ \bar b_2 Q_2)\, e^{ i n \zeta} \big]\nonumber \\
& - \frac{2\,|b_4|^2}{k R } \,\frac{(r^2 + a^2)}{\Sigma} \, \bigg( \frac{\big( \Delta^{2 k n}-1\big)\sin^2 \theta}{(r^2 + a^2 \sin^2 \theta)} + \frac{1}{a^2} \bigg) ~+~ \frac{J}{a^2}\,\frac{(r^2 + a^2)}{\Sigma} + c_1 \,,\label{om1coiff1} \\
\omega_2 ~=~& \frac{r^2}{4 \,k R}\, { \frac{\Delta^{k n}}{\Sigma }} \, \big( (b_1 Q_1+ b_2 Q_2) \, e^{- i n \zeta} + (\bar b_1 Q_1+ \bar b_2 Q_2) \, e^{ i n \zeta} \big) \nonumber \\
& - \frac{2\,|b_4|^2}{ k R }\,\frac{r^2 }{\Sigma} \, \bigg( \frac{\big( \Delta^{2 k n}-1\big)\cos^2 \theta}{(r^2 + a^2 \sin^2 \theta)} - \frac{1}{a^2} \bigg) ~-~ \frac{J}{a^2 }\,\frac{r^2 }{\Sigma} + c_2 \,,\label{om2coiff1} \\
\mathcal{F}~=~ & - \frac{|b_4|^2}{k^2 R^2}\, \frac{1}{(r^2 + a^2 \sin^2 \theta)} \, \big(1 - \Delta^{2 k n}\big)\,, \label{omZ3sol1a}\end{aligned}$$ where $J$, $c_1$ and $c_2$ are constants to be determined.
One can, of course, do gauge transformations of the form (\[gaugeh\]) and set $\omega_r$ or $\omega_\theta$ to zero.
It is amusing to note that if we choose $Q_1 =Q_2$ then (\[breln0\]) implies $b_1 = -b_2$ and every oscillating term cancels from our expression for $\omega$: $$\begin{aligned}
\omega_r ~=~ & \omega_\theta ~=~ 0 \,, \label{omsimple0} \\
\omega_1 ~=~ & - \frac{2\,|b_4|^2}{k R } \,\frac{(r^2 + a^2)}{\Sigma} \, \bigg( \frac{\big( \Delta^{2 k n}-1\big)\sin^2 \theta}{(r^2 + a^2 \sin^2 \theta)} + \frac{1}{a^2} \bigg) ~+~ \frac{J}{a^2}\,\frac{(r^2 + a^2)}{\Sigma} + c_1 \,,\label{omsimple1} \\
\omega_2 ~=~& - \frac{2\,|b_4|^2}{ k R }\,\frac{r^2 }{\Sigma} \, \bigg( \frac{\big( \Delta^{2 k n}-1\big)\cos^2 \theta}{(r^2 + a^2 \sin^2 \theta)} - \frac{1}{a^2} \bigg) ~-~ \frac{J}{a^2 }\,\frac{r^2 }{\Sigma} + c_2 \,,\label{omsimple2} \end{aligned}$$ This is analogous to the completely coiffured black rings and microstate geometries discussed in [@Bena:2013ora; @Bena:2014rea].
Returning to the more general Style 1 coiffuring (with independent $Q_1$ and $Q_2$), we wish to examine the necessary conditions to avoid closed time-like curves[^9]. As in Section \[ss:rndstube\] we require that $\omega$ vanish at infinity and, as $r \to \infty$, one finds that $\beta$ vanishes and $$\omega + \beta ~\to~ \bigg(\frac{J}{a^2} - \frac{2\,|b_4|^2}{k R \,a^2} + c_1 \bigg) d\varphi_1 - \bigg(\frac{J}{a^2} - \frac{2\,|b_4|^2}{k R \,a^2} - c_2 \bigg) d\varphi_2 \,.
\label{ominfty2}$$ At the center of space, $r=0, \theta =0$, the $\varphi_1$ and $\varphi_2$ circles pinch off in the base metric $ds_4^2(\cB)$. At $r=0, \theta =0$ one finds: $$\omega +\beta ~\to~ \bigg(\frac{J}{a^2} - \frac{2\,|b_4|^2}{k R \,a^2} + c_1 \bigg) d\varphi_1 + \big( c_2 -2 k R \big) d\varphi_2\,
\label{cspace2a}$$ which for the absence of CTC’s must vanish. Thus we require that: $$c_1= - 2 \,k R \,, \qquad c_2 ~=~ 2\, k R\,, \qquad J~=~ \frac{2\,|b_4|^2}{k R } +2\,k R \, a^2 \,.
\label{cspace2b}$$ As noted earlier, regularity of the metric near the supertube means that as one approaches $\Sigma =0$, or $r=0, \theta =\frac{\pi}{2}$, the metric must remain smooth. The only potentially singular terms are proportional to $d\varphi_1^2$ but compared to simple supertube of Section \[ss:rndstube\], $\mathcal{F}$ is now finite as one approaches the supertube and so (\[JSTdiv1\]) generalizes to: $$6
- \frac{2}{\sqrt{\cP}}\,\beta_1 \, \big( \omega_1 + \coeff{1}{2}\, \mathcal{F}\, \, \beta_1 \big) ~+~ \sqrt{\cP} \,a^2 \, d\varphi_1^2 \,
\label{nearST3}$$ Collecting the singular terms terms and requiring that they vanish leads to a simple generalization of (\[STreg0\]): $$J ~=~ \frac{1}{4 kR} \, \big[ Q_1 Q_2 + 4 \, |b_4|^2 \big] \,,
\label{STreg3}$$ and combined with (\[cspace2b\]) we obtain $$a^2 ~=~ \frac{1}{8\, k^2 R^2} \, \big[ Q_1 Q_2 - 4\, |b_4|^2 \big] \,,
\label{STreg4}$$ which determines the radius of the supertube in terms of its electric charges.
Of particular significance is that, at infinity, one has $$- \frac{\mathcal{F}}{2} ~\sim~ \frac{|b_4|^2}{2k^2 R^2}\, \frac{1}{r^2} ~\sim~ \frac{Q_P}{r^2} \,,
\label{Z3infty1}$$ which implies that the supertube now carries a momentum charge of $$Q_P~=~ \frac{|b_4|^2}{2k^2 R^2}\,.
\label{QP1}$$ Note also that (\[STreg4\]) implies the following bounds: $$|b_4|^2 ~<~ \frac{Q_1 Q_2}{4} \qquad \Rightarrow \qquad Q_P ~<~ \frac{Q_1 Q_2}{8\,k^2 R^2} ~=~ \frac{Q_1 Q_2}{k^2 R_y^2} \,.
\label{bounds1}$$ More generally, it is instructive to rewrite (\[STreg3\]) and (\[STreg4\]) in terms of the momentum charge: $$J ~=~ 2\, kR \,(a^2 + 2 Q_P )\,, \qquad a^2 ~=~ \frac{Q_1 Q_2 }{8\, k^2 R^2} - Q_P \,.
\label{STreg-c1}$$
For future reference, it is convenient to extract the components of $\omega_1$ and $\omega_2$ that do not contain powers of $\Delta$: $$\begin{aligned}
\hat \omega_1 ~\equiv~ & - \frac{2\,|b_4|^2}{k R } \,\frac{(r^2 + a^2)}{\Sigma} \, \bigg(- \frac{ \sin^2 \theta}{(r^2 + a^2 \sin^2 \theta)} + \frac{1}{a^2} \bigg) ~+~ \frac{J}{a^2}\,\frac{(r^2 + a^2)}{\Sigma} + c_1 \,,\label{om1coiff1a} \\
\hat \omega_2 ~\equiv~& - \frac{2\,|b_4|^2}{ k R }\,\frac{r^2 }{\Sigma} \, \bigg(- \frac{ \cos^2 \theta}{(r^2 + a^2 \sin^2 \theta)} - \frac{1}{a^2} \bigg) ~-~ \frac{J}{a^2 }\,\frac{r^2 }{\Sigma} + c_2 \,.\label{om2coiff1a} \end{aligned}$$ The terms involving powers of $\Delta$ represent higher multipoles arising from the oscillations and, when $k n$ is sufficiently large, these are highly suppressed in the regions $r >> a$. Thus the $\hat \omega_i$ are the ‘higher-multipole-free’ components of the angular momentum. Substituting (\[cspace2b\]) into (\[om1coiff1a\]) and (\[om2coiff1a\]) yields $$\hat \omega_1 ~\equiv~ \frac{2\,|b_4|^2}{k R } \,\frac{a^2 \,\sin^2 \theta \cos^2 \theta }{\Sigma \,(r^2 + a^2 \sin^2 \theta) } ~+~ \frac{J}{\Sigma}\, \sin^2 \theta \,, \qquad
\hat \omega_2 ~\equiv~ - \frac{2\,|b_4|^2}{k R } \,\frac{a^2 \,\sin^2 \theta \cos^2 \theta }{\Sigma \,(r^2 + a^2 \sin^2 \theta) } ~+~ \frac{J}{\Sigma}\, \cos^2 \theta \,.\label{omcoiff1b}$$ Note that the first terms in these expressions vanish as $r^{-4}$ when $r \to \infty$ and hence for sufficiently large $kn$, the asymptotic structure of $\omega$ is determined entirely by $J$: $$\hat \omega ~\sim~ \frac{J}{r^2}\, (\sin^2 \theta d\varphi_1 + \cos^2 \theta d\varphi_2) \,.
\label{omasymp1}$$ Recall from (\[betaform3\]) that one has: $$\beta ~\sim~ \frac{2\, kR a^2}{r^2}\, (\sin^2 \theta d\varphi_1 - \cos^2 \theta d\varphi_2) \,.
\label{betaasymp1}$$ It therefore follows that this configuration has angular momenta (here we switch back to the physical $y$ radius $R_y=2\sqrt{2}R$ for later use) $$J_1 ~=~ \frac{1}{\sqrt{2}}\, (J + 2\, kR a^2) ~=~ \frac{Q_1 Q_2 }{k R_y} \,,
\qquad J_2 ~=~ \frac{1}{\sqrt{2}}\, (J - 2\, kR a^2) ~=~ k R_y Q_P \,.
\label{angmom1a}$$ and $J$ should be identified with $$J_L ~\equiv~ \frac{1}{2}\, (J_1 + J_2) ~=~ \frac{J}{\sqrt{2}}
~=~ \frac{1}{2} \frac{Q_1 Q_2 }{k R_y} + \frac{1}{2} k R_y Q_P .
\label{angmomL1}$$ Also note that $$J_R ~\equiv~ \frac{1}{2}\,\, (J_1- J_2) ~=~ \sqrt{2}\, kR a^2
~=~ \frac{1}{2} \frac{Q_1 Q_2 }{k R_y} - \frac{1}{2} k R_y Q_P .
\label{angmomR1}$$ For later use, we record that in terms of $b_4$, the angular momenta are $$J_L ~=~ \frac{1}{2} \frac{Q_1 Q_2 }{k R_y} + \frac{2|b_4|^2}{k R_y} \,,
\qquad \quad
J_R ~=~ \frac{1}{2} \frac{Q_1 Q_2 }{k R_y} - \frac{2|b_4|^2}{k R_y} \,.
\label{angmomsty1-2}$$ We observe that, compared to the angular momenta of the original supertube solution in Section \[ss:rndstube\], $J_1$ is unchanged, $J_L$ has increased and $J_R$ has decreased. We will interpret this in the CFT shortly.
Supersymmetric BMPV black holes [@Breckenridge:1996is] with macroscopic horizons exist in the regime of parameters $$Q_1 Q_2 Q_P - J_L^2 > 0 \,, \qquad J_R = 0 \,.
\label{overspin}$$ Indeed, see Appendix \[app:BMPV\] and, specifically (\[BMPVbound\]), where we have given the metric of the BMPV black hole in our conventions.
It is useful to parameterize the momentum charge via: Q\_P = c\_p , 0 c\_p<1 , where the upper bound on $c_p$ is a rewriting of . Then using (\[angmomL1\]) we find $$Q_1 Q_2 Q_P - J_L^2 ~=~ - \frac{(1-c_p)^2}{4} \left(\frac{Q_1 Q_2}{k R_y} \right)^2
\label{overspin-2}$$ and so, in terms of the quantum numbers of a BMPV black hole, this geometry is “overspinning” and becomes extremal in the scaling limit: $$Q_P ~\to~ \frac{Q_1 Q_2 }{k^2 R_y^2} \quad \Rightarrow \quad a^2 ~\to~ 0 \,.
\label{scaling1}$$ To understand why our solutions are overspinning, note that the original supertube of Section \[ss:rndstube\] is overspinning ($c_P=0$) and as we add momentum $Q_P$, (\[STreg-c1\]) shows that we must also add a corresponding amount of angular momentum, and that $a$ is adjusted according to (\[STreg-c1\]) such that we obtain (\[overspin\]).
### Coiffuring: Style 2 {#ss:Coiffure2 .unnumbered}
For our second style of coiffuring, we employ the coiffuring technique used in [@Bena:2015bea]. We will see in due course that the holographic dictionary is somewhat simpler for these solutions. The first step is to set $b_2 =0$ and take $n = 2p$ in Eqs.(\[Z12osc1\])–(\[T4osc1\]). The leading $r^{-1}$ singularities are cancelled by taking: $$Q_1 (Q_2 + a^2) b_1 ~=~ b_4^2 \,,
\label{breln3}$$ which fixes the Fourier coefficient $b_1$ in terms of $b_4$.
This leads to the solution: $$\begin{aligned}
\omega_r ~=~ & - \frac{i \,Q_1}{4 \,k R}\, \frac{\Delta^{ 2 k p} \, r }{(r^2 + a^2) } \, \big[ b_1 e^{- 2 i p \zeta} - \bar b_1 e^{ 2 i p \zeta} \big] \,, \qquad
\omega_\theta ~=~ \frac{i \,Q_1\, \Delta^{ 2 k p } }{4 \,k R}\,\tan \theta \, \big( b_1 e^{-2 i p \zeta} - \bar b_1 e^{ 2 i p \zeta} \big) \,, \label{omthsol3} \\
\omega_1 ~=~ & - \frac{ Q_1 \, a^2}{4 \,k R}\, \frac{\Delta^{2 k p} \sin^2\theta}{\Sigma } \, \big( b_1 \, e^{- 2 i p \zeta} + \bar b_1 \, e^{2 i p \zeta} \big)
~-~ \frac{|b_4|^2}{k R } \,\frac{(r^2 + a^2)}{\Sigma} \, \bigg( \frac{\big( \Delta^{2 k p}-1\big)\sin^2 \theta}{(r^2 + a^2 \sin^2 \theta)} + \frac{1}{a^2} \bigg) \nonumber \\
& ~+~ \frac{J}{a^2 }\,\frac{(r^2 + a^2)}{\Sigma} + c_1 \,,\label{om1sol3} \\
\omega_2 ~=~ & \frac{ Q_1 }{4 \,k R}\, \frac{\Delta^{2 k p} \, r^2\cos^2\theta}{\Sigma } \, \big( b_1 \, e^{- 2 i p \zeta} + \bar b_1 \, e^{2 i p \zeta} \big)~-~\frac{|b_4|^2}{k R } \, \frac{r^2 }{\Sigma} \, \bigg( \frac{\big( \Delta^{2 k p}-1\big)\cos^2 \theta}{(r^2 + a^2 \sin^2 \theta)} - \frac{1}{a^2} \bigg) \nonumber \\ &~-~ \frac{J}{a^2 }\,\frac{r^2 }{\Sigma} + c_2 \,,\label{om2sol3} \\
\mathcal{F}~=~ & - \frac{|b_4|^2}{2 \,k^2 R^2}\, \frac{1}{(r^2 + a^2 \sin^2 \theta)} \, \big(1 - \Delta^{2 k p}\big)\,, \label{omZ3sol3}\end{aligned}$$ The analysis of the absence of CTC’s proceeds exactly as before, giving: $$c_1= -2\,k R \,, \qquad c_2 ~=~ 2\, k R\,, \qquad J~=~ \frac{|b_4|^2}{k R } + 2\,k R \, a^2 \,.
\label{cspace3}$$ Regularity at the supertube once again requires (\[nearST3\]) to be finite at $\Sigma =0$. This yields $$J ~=~ \frac{1}{4kR} \, \big[ Q_1 Q_2 + 2\, |b_4|^2 \big] \,,
\label{STreg5}$$ and combined with (\[cspace3\]) we obtain $$a^2 ~=~ \frac{1}{8\, k^2 R^2} \, \big[ Q_1 Q_2 - 2\, |b_4|^2 \big] \,,
\label{STreg6}$$ which, again, determines the radius of the supertube in terms of its electric charges.
At infinity we now have $$-\frac{\mathcal{F}}{2}~\sim~ \frac{|b_4|^2}{4\,k^2 R^2}\, \frac{1}{r^2} \,,
\label{Z3infty2}$$ which implies that the supertube now carries a momentum charge of $$Q_P~=~ \frac{|b_4|^2}{4\,k^2 R^2}\,.
\label{QP2}$$ Since we have set $b_2 =0$ we have, in a sense, half as many oscillations and this leads to halving of various quantities in this style of coiffuring.
As before, the positivity of (\[STreg6\]) places a bound on $|b_4|$ which, in turn, results in the same bound on the momentum charge: $$|b_4|^2 ~\le~ \frac{Q_1 Q_2}{2} \qquad \Rightarrow \qquad Q_P ~\le~ \frac{Q_1 Q_2}{8\,k^2 R^2}
~=~ \frac{Q_1 Q_2}{k^2 R_y^2} \,.
\label{bounds2}$$ More generally, when (\[STreg5\]) and (\[STreg6\]) are rewritten in terms of the momentum charge we obtain exactly the same conditions as in (\[STreg-c1\]): $$J ~=~ 2\, kR \,(a^2 + 2Q_P )\,, \qquad a^2 ~=~ \frac{Q_1 Q_2 }{8\, k^2 R^2} - Q_P \,.
\label{STreg-c2}$$ Furthermore, in terms of $Q_P$ and $J$, the ‘higher-multipole-free’ components of the angular momentum are identical to those of (\[omcoiff1b\]): $$\hat \omega_1 ~\equiv~ \frac{ 4\, k R \, a^2 \, Q_P \sin^2 \theta \cos^2 \theta }{\Sigma \,(r^2 + a^2 \sin^2 \theta) } ~+~ \frac{J}{\Sigma}\, \sin^2 \theta \,, \qquad
\hat \omega_2 ~\equiv~ - \frac{4\, k R \, a^2 \, Q_P \sin^2 \theta \cos^2 \theta }{\Sigma \,(r^2 + a^2 \sin^2 \theta) } ~+~ \frac{J}{\Sigma}\, \cos^2 \theta \,.\label{omcoiff2b}$$ Thus the discussion of the asymptotic angular momenta is the same as in Section \[ss:Coiffure1\], and we again have $$J_L ~=~ \frac{1}{2} \frac{Q_1 Q_2 }{k R_y} + \frac{1}{2} k R_y Q_P \,,
\qquad \quad
J_R ~=~ \frac{1}{2} \frac{Q_1 Q_2 }{k R_y} - \frac{1}{2} k R_y Q_P \,.
\label{angmomsty2}$$ The difference between Style 1 and Style 2 comes when we express the angular momenta in terms of the respective coefficients of the oscillating terms. For Style 2, we obtain $$J_L ~=~ \frac{1}{2} \frac{Q_1 Q_2 }{k R_y} + \frac{|b_4|^2}{k R_y} \,,
\qquad \quad
J_R ~=~ \frac{1}{2} \frac{Q_1 Q_2 }{k R_y} - \frac{|b_4|^2}{k R_y} \,.
\label{angmomsty2-2}$$ In the limit of $k=1$, and taking $b_4$ to be real, our Style 2 solution is the extension to asymptotically flat space of a particular subset[^10] of the solutions constructed in [@Bena:2015bea].
### The lowest harmonics {#ss:lowharms .unnumbered}
Recalling the form of $\Delta$ in (\[Deltadefn\]), $$\Delta ~\equiv~ \frac{a \, \cos \theta}{(r^2 + a^2)^\frac{1}{2}} \,,$$ we see that for low values of $k,n$ and $p$, the powers of $\Delta$ do not fall off strongly at infinity and do not vanish very strongly at the ring ($r=0$, $\theta =\frac{\pi}{2}$). This can potentially lead to apparently singular behavior at the ring and unusual asymptotics at infinity. We now examine this more carefully.
First, note that for $kn =1$ there is an additional singularity at $r=0, \theta =\frac{\pi}{2}$ in the first term of $\omega_1$ in (\[om1coiff1\]), and $\omega_\theta$ and $\omega_2$ both contain terms that oscillate and fall off as as $r^{-1}$. However these terms are absent when $Q_1=Q_2$, and also in the decoupling limit, since they arise from the solution to Source 2 as described in Section \[ss:Layer2\]. Therefore there is a good asymptotically AdS solution for $kn=1$.
Restricting attention now to $kn \ge 2$, in “Style 1” one sees that $\omega_\theta$ and $\omega_2$ both contain terms that oscillate and fall off as as $r^{-k n}$, while $\omega_r$ falls off as $r^{-(k n+1)} dr$. Similarly, in “Style 2”, one sees that $\omega_\theta$ and $\omega_2$ both contain terms that oscillate and fall off as as $r^{-2 k p}$, while $\omega_r$ falls off as $r^{-(2k p+1)} dr$. Since we normally expect the angular momentum to appear as the leading term and fall off as $r^{-2}$ at infinity, the $r^{-2}$ terms may, at first, seem anomalous.
However, these oscillating terms do not present a problem. The most direct way to see this is to observe that they oscillate around the compactified $y$-circle and so average to zero in any measurement of asymptotic charge at infinity in the non-compact space. Such terms have also been encountered in other holographic solutions. Indeed, oscillating terms that fall off as $r^{-2}$ were encountered in [@Mathur:2011gz; @Mathur:2012tj; @Lunin:2012gp] and [@Giusto:2013bda] (see Eq.(5.21g)) where they arose through the action of the underlying [*global*]{} chiral algebra. Consequently, angular momentum modes that oscillate along $y$ and fall off as $r^{-2}$ in flat space represent physical solutions, and upon taking the decoupling limit, the corresponding asymptotically-AdS solutions are dual to well-defined CFT states.
Regularity bounds and CTC’s {#ss:Pbound}
---------------------------
As we have seen, there is a bound on the Fourier coefficients, $|b_4|$, that resulted in a bound on the momentum charge that was independent of the coiffuring style: $$Q_P ~\le~ \frac{Q_1 Q_2}{4\,k^2 R^2} \,.
\label{QPbound}$$ In addition, the coiffuring conditions relate $|b_4|$ to the $|b_A|$ via (\[breln1\]) or (\[breln3\]) so that we have: $$|b_1 b_2 | ~=~\frac{|b_4|^2}{Q_1 Q_2 } ~\le~ \frac{1}{4} \qquad {\rm or} \qquad |b_1| ~=~\frac{|b_4|^2}{Q_1 (Q_2 +a^2) } ~<~ \frac{1}{2}\,,
\label{bbound}$$ depending upon the coiffuring style. These conditions are completely consistent with the bounds that we obtained earlier, (\[babound\]), based upon the regularity of the $Z_A$.
One can also examine the possibility of CTC’s in the ‘intermediate region’ where $ a^2 \ll r^2 \ll Q_X$ for all charges, $Q_1, Q_2$ and $Q_P$. We also assume $k n$ or $k p$ is sufficiently large that we can drop such powers of $\Delta$ everywhere and, in particular, work with the ‘higher-multipole-free’ components, $\hat \omega_i$, of the angular momentum. In this intermediate region we have $Z_I \sim \frac{Q_I}{r^2}$ and the configuration looks like a BMPV black hole. Moreover, this region also contains the scaling limit (\[scaling1\]).
Since $|\mathcal{F}| \gg 1$ in the intermediate region, it is more natural to complete the squares in the metric (\[sixmet\]) by writing $$ds_6^2 ~=~ \frac{1}{\mathcal{F} \, \sqrt{\cP}} \, \big(du + \omega)^2 ~-~\frac{\mathcal{F}}{\sqrt{\cP} } \, \big(dv+\beta + \mathcal{F}^{-1}(du + \omega) \big)^2
~+~ \sqrt{\cP} \, ds_4^2(\cB)\,, \label{sixmet2}$$ If one considers displacements only in the $(v,\varphi_1,\varphi_2)$ directions and chooses $dv$ so that the middle term in (\[sixmet2\]) vanishes then the absence of closed timelike curves (CTC’s) requires: $$- \omega ^2 ~-~ \mathcal{F} \, \cP \, \big( (r^2 + a^2) \sin^2 \theta \, d\varphi_1^2 + r^2 \cos^2 \theta \, d\varphi_2^2 \big) ~\ge~ 0\,. \label{CTC1}$$ Dropping all powers of $\Delta^k$ we can replace $\omega$ by $\hat \omega$. We will retain the $1$’s in the $Z_A$’s, relabelling them by $\varepsilon_0$ so as to keep track of them. The absence of a negative eigenvalue in this two dimensional metric results in an inequality on the determinant that may be simplified to: $$(Q_1 + \varepsilon_0 \,\Sigma) (Q_2 + \varepsilon_0\, \Sigma) Q_P ~-~ J^2 ~\ge~ \frac{4\, k^2 R^2 a^8 \, \sin^2 \theta \cos^2 \theta}{r^2 (r^2 +a^2)} \,, \label{CTC2}$$ where $\varepsilon_0 =1$. Using (\[overspin\]) this identity simplifies to $$\varepsilon_0 \, (Q_1 + Q_2 + \varepsilon_0\, \Sigma) \, Q_P ~\ge~ \frac{a^4 \, (r^2 +a^2\sin^2 \theta) }{r^2 (r^2 +a^2)} \,, \label{CTC-3}$$ which is generically satisfied in the intermediate region. Thus we have solutions without CTC’s[^11] but that have the charges of overspinning BMPV black holes.
Dual CFT states {#Sect:CFT}
===============
The spacetime CFT dual to gravity on AdS$_3\times\bbS^3\times \cM$ is a non-linear sigma model on the moduli space of instantons on $\cM=\bbT^4$ or $K3$ [@Vafa:1995bm; @Douglas:1995bn; @Maldacena:1997re]. As is usual in AdS/CFT duality, the CFT is strongly coupled where gravity is weakly coupled, and vice versa. There is a locus in the moduli space where the target space of the CFT is the symmetric orbifold $\cM^N/S_N$ [@Vafa:1995zh; @Bershadsky:1995qy] (see also the review [@David:2002wn]), and since the BPS spectrum does not change in the passage from weak to strong coupling, one can hope to identify the CFT states in the orbifold theory which, when transported across moduli space to the regime where supergravity is weakly coupled, are dual to our geometries.
In the orbifold theory, the duals to black-hole states are the twisted sectors of the orbifold containing long cycles that permute many copies of $\cM$. Most of the entropy comes from oscillator excitations with fractional moding, and it has proven challenging to construct solutions that map to CFT states involving such fractional oscillators (for some previous examples, see [@Giusto:2012yz; @Chakrabarty:2015foa]). A major motivation for our construction is that it provides a large class of supergravity solutions whose CFT duals involve fractionally-moded oscillators.
In this section we begin with a review of the structure of the symmetric product orbifold CFT – covering both the structure of its supersymmetric ground states (in Section \[ss:symprod twistops\]), and the relation between BPS operators in the CFT and linearized mode operators in supergravity (in Section \[ss:sugra modes\]).
Previous studies have considered [*spectral flow*]{} as a means of introducing momentum charge to the system starting from a two-charge seed solution [@Giusto:2004id; @Giusto:2004ip; @Lunin:2004uu]. In CFT states where all strands have windings which have a common divisor greater than one, there is the possibility to perform [*fractional spectral flow*]{} [@Martinec:2001cf; @*Martinec:2002xq] which can be used to generate three-charge solutions [@Giusto:2012yz; @Chakrabarty:2015foa]. After a brief review of spectral flow in Section \[ss:CFT specflow\], a proposal is made in Section \[ss:CFT dual states\] for the CFT states dual to our geometries, built from fractional spectral flow on a [*subset of strands*]{} of a suitable two-charge BPS seed state. These candidate dual states are shown to carry the appropriate conserved quantum numbers, and reproduce at leading order the selection rules on the vevs of CFT operators dual to supergravity modes. The precise content of the dual CFT states is then specified at the fully non-linear level by finding the CFT states dual to a two-charge supertube profile that yield the CFT states we construct by fractional spectral flow.
For the purpose of comparison, it will be somewhat more convenient to work in the F1-NS5 duality frame, where the background fields are all from the NS sector. NS-R parity is then manifest (it is simply fermion parity in the CFT), and is an additional tool which can be used to characterize states and operators.
states: Twisted sector ground states of the CFT {#ss:symprod twistops}
------------------------------------------------
The vast majority of the states of the symmetric orbifold $(\cM)^N/S_N$ CFT are the twisted sector ground states under the symmetric group. There is an independent twisted sector for each conjugacy class in the orbifold group. In the symmetric group, one may write elements of the group as [*words*]{} consisting of products of (non-overlapping) cyclic permutations of the copies of $\cM$. The conjugacy class of a word is characterized simply by the lengths of all the cycles in the word. Thus the conjugacy class is specified by the number $n_\kappa$ of cycles of length $\kappa$, $\kappa\in\{1,\dots N\}$, and the total length (including cycles of length one) is $\sum_i n_i = N$.
We will mostly focus on $\bbT^4$, and comment on the modifications that result when $K3$ is realized as $\bbT^4/\bbZ_2$, though, of course, the Ramond ground state structure is the same anywhere on the $K3$ moduli space. The sigma model on the $\ell^{\rm th}$ copy of $\cM$ has bosonic fields $X^{(\ell)}_{A\Abar}$ and fermions $\chi^{(\ell)}_{A\alpha},\chibar^{(\ell)}_{A\alphabar}$. These carry labels under a variety of $SU(2)$ symmetries:
- The doublets $\alpha,\alphabar$ of the left and right $(SU(2) \times SU(2))_\cR$ $\cR$-symmetry.
- The doublet $\Abar$ under a custodial $SU(2)_\cC$ which is a global symmetry of the $\cN=(4,4)$ superalgebra. The supercurrents carry spin one-half under $SU(2)_\cC$ as well as under the $\cR$-symmetry.
- The doublet $A$ under an auxiliary $SU(2)_\cA$. This $SU(2)_\cA$ is a symmetry for $\cM=\bbT^4$, but is broken by the holonomy of the connection for $\cM=K3$.
In a given twisted sector cycle, the bosons $X^{(\ell)}_{A\Abar}$ and fermions $\chi^{(\ell)}_{A\alpha},\chibar^{(\ell)}_{A\alphabar}$ of the individual $\bbT^4$ CFTs are cyclically permuted: X\^[()]{}(e\^[2i]{} z) = X\^[(+1)]{}(z) ,=0,…,-1 , where $X^{(\kappa)}\equiv X^{(0)}$; similarly for the fermions $\chi^{(\ell)},\chibar^{(\ell)}$. The twist operator for such a cyclic orbifold is most conveniently expressed in terms of fields that diagonalize the twist action. Define the “clock” fields that are discrete Fourier transforms of these “shift” fields \[Xclock monodromy\] \^[()]{} = \_[=0]{}\^[-1]{} X\^[()]{} ,= 0,…,-1 , and similarly for the fermions $\chiclock^{(\nu)},\chiclockbar^{(\nu)}$. The clock fields diagonalize the cyclic permutation \^[()]{}(e\^[2i]{} z) = e\^[2i /]{} \^[()]{}(z) . A twist operator that implements these boundary conditions is the tensor product of standard $\bbZ_\kappa$ orbifold twist operators $\sigma_{(\nu/\kappa)}$ for each clock sector.[^12] These have dimension $h_{\nu,b} = \nu(\kappa-\nu)/\kappa^2$ for the bosonic twist operators, and $h_{\nu,NS} = (\nu/\kappa)^2$ for NS sector fermion twist operators, or $h_{\nu,R} = (\nu/\kappa - 1/2)^2$ for the R sector fermion twists. Taking the product over all the clock sectors yields the full twist operator for the cycle \^[()]{} = \_[=0]{}\^[-1]{} \_[(/)]{} , h\_=
(-1)/2 , & [NS]{} ;\
/4 , & [R]{} .
These NS sector twist ground state operators are spin $(\kappa-1)/2$ under both left- and right-moving $SU(2)_\cR$ $\cR$-symmetries, as may be seen by bosonizing the clock fermions and building the fermion twist operators as exponentials. Thus, these operators, and the states that they create from the NS sector vacuum, are , breaking half the supersymmetries of each chirality. Additional BPS operators are obtained by combining the lowest-dimension twist operator with the center-of-mass ($\nu=0$) fermion field. Similarly, the $\kappa$-cycle R sector ground state operators preserve one quarter of the Ramond supersymmetries.
The monodromy results in fractional mode expansions for the $\Xclock^{(\nu)}$ $$\begin{aligned}
\partial_z\Xclock^{(\nu)}(z) &= \sum_{m\in\bbZ} \xmode^{(\nu)}_{m+\nu/\kappa} \,z^{-m-\nu/\kappa-1}
\nonumber\\
\chiclock^{(\nu)}(z) &= \sum_{m\in\bbZ} \chimode^{(\nu)}_{m+\nu/\kappa} \,z^{-m-\nu/\kappa} ~,\end{aligned}$$ together with the oscillator commutation relations $$\begin{aligned}
[\xmode^{(\nu)}_{m+\nu/\kappa} , \xmode^{(\kappa-\nu)}_{-m'-\nu/\kappa} ] &= \alpha' (m+\nu/\kappa)\delta_{mm'}
\nn\\
[\chimode^{(\nu)}_{m+\nu/\kappa} , \chimode^{(\kappa-\nu)}_{-m'-\nu/\kappa} ] &= \alpha' (m+\nu/\kappa)\delta_{mm'}
~~~,\end{aligned}$$ where to reduce clutter the tangent space indices on the modes and fields have been suppressed in these expressions.
At this point, one can assemble all the different clock sector modes into a single set of “untwisted” (integer moded) $\bbT^4$ scalar fields $\xcover_{A\Abar}$ and fermions $\chicover_{A\alpha}$, $\chibarcover_{A\alphabar}$ living on the $\kappa$-fold cover of the cylinder. In order that the oscillator commutation relations remain canonical, one must rescale the effective string tension $\alpha'$ by a factor of $\kappa$, to ${\widehat\alpha}'=\alpha'/\kappa$; the fractionated oscillator mode energies are also $\kappa$ times smaller than the energies of the the untwisted oscillator modes.
The covering-space picture makes it clear that the R ground states carry spinor quantum numbers in the target space, since the structure is the same as the worldsheet theory of free perturbative strings. We can label the Ramond ground states for $\bbT^4$ as \[R gd states\] [|]{} ,[|A B ]{} ,[|B ]{} ,[|A]{} ; for $\bbT^4/\bbZ_2$ the fixed points provide sixteen more. One moves around in the space of ground states by the action of the zero modes of the fermions $\chicover_{A\alpha}$, $\chibarcover_{B\alphabar}$, which act as gamma matrices. We will focus on two ground states in particular – the highest weight state ${|++ \rangle}$ of the spin-1/2 multiplet, and the “singlet” combination of the auxiliary $SU(2)_\cA$ bispinor [|00 ]{} \^[AB]{}[|AB ]{} .
The covering space picture also leads to a somewhat more geometrical picture of the states in the Ramond sector. The conformal dimension of the ground state in the twisted sector is determined by the covering space transformation $z\to z^\kappa$, for which the Schwarzian derivative contribution to the stress tensor leads to \[twist base dim\] h\_0\^[()]{} = - . One can then apply any operator $\cO$ of the $\cM=\bbT^4$ (or $K3$) SCFT to this ground state; recalling that energies of the covering space theory are reduced by a factor of $\kappa$ leads to the spectrum \[twistdims\] h\_\^[()]{} = h\_0\^[()]{} + .
The Ramond ground states of $\cM$ are in one-to-one correspondence with the cohomology of $\cM$;[^13] for instance, for $\cM=\bbT^4$ the special spin-1/2 multiplet ${|\alpha\alphabar \rangle}$ is associated to the $(0,0)$, $(0,2)$, $(2,0)$ and $(2,2)$ cohomology, while the $j=0$ states ${|AB \rangle}$ are associated to the $(1,1)$ cohomology. Thus, in the $\kappa$-cycle twisted Ramond sector for $\bbT^4$, the supersymmetric ground states with $h=\kappa/4$ consist of one $(j,\bar j)=(1/2,1/2)$ multiplet and a quartet of singlets. In addition there are representations $(1/2,0)$ and $(0,1/2)$ which correspond to the odd cohomology. The action of the fermion zero modes of $\chi,\chibar$ moves one among these various representations. A similar story holds for $\cM=K3$. If we realize $K3$ as a $\bbT^4/\bbZ_2$ orbifold, we obtain 16 additional singlets from the 2-cohomology associated to the 16 fixed points of the orbifold, however there is no odd cohomology and so no $(1/2,0)$ or $(0,1/2)$ representations.
Under spectral flow, the operators that create these Ramond ground states from the vacuum transform into BPS short multiplet operators in the NS sector. We will discuss spectral flow in more detail below; here we simply wish to note that spectral flow generates from any operator with quantum numbers $(L_0,J_3)=(h,j)$ a related set of flowed operators with quantum numbers \[int spec flow\] L\_0 = h + 2 j s + s\^2 ,J\_3 = j + s ,s12 . If the initial operator is in the R (NS) sector, then spectral flow by integer amounts leads to another R (NS) operator, while spectral flow by odd half-integer amounts leads to NS (R) operators.
Finally, the twist operator for the full word conjugacy class in the symmetric group is given by the product of the twist operators for the $n$-component cycles in the word, for instance \[twistops\] = \_[i=1]{}\^n\_i\^[(\_i)]{} , h\_=
(N-n)/2 , & [NS]{} ;\
N/4 , & [R]{} ,
where we have used the fact that the sum of all the $\kappa_i$ is $N$. In the NS sector, only if all the polarizations $\alpha_i, \alphabar_i$ are aligned is the state BPS, since only then does the $\cR$-charge equal (plus or minus) the scaling dimension. In the R sector, any choice of polarizations will do, and one obtains a large degeneracy of BPS ground states carrying any $\bbS^3$ angular momentum in the tensor product $(\frac12)^{\otimes n}$. Note also that all the Ramond ground states are at zero energy once we include the Casimir energy $E_0=-c/24=-N/4$ for the CFT on a cylindrical geometry. The geometries constructed in Sections \[Sect:sixD\]–\[Sect:MomST\] are dual to CFT states in the Ramond sector, so henceforth we specialize to this sector. On the other hand, linearized excitations are NS sector operators, and so we will be interested in these NS operators when probing a CFT state to see what vevs of the supergravity fields are turned on in the supergravity background dual to this CFT state.
operators: Linearized supergravity modes {#ss:sugra modes}
-----------------------------------------
The spectrum of linearized supergravity on AdS$_3\times \bbS^3$ and its relation to the symmetric product was worked out in [@deBoer:1998ip; @Larsen:1998xm] (see also [@Deger:1998nm; @Kutasov:1998zh]). The bosonic spectrum consists of
- $\bbT^4$: the graviton, 5 self-dual (SD) plus 5 ASD tensors, 16 vectors, and 25 scalars;
- $K3$: the graviton, 5 SD plus 21 ASD tensors, and 105 scalars.
All these fields lie in short multiplets of the $\cN\!=\!(4,4)$ superconformal algebra. The $\cR$-charge of these multiplets is a combination of the spatial momentum on $\bbS^3$ and the tensor structure of the fields. The left-moving $\cR$-charge content of an NS sector short multiplet consists of
[state]{} & j & j’ & h\
[|]{} & n/2 & 0 & n/2\
G\_[-]{}[|]{} & (n-1)/2 & 1/2 & (n+1)/2\
(G\_[-]{})\^2 [|]{} & (n-2)/2 & 0 & (n+2)/2\
.3cm where $j$ is the spin under the $SU(2)_\cR$ $\cR$-symmetry, and $j'$ is the spin under the global (custodial) $SU(2)_\cC$ of the $N=4$ algebra; similarly for the right-moving structure. [Short multiplets may also carry an additional auxiliary $SU(2)_\cA$ quantum number $A,B$ associated to the fermions $\chi_{\alpha A}, \chibar_{\alphabar A}$ for $\bbT^4$. For $K3$, this is the $SU(2)$ for which the connection has holonomy, and so is not generically a good quantum number, however it is an “accidental” symmetry for untwisted states of the $\bbT^4/Z_2$ orbifold locus and we can continue to use this labelling.]{}
Consider the highest weight component of a short multiplet operator \^[()]{}\_[[**m**]{},[**|m**]{}]{} = \_[(\_1...\_[**m**]{}),(\_1...\_[**|m**]{}) ]{} of $\cR$-charge spins $(2j+1,2\bar j+1)=({\bf m},{\bf\bar m})$. Its single descendants are thus \_[(\_2...\_[**m**]{}),(\_2...\_[**|m**]{}) ]{}\^ = G\_[-]{}\^[\_1 ]{} |G\_[-]{}\^[\_1 ]{} \_[(\_1...\_[**m**]{}),(\_1...\_[**|m**]{}) ]{} , where $\Abar,\Bbar$ are custodial $SU(2)_\cC$ indices (not to be confused with the auxiliary $SU(2)_\cA$ labels $A,B$ for the ground states in equation ); the double descendants are \_[(\_3...\_[**m**]{}),(\_3...\_[**|m**]{}) ]{} = (\_G\_[-]{}\^[\_1 ]{}G\_[-]{}\^[\_2 ]{} ) (\_[’’]{}|G\_[-]{}\^[\_1 ’]{}|G\_[-]{}\^[\_2 ’]{}) \_[(\_1...\_[**m**]{}),(\_1...\_[**|m**]{}) ]{} . One also has the helicity $({\bf m}-{\bf\bar m} \pm 2)/2$ fields one gets by taking the double-descendant only on one side.[^14] These comprise the bosonic content of the supermultiplet.[^15]
According to [@deBoer:1998ip; @Larsen:1998xm], the spectrum of $\cN\!=\!(4,4)$ short multiplets for type IIB supergravity on AdS$_3\times\bbS^3\times K3$ is $$\begin{aligned}
\label{K3 short}
&\oplus_{m\ge1}\big[ ({\bf m},{\bf m+2})_S+({\bf m+2},{\bf m})_S
+ ({\bf m+2},{\bf m+2})_S\big] + n_T\big[\oplus_{m\ge2} ({\bf m},{\bf m})_S \big] ~,\end{aligned}$$ where $\bf m$ is the dimension of the $SU(2)_\cR$ representation. These supermultiplets expand into a set of $\bbS^3$ harmonics (ignoring special restrictions at low angular momentum) $$\begin{aligned}
\oplus_{m} \big( & ({\bf m},{\bf m\pm 4}) + 4({\bf m},{\bf m\pm 3}) + (n_T+7) ({\bf m},{\bf m\pm 2})\nn\\
&+(4n_T+8) ({\bf m},{\bf m\pm 1}) + (6n_T+8) ({\bf m},{\bf m}) \big) ~.\end{aligned}$$ The number of ASD tensors $n_T=21$ is dictated by anomaly cancellation. These quantum numbers result from the product of spherical harmonics on the $\bbS^3$ with the representations of the $SO(4)_{\cL}$ little group ([**3**]{},[**3**]{}) + 4([**2**]{},[**3**]{}) + 5([**1**]{},[**3**]{}) + n\_T([**3**]{},[**1**]{}) + 4 n\_T ([**2**]{},[**1**]{}) + 5 n\_T ([**1**]{},[**1**]{}) . Similarly, the spectrum of short multiplets for AdS$_3\times\bbS^3\times\bbT^4$ is [@deBoer:1998ip; @David:2002wn] $$\begin{aligned}
\label{T4 short}
&\oplus_{m\ge1}\big[ ({\bf m},{\bf m+2})_S+({\bf m+2},{\bf m})_S
+ ({\bf m+2},{\bf m+2})_S\big] \nn\\
&\qquad\qquad
+ 5\big[\oplus_{m\ge2} ({\bf m},{\bf m})_S \big]
+ 4\oplus_{m\ge2}\big[ ({\bf m},{\bf m+1})_S+({\bf m+1},{\bf m})_S\big] ~.\end{aligned}$$
There is a one-to-one correspondence between single-particle supergravity modes and $\kappa$-cycle Ramond ground states, by starting with the operators associated to the latter and performing a single unit of spectral flow to the NS sector. The operators associated to a single cycle of the symmetric group correspond to the single-particle modes in supergravity. The cycle winds together $\kappa$ copies of $\cM$, and thus has central charge $c=6\kappa$. Under the spectral flow operation , the $j=1/2$ Ramond operators flow to one $h=j=(\kappa-1)/2$ NS operator (from the $j_3=-1/2$ polarization), and one $h=j=(\kappa+1)/2$ NS operator (from the $j_3=+1/2$ polarization).[^16] Similarly, the $j=0$ operators flow to $h=j=\kappa/2$ operators.
The special spin-1/2 multiplet thus yields $SU(2)_\cR$ representations ${\bf m}=\kappa,\kappa+2$ after spectral flow, while the spin-0 multiplets yield representation ${\bf m}=\kappa+1$ after spectral flow. Combining left- and right-movers yields the spectra , . Note that for $K3$, the Ramond operators have the same fermion parity on left and right, while for $\bbT^4$, the left and right fermion parity can be chosen independently since one can act with fermion zero modes on left and right independently; this is the origin of the short multiplets $({\bf m}, {\bf m\pm 1})_S$ which comprise the harmonic expansion of the vector supermultiplets.
The special spin $(1/2,1/2)$ Ramond ground states are universal, and upon spectral flow to the NS sector are associated to the six-dimensional graviton, dilaton and NS B-field. Their harmonics on the spatial $\bbS^3$ organize themselves into $\cN\!=\!(4,4)$ short multiplets \[graviton sector\] ([ ]{},[ ]{})\_S + ([ ]{},[ +2]{})\_S+([ +2]{},[ ]{})\_S + ([+2]{},[+2]{})\_S comprising the lowest spin chiral primary $\cO^{(\kappa)}_{\bf m,m}$ of the $\kappa$-cycle, which has ${\bf m} = \kappa$, together with the three additional chiral primaries built by tensoring with the $\kappa$-cycle currents $J^+$ and/or $\bar J^+$ on $\cO^{(\kappa)}_{\kappa,\kappa}$. The bosonic content of these multiplets consists of two six-dimensional supergravity supermultiplets. The first, the graviton supermultiplet, contains the graviton plus the self-dual part of the B-field, as well as four more self-dual, six-dimensional, two-form tensor fields from the RR sector. The second, a six-dimensional tensor multiplet, contains the ASD six-dimensional polarizations of the NS B-field, as well as the dilaton and four six-dimensional scalars made from the triplet of RR fields $C_2^+$ (the RR tensor which is self-dual on $\bbT^4$) together with the self-dual combination $v_4 C_0+C_4$ of the RR scalar and $\bbT^4$ four-form. The zero modes of these latter four scalars are moduli in the F1-NS5 duality frame. The ${\bf m=\bar m}$ CFT primaries map to linear combinations of supergravity field modes that diagonalize the linearized field equations.[^17] In general, there are also non-linear corrections to the map between CFT operators and supergravity field modes; in a typical correlator, these corrections are suppressed by powers of the gravitational coupling, but in so-called extremal correlators (where the conformal dimension of one operator is the sum of all the others) these non-linearities can contribute at leading order.
The highest weight, together with the double descendants of the quartet of superfields , yield the harmonic expansion of the six-dimensional graviton, the six-dimensional NS B-field, and the six-dimensional dilaton. The quantum numbers $(h,\bar h, j,\bar j)$ are the resolution of the product of the spatial harmonic and the tensor structure onto states of definite total spin in both $SL(2)$ and $SU(2)$. The graviton $g_{MN}$ and B-field $B_{MN}$ can have their tensor polarizations either along AdS$_3$, $M,N=\mu,\nu$, or along $\bbS^3$, $M,N=a,b$. Of the two indices, one transforms under the left $SL(2)\times SU(2)$, and the other transforms under the right $SL(2)\times SU(2)$. An analysis of [@Kutasov:1998zh] shows that the physical combinations of tensor polarization and $SL(2)\times SU(2)$ spatial harmonic $(h,j)=(\lambda,\ell)$ are those whose total $SL(2)\times SU(2)$ quantum numbers are $(h,j)=(\lambda\pm1,\ell)$ or $(\lambda,\ell\pm 1)$. One can thus trace the six-dimensional polarizations through the field transformation and resolution onto components of definite total spin, in order to match supergravity fields with CFT operators at the linearized level. As discussed above, beyond the leading order in the small field expansion, the map between CFT operators and supergravity modes is non-linear.
The single descendants have the opposite NSR parity, and comprise a quartet of tensor harmonics; this custodial $SU(2)_\cC$ bi-doublet can be decomposed into a scalar and self-dual tensor on $\bbT^4$/$K3$, and thus one obtains the self-dual six-dimensional polarizations of $C_2$, as well as $C_4$ having two legs in six-dimensional and two legs along $\bbT^4$/$K3$. Note that for $\kappa=2$, one finds the four RR moduli of the background; a null vector truncates the representation from above, so that these components are in fact the highest components of the superfield – a multiplet with $h=j=1/2$ is an [*ultrashort*]{} multiplet, and thus perturbing by the single-descendant operators preserves $\cN\!=\!(4,4)$ supersymmetry.
The remaining quartet $\cO^{(\kappa)}_{AB}$ of spin $j=\bar j=\kappa/2$ superfields (which have $\mm=\mmb=\kappa+1$), comprise four additional tensor multiplets containing 4 ASD tensors and 20 scalars. The four tensors appear in the lowest and highest components plus the helicity $\pm 1$ one-sided double-descendants, and are the ASD parts of the RR tensors whose opposite chiralities are in the gravity supermultiplets . These components also include the harmonics of four RR fixed scalars (the ASD combinations of $C_0,C_4$ and $C_2$ with polarization entirely on $\bbT^4/K3$). The single-descendants comprise 16 NS sector scalars – the polarizations of the graviton and B-field along $\bbT^4/K3$. For $\kappa=1$, one has the 16 NS sector moduli of $\bbT^4$ (again these are ultrashort multiplets, so the single-descendant is the highest component). The spectrum is then completed either with the $ ({\bf m},{\bf m\pm1})_S$ vector multiplets for $\bbT^4$; or 16 more $ (\kappa,\kappa)_S$ tensor multiplets for $K3=\bbT^4/\bbZ_2$, with similar content.
We summarize the short multiplet content of the $\kappa$-cycle sector of the gravity sector supermultiplets in the following table, where $(\mm,\mmb) \in \{(\kappa,\kappa),(\kappa,\kappa\!+\!2),(\kappa\!+\!2,\kappa),(\kappa+2,\kappa+2)\}$: \[graviton table\]
[|c|c|c|c|c|]{} [multiplet]{} & (2j+1,2|j+1) & SU(2)\_& [sugra field]{}\
\^[()]{}\_[,]{} & ([,]{}) & [1]{} & G,B,\
(G\_[-]{})\^2 \^[()]{}\_[,]{} & ([-2,]{}) & [1]{} & G,B,\
(|G\_[-]{})\^2 \^[()]{}\_[,]{} & ([,-2]{}) & [1]{} & G,B,\
(G\_[-]{})\^2 (|G\_[-]{})\^2 \^[()]{}\_[,]{} & ([-2,-2]{}) & [1]{} & G,B,\
G\_[-]{}\^[ |G]{}\_[-]{}\^ \^[()]{}\_[,]{} & ([-1,-1]{}) & [13]{} & C\_2\^+,C\_4\^+,C\_0\
The RR six-dimensional tensor fields together with the six-dimensional tensor $B$ field comprise the five self-dual tensors in the six-dimensional $N=(2,0)$ graviton supermultiplet.
The remaining six-dimensional tensor supermultiplets contain the torus moduli fields, and consist of: \[torus moduli table\]
[|c|c|c|c|c|]{} [multiplet]{} & (2j+1,2|j+1) & SU(2)\_& [sugra field]{}\
\^[()AB]{}\_[+1,+1]{} & (+1,+1) & [1]{} & C\_0, C\_2\^-, C\_4\^- [*tensors/scalars*]{}\
(G\_[-]{})\^2 \^[()AB]{}\_[+1,+1]{} & (-1,+1) & [1]{} & C\_2\^-, C\_4\^- [*tensors*]{}\
(|G\_[-]{})\^2 \^[()AB]{}\_[+1,+1]{} & (+1,-1) & [1]{} & C\_2\^-, C\_4\^- [*tensors*]{}\
(G\_[-]{})\^2 (|G\_[-]{})\^2 \^[()AB]{}\_[+1,+1]{} & (-1,-1) & [1]{} & C\_0\^
,C\_2\^-, C\_4\^- [*tensors/scalars*]{}\
G\_[-]{}\^[ |G]{}\_[-]{}\^ \^[()AB]{}\_[+1,+1]{} & (,) & [13]{} & G,B [*moduli*]{}\
These four multiplets contain six-dimensional ASD RR tensors. In all of the tables, the plus/minus superscript on tensors indicates their six-dimensional chirality. The additional 16 ASD tensor supermultiplets of the $K3$ theory arising from the fixed points of $\bbT^4/\bbZ_2$ are similar in content to the above table. It is straightforward to work out the vector multiplets, which only contain transverse vector polarizations and their fermionic superpartners.
Given the foregoing collection of $SL(2,R)\times SU(2)$ highest weights organized into $\cN\!=\!(4,4)$ multiplets, the action of the lowering operators $J^-$, $\bar J^-$ of $SU(2)_L\times SU(2)_R$ and raising operators $L_{-1}$, $\bar L_{-1}$ of $SL(2,R)_L\times SL(2,R)_R$ fills out a complete basis of six-dimensional spatial harmonics of the supergravity fields.
CFT spectral flow to states {#ss:CFT specflow}
---------------------------
We need one more ingredient to specify the class of CFT states dual to the supergravity geometries above. Spectral flow is a coherent deformation of the charge in a CFT with a $U(1)$ current. Any primary field $\mathcal O$ in such a theory can be written \[charged op\] [O]{} = e\^[i2H]{} where the $U(1)$ current is bosonized as $J=i \partial H$, and $\Phi$ is a $U(1)$ singlet operator. Spectral flow is then the deformation along $\alpha$, which leads to a family of operators/states of dimension and charge \[specflow\] h = h\_+ [\^2 ]{} ,q = [2]{} [ ]{} where the normalization of the current is J(z) J(w) \~ , For an $\cN\!=\!(4,4)$ SCFT, the normalization of the $SU(2)$ $\cR$-current $J_3$ is set by the algebra, $\kappa=c/6$, and the $SU(2)$ spin of operators is $j_3=\alpha \kappa$.
One can decompose the $1/4$-BPS twist operators under spectral flow as follows. Consider the NS sector twist field for a cyclic permutation of order $\kappa$, with quantum numbers c=6,h=j\_3=2. One can determine the dimension of the operator $\Phi$ via a spectral flow by an amount $\alpha=-(\kappa-1)/2\kappa$ that strips off the $j_3$ charge; in this way one finds \[gd state dim\] h\_= - 1[4]{} , the dimension of the operator that implements the covering transformation $z\to z^\kappa$. Spectral flow to the R sector shifts the $j_3$ charge by $\kappa/2$, from $j_3=(\kappa-1)/2$ to $j_3=-1/2$. The $U(1)$ charge exponential now carries dimension $1/(4\kappa)$, resulting in the Ramond-sector twist operator dimension $\kappa/4$, equation . Similarly, on $\bbT^4$ one may regard the $\cR$-symmetry singlet states ${|AB \rangle}$ as the result of acting on the state ${|\Phi,\alpha\!=\!0 \rangle}$ by a spectral flow to spin $1/2$ in the [*auxiliary*]{} $SU(2)_\cA$ (for $\bbT^4/\bbZ_2$, there are 16 additional states coming from the fixed points).
One can now obtain new R-sector states from the cyclic twist ground states ${|\alpha\alphabar \rangle}$ and ${|AB \rangle}$ via spectral flow by an amount $s/\kappa$, $s\in \bbZ$. This operation is [*fractional spectral flow*]{} on the $\kappa$-cycle, but [*integer*]{} spectral flow on the covering space; it results in a series of states, for example $$\begin{aligned}
\label{specflow states}
{|++ \rangle}_{\kappa,s}
\quad,\qquad &h_{\kappa,s} = \frac{\kappa^2-1}{4\kappa} + \frac{(s+1/2)^2}{\kappa}
\quad, \hskip -2cm &&j^3_{\kappa,s} = s+\frac12
\nn\\
{|00 \rangle}_{\kappa,s}
\quad, \qquad &h_{\kappa,s} = \frac{\kappa}{4} + \frac{s^2}{\kappa}
\quad, \hskip -2cm &&j^3_{\kappa,s} = s \quad ,\end{aligned}$$ and corresponding operators. For a general conjugacy class in the symmetric group, one has the choice of independent spectral flow on each component cycle.
States that survive the symmetric group quotient have $h-\bar h\in \bbZ$ for each cycle. In the twisted sectors, a cycle of length $\kappa$ has a $\bbZ_\kappa$ projection on its Hilbert space[^18] that assigns charge $\nu/\kappa$ to the $\nu^{\rm th}$ clock sector, and neutrality under this projection guarantees that states have integer momentum, cycle by cycle. To satisfy this requirement, one must have $s^2/\kappa\in\bbZ$ for the ${|00 \rangle}$ state, or $s(s+1)/\kappa\in\bbZ$ for the ${|++ \rangle}$ state.
The generic state in this construction is obtained by taking tensor products of ${|\alpha\alphabar \rangle}_{\kappa,s}$ and ${|AB \rangle}_{\kappa,s}$ chosen independently for each cycle; these states are built from fractional spectral flow under the $J_3$ pertaining to that cycle only, and are subject to the integer momentum constraint on each cycle (and the $\bbZ_2$ quotient for $K3=\bbT^4/\bbZ_2$). Note that the $U(1)$ currents that we are using to spectral flow are not present in the CFT away from the orbifold locus, apart from the overall $U(1)$. Nevertheless, at the orbifold locus they serve to generate states for us that are protected by the BPS property as we move away from the orbifold locus in moduli space, and so we can continue to characterize them through the use of this special property of the orbifold theory.
The states spectrally flowed under $J_3$ have an equivalent description in terms of descendant states in the $SU(2)$ current algebra; for example \[00 spec flow\] [|00 ]{}\_[,s]{} = (J\^+\_[-s/]{})\^[s]{} [|00 ]{}\_ . This relation is straightforward to see in the covering space description, where this state can be thought of in terms of the raising operator $(J^+_{-s})^s$ acting on the current algebra vacuum (recall the moding is rescaled by a factor $\kappa$ on the covering space). The covering space $SU(2)$ current algebra has level one, and is entirely accounted for through bosonization. The operator $(J^+_{-s})^s$ is a Virasoro highest weight operator of spin $s$ and dimension $s^2$, and therefore must be an exponential $\exp[i\sqrt2 \,s\widehat H]$ of the boson $J_3=i\partial \widehat H$ on the covering space, hence indeed implements a spectral flow transformation.
Finally, in the $\bbT^4$ SCFT, spectral flow has a third interpretation in terms of shifting the Fermi sea of the $\chicover_{A\alpha}$, by populating all the modes in the Hilbert space up to and including level $s/
\kappa$. It is straightforward to check that this leads to the shifts in energy and charge.
CFT duals of our superstrata {#ss:CFT dual states}
----------------------------
We now combine the ingredients discussed above to develop our proposal for the CFT duals of our superstrata. We observed below Eq. that our Style 2 solution, in the limit of $k=1$, is the extension to asymptotically flat space of a particular subset of solutions constructed in [@Bena:2015bea]. The proposed dual CFT states [@Bena:2015bea; @Giusto:2015dfa] can be written in terms of spectral flow on chiral primary states. This suggests we look to similar states for candidate CFT duals of our solutions. The fact that the wavenumber in $v$ is a fraction $1/k$ of the wavenumber in $\psi$ suggests that we consider “fractional spectral flow” states of the sort described above.
The orbifold projection on cycles of length $\kappa$ enforces integer momenta on each strand. Consider spectral flow on Ramond sector ${|00 \rangle}_\kappa$ cycles. Integer $h-\bar h$ means that $\alpha^2\kappa\in \bbZ$ in Eq.. One also wants $j_3 = k (h-\bar h)$ so that the state corresponds to a supergravity solution whose fields have a phase dependence which is a multiple of $\zeta=\frac{v}{2R}-k\varphi_2$ (Eq.). Equation then requires $\alpha\kappa = k \alpha^2 \kappa$; thus $\alpha = 1/k$. Therefore $\kappa$ should be a multiple of $k^2$ in order for $h-\bar h\in\bbZ$, i.e. $\kappa=k^2 \hat{p}$ for some integer $\hat{p}$. Then $\alpha=s/\kappa$ gives $s= k \hat{p}$. Thus, one component of the candidate CFT dual for Style 2 coiffuring is fractional spectral flow by an amount $\alpha = 1/k$ on cycles ${|00 \rangle}_{k^2 \hat{p}}$ of length $k^2\hat{p}$.
A second candidate component of the CFT dual arises from spectral flow on Ramond sector cycles ${|++ \rangle}_{\kappa'}$. Applying the same logic as above, one finds the criteria of integer $h-\bar h$ and spectral flow yielding $j_3-\bar j_3 = k (h-\bar h)$ result in a spectral flow by amount $s'=k \hat{n}$, and cycle length $\kappa'=k(k\hat{n}+1)$, for some $\hat{n}\in\bbZ$.
Finally, a third candidate component of the CFT dual uses spectral flow on Ramond sector cycles ${|-- \rangle}_{\kappa''}$. Once again the spectral flow amount is $s''=k\hat{m}$, and the cycle length is $\kappa''=k(k\hat{m}-1)$ for some $\hat{m}\in\bbZ$.
The cycles excited by fractional spectral flow can also be expressed in terms of the action of $J_{-1/k}^+$, and we will find it convenient to do this. For ${|00 \rangle}$ strands this follows from equation , and similarly for ${|\pm\pm \rangle}$ strands using the strand lengths and amounts of spectral flow above. In addition, the supergravity solution is built on a “ground state” which is a supertube of radius $a$ and angular momentum $Q_1Q_2/4kR$, whose CFT dual consists of length $k$ cycles ${|++ \rangle}_k$ whose number is proportional to $a^2$.
A state which combines these supertube strands with the above longer cycles excited via fractional spectral flow has the form \[CFT dual state\] ([|++ ]{}\_k\^
)\^[n\_1]{} \_[,,]{} ( (J\_[-1/k]{}\^+)\^[k]{}[|00 ]{}\^
\_[k\^2]{})\^ ( (J\_[-1/k]{}\^+)\^[k]{}[|++ ]{}\^
\_[k(k+1)]{})\^ ( (J\_[-1/k]{}\^+)\^[k]{}[|– ]{}\^
\_[k(k-1)]{})\^ with appropriate conditions on the strand numbers so that the state carries the same quantum numbers as the supergravity solution, Eqs.–. Of course, one can write the above tensor product with a single index $\hp$, but we will temporarily carry along $\hm$ and $\hn$ to emphasize the differences between the strands. Ultimately, our proposed dual CFT states will be coherent states built from superpositions of the states , as discussed in [@Skenderis:2006ah; @*Kanitscheider:2006zf; @Giusto:2015dfa]. The first check that we have the right class of states is to show that the appropriate supergravity field modes are turned on under a small deformation away from the parent supertube solution (i.e. when the total number of copies taken up by the excited strands is small compared to the total number of copies taken up by the unexcited base supertube strands).
### Expectation values of supergravity mode operators {#expectation-values-of-supergravity-mode-operators .unnumbered}
One can regard the cyclic twist components ${|00 \rangle}_{k^2\hp}$ or ${|++ \rangle}_{k(\knpo)}$ or ${|-- \rangle}_{k(\kmmo)}$ as excitations above the supertube “ground state”. Supergravity field modes turned on by the coiffuring procedure are associated to operators in the CFT having expectation values in the coherent states built from the states . These operators will include those that have a matrix element that annihilates one of the long cycles, and converts it into multiple copies of ${|++ \rangle}_k$. Indeed, the order $k\hp$ anti-cyclic permutation \[annihilator\] (kk,(k-1)k,(k-2)k,…,k) acting on the cycle (1,2,3,4,…,k\^2) results in the tensor product of $k\hp$ cycles of length $k$ (1,…,k) (k+1,…,2k) (2k+1,…,3k) ((k-1)k+1,…,kk) , and similarly an anticyclic permutation of length $\knpo$ can convert a cycle of length $k(\knpo)$ into $\knpo$ cycles of length $k$, and an anticyclic permutation of length $\kmmo$ can convert a cycle of length $k(\kmmo)$ into $\kmmo$ cycles of length $k$.
The operators that mediate the appropriate matrix elements for the state must also soak up the currents that spectrally flow the state from the $1/4$-BPS ground state in this twist sector. Each $k^2\hp$-cycle in this state carries $(J_3,\bar J_3)$ charges $(k\hp,0)$, while the final state has $k\hp$ extra cycles ${|++ \rangle}_{k}$, with charges $(k\hp/2,k\hp/2)$, and so the operator that effects the transition must have charge $(-k\hp/2,k\hp/2)$. Similarly, each $k(\knpo)$-cycle has $(J_3,\bar J_3)=(k\hn+\half,\half)$, while the final state has charge $(\half(\knpo),\half(\knpo))$, and so the operator that mediates the transition must have charge $(-k\hn/2,k\hn/2)$; and each $k(\kmmo)$-cycle has $(J_3,\bar J_3)=(k\hm-\half,-\half)$, while the final state has charge $(\half(\kmmo),\half(\kmmo))$, and so the operator that mediates the transition must have charge $(-k\hm/2,k\hm/2)$.
The BPS twist operator whose conjugacy class contains the $k\hat{p}$-cycle and which also has these $SU(2)_\cR$ quantum numbers, and obeys selection rules of the auxiliary $SU(2)_\cA$, is the NS sector operator \[specflow vev 1\] \_[AB]{}(J\^-\_0)\^[k]{} \^[(k)AB]{}\_[k+1,k+1]{} . This component operator has $(J_3,\bar J_3)=(-k\hp/2,k\hp/2)$ and so carries the appropriate $\cR$-charges to implement the matrix element that sends $(n_1,n_2,n_3,n_4)$ to $(n_1\!+\!k\hp,n_2\!-\!1,n_3,n_4)$. Similarly, the NS sector operator \[specflow vev 2\] (J\^-\_0)\^[k]{} \^[()]{}\_[,]{} has $(J_3,\bar J_3)=(-k\hn/2,k\hn/2)$ and so carries the appropriate quantum numbers to mediate the transition $(n_1,n_2,n_3)\to (n_1\!+\!\knpo,n_2,n_3\!-\!1,n_4)$; and the operator \[specflow vev 3\] (J\^-\_0)\^[k]{} \^[()]{}\_[,]{} has the $SU(2)_\cR$ quantum numbers $(J_3,\bar J_3)=(-k\hm/2,k\hm/2)$ and mediates the transition $(n_1,n_2,n_3,n_4)\to (n_1\!+\!\kmmo,n_2,n_3,n_4\!-\!1)$.[^19]
Ward identities for the conformal group $SL(2,R)\times SL(2,R)$ guarantee a dependence $\exp[i\hp v/2R]$ for the matrix elements mediated by the operator . The $SU(2)\times SU(2)$ $\cR$-symmetry quantum numbers of this operator ensures that the matrix elements have the angular dependence \[ang dep\] \^[k]{}on $\bbS^3$. This angular dependence equates to the fact that the operator is a twisted chiral operator ($SU(2)_{\cR}$ lowest weight on the left and highest weight on the right). Similarly, the operators and are twisted chiral operators that mediate analogous matrix elements, whose coordinate dependences are the same apart from the substitution $\hp\to \hn$ and $\hp\to \hm$, respectively. The leading asymptotic power of $r$ in the matrix element is also dictated by the scale dimension of the operator, and a matching power of $a^{k\hp}$ in the quantity $\Delta^{k\hp}$ comes from the number of supertube $k$-cycles created when a cycle of length $k^2\hp$ is annihilated.
The general solution of type IIB supergravity compactified on $\bbT^4\times \bbS^1$ that preserves the same supercharges as the F1-NS5-P system and is invariant under rotations of $\bbT^4$ has the form
\[ansatzSummary\] $$\begin{aligned}
d s_{10}^2 &=-\frac{2Z_2}{\cP}\,\big(d v+\beta\big)\,\Big[d u+\omega + \frac{\mathcal{F}}{2}\big(d v+\beta\big)\Big]
+Z_2\,d s^2_4 + d \hat{s}^2_{4} ~, \label{10dmetric}\\
e^{2\Phi}&=\frac{Z_2^2}{\cP}\, ,\\
B_2 &= -\frac{Z_2}{\cP}\,(d u+\omega) \wedge(d v+\beta) ~,\\
B_6 &=\widehat{\mathrm{vol}}_{4} \wedge \left[ -\frac{Z_1}{\cP}\,(d u+\omega) \wedge(d v+\beta)\right] +\dots\notag\\
C_0&=\frac{Z_4}{Z_2}\, ,\\
C_2&= \frac{Z_4}{\cP}\,(d u+\omega) \wedge(d v+\beta)+ \dots ~, \\
C_4 &= \frac{Z_4}{Z_2}\, \widehat{\mathrm{vol}}_{4} +\dots ~, \\
$$
with $$\cP \equiv Z_1 \, Z_2 - Z_4^2 ~.
\label{Psimp-2}$$ Here $ds^2_{10}$ is the ten-dimensional string-frame metric, $ds_4^2$ is the metric on the space transverse to the branes, $\Phi$ is the dilaton, $B_p$ and $C_p$ are the NS-NS and RR gauge forms. (It is useful to also list $B_6$, the 6-form dual to $B_2$, to make explicit the appearance of $Z_1$ and $Z_2$ as the magnetic and electric components of the NS B-field.) The flat metric on $\bbT^4$ is denoted by $d \hat{s}^2_4$ and the corresponding volume form by $\widehat{\mathrm{vol}}_{4}$. For further discussion, see [@Skenderis:2006ah; @*Kanitscheider:2006zf; @Giusto:2013rxa; @Bena:2015bea; @Giusto:2015dfa]. In the supergravity solution , the harmonic function $Z_4$ appears in the RR scalars $C_0$ and $C_4$ as well as in the six-dimensional $C_2$ tensor field in the F1-NS5 frame, and carries the quantum numbers leading to the angular dependence . The operator corresponds (in the F1-NS5 duality frame) to the scalar $C_0 v_4 - C_4$ and the six-dimensional tensor $C_2^-$, according to , and its matrix elements carry the appropriate angular dependence. Thus for both coiffuring styles, we expect that there should be a vev of this operator proportional to $b_4$. When one builds coherent states out of the building blocks , one determines the average number of of strands $\bar n_2$ such that it reproduces this vev [@Skenderis:2006ah; @*Kanitscheider:2006zf; @Giusto:2015dfa].
The harmonic functions $Z_{1,2}$ appear in the electric and magnetic components of the six-dimensional NS B-field and the dilaton in this duality frame, with angular dependence of the form , where $\hp=n=p$ for Style 1 and $\hp=n=2p$ for Style 2. The operators , correspond to the supermultiplet containing the six-dimensional NS B-field and the dilaton. Their matrix elements also have angular dependence of the form , with the same replacements for the two coiffuring styles, and imply the corresponding vevs for the dual CFT coherent states.
Thus we have all the ingredients to reproduce the coiffured supergravity solutions of Section \[Sect:MomST\] from the CFT, and it is natural to anticipate that the average numbers of excited ${|00 \rangle}_{k^2\hat{p}}$ strands $\bar{n}_{2,\hat{p}}$ will be related to the coefficient $b_4$, and average number of excited ${|++ \rangle}_{k^2\hat{n}+k}$ and ${|-- \rangle}_{k^2\hat{m}-k}$ strands $\bar{n}_{3,\hat{n}}$, $\bar{n}_{4,\hat{m}}$ will be related to the coefficients $b_{1}$, $b_{2}$. This is indeed what we will find.
The coiffuring construction imposes relations on the mode amplitudes and frequencies in order that the supergravity solution is regular at $r=0$. These restrictions are not apparent in the CFT states in the linearized analysis, which, [*a priori*]{}, allows independent values for the cycle length quantum numbers $(\hp,\hn,\hm)$ and the corresponding amplitudes $(n_{2,\hat{p}},n_{3,\hat{n}},n_{4,\hat{m}})$.
For coiffuring Style 1 in supergravity, one has $n=p$ and the amplitude relations -; for Style 2, one has $n=2p$ together with the amplitude restrictions $b_2=0$ and . For Style 2, at leading order there is no amplitude for $b_1$ (since $b_1\sim b_4^2$) and $b_2=0$, hence there is no linearized vev for the NS B-field; this suggests that Style 2 corresponds to a state with $n_{3,\hat{n}}=n_{4,\hat{m}}=0$ in the CFT. The vev of the B-field at second order in $b_4$ could be accounted for by the non-linearities in the CFT-supergravity mode map, and indeed we will show this to be true in the next subsection.
For Style 1, an amplitude at leading order in $b_4$ is present for all of the $m=n=p$ modes of the NS B-field, but in the CFT it seems that the amplitude of the corresponding vevs can be independently varied – at this point there appears to be no restriction on the relative numbers $(n_{2,\hp},n_{3,\hn},n_{4,\hm})$ of the different kinds of strands at leading order. Again, to understand these restrictions, it is necessary to understand the relation between supergravity modes and CFT fields at the non-linear level.
Instead of trying to carry through the somewhat daunting task of determining the non-linear corrections to the supergravity-CFT map, we will instead proceed somewhat differently, and determine what strands are present in the CFT (and in what amounts) by an analysis of the two-charge solutions on which the coiffured solutions are based.
### Information from two-charge solutions {#information-from-two-charge-solutions .unnumbered}
Our proposed dual CFT states involve fractional spectral flow on a two-charge state, and spectral flow does not change the strand content of a BPS state. Therefore, we can determine the amounts of the various types of strands present (at the fully non-linear level) in both styles of coiffuring, by studying the known map between CFT and supergravity for the two-charge system [@Lunin:2001jy; @Lunin:2002bj; @Skenderis:2006ah; @*Kanitscheider:2006zf; @Kanitscheider:2007wq].
The harmonic functions determining the geometry of a circular F1-NS5 supertube in the decoupling limit are
\[generaltwocharge\] $$\begin{aligned}
\label{Z2profile}
& Z_2 = \frac{Q_2}{L} \int_0^{L} \frac{1}{|x_i -g_i(\v)|^2}\, d\v~, \qquad
Z_4 = - \frac{Q_2}{L} \int_0^{L} \frac{\dot{g}_5(\v)}{|x_i -g_i(\v)|^2} \, d\v \,,\\
\label{Z1profile}
& Z_1 = \frac{Q_2}{L} \int_0^{L} \frac{|\dot{g}_i(\v)|^2+|\dot{g}_5(\v)|^2}{|x_i -g_i(\v)|^2} \, d\v ~,
\\
& A = - \frac{Q_2}{L} \int_0^{L} \frac{\dot{g}_j(\v)\,dx^j}{|x_i -g_i(\v)|^2} \, d\v ~, \quad~~
dB = - *_4 dA~, \quad~~ ds^2_4 = dx^i dx^i~, \\
& \beta = \frac{-A+B}{\sqrt{2}}~, \qquad \omega = \frac{-A-B}{\sqrt{2}}~, \qquad {\mathcal{F}}=0~,\end{aligned}$$
where the dot on the profile functions indicates a derivative with respect to $\v$ and $*_4$ is the dual with respect to the flat transverse $\mathbb{R}^4$ parametrized by $x_i$.
The onebrane charge is given by $$\label{Q1int}
Q_1={Q_2\over L}\int_0^L \bigl(|\dot{g}_i(\v)|^2+|\dot{g}_5(\v)|^2\bigr)d\v ~.$$ The quantities $Q_1$, $Q_2$ are related to quantized onebrane and fivebrane numbers $n_1$, $n_5$ by $$Q_1 = \frac{(2\pi)^4\,n_1\,g_s^2\,\alpha'^3}{V_4}\,,\qquad Q_2 = n_5\,\alpha' ~,
\label{Q1Q5_n1n5}$$ where $V_4$ is the coordinate volume of $\bbT^4$.
The circular supertube profile is is given by \[circular\] g\_1+ig\_2 = a . It will prove convenient to denote $x=x_1+ix_2$, $y=x_3+ix_4$, and parametrize the profile by $\xi\equiv2\pi k\v/L$. Since the supertubes of interest run around the same profile $k$ times, the integral is simply $k$ times the integral over the range $\xi\in(0,2\pi)$. The further change of variables $z=e^{i\xi}$, and the use of $\bar z =1/z$ for an integral along the unit circle in $z$, converts the integrals into contour integrals for which we can use the method of residues, for example \[Hint\] Z\_2= . The poles in the integrand are located at z\_= , where $\tilde{w}=x\bar x+y\bar y+a^2$, and so Z\_2= . Converting from Cartesian coordinates to spherical bipolar ones $$\begin{aligned}
x = \tilde r \sin\tilde\theta e^{i\varphi_1}
~~&,\quad
y = \tilde r \cos\tilde\theta e^{i\varphi_2} \nn\\
\tilde r =\sqrt{r^2 +a^2\sin^2\theta}
~~&,\quad
\cos\tilde\theta = \frac{r\cos\theta}{\sqrt{r^2+a^2\sin^2\theta}}\end{aligned}$$ leads to the correct form of $Z_2$ in the decoupling limit, \[Hanswer\] Z\_2 = = . Next, we introduce $\nu = kp$ for convenience and we add a $g_5$ term to the profile function, g\_5() = - ( v) = (z\^-z\^[-]{}) , \[g5\] where $b_4$ is real, and corresponds to the magnitude of the quantity $b_4$ in the supergravity . The quantity that corresponds to the phase of the supergravity $b_4$ is a shift in $\v$ in . In what follows, for both Style 1 and Style 2, we will take $b_4$ to be real, both for convenience and for ease of comparison to [@Bena:2015bea]. This $g_5$ term in the profile function gives rise to the following contour integral expression for the harmonic function $Z_4$: \[Aint\] Z\_4 = b\_4 . The $z^\nu$ term yields the result b\_4 . The denominator gives again the factor of $\Sigma$; furthermore, one has = , z\_+ z\_- = e\^[2i\_1]{} . One can then rewrite $z_-^\nu$ as \[2-charge Delta\] z\_-\^ = ( )\^[/2]{} (z\_+ z\_-)\^[/2]{} = ()\^[/2]{} e\^[i\_1]{} . The harmonic function depends on the combination $\,\sin\theta \,e^{i\varphi_1}\,$ rather than $\,\cos\theta\, e^{i\varphi_2}\,$ because the linearized supergravity modes getting a vev correspond to chiral rather than twisted-chiral operators in the CFT. The fractional spectral flow operation converts one to the other.
One is looking to match the structure in [@Bena:2015bea] equation (3.11c) which is ()\^[/2]{} = ; this is exactly what is found once the contribution from the $z^{-\nu}$ term in is added. So the seed $Z_4$ is Z\_4 = 2 b\_4 ()\^[/2]{} .
### Style 2 {#style-2 .unnumbered}
For coiffuring Style 2, the harmonic function $Z_1$ exhibited in [@Bena:2015bea] equation (3.11a) for the corresponding two-charge seed solution, translated into our conventions is \[Z1 style2\] Z\_1 = + ()\^ which follows from equation (\[generaltwocharge\]b). Thus we see that Style 2 coiffuring is the result solely of exciting ${|00 \rangle}$ strands of length $\kappa=k\nu=k^2p$; the strength $b_1\propto b_4^2$ of the vev is entirely accounted for by non-linear effects of the ${|00 \rangle}$ strands, and so no additional contribution corresponding to nonzero $n_{3,\hat{n}},n_{4,\hat{m}}$ is necessary. The corresponding two-charge solution is precisely as in [@Bena:2015bea], and the spectral flow that adds the third charge simply turns vevs from chiral to twisted-chiral – under fractional spectral flow, the factor $\,\sin^\nu\theta\,e^{i\nu\varphi_1}\,$ turns into $\,\cos^\nu\theta\,e^{i\nu\varphi_2}\,$, which is what we see in the coiffured harmonic functions of Section \[Sect:MomST\] above.
### Style 1 {#sec:sty1cft .unnumbered}
It remains to match Style 1 to a set of supertube strands in a two-charge solution prior to spectral flow and coiffuring. We expect to at least have strands of length $k(kp+c)$ for $c=\{-1,0,+1\}$, since vevs for the operators - must appear at linear order in $b_4$. The $c=\pm1$ strands correspond to ${|\pm\pm \rangle}$ cycles and so in general will affect the location of the supertube profile in the transverse $\bbR^4$. Introducing ${|++ \rangle}_{k(kp+1)}$ and ${|-- \rangle}_{k(kp-1)}$ strands, the deformation profile becomes \[style 1 profile\] g\_1+ ig\_2 = a z + b\_- z\^[-(kp-1)]{} + b\_+ z\^[kp+1]{} = z ( a+b\_-z\^[-]{} +b\_+z\^) , where $\nu=kp$ and $z=\exp(2\pi i k\v/L)$.
For small amplitude deformation $b_\pm\ll a$, there are no new poles inside the contour of integration, and the pole in the integrand will still be close to $z_{-}$. We can map the profile back to a unit-velocity circular profile (i.e. $g_{1}+ig_2 = a e^{i\u}$) via a single-valued conformal map, at the cost of a Jacobian for the transformation. In general this leads to an infinite series in the expressions for $Z_{1}$ and $Z_2$ if there are only one or two lengths of strand in the profile $g_{1}+ig_2$; since we wish to engineer a finite Fourier series for $Z_{1}$ and $Z_2$, the dual CFT state will have a series of strand lengths involving all possible multiples of $p$.
Working firstly to leading order in $b_\pm$, consider the profile (g\_1+ig\_2)() = a ,() = , () = - + …where we set $b_+=b_-\equiv - i ab/2\nu$ in order that the map is a proper element of ${\it Diff}(\bbS^1)$. Here again $\xi$ serves as a rescaled periodic coordinate which ranges over $[0,2\pi k)$. The motivation for considering such a profile comes from coiffuring – the idea is that coordinate transformations on the supertube worldvolume apply a density perturbation to the round supertube without perturbing its location in space. The fivebrane and onebrane charge densities will no longer be constant along the supertube. Expanding this profile out to leading order in $b$ reproduces . Such a change of variable has no effect on $Z_4$ (which is reparametrization invariant) but it will change $Z_{1}$ and $Z_2$. The integration measure picks up a factor dvdu()\^[-1]{} , = where we have expanded the Jacobian factor to clarify that $\u$ means $\u\big(\xi(\v)\big)$ as above. Similarly the “energy densities” in the numerator of $Z_1$ in pick up a factor | (\_1+i\_2)()|\^2 + |g\_5()|\^2 ()\^[2]{}( | (g’\_[1]{}+ig’\_[2]{})()|\^2 + | g’\_5()|\^2 ) where primes denote derivatives with respect to the argument. Again evaluating the factors of $(d\u/d\v)$ to leading order in $b$ one finds that in the new integration variable $\u$, the integrand of $Z_2$ is modified by a factor of $1-b\sin \nu \u$ in , while the integrand of $Z_1$ gets a factor of $1+b\sin\nu \u$ (in addition to the corresponding factors of $2\pi k/L$). We then find the same sort of integral we encountered in , with the same result. Our primitive approximations give $b_1=-b_2$, which is appropriate for $a^2\ll Q_1,Q_2$; the latter is a consequence of the decoupling limit.
In principle one can proceed order-by-order in a series expansion in $b_I$, $(I=1,2,4)$, working out the non-linear map between the $\v$ coordinate frame in which the strand content is specified, and the $\u$ coordinate frame in which the supertube profile is a constant velocity parametrization of a circle. However, ultimately we are interested in the harmonic functions $Z_{I}$ having a single non-trivial Fourier coefficient. In Style 1, the perturbation to $Z_2$ looks like with $\nu=kp$ and a coefficient $b_2$; and $Z_4$ is the same but with a coefficient $b_4$. These simple forms suggest that the more straightforward route is to work directly in the $\u$ coordinate frame and only implicitly specify the coordinate map via its inverse, \[implicit map\] () = , () = + u. Plugging this into gives exactly the right result for $Z_2$ and $Z_4$ in Style 1, using (g\_1+ig\_2)() = a , g\_5() = - . We have now used up almost all our freedom to specify the state; all that remains are the amplitudes $b$, $b_4$. The integral for $Z_1$ is $$\begin{aligned}
\label{Kint}
Z_1 &= \frac{k^2\R^2}{2\pi Q_2} \int_0^{2\pi} \!d\u \Big(\frac{d\u}{d\xi}\Big)^{-1}\,
\frac{\big(| (g'_{1}+ig'_{2})(\u)|^2 + | g'_5(\u)|^2\big)(d\u/d\xi)^2}{|x-(g'_{1}+ig'_{2})(\u)|^2+|y|^2}
\nn\\
&= \frac{k^2\R^2}{2\pi Q_2}\int_{0}^{2\pi}\!\!d\u\;
\frac{a^2+(2b_4/k\R)^2\cos^2\nu \u}{1-b \sin\nu \u}\,\frac{1}{|x-a e^{i\u}|^2+|y|^2}\end{aligned}$$ where we have used the relation L= . Let us choose b\^2 = = , where the second equality is the analog of the gravity regularity condition . Then the factor in the integrand becomes = (a\^2+(2b\_4/k)\^2)(1+bu) = (1+bu) , where the last equality comes from evaluating the expression . All harmonic functions have only terms that are constant or a single $\bbS^3$ harmonic of the form , with $m=n=p$ and $b_1=-b_2= i b_4/\sqrt{Q_1Q_2}$. These results agree precisely with the decoupling limit $a^2\ll Q_1,Q_2$ of the amplitude relations of Style 1 coiffuring in supergravity.
For these two-charge solutions, the mode amplitude restrictions do not come from requiring regularity of the supergravity solution – all the two-charge solutions are non-singular. Rather, the restriction comes from the somewhat arbitrary requirement that the harmonic functions contain only a single Fourier mode rather than a combination of modes of different wavenumbers.
It is worth reiterating that the map between supergravity and CFT takes place in the $\v$ coordinate frame, which is only implicitly specified above through the relation . In the $\v$ coordinates the solution is very complicated and has in principle all values of $\hm,\hn,\hp$ turned on. The non-zero values of $\hm$, $\hn$ are given by the non-zero Fourier coefficients of $(g_1+ig_2)(\v) = a \exp[ i \u(\xi(\v))]$, \[Fcoeff\] c\_n = \_[0]{}\^[2]{} e\^[-in ]{} e\^[i()]{} = \_[0]{}\^[2]{} e\^[-in()+iu]{} and are predominantly concentrated on the lowest modes. Expanding in $b$, c\_n = \_0\^[2]{} (1+b())e\^[i(1-n)w]{} \_[=0]{}\^1[!]{}()\^ \[eq:Fourier-sty1\] one sees that the only nonzero Fourier coefficients occur for $n=1+q\nu$, $q\in\bbZ$, generalizing . Of these non-zero Fourier coefficients, the positive values of $n$ give the non-zero values of $\hm$, and the negative values of $n$ give the non-zero values of $\hn$. Thus we see that $\hm$ and $\hn$ must be multiples of $p$, the mode number of the supergravity solution. Similarly, the non-zero values of $\hp$ are all multiples of $p$.
In addition, reality of the conformal map implies $c_{1+q\nu}=c_{1-q\nu}^*$, which in turn means that for each $q$, the average numbers of ${|++ \rangle}_{k(qkp+1)}$ and ${|-- \rangle}_{k(qkp-1)}$ strands are equal. Finally, note that in the quantum theory, there is a maximum mode number $N=N_1N_5$ and so one cannot precisely generate Style 1 because the Fourier expansion is necessarily finite; the result will differ at the $1/N$ level.
For $kp=1$, this family of states has a somewhat degenerate limit, since the length of the ${|-- \rangle}_{k(kp-1)}$ strands is zero. This simply means that this particular strand type is absent for $kp=1$, while the other strands remain as described above. In particular, the average numbers of ${|++ \rangle}_{k(qkp+1)}$ and ${|-- \rangle}_{k(qkp-1)}$ strands are equal for $q\ge2$.
### Summary of proposed dual CFT states {#summary-of-proposed-dual-cft-states .unnumbered}
In both Style 1 and Style 2, we start with a two-charge seed solution, determined by a profile function. The general dictionary for two-charge states is discussed in [@Skenderis:2006ah; @*Kanitscheider:2006zf; @Kanitscheider:2007wq; @Giusto:2015dfa]. We now describe how it applies to our two-charge seed states. Given a profile function, the non-zero Fourier coefficients specify the types of strands involved in the dual CFT state, and the values of the Fourier coefficients control the coefficients of the individual terms in the coherent state superposition.
[**Style 2**]{}
As shown in the previous subsection, the Style 2 seed solution is determined by the profile function , : (g\_1+ig\_2)() = a (i v) , g\_5() = - ( v). \[eq:sty2profile\] Since both $g_1+ig_2$ and $g_5$ have only a single Fourier mode, the dual CFT state contains just two types of strands, \[CFT dual state-sty2seed\] [|++ ]{}\_k\^
, [|00 ]{}\^
\_[k\^2 p]{} , where the excited strands are only of one type, given by $\hat{p}=p$.
To form the coherent state, one considers all partitions of the $N_1 N_5$ copies of the CFT into strands of the two above types. Then one forms a sum in which the coefficients of these different partitions are controlled in a specific way by the two non-zero Fourier coefficients of the profile function (for more details, see in particular the discussion in [@Giusto:2015dfa]).
Given this seed two-charge state, we excite all strands except for the ${|++ \rangle}_k$ strands in the way described in Section \[ss:CFT dual states\], so that the resulting three-charge state is composed only of strands of type \[CFT dual state-sty2\] [|++ ]{}\_k\^
, (J\_[-1/k]{}\^+)\^[k p]{}[|00 ]{}\^
\_[k\^2 p]{} . The coefficients in the coherent state sum remain as in the two-charge seed solution.
[**Style 1**]{}
For Style 1, the seed solution is given by the profile function (g\_1+ig\_2)() = a , g\_5() = - \[eq:sty12chseed\] where from we specify the map implicitly through its inverse, \[implicit map-1\] () = , () = + u. Since both $g_1+ig_2$ and $g_5$ have an infinite Fourier series, the types of CFT strands present are those of type [|++ ]{}\_k\^
, [|00 ]{}\^
\_[k\^2]{}, [|++ ]{}\^
\_[k(k+1)]{}, [|– ]{}\^
\_[k(k-1)]{} , where $\hat{m}$, $\hat{n}$, $\hat{p}$ can be any independent multiples of $p$, compatible with the total number of strands being $N_1 N_5 $. The coherent state has many more ingredients, however the coefficients in the superposition are again fully specified by the Fourier coefficients of the profile function . Therefore all the coefficients are determined by the parameter $b_4$ (since $b_4$ fixes $b$ and $a$).
Given this seed two-charge state, we again excite all strands except for the ${|++ \rangle}_k$ strands in the way described in Section \[ss:CFT dual states\], so that the resulting three-charge state is composed only of strands of type \[CFT dual state-gen\] [|++ ]{}\_k\^
, (J\_[-1/k]{}\^+)\^[k]{}[|00 ]{}\^
\_[k\^2]{} , (J\_[-1/k]{}\^+)\^[k]{}[|++ ]{}\^
\_[k(k+1)]{} , (J\_[-1/k]{}\^+)\^[k]{}[|– ]{}\^
\_[k(k-1)]{} , where again the values of $\hat{m}$, $\hat{n}$, $\hat{p}$ are independent multiples of $p$, compatible with the total number of strands being $N_1 N_5 $, and the coefficients in the coherent state sum remain as in the two-charge seed solution.
Finally, note that, at the level of counting free parameters in the solutions, we expect there to be good agreement more generally between coiffured deformations of circular supertubes on the supergravity side, and fractional spectral flows of circular two-charge seed solutions on the CFT side. On the CFT side, one has two functional degrees of freedom – the specification of the profile of the ${|00 \rangle}$ strands embodied in the function $g_5$, and the diffeomorphism $\u(\xi)$ that changes the parametrization of the round supertube. On the supergravity side, the diffeomorphism $\u(\xi)$ corresponds to the charge densities $\rhoone$ and $\rhofour$ discussed in Section \[ss:SIexample\], and the profile of the ${|00 \rangle}$ strands corresponds to the function $Z_4$. In Section \[ss:SIexample\] we saw that in the absence of $Z_4$ there are three functions and two functional constraints, leaving one functional degree of freedom; adding in $Z_4$ gives two functional degrees of freedom, which agrees with the CFT.
There are interesting parallels between the supergravity construction of Section \[Sect:SpecInter\] and the appearance of density fluctuations in the CFT. However the relationship is not direct. In the CFT, the density profile appears in the two-charge seed solutions before applying fractional spectral flow; on the gravity side, the density perturbations were introduced in a spectrally inverted frame, and then a second spectral inversion was applied to transform back to the original frame. The density fluctuations were thus applied to a supertube that does not have a simple, direct relationship to the original D1-D5 CFT. There is also the technical distinction in that the construction of Section \[Sect:SpecInter\] initially involves three apparently independent charge density functions that must then satisfy the constraints of supertube regularity (\[regcondb\]) and (\[regcondc\]), leaving only one independent density function. In this section, the density fluctuation is introduced via a combination of the $g_5$ profile and (in Style 1) a conformal map of the round supertube profile, which, via the Lunin-Mathur map, automatically maintains the supertube regularity conditions. It would be very interesting to investigate this relationship in more detail.
Comparison of conserved charges
-------------------------------
We now compare the angular momenta $J^3$, $\bar{J}^3$ and the momentum charge $Q_P$, and demonstrate the agreement between our supergravity solutions and our proposed dual CFT states. For ease of comparison to the supergravity discussion in Section \[ss:STReg\], we revert to the D1-D5 duality frame.
The discussion that follows requires a certain amount of notation to write the charges explicitly, however the reasons that underlie the agreement can be stated simply.
Firstly, all our momentum excitations can be expressed in terms of the action of powers of $J^+_{-\frac{1}{k}}$. Secondly, for each $\hat{p}$ the average numbers of the strands of length $k^2\hat{p}+k$ and $k^2\hat{p}-k$ are equal, because of their origin as the (real-valued) two-charge density profile. Therefore adding momentum $\hat{p}$ requires, on average, $k^2\hat{p}$ strands of the CFT. This fact leads to the relation between the angular momenta $J^3$, $\bar{J}^3$ and the momentum charge $Q_P$ observed in the supergravity, as we now show explicitly.
### Style 2 {#style-2-1 .unnumbered}
For Style 2, we have a coherent state which is a sum of terms of the form \[CFT dual state-style2\] ([|++ ]{}\_k\^
)\^[n\_1]{} ( (J\_[-1/k]{}\^+)\^[kp]{}[|00 ]{}\^
\_[k\^2p]{})\^[n\_2]{} , where the sum runs over all $n_2$ such that k n\_1 + (k\^2 p) n\_2 &=& N\_1 N\_5, weighted with coefficients as described in the previous subsection.
For Style 2 coiffuring, from and we have on the gravity side Q\_P = , J\_R = - k R\_y Q\_P , J\_L = J\_R + k R\_y Q\_P . In the CFT, the expectation value of the momentum $L_0 - \bar{L}_0$ in the Style 2 state is N\_p &=& p |[n]{}\_2. The total number of strands is $N_1 N_5$; this determines $\bar{n}_1$ in terms of $\bar{n}_2$ (or $N_p$) as |[n]{}\_1 &=& - k N\_p . Then the CFT ${\bar \jmath}^3$ is \^3 = = - N\_p . We convert the supergravity charges to quantized charges using Q\_1 = , Q\_5 = g\_s N\_5 ’ , Q\_P = , = which lead to the useful relations = N\_1 N\_5 , R\_y Q\_P = N\_P . Thus we obtain \^3\_[grav]{} = J\_R = - N\_p which agrees with the CFT. Next, the CFT $j^3$ is j\^3 = + (kp) |[n]{}\_2 = [|]{}\^3 + k N\_p . Comparing to the gravity solution we have \^3\_[grav]{} = J\_L = [|]{}\^3\_[grav]{} + k N\_p which is also in agreement. Then by comparing the momentum charge we obtain the map between $|b_4|^2$ and $\bar{n}_2$: N\_P &=& R\_y Q\_P p |[n]{}\_2 = ( ) .
### Style 1 {#sec:sty1charges .unnumbered}
For Style 1, we first consider $kp > 1$. As described above, the ingredients in the coherent state sum are \[CFT dual state-sty1\] ([|++ ]{}\_k\^
)\^[n\_1]{} \_[p]{} ( (J\_[-1/k]{}\^+)\^[k]{}[|00 ]{}\^
\_[k\^2]{})\^ ( (J\_[-1/k]{}\^+)\^[k]{}[|++ ]{}\^
\_[k(k+1)]{})\^ ( (J\_[-1/k]{}\^+)\^[k]{}[|– ]{}\^
\_[k(k-1)]{})\^ For Style 1 coiffuring, from , and we have on the gravity side Q\_P = , J\_R = - k R\_y Q\_P , J\_L = J\_R + k R\_y Q\_P . In the CFT, each individual element in the coherent state sum has momentum eigenvalue \_[p]{} ( + + ) and so the expectation value of $L_0-\bar{L}_0$ again involves the average numbers of strands, N\_p &=& \_[p]{} ( + + ). Since the total number of strands is $N_1 N_5$, we have k |[n]{}\_1 + \_[p]{} &=& N\_1 N\_5 . Because the density profile function $w(\xi)$ is real, we have the relation on the average numbers &=& . Therefore we have k |[n]{}\_1 + \_[p]{} (k\^2 ) (++) &=& N\_1 N\_5 and so, as for the Style 2 states, $\bar{n}_1$ is given by |[n]{}\_1 &=& - k N\_p . Next, the CFT ${\bar \jmath}^3$ is \^3 = 12 = = - N\_p . The gravity ${\bar \jmath}^3$ is \^3\_[grav]{} = - N\_p , \[eq:sty1jbarcft\] so we find perfect agreement.
The CFT $j^3$ is \^3 = 12 + \_[p]{}(k) (++) = [|]{}\^3 + k N\_p . Comparing to the gravity we have \^3\_[grav]{} = J\_L = [|]{}\^3\_[grav]{} + k N\_p \[eq:sty1j3cft\] and so we again find perfect agreement.
Finally, by comparing the momentum charge we obtain the map between $|b_4|^2$ and the average numbers of excited CFT strands, N\_P &=& R\_y Q\_P \_[p]{} ( + + ) = ( ) .
For $kp=1$, the analysis contains minor differences, however the expressions for $j^3$ and $\bar{\jmath}^3$ in terms of $N_1$, $N_5$, $N_P$ are the same as those given in and , as we show in Appendix \[sec:sty1chargeskp1\]. Thus the conserved charges agree for all values of $k$ and $p$.
Therefore we find exact agreement of conserved charges between gravity and CFT, providing supporting evidence for our proposal. It would be interesting to scrutinize our proposal further with the tools of precision holography [@Skenderis:2006ah; @*Kanitscheider:2006zf; @Kanitscheider:2007wq; @Taylor:2007hs; @Giusto:2015dfa].
Comments on momentum fractionation {#ss:fractionation}
----------------------------------
The fractional spectral flow that we perform results in filled Fermi seas on the excited strands. One way to see this is to observe that the $SU(2)_\cR$ current algebra has the identity \[specflow strand\] (J\_[-1/k]{}\^+)\^[k]{}[|00 ]{}\^
\_[k\^2]{} &=& J\_[-]{}\^+ J\_[-]{}\^+ J\_[-]{}\^+ J\_[-]{}\^+ [|00 ]{}\^
\_[k\^2]{} . Similar expressions apply for the ${|++ \rangle}$ and ${|-- \rangle}$ strands.
So the CFT state can be written in different ways, and in one way of looking at our states, we excite modes with the lowest possible energy compatible with the constraint of integer momentum per strand. Saying this another way, spectral flow creates a state with the lowest possible energy for a given angular momentum, or equivalently maximal angular momentum for a given energy, so that there is no available free energy for thermal excitations of the state.
As one backs away from maximal angular momentum, one has the freedom to excite different modes, and the entropy increases. For instance, if we change one of the current raising operators on the right-hand side of from a $J^+$ to a $J^3$, the angular momentum is decreased by one unit but the energy and momentum remain the same; and there are $kp$ distinct ways to do this. Decrease the angular momentum by one more unit, and we can either have one $J^-$ or two $J_3$ with the rest remaining $J^+$, and there are of order $(kp)^2$ choices; and so on.
Such a deformation away from maximal spin preserves the BPS property of the CFT state. It is interesting to ask what the gravitational description of such excitations will be, and whether they will match those of the CFT. If we change the lowest modes with energy/momentum of order $1/k^2p$, we would expect to have made a change in the geometry in the places with the deepest red-shift. Note that such BPS deformations are not available in the two-charge seed on which the three-charge coiffured solution is based.
Since the CFT state has strands of length of order $k^2p$, there are also non-BPS excitations that have zero momentum and angular momentum, and energy of order $1/k^2p$. Such excitations are also present in the two-charge seed states. In the supergravity, the non-BPS excitations are described at the linearized level by solving wave equations in the superstratum geometry.
The supergravity solutions do not appear to have excitations at the scale $1/k^2p$ suggested by the CFT, however; in general, there seems to be a mismatch between the gap in supergravity and in the CFT. The two-charge seed for Style 2 coiffuring is quite similar to a class of two-charge solutions studied in [@Lunin:2002iz], for which the gap was estimated to be $a/b$ (with $a$ related to the number of ${|++ \rangle}$ strands, $b$ the number of $\bbT^4$ strands including ${|00 \rangle}$ strands). In the CFT, the gap depends only on the length $\kappa$ of the strands and is independent of the relative amounts $a$ and $b$ of the different kinds of strands. A preliminary study of the foregoing three-charge geometries indicates that, similar to the examples of [@Lunin:2002iz], the red-shift depends on the amplitudes $a$ and $b$, and that the deepest red-shifts are not $kp$ times deeper than those of the parent $k$-wound supertube.
In general, one can arrange that the throat in supergravity is deeper and results in a smaller gap than in the orbifold CFT ([*e.g.*]{} supergravity duals to CFT states discussed in [@Lunin:2002iz] having only short cycles but low total angular momentum), and in yet other examples the throat in supergravity is shallower and results in a larger gap than in the orbifold CFT ([*e.g.*]{} the coiffured geometries discussed in this paper when $b$ is finite but much less than $a$). It would be useful to understand better the cause of this discrepancy.
The two-charge seed geometries of Section \[ss:CFT dual states\] offer a qualitative explanation of the gap in supergravity. The dual of the F1-P source in the Lunin-Mathur construction of two-charge geometries [@Lunin:2001fv] is a D1-D5 supertube smeared over the compact directions – the circle parametrized by $y$ and the compactification manifold $\cM$ [@Lunin:2001jy; @Lunin:2002bj; @Lunin:2002iz; @Kanitscheider:2007wq]. When segments of the unsmeared source approach one another, a throat opens and deepens in the geometry. This property explains why the profile results in a red-shift of order $k$ – the supertube source traces the same profile in the transverse space $k$ times in the course of the supertube winding the $y$ circle, and is $k$ times more compact (in ${\mathbb R}^4$ coordinates); as a consequence, the harmonic functions are $k$ times bigger at their maximum, and the throat is $k$ times deeper.
For a small perturbation of this profile, it may be that the oscillations of the profile are $kp$ times faster than the $k$-fold spiral of the supertube, but this is a small perturbative wiggle and does not make the profile $kp$ times more bunched together, and hence the deepest parts of the throat do not exhibit a red-shift $kp$ times deeper. However, as one shifts more of the strands from ${|++ \rangle}$ type to ${|00 \rangle}$ type, the angular momentum is reduced, the source becomes more compact, and the throat deepens.
It remains a puzzle why there is such a mismatch between the behavior of supergravity and that of the CFT for such a coarse property of the geometry. The gap to non-BPS excitations is of course not a robust property of the system, and could change dramatically as one passes from the regime where the CFT is weakly coupled to the regime where it is strongly coupled and gravity is a good approximation. Nevertheless, there are examples (see for instance [@Giusto:2012yz]) where the gap can be matched on both sides of the duality. The presence or absence of strands polarized in the $\bbT^4$ directions appears to be an ingredient which influences whether this quantity agrees between gravity and CFT; it would be useful to understand fully when this comparison does and does not work.
Discussion {#Sect:Discussion}
==========
This work has expanded the construction of superstrata to include momentum-carrying modes in deep AdS$_3$ throats, in which the red-shift at the bottom of the throat is $k$ times that of a singly-wound supertube. Our construction started from a $k$-wound circular supertube geometry. We performed spectral inversion on this solution, then altered its angular momentum by adding charge density fluctuations along the supertube with a wavenumber $kp$ for some integer $p$, without deforming the shape of the supertube. We then brought the solution back to the original frame, where these fluctuations became momentum-carrying excitations.
Our construction also produced the first examples of asymptotically-flat superstrata. We built two classes of solutions, corresponding to two different ways of arranging the Fourier coefficients in order to obtain smooth solutions (with the usual $Z_k$ orbifold singularities at the location of the supertube).
Taking the decoupling limit to obtain the corresponding asymptotically-AdS solutions, we derived a proposal for the dual CFT states, for both classes of solutions. The starting supertube is built from a macroscopic ensemble of cycles of length $k$ in the twisted sector of the symmetric orbifold CFT. The angular excitations in the CFT description are coherent fractional spectral flows on additional cycles of the twisted sector state, whose length is of order $k^2p$. This fractional spectral flow can also be thought of either as acting of order $kp$ times with the fractionally-moded raising operator $J^+_{-1/k}$, or as raising the Fermi seas on these cycles by filling all the fermion modes with positive $\cR$-charge up to a level of order $1/k$.
In our states, the fractionally-moded quanta in the CFT correspond to perfectly regular, local excitations in the supergravity theory and not to non-geometric or multi-valued perturbations. The bulk reflection of the fractional momentum carriers is rather the red-shift of the perturbations down the supertube throat.
A small puzzle that remains is the apparent mismatch in the excitation gap of orbifold CFT states and supergravity geometries discussed in Section \[ss:fractionation\]. A very similar mismatch was previously noted for certain two-charge solutions [@Lunin:2002iz]. In the CFT, the gap is determined by the length of the longest cycles in the twisted sector ground state. In the geometry, the depth of the throat depends on other quantities, such as the relative proportions of the different strands. The supergravity gap can be larger or smaller than the orbifold CFT gap. The gap to non-BPS excitations is not protected in general, so this is not a serious problem for the holographic duality. However there are examples (see for instance [@Giusto:2012yz]) where the gap matches between gravity and CFT. It would be interesting to understand when the gap should agree, and when it should not.
Our solutions do not have all desired features of typical black-hole microstates: Their angular momenta are over-spinning and the throats are not as deep as those of typical states. The corresponding orbifold CFT states contain strands having length of order $k^2 p$, and so $k$ can at most be of order $\sqrt{N_1N_5}$, while the longer wavelength scale $k^2p$ is not apparent in the geometry. Thus we regard the supergravity solutions presented here as a “proof of concept” of a supergravity realization of momentum fractionation on superstrata, much like the solutions in [@Bena:2015bea] are a proof of concept of the existence of superstrata solutions parameterized by arbitrary functions of two variables.
For the future, one would like to improve on both of these (related) features: To lower the angular momenta, and to deepen the throat further. First, regarding the angular momenta, in Section \[ss:fractionation\] we identified CFT excitations that move away from the maximally spinning/overspinning regime by reducing the angular momentum through a change in the polarization of the $\cR$-symmetry currents acting on the two-charge seed. Using this freedom, one can make available some of the free energy to wiggle the throat while remaining BPS. Where in the throat the excitation lies should correlate with the degree of fractionation of the modes whose polarizations are being adjusted in the CFT.
One place to look for these more general solutions on the supergravity side is to consider more generic superstrata, described by arbitrary functions of two variables. In this work we have focused on a sub-class of solutions which are parameterized by functions of one variable. This has been a choice made for technical convenience, to focus on the physics of momentum fractionation in a tractable system. It would be interesting to generalize our solutions to superstrata which are parametrized by functions of two variables and which exhibit momentum fractionation. Looking further ahead, the generic CFT state deformations discussed above, which stay BPS by deforming the polarizations of the spectral flow $\cR$-currents, will correspond to deformations of the supergravity solution that depend on all all three angular variables $(v,\varphi_1,\varphi_2)$.
The next essential step in the study of superstrata is to construct states with deeper throats, that are in a macroscopic scaling regime. Our solutions have throats $k$ times deeper than the first superstrata constructed in [@Bena:2015bea], and so represent progress in this direction. The standard way to obtain a macroscopic scaling solution is to use at least three Gibbons-Hawking centers, but it may also be possible to construct scaling solutions with two centers when the supertubes fluctuate. As we noted above, for technical reasons we have focussed on some very particular modes and this choice of modes meant that whenever we added momentum to the supertube we also added a similar amount of angular momentum. Thus our solutions remained over-spinning or extremal. As a result, we could not access the scaling region that is usually associated with the microstates of a black hole with macroscopic horizon area. In this paper we added charges to the supertube in a manner that precluded us from exploring such deep, scaling geometries.
In addition to the excitations discussed above that lower the angular momenta, more broadly one can consider excitations that either have no angular momentum, or have negative angular momentum. In principle, by using these excitations one can add momentum to the supertube in a way that takes the charges into the BMPV regime. The corresponding black hole would then have a macroscopic horizon and the microstate geometry should then scale and exhibit larger red-shifts and lower holographic energy gaps. This is presently under investigation.
More generally, one may desire to embed superstrata and the kind of twisted-sector structure elucidated here, in multi-centered deep, scaling geometries since this is (as yet) the only known way to access [*typical*]{} twisted-sector CFT states within the supergravity approximation. On a technical level this will be challenging, since it means going beyond two centers and yet our construction has made very heavy use of the flat $\IR^4$ base and the separability of various wave equations in bipolar coordinates. However, this does not mean that it is impossible: The scalar Green functions for charge density fluctuations in generic ambipolar backgrounds were discussed in [@Bena:2010gg], and a three-centered Green function was constructed explicitly. So while this may be very difficult, it is not completely out of reach. Moreover, we hope to find physical arguments that illuminate what the geometries constructed in this paper will probe once they are combined with generic superstrata and embedded in deep, scaling geometries.
Looking further to the future, it would be of great interest to study momentum fractionation in non-supersymmetric microstates, as done in [@Chakrabarty:2015foa]. The recent construction of multi-bubble non-BPS black-hole microstate geometries [@Bena:2015drs] offers the prospect of progress in this direction.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Stefano Giusto, Rodolfo Russo and Masaki Shigemori for helpful discussions. The work of IB and DT was supported by John Templeton Foundation Grant 48222 and by a grant from the Foundational Questions Institute (FQXi) Fund, a donor advised fund of the Silicon Valley Community Foundation on the basis of proposal FQXi-RFP3-1321 (this grant was administered by Theiss Research). The work of EJM was supported in part by DOE grant DE-SC0009924. The work of DT was supported in part by a CEA Enhanced Eurotalents Fellowship. The work of NPW was supported in part by DOE grant DE-SC0011687. For hospitality during the course of this work, EJM and NPW are very grateful to the IPhT, CEA-Saclay; IB, EJM, and DT thank the Centro de Ciencias de Benasque Pedro Pascual; and DT and NPW thank the Yukawa Institute for Theoretical Physics, Kyoto University.
The BMPV black hole {#app:BMPV}
===================
To help establish normalizations, it is useful to give the standard BMPV black-hole metric [@Breckenridge:1996is] in terms of the Ansatz used in this paper. Everything is, of course, $v$-independent and the vector field, $\beta$, and the $\Theta_I$, are set to zero. For a BMPV black hole located at the center of space ($r=0$, $\theta =0$) the $Z_I$ are appropriately-sourced harmonic functions: $$Z_1 ~=~ 1 ~+~ \frac{Q_1}{\Lambda} \,, \qquad Z_2 ~=~ 1 ~+~ \frac{Q_2}{\Lambda} \,, \qquad \mathcal{F} ~=~ - \frac{2Q_3}{\Lambda} \,, \qquad Z_4 ~=~ 0 \,.
\label{BMPV-Zs}$$ The angular momentum vector, $\omega$, is then simply the “harmonic” solution to the homogeneous equation (\[BPSlayer2a\]) with source at the center of space: $$\omega ~=~ \frac{J}{\Lambda^2} \, \big( (r^2 + a^2)\sin^2 \theta \, d \varphi_1 - r^2 \cos^2 \theta\, d \varphi_2 \big) \,.
\label{BMPV-om}$$ Note that as $r \to \infty$ one has $$Z_I ~\sim~ 1+\frac{Q_I}{r^2} \,, \qquad I=1,2,3 \,, \quad\qquad \omega ~\sim~ \frac{J}{r^2} \, \big(\sin^2 \theta \, d \varphi_1 - \cos^2 \theta\, d \varphi_2 \big) \,,
\label{BMPV-Zs-2}$$ which determine the charges and angular momenta of the black hole.
To make the asymptotic analysis of the metric in the vicinity of the center of space using more standard spherical coordinates in the infinitesimal neighborhood of $r=0$, $\theta =0$, one can simply take: $$r ~=~ \lambda\, \sin \chi \,, \qquad \theta ~=~ \frac{\lambda}{a}\, \cos \chi \,.
\label{infsph}$$ and expand to lowest order in $\lambda$. One then finds that the leading part of the metric becomes: $$\begin{aligned}
{ds}_5^2 ~=~ \sqrt{Q_1Q_2} \, \bigg[ \, & \frac{d\lambda^2}{\lambda^2} ~+~ d \chi^2 ~+~\sin^2 \chi \cos^2 \chi \, (d \varphi_1- d \varphi_2)^2 \nonumber \\ & ~+~ \frac{2 Q_3}{Q_1 Q_2} \, \Big(dv - \frac{J}{2Q_3} \, (\cos^2 \chi \, d \varphi_1+\sin^2 \chi\, d \varphi_2) \Big)^2 \nonumber \\ & ~+~ \Big(1 -\frac{J^2}{2 Q_1 Q_2 Q_3} \Big)\, (\cos^2 \chi \, d \varphi_1+\sin^2 \chi\, d \varphi_2)^2\, \bigg] \,.
\label{BMPVnear}\end{aligned}$$ In particular, we see that with our normalizations one must impose the condition: $$J^2 ~\le~ 2 Q_1 Q_2 Q_3 \qquad \Leftrightarrow \qquad J_L^2 ~\le~ Q_1 Q_2 Q_P \,,
\label{BMPVbound}$$ where $J_L = J/ \sqrt{2}$ and $Q_P =Q_3$.
Reduction to five dimensions {#app:5dlimit}
============================
There are two standard ways of reducing the six-dimensional solution, and the system of BPS equations [@Gutowski:2003rg; @Cariglia:2004kk; @Bena:2011dd; @Giusto:2013rxa], to the standard, five-dimensional analogs found in may references (see, for example, [@Bena:2007kg; @Giusto:2012gt]). These two choices of reduction come from different embeddings of the five-dimensional fields in the six-dimensional formulation; we summarize these two standard choices here. The five-dimensional BPS equations are: $$\begin{aligned}
\Theta^{(I)} &~=~& \star_4 \, \Theta^{(I)} \label{5dBPSeqn:1} \,, \\
\nabla^2 Z_I &~=~& \coeff{1}{2} \, C_{IJK} \star_4 (\Theta^{(J)} \wedge
\Theta^{(K)}) \label{5dBPSeqn:2} \,, \\
d\mathbf{k} ~+~ \star_4 d\mathbf{k} &~=~& Z_I \, \Theta^{(I)}\,.
\label{5dBPSeqn:3}\end{aligned}$$ Our goal will be to take $v$-independent, six-dimensional solutions and compactify on an $S^1$ fiber so that the system equations (\[BPSlayer1a\])–(\[BPSlayer1c\]), (\[BPSlayer2a\]) and (\[BPSlayer2b\]) reduce to the five-dimensional system.
Reduction 1
-----------
This is the canonical choice if $\mathcal{F}$ never vanishes and in particular, when $\mathcal{F} \to -1$ at infinity. One can then write the metric globally as $$ds_6^2 ~= \frac{1}{\sqrt{\cP}\, \mathcal{F}} \,(du + \omega)^2 ~-~ \frac{\mathcal{F}}{\sqrt{\cP}}\, \big(dv+\beta + \mathcal{F}^{-1} (du + \omega) \big)^2
~+~ \sqrt{\cP} \, ds_4^2(\cB)\,. \label{sixmet-sq}$$ Upon making the identifications $$\mathcal{F} ~=~ - Z_3\,, \qquad u ~=~t \,, \qquad v~=~t+y \,, \qquad \mathbf{k} ~=~ \omega\,,\qquad \Theta_3 ~=~ d \beta \,,
\label{identifications1}$$ the six-dimensional metric is given by $$ds_6^2 ~= -\frac{1}{Z_3\, \sqrt{\cP}} \, (dt + \mathbf{k})^2 ~+~ \frac{Z_3}{\sqrt{\cP}}\,
\Big[ dv+\beta - Z_3^{-1} (dt+\mathbf{k}) \Big]^2
~+~ \sqrt{\cP} \, ds_4^2(\cB)\,, \label{Bsixmet-sq1}$$ which can also be written as $$ds_6^2 ~= -\frac{1}{Z_3\, \sqrt{\cP}} \, (dt + \mathbf{k})^2 ~+~ \frac{Z_3}{\sqrt{\cP}}\,
\Big[dy + (1-Z_3^{-1}) (dt + \mathbf{k}) + (\beta-\mathbf{k}) \Big]^2
~+~ \sqrt{\cP} \, ds_4^2(\cB)\,. \label{Bsixmet-sq2}$$ Compactifying on the $y$-circle yields an overall warp factor of $(\frac{Z_3}{\sqrt{\cP}})^{1/3}$ on the five-dimensional metric and leads to $$ds_5^2 ~= -\big(Z_3\,\cP \big)^{-\frac{2}{3}} \, (dt + \mathbf{k})^2 ~+~ \big(Z_3\,\cP \big)^{\frac{1}{3}} \, ds_4^2(\cB)\,, \label{Bfivemet-sq}$$ These identifications reduce the six-dimensional BPS system used in this paper directly to the canonical five-dimensional system; this is the origin of how we have chosen to normalize the flux fields like $\Theta_I$. However we have chosen the $t$, $y$ coordinates (\[uvdefn\]), meaning that $\mathcal{F} \to 0$ at infinity, leading to a canonical embedding more closely associated with supertubes. We will now describe this in more detail.
Reduction 2
-----------
In this reduction we use the coordinates (\[uvdefn\]): $$u ~\equiv~ \frac{1}{\sqrt{2}}\, (t-y) \,, \qquad v ~\equiv~ \frac{1}{\sqrt{2}}\, (t+y) \,. \label{Buvdefn}$$ Then as described in , we introduce \[eq:Z3k-2\] Z\_3 = 1- , = , and complete the squares in the metric as in to obtain $$ds_6^2 ~= -\frac{1}{Z_3\sqrt{\cP} } \, (dt + \mathbf{k})^2 \,+\,
\frac{Z_3}{\sqrt{\cP}}\, \left[dy +\left(1- Z_3^{-1}\right) (dt + \mathbf{k}) +\frac{\beta-\omega}{\sqrt{2}} \right]^2 + \sqrt{\cP} \, ds_4^2(\cB)\,.
\label{sixmet-sqty-2}$$
With these identifications one must make the following replacements and re-definitions for the quantities defined in the body of this paper $$\Theta_I ~\to~ \sqrt{2}\, \Theta_I \,, \quad I=1,2,4 \,; \qquad \qquad \Theta_3 ~=~ \sqrt{2} \, d \beta \,.
\label{rescales2}$$ Doing this, the BPS equations (\[BPSlayer1a\])–(\[BPSlayer1c\]), (\[BPSlayer2a\]) and (\[BPSlayer2b\]) reduce to the five-dimensional system –. In particular, the terms arising from the constant in $\mathcal{F} = - 2(Z_3-1)$ cancel in (\[BPSlayer2a\]) against the terms $D\beta +*_4 D\beta$ arising from the replacement $\omega =\sqrt{2} \mathbf{k}-\beta$.
The lowest Style 1 modes {#sec:sty1chargeskp1}
========================
In this appendix we demonstrate the agreement of conserved charges for the lowest possible modes in Style 1, those with $kp=1$, following the analysis for $kp>1$ done in Section \[sec:sty1charges\].
For $kp=1$, the dual CFT state is a particular superposition of states of the Style 1 type , \[CFT dual state-sty1\_app\] ([|++ ]{}\_1\^
)\^[n\_1]{} \_ ( (J\_[-1]{}\^+)\^[|00 ]{}\^
\_)\^ ( (J\_[-1]{}\^+)\^[|++ ]{}\^
\_[+1]{})\^ ( (J\_[-1]{}\^+)\^[|– ]{}\^
\_[-1]{})\^ As explained at the end of Section \[sec:sty1cft\], the average numbers of ${|++ \rangle}_{k(k\hat{p}+1)}$ and ${|-- \rangle}_{k(k\hat{p}-1)}$ strands are equal for $\hat{p}\ge 2$, |[n]{}\_[3,]{} &=& |[n]{}\_[4,]{} 2 , \[eq:kp1balance\] while the excited ${|-- \rangle}$ strands that would be counted by $n_{4,1}$ would have length zero, which does not exist. Therefore we set n\_[4,1]{} &=& 0 . This means that the excited ${|++ \rangle}_2$ strands that are counted by $n_{3,1}$ are not balanced out by corresponding ${|-- \rangle}$ strands. Nevertheless, the conserved charges will work out properly, as we now show.
Since the total number of strands is $N_1 N_5$, using we obtain |[n]{}\_1 + \_ &=& N\_1 N\_5 |[n]{}\_1 + \_ + |[n]{}\_[3,1]{} &=& N\_1 N\_5 and so $\bar{n}_1$ is given by |[n]{}\_1 &=& N\_1 N\_5 - N\_p - |[n]{}\_[3,1]{} . Next, the CFT ${\bar \jmath}^3$ is \^3 = 12 = ( |[n]{}\_1 + |[n]{}\_[3,1]{} ) = N\_1 N\_5 - N\_p , \[eq:jbarkp1-cft\] in perfect agreement with the value of ${\bar \jmath}^3$ computed from the gravity.
The CFT $j^3$ is \^3 &=& 12 + \_ (|[n]{}\_[2,]{}+|[n]{}\_[3,]{}+|[n]{}\_[4,]{}) &=& 12 ( |[n]{}\_1 + |[n]{}\_[3,1]{} ) + \_ (|[n]{}\_[2,]{}+|[n]{}\_[3,]{}+|[n]{}\_[4,]{}) = [|]{}\^3 + k N\_p . \[eq:j3kp1\] which again agrees exactly with the value of $j^3$ computed from the gravity. The momentum charge determines $b_4$ just as for $kn>1$, and so all conserved charges agree.
This agreement shows that comparing conserved charges alone does not put any constraint on the value of $\bar{n}_{3,1}$. Of course, our proposal of Section \[ss:CFT dual states\] fixes $\bar{n}_{3,1}$ unambiguously, since we have specified in principle all coefficients in the coherent state. To scrutinize our proposal further, one would have to perform further holographic tests.
One can see how this agreement works in another way: Relative to the unexcited base supertube ${|++ \rangle}_1$ strands, the difference in conserved charges is as follows. For each excited ${|++ \rangle}_2$ strand, the change in $\bar{\jmath}^3$ is $\Delta \bar{\jmath}^3 = - 1/2$; for $j^3$ we have $\Delta j^3=1-1/2 = 1/2$; and we have $\Delta P=1$. So regardless of the value of $\bar{n}_{3,1}$, the above expressions for $j^3$ and $\bar{\jmath}^3$ in terms of $N_1$, $N_5$, $N_P$ are the same.
[-3mm]{}[-3mm]{}
[^1]: There is a sense in which states obtained by the action of integer-moded generators acting on multi-wound strands can be argued to involve momentum fractionation, however this fractionation is somewhat trivial and does not correspond to degrees of freedom deep inside a throat [@Mathur:2011gz; @Mathur:2012tj; @Lunin:2012gp]. Thus, by “CFT states involving momentum fractionation” we mean states which cannot be written in terms of integer-moded generators acting on R-R ground states.
[^2]: The same is true of the three-charge solutions obtained by integer spectral flow [@Giusto:2004id; @Giusto:2004ip; @Lunin:2004uu].
[^3]: Note that in our conventions $ \mathcal{F}$ is always negative.
[^4]: To see this, let us suppose that such curves are timelike, and let $\cC_1$ be such a curve. $\cC_1$ itself is not necessarily closed; denote the $y$ values at the start and end of the curve by $y_1$ and $y_2$. If $y_2$ is not equal to $y_1$ (modulo $2\pi R_y$), consider $y_2$ as the starting point of a new curve $\cC_2$, similarly defined so that $dy$ is related to the other angles such that the second square vanishes. By iterating, one obtains a sequence of timelike-related points along the $y$ direction, with fixed values of the other coordinates. Since $y$ is periodic, by iterating this procedure one either obtains a CTC or comes arbitrarily close to obtaining a CTC, meaning that the spacetime has ‘almost-closed’ timelike curves and so fails to be ‘strongly causal’ as defined in [@Hawking:1973uf].
[^5]: We define the $d$-dimensional Hodge star $*_d$ acting on a $p$-form to be $$*_d\, (dx^{m_1}\wedge\cdots\wedge dx^{m_p})
~=~ \frac{1}{(d-p)!} \, dx^{n_1}\wedge\cdots\wedge dx^{n_{d-p}}\, \epsilon_{n_1\dots n_{d-p}}{}^{m_1\dots m_p} \,,$$ where we use the orientation $\epsilon^{+-1234} \equiv \epsilon^{vu1234} = \epsilon^{1234} = 1$. These are the conventions used in [@Gutowski:2003rg] and note that they differ from the typical conventions for the Hodge dual.
[^6]: This simplified form is equivalent to (2.9b) of [@Giusto:2013bda].
[^7]: Our spherical bipolar angles $\varphi_1$ and $\varphi_2$ are related to those of [@Bena:2015bea] by $\varphi_1^{\rm here} = \phi^{\rm there}$, $\varphi_2^{\rm here} = \psi^{\rm there}$.
[^8]: Technically, one should restrict to the global diffeomorphisms, $SL(2,\ZZ)$, but if one allows orbifolds it is sometimes convenient to use $GL(2,\ZZ)$.
[^9]: Following standard practice, we show that there are no CTC’s near the supertube and no CTC’s at infinity. This is usually sufficient to guarantee the absence of CTC’s globally.
[^10]: This subset of the solutions in [@Bena:2015bea] is given by taking a single mode of that construction and setting $m = k$ in the notation of that paper.
[^11]: Strictly speaking we have only shown that there are no CTC’s near the supertube, in the intermediate region and at infinity. Again, this should be sufficient to guarantee the absence of CTC’s globally.
[^12]: The full orbifold is of course non-abelian, but for the purpose of describing the spectrum, one can use abelian orbifold terminology.
[^13]: Since topologically twisting the supersymmetry of the sigma model relates the cohomology of the supersymmetry charges to the cohomology of a Dolbeault-type operator on the target space.
[^14]: Note that the action of $G_{-\half}$ lowers the $SU(2)$ spin while raising the $SL(2)$ spin, so that the six-dimensional helicity stays constant; similarly for $\bar G_{-\half}$. Thus the short multiplets with ${\bf m}-{\bf \bar m}=\pm 2$, whose highest weight has spin one in both $SU(2)$ and $SL(2)$, contain the six-dimensional spin-two graviton polarizations.
[^15]: The lowest BPS operator in the short multiplet $(k,k)_S$ has special properties at low $k$. For $k=1$ this operator is the identity operator, and the higher components of the superfield are absent. For $k=2$, the lowest component has dimension $h=\bar h=1/2$, and the double-descendant is null. Not until $k=3$ is there a non-trivial double-descendant operator.
[^16]: Note that the latter operator can also be obtained from the former by tensoring with the center-of-mass current $J^+$ of the $\kappa$-cycle. By $\kappa$-cycle currents, or center-of-mass currents, we mean the total $SU(2)$ currents built of the copies of $\bbT^4$ being sewn together in a particular cycle of length $\kappa$ in a symmetric group word, rather than the total $\cR$-currents of the entire theory.
[^17]: See [@Taylor:2007hs] for a discussion of subtleties in this map.
[^18]: In the $g$-twisted sector, one has a projection by the action of all group elements $h$ that commute with $g$, $hgh^{-1}=g$. This includes $g$ itself, whose action imparts a phase to the fractional modes.
[^19]: The operator is the spectral flow to the NS sector of the Ramond operator corresponding to the state ${|00 \rangle}_{k\hp}$ (flowed in opposite directions on left and right), while the operator is the spectral flow to the NS sector of the Ramond operator corresponding to the state ${|-- \rangle}_{\kppo}$, and the operator corresponds to the NS to R flow of the state ${|++ \rangle}_{\kmmo}$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: '[Does galaxy evolution proceed through the green valley via multiple pathways or as a single population? Motivated by recent results highlighting radically different evolutionary pathways between early- and late-type galaxies, we present results from a simple Bayesian approach to this problem wherein we model the star formation history (SFH) of a galaxy with two parameters, $[t, \tau]$ and compare the predicted and observed optical and near-ultraviolet colours. We use a novel method to investigate the morphological differences between the most probable SFHs for both disc-like and smooth-like populations of galaxies, by using a sample of $126,316$ galaxies $(0.01 < z < 0.25)$ with probabilistic estimates of morphology from Galaxy Zoo. We find a clear difference between the quenching timescales preferred by smooth- and disc-like galaxies, with three possible routes through the green valley dominated by smooth- (rapid timescales, attributed to major mergers), intermediate- (intermediate timescales, attributed to minor mergers and galaxy interactions) and disc-like (slow timescales, attributed to secular evolution) galaxies. We hypothesise that morphological changes occur in systems which have undergone quenching with an exponential timescale $\tau < 1.5~\rm{Gyr}$, in order for the evolution of galaxies in the green valley to match the ratio of smooth to disc galaxies observed in the red sequence. These rapid timescales are instrumental in the formation of the red sequence at earlier times; however we find that galaxies currently passing through the green valley typically do so at intermediate timescales.]{}'
author:
- |
R. J. Smethurst,$^{1}$ C. J. Lintott,$^{1}$ B. D. Simmons,$^{1}$ K. Schawinski,$^{2}$ P. J. Marshall,$^{3,1}$ S. Bamford,$^{4}$ L. Fortson,$^{5}$ S. Kaviraj,$^{6}$ K. L. Masters,$^{7}$ T. Melvin,$^{7}$ R. C. Nichol,$^{7}$ R. A. Skibba,$^{8}$ K. W. Willett$^{5}$\
$^1$ Oxford Astrophysics, Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK\
$^2$ Institute for Astronomy, Department of Physics, ETH Zurich, Wolfgang-Pauli Strasse 27, CH-8093 Zurich, Switzerland\
$^3$ Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, 452 Lomita Mall, Stanford, CA 95616, USA\
$^4$ School of Physics and Astronomy, The University of Nottingham, University Park, Nottingham, NG7 2RD, UK\
$^5$ School of Physics and Astronomy, University of Minnesota, 116 Church St SE, Minneapolis, MN 55455, USA\
$^6$ Centre for Astrophysics Research, University of Hertfordshire, College Lane, Hatfield, Hertfordshire, AL10 9AB, UK\
$^7$ Institute of Cosmology and Gravitation, University of Portsmouth, Dennis Sciama Building, Barnaby Road, Portsmouth, PO1 3FX, UK\
$^8$ Center for Astrophysics and Space Sciences, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA\
\
Accepted 2015 January 22. Received 2015 January 14; in original form 2014 September 17
title: 'Galaxy Zoo: Evidence for Diverse Star Formation Histories through the Green Valley'
---
Introduction
============
Previous large scale surveys of galaxies have revealed a bimodality in the colour-magnitude diagram (CMD) with two distinct populations; one at relatively low mass, with blue optical colours and another at relatively high mass, with red optical colours [@Baldry04; @Baldry06; @Willmer06; @BLB08; @Brammer09]. These populations were dubbed the ‘blue cloud’ and ‘red sequence’ respectively [[@Chester64; @BLE92; @Driver06; @Faber07]]{}. The Galaxy Zoo project [@Lintott11], which produced morphological classifications for a million galaxies, helped to confirm that this bimodality is not entirely morphology driven [[@Strat01; @Salim07; @Sch07; @CHV08; @Bamford09; @Skibba09], detecting larger fractions of spiral galaxies in the red sequence [@Masters10] and elliptical galaxies in the blue cloud [@Sch09] than had previously been detected. ]{}
The sparsely populated colour space between these two populations, the so-called ‘green valley’, provides clues to the nature and duration of galaxies’ transitions from blue to red. This transition must occur on rapid timescales, otherwise there would be an accumulation of galaxies residing in the green valley, rather than an accumulation in the red sequence as is observed [@Arnouts07; @Martin07]. Green valley galaxies have therefore long been thought of as the ‘crossroads’ of galaxy evolution, a transition population between the two main galactic stages of the star forming blue cloud and the ‘dead’ red sequence [@Bell04; @Wyder07; @Schim07; @Martin07; @Faber07; @Mendez11; @Gonc12; @Sch2014; @Pan14].
The intermediate colours of these green valley galaxies have been interpreted as evidence for recent quenching (suppression) of star formation [@Salim07]. Star forming galaxies are observed to lie on a well defined mass-SFR relation, however quenching a galaxy causes it to depart from this relation (@Noeske07 [@Peng]; see Figure \[sfr\_mass\_sub\])
By studying the galaxies which have just left this mass-SFR relation, we can probe the quenching mechanisms by which this occurs. There have been many previous theories for the initial triggers of these quenching mechanisms, including negative feedback from AGN [[@diMatteo05; @Martin07; @Nandra07; @Sch07], mergers [@Darg10a; @Cheung12; @Barro13], supernovae winds [@MFB12], cluster interactions [@Coil08; @Mendez11; @Fang13] and secular evolution [@Masters10; @Masters11; @Mendez11].]{} By investigating the *amount* of quenching that has occurred in the blue cloud, green valley and red sequence; and by comparing the amount across these three populations, we can apply some constraints to these theories.
We have been motivated by a recent result suggesting two contrasting evolutionary pathways through the green valley by different morphological types (@Sch2014, hereafter S14), specifically that late-type galaxies quench very slowly and form a nearly static disc population in the green valley, whereas early-type galaxies quench very rapidly, transitioning through the green valley and onto the red sequence in $\sim 1$ Gyr [@Wong12]. That study used a toy model to examine quenching across the green valley. Here we implement a novel method utilising Bayesian statistics (for a comprehensive overview of Bayesian statistics see either @MacKay or @Sivia) in order to find the most likely model description of the star formation histories of galaxies in the three populations. This method also enables a direct comparison with our current understanding of galaxy evolution from stellar population synthesis (SPS, see section \[models\]) models.
[0.9]{}[r @cccc]{}
--------
[ -]{}
[ -]{}
--------
& All & Red Sequence & Green Valley & Blue Cloud\
Smooth-like ($p_s > 0.5$) &
---------
42453
(33.6%)
---------
&
---------
17424
(61.9%)
---------
&
---------
10687
(44.6%)
---------
&
---------
14342
(19.3%)
---------
\
Disc-like ($p_d > 0.5$) &
---------
83863
(80.7%)
---------
&
---------
10722
(38.1%)
---------
&
---------
13257
(55.4%)
---------
&
---------
59884
(47.4%)
---------
\
Early-type ($p_s \geq 0.8$) &
--------
10517
(8.3%)
--------
&
---------
5337
(18.9%)
---------
&
---------
2496
(10.4%)
---------
&
--------
2684
(3.6%)
--------
\
Late-type ($p_s \geq 0.8$) &
---------
51470
(40.9%)
---------
&
---------
4493
(15.9%)
---------
&
---------
6817
(28.5%)
---------
&
---------
40430
(54.4%)
---------
\
**Total** &
------------
**126316**
(100.0%)
------------
&
---------
28146
(22.3%)
---------
&
---------
23944
(18.9%)
---------
&
---------
74226
(58.7%)
---------
\
\[subs\]
Through this approach, we aim to determine the following:
1. What previous star formation history (SFH) causes a galaxy to reside in the green valley at the current epoch?
2. Is the green valley a transitional or static population?
3. If the green valley is a transitional population, how many routes through it are there?
4. Are there morphology-dependent differences between these routes through the green valley?
This paper proceeds as follows. Section \[data\] contains a description of the sample data, which is used in the Bayesian analysis of an exponentially declining star formation history model, all described in Section \[models\]. Section \[results\] contains the results produced by this analysis, with Section \[diss\] providing a detailed discussion of the results obtained. We also summarise our findings in Section \[conc\]. The zero points of all *ugriz* magnitudes are in the AB system and where necessary we adopt the WMAP Seven-Year Cosmological parameters [@WMAP] with $(\Omega_m, \Omega_{\lambda}, h) = (0.26, 0.73, 0.71)$.
Data
====
Multi-wavelength data {#multi}
---------------------
The galaxy sample is compiled from publicly available optical data from the Sloan Digitial Sky Survey (SDSS; @York00) Data Release 8 [@Aihara11]. Near-ultraviolet (NUV) photometry was obtained from the Galaxy Evolution Explorer (GALEX; @Martin05) and was matched with a search radius of $1''$ in right ascension and declination.
[Observed optical and ultraviolet fluxes are corrected for galactic extinction [@Oh11] by applying the @Cardelli89 law, giving an typical correction of $u-r \sim 0.05$. We also adopt k-corrections to $z=0.0$ and obtain absolute magnitudes from the NYU-VAGC [@Blanton05; @Pad08; @BR07], giving a typical $u-r$ correction of $\sim 0.15$ mag. The change in the $u-r$ colour due to both corrections therefore ranges from $\Delta u-r \sim 0.2$ at low redshift, increasing up to $\Delta u-r \sim 1.0$ at $z \sim 0.25$, which is consistent with the expected k-corrections shown in Figure 15 of @BR07. These corrections were calculated by @Bamford09 for the entire Galaxy Zoo sample. These corrections are a crucial aspect of this work since a $\Delta u-r \sim 1.0$ can cause a galaxy to cross the definition between blue cloud, green valley and red sequence.]{}
We obtained star formation rates and stellar masses from the MPA-JHU catalog (@Kauff03 [@Brinch04]; average values, <span style="font-variant:small-caps;">AVG</span>, corrected for aperture and extinction), which are in turn calculated from the SDSS spectra and photometry.
We further select a sub-sample with detailed morphological classifications, as described below, [to give a volume limited sample in the redshift range $0.01 < z < 0.25$]{}.
Galaxy Zoo 2 Morphological classifications {#class}
------------------------------------------
In this investigation we use visual classifications of galaxy morphologies from the Galaxy Zoo 2[^1] citizen science project [@GZ2], which obtains multiple independent classifications for each galaxy image; the full question tree for each image is shown in Figure 1 of @GZ2.
The Galaxy Zoo 2 (GZ2) project consists of $304, 022$ images from the SDSS DR8 (a subset of those classified in Galaxy Zoo 1; GZ1) all classified by *at least* 17 independent users, with the mean number of classifications standing at $\sim42$. The GZ2 sample is more robust than the GZ1 sample and provides more detailed morphological classifications, including features such as bars, the number of spiral arms and the ellipticity of smooth galaxies. It is for these reasons we use the GZ2 sample, as opposed to the GZ1, allowing for further investigation of specific galaxy classes in the future (see Section \[future\]). The only selection that was made on the sample was to remove objects considered to be stars, artefacts or merging pairs by the users (i.e. with $p_{star/artefact} ~\geq~ 0.8$ or $p_{merger} ~\geq 0.420$; see @GZ2 Table 3 and discussion for details of this fractional limit). Further to this, we required NUV photometry from the GALEX survey, within which $\sim42\%$ of the GZ2 sample were observed, giving a total sample size of $126, 316$ galaxies. [The completeness of this subsample of GZ2 matched to GALEX is shown in Figure \[complete\] with the $u$-band absolute magnitude against redshift for this sample compared with the SDSS data set. Typical Milky Way $L_*$ galaxies with $M_u \sim -20.5$ are still included in the GZ2 subsample out to the highest redshift of $z \sim 0.25$; however dwarf and lower mass galaxies are only detected at the lowest redshifts.]{}
The first task of GZ2 asks users to choose whether a galaxy is mostly smooth, is featured and/or has a disc or is a star/artefact. Unlike other tasks further down in the decision tree, every user who classifies a galaxy image will complete this task (others, such as whether the galaxy has a bar, is dependent on a user having first classified it as a featured galaxy). Therefore we have the most statistically robust classifications at this level.
The classifications from users produces a vote fraction for each galaxy (the debiased fractions calculated by @GZ2 were used in this investigation); for example if 80 of 100 people thought a galaxy was disc shaped, whereas 20 out of 100 people thought the same galaxy was smooth in shape (i.e. elliptical), that galaxy would have vote fractions $p_{s} = 0.2$ and $p_{d} = 0.8$. In this example this galaxy would be included in the *‘clean’* disc sample ($p_d \geq 0.8$) according to [@GZ2] and would be considered a late-type galaxy. [All previous Galaxy Zoo projects have incorporated extensive analysis of volunteer classifications to measure classification accuracy and bias, and compute user weightings (for a detailed description of debiasing and consistency-based user weightings, see either Section 3 of @Lintott09 or Section 3 of @GZ2). ]{}
[The classifications are highly accurate and provide a continuous scale of morphological features, as shown in Figure \[mosaic\], rather than a simple binary classification separating elliptical and disc galaxies. These classifications allow each galaxy to be considered as a probabilistic object with both bulge and disc components.]{} For the first time, we incorporate this advantage of the GZ classifications into a large statistical analysis of how elliptical and disc galaxies differ in their SFHs.
Defining the Green Valley {#defGV}
-------------------------
To define which of the sample of $126, 316$ galaxies were in the green valley, [we looked to previous definitions in the literature defining the separation between the red sequence and blue cloud to ensure comparisons can be made with other works. @Baldry04 used a large sample of local galaxies from the SDSS to trace this bimodality by fitting double Gaussians to the colour magnitude diagram without cuts in morphology.]{} Their relation is defined in their Equation 11 as: $$\label{eqgv}
C'_{ur}(M_{r}) = 2.06 - 0.244 \tanh \left( \frac{M_r + 20.07}{1.09}\right)$$ and is shown in Figure \[CMGV\] by the dashed line in comparison to both the GZ2 subsample (left) and the SDSS data from [@Baldry04]. [This ensures that the definition of the green valley used is derived from a complete sample, rather than from our sample that is dominated by blue galaxies due to the necessity for NUV photometry.]{} Any galaxy within $\pm 1\sigma$ of this relationship, shown by the solid lines in Figure \[CMGV\], is therefore considered a green valley galaxy. The decomposition of the sample into red sequence, green valley and blue cloud galaxies is shown in Table \[subs\] along with further subsections by galaxy type. This table also lists the definitions we adopt henceforth for early-type ($p_s~ \geq~0.8$), late-type ($p_d~ \geq~0.8$), smooth-like ($p_s~ >~0.5$) and disc-like ($p_d~ >~0.5$) galaxies.
Models
======
In the following section, the quenched SFH models are described in Section \[qmod\] and the probabilistic fitting method to the data is described in Section \[stats\].
Quenching Models {#qmod}
----------------
The quenched star formation history (SFH) of a galaxy can be simply modelled as an exponentially declining star formation rate (SFR) across cosmic time ($0 \leq t ~\rm{[Gyr]} \leq 13.8$) as: $$\label{sfh}
SFR =
\begin{cases}
i_{sfr}(t_q) & \text{if } t < t_q \\
i_{sfr}(t_q) \times exp{\left( \frac{-(t-t_{q})}{\tau}\right)} & \text{if } t > t_q
\end{cases}$$ where $t_{q}$ is the onset time of quenching, $\tau$ is the timescale over which the quenching occurs and $i_{sfr}$ is an initial constant star formation rate dependent on $t_q$. A smaller $\tau$ value corresponds to a rapid quench, whereas a larger $\tau$ value corresponds to a slower quench.
We assume that all galaxies formed at a time $t=0~\rm{Gyr}$ with an initial burst of star formation. The mass of this initial burst is controlled by the value of the $i_{sfr}$ which is set as the average specific SFR (sSFR) at the time of quenching $t_q$. [@Peng defined a relation (their equation 1) between the average sSFR and redshift (cosmic time, $t$) by fitting to measurements of the mean sSFR of blue star forming galaxies from SDSS, zCOSMOS and literature values at increasing redshifts [@Elbaz07; @Daddi07]:]{} $$sSFR(m,t) = 2.5 \left( \frac{m}{10^{10} M_{\odot}} \right)^{-0.1} \left(\frac{t}{3.5 {\refchange ~\rm{Gyr}}}\right)^{-2.2} \rm{Gyr}^{-1}.$$ Beyond $z \sim 2$ the characteristic SFR flattens and is roughly constant back to $z\sim6$. The cause for this change is not well understood but can be seen across similar observational data [@Peng; @Gonzalez; @Beth]. Motivated by these observations, the relation defined in @Peng is taken up to a cosmic time of $t=3~\rm{Gyr}~(z \sim 2.3)$ and prior to this a constant average SFR is assumed (see Figure \[sfr\_mass\_col\]). At the point of quenching, $t_{q}$, the models are defined to have a SFR which lies on this relationship for the sSFR, for a galaxy with mass, $m = 10^{10.27} M_{\odot}$ (the mean mass of the GZ2 sample; see Section \[results\] and Figure \[sfr\_mass\_col\]).
Under these assumptions the average SFR of our models will result in a lower value than the relation defined in @Peng at all cosmic times; each galaxy only resides on the ‘main sequence’ at the point of quenching. However galaxies cannot remain on the ‘main sequence’ from early to late times throughout their entire lifetimes given the unphysical stellar masses and SFRs this would result in at the current epoch in the local Universe [@Beth; @Heinis14]. If we were to include prescriptions for no quenching, starbursts, mergers, AGN etc. into our models we would improve on our reproduction of the average SFR across cosmic time; however we chose to initially focus on the simplest model possible.
Once this evolutionary SFR is obtained, it is convolved with the @BC03 population synthesis models to generate a model SED at each time step. The observed features of galaxy spectra can be modelled using simple stellar population techniques which sum the contributions of individual, coeval, equal-metallicity stars. The accuracy of these predictions depends on the completeness of the input stellar physics. Comprehensive knowledge is therefore required of (i) stellar evolutionary tracks and (ii) the initial mass function (IMF) to synthesise a stellar population accurately.
These stellar population synthesis (SPS) models are an extremely well explored (and often debated) area of astrophysics [@Maraston05; @Eminian08; @CGW09; @Falk09; @Chen10; @Kriek10; @MRC11; @Mel12]. In this investigation we chose to utilise the @BC03 *GALEXEV* SPS models, to allow a direct comparison with S14, along with a Chabrier [@Chab03] IMF, across a large wavelength range ($0.0091 < ~\lambda~\rm{[\mu m]}~ < 160 $) with solar metallically (m62 in the @BC03 models; hereafter BC03).
Fluxes from stars younger than $3~$Myr in the SPS model are suppressed to mimic the large optical depth of protostars embedded in dusty formation cloud (as in S14), then filter transmission curves are applied to the fluxes to obtain AB magnitudes and therefore colours. [For a particular galaxy at an observed redshift, $z$, we define the observed time, $t^{obs}$ for that galaxy using the standard cosmological conversion between redshift and time. We utilise the SFH models at this observed time for each individual galaxy to compare the predicted model and observed colours directly.]{}
Figure \[pred\] shows these predicted optical and NUV colours at a time of $t^{obs} = 12.8 ~\rm{Gyr}$ (the average observed time of the Galaxy Zoo 2 sample, $z \sim 0.076$) provided by the exponential SFH model. These predicted colours will be referred to as $d_{c,p}(t_{q}, \tau, t^{obs})$, where c={opt,NUV} and p = predicted. The SFR at a time of $t^{obs}=12.8~\rm{Gyr}$ is also shown in Figure \[pred\] to compare how this correlates with the predicted colours. The $u-r$ predicted colour shows an immediate correlation with the SFR, however the $NUV-u$ colour is more sensitive to the value of $\tau$ and so is ideal for tracing any recent star formation in a population . At small $\tau$ (rapid quenching timescales) the $NUV-u$ colour is insensitive to $t_{q}$, whereas at large $\tau$ (slow quenching timescales) the colour is very sensitive to $t_{q}$. Together the two colours are ideal for tracing the effects of $t_{q}$ and $\tau$ in a population.
[We stress here that this model is not a fully hydrodynamical simulation, it is a simple model built in order to test the understanding of the evolution of galaxy populations. These models are therefore not expected to accurately determine the SFH of every galaxy in the GZ2 sample, in particular galaxies which have not undergone any quenching. In this case the models described above can only attribute a constant star formation rate to these unquenched galaxies. In reality, there are many possible forms of SFH that a galaxy can take, a few of which have been investigated in previous literature; starbursts [@Canalizo01], a power law [@Glazebrook03], single stellar populations [@Trager00; @Sanchez06; @Vazdekis10] and metallicity enrichment [@deLucia14]. Incorporating these different SFHs along with prescriptions for mergers and a reinvigoration of star formation post quench into our models is a possible future extension to this work once the results of this initial study are well enough understood to permit additional complexity to be added.]{}
Probabilistic Fitting {#stats}
---------------------
In order to achieve robust conclusions we conduct a Bayesian analysis [@Sivia; @MacKay] of our SFH models in comparison to the observed GZ2 sample data. This approach requires consideration of all possible combinations of $\theta \equiv (t_{q}, \tau)$. Assuming that all galaxies formed at $t=0~\rm{Gyr}$ with an initial burst of star formation, we can assume that the ‘age’ of each galaxy in the GZ2 sample is equivalent to an observed time, $t^{obs}_{k}$ (see Section \[class\]). We then use this ‘age’ to calculate the predicted model colours at this cosmic time for a given combination of $\theta$: $d_{c,p}(\theta_k, t^{obs}_{k})$ for both optical and NUV $(c={opt,NUV})$ colours. We can now directly compare our model colours with the observed GZ2 galaxy colours, so that for a single galaxy $k$ with optical ($u-r$) colour, $d_{opt, k}$ and NUV ($NUV-u$) colour, $d_{NUV,k}$, the [likelihood $P(d_{k}|\theta_k, t^{obs}_{k})$ is]{}:
$$\begin{gathered}
\label{like}
P(d_{k}|\theta_k, t^{obs}_{k}) = \frac{1}{\sqrt{2\pi\sigma_{opt, k}^2}}\frac{1}{\sqrt{2\pi\sigma_{NUV, k}^2}} \\ \exp{\left[ - \frac{(d_{opt, k} - d_{opt, p}(\theta_k, t_{k}^{obs}))^2}{\sigma_{opt, k}^2} \right]} \\ \exp{\left[ - \frac{(d_{NUV, k} - d_{NUV, p}(\theta_k, t_{k}^{obs}))^2}{\sigma_{NUV, k}^2} \right]}.\end{gathered}$$
We have assumed that $P(d_{opt}|\theta_k, t^{obs}_{k})$ and $P(d_{NUV}|\theta_k, t^{obs}_{k})$ are independent of each other and that the errors on the observed colours are also independent. To obtain the probability of each combination of $\theta$ values the GZ2 data: $P(\theta_k|d_k, t^{obs})$, i.e. how likely is a single SFH model given the observed colours of a single GZ2 galaxy, [we utilise Bayes’ theorem]{}: [$$\label{big}
P(\theta_k|d_k, t^{obs}) = \frac{P(d_k|\theta_k, t^{obs})P(\theta_k)}{\int P(d_k |\theta_k, t^{obs})P(\theta_k) d\theta_k}.$$]{} [We assume a flat prior on the model parameters so that: $$\label{prior}
P(\theta_k) =
\begin{cases}
1 & \text{if } 0 \leq t_q ~\rm{[Gyr]}~ \leq 13.8 ~ \text{ and } ~ 0 \leq \tau ~\rm{[Gyr]}~ \leq 4\\
0 & \text{otherwise.} \\
\end{cases}$$]{}
As the denominator of Equation \[big\] is a normalisation factor, comparison between likelihoods for two different SFH models (i.e., two different combinations of $\theta_k = [t_q, \tau]$) is equivalent to a comparison of the numerators. Calculation of $P(\theta_k|d_k, t^{obs})$ for any $\theta$ is possible given galaxy data from the GZ2 sample. Markov Chain Monte Carlo (MCMC; @MacKay [@Dan; @GW10]) provides a robust comparison of the likelihoods between $\theta$ values; here we choose *emcee*,[^2] a Python implementation of an affine invariant ensemble sampler by [@Dan].
This method allows for a more efficient exploration of the parameter space by avoiding those areas with low likelihood. A large number of ‘walkers’ are started at an initial position where the likelihood is calculated; from there they individually ‘jump’ to a new area of parameter space. If the likelihood in this new area is greater (less) than the original position then the ‘walkers’ accept (reject) this change in position. Any new position then influences the direction of the ‘jumps’ of other walkers. [This is repeated for the defined number of steps after an initial ‘burn-in’ phase. *emcee* returns the positions of these ‘walkers’, which are analogous to the regions of high probability in the model parameter space.]{} The model outlined above has been coded using the *Python* programming language into a package named which has been made freely available to download[^3]. [An example output from this Python package for a single galaxy from the GZ2 sample in the red sequence is shown in Figure \[one\_example\]. The contours show the positions of the ‘walkers’ in the Markov Chain which are analogous to the areas of high probability.]{}
We wish to consider the model parameters for the populations of galaxies across the colour magnitude diagram for both smooth and disc galaxies, therefore we run the package on each galaxy in the GZ2 sample. This was extremely time consuming; for each combination of $\theta$ values which *emcee* proposes, a new SFH must be built, prior to convolving it with the BC03 SPS models at the observed age and then predicted colours calculated from the resultant SED. For a single galaxy this takes up to 2 hours on a typical desktop machine for long Markov Chains. A look-up table was therefore generated at $50 ~t^{obs}$, for $100 ~t_{quench}$ and $100 ~\tau$ values; this was then interpolated over for a given observed galaxy’s age and proposed $\theta$ values at each step in the Markov Chain. This ensures that a single galaxy takes approximately 2 minutes to run on a typical desktop machine. This interpolation was found to incorporate an error of $\pm 0.04$ into the median $\theta$ values found [(the 50th percentile position of the walkers]{}; see Appendix section \[app\_lookup\] for further information).
Using this lookup table, each of the $126,316$ total galaxies in the GZ2 sample was run through on multiple cores of a computer cluster to obtain the Markov Chain positions (analogous to $P(\theta_k|d_k)$) for each galaxy, $k$ (see Figure \[one\_example\]). In each case the Markov Chain consisted of $100$ ‘walkers’ which took $400$ steps in the ‘burn-in’ phase and $400$ steps thereafter, at which point the MCMC acceptance fraction was checked to be within the range $0.25 < f_{acc} < 0.5$ (which was true in all cases). [Due to the Bayesian nature of this method, a statistical test on the results is not possible; the output is probabilistic in nature across the entirety of the parameter space.]{}
[These individual galaxy positions are then combined to visualise the areas of high probability in the model parameter space across a given population (e.g. the green valley).]{} We do this by first discarding positions with a corresponding probability of $P(\theta_k|d_k) < 0.2$ in order to exclude galaxies which are not well fit by the quenching model; for example blue cloud galaxies which are still star forming will be poorly fit by a quenching model (see Section \[qmod\]). [Using this constraint, $2.4\%$, $7.0\%$ and $5.4\%$ of green, red and blue galaxies respectively had *all* of their walker positions discarded. These are not significant enough fractions to affect the results (see Appendix section \[discard\] for more information.)]{} The Markov Chain positions are then binned and weighted by their [corresponding logarithmic posterior probability $\log [P(\theta_k|d_k)]$, provided by the *emcee* package, to further emphasise the features and differences between each population in the visualisation]{}. The GZ2 data also provides uniquely powerful continuous measurements of a galaxy’s morphology, therefore we utilise the user vote fractions to obtain separate model parameter distributions for both smooth and disc galaxies. This is obtained by also weighting by the morphology vote fraction when the binned positions are summed. [We stress that this portion of the methodology is a non-Bayesian visualisation of the combined individual galaxy results for each population.]{}
For example, the galaxy shown in Figure \[one\_example\] would contribute almost evenly to both the smooth and disc parameters due to the GZ2 vote fractions. Since galaxies with similar vote fractions contain both a bulge and disc component, this method is effective in incorporating intermediate galaxies which are thought to be crucial to the morphological changes between early- and late-type galaxies. It was the consideration of these intermediate galaxies which was excluded from the investigation by S14.
Results
=======
Initial Results
---------------
{width="49.75000%"} {width="49.75000%"}
Figure \[sfr\_mass\_sub\] shows the SFR versus the stellar mass for the observed GZ2 sample which has been split into blue cloud, green valley and red sequence populations as well as into the ‘clean’ disc and smooth galaxy samples (with GZ2 vote fractions of $p_d \geq 0.8$ and $p_s \geq 0.8$ respectively). The green valley galaxies are indeed a population which have either left, or begun to leave, the star forming sequence or have some residual star formation still occurring.
The left panel in Figure \[sfr\_mass\_col\] shows a handful of quenching models and how they reproduce the observed relationship between the SFR and the mass of a galaxy, including how at the time of quenching they reside on the star forming sequence shown by the solid black line for a galaxy of mass, $M = 10^{10.27} M_{\odot}$.
The right panel shows how these SFRs translate into the optical-NUV colour-colour plane to reproduce observed colours of green valley and red sequence galaxies. Some of the SFHs produce colours redder than the apparent peak of the red sequence in the GZ2 subsample; however this is not the *true* peak of the red sequence due to the necessity for NUV colours from GALEX (see Section \[class\]).
The majority of the red galaxies in the sample therefore lie towards the *blue end* of the red sequence and have a small amount of residual star formation in order to be detected in the NUV [resulting in a specific subset of the red sequence studied in this investigation. Only $47\%$ of the red sequence galaxies present in the entire Galaxy Zoo 2 sample are matched with GALEX to produce our final sample of $126, 316$ galaxies, as opposed to $72\%$ of the blue cloud and $53\%$ of the green valley galaxies.]{} This limitation should be taken into account when considering the results in the following sections.
[The SFH models were implemented with the package to produce Figures \[red\_s\], \[green\_v\] & \[blue\_c\] for the red sequence, green valley and blue cloud populations of smooth and disc galaxies respectively.]{}
The percentages shown in Figures \[red\_s\], \[green\_v\] & \[blue\_c\] are calculated as the fractions of the combined posterior probability distribution located in each region of parameter space for a given population.
Since the sample contains such a large number of galaxies, we interpret these fractions as broadly equivalent to the percentage of galaxies in a given population undergoing quenching within the stated timescale range. Although this is not quantitatively exact, it is nevertheless a useful framework for interpreting the results of combining the individual posterior probability distributions of each galaxy.
[Also shown in Figure 11 are the median walker positions (the 50th percentile of the Bayesian probability distribution) of each individual galaxy, split into red, green and blue populations also with a hard cut in the vote fraction of $p_d > 0.5$ and $p_s > 0.5$ to show the disc and smooth populations respectively. These positions were calculated without discarding any walker positions due to low probability and without weighting by vote fractions; therefore this may be more intuitive to understand than Figures \[red\_s\], \[green\_v\] & \[blue\_c\].]{}
[Although the quenching timescales are continuous in nature, in this Section we refer to rapid, intermediate and slow quenching timescales which correspond to ranges of [$\tau ~\rm{[Gyr]} < 1.0$, $1.0 < \tau ~\rm{[Gyr]} < 2.0$ and $\tau ~\rm{[Gyr]} > 2.0$]{} respectively for ease of discussion.]{}
The Red Sample {#rs}
--------------
The left panel of Figure \[red\_s\] reveals that [smooth galaxies with red optical colours]{} [show a preference $(49.5\%$; see Figure \[red\_s\])]{} for rapid quenching timescales across all cosmic time resulting in a very low current SFR. [For these smooth red galaxies we see, at early times only, a preference for slow and intermediate timescales in the left panel of Figure \[red\_s\]. Perhaps this is the influence of intermediate galaxies (with $p_s \sim p_d \sim 0.5$), hence why similar high probability areas exist for both the smooth-like and disc-like galaxies in the left and right panels of Figure \[red\_s\]]{}. This is especially apparent considering there are far more of these intermediate galaxies than those that are definitively early- or late-types (see Table \[subs\]). These galaxies are those whose morphology cannot be easily distinguished either because they are at a large distance or because they are an S0 galaxy whose morphology can be interpreted by different users in different ways. @GZ2 find that S0 galaxies expertly classified by @NA10 are more commonly classified as ellipticals by GZ2 users, but have a significant tail to high disc vote fractions, giving a possible explanation as to the origin of this area of probability.
[The right panel of Figure \[red\_s\] reveals that red disc galaxies show similar preferences for rapid $(31.3\%)$ and slow $(44.1\%)$ quenching timescales. The preference for *very* slow ($\tau > 3.0 ~\rm{Gyr}$) quenching timescales (which are not seen in either the green valley or blue cloud, see Figures \[green\_v\] and \[blue\_c\])]{} suggests that these galaxies have only just reached the red sequence after a very slow evolution across the colour-magnitude diagram. Considering their limited number and our requirement for NUV emission, it is likely that these galaxies are currently on the edge of the red sequence having recently (and finally) moved out of the green valley. Table \[subs\] shows that $3.9\%$ of our sample are red sequence clean disc galaxies, i.e. red late-type spirals. This is, within uncertainties, in agreement with the findings of @Masters10, who find $\sim6\%$ of late-type spirals are red when defined by a cut in the $g-r$ optical colour (rather than with $u-r$ as implemented in this investigation) and are at the ‘blue end of the red sequence’.
[Despite the dominance of slow quenching timescales, the red disc galaxies also show some preference for rapid quenching timescales ($31.3\%$), similar to the red smooth galaxies but with a lower probability. Perhaps these rapid quenching timescales can also be attributed to a morphological change, suggesting that the quenching has occurred more rapidly than the morphological change to a bulge dominated system.]{}
Comparing the resultant SFRs for both the smooth- and disc-like galaxies in Figure \[red\_s\] by noticing [where the areas of high probability lie with respect to the bottom panel of Figure \[pred\] (which shows the predicted SFR at an observation time of $t\sim12.8~\rm{Gyr}$, the average ‘observed’ time of the GZ2 population)]{} reveals that red disc galaxies with a preference for slow quenching still have some residual star formation occurring, SFR$~\sim0.105 M_{\odot}yr^{-1}$, whereas the smooth galaxies with a dominant preference for rapid quenching have a resultant SFR$~\sim0.0075 M_{\odot}yr^{-1}$. This is approximately 14 times less than the residual SFR still occurring in the red sequence disc galaxies. Within error, this is in agreement with the findings of @Toj13 who, by using the VErsatile SPectral Analyses spectral fitting code [(VESPA; @Tojero07)]{}, found that red late-type spirals show 17 times more recent star formation than red elliptical galaxies.
These results for the red galaxies investigated here with NUV emission, have many implications for green valley galaxies, as all of these systems must have passed through the green valley on their way to the red sequence.
Green Valley Galaxies {#gv}
---------------------
{width="49.75000%"} {width="49.75000%"}
In Figure \[green\_v\] we can make similar comparisons for the green valley galaxies to those discussed previously for the subset of red galaxies studied. [For the red galaxies, an argument can be made for two possible tracks across the green valley, shown by the bimodal nature of both distributions in $\tau$ with a common area in the intermediate timescales region where the rapid and slow timescales peaked distributions intersect. However in the green valley this intermediate quenching timescale region becomes more significant [(in agreement with the conclusions of @Gonc12)]{}, particularly for the smooth-like galaxies (see the left panel of Figure \[green\_v\]). ]{}
The smooth galaxy parameters favour these intermediate quenching timescales ($40.6\%$) with some preference for slow quenching at early times ($z > 1$). The preference for rapid quenching of smooth galaxies has dropped by over a half compared to the red galaxies, [however this will be influenced by the observability of galaxies undergoing such a rapid quench which will spend significantly less time in the transitional population of the green valley]{}. [Those galaxies with such a rapid decline in star formation will pass so quickly through the green valley they will be detected at a lower number than those galaxies which have stalled in the green valley with intermediate quenching timescales;]{} accounting for the observed number of intermediate galaxies which are present in the green valley [and the dominance of rapid timescales detected for red galaxies for both morphologies.]{}
[The disc galaxies of the green valley now overwhelmingly prefer slow quenching timescales ($47.4\%$) with a similar amount of intermediate quenching compared to the smooth galaxy parameters ($37.6\%$; see Figure \[green\_v\]).]{} There is still some preference for galaxies with a star formation history which results in a high current SFR, suggesting there are also some late-type galaxies that have just progressed from the blue cloud into the green valley.
[If we compare Figure \[green\_v\] to Figure \[red\_s\] [we can see quenching has occurred at later (more recent) cosmic times]{} in the green valley [at least for red galaxies]{} for both morphological types.]{} Therefore both morphologies are tracing the evolution of the red sequence, confirming that the green valley is indeed a transitional population between blue cloud and red sequence regardless of morphology. Currently as we observe the green valley, its main constituents are very slowly evolving disc-like galaxies along with intermediate- and smooth-like galaxies which pass across it with intermediate timescales within $\sim 1.0-1.5~\rm{Gyr}$.
Given enough time ($t\sim4 - 5~\rm{Gyr}$), the disc galaxies will eventually fully pass through the green valley and make it out to the red sequence (the right panel of Figure \[sfr\_mass\_col\] shows galaxies with $\tau > 1.0~\rm{Gyr}$ do not approach the red sequence within $3~\rm{Gyr}$ post quench). This is most likely the origin of the ‘red spirals’.
{width="49.75000%"} {width="49.75000%"}
[If we consider [then that the green valley is a transitional population]{}, then we can expect that the ratio of smooth:disc galaxies that is currently observed in the green valley will evolve into the ratio observed for [the red galaxies with NUV emission investigated]{}. Table \[subs\] shows the ratio of smooth : disc galaxies in the observed red sequence of the GZ2 sample is $62:38$ whereas in the green valley it is $45:55$. [Making the very simple assumptions that this ratio does not change with redshift and that quenching is the only mechanism which causes a morphological transformation, we can infer that $31.2\%$ of the disc-dominated galaxies currently residing in the green valley would have to undergo a morphological change to a bulge-dominated galaxy.]{} We find that the fraction of the probability for green valley disc galaxies occupying the parameter space $\tau < 1.5 ~\rm{Gyr}$ is $29.4\%$, and therefore suggest that quenching mechanisms with these timescales are capable of destroying the disc-dominated nature of galaxies. [This is most likely an overestimate of the mechanisms with timescales that can cause a morphological change because of the observability of those galaxies which undergo such a rapid quench; @Martin07 showed that after considering the time spent in the green valley, the fraction of galaxies undergoing a rapid quench quadruples.]{}]{}
All of this evidence suggests that there are not just two routes for galaxies through the green valley [as concluded by S14]{}, but a continuum of quenching timescales which we can divide into three general regimes: [rapid ($\tau < 1.0 ~\rm{Gyr}$), intermediate ($1.0 < \tau < 2.0~\rm{Gyr}$) and slow ($\tau > 2.0~\rm{Gyr}$). The intermediate quenching timescales reside in the space between the extremes sampled by the UV/optical diagrams of S14; the inclusion of the intermediate galaxies in this investigation (unlike in S14) and the more precise Bayesian analysis, quantifies this range of $\tau$ and specifically ties the intermediate timescales to all variations of galaxy morphology.]{}
Blue Cloud Galaxies {#bc}
-------------------
\[bestfit\] {width="95.00000%"}
Since the blue cloud is considered to be primarily made of star forming galaxies we expect to have some difficulty in determining the most likely quenching model to describe them, as confirmed by Figure \[blue\_c\]. The attempt to characterise a star forming galaxy with a quenched SFH model leads to attribute the extremely blue colours of the majority of these galaxies to fast quenching at recent times (i.e. very little change in the SFR; see the right panel of Figure \[blue\_c\] in comparison with the bottom panel of Figure \[pred\]).
[This is particularly apparent for the blue disc population. Perhaps even galaxies which are currently quenching slowly across the blue cloud cannot be well fit by the quenching models implemented, as they still have high SFRs despite some quenching (although a galaxy has undergone quenching, star formation can still occur in a galaxy, just at a slower rate than at earlier times, described by $\tau$).]{}
There is a very small preference among blue bulge dominated galaxies for slow quenching which began prior to $z \sim 0.5 $. These populations have been blue for a considerable period of time, slowly using up their gas for star formation by the Kennicutt$-$Schmidt law [@Schmidt59; @Kennicutt97]. However the major preference is for rapid quenching at recent times in the blue cloud; this therefore provides some support to the theories for blue ellipticals as either merger-driven [($\sim76\%$; like those identified as recently quenched ellipticals with properties consistent with a merger origin by @McIntosh14) or gas inflow-driven reinvigorated star formation that is now slowly decreasing ($\sim24\%$; such as the population of blue spheroidal galaxies studied by @Kaviraj13).]{} However, we remind the reader that the quenching models used in this work do not provide an adequate fit to the blue cloud population.
The blue cloud is therefore primarily composed of both star forming galaxies with any morphology and smooth galaxies which [are undergoing a rapid quench, presumably after a previous event triggered star formation and turned them blue]{}.
Discussion {#diss}
==========
We have implemented a Bayesian statistical analysis of the star formation histories (SFHs) of a large sample of galaxies morphologically classified by Galaxy Zoo. We have found differences between the SFHs of smooth- and disc-like galaxies across the colour-magnitude diagram in the red sequence, green valley and blue cloud. In this section we will speculate on the question: what are the possible mechanisms driving these differences?
Rapid Quenching Mechanisms {#rapid}
--------------------------
[Rapid quenching is much more prevalent in smooth galaxies than disc galaxies, and the [red galaxies with NUV emission in this study]{} are also much more likely to be characterised by a rapid quenching model than green valley galaxies (ignoring blue cloud galaxies due to their apparent poor fit by the quenching models, see Figure \[blue\_c\]). In the green valley there is also a distinct lack of preference for rapid quenching timescales with $\tau < 0.5~\rm{Gyr}$; [however we must bear in mind the observability of a rapid quenching history declines with decreasing $\tau$. Rapid mechanisms may be more common in the green valley than seen in Figure \[green\_v\], however this observability should not depend on morphology so we can still conclude that rapid quenching mechanisms are detected more for smooth rather than disc galaxies.]{}]{} This suggests that this rapid quenching mechanism causes a change in morphology from a disc- to a smooth-like galaxy as it quickly traverses the colour-magnitude diagram to the red sequence, [supported by the number of disc galaxies that would need to undergo a morphological change in order for the disc : smooth ratio of galaxies in the green valley to match that of the red galaxies (see Section \[gv\])]{}. [From this indirect evidence we suggest that]{} this rapid quenching mechanism is due to major mergers.
Inspection of the galaxies contributing to this area of probability reveals that this does not arise due to *currently* merging pairs missed by GZ users which were therefore not excluded from the sample (see Section \[class\]), but by typical smooth galaxies with red optical and NUV colours that the model attributes to rapid quenching at early times. [Although a prescription for modelling a merger in the SFH is not included in this work we can still detect the after effects (see Section \[future\] for future work planned with ).]{}
One simulation of interest by @Springel05 showed that feedback from black hole activity is a necessary component of destructive major mergers to produce such rapid quenching timescales. Powerful quasar outflows remove much of the gas from the inner regions of the galaxy, terminating star formation on extremely short timescales. @Bell06, using data from the COMBO-17 redshift survey ($0.4 < z < 0.8$), estimate a merger timescale from being classified as a close galaxy pair to recognisably disturbed as $\sim 0.4~\rm{Gyr}$. @Springel05 consequently find using hydrodynamical simulations that after $\sim1~\rm{Gyr}$ the merger remnant has reddened to $u-r \sim 2.0$. This is in agreement with our simple quenching models which show (Figure \[sfr\_mass\_col\]) that within $\sim1~\rm{Gyr}$ the models with a SFH with $\tau < 0.4~\rm{Gyr}$ have reached the red sequence with $u-r ~\ga 2.2$. [This could explain the preference for red disc galaxies with rapid quenching timescales ($31.3\%$), as they may have undergone a major merger recently but are still undergoing a morphological change from disc, to disturbed, to an eventual smooth galaxy (see also @vdW09).]{}
[We reiterate that this rapid quenching mechanism occurs much more rarely in green valley galaxies of both morphologies than [for the subset of red sequence galaxies studied]{}, however does not fully characterise all the galaxies in either the red sequence or green valley.]{} Dry major mergers therefore do not fully account for the formation of any galaxy type at any redshift, supporting the observational conclusions made by @Bell07 [@Bundy07; @Kav14a] and simulations by @Genel08.
Intermediate Quenching Mechanisms {#int}
---------------------------------
[Intermediate quenching timescales are found to be equally prevalent across populations for both smooth and disc galaxies across cosmic time, [particularly in the green valley. Intermediate timescales are the prevalent mechanism for quenching smooth green valley galaxies, unlike the rapid quenching prevalent for red galaxies.]{}]{} We suggest that this intermediate quenching route must therefore be possible with routes that both preserve and transform morphology. It is this result of [another route through the green valley that is in contradiction]{} with the findings of S14.
If we once again consider the simulations of @Springel05, this time without any feedback from black holes, they suggest that if even a small fraction of gas is not consumed in the starburst following a merger (either because the mass ratio is not large enough or from the lack of strong black hole activity) the remnant can sustain star formation for periods of several Gyrs. The remnants from these simulations take $\sim5.5~\rm{Gyr}$ to reach red optical colours of $u-r \sim 2.1$. We can see from Figure \[sfr\_mass\_col\] that the models with intermediate quenching timescales of $1.0 \la ~\tau~\rm{[Gyr]} ~\la 2.0$ take approximately $2.5-5.5~\rm{Gyr}$ to reach these red colours.
We speculate that the intermediate quenching timescales are caused by gas rich major mergers, major mergers without black hole feedback and from minor mergers, the latter of which is the dominant mechanism. This is supported by the findings of @Lotz08 who find that the detectability timescales for equal mass gas rich mergers with large initial separations range from $\sim 1.1-1.9~\rm{Gyr}$, and of @Lotz11, who find in further simulations that as the baryonic gas fraction in a merger with mass ratios of 1:1-1:4 increases, so does the timescale of the merger from $\sim0.2~\rm{Gyr}$ (with little gas, as above for major mergers causing rapid quenching timescales) up to $\sim1.5~\rm{Gyr}$ (with large gas fractions). [Here we are assuming that the morphologically detectable timescale of a merger is roughly the same order as the quenching timescale. However, we must consider the existence of a substantial population of blue ellipticals [@Sch09], which are thought to be post-merger systems with no detectable morphological signatures of a merger but with the merger-induced starburst still detectable in the photometry. This photometry is an indicator for the SFH and therefore should present with longer timescales for the photometric effects of a merger than found in the simulations by @Lotz08 and @Lotz11. Observing this link between the timescale for the morphological observability of a merger and the timescales for the star formation induced by a merger is problematic, as evidenced by the lack of literature on the subject.]{}
@Lotz08 also show that the remnants of these simulated equal mass gas rich disc mergers (wet disc mergers) are observable for $\ga1~\rm{Gyr}$ post merger and [state that they appear “disc-like and dusty" in the simulations, which is consistent with an “early-type spiral morphology"]{}. Such galaxies are often observed to have spiral features with a dominant bulge, suggesting that such galaxies may divide the votes of the GZ2 users, producing vote fractions of $p_s \sim p_d \sim 0.5$. We believe this is why the intermediate quenching timescales are equally dominant for both smooth and disc galaxies across each population in Figures \[red\_s\] and \[green\_v\].
Other simulations (e.g. such as @Rob06 and @Barnes02) support the conclusion that both gas rich major mergers and minor mergers can produce disc-like remnants. Observationally, @Darg10a showed an increase in the spiral to elliptical ratio for merging galaxies ($0.005 < z < 0.1$) by a factor of two compared to the general population. They attribute this to the much longer timescales during which mergers of spirals are observable compared to mergers with elliptical galaxies, [confirming our hypothesis that the quenching timescales $\tau < 1.5 ~\rm{Gyr}$ preferred by disc galaxies may be undergoing mergers which will eventually lead to a morphological change]{}. Similarly, @Casteels13 observe that galaxies ($0.01 < z < 0.09$) which are interacting often retain their spiral structures and that a spiral galaxy which has been classified as having ‘loose winding arms’ by the GZ2 users are often entering the early stages of mergers and interactions.
[$40.6\%$ of the probability for smooth galaxies in the green valley arises due to intermediate quenching timescales (see Figure \[green\_v\]); this is in agreement with work done by @Kav14a [@Kav14b] who by studying SDSS photometry ($z<0.07$) state that approximately half of the star formation in galaxies is driven by minor mergers at $0.5 < z < 0.7$ therefore exhausting available gas for star formation and consequently causing a gradual decline in the star formation rate]{}. This supports earlier work by [@Kav11] who, using multi wavelength photometry of galaxies in COSMOS [@Scoville07], found that $70\%$ of early-type galaxies appear morphologically disturbed, suggesting either a minor or major merger in their history. [This is in agreement with the total percentage of probability with $\tau < 2.0 ~\rm[Gyr]$; $73.9\%$ and $59.3\%$, for the smooth red and green galaxies in Figures \[red\_s\] and \[green\_v\] respectively.]{} [Note that the star formation model used here is a basic one and has no prescription for reignition of star formation post-quench which can also cause morphological disturbance of a galaxy, like those detected by [@Kav11].]{}
[@Darg10a show in their Figure 6 that that beyond a merger ratio of $1:10$ (up to $\sim 1:100$), green is the dominant average galaxy colour of the visually identified merging pair in GZ. These mergers are also dominated by spiral-spiral mergers as opposed to elliptical-elliptical and elliptical-spiral. This supports our hypothesis that these intermediate timescales dominating in the green valley are caused in part by minor mergers. This is contradictory to the findings of @Mendez11 who find the merger fraction in the green valley is much lower than in the blue cloud, however they use an analytical light decomposition indicator ($Gini/M_{20}$; see @Lotz08) to identify their mergers, which tend to detect major mergers more easily than minor mergers. We have discussed the lower likelihood of a green valley galaxy to undergo a rapid quench, which we have attributed to major mergers (see Section \[rapid\]), despite the caveat of the observability and believe that this may have been the phenomenon that @Mendez11 detected.]{}
The resultant intermediate quenching timescales occur due to one interaction mechanism, unlike the rapid quenching, which occurs due to a major merger combined with AGN feedback, and decreases the SFR over a short period of time. Therefore any external event which can cause either a burst of star formation (depleting the gas available) or directly strip a galaxy of its gas, [for example galaxy harassment, interactions, ram pressure stripping, strangulation and interactions internal to clusters, would cause quenching on an intermediate timescale. Such mechanisms would be the dominant cause of quenching in dense environments;]{} considering that the majority of galaxies reside in groups or clusters (@Coil08 find that green valley galaxies are just as clustered as red sequence galaxies), it is not surprising that the majority of our galaxies are considered intermediate in morphology (see Table \[subs\]) [and therefore are undergoing or have undergone such an interaction.]{}
Slow Quenching Timescales {#slow}
-------------------------
Although intermediate and rapid quenching timescales are the dominant mechanisms across the colour-magnitude diagram, together they cannot completely account for the quenching of disc galaxies. S14 concluded that slow quenching timescales were the most dominant mechanism for disc galaxies. [However we show that: (i) intermediate quenching timescales are equally important in the green valley and (ii) rapid quenching timescales are equally important for [red galaxies with NUV emission]{}.]{} There is also a significantly lower preference for smooth galaxies to undergo such slow quenching timescales; suggesting that the evolution (or indeed creation) of typical smooth galaxies is dominated by processes external to the galaxy. [This is excepting galaxies in the blue cloud where a small amount of slow evolution of blue ellipticals is occurring, presumably after a reinvigoration of star formation which is slowly depleting the gas available according to the Kennicutt$-$Schmidt law.]{}
@Bamford09 using GZ1 vote fractions of galaxies in the SDSS, found a significant fraction of high stellar mass red spiral galaxies in the field. As these galaxies are isolated from the effects of interactions from other galaxies, the slow quenching mechanisms present in their preferred star formation histories are most likely due to secular processes (i.e. mechanisms internal to the galaxy, in the absence of sudden accretion or merger events; @KK04 [@Sheth12]). Bar formation in a disc galaxy is such a mechanism, whereby gas is funnelled to the centre of the galaxy by the bar over long timescales where it is used for star formation [[@Masters12; @Saint12; @Cheung13]]{}, consequently forming a ‘pseudo-bulge’ [@Kormendy10; @Simmons13].
If we believe that these slow quenching timescales are due to secular evolution processes, this is to be expected since these processes do not change the disc dominated nature of a galaxy.
Future Work {#future}
-----------
Due to the flexibility of our model we believe that the module will have a significant number of future applications, including the investigation of various different SFHs (e.g. constant SFR and starbursts). Considering the number of magnitude bands available across the SDSS, further analysis will also be possible with a larger set of optical and NUV colours, providing further constraints [and to ensure a more complete sample, containing a larger fraction of typical red sequence galaxies, if the need for NUV photometry was replaced with another band]{}. It would also be of interest to consider galaxies at higher redshift [(e.g. out to $z \sim 1$ with Hubble Space Telescope photometry and the GZ:Hubble project, see @Melvin14 for first results) and consider different redshift bins in order to study the build up of the red sequence with cosmic time. ]{}
With further use of the robust, detailed GZ2 classifications, we believe that will be able to distinguish any statistical difference in the star formation histories of barred vs. non-barred galaxies. This will require a simple swap of $\{p_s, p_d\}$ with $\{p_{bar}, p_{no bar}\}$ from the available GZ2 vote fractions. We believe that this will aid in the discussion of whether bars act to quench star formation (by funnelling gas into the galaxy centre) or promote star formation (by causing an increase in gas density as it travels through the disc) both sides of which have been fiercely argued [@Masters11; @Masters12; @Sheth05; @Ellison11].
Further application of the code could be to investigate the SFH parameters of:
1. Currently merging/interacting pairs in comparison to those galaxies classified as merger remnants, from their degree of morphological disturbance,
2. Slow rotators and fast rotators which are thought to result from dry major mergers on the red sequence [@Em11] and gas rich, wet major mergers [@Em07] respectively,
3. Field and cluster galaxies using the projected neighbour density, $\Sigma$, from @Baldry06.
Conclusion {#conc}
==========
We have used morphological classifications from the Galaxy Zoo 2 project to determine the morphology-dependent star formation histories of galaxies via a Bayesian analysis of an exponentially declining star formation quenching model. We determined the most likely parameters for the quenching onset time, $t_q$ and quenching timescale $\tau$ in this model for galaxies across the blue cloud, green valley and red sequence to trace galactic evolution across the colour-magnitude diagram. We find that the green valley is indeed a transitional population for all morphological types (in agreement with @Sch2014), however this transition proceeds slowly for the majority of disc-like galaxies and occurs rapidly for the majority of smooth-like galaxies in the red sequence. However, in addition to @Sch2014, [our Bayesian approach has revealed a more nuanced result, specifically that the prevailing mechanism across all morphologies and populations is quenching with intermediate timescales]{}. Our main findings are as follows:
1. [The subset of red sequence galaxies with NUV emission studied in this investigation]{} are found to have similar preferences for quenching timescales compared to the green valley galaxies, but occurs at earlier quenching times regardless of morphology (see Figures \[red\_s\] and \[green\_v\]). Therefore the quenching mechanisms currently occurring in the green valley were also active in creating [the ‘blue end of of the red sequence’]{} at earlier times; confirming that the green valley is indeed a transitional population, regardless of morphology.
2. [We confirm that the typical [red galaxy with NUV emission studied in this investigation]{}, is elliptical in morphology and conclude that it has undergone a rapid to intermediate quench at some point in cosmic time, resulting in a very low current SFR (see Section \[rs\].]{}
3. [The green valley as it is currently observed is dominated by very slowly evolving disc-like galaxies along with intermediate- and smooth-like galaxies which pass across it with intermediate timescales within $\sim 1.0-1.5~\rm{Gyr}$ (see Section \[gv\]).]{}
4. [There are many different timescales responsible for quenching, causing a galaxy to progress through the green valley, which are dependant on galaxy type, with the smooth- and disc-like galaxies each having different dominant star formation histories across the colour-magnitude diagram. These timescales can be roughly split into three main regimes; rapid ($\tau < 1.0~$Gyr), intermediate ($1.0 < \tau~$\[Gyr\] $< 2.0$) and slow ($\tau > 2.0~$ Gyr) quenching. ]{}
5. [Blue cloud galaxies are not well fit by a quenching model of star formation due to the continuous high star formation rates occurring (see Figure \[blue\_c\]).]{}
6. [Rapid quenching timescales are detected with a lower probability for green valley galaxies [than the subset of red sequence galaxies studied]{}.]{} We speculate that this quenching mechanism is caused by major mergers with black hole feedback, which are able to expel the remaining gas not initially exhausted in the merger-induced starburst and which can cause a change in morphology from disc- to bulge-dominated. The colour-change timescales from previous simulations of such events agree with our derived timescales [(see Section \[rapid\]). These rapid timescales are instrumental in forming red galaxies, however galaxies at the current epoch passing through the green valley do so at more intermediate timescales (see Figure \[green\_v\]).]{}
7. Intermediate quenching timescales ($1.0 < ~\tau~\rm{[Gyr]}~ < 2.0 $) are found with constant probability across red and green galaxies for both smooth- and disc-like morphologies, the timescales for which agree with observed and simulated minor merger timescales (see Section \[int\]). We hypothesise such timescales can be caused by a number of external processes, including gas rich major mergers, mergers without black hole feedback, galaxy harassment, interactions and ram pressure stripping. The timescales and observed morphologies from previous studies agree with our findings, including that this is the dominant mechanisms for intermediate galaxies such as early-type spiral galaxies with spiral features but a dominant bulge, which split the GZ2 vote fractions (see Section \[int\]).
8. Slow quenching timescales are the most dominant mechanism in the disc galaxy populations across the colour-magnitude diagram. Disc galaxies are often found in the field, therefore we hypothesise that such slow quenching timescales are caused by secular evolution and processes internal to the galaxy (see Section \[bc\]). [We also detect a small amount of slow quenching timescales for blue elliptical galaxies which we [attribute to a reinvigoration of star formation, the peak of which has passed and has started to decline by slowly depleting the gas available (see Section \[bc\]).]{}]{}
9. Due to the flexibility of this model we believe that the module compiled for this investigation will have a significant number of future applications, including the different star formation histories of barred vs non-barred galaxies, merging vs merger remnants, fast vs slow rotating elliptical galaxies and cluster vs field galaxies (see Section \[future\]).
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank the anonymous referee for helpful and insightful comments which improved both the presentation and the discussion of the results presented in this paper.
The authors would like to thank D. Forman-Mackey for extremely useful Bayesian statistics discussions, J. Binney for an interesting discussion on the nature of quenching and feedback in disc galaxies and M. Urry for the assistance in seeing the big picture.
RS acknowledges funding from the Science and Technology Facilities Council Grant Code ST/K502236/1. BDS gratefully acknowledges support from the Oxford Martin School, Worcester College and Balliol College, Oxford. KS gratefully acknowledges support from Swiss National Science Foundation Grant PP00P2\_138979/1. KLM acknowledges funding from The Leverhulme Trust as a 2010 Early Career Fellow. TM acknowledges funding from the Science and Technology Facilities Council Grant Code ST/J500665/1. KWW and LF acknowledge funding from a Grant-in-Aid from the University of Minnesota.
The development of Galaxy Zoo was supported in part by the Alfred P. Sloan Foundation. Galaxy Zoo was supported by The Leverhulme Trust.
Based on observations made with the NASA Galaxy Evolution Explorer. GALEX is operated for NASA by the California Institute of Technology under NASA contract NAS5-98034
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is <http://www.sdss.org/>. The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
This publication made extensive use of the Tool for Operations on Catalogues And Tables (TOPCAT; @Taylor05) which can be found at <http://www.star.bris.ac.uk/~mbt/topcat/>. Ages were calculated from the observed redshifts using the *cosmolopy* package provided in the Python module *astroPy*[^4]; @Rob13). This research has also made use of NASA’s ADS service and Cornell’s ArXiv.
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Testing starpy {#app_test}
==============
In order to test that can find the correct quenching model for a given observed colour, [25 synthesised galaxies were created with known SFHs (i.e. known values of $\theta$) from which optical and NUV colours were generated. These were input into to ensure that the known values of $\theta$ were reproduced, within error, for each of the 25 synthesised galaxies. Figure \[test\_mosaic\] shows the results for each of these 25 synthesised galaxies]{}, with the known values of $\theta$ shown by the red lines. In some cases this red line does not coincide with the peak of the distribution shown in the histograms for one parameter, however in all cases the intersection of the red lines is within the sample contours.
[We find peaks in the histograms across all areas of the parameter space in both dimensions of $[t, \tau]$, this ensures that the results presented in Figures \[red\_s\], \[green\_v\] & \[blue\_c\] arise due to a superposition of extended probability distributions, as opposed to a bimodal distribution of probability distributions across all galaxies.]{}
Using look up tables {#app_lookup}
====================
\[median\_lu\]
[Considering the size of the sample in this investigation of $126,316$ galaxies total, a three dimensional look up table (in observed time, quenching time and quenching rate) was generated using the star formation history function in to speed up the run time. Figure \[lookup\] shows an example of how using the look up table in place of the full function does not affect the results to a significant level. Table \[median\_lu\] quotes the median walker positions [(the 50th percentile of the Bayesian probability distribution) ]{}along with their $\pm 1\sigma$ ranges for both methods in comparison to the true values specified to test . The uncertainties incorporated into the quoted values by using the look up table are therefore minimal with a maximum $\Delta = 0.043$.]{}
Discarding Poorly Fit Galaxies {#discard}
==============================
\[discardnum\]
We discard walker positions returned by with a corresponding probability of $P(\theta_k|d_k) < 0.2$ in order to exclude galaxies which are not well fit by the quenching model; for example blue cloud galaxies which are still star forming will be poorly fit by a quenching model (see Section \[qmod\]). This raises the issue of whether we exclude a significant fraction of our galaxy sample and whether those galaxies reside in a specific location of the colour-magnitude. The fraction of galaxies which had all or more than half of their walker positions discarded due to low probability are shown in Table \[discardnum\].
This is not a significant fraction of either population, therefore this shows that the module is effective in fitting the majority of galaxies and that this method of discarding walker positions ensures that poorly fit galaxies are removed from the analysis of the results. Figure \[discarded\] shows that these galaxies with discarded walker positions are also scattered across the optical-NUV colour-colour diagram and therefore is also effective in fitting galaxies across this entire plane.
{width="90.00000%"}
Observability of quenching galaxies {#observe}
===================================
The numbers of galaxies found undergoing a rapid quench will be underestimated compared to the true value due to their observability, i.e. their time spent in the green valley is extremely short, so detecting a galaxy there is difficult. We considered this time spent in the green valley across our model parameter space of star formation histories and the results are shown below in Figure \[obsplot\].
[^1]: <http://zoo2.galaxyzoo.org/>
[^2]: [dan.iel.fm/emcee/](dan.iel.fm/emcee/)
[^3]: [github.com/zooniverse/starpy](github.com/zooniverse/starpy)
[^4]: <http://www.astropy.org/>
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Terrestrial-type exoplanets orbiting nearby red dwarf stars (M-dwarfs) are among the best targets for atmospheric characterization and biosignature searches in the near future. Recent evolutionary studies have suggested that terrestrial planets in the habitable zone of M-dwarfs are probably tidally locked and have limited surface water inventories as a result of their host stars’ high early luminosities. Several previous climate simulations of such planets have indicated that their remaining water would be transported to the planet’s permanent nightside and become trapped as surface ice, leaving the dayside devoid of water. Here we use a three-dimensional general circulation model with a water cycle and accurate radiative transfer scheme to investigate the surface water evolution on tidally locked terrestrial planets with limited surface water inventories. We show that there is a competition for water trapping between the nightside surface and the substellar tropopause in this type of climate system. Although under some conditions the surface water remains trapped on the nightside as an ice sheet, in other cases liquid water stabilizes in a circular area in the substellar region as a . Planets with 1 bar N[$_2$]{} and atmospheric CO[$_2$]{} levels greater than 0.1 bar retain stable dayside liquid water, even with very small surface water inventories. Our results reveal the diversity of possible climate states on terrestrial-type exoplanets and highlight the importance of surface liquid water detection techniques for future characterization efforts.'
author:
- Feng Ding
- 'Robin D. Wordsworth'
bibliography:
- 'fmspcm.bib'
title: Stabilization of dayside surface liquid water via tropopause cold trapping on arid tidally locked planets
---
Introduction {#sec:intro}
============
Terrestrial planets around M-dwarfs are the first potentially habitable exoplanets for which atmospheric characterization will be possible because of their enhanced transit probability, large transit depth and high planet-star contrast ratio [@seager2010exoplanet]. However, the climate evolution of this type of planet likely differs from that of the Earth in several important aspects [@shields2016habitability]. First, potentially habitable exoplanets around M-dwarfs are probably tidally locked by gravitational tidal torques. In particular, planets with small eccentricities are likely to be trapped in a 1:1 spin-orbit resonance [@Kasting1993habitable; @barnes2017tidal]. In addition, many of these planets may be deficient in water, because they receive an insolation above the runaway greenhouse threshold during their host stars’ pre-main-sequence phase, leading to extensive water loss [@ramirez2014premain; @luger2015waterloss; @tian2015waterloss]. Studies that assumed an Earth-like atmospheric composition have shown that for such planets, water vapor would be transported by the atmospheric circulation to the cold permanent nightside and trapped as surface ice [@heath1999habitability; @joshi2003synchronously; @menou2013watertrap; @yang2014watertrap]. However, the water cycle on these planets over a wider range of atmospheric parameters has not yet been investigated in detail.
Moist general circulation model {#sec:methods}
===============================
Here we investigate the water trapping on tidally locked planets with limited surface water inventories (referred to as “arid planets” in comparison with the “aqua-planet” simulations widely used in previous studies) over a range of atmospheric compositions. We developed a three-dimensional general circulation model (GCM) with a self-consistent water cycle based on our dry GCM that uses a line-by-line approach to describe the radiative transfer for simulating diverse planetary atmospheres [@ding2019fmspcm]. The physical schemes for moist processes are very similar to those used in @merlis2010tidally, including a moist convection scheme, a large-scale condensation scheme and a planetary boundary scheme. Our moist GCM can reproduce the climatology on tidally locked Earth-like aqua-planets in @merlis2010tidally when using the same gray-gas radiative transfer calculation.
To simulate the climate of tidally locked terrestrial planets around M-dwarfs, we use AD Leo’s stellar spectrum. The planet has the same radius and surface gravity as Earth’s and an orbital period of 35 Earth days. Both the eccentricity and obliquity are zero. The incoming stellar radiation above the substellar point is 1200 W m$^{-2}$ . 2000 spectral points and four quadrature points (two upwelling and two downwelling) are used for both shortwave and longwave radiative calculations. The atmosphere is made of N[$_2$]{}, CO[$_2$]{} and H[$_2$]{}O. The column-integrated mass of N[$_2$]{} is equivalent to the mass of a 1 bar N[$_2$]{} atmosphere if the atmosphere was made of N[$_2$]{} alone. We performed simulations with various CO[$_2$]{} levels and found the surface water starts to be converged to the substellar region when the CO[$_2$]{} mass mixing ratio is 0.1. So we chose two CO[$_2$]{} levels to present our results and discuss the two climate regimes: (1) one has a present-day Earth-like CO[$_2$]{} level with the CO[$_2$]{} volume mixing ratio of 400 ppmv (referred to as the “low CO[$_2$]{} run”); (2) the other has a relatively higher CO[$_2$]{} level and the column-integrated mass of CO[$_2$]{} is equivalent to the mass of a 0.1 bar CO[$_2$]{} atmosphere if the atmosphere was made of CO[$_2$]{} alone (referred to as the “high CO[$_2$]{} run”). Thus, for the latter case, CO[$_2$]{} also contributes substantially to the total mass of the atmosphere. The total surface pressure is 1.1 bar with a CO[$_2$]{} volume mixing ratio of 0.063 relative to N[$_2$]{}. In both simulations, the water vapor mixing ratio is controlled by the atmospheric circulation and surface water distribution.
To simulate the surface water evolution, we implemented a bucket water model in which the water depth of the bucket varies with the local precipitation and evaporation (see Appendix \[app:bucket\] for details). The initial condition of the surface water inventory is 1 m of ice uniformly distributed on the nightside surface. The GCM simulations are run until reaching the statistical equilibrium state ($\sim$3000 days) when the absorbed stellar radiation of the climate system is balanced by outgoing longwave radiation and the surface water distribution stops changing with time. Averaged results over the last 600 days are presented.
Equilibrium distribution of surface water inventory and water trapping competition {#sec:equilibrium}
==================================================================================
![Long-term mean surface water depth for the low CO[$_2$]{} run with volume mixing ratio of 400 ppmv (a) and for the high CO[$_2$]{} run with volume mixing ratio of 0.063 (b). The white dot is the substellar point.[]{data-label="fig:depth"}](fig1_bp){width="\columnwidth"}
Fig. \[fig:depth\] shows the distributions of the surface water depth in the equilibrium state of the two simulations, after the surface water evolution algorithm had converged. For the low CO[$_2$]{} run, the surface water is stably trapped on the nightside as an ice sheet (Fig. \[fig:depth\]a). But for the high CO[$_2$]{} run, the initial nightside surface ice migrates towards the substellar area and eventually forms a surrounded by a hyper-arid desert on the dayside. This substellar “oasis” occupies a circular area roughly 20$^\circ$ around the substellar point (Fig. \[fig:depth\]b). The formation of this oasis results from the moist climate dynamics, specifically, the cold trapping effect of water vapor near the substellar tropopause and the associated precipitation. In Fig. \[fig:depth\]b, the center of the oasis is slightly shifted eastward relative to the substellar point, because the upwelling motion west of the substellar point is suppressed by planetary equatorial waves [@yang2013innerhz].
![Zonal mean distribution of the specific humidity in the tidally locked coordinate for the low CO[$_2$]{} run (a) and high CO[$_2$]{} run (b). q$_{sat,n}$ and q$_{sat,t}$ above the color bar are the saturation specific humidity calculated by the conditions on the nightside surface and at the substellar tropopause, respectively. The dotted gray, white and light blue area at the bottom in the two panels mark the distribution of the dry surface, nightside surface ice in (a) and the substellar open water in (b), respectively. The gray dashed contours in the two panels mark the tropospheric overturning circulation where air rises in the substellar region.[]{data-label="fig:q"}](fig3_q){width="\columnwidth"}
By comparing the two simulations with different CO[$_2$]{} levels, we can elucidate how the cold trapping competition between the nightside surface and the substellar tropopause determines the equilibrium distribution of the surface water inventory on a tidally locked arid planet. To better illustrate this competition, we plot the specific humidity (q, in kg/kg) distribution in the tidally locked coordinate [^1] [@koll2015phasecurve; @wordsworth2015heat] in Fig. \[fig:q\] and we use the saturation specific humidity (q$_{sat}$) calculated by the conditions on the nightside surface (q$_{sat,n}$) and at the substellar tropopause (q$_{sat,t}$) to evaluate the strength of water vapor cold trapping. For the low CO[$_2$]{} run, q$_{sat,n}$ $ \approx 3.74\times10^{-9}$, q$_{sat,t}$ $\approx 1.25\times10^{-6}$ and q$_{sat,n}$ $\ll$ q$_{sat,t}$. Therefore, the atmospheric water vapor is constrained by the nightside surface ice sheet and well mixed in the atmosphere without any condensation (Fig. \[fig:q\]a). But for the high CO[$_2$]{} run, q$_{sat,n}$ $\approx 7.2\times10^{-4}$, q$_{sat,t}$ $\approx 1\times10^{-5}$. Precipitation forms within the upwelling branch of the overturning circulation, giving rise to an open water area with the same size as the area where air rises. In Fig. \[fig:q\]b, the atmospheric profile follows the moist adiabat above the open water, but once the air parcel leaves the upwelling branch of the overturning cell condensation ceases, and the water vapor concentration becomes well mixed again. In fact, most water vapor on the nightside is not last saturated exactly at the substellar tropopause but in a region around the tropopause, shown as the green shading area in Fig. \[fig:q\]b. Hence, the nightside atmospheric specific humidity is slightly higher than q$_{sat,t}$ but still well below q$_{sat,n}$. As a result, any nightside surface water inventory slowly sublimates into the atmosphere, explaining the migration of surface water from nightside to the substellar area. This occurs despite the fact that the nightside thermal stratification is very stable due to the strong near-surface temperature inversion.
![Hemispheric-averaged dayside (solid) and nightside (dashed) vertical temperature profiles in the low CO[$_2$]{} (black) and the high CO[$_2$]{} (red) simulations. The hemispheric-averaged day and nightside surface temperatures are marked by crosses and circles, respectively.[]{data-label="fig:t"}](fig4_t){width="0.5\columnwidth"}
In the high CO[$_2$]{} run, the substellar tropopause wins in the competition of water vapor cold trapping mainly because of the radiative effect of CO[$_2$]{}. The ratio of the two saturation specific humidities at the two cold traps can be written as
$$\label{eq:coldtrap}
\frac{q_{sat,t}}{q_{sat,n}} \approx \frac{e_{sat}(T_{trop})}{e_{sat}(T_{sn})} \frac{p_{sn}}{p_{trop}}$$
where $e_{sat}$ is the saturation vapor pressure of water, $T_{trop}$ and $p_{trop}$ are the air temperature and pressure at the substellar tropopause, and $T_{sn}$ and $p_{sn}$ are the surface temperature and pressure on the nightside. Both $p_{sn}$ and $e_{sat}(T_{trop})$ increase a little in the high CO[$_2$]{} run due to the mass contribution and near-infrared absorption of CO[$_2$]{}. But the major impact of the high CO[$_2$]{} level is warming of the nightside surface temperature from 170 K under the present-day CO[$_2$]{} level to 255 K when the CO[$_2$]{} volume mixing ratio is 0.063 (Fig. \[fig:t\]). The saturation surface specific humidity is therefore raised by nearly five orders of magnitude, after which point it exceeds the value around the substellar tropopause (Fig. \[fig:q\]). The enhanced warming on the nightside surface is attributed to the enhanced infrared emissivity of the atmosphere, not only from the high CO[$_2$]{} level, but also from the higher water vapor concentration (Fig. \[fig:q\]). This warming mechanism has been discussed previously in the context of completely dry atmospheres [@wordsworth2015heat; @koll2016heat; @ding2019fmspcm]. Another factor that can potentially affect the nightside surface temperature is turbulent mixing in the planetary boundary layer. However, in our high CO[$_2$]{} run, the sensible heat flux from the atmosphere to the nightside surface is less than 5 W m$^{-2}$, while the infrared radiative flux reaching the nightside surface is $\sim$ 230 W m$^{-2}$. Hence, the nightside surface is mainly in radiative equilibrium and the turbulent heat mixing plays a minor role here.
CO[$_2$]{} budget and stability of the two equilibrium climate states {#sec:co2}
=====================================================================
So far, we have shown the two equilibrium climate states under different but fixed CO[$_2$]{} levels. However, the CO[$_2$]{} partial pressure on Earth is regulated by the silicate-weathering feedback [@walker1981co2], and for many terrestrial exoplanets a similar feedback may be important. Although it is challenging to evaluate the outgassing and weathering of CO[$_2$]{} on exoplanets in general, we can gain insight into the CO[$_2$]{} budget of the two equilibrium climate states qualitatively by using a simple equation for the partial pressure of CO[$_2$]{}. In this simple CO[$_2$]{} budget model, the surface partial pressure of CO[$_2$]{} is determined by outgassing and weathering, and the CO[$_2$]{} weathering rate is primarily determined by the CO[$_2$]{} surface partial pressure and the surface area fraction of liquid water. $$\label{eq:co2}
\frac{dP_{co2}}{dt} = \begin{cases}
\Phi, & P_{co2}<0.07\ \mathrm{bar}\ (\mathrm{nightside\ cold\ trap}) \\
\Phi-W_0e^{k[T_s(P_{co2})-T_0]} A_{frac}. & P_{co2} \ge 0.07\ \mathrm{bar}\ (\mathrm{substellar\ tropopause\ cold\ trap}) \\
\end{cases} \\$$ where $P_{co2}$ is the surface CO[$_2$]{} partial pressure, $\Phi$ is the CO[$_2$]{} outgassing rate, $W_0$ = 70 bar Gyr$^{-1}$, $T_0$ = 288 K are the CO[$_2$]{} weathering rate and surface temperature in the reference state, k = 0.1 K$^{-1}$ is the weathering-temperature rate constant [@abbot2016co2], and Ts, Afrac = 0.03 are the substellar surface temperature and fraction of the substellar liquid water area, respectively. In our CO[$_2$]{} budget model, the weathering rate depends on the surface CO[$_2$]{} partial pressure indirectly through the substellar surface temperature. The direct power law dependence on the surface CO[$_2$]{} partial pressure is ignored here, because recent work based on laboratory experiments that have quantified the dissolution rate of a variety of silicate minerals has indicated that this dependence is weak [@graham2019pressure].
In our high CO[$_2$]{} run, the surface partial pressure of CO[$_2$]{} is 1.1$\times$0.063=0.07 bar. So we use a CO[$_2$]{} partial pressure of 0.07 bar as the threshold above which surface water migrates from the nightside to the substellar area and then weathering occurs. The timescale of the surface water migration is short compared to the weathering timescale, so we assume that the surface water is always in an equilibrium state in this model. When the surface CO[$_2$]{} partial pressure is 0.07 bar, a critical CO[$_2$]{} outgassing rate of $\Phi_c$=52 bar Gyr$^{-1}$ is found in this CO[$_2$]{} budget model by which two climate regimes can be discriminated. If the CO[$_2$]{} outgassing rate is less than $\Phi_c$, the climate on tidally locked arid planets has no stable equilibrium state, and the surface water will oscillate between the nightside ice and substellar water states, in a pattern that resembles the limit cycles predicted to occur for some planets near the outer edge of habitable zone [@haqq2016limitcycle]. For CO[$_2$]{} outgassing rate larger than $\Phi_c$, the substellar water solution becomes a stable climate state because the weathering rate can always balance the outgassing rate through the CO[$_2$]{} partial pressure dependence. Interestingly, the critical value of the CO[$_2$]{} outgassing rate found in our simple formula is close to Earth’s CO[$_2$]{} outgassing rate. In reality, weathering is affected by many complicated processes, including uplift rates, the planetary tectonic regime, surface maficity and other factors [@macdonald2019arc; @oneill2007superearth; @valencia2007superearth], and hence can be expected to vary over a wide range.
Three climate regimes under increased insolation {#sec:evo}
================================================
![Schematic of the long-term climate evolution and possible climate states on a tidally locked arid planet receiving various insolations. The processes that lead to climate transitions between various states are marked by red arrows, e.g., the CO[$_2$]{} cycle (solid) and impact erosion (dashed). S$_1$ is the critical absorbed stellar flux when the substellar liquid water state enters the runaway greenhouse, which is $\sim$ W m$^{-2}$ [@yang2013innerhz]. S$_2$ is the critical absorbed stellar flux when the nightside ice state enters the runaway greenhouse, $\sim$ W m$^{-2}$ in our simulation.[]{data-label="fig:evo"}](drawing2){width="\columnwidth"}
The two simulations we have discussed with varying CO[$_2$]{} levels were performed under an Earth-like insolation level. Because planets closer to their host stars are more observationally favorable, we also investigated how the substellar surface water and nightside surface ice state evolve when the arid planet receives increased insolation. Similar to aqua-planets, arid planets with stable substellar surface water should enter the runaway greenhouse state when the absorbed stellar radiation of the planet exceeds the upper limit of the outgoing longwave radiation that the planet can emit. We did not attempt to simulate this climate transition in our current model because water vapor is a non-dilute component when approaching the runaway greenhouse state for which special treatment is required [@ding2016condensible; @pierrehumbert2016nondilute]. However, based on previous simulations of the runaway greenhouse on tidally locked aqua-planets [@yang2013innerhz], the critical absorbed stellar flux for this transition on Earth-like tidally locked planets can be estimated as S$_1 \sim$ W m$^{-2}$. The nightside surface temperature also increases with the insolation. In our simulations with enhanced insolation and stable nightside ice, the ice begins to irreversibly sublimate into the atmosphere when the absorbed stellar radiation exceeds S$_2 \sim$ W m$^{-2}$.
To summarize, three climate regimes on a tidally locked arid planet can be discriminated by these two critical fluxes S$_1$ and S$_2$ as illustrated in Fig.\[fig:evo\]. When the absorbed stellar radiation of the climate system S$_{abs} <$ S$_1$, the surface water can either distribute on the nightside as ice sheet or in the substellar area. These two states can transfer from one to the other via changes in the carbonate-silicate cycle, as discussed in Section \[sec:co2\]. When S$_1 <$ S$_{abs} <$ S$_2$, surface water can only exist as an ice sheet on the nightside. A runaway greenhouse will occur if the cold trap on the nightside surface is weaker, for example, due to nightside surface warming by CO[$_2$]{}. This is similar to the bi-stable moist climate states discussed for close-in arid exoplanets [@leconte2013bistable]. Finally, when S$_{abs} >$ S$_2$, all water is present in the atmosphere as vapor, and no surface water is possible. On long timescales this atmospheric water vapor would be vulnerable to photodissociation and hydrogen loss to space [@kasting1988runaway].
Conclusion and discussions {#sec:discussions}
==========================
Our results demonstrate that even for tidally locked arid exoplanets with low water inventories, climate solutions exist where dayside surface liquid water is stable. This implies habitable conditions on planets around M-stars may be more common than previously considered. Future research efforts will need to focus on the effects of other volatile cycles and water-rock interactions on the climates of such planets, as well as on developing robust observational techniques to constrain the abundance of surface liquid water remotely [@robinson2010glint; @loftus2019sulfate].
same as in @merlis2010tidally, our GCM is cloud-free to allow a focus on the key physical processes (e.g., the radiative impact of greenhouse gases) without invoking cloud parameterizations. Cloud effects involve various atmospheric processes over a wide range of spatial scales and always need to be parameterized in large-scale climate models, which remain a major problem even for modern climate change simulations [@stephens2005cloud]. Despite the challenges associated with cloud modeling in GCMs, via basic reasoning we can establish that cloud radiative effects would have limited impact on the water trapping competition on arid tidally locked planets. When the surface water is trapped on the nightside as ice, clouds would be scarce in the atmosphere and could not contribute to the migration of surface water towards the substellar region. In the climate state with substellar surface liquid water, the troposphere above the surface water region would be covered by deep convective clouds that are highly reflective. The upper troposphere would be covered by high cirrus clouds that could extend to the nightside, as seen in some aqua-planet climate simulations [@yang2013innerhz]. These cirrus clouds would warm the nightside surface, making the dayside tropopause cold trapping more robust. The substellar surface water only occupies a very small fraction of the planet’s surface area (Figs \[fig:depth\] and \[fig:q\]), where only 11% of incoming radiation is received in total. So even if the deep convective clouds increased the local albedo to 70% above the dayside liquid water region, the total planetary albedo would only be raised by 6%, which would have a limited effect on the global mean surface temperature. Even if inclusion of clouds in the simulations somehow induced a more significant cooling effect, it would only mean that a somewhat higher CO[$_2$]{} mixing ratio would be required to drive the surface water from the nightside to dayside.
In situations where the substellar tropopause cold trap dominates the hydrological cycle, the climate regime we find is a natural extension of Earth’s tropical climate in which the Hadley circulation transports surface moisture towards the equator and then precipitation forms within the upwelling branch of the overturning cell. The stable substellar water region resembles Earth’s tropical rain belt (Intertropical Convergence Zone, ITCZ) and the surrounding dry land resembles Earth’s subtropical deserts. Similar comparisons to our results can be made with Titan, where a hydrological cycle operates with methane as the condensible component. In Titan’s case, the seasonal cycle is much stronger due to the small thermal inertia of the lower boundary and strong seasonal migration of the ICTZ gives rise to Titan’s dry tropics and polar lakes [@mitchell2016titan].
We thank the referee for thoughtful comments that improved the manuscript. The GCM simulations in this paper were carried out on the FASRC Cannon cluster supported by the FAS Division of Science, Research Computing Group at Harvard University. F.D. and R.W. were supported by NASA grants NNX16AR86G and 80NSSC18K0829.
Bucket water model iteration {#app:bucket}
============================
Because the evolution of the surface water inventory is a much slower process than the atmospheric dynamics, we implemented an iteration scheme for the surface water evolution. The time-mean tendency of water depth in the bucket model is evaluated every 200 days and is used to update the surface water distribution by multiplication with a 10-year timestep, similar to the algorithm described in @wordsworth2013mars to explore surface ice evolution on early Mars. To validate the surface water iteration scheme, we performed a simulation for the high CO[$_2$]{} run with different initial surface water distributions. When the surface water is initially distributed in the extratropical region (upper right panel in Fig. \[fig:bp\_time\]), the climate evolves towards the same equilibrium climate state and the same substellar “oasis” forms.
![Evolution of the surface water inventory in the high CO[$_2$]{} run. The total model integration time is 3000 days. The surface water iteration is performed every 200 days with a 10-year timestep. Left and right columns are the snapshots for the two cases with initial surface water on the nightside surface and in the extratropical region, respectively. []{data-label="fig:bp_time"}](fig2_bptime){width="\columnwidth"}
[^1]: <https://github.com/ddbkoll/tidally-locked-coordinates>
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Given a sequence $(\mT_1, \mT_2, \dots)$ of random $d \times d$ matrices with nonnegative entries, suppose there is a random vector $X$ with nonnegative entries, such that $ \sum_{i \ge 1} \mT_i X_i $ has the same law as $X$, where $(X_1, X_2, \dots)$ are i.i.d. copies of $X$, independent of $(\mT_1, \mT_2, \dots)$. Then (the law of) $X$ is called a fixed point of the multivariate smoothing transform. Similar to the well-studied one-dimensional case $d=1$, a function $m$ is introduced, such that the existence of $\alpha \in (0,1]$ with $m(\alpha)=1$ and $m''(\alpha) \le 0$ guarantees the existence of nontrivial fixed points. We prove the uniqueness of fixed points in the critical case $m''(\alpha)=0$ and describe their tail behavior. This complements recent results for the non-critical multivariate case. Moreover, we introduce the multivariate analogue of the derivative martingale and prove its convergence to a non-trivial limit.'
address: |
$^*$ [Uniwersytet Warszawski\
Instytut Matematyki\
ul. Banacha 2\
02-097 Warszawa, Poland]{}\
$^\dagger$ Uniwersytet Wrocławski\
Instytut Matematyczny\
pl. Grunwaldzki 2/4\
50-384 Wrocław, Poland
author:
- 'Konrad Kolesko$^*$, Sebastian Mentemeier$^\dagger$'
title: 'Fixed Points of the Multivariate Smoothing Transform: The Critical Case'
---
Introduction
============
Let $d \ge 2$ and $(\mT_i)_{i \ge 1}$ be a sequence of random $d\times d$-matrices with nonnegative entries. Assume that $$N := \# \{ i \, : \, \mT_i \neq 0 \}$$ is finite a.s. We will presuppose throughout that the $(\mT_i)_{i \ge 1}$ are ordered in such a way that $\mT_i \neq 0$ if and only if $i \le N$. Given a random variable $X \in \Rdnn=[0,\infty)^d$, let $(X_i)_{i \ge 1}$ be i.i.d. copies of $X$ and independent of $(\mT_i)_{i \ge 1}$. Then $ \sum_{i=1}^N \mT_i X_i$ defines a new random variable in $\Rdnn$. If it holds that $$\label{eq:FPE} X \eqdist \sum_{i=1}^N \mT_i X_i,$$ where $\eqdist$ means same law, then we call the law $\law{X}$ of $X$ a fixed point of the multivariate smoothing transform (associated with $(\mT_i)_{i \ge 1}$). By an slight abuse of notation, we will also call $X$ a fixed point.
This notion goes back to Durrett and Liggett [@DL1983]. For $d=1$, they proved (see also [@Liu1998; @ABM2012]) that properties of fixed points are encoded in the function $m(s) := \E \sum_{i=1}^N T_i^s$ (here $(T_i)_{i\ge 1}$ are nonnegative random numbers): If $m(\alpha) =1$ and $m'(\alpha) \le 0$ for some $\alpha \in (0,1]$ and some non-lattice and moment assumptions are satisfied, then there is a fixed point which is unique up to scaling. Conversely, the condition $m(\alpha) =1$, $m'(\alpha) \le 0$ for some $\alpha \in (0,1]$ is also necessary for the existence of fixed points.
Moreover, if $\LTfp(r) = \Erw{e^{-rX}}$ is the Laplace transform of a fixed point, then there is a positive function $L$, slowly varying at $0$, and $K >0$ such that $$\label{eq:slowvar1} \lim_{r \to 0} \frac{1-\LTfp(r)}{L(r) r^\alpha} = K.$$ The function $L$ is constant if $m'(\alpha) < 0$ and $L(t) = (\abs{ \log t} \vee 1)$ if $m'(\alpha)=0$, the latter being called the *critical case*. For $\alpha<1$, the property implies that the fixed points have Pareto-like tails with index $\alpha$, i.e. $ \lim_{t \to \infty} t^{-\alpha}\P{X>t}/L(1/t) \in (0, \infty)$, see [@Liu1998] for details. Tail behavior in the case $\alpha=1$, in which there is no such implication, is investigated in [@Guivarch1990; @Liu1998; @Buraczewski2009]. Existence and uniqueness results in the multivariate setting $d \ge 2$ for the non-critical case have been recently proved in [@Mentemeier2013]. The aim of this note is to provide the corresponding result for the multivariate critical case. In order to so, we will first review necessary notation and definitions from [@Mentemeier2013], in particular introducing the multivariate analogue of the function $m$, as well as a result about the existence of fixed points in the critical case. Following the approach in [@Biggins1997; @Biggins2005b; @Kyprianou1998] we will then prove that a multivariate regular variation property similar to holds for fixed points (with an essentially unique, but yet undetermined slowly varying function $L$), which we use in order to prove the uniqueness of fixed points, up to scalars. Under some extra (density) assumption, we identify the slowly varying function to be the logarithm also in the multivariate case, which allows us to introduce and prove convergence of the multivariate version of the so-called *derivative martingale*, a notion coined in [@Biggins2004]. It appears prominently in the limiting distribution of the minimal position in branching random walk, see [@Aidekon2013; @Aidekon2014; @Biggins2004] for details and further references.
Statement of Results
====================
We start by introducing the assumptions and some notation needed therefore. Write $\Pset(\Rdnn)$ for the set of probability measures on $\Rdnn$ and $\Mset:=M(d\times d,\Rnn)$ for the set of $d \times d$-matrices with nonnegative entries. Given a sequence $\T:=(\mT_i)_{i \ge 1}$ of random matrices from $\Mset$, only the first $N$ of which are nonzero, with $N< \infty$ a.s., we aim to determine the set of fixed points of the mapping $\ST : \Pset(\Rdnn) \to \Pset(\Rdnn),$ $$\ST \eta := \law{\sum_{i=1}^N \mT_i X_i}, \qquad \text{for $(X_i)_{i\ge 1}$ i.i.d.~ with law $\eta$ and independent of $(\mT_i)_{i \ge 1}$}.$$ Without further mention, we assume $(\Omega, \B, \Prob)$ to be a probability space which is rich enough to carry all the occurring random variables.
The weighted branching process and iterations of $\ST$
------------------------------------------------------
Let $\V := \bigcup_{n=0}^\infty \N^n$ be a tree with root $\emptyset$ and Ulam-Harris labeling. We write $\abs{v}=n$ if $v=v_1 \cdots v_n \in \{1, \dots,N\}^n$, $v|k = v_1 \cdots v_k$ for the ancestor in the $k$-th generation and $vi=v_1 \cdots v_n i$ for the $i$-th child of $v$, $i \in \N$.
To each node $v \in \V$ assign an independent copy $\T(v)$ of $\T$ and, given a random variable $X \in \Rdnn$, as well an independent copy $X(v)$ of $X$, such that $(\T(v))_{v \in \V}$ and $(X(v))_{v \in \V}$ are independent. Introduce a filtration by $$\B_n ~:=~ \sigma\bigg( (\T(v))_{\abs{v}<n}\bigg).$$ Upon defining recursively the product of weights along the path from $\emptyset$ to $v$ by $$\mL(\emptyset) := \Id, \qquad \mL(vi) = \mL(v) \mT_i(v),$$ we obtain the iteration formula $$\ST^n \law{X} = \law{\sum_{\abs{v}=n} \mL(v) X(v)},$$ which in terms of Laplace transforms $\LTa(x) = \E \bigg[e^{- \skalar{x,X}} \bigg]$ becomes $$\label{eq:STLT} \ST^n \LTa(x) = \E \bigg[ \prod_{\abs{v}=n} \LTa( \mL(v)^\top x) \bigg], \qquad x \in \Rdnn.$$
Assumptions
-----------
As noted before, we assume $$\tag{A1}\label{A1} \text{the r.v. $N := \# \{ i \, : \, \mT_i \neq 0 \}$ equals $\sup\{ i \, : \, \mT_i \neq 0 \} $ and is finite a.s.}$$ and$$N \ge 1 \text{ a.s. and } 1 < \E N < \infty.$$ This assumption guarantees, that the underlying Galton-Watson tree (consisting of the nodes $v$ with $\mL(v) \neq 0$) is supercritical and allows to define a probability measure $\mu$ on $\Mset$ by $$\label{eq:defmu}
\int \, f(\ma) \, \mu(d\ma) ~:=~ \frac{1}{\E N}\Erw{\sum_{i=1}^N f(\mT_i) }.$$ On the (support of the) measure $\mu$, we will impose the following condition $\condC$:
\[defn:condc\] A subsemigroup $\Gamma \subset \Mset$ satisfies condition $\condC$, if
1. every $\ma$ in $\Gamma$ is [*allowable*]{}, i.e. it has no zero row nor column, and
2. $\Gamma$ contains a matrix with all entries positive $(>0)$.
For the measure $\mu$ as defined in Eq. , we assume $$\tag{A2} \label{A3} \text{ The subsemigroup $[\supp \mu]$ generated by $\supp \mu$ satisfies $\condC$.}$$ Note that if $\ma \in \Mset$ is an allowable matrix, then we can define its action on $\Sp := \Sd \cap \Rdnn$ by $$\ma \as u ~:=~ \frac{\ma u}{\abs{\ma u}}, \qquad u \in \Sp.$$ Furthermore, we need a multivariate analogue of a non-lattice condition: Recall that a matrix $\ma$ with all entries positive has a algebraic simple dominant eigenvalue $\lambda_\ma >0$ with corresponding normalized eigenvector $v_\ma$ the entries of which are all positive. $$\tag{A3} \label{A4} \text{The additive group generated by $\{ \log \lambda_{\ma} \, : \, \ma \in [\supp \, \mu] $ has all entries positive$\}$ is dense in $\R$}$$
Let $\mM, (\mM_n)_{n \in \N}$ be i.i.d. random matrices with law $\mu$, and write $\mPi_n := \prod_{i=1}^n \mM_n$. Then it is shown in [@Mentemeier2013], that the multivariate analogue of the function $m$ is given by $$m(s) := \E[N] \lim_{n \to \infty} \left( \E \norm{\mPi_n}^s \right)^{1/n},$$ which is finite on $$I_\mu := \{ s > 0 \, : \, \E\bigg[ \norm{\mM}^s \bigg]< \infty \}.$$ On $I_\mu$, it is log-convex, and thus the left-handed derivatives $m'(s^-)$ exist.
We assume to be in the critical case, i.e. $$\label{A5}\tag{A4} \text{there is $\alpha \in (0,1] \cap I_{\mu}$ with $m(\alpha)=1$ and $m'(\alpha^-)=0$.}$$
For the multivariate case, the classical *T-log T* condition splits into an upper bound and a lower bound: Introducing $\iota(\ma) := \inf_{u \in \Sp}\abs{\ma u}$, we observe that $\iota(\ma) >0$ for $\ma \in \Mset$, and that for all $u \in \Sp$, $$\iota(\ma) ~ \le ~ \abs{\ma u} ~\le ~ \norm{\ma}.$$ Note that if $\ma$ is invertible, then $\norm{\ma^{-1}}^{-1} \le \iota(\ma)$. $$\tag{A5}\label{A6}
\E \bigg[ \norm{\mM}^\alpha \log(1+ \norm{\mM}) \bigg]< \infty, \qquad \E \bigg[ (1+\norm{\mM})^\alpha \abs{\log \iota(\mM^\top)} \bigg] < \infty \\$$ Sometimes we will impose the stronger condition $$\label{A7} \tag{A6}
\text{There is $c >0$ such that $\P{\iota(\mM^\top) \ge c}=1$,}$$ which together with the first part of implies the second part of .
In the second part of the paper, we will need stronger assumptions on $\mu$, which guarantee that the associated Markov random walk (to be defined below) is Harris recurrent. We will consider the absolute continuity assumption $$\label{A4c}\tag{A3c}
\exists\, {{\ma_0}}\in \interior{\Mset} \, \exists\, \gamma_0, c >0 \text{ s.t. } \P{\mM \in \cdot } ~\ge~ \gamma_0 \, \llam^{d \times d}(\cdot \cap B_c({{\ma_0}})),$$ where $\llam^{d\times d}$ denotes the Lebesgue measure on the set of $d \times d$-matrices, seen as a subset of $\R^{d^2}$. A similar assumption for invertible matrices appears in [@Kesten1973 Theorem 6] and subsequently in [@AM2010]. It is easy to check that implies .
We will consider as well a quite degenerate case, namely $$\label{A4f}\tag{A3f} \text{$\supp \mu$ is finite and consists of rank-one matrices, and \eqref{A4} holds.}$$ Note that an allowable rank-one matrix $\ma$ has all entries positive, the columns are multiples of a vector $v_\ma \in \interior{\Sp}$, and consequently, $\ma \as u = v_\ma$ for all $u \in \Sp$.
We will also impose a stronger moment condition, namely $$\label{A8}\tag{A7}
\E\bigg[ N^{p_0}+{\left(\sum_{i=1}^N\norm{\mT_i}\right)}^{p_1}\bigg]<{\infty}\qquad\text{for some } p_0,p_1\ge1\text{ such that }p_0+p_1>2.$$
Note that implies $$\label{eq:moments_assumption}
\E\bigg[\left(\sum_{i=1}^N\norm{\mT_i}^{\alpha/(1+\delta)}\right)^{1+\delta}\bigg]<{\infty}\text { for small enough }\delta>0.$$ Indeed, for any $0<s<1$ and $p$ such that $1/p=(1-s)/p_0+s/p_1$, using first Jensen’s and then Hölder’s inequality, the random variable $\sum_i^N\norm{\mT_i}^s$ has finite moment of the order $p$: $$\begin{aligned}
\E\bigg[\left(\sum_i^N\norm{\mT_i}^s\right)^{p}\bigg]\le\E\bigg[N^{(1-s)p} \left(\sum_i^N\norm{\mT_i}\right)^{sp}\bigg]
\le(\E N^{p_0})^{\frac{(1-s)p}{p_0}}\left(\E\bigg[\bigg(\sum_i^N\norm{\mT_i}\bigg)^{p_1}\bigg]\right)^{\frac{sp}{p_1}}<{\infty}.\end{aligned}$$
Previous Results
----------------
We have the following existence result in the critical case.
\[prop:existence\] Assume – and – . Then Eq. has a nontrivial fixed point.
Theorem 1.2 in [@Mentemeier2013].
The main contribution of this paper is to prove the uniqueness of this fixed point, and to give asymptotic properties of its Laplace transform. It is convenient to introduce polar coordinates $(r,u) \in [0, \infty) \times \Sp$ on $\Rdnn$. Moreover, we will use that for $s \in I_{\mu}$, the operators $\Ps$ and $\Pst$, being self-mappings of the set $\Cf{\Sp}$ of continuous functions on $\Sp$ and defined by $$\Ps f(u) := \Erw{\abs{\mM u}^s \, f(\mM \as u)}, \qquad \Pst f(u) := \Erw{\abs{\mM^\top u}^s f(\mM^\top \as u)},$$ are quasi-compact with spectral radius equal to $k(s):=(\E\, N)^{-1} m(s)$ and there is a unique positive continuous functions ${H^{s}}\in \Cf{\Sp}$ and unique probability measures $\nus, \nust \in \Pset(\Sp)$ such that $$\begin{aligned}
\label{eq:eigenfunctions}
\Pst {H^{s}}= \frac{m(s)}{\E N} {H^{s}}, \qquad \nus \Ps = \frac{m(s)}{\E N} \nus, \qquad \nust \Pst = \frac{m(s)}{\E N} \nust\end{aligned}$$ and the following relation holds: $$\label{eq:estnus} {H^{s}}(u) ~=~ \int_{\Sp} \skalar{u,y}^s \, \nus(dy) \qquad \text{ for all $u \in \Sp$.}$$ See [@BDGM2014] for details and proofs. Using Eq. , we can extend ${H^{s}}$ to a $s$-homogeneous function on $\Rdnn$, i.e. $${H^{s}}(x) ~:=~ \int_{\Sp} \skalar{x,y}^s \, \nus(dy) ~=~ \abs{x}^s H^s\bigg( \frac{x}{\abs{x}} \bigg), \qquad x \in \Rdnn.$$ Using ${H^{s}}$, we are now going to provide a many-to-one lemma.
Let $u \in \Sp$. Define for $v \in \V$ $$S^u(v) := -\log \abs{\mL(v)^\top u}, \qquad U^u(v) := \mL(v)^\top \as u.$$ Then, by Eq. , we see that $$\begin{aligned}
\overline{\Qs}f(u,t) ~:=&~ \frac{1}{{H^{s}}(u)\, m(s)} \E \left[ \sum_{i=1}^N \, f(U^u(i), t-S^u(i)) e^{-s S^u(i)} {H^{s}}(U^u(i)) \right] \label{def:Q}\\
=&~ \frac{1}{{H^{s}}(u)\, m(s)} \E N \, \E \Bigl[{f(\mM^\top \as u, t+ \log \abs{\mM^\top u}) \, \abs{\mM^\top u}^\alpha {H^{s}}(\mM^\top \as u)}\Bigr] \nonumber\end{aligned}$$ defines a Markov transition operator on $\Sp \times \R$. Let $(U_n, S_n)_n$ be a Markov chain in $\Sp \times \R$ with transition operator $\overline{\Qs[\alpha]}$ and denote the probability measure on the path space $(\Sp \times \R)^\N$ with initial values $(U_0,S_0)=(u,s)$ by $\Prob_{u,s}^\alpha$ and the corresponding expectation symbol by $\E_{u,s}^\alpha$. Most times, we will use the shorthand notations $\Prob_u^\alpha = \Prob_{u,0}^\alpha$ and $\Prob_{\eta}^\alpha = \int \, \Prob_{u,s}^\alpha \, \eta(du,ds)$ for a probability measure $\eta$ on $\Sp \times \R$.
\[prop:many to one\] For all $s \in I_\mu$, $u \in \Sp$, $n \in \N$ and measurable $f : (\Sp \times \R)^{n+1} \to \R$, $$\label{eq:many to one}
\frac{1}{{H^{s}}(u)\, m(s)^n} \, \E \left[ \sum_{\abs{v}=n} \, f\Bigl( (U^u\bigl(v|k \bigr), S^u\bigl(v|k \bigr))_{k \le n}\Bigr) \, e^{-s S^u(v)} {H^{s}}(U^u(v)) \right]
=~ \E_u^s f(U_0, S_0, \cdots, U_n, S_n).$$
Corollary 4.3 in [@Mentemeier2013].
We call $(U_n, S_n)_{n \in \N}$ the [*associated Markov random walk*]{}. It generalizes the concept of the associated random walk in [@DL1983; @Liu1998]. In particular, it holds for all $u \in \Sp$, that $$\lim_{n \to \infty} \frac{S_n}{n} ~=~ 0 \qquad \Prob_u^\alpha\text{-a.s.},$$ see [@BDGM2014 Theorem 6.1]. Moreover, it is shown in [@Buraczewski2014 Lemma 7.1] that $$b(u):= \lim_{n \to \infty} \E_u^\alpha S_n$$ is well defined and continuous, and satisfies $$\label{eq:b} \E_u^\alpha [S_1 { + } b(U_1)] = b(u) .$$
Using Eq. , we obtain that $$\mathcal{W}_n(u) := \sum_{\abs{v}=n} \left[ S^u(v) { + } b(U^u(v)) \right] \, {H^{\alpha}}(U(v)) e^{-\alpha S(v)}$$ defines a martingale with respect to the filtration $\B_n$, which we will show to be the multivariate analogue of the derivative martingale. In fact, $b$ can be considered as the derivative of ${H^{\alpha}}$, see [@Buraczewski2014 (7.9)].
Main Results
------------
Our first result proves that, upon imposing the non-lattice condition and the stronger moment reap. boundedness assumptions –, the fixed point given by Proposition \[prop:existence\] is unique up to scaling, and satisfies an multivariate analogue of the regular variation property .
\[thm:main\] Assume – . Then there is a random measurable function $Z : \Sp \to [0, \infty)$ with $\P{Z(u)>0}=1$ for all $u \in \Sp$, such that $X$ is a nontrivial fixed point of on $\Rdnn$ if and only if its Laplace transform satisfies $$\label{LTofFP} \LTfp(ru) ~:=~ \Erw{e^{-r \skalar{u,X}}} ~=~ \Erw{e^{-r^\alpha K Z(u)}} \qquad \forall u \in \Sp,\, r \in \Rnn$$ for some $K > 0$.
There is an essentially unique positive function $L$, slowly varying at $0$ with $\liminf_{r \to 0} L(r) = \infty$, such that $$\label{eq:regvar}\lim_{r \to 0} \frac{1-\LTfp(ru)}{L(r) \, r^\alpha} ~=~ K {H^{\alpha}}(u).$$
Essentially unique means that if $L_1$ and $L_2$ satisfy Eq. , then $\lim_{r \to 0} L_1(r)/L_2(r)=1$. Depending on the value of $\alpha$, additional information can be extracted from Eq. .
1. If $\alpha <1$, then a Tauberian theorem (see [@Feller1971 XIII.(5.22)]) together with [@BDM2002 Theorem 1.1] implies the following multivariate regular variation property $$\lim_{r \to \infty} \frac{\P{\abs{X} > sr, \ \frac{X}{\abs{X}} \in \cdot}}{\P{\abs{X} > r}} = s^{-\alpha} \nus[\alpha],$$ see [@Mentemeier2013 Section 6] for details.
2. If $\alpha=1$, then $\E \abs{X}=\infty$ for every non-trivial fixed point, see Lemma \[lem:Linfty\]. Moreover, the aperiodicity condition is not needed, see Remark \[rem:aperiodic\]. This is in analogy with the one-dimensional situation, see e.g. [@Liu1998 Corollary 1.5].
Upon imposing the additional assumptions or on $\mu$, we will identify the function $L$ as well as the random variable $Z$.
\[thm:main2\] Assume – [ ]{}, with or instead of . Then $\mathcal{W}_n(u)$ converges a.s. to a nonnegative limit $\mathcal{W}(u)$ with $\P{\mathcal{W}(u)>0}=1$, and a random variable $X \in \Rdnn$ is a nontrivial fixed point of if and only if for some $K>0$, $$\Erw{e^{-r \skalar{u,X}}} ~=~ \Erw{e^{-r^\alpha K \mathcal{W}(u)}} \qquad \forall u \in \Sp,\, r \in \Rnn.$$ Moreover, the slowly varying function $L$ in Eq. can be chosen as (a scalar multiple of) $L(r) = \abs{\log r} \vee 1$.
Structure of the Paper
----------------------
The further organization is as follows: In Section \[sect:preliminaries\], we study the associated Markov random walk, which is recurrent due to the criticality assumption. Under assumptions , a regeneration property known from the theory of Harris recurrent Markov chains will be shown to hold. In Section \[sect:regular variation\], we prove that each fixed point satisfies , which is a main ingredient in the proof of uniqueness in Section \[sect:uniqueness\]. In Section \[sect:L\], we turn to the proof of Theorem \[thm:main2\] and study the behavior of the Laplace transform of the fixed point. We conclude with Section \[sect:derivative martingale\], where the convergence of the derivative martingale is proved.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The main part of this work was done during mutual visits to the Universities of Muenster and Warsaw, to which we are grateful for hospitality. S.M. was partlially supported by the Deutsche Forschungsgemeinschaft (SFB 878). K.K. was partially supported by NCN grant DEC-2012/05/B/ST1/00692.
The Associated Markov Random Walk {#sect:preliminaries}
=================================
In this section, we provide additional information about the associated Markov random walk, in particular about its stationary distribution and recurrence properties. Moreover, we show that it is Harris recurrent and satisfies a minorization condition under the additional assumption .
The Associated Markov Random Walk {#the-associated-markov-random-walk}
---------------------------------
The Markov chain $(U_n, S_n)_n$ constitutes a Markov random walk, i.e. for each $n \in \N$, the increment $S_n - S_{n-1}$ depends on the past only through $U_{n-1}$, this follows from the definition of $\overline{\Qs[\alpha]}$. Such Markov random walks which are generated by the action of nonegative matrices where first studied by Kesten in his seminal paper [@Kesten1973], and very detailed results are given in [@BDGM2014]. For the reader’s convenience, we cite those who are important for what follows. Recall that we denoted the Perron-Frobenius eigenvalue and the corresponding normalized eigenvector of a matrix $\ma \in \interior{\Mset}$ by $\lambda_\ma$ resp. $v_\ma$.
\[prop:UnSn\] Assume – and let $\alpha \in I_\mu$ ($m(\alpha)=1$ is not needed here). For this $\alpha$, assume . Then the following holds:
1. The Markov chain $(U_n)_n$ on $\Sp$ has a unique stationary distribution $\pist[\alpha]$ under $\Prob_u^\alpha$, with density (proportional to) ${H^{\alpha}}$ w.r.t the measure $\nust[\alpha]$. \[Un ergodic\]
2. $\supp \pist[\alpha] = \overline{\{ v_\ma \, : \, \ma \in [\supp \mu] \cap \interior{\Mset} \}}.$
3. For all $u \in \Sp$, \[SLLN\] $$\lim_{n \to \infty} \frac{S_n}{n} = \E_{\pist[\alpha]}^\alpha S_1 = \int_{\Sp}\, \E_u^\alpha S_1 \, \pist[\alpha](du) = \frac{m'(\alpha^{-})}{m(\alpha)} \qquad \Prob_u^\alpha\text{-a.s.}$$
Now assume $1 \in I_\mu$ and that holds for $\alpha=1$. Then $\mb :=\E \mM \in \interior{\Mset}$.
1. $m(1) = (\E N) \lambda_{\mb}$ and ${H^{1}}(u) = \skalar{u,v_{\mb}}$.
2. The derivative of $m$ at 1 can be calculated to $$m'(1^{-})= \int_{\Sp} \, \Erw{\frac{\skalar{\mM u,v_{\mb}}}{\skalar{u,v_{\mb}}} \, \log \abs{\mM u} } \, \pist[1](du)$$
Sections 4 and 6 of [@BDGM2014].
Recurrence of Markov Random Walks
---------------------------------
By Proposition \[prop:UnSn\] , in the critical case $m'(\alpha^{-})=0$ the Markov random walk $(S_n)_n$ is centered in the stationary regime and satisfies a strong law of large numbers. Alsmeyer [@Alsmeyer2001] studied recurrence properties of such Markov random walks, which we will make use of.
\[lem:recurrence\] Assume that - hold. For any open set $A$ with $\pist[\alpha](A) > 0$ and any open interval $B \subset \R$, it holds that $$\label{recurrence}\Prob_{\pist[\alpha]}^\alpha ({(U_n, S_n) \in A \times B \text{ infinitely often}})=1.$$ If the aperiodicity condition is not assumed, then still $$\label{eq:oscillates} \liminf_{n\to \infty}{S_n} ~=~ - \infty, \qquad \limsup_{n \to \infty}{S_n} ~=~ \infty \qquad \Prob_{\pist[\alpha]}^\alpha\text{-a.s.}$$
Let $A$ be any open set $A$ with $\pist[\alpha](A) > 0$. By the strong law of large numbers for Markov chains (see [@Breiman1960]), $$\label{SLLNmarkov} \lim_{n \to \infty} \frac{1}n \sum_{k=1}^n f(U_k) ~=~ \int\, f(x) \, \pist[\alpha](dx) \qquad \Prob_{\pist[\alpha]}^\alpha\text{-a.s.},$$ thus, using $f={\mathds{1}_{A}}$, we obtain that $\Prob_{\pist[\alpha]}^\alpha(U_n \in A \text{ infinitely often})=1 $. Denote the successive hitting times of $A$ by $\tau_n$. Then $(U_{\tau_n}, S_{\tau_n})$ is again a Markov random walk, and $\pi_A :=\pist[\alpha](\cdot \cap A)/\pist[\alpha](A)$ is the stationary probability measure for $U_{\tau_n}$. The aperiodicity assumption implies that $(U_n,S_n)$ are nonarithmetic in the sense of [@Alsmeyer2001], see [@Buraczewski2014] for details. Lemma 1 in [@Alsmeyer2001] gives that $(U_{\tau_n}, S_{\tau_n})$ is nonarithmetic as well. Using with $f={\mathds{1}_{A}}$ again, this gives that $n/\tau_n \to \pi(A)$ a.s. Combining this with the strong law of large numbers in Proposition \[prop:UnSn\], we deduce that $$\lim_{n \to \infty} \frac{S_{\tau_n}}{n} = \lim_{n \to \infty} \frac{S_{\tau_n}}{\tau_n} \, \frac{\tau_n}{n} = \frac{1}{\pist[\alpha](A)} \cdot 0 \qquad \Prob_{\pist[\alpha]}^\alpha\text{-a.s.}.$$
Then Theorem 2 in [@Alsmeyer2001] (for the nonarithmetic case) gives that the recurrence set $$\{ s \in \R \, : \, \text{ for all $\epsilon > 0$, }S_{\tau_n} \in (s-\epsilon, s+\epsilon) \text{ infinitely often } \}$$ equal to $\R$, which shows that $\Prob_{\pist[\alpha]}^\alpha({S_{\tau_n} \in B \text{ infinitely often}})=1$.
In the arithmetic case, the recurrence set is still a closed subgroup of $\R$, which implies the oscillation property.
\[cor:uo\] There is $u_0 \in \interior{\Sp} \cap (\supp \pist[\alpha])$ such that $$\label{recurrence2}\Prob_{u_0}^\alpha ({(U_n, S_n) \in A \times B \text{ infinitely often}})=1, \text{ and }$$ $$\label{eq:oscillates2} \liminf_{n\to \infty}{S_n} ~=~ - \infty, \qquad \limsup_{n \to \infty}{S_n} ~=~ \infty \qquad \Prob_{u_0}^\alpha\text{-a.s.}$$
By Proposition \[prop:UnSn\], $\supp \pist[\alpha]$ consists of the (closure of the) set of normalized Perron-Frobenius eigenvectors of matrices $\ma \in [\supp \mu]$ with all entries strictly positive. By part (2) of $\condC$, this set is nonempty, hence $\interior{\Sp} \cap (\supp \pist[\alpha]) \neq \emptyset$ and even $\pist[\alpha](\interior{\Sp})=1$. On the other hand, Lemma \[lem:recurrence\] implies validity of and for $\pist[\alpha]$-a.e. $u \in \Sp$, hence we can find $u_0 \in \interior{\Sp}$ satisfying the assertions.
Implications of Assumptions and
--------------------------------
In this subsection, we explain how Assumptions and imply that the Markov chain $(U_n)_{n \in \N}$ has an atom (possibly after redefining it on an extended probability space), which can be used to obtain a sequence $(\sigma_n)_{n \in \N}$ of [*regeneration times*]{} for the Markov random walk $(U_n, S_n)$, i.e. stopping times such that$(U_{\sigma_n}, S_{\sigma_n}-S_{\sigma_{n-1}})_{n \in \N}$ becomes an i.i.d. sequence. Namely, we are going to prove the following lemma for the Markov chain $(U_n, Y_n):=(U_n, S_{n}-S_{n-1})$.
\[regenerationlemma\] Assume – and or . On a possibly enlarged probability space, one can redefine $(U_n, Y_n)_{n \geq 0}$ together with an increasing sequence $(\sigma_n)_{n \geq 0}$ of random times such that the following conditions are fulfilled under any $\Prob_u^\alpha$, $u\in \Sp$:
- There is a filtration $\mathcal{G}=(\mathcal{G}_n)_{n \geq 0}$ such that $(U_n, Y_n)_{n \geq 0}$ is Markov adapted and each $\sigma_n$ a stopping time with respect to $\mathcal{G}$, moreover, $\{\sigma_n = k\} \in \mathcal{G}_{k-1}$ for all $n,k \ge 0$. \[R1\]
- The sequence $(\sigma_{n+1}-\sigma_{n})_{n\ge 1}$ is i.i.d. with law $\P[\eta]{{\sigma_1} \in \cdot}$ and is independent of $\sigma_{1}$. \[R2\]
- For each $k\ge 1$, $(U_{\sigma_k+n},Y_{\sigma_k+n})_{n \geq 0}$ is independent of $(U_j, Y_j)_{0 \leq j \leq \sigma_k -1}$ with distribution $\Prob^\alpha_\eta((U_n, Y_n)_{n \geq 0} \in \cdot)$. \[R3\]
- There is $q \in (0,1)$ and $l \in \N$ such that $\sup_{u \in \Sp} \Prob_u^\alpha(\sigma_1 > ln) \le q^n$.
This lemma is quite immediate under condition , for Proposition \[prop:UnSn\], (2) shows that the unique stationary measure $\pist[\alpha]$ for $(U_n)$ under $\Prob_u^\alpha$ is supported on the [*finite*]{} set $\mathds{S} :=\{ v_\ma \, : \, \ma \in \supp \mu\}$ (note that $v_{\ma \mb} = v_{\ma}$ if $\ma$ has rank one, thus the semigroup $[\supp \mu]$ can be replaced by $\supp \mu$.) Moreover, independent of the initial value $u \in \Sp$, $U_1 \in \mathds{S}$ $\Prob_u^\alpha$-f.s., i.e. $\Sp \setminus \mathds{S}$ is uniformly transient for $(U_n)_{n \in \N}$, and thus we can study $(U_n)_{n \in \N}$ on the finite state space $\mathds{S}$. Then, if $(\sigma_n)_{n \in \N}$ is a sequence of successive hitting times of a point $u_0 \in \mathds{S}$, the assertions of the lemma follow from the theory of Markov chains with finite state space.
A crucial point is that we also obtain the independence of $Y_{\sigma_k}$ from $(U_j, Y_j)_{0 \leq j \leq \sigma_k -1}$, thereby strengthening analogous results for invertible matrices, obtained in [@AM2010; @Mentemeier2013a].
From now on, assume . We are going to prove that the chain $(U_n, Y_n)$ satisfies a minorization condition as in [@Athreya1978 Definition 2.2] resp. [@Num1978 (M)]. If $v_{{\ma_0}}\in \Sp$ is the Perron-Frobenius eigenvalue of the matrix $\ma_0$ from , then we have the following result:
\[lem:min1\] For each $u \in \Sp$, $\delta >0$, $$\Prob_u^\alpha(U_n \in B_{\delta}(v_{{\ma_0}}) \text{ infintely often })=1,$$ moreover, if $\tau$ denotes the first hitting time of $B_\delta(v_{{\ma_0}})$, then there is $l \ge 1$ and $q_0 \in (0,1)$ such that $$\sup_{u \in \Sp} \Prob_u^\alpha(\tau > ln) \le q_0^n,$$ i.e. $\tau/l$ is stochastically bounded by a random variable with geometric distribution.
This is proved in [@Kesten1973 p.218-220, proof of I.1], the crucial point being that $v_{{\ma_0}}$ is a strict contraction on $\Sp$ with attractive fixed point $v_{{\ma_0}}$, and small perturbations of ${{\ma_0}}$ still attract to a neighborhood of $v_{{\ma_0}}$, and such matrices are realized with positive probability.
\[lem:min2\] There are $\delta >0$, $\gamma >0$ and a probability measure $\eta$ on $\mathcal{R}:=B_\delta(v_{{\ma_0}}) \times \R$ such that for all $u \in B_\delta(v_{{\ma_0}})$ and all measurable subsets $A \subset B_\delta(v_{{\ma_0}})$, $B \subset \R$ $$\Prob_u^\alpha(U_1 \in A, Y_1 \in B) ~\ge~ \gamma \, \eta(A \times B).$$
We follow the approach in [@AM2010; @Mentemeier2013a].
. Given $c >0$, ${{\ma_0}}\in \interior{\Mset}$, there is $\epsilon >0$ such that for all orthogonal matrices $\matrix{O}$, satisfying $\norm{\matrix{O}-\Id}< \epsilon$, $B_{c/2}({{\ma_0}})\matrix{O} \subset B_c({{\ma_0}})$. [*Proof*]{}: Let $\mb \in B_{c/2}{{{\ma_0}}}$, then, since $\matrix{O}$ is an isometry, $$\norm{\mb \matrix{O}- {{\ma_0}}} \le \norm{\mb\matrix{O} - {{\ma_0}}\matrix{O}} + \norm{{{\ma_0}}\matrix{O} - {{\ma_0}}} \le \norm{\mb - {{\ma_0}}} - \norm{{{\ma_0}}} \norm{\matrix{O} - \Id} \le c/2 + \epsilon \norm{{{\ma_0}}}.$$ . For all $\epsilon >0$ there is $\delta>0$ such that for each $u \in B_\delta(v_{{\ma_0}})$ there exists an orthogonal matrix $\matrix{O}_u$ with $ u= \matrix{O}_u v_{{\ma_0}}$ and $\norm{\matrix{O}_u-\Id} < \epsilon$. [*Source*]{}: [@Mentemeier2013a Lemma 15.1].
. Introduce the finite measure $$\tilde\eta(A \times B) ~:=~ \int_{B_{c/2}({{\ma_0}})} \, {\mathds{1}_{A}}(\ma \as v_{{\ma_0}})\, {\mathds{1}_{B}}(-\log \abs{\ma v_{{\ma_0}}}) \, \llam^{d \times d}(d\ma).$$ Combining Steps 1 and 2 and Assumption , there is $\delta >0$, such that for all $u \in B_\delta(v_{{\ma_0}})$ there exists an orthogonal matrix $\matrix{O}_u$ with $ u= \matrix{O}_u v_{{\ma_0}}$ and $B_{c/2}({{\ma_0}})\matrix{O}_u \subset B_c({{\ma_0}})$. Hence for all $u \in B_\delta(v_{{\ma_0}})$, by Assumption and using that $\llam^{d \times d}$ is invariant under transformations by a matrix with determinant 1 (see [@Mentemeier2013a proof of Prop. 15.2, Step 1] for more details, using the Kronecker product) $$\begin{aligned}
\Prob(\mM^\top \as u \in A, -\log \abs{\mM^\top u} \in B) ~\ge&~ \gamma_0 \, \int_{B_{c/2}({{\ma_0}})\matrix{O}_u} {\mathds{1}_{A}}(\ma \as u)\, {\mathds{1}_{B}}(- \log \abs{\ma u}) \ \llam^{d \times d}(d\ma) \\
~=&~ \gamma_0 \, \int_{B_{c/2}({{\ma_0}})} {\mathds{1}_{A}}(\ma \matrix{O}_u^{-1}\as u)\, {\mathds{1}_{B}}(- \log \abs{\ma \matrix{O}_u^{-1} u}) \ \llam^{d \times d}(d\ma) \\
~=&~ \gamma_0\, \tilde{\eta}(A \times B).\end{aligned}$$
: To obtain a minorization for the shifted measure $\Prob^\alpha_u$, recall that ${H^{\alpha}}$ is bounded from below and above, to obtain that $$\begin{aligned}
\Prob_u^\alpha(U_1 \in A, Y_1 \in B) ~\ge&~ \int_{A \cap{B_\delta(v_{{\ma_0}})}} \int_B \frac{{H^{\alpha}}(w)}{{H^{\alpha}}(u)} e^{-\alpha y} \ \Prob(\mM^\top \as u \in dw, -\log \abs{\mM^\top u} \in dy) \\
\ge&~ \gamma_1\int_{A \cap{B_\delta(v_{{\ma_0}})}} \int_B {{H^{\alpha}}(w)}e^{-\alpha y} \ \tilde{\eta}(dw, dy) ~=:~ \eta(A \times B) \end{aligned}$$ Upon renormalizing $\eta$ to a probability measure, and thereby determining $\gamma$, we obtain the assertion.
Now we are ready to prove Lemma \[regenerationlemma\] under Assumption :
Lemmata \[lem:min1\] and \[lem:min2\] imply that the chain $(U_n, Y_n)_{n \geq 0}$ is $\Bigl(\mathcal{R}, \gamma, \eta, 1\Bigr)$-recurrent in the sense of [@Athreya1978 Definition 2.2]. Then the lemma follows from [@Athreya1978 Lemma 3.1 and Corollary 3.4]. The [*regeneration times*]{} $\sigma_n$ are constructed as follows: Let $(\xi_n)_{n \ge 0}$ be a sequence of i.i.d. Bernoulli(1,$\gamma$) random variables, independent of $(U_n, Y_n)_{n \ge 0}$. Whenever $(U_n, Y_n)$ enters the set $\mathcal{R}$, $(U_{n+1}, Y_{n+1})$ is generated according to $\eta$ if $\xi_n=1$, and according to $(1-\gamma)^{-1}(P-\gamma\eta)$ if $\xi=0$. The total transition probability thus remains $P=\Prob_u^\alpha((U_1, Y_1) \in \cdot)$. Together with Lemma \[lem:min1\], this construction immediately gives that $\sigma_1$ can be bounded stochastically by a random variable with geometric distribution.
Regular Variation of Fixed Points {#sect:regular variation}
=================================
In this section, we show that every fixed point of $\ST$, the existence of which is provided by Proposition \[prop:existence\], satisfies the regular variation property .
Let $\LTfp$ be the Laplace transform of a fixed point of $\ST$ in the critical case $m'(\alpha)=0$. Introduce $$\begin{aligned}
\label{defn:D} D(u,t) :=\ & \frac{1-
\LTfp(e^{-t}u)}{e^{-{\alpha}t}{H^{{\alpha}}}(u)}, \qquad u \in \Sp,\, t \in \R.\end{aligned}$$ Our aim is to study behavior of $D$ as $t$ goes to infinity. Let $u_0$ be given by Corollary \[cor:uo\]. Following the approach in [@Kyprianou1998], we are going to show that $$h_t(u,s) := \frac{D(u,s+t)}{D(u_0,t)} = \frac{1-\LTfp(e^{-(s+t)}u)}{e^{-\alpha s}(1-\LTfp(e^{-t}u_0))} \frac{{H^{\alpha}}(u_0)}{{H^{\alpha}}(u)}$$ converges to $1$ as $t$ tends to infinity. This shows in particular, that $D(u_0,t)$ is slowly varying as $t \to \infty$. We then use the results of [@Mentemeier2013] to deduce that this already implies that $D(u,t)$ is slowly varying for all $u \in \Sp$.
\[lem:subsequence\] For every sequence $(t_k)_{k \in \N}$, tending to infinity, there is a subsequence $(t_n)_{n \in \N}$ such that $h_{t_n}(u,s)$ converges pointwise to a continuous function $h : \Sp \times \R \to [0, \infty)$.
Introduce for $t \in \R$ the function $f_t : \Rdnn \to [0, \infty)$ $$f_t(x):= \frac{1-\LTfp(e^{-t}x)}{1-\LTfp(e^{-t}u_0)}.$$ Since $\LTfp$ is a Laplace transform and $t$ is fixed, it follows (using the multivariate version of the Bernstein theorem, [@Bochner1955 Theorem 4.2.1]), that the derivative of $f_t$ is completely monotone in the multivariate sense, and hence, $$\varphi_t(x):= \exp(-f_t(x))$$ is the Laplace transform of a probability measure on $\Rdnn$, due to [@Feller1971 Criterion XIII.4.2]. Note $\varphi_t(0)=1$, while the limit as $\abs{x} \to \infty$ may be positive, so the corresponding probability measure might have some mass in zero.
Since the set of probability measures is vaguely compact, we deduce that for any sequence $t_k$, tending to infinity, there is a subsequence $t_n$ such that $\varphi_{t_n}$ converges pointwise to the Laplace transform $\varphi$ of a (sub-)probability measure on $\Rdnn$, which is continuous except for maybe in $0$. Since $\varphi_{t_n}(u_0)=e^{-1} > 0$ for all $n$, it follows that $\varphi >0$ on $\Rdnn$, and hence, we obtain that $$\lim_{n \to \infty} f_{t_n}(x) ~=~ f(x) ~:=~ - \log \varphi(x)$$ exists for all $x \in \Rdnn$ with $f$ being continuous on $\Rdnn \setminus\{0\}$.
This implies the pointwise convergence $$\lim_{n \to \infty} h_{t_n}(u,s) ~=~ h(u,s) ~:=~ \frac{f(e^{-s}u)}{e^{-\alpha s}} \, \frac{{H^{\alpha}}(u_0)}{{H^{\alpha}}(u)},$$ where the function $h$ is continuous on $\R \times \Sp$.
\[lem:superharmonic\] Let $t_n$ be a sequence such that $h_{t_n}$ converges to a limit $h$. Then $h$ is superharmonic for $(U_n, V_n)$ under $\Prob_u^\alpha$, i.e. $$h(u,s) \ge \E_u^\alpha \, h(U_1, s+S_1).$$
Using Eq. and a telescoping sum, we obtain (since $\LTfp$ is a fixed point), $$\begin{aligned}
D(u,s+t) &=~\frac{1- \LTfp(e^{-(s+t)}u)}{e^{-\alpha(s+t)}{H^{\alpha}}(u)} \\
&=~ \Erw{\frac{1- \prod_{i=1}^N \LTfp(\mT_i^\top e^{-(s+t)}u)}{e^{-\alpha(s+t)}{H^{\alpha}}(u)}} \\
&=~ \Erw{\sum_{i=1}^N \frac{1- \LTfp(\mT_i^\top e^{-(s+t)}u)}{e^{-\alpha(s+t) }{H^{\alpha}}(u)} \prod_{1 \le j < i} \LTfp(\mT_i^\top e^{-(s+t)}u)} \\\end{aligned}$$ Now divide by $e^{\alpha t}(1-\LTfp(e^{-t}u_0)) / {H^{\alpha}}(u_0)$ to obtain $$\begin{aligned}
h_{t}(u,s)=&~ \frac{{H^{\alpha}}(u_0)}{{H^{\alpha}}(u)} \E\Biggl(\sum_{i=1}^N \frac{1- \LTfp(e^{-S^u(i)-(s+t)}U^u(i))}{(1-\LTfp(e^{-t}u_0)){H^{\alpha}}(U^u(i)) e^{-\alpha S^u(i)}e^{-\alpha s}}e^{-\alpha S^u(i)} {H^{\alpha}}(U^u(i))\\& \hspace{8cm}\times\prod_{1 \le j < i} \LTfp(e^{-S^u(i)-(s+t)}U^u(i))\Biggr) \\
=&~ \frac{{H^{\alpha}}(u_0)}{{H^{\alpha}}(u)} \E\Biggl(\sum_{i=1}^N \ \, \frac{f_{t}\Bigl( e^{-S^u(i)-s}, U^u(i) \Bigr)}{({H^{\alpha}}(U^u(i)) e^{-\alpha (S^u(i)+s)}}e^{-\alpha S^u(i)} {H^{\alpha}}(U^u(i)) \\& \hspace{8cm}\times\prod_{1 \le j < i} \LTfp(e^{-S^u(i)-(s+t)}U^u(i))\Biggr) \\
=&~ \frac{1}{{H^{\alpha}}(u)} \E\Biggl(\sum_{i=1}^N \ \, h_t\Bigl(U^u(i), s+S^u(i) \Bigr) \, e^{-\alpha S^u(i)} {H^{\alpha}}(U^u(i)) \prod_{1 \le j < i} \LTfp(e^{-S^u(i)-(s+t)}U^u(i))\Biggr) \end{aligned}$$ Now consider the subsequential limit $t_n \to \infty$, then the LHS converges by assumption to $h$, while for the RHS, we use Fatou’s lemma and observe that the product tends to $1$, so that we obtain: $$\begin{aligned}
h(u,s) \ge &~ \frac{1}{{H^{\alpha}}(u)} \Erw{\sum_{i=1}^N \ \, h\Bigl(U^u(i), s+S^u(i) \Bigr) \, e^{-\alpha S^u(i)} {H^{\alpha}}(U^u(i)) } \\
=&~ \E_u^\alpha \, h(U_1, s+S_1).\end{aligned}$$
\[lem:limit\] The (subsequential limit) function $h$ is constant and equal to 1 on $\supp \pist[\alpha] \times \R$.
It follows from Lemma \[lem:superharmonic\] that $h(U_n,s+ S_n)$ is a nonnegative supermartingale, which hence converges a.s. as $n \to \infty$. Now assume that $h(u,s) \neq h(w,t)$ for $u,w \in \supp \pist[\alpha]$ and $s,t \in \R$. Since $m'(\alpha)=0$, $(U_n, S_n)$ under $\Prob_{u_0}^\alpha$ is a recurrent Markov Random Walk by Lemma \[lem:recurrence\], thus it visits every neighborhood of $(u,s)$ resp. $(w,t)$ infinitely often. But then, due to the a.s. convergence of $h(U_n, s+S_n)$ and the continuity of $h$, we infer that $h$ has to be constant. Since furthermore $h(u_0,0)=1$, the assertion follows.
\[rem:aperiodic\] Note that here (via Lemma \[lem:recurrence\]) the aperiodicity condition enters. It is not needed if $\alpha=1$, because then $h$ itself is a multivariate Laplace transform, which is in particular monotone. Then using again the a.s. convergence of $h(U_n, s+S_n)$ together with the fact that $S_n$ oscillates (see Eq. ) shows that $h$ has to be constant.
\[lem:slowvar\] It holds that $$\label{eq:convh} \lim_{t \to \infty} \frac{1-\LTfp(e^{-(s+t)}u)}{e^{-\alpha s}(1-\LTfp(e^{-t}u_0))} \frac{{H^{\alpha}}(u_0)}{{H^{\alpha}}(u)}~=~1 \qquad \forall u \in \Sp, \, s \in \R,$$ and the convergence is uniform on compact subsets of $\Sp \times \R$. In particular, the positive function $$\label{eq:defL} L(r) ~:=~ \frac{1-\LTfp(ru_0)}{r^\alpha {H^{\alpha}}(u_0)} \qquad \bigg( =~ D(u_0, - \log r) \bigg)$$ is slowly varying at 0, and $$\label{eq:slowvar} \lim_{r \to 0} \, \sup_{u \in \Sp} \abs{\frac{1-\LTfp(ru)}{L(r) \, r^\alpha} - H^\alpha(u)} ~=~ 0.$$
Combining Lemmata \[lem:superharmonic\] and \[lem:limit\], we obtain that for every sequence $t_k \to \infty$ there is a subsequence $t_n \to \infty$ such that for each $s \in \R$, $$1 ~=~ \lim_{n \to \infty} h_{t_n}(u_0,s) ~=~ \lim_{n \to \infty} \frac{1-\LTfp(e^{-(s+t_n)}u_0)}{e^{-\alpha s}(1-\LTfp(e^{-t_n}u_0))}.$$ Since all subsequential limits are the same, we infer that $\lim_{t \to \infty} h_t(u_0,s)=1$ for all $s \in \R$, which in particular proves the slow variation assertion about the function $L(r)$, for $L(sr)/L(r) = h_{-\log r}(u_0, -\log s)$. Using the estimate $$\left( \min_{1 \le i \le d} (u_0)_i \right) (1-\LTfp(r{\mathbf{1}})) ~\le~(1-\LTfp(ru_0)) ~\le~ (1-\LTfp(r{\mathbf{1}}))$$ (see [@Mentemeier2013 Lemma A.1]), we deduce further that $$0 ~<~ \liminf_{r \to \infty} \frac{1-\LTfp(r{\mathbf{1}})}{L(r)r^\alpha} ~\le~ \limsup_{r \to \infty} \frac{1-\LTfp(r{\mathbf{1}})}{L(r)r^\alpha} ~<~ \infty,$$ i.e., $\LTfp$ is [*$L$-$\alpha$-regular*]{} in the sense of [@Mentemeier2013 Definition 2.1]. Then [@Mentemeier2013 Theorem 8.2] provides us with the first assertion, i.e. the (uniform) convergence in Eq. . Then Eq. is a direct consequence when considering the compact set $\Sp \times \{0\}$.
Uniqueness of Fixed Points {#sect:uniqueness}
==========================
In this section, we are going to finish the proof of Theorem \[thm:main\]. Therefore, we show that the slowly varying function appearing in is essentially unique, and that this property then identifies the fixed points. The approach is the multivariate analogue of [@Biggins1997 Theorem 8.6].
We start with the following lemma, the proof of which we postpone to the end of this section for a better stream of arguments.
\[lem:maxpos\] Assume –, and . Then $$\label{eq:maxpos}
\lim_{n \to \infty} \max_{\abs{v}=n} \norm{\mL(v)}=0 \qquad \Pfs$$
For $u \in \Sp$, we can introduce for $t \in \R$ the [*homogeneous stopping line*]{} $$\slineu[t] ~:=~ \left\{ v \in \tree \, : \, S^u(v) > t, \ S^u(v|k) \le t \, \forall k < \abs{v} \right\}.$$ Since $\max_{\abs{v}=n} \norm{\mL(v)} \to 0$ $\Pfs$ by Lemma \[lem:maxpos\], this stopping line is finite $\Pfs$ and intersects the whole tree ([*is dissecting*]{}).
Let $\LTfp$ be a fixed point of $\ST$. Define $$M_n(x) := \prod_{\abs{v}=n} \, \LTfp(\mL(v)^\top x), \qquad x \in \Rdnn.$$ By Eq. , this constitutes a bounded martingale w.r.t. $\B_n$ for every $x$ and we call its $\Pfs$ limit $M(x) \in [0, \infty)$ the [*disintegration*]{} of the fixed point $\LTfp$. Setting $$Z(x) ~:=~ - \log M(x),$$ the martingale property together with boundedness implies that $\LTfp(x)=\E \exp(-Z(x))$ for all $x \in \Rdnn$. [Following the proof of [@AM2010a Lemma 4.1], one can show that $M(\cdot,\omega)$ is a Laplace transform for $\Prob$-a.e. $\omega \in \Omega$, and that $M$ is jointly measurable on $\Sp \times \Omega$. This implies the same for $Z$.]{}
\[prop:dis\] Assume – . Let $\LTfp$ be a fixed point of $\ST$ with disintegration $M$. Let $F : \Rdnn \to [0,\infty)$ be a nonnegative measurable function with $\lim_{s \to 0} \sup_{u \in \Sp} \abs{F(su)-\gamma}=0$ for some $\gamma \ge 0$. Then the following holds:
1. $\lim_{n \to \infty} \, \sum_{\abs{v}=n} F(\mL(v)^\top x) (1 - \LTfp(\mL(v)^\top x)) = \gamma Z(x)$ $\Pfs$\[as2\]
2. For all $u \in \Sp$, $r \in \Rp$, $Z(ru) = r^\alpha Z(u)$.\[as3\]
3. $\LTfp(ru) = \E e^{-r^\alpha Z(u)}$ for all $u \in \Sp$, $r \ge 0$.\[as4\]
4. $ Z(u) \in (0, \infty)$ $\Pfs$.\[as5\]
5. \[eq:Mt\] $\lim_{t \to \infty} \, \sum_{v \in \slineu[t]} \left(1 - \LTfp(e^{-S^u(v)} U^u(v)) \right) ~=~ Z(u)$ $\Pfs$ for all $u \in \Sp$.
Using Lemma \[lem:maxpos\], the proof of Assertion is the same as for [@Mentemeier2013 Lemma 7.3] and therefore omitted. By Lemma \[lem:slowvar\], for all $r \in \Rp$ and $u \in \Sp$, the function $F(su):= \frac{1- \LTfp(rsu)}{1-\LTfp(ru)}$ converges uniformly to $r^\alpha$. Thus we obtain by an application of . Then is an immediate consequence of $\LTfp(x)=\E \exp(-Z(x))$.
Reasoning as in the proof of [@BDGM2014 Theorem 2.7, Step 6], we see that for any nontrivial fixed point $X$ of $\ST$, $\P{X=0}=0$, and consequently $Z(u) > 0$ $\Pfs$. On the other hand, since $\LTfp$ is the Laplace transform of a random variable on $\Rdnn$, $Z(u) < \infty$ $\Pfs$
The subsequent lemma is where we use assumption . Using the definition of $\mu$, it implies that with $c' := - \log c$ $$\begin{aligned}
\P{S^u(i) > c' \ \forall\, 1 \le i \le N} ~\le&~ \E \bigg[ \sum_{i=1}^N {\mathds{1}_{}}\big( S^u(i) > c' \big) \bigg] \\
=&~ (\E N) \E \bigg[ {\mathds{1}_{}}\big( -\log \abs{\mM^\top u} > c' \big) \bigg] ~=~ (\E N) \P{\abs{\mM^\top u} < c} \\
\le&~ (\E N) \P{\iota(\mM^\top) < c} ~=~0.\end{aligned}$$ In other words, the increments of $S(vi) - S(v)$ are $\Pfs$ bounded by $c'$.
\[lem:LMt\] Assume –. Let $\LTfp$ be a nontrivial fixed point of $\ST$ with associated slowly varying function $L$ given by Eq. . Then $$\label{eq:LMt} \lim_{t \to \infty} L(e^{-t}) \, \sum_{v \in \slineu[t]} {H^{\alpha}}(U^u(v)) e^{- \alpha S^u(v)} ~=~ Z(u) \quad \Pfs[u]$$
By Lemma \[lem:slowvar\], $$\lim_{t \to \infty} \frac{1 - \LTfp(e^{-s-t}y)}{{H^{\alpha}}(y) e^{-\alpha(s+t)} L(e^{-t}) } = 1,$$ and the convergence is uniform on compact sets for $(y,s)$. In particular, it is uniform on the set $\Sp \times [0,c']$. Now applying this result with $s=S^u(v)-t$ and $y = U^u(v)$ with $v \in \slineu[t]$ and using that $$0 ~<~ S^u(v)-t ~\le~ S^u(v)-S^u(v|(\abs{v}-1)) \in [0,c']$$ by Assumption , we deduce from Proposition \[prop:dis\], that $$\begin{aligned}
Z(u) ~=&~ \lim_{t \to \infty} \, \sum_{v \in \sline[t]} L(e^{-t}) {H^{\alpha}}(U^u(v)) e^{- \alpha S^u(v)} \frac{1 - \LTfp(e^{-(S^u(v)-t)-t}U^u(v))}{{H^{\alpha}}(U^u(v)) e^{-\alpha(S^u(v)-t+t)} L(e^{-t}) }\\
=&~ \lim_{t \to \infty} L(e^{-t}) \, \sum_{v \in \sline[t]} {H^{\alpha}}(U^u(v)) e^{- \alpha S^u(v)} \qquad \Pfs\end{aligned}$$
The idea of this proof follows that of [@Biggins1997 Theorem 8.6]. There an assumption similar to is avoided by using the theory of general branching processes, see [@Jagers1975; @Nerman1981]. A similar approach is taken in [@Mentemeier2013] in the non-critical case, a crucial ingredient of which is an application of Kesten’s renewal theorem [@Kesten1974 Theorem 1]. In the critical case, a variant of Kesten’s renewal theorem for driftless Markov random walks, or a strong theory of Wiener-Hopf factorization seems to be needed in order to proceed along similar lines.
Now we are ready to prove our main result.
: By Proposition , there is a nontrivial fixed point of $\ST$ with LT $\LTfp$, say. By Proposition \[prop:dis\], for each $u \in \Sp$, there is a random variable $Z(u)$ with $\P{Z(u)>0}=1$ and such that $\LTfp(ru)=\E [\exp(-r^\alpha Z(u)) ]$ for all $r \in [0,\infty)$. Define $L(r)$ by , choosing a suitable $u_0$.
: Let now $\LTfp_2$ be the Laplace transform of a different nontrivial fixed point, with corresponding disintegration $M_2$ and $Z_2$, and slowly varying function $L_2$, defined by , using the same $u_0$ as before. Recall that $Z(u)$ and $Z_2(u)$ are $\Pfs$ positive and finite by by Proposition \[prop:dis\], for each $u \in \Sp$. Then we have by Lemma \[lem:LMt\] that $\Pfs$, $$\lim_{t \to \infty} \frac{Z_2(u)}{Z(u)} = \lim_{t \to \infty} \frac{L_2(e^{-t}) \, \sum_{v \in \slineu[t]} {H^{\alpha}}(U^u(v)) e^{- \alpha S^u(v)} }{L(e^{-t}) \, \sum_{v \in \slineu[t]} {H^{\alpha}}(U^u(v)) e^{- \alpha S^u(v)} } = \lim_{t \to \infty} \frac{L_2(e^{-t})}{L(e^{-t})}.$$ First, fixing $u \in \Sp$, this proves that the limit of the right hand side exists and equals some $K \in (0, \infty)$. Then, using the equation again for general $u$, we obtain $Z_2(u) = K Z(u)$ $\Pfs$. Consequently, $$\LTfp_2(ru)= \Erw{e^{-r^\alpha Z_2(u)}} = \Erw{e^{-r^\alpha K Z(u)}} = \LTfp(K^{1/\alpha}ru),$$ which proves Eq. \[LTofFP\].
: Fix $L$ to be the slowly varying function corresponding to $\LTfp$. Then Eq. follows from Eq. for this particular $\LTfp$, and moreover, $$\lim_{r \to 0} \frac{1-\LTfp_2(ru)}{r^\alpha L(r)} ~=~ \lim_{r \to 0} \frac{K(1-\LTfp(K^{1/\alpha}ru))}{K r^\alpha L(K^{1/\alpha}r)} \frac{L(K^{1/\alpha}r)}{L(r)} ~=~ K H^\alpha(u)$$ The final assertion about $\limsup_{r \to 0} L(r)$ will be proved in Lemma \[lem:Linfty\].
Proof of Lemma \[lem:maxpos\]
-----------------------------
Using Proposition \[prop:many to one\], one shows that for all $u \in \Sp$, $$W_n(u) ~:=~ \sum_{\abs{v}=n} H^\alpha(\mL(v)^\top u) ~=~ \sum_{\abs{v}=n} \int_{\Sp} \, \skalar{\mL(v)^\top u,y}^\alpha \, \nus[\alpha](dy)$$ defines a nonnegative martingale w.r.t. the filtration $\B_n$. Its $\Pfs$ limit $W(u)$ appears prominently in the non-critical case, where every fixed point has a Laplace transform of the form $\LTa(ru)=\E \exp(-K r^\alpha W(u))$, see [@Mentemeier2013 Theorem 1.2]. In the critical case, its limit is trivial:
\[prop:W\] Assume – and and . Then $W(u) = 0$ $\Pfs$ for all $u \in \Sp$.
Since $W(u)$ as the limit of a nonnegative martingale is again nonnegative, it suffices to show that $\E W(u)=0$. It even suffices to show that $\E W(u_0)=0$ for one $u_0 \in \interior{\Sp}$, for due to nonnegativity $$\label{eq:eq1}\skalar{ u,\mL(v) y} ~\le~ \skalar{ {\mathbf{1}}, \mL(v) y} ~\le~ \frac{1}{\min_i \, (u_0)_i} \skalar{u_0, \mL(v) y}$$ and hence $W_n(u) \le c W_n(u_0)$ for $c=[\min_{i} (u_0)_i]^{-1}$.
It is shown in [@Biggins2004 Theorem 2.1 (iii)], that $\E W(u_0)=0$ follows from $\limsup_{n \to \infty} {H^{\alpha}}(U_n)e^{\alpha S_n} = \infty$ $\Prob_{u_0}^\alpha$-a.s. But the latter is a direct consequence of , together with the strict positivity of ${H^{\alpha}}$.
Let as before $u_0 \in \interior{\Sp}$ and set $c=[\min_{i} (u_0)_i]^{-1} < \infty$. Recalling Eq. \[eq:eq1\] and the definition of $W_n(u_0)$, we have $$c W_n(u) ~\ge~ \sum_{\abs{v}=n} \int_{\Sp} \abs{\mL(v)y}^\alpha \, \nus[\alpha](dy).$$ By [@BDGM2014 Corollary 4.7], there is a constant $C$ such that for any allowable $\ma$, $\norm{\ma}^\alpha \le C \int_{\Sp} \, \abs{\ma y}^\alpha \, \nus[\alpha](dy),$ hence $$\frac{C}{c_u} \sqrt{d} W_n(\eins) ~\ge~ \sum_{\abs{v}=n} \norm{\mL(v)}^\alpha ~\ge~ \max_{\abs{v}=n}\, \norm{\mL(v)}^\alpha,$$ and the assertion follows.
\[lem:Linfty\] Under the assumptions of Theorem \[thm:main\], $\limsup_{r \to \infty} L(r)=0$. If $\alpha=1$, then $\E\abs{X}=\infty$ for every nontrivial fixed point $X$.
Suppose that $\limsup_{r \to 0} L(r) \le C < \infty$. By an extension of Prop. \[prop:dis\], , $$\begin{aligned}
Z(u) ~=&~ \lim_{n \to \infty} \sum_{\abs{v}=n} L(\abs{\mL(v)^\top u}) \, H^\alpha(\mL(v)^\top u) \frac{1-\LTfp(\mL(v)^\top u)}{L(\abs{\mL(v)^\top u}) \, H^\alpha(\mL(v)^\top u)} \\ ~\le&~ C \lim_{n \to \infty} \sum_{\abs{v}=n} H^\alpha(\mL(v)^\top u) ~=~ C W(u) =0 \end{aligned}$$ by Proposition \[prop:W\], which gives a contradiction.
If now $\alpha=1$, then $$\lim_{r \to 0} \frac{1-\LTfp(ru)}{r} ~=~ \skalar{u, \E X},$$ being finite or not. Combining this with Eq. implies that $$\lim_{r \to 0} L(r) ~=~ \frac{\skalar{u, \E X}}{K {H^{\alpha}}(u)},$$ hence $\E \abs{X} =\infty$, since $\limsup_{r \to 0} L(r)=\infty$.
Determining the Slowly Varying Function {#sect:L}
=======================================
In this section, we work under one of the additional assumptions or , together with . We want to identify the slowly varying function $L$, which was (given a nontrivial fixed point $\LTfp$ and a reference point $u_0 \in \supp \nust[\alpha]$) defined in Eq. to be $$L(r) ~=~ \frac{1-\LTfp(ru_0)}{r^\alpha H^\alpha (u_0)} ~=~ D(u_0, - \log r).$$ We are going to show that $$\label{eq:Dsv} \lim_{t \to \infty} \frac{D(u_0, t)}{t} = K' \in (0,\infty),$$ which gives that $\lim_{r \to 0}L(r)/\abs{\log r} =K'$, i.e. we may choose the slowly varying function to be a scalar multiple of $\abs{\log r} \vee 1$.
The basic idea to prove Eq. comes from [@DL1983] and is by using a renewal equation satisfied by (the one-dimensional analogue of) $D(u_0,t)$. In the present multivariate situation, we obtain a Markov renewal equation for a drift-less Markov random walk. By a clever application of the regeneration lemma, we can reduce this again to a (one-dimensional) renewal equation for a drift-less random walk, for which enough theory is known to solve it.
The Renewal Equation
--------------------
In this subsection we present the Markov renewal equation for $D(u,t)$ and show how, using Lemma \[regenerationlemma\], it can be replaced by a one-dimensional renewal equation.
\[lem:link\_D\_G\] Assume – and . Then the following renewal equation holds $$\label{eq:renewal_D}
D(u,t) = \E_u^{\alpha}D(U_1, t+ S_1) - G(u,t),$$ where $$\begin{aligned}
\label{defn:G} G(u,t) :=\ & \frac{e^{{\alpha}t}}{{H^{{\alpha}}}(u)}\Erw{\prod_{i=1}^N \LTa(e^{-t}\mT_i^\top u) + \sum_{i=1}^N\left(
1- \LTa(e^{-t}\mT_i^\top u) \right) -1}.\end{aligned}$$
Lemma 9.6 in [@Mentemeier2013a], note there the different notation $V_1=-S_1$.
\[lem:properties\_of\_G\] Assume – and .Then
1. $G(u,t) \ge 0$ for all $(u,t) \in \Sp \times \R$.
2. For all $u \in \Sp$, $t \mapsto e^{-\alpha t}G(u,t)$ is decreasing.
Lemma 9.7 in [@Mentemeier2013a], being a straightforward generalization of [@DL1983 Lemma 2.4].
From now on, assume that the assumptions of the Regeneration Lemma, Lemma \[regenerationlemma\] are satisfied, i.e. there is a sequence of stopping times $(\sigma_n)_{n \in \N}$ and a probability measure $\eta$ on $\Sp \times \R$ such that in particular holds.
For any nonnegative measurable function $F$ on $\Sp\times\R$ we define $\hat{F}:\R\mapsto\R$ by $$\begin{aligned}
\label{def:projection}
\hat{F}(t) ~:=~\E_\eta^\alpha \, {F(U_{\sigma_1-1},t+S_{\sigma_1-1})}.\end{aligned}$$ Moreover, under each $\Prob_u^\alpha$, let $(V_n)_{n \in \N}$ be a zero-delayed random walk with increment distribution $\Prob_\eta^\alpha(S_{\sigma_1-1} \in \cdot)$, independent of all other occurring random variables. Note that $V_n$ is a drift-less random walk.
For any nonnegative measurable function $F$ on $\Sp\times\R$ and $k\ge0$, the following equation holds $$\E_\eta^\alpha \left[F(U_{\sigma_{k+1}-1}, S_{\sigma_{k+1}-1})\right]=\E_\eta^\alpha {\hat{F}(V_k)}$$
We prove by induction. By the definition of $\hat{F}$, the equation holds for $k=0$. Suppose now that it holds for some $k \ge 0$. Then $$\begin{aligned}
\Ex[\eta]{F(U_{\sigma_{k+2}-1}, S_{\sigma_{k+2}-1})}~=~\Ex[\eta]{\E_\eta^\alpha \bigg[F(U_{\sigma_{k+2}-1}, S_{\sigma_{1}-1}+(S_{\sigma_{k+2}-1}- S_{\sigma_{1}-1}))|\F_{\sigma_{1}-1}\bigg]}\\
~=~\Ex[\eta]{{\E^\alpha_{\eta}}' \bigg[F(U'_{\sigma_{k+1}-1}, S_{\sigma_{1}-1}+S'_{\sigma_{k+1}-1})\bigg]}
~=~\Ex[\eta]{{\E_\eta^\alpha}' \bigg[{\hat{F}(S_{\sigma_{1}-1}+V'_k)}\bigg]}=\Ex[\eta]{\hat{F}(V_{k+1})},
\end{aligned}$$ where from Lemma \[regenerationlemma\] is used in the second equality and we denote by $(U'_n,S'_n), V_k$ an independent copy of $(U_n,S_n), V_k$ with corresponding expectation ${\E_\eta^\alpha}'$.
Now we can formulate the univariate renewal equation, corresponding to Eq. .
\[lem:renewal\_hatD\] For $g(t)=\Ex[\eta]{\sum_{i=0}^{{\sigma}_1-2}G(U_i,t+V_1+S_i)}$ we have $$\label{eq:renewal_hatD}
\hat D(t)=\E_\eta^\alpha\hat D(t+V_1)-g(t).$$
Let $$M_n=D(U_n,t+S_n)-\sum_{i=0}^{n-1}G(U_i,t+S_i).$$ Since $(U_n,S_n)$ is a Markov chain, the Markov renewal equation implies that $M_n$ is a $\Prob_u^\alpha$-martingale (with respect to the filtration $\mathcal{G}_n$) for each $u \in \supp \nust[\alpha]$. Since $\tau=\sigma_1-1$ is a stopping time by , the optional stopping theorem implies that
$$\label{eq:a}D(u,t+s)=\Ex[(u,s)]{D(U_{{\sigma}_1-1},t+S_{{\sigma}_1-1})-\sum_{i=0}^{{\sigma}_1-2}G(U_i,t+S_i)}$$
and $$\label{eq:c} D(u,t+s)=\Ex[(u,s)]{D(U_{{\sigma}_2-1},t+S_{{\sigma}_2-1})-\sum_{i=0}^{{\sigma}_2-2}G(U_i,t+S_i)}.$$ Equating the right hand sides of and and integrating with respect to $\eta$, we obtain $$\begin{aligned}
\hat D(t)=\Ex[\eta]{D(U_{{\sigma}_1-1},t+S_{{\sigma}_1-1})}=\Ex[\eta]{D(U_{{\sigma}_2-1},t+S_{{\sigma}_2-1})-\sum_{i={\sigma}_1-1}^{{\sigma}_2-2}G(U_i,t+S_i)}\\
=\E_\eta^\alpha \hat D(t+V_1)-\Ex[\eta]{\sum_{i=0}^{{\sigma}_1-2}G(U_i,t+V_1+S_i)}.
\end{aligned}$$
Solving the Renewal Equation
----------------------------
In this subsection, we will show that $\lim_{t \to \infty} D(u_0,t)/t=1$. Before we can use the renewal equation, we first have to consider some technicalities, e.g. direct Riemann integrability of $g$. We start by considering moments of $V_1$.
\[lem:exp\_moment\] Assume additionally -. Then there exists $\delta>0$ such that $\E_\eta^\alpha {e^{\delta |V_1|}}<{\infty}$.
We proof the boundedness of $\E_\eta^\alpha {e^{-\delta V_1}}$ and $\E_\eta^\alpha {e^{\delta V_1}}$ separately, starting with the first one.
Property (R4) implies that there exists $\delta_0$ such that $\sup_u\Ex[u]{e^{\delta_0 (\sigma-1)}}<{\infty}$. Due to Assumption , there is ${\varepsilon}>0$ such that $m(\alpha+{\varepsilon})\le e^{\delta_0}$. Observe that there is $C_\epsilon <\infty$ such that $$\frac{e^{-{\varepsilon}S_n}}{m(\alpha+{\varepsilon})^n} ~\le~ C_{\varepsilon}\frac{{H^{\alpha}}(u)}{{H^{\alpha+{\varepsilon}}}(u)} \frac{{H^{\alpha+{\varepsilon}}}(U_n)}{{H^{\alpha}}(U_n)} \frac{e^{-{\varepsilon}S_n}}{m(\alpha+{\varepsilon})^n},$$ and the right hand side is a martingale under $\Prob_u^\alpha$ with expectation $C_\epsilon$ due to Proposition \[prop:many to one\]. Therefore, the optional stopping theorem and the Fatou lemma imply $$\E_u^\alpha \left( \frac{e^{-{\varepsilon}S_{{\sigma-1}}}}{m(\alpha+{\varepsilon})^{{\sigma-1}}} \right) \le\lim_{n \to \infty}\E_u^\alpha \left( \frac{e^{-{\varepsilon}S_{(\sigma-1)\wedge n}}}{m(\alpha+{\varepsilon})^{(\sigma-1)\wedge n}} \right) \le C_{{\varepsilon}}.$$ The choice of ${\varepsilon}$ gives us $\sup_u\Ex[u]{m(\alpha+{\varepsilon})^{\sigma-1}}<{\infty}$, hence by the Cauchy-Schwartz inequality, $$(\Ex[u]{e^{-\frac{{\varepsilon}}{2} S_{\sigma-1}}})^2 \le\Ex[u]{e^{-{\varepsilon}S_{\sigma-1}}/m(\alpha+{\varepsilon})^{\sigma-1}} \Ex[u]{m(\alpha+{\varepsilon})^{\sigma-1}}$$ is bounded uniformly in $u$. Choose $\delta= \min\{\delta_0,\epsilon/2\}$.
For the second part recall that assumption implies that the increments of $S_n$ are bounded from above by $-\log c$. Therefore, $$\sup_u\Ex[u]{e^{\delta S_{\sigma-1}}}\le \sup_u\Ex[u]{{(1/c)^{\delta_0 (\sigma-1)}}}<{\infty}.$$ Integrating with respect to $\eta$ finishes the proof.
Before proving that $g(t)$ is dRi, we need the following consequence of the slow variation of $D(u_0,t)$ (for $t \to \infty$).
\[lem:bounds\_on\_LTa\] Let $d^*(t)=\sup_{u\in \Sp} D(t,u)$. Then for all $0<{\varepsilon}<\alpha$, there is $C >0$, such that for $t\ge0$ and any $s$ $$\begin{aligned}
\label{eqn:bound_1-LTa}
d^*(s)&\le Ce^{{\varepsilon}s}, \\
\label{eqn:bound_1-LTa_2}
\frac{d^*(t+s)}{L(e^{-t})}&\le C e^{{\varepsilon}|s|} .\end{aligned}$$
Since the ratio $D(t,u)/L(e^{-t})$ is bounded it suffice to show the above inequalities with $L(e^{-t})$ instead of $d^*(t)$. Potter’s theorem [@BGT1987 Theorem 1.5.6], applied to the slowly varying function $L$ proves that $$\begin{aligned}
\label{eq:potter}
\frac{L(e^{-x})}{L(e^{-y})}\le C e^{{\varepsilon}|x-y|},\end{aligned}$$ for any positive $x,y$. Using also the trivial bound $L(e^{-t})\le C e^{\alpha t}$ we get . In order to show we use in the case when $t+s\ge0$. When $t+s\le0$ we have $$\begin{aligned}
\frac{L(e^{-t-s})}{L(e^{-t})}=\frac{L(e^{-t-s})}{L(1)}\frac{L(1)}{L(e^{-t})}\le Ce^{\alpha(t+s)}e^{{\varepsilon}t}\le Ce^{{\varepsilon}|s|}.\end{aligned}$$
\[lem:g\_is\_dRi\] Assume in addition and . Then the function $g(x)$ is nonnegative and directly Riemann integrable.
Referring to Lemma \[lem:properties\_of\_G\], $G$ is nonnegative and $t \mapsto e^{-\alpha t} {G}(t)$ is decreasing, hence the same holds for $g$. For such functions, a sufficient condition for direct Riemann integrability is that ${g} \in \Lp[1]{\R}$, see [@Goldie1991 Lemma 9.1]. Since moreover, by Lemma \[regenerationlemma\], $\E\,\sigma_1<{\infty}$, it suffices to show the integrability of ${g}^* : t \mapsto \sup_{u \in \Sp} G(u,t)$.
Set $h(x):=e^{-x}+x-1$. Since $h$ is positive for $x\ge0$, we have $\LTa(e^{-t}\mT_i^\top u)\le e^{(1-\LTa(e^{-t}\mT_i^\top u))}$. Therefore $$\begin{aligned}
\int g^*(t)dt~=&~\int \sup_{u \in \Sp} \frac{e^{{\alpha}t}}{{H^{{\alpha}}}(u)}\Erw{\prod_{i=1}^N \LTa(e^{-t}\mT_i^\top u) + \sum_{i=1}^N\left(
1- \LTa(e^{-t}\mT_i^\top u) \right) -1}dt \\
\le&~ C\int \sup_{u \in \Sp} {e^{{\alpha}t}}\Erw{e^{\sum_{i=1}^N (1-\LTa(e^{-t}\mT_i^\top u))} + \sum_{i=1}^N\left(1- \LTa(e^{-t}\mT_i^\top u) \right) -1}dt\\
=&~ C\int \sup_{u \in\Sp} {e^{{\alpha}t}}\Erw{h\left(\sum_{i=1}^N(1-\LTa(e^{-t}\mT_i^\top u)) \right)} dt.\end{aligned}$$ Using Lemma \[lem:bounds\_on\_LTa\], boundedness of $H^{\alpha}$ and fact that $h(x)$ is increasing, comparable with $\min(x,x^2)$ on the positive half line, the later can be bounded by $$\begin{aligned}
\int \sup_{u \in\Sp} \, {e^{{\alpha}t}}\Erw{h\left(\sum_{i=1}^Ne^{({\varepsilon}-\alpha)t}\|\mT_i^\top u\|^{\alpha-{\varepsilon}} \right)} dt
\le~ C\,\Erw{\int {e^{{\alpha}t}}h\left(e^{({\varepsilon}-\alpha)t}\sum_{i=1}^N\|\mT_i\|^{\alpha-{\varepsilon}} \right) dt} \\
\le~ C\,\Erw{\left(\sum_{i=1}^N\|\mT_i\|^{\alpha-{\varepsilon}}\right)^{\frac{\alpha}{\alpha-{\varepsilon}}}\int {e^{ \frac{{\alpha}}{\alpha-{\varepsilon}}s}}h\left(e^{-s} \right) ds}
<{\infty},\end{aligned}$$ by , provided $\frac{\alpha}{\alpha-{\varepsilon}}<1+\delta<2$.
Now we show that the identification of $\hat{D}$ indeeds identifies $L(r)=D(u_0,-\log r)$.
\[lem:compare\_D\] Assume that - then $\lim_{t \to \infty} \hat D(t)/D(u_0,t)=1$. In particular, ${\hat{D}(t+s)}/{\hat{D}(t)}$ converge to 1 as $t$ goes to infinity.
Recalling the definition of $h_t$ from Section \[sect:regular variation\], we have that $$\begin{aligned}
\hat D(t)/D(u_0,t)=\Ex[\eta]{\frac{D(U_{{\sigma}_1-1},t+S_{{\sigma}_1-1})}{D(u_0,t)}} ~=~ \Ex[\eta]{h_t(U_{{\sigma}_1-1},S_{{\sigma}_1-1})}
\end{aligned}$$
Using Lemma \[lem:slowvar\], $\lim_{t \to \infty} h_t \equiv 1$. Lemmata \[lem:exp\_moment\] and \[lem:bounds\_on\_LTa\] allow us to apply the dominated convergence theorem to obtain the assertion.
Now we can identify the slowly varying function.
\[thm:Llog\] Assume that a function $\hat{D}$, such that ${\hat{D}(t+s)}/{\hat{D}(t)}\to1$ satisfies renewal equation with a directly Riemann integrable function $g$ and a nonarithmetic random variable $V_1$ such that $\Ex[\eta]{e^{\delta |V_1|}}<{\infty}$ for some positive $\delta$. Then $\lim_{t \to \infty} \hat D(t)/t $ exists and it is positive.
The proof is almost the same as the proof of Theorem 2.18 in [@DL1983]. Note that, although in [@DL1983] the derivative of $\hat D$ is used this can be easily avoided.
The Derivative Martingale {#sect:derivative martingale}
=========================
In this section, we finish the proof of Theorem \[thm:main2\], by proving the convergence of $$\mathcal{W}_n(u) = \sum_{\abs{v}=n} \left[ S(v) { + } b(U(v)) \right] \, {H^{\alpha}}(U(v)) e^{-\alpha S(v)}$$ to a nontrivial limit, which constitutes the exponent of fixed points. The assertions of Theorem \[thm:main2\] are contained in the Theorem below, except for the identification of the slowly varying function, which was given in Section \[sect:L\], in particular in Theorem \[thm:Llog\].
Under Assumptions – and or instead of , the martingale $\mathcal{W}_n(u) $ for each $u \in \Sp$ has a nonnegative, nontrivial limit $\mathcal{W}(u)$, and $\LTfp(ru):= \Erw{e^{-r^\alpha \mathcal{W}(u)}}$ is a fixed point of $\ST$.
Let $M(u)$ be the disintegration of the (up to scaling) unique fixed point of $\ST$ (described in Theorem \[thm:main\]). By Theorem \[thm:Llog\], combined with Eq. from Lemma \[lem:slowvar\], there is $K' \in (0,\infty)$ such that $$\lim_{r \to 0} \sup_{u \in \Sp} \abs{\frac{1 - \LTfp(r u) }{r^\alpha {H^{\alpha}}(u) K'\abs{\log(r)}} - 1} ~=~0.$$ Then by from Proposition \[prop:dis\], $$\lim_{n \to \infty} \, \sum_{\abs{v}=n} K' S^u(v) {H^{\alpha}}(U^u(v)) e^{-\alpha S^u(v)} \frac{1 - \LTfp(e^{-S^u(v)} U^u(v))}{K' S^u(v) {H^{\alpha}}(U^u(v)) e^{-\alpha S^u(v)}} ~=~ Z(u) \quad \Pfs$$ As a continuous function on $\Sp$, $u \mapsto b(u)$ is bounded, and by Lemma \[lem:maxpos\], $$\lim_{n \to \infty} \sup_{\abs{v}=n} \abs{\frac{S^u(v) + b(U^u(v))}{S^u(v)}-1}=0.$$ Therefore, we can replace $S^u(v)$ by $S^u(v) + b(U^u(v))$, and obtain $$\lim_{n \to \infty} \,\sum_{\abs{v}=n} \big[S^u(v)+ b(U^u(v)) \big] {H^{\alpha}}(U^u(v)) e^{-\alpha S^u(v)} ~=~ K' Z(u) \quad \Pfs$$ This shows the $\Pfs$ convergence of $\mathcal{W}_n(u)$ to $\mathcal{W}(u):=K'Z(u)$. Then $\P{\mathcal{W}(u)>0}=1$ by of Proposition \[prop:dis\]. That $\LTfp(ru)= \Erw{e^{-r^\alpha \mathcal{W}(u)}}$ is a fixed point follows immediately, since $\Erw{e^{-r^\alpha K' Z(u)}}$ is a fixed point for any $K'>0$.
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| {
"pile_set_name": "ArXiv"
} |
Frequent polarization reversals, or spin-flips, of a stored polarized beam in a high energy scattering asymmetry experiments may greatly reduce systematic errors of spin asymmetry measurements. A spin-flipping technique is being developed by using rf magnets running at a frequency close to the spin precession frequency, thereby creating spin-depolarizing resonances; the spin can then be flipped by ramping the rf magnet’s frequency through the resonance. We studied, at the Indiana University Cyclotron Facility Cooler Ring, properties of such rf depolarizing resonances in the presence of a nearly-full Siberian snake and their possible application for spin-flipping. By using an rf-solenoid magnet, we reached a 98.7$\pm$1$\%$ efficiency of spin-flipping. However, an rf-dipole magnet is more practical at high energies; hence, studies of spin-flipping by an rf-dipole are underway at IUCF. I would like to thank Professor Alan Krisch for his valuable advice, enouragement and support. I also thank my dissertation committee members their helpful comments and suggestions to this thesis. I thank my fellow collaborators from Michigan, Indiana, Protvino and Brookhaven for many hours they devoted to this experiment. I would especially like to thank Rick Phelps for teaching me a lot about how the experiment works.
I would like to thank the IUCF technical staff for their hard work on successful experiment operation. In particular, I want to thank the operators headed by Gary East with help from Terry Sloan for giving us beam when we needed it, the cryogenics experts Kevin Komisarcik and John Vanderwerp for keeping our snake cool, and the polarizaed ion source group headed by Vladimir Derenchuk for making polarized protons out of hydrogen gas.
Finally, I would like to thank my parents, my big brother and my little sister for their love and support, and my wife Svetlana for always being there when I needed her.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We construct the HNN extension of discrete quantum groups, we study their representation theory and we show that an HNN extension of amenable discrete quantum groups is $K$-amenable.'
---
[**K-amenability of HNN extensions of amenable discrete quantum groups**]{}
[Pierre Fima$^{(1,2)}$ ]{}
Introduction
============
The notion of $K$-amenability for discrete groups was introduced by Cuntz [@Cu83] in order to give a simpler proof of a result of Pimsner and Voiculescu [@PV82] calculating the $K$-theory of the reduced $C^*$-algebra of a free group. Cuntz proved that the free product of $K$-amenable discrete groups is $K$-amenable. Julg and Valette [@JV84] extended the notion of $K$-amenability to the locally compact case and proved the $K$-amenability of locally compact groups acting on trees with amenable stabilizers. By Bass-Serre theory [@Se83], this includes the case of amalgamated free products and HNN extensions of amenable discrete groups. Then, Pimsner [@Pi86] proved the $K$-amenability of locally compact groups acting on trees with $K$-amenable stabilizers.
At the quantum side, Skandalis [@Sk88] defined a notion of $K$-theoretic nuclearity for $C^*$-algebras analogous to Cuntz’s $K$-theoritic amenability and Germain [@Ge96] proved that the free product of unital separable $K$-nuclear $C^*$-algebras in $K$-nuclear.
In the 1980’s, Woronowicz [@Wo87], [@Wo88], [@Wo95] introduced the notion of compact quantum groups and generalized the classical Peter-Weyl representation theory. In this paper we consider discrete quantum groups as dual of compact quantum groups. Wang [@Wa95] introduced the amalgamated free product construction in the setting of Woronowicz and the free orthogonal and unitary quantum groups. The construction of free orthogonal and unitary quantum groups was generalized by Van Daele and Wang [@VW96].
Baaj and Skandalis developed [@BS89] the equivariant $KK$-theory with respect to coactions of Hopf $C^*$-algebras and the general theory of locally compact quantum groups was done by Kustermans and Vaes [@KV00]. Vergnioux [@Ve04] developed the equivariant $KK$-theory for locally compact quantum groups and proved the $K$-amenability of amalgamated free products of discrete amenable quantum groups. Voigt [@Vo11] proved the $K$-amenability of free orthogonal quantum groups and Vergnioux and Voigt [@VV11] proved the $K$-amenability of free products of free orthogonal and unitary quantum groups.
The goal of this paper is to prove $K$-amenability of HNN extensions of discrete amenable quantum groups. The HNN construction of a given group $H$ is a group $\Gamma$ in which $H$ embeds in such a way that two given isomorphic subgroups of $H$ are conjugate. More precisely, given a subgroup $\Sigma< H$ and an injective homomorphism $\theta\,:\,\Sigma\rightarrow H$, the HNN extension is defined by $\Gamma=\langle H,t\,:\,\theta(\sigma)=t\sigma t^{-1}\,\,\forall\sigma\in\Sigma\rangle$. The name HNN is given in honor to G. Higman, B. H. Neuman and H. Neumann who were the first authors to consider this construction in [@HNN49]. This construction was developed by Ueda [@Ue05] in the setting of von Neumann algebras and $C^*$-algebras. Another approach was given by the author and S. Vaes [@FV12] in the setting of tracial von Neumann algebras. In this paper, we follow this approach to construct the HNN extension of discrete quantum groups, we study its representation theory and we prove the $K$-amenability in the case where the given starting quantum group is amenable.
This paper is organized as follows. The section $2$ is a preliminary section in which we fix some notations and recall some basic definitions and results about quantum groups and $K$-amenability. In section $3$ we give a detailed description of HNN extensions of $C^*$-algebras. In section $5$ we construct HNN extensions of discrete quantum groups and study their representation theory. Finally, we proved the $K$-amenability of HNN extensions of discrete amenable quantum groups in section $5$.
Preliminaries
=============
All $C^*$-algebras are supposed to be separable and unital and all Hilbert $C^*$-modules are supposed to be separable. Let $A$ be a $C^*$-algebra and $H$ a Hilbert $A$-module. The $A$-valued scalar product is denoted by $\langle .,.\rangle$ and is supposed to be linear in the second variable. The $C^*$-algebra of adjointable maps on $H$ is denoted by $\mathcal{L}_A(H)$. We will use the same symbol ${\otimes}$ to denote the tensor product of Hilbert $C^*$-modules and the minimal tensor product of $C^{*}$-algebras. We use the symbol $\odot$ to denote the algebraic tensor product of vector spaces. We will use freely the leg numbering notation.
A *compact quantum group is a pair* ${\mathbb{G}}=(A,\Delta)$, where $A$ is a unital $C^{*}$-algebra, $\Delta$ is unital \*-homomorphism from $A$ to $A{\otimes}A$ satisfying $(\Delta{\otimes}{\text{id}})\Delta=({\text{id}}{\otimes}\Delta)\Delta$ and $\Delta(A)(A{\otimes}1)$ and $\Delta(A)(1{\otimes}A)$ are dense in $A{\otimes}A$.
We denote by $C({\mathbb{G}})$ the $C^{*}$-algebra $A$. The major results in the general theory of compact quantum groups are the existence and uniqueness of the Haar state and the Peter-Weyl representation theory.
Let ${\mathbb{G}}$ be a compact quantum group. There exists a unique state $\varphi$ on $C({\mathbb{G}})$ such that $({\text{id}}{\otimes}\varphi)\Delta(a)=\varphi(a)1=(\varphi{\otimes}{\text{id}})\Delta(a)$ for all $a\in C({\mathbb{G}})$. The state $\varphi$ is called the [Haar state]{} of ${\mathbb{G}}$.
The Haar state need not be faithful. When the Haar state is faithful we say that ${\mathbb{G}}$ is *reduced*. Let $C_{\text{red}}({\mathbb{G}}):=C({\mathbb{G}}) / I$ be the reduced $C^*$-algebra of ${\mathbb{G}}$, where $I=\{x\in A\,|\,\varphi(x^{*}x)=0\}$. $C_{\text{red}}({\mathbb{G}})$ has a canonical structure of compact quantum group, called the *reduced compact quantum group of* ${\mathbb{G}}$.
A unitary representation of dimension $n$ of ${\mathbb{G}}$ is a unitary $u\in M_n({\mathbb{C}}){\otimes}C({\mathbb{G}})$ such that $({\text{id}}{\otimes}\Delta)(u)=u_{12}u_{13}$. If $u\in M_n({\mathbb{C}}){\otimes}C({\mathbb{G}})$ and $v\in M_k({\mathbb{C}}){\otimes}C({\mathbb{G}})$ we define their *tensor product* by $$u{\otimes}v=u_{13}v_{23}\in M_n({\mathbb{C}}){\otimes}M_k({\mathbb{C}}){\otimes}C({\mathbb{G}}).$$ An *intertwiner* between $u$ and $v$ is a linear map $T\,:\,{\mathbb{C}}^n\rightarrow {\mathbb{C}}^k$ such that $(T{\otimes}1)u=v(T{\otimes}1)$. The unitary representations $u$ and $v$ are called *unitarily equivalent* if there exists a unitary intertwiner between $u$ and $v$. We call $u$ *irreducible* if the only intertwiners between $u$ and $u$ are the scalar multiples of the identity.
Every unitary representation is unitarily equivalent to a direct sum of irreducible unitary representations.
We denote by ${{\rm Irr}}({\mathbb{G}})$ the set of (equivalence classes) of irreducible unitary representations of a compact quantum group ${\mathbb{G}}$. For each $x\in{{\rm Irr}}({\mathbb{G}})$ we choose a representative $u^{x}\in M_{n_x}{\otimes}C({\mathbb{G}})$. The class of the trivial representation is denoted by $1$.
We denote by $\mathcal{C}({\mathbb{G}})$ the linear span of the coefficients of the $u^x$ for $x\in{{\rm Irr}}({\mathbb{G}})$. It is a unital dense \*-subalgebra of $C({\mathbb{G}})$. Let $C_{\text{max}}({\mathbb{G}})$ be the maximal $C^{*}$-completion of the unital \*-algebra $\mathcal{C}({\mathbb{G}})$. $C_{\text{max}}({\mathbb{G}})$ has a canonical structure of a compact quantum group called the *maximal quantum group* of ${\mathbb{G}}$. Observe that we have a canonical surjective morphism $\lambda\,:\,C_{\text{max}}({\mathbb{G}})\rightarrow C_{\text{red}}({\mathbb{G}})$ which is the identity on $\mathcal{C}({\mathbb{G}})$. ${\mathbb{G}}$ is called *amenable* if $\lambda$ is an isomorphism. ${\mathbb{G}}$ is called $K$*-amenable* if there exists $\alpha\in {\rm KK}(C_{\text{red}}({\mathbb{G}}),{\mathbb{C}})$ such that $\lambda^*(\alpha)=[\epsilon]\in{\rm KK}(C_{\text{max}}({\mathbb{G}}),{\mathbb{C}})$ where $\epsilon\,:\,C_{\text{max}}({\mathbb{G}})\rightarrow{\mathbb{C}}$ is the trivial representation i.e., $({\text{id}}{\otimes}\epsilon)(u^x)=1$ for all $x\in{{\rm Irr}}({\mathbb{G}})$.
[HNN]{} extensions of $C^*$-algebras. {#HNNCstar}
=====================================
**The reduced HNN extension**
The reduced HNN extension was introduced in [@Ue05]. Here, we follow the approach of [@FV12].
Let $B\subset A$ be a unital $C^*$-subalgebra of the unital $C^*$-algebra $A$ and $\theta\,:\,A\rightarrow B$ be an injective $*$-homomorphism. Define, for $\epsilon\in\{-1,1\}$, $$B_{\epsilon}=\left\{\begin{array}{lcl}
B&\text{if}&\epsilon=1,\\
\theta(B)&\text{if}&\epsilon=-1.
\end{array}\right.$$ We define $\theta^{\epsilon}\,:B_{\epsilon}\rightarrow B_{-\epsilon}\subset A$ in the obvious way.
We suppose that there exist conditional expectations $E_\epsilon\,:A\rightarrow B_{\epsilon}$ for $\epsilon\in\{-1,1\}$. For $\epsilon=1$, we denote by $(H_{1},\pi_{1},\eta_{1})$ the G.N.S. construction associated to $E_{1}$ i.e. $H_{1}$ is the Hilbert $B$-module obtained by separation and completion of $A$ for the $B$-valued scalar product $\langle x,y\rangle= E_{1}(x^*y)$, $x,y\in A$, the right action of $B$ is given by right multiplication, $\pi_{1}$ is the representation of $A$ on $H_{1}$ given by left multiplication and $\eta_{1}$ is the image of $1$ in $H_{1}$. For $\epsilon=-1$, we denote by $(H_{-1},\pi_{-1},\eta_{-1})$ the “G.N.S. construction” associated to $\theta^{-1}\circ E_{-1}$ i.e. $H_{-1}$ is the Hilbert $B$-module obtained by separation and completion of $A$ for the $B$-valued scalar product $\langle x,y\rangle= \theta^{-1}\circ E_{-1}(x^*y)$, $x,y\in A$, the right action of $b\in B$ is given by the right multiplication by $\theta(b)$, $\pi_{-1}$ is the representation of $A$ on $H_{-1}$ given by left multiplication and $\eta_{-1}$ is the image of $1$ in $H_{-1}$.
Observe that, for $\epsilon\in\{-1,1\}$, the map $(b\mapsto \pi_{\epsilon}(b)\eta_{\epsilon})$ is faithful on $B_{\epsilon}$ (hence $\pi_{\epsilon}|_{B_{\epsilon}}$ is also faithful). Although the representation $\pi_{\epsilon}$ may be not faithful on $A$ we will simply write $a\xi$ for $\pi_{\epsilon}(a)\xi$ when $\xi\in H_{\epsilon}$ and $a\in A$. We will also use the notation $\widehat{a}=\pi_{\epsilon}(a)\eta_{\epsilon}\in H_{\epsilon}$ for $a\in A$.
Observe that the submodule $\eta_{\epsilon} B$ is orthogonally complemented in $H_{\epsilon}$. Denote by $H_{\epsilon}^{\circ}$ the orthogonal complement of $\eta_{\epsilon} B$ in $H_{\epsilon}$ (it is the closure of $\{x\eta_{\epsilon}\,:\,E_{\epsilon}(x)=0\}$). One has $H_{\epsilon}=\eta_{\epsilon} B\oplus H_{\epsilon}^{\circ}$ and $B_{\epsilon}H_{\epsilon}^{\circ}= H_{\epsilon}^{\circ}$.
For $n\geq 1$ and $\epsilon_1,\ldots,\epsilon_n\in\{-1,1\}$ define $K_0=H_{-\epsilon_1}$, $K_n=H_1$ and, for $n\geq 2$ and $1\leq i\leq n-1$, $$K_i=\left\{\begin{array}{lcl}
H_{-\epsilon_i} &\text{if}&\epsilon_i=\epsilon_{i+1},\\
H_{\epsilon_i}^{\circ}&\text{if}&\epsilon_i\neq\epsilon_{i+1}.\end{array}\right.$$
For $i=0,\ldots,n$, we view all the $K_i$ as a Hilbert $B$-module as explained before. For $i=1,\ldots,n$ we have a representation $\rho_i\,:\,B\rightarrow \mathcal{L}_B(K_i)$ defined by, if $\xi\in K_i$ and $b\in B$, $$\rho_i(b)\xi=\left\{\begin{array}{lcl}
b\xi&\text{if}&\epsilon_i=1,\\
\theta(b)\xi&\text{if}&\epsilon_i=-1.\end{array}\right.$$
Define the Hilbert $B$-module $\mathcal{H}_{\epsilon_1,\ldots,\epsilon_n}=K_0\underset{\rho_1}{{\otimes}}\ldots\underset{\rho_n}{{\otimes}} K_n$. The left action of $A$ on $K_0$ by left multiplication induces a left action of $A$ on $\mathcal{H}_{\epsilon_1,\ldots,\epsilon_n}$ in the obvious way.
We define the Hilbert $B$-module $\mathcal{H}$ by the orthogonal direct sum $$\mathcal{H}=H_1\oplus\bigoplus_{n\geq 1,\,\epsilon_1,\ldots,\epsilon_n\in\{-1,1\}}\mathcal{H}_{\epsilon_1,\ldots,\epsilon_n},$$ with the left action of $A$ given by the direct sum of the left actions of $A$ on the Hilbert $B$-modules $\mathcal{H}_{\epsilon_1,\ldots,\epsilon_n}$ and the left action of $A$ on $H_1$. We denote this action by $\pi\,:\, A\rightarrow \mathcal{L}_B(\mathcal{H})$.
Observe that $\pi|_B$ is faithful. Also, if $\pi_1$ is faithful then $\pi$ is faithful.
Let $\epsilon\in\{-1,1\}$. We define an operator $u^{\epsilon}$ on $\mathcal{H}$ in the following way.
- If $\xi\in H_1$ we define $u^{\epsilon}\xi=\widehat{1}{\otimes}\xi\in\mathcal{H}_{\epsilon}$.
- If $\xi\in \mathcal{H}_{\epsilon_1,\ldots,\epsilon_n}$ with $n\geq 1$ and $\epsilon_1=\epsilon$ we define $u^{\epsilon}\xi=\widehat{1}{\otimes}\xi\in \mathcal{H}_{\epsilon,\epsilon_1,\ldots,\epsilon_n}$.
- If $\xi=\widehat{a}{\otimes}\xi_0\in\mathcal{H}_{\epsilon_1}$ with $\epsilon_1\neq\epsilon$ and $\widehat{a}\in K_0=H_{\epsilon}$, $\xi_0\in K_1=H_1$ we define $$u^{\epsilon}(\widehat{a}{\otimes}\xi_0)=\left\{
\begin{array}{llcl}
\widehat{1}{\otimes}\widehat{a}{\otimes}\xi_0&\in\mathcal{H}_{\epsilon,\epsilon_1}&\text{if}&E_{\epsilon}(a)=0,\\
\theta^{\epsilon}(a)\xi_0&\in H_{1}&\text{if}&a\in B_{\epsilon}.\\
\end{array}\right.$$
- If $\xi=\widehat{a}{\otimes}\xi_0\in\mathcal{H}_{\epsilon_1,\ldots,\epsilon_n}$ with $n\geq 2$, $\epsilon_1\neq\epsilon$ and $\widehat{a}\in K_0$, $\xi_0\in K_1\underset{\rho_2}{{\otimes}}\ldots\underset{\rho_n}{{\otimes}} K_n$ (which is a sub-$B$-module of $\mathcal{H}_{\epsilon_2,\ldots,\epsilon_n}$) we define $$u^{\epsilon}(\widehat{a}{\otimes}\xi_0)=\left\{
\begin{array}{llcl}
\widehat{1}{\otimes}\widehat{a}{\otimes}\xi_0&\in\mathcal{H}_{\epsilon,\epsilon_1,\ldots,\epsilon_n}&\text{if}&E_{\epsilon}(a)=0,\\
\theta^{\epsilon}(a)\xi_0&\in\mathcal{H}_{\epsilon_2,\ldots,\epsilon_n}&\text{if}&a\in B_{\epsilon}.\\
\end{array}\right.$$
It is easy to check that $u^{\epsilon}$ commutes with the right action of $B$ and extends to a unitary on the Hilbert $C^*$-module $\mathcal{H}$ such that $(u^{\epsilon})^*=u^{-\epsilon}$ so that the superscript $\epsilon$ really means “to the power $\epsilon$”. We denote by $u$ the unitary $u^1$. One can also easily check the following formula : $$u\pi(b)u^*=\pi(\theta(b))\quad\text{for all}\quad b\in B.$$
Although it is not necessary, we will assume, to simplify notations and for the rest of this section, that $E_{\epsilon}$, for $\epsilon\in\{-1,1\}$, is G.N.S. faithful i.e., $\pi_{\epsilon}$ is faithful. Hence, $\pi$ is faithful and we may and will assume that $A\subset\mathcal{L}_B(\mathcal{H})$ and $\pi={\text{id}}$. The preceding relation becomes $ubu^*=\theta(b)$ for all $b\in B$.
The *reduced* HNN *extension* ${{\rm HNN}}(A,B,\theta)$ is the $C^*$-subalgebra of $\mathcal{L}_B(\mathcal{H})$ generated by $A$ and $u$: $${{\rm HNN}}(A,B,\theta):=\langle A,u\rangle\subset\mathcal{L}_B(\mathcal{H}).$$
Let $P={{\rm HNN}}(A,B,\theta)$. An operator $x\in P$ of the form $x=x_0u^{\epsilon_1}x_1\ldots u^{\epsilon_n}x_n$ with $n\geq 1$, $x_i\in A$ and $\epsilon_i\in\{-1,1\}$ will be called *reduced* if for all $1\leq i\leq n-1$ we have $E_{\epsilon_i}(x_i)=0$ whenever $\epsilon_{i+1}\neq\epsilon_i$. Observe that our terminology is different from the one adopt in [@FV12]: we do not allow $n=0$ in the definition of a reduced operator.
Let $\Omega=\eta_1\in H_1\subset\mathcal{H}$. Observe that $\Omega$ is $B$-central. Namely, $b\Omega=\Omega b$ for all $b\in B$. Let $x=x_0u^{\epsilon_1}\ldots u^{\epsilon_n}x_n$ be a reduced operator. One has $$\begin{aligned}
\label{gns}
x\Omega=\hat{x}_0{\otimes}\ldots{\otimes}\hat{x}_n\in\mathcal{H}_{\epsilon_1,\ldots,\epsilon_n}.\end{aligned}$$ It follows that the integer $n$ (and the sequence $\epsilon_1,\ldots,\epsilon_n$) only depends on the operator $x$. The integer $n$ is called the *length* of the reduced operator $x$.
Let $\mathcal{P}$ be the vector subspace of $P$ spanned by the reduced operators and $A$. By the relation $\theta(b)=ubu^*$ for $b\in B$, it is easy to check that $\mathcal{P}$ is a \*-subalgebra of $P$. Moreover, by definition of the HNN extension, $\mathcal{P}$ is dense in $P$.
Define, for $x\in P$, $E_B(x)=\langle\Omega,x\Omega\rangle\in B$. It is easily seen that $E_B$ a conditional expectation onto $B$ satisfying $E_B|_A=E_1$. Moreover, using $(\ref{gns})$, we see that, for all reduced operator $x\in P$, one has $E_B(x)=0$. Equation $(\ref{gns})$ also implies that $\overline{P\Omega}=\mathcal{H}$ hence, $(\mathcal{H},{\text{id}},\Omega)$ is the GNS construction of $E_B$.
Define, for $x\in P$, $E_{\theta(B)}(x)=uE_B(u^*xu)u^*\in \theta(B)$. Again, it is easy to check that $E_{\theta(B)}$ a conditional expectation onto $\theta(B)$ satisfying $E_{\theta(B)}|_A=E_{-1}$.
The reduced HNN extension $P$ satisfies the following universal property.
\[universal\] Let $C$ be a unital $C^*$-algebra with a unital faithful $*$-homomorphism $\rho\,:\,A\rightarrow C$. Suppose that there exists a unitary $w\in C$ and a conditional expectation $E'$ from $C$ to $\rho(B)$ such that:
1. $C$ is generated by $\rho(A)$ and $w$.
2. $w\rho(b)w^*=\rho(\theta(b))$ for all $b\in B$ and $E'\circ\rho=\rho\circ E_1$.
3. For all $n\geq 1$, $\epsilon_1,\ldots,\epsilon_n\in\{-1,1\}$ one has $E'(\rho(x_0)w^{\epsilon_1}\ldots w^{\epsilon_n}\rho(x_n))=0$ for all $x_i\in A$ such that $E_{\epsilon_i}(x_i)=0$ whenever $\epsilon_i\neq \epsilon_{i+1}$, $1\leq i\leq n-1$.
4. $E'$ is G.N.S. faithful i.e., for all $x\in C$, if $E'(y^*x^*xy)=0$ for all $y\in C$, then $x=0$.
Then, there exists a unique $*$-isomorphism $\widetilde{\rho}\,:\, P\rightarrow C$ such that $$\widetilde{\rho}(u)=w\quad\text{and}\quad\widetilde{\rho}(a)=\rho(a)\,\,\text{for all}\,\,a\in A.$$ Moreover, $\widetilde{\rho}$ intertwines $E'$ and $E_B$.
Since $P$ is generated by $A$ and $u$, the uniqueness is obvious. Let $(H',\rho',\eta')$ be the GNS construction of $\rho^{-1}\circ E'$ i.e, $H'$ as a Hilbert $B$-module obtained by separation and completion of $C$ for the $B$-valued scalar product $\langle x,y\rangle= \rho^{-1}\circ E'(x^*y)$, the right action of $b\in B$ is given by the right multiplication by $\rho(b)$, $\rho'$ is the representation of $C$ given by left multiplication and $\eta'$ is the image of $1$ in $H'$. By $4$, $\rho'$ is faithful so we may and will assume that $C\subset\mathcal{L}_B(H')$ and $\rho'={\text{id}}$. Define $V\,:\,\mathcal{H}\rightarrow H'$ by $Va\Omega=\rho(a)\eta'$ for $a\in A$ and, for $x=x_0u^{\epsilon_1}\ldots u^{\epsilon_n}x_n\in P$ a reduced operator, $Vx\Omega=\rho(x_0)w^{\epsilon_1}\ldots w^{\epsilon_n}\rho(x_n)\eta'$. It is easy to check that $V$ extends to a unitary $V\in\mathcal{L}_B(\mathcal{H},H')$ such that $VaV^*=\rho(a)$ for all $a\in A$ and $VuV^*=w$. Then, $\widetilde{\rho}(x)=VxV^*$ does the job.
We construct now a conditional expectation from $P$ to $A$. Let $Q\in\mathcal{L}_B(\mathcal{H})$ be the projection onto the Hilbert sub-B-module $H_1$ of $\mathcal{H}$. Then, it is easy to check that the formula $E_A(x)=QxQ\in\mathcal{L}_B(Q\mathcal{H})=\mathcal{L}_B(H_1)$ defines a conditional expectation from $P$ to $A\subset\mathcal{L}_B(H_1)$ satisfying: $$E_A(x)=0\quad\text{for all reduced operator $x\in P$}.$$ Moreover, $E_{\epsilon}\circ E_A=E_{B_{\epsilon}}$ for $\epsilon\in\{-1,1\}$.
**The maximal HNN extension**
The maximal (or full, or universal) HNN extension was also introduced in [@Ue05]. We keep the same notations as before and we still assume that we have conditional expectations with faithful G.N.S. constructions from $A$ to $B_{\epsilon}$ for $\epsilon\in\{-1,1\}$. The maximal HNN extension is the unital $C^*$-algebra $P_m$ generated by $A$ and a unitary $w\in P_m$ such that $wbw^*=\theta(b)$ for all $b\in B$ and satisfying the universal property that whenever $C$ is a unital $C^*$-algebra with a unitary $u\in C$ and a $*$-homomorphism $\rho\,:\,A\rightarrow C$ such that $u\rho(b)u^*=\rho(\theta(b))$ for all $b\in B$ there exists a unique $*$-homomorphism $\widetilde{\rho}\,:\,P_m\rightarrow C$ such that $\widetilde{\rho}|_A=\rho$ and $\widetilde{\rho}(w)=u$. Such a $C^*$-algebra is obviously unique (up to a canonical isomorphism) and is denoted by ${\rm HNN}_{\text{max}}(A,B,\theta)$.
HNN extensions of Compact Quantum Groups {#HNNCQG}
========================================
We consider two reduced compact quantum groups ${\mathbb{G}}_A=(A,\Delta_A)$ and $\mathbb{G}_B=(B,\Delta_B)$. We denote by $\varphi_A$ and $\varphi_B$ the Haar (faithful) states on $A$ and $B$ respectively.
We suppose that $\theta\,:\,B\rightarrow A$ is an injective unital \*-homomorphism which intertwines the comultiplications. Hence, $\theta(B)$ is a Woronowicz $C^*$-subalgebra of $A$. By [@Ve04], $\varphi_A\circ\theta=\varphi_B$ and there exists a unique conditional expectation $E_{\theta}\,:\, A\rightarrow\theta(B)$ such that $\varphi_A=\varphi_B\circ\theta^{-1}\circ E_{\theta}$. Since $\varphi_A$ is faithful, $E_{\theta}$ is faithful. In particular, $E_{\theta}$ is G.N.S. faithful. This conditional expectation is also characterized by the following invariance property: $$({\text{id}}{\otimes}E_{\theta})\circ\Delta_A=(E_{\theta}{\otimes}{\text{id}})\circ\Delta_A=\Delta_B\circ\theta^{-1}\circ E_{\theta}=\Delta_A\circ E_{\theta}.$$
We suppose that $B\subset A$ is a Woronowicz $C^*$-subalgebra. We can apply the preceding discussion to the map $\theta={\text{id}}$. In particular, we have a G.N.S. faithful conditional expectation $E_1\,:\,A\rightarrow B$. Let $\theta\,:\,B\rightarrow A$ be an embedding which intertwines the comultiplications. Once again, the preceding discussion applies to $\theta$ and we have a G.N.S. faithful conditional expectation $E_{-1}\,:\,A\rightarrow \theta(B)$. We will freely use the notations and results of section \[HNNCstar\]. Define $P={{\rm HNN}}_{\text{red}}(A,B,\theta)=\langle A, u\rangle$.
From the hypothesis, we also have canonical inclusions $C_{\text{max}}({\mathbb{G}}_B)\subset C_{\text{max}}({\mathbb{G}}_A)$ which intertwine the comultiplications. Also, since the injective morphism $\theta\,:\,\mathcal{C}({\mathbb{G}}_B)\rightarrow\mathcal{C}({\mathbb{G}}_A)$ intertwines the comultiplications we have a canonical injective morphism $\theta\,:\,C_{\text{max}}({\mathbb{G}}_B)\rightarrow C_{\text{max}}({\mathbb{G}}_A)$ which intertwines the comultiplications.
Define $P_{m}:={\rm HNN}_{\text{max}}(C_{\text{max}}({\mathbb{G}}_A),C_{\text{max}}({\mathbb{G}}_B),\theta)=\langle C_{\text{max}}({\mathbb{G}}_A), w\rangle$. By the universal property, there exists a unique $*$-homomorphism $\Delta_m\,:\, P_m\rightarrow P_m{\otimes}P_m$ such that $$\Delta(w)=w{\otimes}w\quad\text{and}\quad\Delta_m(a)=\Delta_A(a)\,\,\,\,\forall a\in C_{\text{max}}({\mathbb{G}}_A).$$ $P_m$ is generated, as a $C^*$-algebra, by the elements $v^x_{i,j}$ for $x\in{\rm Irr}({\mathbb{G}}_A)$ and $1\leq i,j\leq \text{dim}(x)$ and by $w$ for which it is easy to check that the conditions of [@Wa95 Definition 2.1’] are satisfied. Hence, ${\mathbb{G}}_m=(P_m,\Delta_m)$ is a compact quantum group.
Let us denote by $\lambda$ the canonical surjective morphism from $C_{\text{max}}({\mathbb{G}}_A)$ to $A$. By the universal property, we have a unique $*$-homomorphism, still denoted by $\lambda$, from $P_m$ to $P$ such that $$\lambda(w)=u\quad\text{and}\quad\lambda(a)=\lambda(a)\,\,\,\text{for all}\,\,a\in C_{\text{max}}({\mathbb{G}}_A).$$
We view ${{\rm Irr}}({\mathbb{G}}_B)$ as a subset of ${{\rm Irr}}({\mathbb{G}}_A)$ and we also view ${{\rm Irr}}({\mathbb{G}}_A)$ as a subset of ${{\rm Irr}}({\mathbb{G}}_m)$. The map $\theta$ induces an injective map, still denoted by $\theta$, from ${{\rm Irr}}({\mathbb{G}}_B)$ to ${{\rm Irr}}({\mathbb{G}}_A)$. For $\epsilon\in\{-1,1\}$ we define $${{\rm Irr}}({\mathbb{G}}_A)_{\epsilon}=\left\{\begin{array}{lcl}
{{\rm Irr}}({\mathbb{G}}_A)\setminus{{\rm Irr}}({\mathbb{G}}_B)&\text{if}&\epsilon=1,\\
{{\rm Irr}}({\mathbb{G}}_A)\setminus\theta({{\rm Irr}}({\mathbb{G}}_B))&\text{if}&\epsilon=-1.\end{array}\right.$$ Observe that $w\in P_m$ is a irreducible representation of ${\mathbb{G}}_m$ of dimension $1$.
Let $v$ be a unitary representation of ${\mathbb{G}}_m$. We call $v$ *reduced* if $v$ is of the form $v=v^{x_0}{\otimes}w^{\epsilon_1}{\otimes}\ldots{\otimes}w^{\epsilon_n}{\otimes}v^{x_n}$ where $n\geq 1$, $x_k\in{{\rm Irr}}({\mathbb{G}}_A)$ and $\epsilon_k\in\{-1,1\}$ are such that, for all $1\leq k\leq n-1$, $x_k\in{{\rm Irr}}({\mathbb{G}}_A)_{\epsilon_k}$ whenever $\epsilon_k\neq\epsilon_{k+1}$.
\[rep\] The following holds.
1. The Haar state is given by $\varphi_m=\varphi_A\circ E_A\circ\lambda$.
2. Every non-trivial irreducible unitary representation of ${\mathbb{G}}_m$ is unitarily equivalent to a subrepresentation of a reduced representation or to an irreducible representation of ${\mathbb{G}}_A$. Hence, ${{\rm Irr}}({\mathbb{G}}_A)$ and $w$ generate the representation category of ${\mathbb{G}}_m$.
3. The reduced $C^*$-algebra of ${\mathbb{G}}_m$ is $P$, the maximal one is $P_m$.
$1.$ Let $\mathcal{P}_m\subset P_m$ be the linear span of the coefficients of the reduced representations. It is easy to see that the linear span of $\mathcal{P}_m$ and $A$ is a dense $*$-subalgebra of $P_m$. Moreover, $\Delta_m(\mathcal{P}_m)\subset\mathcal{P}_m\odot\mathcal{P}_m$ and $\lambda(\mathcal{P}_m)$ is contained in the linear span of the reduced operators in $P$. Hence, $E_A\circ\lambda(\mathcal{P}_m)=\{0\}$. It follows that, for all $x\in\mathcal{P}_m$, $({\text{id}}{\otimes}\varphi_m)\Delta_m(x)=(\varphi_m{\otimes}{\text{id}})\Delta_m(x)=0=\varphi_m(x)1$. Hence, it suffices to check the invariance property for $x$ a coefficient of a irreducible representation of ${\mathbb{G}}_A$ for which it is obvious.
$2.$ Since the linear span of the coefficients of the reduced representations and the coefficients of the irreducible representations of ${\mathbb{G}}_A$ is dense in $P_m$, the result follows from the general theory.
$3.$ Since the morphism $\lambda$ is surjective and the state $\varphi_A\circ E_A$ is faithful on $P$, it follows from $1$ that the reduced $C^*$-algebra of ${\mathbb{G}}_m$ is $P$. Moreover, it follows from $2$ that $\mathcal{C}({\mathbb{G}}_m)$ is equal to the linear span of $\mathcal{P}_m$ and $\mathcal{C}({\mathbb{G}}_A)$. Hence, $C_{\text{max}}({\mathbb{G}}_m)$ is generated, as a $C^*$-algebra, by $\mathcal{C}({\mathbb{G}}_A)$ and $w$. By the universal property of $C_{\text{max}}({\mathbb{G}}_A)$, we have a $*$-homomorphism from $C_{\text{max}}({\mathbb{G}}_A)$ to $C_{\text{max}}({\mathbb{G}}_m)$ which is the identity on $\mathcal{C}({\mathbb{G}}_A)$. Since the relation $\theta(b)=wbw^*$ holds in $C_{\text{max}}({\mathbb{G}}_m)$ for all $b\in C_{\text{max}}({\mathbb{G}}_B)$, we have a surjective homomorphism from the HNN extension $P_m$ to $C_{\text{max}}({\mathbb{G}}_m)$ which is the identity of $\mathcal{C}({\mathbb{G}}_m)$. It follows that $P_m=C_{\text{max}}({\mathbb{G}}_m)$.
One could have have constructed first the reduced compact quantum group and prove that the maximal one is ${\mathbb{G}}_m$. Indeed, one can prove directly, at the reduced level, that there exists a unique $*$-homomorphism $\Delta\,:\,P\rightarrow P\otimes P$ such that $$\Delta(u)=u\otimes u\quad\text{and}\quad\Delta(x)=\Delta_A(x)\,\,\forall x\in A.$$ To prove that, it suffices to consider the $C^*$-subalgebra $C$ of $P{\otimes}P$ generated by $\Delta_A(A)$ and $u{\otimes}u$, to view $\rho=\Delta_A$ as a unital faithful $*$-homomorphism from $A$ to $C$, and to check the hypothesis of Proposition \[universal\] with the conditional expectation $E'=({\text{id}}{\otimes}E_1)|_C$. One can also check easily that $\varphi=\varphi_A\circ E_A$ is $\Delta$-invariant. It follows from the general theory that $(P,\Delta)$ is a reduced compact quantum group. Moreover, one can show that the maximal $C^*$-algebra of $(P,\Delta)$ is $P_m$ by studying the representations, as in the proof of Theorem \[rep\].
Let $N<G$ be a non-trivial closed normal subgroup of a compact group $G$ and define $K=G/N$. Let $\theta\,:\,G\rightarrow K$ be a continuous surjective group homomorphism. View $C(K)\subset C(G)$ as $N$-right invariant functions and define the injective $*$-homomorphism $C(K)\rightarrow C(G)$ by composing with $\theta$. Then, one can perform the HNN construction to get a compact quantum group which is non-commutative and non-cocommutative whenever $G$ is non-commutative.
K-amenability
=============
This section contains the proof of the following theorem.
\[Kamenability\] An HNN extension of amenable discrete quantum groups is K-amenable.
Let $P=\text{HNN}(A,B,\theta)=\langle A, u\rangle$ be a reduced HNN extension of $C^*$-algebras. Let $E_A$ and $E_B$ be the conditional expectations from $P$ to $A$ and $B$ respectively.
\[iso\] Let $x=x_0u^{\epsilon_1}\ldots u^{\epsilon_n}\in P$ be a reduced word in $P$.
1. If $\epsilon_n=-1$ then $E_A(x^*x)=\theta\circ E_B((xu)^*xu)$.
2. If $\epsilon_n=1$ then $E_A(x^*x)=E_B(x^*x)$.
$1.$ We prove it by induction on $n$. If $n=1$, write $x=x_0u^*$, then $$E_A(x^*x)=E_A(ux_0^*x_0u^*)=E_A(uE_B(x_0^*x_0)u^*)=\theta\circ E_B(x_0^*x_0)=\theta\circ E_B((xu)^*xu).$$ Suppose that $1$ holds for $n$. Let $x=x_0u^{\epsilon_1}\ldots u^{\epsilon_n}x_nu^*$ be a reduced word. In the following computation we use the notation $E_{-1}=E_{\theta(B)}$ and $E_1=E_B$. One has $$\begin{aligned}
E_A(x^*x)&=&E_A(ux_n^*u^{-\epsilon_n}\ldots u^{-\epsilon_1}x_0^*x_0u^{\epsilon_1}\ldots u^{\epsilon_n}x_nu^*)\\
&=&E_A(ux_n^*u^{-\epsilon_n}\ldots \theta^{-\epsilon_1}(E_{-\epsilon_1}(x_0^*x_0))\ldots u^{\epsilon_n}x_nu^*)=E_A(y^*y),\end{aligned}$$ where $y=x_1'u^{\epsilon_2}\ldots u^{\epsilon_n}x_nu^*$ and $x_1'=\sqrt{\theta^{-\epsilon_1}(E_{-\epsilon_1}(x_0^*x_0))}x_1$. Observe that $y$ is reduced of length $n$ and ends with $u^*$. By the induction hypothesis one has $E_A(y^*y)=\theta\circ E_B((yu)^*yu)$. But $$\begin{aligned}
E_B((yu)^*yu)&=&E_B(x_n^*u^{-\epsilon_n}\ldots \theta^{-\epsilon_1}(E_{-\epsilon_1}(x_0^*x_0))\ldots u^{\epsilon_n}x_n)\\
&=&E_B(x_n^*u^{-\epsilon_n}\ldots u^{-\epsilon_1}x_0^*x_0u^{\epsilon_1}\ldots u^{\epsilon_n}x_n)=E_B((xu)^*xu).\end{aligned}$$ This finishes the proof of $1$. The proof of $2$ is similiar.
We suppose that ${\mathbb{G}}_A=(A,\Delta_A)$ and ${\mathbb{G}}_B=(B,\Delta_B)$ are two reduced compact quantum groups such that the inclusion $B\subset A$ and the $*$-homomorphism $\theta$ intertwine the comultiplications. Assume that ${\mathbb{G}}_A$ is coamenable and let $\epsilon\,:\, A\rightarrow{\mathbb{C}}$ be the counit. Hence, $P$ is the reduced $C^*$-algebra of the HNN extension compact quantum group. Observe that $\epsilon\circ \theta=\epsilon$. Let $(H,\pi,\xi)$ be the GNS construction of the state $\epsilon\circ E_A$ on $P$ and $(K,\rho,\eta)$ be the GNS construction of the state $\epsilon\circ E_B$ on $P$. Define $$H_0={\mathbb{C}}\xi,\,\,H_{\pm 1}=\overline{\text{Span}}\{\pi(x)\xi\,:\,\,x=x_0u^{\epsilon_1}\ldots u^{\epsilon_n}\,\,\text{is a reduced word with}\,\epsilon_n=\pm 1\}.$$ Observe that the spaces $H_0, H_{-1}, H_1$ are pairwise orthogonal. Moreover, since $\pi(a)\xi=\epsilon(a)\xi$ for all $a\in A$, one has $H=H_0\oplus H_{-1}\oplus H_1$. We also define $$K_{-1}=\overline{\text{Span}}\{\rho(x)\eta\,:\,x\in A\,\,\text{or}\,\,x=x_0u^{\epsilon_1}\ldots u^{\epsilon_n}x_n\,\,\text{is a reduced word with}\,(\epsilon_n=- 1)\,\text{or}\,(\epsilon_n=1\,\text{and}\, E_B(x_n)=0)\},$$ and $K_{1}=\overline{\text{Span}}\{\rho(x)\eta\,:\,x=x_0u^{\epsilon_1}\ldots u^{\epsilon_n}\,\,\text{is a reduced word with}\,\epsilon_n=1\}$. Observe that $K_{-1}$ and $K_1$ are orthogonal subspaces. Moreover, since $\rho(b)\eta=\epsilon(b)\eta$ for all $b\in B$, one has $K=K_{-1}\oplus K_1$.
By lemma \[iso\] (and since $\epsilon\circ\theta=\epsilon$) we have isometries $F_i\,:\, H_i\rightarrow K_i$, for $i=\pm 1$, defined by, for $x=x_0u^{\epsilon_1}\ldots u^{\epsilon_n}$ a reduced word in $P$, $$F_{-1}(\pi(x)\xi)=\rho(xu)\eta\,\,\text{if}\,\,\epsilon_n=-1\quad\text{and}\quad F_1(\pi(x)\xi)=\rho(x)\eta\,\,\text{if}\,\,\epsilon_n=1.$$ Since $F_{-1}$ and $F_1$ are clearly surjective, they are unitaries. Hence, we get a unitary $$F=F_{-1}\oplus F_1\,:\,H_{-1}\oplus H_1\rightarrow K_{-1}\oplus K_1.$$
We define the *Julg-Valette operator* $\mathcal{F}\,:\,H\rightarrow K$ by $\mathcal{F}|_{{\mathbb{C}}\xi}=0$ and $\mathcal{F}|_{H_{-1}\oplus H_1}=F$. Hence, $\mathcal{F}$ is a partial isometry with $\mathcal{F}\mathcal{F}^*=1$ and $\mathcal{F}^*\mathcal{F}=1-p$ where $p$ is the orthogonal projection onto the one dimensional subspace ${\mathbb{C}}\xi$.
\[compact\] The following holds.
1. For all $a\in A$, $\mathcal{F}\pi(a)=\rho(a)\mathcal{F}$.
2. $\mathcal{F}\pi(u)-\rho(u)\mathcal{F}$ is a rank one operator with image ${\mathbb{C}}\rho(u)\eta$.
3. $\mathcal{F}\pi(u^*)-\rho(u^*)\mathcal{F}$ is a rank one operator with image ${\mathbb{C}}\eta$.
$1.$ Let $a\in A$. One has $\mathcal{F}\pi(a)\xi=\epsilon(a)\mathcal{F}\xi=0=\rho(a)\mathcal{F}\xi$ and, for $x=x_0u^{\epsilon_1}\ldots u^{\epsilon_n}$ a reduced word, $$\mathcal{F}\pi(a)\pi(x)\xi=\mathcal{F}\pi(ax)\xi=\left\{\begin{array}{lcl}
\rho(axu)\eta=\rho(a)\mathcal{F}\pi(x)\xi &\text{if}&\epsilon_n=-1,\\
\rho(ax)\eta=\rho(a)\mathcal{F}\pi(x)\xi &\text{if}&\epsilon_n=1.\end{array}\right.$$
$2.$ Let $x=x_0u^{\epsilon_1}\ldots u^{\epsilon_n}$ be a reduced word. If $n\geq 2$ then it is easy to see that $ux$ can be written has a reduced word or a sum of two reduced words that end with $u^{\epsilon_n}$. Hence, $$\mathcal{F}\pi(u)\pi(x)\xi=\mathcal{F}\pi(ux)\xi=\left\{\begin{array}{lcl}
\rho(uxu)\eta=\rho(u)\mathcal{F}\pi(x)\xi &\text{if}&\epsilon_n=-1,\\
\rho(ux)\eta=\rho(u)\mathcal{F}\pi(x)\xi &\text{if}&\epsilon_n=1.\end{array}\right.$$ If $n=1$ and $x=x_0u^{\epsilon}$. When ($\epsilon=1$) or ($\epsilon=-1$ and $E_B(x_0)=0$) we see that $ux$ can be written as a reduced word that ends with $u^{\epsilon}$. As before, we conclude that $\mathcal{F}\pi(u)\pi(x)\xi=\rho(u)\mathcal{F}\pi(x)\xi$ in this case. Hence, the operator $\mathcal{F}\pi(u)-\rho(u)\mathcal{F}$ vanishes on the subspace $L$ where $$\begin{aligned}
L^{\bot}&=&\overline{\text{Span}}\{\xi,\pi(x_0u^*)\xi\,:\,x_0\in B\}=\overline{\text{Span}}\{\xi,\pi(u^*\theta(x_0))\xi\,:\,x_0\in B\}\\
&=&\overline{\text{Span}}\{\xi,\epsilon\circ\theta(x_0)\pi(u^*)\xi\,:\,x_0\in B\}={\mathbb{C}}\xi\oplus{\mathbb{C}}\pi(u^*)\xi.
\end{aligned}$$ Since $(\mathcal{F}\pi(u)-\rho(u)\mathcal{F})(\pi(u^*)\xi)=\mathcal{F}\xi-\rho(u)\eta=-\rho(u)\eta$ and $(\mathcal{F}\pi(u)-\rho(u)\mathcal{F})\xi=\rho(u)\eta$, this finishes the proof of $2$. The proof of $3$ is similar.
Since $P$ is the closed linear span of the reduced words and $A$, Lemma \[compact\] implies that $\mathcal{F}\pi(x)-\pi(x)\mathcal{F}$ is a compact operator for all $x\in P$. Hence, the triple $(\pi,\rho,\mathcal{F})$ defines an element $\alpha\in{\rm KK}(P,{\mathbb{C}})$. To prove Theorem \[Kamenability\], it suffices to show that $\lambda^*(\alpha)=[\epsilon]$ in ${\rm KK}(P_m,{\mathbb{C}})$, where $P_m$ be the maximal $C^*$-algebra of the HNN extension i.e, the maximal HNN extension, and $\epsilon$ is the trivial representation of $P_m$.
Define $\widetilde{K}=K\oplus{\mathbb{C}}\Omega$, where $\Omega$ is a norm one vector, with the representation $\widetilde{\rho}=\rho\circ\lambda\oplus\epsilon$ of $P_m$. Define the unitary $\widetilde{\mathcal{F}}\,:\,H\rightarrow\widetilde{K}$ by $$\widetilde{\mathcal{F}}\xi=\Omega\quad\text{and}\quad\widetilde{\mathcal{F}}|_{H_{-1}\oplus H_1}=F.$$ The triple $(\widetilde{\pi}\circ\lambda,\widetilde{\rho},\widetilde{\mathcal{F}})$, where $\widetilde{\pi}=\pi\circ\lambda$, defines an element $\gamma\in{\rm KK}(P_m,{\mathbb{C}})$ satisfying $\gamma=\lambda^*(\alpha)-[\epsilon]$. It suffices to show that $(\widetilde{\pi},\widetilde{\rho},\widetilde{\mathcal{F}})$ is homotopic to a degenerated triple.
Define the unitary $v\in\mathcal{B}(\widetilde{K})$ by $$v\eta=\Omega,\quad v\Omega=\eta,\quad v\rho(x)\eta=\rho(x)\eta\,\,\text{for}\,x\in P\,\text{with}\, E_B(x)=0.$$
\[homotopy\] Write $P_m=\langle A,w\rangle$. The following holds.
1. $\widetilde{\mathcal{F}}\widetilde{\pi}(a)\widetilde{\mathcal{F}}^*=\widetilde{\rho}(a)$ for all $a\in A\subset P_m$.
2. $\widetilde{\mathcal{F}}\widetilde{\pi}(w)\widetilde{\mathcal{F}}^*=\widetilde{\rho}(w)v$.
3. $v\widetilde{\rho}(b)v^*=\widetilde{\rho}(b)$ for all $b\in B$.
$1.$ Let $a\in A$. One has $\widetilde{\mathcal{F}}\pi(a)\widetilde{\mathcal{F}}^*\Omega=\widetilde{\mathcal{F}}\pi(a)\xi=\epsilon(a)\widetilde{\mathcal{F}}\xi=\epsilon(a)\Omega=\widetilde{\rho}(a)\Omega$. Since $\widetilde{\mathcal{F}}|_{H_{-1}\oplus H_1}=\mathcal{F}|_{H_{-1}\oplus H_1}$ we find, using assertion $1$ of Lemma \[compact\], that $$\widetilde{\mathcal{F}}\pi(a)\widetilde{\mathcal{F}}^*|_K=\rho(a)|_K=\widetilde{\rho}(a)|_K.$$ This concludes the proof of $1$.
$2.$ Since $\widetilde{\pi}(w)=\pi(u)$, it suffices to prove the following.
- $\widetilde{\mathcal{F}}\pi(u)\widetilde{\mathcal{F}}^*\Omega=\widetilde{\rho}(w)v\Omega$.
- $\widetilde{\mathcal{F}}\pi(u)\widetilde{\mathcal{F}}^*\eta=\widetilde{\rho}(w)v\eta$.
- $\widetilde{\mathcal{F}}\pi(u)\widetilde{\mathcal{F}}^*\rho(x)\eta=\widetilde{\rho}(w)v\rho(x)\eta$ for all $x\in P$ such that $E_B(x)=0$.
Since $\widetilde{\rho}(w)v\Omega=\widetilde{\rho}(w)\eta=\rho(u)\eta$, the first point follows from the computation: $$\widetilde{\mathcal{F}}\pi(u)\widetilde{\mathcal{F}}^*\Omega=\widetilde{\mathcal{F}}\pi(u)\xi=F\rho(u)\eta=\rho(u)\eta.$$ Since $w\in P_m$ is an irreducible unitary representation we get $\epsilon(w)=1$ and $\widetilde{\rho}(w)v\eta=\widetilde{\rho}(w)\Omega=\epsilon(w)\Omega=\Omega$. Hence, the second point follows from the computation: $$\widetilde{\mathcal{F}}\pi(u)\widetilde{\mathcal{F}}^*\eta=\widetilde{\mathcal{F}}\pi(u)F^*\eta=\widetilde{\mathcal{F}}\pi(u)\pi(u^*)\xi=\widetilde{\mathcal{F}}\xi=\Omega.$$ For the last point we separate the different cases. First, observe that, for $x\in P$ with $E_B(x)=0$, one has $\widetilde{\rho}(w)v\rho(x)\eta=\widetilde{\rho}(w)\rho(x)\eta=\rho(u)\rho(x)\eta=\rho(ux)\eta$.
**Case 1:** *If $x\in A$ and $E_B(x)=0$*. Then, since $uxu^*$ is reduced, $$\widetilde{\mathcal{F}}\pi(u)\widetilde{\mathcal{F}}^*\rho(x)\eta=\widetilde{\mathcal{F}}\pi(u)F^*\rho(x)\eta
=\widetilde{\mathcal{F}}\pi(u)\pi(xu^*)\xi=F\pi(uxu^*)\xi=\rho(ux)\eta.$$ For the other case i.e. when $E_A(x)=0$, we can assume that $x=x_0u^{\epsilon_1}\ldots u^{\epsilon_n}x_n$ is reduced. We separate again in different cases.
**Case 2:** *If $x=x_0u^{\epsilon_1}\ldots u^{\epsilon_n}x_n$ is reduced with $\epsilon_n=-1$*. One has $$\widetilde{\mathcal{F}}\pi(u)\widetilde{\mathcal{F}}^*\rho(x)\eta=\widetilde{\mathcal{F}}\pi(u)F^*\rho(x)\eta
=\widetilde{\mathcal{F}}\pi(uxu^*)\xi.$$ Since $\epsilon_n=-1$, $uxu^*$ can be written as a reduced word or the sum of two reduced words that end with $u^*$. Hence, $\widetilde{\mathcal{F}}\pi(u)\widetilde{\mathcal{F}}^*\rho(x)\eta=F\pi(uxu^*)\xi=\rho(ux)\eta$.
**Case 3:** *$\epsilon_n=1$*. If $x_n\in B$, since $\rho(b)\eta=\epsilon(b)\eta$ we may assume that $x_n=1$. Then, $$\widetilde{\mathcal{F}}\pi(u)\widetilde{\mathcal{F}}^*\rho(x)\eta=\widetilde{\mathcal{F}}\pi(u)F^*\rho(x)\eta
=\widetilde{\mathcal{F}}\pi(ux)\xi.$$ Since $\epsilon_n=1$, $ux$ can be written as a reduced word or the sum of two reduced words that end with $u$. Hence, $$\widetilde{\mathcal{F}}\pi(u)\widetilde{\mathcal{F}}^*\rho(x)\eta=F\pi(ux)\xi=\rho(ux)\eta.$$ If $E_B(x_n)=0$ then $\widetilde{\mathcal{F}}\pi(u)\widetilde{\mathcal{F}}^*\rho(x)\eta=\widetilde{\mathcal{F}}\pi(u)F^*\rho(x)\eta
=\widetilde{\mathcal{F}}\pi(uxu^*)\xi$. Since $\epsilon_n=1$ and $E_B(x_n)=0$, $uxu^*$ can be written as a reduced word or the sum of two reduced words that end with $u^*$. Hence, $$\widetilde{\mathcal{F}}\pi(u)\widetilde{\mathcal{F}}^*\rho(x)\eta=F\pi(uxu^*)\xi=\rho(ux)\eta.$$
$3.$ Let $b\in B$. One has $v\widetilde{\rho}(b)v^*\Omega=v\rho(b)\eta=\epsilon(b)v\eta=\epsilon(b)\Omega=\widetilde{\rho}(b)\Omega$. Moreover, $$v\widetilde{\rho}(b)v^*\eta=v\widetilde{\rho}(b)\Omega=\epsilon(b)v\Omega=\epsilon(b)\eta=\rho(b)\eta
=\widetilde{\rho}(b)\eta.$$ Eventually, for $x\in P$ with $E_B(x)=0$, one has $E_B(bx)=0$. Hence, $v\widetilde{\rho}(b)v^*\rho(x)\eta=v\rho(bx)\eta=\rho(b)\rho(x)\eta$.
*End of the proof of Theorem \[Kamenability\].* By Lemma \[homotopy\], $v\in\widetilde{\rho}(B)'\cap\mathcal{B}(\widetilde{K})$. Let $a\in\widetilde{\rho}(B)'\cap\mathcal{B}(\widetilde{K})$ be the unique self-adjoint element with spectrum $[-\pi,\pi]$ such that $v=e^{ia}$ and define, for $s\in{\mathbb{R}}$, $v_s=e^{isa}$. It follows that $v_s$ is a continuous one-parameter group of unitaries in $\widetilde{\rho}(B)'\cap\mathcal{B}(\widetilde{K})$. For $s\in{\mathbb{R}}$ define the unitary $w_s=\widetilde{\rho}(w)v_s\in\mathcal{B}(\widetilde{K})$. Observe that, for all $b\in B$ and all $s\in{\mathbb{R}}$, $$w_s\widetilde{\rho}(b)w_s^*=\widetilde{\rho}(w)v_s\widetilde{\rho}(b)v_s^*\widetilde{\rho}(w^*)=\widetilde{\rho}(w)\widetilde{\rho}(b)\widetilde{\rho}(w^*)=\widetilde{\rho}(wxw^*)=\widetilde{\rho}(\theta(x)).$$ By the universal property of $P_m$, for each $s\in{\mathbb{R}}$, there exists a unique representation $\rho_s$ of $P_m$ on $\widetilde{K}$ such that $$\rho_s(w)=w_s\quad\text{and}\quad\rho_s(a)=\widetilde{\rho}(a)\,\,\text{for}\,a\in A.$$ The family of triples $x_s=(\widetilde{\pi},\rho_s,\widetilde{F})$, for $s\in{\mathbb{R}}$, defines an homotopy between $x_0=(\widetilde{\pi},\widetilde{\rho},\widetilde{F})$ and $x_1$ which is degenerate by Lemma \[homotopy\].
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We compute the distribution of the decay rates (also referred to as residues) of the eigenstates of a disordered slab from a numerical model. From the results of the numerical simulations, we are able to find simple analytical formulae that describe those results well. This is possible for samples both in the diffusive and in the localised regime. As example of a possible application, we investigate the lasing threshold of random lasers.'
author:
- 'M. Patra'
bibliography:
- 'paper.bib'
title: Decay Rate Distributions of Disordered Slabs and Application to Random Lasers
---
Introduction
============
A very successful approach to describe disordered materials is supplied by random-matrix theory, see Refs. for reviews. While one can put the beginning of random-matrix theory at Wigner’s surmise for describing the scattering spectra of heavy atomic nuclei,[@wigner:56a] its theoretical foundations were laid only much later.[@bohigas:84a] It was very successfully applied to electronic transport in disordered wires and mesoscopic quantum dots, and recently these methods have been adopted to model (quantum) transport of optical radiation in media with spatially fluctuating dielectric constant.[@beenakker:98a; @beenakker:98b; @patra:98a; @patra:99a]
In the theoretical treatment of disordered materials, two particular geometries are of special importance, namely the disordered slab and the chaotic cavity (see Fig. \[figGeometrien\]). The principal difference between the geometries is easily explained: A chaotic cavity is an object in which the dynamics is chaotic due to the shape of the cavity or due to scatterers placed at random positions. The size of the opening is small compared to the total surface area of the cavity. Particles (electrons, photons) are then “trapped” inside the structure for a time that is long enough to ergodically explore the entire cavity. In a disordered slab, particles cannot be trapped that efficiently. They can no longer explore the entire volume ergodically but they still stay long enough to explore the entire cross-section of the sample, thus still making a random-matrix description possible. In order to call this geometry a “wire” or a “waveguide” its length should be much larger than its width. To be able to apply the theory only the much weaker criterion that the length is sufficiently larger than the mean-free path of the medium has to be fulfilled.
Two different aspects are of special important in the theory of disordered media, namely transport properties and resonances. The transport properties (moments of the eigenvalues of the transmission and reflection matrices) are known for the disordered slab in the limit that it is wide,[@brouwer:98a] for the chaotic cavity with an opening that is so small that only one or two modes can propagate through it,[@brouwer:95b; @brouwer:97a] or a chaotic cavity with a wide opening.[@beenakker:98b]
Less is known about the poles of such systems. (The eigenvalues of the Hamiltonian correspond to poles of the scattering matrix, and these show up as resonances in a scattering experiment. Hence the somewhat inconsistent nomenclature.) The beginning of random-matrix theory can be put at the moment when Wigner surmised the eigenvalue distribution for a closed chaotic cavity.[@wigner:56a; @mehta:90; @beenakker:97a] Here we are interested in open systems, where the eigenvalues acquire an imaginary part. (The imaginary part is referred to as residue.) It determines the decay rate of the (quasi-)eigenstate of the system. For chaotic cavities with broken time-reversal symmetry, the decay rate distribution is known analytically for an opening of arbitrary size.[@fyodorov:97a] The distribution for the more important case of preserved time-reversal symmetry[^1] is not known but can be approximated by a cavity with broken symmetry and an opening of half the real size.
Information on the residues of a disordered slab is very limited, and only the scaling behaviour of the large residue-tail in the localised regime was determined recently.[@titov:00a; @terrano:00a] This deficiency is felt especially strong in the random-laser community since the location of the residues gives the lasing threshold of an optical system, and most experimental setups resemble a disordered slab much more than they resemble a cavity. This paper fills this gap by presenting the results of numerical simulations. The quality of the numerical decay rate distributions is good enough that it allows us to arrive at analytical formulae for the distribution function, including its dependence on the parameters of the system. The idea to use high-quality simulations to arrive at formulae is not completely new as the distribution of the scattering strengths of chaotic cavities was found in the same way.[@beenakker:98b]
This paper is organised as follows: First, we introduce the Anderson Hamiltonian used the describe the disordered slab. In Sec. \[secEigenSolver\] we show how the eigenvalues of that Hamiltonian can be computed in a efficient numerical way. Depending on the length of the slab, it can be in either the diffusive or in the localised regime. We will analyse the decay rate distributions for both regimes separately, first in Sec. \[secDiffusive\] for the diffusive and then in Sec. \[secLocalised\] for the localised regime. Until that point all results are completely general and can be applied to electronic and photonic systems. In Sec. \[secThreshold\] we specialise on the lasing threshold in amplifying disordered media. We distinguish between the diffusive and the localised regimes (Secs. \[secThresholdDiffusive\] and \[secThresholdLocalised\]). We conclude in Sec. \[secConclusions\].
Anderson Hamiltonian for a disordered slab {#secModell}
==========================================
We consider a two-dimensional slab of length $L$ and width $N$. The slab is described by an Anderson-type lattice Hamiltonian with spacing $1$. In the Anderson model, transport is modelled by nearest-neighbour hopping between lattice sites. Without loss of generality we can set the hopping rate to $1$. With a spatially varying potential $P(x,y)$ the Hamiltonian becomes
\[Hdef\] $$\begin{aligned}
\mathcal{H}_{(x,y),(x,y)} &= P(x,y) &(y \ne 1, L) \\
\mathcal{H}_{(x,y),(x,y)} &= P(x,y) - i \kappa &(y = 1, L) \label{Hdefkappa} \\
\mathcal{H}_{(x,y),(x+1,y)} &= 1 &(x<W) \\
\mathcal{H}_{(x,y),(x-1,y)} &= 1 &(x>1) \\
\mathcal{H}_{(x,y),(x,y+1)} &= 1 &(y< L) \\
\mathcal{H}_{(x,y),(x,y-1)} &= 1 &(y>1) \end{aligned}$$
All other elements are zero. $x$ runs from $1$ to $N$, and $y$ from $1$ to $L$.
The real part $E$ of the eigenvalues of $\mathcal{H}$ in the limit of large $N$ and $L$ is confined to the interval $[-4;4]$. (If the average of $P(x,y)$ is nonzero, the interval is simply shifted by that average. If $P(x,y)$ is fluctuating as a function of $x$ and $y$ — like it does for a disordered medium — the interval becomes a bit wider.) For electronic systems, $E$ gives the energy of the eigenstate, and Eq. (\[Hdef\]) hence describes a slab with a conduction band of width $8$. For photonic systems, the real part of the eigenvalue gives the eigenfrequency. For both systems, the imaginary part $\gamma$ of the eigenvalue gives the decay rate of the eigenstate. (Actually not $\gamma$ but rather $\gamma/2$ is the decay rate but for the ease of notation we will continue to refer to $\gamma$ simply as the decay rate.)
$\kappa$ in Eq. (\[Hdefkappa\]) quantifies the strength of the coupling between the slab and the outside.[@fyodorov:97a] Using the units introduced above, $\kappa$ has the value $\sin^2 k$ where $k$ is the wavevector at the energy at which particles are injected respectively emitted. This quantity is proportional to the velocity of the particle perpendicular to the interface. In the centre of the band $\sin k=1$ whereas at the edges $\sin k=0$.
If $\kappa$ is chosen to be constant (i.e. not to depend on energy) ideal outcoupling can be described only for one specific value of the energy. We will do this since otherwise solving the Hamiltonian no longer is a standard eigenvalue problem, and set $\kappa\equiv 1$, hence modelling ideal coupling at the centre of the band.[^2] Working at the centre of the conduction bands offers the advantage that the width $N$ of the sample is identical to the number $N$ of propagating modes, and thus allows the describe the largest number of propagating modes for given size of the Hamiltonian (i.e. given numerical work). It is possible to include energy dependent coupling terms[@terraneo:00a] but it should be stressed that a constant $\kappa$ is more efficient and gives completely correct results as long as only eigenvalues near the respective energy are considered. We set $\kappa=1$, meaning ideal coupling at the centre of the conduction band.
It should be stressed that – even though we are modelling a two-dimensional system — the results are valid for three-dimensional systems as long as $L\gtrsim N$. A particle that is injected into the slab ergodically explores the entire cross-section of the sample before being emitted again, and hence loose its memory of which sites are “connected” by hopping terms. The sites can then be re-arranged, e.g. in a three-dimensional structure. Only for very short samples, $L\lesssim N$, this is not possible but for such samples already applying the Anderson model (i.e. only allowing nearest-neighbour hopping) is very questionable. The only “real” restriction that can limit the application of our results to certain photonic three-dimensional systems is that particles can leave the sample only at the front and at the back — and not through the “sides”.
In the formulation of Eq. (\[Hdef\]) the matrix $\mathcal{H}$ has double indices but these are easily removed by considering $\mathcal{H}_{n n'}$ with $n=x+(y-1) N$. (It would not make sense to set $n=y+(x-1) L$ since usually $L\gg
N$, and we want to work with a band matrix that is as small as possible.) This results is a matrix of the form as depicted in Fig. \[figmatrix\]. It is a banded $L N \times L N$ matrix with band width $2 N + 1$. Also within the band most elements are zero (since usually $N\gg 1$). The matrix is symmetric but non-Hermitian as there are complex numbers on the diagonal.
The model (\[Hdef\]) has been widely used since an efficient way to compute the transmission through such a slab is known.[@baranger:91a] The method of recursive Greens functions allows to compute the entire scattering matrix, hence all linear transport properties, in a time of order $\mathcal{O}(L N^3)$ and with only minimal storage requirements $\mathcal{O}(N^2)$. No explicit reference to the Hamiltonian $\mathcal{H}$ is made, so that eigenvalues cannot be computed by this method.
Computing eigenvalues of symmetric complex non-Hermitian banded matrices {#secEigenSolver}
========================================================================
Since the Hamiltonian $\mathcal{H}$ from Eq. (\[Hdef\]) is both banded and sparse one might be tempted to use an eigensolver for sparse matrices to compute the eigenvalues of Eq. (\[Hdef\]). A sparse eigenvalue routine needs to be able to solve the equation $$( \mathcal{H} - \mu {\mathbbm{1}}) \vec{x} = \vec{y}
\label{inviteration}$$ for the unknown vector $\vec{x}$ for arbitrary $\mu$ and $\vec{y}$. In particular, the eigensolver needs to set $\mu$ close to an eigenvalue of $\mathcal{H}$ so that the matrix $\mathcal{H} - \mu {\mathbbm{1}}$ is ill-conditioned. A numerical solution of Eq. (\[inviteration\]) is then difficult and very expensive. Furthermore, only one eigenvalue is found at a time, and control of which eigenvalue the algorithm will converge to is difficult. (Algorithms for sparse matrices always use inverse iteration so that the corresponding eigenvector will be returned without additional effort but the eigenvector is of no use for us.) Using an algorithm for banded matrices is thus the better alternative.
There exist a number of algorithms for real symmetric or complex Hermitian band matrices. Both problems are characterised by real eigenvalues, so that they are conceptually identical. Only one algorithm for computing an eigenvalue (plus the corresponding eigenvector) of a general complex band matrix has been published.[@schrauf:91a] It uses inverse iteration, so it is of limited use here.
We thus had to implement our own eigenvalue solver. The eigenrepresentation of $\mathcal{H}$ in terms of the diagonal matrix $\Lambda = \mathrm{diag}(\lambda_1,\ldots,\lambda_N)$ of the eigenvalues $\lambda_i$ of A and the matrix $U$ of eigenvectors is $$\mathcal{H} = U \Lambda U^{-1} \;.
\label{AewDecomp}$$ We now observe that for symmetric, that includes *complex* symmetric, $\mathcal{H}$ it is always possible to chose eigenvectors such that $U
U^T = {\mathbbm{1}}$. If $U$ would be a real matrix, one would call $U$ orthogonal but since it is complex there is no special name for the property $U U^T = {\mathbbm{1}}$.
Algorithms for diagonalising a real symmetric matrix $A$ implicitly decompose $A$ as $$A = Q D Q^T\;,\qquad Q Q^T = {\mathbbm{1}}$$ with the matrix $D$ of eigenvalues of $A$. It is therefore possible to adapt such an algorithm for our needs. Most algorithms for banded matrices first reduce $A$ to tridiagonal form $A'$ by transformations of the form $A'=Q' A Q'^T$, and we will adopt this strategy. (A matrix is called tridiagonal if only the diagonal and its neighbouring elements are nonzero. If $A$ would be real, the transformation $A\to A'$ would be called a similarity transformation.) For a band matrix this is possible in an efficient way since it is not necessary to compute (and thus store) the full matrices $Q'$, and by annihilating the elements of the matrix $A$ in a clever order, the band structure is kept intact in all steps.[@kaufman:84a; @kaufman:00a]
The reduction of the complex matrix $\mathcal{H}$ to tridiagonal shape is done by straight-forward adaptation of this algorithm from real to complex numbers, where care needs to be taken that the dot product $\vec{x} \cdot \vec{y} = \sum_i x_i y_i$ is used and not the dot product $\vec{x} \cdot \vec{y} = \sum_i \overline{x}_i
y_i$ normally used for complex vectors. (The overbar marks the complex conjugate.)
To compute the eigenvalues of the tridiagonal matrix for the real symmetric or complex Hermitian case, methods that isolate eigenvalues in disjunct intervals are used (“divide and conquer” and similar methods[@golub:89a]). Such methods work for both of these cases as all eigenvalues are real and can thus be ordered. This no longer is possible here as the eigenvalues are complex. We therefore use the QR respectively QL method.[@lapack:99]
For an $K\times K$ banded matrix with band width $W$ the time needed to compute the eigenvalues increases as $\mathcal{O}(K^2 W)$ whereas the storage requirements increase as $\mathcal{O}(K W)$. In terms of the dimensions $L$ and $N$ of the disordered slab, this means that the time increases as $\mathcal{O}(L^2 N^3)$ and the storage space as $\mathcal{O}(L
N^2)$. For given computational resources, both scalings impose an upper limit on the system size that can feasible be treated. For typical values of the ratio $L$ and $N$, and for “realistic” computer equipment, the time limit is reached somewhat earlier than the memory limit.[^3]
With respect to a similar algorithm for full matrices one wins a factor $L$ (usually of order several hundred) for both time and memory by using the banded algorithm, thus allowing to treat system that could not be treated otherwise. Still, the work presented in this paper is a big numerical challenge. To arrive at the results, of the order of 100000 hours of cpu time on fast PC’s were needed.
Numerical simulations
=====================
Disorder is modelled by assigning random values to $P(x,y)$. It is assumed that those random numbers are uniformly distributed in the interval $[-w;w]$ so that $w$ measures the amount of disorder.
We only consider eigenvalues near the centre of the conduction band as the assumption of ideal coupling is only valid there. For numerical reasons it is essential that the centre of the conduction band is at $E=0$, i.e. one is not allowed to add an offset to $P(x,y)$.[^4]
We hence chose a window $[-d;d]$ and only include eigenvalues in the further analyses when their real part is inside that window. If the window is too large, systematic errors are introduced while too small a window leads to bad statistics. As can be seen from Fig. \[windowsizefig\], for of $d=0.1$ the distribution function already agrees with the distribution function for $d=0.01$ but has much better statistics. $d=0.5$ and $d=1.0$ gives significant systematic deviations. For this reason, all results presented in this paper assume a window with $d=0.1$.
The formulation of the model in Sec. \[secModell\] is in terms of generic units. Contact with a microscopic model or an experiment is best made in terms of the mean free path. It can be computed from the length-dependence of the transmission probability $T$ through the sample. In the diffusive regime, $l\lesssim L\ll N l$, it is given by [@beenakker:97a] $$\frac{1}{T} = 1 + \frac{L}{l} \;.$$ The mean free path can be computed by fitting $T(L)$ to this functional form.
The transmission probability has been computed using the method of recursive Green’s functions[@baranger:91a] for variable disorder strength $w$. As Fig. \[freiwegfig\] shows, the numerically computed mean free path $l$ is for the range of $w$ in question in very good approximation given by $$l = \frac{6}{w^{3/2}} \;.
\label{freiwegeq}$$ (Computed for each value of $w$ from $50$ samples with $L=2,4,\ldots,98,100$ and $N=50$.) In the following, we will no longer make explicit reference to $w$ but rather give the more intuitive mean-free path $l$.
Diffusive regime {#secDiffusive}
================
For a sample length $L$ with $l\lesssim L \ll N l$ the sample is said to be in the diffusive regime. It is immediately obvious that the diffusive regime can only be observed in sufficiently wide samples, $N\gg 1$.
For chaotic cavities with broken time-reversal symmetry an analytical result for the decay rate distribution has been given by Fyodorov and Sommers.[@fyodorov:97a] We start from their result and rescale it, $$\mathcal{P}(y) = \frac{1}{y^2 M!} \int_0^{M y} x^M e^{-x} d x
= \frac{1}{y^2} \left[ 1 - e^{-M y} \sum_{k=0}^M \frac{M^k}{k!}
y^k \right] \;.
\label{PyAnsatz}$$ $\mathcal{P}(y)$ is normalised to one and in our scaling is for all $M>1$ peaked near a value of $y$ of order $1$. In the original formulation for a chaotic cavity, $M$ is the number of modes propagating through the opening of the cavity.
In the following we will argue that the decay rate distribution $P(g)$ can be written in the form (\[PyAnsatz\]) as $$P(\gamma)=\frac{1}{\gamma_0}
\mathcal{P}\bigl(\frac{\gamma}{\gamma_0}\bigr)$$ with some scaling factor $\gamma_0$ and some effective number of modes $M\ne N$. In Fig. \[PyGueltigFig\] a comparison between the analytical suggestion and a simulation is given, and the agreement is striking. The horizontal axis has been plotted logarithmically since this results in both the differences between the $P(y)$ for different $N$ becoming easier to recognise and in giving a more prominent place to the small-$\gamma$ tail of $P(\gamma)$. In most applications, including the random laser discussed later in this paper, one is much more interested in small $\gamma$ than in large $\gamma$.
The results of the simulations are fitted “by eye” against the functional form (\[PyAnsatz\]), resulting in one pair of values for $\gamma_0$ and $M$ for each set of parameters. Especially at very small $\gamma$, there are sometimes numerical errors that introduce artifacts into the numerical histogram so that using an automatic fitting algorithm is not feasible. (Usually we computed 500–1500 realisations for each parameter set.) From our simulations, we find that the scaling factor $\gamma_0$ only depends on the length $L$ of the sample and its mean free path $l$ but not on its width $N$, and seems to be given by $$\gamma_0 = \frac{2 l}{L^2} \;.
\label{g0Gleichung}$$ As Fig. \[g0Grafik\] shows, the agreement between the result of the numerical simulations and Eq. (\[g0Gleichung\]) is good, and all major deviations are for small $L$ where universal scaling is expected to be worse than for larger $L$. The model equations set the speed of propagation to $1$ but it is obvious that for some other choice for the propagation speed $c$ one has to change Eq. (\[g0Gleichung\]) to $\gamma_0 = 2 c l / L^2$.
While the determination of $\gamma_0$ is very precise, there is a somewhat larger error involved in determining $M$ by fitting the analytical form to the results of numerical simulations. First, we only fitted against integer $M$, though in principle a generalisation of Eq. (\[PyAnsatz\]) to noninteger $M$ is possible, see Eq. (\[PvertGamma\]). Secondly, if $M \gtrsim 25$, the difference between $P(y)$ for $M$ and for $M+1$ becomes too small to tell with certainty which of these two values describes the numerical result better. Thirdly and finally, even with 500-1500 samples for each set of parameter values, there are still some fluctuations in the numerically computed histogram for the decay rate distribution that in some cases make the decision on the right $M$ a bit difficult. Considering all of this, one should allow for an error of $1$ for $M$, and even of $2$ for $M\gtrsim 25$.
We have computed $M$ for a series of samples with increasing length for three different widths $N$. As Fig. \[Mgrafik\] shows, the effective number $M$ of modes is well approximated by $$M = \frac{N}{1 + L / (6 l)} \;.
\label{Mgleichung}$$ The agreement between this suggested analytical form and the numerical simulation becomes better as the width $N$ of the sample is increased. From the simulations it is obvious that the functional form Eq. (\[Mgleichung\]) is correct but there still is the (small) possibility that the factor $6$ might need to be replaced by a slightly smaller value. To answer this question with certainty, we would need to increase both $L$ and $N$ significantly. Unfortunately, such simulations are outside the present time and memory constraints.
Equations (\[PyAnsatz\]–\[Mgleichung\]) give a good description of the decay rate distribution of a disordered slab in the diffusive regime, provided the slab is sufficiently wide. Since the transversal length scales are set by microscopic quantities (wave length of the light for optical systems, Fermi wave length for electronic systems), all macroscopic objects are “wide”.
Localised regime {#secLocalised}
================
If the length $L$ of a disorder medium is increased, the phenomenon of localisation sets in once $L\gtrsim N l$ (see Ref. for a review). In the localised regime the probability of transmission $T$ through the sample is reduced significantly and decays exponentially with the length $L$ of the sample. The length scale $\xi$ is called the localisation length, and can be computed from an ensemble of disordered slabs by computing the average of the logarithm of the transmission as a function of the length of the samples, hence $$-L/\xi = \langle \ln T(L) \rangle \;.$$ One should note that this is not identical to fitting the transmission to $\langle T(L)
\rangle \propto \exp( -L/\xi )$ since the large sample-to-sample fluctuations of $T(L)$ in the localised regime would give a value for $\xi$ that is off by a factor $4$. The localisation length can also be computed analytically from the mean-free path using the DPMK equation, with the result[@efetov:83b; @efetov:83c; @dorokhov:82a; @dorokhov:82b; @dorokhov:83a; @dorokhov:83b] $$\xi = \frac{N+1}{2} l \;,
\label{XiVorhersage}$$ and agrees well with our numerical results.
It is generally accepted that the distribution of the decay rates $\gamma$ (at least for small $\gamma$) in the localised regime is log-normal, i.e., $\ln
\gamma$ is distributed according to a Gaussian distribution. Recent interest has rather been in the large-$\gamma$ tail which was shown to follow a power-law.[@titov:00a; @terrano:00a] In a log-normal distribution, most of the weight lies in the right tail, so those papers give a sufficient description for most of the eigenmodes. In the context of applications to random lasers we are, however, interested in the small decay rate tail, hence in the log-normal distribution.
To our knowledge, there is only a single paper by Titov and Fyodorov that gives explicit expressions for the parameters of that log-normal distribution.[@titov:00a] However, their analytical results are for a somewhat different system so it is difficult to tell whether they agree or disagree with our findings. We will return to this aspect at the end of this section. First, we want to present the results of our numerical simulations.
Using the log-normal ansatz, the distribution of the decay rates $\gamma$ is $$P(\gamma) = b \exp\left(-\frac{(\log \gamma - \log
\gamma_0)^2}{\sigma^2} \right) \;.
\label{pLokalAnsatz}$$ The numerically computed histograms indeed follow this form, see Fig. \[logbeispiel\], except for the large-$\gamma$ tail — as already mentioned above but this deviation is only seen in a log-log plot.
Fig. (\[figlocmax\]) shows in the left the numerically computed $\gamma_0$ as a function of $L$ for $N=15$. Also displayed is the localisation length $\xi$ computed numerically from the transmission, being in good agreement with the analytical prediction (\[XiVorhersage\]). The quality of the data is good enough to say with confidence that $\gamma_0$ decays exponentially with a length scale that is somewhat larger than $\xi$. Fig. (\[figlocmax\]) shows in the right the value of $\gamma_0$ also for two other values of $N$, and all three cases are well-described by introducing a numerical fitting factor $a$, $$\gamma_0 \propto \exp\left(-\frac{L}{a \xi}\right) \quad
\mathrm{with}\quad a=1.12 \;.
\label{ylocposprop}$$ It is known that working at the centre of the conduction band when in the localised regime can introduce certain artefacts, especially in analytical approaches. Among other, the localisation length at the band centre can differ by approximately $10\,\%$ from the value outside the centre.[@kappus:81a] We have defined $\xi$ based on the transmission through the sample (at an energy corresponding to the band centre), and in transmission resonances at all energies can contribute. A numerical prefactor $a$ that differs by about $10\,\%$ from $1$ thus does not come as a complete surprise.
We still need to compute the proportionality factor appearing in Eq. (\[ylocposprop\]). For this purpose we need to plot the ratio of the numerically computed $\gamma_0$ and the right-hand side of Eq. (\[ylocposprop\]) for different values of $N$. We did this for $L=71.55~l$. Since this is a very expensive operation, we have computed a large number of samples only for $N=10,15,20$ so that their statistics is better than for the other values of $N$. An estimate of the error for these “better” data points has been included in the figure. This allows us to conclude that $$\gamma_0 = \frac{a}{N^2} \exp\left(-\frac{L}{a \xi}\right) \quad
\mathrm{with}\quad a=1.12 \;.
\label{ylocpos}$$ It should be noted that this equation contains two numerical coefficients, and there is no obvious reason why they should be identical. Still, we find that they both are approximately $a=1.12$.
Re-introducing “physical units” into Eq. (\[ylocpos\]) is a bit more difficult than it was for Eq. (\[g0Gleichung\]) where it was obvious that one simply has to multiply by the velocity of propagation $c$. Here one has to multiply by $c/\Delta$ where $\Delta$ is the perpendicular grid spacing. Due to the assumption of one propagating mode per (lateral) grid point made in Sec. \[secModell\], $\Delta$ is not arbitrary but has a well-defined physical meaning. For the electronic case, $\Delta=\pi/k_{\mathrm{F}}$ with $k_{\mathrm{F}}$ the wave vector at the Fermi level, and for the photonic case $\Delta=2\lambda/\pi$ with $\lambda$ the wave length of the light (hence $c/\Delta = 1/(4\nu)$).
\
Determining the width $\sigma$ of the distribution is more difficult since we can only use the left wing of the distribution — the right wing eventually turns into a power-law tail and thus no longer follows a log-normal distribution. Once again, we have accumulated more data for $N=10,15,20$ so that some indication of the error is possible for those three data points.
From our data, we propose the formula $$\sigma = \frac{2}{3} \left(\frac{L}{a \xi}\right)^{2/3} \;,
\label{eqsigma}$$ where $a=1.12$ has the same value as in Eq. (\[ylocpos\]). As Fig. \[figsigma\] shows, there clearly is no disagreement between the numerical data and Eq. (\[eqsigma\]). However, please remember that the $\frac{2}{3}$ should be thought of as a fitting factor that might not be exactly $2/3$ but perhaps rather $0.67$ or some other numerical factor.
Since the distribution is log-normal only for not too large $\gamma$ (remember the power-law tail for large $\gamma$) the normalisation is nontrivial \[$P(\gamma)$ is not normalised to $1$ any longer!\] and cannot be computed from $\gamma_0$ and $\sigma$. The constant $a$ in Eq. (\[pLokalAnsatz\]) is directly equal to the height of the peak of the numerically computed $P(\gamma)$. Since the total area underneath the numerically computed $P(y)$ (and hence its normalisation) is dominated by the large-$\gamma$ tail, $a$ has a relatively large error. Taking all the available data, the most likely value is $$b=\frac{1}{N^2} \exp\left(\frac{L}{a\xi}\right) \;.$$ This value has been determined from a large number of simulations that for space reasons cannot be presented here. Unfortunately the quality of the data is not good enough to decide whether an additional prefactor $a=1.12$ should appear.
At the present it is not possible to tell whether our results agree with the ones put forward by Titov and Fyodorov.[@titov:00a] In particular, they arrive at $$\gamma_0 \propto \exp\left(-\frac{3 L'}{\xi}\right) \;,
\label{EQmisha}$$ whereas our finding (\[ylocpos\]) was $\gamma_0 \propto \exp(-L/1.12\xi)$. There are two obvious differences between the model used by them and the model employed by us. First, for numerical reasons we work at the centre of the conduction band while they work near (but sufficiently far away from) the band edges. This might explain the factor $a=1.12$ that we have to introduce. Secondly and probably more importantly, they consider a system of length $L'$ that is closed at one end whereas our systems have length $L$ and are open at both ends. It is obvious that a half-closed system of length $L'$ corresponds to an open system of length $L>L'$. Eq. (\[EQmisha\]) suggests that those two systems could be mapped into each other by setting $L\approx 3 L'$ but there is no further evidence to support this claim.
Lasing threshold of a random laser {#secThreshold}
==================================
A random laser is a laser where the necessary feedback is not due to mirrors at the ends of the laser but due to random scattering inside the medium.[@wiersma:95a; @wiersma:97a; @beenakker:98b] We model the random laser as a disordered slab containing a dye that is able to amplify the radiation in a certain frequency interval with rate $1/\tau_{\mathrm{a}}$. The lasing threshold is the amplification rate at which the intensity of the emitted radiation diverges in a linear model. If saturation effects are included, the emitted intensity increases abruptly but finitely at crossing the lasing threshold.
The lasing threshold is given by the value of the smallest decay rate of all eigenmodes in the amplification window.[@misirpashaev:98a] (Remember that $\gamma$ actually is twice the decay rate. On the other hand, also $1/\tau_{\mathrm{a}}$ enters the relevant formulae only with a prefactor $1/2$. $\gamma$ thus indeed gives the necessary amplification rate $1/\tau_{\mathrm{a}}$.) This is easily understood since this simply means that in the mode with the smallest decay rate the photons are created faster by amplification than they can leave the sample (=decay). It, however, also follows from a complete quantum mechanical analysis.[@schomerus:00a; @patra:99a]
The distribution of the decay rate has been computed in this paper. A certain number $K$ of modes will be in the frequency window where amplification is possible. The lasing threshold is given by the smallest $\gamma$ of these $K$ modes. In a simple picture that is valid once $K\gg 1$ we can assume that the $K$ different $\gamma$’s are distributed independently according to $P(\gamma)$.[@misirpashaev:98a] The distribution $\tilde{P}(\gamma)$ of the smallest mode and hence of the lasing threshold then becomes $$\tilde{P}(\gamma) = K P(\gamma)\left[ 1-\int_0^{\gamma}
P(\gamma') d\gamma' \right]^{K-1} \;.
\label{eqtreshold1}$$
For $K\ne 1$, the distribution $\tilde{P}(\gamma)$ of the lasing threshold is not longer identical to the distribution $P(\gamma)$ of the decay rate of each individual mode. In particular, not only the precise form of these two distribution will be different, but also the “typical” value of the lasing threshold can be different from the “typical” decay rate $\gamma_0$. Interestingly, for chaotic cavities in the diffusive regime it was found that the latter two quantities differ only insignificantly[@frahm:00a; @schomerus:00a] which might seem counter-intuitive. A slab geometry is more “complicated” in that the scaling $K\propto N$ “tries” to lower the lasing threshold with increasing $N$.
For $K\gg 1$ the distribution $\tilde{P}(\gamma)$ is sharply peaked around its maximum. The position $\gamma_{\mathrm{m}}$ of the maximum is given by the solution of the equation $d \tilde{P}(y_{\mathrm{m}})/d\gamma_{\mathrm{m}}=0$, hence $$0 = \frac{d P(\gamma_{\mathrm{m}})}{d\gamma_{\mathrm{m}}}
\left[ 1-\int_0^{\gamma_{\mathrm{m}}} P(\gamma') d\gamma'
\right]
- (K-1) [ P (\gamma_{\mathrm{m}})]^2 \;.
\label{eqtreshold2}$$ While Eq. (\[eqtreshold1\]) is difficult to compute numerically due to the large exponent $K-1\gg 1$, in Eq. (\[eqtreshold2\]) this exponent no longer appears.
Eq. (\[eqtreshold2\]) depends on $P(\gamma)$ which in turn depends on the dimensions $L$ and $N$ of the system. In assuming that the number of propagating modes is equal to the width $N$ of the sample we already have made the assumption that the width (and hence also the length) is measured in units of $\lambda/2$. (The “$2$” accounts for polarisation.) The total number of modes in the sample thus is $L N$. We assume that a fraction $f$ of them is inside the amplification window of the dye, hence $K=f N L$. For simplicity we neglect complications as the shape of the mode profile. (It is easily incorporated into the numerics and we refrain from doing this just to prevent having to introduce even more parameters.) $f$ depends only on the chemical properties of the dye and not on the dimensions of the sample.
In the following we will show how to compute the most likely lasing threshold for samples in both the diffusive and in the localised regime.
Lasing threshold in the diffusive regime {#secThresholdDiffusive}
----------------------------------------
The change of the lasing threshold with increasing system size is influenced by a subtle interplay between $L$ and $N$ in determining the distribution $P(\gamma)$ and in determining the number $K=f N L$ of total modes.
If $K\gg 1$ the lasing mode has a decay rate in the low-$\gamma$ tail of $P(\gamma)$ (i.e. $\gamma<\gamma_0$ or $y<1$). The weight of this tail is $$\int_0^1 P(y) d y = \frac{M^{M-1}}{(M-1)!} e^{-M} \;,
\label{P0bis1}$$ and goes to zero as $M$ becomes larger. For $M\to\infty$ the tail disappears completely, as is already obvious from the asymptotic form of the distribution, $$P_{M\to\infty}(y)=\left\{ \begin{aligned}
& 0 & (y<1) \\
& 1/y^2 & (y\ge 1)
\end{aligned} \right.$$ With increasing $N$ and hence increasing $M$, the probability that a given mode has a small $y$ thus decreases rapidly. On the other hand, we are interested in the smallest decay rate out of $K$ modes, and $K$ increases linearly with $N$. This are two counter-acting effects, and it is not obvious which of these two is stronger.
The effect of an increase of the system size $L$, on the contrary, is obvious. First, the average decay rate $\gamma_0$ decreases according to Eq. (\[g0Gleichung\]). Secondly, $M$ decreases from Eq. (\[Mgleichung\]), leading to even smaller values for $\gamma$ of the lasing mode.
There have been some analytical attempts to compute the lasing threshold for a chaotic cavity [@frahm:00a; @schomerus:00a]. For large $M$, the small-$y$ tail of Eq. (\[PyAnsatz\]) was approximated by $$P(y) \approx \frac{1}{2 M}
\left[ 1 + \mathrm{erf}\bigl( \sqrt{M/2} [y-1]
\bigr) \right] \;.
\label{Pyapproximation}$$ This allows to arrive at scaling laws of the lasing threshold for variable $M$ at fixed $K$. Unfortunately, the difference between two counter-acting effects of an increase in $N$ are so small that Eq. (\[Pyapproximation\]) is a bit too crude for our needs.
We thus have to revert to a numerical procedure. Eq. (\[PyAnsatz\]) can be rewritten using the incomplete Gamma function $$\Gamma(a,x)=\int_x^{\infty} t^{n-1} e^{-t} d t \;.$$ $\Gamma(a,0)$ reduces to the well-known Gamma function $\Gamma(a)$. For numerical reasons it is advisable to introduce the regularised Gamma function $Q(a,x)=\Gamma(a,x)/\Gamma(a)$. Fast numerical algorithms to compute $Q(a,x)$ exist. \[Please note that in the literature the definitions of the regularised Gamma function sometimes disagree in that our $Q(a,x)$ is denoted as $1-Q(a,x)$.\] Now we can express Eq. (\[PyAnsatz\]) and its derivative and integral as
\[PvertGamma\] $$\begin{aligned}
P(y) &= \frac{1}{y^2} \bigl[1 - Q(M+1, M y)\bigr] \;, \\
\frac{d P(y)}{d y} &= \frac{(M y)^M}{y^2 \Gamma(M)} e^{-M y}
- \frac{2}{y^3}\bigl[1- Q(M+1, M y)\bigr] \;, \\
\int_0^y P(y') d y' &= \frac{1}{y} \bigl[Q(M+1,M y)-1\bigr] + 1
- Q(M,M y) \;,\end{aligned}$$
so that Eq. (\[eqtreshold2\]) can be evaluated efficiently.
Lasing threshold in the localised regime {#secThresholdLocalised}
----------------------------------------
From Eq. (\[pLokalAnsatz\]) we directly arrive at
$$\begin{aligned}
\frac{d P(\gamma)}{d \gamma} &=
-2 b \frac{\ln\gamma-\ln\gamma_0}{\gamma\sigma^2}
\exp\left[
-\frac{(\ln\gamma-\ln\gamma0)^2}{\sigma^2}\right]
\;, \\
\int_0^{\gamma} P(\gamma') d \gamma' &=
\frac{b \sqrt{\pi} \sigma \gamma_0}{2}
e^{\sigma^2/4} \left[1+\mathrm{erf}\left(
\frac{2 \ln( \gamma / \gamma_0)-\sigma^2}{2\sigma}\right)
\right] \;.\end{aligned}$$
A further simplification is not possible, and we did not manage to find suitable approximations. Also for the localised regime we thus are restricted to a numerical evaluation.
Numerical results
-----------------
The lasing threshold is computed numerically from Eq. (\[eqtreshold2\]), using the formulae from Secs. \[secThresholdDiffusive\] and \[secThresholdLocalised\]. Into the formulae presented there, we have to insert the correct dependence of the $\gamma_0$, $M$, $\sigma$, etc. on $L$ and $N$ that was presented earlier in this paper. Despite this complication the numerical calculation is straight forward as Eq. (\[eqtreshold2\]) possesses a single root only. Since this root has a change of sign, it is easily found numerically.
Fig. \[figLaserschwelle\] shows the results for both the diffusive and the localised regimes, for both $f=0.1~l$ and $f=0.001~l$. (The mean-free path appears as a factor since the figure is in units $L/l$ and not $L$.) The formulae found in this paper are valid deep in the diffusive regime respectively deep in the localised regime. Near the cross over, hence near the line $L\approx N l$, this condition is not fulfilled. The numerical values near the diagonal line in Fig. \[figLaserschwelle\] should thus be viewed with caution.
As can be seen from the figure, in the diffusive regime with $N\gg L/l$ the lasing threshold becomes almost independent of the width $N$ of the sample (for sufficiently large $N$), and the most likely value of the lasing threshold is about $$\gamma_{\mathrm{m}} \approx \frac{2 c l}{L^2} \;,
\label{gammaMdiffuse}$$ hence the value given by Eq. (\[g0Gleichung\]). This means that even though $K\gg 1$, $P(y)$ for $y<1$ is already so small that it dominates over the large value of $K$. Differences between this simple approximation and the precise numerical result appear for finite $N$, with the size of this difference depending on $f$. However, for designing experiments it is obvious from the results presented here that the only feasible way to lower the lasing threshold of a random laser in the diffusive regime is increasing its length, not modifying its width.
As Fig. \[figLaserschwelle\] shows, also in the localised regime there is only a small dependence on $f$. This means that in a log-normal distribution the weight of the left tail is so small that unless $K$ is exponentially large $\gamma_{\mathrm{m}}$ cannot become much smaller than the position $\gamma_0$ of the peak of the distribution. The difference to the diffusive regime is that the lasing threshold can be lowered efficiently not only by increasing the length but also decreasing the width $N$ and hence driving the system farther into localisation.
It is no surprise that samples in the localised regime generally have a lower lasing threshold than samples in the diffusive regime. We have shown that also the diffusive samples can have an “acceptably small” lasing threshold as it is trivial to make them very long (since there is no need to care much about their width). For both the diffusive and the localised regime, the typical decay rates of a single mode are comparable to the lasing threshold.
Conclusions {#secConclusions}
===========
We have numerically computed the distributions of the residues (or decay rates) of a disordered slab. The slab has length $L$, mean free path $l$, width respectively cross-sectional area $N$ ($N$ is given as number of propagating channels) and velocity of propagation $c$. We were able to “guess” simple analytical formulae that are able to describe the numerical results well.
For a sample in the diffusive regime ($L\lesssim N l$) we found in Eqs. (\[PyAnsatz\]–\[Mgleichung\])
\[ergebnis1\] $$\begin{aligned}
P(\gamma)&=\frac{L^2}{2 l c}
\mathcal{\mathcal{P}}\bigl(\frac{\gamma L^2}{2 l c}\bigr) \;, \\
\mathcal{P}(y) &= \frac{1}{y^2} \Bigl[1 -
\frac{\Gamma(1+\frac{N}{1 + L/6 l},\frac{N y}{1 + L/6 l})}{
\Gamma(1+\frac{N}{1 + L/6 l})} \Bigr] \;,\end{aligned}$$
where $\Gamma(a,x)$ is the incomplete Gamma function. The agreement between the numerical results and the proposed formulae is good, and there is the possibility that Eq. (\[ergebnis1\]) might become exact in the limit $L/l\gg N \gg 1$. However, there is only numerical and no analytical evidence to back this claim.
For a sample in the localised regime ($L \gtrsim N l$) with localisation length $\xi=(N+1)l/2$ we found in Sec. \[secLocalised\] $$\begin{gathered}
P(\gamma) = \frac{1}{N^2} \exp\left(\frac{L}{a\xi}
-\frac{(\log \gamma - \log
\gamma_0)^2}{\sigma^2} \right) \;, \quad a=1.12 \;, \nonumber\\
\gamma_0 = \frac{a}{N^2} \exp\left(-\frac{L}{a \xi}\right) \;,\quad
\sigma = \frac{2}{3} \left(\frac{L}{a \xi}\right)^{2/3} \;.
\label{ergebnis2}\end{gathered}$$ The quality of the simulations results in the localised regime is somewhat less than in the diffusive regime. For this reason, Eq. (\[ergebnis2\]) should be understood as an approximate fit only, and it very probably differs from the exact relation, especially outside the band centre.
These results can be applied to both electronic and photonic systems. For photonic systems we have shown that under realistic assumptions the lasing threshold of a random laser is close to $\gamma_0$ both in the diffusive and in the localised regime. Eqs. (\[ergebnis1\]) and (\[ergebnis2\]) thus not only give the distribution of the decay rate of each individual mode but also a good estimate of the lasing threshold, i.e. the smallest decay rate of a large number of modes.
Valuable discussions with C.W.J. Beenakker are acknowledged.
[^1]: Optical experiments always preserve time-reversal symmetry unless a magneto-optical effect is included. For electric systems time-reversal symmetry can be broken by applying a large magnetic field to the sample. (Such fields are created routinely in experiments.)
[^2]: It is not possible to have more than ideal coupling. For $\kappa<1$ the loss rates are smaller than for $\kappa=1$, so this is easily identified as “sub-ideal”. For $\kappa>1$ the loss rates split into two separate parts: Most become smaller, as for $\kappa<1$, while a few loss rates become very large, thereby fulfilling the requirement that the average loss rate has to be proportional to $\kappa$. We should note that this somewhat counter-intuitive behaviour is also observed for chaotic cavities.[@fyodorov:97a]
[^3]: On a modern computer a single diagonalisation for a $L=700$, $N=70$ systems takes about $2$ days and uses $256$ Mbytes of memory. While this memory requirement frequently is no problem, the computing time usually is. Remember that the task is to compute the distribution of the decay rates. Hence, many matrices with different realisations of the random potential $P(x,y)$ have to be diagonalised — not just a single matrix. However, the restrictions imposed by time and memory are of the same order of magnitude.
[^4]: The algorithms will return eigenvalues $z'$ that have a very small but finite deviation $|z-z'|$ from their correct value $z$. Since we are primarily interested in the imaginary part of the eigenvalue and want it to be as precise as possible the magnitude of the real part has to be as small as possible.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
------------------------------------------------------------------------
In the molecular picture the hidden-charm, pentaquark-like $P_c(4450)$ resonance is a $\bar{D}^* \Sigma_c$ bound state with quantum numbers $I=\tfrac{1}{2}$ and $J^P = \tfrac{3}{2}^-$. If this happens to be the case, it will be natural to expect the existence of $\bar{D}^* \bar{D}^* \Sigma_c$ three-body bound states. The most probable quantum numbers for a bound $\bar{D}^* \bar{D}^* \Sigma_c$ trimer are the isoscalar $J^P = \tfrac{1}{2}^+$, $\tfrac{5}{2}^+$ and the isovector $J^P = \tfrac{3}{2}^+$, $\tfrac{5}{2}^+$ configurations. Calculations within a contact-range theory indicate a trimer binding energy $B_3 \sim 3-5\,{\rm MeV}$ and $14-16\,{\rm MeV}$ for the isoscalar $\tfrac{1}{2}^+$ and $\tfrac{5}{2}^+$ states and $B_3 \sim 1-3\,{\rm MeV}$ and $3-5\,{\rm MeV}$ for the isovector $\tfrac{3}{2}^+$ and $\tfrac{5}{2}^+$ states, respectively, with $B_3$ relative to the $\bar{D}^* P_c(4450)$ threshold. These predictions are affected by a series of error sources that we discuss in detail.
author:
- Manuel Pavon Valderrama
title: 'The $\bar{D}^* \bar{D}^* \Sigma_c$ Three-Body System'
---
Introduction
============
The discovery of two hidden charm pentaquark-like resonances by the LHCb [@Aaij:2015tga], the $P_c(4380)$ and $P_c(4450)$, begs the question of what is their nature. These resonances are indeed different: while the $P_c(4380)$ is relatively broad, the $P_c(4450)$ is narrow and is located close to a few meson-baryon thresholds, namely the $\bar{D}\Lambda_{c}(2595)$, $\bar{D}^*\Sigma_c$, $\bar{D}^*\Sigma_c^*$ and $J/\psi \, p$ thresholds. This coincidence makes the $P_c(4450)$ a good candidate for a hadronic molecule, a type of exotic hadron theorized a few decades ago [@Voloshin:1976ap; @DeRujula:1976qd]. In particular the $P_c(4450)$ has been suggested to be a $\bar{D}^* \Sigma_c$ [@Roca:2015dva; @He:2015cea; @Xiao:2015fia], a $\bar{D}^* \Sigma_c^*$ molecule [@Chen:2015loa; @Chen:2015moa] (in these two cases probably with a small admixture of $\bar{D} \Lambda_{c}(2595)$ [@Burns:2015dwa; @Geng:2017hxc]), and a $\chi_{c1} \, p$ molecule [@Meissner:2015mza]. Of these possibilities, the most natural one probably is $\bar{D}^* \Sigma_c$. First, the interaction between a heavy baryon and antimeson is expected to be mediated by the exchange of light mesons, providing a mechanism to justify the existence of attraction. Second ,the binding energy of the $\Sigma_c \bar{D}^*$, $B_2 = 12 \pm 3 \,{\rm MeV}$, translates into a bound state that is not excessively compact, which is compatible with the molecular hypothesis. The expected size is of the order of $1 / \sqrt{2\mu B_2} \sim 1.2\,{\rm fm}$, with $\mu$ is the reduced mass of the molecule. Besides the molecular hypothesis there are other competing explanations for the nature of the $P_c^*$, which include a genuine pentaquark [@Diakonov:1997mm; @Jaffe:2003sg; @Yuan:2012wz; @Maiani:2015vwa; @Lebed:2015tna; @Li:2015gta; @Wang:2015epa], threshold effects [@Guo:2015umn; @Liu:2015fea] (for a detailed discussion see Ref. [@Bayar:2016ftu]), baryocharmonia [@Kubarovsky:2015aaa], a molecule bound by [*colour chemistry*]{} [@Mironov:2015ica] or a soliton [@Scoccola:2015nia]. The molecular hypothesis relies on the quantum numbers of the pentaquark to be $J^P = \tfrac{3}{2}^{-}$ or alternatively $\tfrac{5}{2}^-$. Experimentally $J^P$ is not uniquely determined yet, with $\tfrac{3}{2}^{+}$ and $\tfrac{5}{2}^+$ also probable [@Aaij:2015tga]. Until the quantum numbers of the $P_c^*$ are known, the discussion about its nature will remain theoretical.
If the $P_c(4450)$ — the $P_c^*$ from now on — is indeed a $J^P=\tfrac{3}{2}^-$ $\bar{D}^* \Sigma_c$ bound state, its location will provide useful information about the dynamics of this two-body system. This information can be used to deduce the existence of new molecules. For instance, from heavy quark spin symmetry it is sensible to expect the existence of a $J^P=\tfrac{5}{2}^-$ $\bar{D}^* \Sigma_c^*$ partner of the $P_c^*$ with a mass of $5515\,{\rm MeV}$ [@Liu:2018zzu]. In this manuscript we argue that if the $P_c^*$ turns out to be molecular the dynamics of the $\bar{D}^* \Sigma_c$ two-body system imply the existence of a series of $\bar{D}^* \bar{D}^* \Sigma_c$ three-body bound states. Concrete calculations in a contact-range theory lead to the following predictions:
1. a $J=\tfrac{1}{2}$, $I=0$ trimer with $B_3 \sim 3-5 \,{\rm MeV}$,
2. a $J=\tfrac{3}{2}$, $I=1$ trimer with $B_3 \sim 1-3 \,{\rm MeV}$,
3. a $J=\tfrac{5}{2}$, $I=0$ trimer with $B_3 \sim 14-16 \,{\rm MeV}$,
4. a $J=\tfrac{5}{2}$, $I=1$ trimer with $B_3 \sim 3-5 \,{\rm MeV}$,
where the trimer binding energy $B_3$ is relative to the hadron-dimer threshold, i.e. the $\bar{D}^* P_c^*$ threshold. There are a series of uncertainties related to the previous predictions of which the most crucial one is whether the $P_c^*$ is really a $\bar{D}^* \Sigma_c$ bound state. Until the nature of the $P_c^*$ is settled, these trimers will remain a theoretical possibility. Conversely the experimental production of these trimers or their detection in the lattice will strongly suggest a molecular nature for the $P_c^*$. It is worth noticing that analogous trimer predictions have been made for other molecular candidates. For instance, from the hypothesis that the $X(3873)$ is a $D^*\bar{D}$ molecule it is possible to theorize about $D^* \bar{D} K$, $D^* \bar{D}^* K$ [@Ren:2018pcd], $D^* D^* \bar{D}$ and $D^* D^* \bar{D}^*$ trimers [@Valderrama:2018sap]. Within more phenomenological approaches there has also been a growing interest in the possibility of three-body systems in the heavy sector, for instance $D^* \bar{D}^* \rho$ [@Bayar:2015oea], $B^* \bar{B}^* \rho$ [@Bayar:2015zba], $B D D$, $B D \bar{D}$, $B^* D \bar{D}$, etc. [@Dias:2017miz; @Dias:2018iuy], just to mention a few recent works.
The manuscript is structured as follows: in Sect. \[sec:Faddeev\] we write the Faddeev equations for the $\bar{D}^* \bar{D}^* \Sigma_c$ system. In Sect. \[sec:Efimov\] we consider the $\bar{D}^* \bar{D}^* \Sigma_c$ system in the unitary limit, i.e. when the binding energy of the $\bar{D}^* \Sigma_c$ system approaches zero. In Sect. \[sec:PcD\] we present our predictions of three body states. Finally in Sect. \[sec:Conclusions\] we summarize the results of this manuscript.
Faddeev Equations for the $\Sigma_c \bar{D}^* \bar{D}^*$ System {#sec:Faddeev}
===============================================================
In this section we write the Faddeev decomposition and equations for the $\bar{D}^* \bar{D}^* \Sigma_c$ three body system. We will consider the S-wave three body states only, as they have the highest chances of binding. We will make the following assumptions:
- the $\bar{D} \bar{D}$, $\bar{D} \bar{D}^*$ and $\bar{D}^* \bar{D}^*$ pairs do not interact,
- the $\bar{D}^* \Sigma_c$ only interacts in the $P_c^*$ channel,
- the $\bar{D}^* \Sigma_c$ interaction in the $P_c^*$ channel can be described in terms of a contact-range interaction.
This leads to a considerable simplification of the Faddeev equations. Equivalently we can consider the $\bar{D}^* \bar{D}^* \Sigma_c$ system from the effective field theory point of view. In this case we will say that the $\bar{D}^* \Sigma_c$ contact-range interaction is leading order, while all the other interactions are subleading and can be ignored at leading order. We will review the previous set of assumptions at the end of the manuscript.
The Equations for $\bar{D}^* \bar{D}^* \Sigma_c$
------------------------------------------------
We begin by writing the three body wave function in terms of Faddeev components for the $\bar{D}^{*} \bar{D}^{*} \Sigma_c$ system $$\begin{aligned}
\Psi_{3B} &=& \sum_{\beta}\,
\left[ \phi_{\beta}(\vec{k}_{23},\vec{p}_1) + \zeta_{\beta}\,
\phi_{\beta}(\vec{k}_{31},\vec{p}_2)
\right] \, \nonumber \\ && \quad \times
| S_{12} \otimes \frac{1}{2} \rangle_S\,| I_{12} \otimes 1 \rangle_I
\, ,\end{aligned}$$ where we have labeled the two $D^*$ mesons as particles $1$ and $2$ and the $\Sigma_c$ as particle $3$. The summation index $\beta = (S_{12}, I_{12})$ refers to the spin and isospin of the $\bar{D}^{*} \bar{D}^{*}$ subsystem. The spin and isospin piece of the wave function is indicated by $$\begin{aligned}
| S_{12} \otimes \frac{1}{2} \rangle_S\,| I_{12} \otimes 1 \rangle_I \, , \end{aligned}$$ where $S$ and $I$ are the total spin and isospin of the $\bar{D}^* \bar{D}^* \Sigma_c$ three body system. The sign $\zeta_{\beta} = \pm 1$ indicates whether the spatial part of the $\bar{D}^{*} \bar{D}^{*}$ wave function is symmetric or antisymmetric. We will ignore the $\zeta_{\beta} = -1$ configurations: they require the orbital angular momentum of the $12$ subsystem to be $L_{12} \geq 1$, which implies that they are suppressed for the positive parity states (they depend on the P-wave $\bar{D}^* \Sigma_c$ interaction). We define the Jacobi momenta as follows $$\begin{aligned}
\vec{k}_{ij} &=& \frac{m_j \vec{k}_i - m_i \vec{k}_j}{m_i + m_j} \, , \\
\vec{p}_{k} &=& \frac{1}{M_T}\,\left[ (m_i + m_j)\,\vec{k}_k -
m_k\,(\vec{k}_i + \vec{k}_j) \right] \, , \end{aligned}$$ with $m_1$, $m_2$, $m_3$ the masses of particles $1$, $2$, $3$, $M_T = m_1 + m_2 + m_3$ the total mass and $ijk$ an even permutation of $123$. In particular we have $m_1 = m_2 = m(\bar{D}^{*})$ and $m_3 = m(\Sigma_c)$. At this point it will be helpful to make the following observation about the Faddeev components: in principle there are up to three Faddeev components for each channel $\beta$. Of these, the first two Faddeev components are symmetric or antisymmetric (as indicated by the sign $\zeta_{\beta}$) because particles $1$ and $2$ are identical: $\phi_{\beta}(\vec{k},\vec{p})$ denotes these components. Finally the third Faddeev component vanishes because we have considered that the $\bar{D}^{*} \bar{D}^{*}$ subsystem does not interact.
We assume a contact-range $\bar{D}^* \Sigma_c$ potential of the type $$\begin{aligned}
V_{\Sigma_c \bar{D}^*} = C_{\sigma \tau} g(k) g(k') \, , \end{aligned}$$ where $C_{\sigma \tau}$ is a coupling constant and $g(p)$ is a regulator function. The subscripts $\sigma$ and $\tau$ indicate the spin and isospin channel of the $\bar{D}^* \Sigma_c$ subsystem, i.e. $\sigma = J_{23}$ and $\tau = I_{23}$. The $\bar{D}^* \Sigma_c$ T-matrix can be written as $$\begin{aligned}
T_{23} = t_{\sigma \tau}(Z) g(k) g(k') \, ,\end{aligned}$$ depending on the spin and isospin channel, where $Z$ represents the energy of the few-body system with respect to the few-body mass threshold. With this two-body T-matrix, the Faddeev component $\phi_{\beta}$ admits the ansatz $$\begin{aligned}
\phi_{\beta}(k,p) =
\frac{g(k)}{Z - \frac{k^2}{2 \mu_{23}} - \frac{p^2}{2 \mu_1}} a_{\beta}(p)
\, ,\end{aligned}$$ with the reduced masses defined as $$\begin{aligned}
\frac{1}{\mu_{ij}} &=& \frac{1}{m_i} + \frac{1}{m_j} \, , \label{eq:mu_ij} \\
\frac{1}{\mu_{k}} &=& \frac{1}{m_k} + \frac{1}{m_i + m_j} \label{eq:mu_k} \, .\end{aligned}$$ From the previous, we find that $a_{\beta}(p)$ follows the integral equation $$\begin{aligned}
a_{\beta}(p_1) &=&
\left[
\sum \lambda^{\beta \gamma}_{\sigma \tau} t_{\sigma \tau}(Z_{23})\, \right]
\nonumber \\
&\times&
\int \frac{d^3 p_2}{(2\pi)^3}\,
B^0_{12}(\vec{p}_1, \vec{p}_2)\,a_{\gamma}(p_2) \, ,\end{aligned}$$ where $\beta$ and $\gamma$ refer to the $S_{12}$ and $I_{12}$ spin and isospin channels and $Z_{23} = Z -\tfrac{p_1^2}{2 m_1}\,\tfrac{M_T}{m_2 + m_3}$. The driving term $B^0_{12}$ is given by $$\begin{aligned}
B^0_{ij} (\vec{p}_i, \vec{p}_j) =
\frac{g(q_i)\,g(q_j)}
{Z - \frac{p_1^2}{2 m_1} - \frac{p_2^2}{2 m_2} - \frac{p_3^2}{2 m_3}} \, ,\end{aligned}$$ with $\vec{p}_1 + \vec{p}_2 + \vec{p}_3 = 0$ and $$\begin{aligned}
\vec{q}_k = \frac{m_j \vec{p}_i - m_i \vec{p}_j}{m_j + m_i} \, .\end{aligned}$$ The integral equation can be discretized, in which case finding the bound state solution reduces to an eigenvalue problem.
Regarding the $\bar{D}^* \Sigma_c$ potential, the only $\sigma \tau$ channel for which we know the interaction is the $P_c^*$ channel, i.e. $\sigma = \frac{3}{2}$ and $\tau = \frac{1}{2}$. For simplicity we will only consider the interaction in the $P_c^*$ channel and neglect the interaction in all other channels $$\begin{aligned}
t_{\sigma \tau}(Z) = t_{P_c^*}(Z)\,
\delta_{\sigma \frac{3}{2}} \delta_{\tau \frac{1}{2}} \, .\end{aligned}$$ Equivalently, in the effective field theory language this means that we are considering the $P_c^*$ interaction as leading order and the interaction in all the other channels as subleading corrections. This point is important as it will entail a simplification in the Faddeev equations: coupled-channel equations will become single-channel ones.
To understand this simplification we first notice that there are nine possible quantum numbers for the S-wave $\bar{D}^* \bar{D}^* \Sigma_c$ system: three spin states $J = \frac{1}{2}$, $\frac{3}{2}$, $\frac{5}{2}$ compounded by three isospin states $I=0$, $1$, $2$. Seven of these configurations involve a single channel, i.e. a single possible $\beta = (S_{12},I_{12})$ configuration for the spin and isospin of the $\bar{D}^* \bar{D}^*$ subsystem. Then there are two quantum numbers that involve coupled channels: $J=\frac{1}{2}$, $\frac{3}{2}$ with $I=1$. Now if we ignore all the $\bar{D}^* \Sigma_c$ interactions with the exception of the $P_c^*$ channel, we can simply make the substitution $$\lambda^{\beta \gamma}_{\sigma \tau} \,t_{\sigma \tau}\to
\lambda^{\beta \gamma}_{P_c^*} \, t_{P_c^*} \, .$$ The matrix $\lambda^{\beta \gamma}_{P_c^*}$ can be diagonalized, in which case the coupled-channel equation reduces to a single-channel one: $$\begin{aligned}
a(p_1) &=&
\lambda\, t_{P_c^*}(Z_{23})\,
\int \frac{d^3 p_2}{(2\pi)^3}\,
B^0_{12}(\vec{p}_1, \vec{p}_2)\,a(p_2) \, , \nonumber \\
\label{eq:PcD-trimer}\end{aligned}$$ where $\lambda$ is one of the eigenvalues of $\lambda^{\beta \gamma}_{P_c^*}$. In fact this is the equation that we will use in all cases to find the binding energies of the $\bar{D}^* \bar{D}^* \Sigma_c$ trimers. The factors $\lambda$ for the different trimer quantum numbers can be found in Table \[tab:lambda1-spin\].
$J^P$ $I$ $\beta = (S_{12}, I_{12})$ $\lambda_{\frac{1}{2},\frac{1}{2}}$ $\lambda_{\frac{1}{2},\frac{3}{2}}$ $\lambda_{\frac{3}{2},\frac{1}{2}}$ $\lambda_{\frac{3}{2},\frac{3}{2}}$ $\lambda$
----------------- ----- ---------------------------- ------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------- ---------------------
$\frac{1}{2}^+$ $0$ $(0,1)$ $\frac{1}{3}$ $0$ $\frac{2}{3}$ $0$ $\frac{2}{3}$
$\frac{1}{2}^+$ $1$ $\{ (0,1), (1,0) \}$ $\left( \begin{smallmatrix} \frac{2}{9} & \frac{2}{9} \\ \frac{2}{9} & \frac{2}{9} \\ \end{smallmatrix} \right)$ $\left( \begin{smallmatrix} \frac{1}{9} & -\frac{2}{9} \\ -\frac{2}{9} & \frac{1}{9} \\ \end{smallmatrix} \right)$ $\left( \begin{smallmatrix} \frac{4}{9} & -\frac{2}{9} \\ -\frac{2}{9} & \frac{1}{9} \\ \end{smallmatrix} \right)$ $\left( \begin{smallmatrix} \frac{2}{9} & \frac{2}{9} \\ \frac{2}{9} & \frac{2}{9} \\ \end{smallmatrix} \right)$ $\frac{5}{9}$, $0$
$\frac{3}{2}^+$ $0$ $(2,1)$ $\frac{5}{6}$ $0$ $\frac{1}{6}$ $0$ $\frac{1}{6}$
$\frac{3}{2}^+$ $1$ $\{(1,0),(2,1)\}$ $\left( \begin{smallmatrix} \frac{1}{18} & \frac{\sqrt{10}}{18} \\ \frac{\sqrt{10}}{18} & \frac{5}{9} \\ \end{smallmatrix} \right)$ $\left( \begin{smallmatrix} \frac{1}{9} & -\frac{\sqrt{10}}{18} \\ -\frac{\sqrt{10}}{18} & \frac{5}{18} \\ \end{smallmatrix} \right)$ $\left( \begin{smallmatrix} \frac{5}{18} & -\frac{\sqrt{10}}{18} \\ -\frac{\sqrt{10}}{18} & \frac{1}{9} \\ \end{smallmatrix} \right)$ $\left( \begin{smallmatrix} \frac{5}{9} & -\frac{\sqrt{10}}{18} \\ -\frac{\sqrt{10}}{18} & \frac{1}{18} \\ \end{smallmatrix} \right)$ $\frac{7}{18}$, $0$
$\frac{5}{2}^+$ $0$ $(2,1)$ $0$ $0$ $1$ $0$ $1$
$\frac{5}{2}^+$ $1$ $(2,1)$ $0$ $0$ $\frac{2}{3}$ $\frac{1}{3}$ $\frac{2}{3}$
The $\bar{D}^* \bar{D}^* \Sigma_c$ System in the Unitary Limit {#sec:Efimov}
==============================================================
Here we will consider the unitary limit of the previous set of Faddeev equations. The unitary limit refers to the limit in which a two-body system is bound at threshold, which is interesting from a theoretical perspective because of its relation with the Efimov effect [@Efimov:1970zz]. Actually the $\bar{D}^* \Sigma_c$ system in the $P_c^*$ channel is far from the unitary limit: its expected size $1/\sqrt{2 \mu_{23} B_2} \sim 1.2\,{\rm fm}$ is comparable with the typical hadronic size ($0.5-1.0\,{\rm fm}$). Yet from a theoretical point of view the discussion about the unitary limit is relevant because of the following reasons: (i) the relation between the Efimov effect [@Efimov:1970zz], Thomas collapse [@Thomas:1935zz] and the requirement of three body forces to compensate for the later [@Bedaque:1998kg; @Bedaque:1998km] and (ii) the $\bar{D}^* \Sigma_c$ system might be closer to the unitary limit for quantum numbers different than the ones of the $P_c^*$ channel.
As already seen, the eigenvalue equation of the $\bar{D}^* \bar{D}^* \Sigma_c$ system always reduces to $$\begin{aligned}
a(p_1) = \lambda \, \tau(Z_{23}) \int \frac{d^3 p_2}{(2\pi)^3}\,
B^0_{12}(\vec{p}_1, \vec{p}_2)\, a(p_2) \, ,
\label{eq:3B-isospin}\end{aligned}$$ with $\lambda$ the factor listed in Table \[tab:lambda1-spin\], which ranges from $\tfrac{1}{6}$ to $1$ for the isoscalar and isovector trimers. Now we take the unitary limit, where we have $$\begin{aligned}
\tau(Z_{23}) &\to& - \frac{2 \pi}{\mu_{23}}\,\sqrt{\frac{\mu_{23}}{\mu_1}}
\frac{1}{p_1} \, , \\
\int \frac{d^2 p_2}{4 \pi}\,B^0_{12} &\to& -\frac{m_3}{2 p_1 p_2}\,
\log{\left[ \frac{p_1^2 + p_2^2 + \frac{2 m}{m + m_3}\,p_1 p_2}
{p_1^2 + p_2^2 - \frac{2 m}{m + m_3}\,p_1 p_2} \right]}
\, , \nonumber \\\end{aligned}$$ with $m = m(D^*)$, $m_3 = m(\Sigma_c)$ and where $\mu_{23}$ and $\mu_1$ are defined in Eqs. (\[eq:mu\_ij\]) and (\[eq:mu\_k\]). From this we arrive to $$\begin{aligned}
p_1\,a(p_1) &=& \frac{\lambda}{2\pi}\,
\frac{m_3}{\mu_{23}}\sqrt{\frac{\mu_{23}}{\mu_1}}\,\frac{1}{p_1}\,
\int_0^{\infty} dp_2\,p_2 a(p_2)\, \nonumber \\
&& \quad \times
\log{\left[ \frac{p_1^2 + p_2^2 + \frac{2 m}{m + m_3}\, p_1 p_2}{p_1^2 + p_2^2 -\frac{2 m}{m + m_3}\, p_1 p_2} \right]}
\, ,\end{aligned}$$ which after the change of variable $p^2 a(p) = b(p)$ transforms into $$\begin{aligned}
b(p) &=& \frac{\lambda}{2\pi}\,
\frac{m_3}{\mu_{23}}\sqrt{\frac{\mu_{23}}{\mu_1}}\,
\int_0^{\infty} dx \,\frac{b(x p)}{x}\,
\nonumber \\ && \quad \times
\log{\left[ \frac{1 + x^2 + \frac{2 m}{m + m_3}\,x}{1 + x^2 - \frac{2 m}{m + m_3}\,x} \right]}
\, .\end{aligned}$$ This equation admits power-law solutions of the type $b(p) = p^s$, in which case we end up with an eigenvalue equation for $s$ $$\begin{aligned}
1 &=&
\frac{\lambda}{2\pi}\,
\frac{m_3}{\mu_{23}}\sqrt{\frac{\mu_{23}}{\mu_1}}\,
\int_0^{\infty} dx \,x^{s-1}\,
\nonumber \\ && \quad \times
\log{\left[ \frac{1 + x^2 + \frac{2 m}{m + m_3}\,x}
{1 + x^2 - \frac{2 m}{m + m_3}\,x} \right]}
\, .\end{aligned}$$ This integral can be evaluated analytically [@Helfrich:2010yr; @Helfrich:2011ut], in which case we arrive at the eigenvalue equation $$\begin{aligned}
1 = \lambda\,J_{\rm Efimov}(s, \alpha) \, , \label{eq:efimov}\end{aligned}$$ where $J_{\rm Efimov}$ is given by $$\begin{aligned}
J_{\rm Efimov}(s, \alpha) = \frac{1}{\sin{2 \alpha}}\,\frac{2}{s}\,
\frac{\sin{\alpha s}}{\cos{\frac{\pi}{2} s}} \, ,\end{aligned}$$ and with the angle $\alpha$ determined as $$\begin{aligned}
\alpha = \arcsin{\left( \frac{1}{1 + \frac{m_3}{m}} \right)} \, ,\end{aligned}$$ with $m$ and $m_3$ the masses of the charmed meson and baryon, respectively.
The Efimov effect [@Efimov:1970zz] happens when the eigenvalue equation, i.e. Eq. (\[eq:efimov\]), admits complex solutions of the type $s = \pm i s_0$, which leads to a $b(p)$ that oscillates: $$\begin{aligned}
b(p) \propto
\sin{\left[ s_0 \log{(\frac{p}{\Lambda_3})} + \varphi \right]} \, ,\end{aligned}$$ with $\Lambda_3$ a momentum scale and $\varphi$ a phase. From the form of $b(p)$ we deduce that the system is invariant under the discrete scaling transformation $p \to e^{\pi/s_0} p$, which for the case of the binding energy reads as $B_3 \to e^{2 \pi/s_0} B_3$. The condition for having the Efimov geometric spectrum is $$\begin{aligned}
\lambda \geq \lambda_c = \frac{\sin{2 \alpha}}{2 \alpha} \, ,\end{aligned}$$ which for the $\bar{D}^* \bar{D}^* \Sigma_c$ system give us $\lambda_c \simeq 0.861$. From this we deduce that the $J=\frac{5}{2}$, $I=0$ trimer is potentially Efimov-like, while all the others are not (but only for the set of assumptions laid out at the beginning of Sect.\[sec:Faddeev\]). For the Efimov-like trimer we have $s_0 \simeq 0.363$, from which the discrete scaling factor is $e^{\pi/s_0} \simeq 5711$ for momenta and $e^{2 \pi/s_0} \simeq 3.262 \cdot 10^7$ for the trimer binding energy [^1]. Even with a molecular $P_c^*$ close to the unitary limit, the factors are too big to be realistically observed. Thus the analysis presented here is mostly academical, except for one thing: if the forces binding the trimer are short-ranged, the possibility of the Efimov effect is related to the necessity of a three body force for the system to be properly renormalized [@Bedaque:1998kg; @Bedaque:1998km]. The reason is the relation between the Efimov effect and the Thomas collapse [@Thomas:1935zz]: that is, as the range of the two-body potential shrinks, the trimer binding energy grows, eventually diverging. However for the $\bar{D}^* \bar{D}^* \Sigma_c$ trimer, the observation of this collapse requires interaction ranges that are orders of magnitude smaller than the typical hadron size.
Finally, though the Efimov effect is absent in the $\bar{D}^* \bar{D}^* \Sigma_c$ trimers under the assumptions we are making, it might still happen in other heavy hadron systems. For instance, the ${B}^* {B}^* \Sigma_c$ system might be a better candidate for the Efimov effect because the associated discrete scaling factors are larger: for $m= m(B^*)$, $m_3 = m(\Sigma_c)$ and $\lambda = 1$ we obtain $s_0 \simeq 0.652$ and $e^{\pi/s_0} \simeq 123.4$.
Predictions {#sec:PcD}
===========
The basic building block of the three body calculation is the $\bar{D}^* \Sigma_c$ interaction in the channel with the quantum numbers of the $P_c^*$: $J^P=\frac{3}{2}^-$ and $I=\frac{1}{2}$. We will describe the $\bar{D}^* \Sigma_c$ system in terms of a contact-range potential of the type $$\begin{aligned}
V(\bar{D}^* \Sigma_c) =
C(\Lambda)\,f(\frac{k}{\Lambda})\,f(\frac{k'}{\Lambda}) \, ,\end{aligned}$$ where $C(\Lambda)$ is a coupling constant and $f(\Lambda)$ a regulator function. Here we will use a Gaussian regulator $f(x) = e^{-x^2}$ and a cut-off window $\Lambda = 0.5-1.0\,{\rm GeV}$. The regulator choice is arbitrary, where we have chosen a Gaussian regulator mostly for convenience and compatibility with previous works [@Valderrama:2018sap; @Liu:2018zzu] (we will comment on the regulator dependence latter). The cut-off choice corresponds to the expected momenta at which the description of the $P_c^*$ as a $\bar{D}^* \Sigma_c$ bound state will break down. In the molecular interpretation the $P_c^*$ is a $\bar{D}^* \Sigma_c$ bound state with a binding energy of $B_2 = 12 \pm 3\,{\rm MeV}$. This binding energy is simply the difference $m(D^*) + m(\Sigma_c) - m(P_c^*)$, where we take the isospin symmetric limit for the $D^*$ and $\Sigma_c$ masses and with the uncertainty basically corresponding to the experimental error in $m(P_c^*)$. We can determine the strength of the coupling $C(\Lambda)$ from the condition of reproducing the binding energy $B_2$. For this we use the two-body eigenvalue equation $$\begin{aligned}
1 + C(\Lambda)\,\int \frac{d^3 q}{(2 \pi)^3}\,
\frac{f^2(\frac{q}{\Lambda})}{B_2 + \frac{q^2}{2 \mu_{23}}} = 0 \, ,\end{aligned}$$ where $B_2$ is the two-body binding energy and $\mu_{23}$ the reduced mass of the $\bar{D}^* \Sigma_c$ system.
Once the contact-range coupling is obtained from the two-body eigenvalue equation, we can solve the three-body eigenvalue equation with $C_{\frac{3}{2} \frac{1}{2}} = C(\Lambda)$, $g(k) = f(k/\Lambda)$ and the appropriate factor $\lambda$. Concrete calculations lead to the predictions of Table \[tab:trimers\], where the binding energy $B_3$ is shown for different trimer configurations. The binding energy $B_3$ is defined with respect to the dimer-particle threshold, i.e. the mass of the trimers is $$\begin{aligned}
M = 2 m + m_3 - B_2 - B_3 \, ,\end{aligned}$$ with $m$ and $m_3$ the mass of the $\bar{D}^*$ meson and $\Sigma_c$ baryon, respectively. The binding energy does not directly depend on the quantum numbers of the trimer, but indirectly by means of the factor $\lambda$ as can be appreciated in Table \[tab:trimers\]. This dependence on the coefficient $\lambda$ is shown explicitly in Figure \[fig:B3-lambda\].
$J^P$ $I$ $B_3(\Lambda = 0.5\,{\rm GeV})$ $B_3(\Lambda = 1.0\,{\rm GeV})$ $\lambda$
----------------- ----- --------------------------------- --------------------------------- ----------------
$\frac{1}{2}^+$ $0$ $4.8^{+1.5}_{-1.4}$ $3.1^{+1.1}_{-1.0}$ $\frac{2}{3}$
$\frac{1}{2}^+$ $1$ $2.6^{+1.0}_{-0.9}$ $1.3^{+0.6}_{-0.5}$ $\frac{5}{9}$
$\frac{3}{2}^+$ $0$ - - $\frac{1}{6}$
$\frac{3}{2}^+$ $1$ $0.5^{+0.3}_{-0.2}$ - $\frac{7}{18}$
$\frac{5}{2}^+$ $0$ $14 \pm 3$ $16^{+3}_{-4}$ $1$
$\frac{5}{2}^+$ $1$ $4.8^{+1.5}_{-1.4}$ $3.1^{+1.1}_{-1.0}$ $\frac{2}{3}$
: Predictions for the binding energy $B_3$ (in units of MeV) of the $\bar{D}^* \bar{D}^* \Sigma_c$ trimers for different quantum numbers and for a cut-off $\Lambda = 0.5-1.0\,{\rm GeV}$, where $B_3$ is relative to the $\bar{D}^* P_c^*$ threshold (i.e. $B_3 > 0$ indicates that the trimer binds). The errors in $B_3$ are a consequence of the uncertainty in the $\bar{D}^* \Sigma_c$ binding energy, $B_2 = 12 \pm 3\,{\rm MeV}$. []{data-label="tab:trimers"}
The trimer binding energies are affected by a series of uncertainties, which we will discuss below. The results of Table \[tab:trimers\] already contain two error sources, the binding energy of the $P_c^*$ ($B_2 = 12 \pm 3\,{\rm MeV}$) and the cut-off window ($\Lambda = 0.5-1.0\,{\rm GeV}$). Assuming that the $P_c^*$ is indeed molecular, the next most important source of uncertainty is the the choice of which interactions are leading and subleading. Previously we have assumed that the only leading order interaction is the $\bar{D}^* \Sigma_c$ short-range potential in the $P_c^*$ channel. We will review this assumption in detail in the following lines, where the different possibilities will be named scenarios A, B and C.
We begin with the $\bar{D}^* \bar{D}^*$ interaction. Törnqvist pointed out [@Tornqvist:1993ng] that the flavour exotic configurations of this system are less likely to bind in general. But this conclusion is probably incomplete because it relies on one-pion exchange while ignoring the short-range contributions to the $\bar{D}^* \bar{D}^*$ interaction. In this regard several works [@Carlson:1987hh; @Gelman:2002wf; @Vijande:2009kj; @Junnarkar:2018twb] ([@Karliner:2017qjm; @Eichten:2017ffp; @Mehen:2017nrh]) have indicated the possibility of a isoscalar flavour exotic $1^+$ tetraquark below (above) the $\bar{D} \bar{D}^*$ threshold. If close enough to the $\bar{D}^* \bar{D}^*$ threshold, it might contribute to the dynamics of this two-body system, suggesting a strong attraction in the $J=1$, $I=0$ channel. In turns out that this contribution will reduce/increase the binding of the isovector $J^P = \tfrac{1}{2}^+$/$\tfrac{3}{2}^+$ trimer. The reason for this reduction/increase of the binding energy are the $\lambda^{\beta \gamma}_{\sigma \tau}$ coefficients in Table \[tab:lambda1-spin\]: for the isovector $J^P = \tfrac{1}{2}^+$/$\tfrac{3}{2}^+$ three-body state, attraction in the isoscalar $\bar{D}^* \bar{D}^*$ subsystem forces the trimer into a configuration that has less/more overlap with the $P_c^*$ channel of the $\bar{D}^* \Sigma_c^*$ subsystem. Scenario A will represent the possibility of a shallow isoscalar $J=1$ $\bar{D}^* \bar{D}^*$ bound state located at threshold, where predictions for the binding energy of the two affected trimers can be found in Table \[tab:trimer-landscape\].
Next we consider the $\bar{D}^* \Sigma_c$ interaction in channels different than the $P_c^*$ ($I=\tfrac{1}{2}$, $J^P = \frac{3}{2}^-$). Phenomenology, in particular the hidden-gauge model [@Wu:2010jy], indicates that the interaction in the $I=\tfrac{1}{2}$, $J^P = \tfrac{1}{2}^-$ channel could very well be as attractive as in the $P_c^*$ channel. Scenario B will refer to this possibility. If this is the case the three isoscalar trimers will be degenerate, having the same binding energy as the isoscalar $J^P = \tfrac{5}{2}^+$ trimer, see Table \[tab:trimer-landscape\]. The $\bar{D}^* \Sigma_c^*$ interaction in the $I=\tfrac{3}{2}$ channel has received less attention, though it could also be strong and attractive [@Chen:2015loa]. Scenario C will consider that there is a $I=\tfrac{3}{2}$ $\bar{D}^* \Sigma_c^*$ bound state at threshold for both $J^P = \tfrac{1}{2}^-$ and $\tfrac{3}{2}^-$ (while also assuming scenario B). In this scenario the isovector trimers will be degenerate and more bound than expected, see Table \[tab:trimer-landscape\]. The degeneration is again a consequence of the $\lambda^{\beta \gamma}_{\sigma \tau}$ coefficients of Table \[tab:lambda1-spin\]: the isovector trimers can always be in a configuration with the same relative contribution from the $I=\tfrac{1}{2}$ and $\tfrac{3}{2}$ $\bar{D}^* \Sigma_c^*$ channels. In addition the hypothesis of $\bar{D}^* \Sigma_c^*$ spin degeneracy in scenario B means that the trimers are effectively decoupled of the isoscalar $J=1$ $\bar{D}^* \bar{D}^*$ interaction. That is, the combination of scenarios A and C leads to the same predictions as scenario C alone, see again Table \[tab:trimer-landscape\].
{width="8.5cm"}
\[fig:B3-lambda\]
Another aspect of the $\bar{D}^* \Sigma_c$ interaction is its long-range piece, which is given by one-pion exchange (OPE). Here we have considered OPE to be subleading. This assumption can be analyzed with the formalism of Refs. [@Valderrama:2012jv; @Lu:2017dvm], which investigated the range of momenta for which OPE is perturbative in the heavy meson-meson and heavy baryon-baryon systems. By adapting the methods of Refs. [@Valderrama:2012jv; @Lu:2017dvm] to the heavy meson-baryon case, i.e. to $\bar{D}^* \Sigma_c$, we arrive at the conclusion that in the $P_c^*$ channel OPE is perturbative for $p < \Lambda_{\rm OPE} \simeq 270-450\,{\rm MeV}$ in the chiral limit ($m_{\pi} = 0$, with $m_{\pi}$ the pion mass). The uncertainty is a consequence of the axial coupling constant of the pion with the charmed meson $\Sigma_c$, which is not known experimentally, see Ref. [@Cheng:2015naa] for a recent summary. For the physical pion mass, $m_{\pi} \simeq 140\,{\rm MeV}$, we expect $\Lambda_{\rm OPE} \simeq 410-710\,{\rm MeV}$, see Refs. [@Valderrama:2012jv; @Lu:2017dvm] for a detailed explanation. If we take into account that the binding momentum of a molecular $P_c^*$ is about $160\,{\rm MeV}$, it is natural to expect OPE to be subleading, although it will provide important corrections to the leading order description. The binding momentum of the $\tfrac{5}{2}^+$ trimer, the most bound three-body state we are predicting, is of the same order: $\sqrt{2 \mu_1 B_3} \sim 200-210\,{\rm MeV}$ (or, in terms of size $1 / \sqrt{2 \mu_1 B_3} \sim 0.9-1.0\,{\rm fm}$). Owing to the similarity in the scales involved, we expect the conclusions derived in the two-body sector to apply in the three-body sector, in agreement with our original assumption.
Scenario $J^P$ $I$ $B_3$
---------- ----------------- ----- -----------
A $\frac{1}{2}^+$ $1$ $0.5-1.9$
A $\frac{3}{2}^+$ $1$ $0.1-0.9$
B $\frac{1}{2}^+$ $0$ $14-16$
B $\frac{3}{2}^+$ $0$ $14-16$
C/A+C $\frac{1}{2}^+$ $1$ $6.7-7.3$
C/A+C $\frac{3}{2}^+$ $1$ $6.7-7.3$
C/A+C $\frac{5}{2}^+$ $1$ $6.7-7.3$
: Different scenarios for the predictions for the binding energy $B_3$ of the $\bar{D}^* \bar{D}^* \Sigma_c$ trimers. Scenario A refers to the existence of a $\bar{D}^* \bar{D}^*$ bound state at threshold. Scenario B is when the $I=\frac{1}{2}$ and $J^P = \frac{1}{2}^-$ and $\frac{3}{2}^-$ $\bar{D}^* {\Sigma}_c^*$ are degenerate. Scenario C assumes that the $I=\frac{3}{2}$ $\bar{D}^* {\Sigma}_c^*$ interaction is strong, generating a bound state at threshold for both $J^P = \frac{1}{2}^-$ and $\frac{3}{2}^-$. For each scenario only the quantum numbers affected are shown. []{data-label="tab:trimer-landscape"}
Finally, though the isoscalar $\tfrac{5}{2}^+$ trimer depends only on the interaction in the $P_c^*$ channel, it actually contains an additional source of uncertainty. The Efimov effect [@Efimov:1970zz] can happen in this trimer, as shown by the analysis of the $\bar{D}^* \bar{D}^* \Sigma_c$ system in the unitary limit, see Sect. \[sec:Efimov\]. The $\bar{D}^* \Sigma_c$ two-body system in the $P_c^*$ channel is actually far from the unitary limit, but the analysis is still relevant because of the relation between the Efimov effect and Thomas collapse [@Thomas:1935zz]. The idea is that, even though a molecular $P_c^*$ is not in the unitary limit, for $\Lambda \to \infty$ the isoscalar $\tfrac{5}{2}^+$ trimer will collapse, i.e. its binding energy will diverge ($B_3 \to \infty$). This collapse can be prevented with the inclusion of a three-body force, without which the trimer binding energy predictions will not be formally cut-off independent [@Bedaque:1998kg; @Bedaque:1998km]. In practice, owing the large discrete scaling factor of $e^{\pi/s_0} \simeq 5711$, the divergence of the trimer binding requires fantastically large cut-offs to be noticed. For instance, the cut-off required for the first nonphysical isovector $J^P = \tfrac{5}{2}^+$ trimer to appear is $\Lambda \sim 62\,{\rm GeV}$, which is pretty large. Thus it is not surprising that the cut-off uncertainty for this trimer is not particularly big, about $2\,{\rm MeV}$ in the $\Lambda = 0.5-1.0\,{\rm GeV}$ cut-off window. Alternatively we can check the model dependence of the $J^P = \tfrac{5}{2}^+$ trimer prediction by using a different regulator, for instance a delta-shell in coordinate space $$\begin{aligned}
V(r; \bar{D}^* \Sigma_c) =
C(R_C)\,\frac{\delta(r - R_c)}{4 \pi R_c^2} \, ,\end{aligned}$$ where $R_c$ is a cut-off radius which we take of the order of the typical hadronic size: $R_c = 0.5-1.0\,{\rm fm}$. The delta-shell potential is actually a separable interaction, where its momentum space form is $$\begin{aligned}
V(\bar{D}^* \Sigma_c) =
C(R_C)\,\frac{\sin(k R_c)}{k R_c}\,\frac{\sin(k' R_c)}{k' R_c} \, .\end{aligned}$$ With this regulator the prediction for the location of the $\tfrac{5}{2}^+$ trimer binding is $B_3 = 14 \pm 3\,{\rm MeV}$ ($17^{+3}_{-4}\,{\rm MeV}$) for $R_c = 1.0\,{\rm fm}$ ($0.5\,{\rm fm}$), i.e. almost identical to the Gaussian regulator predictions. Despite the previous checks caution is advised because the formal requirement of a three-body force, even if the related divergence happens at really large cut-offs, indicates the existence of systematic uncertainties that are not being taken into account.
Conclusions {#sec:Conclusions}
===========
The hypothesis that the $P_c^*$ is a $I=\tfrac{1}{2}$, $J^P = \tfrac{3}{2}^{-}$ $\bar{D}^* \Sigma_c$ molecule implies the existence of a few $\bar{D}^* \bar{D}^* \Sigma_c$ trimers. Calculations in a contact-range theory indicate that the most bound of these trimers has the quantum numbers $I=0$, $J^P = \tfrac{5}{2}^+$ and a binding energy $B_3 \sim 14-16\,{\rm MeV}$. There are other three or four trimer configurations that are likely to bind, but they are expected to be considerably less bound: two states with $B_3 \sim 3-5\,{\rm MeV}$ with quantum numbers $I=0$, $J^P = \frac{1}{2}^+$ and $I=1$, $J^P = \frac{5}{2}^+$, a state with $B_3 \sim 1-3\,{\rm MeV}$ and quantum numbers $I=1$, $J^P = \frac{1}{2}^+$ and maybe a state on the verge of binding with $I=1$, $J^P = \frac{3}{2}^+$, see Table \[tab:trimers\] for details. These predictions are affected by a series of uncertainties, which mostly stem from the fact that we do not know too much about the $\bar{D}^* \Sigma_c$ interaction except in the $P_c^*$ channel (and even this is dependent on the nature of the $P_c^*$). These uncertainties are taken into account on the basis of considering different hypothesis about the $\bar{D}^* \bar{D}^*$ and $\bar{D}^* \Sigma_c$ interactions, which are summarized in Table \[tab:trimer-landscape\]. From the previous considerations we arrive to the conclusion that the most solid prediction is that of the isoscalar $J^P = \frac{5}{2}^+$ trimer.
We have also considered the $\bar{D}^* \bar{D}^* \Sigma_c$ system in the unitary limit, i.e. when the $P_c^*$ is located at the $\bar{D}^* \Sigma_c$ threshold. In this situation the $J^P = \frac{5}{2}^+$ trimer will be Efimov-like and will have a geometric spectrum with a scaling factor of $3.3 \cdot 10^7$ for the binding energies. This scaling factor is fantastically large, which implies that there would be no practical way to observe it even if we could tune the $\bar{D}^* \Sigma_c$ scattering length. Yet the possibility of the Efimov effect in the isoscalar $J^P = \frac{5}{2}^+$ trimer implies that a three-body force should be included in the calculations at leading order. The practical importance of this three-body force might be tangential, as reflected by the mild cut-off dependence of the isoscalar $J^P = \frac{5}{2}^+$ trimer binding in the cut-off window chosen in this work, $\Lambda = 0.5-1.0\,{\rm GeV}$.
Without knowing the nature of the $P_c^*$, the predictions of this work will remain theoretical: if the $P_c^*$ is not molecular but a compact pentaquark instead the $\bar{D}^* \bar{D}^* \Sigma_c$ trimers are not expected to bind. The experimental production of these trimers is expected to be difficult, with the lattice probably providing a more convenient way to investigate them. Finally we notice that the existence of $\Sigma_c \Sigma_c \bar{D}^*$ bound trimers is also likely, but their location will be subjected to even larger uncertainties as a consequence of the $\Sigma_c \Sigma_c$ interaction, which is not particularly well-known but probably strong.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank Li-Sheng Geng, Kanchan P. Khemchandani and Alberto Martinez Torres for discussions and a critical and careful reading of this manuscript. This work is partly supported by the National Natural Science Foundation of China under Grants No.11522539, No.11735003, the Fundamental Research Funds for the Central Universities and the Thousand Talents Plan for Young Professionals.
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[^1]: Notice that the discrete scaling factor for the $\bar{D}^* \bar{D}^* \Sigma_c$ system is different than in the three boson system, where the Efimov effect was originally proposed. In the three boson system $s_0 \simeq 1.00624$ and $e^{\pi/s_0} \simeq 22.7$, which can be easily reproduced from Eq. (\[eq:efimov\]) by simply setting $\lambda = 2$ and $m = m_3$. For $\lambda = 1$ and $m = m_3$ the factor for the standard heteronuclear Efimov effect with equal masses (i.e. a system of three particles with identical masses but only two resonantly particle pairs) $s_0 \simeq 0.4137$ and $e^{\pi/s_0} \simeq 1986.1$, is reproduced. Details on how the Efimov effect manifest in different systems can be consulted in Refs. [@Hammer:2010kp; @Naidon:2016dpf]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
The boundary of two mixed Bose-Einstein condensates interacting repulsively was considered in the case of spatial separation at zero temperature. Analytical expressions for density distribution of condensates were obtained by solving two coupled nonlinear Gross-Pitaevskii equations in cases corresponding weak and strong separation. These expressions allow to consider excitation spectrum of a particle confined in the vicinity of the boundary as well as surface waves associated with surface tension.\
`PACS Number(s): 03.75.Fi`
address: 'Department of Physics, The University of Texas at Austin, Austin, Texas, 78712'
author:
- 'R.A. Barankov'
title: 'Boundary of two mixed Bose-Einstein condensates'
---
Introduction
============
The experimental realization of Bose Einstein condensation in trapped dilute gases [@Anderson; @Davis; @Bradley] has allowed to investigate variety of properties of quantum fluids both theoretically and experimentally. In the recent years, it has become possible to produce and explore mixtures of Bose-Einstein condensates corresponding to different internal states [@Myatt; @Hall1; @Hall2].
Theoretical treatment of mixtures [@Timmermans; @Ao; @Pu; @Shi] assures that depending on the relative strength of interactions inside each condensate and between them it is possible to observe spatial separation. Experimental realization of such systems [@Myatt; @Hall1; @Hall2] has given an opportunity to study both equilibrium properties and dynamics of separation. Although, structure of the boundary has been qualitatively analyzed in [@Timmermans; @Ao] an analytical distribution of densities has not been derived yet. Authors of [@Ao] discussed asymptotic behavior of densities far from the boundary, and on the basis of these estimations analyzed surface tension.
In the present paper, we explore the boundary between two repulsively interacting condensates at zero temperature in two limit cases corresponding weak and strong separation of the condensates. We show that in the case of weak separation it is possible to derive equations for densities, and using iteration method solve them analytically. We show that asymptotic behavior of our solution coincides with those predicted in [@Ao]. For the sake of completness, we also provide the solutions of these equations in the case of strong separation. Importance of the solution lies in the possibility to explore quantitatively different types of excitations on the boundary. The structure of the boundary in both cases allows to consider one-particle excitations as well as surface waves associated with the boundary. Obtained expressions for dispersion relation of surface waves can be used to explore that phenomenon experimentally.
The Hamiltonian describing the mixture of two weakly interacting Bose-Einstein condensates can be written in the form $$\label{ham}
\begin{array}{r}
H=\sum\limits_{i=1,2}\int d{\bf r} \Psi_{i}^{+} \left[-\frac{\hbar^{2}
\nabla^{2}}{2 m_{i}}+V_{i}({\bf r})+
\frac{u_{i}}{2}\Psi_{i}^{+}\Psi_{i}\right]\Psi_{i}\\
+u_{12}\int d{\bf r}\Psi_{1}^{+}\Psi_{2}^{+}\Psi_{1}\Psi_{2}
\end{array}$$ Here, $u_{i}=4 \pi \hbar^{2} a_{i}/m_{i}>0$ characterizes the interaction inside each condensate, $u_{12}=2 \pi \hbar^{2} a_{12} (m_{1}+m_{2})/(m_{1}m_{2})>0$ – the intercondensate interaction, $m_{i}$ – mass of a particle of each condensate, $a_{i}$, $a_{12}$ – corresponding scattering lengths, $V_{i}(\bf r)$ – external trapping potentials. Theoretical treatment of the mixtures [@Timmermans; @Ao] has shown that separation takes place when $u_{12}/\sqrt{u_{1} u_{2}}>1$.
Starting with the Hamiltonian (\[ham\]) we get Gross-Pitaevskii equations for condensate wave functions: $$\label{nonsteqs}
\begin{array}{r}
i\hbar\frac{\partial\Psi_{1}}{\partial t}=\left(-\frac{\hbar^{2}
\nabla^{2}}{2 m_{1}}+V_{1}({\bf r})+u_{1}|\Psi_{1}|^{2}+
u_{12}|\Psi_{2}|^{2}\right)\Psi_{1},
\\
i\hbar\frac{\partial\Psi_{2}}{\partial t}=\left(-\frac{\hbar^{2}
\nabla^{2}}{2 m_{2}}+V_{2}({\bf r})+u_{2}|\Psi_{2}|^{2}+
u_{12}|\Psi_{1}|^{2}\right)\Psi_{2}
\end{array}$$
As we are interested in studying of stationary solutions of these equations, then assuming as usual $\Psi_{j} \propto \exp(-i\mu_{j} t)$, where $\mu_{j}$ – chemical potentials of the condensates, we obtain two coupled nonlinear equations for densities of gases $n_{i}({\bf r})=|\Psi_{i}({\bf r})|^{2}$: $$\label{steqs}
\begin{array}{l}
\mu_{1}=-\frac{\hbar^{2}}{2 m_{1}}\frac{\nabla^{2}\sqrt{n_{1}}}
{\sqrt{n_{1}}}+V_{1}({\bf r})+u_{1}n_{1}+u_{12}n_{2},
\\
\mu_{2}=-\frac{\hbar^{2}}{2 m_{2}}\frac{\nabla^{2}\sqrt{n_{2}}}
{\sqrt{n_{2}}}+V_{2}({\bf r})+u_{2}n_{2}+u_{12}n_{1}
\end{array}$$
These equations are essentially nonlinear, so to find solutions we need to make some simplifications. We assume that $V_{1}({\bf r})=V_{2}({\bf r})$ and consider the case when the size of a boundary between condensates much less than characteristic length of the trap. In the case of a parabolic trap potential and Thomas-Fermi regime, it means that $d\ll R_{TF}$, where $d$ is the size of the boundary, and $R_{TF}$ is the Thomas-Fermi radius of the atomic cloud. Physically, it helps to avoid the effect of the potential on the form of the boundary. To simplify calculations further, we also suppose that separation takes place in one dimension, $z$. This allows us to write the system of equations (\[steqs\]) in the following form: $$\label{eqs}
\begin{array}{l}
\mu_{1}=-\frac{\hbar^{2}}{2 m_{1}\sqrt{n_{1}}}\frac{d^{2}}{dz^{2}}
\sqrt{n_{1}}+u_{1}n_{1}+u_{12}n_{2},
\\
\mu_{2}=-\frac{\hbar^{2}}{2 m_{2}\sqrt{n_{2}}}\frac{d^{2}}{dz^{2}}
\sqrt{n_{2}}+u_{2}n_{2}+u_{12}n_{1}
\end{array}$$
Although, there is no explicit trapping potential in these equations, we have to impose external conditions on the solutions, that implicitly take it into account. Let us assume that separation takes place along the $z$ direction, and the condensate with the label “$1$" is to the right, and the one with the label “$2$" is to the left of the boundary. Then, asymptotically we require: $$\label{conds}
\begin{array}{l}
n_{1}(z\to +\infty)\to n_{10},\, n_{1}(z\to -\infty)\to 0,\\
n_{2}(z\to -\infty)\to n_{20},\,n_{2}(z\to +\infty)\to 0,
\end{array}$$ where $n_{10}$, $n_{20}$ – equilibrium densities of condensates far from the boundary. Substituting these conditions in equations (\[eqs\]) for densities, and using the obvious condition of equilibrium we obtain $$\label{prconds}
\begin{array}{l}
\mu_{1}=u_{1}n_{10},\,\mu_{2}=u_{2}n_{20},\\
P_{1}=u_{1}n_{10}^{2}/2=P_{2}=u_{2}n_{20}^{2}/2
\end{array}$$ Here, we used well-known expression for the pressure of a homogeneous weakly-interacting Bose gas. That condition connects equilibrium densities of condensates far from the boundary.
To reduce the number of parameters in equations (\[eqs\]) we notice that it is possible to exclude difference of masses by the change: $$\label{mass}
\begin{array}{l}
u_{1}^{*}=u_{1}m_{1}/m_{2},\,u_{2}^{*}=u_{2}m_{2}/m_{1},\\
n_{1}^{*}=n_{1}\sqrt{m_{2}/m_{1}},\,n_{2}^{*}=n_{2}\sqrt{m_{1}/m_{2}},\\
\mu_{1}^{*}=u_{1}^{*}n_{10}^{*},\,\mu_{2}^{*}=u_{2}^{*}n_{20}^{*},\\
m^{*}=\sqrt{m_{1}m_{2}}
\end{array}$$
We get the same equations (\[eqs\]) and conditions (\[conds\]), (\[prconds\]) but for quantities with stars and with the same mass $m^{*}$. To simplify notations, in the following discussion we omit stars. The generalization for different masses can be easily done by following the above rules.
We can solve equations for densities (\[eqs\]) with conditions (\[conds\]),(\[prconds\]) analytically in two limit cases for weak separation when $\Delta=u_{12}/\sqrt{u_{1} u_{2}}-1\ll 1$, and strong separation when $\Delta\gg 1$. Notice, that both parameters $u_{12}$ and $\Delta$ are not affected by the above procedure of mass difference excluding.
Weak separation
===============
Let us consider weak separation when condition $\Delta=u_{12}/\sqrt{u_{1} u_{2}}-1\ll 1$ is satisfied. We expect that in the simplest case when $u_{1}=u_{2}$ the whole density of a gas $n(z)=n_{1}(z)+n_{2}(z)$ is approximately constant. That is why, to find the solution it is natural instead of $n_{1}$, $n_{2}$ to introduce other quantities and solve equations (\[eqs\]) using small parameter $\Delta$. Consider the functions: $$\label{new}
\begin{array}{l}
\rho=(u_{1}/u_{2})^{1/4}n_{1}+(u_{2}/u_{1})^{1/4}n_{2},\\
g=\left[(u_{1}/u_{2})^{1/4}n_{1}-(u_{2}/u_{1})^{1/4}n_{2}\right]/\rho
\end{array}$$
Conditions (\[conds\]), (\[prconds\]) give us simple asymptotic behavior of these functions: $$\label{condsnew}
\begin{array}{l}
\rho(z\to\pm\infty)\to\rho_{0}=\sqrt{n_{10}n_{20}},\\
g(z\to\pm\infty)\to\pm 1
\end{array}$$
Densities of condensates “1" and “2" are easily obtained if functions $\rho$ and $g$ are known: $$\begin{array}{l}
n_{1}=(u_{2}/u_{1})^{1/4}\rho[1+g]/2,\\
n_{2}=(u_{1}/u_{2})^{1/4}\rho[1-g]/2
\end{array}$$
It is straightforward to derive equations for $\rho$ and $g$: $$\label{eqsnew}
\begin{array}{r}
\frac{\sqrt{\rho}^{\,\prime\prime}}{\sqrt{\rho}}-
\frac{\rho}{\rho_{0}}\left[1+\alpha g +\frac{\Delta}{2}
(1-g^{2})\right]\\
=\frac{g^{\prime\,2}}{4(1-g^{2})}-1-\alpha g, \\
\frac{g^{\prime\prime}}{1-g^{2}}+\frac{2\sqrt{\rho}^{\,\prime}
g^{\prime}}{\sqrt{\rho}(1-g^{2})}+\frac{g g^{\prime\,2}}
{(1-g^{2})^{2}}+2\alpha\\
=\frac{\rho}{\rho_{0}}\left[2\alpha+\Delta(\alpha-g)\right]\\
\end{array}$$ Here, $f^{\prime}=\xi_{0}\frac{d f}{d z}$, and we also introduced $\alpha=\frac{\sqrt{u_{1}}-\sqrt{u_{2}}}{\sqrt{u_{1}}+\sqrt{u_{2}}}$ and $1/\xi_{0}^{2}=m(u_{1}u_{2})^{1/4}(\sqrt{u_{1}}+\sqrt{u_{2}})
\rho_{0}/\hbar^{2}$.
To find asymptotic solutions of equations (\[eqsnew\]) in the case of $\Delta\ll 1$ we use iteration method. Let us suppose that terms with derivatives of $\rho$ are much smaller than the others. We will justify this assumption in the end of calculations.
Neglecting terms with derivatives of $\rho$, from equations (\[new\]) we get $$\begin{array}{l}
\rho/\rho_{0}=1-\frac{g^{\prime\,2}+2\Delta(1-g^{2})^{2}}{4(1-g^{2})
(1+\alpha g)},\\
\frac{g^{\prime\prime}}{1-g^{2}}+\frac{2g+\alpha(1+ g^{2})}{2(1-g^{2})
(1+\alpha g)}g^{\prime\,2}+\frac{\Delta (1-\alpha^{2})g}{1+\alpha g}=0
\end{array}$$
The equation for $g$ can be solved by substitution $g^{\prime}=f$, so taking into account conditions (\[condsnew\]) we get that $g$ is the solution of the equation $$\label{eqsg}
g^{\prime}=\sqrt{\Delta(1-\alpha^{2})}\frac{(1-g^{2})}{\sqrt{1+\alpha g}}$$
At this point it is clear that assumptions we made were correct. Namely, we obtain that $\sqrt{\rho}^{\,\prime}\propto \Delta^{3/2}$, $\sqrt{\rho}^{\,\prime\prime}\propto \Delta^{2}$, $g^{\prime}\propto\sqrt{\Delta}$, so neglected terms by $\Delta$ times less than the others.
Now we can write down solutions for both $\rho$ and $g$ in the parametric form $$\label{solutions}
\begin{array}{r}
1-\frac{\rho}{\rho_{0}}=\frac{\Delta}{4}\frac{1-g^{2}}{(1+\alpha g)^{2}}
[3-\alpha^{2}+2\alpha g],\\
\frac{z-z_{0}}{\xi_{0}/\sqrt{\Delta(1-\alpha^{2})}}=\frac{\sqrt{1+\alpha}}{2}
\,\ln|\frac{\sqrt{1+\alpha g}+\sqrt{1+\alpha}}{\sqrt{1+\alpha g}-
\sqrt{1+\alpha}}|\\
-\frac{\sqrt{1-\alpha}}{2}\ln|\frac{\sqrt{1+\alpha g}+\sqrt{1-\alpha}}
{\sqrt{1+\alpha g}-\sqrt{1-\alpha}}|
\end{array}$$ Here, the second equation is the solution of (\[eqsg\]). In these expressions an arbitrary constant $z_{0}$ defines the position of the boundary, and in the case of finite geometry with given average number of particles in condensates can be obtained from the condition of equal pressures (\[prconds\]). There is also physically obvious symmetry $z-z_{0}\to z_{0}-z$, $g\to-g$, $\alpha\to-\alpha$.
Finally, the densities of condensates have the following form: $$\label{dens}
\begin{array}{l}
n_{1}=\frac{n_{10}}{2}\left(1-\frac{\Delta}{4}\frac{1-g^{2}}
{(1+\alpha g)^{2}}[3-\alpha^{2}+2\alpha g]\right)\left[1+g\right],\\
n_{2}=\frac{n_{20}}{2}\left(1-\frac{\Delta}{4}\frac{1-g^{2}}
{(1+\alpha g)^{2}}[3-\alpha^{2}+2\alpha g]\right)\left[1-g\right]
\end{array}$$ With the parametric equation for $g$, these densities are the main result of the paper. On fig With the use of the desribed method, it is possible to derive expressions for densities in the next orders of the small parameter $\Delta$ in the form of asymptotic series. As follows from the above estimations of neglected terms in equations (\[eqsnew\]) the next order is proportional to $\Delta^{2}$. The typical dependence of densities on the distance from the boundary is shown in Fig. \[fig:well\] and Fig. \[fig:nowell\].
It is interesting to notice that the whole density $n=n_{1}+n_{2}$ at some values of parameters has a well on the boundary as shown in Fig. \[fig:well\]. In the case of weak separation we see that it is broad and shallow. This result is a consequence of the interparticle interactions. The interparticle potential for each condensate acts in some sense as a wall, so we expect that probability of a particle to be close to the boundary decreases, which means lower density. The existence of such a well allows to consider the possibility of confining of a particle of another sort in the vicinity of the boundary, which is discussed below.
Although, we can not further simplify solutions (\[solutions\]), it is interesting to derive asymptotic behavior of these functions in particular limits: $$\label{asymp}
\begin{array}{l}
1-g(z\to +\infty)\propto \exp[-\frac{2 z \sqrt{\Delta}}{\xi_{2}}],\\
1+g(z\to -\infty)\propto \exp[\frac{2 z \sqrt{\Delta}}{\xi_{1}}],\\
g(z\to z_{0})\to \frac{(z-z_{0})\sqrt{\Delta(1-\alpha^{2})}}{\xi_{0}}
\end{array}$$ Here, $\xi_{i}=\hbar/\sqrt{2m_{i}\mu_{i}}$ – correlation lengths of condensates, defined by chemical potential $\mu_{1}$ and $\mu_{2}$, and we used (\[mass\]) to include mass difference. The asymptotic behavior far from the boundary of each condensate is defined by its correlation length. As we see, the size of the boundary can be approximated as $d\sim (\xi_{1}+\xi_{2})/\sqrt{\Delta}$ and for $\Delta\ll 1$ appears to be much larger than correlation lengths.
Solutions have the simplest form when $\alpha=0$: $$\label{simdens}
\begin{array}{l}
n_{1}(z)=\frac{n_{0}}{2}\left(1-\frac{3\Delta}{4 \cosh^{2}
[\frac{z\sqrt{\Delta}}{\xi_{0}}]}\right)
\left(1+\tanh[\frac{z\sqrt{\Delta}}{\xi_{0}}]\right),\\
n_{2}(z)=\frac{n_{0}}{2}\left(1-\frac{3\Delta}{4 \cosh^{2}
[\frac{z\sqrt{\Delta}}{\xi_{0}}]}\right)
\left(1-\tanh[\frac{z\sqrt{\Delta}}{\xi_{0}}]\right)\\
\end{array}$$ Here, we choose $z_{0}=0$.
Strong separation
=================
To analyze the case of strong separation $\Delta=u_{12}/\sqrt{u_{1}u_{2}}\gg 1$ (we use the same notation but for another quantity) we start from density equations (\[eqs\]). In this case we expect that density on the boundary will be approximately zero because interparticle interactions make it almost impossible for one condensate to penetrate inside the other. To estimate the density of condensates on the boundary we can use the fact that second derivatives of wave functions should be approximately zero there. That makes system (\[eqs\]) a set of two linear equations with solution: $$\begin{array}{l}
n_{1B}=\frac{n_{10}}{\Delta+1}\approx\frac{n_{10}}{\Delta}\ll n_{10},\\
n_{2B}=\frac{n_{20}}{\Delta+1}\approx\frac{n_{20}}{\Delta}\ll n_{20}
\end{array}$$
This allows us in the zero approximation to use simple conditions for the densities $n_{1}(z\le0)=n_{2}(z\ge0)=0$. Then equations (\[eqs\]) have simple form: $$\begin{array}{l}
\mu_{1}=-\frac{\hbar^{2}}{2 m_{1}\sqrt{n_{1}}}\frac{d^{2}}{dz^{2}}
\sqrt{n_{1}}+u_{1}n_{1},\,\mbox{ for } z\ge0,\\
\mu_{2}=-\frac{\hbar^{2}}{2 m_{2}\sqrt{n_{2}}}\frac{d^{2}}{dz^{2}}
\sqrt{n_{2}}+u_{2}n_{2},\,\mbox{ for } z\le0
\end{array}$$
Solutions are easily obtained: $$\label{strong}
\begin{array}{l}
n_{1}(z\ge 0)=n_{10}\tanh^{2}[\frac{z}{\sqrt{2}\xi_{1}}],\\
n_{2}(z\le 0)=n_{20}\tanh^{2}[\frac{z}{\sqrt{2}\xi_{2}}]
\end{array}$$ Here, $n_{10}$ and $n_{20}$ are connected by the condition of equal pressures (\[prconds\]), and we choose the position of the boundary at $z=0$.
The size of the boundary in this case is approximately $d\approx 2\sqrt{2}(\xi_{1}+\xi_{2})$. The dependence of the densities on the distance from the boundary is shown in Fig. \[fig:strong\]. As in the previous section, we see that there is again a well in the whole density but in the case of strong separation it becomes narrower and deeper in comparison with the one for weak separation.
One-particle excitations on the boundary
========================================
The existence of a well in the whole density allows to consider confining of a particle of another sort in the vicinity of the boundary. As a general property of a quantum-mechanical motion in a one-dimensional well there always exists such a confined state. As an example, we consider the simplest case when $\alpha=0$ and a particle of another sort interacts with both condensates repulsively with the same constant $\lambda$. The Schrödinger equation for the wave function of a particle with the mass $M$ has the form: $$\label{part}
\left[-\frac{\hbar^{2}}{2 M}\frac{d^{2}}{d\,z^{2}}+\lambda n(z)\right]
\phi=E \phi$$
We can solve equation (\[part\]) for weak and strong separation cases simultaneously. It has universal form $$\frac{d^{2}\phi}{d\,z^{2}}+\frac{2 M}{\hbar^{2}}\left(\epsilon+
\frac{U_{0}}{\cosh^{2}[\beta z]}\right)\phi=0,$$ where $\epsilon=E-\lambda n_{0}$, and $U_{0}=3\Delta\lambda n_{0}/4$, $\beta=\sqrt{\Delta}/\xi_{0}$ for weak; and $U_{0}=\lambda n_{0}$, $\beta=1/(\sqrt{2}\xi_{0})$ for strong separation. The spectrum of energy $\epsilon$ is well-known: $$\begin{array}{l}
\epsilon_{j}=-\mu\frac{\Delta m}{4M}\left[-(1+2j)+\sqrt{1+3
\frac{M \lambda}{m u}}\right]^{2} \,\mbox{``weak"},\\
\epsilon_{j}=-\mu\frac{m}{8M}\left[-(1+2j)+\sqrt{1+8
\frac{M \lambda}{m u}}\right]^{2} \,\mbox{``strong"}
\end{array}$$ where $\mu=un_{0}$ – chemical potential, $j=0,1,\dots$ and the condition that an expression in $[\dots]$ is positive defines the upper limit for $j$. There is always at least one state with $j=0$.
Surface waves on the boundary
=============================
There is also another type of excitations associated with the boundary. As we see condensate density distributions give rise to nonzero surface tension which was previously analyzed in [@Ao], and we use the expression for surface tension derived there: $$\label{gentension}
\sigma=\frac{1}{2}\int\limits_{-\infty}^{+\infty}dz\sum\limits_{i=1,2}
\frac{\hbar^{2}}{2 m_{i}} \left(\frac{d\sqrt{n_{i}}}{dz}\right)^{2}$$
Substituting expressions (\[dens\]) for densities in the case of weak separation and taking into account only first nonzero order in $\Delta$ we obtain for surface tension $$\label{tensionweak}
\sigma_{w}=\frac{P\sqrt{\Delta}}{6}\left[
\begin{array}{r}
(\xi_{1}+\xi_{2})\frac{2\alpha^{2}-1+\sqrt{1-\alpha^{2}}}
{\alpha^{2}}\\
-(\xi_{1}-\xi_{2})\frac{1-\sqrt{1-\alpha^{2}}}{\alpha}
\end{array}
\right],$$ where $\alpha=\left(m_{1}\sqrt{u_{1}}-m_{2}\sqrt{u_{2}}\right)/
\left(m_{1}\sqrt{u_{1}}+m_{2}\sqrt{u_{2}}\right)$ and we used (\[mass\]) to include mass difference; $\xi_{1}$, $\xi_{2}$ – correlation lengths of condensates; $P$ – the pressure given by (\[prconds\]). When $\alpha=0$, $\sigma_{w}=P\sqrt{\Delta}(\xi_{1}+\xi_{2})/4$
In the case of strong separation we use expressions (\[strong\]) to get surface tension $$\label{tensionstrong}
\sigma_{s}=\frac{P\sqrt{2}}{3}(\xi_{1}+\xi_{2})$$
Let us notice that expressions (\[tensionweak\]) and (\[tensionstrong\]) differ from the ones obtained in [@Ao]. Although, in qualitative sense our expressions coincide with those of [@Ao], using our method of solution we can get a general expression applicable for variety of parameters, and for example retrieve the correct numerical factor for the case considered in [@Ao]. As follows from the general expression for surface tension (\[gentension\]), we need to know the behavior of densities not only far but also in the vicinity of the boundary. That is why, estimate character of expressions for densities in [@Ao] could only give qualitative answer for surface tension.
For velocities smaller than the speed of sound we can consider gas as incompressible, so it is possible to write down the usual hydrodynamic equations and find dispersion relation of surface waves on the boundary associated with surface tension. Suppose that we have condensates in a “box" and the position of the boundary is defined by the distances $L_{1}$ and $L_{2}$ from the “box" walls, where labels “1", “2" correspond to the side with the same condensate, and the box sizes for other directions are $L_{x}$, $L_{y}$. Taking into account the fact that velocities are zero on the walls we get the dispersion relation $$\label{wave}
\omega^{2}(k)=\frac{\sigma\,k^{3}}{\rho_{1}\coth(k L_{1})+
\rho_{2}\coth(k L_{2})},$$ where $\sigma$ is the surface tension; $\rho_{10}=m_{1}n_{10}$, $\rho_{20}=m_{2}n_{20}$ – mass densities; wave vector along the boundary $k=\sqrt{(\pi n_{x}/L_{x})^{2}+(\pi n_{y}/L_{y})^{2}}$, $n_{x}, n_{y}=0,1\dots$, not $n_{x}=n_{y}=0$; $L_{1}$, $L_{2}$ – sizes of condensates in the direction perpendicular to the boundary. We assume that $L_{1}$, $L_{2}$ are much larger than the size of the boundary $d$.
For dispersion relation we consider two limit cases corresponding to wavelengths $\lambda\ll L_{i}$ ($k L_{i}\gg 1$) and $\lambda\gg L_{i}$ ($k L_{i}\ll 1$). Let us notice that the second case is possible only if $L_{i}\ll L_{x,y}$.
In the first case we obtain $$\omega(k)=\left(\frac{\sigma}{\rho_{10}+\rho_{20}}\right)^{1/2}k^{3/2}$$
In the long wavelength limit we get $$\omega(k)=\left(\frac{\sigma L_{1} L_{2}}{\rho_{10}L_{2}+\rho_{20}L_{1}}
\right)^{1/2}k^{2}$$
As we see, for weak separation $\omega\propto \Delta^{1/4}$, and surface waves are relatively “soft" in that case, which enables to consider them as a dissipative channel for other condensate excitations.
Conclusion
==========
We performed the analysis of the boundary of two Bose-Einstein condensates interacting repulsively in the limit cases corresponding weak and strong separation at zero temperature.
For weak separation we obtained solutions (\[dens\]) of two coupled nonlinear Gross-Pitaevskii equations using small parameter $\Delta$. The solutions show that the penetration depth of the condensate “$i$" inside the other is estimated as $\xi_{i}/\sqrt{\Delta}$, so that the size of the boundary $d\sim (\xi_{1}+\xi_{2})/\sqrt{\Delta}$ is much larger than correlation lengths, which was obtained experimentally [@Hall1]. There is also a well in the full density profile, which is the consequence of the wave function behavior near the boundary. In general, the proposed method of obtaining density distributions for the weak separation case if desired can be expanded to obtain expressions for the next orders of $\Delta$.
We also considered the case of strong condensate separation but restricted ourselves to the zero order approximation. In that case the size of the boundary is $d\sim (\xi_{1}+\xi_{2})$, and the whole density of gases goes approximately to zero on the boundary.
The existence of the well in the density profile at some parameters allowed us to consider one-particle excitations on the boundary. Using the expressions for densities we found excitation spectrum of a particle in the simplest case when constants of interaction of the particle with condensates are the same, and distribution of densities on the boundary corresponds to $\alpha=0$. The generalization to other cases can be easily done with the use of expressions (\[dens\]),(\[strong\]) for density distributions. Let us notice that the existence of a potential well depends on the interaction constants of a particle with condensates as well as the relation between interaction constants of the condensates. Observation of such confined states can be possible only if temperature are smaller than the potential well.
It was shown that there also exist collective excitations associated with surface tension. The expressions for surface tension were obtained for both weak and strong separation. The dispersion relation for surface waves was analysed in the case where condensates fill finite volumes. The dispersion relation has different forms in two cases corresponding to short- and long-wavelength limit. In the case of weak separation “soft" surface modes can be a dissipative channel of other condensate excitations.
Author gratefully acknowledges valuable discussions with Yu.M. Kagan and other members of the Theoretical Division of RRC “Kurchatov Institute", Russia.
M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell, Science [**269**]{}, 1989 (1995) K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, W. Ketterle, Phys.Rev.Lett. [**75**]{}, 3969 (1995) C.C. Bradley, C.A. Sackett, R.G. Hulet, Phys.Rev.Lett. [**78**]{}, 985 (1997) C.J. Myatt, E.A. Burt, W. Ghirst, E.A. Cornell, C.E. Wieman, Phys.Rev.Lett. [**78**]{}, 586 (1997) D.S. Hall, M.R. Matthews, J.R. Ensher, C.E. Wieman, E.A. Cornell, Phys.Rev.Lett. [**81**]{}, 1539 (1998) D.S. Hall, M.R. Matthews, C.E. Wieman, E.A. Cornell, Phys.Rev.Lett. [**81**]{}, 1543 (1998) E. Timmermans, Phys.Rev.Lett. [**81**]{}, 5718 (1998) P. Ao, S.T. Chui, Phys.Rev.A [**58**]{}, 4836 (1998) H. Pu, N.P. Bigelow, Phys.Rev.Lett. [**80**]{}, 1134 (1998) H. Shi, W.-M. Zheng, S.-T. Chui, Phys.Rev.A [**61**]{}, 063613 (2000)
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Scintillation properties of pure CsI crystals used in the shower calorimeter being built for precise determination of the $\pi^+$$\rightarrow$$\pi^0e^+\nu_e$ decay rate are reported. Seventy-four individual crystals, polished and wrapped in Teflon foil, were examined in a multiwire drift chamber system specially designed for transmission cosmic muon tomography. Critical elements of the apparatus and reconstruction algorithms enabling measurement of spatial detector optical nonuniformities are described. Results are compared with a Monte Carlo simulation of the light response of an ideal detector. The deduced optical nonuniformity contributions to the FWHM energy resolution of the PIBETA CsI calorimeter for the $\pi^+$$\rightarrow$$e^+\nu$ 69.8$\,$MeV positrons and the monoenergetic 70.8$\,$MeV photons were 2.7$\,$% and 3.7$\,$%, respectively. The upper limit of optical nonuniformity correction to the 69.8$\,$MeV positron low-energy tail between 5$\,$MeV and 55$\,$MeV was $+\,$0.2$\,$%, as opposed to the $+\,$0.3$\,$% tail contribution for the photon of the equivalent total energy. Imposing the 5 MeV calorimeter veto cut to suppress the electromagnetic losses, [GEANT]{}-evaluated positron and photon lineshape tail fractions summed over all above-threshold ADCs were found to be 2.36$\pm\,$0.05(stat)$\pm\,$0.20(sys)$\,$% and 4.68$\pm\,$0.07(stat)$\pm\,$0.20(sys)$\,$%, respectively.
[PACS Numbers: 87.59.F; 29.40.Mc; 24.10.Lx]{}
[*Keywords:*]{} Computed tomography; Scintillation detectors; Monte Carlo simulations
address:
- 'Department of Physics, University of Virginia, Charlottesville, VA 22901, USA'
- 'Institute Rudjer Bošković, Bijenička 46, HR-10000 Zagreb, Croatia'
- 'Physics Department, Hampton University, Hampton, VA 23668, USA'
- 'Paul Scherrer Institut, Villigen PSI, CH-5232, Switzerland'
- 'Institut für Teilchenphysik, Eidgenössische Technische Hochschule Zürich, CH-8093 Zürich, Switzerland'
- 'Deutsches Elektronen Synchrotron, D-22603 Hamburg, Germany'
- 'Department of Physics and Astronomy, Arizona State University, Tempe, AZ 85281, USA'
- 'Department of Physics, University of Massachusetts, Amherst, MA 01003, USA'
- 'Institute for High Energy Physics, Tbilisi State University, 380086 Tbilisi, Georgia'
- 'Jet Propulsion Laboratory, Pasadena, CA 91109, USA'
author:
- 'E. Frlež'
- 'I. Supek'
- 'K. A. Assamagan'
- 'Ch. Brönnimann'
- 'Th. Flügel'
- 'B. Krause'
- 'D. W. Lawrence'
- 'D. Mzavia'
- 'D. Počanić'
- 'D. Renker'
- 'S. Ritt'
- 'P. L. Slocum'
- 'N. Soić'
title: |
Cosmic muon tomography of pure\
cesium iodide calorimeter crystals
---
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Introduction
============
The PIBETA collaboration has proposed an experimental program [@Poc88] with the aim of making a precise determination of the $\pi^+$$\rightarrow$$\pi^0e^+\nu_e$ decay rate at the Paul Scherrer Institute (PSI). The proposed technique is designed to achieve an overall level of accuracy in the range of $\sim\,$0.5$\,$%, improving thus on the present branching ratio uncertainty of 4$\,$% [@McF85]. The (1.025$\pm\,$0.034)$\times$$10^{-8}$ pion beta decay ($\pi\beta$) branching ratio will be remeasured relative to the $10^{4}$ times more probable $\pi^+$$\rightarrow$$e^+\nu_e$ decay rate, that is known with the combined statistical and systematic uncertainty of $\sim\,$0.40$\,$% [@Cza93; @Bri94].
The Standard Model description of the $\pi\beta$ decay and its radiative corrections [@Sir78] enables a stringent test of the conserved vector current (CVC) hypothesis and the unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix. The most accurate extraction of the CKM matrix element $V_{ud}$, based on the superallowed Fermi transitions in nuclei, involves the theoretical nuclear overlap corrections that dominate the 0.3$\,$% total uncertainty [@Sir89]. Recent measurements of nine different nuclear transition rates confirm the CVC hypothesis at the level of 4$\times$$10^{-4}$, but violate the three-generation CKM unitarity condition by more than twice the estimated error [@Tow95b]. Complementary neutron decay experiments yield a result that also differs from unitarity by over two standard deviations, but in the opposite sense [@Tow95b]. It is of fundamental interest to measure the branching ratio of the $\pi^+$$\rightarrow$$\pi^0e^+\nu_e$ decay at the 0.5$\,$% level or better to check the consistency with the nuclear and neutron $\beta$ decay values and the unitary prediction of the Minimal Standard Model (MSM). Moreover, the precisely measured $\pi\beta$ branching ratio could be used to constrain masses and couplings of additional neutral gauge bosons in grand unified theories. If such couplings are directly discovered at future collider experiments, precision low-energy data will be an essential ingredient in extending the MSM framework [@Mar87].
The central part of the PIBETA detector system is a high-resolution, highly segmented fast shower calorimeter surrounding the active stopping target in a near-spherical geometry. The requirements imposed on the calorimeter design were:
- high energy resolution for effective suppression of background processes;
- high segmentation and fast timing response to handle the high event rates;
- acceptable radiation, mechanical and chemical resistance;
- compact geometry to simplify the operation and reduce the cost.
These design constraints are met in a spherical detector with individual calorimeter modules made from undoped cesium iodide (CsI) crystals. The nonactivated alkali iodides have been known for more than thirty years to exhibit a near-ultraviolet emission component when excited by ionizing radiation [@Mur65; @Bat77]. The pure CsI material was reintroduced as a fast, rugged, and relatively inexpensive scintillator material by Kobayashi et al. [@Kob87] and Kubota et at. [@Kub88a; @Kub88b]. The scintillation characteristics of pure CsI crystals have been reported subsequently by a number of experimental groups. These investigations have covered light readout techniques, scintillation decay times, origins of different emission components, crystal light yields, and energy and timing resolutions [@Kes91; @Woo90; @Sch90; @Utt90; @Gek90; @Ham95], radiation resistances [@Woo92; @Wei93], wrapping and tuning methods and the uniformity of light responses [@Bro95; @Dah96]. The design and performance of pure CsI electromagnetic calorimeters have been recently described in Refs. [@Mor89] (NMS at LAMPF/BNL), [@Ray94; @Kess96] (KTeV at Fermilab), and [@Dav94] (PHENIX at RHIC).
The PIBETA calorimeter geometry shown in Fig. \[fig:ball\] is obtained by the class II geodesic triangulation of an icosahedron [@Ken76]. Selected geodesic breakdown results in 220 truncated hexagonal, pentagonal and trapezial pyramids covering the total solid angle of 0.77$\times$$4\pi$ sr. Additional 20 crystals cover two detector hemispheres open to the beam and act as shower leakage vetoes. The inner radius of the crystal ball is 26$\,$cm, and the axial module length is 22$\,$cm, corresponding to 12 CsI radiation lengths ($X_0$=1.85$\,$cm [@PGP]). There are nine different module shapes: four irregular hexagonal truncated pyramids (we label them HEX–A, HEX–B, HEX–C, and HEX–D), one regular pentagonal (PENT) and two irregular half-hexagonal truncated pyramids (HEX–D1 and HEX–D2), and two trapezohedrons which function as vetoes (VET–1, VET–2). The volumes of our CsI crystal detector modules vary from 797$\,$cm$^3$ (HEX–D1/2) to 1718$\,$cm$^3$ (HEX–C). Dimensioned drawings of the HEX–A and HEX–D1 crystal shapes are shown as examples in two panels of Fig. \[fig:shape\].
All key components of the complete PIBETA detector have been prototyped and built and have met the required specifications [@Ass95]. All the detector components were delivered to PSI by mid-1998. The final assembly of the apparatus was followed by the in-beam commissioning and calibration. “Production” measurements are scheduled for 1999, and will extend over two more years to complete the first phase of the project, a $\sim\,$0.5$\,$% measurement of the $\pi^+$$\rightarrow$$\pi^0e^+\nu$ decay rate.
The quality of the delivered CsI calorimeter crystals was controlled in the following program of measurements:
1. check precisely the crystal physical dimensions against the specifications;
2. study the effects of different crystal surface treatments and wrapping configurations on the energy and timing resolution and on the uniformity of light response;
3. determine the percentage of the fast scintillation component in the light output;
4. measure the contribution of detector photoelectron statistics to energy resolution;
5. find the temperature dependence of a detector’s ADC readings;
6. determine axial and transverse optical nonuniformities for each crystal.
The ultimate goal of the program was to provide the input parameters for a Monte Carlo simulation of the calorimeter response and to deduce the effects of CsI crystal light yields and optical nonuniformities on the energy lineshapes of detected photons and positrons ranging in energy between 10$\,$MeV and 120$\,$MeV.
The spectra generated by cosmic muons have been used to examine the uniformity of light response of two large (24$\times$36$\,$cm) cylindrical NaI(Tl) calorimeter crystals in the earlier work of Dowell et al. [@Dow90]. Their apparatus relied on plastic scintillator hodoscopes with 3.8$\times$3.8$\,$cm$^2$ cross section for the charged particle tracking. The precision of the trajectory reconstruction was therefore limited to $\sim\,$4$\,$cm. Such positional resolution was deemed inadequate for our application because the PIBETA CsI crystals have an axial length of 22$\,$cm with front (back) surface side lengths of less than 4$\,$cm (7$\,$cm).
Most of our measurements were done with the tomography apparatus built around three pairs of multiwire drift chambers (MWDC) using cosmic muons as the probe. One dozen PIBETA CsI detectors were examined in a 350$\,$MeV/c minimum ionizing muon beam of the PSI $\pi$E1 area, as well as in the 405$\,$MeV/c penetrating pion and stopping proton beams in the $\pi$M1 channel. Selected CsI crystals were also scanned with a 662 keV $^{137}$Cs $\gamma$-source. The light yield nonuniformities measured in the beam and with a radioactive-source scan method were compared with more precise cosmic muon tomography data. The experimental apparatus used for a $\gamma$-source crystal scans and the subsequent data reduction were simple enough to be easily adopted by the crystal manufacturers, thus accelerating the subsequent quality control cycle.
Crystal Production, Mechanical Quality Control and Surface Treatment
====================================================================
Pure CsI crystals produced for the PIBETA calorimeter came from two different sources. Twenty-five crystals were grown and cut to specified shapes in the Bicron Corporation facility in Newbury, Ohio. The rest of the scintillators were grown in the Institute for Single Crystals in Kharkov (AMCRYS), Ukraine. Preliminary examination of fifty-five AMCRYS CsI crystals, including the mechanical and physical tests, was done in the High Energy Institute of Tbilisi University, Georgia.
Manufacturing tolerances of the crystals were specified for the linear dimensions ($+$150 $\mu$m/$-$50 $\mu$m) and for the angular deviations ($+$0.040$^\circ$/$-$0.013$^\circ$). Geometrical dimensions of the machined crystals were measured upon delivery at PSI using the computer-controlled distance-measuring device [*WENZEL Precision*]{}. The machine was programmed to automatically probe the surfaces of a subject crystal with a predefined shape. Each crystal surface plane was scanned with a touch head at six points and the equations of planes were found through these surveyed points. Body vertices, measured with an absolute precision of 2 $\mu$m and reproducible within 20 $\mu$m, were then compared with the expected theoretical values. Those crystals that failed the imposed geometrical tolerances were returned to the manufacturer for reuse as raw crystal growing material.
After physical measurements crystal surfaces were polished with a mixture of 0.2 $\mu$m aluminum oxide powder and etylenglycol. Next came a measurement of the light output of the fast CsI scintillation component (F) that is completely decaying in the first 100 ns, relative to the total CsI signal (T), integrated in a 1 $\mu$s ADC gate. These measurements were made with unwrapped crystals using air-gap coupled photomultiplier tubes (PMTs) and a Tektronix TDS 744 digital oscilloscope. Only the crystals with a fast-to-total component ratio (F/T) better than 0.7 were accepted. The mean value of fast-to-total ratio for all accepted CsI crystals was 0.788, as shown in Fig. \[fig:ft\].
EMI photomultiplier tubes 9821QKB [@EMI] with 75$\,$mm diameter cathode were glued to the back faces of hexagonal and pentagonal CsI crystals using a 300 $\mu$m layer of silicone Sylgard 184 elastomer (Dow Corning RTV silicon rubber plus catalyst). Smaller half-hexagonal and trapezial detector modules were equipped with two inch EMI 9211QKA phototubes [@EMI]. The resulting crystal–photomultiplier tube couplings were strong and permanent, but could be broken by application of a substantial tangential force. The PMT quartz window transparency, peaking at $\sim\,$380 nm [@EMI], is approximately matched to the spectral excitation of a pure CsI fast scintillation light component with a maximum room temperature emission at $\sim\,$310 nm [@Woo90]. The PMT high voltage dividers were modified EMI-recommended bases designed and built at the University of Virginia. These dividers minimized the so-called “super”-linearity exhibited by many PMTs well below the onset of saturation. The maximum PMT nonlinearity measured with a pair of light-emitting diodes was less than 2$\,$% over the full dynamic range expected in the $\pi\beta$ decay rate measurement [@Col95].
Four different wrapping materials for lateral crystal surfaces were investigated in wrapping and tuning studies with 405$\,$MeV/c pion and proton beams in PSI $\mu$M1 area:
1. one to five layers of a 38$\,$$\mu$m PTFE Teflon sheet (CF$_2$ monomer, $X_0$=16.0$\,$cm);
2. one to five layers of 30$\,$$\mu$m Mylar sheet (C$_5$H$_4$O$_2$, $X_0$=28.7$\,$cm);
3. a 250 $\mu$m thick Tyvek fleece (polyethylene CH$_2$, $X_0$=131.3$\,$cm);
4. a 110$\,$$\mu$m thick Millipore filter (polyvinylidene fluoride PVDP, $X_0$=19.0$\,$cm) with 0.22$\,$$\mu$m pores;
5. a wavelength shifter lacquer treatment of crystal surfaces.
In all cases a primary diffuse reflector was protected by an additional 20 $\mu$m thick aluminized Mylar cover. These preliminary tests showed more uniform axial light collection from the crystal front section when a polished front crystal surface was covered with a black paper sheet. A similar tuning method, using black paper strips at the front section of 5$\times$6$\times$36$\,$cm$^3$ CsI(Tl) crystals in order to improve the uniformity of light collection, was recently described in Ref. [@Dah96].
The maximum fast-to-total ratio was always achieved with unwrapped crystals because all applied wrapping materials absorb more ultraviolet light which dominates the fast component, than visible light. A two-layered Teflon cover was found to be superior in delivering $\ge$20$\,$% more fast scintillation light component than the other reflectors. It was also comparatively better in not degrading the F/T ratio by more than $\sim\,$2$\,$%. These differences in measured light output were reproducible in the beam tests with $\sim\,$1$\,$% event statistics and estimated 2$\,$% systematic uncertainty. They are subsequently confirmed in more controlled cosmic muon tomography measurements.
On the basis of these findings the adopted standard wrapping configuration for all studied CsI crystals consisted of: (a) the lateral surfaces being wrapped with two layers of Teflon foil plus one layer of aluminized Mylar sheet, and (b) the front crystal surface covered with the black paper template. The back crystal surface with the glued phototube was left uncovered.
Light yield measurements were repeated for a subset of crystals after a six month period to confirm that no appreciable changes occurred as a result of degradation of the surface reflectivities and wrapper material qualities. The measured light output was typically within 5$\,$% of the originally measured value.
The most successful surface treatment of CsI crystals, however, involved painting the lateral crystal surfaces with a waveshift lacquer. We recently studied this method on a large sample of CsI detectors. The light response and uniformity properties were noticeably improved, resulting in $\sim\,$20$\,$% better energy resolutions for the 70$\,$MeV monoenergetic $e^\pm$ and 50–82$\,$MeV tagged photons. The possible degradation and change in the crystal surface reflectivities that could be difficult to account for in the multi-year long precision experiment like the PIBETA is also expected to be arrested by such a treatment. These measurements will be covered in detail in a forthcoming paper [@Frl98].
Tomographic Apparatus
=====================
A simplified sketch of the tomographic apparatus layout is illustrated in a [GEANT]{} rendering showing a few simulated cosmic muon trajectories on Fig. \[fig:drift\]. Three identical delay-line readout drift chambers were used to define the cosmic muon tracks intersecting CsI crystals. The chambers were built at the Los Alamos Meson Physics Facility (LAMPF) by the M. Sadler group from the Abilene Christian University. This type of drift chambers has been used reliably for years with several LAMPF spectrometers [@Ate81; @Mor82; @Ran81; @Ran82]. Each chamber consists of two signal and three ground planes with the nominal active area of 60$\times$60$\,$cm$^2$. The horizontally oriented “$x$” and “$y$” signal planes are two orthogonal sets of alternating cathode and anode wires evenly spaced at 0.4064$\,$cm.
The light-tight aluminum box had two drift chamber pairs mounted on its top plate, and one pair fixed below the bottom side. Distances between the centers of chamber pairs 1–3 and 2–3 were 24$\,$cm and 27$\,$cm, respectively. The dark box could accommodate up to six CsI calorimeter modules at one time. It had feedthrough connectors for twelve signal and twelve high voltage cables as well as six temperature sensor lines and six LED calibration signal cables. A pair of 1$\,$cm thick plastic scintillators were placed directly below the apparatus and separated from the chamber system by a 5$\,$cm layer of lead bricks shielding out the soft cosmic ray component. The Monte Carlo simulation (see Sec. \[mcd\])showed that hard cosmic muons penetrating the frames of the apparatus, CsI crystals and shielding material and triggering two scintillators had a smooth energy threshold starting at $\sim\,$120$\,$MeV. The zenithal angle range of accepted cosmic muon trajectories intersecting all three MWDCs and at least one CsI detector was $\pm\,$45$^\circ$.
The chambers were operated with anode wires held at positive voltage between 2400–2600 V. The chamber cathode wires were grounded. The gas mixture was 65$\,$% argon, 35$\,$% isobutane plus 0.1$\,$% isopropyl alcohol. Detection efficiencies of individual chambers for penetrating cosmic muons exceeded 90$\,$%, with the combined six-chamber efficiency routinely surpassing 50$\,$%.
Anode wires were attached directly to a fast 2.5 ns/cm delay-line. Signals from both ends of the delay line were amplified twenty-fold, discriminated at a threshold of 10 mV and connected to two channels of a time-to-digital converter (LCR 2229A TDC). The time difference in the two TDC readings identified the fired anode wire. Cathode lines defining the alternate field [@Wal78; @Ers82] were connected to one “odd” (O) and one “even” (E) line, and the signals on these lines were processed by a custom-made electronics unit which added and subtracted the analog pulses [@Bro79]. The electronic sum of cathode pulses (O$+$E) was discriminated and the resulting delayed signal determined the drift position timing, and was used to stop another TDC channel. The difference of the cathode signals (O$-$E) was digitized with an analog-to-digital converter (LCR 2249A ADC). That information was used to discriminate between the events that produced the ionization tracks left and right of the given anode wire. The 100 ns integration gate timing was defined by the (O+E) logic pulse timing. Four measurements were therefore required to find an intersection between a cosmic muon track and one drift chamber plane, namely three TDC values and one ADC value. The schematic diagram of the electronic logic is shown in Fig. \[fig:tmelec1\].
Advantages of the system were good charged particle track resolution ($\sim\,$0.5$\,$mm root-mean-square in the horizontal plane, Fig. \[fig:deviation\]), stability to fluctuations in the outside environmental parameters (humidity, temperature and pressure) and low cost of the associated electronics logic and readout. Limited counting rate and restriction to single hit events in the chambers did not represent a drawback in this tomographic application. The apparatus was operated in an air-conditioned room with a controlled humidity level kept below 30$\,$%. The temperature at six points inside the dark box as well as absolute time were recorded for every triggered event. The temperature range recorded inside the dark box during three years of data taking was (22$^\circ$$\pm\,$3$^\circ\,$)C. The typical temperature gradient was 0.4$^\circ\,$C/day, leading to average absolute temperature variation of 0.2$^\circ\,$C and average CsI ADC gain drift of $\sim\,$0.4$\,$% in a single data taking run (fixed at 250000 triggers, $\sim\,$6 hours). Measured light output variations of the CsI crystals and the PMT gain drifts caused by daily temperature cycles were compensated for in the replay analysis (see Sec. \[raw\]).
Data Acquisition System
=======================
The computer code used for data acquisition was HIX (Heterogeneous Information Exchange), a data acquisition system developed by S. Ritt at PSI, originally intended for use in small and medium-size nuclear physics experiments [@Rit95].
The “frontend” 486 personal computer was connected to a CAMAC crate via a HYTEC 1331 interface that read ADC, TDC, scalar, and temperature sensor units. The C program running under MS-DOS accessed data and sent it over the network to a VAX 3100 server. A simple communication program on a VAXstation received data from the frontend computer and stored it in a global section buffer. Buffered data were passed by the Logger application to a user-written Analyzer program.Analyzed events were subject to predefined cuts, filled in predefined histograms, and stored in a raw data stream directed simultaneously to a hard disk drive and an 8-mm tape system. The experiment was controlled from a PC computer by a Microsoft Windows control program. A windows-based graphical user interface to the program allowed starting, pausing and stopping data acquisition, as well as online inspection of individual raw data words, calculated data words, scalers, assorted efficiencies, one and two-dimensional histograms, and gate, box and Boolean tests. Different PC computers on the Ethernet could make a connection to the VAX server called Link and access the same information remotely.
During data acquisition only those events for which at least two $x$ and two $y$ drift chambers had nonzero ADC and TDC values were written to an disk-resident ASCII Data Summary File (DSF). One good event contained 55 raw data words specifying the response of the MWDCs and CsI detectors, instantaneous temperatures and absolute times. That was a full set of observables for which a cosmic muon trajectory could be unambiguously reconstructed. The raw trigger rate was $\sim\,$13 Hz with six CsI crystals in the dark box. The average DSF event rate was $\sim\,$4 Hz. Individual runs were stopped and restarted automatically after 250,000 collected triggers. A total of about 10$^5$ DSF events per crystal were typically collected in one week of data acquisition. Tomography data for one set of six crystals were usually collected over a two week period.
Drift Chamber Calibration
=========================
Two different trigger configurations were used interchangeably during the data collection. Drift chamber calibration was done with a trigger requiring a two-scintillator coincidence and good ADC and TDC data for at least one drift chamber. In the tomography data acquisition mode, the triple coincidence between two tag scintillators and one CsI detector was required, accompanied by at least one good pair of $x$ and $y$ chamber hits. The rare accidental coincidences involving signals in more than one CsI detector were eliminated from the DSF records in the offline analysis.
The hit anode wire number $n_A$ was calculated from $$n_A={{(T_1-T_2+N\cdot D)}\over {2D}},$$ where $T_1$ and $T_2$ were the TDC values for two ends of the anode delay line, $N$ was the number of wires in a chamber (between 71 and 73), and $D$ was the $\sim\,$2.05 ns time delay between the adjacent wires. Truncated wire position $x_T$ was defined as the nearest integer multiple $n_A$ of the 0.8128$\,$cm wire separation $g$: $$x_T={\tt NINT}(n_A\cdot g).$$
The anode delay time difference $(T_1-T_2)$ depended nonlinearly on the anode position. Linearized anode wire positions for each chamber were approximated by a polynomial expansion $$x_A=\sum_{i=0}^m c_i\cdot (T_1-T_2)^i,$$ where the order $m$ of the fit with $\chi^2$ per degree of freedom $\chi^2/(m-1)$$\approx\,$1 depended on the particular delay line. For our chambers the orders of the fits were 3, 4, or 5. The coefficients $c_i$ were optimized with the program [MINUIT]{} [@Jam89] that minimized the sum of squared deviations $(x_T-x_A)^2$ for each chamber in every run.
The drift distance $d_x$ was determined from the drift timing $T_x$ using the calibration lookup table $f(T_x)$ $$d_x=f(T_x)\cdot T_x+\Delta T_{x},$$ where $\Delta T_{x}$ was the drift time offset. The lookup table was calculated assuming that incoming cosmic muons are distributed uniformly over the equidistantly spaced wires.
The final hit position $x$ was given by expression: $$x=x_T+(-1)^n\cdot sign({\rm ADC_{O-E}+ADC_0})\cdot d_x+x_{\rm OFF},$$ where $x_{\rm OFF}$ was the chamber $x$ coordinate offset, and ${\rm ADC_{O-E}}$ and ${\rm ADC_{0}}$ were the (O$-$E) ADC signal and associated offset, respectively.
The individual chamber phases $n$ in the Eq. (5) and absolute horizontal coordinate offsets $x_{\rm OFF}$ and $y_{\rm OFF}$ between six chambers were adjusted with help of two-dimensional histograms $(x_1-x_3)$ vs $(x_2-x_3)$ and $(y_1-y_3)$ vs $(y_2-y_3)$ [@McN87; @Sup89].
Raw Data Reduction {#raw}
==================
Data summary files were analyzed offline by compressing the DSF event data into [PAW]{} Ntuples [@Bru93]. Drift time-to-distance lookup tables and assorted chamber calibration constants, as well as CsI detector ADC pedestals and ADC temperature corrections, were determined for each run separately. The high voltages of the CsI detector phototubes were selected to give $\sim\,$8 ADC channels/MeV scale. The FWHMs of the pedestal peaks were typically 2 ADC channels. Therefore, the pedestal widths corresponded to an energy deposition of 0.25$\,$MeV, 24 $\mu$m pathlength or 0.5$\,$% of the most probable energy deposition by a minimum ionizing charged particle. Drifts in the pedestal position over a one month period amounted to less than 2 ADC channels.
The precise horizontal coordinate offsets of six CsI crystals inside the dark box were determined in the next stage of analysis. Using the preliminary offsets read from the plastic template on which the crystals were laying, between 5 and 10 percent of the reconstructed cosmic muon trajectories were found not to intersect any of the predefined detector volumes. After adjusting the $x$ and $y$ translation software offset parameters of the CsI modules by maximizing the number of tracks intersecting individual crystal volumes with nonzero ADCs, the real number of events undergoing scattering in the apparatus or having the improperly reconstructed tracks was shown to be below 1$\,$%. That percentage was in agreement with the Monte Carlo simulation of cosmic muons interacting with the experimental apparatus, showing the most probable muon scattering angle of 0.25$^\circ$ and the 2.5$^\circ$ mean scattering angle. The Monte Carlo root-mean-square of the scattering angle for the accepted events was 0.66$^\circ$, Fig. \[fig:ths\]. That value translates into an average pathlength uncertainty of 1$\,$mm. The estimated error in finding the correct crystal position inside the dark box was smaller than $\sim\,$0.5$\,$mm, Fig. \[fig:position22\].
The energy deposited in CsI crystal by cosmic muons along the fixed pathlength is a broadened distribution due to the statistical nature of energy transfers. Maximal measured pathlengths were up to 12$\,$cm, while the average pathlength was $\sim\,$6$\,$cm. A Gaussian model for the energy loss distribution of the minimum ionizing particles in CsI thicknesses of less than 12$\,$cm is not a good approximation. The distribution of energy losses is asymmetric, with a long high energy tail; the most probable energy loss is smaller than mean energy loss [@Leo87].
Light yield temperature coefficients for the individual CsI detectors were determined by the “robust” estimation [@Hub81] of the ADC values per unit pathlength as a linear function of temperature recorded inside the dark box. The least-squares condition assumes normally distributed measurements with constant standard deviations and is therefore not appropriate in this application, as pointed out above. Typically, more than $10^5$ cosmic muon events, collected over at least one week and spanning the temperature range of $\sim\,$2$^\circ$C, were used as an input experimental data set. The [FORTRAN]{} subroutine [MEDFIT]{}, documented in the Ref. [@Pre86], was adopted as the fitting procedure by imposing the requirement of minimum absolute deviation between the measured and calculated ADC values per unit path that were dependent linearly on the temperature variable. The average light output temperature coefficient for 74 different CsI detectors extracted by that method was $-1.4\pm\,1.4$$\,$%/$^\circ$C, both for fast and total scintillation light components. These numbers are in good agreement with the previously reported value of $-$1.5$\,$%/$^\circ$C [@Kob87], but the spread of extracted coefficients for different CsI detectors was large: $\pm\,$4$\,$%/$^\circ$C (Fig. \[fig:tf\]). We point out that the method did not allow us to separate the temperature dependence of the crystal light outputs from the temperature instabilities of the phototubes and high voltage dividers as well as the temperature instabilities of the ADC modules themselves. The LeCroy catalogue specification [@LCR95] lists the typical temperature coefficient of a LCR 2249A ADC unit as zero, and the maximum coefficient up to $\pm\,$3$\,$% for an ADC gate longer than 100 ns and an average ADC reading of 75 pC (about a quarter of a full 256 pC scale).
The raw ADC values were corrected for the temporal temperature variations and written into the revised DSF files used in subsequent analysis.
Detector Photoelectron Statistics
=================================
The contribution of the photoelectron statistics to the energy resolution of CsI crystals was determined by a photodiode-based system. Six individual detectors (CsI crystals with photomultiplier tubes and high voltage dividers) having identical light emitting diodes (LEDs) coupled to their back sides were placed inside the cosmic muon tomography apparatus. The LEDs were pulsed at a 10 Hz rate using a multichannel diode driver with continuously adjustable output voltage. One split output signal from each channel of the driver generated a 100 ns wide ADC gate. The LED light in a CsI detector produced fast ($\sim\,$20 ns FWHM) PMT anode pulses whose integrated values depended on the driving voltage and were equivalent to the fixed cosmic muon energy depositions between 10$\,$MeV and 100$\,$MeV.
A total of five different LED amplitudes were used in measurements of each CsI detector. Examples of the LED spectra are shown in Fig. \[fig:set2\]. The ADC pedestals were recorded simultaneously during the run. The intensity of LED light was cross-calibrated against the cosmic muon spectra peaks in CsI crystals. These muons were tracked in tomography drift chambers and both their pathlengths and energy depositions in the crystals could be easily calculated. That calculation enabled the establishment of the absolute energy scale in MeV.
The variance $\sigma_E^2$ of a photodiode peak depended upon the mean number of photoelectrons $\bar N_{pe}$ on the photocathode created per unit energy deposition in the crystal: $$\sigma_E^2=\sum_i\sigma_i^2+\bar E/\bar N_{pe},
\label{eq:led}$$ where $\bar E$ was the LED spectrum peak position and $\sigma_i^2$’s were assorted variances, such as instabilities of the LED driving voltage and temporal pedestal variations. Five measured points ($\sigma_E^2$,$\bar E$) were fitted with a linear function (\[eq:led\]) and the mean number of photoelectrons per MeV $\bar N_{\rm pe}$ for each CsI detector was determined. The statistical error of the least-squares fits was typically less than one photoelecton/MeV. Using the previously extracted light output temperature coefficients all $\bar N_{\rm pe}$ values were scaled to the 18$^\circ$C point. That value was the designed operating temperature of the PIBETA calorimeter.
The number of extracted photoelectrons per MeV for the fast scintillation component fell into the 20–130 range as shown in Fig. \[fig:npf\]. The hexagonal and pentagonal crystals equipped with three inch phototubes averaged 73 photoelectrons/MeV, while the half-hexagonal and veto detectors with two inch phototubes had a mean of 33 photoelectrons/MeV. The 73/33 ratio is very well explained by the (3/2)$^2$ ratio of photosensitive areas for two different photocathode sizes. The measured photoelectron statistics were in agreement with the 100 ns ADC gate values of $\bar N_{\rm pe}$=(20–260) for large ($\sim\,$10$\,$cm) pure CsI crystals equipped with two or three inch PMTs reported in the past in Refs. [@Woo90; @Sch90; @Utt90; @Mor89].
The quantum efficiency of our bialkali photocathodes for pure CsI scintillation light was 23$\,$% [@EMI], the average light collection probability for our detector shapes 23$\,$% (Sec. \[tks\]) and the fraction of deposited energy converted into the scintillation light was about 12$\,$%. Therefore, starting from the mean number of 73 photoelectrons/MeV, we calculated that 1$\,$MeV energy deposited in the CsI crystal produced on average about 10$^4$ scintillation photons.
Monte Carlo Description
=======================
Tomography System {#mcd}
-----------------
Geometrical layout of the experimental apparatus was defined using the [GEANT]{} detector description and simulation tools [@Bru94]. The active elements of the simulated apparatus were an aluminum box housing six CsI modules, six multiwire drift chambers, two scintillator planes and a layer of lead brick shielding, all shown in Fig. \[fig:drift\]. Generated events were muons with the energy, angular and charge distribution of a hard cosmic ray component at sea level. The zenithal angle $\theta_z$ distribution of muons at the ground was assumed to be proportional to $\cos^2\theta_z$; the momentum spectrum between 0.1 and 10$^3$ GeV/c and the energy-dependent ratio of number of positive to number of negative cosmic muons was parameterized from the data given in Refs. [@PGP; @Ros48].
The simulation trigger was defined by requiring a minimum energy deposition of more than 0.2$\,$MeV in each scintillator plane. This threshold was about one-tenth of the minimum-ionizing peak in the triggered scintillator and corresponded to the discriminator level used in the data acquisition electronics.
Penetrating cosmic muons and generated secondary particles were tracked through the apparatus, and energy depositions, pathlengths in CsI crystals and hits in the drift chambers were digitized. All physical processes were turned on in the [GEANT]{} programs with the cutoff energies of 0.2$\,$MeV for charged particles and photons. Inspection of simulated energy deposition spectra showed deviations from theoretical Vavilov distributions [@Leo87] expected in the planar detector geometries. The differences were caused by multiple scattering in the irregularly shaped crystals, and were particularly prominent for the events with shorter pathlengths close to the crystal edges. The simulated ADC histograms revealed that triggering cosmic muons deposit on average 10.33$\,$MeV/cm in a CsI detector, with the most probable energy loss 5.92$\,$MeV/cm.
Cosmic muon spectra in CsI detectors were described in a satisfactory way ($\chi^2/(m-1)$$\approx\,$1.2) by the combination of a Gaussian distribution and a falling exponential tail: $${A}(\epsilon)=\theta(\epsilon_0-\epsilon)e^{ -{1\over 2}[{{(
\epsilon-\bar\epsilon})/{\sigma_\epsilon}}]^2 }+\theta(\epsilon-
\epsilon_0)e^{\alpha-\beta\epsilon},$$ where $\epsilon$ is energy (in MeV) deposited in one full CsI module. Parameters $\bar\epsilon$, $\sigma_\epsilon$, $\epsilon_0$, $\alpha$, and $\beta$ are extracted from the [GEANT]{} spectra by imposing the least squares constraint to the fit and leaving the pathlength $d$ (cm) as free parameter: $$\begin{aligned}
\bar\epsilon(d)&=&5.079+0.1876d-9.9390\cdot 10^{-3}d^2, \\
\sigma_\epsilon(d)&=&0.3545+7.2329\cdot 10^{-3}d-3.4148\cdot 10^{-4}d^2,
\\
\epsilon_0(d)&=&5.1800+0.1989d-1.0297\cdot 10^{-2}d^2, \\
\alpha(d)&=&4.0581+0.4465d-1.5760\cdot 10^{-2}d^2, \\
\beta(d)&=&0.7952+5.0313\cdot 10^{-1}d-1.5240\cdot 10^{-3}d^2, \
\label{eq:mc}\end{aligned}$$
and
$$\begin{aligned}
\theta(\epsilon_0-\epsilon)=\cases{ 0,& if $\epsilon_0<0$,\cr
1,& otherwise.\cr }\end{aligned}$$
These Monte Carlo spectra, broadened with photoelectron statistics, were used to describe our cosmic muon lineshapes produced by the optically uniform CsI detectors.
Light Collection Simulation: [TkOptics]{} Code {#tks}
----------------------------------------------
Propagation of scintillating photons through a uniform detector with ideal reflecting dielectric surfaces and different wrapping materials was studied using the [TkOptics]{} simulation program [@Wri92; @Wri94]. The code is a library of [FORTRAN]{} subroutines with an X-Windows-based user interface written with [Tck/Tk]{} toolkit [@Ous94]. The program can simulate the light output response of an arbitrarily shaped scintillation detector with given bulk and surface optical properties.
The polygonal detector shape is defined by the coordinates of its vertices. The detector attributes are reflector types of lateral and front crystal surfaces and wrapper material, crystal surface-wrapper gap distances and characteristics of the photomultiplier tube and a phototube-crystal joint coupling. The program handles normal dielectric, specular, diffuse and partially absorbent reflector surfaces with arbitrary diffuse fraction, roughness, and specular $r_s$ and diffuse reflectivity $r_d$. Predefined bulk properties of a detector are the medium scattering and attenuation length as well as refractive index. A photomultiplier tube is specified by the diameter, position and quantum efficiency of the photocathode.
Different choices of initial scintillating photon distribution are possible, the default being a uniform starting distribution throughout the scintillating volume. A working volume is a box divided into elementary cubic cells of fixed size. Output menu options include the initial and endpoint photon coordinates, and direction vectors and timing distributions of the detected scintillation photons organized in a [PAW]{} Ntuple. Every elementary cell is flagged as a bulk or edge crystal cell. Center coordinates of the cells and a fraction of photons generated in every cell, and subsequently detected on the PMT sensitive surface, are always recorded. Results of high statistics runs with $10^7$ photons generated uniformly through the detector volume were plotted to show the number of photoelectrons as a function of scintillation source position inside the crystal. The relative statistical errors of calculated light collection probabilities for the bulk crystal cells were $\le\,$2$\,$%.
The bulk attenuation length and the light scattering length of the near-ultraviolet emission component inside the CsI crystals were set to 150$\,$cm and 200$\,$cm, respectively [@Frl89]. The index of refraction for the CsI medium increases from 1.82 for the blue-green light to about 2.08 for ultraviolet light [@Frl89].
The simulated light collection probabilities $P(x,z)$ that depend on axial $z$ and transverse $x$ positions are shown integrated in the vertical coordinate ($y$) in Figs. \[fig:hexa\_9\] and \[fig:hh1d\_9\]. in the form of “lego” plots for one wrapping configuration and two different crystal shapes. The coordinate system is defined with the $z$=0$\,$cm origin at the front face of the crystal and a photocathode window at the $z$=22$\,$cm plane.
We find that simulated light response of the ideal hexagonal or pentagonal PIBETA CsI detector with specular lateral surfaces and a diffuse wrapping material can be described with the following parameters:
1. The average photon collection probability $P$ for a three inch photocathode is about 23$\,$% for ideal $r_s$$=$1.0 crystal surfaces and a $r_d$$=$0.9 diffuse wrapper, decreasing by half, to 12$\,$% for $r_s$$=0.8$.
2. The axial light collection probability variation through the first 10 centimeters is in the $\pm\,5\,$% range, with a positive slope $dP/dz\ge 0$ for higher specular and diffuse reflectivities, namely for $r_{s,d}$$\ge$0.9.
3. The axial detected light variation in the back crystal half ($z$$\ge$10$\,$cm) is up to $-30\,$%.
4. The transverse light response referenced to the light yield at the crystal axis typically increases towards the crystal surfaces by up to $+\,$5$\,$% for $z$$\le$10$\,$cm, but is generally declining away from the central axis by $-\,$30$\,$% at the $z$=18$\,$cm plane.
5. The root-mean-square of a three-dimensional light nonuniformity is between 2.5$\,$% and 3.5$\,$% for $z$$\le$10$\,$cm, compared to a $\sim\,$20$\,$% root-mean-square for a $z$$\ge10$$\,$cm volume, where a large spread is caused by the inefficient light collection from crystal back corners.
For lateral surfaces painted with a $r_d$$\approx\,$0.9 diffuse substance without an air gap we find that the average photon collection probability is lower, $\sim\,$17$\,$%, and the axial light collection nonuniformity as a function of axial position is always positive and is usually larger than $+\,$10$\,$% in a 22$\,$cm long detector, making the root-mean-square of the 3D nonuniformity function $\ge\,$20$\,$%.
The Monte Carlo results for the optically uniform half-hexagonal and trapezial CsI crystals with the same range of optical properties predict smaller light yields and somewhat higher light collection nonuniformities:
1. The average photon collection probability with a two inch photocathode is about 12$\,$%.
2. The typical axial nonuniformity is positive in front, $+\,$5$\,$%/10 cm, and negative in the back crystal half, with a $-30\,$%/10$\,$cm variation,
3. The simulated transverse light output from the front crystal half is very similar to one for the full crystal shapes, increasing up to $+\,$3$\,$% away from the detector central axis, but decreasing by almost a third in the back corners of the crystal.
4. The root-mean-square of detected light output varying between 3.0$\,$% and 4.5$\,$%, in the front ($z$$\le$10$\,$cm) and back ($z$$\ge$10$\,$cm) crystal halves, respectively.
These light collection probability distributions calculated for optically homogeneous crystals explain the major features of the measured optical nonuniformities presented below.
Measured Nonuniformities of CsI Detector Light Responses
========================================================
The three-dimensional (3D) spatial distribution of the light output of a scintillation detector can be specified by giving the number of photoelectrons $N_{\rm pe}(x,y,z)$ produced by 1$\,$MeV energy deposition at point ($x,y,z$) (“3D light nonuniformity function”). In the following discussion we limit ourselves to the linear one-dimensional variations of the detected light output separable in the axial and transverse directions: $$\begin{aligned}
N_{\rm pe}(z,x)\propto\cases{
N_1+a_{z1}\cdot z+a_t(z)\cdot x,& $z\le10\ {\rm cm}$, and $x=\pm15\
{\rm cm}$ \cr
N_2+a_{z2}\cdot z+a_t(z)\cdot x,& $z\ge10\ {\rm cm}$, and $x=\pm15\
{\rm cm}$\cr
}\end{aligned}$$ where $a_z$ ($a_t$) is the linear optical nonuniformity coefficient in the $z$ ($x$) coordinate, and the coordinate system origin is at the center of the detector front face, as explained in Sec. \[tks\].
A simple and straightforward parameterization of the detector light response nonuniformity can be made on the basis of scatter plots showing the light output per unit pathlength as a function of longitudinal or transverse cosmic muon-CsI crystal intersection coordinates. In this analysis we take only cosmic muon events with almost perpendicular trajectories ($\theta_z$$\ge$85$^\circ$), so that the measured two-dimensional light outputs are averaged over the pathlengths and over $\sim\,1\,$cm$^2$ vertical cross sections. The cosmic muons deposit the energy along the well-defined ionization track of the length $d$ in the crystal. This description will somewhat underestimate the real light collection probabilities (see Sec. \[137s\]) due to the integration of a three-dimensional light nonuniformity function along the charged particle track. Our work on the fully 3D reconstruction of the scintillator light response will be presented in a forthcoming publication [@Frl96].
Fig. \[fig:expected6\] shows the axial variation of normalized ADC values as a function of distance of energy deposition from the front crystal face for the six representative crystals. The axial positions were calculated by averaging two $z$ values of the cosmic muon track intersection with the crystal surfaces.
Panels on Figs. \[fig:expected7a\] and \[fig:expected7c\] show the transverse dependence of measured light output per unit pathlength, where the independent variable is the distance from the crystal axis measured in the horizontal plane. The selected scatter plots show the data points for the axial slices at $z_0$=6$\pm$1$\,$cm and $z_0$=18$\pm$1$\,$cm.
We have chosen to parameterize the axial light output nonuniformity with two piecewise linear functions: the light output nonuniformity coefficient $a_{z1}$($z$$\le$10$\,$cm) for the front half of the crystal, and the light nonuniformity coefficient $a_{z2}$($z$$\ge$10$\,$cm) for the back half of the crystal, both values expressed in %/cm. The transverse light output variation at a fixed axial distance $z_0$, $a_t(z_0)$, is described by the average value of a change in luminosity left and right of the crystal axis.
Several general features are readily noticeable: the spread of measured points due to energy deposition straggling, the gradual variation in the collected light along and perpendicular to the detector axis, and the decrease in detected light when the photon generation occurs close to lateral detector surfaces and, in particular, near the back corners of the detector volumes.
The calculated ADC/pathlength data points shown on the panels of Figs. \[fig:expected6\]–\[fig:expected7c\] have been fitted with linear functions imposing the “robust” condition of the minimum absolute deviation between the measured and calculated values, Eqs. 14. The normalized light output scatter plots in the axial and transverse coordinates, as well as two-dimensional ADC/$d$ distributions in the $x$-$z$ bins and associated linear light output nonuniformity coefficients have been documented for all studied CsI crystals and are available for inspection at the PIBETA WWW site [@pb]. The average scintillation properties of all the measured CsI crystals are listed in Tables \[tab1\] and \[tab2\].
In summary, for a set of fifty-nine full hexagonal and pentagonal crystals we find:
1. The average axial light nonuniformities are $a_{z1}$($z\le10$$\,$cm)=$-$0.1$\,$%/cm and $a_{z2}$($z\ge10$$\,$cm)=$-$1.3$\,$%/cm, respectively (see Fig. \[fig:a1a2\]).
2. Distribution of nonuniformity coefficients could be described by a Gaussian with $\sigma_{a_{z1}}$$\approx\,$1.3$\,$%/cm.
3. The average transverse light output is flat for $z$$\le$10$\,$cm, and the typical light variation is $-15\,$% at $z$=18$\,$cm.
4. The Kharkov-grown crystals on average have twice the axial optical nonuniformity of the Bicron-grown crystals.
For fifteen half-hexagonal and trapezial crystal shapes we find:
1. The average axial nonuniformities are $a_{z1}$($z$$\le$10 cm)=$-\,$0.3$\,$%/cm and $a_{z2}$($z$$\ge$10$\,$cm)=$-$1.0$\,$%/cm.
2. The transverse light output variation is limited in the front crystal half, but increases up to $-\,$30$\,$% in the back of the crystal.
With the insights gained in the simulation of the light collection probabilities (Sec. \[tks\]) we conclude that the $\pm\,$2$\,$%/cm spread of the nonuniformity coefficients corresponds to an equivalent range of 0.8–1.0 in crystal surface reflectivities, and/or 100–250$\,$cm range in CsI attenuation lengths.
Light output uniformity scans of CsI crystals with $^{137}$Cs gamma source {#137s}
==========================================================================
A light-tight plywood box was made to house a single CsI detector and associated measurement apparatus described below. A 0.662$\,$MeV $^{137}$Cs gamma source was placed on a moveable support next to the detector and collimated by a 20$\times$10$\times$5$\,$cm$^3$ lead brick with a 6$\,$mm collimator hole. The photomultiplier analog signal from the detector was processed with an ORTEC 454 timing amplifier with a gain of 30 and a 50 ns integration time constant. The amplifier output was digitized with a peak sensing ORTEC AD811 ADC unit. The same signal was discriminated and produced the trigger rate of about 5 kHz with a $^{137}$Cs source present. The background rate, without the source present, was 50–100 Hz.
Pedestal runs were taken with a clock trigger and used to properly correct offsets of the ADC spectra. Temperature variation during one run was less than 0.3$^\circ$C, typically causing the overall light output variation of less than 0.5$\,$% in a ADC spectrum that was gated with a 100 ns window. Several runs were taken with the source removed to find the shape of the background spectrum. It was confirmed that the background spectrum does not depend on the position of the lead collimator, so the same background lineshape was used in the analysis for all source positions.
The collimated $^{137}$Cs source was placed by remote control in turn at five points along the axis of each crystal at 2, 6, 10, 15 and 20$\,$cm from the front face of the crystal.
Following background subtraction, recorded spectra were fitted with a sum of a Gaussian and exponential function, Fig. \[fig:cs137\]. The peak position and the FWHM for each spectrum were extracted with the statistical uncertainty lower than 0.3$\,$%.
The dependence of the peak position on the placement of the source is illustrated in Fig. \[fig:cs137f\] and follows the trend of the tomography results. The [GEANT]{} simulations revealed that the 662 keV gamma rays could probe all the volume of CsI crystal, but the shower energy deposited at the central axis is only about one-sixth of the energy deposition near the crystal surface. The cosmic muons transfer the energy uniformly along their tracks in CsI material: the scintillation volume over which the measured light output is integrated is therefore larger, and inherent averaging leads to smaller extracted optical nonuniformity coefficients. This feature is borne out in the panels of Fig. \[fig:cs137\] where the tomography data for three different crystals yield consistently smaller axial light nonuniformities when compared with the radioactive source measurements.
The simple $^{137}$Cs scans with the described apparatus were used to evaluate the light collection nonuniformities for up to six crystals per day.
Radioactive source scan measurements similar to ours are described in Refs. [@Bro95; @Dow90].
A [GEANT]{} simulation of the PIBETA calorimeter response to 10–120$\,$MeV positrons and photons
================================================================================================
We have studied the simulated PIBETA calorimeter response to the 69.8$\,$MeV $\pi^+$$\rightarrow$$e^+\nu$ positrons and to the (69.8+2$m_{e^+}$) MeV photons, where $m_{e^+}$ is the positron rest mass. The goal was to find the intrinsic difference between the responses to monoenergetic positrons and photons with identical total incident energies, emphasizing the correct modeling of the low-energy tail below the edge of the Michel $\mu$$\rightarrow$$e^+\nu\nu$ spectrum at 52.8$\,$MeV.
The 0.46$\,$% accuracy of the new TRIUMF measurement [@Bri94] of the $\pi^+$$\rightarrow$$e^+\nu$ branching ratio was limited by event statistics and systematic uncertainties in evaluating the low-energy tail of the positron peak. The $\pi^+$$\rightarrow$$e^+\nu\gamma$ positrons and the forward-peaked bremsstrahlung gammas were detected in a 46-cm-diameter$\times$51-cm-long NaI(Tl) crystal “TINA”. The tail correction of 1.44$\pm\,$0.24$\,$% for the energy region from 0$\,$MeV to 52.3$\,$MeV was determined by subtracting the measured Michel positron spectrum from the detector positron response functions measured with the 20–85$\,$MeV $e^+$ monoenergetic beams. An additional tail component of $\sim\,$0.4$\,$% due to radiative processes was estimated by Monte Carlo method. The TRIUMF group apparently made no attempt to account for the potential difference in the scintillator light response of positrons and photons and neglected potential nonlinearities of the energy scale. Both effects could arise from the light collection nonuniformities of the NaI(Tl) detector.
In the recent PSI $\pi^+$$\rightarrow$$e^+\nu$ experiment [@Cza93] the quoted 0.29$\,$% systematic uncertainty of the extracted branching ratio was also dominated by the 1.64$\pm\,$0.09$\,$% electromagnetic loss fraction as well as by the 0.95$\pm\,$0.19$\,$% contribution from the photonuclear reactions. The $\sim\,$4$\pi$ calorimeter used in that measurement was made of 132 hexagonally shaped BGO crystals, with the light yield claimed to be homogeneous to within 1.5$\,$% over the whole 20$\,$cm length of every crystal.
The lineshapes of positrons and photons in our PIBETA calorimeter and their dependence on the optical properties of 240 constituent CsI crystals were predicted using the [GEANT]{} detector description and simulation tools. The code defined the geometry and tracking media for the complete PIBETA detector [@Pib95]. The inner detector region was occupied by two cylindrical wire chambers and a segmented plastic veto detector. The incident particles, monoenergetic positrons and photons, were generated in the center of the crystal sphere.
Photonuclear reactions in the active detectors as well as in the surrounding passive material were modeled in a user subroutine [@Frl95] that was added to the default [GEANT]{} version [3.21]{}. The probabilities of photonuclear reactions were calculated using the published cross sections from the $\gamma N$ reaction thresholds up to the energy of 120 MeV [@Ahr75; @Lep81; @Heb76; @Ber69; @Bra66; @Jon68].
The 240 individual CsI module shapes were specified, taking into account the mechanical tolerances of the physical CsI crystals, with Teflon and aluminized Mylar wrappings filling the 200 $\mu$m intermodule gaps. The irregular CsI modules had to be constructed from up to six [GEANT]{} generalized trapezoidal wedges. Every CsI crystal volume was considered a sensitive detector with the associated luminosity $\bar N_{\rm pe}$ expressed as the number of photoelectrons per MeV, and two axial and one transverse light collection nonuniformity coefficients, $a_{z1}$, $a_{z2}$, and $a_t$, respectively. A set of seventy-four detector luminosity and nonuniformity values, extracted in the tomography analysis, were initialized in the [GEANT]{}-accessible database. The optical parameters of the remaining 163 modules were drawn randomly from the distributions in Figs. \[fig:npf\] and \[fig:a1a2\]. The assumption was that the crystals produced in the future will be of the same optical quality as the ones that are already delivered.
We have always simulated at least 10$^5$ events for every fixed set of the calorimeter parameters ($\bar N_{\rm pe}$,$a_{z1}$,$a_{z2}$,$a_t$). The low-threshold trigger was defined by requiring a sum of the ADC readings exceed 5$\,$MeV of the light-equivalent energy for one calorimeter supercluster containing 32 crystals. The integrated detector acceptances with a low-level trigger were 85.8$\,$% for $\sim\,$70$\,$MeV positrons and 85.5$\,$% for $\sim\,$70$\,$MeV photons, so all extracted quantities had the relative statistical uncertainties of $\sim\,$0.2$\,$%. The response of the calorimeter was parameterized by the FWHM of the simulated ADC spectrum and the tail contribution being between the adjustable low level (default [LT]{} being 5$\,$MeV) and high level thresholds (two default values [HT$_1$]{} and [HT$_2$]{} being 54$\,$MeV and 55$\,$MeV, respectively).
For the purposes of comparison and gain normalization we first studied the response of an ideal, homogeneous calorimeter, with realistic CsI crystal light outputs. The “software” gain factor of every individual CsI detector was determined from the fitted peak positions of the simulated ADC spectrum sum over the crystal with maximum energy deposition and its nearest neighbors. The peak value positions were found after smoothing the histograms with a multiquadric function, to improve on the limited simulation statistics. The ratios of peak positions in the ideal, homogeneous calorimeter and the corresponding values for the physical, nonuniform calorimeter, were, by definition, the individual detector gain corrections. The values of these ratios were refined in three steps of the iterative procedure. The extracted energy resolution of the nonuniform detector in principle depends on the convergence of the gain-matching process and the resulting uncertainties of the individual detector gains. That gain-matching process was equivalent to the PMT high voltage adjustments in the real experiment that are effected to obtain the best energy resolution with the modular detector. We estimate that our simulation procedure fixes the software detector gains with the accuracy of $\sim\,$1$\,$%.
The set of gain constants depended on the optical properties of our CsI crystals as defined in our [GEANT]{} database, but also upon the incident particle chosen for the Monte Carlo calibration runs and its energy because of differences in shower developments of photons and positrons. The gain normalizations of the calorimeter detectors were calculated by aligning the simulated $\pi^+$$\rightarrow$$e^+\nu$ positron peak positions. The same procedure for matching the CsI detector gains was used in the real data-taking runs. Due to the gain renormalization, the influence of transverse light nonuniformities smaller than 10$\,$% on the ADC spectrum peak positions and the ADC lineshape could be neglected in comparison with the axial light nonuniformity effects. The simulated ADC values were summed over all detectors with the energy deposition above the 1$\,$MeV TDC threshold. We also examined the calculated ADC sums for the clusters which contained the crystal with the maximum energy deposition and its nearest neighbors. The full-widths at half maximum for these spectra are labeled in the following figures and tables as FWHM$_{(220)}$ and FWHM$_{\rm (NN)}$, respectively. No event-to-event uncertainties of the ADC pedestal values were assumed in the simulation. Stability of the PIBETA electronics tested under real experimental conditions in 1996 calibration runs and the quality of the algorithms used for the first and second pedestal correction reduces the pedestal peak root-mean-square to $\sim\,$8 channels, equivalent to $\sim\,$0.3$\,$MeV.
The [GEANT]{}-calculated resolution of the PIBETA calorimeter consisting of 220 CsI detector crystals and 20 veto crystals was parameterized by the fractional full width at half maximum:
$$\begin{aligned}
{ {\rm FWHM_{(220)}}\over {E_{e^+}} }(\%)&=&
{ {2.36\cdot\sigma_{E_{e^+}} }\over {E_{e^+}} }= \\
&=&\cases{ {{(54.69+1.197 E_{e^+}+0.4656\cdot 10^{-2}E_{e^+}^2)}
/ {E_{e^+}^{0.8030}} }\cr
{ {(37.19+0.060 E_{e^+}+0.2734\cdot 10^{-2}E_{e^+}^2)}
/ {E_{e^+}^{0.5256}} }\cr} \\end{aligned}$$
for the positrons with the most probable peak energy $E_{e^+}$ MeV and $$\begin{aligned}
{ {\rm FWHM_{(220)}}\over {E_\gamma} }(\%)=\cases{
{ {(37.07+0.5507 E_\gamma+0.3233\cdot 10^{-2}E_\gamma^2)}
/ { E_\gamma^{0.6549}} }\cr
{ {(40.52+0.9437\cdot E_\gamma+0.4931\cdot 10^{-2}E_\gamma^2)}
/ { E_\gamma^{0.6942}} }\cr}\end{aligned}$$ for the monoenergetic photons with the detected peak energy $E_\gamma$ MeV (Fig. \[fig:fit\_fwhm\_220\]). The top lines refer to the cases of an optically uniform detector ($a_z$=0) while the bottom equations describe the predicted response of the nonuniform calorimeter ($a_z$=[TOMOGRAPHY]{}).
The average fractional energy resolution FWHM$_{(220)}$ of 5.3$\,$% (6.0$\,$%) was achieved for the $\sim\,$70$\,$MeV kinetic energy positrons in the optically uniform (nonuniform) calorimeter, as compared to 5.7$\,$% (6.8$\,$%) resolution for the equivalent energy photons. That was the resolution found with no cuts applied on the light-equivalent energy generated in the calorimeter vetoes. Requiring less than 5$\,$MeV detected in the veto shield decreased the statistics by about 10$\,$% and suppressed the low-energy tail, but did not improve the energy resolution at the peak position.
Limiting the ADC sums to the nearest-neighbor crystals (six or seven crystal clusters) the fractional FWHM$_{\rm (NN)}$ energy resolution could be parameterized by: $$\begin{aligned}
{ {\rm FWHM_{\rm (NN)}}\over {E_{e^+}} }(\%)=
\cases{ {{(69.19+3.057 E_{e^+}+0.4198\cdot 10^{-2}E_{e^+}^2)}
/ {E_{e^+}^{0.8560}} }\cr
{ {(68.13+2.479 E_{e^+}+0.3368\cdot 10^{-2}E_{e^+}^2)}
/ {E_{e^+}^{0.8018}} }\cr} \end{aligned}$$ for the positrons with the detected peak energy $E_{e^+}$ MeV and $$\begin{aligned}
{ {\rm FWHM_{\rm (NN)}}\over {E_\gamma} }(\%)=\cases{
{ {(43.16+2.318 E_\gamma+0.4148\cdot 10^{-2}E_\gamma^2)}
/ { E_\gamma^{0.7723}} }\cr
{ {(39.57+2.302 E_\gamma+0.1329\cdot 10^{-2}E_\gamma^2)}
/ { E_\gamma^{0.7199}} }\cr}\end{aligned}$$ for the monoenergetic photons with the detected peak energy $E_\gamma$ MeV (Fig. \[fig:fit\_fwhm\_nn\]).
The average FWHM$_{\rm (NN)}$ for the 70$\,$MeV positrons and gammas in nearest-neighbor nonuniform crystal clusters was 8.7$\,$% (6.1$\,$MeV) and 9.9$\,$% (6.8$\,$MeV), respectively. Imposing the 5$\,$MeV cut on the detector veto signals improves these resolutions only marginally. These numbers should be compared with the response width of the ideal uniform nearest-neighbor clusters: 8.2$\,$% (5.7$\,$MeV) and 8.7$\,$% (6.2$\,$MeV).
The percentage of the events in the tail between the preset low and high energy threshold was tracked in the same [GEANT]{} simulation.
The listing of the low-energy tail contributions for 69.8$\,$MeV $e^+$ and 70.8$\,$MeV $\gamma$ in Table \[tab3\] shows that in the optically uniform calorimeter they differ by 1.8$\,$%. The optical nonuniformity does not change the positron tail contribution, but increases the photon tail by $\sim\,$0.2$\,$%. The imposition of the 5$\,$MeV veto-shield cut decreases the electromagnetic leakage to 2.4$\,$% for positrons and 4.7$\,$% for photons, a $e^+$-$\gamma$ tail difference of $+\,$2.3$\,$%.
Table \[tab4\] shows the low-energy tail components for the simulated nearest-neighbor cluster ADC sums: with no applied cuts the contributions were 4.7$\,$% and 6.7$\,$% for $e^+$’s and $\gamma$’s, respectively. The veto shielding cuts decreased both fractions by about 0.3$\,$%.
If the positron originates from a radiative $\pi^+$$\rightarrow$$e^+\nu\gamma$ decay, its average Monte Carlo peak position is essentially unchanged. Its tail contribution is unaffected if simulated ADC sums extend over full calorimeter but are increased by $\sim\,$1.4$\,$% if only the nearest-neighbor clusters are summed. The radiative decay matrix elements used in the calculation were taken from Ref. [@Bro64]. All quoted numbers have an approximate systematic uncertainty of $\sim\,$0.2$\,$%.
The detector response to positrons and photons in the energy range between 10$\,$MeV and 120$\,$MeV is also nonlinear because the volume distribution of the shower energy deposition depends on the incident particle type and the energy. Our [GEANT]{} studies showed that the magnitude and energy dependence of these nonlinearities does not change significantly because of CsI crystal light collection nonuniformity. Results of the calculations, both for the case of a homogeneous and optically nonhomogeneous calorimeter, are displayed in Figs. \[fig:fit\_nonlin\_220\] and \[fig:fit\_nonlin\_nn\]. Nonlinearities of the detector response in the covered energy range for positrons and photons are very close in magnitude and shape of energy variation and amounted to $\sim\,$1.5$\,$% for the ADC sums over 220 crystals and up to $\sim\,$2.8$\,$% if the simulated ADC sums were restricted to the over-the-threshold ADCs of nearest neighbors.
Conclusion
==========
We have measured the optical properties of seventy-four pure cesium iodide crystals that were polished and wrapped in the diffuse Teflon reflector. The results are summarized separately for the full-sized and half-sized crystals in Tables \[tab1\] and \[tab2\].
The deduced light yields parameterized by two axial and one transverse light collection nonuniformity coefficients constitute a minimum set of parameters necessary for a realistic Monte Carlo simulation of the modular CsI calorimeter.
The predicted energy resolutions FWHM$_{(220)}$ for $\sim\,$70$\,$MeV positrons and photons in the full PIBETA calorimeter with the ideal, optically uniform CsI modules with specified luminosities were shown to be close, 3.7$\,$MeV and 4.0$\,$MeV, respectively. The upper limit of the low energy tail contributions in the region between 5$\,$MeV and 55$\,$MeV were calculated to be 6.9$\,$% and 8.7$\,$% for the positrons and gammas in an optically homogeneous detector, respectively. After applying the 5$\,$MeV calorimeter veto cut, these tail corrections decrease to 2.2$\,$% and 4.5$\,$%.
The average deduced axial light nonuniformities of real CsI crystals wrapped in a Teflon sheet had negative slopes, $-\,$0.18$\,$%/cm and $-\,$1.6$\,$%/cm, for the front and back half of the crystal volume, respectively. The corresponding [GEANT]{} simulation of the nonuniform PIBETA apparatus for 68.9$\,$MeV $e^+$ and 70.8$\,$MeV $\gamma$’s shows that both energy responses will be broadened to the average FWHM$_{(220)}$ of 5.9$\,$% and 6.9$\,$% while the associated tail contributions will change only for photon spectra, increasing the low-energy tail by $\sim\,$0.3$\,$%. The simulated calorimeter ADC spectra are shown in Figs. \[fig:tails1\] and \[fig:tails2\].
The nonlinearity of the measured energy scale caused by the optical nonuniformity is $\le\,$2.8$\,$% throughout the relevant $e^+$/$\gamma$ energy range. This spread is consistent with the precision of energy calibration required to extract the tail corrections with the systematic uncertainty of $\sim\,$0.2$\,$%.
The predicted ADC spectra of the monoenergetic positrons, electrons and tagged photons in the energy range 10–70$\,$MeV will be compared with the measured responses of the partial CsI calorimeter arrays in a forthcoming publication [@Frl97].
Acknowledgements
================
The authors wish to thank Micheal Sadler of the Abilene Christian University (ACU) for lending us the drift chamber tomography apparatus. Derek Wise, also of ACU, has helped with the cosmic muon tomography measurements. Roger Schnyder of Paul Scherrer Institut has maintained and repaired the faulty electronics modules. Their help is gratefully acknowledged.
This work is supported and made possible by grants from the US National Science Foundation and the Paul Scherrer Institute.
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--------------------------------------------- ---------- ----------
Fast-to-Total Ratio (100 ns/1 $\mu$s gate) 0.835 0.739
\# Photoelectrons/MeV (100 ns ADC gate) 83.4 65.1
\# Photoelectrons/MeV (1 $\mu$s ADC gate) 99.9 88.0
Fast Light Temp. Coefficient (%/$^\circ$C) $-1.20$ $-1.61$
Total Light Temp. Coefficient (%/$^\circ$C) $-1.25$ $-1.40$
Axial Nonuniformity Coefficient (%/cm),
$z$$\le$10$\,$cm, 100 ns ADC gate $-0.200$ $-0.124$
Axial Nonuniformity Coefficient(%/cm),
$z$$\ge$10$\,$cm, 100 ns ADC gate $-1.62$ $-1.61$
Transverse Nonuniformity Coefficient(%/cm),
$z_0$$=$6$\,$cm, 100 ns ADC gate $-0.400$ $-0.500$
Transverse Nonuniformity Coefficient(%/cm),
$z_0$$=$18$\,$cm, 100 ns ADC gate $-1.10$ $-1.20$
--------------------------------------------- ---------- ----------
: Average scintillation properties of hexagonal and pentagonal PIBETA CsI calorimeter shapes PENTAs, HEX–As, HEX–Bs, HEX–Cs and HEX–Ds (59 crystals). All crystals were polished and wrapped in two layers of a Teflon membrane and one layer of aluminized Mylar film. The light yields are normalized to the temperature of 18$^\circ$C. All other parameters were measured at the average laboratory room temperature of 22$^\circ$C.[]{data-label="tab1"}
--------------------------------------------- ---------- ----------
Fast-to-Total Ratio (100 ns/1 $\mu$s gate) 0.806 0.728
\# Photoelectrons/MeV (100 ns ADC gate) 34.5 31.5
\# Photoelectrons/MeV (1 $\mu$s ADC gate) 42.8 43.3
Fast Light Temp. Coefficient (%/$^\circ$C) $-2.08$ $-0.63$
Total Light Temp. Coefficient (%/$^\circ$C) $-1.78$ $-0.69$
Axial Nonuniformity Coefficient (%/cm),
$z$$\le$10$\,$cm, 100 ns ADC gate $-0.326$ $-0.500$
Axial Nonuniformity Coefficient(%/cm),
$z$$\ge$10$\,$cm, 100 ns ADC gate $-1.00$ $-1.83$
Transverse Nonuniformity Coefficient(%/cm),
$z_0$$=$6$\,$cm, 100 ns ADC gate $-0.755$ $-0.827$
Transverse Nonuniformity Coefficient(%/cm),
$z_0$$=$18$\,$cm, 100 ns ADC gate $-4.10$ $-5.14$
--------------------------------------------- ---------- ----------
: Average scintillation properties of half-hexagonal and trapezial PIBETA CsI calorimeter shapes HEX–D1/2s and VETO–1/2s (15 crystals). All crystals were polished and wrapped in two layers of Teflon foil and one layer of aluminized Mylar.[]{data-label="tab2"}
-------------------------------- ---------------- ---------------- ---------------- ----------------
Peak Position (MeV) 68.87$\pm$0.03 68.90$\pm$0.03 70.10$\pm$0.03 69.97$\pm$0.03
FWHM$_{(220)}$ (MeV) 3.66$\pm$0.03 4.10$\pm$0.03 3.98$\pm$0.03 4.76$\pm$0.03
5$\le$Events$\le$54$\,$MeV (%) 6.44$\pm$0.09 6.46$\pm$0.09 8.20$\pm$0.10 8.47$\pm$0.10
5$\le$Events$\le$55$\,$MeV (%) 6.89$\pm$0.09 6.98$\pm$0.09 8.86$\pm$0.10 9.11$\pm$0.11
Peak Position (MeV) 68.88$\pm$0.03 68.92$\pm$0.03 70.07$\pm$0.03 69.99$\pm$0.03
FWHM$_{(220)}$ (MeV) 3.67$\pm$0.03 4.09$\pm$0.03 3.99$\pm$0.03 4.76$\pm$0.03
5$\le$Events$\le$54$\,$MeV (%) 1.91$\pm$0.05 2.02$\pm$0.05 3.92$\pm$0.07 4.15$\pm$0.07
5$\le$Events$\le$55$\,$MeV (%) 2.24$\pm$0.05 2.36$\pm$0.05 4.46$\pm$0.07 4.68$\pm$0.07
-------------------------------- ---------------- ---------------- ---------------- ----------------
: The predicted energy resolutions and tail contributions for 69.8$\,$MeV $e^+$ and 70.8$\,$MeV $\gamma$’s events in the full PIBETA calorimeter. The light output in photoelectrons/MeV and the linear axial light collection nonuniformities measured for individual CsI crystals ($a_{z1,z2}$=[TOMOGRAPHY]{}) were used in a [GEANT]{} simulation. The values for the perfect optically homogeneous crystals ($a_{z1,z2}$=0) were shown for comparison.[]{data-label="tab3"}
-------------------------------- ---------------- ---------------- ---------------- ----------------
Peak Position (MeV) 67.74$\pm$0.03 67.85$\pm$0.03 68.80$\pm$0.03 68.65$\pm$0.03
FWHM$_{\rm (NN)}$ (MeV) 5.22$\pm$0.03 5.90$\pm$0.03 6.01$\pm$0.03 6.80$\pm$0.03
5$\le$Events$\le$54$\,$MeV (%) 3.83$\pm$0.07 4.01$\pm$0.07 5.47$\pm$0.08 5.78$\pm$0.08
5$\le$Events$\le$55$\,$MeV (%) 4.54$\pm$0.07 4.73$\pm$0.08 6.33$\pm$0.09 6.67$\pm$0.09
Peak Position (MeV) 67.75$\pm$0.03 67.83$\pm$0.03 68.80$\pm$0.04 68.64$\pm$0.03
FWHM$_{\rm (NN)}$ (MeV) 5.52$\pm$0.03 5.83$\pm$0.03 5.92$\pm$0.03 6.76$\pm$0.03
5$\le$Events$\le$54$\,$MeV (%) 3.53$\pm$0.07 3.73$\pm$0.07 5.28$\pm$0.08 5.59$\pm$0.08
5$\le$Events$\le$55$\,$MeV (%) 4.19$\pm$0.07 4.39$\pm$0.07 6.10$\pm$0.09 6.42$\pm$0.09
-------------------------------- ---------------- ---------------- ---------------- ----------------
: The predicted energy resolutions and tail contributions for 69.8$\,$MeV $e^+$ and 70.8$\,$MeV $\gamma$’s events in the PIBETA clusters containing the crystal with maximum energy deposition and its nearest neighbors.[]{data-label="tab4"}
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Geometric, topological and graph theory modeling and analysis of biomolecules are of essential importance in the conceptualization of molecular structure, function, dynamics, and transport. On the one hand, geometric modeling provides molecular surface and structural representation, and offers the basis for molecular visualization, which is crucial for the understanding of molecular structure and interactions. On the other hand, it bridges the gap between molecular structural data and theoretical/mathematical models. Topological analysis and modeling give rise to atomic critical points and connectivity, and shed light on the intrinsic topological invariants such as independent components (atoms), rings (pockets) and cavities. Graph theory analyzes biomolecular interactions and reveals biomolecular structure-function relationship. In this paper, we review certain geometric, topological and graph theory apparatuses for biomolecular data modeling and analysis. These apparatuses are categorized into discrete and continuous ones. For discrete approaches, graph theory, Gaussian network model, anisotropic network model, normal mode analysis, quasi-harmonic analysis, flexibility and rigidity index, molecular nonlinear dynamics, spectral graph theory, and persistent homology are discussed. For continuous mathematical tools, we present discrete to continuum mapping, high dimensional persistent homology, biomolecular geometric modeling, differential geometry theory of surfaces, curvature evaluation, variational derivation of minimal molecular surfaces, atoms in molecule theory and quantum chemical topology. Four new approaches, including analytical minimal molecular surface, Hessian matrix eigenvalue map, curvature map and virtual particle model, are introduced for the first time to bridge the gaps in biomolecular modeling and analysis. Emphasis is given to the connections of existing biophysical models/methods to mathematical theories, such as graph theory, Morse theory, Poicaré-Hopf theorem, differential geometry, differential topology, algebraic topology and geometric topology. Potential new directions and standing open problems are briefly discussed.'
author:
- |
Kelin Xia$^1$ [^1] and Guo-Wei Wei$^{2,3}$ [^2]\
$^1$Division of Mathematical Sciences, School of Physical and Mathematical Sciences,\
Nanyang Technological University, Singapore 637371\
$^2$Department of Mathematics\
Michigan State University, MI 48824, USA\
$^3$Department of Biochemistry & Molecular Biology\
Michigan State University, MI 48824, USA\
title: 'A review of geometric, topological and graph theory apparatuses for the modeling and analysis of biomolecular data '
---
Key words: Molecular bioscience, Molecular biophysics, Graph theory, Graph spectral theory, Graph Laplacian, Differential geometry, Differential topology Persistent homology, Morse theory, Conley index, Poincaré-Hopf index, Laplace-Beltrami operator, Dynamical system.
[ ]{}
Introduction
============
Life science is regarded as the last forefront in natural science and the 21$^{\rm st}$ century will be the century of biological sciences. Molecular biology is the foundation of biological sciences and molecular mechanism, which is governed by all the valid mechanics, including quantum mechanics when it is relevant, and is the ultimate truth of life science. One trend of biological sciences in the 21$^{\rm st}$ century is that many traditional disciplines, such as epidemiology, neuroscience, zoology, physiology and population biology, are transforming from macroscopic and phenomenological to molecular-based sciences. Another trend is that biological sciences in the 21$^{\rm st}$ century are transforming from qualitative and descriptive to quantitative and predictive, as many other disciplines in natural science have done in the past. Such a transformation creates unprecedented opportunities for mathematically driven advances in life science [@Wei:2016].
Biomolecules, such as proteins and the nucleic acids, including DNA and RNA, are essential for all known forms of life, such as animals, fungi, protists, archaea, bacteria and plants. Indeed, proteins perform a vast variety of biological functions, including membrane channel transport, signal transduction, organism structure supporting, enzymatic catalysis for transcription and the cell cycle, and immune agents. In contrast, nucleic acids function in association with proteins and are essential players in encoding, transmitting and expressing genetic information, which is stored through nucleic acid sequences, i.e., DNA or RNA molecules and transmitted via transcription and translation processes. The understanding of biomolecular structure, function, dynamics and transport is a fundamental issue in molecular biology and biophysics. A traditional dogma is that sequence determines structure, while structure determines function [@Anfinsen:1973]. This, however, has been undermined by the fact that many intrinsically disordered proteins can also be functional [@Onuchic:1997; @White:1999; @Schroder:2005; @Chiti:2006]. Disordered proteins are associated with sporadic neurodegenerative diseases, including Alzheimer’s disease, Parkinson’s disease and mad cow disease [@Chiti:2006; @Uversky:2008]. Randomness in disordered proteins is a consequence of protein flexibility, which is an intrinsic protein function. In general, the understanding of protein structure-function relationship is also crucial for shedding light on protein specification, protein-protein interactions, protein-drug binding that are essential to drug design and discovery, and improving human health and wellbeing.
Much of the present understanding of biomolecular structures and functions and their relationship come from experimental data that are collected from a number of means, such as macromolecular X-ray crystallography, nuclear magnetic resonance (NMR), cryo-electron microscopy (cryo-EM), electron paramagnetic resonance (EPR), multiangle light scattering, confocal laser-scanning microscopy, scanning capacitance microscopy, small angle scattering, ultra fast laser spectroscopy, etc. The major players for single macromolecules are X-ray crystallography and NMR. For example, advanced X-ray crystallography technology is able to offer decisive structural information at Armstrong and sub-Armstrong resolutions, while an important advantage of NMR experiments is that they are able to provide biomolecular structural information under physiological conditions. The continuously effort in the past few decades has made X-ray crystallography and NMR technologically relatively well developed, except for their use in special circumstances, such as the study of membrane proteins. However, theses approaches are not directly suitable for proteasomes, subcellular structures, organelles, cells and tissues, whose study has become increasingly popular in structural biology. Currently, a unique experimental tool for imaging subcellular structures, organelles, multiprotein complexes and even cells and tissues is cryo-EM [@Volkmann:2010].
The rapid advances of experimental technology in the past few decades have led to the accumulation of vast amount of three-dimensional (3D) biomolecular structural data. The [Protein Data Bank (PDB)](http://www.rcsb.org/pdb/home/home.do) has collected more than one hundred twenty thousands of 3D biomolecular structures. Biomolecular geometric information holds the key to our understanding of biomolecular structure, function, and dynamics. It has wide spread applications in virtual screening, computer-aid drug design, binding pocket descriptor, quantitative structure activity relationship, protein design, RNA design, molecular machine design, etc. In general, the availability of biomolecular structural data has paved the way for the transition from the traditional qualitative description to quantitative analysis and prediction in biological sciences. An essential ingredient of quantitative biology is geometric, topological and graph theory modeling, analysis and computation. Aided by increasingly powerful high performance computers, geometric, topological and graph theory modeling, analysis and computation have become indispensable apparatuses not only for the visualization of biological data, but also filling the gap between biological data and mathematical models of biological systems [@ZYu:2008; @ZYu:2008b; @XFeng:2012a; @XFeng:2013b; @KLXia:2014a; @MXChen:2012; @Rocchia:2002; @PMach:2011; @XShi:2011; @JLi:2013; @Decherchi:2013; @LiLin:2014].
One of the simplest molecular geometric models, or molecular structural models, is the space-filling Corey-Pauling-Koltun (CPK) theory, which represents an atom by a solid sphere with a van der Waals (VDW) radius [@Koltun:1965]. The outer boundary of CPK model gives rise to the van der Waals (vdW) surface, which is composed of piece-wise unburied sphere surfaces. Solvent accessible surface (SAS) and solvent-excluded surface (SES) have also been introduced to create smooth molecular surfaces by rolling a probe molecule over the vdW surface [@Richards:1977; @Connolly85]. SESs have widely been applied to protein folding [@Spolar], protein surface topography [@Kuhn], protein-protein interactions [@Crowley], DNA binding and bending [@Dragan], macromolecular docking [@Jackson], enzyme catalysis [@LiCata], drug classification [@Bergstrom], and solvation energies [@Raschke]. The SES model also plays a crucial role in implicit solvent models [@Baker:2005; @DuanChen:2011a], molecular dynamics simulations [@Geng:2011] and ion channel transports [@DuanChen:2011a; @QZheng:2011a; @QZheng:2011b]. Computationally, efficient algorithms for computing SES are developed or introduced, such as alpha-shapes [@WYChen:2010] and marching tetrahedra [@SLChan:1998]. A popular software for the Lagrangian representation of SESs, called MSMS, has been developed [@Sanner:1996]. Recently, a software package, called Eulerian solvent excluded surface (ESES), for the Eulerian representation of SESs [@ESES:2015], has also been developed. However SAS and SES are still not differentiable and have geometric singularities, i.e., cusps and tips. To construct smooth surface representation of macromolecules, Gaussian surface (GS) has been proposed to represent each atom by a C$^{\infty}$ Gaussian function, while accounting their overlapping properties [@Grant:1995]. Differential geometry theory of surfaces provides a natural approach to describe biomolecular surfaces and boundaries. Utilizing the Euler-Lagrange variation, a differential geometry based surface model, the minimal molecular surface (MMS), has been introduced for biomolecular geometric modeling [@Bates:2006; @Bates:2008]. Differential geometry based variational approach has been widely applied to biophysical modeling of solvation [@Wei:2009; @ZhanChen:2010a; @ZhanChen:2010b; @BaoWang:2015a], ion channel [@Wei:2012; @DuanChen:2012a; @DuanChen:2012b] and multiscale analysis [@Wei:2009], in conjugation with other physical models, such as electrostatics, elasticity and molecular mechanics [@Wei:2013]. Geometry modeling and annotation of biomolecular surfaces together with physical features, such as electrostatics and lipophilicity, provide some of the best predictions of biomolecular solvation free energies [@BaoWang:2016FFTS; @BaoWang:2016HPK], protein-drug binding affinities [@BaoWang:2016FFTB], protein mutation energy changes [@ZXCang:2016a] and protein-protein interaction hot spots [@Darnell:2008; @Demerdash:2009].
Theoretical modeling of the structure-function relationship of biomolecules is usually based on fundamental laws of physics, i.e., quantum mechanics (QM), molecular mechanism (MM), continuum mechanics, statistical mechanics, thermodynamics, etc. QM methods are indispensable for chemical reactions and protein degradations [@Warshel:1976; @Cui:2002; @YZhang:2009a]. Molecular dynamics (MD) [@McCammon:1977] is a powerful tool for the understanding of the biomolecular conformational landscapes and elucidating collective motion and fluctuation. Currently, MD is a main workhorse in molecular biophysics for biomolecular modeling and simulation. However, all-electron or all-atom representations and long-time integrations lead to such an excessively large number of degrees of freedom that their application to real-time large-scale dynamics of large proteins or multiprotein complexes becomes prohibitively expensive. For instance, current computer simulations of protein folding take many months to come up with a very poor copy of what Nature administers perfectly within a tiny fraction of a second. Therefore, in the past few decades, many graph theory based biomolecular models, including normal mode analysis (NMA) [@Go:1983; @Tasumi:1982; @Brooks:1983; @Levitt:1985], elastic network model (ENM) [@Bahar:1997; @Bahar:1998; @Atilgan:2001; @Hinsen:1998; @Tama:2001; @LiGH:2002] become very popular for understanding protein flexibility and long time dynamics. In these models, the diagonalization of the interaction matrix or Hamiltonian of a protein is a required procedure to obtain protein eigenmodes and associated eigenvalues. The low order eigenmodes can be interpreted as the slow motions of the protein around the equilibrium state and the Moore-Penrose pseudo-inverse matrix can be used to predict the protein thermal factors, or B-factors. However, NMA interaction potentials are quite complicated. Tirion simplified its complexity by retaining only the harmonic potential for elasticity, which is the dominant term in the MD Hamiltonian [@Tirion:1996]. Network theory [@Flory:1976] has had a considerable impact in protein flexibility analysis. The combination of elasticity and coarse-grained network gives rise to elastic network model (ENM) [@Hinsen:1998]. In this spirit, Gaussian network model (GNM) [@Bahar:1997; @Bahar:1998; @QCui:2010] and anisotropic network model (ANM) [@Atilgan:2001] have been proposed. Yang et al. [@LWYang:2008] have shown that the GNM is about one order more efficient than most other flexibility approaches. The above graph theory based methods have been improved in a number of aspects, including crystal periodicity and cofactor corrections [@Kundu:2002; @Kondrashov:2007; @Hinsen:2008; @GSong:2007], and density - cluster rotational - translational blocking [@Demerdash:2012]. These approaches have many applications in biophysics, including stability analysis [@Livesay:2004], molecular docking [@Gerek:2010], and viral capsid analysis [@Rader:2005; @Tama:2005]. In particular, based on spectral graph theory that the behavior of the second eigenmode can be used for clustering, these methods have been utilized to unveil the molecular mechanism of the protein domain motions of hemoglobin [@CXu:2003], F1 ATPase [@WZheng:2003; @QCui:2004], chaperonin GroEL [@Keskin:2002; @WZheng:2007] and the ribosome [@Tama:2003; @YWang:2004]. The reader is referred to reviews for more details [@JMa:2005; @LWYang:2008; @Skjaven:2009; @QCui:2010].
Note that ENM type of methods is still too expensive for analyzing subcellular organelles and multiportein complexes, such as HIV and Zika virus, and molecular motors, due to their matrix decomposition procedure which is of the order of ${\cal O}(N^3)$ in computational complexity, where $N$ is the number of network nodes, or protein atoms. An interesting and important mathematical issue is how to reduce the computational complexity of ENM, GNM and ANM for handling excessively large biomolecules. Flexibility-rigidity index (FRI) has been developed as a more accurate and efficient method for biomolecular graph analysis [@KLXia:2013d; @Opron:2014]. In particular, aided with a cell lists algorithm [@Allen:1987], the fast FRI (fFRI) is about ten percent more accurate than GNM on a test set of 364 proteins and is orders of magnitude faster than GNM on a set of 44 proteins, due to its ${\cal O}(N)$ computational complexity. It has been demonstrated that fFRI is able to predict the B-factors of an entire HIV virus capsid with 313,236 residues in less than 30 seconds on a single-core processor, which would require GNM more than 120 years to accomplish if its computer memory were not a problem [@Opron:2014].
Topological analysis of molecules has become very popular since the introduction of the theory of atoms in molecules (AIM) for molecular electron density data by Bader and coworkers [@Bader:1985; @Bader:1990]. AIM was proposed to quantitatively define the atomic bonds and interatomic surfaces (IASs) by employing a topology based partition of electron density. It has two main strands: the scalar field topology of molecular electron density maps and the scalar field topology of the local Laplacian of electron density [@Popelier:2000; @Popelier:2005]. The former characterizes chemical bonds and atoms, and the latter provides a new procedure to analyze electron pair localization. Electron localization function (ELF) [@Silvi:1994] was proposed for the study of electron pairing. ELF utilizes the gradient vector field topology to partition the electron density map into topological basins. In fact, a general theory called quantum chemical topology (QCT) [@Popelier:2005] has been developed for the topological analysis of electron density functions. Apart form the above mentioned AIM and ELF, QCT also includes the electrostatic potential [@Leboeuf:1999], electron localizability indicator (ELI) [@Kohout:2004], localized orbital locator (LOL) [@Schmider:2000], the virial field [@Keith:1996], the magnetically induced current distribution [@Keith:1993], the total energy (catchment regions) [@Mezey:1981] and the intracule density [@Cioslowski:1999]. QCT has proved to be very effective in analyzing interactions between atoms in molecular systems, particularly the covalent interactions and chemical structure of small molecular systems. Many software packages have been developed for QCT analysis [@Biegler:2002; @Henkelman:2006].
The essential idea behind QCT is the scalar field topology analysis [@Beketayev:2011; @Gunther:2014], which includes several major components such as critical points (CPs) and their classification, zero-flux interface (interatomic interface), gradient vector field, etc. In fact, when vector field topology is applied to the gradient of a scalar function, it coincides with scalar field topology. Mathematically, this topological analysis is also known as the Morse theory, which describes the topological structure of a closed manifold by means of a nondegenerate gradient vector field. Morse theory is a powerful tool for studying the topology of molecular structural data through critical points of a Morse function. A well-defined Morse function needs to be differentiable and its CPs are isolated and non-degenerated. It can be noticed that all the above-mentioned scalar fields in QCT are Morse functions and more can be proposed as long as they satisfy the Morse function constraints. In the Morse theory, critical points are classified into minima, maxima, and saddle points based on their indices. In AIM, the three types of CPs are associated with chemical meanings. A maximal CP is called a nucleic critical point (NCP). A minimal CP is related with cage critical point (CCP). Finally, saddle points can be further classified into two types, i.e., bond critical points (BCPs) and ring critical points (RCPs). Current research issues in QCT include how to reduce the computational complexity and extend this approach for biomolecules [@Gillet:2012; @Gunther:2014]. Additionally, its connection to scalar field topology and vector field topology needs to be further clarified so that related mathematical theories, including Poincaré - Hopf theorem [@Poincare:1890], Conley index theory [@Conley:1978], Floer homology, etc., and algorithms developed in computer science can be better applied to molecular sciences.
In additional to differential topology, algebraic topology, specifically, persistent homology, has drawn much attention in recent years. Persistent homology has been developed as a new multiscale representation of topological features. The 0th dimensional version was originally introduced for computer vision applications under the name “size function" [@Fro90; @Frosini:1999] and the idea was also studied by Robins [@Robins:1999]. Persistent homology theory was formulated, together with an algorithm given, by Edelsbrunner et al. [@Edelsbrunner:2002], and a more general theory was developed by Zomorodian and Carlsson [@Zomorodian:2005]. There has since been significant theoretical development [@BH11; @CEH07; @CEH09; @CEHM09; @CCG09; @CGOS11; @Carlsson:2009theory; @CSM09; @SMV11; @zigzag], as well as various computational algorithms [@OS13; @DFW14; @Mischaikow:2013; @javaPlex; @Perseus; @Dipha]. Often, persistent homology can be visualized through barcodes [@CZOG05; @Ghrist:2008], in which various horizontal line segments or bars are the homology generators that survive over filtration scales. Persistence diagrams are another equivalent representation [@edelsbrunner:2010]. [ Computational homology and persistent homology have been applied to a variety of domains, including image analysis [@Carlsson:2008; @Pachauri:2011; @Singh:2008; @Bendich:2010; @Frosini:2013], chaotic dynamics verification [@Mischaikow:1999; @kaczynski:mischaikow:mrozek:04], sensor network [@Silva:2005], complex network [@LeeH:2012; @Horak:2009], data analysis [@Carlsson:2009; @Niyogi:2011; @BeiWang:2011; @Rieck:2012; @XuLiu:2012], shape recognition [@DiFabio:2011; @AEHW06] and computational biology [@Kasson:2007; @Gameiro:2014; @Dabaghian:2012; @Perea:2015a; @Perea:2015b].]{} Compared with traditional computational topology [@Krishnamoorthy:2007; @YaoY:2009; @ChangHW:2013] and/or computational homology, persistent homology inherently has an additional dimension, the filtration parameter, which can be utilized to embed some crucial geometric or quantitative information into topological invariants. The importance of retaining geometric information in topological analysis has been recognized [@Biasotti:2008], and topology has been advocated as a new approach for tackling big datasets [@BVP15; @BHPP14; @Fujishiro:2000; @Carlsson:2009; @Ghrist:2008]. Most recently, persistent homology has been developed as a powerful tool for analyzing biomolecular topological fingerprints [@KLXia:2014c; @KLXia:2015d; @KLXia:2015e], quantitative fullerene stability analysis [@KLXia:2015a], topological transition in protein folding [@KLXia:2015c], cryo-EM structure determination [@KLXia:2015b], and in conjugation with machine learning for protein classification [@ZXCang:2015] and protein-ligand/drug binding affinity prediction [@ZXCang:2016b]. Differential geometry based topological persistence [@BaoWang:2016a] and multidimensional persistence [@KLXia:2015c] have also been developed for biomolecules analysis and modeling. It is worthy to mention that persistent topology along is able to outperform all the eminent methods in computational biophysics for the blind binding affinity prediction of protein-ligand complexes from massive data sets [@ZXCang:2016b].
The objective of this paper is threefold. First, the main objective is to provide a review of some widely used geometric, topological and graph theory apparatuses for the modeling and analysis of biomolecular data. We keep our description concise, elementary and accessible to upper level undergraduate students in mathematics and most researchers in computational biophysics. We point out some open problems and potential topics in our discussions. Our goal is to provide a reference for mathematicians who are interested in mathematical molecular bioscience and biophysics (MMBB), an emergent field in mathematics [@Wei:2016], and for biophysicists and theoreticians who are interested in mathematical foundations of many theoretical approaches in molecular biology and biophysics. Obviously, our topic selection is limited by our knowledge, experience and understanding, and for the same reason, we might have missed many important results and references on the selected topics as well. Additionally, inspired by the success of QCT and persistent homology, the density filtration for Hessian matrix eigenvalue maps and molecular curvature maps has been introduced. Both maps are constructed from molecular rigidity density obtained via a discrete to continuum mapping (DCM) technique, which transfers atomic information in a molecule to atomic density distribution, a continuous scalar function. In this approach, a series of isosurfaces are generated and systematically studied for eigenvalue and curvature maps. Geometric and topological (Geo-Topo) fingerprints are identified to characterize unique patterns within eigenvalue and curvature maps, specifically, the maps of three eigenvalues derived from local Hessian matrix at each location of the rigidity density and the maps of Gaussian, mean, maximal and minimal curvatures computed everywhere of the rigidity density. Topological properties of eigenvalue and curvature maps are classified by their critical points. The evolution of isosurfaces during the filtration process is found to be well characterized by CPs. Different behaviors are found in different types of maps. Persistent homology analysis is also employed for eigenvalue map analysis to reveal intrinsic topological invariants of three Hessian matrix eigenvalues. Finally, a new minimal molecular surface, called analytical minimal molecular surface (AMMS) via the zero-value isosurface of the mean curvature map, has been introduced. It is found that this new surface definition can capture the topological property, such as the inner bond information. It also offers an efficient geometric modeling of biomolecules.
The rest of this paper is organized as following: Section \[sec:discrete\] is devoted to a review of some discrete mathematical apparatuses, namely, graph theory and persistent homology, for the analysis and modeling of biomolecular data. More specifically, we illustrate the applications of graph theory, Gaussian network model, anisotropic network model, normal mode analysis, flexibility and rigidity index, spectral graph theory, and persistent homology to biomolecular data analysis, such as protein B-factor prediction, domain separation, anisotropic motion, topological fingerprints, etc. The review of some continuous geometric and topological apparatuses for scalar field topology and geometry are given in Section \[sec:continuous\]. We examine the basic concepts in differential geometry, biomolecular surfaces, curvature analysis, and theory of atoms in molecules. Discrete to continuum mapping and two algorithms for curvature evaluation are discussed. Further, we introduce two new approaches, i.e., Hessian matrix eigenvalue maps and curvature maps, for geometric-topological fingerprint analysis of biomolecular data. The relation between scalar field geometry and topological CPs are discussed in detail. A new analytical minimal molecular surface is also introduced. Virtual particle model is proposed to analyze the anisotropic motions of continuous scalar fields, such as cryo-EM maps. Finally, persistent homology analysis for eigenvalue scalar field is discussed. This paper ends with some concluding remarks.
Discrete apparatuses for biomolecules {#sec:discrete}
=====================================
One of the major challenges in the biological sciences is the prediction of protein functions from protein structures. One function prediction is about protein flexibility, which strongly correlates with biomolecular enzymatic activities, such as allosteric transition, ligand binding and catalysis, as well as protein stiffness and rigidity for structural supporting. For instance, in enzymatic processes, protein flexibility enhances protein-protein interactions, which in turn reduce the activation energy barrier. Additionally, protein flexibility and motion amplify the probability of barrier crossing in enzymatic chemical reactions. Therefore, the investigation of protein flexibility at a variety of energy spectra and time scales is vital to the understanding and prediction of other protein functions. Currently, the most important technique for protein flexibility analysis is X-ray crystallography. Among more than one hundred twenty thousand structures in the protein data bank (PDB), more than eighty percent structures are collected by X-ray crystallography. The Debye-Waller factor, or B-factor, can be directly computed from X-ray diffraction or other diffraction data. In the PDB, biomolecular structures are recorded in terms of (discrete) atomic types, atomic positions, occupation numbers, and B-factors. Although atomic B-factors are directly associated with atomic flexibility, they can be influenced by variations in atomic diffraction cross sections and chemical stability during the diffraction data collection. Therefore, only the B-factors for specific types of atoms, say C$_\alpha$ atoms, can be directly interpreted as their relative flexibility without corrections. Another important method for accessing protein flexibility is nuclear magnetic resonance (NMR) which often provides structural flexibility information under physiological conditions. NMR spectroscopy allows the characterization of protein flexibility in diverse spatial dimensions and a large range of time scales. About six percent of structures in the PDB are determined by electron microscopy (EM) which does not directly offer the flexibility information at present. Therefore, it is important to have mathematical or biophysical methods to predict their flexibility.
Graph theory related methodologies {#sec:graph}
----------------------------------
With the development of experimental tools, vast amount of data for biomolecular structures and interaction networks are available and provide us with unprecedented opportunities in mathematical modeling. The graph theory and network models have been widely used in the study of biomolecular structures and interactions and found many applications in drug design, protein function analysis, gene identification, RNA structure representation, etc. [@DasGupta2016; @Gan:2004rag; @Fera:2004rag] Generally speaking, biomolecular graph and network models can be divided into two major categories, namely, abstract graph/network models, which include biomolecular interaction-network models, and geometric graph/network models, where the distance geometry plays an important role.
The geometric graph models or biomolecular structure graph models construct unique graphs based on biomolecular 3D structural data. The graph theory is then employed to analyze biomolecular properties in four major aspects: flexibility and rigidity analysis, protein mode analysis, protein domain decomposition and biomolecular nonlinear dynamics. Many other network based approaches, including GNM [@Bahar:1997; @Bahar:1998] and ANM [@Atilgan:2001], have been developed for protein flexibility analysis. More recently, FRI has been proposed as a matrix-decomposition-free method for flexibility analysis, [@KLXia:2013d; @Opron:2014]. The fundamental assumptions of the FRI method are as follows. Protein functions, such as flexibility, rigidity, and energy, are fully determined by the structure of the protein and its environment, and the protein structure is in turn determined by the relevant interactions. Therefore, whenever a native protein structure is available, there is no need to analyze protein flexibility and rigidity by tracing back to the protein interaction Hamiltonian. Consequently, the FRI bypasses the ${\cal O}(N^3)$ matrix diagonalization. In fact, FRI does not even require the 3D geometric information of the protein structure. It assesses graphic connectivity of the protein distance geometry and analyzes the geometric compactness of the protein structure. It can be regarded as a kernel generalization of the local density model [@Halle:2002; @DWLi:2009; @CPLin:2008]. Another very important application of biomolecular structure graph model is the protein mode analysis. As stated above, the low order eigenmodes provide information of the protein dynamics at equilibrium state. Normal mode analysis (NMA) plays important roles in mode analysis. However, its potential function involves too many interactions and it is very inefficient for large biomolecular systems. Anisotropic network model (ANM) has dramatically reduced the complexity of the potential function by representing the biological macromolecule as an elastic mass-and-spring network. In the network each node is a C$_\alpha$ atom of the associated residue and springs represent the interactions between the nodes. The overall potential is the sum of harmonic potentials between interacting nodes. To describe the internal motions of the spring connecting two atoms, there is only one degree of freedom. Qualitatively, this corresponds to the compression and expansion of the spring in a direction given by the locations of the two atoms. In other words, ANM is an extension of the GNM to three coordinates per atom, thus accounting for directionality.
The biomolecular structure graph models can also be used in protein domain decomposition. The biomolecular structural domains are stable and compact units of the structure that can fold independent of the rest of the protein and perform a specific function. A domain usually contains a hydrophobic core and a protein is usually formed by the combination of two or several domains. There are many methods that decompose a protein structure into domains [@alexandrov:2003; @Guo:2003; @holm:1996; @SKundu:2004; @murzin:1995; @orengo:1997; @veretnik:2007]. Some of them are done manually through structure visualization. One of them is the structural classification of proteins (SCOP) database, where data are largely manually classified into protein structural domains based on similarities of their structures and amino acid sequences. With the surge of protein structure data, efficient and robust computational algorithms are developed. They have demonstrated a high level of consistency and robustness in the process of partitioning a structure into domains. Graph theory is also used in RNA structure analysis, particularly in RNA motif representation and RNA classification[@Gan:2004rag; @Fera:2004rag; @Kim2004:candidates]. Spectral graph theory is widely used for clustering. The essential idea is to study and explore graphs through the eigenvalues and eigenvectors of matrices naturally associated with these graphs.
Molecular nonlinear dynamics (MND) models can be naturally derived from biomolecular graph models [@KLXia:2014b]. Essentially, each node in the graph is an atom and represented by a nonlinear oscillator. These nonlinear oscillators are connected through the graph connectivity. In this manner, one can study protein structure and function through the nonlinear dynamics theory widely used in chaos, synchronization, stability, pattern formation, etc.
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![Illustration of a graph. The associate adjacent, weight and Laplacian matrix can be found in Eq. (\[eq:graph\_matrix\]). []{data-label="fig:GRAPH"}](GRAPH.png "fig:"){width="30.00000%"}
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### Elementary graph theory {#sec:Graph}
Graph theory deals with a set of discrete vertices (or atoms) and their connectivity (or bonds). Normally, an undirected graph $G$ can be denoted as a pair $G(V,E)$, where $V= \{v_i;i=1,2,...,N \}$ denotes its set of $N$ vertices (or protein atoms), $N=|V|$. Here $E=\{e_i=(v_{i_1},v_{i_2});1\leq i_1 \leq N, 1\leq i_2 \leq N \}$ denotes its set of edges, which can be understood as certain covalent or noncovalent bonds among atoms in a molecule. Each edge in $E$ is an unordered pair of vertices, with the edge connecting distinct vertices $v_{i_1}$ and $v_{i_2}$ written as $e_i=(v_{i_1},v_{i_2})$. Then the adjacency matrix $A$ of $G$ is given by [@ChungOverview; @Mohar:1991laplacian; @Mohar:1997some; @von:2007tutorial] $$\begin{aligned}
\label{eq:couple_matrix28}
A_{ij}=\begin{cases} \begin{array}{ll}
1 & (v_i,v_j) \in E\\
0 & (v_i,v_j) \not \in E.\\
\end{array}
\end{cases}\end{aligned}$$ The degree of a vertex $v_i$ is defined as $d_i=\sum_{i \neq j}^N A_{ij}$, which is the total number of edges that are connected to node $v_i$. The degree matrix $D$ can be defined as $$\begin{aligned}
\label{eq:couple_matrix27}
D_{ij}=\begin{cases} \begin{array}{ll}
\sum_{i \neq j}^N A_{ij} & i=j\\
0 & i \neq j.
\end{array}
\end{cases}\end{aligned}$$ With these two matrices, one can defined Laplacian matrix as $L=D-A$. The Laplacian matrix is also known as admittance matrix, Kirchhoff matrix or discrete Laplacian. It is widely used to represent a graph. More specifically, it can be expressed as, $$\begin{aligned}
\label{eq:couple_matrix26}
L_{ij}=\begin{cases} \begin{array}{ll}
-1 & i \neq j~{\rm and} ~ (v_i,v_j) \in E\\
-\sum_{i \neq j}^N L_{ij} &i=j\\
0 &{ \rm otherwise.}
\end{array}
\end{cases}\end{aligned}$$ For example, the adjacency, degree and Kirchhoff matrices for the graph in Fig. \[fig:GRAPH\] are, respectively [$$\begin{aligned}
\label{eq:graph_matrix}
A=\left( \begin{array}{llllllll}
0 &1 &0 &0 &0 &0 &0 &1 \\
1 &0 &1 &0 &0 &0 &0 &0 \\
0 &1 &0 &1 &0 &0 &1 &0 \\
0 &0 &1 &0 &1 &0 &1 &0 \\
0 &0 &0 &1 &0 &1 &0 &0 \\
0 &0 &0 &0 &1 &0 &1 &0 \\
0 &0 &1 &1 &0 &1 &0 &1 \\
1 &0 &0 &0 &0 &0 &1 &0
\end{array}
\right),
D=\left( \begin{array}{llllllll}
2 &0 &0 &0 &0 &0 &0 &0 \\
0 &2 &0 &0 &0 &0 &0 &0 \\
0 &0 &3 &0 &0 &0 &0 &0 \\
0 &0 &0 &3 &0 &0 &0 &0 \\
0 &0 &0 &0 &2 &0 &0 &0 \\
0 &0 &0 &0 &0 &2 &0 &0 \\
0 &0 &0 &0 &0 &0 &4 &0 \\
0 &0 &0 &0 &0 &0 &0 &2
\end{array}
\right),
L=\left( \begin{array}{llllllll}
2 &$ -1$ &0 &0 &0 &0 &0 &$-1$ \\
$ -1$ &2 &$-1$ &0 &0 &0 &0 &0 \\
0 &$ -1$ &3 &$-1$ &0 &0 &$-1$ &0 \\
0 &0 &$-1$ &3 &$-1$ &0 &$-1$ &0 \\
0 &0 &0 &$-1$ &2 &-1 &0 &0 \\
0 &0 &0 &0 &$-1$ &2 &$-1$ &0 \\
0 &0 &$-1$ &$-1$ &0 &$-1$ &4 &$-1$ \\
$-1$ &0 &0 &0 &0 &0 &$-1$ &2
\end{array}
\right)\end{aligned}$$ ]{} The Laplacian matrix has several basic properties. It is a symmetric and semi-positive definite. The rank of the Laplacian matrix is $N-N_0$ with $N_0$ the number of connected components. Its second smallest eigenvalue is known as the algebraic connectivity (or Fiedler value)[@ChungOverview; @von:2007tutorial].
More generally, one can assign weights to edges to construct a weighted graph $G(V,E,W)$. Here $G(V,E)$ is the associated unweighted graph, and $W=\{w_{ij}; 1\leq i \leq N, 1\leq j \leq N, w_{ij}\geq 0\}$ is the weighted adjacent matrix. The weight is also known as pairwise distance or pairwise affinity. The new degree of vertex $v_i$ is $d_i=\sum_{j=1}^N w_{ij}$. The weighted degree matrix $D$ and weighted Laplacian matrix $L$ can be defined accordingly.
Normally, a graph structure is not given. Instead, one may have the information of nodes and general weight functions. In this case there are several general ways to construct a graph[@von:2007tutorial]:
1. $\epsilon$-neighborhood graph: connect all points whose pairwise distances are smaller than $\epsilon$;
2. $k$-nearest neighbor graph: connect vertex $v_i$ with vertex $v_j$, if $v_j$ is among the $k$-nearest neighbors of $v_i$; and
3. fully connected graph: connect all points with positive similarity with each other.
In biomolecular structure graph models, coordinates for atoms in molecules are available. Therefore, distances and distance-based functions can be used to construct structure graphs. The simplest way is to use a cutoff distance $r_c$ and build up edges between atoms or residues within the cutoff distance only. This approach has been used in GNM, which is an important tool for the study of protein flexibility and rigidity.
To unify the notation, in the following discussion, one can consider an $N$-particle representation of a biomolecule. Here a particle can be an ordinary atom in a full atomic representation or a C$_\alpha$ atom in a coarse-grained representation. One can denote $\{ {\bf r}_{i}| {\bf r}_{i}\in \mathbb{R}^{3}, i=1,2,\cdots, N\}$ the coordinates of these particles and $r_{ij}=\|{\bf r}_i-{\bf r}_j\|$ the Euclidean space distance between $i$th and $j$th particles. More specifically, the coordinate is a position vector ${\bf r}_i=(x_i, y_i, z_i)$.
### Gaussian network model (GNM) {#sec:GNM}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Illustration of a weighted graph Laplacian matrix for HIV capsid protein 1E6J. Left: six subdomains. Right: correlation map for residue C$_{\alpha}$ atoms indicating domain separations.[]{data-label="fig:correlation_matrix"}](correlation_matrix.png "fig:"){width="60.00000%"}
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Gaussian network model (GNM) [@Bahar:1997; @Bahar:1998; @QCui:2010; @LiGH:2002; @Yang:2006] can be viewed as a special graph model using the Kirchhoff matrix. It was proposed for biomolecular flexibility and long-time scale dynamics analysis, particularly, the prediction of the Debye-Waller factor or B-factor. Experimentally, B-factor is an indication of the relative thermal fluctuations of different parts of a structure. Atoms with small B-factors belong to a part of the structure that is very rigid. Atoms with large B-factors generally belong to part of the structure that is very flexible. The B-factor information can be found in the structural data downloaded from the Protein Data Bank (PDB).
As stated above, the graph or network in GNM is constructed by using a cutoff distance $r_c$. If the distance between two atoms are less than the cutoff distance, an edge is formed between them. Otherwise, no edge is built. The corresponding discrete Laplacian matrix describes the relative connectivity within a protein structure, and thus, it is also called a connectivity matrix. $$\begin{aligned}
\label{eq:couple_matrix25}
L_{ij}=\begin{cases} \begin{array}{ll}
-1 & i \neq j~{\rm and} ~ r_{ij} \leq r_c\\
-\sum_{i \neq j}^N L_{ij} &i=j\\
0 &{ \rm otherwise}
\end{array}.
\end{cases}\end{aligned}$$
In a nutshell, the GNM prediction of the $i$th B-factor of the biomolecule can be expressed as [@Bahar:1997; @Bahar:1998; @JKPark:2013] $$\begin{aligned}
\label{eqn:GNM}
B_i^{\rm GNM}=a \left(L^{-1} \right)_{ii}, \forall i=1,2,\cdots, N,\end{aligned}$$ where $a$ is a fitting parameter that can be related to the thermal energy and $\left(L^{-1} \right)_{ii}$ is the $i$th diagonal element of the Moore-Penrose pseudo-inverse of graph Laplacian matrix $L$. More specifically, $\left(L^{-1} \right)_{ii}=\sum_{k=2}^N \lambda_k^{-1}\left[{\bf q}_k {\bf q}_k^T \right]_{ii}$, where $T$ denotes the transpose and $\lambda_k$ and ${\bf q}_k$ are the $k$th eigenvalue and eigenvector of $\Gamma$, respectively. The summation omits the first eignmode whose eigenvalue is zero.
### Anisotropic network model (ANM) {#sec:ANM}
In Gaussian network model [@Bahar:1997; @Bahar:1998; @QCui:2010; @LiGH:2002; @Yang:2006], only the distance information is used with no consideration about the anisotropic properties in different directions. It should be noticed that in GNM, the Kirchhoff matrix is of the dimension $N*N$ with $N$ being the total number of atoms. In order to introduce the anisotropic information, one has to discriminate the distance between atoms in three different directions. To this end, at each label of $ij$, a local $3*3$ Hessian matrix is constructed [@Atilgan:2001] $$\begin{aligned}
\label{eq:multi-kirchoff1}
H_{ij} = -\frac{1}{r_{ij}^2}\left[ \begin{array}{ccc}
(x_j-x_i)(x_j-x_i) &(x_j-x_i)(y_j-y_i) &(x_j-x_i)(z_j-z_i)\\
(y_j-y_i)(x_j-x_i) &(y_j-y_i)(y_j-y_i) &(y_j-y_i)(z_j-z_i)\\
(z_j-z_i)(x_j-x_i) &(z_j-z_i)(y_j-y_i) &(z_j-z_i)(z_j-z_i)
\end{array}\right] ~ \forall ~ i \neq j ~{\rm and} ~ r_{ij}\leq r_c.
\end{aligned}$$ As the same in the GNM, the diagonal part is the negative summation of the off diagonal elements: $$\begin{aligned}
\label{eq:multi-kirchoff1_diagonal}
H_{ii} = -\sum_{i\neq j} H_{ij}.
\end{aligned}$$ This approach, called anisotropic network model (ANM), can be used to generate the anisotropic motion of biomolecules. It is noticed that the dimension of the Hessian matrix is no longer $N*N$, instead it is $3N * 3N$. The dimension of an eigenvector is $3N$. Therefore, for each atom, one now has a vector associated with it, which gives an direction in the ${\mathbb R}^3$. The norm of this vector gives a relative amplitude. This eigenvector is also called eigenmode. It describes the relative motion of the protein near its equilibrium state.
#### Generalized GNM and generalized ANM
In both Gaussian network model and anisotropic network model, a cutoff distance is used to construct their connectivity matrices, i.e., Laplacian matrix and Hessian matrix, respectively. However, physically, the correlation between any two particles normally decays with respect to distance. To account for this effect, a correlation function $\Phi(r_{ij}; \eta_{ij}) $ is introduced. In general, it is a real-valued monotonically decreasing radial basis function satisfying [@KLXia:2013d; @Opron:2014], $$\begin{aligned}
\label{eq:couple_matrix1-1}
\Phi( r_{ij};\eta_{ii})&=&1 \\ \label{eq:couple_matrix1-12}
\Phi( r_{ij};\eta_{ij})&=&0 \quad {\rm as }\quad r_{ij} \rightarrow\infty.\end{aligned}$$ In this function, the parameter $\eta_{ij}$ is a characteristic distance between particles $v_i$ and $v_j$. It can also be simplified to atomic parameter $\eta_{j}$, which depends only on the atomic type. In coarse-grained models, only $C_{\alpha}$ atom is considered. Therefore, one can further simplify it to a constant $\eta$. This parameter can also be viewed as a resolution parameter.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![ Protein domain separation of protein 3PGK C$_\alpha$ atoms using a new spectral clustering method, gGNM. The separation is carried out with the eigenvector corresponding to the second lowest eigenvalue.[]{data-label="fig:cp1"}](3PGK.png "fig:"){width="40.00000%"}
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Delta sequences of the positive type discussed in an earlier work [@GWei:2000] are all good choices. For example, one can use generalized exponential functions $$\begin{aligned}
\label{eq:couple_matrix1}
\Phi(r_{ij};\eta_{ij}) = e^{-\left(r_{ij}/\eta_{ij}\right)^\kappa}, \quad \kappa >0\end{aligned}$$ or generalized Lorentz functions $$\begin{aligned}
\label{eq:couple_matrix24}
\Phi(r_{ij};\eta_{ij}) = \frac{1}{1+ \left( r_{ij}/\eta_{ij}\right)^{\upsilon}}, \quad \upsilon >0.
\end{aligned}$$ Using these correlation functions, one can obtain a weighted graph representation or weighted graph Laplacian as [@KLXia:2015f], $$\begin{aligned}
\label{eq:couple_matrix23}
L_{ij}=\begin{cases} \begin{array}{ll}
-\Phi(r_{ij};\eta_{ij}) & i \neq j\\
- \sum_{i \neq j}^N L_{ij} &i=j\\
\end{array}.
\end{cases}\end{aligned}$$ It is found that the weighted graph can deliver a better prediction of B-factors. This weighted graph approach is called the generalized GNM (gGNM). Figure \[fig:cp1\] shows the protein domain separation obtained with gGNM.
The local Hessian matrix in Eq. (\[eq:multi-kirchoff1\]) can also be generated to consider the distance effect to obtain a generalized form [@KLXia:2015f], $$\begin{aligned}
\label{eq:multi-kirchoff12}
H_{ij} = -\frac{\Phi( r_{ij};\eta_{ij})}{r_{ij}^2}\left[ \begin{array}{ccc}
(x_j-x_i)(x_j-x_i) &(x_j-x_i)(y_j-y_i) &(x_j-x_i)(z_j-z_i)\\
(y_j-y_i)(x_j-x_i) &(y_j-y_i)(y_j-y_i) &(y_j-y_i)(z_j-z_i)\\
(z_j-z_i)(x_j-x_i) &(z_j-z_i)(y_j-y_i) &(z_j-z_i)(z_j-z_i)
\end{array}\right] ~ \forall ~ i \neq j.
\end{aligned}$$ Again the diagonal part is the negative summation of the off-diagonal elements the same as Eq. (\[eq:multi-kirchoff1\_diagonal\]). Note that Hinsen [@Hinsen:1998] has proposed a special case: $\Phi( r_{ij};\eta_{ij} )= e^{-\left(\frac{r_{ij}}{\eta }\right)^2}$, where $\eta$ is a constant. It was shown that gGNM and generalized anisotropic network model (gANM) outperform the original GNM and ANM respectively in B-factor predictions [@KLXia:2015f]. Figure \[fig:2ABH\] illustrates an eigenmode for protein 2ABH obtained with gANM. There will be a continuous interest in design new and optimal graph theory approaches for biomolecular analysis.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Illustration of eigenmode for protein 2ABH. The eigenmode can be used to describe the biomolecular dynamics near equilibrium state. The eigenmode is generated by the generalized anisotropic normal model method.[]{data-label="fig:2ABH"}](2ABH.png "fig:"){width="30.00000%"}
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### Normal mode analysis (NMA) and quasi-harmonic analysis
Normal mode analysis (NMA) is one of the major tools for the study of biomolecular motions [@Go:1983; @Levitt:1983normal; @Brooks:1983harmonic; @Lopez:2014normal; @Hayward:2008normal]. It is found that protein normal modes with the largest fluctuation or the lowest frequency are functionally relevant. Mathematically, NMA has its root in harmonic analysis. It assumes that conformational energy surface at an energy minimum can be approximated by some harmonic functions.
In normal mode analysis, one usually needs the atomic coordinates and a force field describing the interactions between constituent atoms. Typically, there are three major steps in applying NMA [@Hayward:2008normal]. Firstly, one needs to carry out molecular dynamics simulations to minimize the conformational potential energy to obtain an equilibrium state. Secondly, one needs to calculate the second derivatives of the potential energy to construct the Hessian matrix. Finally, one needs to perform the eigenvalue decomposition of the Hessian matrix.
Originally, NMA uses exactly the same force fields as used in molecular dynamics simulations. Due to the computational inefficiency, elastic network models (ENMs) was proposed. Generally speaking, ENM is just the NMA with a simplified force field and associated coarse-grained representation. It has two major advantages. Firstly, there is no need for energy minimization as the distances of all of the elastic connections are taken to be at their minimal energy lengths. Secondly, due to the coarse-grained representation, the eigenvalue decomposition is much efficient. Even through ENM is a much simplified model, it is found that ENM is able to reproduce the NMA results with a respectable degree of similarity.
#### Standard NMA
For a mechanical system consisting of $N$ atoms ${\bf r}=({r}_1,{ r}_2,\cdots, { r}_{3N})$, its Hamiltonian $\mathcal{H}({\bf r})$ is given by the sum of kinetic energy $\mathcal{K}({\bf r})$ and potential energy $\mathcal{U}({\bf r})$: $$\begin{aligned}
\mathcal{H}({\bf r})=\mathcal{K}({\bf r})+\mathcal{U}({\bf r}).\end{aligned}$$
If the structure has an equilibrium conformation ${\bf r}^0=({r}_1^0,{ r}_2^0,...,{ r}_{3N}^0)$, one can have the Taylor expansion of the potential energy $$\begin{aligned}
\mathcal{U}({\bf r}) \approx \mathcal{U}({\bf r}^0)+ \sum_{i}^{3N}\frac{\partial \mathcal{U}}{\partial r_i} \Big| _{\bf r=r^0}(r_i-r_i^0)+ \frac{1}{2}\sum_{i}^{3N}\sum_{j}^{3N}\frac{\partial^2 \mathcal{U}}{\partial r_i \partial r_j} \Big|_{\bf r=r^0}(r_i-r_i^0)(r_j-r_j^0)+\cdots.\end{aligned}$$ Since the biomolecular system achieves a minimum of the energy at the equilibrium conformation ${\bf r}^0$, the related derivative functions $\frac{\partial \mathcal{U}}{\partial r_i}\Big| _{\bf r=r^0}$ vanishes. If one uses the mass-weighted coordinates $X_i=m_i^{\frac{1}{2}} (r_i-r_i^0)$, the potential function becomes $$\begin{aligned}
&& \mathcal{U}({\bf r}) \approx \frac{1}{2}\sum_{i}^{3N}\sum_{j}^{3N}\frac{\partial^2 \mathcal{U}}{\partial r_i \partial r_j} \Big|_{\bf r=r^0}(r_i-r_i^0)(r_j-r_j^0) \\ \nonumber
&& \qquad \approx \frac{1}{2}\sum_{i}^{3N}\sum_{j}^{3N}\frac{\partial^2 \mathcal{U}}{\partial r_i \partial r_j} \Big|_{\bf r=r^0}(r_i-r_i^0)(r_j-r_j^0).\end{aligned}$$ The related kinetic energy is $$\begin{aligned}
\mathcal{K}({\bf r})=\frac{1}{2}\sum_{i}^{3N}m_i\left(\frac{dr_i}{dt}\right)^2=\frac{1}{2}\sum_{i}^{3N}\dot{X}_i^2.\end{aligned}$$ The Hamiltonian is $$\begin{aligned}
\mathcal{H}({\bf r}) \approx \frac{1}{2}\sum_{i}^{3N}\sum_{j}^{3N} \dot{X}_i^2+\frac{1}{2}\sum_{i}^{3N}\sum_{j}^{3N}\frac{\partial^2\mathcal{ U}}{\partial X_i \partial X_j} \Big|_{ X=X^0}(X_i-X_i^0)(X_j-X_j^0),\end{aligned}$$ where $\dot X$ indicates the derivative of $ X$ with respect to time.
It can be seen that the oscillatory motions in this system are coupled and thus the movement of one atom depends on that of others. However, one can decompose the motion into independent harmonic oscillators with an appropriate normal mode coordinates. This is done by the eigenvalue decomposition of the Hessian matrix $H=Q\Lambda Q^T$. Here $H$ is obtained from the second order derivative of the potential function. The matrix $Q=\{{ {\bf q}_1,{\bf q}_2,...,{\bf q}_{3N}} \}$ contains the eigenvectors and $Q^T Q=I$. The diagonal matrix $\Lambda$ contains the corresponding eigenvalues. The mass-weighted Cartesian and normal mode coordinates are linearly related by $X=QY$. Finally, the Hamiltonian is expressed in the form, $$\begin{aligned}
\mathcal{H} \approx \frac{1}{2}\sum_{i}^{3N} \dot{Y}_i^2+\frac{1}{2}\sum_{i}^{3N}\lambda_i {Y}_i^2.\end{aligned}$$
#### Essential dynamics and quasi-harmonic analysis
Due to the complexity of biomolecular systems, it is notoriously difficult to carry out molecular dynamics simulations over the relevant biological time scales. However, it has been found that the vast majority of protein dynamics can be described by a surprisingly low number of collective degrees of freedom. In this manner, a principal components analysis (PCA) is often employed to analyze the simulation results[@Garcia:1992large; @Kitao:1991effects]. Mathematically, similar to NMA, PCA also employs the eigenvalue decomposition as it assume that that the major collective modes of fluctuation dominate the functional dynamics. In contrast to NMA, PCA of a molecular dynamics simulation trajectory does not rest on the assumption of a harmonic potential. Modes in PCA are usually sorted according to variance rather than frequency. As the collective motion is highly related to biomolecular functions, the dynamics in the low-dimensional subspace spanned by these modes was termed “essential dynamics” [@Amadei:1993essential]. A major advantage of PCA is that individual modes can be visualized and studied separately.
PCA on the mass weighted MD trajectory is also called quasi-harmonic analysis [@Brooks:1995harmonic], which typically consists of three steps [@Hayward:2008normal]. Firstly, one can superimpose all biomolecular configurations from the simulation trajectory to remove the internal rotation and translation. Secondly, one can perform an average over the regularized trajectory to construct a covariance matrix. Thirdly, an eigenvalue decomposition is employed on the covariance matrix. The original trajectory can then be analyzed in terms of principal components.
For a $3N$-dimensional vector trajectory $\bf {r}(t)$, the correlation between atomic motions can be expressed in the covariance matrix $C$: $$\begin{aligned}
C={\rm cov}({\bf r})=<\left({\bf r}(t)-<{\bf r}(t)>\right)\cdot\left({\bf r}(t)-<{\bf r}(t)>\right)>\end{aligned}$$ where ${\rm cov}$ is the statistical covariance and $< >$ denote the average over time. The correlation matrix is symmetric and can be diagonalized by an orthogonal transformation, $$\begin{aligned}
C=Q'\Lambda (Q')^T\end{aligned}$$ with $Q'=\{{\bf q}'_1,{\bf q}'_2,...,{\bf q}'_{3N} \}$ being eigenvectors. The original configurations can be projected into principal components ${\bf q}_i$, i.e., ${q}'_i(t)=({\bf r}(t)-<{\bf r}(t)>)\cdot {\bf q}_i$. For visualization, one can transform principal components into the Cartesian coordinates: $ {\bf r}'(t)={q}'_i(t){\bf q}_i+<{\bf r}(t)>$.
### Flexibility rigidity index (FRI) {#sec:FRI}
Due to the involved matrix diagonalization, the computational complexity of GNM is of the order of ${\cal O}(N^3)$, which is intractable for large biomolecules, such as viruses and subcellular organelles. Therefore, it is both important and desirable to have a method whose computational complexity scales as ${\cal O}(N^2)$ or better, as ${\cal O}(N)$. This order reduction is a standard mathematical issue and is mathematically challenging. However, by examining Eq. (\[eqn:GNM\]), one notices that what is used in the GNM theoretical prediction is the diagonal elements of the inverse of the graph Laplacian matrix. Mathematically, a good approximation is given by the inverse of the diagonal elements of the graph Laplacian matrix, providing that the matrix is diagonally dominant. Flexibility rigidity index (FRI) [@KLXia:2013d; @Opron:2014] is such a method and has several major characteristics. FRI provides a more straightforward and computationally-efficient way to predict B-factors. A major advantage of the FRI method is that it does not resort to mode decomposition and its computational complexity can be reduced to ${\cal O}(N)$ by means of the cell lists algorithm used in fast FRI (fFRI) [@Opron:2014]. The fundamental assumptions of the FRI method are as follows. Protein functions, such as flexibility, rigidity, and energy, are fully determined by the structure of the protein and its environment, and the protein structure is in turn determined by the relevant interactions. Therefore, whenever the protein structural data is available, there is no need to analyze protein flexibility and rigidity by tracing back to the protein interaction Hamiltonian. Consequently, the FRI bypasses the ${\cal O}(N^3)$ matrix diagonalization.
In a nutshell, the FRI prediction of the $i$th B-factor of the biomolecule can be given by [@KLXia:2013d; @Opron:2014] $$\begin{aligned}
\label{eqn:FRI}
B_i^{\rm FRI}=a \frac{1}{\sum_{j,j\neq i}^N w_j\Phi(r_{ij};\eta_{ij})} + b, \forall i=1,2,\cdots, N,\end{aligned}$$ where $a$ and $b$ are fitting parameters, $f_i=\frac{1}{\sum_{j,j\neq i}^N w_j\Phi(r_{ij};\eta_{ij})}$ is the $i$th flexibility index and $$\begin{aligned}
\label{rigidity1}
\mu_i=\sum_{j,j\neq i}^N w_j\Phi(r_{ij};\eta_{ij})\end{aligned}$$ is the $i$th rigidity index. Here, $w_j$ is an atomic number depended weight function that can be set to $w_j=1$ and $\eta_{ij}=\eta$ for a C$_{\alpha}$ network. The correlation function $\Phi( r_{ij};\eta)$ can be chosen from any monotonically decreasing function satisfying Eqs. (\[eq:couple\_matrix1\]) and (\[eq:couple\_matrix2\]). FRI was shown to outperform GNM and ANM in B-factor predictions based on hundreds of biomolecules [@KLXia:2013d; @Opron:2014].
#### Multiscale FRI {#sec:MFRI}
Biomolecules are inherently multiscale in nature due to their multiscale interactions. For example, proteins involve covalent bonds, hydrogen bonds, van der Walls bonds, electrostatic interactions, dipolar and quadrupole interactions, hydrophobic interactions, domain interactions, and protein-protein interactions. Therefore, their thermal motions are influenced by the multiscale interactions among their particles. Multiscale FRI (mFRI) was proposed to capture biomolecular multiscale behavior [@Opron:2015a]. Essential idea is to build multiscale kernels, i.e., kernels parametrized at multiple scales. Multiscale flexibility index can be expressed as $$\begin{aligned}
\label{eq:flexibility3}
f^{n}_i = \frac{1}{\sum_{j=1}^N w^{n}_{j} \Phi^{n}( \|{\bf r}_i - {\bf r}_j \|;\eta^{n} )},
\end{aligned}$$ where $w^{n}_{j}$, $\Phi^{n}( \|{\bf r}_i - {\bf r}_j \|;\eta^{n}) $ and $\eta^{n}$ are the corresponding quantities associated with the $n$th kernel. Then, one organizes these kernels in a multi-parameters minimization procedure $$\begin{aligned}
\label{eq:regression2}
{\rm Min}_{a^{n},b} \left\{ \sum_i \left| \sum_{n}a^n f^{n}_i + b-B^e_i\right|^2\right\}\end{aligned}$$ where $\{B^e_i\}$ are the experimental B-factors. In principle, all parameters can be optimized. For simplicity and computational efficiency, one only needs to determine $\{a^n\}$ and $b$ in the above minimization process. For each kernel $\Phi^n$, $w^n_j$ and $\eta^n_j$ will be selected according to the type of particles.
Specifically, for a simple C$_\alpha$ network (graph), one can set $w^n_j=1$ and choose a single kernel function parametrized at different scales. The predicted B-factors can be expressed as $$\begin{aligned}
\label{eq:flexibility4}
B^{\rm mFRI}_i = b+ \sum_{n=1}\frac{a^n}{\sum_{j=1}^N \Phi( \|{\bf r}_i - {\bf r}_j \|;\eta^{n} )}.
\end{aligned}$$ The difference between Eqs. (\[eq:flexibility3\]) and (\[eq:flexibility4\]) is that, in Eqs. (\[eq:flexibility3\]), both the kernel and the scale can be changed for different $n$. In contrast, in Eq. (\[eq:flexibility4\]), only the scale is changed. One can use a given kernel, such as $$\begin{aligned}
\label{eq:couple_matrixn}
\Phi(\|{\bf r} - {\bf r}_j \|;\eta^n) = \frac{1}{1+ \left( \|{\bf r} - {\bf r}_j \|/\eta^n\right)^{3}},
\end{aligned}$$ to achieve good multiscale predictions. It was demonstrated that mFRI is about 20% more accurate than GNM in the B-factor predictions [@Opron:2015a]. Parameters learned from mFRI were incorporated in GNM and ANM to create multiscale GNM (mGNM) and multiscale ANM (mANM) [@Opron:2015a].
#### Consistency between GNM and FRI {#sec:MFRI2}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![A comparison of B-factor prediction of protein 1CLL by various models, including flexibility rigidity index (FRI), Gaussian network model (GNM), generalized GNM (gGNM) and multiscale FRI (mFRI). Experimental results (Exp.) are given as a reference.[]{data-label="fig:bfactor"}](bfactor.png "fig:"){width="70.00000%"}
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To further explore the relation between GNM and FRI, let us examine the parameter limits of generalized exponential functions (\[eq:couple\_matrix1\]) and generalized Lorentz functions (\[eq:couple\_matrix2\]) $$\begin{aligned}
\label{eq:Asmpt1}
e^{-\left( r_{ij} /\eta\right)^\kappa} \rightarrow \Phi( r_{ij};r_c) & {\rm as } & \kappa\rightarrow\infty\\ \label{eq:Asmpt2}
\frac{1}{1+ \left( r_{ij} /\eta\right)^{\upsilon}} \rightarrow \Phi( r_{ij};r_c) & {\rm as } &\upsilon\rightarrow\infty,\end{aligned}$$ where $r_c=\eta$ and $\Phi( r_{ij};r_c) $ is the ideal low-pass filter (ILF) used in the GNM Kirchhoff matrix $$\begin{aligned}
\label{eqn:IdealLF}
\Phi( r_{ij};r_c) = \begin{cases}\begin{array}{ll}
1, & r_{ij} \leq r_c \\
0, & r_{ij} > r_c \\
\end{array}
\end{cases}.\end{aligned}$$ Relations (\[eq:Asmpt1\]) and (\[eq:Asmpt2\]) unequivocally connect FRI correlation functions to the GNM Kirchhoff matrix.
It has been observed that GNM-ILF and FRI-ILF provide essentially identical predictions when the cutoff distance is equal to or larger than 20Å[@KLXia:2015f]. This phenomenon indicates that when the cutoff is sufficiently large, the diagonal elements of the gGNM inverse matrix and the direct inverse of the diagonal elements of the FRI correlation matrix become linearly strongly dependent. To understand this dependence at a large cutoff distance, an extreme case is considered when the cutoff distance is equal to or even larger than the protein size so that all the particles within the network are fully connected. In this situation, one can analytically calculate $i$th diagonal element of the GNM inverse matrix $$\begin{aligned}
\label{eqn:proprotion0}
\left(L (\Phi( r_{ij};r_c\rightarrow\infty)) \right)_{ii} = \frac{N-1}{N^2},\end{aligned}$$ and the FRI inverse of the $i$th diagonal element $$\begin{aligned}
\label{eqn:proprotion01}
\frac{1}{\sum_{j,j\neq i}^N \Phi(r_{ij};r_c\rightarrow\infty)}= \frac{1}{N-1}.\end{aligned}$$ These two expressions have the same asymptotic behavior as $N\rightarrow\infty$, which explains numerical results. However, mathematically, it is still an open problem to estimate the error bound between gGNM and FRI methods, i.e., the difference between the direct inverse of a diagonal element of a given weighted graph Laplacian matrix and the diagonal element of its matrix inverse.
A comparison of the performances of GNM, FRI, gGNM and mFRI is illustrated in Fig. \[fig:bfactor\]. The hinge around 75th residue is well captured by mFRI. Indeed, mFRI has the best accuracy in B-factor prediction based on the test of 364 proteins [@Opron:2015a]. The possible application to FRI in other systems, such as social networks, genetic networks, cellular networks and tissue networks, is still an open problem.
#### Anisotropic FRI
The anisotropic FRI (aFRI) has been proposed for mode analysis [@Opron:2014]. In this model, depending on one’s interest, the size of the Hessian matrix can vary from $3\times 3$ for a completely local aFRI to $3N\times 3N$ for a completely global aFRI. To construct such a Hessian matrix, one can partition all $N$ atoms in a molecule into a total of $M$ clusters $\{c_1,c_2,\dots,c_M\}$. Each cluster $c_k$ with $k=1,\dots,M$ has $N_k$ atoms so that $N=\sum_{k=1}^{M} N_k$. For convenience, one can denote $$\begin{aligned}
\label{eq:Anisorigidity1}
\Phi^{ij}_{uv} = \frac{\partial}{\partial u_i} \frac{\partial}{\partial v_j} \Phi( \|{\bf r}_i - {\bf r}_j \|; \eta_{ij} ), \quad u,v= x, y, z; i,j =1,2,\cdots,N.\end{aligned}$$ Note that for each given $ij$, one can define $\Phi^{ij}=\left( \Phi^{ij}_{uv} \right)$ as a local anisotropic matrix $$\label{eq:afri_local_Hessian}
\Phi^{ij}=\left(
\begin{array}{ccc}
\Phi^{ij}_{xx} & \Phi^{ij}_{xy}& \Phi^{ij}_{xz}\\
\Phi^{ij}_{yx} & \Phi^{ij}_{yy}& \Phi^{ij}_{yz}\\
\Phi^{ij}_{zx} & \Phi^{ij}_{zy}& \Phi^{ij}_{zz}
\end{array}
\right).$$
In the anisotropic flexibility and rigidity (aFRI) approach, a flexibility Hessian matrix ${\bf F}^{1}(c_k)$ for cluster $c_k$ is defined by $$\begin{aligned}
\label{eq:Anisoflexibility}
{\bf F}^{1}_{ij}(c_k) =& - \frac{1}{w_{j}} {\rm adj}(\Phi^{ij}), &\quad i,j \in c_k; i\neq j; u,v= x, y, z \\ \label{eq:Anisoflexibilityy3}
{\bf F}^{1}_{ii}(c_k)=& \sum_{j=1}^N \frac{1}{w_{j}} {\rm adj}(\Phi^{ij}), &\quad i \in c_k; u,v= x, y, z \\ \label{eq:Anisoflexibility4}
{\bf F}^{1}_{ij}(c_k)=& 0, &\quad i,j \notin c_k; u,v= x, y, z,\end{aligned}$$ where ${\rm adj}(\Phi^{ij})$ denotes the adjoint of matrix $\Phi^{ij}$ such that $\Phi^{ij} {\rm adj}(\Phi^{ij})=| \Phi^{ij}|I$, here $I$ is identity matrix.
Another representation for the flexibility Hessian matrix ${\bf F}^{2}(c_k)$ can be defined as follows $$\begin{aligned}
\label{eq:Anisoflexibility5}
{\bf F}^{2}_{ij}(c_k) =& - \frac{1}{w_{j}} |\Phi^{ij}|(J_{3} - \Phi^{ij}), &\quad i,j \in c_k; i\neq j; u,v= x, y, z \\ {\bf F}^{2}_{ii}(c_k)=& \sum_{j=1}^N \frac{1}{w_{j}} |\Phi^{ij}|(J_3 - \Phi^{ij}), &\quad i \in c_k; u,v= x, y, z \\ {\bf F}^{2}_{ij}(c_k)=& 0, &\quad i,j \notin c_k; u,v= x, y, z,
\end{aligned}$$ where $J_3$ is a $3\times3$ matrix with every element being one.
One can achieve $3N_k$ eigenvectors for $N_k$ atoms in cluster $c_k$ by diagonalizing ${\bf F}^{\alpha}(c_k)$, $\alpha=1,2$. Note that, the diagonal part ${\bf F}^{\alpha}_{ii}(c_k)$, $\alpha=1,2$, has inherent information of all atoms in the system. As a result, the B-factors can be predicted by the following form: $$\begin{aligned}
\label{eq:Anisoflexibility2}
f_i^{\rm AF_\alpha} &=&{\rm Tr} \left({\bf F}^{\alpha}(c_k)\right)^{ii}, \\
&=& \left({\bf F}^{\alpha}(c_k)\right)^{ii}_{xx}+ \left({\bf F}^{\alpha}(c_k)\right)^{ii}_{yy}+ \left({\bf F}^{\alpha}(c_k)\right)^{ii}_{zz}, \quad \alpha=1,2.\end{aligned}$$ It was found that aFRI is much more accurate than ANM in protein B-factor prediction [@Opron:2014]. The anisotropic cluster analysis was found to play a significant role in study the local motion of RNA polymerase II translocation [@Opron:2016a].
### Spectral graph theory {#sec:spectral}
Spectral graph theory [@ChungOverview; @Mohar:1991laplacian; @Mohar:1997some; @von:2007tutorial] concerns the study and exploration of graphs through the eigenvalues and eigenvectors of matrices naturally associated with those graphs [@shi:2000; @meila:2001; @ng:2002; @azran:2006; @zelnik:2005]. Therefore, widely used GNM and ANM methods make use of spectral graph theory.
For a given graph $G(V,E)$ with $N$ nodes, one is interested in its matrix representation. Matrices $A$ and $D$ correspond to weighted adjacent matrix and weighted degree matrix, respectively. With this notation, one has the unnormalized graph Laplacian $$L=D-A.$$ It is often called admittance matrix, Kirchhoff matrix or discrete Laplacian. The graph matrix has several interesting properties. Firstly, it is symmetric and positive semi-definite. Secondly, it has $N$ non-negative, real-valued eigenvalues. Thirdly, the smallest eigenvalue is 0 and the corresponding eigenvector is constant vector $\mathds{1}$. Fourthly, for every vector ${\bf c} \in \mathbb{R}^N$, one has ${\bf c}^T L {\bf c}=\frac{1}{2}\sum_{i,j}w_{ij}(c_i-c_j)^2$, which can be derived from $$\begin{aligned}
\label{eq:couple_matrix22}
&& {\bf c}^T L {\bf c}= {\bf c}^TD{\bf c} -{\bf c}^T A{\bf c}= \sum_{i}c_i^2 d_i- \sum_{i,j}c_i c_i w_{ij}\\\nonumber
&& =\frac{1}{2}\left( \sum_{i}c_i^2 d_i-2\sum_{i,j}c_i c_j w_{ij}+ \sum_{j}c_j^2 d_j \right)=\frac{1}{2}\sum_{i,j}w_{ij}(c_i-c_j)^2,\end{aligned}$$ where $d_i=\sum_{j=1}^N w_{ij}$ is the degree of the $i$th vertex.
In spectral graph theory, two other kinds of Laplacian matrices [@ChungOverview] are also widely used. They are the normalized Laplacian matrix $$L_{\rm sym}=I-D^{-\frac{1}{2}}AD^{-\frac{1}{2}},$$ and random-walk normalized Laplacian matrix [@meila:2001; @Lovasz:1993; @Aldous:2002] $$L_{\rm rw}=I-D^{-1}A,$$ where $I$ is an identity matrix. Three different Laplacian matrices are tightly related to different ways of graph decompositions.
#### Graph decomposition and graph cut
A protein may have different domains. Identifying protein domains and analyzing their relative motions are important for studying protein functions. A protein complex involves different proteins. The study of protein complex can often be formulated as a graph decomposition problem as well.
In many situations, for a given graph $G(V,E)$, one wants to partition it into subgraphs, so that nodes within a subgroup are of similar properties and nodes in different subgroups are of different properties. Mathematically, if the weight is a measurement of similarity, i.e., large weight means a great similarity, an optimized partition means that edges within the same subgroup should have large weights and edges across subgroups should have small weights. State differently, one wants to find a way to cut the graph so that it will minimize the weights of edges connecting vertices in different subgroups.
Let us first consider a simple situation, divide $G$ into two subgroup $G_1$ and $\bar{G_1}$. The notation $\bar{G_1}$ denotes the complementary of ${G_1}$. One can define a partition ${\bf c}=\{c_i; i=1,2,...,N\}$ as, $$\begin{aligned}
\label{eq:couple_matrix2}
c_i=\begin{cases} \begin{array}{ll}
1 & {\rm if} ~i \in G_1,\\
-1 & {\rm if} ~i \in \bar{G_1}.\\
\end{array}
\end{cases}\end{aligned}$$ Therefore, if two nodes $v_i$ and $v_j$ are in the same subgroup, one will have $(c_i-c_j)^2=0$, otherwise $(c_i-c_j)^2=4$. In this way, a ${\rm Cut}(G_1,\bar{G_1})$, which is the total weights of edges connecting two subgroups, can be defined as following, $$\begin{aligned}
\label{eq:couple_matrix222}
&& {\rm Cut}(G_1,\bar{G_1})={\bf c}^T L {\bf c}= {\bf c}^TD{\bf c} -{\bf c}^T A{\bf c}= \sum_{i}c_i^2 d_i- \sum_{i,j}c_i c_i w_{ij}\\\nonumber
&& =\frac{1}{2}\left( \sum_{i}c_i^2 d_i-2\sum_{i,j}c_i c_j w_{ij}+ \sum_{j}c_j^2 d_j \right)=\frac{1}{4}\sum_{i,j}w_{ij}(c_i-c_j)^2.\end{aligned}$$ It can be also noticed that if one defines $W(G_1,\bar{G_1})=\sum_{i\in G_1,j \in \bar{G_1}} w_{ij}$, then one has $ {\rm Cut}(G_1,\bar{G_1})=W(G_1,\bar{G_1})$.
To obtain an optimized partition means to minimize the value of $ {\rm Cut}(G_1,\bar{G_1})$. However, the way of cut stated above does not consider the size of the subgraphs. In this way, the cut or graph composition can be very uneven in terms of the number of nodes in subgroup. For example, one extreme situation is that one of the subgraph may only have a few nodes (i.e., one or two nodes), while the other subgroup may have all the rest of nodes. To avoid this problem, three commonly defined cuts, namely, ratio cut, normalized cut and min-max cut, are proposed in the literature [@Hagen:1992new; @shi:2000; @Ding:2001min]. To facilitate the description, one can define $|G|$ as the total number of nodes in graph $G$ and ${\rm vol}(G)$ is the summation of all weights in $G$, i.e., ${\rm vol}(G)=\sum_{ij}w_{ij}$.
- Ratio cut is defined as [@Hagen:1992new] $$\begin{aligned}
{\rm Rcut}(G_1,\bar{G_1})=\frac{W(G_1,\bar{G_1})}{|G_1|} + \frac{W(G_1,\bar{G_1})}{|\bar{G_1}|}.\end{aligned}$$
- Normalized cut is defined as [@shi:2000] $$\begin{aligned}
{\rm Ncut}(G_1,\bar{G_1})=\frac{W(G_1,\bar{G_1})}{W(G_1,G_1)+W(G_1,\bar{G_1})} + \frac{W(G_1,\bar{G_1})}{W(\bar{G_1},\bar{G_1})+W(\bar{G_1},G_1)}.\end{aligned}$$
- Min-Max cut is given by [@Ding:2001min] $$\begin{aligned}
{\rm Mcut}(G_1,\bar{G_1})=\frac{W(G_1,\bar{G_1})}{W(G_1,G_1)} + \frac{W(G_1,\bar{G_1})}{W(\bar{G_1},\bar{G_1})}.\end{aligned}$$
These decompositions have found many applications in image segmentation [@shi:2000]. However, the impact of these cuts to protein domain partition is yet to be examined. An important issue is how to cut a given biomolecule to elucidate its biological function and predict its chemical and biological behavior.
#### Ratio cut and Laplaician matrix
To solve the optimization problem $$\begin{aligned}
\min \limits_{G_1 \subset G} {\rm Rcut}(G_1,\bar{G_1}),\end{aligned}$$ one can define the vector $\bf {c}$ as $$\begin{aligned}
\label{eq:indicator}
c_i=\begin{cases} \begin{array}{ll}
\sqrt{\frac{\bar{|G_1|}}{|G_1|}} & {\rm if} ~i \in G_1,\\
-\sqrt{\frac{|G_1|}{\bar{|G_1|}}} & {\rm if} ~i \in \bar{G_1}.\\
\end{array}
\end{cases}\end{aligned}$$ One can have $$\begin{aligned}
{\bf c}^T L {\bf c}&=&\frac{1}{2}\sum_{i,j}w_{ij}(c_i-c_j)^2 \\\nonumber
&=& \frac{1}{2}\sum_{i\in G_1,j \in \bar{G_1}} \left( \sqrt{\frac{\bar{|G_1|}}{|G_1|}} + \sqrt{\frac{|G_1|}{\bar{|G_1|}}} \right)^2 + \frac{1}{2}\sum_{i\in \bar{G_1},j \in G_1} \left( -\sqrt{\frac{\bar{|G_1|}}{|G_1|}} - \sqrt{\frac{|G_1|}{\bar{|G_1|}}} \right)^2 \\\nonumber
&=& W(G_1,\bar{G_1}) \left( \frac{\bar{|G_1|}}{|G_1|} + \frac{|G_1|}{\bar{|G_1|}} +2 \right) \\\nonumber
&=& W(G_1,\bar{G_1}) \left( \frac{\bar{|G_1|}+|G_1|}{|G_1|} + \frac{|G_1|+\bar{|G_1|}}{\bar{|G_1|}} \right) \\\nonumber
&=& |G| {\rm Rcut}(G_1,\bar{G_1})\end{aligned}$$ One also has $$\begin{aligned}
\sum_{i} c_i= \sum_{i\in G_1} \sqrt{\frac{\bar{|G_1|}}{|G_1|}} - \sum_{i \in \bar{G_1}}\sqrt{\frac{|G_1|}{\bar{|G_1|}}}=0\end{aligned}$$ Therefore, the vector ${\bf c}$ is orthogonal to the vector with common components (leading eigenvector of the Laplacian matrix). It is noted that $$\begin{aligned}
\|f\|^2= \sum_{i} c_i^2= \sum_{i\in G_1} \frac{\bar{|G_1|}}{|G_1|} - \sum_{i \in \bar{G_1}}\frac{|G_1|}{\bar{|G_1|}}=N.\end{aligned}$$ In this way, the minimization is equivalent to $$\begin{aligned}
\label{Eq:min}
\{ \min {\bf c}^T L {\bf c}~|~ {\bf c}~ {\rm satisfies ~ Eq. ~(\ref{eq:indicator})};~ {\bf c} \bot \mathds{1};~ \|f\|^2=N \}.\end{aligned}$$ As the entries of the solution vector are only allowed to take two particular values, Eq. (\[Eq:min\]) is a discrete optimization problem. One can discharge the discreteness condition and allow the vector to take any arbitrary values. This results in a relaxed problem $$\begin{aligned}
\{ \min \limits_{{\bf c}\in \mathbb{R}^N}{\bf c}^T L {\bf c}~|~ {\bf c} \bot \mathds{1};~ \|f\|^2=N \}.\end{aligned}$$ From the Rayleigh-Ritz theorem, it can be seen that the solution of this problem is the second smallest eigenvector ${\bf q}_2$ of the Laplacian matrix. In order to obtain a partition of the graph, one can choose, $$\begin{aligned}
\label{eq:cluster_eigenv2}
\begin{cases} \begin{array}{ll}
i \in G_1 (c_i= 1) &{\rm if} ~ ({\bf q}_2)_i\geq 0\\
i \in \bar{G_1} ( c_i=-1) &{\rm if} ~({\bf q}_2)_i<0
\end{array}.
\end{cases}\end{aligned}$$ Mathematically, the second smallest eigenvalue $\lambda_2$ is known as the algebraic connectivity (or Fiedler value) of a graph. The corresponding eigenvector of the second eigenvalue offers a near optimized partition. It becomes an interesting issue to design certain weighted Laplacian matrix so that a protein domain partition is optimal with respect to protein functions. Transferring the discrete Laplacian matrix to a continuous Laplacian operator and casting the domain separation problem into an optimization one are promising approaches. Certainly, these issues are also biologically significant in the exploration of protein structure-function relationship.
#### Normalized cut and normalized Laplacian matrix
One can define the vector ${\bf c}$ as $$\begin{aligned}
\label{eq:normalized_indicator}
c_i=\begin{cases} \begin{array}{ll}
\sqrt{\frac{{\rm vol}(\bar{G_1})}{{\rm vol}(G_1)}} & {\rm if} ~i \in G_1\\
-\sqrt{\frac{{\rm vol}(G_1)}{{\rm vol}(\bar{G_1})}} & {\rm if} ~i \in \bar{G_1}
\end{array}.
\end{cases}\end{aligned}$$ One can have $$\begin{aligned}
{\bf c}^T L {\bf c}&=&\frac{1}{2}\sum_{i,j}w_{ij}(c_i-c_j)^2 \\\nonumber
&=& \frac{1}{2}\sum_{i\in G_1,j \in \bar{G_1}} \left( \sqrt{\frac{{\rm vol}(\bar{G_1})}{{\rm vol}(G_1)}} + \sqrt{\frac{{\rm vol}(G_1)}{{\rm vol}(\bar{G_1})}} \right)^2 + \frac{1}{2}\sum_{i\in \bar{G_1},j \in G_1} \left( - \sqrt{\frac{{\rm vol}(\bar{G_1})}{{\rm vol}(G_1)}} - \sqrt{\frac{{\rm vol}(G_1)}{{\rm vol}(\bar{G_1})}} \right)^2 \\\nonumber
&=& |G| {\rm Ncut}(G_1,\bar{G_1}).\end{aligned}$$ Additionally, it is easy to see that $$\begin{aligned}
(D {\bf c})^T\mathds{1}=\sum_{i} d_i c_i= \sum_{i\in G_1} d_i \sqrt{\frac{{\rm vol}(\bar{G_1})}{{\rm vol}(G_1)}} - \sum_{i \in \bar{G_1}} d_i \sqrt{\frac{{\rm vol}(G_1)}{{\rm vol}(\bar{G_1})}}=0\end{aligned}$$ This means that vector ${\bf c}$ is orthogonal to constant one vector. Moreover, one can evaluate $$\begin{aligned}
{\bf c}^T D {\bf c} = \sum_{i} d_i c_i^2= \sum_{i\in G_1} d_i \frac{{\rm vol}(\bar{G_1})}{{\rm vol}(G_1)} + \sum_{i \in \bar{G_1}} d_i \frac{{\rm vol}(G_1)}{{\rm vol}(\bar{G_1})}={\rm vol}(G)\end{aligned}$$ In this way, the minimization process is equivalent to $$\begin{aligned}
\{\min {\bf c}^T L {\bf c}~|~ {\bf c}~ {\rm satisfies ~ Eq.~ (\ref{eq:normalized_indicator}) };~ D{\bf c} \bot \mathds{1};~ {\bf c}^T D {\bf c}={\rm vol}(G) \}\end{aligned}$$ This is also a discrete optimization problem, because the entries of the solution vector are only allowed to take two particular values. By discarding the discreteness condition and allowing the vector to be any arbitrary values, one results in a relaxed problem $$\begin{aligned}
\{ \min \limits_{{\bf c} \in \mathbb{R}^N}{\bf c}^T L {\bf c}~|~ D{\bf c} \bot \mathds{1};~ {\bf c}^T D {\bf c}={\rm vol}(G) \}\end{aligned}$$ Now one can substitute ${\bf c}'=D^{\frac{1}{2}}{\bf c}$. After substitution, the problem is $$\begin{aligned}
\{ \min \limits_{{\bf c}' \in \mathbb{R}^N}({\bf c'})^T D^{-\frac{1}{2}}L D^{\frac{1}{2}} {\bf c'}~|~ {\bf c}' \bot D^{\frac{1}{2}} \mathds{1};~ \|{\bf c'}\|={\rm vol}(G) \}.\end{aligned}$$ It can been seen that $D^{-\frac{1}{2}}L D^{\frac{1}{2}}=L_{\rm sym}$ and $D^{\frac{1}{2}} \mathds{1}$ is the first eigenvector of $L_{\rm sym}$. From the Rayleigh-Ritz theorem, it can be seen that the solution of this problem is the second smallest eigenvalue of $L_{\rm sym}$.
#### Graph Laplacian and continuous Laplace operator
It has been found that there is a connection between graph Laplacian and the continuous Laplace operator [@Belkin:2003; @Lafon:2004; @Belkin:2005; @Hein:2005graphs; @Gine:2006empirical]. Roughly speaking, if one chooses $w_{ij}=\frac{1}{r_{ij}^2}$ with $r_{ij}$ as the distance between node $v_i$ and node $v_j$, one can have $$\begin{aligned}
w_{ij}(c_i-c_j)^2=\left(\frac{c_i-c_j}{r_{ij}}\right)^2.\end{aligned}$$ The term $\frac{c_i-c_j}{r_{ij}}$ can be roughly viewed as a discretization of $\nabla c({\bf r})$. In this way, there is a connection between graph Laplacian and the continuous Laplace operator through this functional formulation, $$\begin{aligned}
{\bf c}^T L {\bf c}=<{\bf c}, L {\bf c}> \approx \int |\nabla c({\bf r})|^2 d{\bf r}.\end{aligned}$$
More specifically, one can set $w_{ij}= \Phi(r_{ij},\eta_{ij})$ as defined in Eqs. (\[eq:couple\_matrix1-1\]) and (\[eq:couple\_matrix1-12\]) $$\begin{aligned}
L_N { c({\bf r}_i)}=c({\bf r}_i) \sum_j \Phi(r_{ij},\eta_{ij})- \sum_j c({\bf r}_i) \Phi(r_{ij},\eta_{ij}),\end{aligned}$$ where $c({\bf r}_i)=c_i$ and ${\bf r}_i$ is the coordinate of $i$th node. This operator can be naturally extended to an integral operator $$\begin{aligned}
L_N {c({\bf r})}=c({\bf r}) \sum_j \Phi(|{\bf r}-{\bf r}_j|,\eta_{ij}) - \sum_j c({\bf r})\Phi(|{\bf r}-{\bf r}_j|,\eta_{ij}).\end{aligned}$$
If data points are sampled from a uniform distribution on a $k$-dimensional manifold $\mathcal{M}$, let set $w_{ij}= \Phi(r_{ij},\eta_{ij})
=e^{-\frac{r_{ij}^2}{4t}}$, with $t=t_N=N^{-\frac{1}{k+2+\alpha}}$, where $\alpha >0$, and assume $c(\bf {r}) \in C^{\infty}(\mathcal{M})$. Belkin [@Belkin:2005] found that there is a constant $C$, such that in probability [@Belkin:2003; @Belkin:2005] $$\begin{aligned}
\lim \limits_{N \rightarrow \infty}C\frac{(4\pi t_N)^{-\frac{k+2}{2}}}{N} L_N^{t_N}c({\bf r})=\Delta_{\mathcal{M}} c({\bf r}).\end{aligned}$$
The continuous Laplace operator has been widely utilized in many biophysical models, such as Poisson-Boltzmann theory for electrostatics [@Holst:1994], Laplace-Beltrami equation for molecular surface modeling [@Bates:2008; @Wei:2009], Poisson-Nernst-Planck equation for ion channel modeling [@Hyon:2010; @Wei:2012], and elasticity equation for macromolecular conformational change induced by electrostatic forces [@Zhou:2008d]. Obviously, these issues are associated with a graph problem. The modeling of biomolecular structure, function, dynamics and transport by combining graph theory and partial differential equation (PDE) is an open problem.
#### Modularity
Modularity is total summation of the weights within the group minus the expected one in an equivalent network with weight randomly placed.
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![An illustration of modularity matrix, second-eigenvector formed matrix and domain indication matrix. The protein network for 2ABH C$_{\alpha}$ is constructed by using Gaussian network model with cut off distance 23 Å. (a) The illustration of modularity matrix in Eq. (\[eq:modularity\_matrix\]); (b) The illustration of the matrix formed by the second eigenvector, i.e., ${\bf q}_2 {\bf q}_2^T$; (c) The illustration of the index matrix ${\bf c} {\bf c}^T$. The vector ${\bf c}$ is generated from ${\bf q}_2$ by $\{c_i=1;~if~({\bf q}_2)_i \geq0\}$ and $\{c_i=-1;~if~({\bf q}_2)_i<0\}$. The parameter $\gamma=1$ is used in the modularity model.[]{data-label="fig:2abh_domain"}](2abh_domain.png "fig:"){width="99.00000%"}
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Mathematically, the modularity is defined as [@Newman:2004; @Newman:2006; @Fortunato:2010; @Jain:2010] $$\begin{aligned}
Q=\frac{1}{2{\rm vol}(G)} \sum_{ij} \left(A_{ij}-\gamma \frac{d_i d_j}{{\rm vol}(G)}\right) (\delta(c_i,c_j)+1)=\frac{1}{2{\rm vol}(G)} \sum_{ij} \left(A_{ij}- \gamma \frac{d_i d_j}{{\rm vol}(G)}\right) \delta(c_i,c_j),\end{aligned}$$ where $\gamma$ is a resolution parameter, which is designed to change the scale at which a network is clustered [@Fortunato:2010]. Here $\delta(c_i,c_j)=1$ if $c_i$ and $c_j$ are in the same subgroup ($c_i=c_j$), otherwise it equals to 0 ($c_i \neq c_j$). The term $A_{ij}$ is the weighted adjacent matrix component.
The modularity matrix is defined as: $$\begin{aligned}
\label{eq:modularity_matrix}
B_{ij}=A_{ij}-\frac{d_i d_j}{{\rm vol}(G)},\end{aligned}$$ then the above equation can be further simplified as $$\begin{aligned}
Q=\frac{1}{2{\rm vol}(G)}{\bf c}^T B {\bf c}.\end{aligned}$$
Again one assume that graph $G$ can be divided into two parts $G_1$ and $\bar{G_1}$ $$\begin{aligned}
&& Q =\frac{1}{{\rm vol}(G)}\left[ \left({\rm vol}(G)-\sum_{c_i \neq c_j}w_{ij} \right) -\frac{\gamma}{{\rm vol}(G)} \left(\sum_{c_i=c_j} d_i d_j \right) \right] \\ \nonumber
&& \quad =1-\frac{1}{{\rm vol}(G)} \left({\rm Cut}(G_1,\bar{G_1}) + \frac{\gamma}{{\rm vol}(G)} {\rm vol}(G_1)^2\right) \\ \nonumber
&& \quad =1-\gamma-\frac{1}{{\rm vol}(G)} \left({\rm Cut}(G_1,\bar{G_1}) -\frac{\gamma}{{\rm vol}(G)}{\rm vol}(G_1){\rm vol}(\bar{G_1})\right).\end{aligned}$$ One can define the total variation (TV): $|c|_{\rm TV}=\frac{1}{2} \sum_{i,j}w_{ij}|c_i-c_j|$, weighted $\ell_2$-norm $\|c\|^2_{\ell_2}= \sum_{i}d_{i}|c_i|^2$, and mean ${\rm mean}(c)=\frac{1}{{\rm vol}(G)} \sum_{i}d_{i}|c_i|$.
One can also define $c$ to be a function $\chi_{G_1}: G \rightarrow \{0,1\}$. This is the indicator function of a subsect $G_1 \subset G$. In this manner, one has $$\begin{aligned}
&& \quad |c|_{\rm TV}-\gamma\|c-{\rm mean}(c)\|_{\ell_2}^2 \\ \nonumber
&&= |\chi_{G_1}|_{\rm TV} -\gamma \|\chi_{G_1}-{\rm mean}(\chi_{G_1})\| \\ \nonumber
&&={\rm Cut}(G_1,\bar{G_1})-\gamma \left( \sum_i d_i \left| \chi_{G_1}-\frac{{\rm vol}(G_1)}{{\rm vol}(G)} \right|^2 \right) \\ \nonumber
&&={\rm Cut}(G_1,\bar{G_1})-\gamma \left( {\rm vol}(G_1)\left(1-\frac{{\rm vol}(G_1)}{{\rm vol}(G)}\right)^2+{\rm vol}(\bar{G_1}) \left(\frac{{\rm vol}(G_1)}{{\rm vol}(G)}\right)^2 \right) \\ \nonumber
&&={\rm Cut}(G_1,\bar{G_1})- \frac{\gamma}{{\rm vol}(G)} {\rm vol}(G_1) {\rm vol}(\bar{G_1}).\end{aligned}$$ Figure \[fig:2abh\_domain\] illustrates the modularity matrix, second-eigenvector formed matrix and domain indication matrix for protein 2ABH. Figure \[fig:2abh\_modularity2\] shows the protein domain decomposition using a modularity matrix.
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![An illustration of graph decomposition with modularity based eigenvectors. The protein network of 2ABH C$_{\alpha}$ atoms is constructed by using the Gaussian network model with cut off distance 23 Å. Based on this network, modularity matrix is constructed with parameter $\gamma=1$. The modularity eigenvector corresponding to the second lowest eigenvalue is used for protein domain decomposition. []{data-label="fig:2abh_modularity2"}](2abh_modularity.png "fig:"){width="40.00000%"}
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One can have the theorem: maximizing the modularity functional $Q$ over all the partitions is equivalent to minimize$|c|_{\rm TV}-\gamma\|c-{\rm mean}(c)\|_{\ell_2}^2$ [@Hu:2013method] . Essentially, this is equal to a balance cut problem $$\begin{aligned}
\min \frac{{\rm Cut}(G_1,\bar{G_1})}{{\rm vol}(G_1){\rm vol}(\bar{G_1})}\end{aligned}$$
One can let $c=(c^1,c^2)$ by $c: G\rightarrow V^2$ with $V^2=\{(1,0),(0,1)\}$ [@Hu:2013method] . In this way, for each $c_i$, it has only a single entry equals 1. The minimizing problem can be solved through the following equation [@Merkurjev:2013; @Hu:2013method] $$\begin{aligned}
\frac{\partial c}{\partial t}=-(L c^1,L c^2)-\frac{1}{\epsilon^2}\nabla W_{\rm multi}(c)+\frac{\delta}{\delta c}\left( \gamma \| c-{\rm mean}(c) \| \right).\end{aligned}$$ Here $\nabla W_{\rm multi}(c)$ is the composition of function $W_{\rm multi}$ and $c$. Normally, the function $W_{\rm multi}$ is a multi-well potential [@Merkurjev:2013].
The field of spectral graph, modularity and related variation formation for biomolecular systems is completely open. There is much to be done on this interesting field. For example, one can formulate molecular design, such as drug design, protein design, and the design of protein-DNA and/or protein-RNA complexes, as graph cut problems. In drug design, one would like to optimize the protein-drug binding affinity for a given drug candidate and its target protein. If the selection of graph notes is also a part of optimization, one then would like to optimize protein-drug binding affinity, drug target selectivity, drug pharmacokinetics, drug toxicity, etc. To be more specific, one needs to consider a minimization process, $$\begin{aligned}
\label{eq:cutfreeenergy}
\min \limits_{{\bf r} \in {\mathbb R}^{3N}} {\rm Cut}(G_1({\bf r}),\bar{G_1}({\bf r}))+\gamma \Delta F ({\bf r}).\end{aligned}$$ Here $N$ is the total number of atoms in the complex, the parameter $\gamma$ is a scale parameter and $ \Delta F ({\bf r})$ is the free energy change. The function $G({\bf r})$ is denoted as the graph representation of protein-protein or protein-ligand complexes. The functions $G_1({\bf r})$ and $\bar{G_1}({\bf r})$ are graph models for the corresponding protein or ligand. They are all position-dependent and the minimization process is to find the best fitting configuration so that one can achieve a cut that minimizes the free energy change $\Delta F ({\bf r})$. In fact, biomolecular free energy minimization often leads to PDE based models for solvation, ion channel, membrane protein interaction, molecular machine assembly, to name only a few [@Wei:2009]. It should be noticed that in Eq. (\[eq:cutfreeenergy\]), all the three types of graph cuts might be used. This approach can be combined with techniques in other mathematical disciplines, such as those in dynamical systems, stochastic analysis, and differential equation, to address complex design problems, namely, drug design, protein design, RNA design, molecular machine design etc, in biomolecular systems.
### Molecular nonlinear dynamics {#sec:MND}
To introduce molecular nonlinear dynamics, one can consider a folding protein that constitutes $N$ particles and has the spatiotemporal complexity of ${\mathbb{R}}^{3N}\times \mathbb{R}^{+} $. Assume that the molecular mechanics of the protein is described by molecular nonlinear dynamics having a set of $N$ nonlinear oscillators of dimension ${\mathbb{R}}^{nN}\times \mathbb{R}^{+}$, where $n$ is the dimensionality of a single nonlinear oscillator. Let us consider an $n\times N$-dimensional nonlinear system for $N$ interacting chaotic oscillators [@KLXia:2014b] $$\begin{aligned}
\label{eq:couple_matrix}
\frac{d{\bf u}}{dt} &=& {\bf F}({\bf u})+ {G}{\bf u}, ~~~\end{aligned}$$ where ${\bf u}=({\bf u}_1,{\bf u}_2,\cdots, {\bf u}_N )^T$ is an array of state functions for $N$ nonlinear oscillators, ${\bf u}_j=(u_{j1},u_{j2}, \cdots, u_{jn})^T$ is an $n$-dimensional nonlinear function for the $j$th oscillator, $ {\bf F}({\bf u})=(f({\bf u}_1), f({\bf u}_2), \cdots, f({\bf u}_N))^T$ is an array of nonlinear functions of $N$ oscillators, and ${G}=\varepsilon L \otimes \Gamma$. Here, $\varepsilon$ is the overall interaction strength, $L$ is an $N\times N$ weighted Laplacian matrix and $\Gamma$ is an $n\times n$ linking matrix. Essentially, for each node in the biomolecular graph, one has an $n$-dimensional nonlinear oscillator. These oscillators are connected by the Laplacian matrix and a fixed $n\times n$ linking matrix.
For example, one can choose a set of $N$ Lorenz attractors [@Lorenz:1963] and a simple $3*3$ link matrix as following: $\mathbf{u}_i=(u_{i1},u_{i2},u_{i3})^T$, $$\begin{aligned}
\begin{cases} \nonumber
\frac{du_{i1}}{dt}=\alpha(u_{i2}-u_{i1}) \\ \label{oscillator}
\frac{du_{i2}}{dt}=\gamma u_{i1}-u_{i2}-u_{i1}u_{i3} \\ \nonumber
\frac{du_{i3}}{dt}=u_{i1}u_{i2}-\beta u_{i3}, i= 1, 2, \cdots, N
\end{cases},~\Gamma= \left( \begin{array}{ccc}
0 & 0 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0
\end{array}
\right).\end{aligned}$$ If the Laplacian matrix shown in Fig. \[fig:GRAPH\] is used as the connectivity matrix, the nonlinear dynamic system for the first node is $$\begin{aligned}
\begin{cases} \nonumber
\frac{du_{11}}{dt}=\alpha(u_{12}-u_{11}) \\ \label{oscillator2}
\frac{du_{12}}{dt}=\gamma u_{11}-u_{12}-u_{11}u_{13}+\varepsilon (2u_{11}-u_{21}-u_{81})\\ \nonumber
\frac{du_{13}}{dt}=u_{11}u_{12}-\beta u_{13}
\end{cases}.\end{aligned}$$
#### Stability analysis {#Sec:Stability}
One can use the FRI kernel weighted Laplacian matrix to define the driving and response relation of nonlinear chaotic oscillators. Due to the synchronization of chaotic oscillators, an $N$-time reduction in the spatiotemporal complexity can be achieved, leading to an intrinsically low dimensional manifold (ILDM) of dimension ${\mathbb{R}}^{n}\times \mathbb{R}^{+}$. Formally, the $n$-dimensional ILDM is defined as $$\begin{aligned}
{\bf u}_1(t)={\bf u}_2(t)=\cdots = {\bf u}_N(t)={\bf s} (t),\end{aligned}$$ where ${{\bf s} (t)}$ is a synchronous state or reference state.
To understand the stability of the ILDM of protein chaotic dynamics, one can define a transverse state function as ${\bf w}(t)={\bf u}(t)-{\bf S}(t)$, where ${\bf S}(t)$ is a vector of $N$ identical components $({\bf s}(t),{\bf s}(t),\cdots, {\bf s}(t) )^T$. Obviously, the invariant ILDM is given by ${\bf w}(t)={\bf u}(t)-{\bf S}(t)={\bf 0}$. Therefore, the stability of the ILDM can be analyzed by $\frac{d {\bf w}(t)}{dt} =\frac {d {\bf u}(t)}{dt} -\frac {d {\bf S}(t)}{dt}$, which can be studied by the following linearized equation [@Pecora:1997; @GHu:1998] $$\begin{aligned}
\label{eqn:trans}
\frac{d{\bf w}}{dt} = ({\bf DF} ({\bf s}) + {G}){\bf w}, ~~~\end{aligned}$$ where ${\bf DF}({\bf s})$ is the Jacobian of ${\bf F}$.
To further analyze the stability of Eq. (\[eqn:trans\]), [one can diagonalize connectivity matrix ${L}$ ]{} $$\begin{aligned}
\label{eqn:trans2}
{L}{\bf \varphi}_j(t) =\lambda_j {\bf \varphi}_j(t), \quad j=1,2,\cdots, N,\end{aligned}$$ [where $\{{\bf \varphi}_j\}_{j=1}^N$ are eigenvectors and ${\lambda_j}_{j=1}^N$ are the associated eigenvalues. These eigenvectors span a vector space in which a transverse state vector has the expansion [@Pecora:1997; @GHu:1998]]{} $$\begin{aligned}
{\bf w}(t)=\sum_{j}{\bf q}_j(t)\phi_j(t).\end{aligned}$$ [Therefore, the stability problem of the ILDM is equivalent to the following stability problem]{} $$\begin{aligned}
\label{eqn:trans3}
\frac{d{\bf q}_j(t)}{dt} &=& ( Df ({\bf s}) + \varepsilon \lambda_j\Gamma){\bf q}_j(t), \quad j=1,2,\cdots,N,\end{aligned}$$ where $Df ({\bf s})$ is the diagonal component of ${\bf DF} ({\bf s})$. The stability of Eq. (\[eqn:trans3\]) is determined by the largest Lyapunov exponent $L_{\rm max}$, namely, $L_{\rm max} < 0$, which can be decomposed into two contributions $$L_{\rm max}=L_{\rm f}+L_{\rm c},$$ where $L_{\rm f}$ is the largest Lyapunov exponent of the original $n$ dimensional chaotic system $\frac{d{\bf s}}{dt} = { f}({\bf s})$, which can be easily computed for most chaotic systems. Here, $L_{\rm c}$ depends on $\lambda_j$ and $\Gamma$. The largest eigenvalue $\lambda_1$ equals 0, and its corresponding eigenvector represents the homogeneous motion of the ILDM, and all of other eigenvalues $\lambda_j, j=2,3,\cdots, N$ govern the transverse stability of the ILDM. Let us consider a simple case in which the linking matrix is the unit matrix ($\Gamma={\bf I}$). Then stability of the ILDM is determined by the second largest eigenvalue $\lambda_{2}$ (algebraic connectivity, or Fiedler value), which enables us to estimate the critical interaction strength $\varepsilon_{c}$ in terms of $\lambda_{2}$ and $L_{f}$ [@KLXia:2014b], $$\label{eq:expect}
\varepsilon_{c}=\frac{L_f}{- \lambda_{2}}.$$ The dynamical system reaches the ILDM when $\varepsilon > \varepsilon_{c}$ and is unstable when $\varepsilon \leq \varepsilon_{c}$. The eigenvalues of protein connectivity matrices are obtained with a standard matrix diagonalization algorithm. Molecular nonlinear dynamics has been recently developed as an efficient means for protein B-factor prediction [@KLXia:2014b]. It can be potentially used for protein domain separation without resorting to the matrix diagonalization.
The availability of more than a hundred thousands of interacting protein networks in the PDB provides living example problems for analyzing dynamical systems. Indeed, the connection between dynamical systems and spectral graph theory in mathematics, and the structure and function of macromolecules gives rise to exciting opportunities to further study dynamical systems and graph theory, and better analyze biomolecules.
Persistent homology {#sec:PHA}
--------------------
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![An illustration of topological change over different dense thresholds for a benzene molecule. From left to right, the density threshold is decreased.[]{data-label="fig:density"}](density.png "fig:"){width="60.00000%"}
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As a branch of algebraic topology, persistent homology is a topological approach that utilizes algebraic algorithms to compute topological invariants in data[@Carlsson:2009; @Niyogi:2011; @BeiWang:2011; @Rieck:2012; @XuLiu:2012]. It is a working horse in the popular topological data analysis and has found its success in qualitative characterization or classification. Specifically, persistent homology describes geometric features of biomolecular date with topological invariants that persist over the systematic change of a scale parameter relevant to topological events. Such a change is called filtration. Figure \[fig:density\] illustrates the change of the topology of benzene molecule over different density thresholds. The idea is to capture topological structures continuously over a range of spatial scales during the filtration process. Unlike computational homology that is based on truly metric free or coordinate free representations, persistent homology can embed geometric information from protein data into topological invariants so that “birth" and “death" of geometric features, such as circles, rings, loops, pockets, voids and cavities can be monitored by topological measurements during the filtration process[@Edelsbrunner:2002; @Zomorodian:2005]
### Simplicial homology and persistent homology {#sec:SimplicialHomology}
An essential ingredient of persistent homology is simplicial homology which is built on simplicial complex. Simplicial complex is a finite set that consists of discrete vertices (nodes or atoms in a protein), edges (line segments or bonds in a biomolecule), triangles, and their high dimensional counterparts. Simplicial homology can be defined on simplicial complex to analyze and extract topological invariants. Then a filtration process is used to establish topological persistence from simplicial homology analysis[@Edelsbrunner:2002; @Zomorodian:2005].
#### Simplicial complex
A key component of simplicial complex $K$ is a $k$-simplex, $\sigma^k$, defined as the convex hall of $k+1$ affine independent nodes in $\mathbb{R}^N$ ($N>k$). Let $v_0,v_1,v_2,\cdots,v_k$ be $k+1$ affine independent points (or atoms in a biomolecule) and express a $k$-simplex $\sigma^k=\{v_0,v_1,v_2,\cdots,v_k\}$ as $$\begin{aligned}
\label{eq:couple_matrix12}
\sigma^k=\left\{\tau_0 v_0+\tau_1 v_1+ \cdots +\tau_k v_k \mid \sum^{k}_{i=0}\tau_i=1;0\leq \tau_i \leq 1,i=0,1, \cdots,k \right\}.\end{aligned}$$ Moreover, let us define an $i$-dimensional face of $\sigma^k$ as the convex hall formed by the nonempty subset of $i+1$ vertices from $\sigma^k$ ($k>i$). Clearly, a 0-simplex is a vertex, a 1-simplex is an edge, a 2-simplex is a triangle, and a 3-simplex represents a tetrahedron. One can also define the empty set as a (-1)-simplex.
A simplicial complex is constructed to combine these geometric components, including vertices, edges, triangles, and tetrahedrons together under certain rules. More specifically, a simplicial complex $K$ is a finite set of simplicies that satisfy two conditions. One is that any face of a simplex from $K$ is also in $K$ and the other is that the intersection of any two simplices in $K$ is either empty or shared faces. The dimension of a simplicial complex is defined as the maximal dimension of its simplicies. The underlying topological space $|K|$ is a union of all the simplices of $K$, i.e., $|K|=\cup_{\sigma^k\in K} \sigma^k$. Further, the concept of chain is introduced to associate this topological space with algebra groups.
#### Homology
One can denote a linear combination $\sum^{k}_{i}\alpha_i\sigma^k_i$ of $k$-simplex $\sigma^k_i$ as a $k$-chain $[\sigma^k]$. The coefficients $\alpha_i$ can be chosen from different fields, such as rational field $\mathbb{Q}$, real number field and complex number field, and from integers $\mathbb{Z}$ and prime integers $\mathbb{Z}_p$ with prime number $p$. For simplicity, one can consider the coefficients $\alpha_i$ are chosen from $\mathbb{Z}_2$, for which the addition operation between two chains is the modulo 2 addition for the coefficients of their corresponding simplices. The set of all $k$-chains of simplicial complex $K$ and the addition operation form an Abelian group $C_k(K, \mathbb{Z}_2)$. Therefore, the homology of a topological space is represented by a series of Abelian groups.
A boundary operation $\partial_k$ is defined as $\partial_k: C_k \rightarrow C_{k-1}$. Without orientation, the boundary of a $k$-simplex $\sigma^k=\{v_0,v_1,v_2,\cdots,v_k\}$ is $$\begin{aligned}
\partial_k \sigma^k = \sum^{k}_{i=0} \{ v_0, v_1, v_2, \cdots, \hat{v_i}, \cdots, v_k \},\end{aligned}$$ where the notation $\{v_0, v_1, v_2, \cdots ,\hat{v_i}, \cdots, v_k\}$ means that the $(k-1)$-simplex is generated by eliminating vertex $v_i$ from the sequence. When a boundary operator is applied twice, any $k$-chain will be mapped to a zero element, i.e., $\partial_{k-1}\partial_k= \emptyset$. As a special case, one has $\partial_0= \emptyset$. The $k$th cycle group $Z_k$ and the $k$th boundary group $B_k$ are the subgroups of $C_k$ and can be defined by means of the boundary operator, $$\begin{aligned}
&& Z_k={\rm Ker}~ \partial_k=\{c\in C_k \mid \partial_k c=\emptyset\}, \\
&&{ B_k={\rm Im} ~\partial_{k+1}= \{ c\in C_k \mid \exists d \in C_{k+1}: c=\partial_{k+1} d\}.}\end{aligned}$$ An element in the $k$th cycle group $Z_k$ or the $k$th boundary group $B_k$ is called the $k$th cycle or the $k$th boundary. One has $B_k\subseteq Z_k \subseteq C_k$ since the boundary of a boundary is always empty $\partial_{k-1}\partial_k= \emptyset$. Geometrically, the $k$th cycle is a $k$ dimensional loop or hole.
With all the above definitions, one can define the homology group. Specifically, the $k$th homology group $H_k$ is defined as the quotient group of the $k$th cycle group $Z_k$ and $k$th boundary group $B_k$: $H_k=Z_k/B_k$. Two $k$th cycle elements are called homologous if they are different by a $k$th boundary element. The $k$th Betti number represents the rank of the $k$th homology group, $$\begin{aligned}
\beta_k = {\rm rank} ~H_k= {\rm rank }~ Z_k - {\rm rank}~ B_k.\end{aligned}$$ From the fundamental theorem of finitely generated Abelian groups, the $k$th homology group $H_k$ can be given as a direct sum, $$\begin{aligned}
H_k= {Z}\oplus \cdots \oplus {Z} \oplus {Z}_{p_1}\oplus \cdots \oplus {Z}_{p_n}= {Z}^{\beta_k} \oplus {Z}_{p_1}\oplus \cdots \oplus {Z}_{p_n},\end{aligned}$$ where $\beta_k$ is the rank of the subgroup and is $k$th Betti number. Here $ {Z}_{p_i}$ is torsion subgroup with torsion coefficients $\{p_i| i=1,2,...,n\}$, the power of prime number.
Topologically, cycle element in $H_k$ forms a $k$-dimensional loop or ring that is not from the boundary of a higher dimensional chain element. The geometric meanings of Betti numbers in $\mathbb{R}^3$ are the follows: $\beta_0$ represents the number of isolated components (i.e., protein atoms), $\beta_1$ is the number of one-dimensional loop or ring, and $\beta_2$ describes the number of two-dimensional voids or cavities. Together, the Betti number sequence [ $\{\beta_0,\beta_1,\beta_2,\cdots \}$]{} gives the intrinsic topological property of biomolecular data.
#### $\check{\rm C}$ech complex, Rips complex and alpha complex
A key concept for the construction of simplicial complex from a point set of a given topological space is nerve. Basically, given an index set $I$ and open set ${\bf U}=\{U_i\}_{i\in I}$ that is a cover of a point set $X \in \mathbb{R}^N$, i.e., $X \subseteq \{U_i\}_{i\in I}$, the nerve [**N**]{} of [**U**]{} should satisfy two basic conditions. One is that $\emptyset \in {\bf N}$. Additionally, if $\cap_{j \in J} U_j \neq \emptyset $ for $J \subseteq I $, then $J \in {\bf N}$. Usually, for a given biomolecular dataset, the simplest way to construct a cover is to assign a ball of certain radius around each atom. If the biomolecular dataset is dense enough and the radius is large enough, then the union of all the balls has the capability to recover the underlying space for the biomolecule.
The nerve of a cover of the biomolecule constructed from the union of atomic balls is a $\check{\rm C}$ech complex for the biomolecule. More specifically, for a biomolecular dataset $X \in \mathbb{ R}^N$, one defines a cover of closed atomic balls ${\bf B}=\{B (x, r)\mid x \in X \}$ with radius $r$ and centered at $x$. The $\check{\rm C}$ech complex of $X$ with radius $r$ is denoted as $\mathcal{C}(X,r)$, which is the nerve of the closed ball set [**B**]{}, $$\begin{aligned}
\mathcal{C}(X,r) = \left\{ \sigma \mid \cap_{x \in \sigma} B (x,r) \neq \emptyset \right\}.\end{aligned}$$ One can relax $\check{\rm C}$ech complex conditions to generate a Vietoris-Rips complex, in which, a simplex $\sigma$ is constructed if the largest distance between any two atoms is at most $2r$. One can denote $\mathcal{R}(X,r)$ the Vietoris-Rips complex, or Rips complex [@Edelsbrunner:1994]. There is a sandwich relation for these abstract complexes, $$\begin{aligned}
\label{eq:SandwichRelation}
\mathcal{C}(X,r)\subset \mathcal{R}(X,r) \subset \mathcal{C}(X,\sqrt{2}r).\end{aligned}$$ In practical applications, Rips complex is preferred due to its computational convenience.
Another important geometric concept in computational geometry is alpha complex. Let $X$ be a biomolecular dataset in Euclidean space $\mathbb{R}^d$ and define the Voronoi cell of a point $x \in X$ as $$\begin{aligned}
V_x = \{ u\in R^d \mid |u-x|\leq |u-x'|, \forall x'\in X \}.\end{aligned}$$ Then the collection of all Voronoi cells for the biomolecule forms a Voronoi diagram. Further, the nerve of the biomolcular Voronoi diagram generates a Delaunay complex.
One can define $R(x,r)$ as the intersection of Voronoi cell $V_x$ with ball $B(x,\epsilon)$, i.e., $R(x,r)= V_x \cap B(x,r)$. The alpha complex $\mathcal{A}(X,r)$ of the dataset $X$ is defined as the nerve of cover $\cup_{x\in X} R(x,r)$, $$\begin{aligned}
\mathcal{A}(X,r) = \left\{ \sigma \mid \cap_{x \in \sigma} R (x,r) \neq \emptyset \right\}.\end{aligned}$$ Therefore, an alpha complex is a subset of the Delaunay complex.
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![Barcode representation of persistent homology analysis for fullerene molecule $C_{60}$.[]{data-label="fig:c60"}](c60.png "fig:"){width="60.00000%"}
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#### General filtration processes
In order to construct a simplicial homology from a dataset, a special parameter (like the radius $r$ mentioned above) is commonly used. However, to find a “suitable" value for this parameter, so that it can reveal the underlying manifold, is not straightforward. An elegant alternative approach is to carry out a filtration process [@Bubenik:2007; @edelsbrunner:2010; @Dey:2008; @Dey:2013; @Mischaikow:2013]. A suitable filtration is vital to the resulting persistent homology, which will be described in the following section. In practice, two commonly used filtration algorithms are the Euclidean-distance or correlation matrix based and density based ones. These basic filtration algorithms can be modified to achieve different goals in data analysis [@KLXia:2014c; @KLXia:2015c; @KLXia:2015d].
In the Euclidean-distance based filtration, one associates each atom with an ever-increasing radius to form an ever-growing ball for each atom. Various aforementioned complex construction algorithms can be utilized to identify the corresponding complexes. This filtration process can be formalized by the use of a distance matrix $\{d_{ij}\}$. Here the matrix element $d_{ij}$ represents the distance between atom $i$ and atom $j$. For diagonal terms, one assumes $d_{ij}=0$. With a filtration threshold $\varepsilon$, a 1-simplex is generated between atoms $i$ and $j$ if $d_{ij}\leq\varepsilon$. Higher dimensional complexes can also be created similarly.
Another important filtration process is the density based filtration process. In this process, the filtration goes along the increase or decrease of the density value. In this way, a series of isosurfaces are generated. Morse complex is used for the characterization of their topological invariants.
#### Persistent homology {#persistent-homology}
Persistent homology is an elegant mathematical theory to describe topological invariants from a series of topological spaces in various scales, that are generated by the filtration process. Persistent homology concerns a family of homologies, in which the connectivity of the given dataset is systematically reset according to a (scale) parameter. For a simplicial complex $K$, the filtration is defined as a nested sub-sequence of subcomplexes, $$\begin{aligned}
\varnothing = K^0 \subseteq K^1 \subseteq \cdots \subseteq K^m=K.\end{aligned}$$ The introduction of filtration leads to the creation of persistent homology. When the filtration parameter is a scale parameter, simplicial complexes generated from a filtration give a multiscale representation of the corresponding topological space, from which related homology groups can be evaluated to reveal topological features of the given dataset. Furthermore, the concept of persistence is introduced to measure the persistent length of topological features. The $p$-persistent $k$th homology group $K^i$ is $$\begin{aligned}
H^{i,p}_k=Z^i_k/(B_k^{i+p}\bigcap Z^i_k).\end{aligned}$$ Through the study of the persistent pattern of these topological features, the so called persistent homology is capable of capturing the intrinsic properties of the underlying protein topological space solely from the protein atomic coordinates.
To visualize the persistent homology results, many elegant representation methods have been proposed, including persistent diagram[@Edelsbrunner:2008persistent], persistent barcode[@Ghrist:2008], persistent landscape [@Bubenik:2015statistical], etc. In this paper, a barcode representation is used. The persistent barcode of an Euclidean-distance based filtration process of a fullerene molecular $c_{60}$ is shown in Fig. \[fig:c60\].
The combination of optimization and persistent homology was discussed in a recent work for biomolecular data analysis [@BaoWang:2016a]. The essentially idea to create an object functional for extracting certain geometric features in data. Then the use of variational principle to result in a differential equation, which is subsequently utilized to filtrate the biomolecular data. In this work, the minimization of the surface energy of biomolecules was the objective, which leads to the Laplace-Beltrami flow for filtration. In this manner, one can have connected persistent homology to other important mathematical subjects, such as partial differential equations, optimizations, and differential geometry [@BaoWang:2016a].
Object-oriented persistent homology is expected to play an important role in massive data analysis. In particular, this approach can be combined with a deep learning strategy to automatically extract desirable information in a semi-supervised or unsupervised learning framework.
### Multiscale persistent homology {#sec:mPHA}
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![Illustration of multiscale properties in an icosahedral viral particle capsid, which consists many hexagons and pentagons as shown on the left chart. A pentagon shown on the second left consists of five proteins. For a protein shown on second right, there are many residues indicated by different colors. Finally, for residue shown on the right, it has many atoms, including hexagonal and pentagonal rings. []{data-label="fig:Multiscale_illustration"}](Multiscale_illustration.png "fig:"){width="90.00000%"}
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The emergence of complexity in self-organizing biological systems frequently requires more comprehensive topological descriptions. Therefore, multiscale persistent homology, multiresolution persistent homology and multidimensional persistence become valuable for biological systems as well as many other complex systems.
It is noted that there is no need for a simplicial complex to be built exclusively on a set of atoms. It can be constructed for a biomolecule from its coarse-grained representation, namely, a set of amino acid residues. For protein complexes, such as a viral particle as shown in Fig. \[fig:Multiscale\_illustration\], a basic vortex in a simplicial complex can be either a protein, a residue (i.e., C$_{\alpha}$ representation) or an atom. Each of these selections gives rise to a different persistent homology at a given scale. To demonstrate the utility of persistent homology for cryo-EM data analysis, a microtubule intermediate structure, protein complex 1129 from electron microscopy data (EMD) is considered [@KLXia:2015b]. In this study, each point in the simplicial complex is a protein (tubulin).
An interesting problem in persistent homology studies is how to select appropriate scales and appropriate types of molecular information in filtration to better analyzing the structure, function and dynamics of subcellular organelles, molecular motors and multiprotein complexes.
### Topology based quantitative modeling
Traditional topological analysis analyzes data in terms of topological invariants, such as Euler characteristic, winding number, Betti numbers and so on, thus leads to so much reduction that the resulting information is hardly useful for complex real world problems. Geometric tools often become computationally intractable for macromolecules and their interactions due to the involved high degrees of freedom. As one can see, persistent homology embeds geometric information in topological invariants and bridges the gap between geometry and topology. However, persistent homology has been mainly employed for qualitative analysis, namely, characterization and classification. Only recently, persistent homology has been devised for quantitative analysis, mathematical modeling, and physical prediction [@KLXia:2014c; @KLXia:2015c; @KLXia:2015d; @BaoWang:2016a]. It has been shown that the length of intrinsic Betti 2 bar provides an excellent model for fullerene thermal energies [@BaoWang:2016a]. It has also been shown that accumulated Betti 0 and Betti 1 bar lengths offer highly accurate predictions of unfolding protein bond and total energies, respectively [@KLXia:2015c]. It is expected that persistent homology will continue to play a significant role in quantitative modeling and prediction.
Additionally, earlier work regard short-lived barcodes or topological invariants as [*noise*]{}. It has also been pointed out that, for biomolecular datsets, these short-lived topological invariants or non-persistent topological features, are part of topological fingerprints and have meaningful biophysical interpretations [@KLXia:2014c].
In general, topology based quantitative modeling and analysis will be a new trend in molecular bioscience and biophysics. Particularly, topological features are ideally suitable for machine learning based quantitative predictions of biomolecular functions.
Continuous apparatuses for biomolecules {#sec:continuous}
=======================================
As stated above, the topological study of scalar fields, particularly electron density data, has advanced the understanding of molecules tremendously. The theory of atoms in molecules (AIM) has provided an elegant and feasible partition of electron density field, so that one can mathematically rigorously define the atoms in molecule [@Bader:1985; @Bader:1990]. AIM can be generalized into a more general theory called quantum chemical topology (QCT), which employs the topological analysis in AIM for the study of other physically meaningful scalar fields [@Popelier:2005]. Additionally, geometric modeling and analysis, particularly the surface curvature analysis, has contributed a lot to the molecular visualization and structure characterization [@Whitley:2012]. Thanks to the geometric analysis, the establishment and further deeper understanding of structure-function relationship have been achieved. In this section, a brief discuss of geometric and topological analysis of scalar fields is presented.
Continuum representation of biomolecules plays an important role in their modeling, analysis and simulation. Volumetric biomolecular data are typically obtained from cryo-EM maps, quantum mechanical simulations and mathematical models that transform discrete datasets originally generated by X-ray crystallography or other means into continuous ones. Therefore, continuous mathematical approaches for analysis and modeling of biomolecules in the volumetric data form are as important as their discrete counterparts.
Geometric representation {#Sec:GeometricRep}
------------------------
Geometric modeling is a crucial ingredient of biophysics. Due to increasingly powerful high performance computers, geometric modeling has become an essential apparatus for biomolecular surface representation, visualization, surface and volumetric meshing, area and volume estimation, curvature analysis and filling the gap between macromolecular structural information and their theoretical models [@ZYu:2008; @XFeng:2012a; @XFeng:2013b; @KLXia:2014a; @JLi:2013; @Quine:2006intensity]. The visualization of macromolecules sheds light on biomolecular structure, function and interaction, including ligand-receptor binding sites, protein specification, drug binding, macromolecular assembly, protein-nucleic acid interactions, protein-protein binding hot spots, and enzymatic mechanism [@GRASP2; @Rocchia:2002; @NKWH07; @Decherchi:2013].
#### Non-smooth biomolecular surface representations
A number of molecular surface models has been proposed. Among them, the van der Waals surface (vdWS) is defined as the union of the atomic surfaces under a given atomic radius for each type of atoms. Solvent accessible surface (SAS) is defined as the trajectory of the center of a probe sphere moving around the van der Waals surface [@Lee:1971]. Because vdWs and SASs are non-smooth at intersection areas where two or more atoms join together, solvent excluded surface (SES) was introduced to generate relatively more smooth surfaces [@Richards:1977; @Connolly85]. The SES can be obtained by tracing the inward moving surface of a probe sphere rolling around the vdW surface. Connolly divided SES into two major parts, the contact areas formed by the subsets of the vdWs surface and the re-entrant surfaces, which contain toroidal patches and concave spherical triangles.
#### Smooth biomolecular surface representations
The SES of proteins admits geometric singularities, such as tips, sharp edges and self-intersecting surfaces [@Sanner:1996; @Yu:2007a] The construction of smooth biomolecular surfaces has been of considerable interest [@Blinn:1982; @Duncan:1993; @QZheng:2012; @ZYu:2008; @MXChen:2011]. The rigidity index in Eq. (\[rigidity1\]) has been extended into a continuous rigidity density [@KLXia:2013d; @Opron:2014] $$\begin{aligned}
\label{eq:rigidity3}
\mu^1(\mathbf{r})=\sum_{\substack{j=1}}^{N} w_{j} \Phi\left(\|\mathbf{r}-\mathbf{r}_j\|;\eta_{j}\right).\end{aligned}$$ Rigidity density (\[eq:rigidity3\]) serves as an excellent representation of molecular surfaces [@KLXia:2015e]. Gaussian surface was proposed with the Gaussian kernel [@Zap; @Grant:2007; @LLi:2013; @LinWang:2015]. Recently, Gaussian surface has been extended to a new class of surface densities equipped with a wide variety of FRI correlation kernels ($\Phi\left(\|\mathbf{r} -\mathbf{r}_j\|;\eta_{ j}\right)$) [@DDNguyen:2016b] $$\begin{aligned}
\label{rigidity2}
\mu^2(\mathbf{r}) =1-\prod_{\substack{j=1}}\left[1-w_{ j}\Phi\left(\|\mathbf{r} -\mathbf{r}_j\|;\eta_{ j}\right)\right].\end{aligned}$$ Two rigidity densities $\mu^\alpha(\mathbf{r}), ~\alpha=1,2$ may behave very differently. Therefore, one can normalize these densities by their maximal values $$\begin{aligned}
\label{normalization}
\bar{\mu}^{\alpha}(\mathbf{r})=\frac{\mu^{\alpha}(\mathbf{r})}{\max\limits_{\mathbf{r}\in \mathbb{R}^3} \mu^\alpha(\mathbf{r})}, \quad \alpha =1,2.\end{aligned}$$ As a result, the behaviors of two rigidity surfaces can be compared.
#### Discrete to continuum mapping
Many geometric and topological apparatuses are invented for continuous volumetric data. A typically examples include differential geometry and differential topology that deal with differentiable functions on differentiable manifolds. Cryo-EM data and electron quantum densities can be directly treated by mathematical tools devised from differentiable manifolds. However, a large variety of discrete macromolecular data originate from X-ray crystallography, NMR etc are not directly differentiable. Therefore, it desirable to transform discrete biomolecular datasets into continuous ones.
The rigidity densities defined in Eq. (\[eq:rigidity3\]) is differentiable for $\eta_{ j}>0$. Therefore, rigidity densities also serve as a discrete to continuum mapping. The resolution parameter can be exploited for generating multidimensional persistence as illustrated in Section \[sec:resultionPH\]. As a result, many mathematical techniques developed for continuous datasets can be employed to analyze discrete biomolecular datasets, such as X-ray crystallography data.
Additionally, the normalized rigidity density $\bar{\mu}^{1}(\mathbf{r})$ given in Eq. (\[normalization\]) can be used as solute domain indicators for implicit solvent models [@Holst:1994; @Baker:2001; @Geng:2007a; @Geng:2011], such as those used in the differential geometry based Poisson-Boltzmann theory [@Wei:2009; @ZhanChen:2010a; @ZhanChen:2010b].
Mathematically, the aforementioned discrete to continuum mapping is an interpolation using kernels. Many other techniques, such as splines, polynomials, wavelets, and Padé approximation can be used as well. Currently, there is little numerical analysis of the mapping in biomolecular context and further mathematical study is needed to improve the stability and efficiency of the mapping for large data sets.
Multiresolution and multidimensional persistent homology {#sec:resultionPH}
--------------------------------------------------------
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![Multiresolution representations of protein 1DYL. Different values of resolution parameter $\eta$ are used to generate rigidity density functions in different scales.[]{data-label="fig:1DYL"}](1DYL.png "fig:"){width="80.00000%"}
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Although persistent homology was originally built over simplicial complex of discrete data sets, homology and persistent homology in the cubical complex setting have been developed [@Kaczynski:2004; @Strombom:2007]. Therefore, one can apply these techniques to volumetric datasets directly, particularly, with an available software package, [Perseus](http://www.sas.upenn.edu/~vnanda/perseus/index.html) [@Mischaikow:2013]. The reader is referred to the literature for more comprehensive discussion and treatment [@Kaczynski:2004; @Strombom:2007].
Recently, persistent homology analysis of macromolecular volumetric datasets have been demonstrated [@KLXia:2014c; @KLXia:2015a; @KLXia:2015b]. In this paper, the multiresolution persistent homology [@KLXia:2015d; @KLXia:2015e] and multidimensional persistent homology [@KLXia:2015c] developed for volumetric macromolecular datasets are discussed.
#### Multiresolution persistent homology
As stated earlier, the basic idea of persistent homology is to exploit the topological changes of a given dataset at different scales of representations [@KLXia:2014c; @KLXia:2015a; @KLXia:2015b]. For the geometric representation given by Eq. (\[eq:rigidity3\]), rigidity density $\mu^1(\mathbf{r})$ depends on the resolution parameter $\eta$. The resolution parameter can be turned to emphasize the molecular features of scale $\eta$, see Fig. \[fig:1DYL\]. Resolution based continuous coarse-grained representations can be constructed for excessively large datasets in the spirit of wavelet multiresolution analysis. This approach is particularly valuable for representing viruses, protein complexes and subcellular organelles.
Since the geometric representation is controlled by the resolution parameter $\eta$ in Eq. (\[eq:rigidity3\]), one can develop multiresolution persistent homology (MPH) for macromolecular analysis. The essential idea is to match the scale of interest with appropriate resolution in the topological analysis. In contrast to the original persistent homology that is based on a uniform resolution of the point cloud data over the filtration domain, the MPH provides a mathematical microscopy of the topology at a given scale through a corresponding resolution. MPH can be utilized to reveal the topology of a given geometric scale and employed as a topological focus of lens. It becomes powerful when it is applied in conjugation with the data that has a multiscale nature, such as a multiprotein complex as shown in Fig. \[fig:Multiscale\_illustration\]. In this case, MPH can be used to extract the topological fingerprints either at atomic scale, residue scale, alpha helix and beta sheet scale, domain scale or at the protein scale.
Another very interesting multiresolution model is Mapper [@singh:2007; @Carlsson:2009]. This method is proposed for qualitative analysis, simplification and visualization of high dimensional data sets. It manages to reduce the complexity by using fewer points which can capture topological and geometric information at a specified resolution. Interestingly, Mapper also uses kernel functions. However, it should be noticed that it does not utilize the resolution parameter in the filtration process for persistent homology. It is expected that related subjects, such co-homology, Floer homology, Sheaf and K-theory, will find interesting applications in biomolecular systems.
#### Multidimensional persistent homology
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![An illustration of multidimensional persistent homology analysis of protein 2YGD. Left chart: protein 2YGD; Right chart: two-dimensional persistence of 2YGD. The horizontal axis denotes density and the vertical axis represents the logarithmic values of persistent Betti numbers.[]{data-label="fig:2ygd"}](2ygd.png "fig:"){width="80.00000%"}
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There have been considerable interest in developing multidimensional persistent homology or multidimensional persistence. Here, two classes of multidimensional persistence algorithms for biomolecular data are discussed. One class of multidimensional persistence is generated by repeated applications of 1D persistent homology to high-dimensional data, such as those from protein folding, molecular dynamics, geometric partial differential equations, etc. The resulting high-dimensional persistent homology is a pseudo-multidimensional persistence and has been applied to protein folding analysis to identify topological transitions [@KLXia:2015c].
Another class of multidimensional persistence is created by the geometric representation given in Eq. (\[eq:rigidity3\]). As $\eta$ is an independent variable that can modify the geometry and topology of the underlying data set, one can carry out filtration with respect to both the original density and the resolution $\eta$ to obtain genuine 2D persistent homology [@KLXia:2015c]. Indeed, each $\eta$ value leads to a family of new simplicial complexes. Similarly, for an $N$-dimensional persistent homology, $N$-independent variables should be introduced for the filtration. Higher dimensional persistence has been demonstrated for macromolecular data [@KLXia:2015c]. Resolution induced persistent homology and multidimensional persistence have been applied to many biomolecular systems, including protein flexibility analysis, protein folding characterization, topological denoising, noise removal from cryo-EM data, and analysis of fullerene molecules. An example of multidimensional persistent homology is depicted in Fig. \[fig:2ygd\]. Clearly, the maps of multidimensional persistent homology given in Fig. \[fig:2ygd\] can be employed for deep learning, which is an open field.
Basically each independent parameter that regulates the filtration process contributes a genuine persistent dimension. When the dimension is higher than two, the result representation is no longer straightforward. Additionally, how to make use of multidimensional persistence in realistic applications is also an interesting problem.
Differential geometry theory of surfaces {#sec:DGA}
----------------------------------------
Differential geometry has fruitful applications in physics, particularly, general relativity and has found its success in biomolecular systems as well [@Bates:2008; @Wei:2009; @ZhanChen:2010a]. As stated earlier, in biophysical modeling, surface representation is a crucial subject and commonly used surface definitions lead to geometric singularities. Gaussian surface and general FRI rigidity surface are based on simple geometric ideas. In contrast, differential geometry theory of surfaces gives rise to natural description macromolecular surfaces [@Bates:2008; @Wei:2009; @ZhanChen:2010a]. This approach becomes powerful when it is combined with variation calculus for biomolecular modeling as shown in Section \[sec:scalar\_field\_geometry\]. In this section, a brief introduction is given about the differential geometry theory of surfaces using the notations and definitions from Ref. [@Kuhnel:2015].
#### Surface elements and immersion
For an open set $U \subset \mathds{R}^2$, a parametrized surface element is an immersion $f: U \rightarrow \mathds{R}^3$. Here $f$ is also known as a parametrization. One can call the elements of $U$ as parameters and their images under $f$ as points. If the rank of map $f$ is maximal, $f$ is an immersion. The point where the rank is not maximal, it is called a singular point or singularity [@Kuhnel:2015].
One can use the following notations for a parametrized surface element $f:U\rightarrow \mathds{R}^3, u \in U, p=f(u)$. $T_uU$ is the tangent space of $U$ at $u$, $T_uU=\{u\}\times \mathds{R}^2 $; $T_p\mathds{R}^3$ is the tangent space of $\mathds{R}^3$ at $p$, $T_p\mathds{R}^3=\{p\}\times \mathds{R}^3 $; $T_uf$ is the tangent space of $f$ at $p$, $T_uf:=Df|_u(T_uU) \subset T_{f(u)} \mathds{R}^3 $; and $\perp_uf$ is the normal space of $f$ at $p$ $T_uf \oplus \perp_uf =T_{f(u)} \mathds{R}^3 $. The element of $T_uf$ is called tangent vector and the element of $\perp_uf$ is the normal vector [@Kuhnel:2015].
#### First fundamental form
The first fundamental form $I$ is the inner product between two tangent vectors $X,Y$ in tangent planes $T_uf$, i.e., $I(X,Y):=<X,Y>$. For coordinate systems $f(u,v)=\left( x(u,v),y(u,v),z(u,v) \right)$, the first fundamental form can be described by the following tensor matrix [@Kuhnel:2015] $$(g_{ij})=\left( \begin{array}{cc}
E(u,v) & F(u,v) \\
F(u,v) & G(u,v)
\end{array}
\right)= \left(
\begin{array}{cc}
I(\frac{\partial f}{\partial u},\frac{\partial f}{\partial u}) & I(\frac{\partial f}{\partial u},\frac{\partial f}{\partial v}) \\
I(\frac{\partial f}{\partial u},\frac{\partial f}{\partial v}) & I(\frac{\partial f}{\partial v},\frac{\partial f}{\partial v}) \end{array}
\right)= \left(
\begin{array}{cc}
<\frac{\partial f}{\partial u},\frac{\partial f}{\partial u}> & <\frac{\partial f}{\partial u},\frac{\partial f}{\partial v}> \\
<\frac{\partial f}{\partial u},\frac{\partial f}{\partial v}> & <\frac{\partial f}{\partial v},\frac{\partial f}{\partial v}>
\end{array}
\right)$$ Also one can have the line element $$ds^2=E(u,v)du^2 +2F(u,v)dudv+G(u,v)dv^2$$ and the surface area $$dA=\sqrt{g}dudv,$$ where $g= {\rm Det}(g_{ij})$ is the determinant.
#### Gauss map
Since each plane is essentially determined by its normal vector, the curvature of the surface can be studied by the variation of the normal vector, i.e., Gauss map. For a surface element $f: U \rightarrow \mathds{R}^3$, the Gauss map is $v:U \rightarrow S^2$ and is defined by the formula [@Kuhnel:2015] $$v(u_1,u_2):=\frac{\frac{\partial f}{\partial u_1} \times \frac{\partial f}{\partial u_2}}{|\frac{\partial f}{\partial u_1} \times \frac{\partial f}{\partial u_2}|},$$ where $S^2$ denotes the unite sphere $S^2=\{(x,y,z)\in \mathds{R}^3 | x^2+y^2+z^2=1\}$.
#### Weingarten map
Let $f: U \rightarrow \mathds{R}^3$ be a surface element with Gauss map $v:U \rightarrow S^2 \in \mathds{R}^3$, and for every $u \in U$ the image plane of the linear map $Dv|_u:T_uU \rightarrow T_{v(u)}\mathds{R}^3$ is parallel to the tangent plane $T_uf$. By canonically identifying $T_{v(u)}\mathds{R}^3 \cong \mathds{R}^3 \cong T_{f(u)}\mathds{R}^3 $, one can have $Dv$ at every point as the map $Dv|_u:T_uU \rightarrow T_uf$. Moreover, by restricting to the image, one may view the map $Df|_u$ as a linear isomorphism $Df|_u:T_uU \rightarrow T_uf$. In this sense the inverse mapping $(Df|_u)^{-1}$ is well-defined and is also an isomorphism. The map $L:=-Dv\circ (Df)^{-1}$ defined point-wisely by $$L_u:=-(Dv|_u)\circ (Df|_u)^{-1}: T_uf \rightarrow T_uf$$ is called a Weingarten map or the shape operator of $f$. This map is independent of the parametrization of $f$, and it is self-adjoint with respect to the first fundamental form I.
#### Second and third fundamental form
Let $f: U \rightarrow \mathds{R}^3$ be a surface element with Gauss map $v:U \rightarrow S^2 \in \mathds{R}^3$. With the shape operator $L$ and tangent vectors $X$ and $Y$, the second fundamental $II$ is given by $$II(X,Y):=I(LX,Y)$$ and the third fundamental form is $$III(X,Y):=I(L^2X,Y)=I(LX,LY).$$ $II$ and $III$ are symmetric bilinear forms on $T_uf$ for every $u \in U$. The three fundamental forms $I$,$II$ and $III$ have a relation as, $$III-{\rm Tr}(L)II+{\rm Det}(L)I=0.$$ In coordinates $f(u,v)=\left( x(u,v),y(u,v),z(u,v) \right)$, three fundamental forms can be expressed as $$\begin{aligned}
&&I: g_{ij}=<\frac{\partial f}{\partial u_i},\frac{\partial f}{\partial u_j}>; \\
&&II: h_{ij}=<v,\frac{\partial^2 f}{\partial u_i \partial u_j}>=-<\frac{\partial v}{\partial u_i}, \frac{\partial f}{\partial u_j}>; \\
&&III: e_{ij}=<\frac{\partial v}{\partial u_i},\frac{\partial v}{\partial u_j}>.\end{aligned}$$
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![Illustration of Gaussian curvature (left side) and mean curvature (right side) for an HIV-1 gp 120 trimer structure (EMD-5020).[]{data-label="fig:curvature_g_k"}](curvature_g_k.png "fig:"){width="50.00000%"}
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#### Principal curvature
For a unite tangent vector $X \in T_uf$ and $I(X,X)=1$, it is a principal curvature direction for $f$ if $II(X,X)$ has a stationary value for $X$, or $X$ is an eigenvalue of the Weingarten map $L$. Further, the corresponding eigenvalue $\kappa$ is the principal curvature.
#### Gaussian and mean curvature
Gaussian curvature is the determinate $K={\rm Det}(L)=\kappa_1\cdot \kappa_2$ and mean curvature is the average value $H=\frac{1}{2}{\rm Tr}(L)=\frac{1}{2}(\kappa_1+\kappa_2)$. Gaussian and mean curvatures can be expressed in local coordinates as $$\begin{aligned}
&&K=\frac{ {\rm Det}(h_{ij})}{{\rm Det}(g_{ij})}=\frac{h_{11}h_{22}-h_{12}^2}{g_{11}g_{22}-g_{12}^2},\\
&&H=\frac{1}{2}\Sigma_i h^i_i=\frac{1}{2}\Sigma_{i,j} h_{ij} g^{ij}= \frac{1}{2 {\rm Det}(g_{ij})}(h_{11}g_{22}-2h_{12}g_{12}+h_{22}g_{11}).\end{aligned}$$ Gaussian and mean curvatures of an HIV virus fragment are illustrated in Fig. \[fig:curvature\_g\_k\]. In biophysics, the region with negative Gaussian is often associated membrane-protein interaction sites, while the region with negative mean curvatures on a protein surface is commonly regarded as a potential-ligand or protein-drug bonding site [@KLXia:2014a].
Differential geometry modeling and computation {#sec:scalar_field_geometry}
----------------------------------------------
With the advance of experimental technology, more than a hundred thousand of 3D macromolecular structural data has been accumulated. Differential geometry based surface modeling and curvature measurement are of essential importance to the geometric description and feature recognition of these 3D structural data [@Wei:2005; @Xu:DSM:2006]. In this section, a brief review is given to the differential geometry based modeling of macromolecular surfaces. Two algorithms for curvature calculations are also discussed.
### Minimal molecular surface {#Sec:MMSgeneration}
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![Comparison of the solvent excluded surface (Left chart) and the minimal molecular surface (Right chart) of protein 1PPL. The minimal molecular surface is free from geometric singularities.[]{data-label="fig:MMS"}](MMS.png "fig:"){width="50.00000%"}
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Minimal molecular surface (MMS) was introduced to construct surfaces free of geometric singularities via the variational principle [@Bates:2006; @Bates:2008]. In this approach, a hypersurface $S$ is defined to represent the biomolecular surface. Basically, one assigns each point with coordinate $(x,y,z)$ a value $S(x,y,z)$, which represents the domain information. It can be viewed as a characteristic function of the macromolecular domain. By using geometric measure theory, the surface energy functional can be expressed as [@Wei:2009] $$\begin{aligned}
\label{eq2surface}
G_{\rm{surface}}= \gamma {\rm{Area}}= \int_{{\mathbb R}^3} \gamma |\nabla S | d{\bf{r}},\end{aligned}$$ where $\gamma$ is the surface tension. As it is convenient for us to set up the total free functional as a 3D integral in ${\mathbb R}^3$, one can make use of the concept of a mean surface area [@Wei:2009; @ZhanChen:2011a] and the coarea formula [@coarea] on a smooth surface $$\begin{aligned}
\label{eqarea}
{\rm{Area}}=\int_0^1 \int_{S^{-1}(c)\bigcap\Omega} d\sigma dc = \int_{\Omega} | \nabla S({\bf{r}}) | d{\bf{r}}, \quad \Omega \subset {\mathbb R}^3.\end{aligned}$$ [Here the value of hypersurface function $S$ is distributed between 0 and 1, $S^{-1}(c)$ represents the inverse function of $S$ and $\Omega$ is defined as the whole domain.]{} The variation of Eq. (\[eq2surface\]) with respect to $S$ leads to the vanishing of surface-tension weighted mean curvature, $\nabla\cdot\left(\gamma \frac{\nabla S}{|\nabla S|}\right)=0$. The energy minimization of Eq. (\[eq2surface\]) can be realized by the introduction of an artificial time to obtain a generalized Laplace-Beltrami equation $$\begin{aligned}
\label{MeanCF}
\frac{\partial S}{\partial t}&=&|\nabla S|\nabla\cdot\left(\frac{\gamma\nabla S}{|\nabla S|}\right),\end{aligned}$$ The final MMS, which is free of geometric singularity, is obtained by extracting an iso-surface from the steady state hypersurface function. During each iteration, one can keep the value of $S$ in the van der Waals surface enclosed domain unchanged. Figure \[fig:MMS\] illustrates the difference between the solvent excluded surface and the minimal molecular surface of protein 1PPL.
In the earlier works, sophisticated computational algorithms have been developed to accelerate the construction of MMSs for large biomolecules [@Bates:2009; @ZhanChen:2010a; @ZhanChen:2010b; @XFeng:2012a]. Differential geometry based molecular surface modeling was extended to solvation analysis [@Wei:2009; @ZhanChen:2010a], including level set approaches [@Cheng:2007e; @Cheng2:2009], and ion channel transport [@Wei:2009; @Wei:2012; @Wei:2013; @DuanChen:2013]. Numerical aspects were examined in the literature [@SZhao:2011a; @SZhao:2014a]. Since these approaches work very well for biomolecular structure, function and dynamics, they will attract much attention in the future. However, a more detailed discussion of these issues is beyond the scope of the present review.
### Scalar field curvature evaluation {#sec:algorithm}
For a given set of volumetric biomolecular data, the efficient and accurate computation of curvatures is needed. The evaluation of curvature properties from iso-surface embedded volumetric data has been studied in geometric modeling, although the related techniques have not received much attention in computational biophysics. In this section, two popular algorithms are reviewed. Many other elegant methods, including isophote surface based curvature evaluation [@Verbeek:1993], Sander-Zucker approach [@Sander:1990], Direct surface mapping based approach [@Stokely:1992], piece-wise linear manifold techniques [@Stokely:1992], etc, are often used in the computer science community.
#### Algorithm I
Essentially, the first and second fundamental forms in the differential geometry are involved in the definition and evaluation of the curvatures. a brief discussion of the mathematical background is given [@Soldea:2006; @Bates:2008].
The surface of interest can be extracted from a level set with iso-value $S_0$, i.e., $S(x,y,z)=S_0$. One can assume $S$ to be non-degenerate, i.e., the norm of the gradient is non-zero at $S(x,y,z)=S_0$. Without loss of generality, one can assume that the projection onto $z$ is non-zero as well. Then the implicit function theorem states that locally, there exists a function $z=f(x,y)$, which parametrizes the surface as ${\bf S}(x,y)=(x,y,f(x,y))$. One can express the iso-value relation as $S(x,y,f(x,y))=S_0$. The differentiation with respect to $x$ and $y$ variables leads to two more equations $$\begin{aligned}
\nonumber
S_x(x,y,f(x,y))+S_z(x,y,f(x,y))f_x(x,y)=0,\\\nonumber
S_y(x,y,f(x,y))+S_z(x,y,f(x,y))f_y(x,y)=0,\end{aligned}$$ where $f_x(x,y)$ and $f_y(x,y)$ can be given by $$\begin{aligned}
\nonumber
f_x(x,y)=-\frac{S_x(x,y,z)}{S_z(x,y,z)}; \quad {\rm and} \quad f_y(x,y)=-\frac{S_y(x,y,z)}{S_z(x,y,z)}.\end{aligned}$$ One can define $E(x,y,z), F(x,y,z), G(x,y,z), L(x,y,z), M(x,y,z)$ and $ N(x,y,z)$ below to be the coefficients in the first and second fundamental forms. For simplicity, one can hide parameter labels. Their values for surface function ${\bf S}=(x,y,f)$ can be given as $$\begin{aligned}
\nonumber
E &=& \langle {\bf S}_x , {\bf S}_x\rangle=1+f_x^2=1+\frac{S_x^2}{S_z^2};\\\nonumber
F &=& \langle {\bf S}_x , {\bf S}_y\rangle=f_xf_y=\frac{S_x S_y}{S_z^2};\\\nonumber
G &=& \langle {\bf S}_y , {\bf S}_y\rangle=1+f_y^2=1+\frac{S_y^2}{S_z^2};\\\nonumber
L &=& \langle {\bf S}_{xx} , {\bf n}\rangle=\frac{2S_xS_zS_{xz}-S_x^2S_{zz}-S_z^2S_{xx}}{g^{\frac{1}{2}}S_z^2};\\\nonumber
M &=& \langle {\bf S}_{xy} , {\bf n}\rangle=\frac{S_xS_zS_{yz}+S_yS_zS_{xz}-S_xS_yS_{zz}-S_z^2S_{xy}}{g^{\frac{1}{2}}S_z^2};\\\nonumber
N &=& \langle {\bf S}_{yy} , {\bf n}\rangle=\frac{2S_yS_zS_{yz}-S_y^2S_{zz}-S_z^2S_{yy}}{g^{\frac{1}{2}}S_z^2}.\end{aligned}$$ The Gaussian curvature can be expressed as the ratio of the determinants of the second and first fundamental forms, $$\begin{aligned}
\nonumber
&&K=\frac{2S_xS_yS_{xz}S_{yz}+2S_xS_zS_{xy}S_{yz}+2S_yS_zS_{xy}S_{xz}}{g^2}-\frac{2S_xS_zS_{xz}S_{yy}+2S_yS_zS_{xx}S_{yz}+2S_xS_yS_{xy}S_{zz}}{g^2} \\\nonumber
&& \qquad \left. +\frac{S_z^2S_{xx}S_{yy}+S_x^2S_{yy}S_{zz}+S_yS_{xx}S_{zz}}{g^2} -\frac{S_x^2S_{yz}^2+S_y^2S_{xz}^2+S_z^2S_{xy}^2}{g^2} \right..\end{aligned}$$
Similarly, the mean curvature is given as the average second derivative with respect to the normal direction, $$\begin{aligned}
\label{eq:meanC}
H=\frac{2S_xS_yS_{xz}+2S_xS_zS_{xz}+2S_yS_zS_{yz}-(S_y^2+S_z^2)S_{xx}-(S_x^2+S_z^2)S_{yy}-(S_x^2+S_y^2)S_{zz}}{2g^{\frac{3}{2}}}.\end{aligned}$$ Note that curvature expressions become analytical when the implicit surface is given by the FRI rigidity density, Eq. (\[eq:rigidity3\]).
#### Algorithm II
An alternative algorithm for the curvature extraction from volumetric data is the Hessian matrix method [@Kindlmann:2003]. For volumetric data $S(x,y,z)$, one defines the surface gradient ${\bf g}$ and surface norm ${\bf n}$. $$\begin{aligned}
&& {\bf g} = \nabla S= (S_x,S_y,S_z)^T;\\
&& {\bf n = -\frac{g}{|g|}}.\end{aligned}$$ Here ${T}$ denoting the transpose. One further calculates the matrix $\nabla {\bf n}^T$, which can be expressed as $$\begin{aligned}
&& \nabla {\bf n}^T=- \nabla ( { \frac{{\bf g}}{|{\bf g}|}})=-( { \frac{\nabla {\bf g}^T}{|{\bf g}|}-\frac{{\bf g} \nabla^T |{\bf g}|}{|{\bf g}|^2}}) \\\nonumber
&& \qquad =-\frac{1}{|{\bf g}|}({H}-\frac{{\bf g} \nabla^T ({\bf g}^T{\bf g})^{\frac{1}{2}}}{|{\bf g}|})=-\frac{1}{|{\bf g}|}({H}-\frac{{\bf g} \nabla^T ({\bf g}^T{\bf g})}{2|{\bf g}|({\bf g}^T{\bf g})^{\frac{1}{2}}}) \\\nonumber
&& \qquad =-\frac{1}{|{\bf g}|}({H}-\frac{{\bf g}(2{\bf g}^TH)}{2|{\bf g}|^2})=-\frac{1}{|{\bf g}|}({ I}-\frac{{\bf g}{\bf g}^T}{|{\bf g}|^2}){H} \\\nonumber
&& \qquad =-\frac{1}{|{\bf g}|}({ I}-{\bf n}{\bf n}^T){H}=-\frac{1}{|{\bf g}|} {PH},\end{aligned}$$ where ${I}$ is the identity matrix, matrix $P=I-{\bf nn}^T$ and Hessian matrix ${H}$ is given by $$\begin{aligned}
{H}=\left[
\begin{array}{ccc}
\frac{\partial^2 S}{\partial^2 x} &\frac{\partial^2 S}{\partial x\partial y} &\frac{\partial^2 S}{\partial x\partial y} \\
\frac{\partial^2 S}{\partial x\partial y} &\frac{\partial^2 S}{\partial^2 y} &\frac{\partial^2 S}{\partial y\partial z}\\
\frac{\partial^2 S}{\partial x\partial z} &\frac{\partial^2 S}{\partial y\partial z} &\frac{\partial^2 S}{\partial^2 z}
\end{array}
\right].\end{aligned}$$ Geometrically, ${\bf nn}^T$ project ${\bf n}$ onto a one-dimensional span of ${\bf n}$. Here $I-{\bf nn}^T$ further projects onto the orthogonal space complement to the span of ${\bf n}$, which is the tangent plane.
In general, both ${ P}$ and ${H}$ are symmetric but ${\bf \nabla n}^T$ is not. If ${\bf q_1}$ lies in the tangent plane, $P{\bf q_1}={\bf q_1}$ and ${\bf q_1}^T P={\bf q_1}^T$. Therefore, for ${\bf q_2}$ and ${\bf q_1}$ in the tangent plane, one has $$\begin{aligned}
{\bf q_1}^T PH {\bf q_2=q_1}^T H {\bf q_2=q_2}^T H{\bf q_1=q_2}^T PH{\bf q_1}\end{aligned}$$
The restriction of $\nabla {\bf n}^T=-\frac{1}{|{\bf g}|} { PH}$ to the tangent plane is symmetric and thus there exists an orthonormal basis ${\bf p_1,p_2}$ for the tangent plane in which ${\bf n}^T$ is a 2\*2 diagonal matrix. This basis can be easily extended to an orthonormal basis for all $\{{\bf p_1}, {\bf p_2}, {\bf n} \}$. In this basis, the derivative of the surface normal is given by $$\begin{aligned}
{\bf \nabla n}^T=\left[
\begin{array}{ccc}
\kappa_1 &0 &\sigma_1 \\
0 &\kappa_2 &\sigma_2\\
0 &0 &0
\end{array}
\right].\end{aligned}$$ The diagonal term in the bottom row is zero because no change in normal ${\bf n}$ can lead to a change in length. Motion along ${\bf p_1}$ and ${\bf p_2}$ results in the change of ${\bf n}$ along the same directions, with a ratio of ${\kappa_1}$ and ${\kappa_2}$ respectively. Here ${\bf p_1}$ and ${\bf p_2}$ are the principal curvature directions, while ${\kappa_1}$ and ${\kappa_2}$ are the principal curvatures. When there is a change in normal ${\bf n}$, it tilts according to ${\sigma_1}$ and ${\sigma_2}$. One can further multiply ${\bf \nabla n}^T$ by ${ P}$ to diagonalize the matrix $$\begin{aligned}
G={\bf \nabla n}^T{ P}= {\bf \nabla n}^T \left[
\begin{array}{ccc}
1 &0 &0 \\
0 &2 &0\\
0 &0 &0
\end{array}
\right]=\left[
\begin{array}{ccc}
\kappa_1 &0 &0 \\
0 &\kappa_2 &0\\
0 &0 &0
\end{array}
\right].\end{aligned}$$
The surface curvature measurements are based on geometry tensor ${ G}$. In a volumetric data set or a scalar field, ${G}$ is known in terms of the Cartesian basis $(x, y, z)$. Matrix invariants provide the leverage to extract the desired curvature values $\kappa_1$ and $\kappa_2$ from ${ G}$, regardless of the coordinate frame of the principal curvature direction. The trace of ${ G}$ is $\kappa_1+ \kappa_2$. The Frobenius norm of ${ G}$, notated $|{ G}|_F$ and defined as $\sqrt{{\rm Tr}({ GG}^T)}$, is $\sqrt{\kappa_1^2+\kappa_2^2}$. Then $\kappa_1$ and $\kappa_2$ are found with the quadratic formula.
The two principal curvatures can be evaluated by the following procedure.
1. Calculate matrix $P=I-{\bf nn}^T$
2. Evaluate matrix $G=I-\frac{PHP}{|{\bf g}|}$, $$\begin{aligned}
{ G}=(g_{ij})_{(i,j=1,3)}\end{aligned}$$
3. Compute the trace $t$ and Frobenius norm $f$ of matrix ${ G}$; $$\begin{aligned}
&& t=g_{11}+g_{22}+g_{33};\\
&& f=\|{ G} \|=\sqrt{\sum_i \sum_j g_{ij}^2};\\
&& \kappa_1=\frac{t+\sqrt{2f^2-t^2}}{2};\\
&& \kappa_2=\frac{t-\sqrt{2f^2-t^2}}{2}.\end{aligned}$$
When the two principal curvatures are available, the Gaussian curvature $K$ and mean curvature $H$ can be obtained as $$\begin{aligned}
&&K=\kappa_1 \kappa_2;\\
&&H=\frac{\kappa_1+\kappa_2}{2}.\end{aligned}$$ Essentially, the Hessian matrix method generates the same results as Algorithm I derived from the first and second fundamental form. However, for FRI rigidity density given in Eq. (\[eq:rigidity3\]), Algorithm I is preferred as it is analytical and without matrix diagonalization.
### Analytical minimal molecular surface {#sec:density2}
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![An illustration of analytical minimal molecular surfaces generated at different $\sigma$ values. From [ (a)]{} to [ (d)]{}, the $\sigma$ is chosen as 0.5, 0.6, 0.7 and 0.8, respectively. For all figures, isosurfaces are extracted at mean curvature isovalue of 0.001. []{data-label="fig:minimal_surface"}](minimal_surface.png "fig:"){width="50.00000%"}
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![An illustration of analytical minimal molecular surfaces generated at different $\sigma$ values. From [ (a)]{} to [ (d)]{}, the $\sigma$ is chosen as 0.5, 0.6, 0.7 and 0.8, respectively. For all figures, isosurfaces are extracted at mean curvature isovalue of 0.001. []{data-label="fig:minimal_surface"}](ms_scale.png "fig:"){width="60.00000%"}
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Minimal surfaces are ubiquitous in nature and are a fascinating topic for centuries. The Euler-Lagrange minimization of the hypersurface was proposed to generate minimal molecular surface (MMS) for biomolecules [@Bates:2008]. In general, MMS is a result of vanishing mean curvature and is free of geometric singularity. It is a powerful concept in biophysical modeling [@Bates:2008; @Wei:2009]. Nevertheless, it is still much more computationally expensive to construct MMS than to generate SES. In this work, a new minimal molecular surface, called analytical minimal molecular surface (AMMS) is proposed. In this approach, the rigidity density representation of biomolecular data by Eq. (\[eq:rigidity3\]) is employed. Then the mean curvature can be [*analytically*]{} computed by using Eq. (\[eq:meanC\]). Finally, AMMS can be constructed by setting the isosurface of the mean curvature to zero (or practically, a number very close zero).
One can consider the hexagonal ring and a fullerene C$_{20}$ molecule in the study of AMMS. The coordinates used for particles in the hexagonal ring are (0.000, 1.403, 0.000; -1.215, 0.701, 0.000; -1.215, -0.701, 0.000; 0.000, -1.403, 0.000; 1.215, -0.701, 0.000; and 1.215, 0.701, 0.000). The fullerene C$_{20}$ molecule has a highly symmetric cage structure made of 12 pentagons. Its atomic coordinates are (1.569, -0.657, -0.936; 1.767, 0.643, -0.472; 0.470, -0.665, -1.793; 0.012, 0.648, -1.826; 0.793, 1.467, -1.028; -0.487, -1.482, -1.216; -1.564, -0.657, -0.895; -1.269, 0.649, -1.277; -0.002, -1.962, -0.007; -0.770, -1.453, 1.036; -1.758, -0.638, 0.474; 1.288, -1.450, 0.163; 1.290, -0.660, 1.305; 0.012, -0.646, 1.853; 1.583, 0.645, 0.898; 0.485, 1.438, 1.194; -0.503, 0.647, 1.775; -1.606, 0.672, 0.923; -1.296, 1.489, -0.166; -0.010, 1.973, -0.006). The fullerene rigidity density is given by Eq. (\[eq:rigidity3\]). In both cases, the parameters $w_i$ and $\sigma_i$ are set to $1$ and $0.7$, respectively.
When the mean curvature isovalue equals to zero, the resulting surfaces are usually composed by several non-intersecting surfaces perpendicular to atomic bonds near the BCPs. When loosing the condition a little bit by setting the isovalue to 0.001 (or some other very small positive value), a better AMMS can be generated. Figure \[fig:minimal\_surface\] depicts the AMMS for the hexagonal ring and C$_{20}$ molecule. It is found that for the hexagonal ring, each atom is enclosed by a surface segment. These segments are tightly close to each other and only connect near BCPs. For the C$_{20}$ molecules, the surface appears to be better connected. However, a careful examination reveals the separation as well. Obviously, these surfaces are not minimal surfaces.
To generate better understanding of the minimal molecular surface generation, one can explore several $\sigma$ values for the hexagonal ring. Figure \[fig:minimal\_surface\] illustrates the results. From [ (a)]{} to [ (d)]{}, $\sigma$ is chosen as 0.5, 0.6, 0.7 and 0.8, respectively. All the isosurfaces are extracted at mean curvature equals to 0.001. It is seen that a relatively large $\sigma$ value produces a better minimal surface. The gaps between atomic segments are gradually narrow as the increase of $\sigma$ value and finally disappear, which gives rise to a good quality AMMS.
Scalar and vector field topology {#sec:scalar_field_topology}
--------------------------------
Topological approaches have become an integral part in data analysis, visualization, and mathematical modeling for volumetric as well as for vector valued data sets. In fact, the results of scalar and vector field topology coincide each other for gradient vector fields, although the respective mathematical approaches are originated from different fields. Vector field topology is usually developed for analyzing the streamlines of fluid flow generated by the velocity vector during the flow evolution. These techniques can be applied to macromolecular analysis for dealing with cryo-EM data and improving the structural construction.
Three dimensional scalar field data, particular the molecular structural data are widely available from many data sources. To decipher useful information from these data requires highly efficient methods and algorithms. Mathematical approaches, including topological tools, such as fundamental groups, homology theory, contour tress, Reeb graphs [@Dey:2013], Morse theory [@harker:mischaikow:mrozek:nanda; @Mischaikow:2013], etc, have a great potential for revealing the intrinsic connectivity or structure-function relationship. Topological data analysis (TDA) has gained much attention in the past decade. The traditional atoms in molecules (AIM) analysis [@Bader:1985; @Bader:1990] can be viewed as an application of TDA to electron density analysis. Historically, the topological study of scalar fields, particularly electron densities, has advanced the understanding of molecules and their chemical and biological functions tremendously. The theory of AIM developed by Bader and his coworkers has provided an elegant and feasible approach to define atoms in molecule. AIM can be generalized into a more general theory called quantum chemical topology (QCT), which applies the topological analysis in AIM to the study of other physically meaningful scalar fields [@Popelier:2005]. Mathematically, this topological analysis used in AIM is known as Morse theory. In general, all scalar fields used in QCT are some kind of Morse functions and more can be proposed as long as they satisfy the Mores function constraints. On the other hand, geometric modelings and analysis contribute a lot to the molecular visualization and structure-function relationship. Various surface definitions from geometric modelings facilitate the visualization and characterization of molecules. Geometric analysis has contributed enormously to the understanding of structure-function relationship. Among the geometric analysis, surface curvature analysis has provide great insights into solvation analysis, protein-protein interaction, drug design, etc. [@KLXia:2014a; @DDNguyen:2016c]
In the QCT theory, researchers explore atomic or molecular properties through the analysis of various properties at critical points. In geometric analysis, curvature estimation is normally done on a special surface, such as SES, SAS, Gaussian surface, FRI rigidity surface, etc. Even though these two approaches are very efficient and powerful in capturing and characterizing atomic and molecular information, enormous information embedded in the electron density scalar field has not been fully utilized. Question is how to improve geometric and topological analysis for biomolecules. Note that in persistent homology analysis, physical properties are analyzed by a systematic filtration process. Therefore, for a given scalar field, a better understanding can be achieved if the topological analysis includes not only some critical points or a special iso-surface, but also a series of iso-surfaces derived from a systematic evolution of iso-values. Further, more information can be extracted if one considers not only simply scalar fields, such as electron density, electron density Laplacian, etc, but also geometric properties like individual eigenvalues and various curvatures. These approaches are developed in the present work.
In the following, a brief review of some very basic concepts in AIM is given. Their connection with mathematical theories is discussed. Then, some new approaches are presented. Examples of applications are provided.
### Critical points and their classification {#sec:CP}
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![An illustration of four types of critical points, i.e., nucleic critical point (NCP), bond critical point (BCP), ring critical point (RCP) and cage critical point (CCP). [**(a)**]{} The demonstration of flowlines and CPs for a benzene molecule (Hydrogen atoms are not considered for simplicity). [**(a)**]{} The CPs for a Cubane (Hydrogen atoms are not considered for simplicity). []{data-label="fig:cp"}](cp.png "fig:"){width="60.00000%"}
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Three dimensional molecular electron density data can be collected through experimental tools, like electron microscopy, cryo-electron microscope, etc and also from theoretical models in quantum mechanics. One can denote electron density function $\rho$ and its domain $\Omega$. A quantitative method to analyze the topology of $\rho$ is to consider its first derivative, i.e., gradient $\nabla \rho$. At certain points, called critical points (CPs), this gradient vanishes. The characteristics of these points is determined by the second derivatives, which form the Hessian of electron density $\rho$. By diagonalization of the Hessian matrix, one can obtain three eigenvalues $\gamma_1 \leq \gamma_2 \leq \gamma_3$. Their sum equals to the Laplacian of electron density $\rho$. That is, locally [@Bader:1990] $$\begin{aligned}
\nabla^2 \rho= \gamma_1+\gamma_2 + \gamma_3=\frac{\partial^2 \rho}{\partial^2 x} +\frac{\partial^2 \rho}{\partial^2 y} +\frac{\partial^2 \rho}{\partial^2 z}.\end{aligned}$$ It can be noticed that the Hessian matrix is symmetric, therefore all the eigenvalues are real. Based on the positive and negative sign of these three eigenvalues, one defines rank and signature to characterize critical points. The rank of a CP is the number of non-zero eigenvalues, and a signature is the algebraic sum of the signs (+1 or -1) of the eigenvalues. In general, a CP is non-degenerate, meaning its three eigenvalues are non-zero (Rank$=3$). A degenerate critical point is unstable in the sense that even a small change in the function will cause it either to vanish or bifurcate into a non-degenerate CP.
Based on the rank and signature value, non-degenerate CPs in three dimensional scalar field can be classified into four basic types, namely, nucleic critical point (NCP), bond critical point (BCP), ring critical point (RCP) and cage critical point (CCP). Table \[tb:cp\] lists these values. An illustration of these four types of CPs can be found in Figure \[fig:cp\]. An NCP is a nucleic center of an atom. It is represented by a cyan-color point. A BCP is a bond center between two atoms and is marked with a red-color point. A RCP is usually found at the center of a ring structure and is colored in blue. A CCP is known as a cage critical point, and can only be found in the center of a cage structure. In general, an NCP is a local maximal point. BCP and RCP both are saddle points. A CCP is a local minimal point. In Sections \[sec:eigenvalue\] and \[sec:T\_C\], it will be demonstrated that these properties can be used to systematically characterize and analyze the eigenvalue and curvature isosuface information.
Topologically, the general structure connectivity can be characterized by the number of CPs. This is stated in the famous Poincaré-Hopf theorem as following: $$\begin{aligned}
\label{Eq:ph}
N_n-N_b+N_r-N_c=\chi (\rho),\end{aligned}$$ where $N_n$, $N_b$, $N_r$ and $N_c$ are numbers of NCPs, BCPs, RCPs and NCPs, respectively. Here $\chi(\rho)$ is the Euler characteristic. The Poincaré index is defined on a vector field. In the AIM theory, one studies the gradient vector field of electron density. Poincaré indices for NCP, BCP, RCP and NCP are $ 1, -1, 1$ and $-1$, respectively. More properties of this gradient vector field are discussed in the following section.
Equation (\[Eq:ph\]) can also be derived from simplicial complex analysis [@KLXia:2014c]. Essentially, NCP can be viewed as a point (0-simplex); BCP corresponds to a straight line (1-simplex); RCP is for a polygon (2-simplex); CCP is then for a polyhedron (3-simplex). In this manner, Euler characteristic can be directly employed and the above equation can be obtained as well.
Rank Signature Poincaré index Simplex Property
----- ------ ----------- ------------------ ----------- --------------
NCP 3 $-3$ 1 0-simplex local maxima
BCP 3 $-1$ $-1$ 1-simplex saddle
RCP 3 1 1 2-simplex saddle
CCP 3 3 $-1$ 3-simplex local minima
: The critical point can be classified into four basic types including: nucleic critical point (NCP), bond critical point (BCP), ring critical point (RCP) and cage critical point (CCP), as demonstrated in Fig. \[fig:cp\].
\[tb:cp\]
### Vector field topology
The gradient $\nabla \rho$ on the entire domain $\Omega$ forms a vector field. The topological property of this vector field can be explored by tools borrowed from dynamic systems or the mathematical analysis of fluid flows. These tools include integral line, separatrix, and Poincaré index, etc.
An integral line is a curve $l(t)$ of a function $f({\bf r})$ that satisfies $\frac{\partial l}{\partial t}=\nabla f(l(t))$. Its origin and destination can be defined as $${\rm org}(l)=\underset{t \rightarrow - \infty}{\lim} l(t)$$ and $${\rm dest}(l)=\underset{t \rightarrow \infty}{\lim} l(t).$$ Integral lines satisfy the following conditions:
1. Two integral lines are either disjoint or the same.
2. Integral lines cover all the manifold.
3. The limits, org$(l)$ and dest$(l)$, are critical points.
In general, integral lines represent the gradient flow between critical points. All the integral lines that share the same $org$ form a region called atomic basin. The whole electron density domain is subdivided into many atomic basins. The interface between these attraction basins is called inter-atomic surface (IAS). Mathematical, IAS is the separatrix of a gradient vector field. Each attraction basin includes one and only one atom. This is known as the quantum topological definition of an atom in a molecule.
IAS also satisfies the zero-flux condition $\nabla f({\bf r}) \cdot {\bf n( r)}=0$. Here $\nabla f({\bf r}) $ is a gradient vector and ${\bf n(r)}$ is the normal vector to the IAS.
#### Morse theory
It should be noticed that basic concepts like critical point, degeneracy, critical point classification, basin, etc. are the essential part of a mathematical tool called Morse theory [@harker:mischaikow:mrozek:nanda; @Mischaikow:2013]. In general, the atom in molecular method can be viewed as an application of Morse theory in molecular density field analysis. More specifically, various critical points, including NCP, BCP, RCP, and CCP, are the counterparts of peak point, saddle-1 point, saddle-2 point and valley point, respectively. The separatrix of a gradient vector field just provides a Morse decomposition of the underlying molecular scale field manifold [@harker:mischaikow:mrozek:nanda; @Mischaikow:2013].
### Topological characterization of chemical bonds {#sec:chemical_bond}
Chemical properties of molecular systems are profoundly determined or influenced by their atomic covalent bonds and noncovalent interactions. Usually, covalent bonds are much stronger and determine the structural integrity of a molecule. Noncovalent interactions are comparably weak but play important roles in macromolecular assembly, protein folding, macromolecular function, etc. Traditionally, the Laplacian of electron density can be used to interpret noncovalent interactions of a molecular system. Recently, signed electron density and reduced gradient, two scalar fields derived from electron density, have drawn much attention in quantum chemistry since they enable a qualitative visualization of noncovalent interactions even in complex molecular systems [@Johnson:2010; @contreras2011analysis; @contreras2011nciplot; @Gillet:2012; @Gunther:2014]. These approaches are reviewed below.
#### The Laplacian of electron density
The Laplacian of electron density $\rho ({\bf r})$ can be used to indicate the electron density concentration and depletion. Essentially, the density is locally concentrated where $\nabla^2 \rho({\bf r})<0$, and locally depleted where $\nabla^2 \rho({\bf r})>0$. In this manner, if one defines the function $L({\bf r})=-\nabla^2 \rho({\bf r})$, the maximum in $L({\bf r})$ denotes the maximum in the concentration of the density. The minimum in $L({\bf r})$ implies a depletion of density. The Laplacian of the electronic charge distribution, $L({\bf r})$, demonstrates the presence of local concentrations of charge in the valence shell of an atom in a molecule. These local maxima faithfully duplicate in number, location, and size of the spatially localized electron pairs of the valence shell electron pair repulsion (VSEPR) model. Thus the Laplacian of the charge density provides a physical basis for the Lewis and VSEPR models [@Bader:1988].
#### Identifying noncovalent interactions
The study of the Hessian matrix and its three eigenvalues has yielded many intriguing results [@Bader:1990]. It has been found that all eigenvalues ($\gamma_1({\bf r})$, $\gamma_2({\bf r})$ and $\gamma_3({\bf r})$) are negative in the vicinity of the nuclei centers. Away from these centers, the largest eigenvalue $\gamma_3({\bf r})$ becomes positive, and varies along the internuclear axis representing covalent bonds. It is also found that $\gamma_1({\bf r})$ and $\gamma_2({\bf r})$ describe the density variation orthogonal to this internuclear axis. More specially, $\gamma_1({\bf r})$ is always negative even it is away from the nuclei. While $\gamma_2({\bf r})$ can be either positive, meaning attractive interactions concentrating electron charge perpendicular to the bond, or negative, meaning repulsive interactions causing density depletion. Using this localized information, the signed electron density $\widetilde{\rho}({\bf r})$ is defined as $$\begin{aligned}
\label{Eq:sed}
\widetilde{\rho} ({\bf r})= {\rm sign}(\gamma_2({\bf r}))\rho({\bf r}).\end{aligned}$$ The signed electron density additionally enables the differentiation of attracting and repulsive interactions.
To further reveal weak noncovalent interactions, the reduced gradient is introduced as following [@Gillet:2012; @Gunther:2014] $$\begin{aligned}
\label{Eq:reduced_gradient}
s({\bf r})=\frac{1}{2(3\pi^2)^{\frac{1}{3}}}\frac{|\nabla \rho({\bf r})|}{\rho({\bf r})^\frac{4}{3}}.\end{aligned}$$ The reduced density gradient describes the deviation in atomic densities due to interactions and has found interesting applications in in analyzing biomolecular structure and function [@Johnson:2010; @Gillet:2012; @Gunther:2014].
Geometric-topological (Geo-Topo) fingerprints of scalar fields {#sec:GTF}
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### Geo-Topo fingerprints of Hessian matrix eigenvalue maps {#sec:eigenvalue}
The QCT and AIM analyses of molecules are limited to special locations and given iso-surfaces. In this work, the Hessian eigenvalue analysis of a molecular density is considered, not only for a few critical points, but also for the whole domain. Additionally, the topology over a series of isosurfaces derived from a systematic evolution of isovalues are analyzed. More specifically, in the topological persistence of Hessian matrix eigenvalues, the topological properties of a series of isosurfaces, generated by varying the isovalue from the smallest to the largest, are systematically studied. Therefore, the isosurface value behaves like a filtration parameter. The topological transitions in this series of isosurfaces are emphasized. Another very important aspect of this analysis is to analyze the behavior of isosurfaces through its relation with the four types of CPs. It should also be noticed that these methods are greatly different from other interatomic surface methods [@Pendas:2003; @Popelier:1996], as these models focus on a special surface.
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![ Hessian matrix eigenvalue maps of the hexagonal ring model at cross section $Z=0$. The behaviors of three eigenvalues, i.e., $\gamma_1$, $\gamma_2$ and $\gamma_3$, are illustrated in [ (a)]{}, [ (b)]{} and [ (c)]{}, respectively. []{data-label="fig:C6_eigen_2d"}](C6_eigen_2d.png "fig:"){width="80.00000%"}
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![ Eigenvalue maps obtained from different isovalues (or level-set values) for a cubic structure. [ (a)]{} The isosurfaces for the first eigenvalue. The isovalues from [ ($a_1$)]{} to [ ($a_4$)]{} are -3.0, -1.5, 0.1 and 0.9. [ (b)]{} The isosurfaces for the second eigenvalue. The isovalues from [ ($b_1$)]{} to [ ($b_4$)]{} are -1.0, 0.5, 1.0 and 1.5. [ (c)]{} The isosurfaces for the third eigenvalue. The isovalues from [ ($c_1$)]{} to [ ($c_4$)]{} are -1.0, 1.5, 2.0 and 2.5. []{data-label="fig:Cubane_eigen_3d"}](C6_eigen_3d.png "fig:"){width="60.00000%"}
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![ Eigenvalue maps obtained from different isovalues (or level-set values) for a cubic structure. [ (a)]{} The isosurfaces for the first eigenvalue. The isovalues from [ ($a_1$)]{} to [ ($a_4$)]{} are -3.0, -1.5, 0.1 and 0.9. [ (b)]{} The isosurfaces for the second eigenvalue. The isovalues from [ ($b_1$)]{} to [ ($b_4$)]{} are -1.0, 0.5, 1.0 and 1.5. [ (c)]{} The isosurfaces for the third eigenvalue. The isovalues from [ ($c_1$)]{} to [ ($c_4$)]{} are -1.0, 1.5, 2.0 and 2.5. []{data-label="fig:Cubane_eigen_3d"}](Cubane_eigen_3d.png "fig:"){width="60.00000%"}
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To illustrate the idea, two toy models, i.e., the hexagonal ring discussed earlier and a cubic structure are considered. The coordinates for the cubic structure are set to (1.245, 0.537, -0.073; 0.924, -0.995, 0.024; -0.123, -0.704, 1.155; 0.199, 0.828, 1.058; 0.123, 0.704, -1.155; -0.924, 0.995, -0.024; -1.245, -0.537, 0.073; and -0.199, -0.828, -1.058). The discrete to continuum mapping, Eq. (\[eq:rigidity3\]), is used to generate rigidity density. The parameters $\sigma$ and $w_i$ are chosen as $0.7$ Å and $1$ for all particles. In this approach, Hessian matrix is evaluated at each point of the computational domain and its eigenvalue is obtained everywhere as well, which forms an eigenvalue scalar field.
To have a general idea of the basic distribution of eigenvalues, one can study the Hessian matrix eigenvalue behavior of the hexagonal ring in a two-dimensional plane $Z=0$, as all its particles are located within this special plane. Results are illustrated in Fig. \[fig:C6\_eigen\_2d\]. Three eigenvalues, i.e., $\gamma_1$, $\gamma_2$ and $\gamma_3$, are demonstrated in subfigure [**(a)**]{}, [**(b)**]{} and [**(c)**]{}, respectively. It can be seen that for regions near NCPs, all three eigenvalues are negative. For regions near BCPs, $\gamma_1$ is always negative. While $\gamma_2$ is negative in the very closed neighborhood and gradual increases to be positive further away. Finally, $\gamma_3$ is always positive. For regions near RCPs, all three eigenvalues are positive. These results are consistent with findings in AIM [@Bader:1990].
To obtain more geometric insights, particularly eigenvalue behaviors on all four types of CPs, the cubic structure is considered. In this case, the CCP can be identified as having the positive $\gamma_1$ and $\gamma_2$.
As stated above, this approach is to extract a series of isosufaces of Hessian matrix through a filtration process. One can carefully observe these eigenvalue isosurface patterns to detect topological transitions. The geometric information is further analyzed and its relation with the four types of CPs is summarized into several unique characteristics, called Geo-Topo fingerprints. Using hexagonal ring and cubic structure models, the Geo-Topo fingerprints for eigenvalues are revealed.
Figures \[fig:C6\_eigen\_3d\] and \[fig:Cubane\_eigen\_3d\] demonstrate four unique patters for each eigenvalue. The subscripts $1$ to $4$ indicate four representative eigenvalue isovalues, from the small to the large. The notations $({\bf a})$, $({\bf b})$ and $({\bf c})$ represent eigenvalue $\gamma_1$, $\gamma_2$ and $\gamma_3$, respectively. As stated above, the filtration process delivers a series of isosurfaces. To avoid confusion, the filtration process always goes from the smallest value to the largest one. Only four representative isosurfaces that capture some unique topological features within the eigenvalue isosurface series, are selected. Through the comparison of these patterns, some common features can be extracted.
1. For all the three eigenvalues, regions around NCPs are always concentrated with negative values.
2. For $\gamma_1$, as the filtration goes, negative isosurfaces first appears near NCPs. Then they enlarge to incorporate regions near BCPs, being still negative. The regions near RCPs can be analyzed from two different perspectives: i.e., along the ring plane and perpendicular to the ring plane. Along the ring plane, $\gamma_1$ values gradually increase to about 0 as the eigenvalue isosurface propagate to RCPs. There is a sudden topological transition for the isosuface when its value passes through 0. When becoming positive, eigenvalue isosurfaces form ellipsoids surfaces perpendicular to the ring plane near all RCPs. These ellipsoids usually appear in pairs and symmetric to each other along the ring plane as indicated in Fig. \[fig:C6\_eigen\_3d\] $({\bf a})$. Finally, positive isosurface appears near the regions of CCPs.
3. For $\gamma_2$, as the filtration process goes, negative isosurfaces first appear near NCPs and then enlarge to incorporate regions near BCPs just like $\gamma_1$. However, positive isosurfaces appear much earlier. They occupy the regions around RCPs and CCPs. More interestingly, they form a loop around each bond near its BCP. To be more precise, these loops are perpendicular to atom bonds at BCPs and atom bonds pass through them at their centers. The isosurfaces gradually shrink and reduce to regions around RCPs as their values grows.
4. For $\gamma_3$, its negative isosurfaces concentrate only in regions around NCPs. Positive isosurfaces form a sphere slice around each atom. Isosurface with relative small positive value also appears around the RCPs and CCPs. As the filtration goes further, positive isosurfaces concentrate around regions near BCPs.
### Geo-Topo fingerprints of curvature maps {#sec:T_C}
In this work, the topological analysis of curvature maps are developed. One can still consider the hexagonal ring and the cubic structure discussed in the last section. The discrete to continuum mapping is carried out to generate the FRI rigidity density. Then curvatures are evaluated at all of the molecular FRI rigidity isosurfaces (or every point in the computational domain) to form a curvature map. At each curvature isovalue, topological analysis is applied. Here, topological analysis is twofold. One type of topological analysis is to identify topological critical points (i.e., 0-simplex, 1-simplex, 2-simplex, etc.). The other type is to carry out persistent homology analysis to track topological invariants during the density filtration of the curvature map.
#### Gaussian and mean curvature maps {#sec:K_H}
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![Isosurfaces of Gaussian and Mean curvature maps for a cubic structure. [ (a)]{} The isosurfaces built from same Gaussian curvature. The isovalues from [ ($a_{1}$)]{} to [ ($a_{4}$)]{} are -20.0, -2.0, 10.0 and 20.0. [ (b)]{} The isosurfaces built from same Mean curvature. The isovalues from [ ($b_{1}$)]{} to [ ($b_{4}$)]{} are -2.0, -1.0, 3.0 and 4.0. []{data-label="fig:Cubane_gaussian_mean_3d"}](C6_gaussian_mean_3d.png "fig:"){width="60.00000%"}
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![Isosurfaces of Gaussian and Mean curvature maps for a cubic structure. [ (a)]{} The isosurfaces built from same Gaussian curvature. The isovalues from [ ($a_{1}$)]{} to [ ($a_{4}$)]{} are -20.0, -2.0, 10.0 and 20.0. [ (b)]{} The isosurfaces built from same Mean curvature. The isovalues from [ ($b_{1}$)]{} to [ ($b_{4}$)]{} are -2.0, -1.0, 3.0 and 4.0. []{data-label="fig:Cubane_gaussian_mean_3d"}](Cubane_gaussian_mean_3d.png "fig:"){width="60.00000%"}
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It is seen from the above analysis that with four types of CPs, Geo-Topo fingerprints are extracted and the behaviors of the Hessian matrix eigenvalue isosurfaces can been characterized very well. However, before the employment of this technique in Gaussian and mean curvature isosurface analysis, it is helpful to review a little bit more the four types of CPs. As stated in Section \[sec:CP\], NCPs and CCPs are locally maximal and locally minimal, respectively. Actually, for NCPs, all three eigenvalues have negative signs, while for CCPs, all three eigenvalues have positive signs. BCPs and RCPs are saddle points, which means that their eigenvalues have both positive and negative values. Geometrically, the curvature is isotropic near NCPs and CCPs, but anisotropic near BCPs and RCPs. So in this analysis, the isosurfaces near BCPs and RCPs can be divided into two types, namely, A-type or V-type. A-type isosurface means it is along atomic bonds or ring planes. V-type isosurface means it is vertical or perpendicular to the atomic bonds or ring planes. For instance, the isosurface near the RCP in Fig. \[fig:C6\_gaussian\_mean\_3d\] [**($a_1$)**]{} is an A-type, but it becomes a V-type in Fig. \[fig:C6\_gaussian\_mean\_3d\] [**($a_3$)**]{}. Another important property is that A-type and V-type isosurfaces are usually with different signs. That is, if A-type of isosurface is obtained from a positive isovlaue, the associated V-type isosurface can only be obtained from a negative isosurface. With this notation, one can extract some Geo-Topo fingerprints for Gaussian and mean curvatures. The basic results are summarized as below.
1. For a Gaussian curvature field, it has negative A-type isosurfaces near RCPs and negative V-type isosurfaces near BCPs. In contrast, positive isosurfaces enclose regions near BCPs and CCPs. Positive V-type isosurfaces can be found in RCPs and positive A-type isosurfaces are found near BCPs. Finally, positive isosurfaces are found in NCPs.
2. For a mean curvature field, it has negative isosurfaces near RCPs and CCPs. Negative V-type isosurfaces can be found near BCPs. Positive isosurfaces are found in NCPs. Finally, A-type positive isosurfaces can be found near BCPs.
#### Maximal and minimal curvature maps {#sec:k1k2}
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![Isosurfaces for maximal and minimal curvature maps for a cubic structure. [**(a)**]{} The isosurfaces built from same maximal curvature. The isovalues from [**($b_{1}$)**]{} to [**($b_{4}$)**]{} are -2.0, 1.0, 3.0 and 4.0. [**(b)**]{} The isosurfaces built from same minimal curvature. The isovalues from [**($a_{1}$)**]{} to [**($a_{4}$)**]{} are -10.0, -4.0, 3.0 and 4.0. []{data-label="fig:Cubane_k1k2_3d"}](C6_k1k2_3d.png "fig:"){width="60.00000%"}
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![Isosurfaces for maximal and minimal curvature maps for a cubic structure. [**(a)**]{} The isosurfaces built from same maximal curvature. The isovalues from [**($b_{1}$)**]{} to [**($b_{4}$)**]{} are -2.0, 1.0, 3.0 and 4.0. [**(b)**]{} The isosurfaces built from same minimal curvature. The isovalues from [**($a_{1}$)**]{} to [**($a_{4}$)**]{} are -10.0, -4.0, 3.0 and 4.0. []{data-label="fig:Cubane_k1k2_3d"}](Cubane_k1k2_3d.png "fig:"){width="60.00000%"}
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One can also carry out a throughout investigation of maximal and minimal curvature maps. Isosurfaces near NCPs, BCPs, RCPs and CCPs are analyzed in the same manner as they were done in the above two sections. Figures \[fig:C6\_k1k2\_3d\] and \[fig:Cubane\_k1k2\_3d\] illustrate results for the hexagonal ring and the cubic structure, respectively. Main results about their Geo-Topo fingerprints are summarized below:
1. For maximal curvature map, it has negative isosurfaces near CCPs and negative V-type isosurfaces near RCPs. Positive isosurfaces enclose regions near BCPs and NCPs. Positive A-type isosurfaces can be found in RCPs.
2. For minimal curvature map, it has negative isosurfaces near RCPs and negative V-type isosurfaces near BCPs. Negative isosurfaces also enclose region near CCPs. Positive isosurfaces are found in NCPs. Finally, A-type positive isosurfaces can be found near BCPs.
It can be noticed that all curvature representations, i.e., Gaussian curvature, mean curvature and two principal curvatures, are quite different from Hessian matrix eigenvalue distributions. In general, negative curvatures usually do not occur near NCPs, this is particularly true for maximal and mean curvatures. They are more likely to appear near BCPs, RCPs and CCPs. In contrast, positive isosurfaces are concentrated in the atomic basin (i.e., NCPs).
NCP BCP RCP CCP
------------ ----- ------------------------ ------------------------ -----
$\gamma_1$ N N P P
$\gamma_2$ N N; P-Loop P P
$\gamma_3$ N P P P
$K$ P A-type (P); V-type (N) V-type (P); A-type (N) P
$H$ P A-type (P); V-type(N) N N
$\kappa_1$ P P A-type (P); V-type (N) N
$\kappa_2$ P A-type (P); V-type (N) N N
: Geo-Topo fingerprints for eigenvalue and curvature maps. For simplicity, notations “P” and “N” mean positive and negative, respectively. Loop means the ring structure around the bond (see Section \[sec:eigenvalue\] for detail). A-type and V-type means the isosurface is [*along with*]{} or [*vertical to*]{} atomic bonds or ring planes. Four types of critical points (CPs) are nucleic critical point (NCP), bond critical point (BCP), ring critical point (RCP) and cage critical point (CCP). Here $\gamma_1$, $\gamma_2$ and $\gamma_3$ are three Hessian matrix eigenvalues. Gaussian and mean curvatures are represented by $K$ and $H$, respectively. The two principal curvatures are maximal curvature $\kappa_1$ and minimal curvature $\kappa_2$.
\[tb:geometric\_fingerprint\]
To have a more general understanding of the properties of eigenvalue and curvature maps, all of the above-mentioned results are summarized in Table \[tb:geometric\_fingerprint\]. To show the consistency of results, one can take some simply tests. For instance, one can compare the results of $\kappa_1$ and $\kappa_2$ with $K$. Since $K=\kappa_1 \kappa_2$, their signs of isosurface at four types of CPs should match with each other. For NCP, $\kappa_1$ and $\kappa_2$ are both positive, and multiplying together yields a positive $K$, exactly as it is found in the table. For all other three types of CPs, they all match very well.
Eigenvector field analysis {#sec:Tensor field}
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![The 4th to 11th eigenmodes of protein 2XHF. The eigenmodes are evaluated based on the rigidity density of 2XHF. A threshold value as 60% of the Maximum density is chosen. []{data-label="fig:Density_2XHF"}](2XHF_mode.png "fig:"){width="80.00000%"}
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Tensor fields widely exist in natural world. For biomolecules, researchers are particular interested in their motions. As stated in the previous sections, various methods, including molecular dynamics, anisotropic network model (ANM), anisotropic FRI (aFRI), etc, have been developed to explore the dynamics of the biomolecular systems. However, all the above mentioned approaches rely on the discrete representation and can not be used in the study of motions of continuous biomolecular density function or data, particularly electron density data, such as cryo-electron microscopy (cryo-EM) maps. To analyze the motion of density profiles, a virtual particle model (VPM) is proposed to systematically explore the anisotropic motions of continuous density.
### Virtual particle model
Both ANM and aFRI depend on their graph or discrete representation of biomolecules. It is not obvious how to construct a continuum model for a continuous density function to characterize its anisotropic motions. Previously, vector quantization (VQ) algorithm [@Gray:1984vector] is employed to decompose the electron density map of a biological molecule into a set of finite Voronoi cells. It is then combined with ANM to explore the dynamic of the cryo-EM data [@Tama:2002exploring; @Ming:2002describe].
In this section, virtual particles are introduced and defined for each small volume or element of a density data. To be more specific, the domain of the density function can be discretized into many elements. In general, the discretization can be non-uniformed and the elements may have different sizes. One can associate each element with a virtual particle, which is centered at the element center but having a continuous density profile. One can assume that all virtual particles are correlated with each other, but the correlations between them decrease with the distance or follow prescribed relations. The anisotropic motions of virtual particles are obtained. Similar to ANM and aFRI approaches, such anisotropic motions are evaluated from the eigenmodes of the anisotropic Kirchhoff matrix.
One can assume that the density function of interest is given by $\mu({\bf r})$, which can be either a cryo-EM density map or a rigidity density computed from atomic coordinates by using the discrete to continuum mapping as shown in Section \[Sec:GeometricRep\]. One can consider two virtual particles centered at ${\bf r}_I$ and ${\bf r}_J$ and enclosed by the volume elements of $\Omega_I$ and $\Omega_J$, respectively. A special scaling parameter $\gamma({\bf r}_I,{\bf r}_J,\Omega_1,\Omega_2,\eta_{IJ})$ is proposed as following: $$\begin{aligned}
\label{rigidity_potential8}
\gamma({\bf r}_I,{\bf r}_J,\Omega_I,\Omega_J,\eta_{IJ})=\left(1+a\int_{\Omega_I}\mu({\bf r}) d{\bf r}\right)\left(1+a\int_{\Omega_J}\mu({\bf r}) d{\bf r}\right)
\Phi(|{\bf r}_I-{\bf r}_J|, \eta_{IJ}),\end{aligned}$$ where the coefficient $a$ is a normalization factor, $\Phi(|{\bf r}_I-{\bf r}_J|,\eta_{IJ})$ is a FRI correlation function and $ \eta_{IJ}$ is the characteristic length of elements. In contrast, the $\eta_j$ in the discrete to continuum mapping is the characteristic length of atomic distances.
Three realizations of VPM by constructing three anisotropic Kirchhoff matrices are proposed. First, one can modify ANM to construct a VPM anisotropic Kirchhoff matrix. For each matrix element $H_{IJ}$, a local $3\times3$ Hessian matrix for ANM in Eq. (\[eq:multi-kirchoff1\]) is formed as $$\begin{aligned}
\label{eq:multi-kirchoff128}
H_{IJ} = -\frac{\gamma({\bf r}_I,{\bf r}_J,\Omega_I,\Omega_J,\eta_{IJ})}{r^2_{IJ}}\left[ \begin{array}{ccc}
(x_J-x_I)(x_J-x_I) &(x_J-x_I)(y_J-y_I) &(x_J-x_I)(z_J-z_I)\\
(y_J-y_I)(x_J-x_I) &(y_J-y_I)(y_J-y_I) &(y_J-y_I)(z_J-z_I)\\
(z_J-z_I)(x_J-x_I) &(z_J-z_I)(y_J-y_I) &(z_J-z_I)(z_J-z_I)
\end{array}\right] ~ \forall ~ I \neq J.
\end{aligned}$$ The diagonal elements are constructed following Eq. (\[eq:multi-kirchoff1\_diagonal\]). The test indicates this ANM based VPM works very well. However, the demonstration of this test is skipped.
Additionally, one can modify the aFRI to generate two other realizations of VPM. It is very natural for one to derive the continuous aFRI by making use of the local anisotropic matrix $\Phi^{IJ}$ defined in Eqs. (\[eq:Anisorigidity1\]) and (\[eq:afri\_local\_Hessian\]). However, in general applications, the correlation $\Phi(\|{\bf r}_I-{\bf r}_J \|; \eta_{IJ})$ can be more specifically defined to describe the interaction between each pair of virtual particles. To construct the final matrix, one can multiply the scaling parameter to the local flexibility Hessian matrix ${\bf F}^{1}$ and ${\bf F}^{2}$, the corresponding two generalized Hessian matrices can be expressed as following: $$\begin{aligned}
\label{eq:Anisoflexibility8}
{\bf F}^{1}_{IJ} =& - \frac{1}{w_{J}} {\rm adj}(\Phi^{IJ}) \gamma({\bf r}_I,{\bf r}_J,\Omega_I,\Omega_J,\eta_{IJ}), &\quad I\neq J; \\ \label{eq:Anisoflexibilityy4}
{\bf F}^{1}_{II}=& \sum_{J=1}^N \frac{1}{w_{J}} {\rm adj}(\Phi^{IJ}) \gamma({\bf r}_I,{\bf r}_J,\Omega_I,\Omega_J,\eta_{IJ}), &\quad I=J\end{aligned}$$ and $$\begin{aligned}
\label{eq:Anisoflexibility58}
{\bf F}^{2}_{IJ} =& - \frac{1}{w_{J}} |\Phi^{IJ}|(J_{3} - \Phi^{IJ}) \gamma({\bf r}_I,{\bf r}_J,\Omega_I,\Omega_J,\eta_{IJ}), &\quad I\neq J; \\ {\bf F}^{2}_{II}=& \sum_{J=1}^N \frac{1}{w_{J}} |\Phi^{IJ}|(J_3 - \Phi^{IJ}) \gamma({\bf r}_I,{\bf r}_J,\Omega_I,\Omega_J,\eta_{IJ}), &\quad I=J,\end{aligned}$$ where ${\rm adj}(\Phi^{IJ})$ denotes the adjoint of matrix. Here $J_3$ is a $3\times3$ matrix with every element being one.
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![The first three nontrivial eigenmodes of Cyo-EM data EMD 8295. A threshold value of 0.08 is used in the model to map out the biomolecule. Modes 4, 5 and 6 are demonstrated in (a), (b) and (c), respectively. []{data-label="fig:EM8295"}](EM8295.png "fig:"){width="80.00000%"}
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![The first three nontrivial eigenmodes of Cyo-EM data EMD 1590. A threshold value of 0.05 is used in the model. Modes 4, 5 and 6 are demonstrated in (a), (b) and (c), respectively. []{data-label="fig:EM1590"}](EM1590.png "fig:"){width="80.00000%"}
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It should be noticed that clustering idea in the discrete aFRI matches with the idea of discretization of the continuous density function very well. However, only the simple global form is highlighted in Eqs. (\[eq:Anisoflexibility8\]) and (\[eq:Anisoflexibility58\]).
The proposed model is tested to evaluate the anisotropic motion of density volumetric data. One only needs to consider those density elements in the anisotropic motion analysis that their density values are larger than a threshold, which is the suggested value for biomolecular visualization. This truncation dramatically reduces the number of entries in final Kirchhoff matrix.
It has been verified that the VPM is able to recover ANM and aFRI when the elements are very small and each element contains at most one real particle (say a C$_\alpha$). The demonstration of this result is omitted.
The proposed method is illustrated with three examples, i.e., protein 2XHF, cryo-EM maps EMD8295 and EMD3266. For protein 2XHF, the original data contain discrete atomic coordinates. The discrete to continuum mapping is used to generate rigidity density $\mu({\bf r})$. In this transformation, the Lorentz kernel as in Eq. (\[eq:couple\_matrix24\]) is chosen with $\epsilon=2$ and $\eta_j=1.0$ Å. To construct VPM Hessian matrix, a threshold is chosen to be $40\%$ of the maximal density value. One can discretize the computational domain by using the element size of 3.0 Å. In the mode analysis, the second aFRI form ${\bf F}^{2}$ is used. The Lorentz kernel in Eq. (\[eq:couple\_matrix24\]) is chosen with $\epsilon=2$ and $\eta_{IJ}=12$Å. Mode 4 to mode 11 are illustrated in Fig. \[fig:Density\_2XHF\]. The results are similar to those obtained with discrete aFRI [@DDNguyen:2016b].
For cryo-EM density map EMD8295, the data have a dimension of $326.4*326.4*326.4$Å$^3$. A mesh of 40\*40\*40 is used to discretize domain. The threshold of 0.08 is used to result in a total number of 639 nonzero elements in the matrix. Again the second aFRI form ${\bf F}^{2}$ with the Lorentz kernel is used. The parameter used are $\epsilon=2$ and $\eta_{IJ}=40$ Å. Modes 4, 5 and 6 are illustrated in Fig. \[fig:EM8295\].
For EMD1590, the region is of the size $436*436*436$Å$^3$. A mesh of 25\*25\*25 is employed for the discretization. The biomolecular domain is chosen by using the threshold of 0.05 and there are 394 nonzero elements in the final matrix. The second aFRI form ${\bf F}^{2}$ with the Lorentz kernel is used with $\epsilon=2$ and $\eta_{IJ}=60$ Å in matrix construction. Modes 4, 5 and 6 are illustrated in Fig. \[fig:EM1590\].
It can be noticed that VPM can be applied to other systems, such as stability analysis of cells, tissues, and some elastic systems with appropriate definitions of correlations functions. However, this aspect is beyond the scope of the present review.
### Eigenvector analysis
Mathematically, vector field can be analyze by Poincaré index [@Poincare:1890], winding number, Morse index [@harker:mischaikow:mrozek:nanda; @Mischaikow:2013] and more interestingly, the Conley index [@Conley:1978; @Mischaikow:2002; @Chen:2012morse; @Manolescu:2013conley]. The essential idea of all these methods is to explore the behavior of the vectors around critical points.
To give a brief introduction of Conley index, one can consider a manifold $M$ and its associated vector field. If one expresses the vector field in terms of a differential equation $\dot{x}=V(x)$. The solution can be expressed as a function $\phi: {\bf R}\times M \rightarrow M$. This solution is a flow that satisfies $\phi(0,x)=x$ for all $x \in M$. One can also define the trajectory as $\phi ({\bf R},x):=\bigcup_{t \in {\bf R}} \phi(t,x)$. With this setting, an invariant set $S \subset M$ is $\phi (R,S)=S$. Two basic types of invariant sets are fixed points and periodic points, see Table \[tb:conley\]. One can define an isolating neighborhood $N \subset M$ as for every $x \in \partial N$, there exists $\epsilon > 0$, such that one has either $\phi((-\epsilon,0),x)\bigcap N =\emptyset$ or $\phi((0,\epsilon),x)\bigcap N =\emptyset$. The exit set of an isolating block $N$ is $L=\{{x \in \partial N |\phi((0, \epsilon),x)\bigcap N =\emptyset}\}$. The pair $(N,L)$ is called an index pair. The Conley index of an invariant set $S$ is the relative homology of the index pair $(N,L)$, i.e., $CH_*(S):=H_*(N,L)$.
Cases Conely index
--------------------------- --------------
Attracting fixed point (1 0 0)
Saddle fixed point (0 1 0)
Repelling fixed point (0 0 1)
attracting periodic orbit (1 1 0)
Repelling periodic orbit (0 1 1)
: Conley index for various types of invariant sets
\[tb:conley\]
It should be noticed that the VPM eigenvector field is different from the traditional vector field. For an individual eigenvector, one has local vectors associated all virtual particles. All these local vectors join together to form a unique global vector field, which is able to capture the collective motions of the biomolecule of interest. However, locally, various fixed points can also be identified as demonstrated in Figs. \[fig:Density\_2XHF\], \[fig:EM8295\] and \[fig:EM1590\]. Rigorous analysis of these eigenvectors is still an open problem.
Demonstrations {#sec:theory}
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In this section, the utility of Geo-Topo algorithms, namely, topological analysis of Hessian matrix eigenvalue and curvature maps is illustrated by a few case studies. Additionally, the persistent homology analysis of molecular Hessian matrix eigenvalue maps is also demonstrated.
### Case studies {#sec:density}
Three molecules, a fullerene, an alpha helix and beta sheet, to illustrate Geo-Topo methods are considered.
#### Fullerene C$_{20}$
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![ Hessian matrix eigenvalue surfaces obtained from different isovalues (or level-set values). [ (a)]{} The isosurfaces for the first eigenvalue. The isovalues from [ ($a_{1}$)]{} to [ ($a_{4}$)]{} are -3.0, -2.0, -1.0 and 0.1. [ (b)]{} The isosurfaces for the second eigenvalue. The isovalues from [ ($b_{1}$)]{} to [ ($b_{4}$)]{} are -1.0, -0.01, 0.5 and 1.0. [ (c)]{} The isosurfaces for the third eigenvalue. The isovalues from [ ($c_{1}$)]{} to [ ($c_{4}$)]{} are -1.0, 1.0, 1.8 and 2.0. []{data-label="fig:C20_eigen_3d"}](C20_eigen_3d.png "fig:"){width="60.00000%"}
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Using the proposed Geo-Topo fingerprint, one can study more complicated structures. The first one is the fullerene C$_{20}$ considered in last section. Again, it rigidity density is described by Eq. (\[eq:rigidity3\]) and parameter $w_j$ and $\eta_j$ are set to $1$ and $0.7$, respectively.
First the Hessian matrix eigenvalue isosurfaces of C$_{20}$ is studied. Figure \[fig:C20\_eigen\_3d\] has illustrated four representative isosurfaces for each eigenvalue. Subscripts $1$ to $4$ indicate four isovalues from small to large. The notations $({\bf a})$ to $({\bf c})$ represent eigenvalues $\gamma_1$ to $\gamma_3$. One can see that their isosurface behaviors are consistent with descriptions of the Geo-Topo fingerprints summarized in Table \[tb:geometric\_fingerprint\]. State differently, the Geo-Topo fingerprints are able to capture the essential Geo-Topo properties of C$_{20}$ isosurfaces.
More specifically, for $\gamma_1$, negative isosurfaces enclose all NCPs and BCPs. Anisotropic isosurfaces are found near RCPs with negative A-type and positive V-type behaviors. CCPs are enclosed with positive isosurfaces. For $\gamma_2$, negative isosurfaces enclose all NCPs and bond regions of BCPs. Positive loops can be found around BCP bonds. RCPs and CCPs are enclosed by positive isosurfaces. For $\gamma_3$, negative isosurfaces [*only*]{} enclose atomic basin of NCPs, whereas BCPs, RCPs and CCPs are all enclosed by positive isosurfaces.
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![Isosurfaces for Gaussian and mean curvature maps of C$_{20}$. [ (a)]{} The isosurfaces built from the Gaussian curvature of C$_{20}$. The isovalues from [ ($a_{1}$)]{} to [ ($a_{4}$)]{} are -2.0, 2.0, 3.0 and 5.0. [ (b)]{} The isosurfaces built from the Mean curvature of C$_{20}$. The isovalues from [ ($b_{1}$)]{} to [ ($b_{4}$)]{} are -1.0, 0.001, 1.0 and 2.0. []{data-label="fig:C20_gaussian_mean_3d"}](C20_k1k2_3d.png "fig:"){width="60.00000%"}
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![Isosurfaces for Gaussian and mean curvature maps of C$_{20}$. [ (a)]{} The isosurfaces built from the Gaussian curvature of C$_{20}$. The isovalues from [ ($a_{1}$)]{} to [ ($a_{4}$)]{} are -2.0, 2.0, 3.0 and 5.0. [ (b)]{} The isosurfaces built from the Mean curvature of C$_{20}$. The isovalues from [ ($b_{1}$)]{} to [ ($b_{4}$)]{} are -1.0, 0.001, 1.0 and 2.0. []{data-label="fig:C20_gaussian_mean_3d"}](C20_gaussian_mean_3d.png "fig:"){width="60.00000%"}
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Figures \[fig:C20\_k1k2\_3d\] and \[fig:C20\_gaussian\_mean\_3d\] illustrates four representative isosurfaces for Gaussian, mean and two principal curvatures. Here the detailed analysis is omitted. However, just as eigenvalue isosurfaces, the curvature isosurfaces can be well-described by the Geo-Topo fingerprints summarized in Table \[tb:geometric\_fingerprint\].
#### An $\alpha$-helix structure
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![An illustration of second eigenvalue, third eigenvalue and mean curvature of a coarse-grained representation (C$_{\alpha}$) of an alpha helix. [ (a)]{} The isosurfaces for the second eigenvalue. The isovalues from [ ($a_{1}$)]{} to [ ($a_{4}$)]{} are -2.0, -0.05, 0.05 and 0.1. [ (b)]{} The isosurfaces for the third eigenvalue. The isovalues from [ ($b_{1}$)]{} to [ ($b_{4}$)]{} are -0.05, 0.2, 0.3 and 0.35. [ (c)]{} The isosurfaces for the mean curvature. The isovalues from [ ($c_{1}$)]{} to [ ($c_{4}$)]{} are -1.0, -0.1, 0.1 and 3.0. []{data-label="fig:1c26_helix"}](1c26_helix.png "fig:"){width="60.00000%"}
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So far, all cases examined are about highly symmetric molecular structures. In this part, the Geo-Topo analysis is applied to irregular biomolecular structures, particularly protein structure. It is known that there are four distinct levels of protein structure, namely primary structure, which is a sequence of amino acids in the polypeptide chain; secondary structure, which is an $\alpha$-helix or a $\beta$-strand; tertiary structure, which refers to the three-dimensional structure of a monomeric and multimeric protein molecule; and quaternary structure, which is the three-dimensional structure of a multi-subunit protein complex. The Geo-Topo analysis of proteins for protein secondary structures is demonstrated.
First, one can consider one of protein secondary structures, i.e., an $\alpha$-helix segment. This segment is extracted from protein with ID 1C26. The coarse-grained (CG) representation is employed and 19 C$_{\alpha}$ atoms from the $335$th residue to $353$th residue in chain A are used. The distance between two adjacent C$_{\alpha}$ atoms are about $3.8$ Å and the $\eta$ used in the CG rigidity density model is $2.0$ Å.
In stead of listing all results of eigenvalues and curvatures, only three geometric parameters of interest are examined, including eigenvalue $\gamma_2$, eigenvalue $\gamma_3$ and mean curvature. For each of them, four representative isosurfaces are extracted. The results are illustrated in Fig. \[fig:1c26\_helix\].
It can be found that results are very consistent with the Geo-Topo fingerprints. For $\gamma_2$, positive loops around atomic bonds are found. For $\gamma_3$, large eigenvalues are still concentrated in regions near BCPs. For mean curvature, one can still find positive A-type and negative V-type isosurfaces near BCPs. Also, large positive values are concentrates around NCPs and indicate the topological connectivity at Figs. \[fig:1c26\_helix\](a$_1$), (c$_3$) and (c$_4$).
It also should be noticed that unlike the regular symmetric structures studied in previous sections, RCPs and CCPs are more complicated in $\alpha$-helix structure. Moreover, since the characteristic distance $\eta$ is chosen as $2.0$ Å, bond effect (or topological connectivity) is observed not only between adjacent two atoms but also between atoms in close distance, as indicated in Fig. \[fig:1c26\_helix\]${ (a_3)}$. At meantime, strongest topological connectivity (largest $\gamma_2$ isovalues) is as usual found between adjacent two atoms as indicated in Fig. \[fig:1c26\_helix\]${(a_4)}$. Interestingly, in the case of $\gamma_3$, positive isosurface forms a strip that is parallel to the backbone of the $\alpha$-helix as demonstrated in Fig. \[fig:1c26\_helix\]${ (b_3)}$. The largest isovalues are concentrated around BCPs not between adjacent two atoms, but neighboring two atoms in adjacent two circles as illustrated in Fig. \[fig:1c26\_helix\]${ (b_4)}$. However, if characteristic distance $\eta$ was chosen as $1.5$ Å, the largest isovalues of $\gamma_3$ would move to BCP regions between adjacent two atoms. This is due to the multiscale nature of the model. Further detailed discussion of this multiscale property is beyond the scope of this paper.
#### A $\beta$-sheet structure
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![An illustration of second eigenvalue, third eigenvalue and mean curvature of a coarse-grained representation (C$_{\alpha}$) of a $\beta$-sheet. [ (a)]{} The isosurfaces for the second eigenvalue. The isovalues from [ (a$_{1}$)]{} to [ (a$_{4}$)]{} are -0.3, -0.05, 0.05 and 0.1. [ (b)]{} The isosurfaces for the third eigenvalue. The isovalues from [ (b$_{1}$)]{} to [ (b$_{4}$)]{} are -0.1, 0.2, 0.3 and 0.35. [ (c)]{} The isosurfaces for the mean curvature. The isovalues from [ (c$_{1}$)]{} to [ (c$_{4}$)]{} are -1.0, -0.1, 0.5 and 3.0. []{data-label="fig:4uw4_sheet"}](4uw4_sheet.png "fig:"){width="60.00000%"}
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Another important protein secondary structure is $\beta$-sheet. In this part, three adjacent $\beta$-strands from protein with ID 4UW4 are considered. Again, the CG representation is used and three stands include residues from $575$ to $586$, $589$ to $600$ and $603$ to $615$ in chain A. Just as the analysis in the $\alpha$-helix structure, the characteristic distance $\eta$ is chosen as $2.0$ Å. One can examine the second eigenvalue $\gamma_2$, the third eigenvalue $\gamma_3$ and mean curvature in the Geo-Topo analysis. For each of them, four representative isosurfaces are extracted. The results are illustrated in Fig. \[fig:4uw4\_sheet\].
Just as the $\alpha$-helix case, results for $\beta$-sheet are also very consistent with the Geo-Topo fingerprints. Positive loops around BCPs for $\gamma_2$ can be observed. For $\gamma_3$, small negative isosurfaces indicate NCPs. Large $\gamma_3$ eigenvalues are concentrated in regions near BCPs. Positive A-type and negative V-type isosurfaces near BCPs are found in mean curvature. Large positive values are concentrates around NCPs. Further, as the characteristic distance is chosen as $2.0 $ Å, the bond connection between the stands or sheets are amplified. It can be clearly observed in Fig. \[fig:4uw4\_sheet\] ${ (b_4)}$.
### Persistent homology for scalar field analysis {#sec:PHA_scalar_field}
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![Barcodes for three eigenvalue maps of benzene molecule. From [ (a)]{} to [ (c)]{}, the barcodes are for $\lambda_1$, $\lambda_2$ and $\lambda_3$, respectively. In each subfigure, from top to bottom, the results are for $\beta_0$, $\beta_1$ and $\beta_2$, respectively. []{data-label="fig:C6_eigen_ph"}](C6_eigen_ph.png "fig:"){width="90.00000%"}
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Finally, the topological persistence in the scalar fields generated by Hessian matrix eigenvalues and curvatures is studied. The hexagonal ring is used for persistent homology analysis. The barcodes for eigenvalues $\lambda_1$, $\lambda_2$ and $\lambda_3$ are demonstrated in Figs. \[fig:C6\_eigen\_ph\] (a)-(c). In each subfigure, results for $\beta_0$, $\beta_1$ and $\beta_2$ are presented. The filtration goes from the smallest value to the largest in the persistent homology analysis.
For $\lambda_1$, at very small values, it has $6$ $\beta_0$ bars, i.e., $6$ independent components. Topologically, it represents small $\lambda_1$ negative values concentrating around $6$ NCPs. As the filtration progresses, a loop is formed, which leads to a bar in $\beta_1$. Further down the filtration process, $\beta_0$ isosurface begins to shrink to two balls perpendicular to the RCP. This contributes two bars in $\beta_2$.
For $\lambda_2$, again $6$ $\beta_0$ bars are found in the earliest stage of the filtration. Here $6$ independent components quickly combine to form a loop as filtration progresses. More interesting, individual loops around BCPs form when filtration value is around the range from 0 to 1. Finally, isosurfaces shrink into the RCP.
For $\lambda_3$, smallest negative values are concentrated around NCPs and contribute $6$ $\beta_0$ bars at the earliest stage of the filtration. Once the isosurface value becomes positive, a new component emerges due to the generation of a new isosurface at the boundary region. This new isosurface also contributes to a long persisting $\beta_2$ bar. Further down the filtration, $6$ more $\beta_0$ bars occur, each representing a very narrow ring region around atom bonds. These regions are relatively small and quickly disappear. After that, $6$ “hat" regions attaching to the original NCP isosurfaces emerge, contributing to $6$ small loops. They quickly become detached and shrink away. At same time, $6$ isosurfaces form near the BCP and gradually disappear. Together they contribute 12 independent $\beta_2$ bars.
It can be seen that barcodes in Figs. \[fig:C6\_eigen\_ph\] (a)-(c) give a detailed account of the full spectrum of the isosurface evolution process in Figs. \[fig:C6\_eigen\_3d\] (a$_1)$-(a$_4)$.
Concluding remarks
==================
Every field in natural science, engineering, medicine, finance and social sciences becomes quantitative when it is getting mature. Mathematics is essential for all quantitative fields. Being regarded the last scientific forefront, biological sciences, particularly molecular biology and structural biology, have accumulated gigantic among of data in terms of biomolecular structures, activity relations and genetic sequences in the past few decades and are transforming from qualitative and phenomenological to quantitative and predictive. Such a transformation offers unprecedented opportunities for mathematically driven advances in biological sciences [@Wei:2016].
Geometry, topology, and graph theory are some of the core mathematics and have been naturally playing a unique role in molecular biology and molecular biophysics. In this paper, we present a brief review of geometric, topological, and graph theory apparatuses that are important to the contemporary molecular biology and biophysics. We first discuss the discrete methods and models, including graph theory, Gaussian network models, anisotropic network model, normal mode analysis, flexibility-rigidity index, molecular nonlinear dynamics, spectral graph theory and persistent homology for biomolecular modeling and analysis. Additionally, we describe continuous algorithms and theories, including, discrete to continuum mapping, multidimensional persistent homology for volumetric data sets, geometric modeling of biomolecules, differential geometry theory of surfaces, curvature analysis, atoms in molecule theory, and quantum chemical topology theory. Attention is paid to the connections between existing biophysical approaches and standard mathematical subjects, such as, Morse theory, Poincaré Hopf index, differential topology, etc. Open problems and potential new directions are point out in discussions.
Two new models, namely the analytical minimal molecular surface and virtual particle model, and two new methods, i.e., Hessian eigenvalue map and curvature map, are introduced for biomolecular modeling and analysis. These new approaches were inspirited by the subject under review during our preparation of this review. For simplicity, only the proof-of-principle applications are demonstrated for all methods, models, theories and algorithms covered in this review.
Selected mathematical topics in geometry, topology, and graph theory are based on our limited knowledge and understanding of mathematics and molecular biophysics. Many subjects in geometry, topology, and graph theory that have found much success in molecular biology and biophysics have not been covered in this review. One of these subjects is knot theory, particularly the DNA knot theory, which is also a very important ingredient of topological modeling of biomolecules [@vologodskii:1992; @pohl:1980; @fuller:1971; @ABates:2005; @sumners:1992; @darcy:2001; @arsuaga:2002; @buck:2007; @IKDarcy:2013; @RBrasher:2013; @Schlick:1992trefoil]. Knot theory is an area of geometric topology that deals with knots and links. Mathematically, a knot is an embedding of circles or its homeomorphisms in the three-dimensional (3D) Euclidean space, ${\mathbb R}^3$. Physically, DNA, as a genetic material, exists usually in two very long strands that intertwine to form chromatins, bind with histones to build nucleosomes, tie into knots, and are subjected to successive coiling before package into chromosomes. The loss of knots in chromosomes can cause Angelman and Prader-Willi syndromes. DNA knot theory has been a very important topic in applied topology. However, in this review, focus is given to geometric and topological methods and models for atomistic biomolecular data. Therefore, DNA knot theory is not covered.
Another relevant subject that have not been considered in this review is biomolecular interaction network models, including protein interaction networks, metabolic networks and transcriptional regulatory networks. Obviously, part of the mathematical foundation of these models is graph theory. Protein interaction networks are designed to study protein-protein interactions [@rain:2001; @giot:2003]. Normally, in these models, proteins are represented as nodes and physical interaction between them are represented by edges. To make the network more reliable, data from different sources are combined together and different rules are applied for the identification of protein interactions. A metabolic network considers all metabolic and physical processes happened within a cell [@jeong:2000; @overbeek:2000]. This network comprises the chemical reactions of metabolism, the metabolic pathways, as well as the regulatory interactions that guide these reactions. Transcriptional regulatory networks describe the regulatory interactions between genes [@lee:2002; @salgado:2006]. In this network, each gene is represented by a node and the regulation relations are represented by edges. The exclusion of this subject is also due to our focus on atomistic biomolecular data.
Topological graph theory concerns immersions of graphs as well as the embedding of graphs in surfaces, spatial embeddings of graphs, and graphs as topological spaces. It has had much success in the mathematical modeling of DNA recombination, DNA-RNA interactions, protein folding and protein-protein interactions [@Angeleska:2009]. The exclusion of this subject is due to our insufficient knowledge and understanding. For the same reason, fascinating applications of combinatorics, algebra and tiling theory in the modeling of viral capsid self assembly [@Jonoska:2009; @Twarock:2008; @Angeleska:2009; @SHarvey:2013; @CHeitsch:2014; @Sadre-Marandi:2014] have not been covered in our review.
A continuous, differentiable curve can be embedded in a three-dimensional Euclidean space and its kinematic properties, such as the derivatives of tangent, normal, and binormal unit vectors, can be described by Frenet-Serret formulas in differential geometry [@crenshaw1993orientation]. Discrete Frenet-Serret frame offers an efficient description of amino-acid and/or nucleic acid chains [@quine2004mathematical; @hu2011discrete]. We believe that discrete Frenet-Serret frame can be easily applied to RNA chains, microtubules, nucleosomes, chromatins, active chromosomes and metaphase chromosomes.
An emergent approach is to combine machine learning with geometry, topology and/or graph theory for analyzing biomolecular data [@ZXCang:2015; @Kovacev-Nikolic:2016]. Machine learning is a cutting edge computer science and statistical tool originally developed for pattern recognition and artificial intelligence. Its combination with mathematical apparatuses leads to extremely powerful approaches to massive biomolecular data challenges, such as the blind predictions of solvation free energies [@BaoWang:2016FFTS; @BaoWang:2016HPK] and protein-ligand binding affinity prediction [@BaoWang:2016FFTB]. For example, topological learning algorithm that utilizes exclusively persistent homology and machine learning for protein-ligand binding affinity predictions outperforms all the existing eminent methods in computational biophysics over massive binding data sets [@ZXCang:2016b]. However, this subject is at its early stage and is evolving too fast to have a conclusive review at present.
It is worth mentioning that the geometric, topological and graph theory apparatuses discussed in this review can be employed in conjugation with partial differential equation (PDE), which is widely used in computational biophysics, to model biomolecular systems. Certainly geometric modeling is often a prerequisite in the PDE models of electrostatics, solvation, ion channels, membrane-protein interactions, and biomolecular elasticity [@Wei:2009]. As mentioned in Section \[sec:spectral\], the graph cut problem can be formulated as a free energy minimization. Such a formulation makes it possible to combine spectral graph theory with PDE approaches for a wide range of biophysical modeling for biomolecular systems, including solvation, ion channel, biomembrane, protein-ligand binding, protein-protein interaction and protein-nucleic acid interaction. Finally, connection between algebraic topology and differential geometry, including Laplace-Beltrami operator, has been made [@BaoWang:2016a]. Essentially, one defines an object function to optimize certain biophysical properties, which leads to a Laplace-Beltrami operator that generates a multiscale representation of the initial data and offers an object-oriented filtration process for persistent homology. The resulting differential geometry based object-oriented persistent homology is able to preserve desirable geometric features in the evolutionary filtration and enhances the corresponding topological persistence. However, how to design object-oriented persistent homology to automatically extract desirable features in the original biomolecular data during the filtration process is still an open problem.
Indeed, the application aspects of geometry, topology and graph theory have become a driven force for the development of abstract geometry, topology, homology, graph theory in recent years. It is expected that a versatile variety of pure mathematics concepts, methods and techniques will find their cutting edges in the transcend description of biomolecular structure, function, dynamics and transport. This health interaction between mathematics and molecular bioscience will benefit both fields and attract young researchers for generations to come.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by a Start-Up Grant from The Nanyang Technological University (KLX), NSF IIS- 1302285 (GWW), NSF DMS-1160352 (GWW), NIH R01GM-090208 (GWW) and MSU Center for Mathematical Molecular Biosciences Initiative (GWW).
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[^1]: E-mail: xiakelin@ntu.edu.sg
[^2]: E-mail: wei@math.msu.edu
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'David Martínez-Delgado, A. Katherina Vivas$^{2}$, Eva K. Grebel$^{1}$, Carme Gallart$^{3,4}$, Adriano Pieres$^{5,6}$, Cameron P. M. Bell$^{7}$, Paul Zivick$^{8}$, Bertrand Lemasle$^{1}$, L. Clifton Johnson$^{9}$, Julio A. Carballo-Bello$^{10,11}$, Noelia E. D. Noël$^{12}$, Maria-Rosa L. Cioni$^{7}$, Yumi Choi$^{13, 14}$, Gurtina Besla$^{13}$, Judy Schmidt$^{15}$, Dennis Zaritsky$^{13}$, Robert A. Gruendl$^{16,17}$, Mark Seibert$^{18}$, David Nidever$^{19}$, Laura Monteagudo$^{3,4}$, Mateo Monelli$^{3,4}$, Bernhard Hubl$^{20}$, Roeland van der Marel$^{21, 22}$, Fernando J. Ballesteros$^{23}$, Guy Stringfellow$^{24}$, Alistair Walker$^{2}$, Robert Blum$^{19}$, Eric F. Bell$^{25}$, Blair C. Conn$^{26}$, Knut Olsen$^{19}$, Nicolas Martin$^{27,28}$, You-Hua Chu$^{29,30}$, Laura Inno$^{28}$, Thomas J. L. Boer$^{12}$, Nitya Kallivayalil$^{8}$, Michele De Leo$^{12}$, Yuri Beletsky$^{31}$, Ricardo R. Muñoz $^{32}$'
title: |
On the nature of a shell of young stars in the outskirts\
of the Small Magellanic Cloud
---
[Understanding the evolutionary history of the Magellanic Clouds requires an in-depth exploration and characterization of the stellar content in their outer regions, which ultimately are key to tracing the epochs and nature of past interactions.]{} [We present new deep images of a shell-like over-density of stars in the outskirts of the Small Magellanic Cloud (SMC). The shell, also detected in photographic plates dating back to the fifties, is located at $\sim 1.9\degr$ from the center of the SMC in the north-east direction.]{} [The structure and stellar content of this feature were studied with multi-band, optical data from the Survey of the MAgellanic Stellar History (SMASH) carried out with the Dark Energy Camera on the Blanco Telescope at Cerro Tololo Inter-American Observatory. We also investigate the kinematic of the stars in the shell using the [*Gaia*]{} Data Release 2.]{} [The shell is composed of a young population with an age $\sim 150$ Myr, with no contribution from an old population. Thus, it is hard to explain its origin as the remnant of a tidally disrupted stellar system. The spatial distribution of the young main-sequence stars shows a rich sub-structure, with a spiral arm-like feature emanating from the main shell and a separated small arc of young stars close to the globular cluster NGC 362. We find that the absolute $g$-band magnitude of the shell is M$_{g,shell} = -10.78\pm 0.02$, with a surface brightness of $\mu_{g,shell} = 25.81\pm 0.01$ mag arcsec$^{-2}$]{} [We have not found any evidence that this feature is of tidal origin or a bright part of a spiral arm-like structure. Instead, we suggest that the shell formed in a recent star formation event, likely triggered by an interaction with the Large Magellanic Cloud and/or the Milky Way, $\sim$ 150 Myr ago.]{}
Introduction
============
The Magellanic Clouds are the largest satellites of the Milky Way (MW), and the only irregular galaxies in its immediate surroundings. The existence of one or even two such gas-rich, massive satellites close to a Milky-Way-sized halo has been shown to be quite rare (e.g., Busha et al. 2011; González et al. 2013; Rodr[í]{}guez-Puebla [*et al.*]{}2013; Boylan-Kolchin [*et al.*]{} 2011; Patel, Besla & Sohn 2017). There is strong evidence from the motions and distributions of the stellar and gaseous constituents of the Magellanic Clouds that interactions between them have occurred in the past. High-precision proper motion measurements with the Hubble Space Telescope revealed that, contrary to earlier belief, the Magellanic Clouds are probably just completing their first passage around the Milky Way (Besla [*et al.*]{} 2007, 2010; Kallivayalil [*et al.*]{} 2006a, 2006b, 2013; Piatek [*et al.*]{} 2008; Bekki 2011).
Clearly, the past interactions between the Clouds have left their marks. One of the most obvious manifestations of this interaction is the gaseous Magellanic Stream (Mathewson et al. 1974), whose trailing and leading arms have since been traced over more than $200\degr$ across the sky (Putman et al. 1998; Nidever et al. 2010). Both tidal and ram pressure stripping origins have been suggested for the Stream (see D’Onghia & Fox 2016 for a review). The mass of ionized and atomic Magellanic gas found outside of the Clouds exceeds the remaining H[i]{} mass in both Clouds combined (Fox et al. 2014) and is of a similar order of magnitude as the [*total*]{} mass of the Small Magellanic Cloud (SMC).
{width="100.00000%"}
Another prominent feature is the gaseous Magellanic Bridge, which connects the two Clouds. The Bridge also contains stars, specifically an irregularly distributed young stellar population with ages of up to several hundred Myr (e.g., Irwin et al. 1985; Skowron et al. 2014), associations, and star clusters (e.g., Bica et al. 2008). Unequivocal evidence of intermediate-age stars was also found in the Magellanic Bridge area (Noël et al. 2013; Noël et al. 2015) as well as unambiguous evidence of a tidal stripping scenario (Carrera et al. 2017). The density distribution of these inter-cloud stars suggests that they belong to the extended outer regions of the two Clouds (e.g., Skowron et al. 2014; Jacyszyn-Dobrzeniecka et al. 2017; Wagner-Kaiser & Sarajedini 2017). This would support a scenario where the Bridge formed in a close encounter between the Clouds some 250 Myr ago (D’Onghia & Fox 2016 and references therein). On the other hand, Belokurov et al. (2017) report the discovery of a separate, off-set old stellar bridge between the Clouds. However, Jacyszyn-Dobrzeniecka et al. (2019) found the existence of this old bridge as controversial, based on the analysis of the distribution of RR Lyrae stars in the OGLE data.
Moreover, the outer regions of both Clouds show distortions, clumps, arcs, and related overdensities, both in the direction of the Bridge and elsewhere (e.g., Nidever et al. 2013; Casetti-Dinescu et al.2014; Besla et al. 2016; Belokurov & Koposov 2016; Mackey et al.2016; Belokurov et al. 2017; Pieres et al. 2017; Subramanian et al. 2017; Carrera et al. 2017; Mackey et al. 2018; Choi et al. 2018a; Choi et al. 2018b). The last close encounter between the two Clouds some 100 to 300 Myr ago is not only consistent with the young ages in the Bridge, but also with a peak in the age distribution of young clusters in both Clouds (Glatt et al. 2010) and with the bimodal ages of Cepheids (Ripepi et al. 2017).
The SMC has been particularly affected by past interactions and shows a highly distorted, amorphous structure in its H[i]{} distribution (e.g., Stanimirovic et al. 2004) and in its younger populations, while its old populations are symmetrically and regularly distributed (e.g. Cioni, Habing & Israel 2000; Zaritsky et al. 2000; Haschke et al. 2012; Jacyszyn-Dobrzeniecka et al. 2017; Muraveva et al. 2018). The SMC has a large line-of-sight depth (e.g., Caldwell & Coulson 1986; Mathewson et al. 1988; Crowl et al. 2001; Subramanian & Subramaniam 2012; Haschke et al.2012; Nidever et al. 2013) and its old stellar population is very extended (Noël & Gallart 2007; Nidever et al. 2011). From their study of the 3-D structure of the SMC using Cepheid stars, Ripepi et al. (2017) found 25-30 kpc (see also Scowcroft et al. 2016; Jacyszyn-Dobrzeniecka et al. 2016). Apart from repeated disruptive encounters between the Clouds, it has been suggested that the complex structure of the SMC may also be due to a dwarf-dwarf merger in the distant past (Bekki & Chiba 2008).
{width="80.00000%"}
One important approach towards understanding the evolutionary history of the Magellanic Clouds is through deep, multi-colour mapping of the Clouds, especially of their neglected outskirts, which can still contain clues about the times and nature of past interactions. As part of our deep, wide-field imaging survey of faint tidal structures around the Magellanic Clouds and other Milky Way satellites (Besla et al. 2016; Martínez-Delgado et al. in preparation) using telephoto lenses, we have detected a coherent shell-like over-density feature embedded in the diffuse “outer arm B” (see de Vaucouleurs & Freeman 1972) and previously visible in photographic studies of the Clouds dating back to the 1950s (e.g. see Figure 12b in de Vaucouleurs & Freeman 1972 and Fig. \[fig-panel\] in this work). Albers et al. (1987) also found evidence of shell-type structures on the north-east side of the SMC based on the analysis of star counts from scanned photographic plates, suggesting they could be associated to faint, spiral arm structure in this region of the SMC. The shell-like feature was also previously noted as “a distinct linear feature perpendicular to the main elongation of the SMC” in a stellar density map using only upper main-sequence stars selected from the Magellanic Cloud Photometric Survey in Zaritsky et al. (2000, their Fig. 2). An early color-magnitude diagram of the north-east outer regions of the SMC by Brueck & Marsoglu (1978) concluded the presence of a young population ($\sim$ 60 Myr) and the association of three star clusters in the [*outer arm B*]{}, that were interpreted as the evidence of a recent burst of star formation in this SMC region (Bruck 1980). In this paper, we study the structure, stellar and gas content, kinematics and possible formation scenarios of this over-density by means of its resolved stellar populations as traced by the imaging data obtained in the Survey of the MAgellanic Stellar History (SMASH, Nidever et al. 2017) and the proper motions available from the [*Gaia*]{} Data Release 2 (DR2) (Gaia Collaboration et al. 2018).
The Data
========
Canon 200 lens data
-------------------
In our data, the SMC over-density was first detected in deep images taken with an equipment consisting of a Canon EF 200 mm f/2.8L II USM and a SBIG STL-11000M CCD camera during two observing runs in 2009 August and September at the European Southern Observatory (La Silla, Chile). This setup provided a total field of view (FoV) of $10\degr \times 7\degr$ and a pixel scale of 9.27 arcsec pixel$^{-1}$. A set of 58 individual exposures of 300 sec were taken in a Luminance filter (see, e.g., Fig. 1 in Martínez-Delgado et al. 2015), with a total exposure time of 290 minutes. This first image is showed in Fig. 1 ([*panel B*]{}), with the SMC over-density marked with label [*d*]{}.
Standard data reduction procedures for bias subtraction and flat fielding were carried out using the CCDRED package in the Image Reduction and Analysis Facility (IRAF[^1]). A detailed analysis of these observations is presented in Martínez-Delgado et al. (in preparation).
To ensure that this shell-like feature was not an artifact or a reflection, a confirmation wide-field image of the SMC was taken with a different Canon EF 200 mm f/2.8L lens attached to a Canon E0S6D (ISO1600) camera in 2015 October 10 from Hacienda Los Andes (Chile). Figure \[fig-canon\] shows the resulting image, with a total exposure time of 156 minutes (obtained by combining 78 individual images with an exposure time of two minutes). Astrometry of this image was obtained using the astrometric calibration service [Astrometry.net](Astrometry.net) (Lang et al. 2010).
SMASH data
----------
{width="50.00000%"}{width="50.00000%"}
SMASH (Nidever et al. 2017) was carried out using the Dark Energy Camera (DECam, Flaugher et al. 2015) on the 4-m Blanco Telescope at Cerro Tololo Inter-American Observatory, Chile. The survey, covering 480 deg$^2$ of the sky, aims to explore the extended stellar populations of the Clouds and their interaction history. For our current work we use a small subset of the SMASH observations. The region of the observed shell in the Canon images is well covered by SMASH in Fields 9 (central coordinates (J2000): $\alpha=$01:01:27.40, $\delta=$-70:43:05.51), 14 ($\alpha=$01:20:26.78, $\delta=$-71:15:32.76), and 15 ($\alpha=$01:24:33.52, $\delta=$-72:49:30.00), which are part of a ring of fields surrounding the central part of the SMC. Each DECam field covers a FoV of 3 deg$^2$. Each of these fields was observed in the [*ugriz*]{} bands, with deep observations of 999s in $u$, $i$, and $z$, and 801s in $g$ and $r$. In addition, for each field we obtained three shorter exposures (60s) with large offsets to tie all the chips together photometrically and to allow us to cover some of the gaps between the CCDs. Details on the data processing and photometry can be found in Nidever et al. (2017).
The photometry of the DECam images was carried out using the photometry program DAOPHOT (Stetson 1987) as incorporated into IRAF. In order to identify point sources in the photometry we imposed the following cuts in the DAOPHOT parameters: $-1.0 < {\rm SHARP} <1.0$ and $\chi^2<3.0$. We also required that the Sextractor (Bertin & Arnouts 1996) stellar probability index was ${\rm PROB} >0.8$. Since the Fields 9, 14, and 15 cover a contiguous part of the sky and overlap with each other, we took care to avoid double entries in the final combined catalog. This leaves us with 2,651,378 stars.
A shell in the outer region of the SMC
======================================
The images of the SMC obtained with two different Canon 200 lenses (Fig.1B and Fig. 2) revealed a clear shell-like feature situated at $1.9\degr$ from the centre of the galaxy, just above the bar of the SMC (Figure 2). The shape of this over-density is similar to those of the tidal features recently reported in the outskirts of the northern LMC (Mackey et al. 2016; Besla et al. 2016) and the SMC (Pieres et al. 2017)[^2]. In this section, we explore the spatial extent, structure, and stellar populations by analysing the deep photometry obtained in the SMASH survey.
Spatial distribution as traced with different stellar populations
-----------------------------------------------------------------
{width="49.00000%"} {width="49.00000%"}
Figure \[fig-cmd\] shows a $g$ versus $(g-i)$ color-magnitude diagram (CMD) of our final catalog. The features of the SMC are clearly seen in this rich diagram, and it shows the wide range of ages of the stellar populations that are typical for this kind of dwarf irregular galaxy. The main sequence (MS) is very extended, tracing the SMC’s young populations. The sub-giant branch (SGB) and a wide red giant branch (RGB) are produced by older populations with typical ages of 1 Gyr or more. The vertical red clump and the red clump (RC) at $19.2\lesssim g \lesssim 20$ are additional obvious features in the diagram, which trace intermediate-age populations. For a detailed exploration of the star formation history of the SMC from wide-field, multi-colour, ground-based imaging we refer the reader to Rubele et al. (2018) and from very deep HST imaging to Cignoni et al. (2012, 2013).
In a first approximation, we selected stars in the main features described above, which are labeled in Figure \[fig-cmd\] as boxes 1 (upper MS), 2 (RC) and 3 (lower part of the RGB). Since all of these features are located in the brighter part of the CMD, the spatial coverage is uniform. This is not necessarily true in the lower part of the CMD since the deepest SMASH exposures were not dithered and hence the gaps among CCDs would be noticeable.
To investigate the nature of the potential shell, we constructed density maps of the region based on stars in different parts of the CMD, as shown in Figure \[fig-maps\]. In the left panel we show a map containing stars in the RC and RGB of the SMC ($\sim 100,000$ stars from boxes 2 and 3 of Figure \[fig-cmd\]). The distribution of these stars, which trace the intermediate-age and old populations of the SMC, is quite uniform in the region, showing only a smoothly increasing density toward the centre of the SMC located in the lower right part of the panel, at $\alpha_{J2000}=00$:52:38, $\delta_{J2000}=-72$:48:01. The globular cluster NGC 362 (top right position) is clearly visible in this map since part of its MS population overlaps with our selection box for the SMC RGB stars.
The smooth distribution of the density of stars in this map suggests that there is no significant extinction in this part of the sky that may cause an apparent shell-like feature. This is also supported by the extinction map of the SMC of Haschke et al. (2011, their fig. 4), which shows generally low extinction in the SMC and values of $E(B-V) \simeq
0.05$ mag in the northeastern part of the SMC. The lack of an over-density in the upper panel of Figure \[fig-maps\] is furthermore consistent with Zaritsky et al. (2000) who also found no evidence of the northeastern over-density in their stellar density maps obtained using giants and red clump stars as tracers (see their Fig. 3).
The region has a very different appearance when plotting only young stars belonging to the upper MS of the SMC population, as selected by Box 1 in Figure \[fig-cmd\]. In this map, made with $\sim 70,000$ stars, the shell is clearly visible (labeled [*a*]{} in the right panel of Fig \[fig-maps\]) and a rich structure is associated with it. Besides the main shell, we find a spiral-arm-like feature of young stars attached to the shell (labeled [*b*]{} in Fig \[fig-maps\]) and a separated small arc situated $\sim$ 30$\arcmin$ West from the globular cluster NGC 362 (labeled [*c*]{} in the right panel of Fig \[fig-maps\]). This last feature has no counterpart in the old population maps, so we conclude that it is not part of a tidal tail from this cluster (see also Carballo-Bello 2019). Instead, it is very interesting that two of the young open clusters discussed in Sec. 3.3 (see Table \[tab-cluster\]) are embedded in the Southern extreme of this small structure (see Fig.\[fig-cluster\]). The structures [*a*]{} and [*b*]{} (Figure 4, right panel) are also clearly visible in the GALEX image plotted in Fig. \[fig-galex\], which traces mainly hot young stars, showing an excellent agreement with the position and morphology of these features as traced in our stellar density maps.
{width="80.00000%"}
{width="60.00000%"}\
Our CMD selection also includes a known streamer of young stars into the Bridge at $\delta \sim$ -73.3 $\deg$, which is also evident in the East side of the SMC in our Canon 200 image in Figure \[fig-canon\]. This is also the origin of some negative features imprinted in this region in the left panel of Figure 4 at $\delta \sim$ -73.3 $\deg$. This is due to the incompleteness in these more crowded areas, since old red stars are harder to detect near bright blue star formation regions.
Stellar content
---------------
Fig. \[fig-maps\] indicates that the shell contains a rich population of young MS stars with a range of ages, which is much less prominent in the surrounding areas. To study the characteristics of the stellar populations present in the shell, we will use de-reddened $g_0$, $(g-i)_0$ CMDs and the corresponding colour functions (CF). Given the spatially variable nature of the foreground Galactic reddening, we use the reddening maps of Schlegel, Finkbeiner, & Davis (1998) and combine these with the revised extinction coefficients of Schlafly & Finkbeiner (2011) to de-redden each star individually. We adopt the recommended “mean” distance modulus of 18.96 to the SMC as advocated by de Grijs & Bono (2015).
Given the areal coverage of the SMASH survey in conjunction with different environmental effects in each field across the survey – such as distinct SFHs, differential reddening, crowding effects, depth, etc. – we opt to compare the de-reddened $g_0$, $(g-i)_0$ CMD and CFs of the shell region against two different “control” fields: ConF1 is located close to the shell (see left panel in Fig. \[fig-maps\]), and ConF2 is on the opposite side of the SMC (not shown in Fig. \[fig-maps\]). We aim to investigate the characteristics of the stellar content in each field, and identify differences in the stellar populations between them.
In Fig. \[fig-sfh\] we plot the CMD of the stars in the shell box (upper left panel) and in ConF1 (upper right panel). Box boxes have the same area. Isochrones from the BaSTI version 5.0.1 library (Pietrinferni et al. 2004[^3]) of ages of 30 Myr, 150 Myr, 250 Myr, 800 Myr, 2.5 Gyr, 6.0 Gyr (Z=0.002) and 13.5 Gyr (Z=0.001) have been plotted as reference. A value of Z=0.002 is consistent with that derived from H[ii]{} regions, young stars, and Cepheids in the SMC (Russell & Dopita 1992; Romaniello et al. 2009; Lemasle et al. 2017). Although the SMASH survey utilizes the 4-m Blanco telescope and DECam imager, who filter system is described in Abbott et al. (2018), the photometric calibration relies on the use of standard stars in the Sloan Digital Sky Survey (SDSS; see Nidever et al. 2017) and as such we adopt the BaSTI isochrones in the SDSS [*ugriz*]{} system for our comparison.
From the two upper panels of Fig. \[fig-sfh\] it seems that the shell sample contains a significantly higher number of luminous MS stars than the same region of the CMD in the “control” field ConF1 sample, particularly in the age range 150-250 Myr. It is interesting, however, that the CMD of ConF1 contains a larger quantity of stars younger that 150 Myr, which are very scarce in the shell sample: from the comparison with the isochrones it can be concluded that very little star formation has taken place in the shell during the last 150 Myr. A step in the density of stars in the main sequence can also be observed, both in the shell and in the ConF1 CMD at the approximate position of the 2.5 Gyr isochrone, possibly indicating an enhanced period of star formation at intermediate ages. This enhancement is well-known and corresponds to a common LMC/SMC burst epoch at about 1.5-3 Gyr ago (e.g. see Harris & Zaritsky 2002; Weisz et al. 2013).
The larger number of young, bright objects in the shell results in a $g$-band surface brightness for the shell sample that is more than 0.5 mag arcsec$^{-2}$ brighter than the ConF1 control field sample (cf. $\mu_{g,shell} = 25.81\pm 0.01$ and $\mu_{g,CF1} = 26.68 \pm 0.01$ mag arcsec$^{-2}$). We also notice this increased surface brightness in the $i$-band but with a smaller difference (cf. $\mu_{i,shell} = 25.55 \pm 0.01$ and $\mu_{i,cont} = 26.12\pm 0.01$ mag arcsec$^{-2}$). For the “mean” SMC distance modulus of 18.96, we determine that the absolute $g$- and $i$-band magnitudes of the shell sample are M$_{g,shell} = -10.78\pm 0.02$ and M$_{i,shell} = -11.05 \pm 0.02$ mag, respectively.
The lower left panel of the Fig. \[fig-sfh\] shows the CMD of a synthetic population computed using the BasTI on-line Stellar Population Synthesis Program[^4], using solar scaled overshooting models. We have assumed a constant star formation rate (SFR) from 13.5 Gyr to 30 Myr ago (the latter is the young age limit for the BasTI library), and a simplified chemical enrichment law, approximately consistent with that obtained by Carrera et al. (2008) from Ca II triplet spectroscopy, that is, \[Fe/H\]=-0.99 ($\sigma$=0.3) for the second half of the galaxy’s life (age $<$6.75 Gyr), and \[Fe/H\]=-1.29 ($\sigma$=0.1) for the first half (age$>$6.75 Gyr). Other parameters of the model are a binary fraction of $\beta$=0.4 with a mass ratio q$>$0.5 and a Kroupa et al. (1993) IMF. Different colours for the synthetic stars have been used to highlight the position in the CMD of stars with different ages. The same isochrones as in the observed CMDs were plotted as reference. Finally, the lower right panel of Fig. \[fig-sfh\] shows the luminosity function of the shell, control fields, and that of the synthetic CMDs. These luminosity functions allow us to conclude that the observed CMDs are basically complete down to $M_i$=3 or $i_0$=21.96. The horizontal lines indicate this magnitude limit, which will be used to compute the CFs.
In Fig. \[fig-cfs\], we use the CFs to further analyze the stellar population content of the shell and the differences to the SMC field populations at similar galactocentric radius (for an introduction to the use of the CF for stellar population analysis, see Gallart et al. 2005; see also Noël et al. 2007 for an application to study SMC field stellar populations). We compare the shell CF with the CF of the same control field shown in Fig. \[fig-maps\] (ConF1) and control field ConF2 located at the opposite side of the SMC. The upper panel of Fig. \[fig-cfs\] shows, in black and different line types, the $(g-i)_0$ observed CFs for the shell and the two control fields mentioned above. An absolute magnitude cut of $M_i$=3 has been adopted to ensure a high level of completeness in the photometric data used to calculate the CFs (see Fig. \[fig-sfh\]). The orange solid line shows the CF of the synthetic CMD with a constant star formation rate at all times. The CF of the synthetic CMD was scaled such that the number of stars in a box on its red clump matches that in a box in the same location in the shell CMD.
![ *Upper panel*: CFs of the shell and control fields (for $M_i<3$), with CFs for a synthetic population with constant SFH superimposed. *Lower panel*: the same CFs for the observed fields, together with the CF for three populations in different age ranges, as labelled (see text for details).[]{data-label="fig-cfs"}](fig6.pdf){width="49.00000%"}
The lower panel of Figure \[fig-cfs\] displays the same observed CFs, together with CFs of the synthetic population in three limited age ranges: 0.03 to 0.8 Gyr, 1.5 to 6 Gyr, and older than 8.0 Gyr. In this case, the normalization of the synthetic CFs is arbitrary, and chosen to approximately match the features present in the observed CF of the shell. The position and width of the blue maximum in the shell CF (approximately centered at $(g-i)_0 \simeq -0.5$) suggests a very recent star formation event. The second maximum approximately centered at $(g-i)_0 \simeq 0$ is well reproduced by a population spanning the age range 1.5 to 6 Gyr. The third maximum corresponds to the position of the red clump. These results on the age range that is contributing to each feature of the CF are in good agreement with the conclusions we reached from the comparison of the CMDs with isochrones. The comparison of the observed CFs indicates significant differences in the stellar content of the shell and the two control fields. The very prominent blue maximum in the shell CF has very low counts in both ConF1 and ConF2. The difference in the height of the second, intermediate colour maximum in the shell and the control fields is not striking, but a difference is nonetheless evident. Taking into account the information on the lower panel of Fig. \[fig-cfs\] regarding the ages contributing to each feature on the CF (complemented by the isochrone information), we conclude that star formation has been very active in the shell in the last $\simeq$ 1 Gyr and much less active in the control fields. In the intermediate-age range 1.5-6 Gyr ago, star formation was also more active in the shell field than in the control fields. The comparison with the CF of a population with a constant star formation rate (orange line in the upper panel of Fig. \[fig-cfs\]) discloses that star formation in the shell field has been enhanced (compared to an average constant SFH) in the second half of the galaxy’s life, while star formation in both control fields has been depressed in the last $\simeq$ 1 Gyr compared to a constant star formation rate.
Our analysis of the total stellar content toward the shell field, through the comparison of the CMD with isochrones and the comparison between observed and synthetic CFs, reveals a very complex stellar population, with stars of all ages contributing to the CMD of the shell field. A strong enhancement of the recent star formation rate is, however, indicated by the height of the bluest peak of the CF. Even though a broad age range seems to be necessary to reproduce the width of that peak, the details of this age range and the precise age composition cannot be accurately constrained by this kind of simple analysis. A full SFH derivation, which is beyond the scope of this paper, would be necessary. Additional considerations brought up by the analysis of the young cluster (see Sec.3.3) and Cepheid population (see Sec. 3.4) in that area of the SMC indicate that a conspicuous burst of star formation around 150-250 Myr contributed outstandingly to the young population of the shell and may be the origin of the features that identify the shell morphology and that are visible in the right panel of Figure \[fig-maps\].
Young Star Clusters
-------------------
Cluster Name RA DEC log(age/yr) Ref.
-------------- -------- --------- ------------- ------
B88 14.233 -70.773 8.10 1
HW33 14.346 -70.809 8.10 2
B139 17.617 -71.561 8.30 1
HW64 17.687 -71.338 8.25 1
IC1655 17.971 -71.331 8.30 1
IC1660 18.158 -71.761 8.20 1
L95 18.687 -71.347 8.30 1
NGC458 18.717 -71.550 8.15 3
HW73 19.108 -71.326 8.15 1
: Young Star Clusters in the SMC shell. RA and DEC are in J2000 system. References: (1) Glatt et al. 2010, (2) Piatti et al. 2014, (3) Alcaino et al. 2003.[]{data-label="tab-cluster"}
![*Top panel*: MS stellar density map showing the locations of nine young ($<$1 Gyr) star clusters (blue circles) associated with the shell feature, defined here spatially by the red polygon. *Bottom panel*: The combined CMD of the nine young star clusters, showing their collective compatibility with a single age of $\sim$160 Myr. We overplot a $\sim$160 Myr PARSEC isochrone (log(age/yr)=8.2, Z=0.002, $A_V$=0.25 mag, (m-M)=18.96) for reference.[]{data-label="fig-cluster"}](fig7.pdf){width="49.00000%"}
A number of young star clusters lie spatially coincident with the northeastern shell feature, with the most prominent clusters appearing as visible over-densities in the upper MS star count map in Fig. \[fig-maps\]. We identify nine young ($<$1 Gyr) star clusters from the Bica et al. (2008) catalog that lie within a $\sim2.2$ square degree search region. We display their location in Figure \[fig-cluster\] (upper panel), and list their properties in Table 1. For this census, we exclude six older (age$>$1 Gyr) clusters from further discussion, and ignore 5-10 diffuse associations and low significance catalog entries. This sample features three relatively massive clusters ($\sim10^4 M_{\sun}$; NGC458, IC1655, IC1660), as well as six other less massive systems[^5].
A striking feature of this young cluster sample is their uniformity in age. As determined from isochrone comparisons to observed CMDs (Alcaino et al. 2003; Glatt, Grebel & Koch 2010; Piatti 2014), the cluster ages pile up around $\sim$ 160 Myr \[log(age/yr)=8.2$\pm$0.1\] and appear consistent with this single age, within current fitting uncertainties. We demonstrate the agreement with a single age by over-plotting an isochrone with log(age/yr)=8.2 on top of a summed SMASH CMD created for the nine cluster sample (see lower panel in Fig. \[fig-cluster\]) . This synchronization in cluster ages suggests (or, further reinforces the conclusion) that an important enhancement of the star formation rate took place $\sim$ 160 Myr ago. This epoch broadly agrees with the SMC-wide peak of cluster ages observed by Glatt et al. (2010) as well as the putative age of the most recent LMC-SMC interaction.
Cepheids
--------
OGLE IV (Udalsky et al. 2015; Soszyński et al. 2016) uncovered almost 5,000 classical Cepheids in the SMC and several tens of them spatially overlap with the shell feature (see Fig. \[fig-ceph1\]). Since Classical Cepheids are young supergiants, their presence in quite large numbers underlines significant star formation in the last few hundred Myr.
Cepheid ages can be computed for individual stars using period-age relations derived from population models (for instance, Bono et al. 2005). Using these relations, the ages vary from $\approx$15 to $\approx$500 Myr[^6] for SMC Cepheids. The age distribution of SMC Cepheids is known to be bimodal, with two peaks at $\approx$110–130 Myr and $\approx$220–230 Myr separated by a minimum at $\approx$150 Myr (e.g., Inno et al. 2015; Subramanian et al. 2015; Jacyszyn-Dobrzeniecka et al. 2016; Ripepi et al. 2017). The former peak has been associated with star formation triggered by the most recent interaction between the LMC and the SMC. Cepheids lying in the shell region span a relatively wide age range that clearly peaks at 100–130 Myr (see Fig. \[fig-ceph2\]). Such ages are in good agreement with current results for the MS sample and match very well the ages derived for the young clusters in the vicinity of the shell[^7]. Cepheids therefore support a scenario where the shell population is dominated by young stars formed during a recent star formation event, possibly related to the interaction between both Magellanic Clouds.
![Classical Cepheids in the OGLE database (red dots) over-plotted on the shell region.[]{data-label="fig-ceph1"}](fig8.pdf "fig:"){width="49.00000%"} \[ceph\]
Stellar Kinematics {#sec-kin}
------------------
Using the *Gaia* DR2 data (Gaia Collaboration et al. 2018), proper motions for thousands of stars in the SMC outer region have recently became available. We use this catalog to investigate the kinematic signature from the shell-like feature. From the DR2 database, we select stars surrounding the SMC and apply a series of astrometric cuts. We start by applying a parallax cut of $\omega < 0.2$ mas in order to remove foreground MW stars. Next we use a cut to the renormalized unit weight error (as described in the *gaia* technical note GAIA-C3-TN-LU-LL-124-01) of 1.40 and a cut for the color excess of the stars (as described in Lindegren et al. 2018 by Equation C.2). As astrometric precision has a strong relationship with the magnitude of the stars, we additionally apply a cut of $\mathrm{G} < 18$. Finally, to trim down potential MW contamination, we cut out an area with radius equal to 3 mas yr$^{-1}$ around the systemic proper motion (PM) of the SMC, which we will take to be $\mu_{\alpha^{*},0,\mathrm{SMC}}, \mu_{\delta^{*},0,\mathrm{SMC}}$ = $0.797 \pm 0.030, -1.220 \pm 0.030$ mas yr$^{-1}$ (Helmi, et al. 2018). This leaves us our final selection of stars (seen in upper Figure \[fig-gaia1\]), where the shell-like over-density can be clearly seen (marked by the purple rectangle).
The CMD of the shell region obtained from the Gaia data (bottom panel of Figure \[fig-gaia1\], expanded down to G $= 20$ for greater context) displays clearly the MS, the red supergiants (RSG), the RC, and the RGB features. Combining the requirement of G $< 18$ and the apparent locations of the stellar features, we created masks for each feature. The masks were then applied to the full sample of SMC stars and the resulting spatial distributions were plotted (upper Fig. \[fig-gaia2\]). Similar to the analysis earlier in the paper, the RGB had no apparent correlation with the shell-like region, but both the MS and RSG display over-densities in the location of the shell. Correspondingly, we select these two sequences from the shell region for our kinematic analysis.
![Age distribution for Cepheids located in the vicinity of the shell. Ages have been computed with the period-age relations for fundamental and first overtone pulsators of Bono et al. (2005) at Z=0.004.[]{data-label="fig-ceph2"}](fig9.pdf "fig:"){width="45.00000%"} \[ceph\_ages\]
![*Top panel:* Star map made with all the [*Gaia*]{} DR2 (Gaia Collaboration et al. 2018) sources with <span style="font-variant:small-caps;">visibility\_periods\_used</span> $\geq$ 5 and <span style="font-variant:small-caps;">phot\_bp\_rp\_excess\_factor</span> < 1.5 in a 400arcmin $\times$ 400arcmin box centered in the SMC. For orientation reference, the LMC is located down and to the left of the plot. The previously described shell and new features showed in Fig.4 (right panel) are clearly visible in the $Gaia$ data. *Bottom panel*: CMD of an astrometrically selected sample of *Gaia* sources with G $<$ 18 and $|{\mu}| < 5$ mas yr$^{-1}$ (marked in gray) with all sources selected within the shell region (the exact area selected can be seen in the top panel of Figure \[fig-gaia2\]) over-plotted in purple. The masks for selecting different CMD features are also over-plotted: main sequence (MS, blue), red supergiants (RSG, green), red clump (orange) and red giant branch (red).[]{data-label="fig-gaia1"}](shell_stars3.png "fig:"){width="50.00000%"} ![*Top panel:* Star map made with all the [*Gaia*]{} DR2 (Gaia Collaboration et al. 2018) sources with <span style="font-variant:small-caps;">visibility\_periods\_used</span> $\geq$ 5 and <span style="font-variant:small-caps;">phot\_bp\_rp\_excess\_factor</span> < 1.5 in a 400arcmin $\times$ 400arcmin box centered in the SMC. For orientation reference, the LMC is located down and to the left of the plot. The previously described shell and new features showed in Fig.4 (right panel) are clearly visible in the $Gaia$ data. *Bottom panel*: CMD of an astrometrically selected sample of *Gaia* sources with G $<$ 18 and $|{\mu}| < 5$ mas yr$^{-1}$ (marked in gray) with all sources selected within the shell region (the exact area selected can be seen in the top panel of Figure \[fig-gaia2\]) over-plotted in purple. The masks for selecting different CMD features are also over-plotted: main sequence (MS, blue), red supergiants (RSG, green), red clump (orange) and red giant branch (red).[]{data-label="fig-gaia1"}](cmd.jpg "fig:"){width="50.00000%"}
![*Top panel:* Spatial plot of our selected *Gaia* sample (marked in grey), discussed in Section \[sec-kin\], with the main sequence (MS, blue) and red supergiant (RSG, green) sequences shown in Figure \[fig-gaia1\] overplotted. The shell feature can be seen towards the center-left area of the plot in addition to significant substructure in the two sequences spread throughout the SMC. This feature has been marked with a purple box, and all stars that fall within the box are examined below in addition to being overplotted in the CMD for the SMC (seen in the bottom panel of Figure \[fig-gaia1\]). *Bottom panel:* Combined histogram of the residual proper motion vector angles of the MS and RSG populations after removing the systemic motion of the SMC and correcting for viewing perspective. Only stars that fall within the marked box in the top of Figure 12 are displayed, indicated by the same colors as above. The angle measurement is defined so that a residual vector pointing vertically in the spatial plot corresponds to an angle of $0^{\circ}$, and the angle increases in a counter- clockwise direction. A clear preference for a mean residual vector angle of $\sim 70-80^{\circ}$ can be seen, which roughly points radially outwards from the center of the SMC.[]{data-label="fig-gaia2"}](pos_pa.jpg){width="50.00000%"}
For this portion of the analysis, we first subtract the systemic PM from the PM of each star in the shell region. As the feature is noticeably extended on the night sky, we also calculate and remove the viewing perspective for each source, as outlined in van der Marel et al. (2002). These residual proper motions are converted into a Cartesian frame assuming the kinematically-derived center of ($\alpha_{J2000}$, $\delta_{J2000}$) = (16.25$^\circ$,$-$72.42$^\circ$), using the transformations from the Gaia Collaboration (Helmi et al. 2018). The position angle (PA, which in the context of this analysis is defined as the vector angle of the residual proper motions) is calculated for each source, defined where $0^{\circ}$ points vertically upwards in the spatial plot in Figure \[fig-gaia2\] and increases in a counter-clockwise direction. We compile all of the PAs for all sources into a histogram for easier interpretation (seen in lower panel of Figure \[fig-gaia2\]). The PAs appear to peak around $70-80^{\circ}$, which points roughly radially outwards from the center of the SMC. The scatter from $0^{\circ}$ to $360^{\circ}$ is expected as the average residual is on the order of $\sim 0.1$ mas yr$^{-1}$ with average errors of comparable magnitude, underscoring that the large peak in PA must be a real signal. This coherent radially outward motion of stars in the SMC is consistent with prior studies of the internal kinematics of the SMC (e.g. Zivick et al. 2018).
H[i]{} and H$\alpha$ emission
-----------------------------
Using the atomic hydrogen (H[i]{}) emission maps from the Galactic All-Sky Survey (GASS) third release[^8] (Kalberla & Haud 2015), we scanned the velocity channels available in that survey (from $-495$ km s$^{-1}$ out to +495 km s$^{-1}$). The H[i]{} gas emission within one square degree centered on $\alpha = 1$h10min, $\delta = -71.5{\degr})$ reaches a maximum in the channel $v = 198$ km s$^{-1}$. more specifically, the velocities of the gas within that region are close to a Gaussian distribution, with the maximum in $v = 198.3$ km s$^{-1}$ and a dispersion $\sigma = 13.6$ km s$^{-1}$. This velocity is very similar to the mean velocity of the stars measured by Evans & Howarth (2008)(172 km s$^{-1}$) with a dispersion of 30 km s$^{-1}$. The leftmost panel of Fig. \[V198stars\] shows the emission for this specific velocity channel in a log scale. The second panel of Fig. \[V198stars\] is the distribution of the mean gas temperature bounded by a larger range of velocities (184-211 km s$^{-1}$) in a linear scale, which is very similar to the previous panel. The rightmost panel shows the gas in the respective velocity channel, in a zoomed region close to the shell discussed in this paper. The young MS stars are over-plotted on the gas density map as white dots. A visual inspection on the last panel suggests a shift between the projected location of the young stars and the gas distribution.
As we do not have constraints to the distance of the H[I]{} gas cloud in the Fig. \[V198stars\], our few arguments are similarities between the velocities and shape of the H[I]{} gas cloud and the shell-like feature of young stars in the Fig. \[fig-maps\]. Assuming that the ‘Z’ shaped gas cloud was forming stars in a recent past (150-250 Myr ago), the velocities of the young stars and of the gas cloud could provide insights on what is the dominant process (tidal stripping or ram pressure) for decoupling the kinematics of the stars and the gas, at least in that region. However, if the gas cloud formed the shell-like feature, it is not clear what mechanism triggered such an intense star formation episode $\sim$200 Myr ago and why it is no longer acting to form stars. An evidence of the quiescence of the gas in forming stars is the lack of MS stars between the current position of the gas cloud and the position of the shell feature. Nevertheless, we must take into account the possibility of large distances in the line-of-sight between the gas feature and the young stars, being two completely separate substructures.
A comparison of the stellar populations map to the H$\alpha$ map (Fig. \[fig-halfa\]) from the Magellanic Clouds Emission Line Survey (MCELS; Winkler, Rathore, & Smith 1999) shows a system of filaments that roughly forms a shell with a radius of $0.75\degr$ centered at $\alpha_{J2000} = 01$:06, $\delta_{J2000} = -71$:45. This interstellar material is reminiscent of supergiant shells similar to those catalogued in the Magellanic Clouds by Meaburn (1980). Such shells are believed to be driven by the collective action of winds from multiple OB associations and supernovae. When the H$\alpha$ images are compared to the H[i]{} ATCA+Parkes observations, there appear to be faint (3-5$\sigma$ significance) H[i]{} emission patches coincident with the H$\alpha$ filaments with velocities between $\sim$125-140. We note that this putative shell does not coincide with those originally identified by Stanimirovic et al. (1999) using the same data set.
If this emission corresponds to the limb-brightened edge of a supergiant shell, the lack of detection of a coherent structure including envelopes expanding along the line-of-sight preclude a detailed kinematic comparison with the underlying stellar population. The velocities of the faint H[i]{} emission are consistent with the gas (and driving stellar population) being part of the SMC main body and not the material being drawn out by the interaction with the LMC. Based on the size, 750 pc (0.75at 60 kpc) and assuming this interstellar structure arose from the collective stellar energy feedback from a population originating 150 Myr ago, we expect an expansion velocity of 5 km s$^{-1}$ which is low but not unreasonable compared to other Magellanic supergiant shells (15 to 30 km s$^{-1}$, Book et al. 2008).
Discussion
==========
Our analysis of the resolved stellar populations of the elongated over-density detected in our Canon 200 images reveals that it is the brighter optical part of a more extended structure mainly traced by blue, young stars distributed in an intricate and complex structure. The structure shows a further, outer arc-like feature observed in the projected proximity of the Galactic globular cluster NGC 362 (see bottom panel of Figure \[fig-maps\]). The over-density also contains nine young star clusters with ages tightly clustered at 175 Myr (see Table 1).
We do not detect any counterpart of this over-density in the distribution of the older stars in this area of the SMC. This dominant young age of our SMC over-density and the lack of an old stellar remnant in the stellar density map plotted in Fig. \[fig-maps\] (left panel) suggest that this feature is neither of tidal origin nor the stellar remnant of a tidally disrupted, lower-mass system as observed in some other dwarf galaxies, e.g., in NGC 4449 (Martínez-Delgado et al. 2012) or in Andromeda II (Amorisco et al. 2014). While recently a large number of new, ultra-faint dwarf galaxies were discovered near the Magellanic System (Drlica-Wagner et al. 2015, 2016), these systems are exclusively devoid of gas and contain very old stellar populations. The existence of a former, more gas-rich small dwarf irregular galaxy is not excluded, but we would expect to also see a clear over-density in the old stellar population map if such a system were to merge. With the exception of tidal dwarf galaxies, there are no nearby dwarf galaxies known that do not contain old populations (Grebel & Gallagher 2004). Thus, the nature of this over-density seems to be different from the one discovered at $8\degr$ north of the SMC by Pieres et al. (2017), which is mainly composed of intermediate-age stars and without significant H[i]{} gas. Although morphologically similar in appearance to the LMC substructure described by Mackey et al. (2016), the young age of the SMC feature distinguishes it from that older and much larger LMC over-density.
Our stellar density maps of the shell region (Figure 4 and Figure 11) and the GALEX data (Figure 5) confirm with higher resolution the asymmetric structure of young stars originally found by Zaritsky et al. (2000) in the SMC outskirts, which contrasts with the smooth distribution of the older stellar populations. This lack of substructure in the older stellar populations led these authors to conclude that the dominant physical mechanism in determining the current appearance of the SMC must be recent star formation possibly triggered by a hydrodynamic interaction of the SMC with a second gas-rich object (e.g., a hot Milky Way halo or the outer gaseous envelope of the LMC). The contrast between the small-scale spatial structures in the young populations and the smooth distribution of the intermediate-age and old populations are even starker in these new data. This finding strengthens our argument (see also Zaritsky et al. 2000) that a purely tidal origin for the feature, which would have affected stars of all ages similarly, is unlikely. Instead, the origin of the feature is most likely hydrodynamical in nature. Given the coherence of the feature, the absence of an old stellar remnant, and the lack of evidence for a strong shock in the H[i]{} distribution, we disfavour models involving a recent accretion event.
The elongated shape, estimated absolute magnitude (Sec. 3.2), and position of the over-density in the outer region of the main body of the SMC could also suggest a tidal dwarf galaxy as another possible formation scenario for the shell-like structure. The stars in such objects are formed from gas stripped in past encounters (Elmegreen, Kaufman & Thomasson 1993) and/or consist of stars that originally formed in the more massive galaxies participating in the interaction (e.g., Duc 2012). Key requirements for tidal dwarfs include that they are made from recycled material, are gravitationally bound, and have decoupled from their former parent (Duc 2012). They may be able to survive for at least 3 Gyr in spite of their lack of dark matter (Ploeckinger et al. 2014). In our case, the over-density appears to be connected with the SMC and there is no evidence to suggest that it is a separate entity. Hence we also discount the possibility of a tidal dwarf galaxy.
A conspicuous feature in the stellar density map is a spiral-arm-like feature emanating from the over-density (labelled [*b*]{} in Figure 4). Because this stellar arm is only seen in young stars, it is unlikely that it was produced via tidal effects. Instead, the arc may consist of young stars formed as a result of a low pitch angle spiral density wave in the outer gas. Such features are common, but challenging to detect, in disks well beyond the optical radius (Ferguson et al. 1998; Herbert-Fort et al. 2012). The presence of a disk population would imply that the young stellar populations of the SMC must have some bulk rotation. However, initial [*Gaia*]{} DR2-based proper motions and rotation measurements of the SMC do not support significant rotation for the young SMC population (van der Marel & Sahlmann 2016), suggesting that, if the density wave scenario is correct, then the observed offset is dominated by the progression of the pattern rather than motion of material through it. In addition, the hypothesis of a density wave also imply an expected offset between current (H$\alpha$) and past star formation. We find not detectable offset between the H$\alpha$ emission and GALEX sources (see Fig. \[fig-halfa\]) which, given the timescales these tracers are sensitive to (10$^7$ vs. 10$^8$ yrs), places an upper limit on the pattern speed. Within the new [*Gaia*]{} DR2 catalog, we find among the bright stars the same shell feature in the young stellar populations. Examining this subset of SMC stars reveals coherent motion among the stars but in an outward radial direction. Given the direction of motion, it continues to appear unlikely that the SMC possesses stellar rotation, at least rotation that lays primarily in the plane of the sky as with the neighboring LMC. Though, as the vast majority of these stars do not possess radial velocities, we are unable to fully assess if this motion may be due to rotation with a large inclination. A more holistic analysis of the SMC, including searching for other kinematic substructure, will be required to improve constraints on the degree of stellar rotation. Our analysis of the stellar content of the shell (see Sec. 3.2) indicates that the main difference between the stellar population of the shell and the surrounding control fields is a recent period of enhanced star formation in the last $\simeq$ 1 Gyr, likely peaking at $\sim$ 150 Myr, as indicated by the clusters and Cepheids age distribution. The comparison of the CFs of the shell and the control fields also provides hints of a comparatively enhanced star formation rate in the total stellar content of this region at intermediate ages. If we compare these hints on the SFH of the shell and the control fields with the SFHs of the fields analyzed by Noël et al. (2009), it can be seen that the SFH of the shell field may present similarities with the SFHs of fields located in the Wing area of the SMC such as [*qj0112*]{}, [*qj0111*]{}, and [*qj0116*]{}, while the SFH of the two control fields may be alike to that of the remaining fields analyzed by Noël et al. (2009), where SFH has been very low in the last $\simeq$ 1 Gyr. The qualitative hints on the SFH of the shell region obtained through the CF analysis are also consistent with the SFH for fields 1 and 4 (located in the bar and the wing area, respectively) analyzed by Cignoni et al. (2012), which present a strong enhancement from $\simeq$ 5 Gyr ago to the present time, and a strong peak at a very recent epoch, $<$ 200 Myr ago, and with the information on detailed SFH maps presented by Rubele et al. (2015), which indicate that the Wing is $<$ 200 Myr old, that metal-poor gas was injected $>$ 1 Gyr ago resulting in the formation of intermediate-age stars, and that the majority of the SMC mass resulted from a star formation episode 5 Gyr ago. The SFH of the Wing field, then, could be then similar to that of the SMC central body and Wing area, but characterized by an enhanced star formation rate in the last 200 Myr with respect to the past average star formation rate. However, since the CMD toward the shell region contains also a component of the field SMC population, it is possible that the intermediate-age populations that make up the CF peak at ($g-i)\sim$ 0 in Fig. 6 are part of that SMC underlying population. Therefore, there are two possible scenarios to explain the nature of the shell: i) a shell composed only of young stars resulting from a huge star forming region active $\sim$ 150 Myr ago; or ii) the shell is a region that had enhanced episodes of star formation at different epochs, with the most striking one happening $\sim$ 150 Myr ago.
It is, thus, interesting to consider the young age of our SMC substructure in the context of the putative recent interaction with the LMC about 100 to 300 Myr ago, which left its signature in the young stellar populations in the Magellanic Bridge (e.g., Skowron et al. 2014) and in the age distribution of young populous star clusters in both Clouds (e.g., Glatt et al. 2010). Recently, Zivick et al. (2018) used mostly [*Hubble Space Telescope*]{} proper motion data to show that the LMC and SMC have had a head-on collision in the recent past, and it is able to constrain both the timescale and the impact parameter of this collision. Based on their measured proper motions and considering the allowed range of masses for the LMC and the Milky Way, they foud that in 97% of all the considered cases, the Clouds experienced a direct collision with each other 147 $\pm$ 33 Myr ago, with a mean impact parameter of 7.5 $\pm$ 2.5 kpc. There is also further evidence of this recent head-on collision in the [*Gaia*]{} DR2 proper motions along the Bridge (Zivick et al. 2019). The age of our feature falls into this age range as well. Moreover, the three-dimensional structure of young populations traced by Cepheids (with ages of about 15–500 Myr) shows a highly asymmetric distribution throughout the SMC: the distribution of classical Cepheids is elongated over 15–20 kpc (e.g., Scowcroft et al. 2016; Jacyszyn-Dobrzeniecka et al. 2016), the NE region being younger, closer to the Sun than the SW region (e.g., Haschke et al. 2012; Subramanian et al. 2015; Ripepi et al. 2017). Haschke et al. (2012) speculate that the displacement and compression of the gas of the SMC through tidal and ram pressure effects caused by the interaction between the Magellanic Clouds (and with the Milky Way) may have locally enhanced star formation, creating some of the irregular, asymmetric features such as the over-density described in the current paper. Among others, Inno et al.(2015) and Jacyszyn-Dobrzeniecka et al.(2016) also propose that the concomitance of the extensive Cepheid formation in the LMC $\approx$140 Myr ago and of the younger episode of Cepheid formation in the SMC may be related to the interaction between the Clouds. Ripepi et al. (2017) elaborate on this scenario and suggest that the relatively young Cepheids ($<$140 Myr) that dominate in the NE region have formed after the dynamical interaction that created the Bridge, from gas already shifted by the interaction.
Unfortunately, there is insufficient resolution in any existing numerical simulations of the LMC-SMC interaction to see the fine structure of our stellar density map in Fig. 4. Our results clearly motivate more detailed studies of the internal structure and star formation induced in the SMC by the LMC-SMC (and the MW) interaction. In particular, if the stellar arc is a spiral structure, these data likely disfavor models where the SMC is originally modeled as a non-rotating spheroid. This result further highlights the discrepant kinematics and spatial distribution of the SMC younger and older stellar populations, the origin of which is currently unknown.
We thank David Hogg for his help with the astrometry solution of the image used in this work. DMD thanks Prof. Ken Freeman for a discussion about the detection of the SMC shell in the photographic plates of the Clouds taken in the 1950s. DMD also thanks the hospitality and fruitful discussion about this work with the European Southern Observatory Garching headquarters staff during his stay as part of the ESO visitor program in September 2017. DMD, EKG, BL and LI acknowledge support by Sonderforschungsbereich (SFB) 881 “The Milky Way System” of the German Research Foundation (DFG), particularly through sub-projects A2, A3 and A5. DMD acknowledges support from the Spanish MINECO grant AYA2016-81065-C2-2. This project used data obtained with the Dark Energy Camera (DECam), which was constructed by the Dark Energy Survey (DES) collaboration. M-RC and CB acknowledge support from the European Research Council (ERC) under the European Union$'$s Horizon 2020 research and innovation program (grant agreement No. 682115). BCC acknowledges the support of the Australian Research Council through Discovery project DP150100862. T.d.B. acknowledges support from the European Research Council (ERC StG-335936). This work has been supported by the Spanish Ministry of Economy and Competitiveness (MINECO) under grant AYA2014-56795-P. MS acknowledges support from the ADAP grant NNX14AF81G. R.R.M. acknowledges partial support from project BASAL AFB-$170002$ as well as FONDECYT project N$^{\circ}1170364$.Y.C. acknowledges support from NSF grant AST 1655677.
Based on observations at Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. This project used data obtained with the Dark Energy Camera (DECam), which was constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Funda[ç]{}[ã]{}o Carlos Chagas Filho de Amparo [à]{} Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cient[í]{}fico e Tecnol[ó]{}gico and the Minist[é]{}rio da Ci[ê]{}ncia, Tecnologia e Inovac[ã]{}o, the Deutsche Forschungsgemeinschaft, and the Collaborating Institutions in the Dark Energy Survey. This work has made use of data from the European Space Agency (ESA) mission [*Gaia*]{} ([<https://www.cosmos.esa.int/gaia](<https://www.cosmos.esa.int/gaia)), processed by the [*Gaia*]{} Data Processing and Analysis Consortium (DPAC, [<https://www.cosmos.esa.int/web/gaia/dpac/consortium](<https://www.cosmos.esa.int/web/gaia/dpac/consortium)). Funding for the DPC has been provided by national institutions, in particular the institutions participating in the [*Gaia*]{} Multilateral Agreement.
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[^1]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
[^2]: Unfortunately, the position of this shell in the SMC halo (situated at 8 $\deg$ north from the SMC) is outside of the sky area covered in our photographic survey, including the Canon 50 images described in Besla et al. (2016).
[^3]: <http:/albinone.oa-teramo.inaf.it>
[^4]: <http://basti.oa-teramo.inaf.it/BASTI/WEB_TOOLS/synth_pop2/>
[^5]: The association of three clusters with the position of this feature was previously mentioned by Brueck & Marsoglou (1978). However, only NGC 458 and L90 actually fall on the shell, while HW62 is clearly offset off the stellar over-density towards the SMC center. The ages assigned to the clusters by these authors, based on photographic plates, have been revisited with our modern observations yielding 160 Myr (instead of 20-50 Myr).
[^6]: Models that include rotation during the Main Sequence phase lead to Cepheid ages increased by 50 to 100%, depending on the period (Anderson et al. 2016)
[^7]: However, it is important to take into account that the raw age distribution of Cepheids cannot be directly interpreted as an age distribution of the underlying star formation because it is convolved with both the stellar IMF and the lifetime of the star within the instability strip during which it would be identified as a Cepheid variable.
[^8]: <https://www.astro.uni-bonn.de/hisurvey/gass/>
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we report our multiwavelength observations of the vertical oscillation of a coronal cavity on 2011 March 16. The elliptical cavity with an underlying horn-like quiescent prominence was observed by the Atmospheric Imaging Assembly (AIA) on board the *Solar Dynamics Observatory* (*SDO*). The width and height of the cavity are 150$\arcsec$ and 240$\arcsec$, and the centroid of cavity is 128$\arcsec$ above the solar surface. At $\sim$17:50 UT, a C3.8 two-ribbon flare took place in active region 11169 close to the solar western limb. Meanwhile, a partial halo coronal mass ejection (CME) erupted and propagated at a linear speed of $\sim$682 km s$^{-1}$. Associated with the eruption, a coronal extreme-ultraviolet (EUV) wave was generated and propagated in the northeast direction at a speed of $\sim$120 km s$^{-1}$. Once the EUV wave arrived at the cavity from the top, it pushed the large-scale overlying magnetic field lines downward before bouncing back. At the same time, the cavity started to oscillate coherently in the vertical direction and lasted for $\sim$2 cycles before disappearing. The amplitude, period, and damping time are 2.4$-$3.5 Mm, 29$-$37 minutes, and 26$-$78 minutes, respectively. The vertical oscillation of the cavity is explained by a global standing MHD wave of fast kink mode. To estimate the magnetic field strength of the cavity, we use two independent methods of prominence seismology. It is found that the magnetic field strength is only a few Gauss and less than 10 G.'
author:
- 'Q. M. Zhang'
- 'H. S. Ji'
title: Vertical oscillation of a coronal cavity triggered by an EUV wave
---
Introduction {#sec:intro}
============
Solar prominences are cool and dense plasma structures suspending in the corona [@lab10; @mac10; @par14]. The routine observation of prominences has a long history. They can be observed in radio, H$\alpha$, Ca[ii]{}H, He[i]{}10830 [Å]{}, extreme-ultraviolet (EUV), and soft X-ray (SXR) wavelengths [e.g., @gop03; @ji03; @ber08; @zqm15; @ste15; @wang16]. When prominences appear on the disk in the H$\alpha$ full-disk images, they look darker than the surrounding area due to the absorption of the background radiation. Therefore, they are also called filaments [@mar98]. Prominences (or filaments) can form in active regions (ARs), quiet region, and polar region [e.g., @hei08; @su15; @yan15]. It is generally believed that the gravity of prominences should be balanced by the upward magnetic tension force of the dips in sheared arcades [@jor92; @mar01; @xia12] or magnetic flux ropes [MFRs; @aul98; @hil13; @xia16; @yan18].
Prominences tend to oscillate after being disturbed [@oli92; @tri09; @arr12; @luna18 and references therein]. According to the amplitude, prominence oscillations can be classified into the small-amplitude [e.g., @oka07; @ning09] and large-amplitude [@chen08] types. According to the period, prominence oscillations can be divided into the short-period ($\leq$10 min) and long-period ($\geq$40 min) [@lin07; @zqm17a] types. A more reasonable classification is based on the direction of oscillation. For longitudinal oscillations, the prominence material oscillates along the threads with small angles of $10^{\circ}-20^{\circ}$ between the threads and spine [@jing03; @zqm12; @li12; @luna12; @luna14]. For transverse oscillations, the direction is horizontal (vertical) if the whole body of the prominence oscillates parallel (vertical) to the solar surface [@hyd66; @ram66; @kle69; @shen14a]. Generally speaking, the amplitudes and periods of longitudinal oscillations are larger than those of transverse oscillations. Occasionally, two types of oscillations coexist in a single event, showing a very complex behavior [@gil08; @shen14b; @zqm17b]. Recently, state-of-the-art magnetohydrodynamics (MHD) numerical simulations have thrown light on the triggering mechanisms, restoring forces, and damping mechanisms of prominence oscillations [@zqm13; @ter13; @ter15; @luna16; @zhou17; @zhou18].
Coronal cavities are dark structures as a result of density depletion in the white light (WL), EUV, and SXR wavelengths [@vai73; @gib06; @vas09]. The densities of cavities are 25%$-$50% lower than the adjacent streamer material [@mar04; @ful08; @ful09; @sch11]. Cavities show diverse shapes, such as semi-circular, circular, elliptical, and teardrop [@for13; @kar15a]. The lengths, heights, widths of cavities are 0.06$-$2.9 $R_{\odot}$, 0.08$-$0.5 $R_{\odot}$, and 0.09$-$0.4 $R_{\odot}$, respectively [@kar15a]. With a broader differential emission measure distribution, the average temperatures (1$-$2 MK) of cavities are slightly higher than the streamers [@hud99; @hab10; @ree12]. Both observations and numerical experiments indicate that the magnetic structure of cavities could be excellently described by a twisted/helical MFR [@gib10; @dove11; @fan12; @bak13; @rach13; @chen18]. At the bottom of cavities, there are denser filament plasmas drained down by gravity [@low95; @reg11]. Spectroscopic observations reveal that there are continuous whirling/spinning motion [@wang10] and large-scale flows with Doppler speeds of 5$-$10 km s$^{-1}$ in prominence cavities [@sch09]. The stable structures can last for days or even weeks to months [@kar15b]. After being strongly disturbed or a certain type of MHD instability is activated, a cavity may experience loss of equilibrium and erupt to drive a coronal mass ejection [CME; @ill85].
For the first time, @liu12 reported the transverse (horizontal) oscillation of a twisted MFR cavity and its embedded filament after the arrival of the leading EUV wave front at a speed of $\geq$1000 km s$^{-1}$. The average amplitude, initial velocity, period, and *e*-folding damping time are $\sim$2.3 Mm, $\sim$8.8 km s$^{-1}$, $\sim$27.6 minutes, and $\sim$119 minutes, respectively. Sometimes, MFRs carrying prominence material undergo global vertical oscillations with periods of hundreds of seconds, which are explained by the standing wave of fast kink mode [@kim14; @zhou16]. However, the vertical oscillation of a cavity as a result of the interaction between a coronal EUV wave and the cavity has rarely been observed and investigated.
In this paper, we report our multiwavelength observations of the vertical oscillation of a coronal cavity triggered by an EUV wave on 2011 March 16. The wave was caused by the eruption of a C3.8 two-ribbon flare and a partial halo CME at the remote AR 11169. The paper is structured as follows. Data analysis is described in detail in Section \[sec:data\]. Results are shown in Section \[sec:result\]. Estimation of the magnetic field strength of the cavity is arranged in Section \[sec:discuss\]. Finally, we give a brief summary in Section \[sec:summary\].
Instruments and data analysis {#sec:data}
=============================
Located north to NOAA AR 11169 (N17W74), the coronal cavity (N62W75) with an underlying prominence was continuously observed by the Global Oscillation Network Group (GONG) in H$\alpha$ line center (6562.8 [Å]{}) and by the Atmospheric Imaging Assembly [AIA; @lem12] on board the *Solar Dynamics Observatory* [*SDO*; @pes12] in UV (1600 [Å]{}) and EUV (171, 193, 211, and 304 [Å]{}) wavelengths. The photospheric line-of-sight (LOS) magnetograms were observed by the Helioseismic and Magnetic Imager [HMI; @sch12] on board *SDO*. The level\_1 data from AIA and HMI were calibrated using the standard *Solar Software* (*SSW*) programs *aia\_prep.pro* and *hmi\_prep.pro*. The full-disk H$\alpha$ and AIA 304 [Å]{} images were coaligned with a precision of $\sim$1$\farcs$2 using the cross correlation method. The CME was observed by the C2 WL coronagraph of the Large Angle Spectroscopic Coronalgraph [LASCO; @bru95] on board *SOHO*[^1]. The LASCO/C2 data were calibrated using the *SSW* program *c2\_calibrate.pro*. The CME was also observed by the COR1[^2] with a field of view (FOV) of 1.3$-$4.0 $R_{\odot}$ on board the *Solar TErrestrial RElations Observatory* [*STEREO*; @kai05]. The ahead satellite (hereafter STA) and behind satellite (hereafter STB) had separation angles of $\sim$88$^{\circ}$ and $\sim$95$^{\circ}$ with respect to the Sun-Earth direction on 2011 March 16. The coronal EUV wave associated with the CME was observed by AIA and the Extreme-Ultraviolet Imager (EUVI) in the Sun Earth Connection Coronal and Heliospheric Investigation package [SECCHI; @how08]. EUVI observes the Sun in four wavelengths (171, 195, 284, and 304 [Å]{}). Calibrations of the COR1 and EUVI data were performed using the *SSW* program *secchi\_prep.pro*. The deviation of STA north-south direction from the solar rotation axis was corrected. The EUV flux of the C3.8 flare in 1$-$70 [Å]{} was recorded by the Extreme Ultraviolet Variability Experiment [EVE; @wood12] on board *SDO*. The SXR flux of the flare in 1$-$8 [Å]{} was recorded by the *GOES* spacecraft. To investigate the hard X-ray (HXR) source of the flare, we made HXR images at different energy bands (6$-$12 keV and 12$-$25 keV) observed by the *Reuven Ramaty High-Energy Solar Spectroscopic Imager* [*RHESSI*; @lin02]. The HXR images were generated using the CLEAN method with integration time of 10 s. The observational parameters, including the instrument, wavelength, time, cadence, and pixel size are summarized in Table \[tab:para\].
[ccccc]{} LASCO & WL & 19:00$-$23:36 & 720 & 11.4\
COR1 & WL & 19:00$-$20:30 & 300 & 15.0\
HMI & 6173 & 17:30$-$23:30 & 45 & 0.6\
AIA & 1600 & 17:30$-$23:30 & 24 & 0.6\
AIA & 171$-$304 & 17:30$-$23:30 & 12 & 0.6\
EUVI & 171 & 17:30$-$23:30 & 150 & 1.6\
EUVI & 195, 284 & 17:30$-$23:30 & 300 & 1.6\
GONG & 6563 & 17:30$-$23:30 & 60 & 1.0\
*GOES* & 1$-$8 & 17:30$-$23:30 & 2.05 &\
EVE & 1$-$70 & 17:30$-$23:30 & 0.25 &\
*RHESSI* & 6$-$25 keV & 17:30$-$23:30 & 10 & 4\
Results {#sec:result}
=======
Flare and CME {#s-flare}
-------------
Figure \[fig1\] shows the H$\alpha$ and EUV images before the flare. In panels (a) and (b), the arrows point at the quiescent prominence resembling a horn structure. Above the prominence, there is a dark void with depleted EUV emissions (see panels (c-e)). This is the typical coronal cavity showing an elliptical shape. The height and width of the cavity are $\sim$240$\arcsec$ and $\sim$150$\arcsec$, and the centroid of the cavity has a height of $\sim$128$\arcsec$, respectively. In panel (f), the HMI LOS magnetogram features AR 11169 close to the western solar limb. The distance between the AR and cavity is $\sim$300$\arcsec$.
In Figure \[fig2\], the bottom panel shows the temporal evolutions of the EUV irradiance and SXR flux of the flare. It is clear that the EUV and SXR intensities of the flare started to rise gradually at $\sim$17:50 UT and reached the peak values around 21:00 UT, which was followed by a long main phase.
The flare observed in H$\alpha$, EUV, and UV wavelengths near the peak time are displayed in Figure \[fig3\]. An online animation (*20110316.mov*) is a combination of panels (c-e) observed by *SDO*/AIA. With a time cadence of 240 s, the animation starts from 17:44 UT on 2011 March 16 to 00:00 UT on 2011 March 17, featuring the vertical oscillation of the dark cavity with an embedded prominence. The flare is a typical two-ribbon flare with bright ribbons and semi-circular post flare loops in AR 11169. In panel (b), an above-the-loop-top HXR source is successfully reconstructed at 6$-$12 keV and 12$-$25 keV energy bands. The intensity contours of the source are drawn with green and yellow lines.
In Figure \[fig4\], the top and bottom panels demonstrate the WL images of the flare-related CME in the FOVs of LASCO/C2 and STA/COR1, respectively. With the central position angle (CPA) and angular width being 268$^{\circ}$ and 184$^{\circ}$, the CME appeared first in the FOV of C2 at $\sim$19:12 UT. During its forward propagation at a linear speed of $\sim$682 km s$^{-1}$, the CME underwent lateral expansion and evolved into a typical three-part structure, which consists of a bright leading edge, a dark cavity, and a bright core [@ill85]. Evolution of the CME in the FOV of STA/COR1 is similar to that in C2, except for a different CPA due to the different perspective of STA.
In Figure \[fig2\](a), we plot the temporal evolution of the CME height with green diamonds. A quadratic polynomial ($h=h_{0}+v_{0}t+at^2/2$) is applied to fit the temporal evolution of the CME height, which is represented by the green dashed line. The initial velocity $v_{0}=24.3$ km s$^{-1}$ at 17:50 UT and the constant acceleration $a=62.7$ m s$^{-2}$.
Coronal EUV Wave {#s-wave}
----------------
The fast and wide CME generated a coronal EUV wave propagating on the solar surface. To illustrate the wave more clearly, we take the EUV images at 17:30 UT as base images and obtain base-difference images at the following times. Figure \[fig5\] demonstrate eight snapshots of the base-difference images in AIA 171 [Å]{} during 18:00$-$19:24 UT. It is seen that as the CME proceeds, a bright EUV wave front forms when the lateral magnetic field lines of the CME are stretched and pressed, which is indicated by the arrow in panel (d). Behind the EUV wave front, there is a dark dimming region where the electron density decreases significantly after the impulsive expulsion of material carried by the CME [@zqm17c].
In order to investigate the evolution of the EUV wave, we select a curved slice (S1) in Figure \[fig5\](h). The long slice with a length of 1220$\arcsec$ starts from AR 11169 and passes through the cavity. Concentric with the solar limb, S1 has a height of 201$\arcsec$. Figure \[fig6\] shows the time-slice diagrams of S1 in 211, 193, and 171 [Å]{}. In panels (a) and (b), the bright inclined feature indicates the propagation of EUV wave in the northeast direction during 18:40$-$19:00 UT, which is overlaid by white dashed lines. The slopes of the dashed lines are equal to the velocities ($\sim$120 km s$^{-1}$) of EUV wave. It is obvious that the EUV wave reached and interacted with the dark cavity. In panel (c), the dashed line indicates the slow lateral expansion of the CME at a speed of $\sim$3 km s$^{-1}$ before 18:40 UT in 171 [Å]{}.
The EUV wave was also observed by STA/EUVI in various wavelengths. Like in Figure \[fig5\], we take the EUV images at 17:30 UT as base images and obtain base-difference images at the following times. Eight snapshots of the base-difference images in EUVI 195 [Å]{} during 18:20$-$18:55 UT are displayed in Figure \[fig7\]. It is clear that as the CME proceeds, the bright EUV wave front propagates in the northeast direction.
In Figure \[fig7\](b), we select a second slice (S2) with a length of 418 Mm, which starts from AR 11169 and extends in the same direction as the EUV wave. The time-slice diagrams of S2 in 171, 195, and 284 [Å]{} are plotted in Figure \[fig8\]. Like in Figure \[fig6\], the bright inclined feature indicates the propagation of EUV wave in the northeast direction during 18:30$-$18:50 UT, which is overlaid by the yellow dashed lines. The slopes of the dashed lines are equal to the velocity ($\sim$114 km s$^{-1}$) of EUV wave, which is close to the values in the FOV of AIA. The CME underwent a slow lateral expansion at a speed of $\sim$5.5 km s$^{-1}$ before $\sim$18:20 UT.
Vertical Oscillation of the Cavity {#s-osci}
----------------------------------
From the online animation (*20110316.mov*), we found clear evidence for vertical oscillation of the cavity after the arrival of the EUV wave. In Figure \[fig1\](e), we select a third slice (S3). With a length of 404$\farcs$5, S3 starts from the solar surface and passes through the cavity, covering the large-scale envelope coronal loops. The time-slice diagrams of S3 in 211, 193, and 171 [Å]{} are plotted in Figure \[fig9\]. It is obvious that after the arrival of the EUV wave at $\sim$19:00 UT, the overlying coronal loops are pressed down before bouncing back gradually to their initial states. The velocities of the downward motion are $\sim$387, $\sim$344, and $\sim$149 km s$^{-1}$ in 211, 193, and 171 [Å]{}, respectively.
Oscillation of the cavity with small amplitudes could not be distinctly displayed. However, when we magnify the regions within the white boxes of Figure \[fig9\], things are different. Close-ups of the regions with a better contrast are shown in Figure \[fig10\]. Now, the damping oscillations lasting for about 2 cycles are clear. We mark the positions of the cavity manually with white plus symbols. In order to obtain parameters of the oscillations, we performed curve fittings using the standard *SSW* program *mpfit.pro* and the same function as that described in previous literatures [@zqm12; @zqm17a; @zqm17b]: $$\label{eqn-1}
y=y_0+bt+A_0\sin(\frac{2\pi}{P}t+\phi_0)e^{-t/\tau},$$ where $y_0$, $A_0$, and $\phi_0$ represent the initial position, amplitude, and phase. $b$, $P$, and $\tau$ stand for the linear velocity of cavity, period, and damping time of the oscillations.
The results of curve fittings are shown in Figure \[fig11\], indicating that the function in Equation \[eqn-1\] can excellently describe the vertical oscillation of the cavity. The parameters in different wavelengths are labeled in each panel and listed in Table \[tab:fitting\] for a easier comparison. The amplitudes, periods, and damping times are 2.4$-$3.5 Mm, 29$-$37 minutes, and 26$-$78 minutes, respectively. The amplitudes are consistent with the values for the vertical oscillation of MFRs [@kim14; @zhou16]. The periods are close to the values for the vertical oscillation of prominences [@hyd66; @gil08; @bocc11]. Since the prominence-cavity system consists of multithermal plasma, the initial positions of oscillation in different wavelengths, which are labeled with white circles in Figure \[fig1\](c-e), agree with their formation temperatures. In other words, hot plasmas oscillate at higher altitudes and cool plasmas oscillate at lower altitudes, implying that the prominence-carrying cavity oscillates as a whole body after the arrival of the EUV wave. The positive values of $b$ in the third column of Table \[tab:fitting\] suggest that the cavity ascended slowly at a speed of 1$-$2 km s$^{-1}$ during the oscillation.
Using the simple formula $v=ds/dt$, the velocities of the vertical oscillations in 211, 193, and 171 [Å]{} are calculated and displayed in Figure \[fig12\]. The velocity amplitudes of the oscillations are less than 10 km s$^{-1}$, which are close to the values previously reported by @shen14a and @zhou16.
[ccccccc]{} 211 & 84.89 & 1.04 & 2.39 & 0.48 & 29.4 & 77.8\
193 & 82.77 & 1.43 & 3.51 & 2.34 & 36.6 & 66.2\
171 & 44.81 & 1.74 & 2.57 & 1.50 & 29.9 & 25.9\
Discussion {#sec:discuss}
==========
How is the Oscillation Triggered?
---------------------------------
Despite of substantial observations and investigations since the discovery of prominence oscillations [e.g., @hyd66; @ram66], the triggering mechanisms of prominence oscillations are far from clear. The longitudinal oscillations can be triggered by microflares or subflares [@jing03; @zqm12], shock waves [@shen14b], magnetic reconnections in the filament channels [@zqm17a], and coronal jets at the legs of filaments [@luna14] or from a remote AR [@zqm17b]. The transverse oscillations, including horizontal and vertical oscillations, are usually triggered by EUV or Moreton waves [e.g., @eto02; @liu12; @shen14a; @shen17]. @liu12 reported the transverse oscillation of a cavity triggered by the arrival of a coronal EUV wave from the remote AR 11105 on 2010 September 8-9. The oscillation is explained by the fast kink-mode wave. In our study, the oscillation of the prominence-carrying cavity is also triggered by a coronal EUV wave from AR 11169. Timeline of the whole events is drawn in Figure \[fig13\]. On the one hand, the direction of oscillation is vertical (see Figure \[fig10\]), which is different from the situation in @liu12. This is most probably due to the different location of interplay. As shown in the schematic cartoon of @liu12, the EUV wave front collides with the cavity laterally. In our case, the EUV wave collides with the cavity from the top, which is supported by the fast downward motion and bouncing back of the overlying magnetic field lines above the cavity (see Figure \[fig9\]). Horizontal oscillation is not found from the time-slice diagrams of S1 (see Figure \[fig6\]). On the other hand, the speed ($\sim$120 km s$^{-1}$) of the EUV wave from AR 11169 accounts for only 10% of that in @liu12. This is probably due to the different component of EUV wave. On 2010 September 8, the fast component of the global EUV wave interacting with the cavity is interpreted by a fast MHD wave. Considering the absence of fast component EUV wave in the time-slice diagrams of S1 and S2 (see Figure \[fig6\] and Figure \[fig8\]), the EUV wave in our case is most probably the slow component of CME-caused restructuring [@chen02; @chen05].
In Figure \[fig14\], we draw a schematic cartoon to illustrate the interaction between the cavity and the EUV wave. In AR 11169 close to the solar limb, a C3.8 two-ribbon flare (red arcades) and a CME (green line and blue circle) take place. Owing to the stretch and lateral expansion of the magnetic field lines constraining the CME, a coronal EUV wave (black lines) is generated and propagates in the northeast direction. As soon as the EUV wave reaches the cavity from top, it presses the overlying magnetic field lines (purple lines) above the cavity, leading to fast downward motion of the magnetic loops. At the same time, the cavity (dark ellipse) and the prominence at the bottom start to oscillate in the vertical direction. Due to the quick attenuation, the oscillation lasts for $\sim$2 cycles and disappears.
Magnetic Field Strength of the Cavity
-------------------------------------
The restoring force of vertical prominence oscillation is another key issue that should be addressed. Recently, @zhou18 performed three-dimensional (3D) ideal MHD simulations of prominence oscillations along a MFR in three directions. The vertical oscillation has a period of $\sim$14 minutes, which can be well explained using a slab model [@diaz01]. The magnetic tension force serves as the dominant restoring force. Considering that the periods of oscillation at different heights are approximately equal, we conclude that the cavity experienced a global vertical oscillation of fast kink mode [@zhou16]. The magnetic field strength of the cavity can be roughly estimated as follows: $$\label{eqn-2}
B_c=2L_c\sqrt{4\pi\rho_c}/P,$$ where $L_c$ and $\rho_c$ represent the total length and density of the cavity, and $P$ denotes the period of oscillation. Since the length of cavity could not be measured directly, we take $L_c$ in the range of 0.06$-$2.9 $R_{\odot}$ [@kar15a]. We adopt the typical density of $n_{e}=1.0\times10^8$ cm$^{-3}$ ($\rho_c=1.67\times10^{-16}$ g cm$^{-3}$) for the coronal cavity at a height of 1.13 $R_{\odot}$ [@ful09; @sch11]. Using the fitted period of $\sim$30 minutes (see Table \[tab:fitting\]), $B_c$ is calculated to be 0.2$-$10 G, which is consistent with the value of 6 G in @liu12.
Another way of estimating the magnetic field strength of prominence at the bottom of cavity is described by @hyd66: $$\label{eqn-3}
B_{r}^2=\pi\rho_{p} r_{0}^2(4\pi^2 P^{-2}+\tau^{-2}),$$ where $B_r$, $\rho_{p}$, and $r_0$ are the radial component of magnetic field, density, and scale height of the prominence, $P$ and $\tau$ are the period and damping time of the oscillation. Equation \[eqn-3\] can be simplified: $$\label{eqn-4}
B_{r}^2=4.8\times10^{-12}r_{0}^2(P^{-2}+0.025\tau^{-2}),$$ if we assume $\rho_p=4\times10^{-14}$ g cm$^{-3}$, which is two orders of magnitude higher than $\rho_c$, and $r_0=3\times10^9$ cm [@shen17]. Using the fitted period of $\sim$30 minutes and damping time of 25.9 minutes (see Table \[tab:fitting\]), $B_r$ is calculated to be 3.8 G, which is close to the values reported in previous literatures [@hyd66; @shen14a; @shen17]. In brief, the magnetic field strength of the prominence-carrying cavity in our case is a few Gauss and less than 10 G.
Summary {#sec:summary}
=======
In this paper, we report our multiwavelength observations of the vertical oscillation of a coronal cavity on 2011 March 16. The main results are summarized as follows:
1. [In EUV wavelengths, the width and height of the elliptical cavity are $\sim$150$\arcsec$ and $\sim$240$\arcsec$. The centroid of the cavity has a height of $\sim$128$\arcsec$ (0.13 $R_{\odot}$) above the solar surface. At the bottom of the cavity, there is a horn-like quiescent prominence.]{}
2. [At $\sim$17:50 UT, a C3.8 two-ribbon flare and a partial halo CME took place in AR 11169 close to the western limb. The EUV and SXR intensities of the flare rose gradually to the peak values at $\sim$21:00 UT. The linear velocity, central position angle, and angular width of the CME are 682 km s$^{-1}$, 268$^{\circ}$, and 184$^{\circ}$ in the LASCO/C2 FOV.]{}
3. [During the impulsive phase of the flare and acceleration phase of the CME, a coronal EUV wave was generated and propagated in the northeast direction at a speed of $\sim$120 km s$^{-1}$. The EUV wave is interpreted by the field line stretching and restructuring caused by the eruption of CME.]{}
4. [Once the EUV wave arrived at the cavity, it interacted with the large-scale overlying magnetic field lines from the top, resulting in quick downward motion and bouncing back of the overlying loops. Meanwhile, the cavity started to oscillate coherently in the vertical direction. The amplitude, period, and damping time are 2.4$-$3.5 Mm, 29$-$37 minutes, and 26$-$78 minutes, respectively. The oscillation lasted for $\sim$2 cycles before disappearing.]{}
5. [The vertical oscillation of the cavity is explained by a global standing MHD wave of fast kink mode. Using two independent methods of prominence seismology, we carry out a rough estimation of the magnetic field strength of the cavity, which is a few Gauss and less than 10 G. Additional case studies using the multiwavelength and high-resolution observations are required to investigate the interaction between a global EUV wave and a prominence-cavity system. MHD numerical simulations are worthwhile to gain an insight into the restoring forces and damping mechanisms of transverse prominence oscillations.]{}
We would like to thank Y. N. Su, D. Li, T. Li, R. S. Zheng, and Y. D. Shen for fruitful and valuable discussions. *SDO* is a mission of NASAs Living With a Star Program. AIA and HMI data are courtesy of the NASA/*SDO* science teams. This work utilizes GONG data from NSO, which is operated by AURA under a cooperative agreement with NSF and with additional financial support from NOAA, NASA, and USAF. QMZ is supported by the Youth Innovation Promotion Association CAS, NSFC (No. 11333009, 11790302, 11773079), the Fund of Jiangsu Province (BK20161618 and BK20161095), “Strategic Pilot Projects in Space Science” of CAS (XDA15052200), and CAS Key Laboratory of Solar Activity, National Astronomical Observatories (KLSA201716).
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[^1]: http://cdaw.gsfc.nasa.gov/CME\_list/
[^2]: http://cor1.gsfc.nasa.gov/catalog/cme/2011/
| {
"pile_set_name": "ArXiv"
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[**Robust estimator of distortion risk premiums for heavy-tailed losses**]{}
Brahim Brahimi[[^1] and ]{}Zoubir Kenioua
[*Laboratory of Applied Mathematics, Mohamed Khider University, Biskra, Algeria*]{}
**Abstract**
We use the so-called t-Hill tail index estimator proposed by [Fab01]{}, rather than Hill’s one, to derive a robust estimator for the distortion risk premium of loss. Under the second-order condition of regular variation, we establish its asymptotic normality. By simulation study, we show that this new estimator is more robust than of [@NeMe09] both for small and large samples.
**Keywords:** Distortion risk premiums; Extreme values; Tail; Robustness.
**AMS 2010 Subject Classification:** 62G10; 62G32.
**Introduction\[sec1\]**
========================
In many important applications in Finance, Actuarial Science, Hydrology, Insurance, one of most crucial topics is the determination of the amounts of losses of a heavy-tailed risks. In literature there are many possible definitions of risk according to the investment objectives, so in some sense risk itself is a subjective concept as well as the risk measure for an investor. Thus, the existence of a unique risk that solves the investor’s problems is not confirmed. However, one must distinguish between risks. The concept of coherence due to the paper of [@ArDeEbHe99] which categorize risks by good or bad risks. For this reason and to improve the performance of investor’s strategies we may identify those risk measures and the appearance of the risk himself heavy tails, asymmetries,... Most of this risk measures, used are special cases of Wang’s distortion premium [@Wang96], defined as follows $$\Pi \lbrack \psi ;F]=\int_{0}^{\infty }\psi (1-F(x))dx, \label{eq1}$$ where $\psi :[0,1]\rightarrow \lbrack 0,1]$ is a non-decreasing function called distortion function, such that $\psi (0)=0$ and $\psi (1)=1.$ The distortion functions $\psi $ are concave, which makes the corresponding distortion premiums $\Pi \lbrack \psi ;F]$ coherent [@ArDeEbHe99] as proved by [@WiHa99]. In this paper, we suppose that the distortion functions $t\longmapsto \psi \left( t^{-1}\right) $ is regularly varying at infinity with index of regular variation $\rho \geq 1,$ such that $$\psi \left( t^{-1}\right) =t^{-1/\rho }\mathcal{L}_{\psi }\left( t\right) ,
\label{g}$$ where $t\longmapsto \mathcal{L}_{\psi }\left( t\right) $ is slowly varying as infinity, that is $\mathcal{L}_{\psi }\left( tx\right) /\mathcal{L}_{\psi
}\left( t\right) \rightarrow 1$ as $t\rightarrow \infty ,$ for any $x>0.$ Note that a non negative random variable (rv) $X$ with finite mean and a cumulative distribution function (cdf) $F$ is called heavy-tailed, if $1-F$ is regularly varying with index $-1/\gamma <0$ ( notation: $F\in \mathcal{RV} _{\left( -1/\gamma \right) })$, that is $$\lim_{t\rightarrow \infty }\frac{1-F\left( tx\right) }{1-F\left( t\right) } =x^{-1/\gamma },\text{ for }x>0. \label{first-condition}$$ In particular, the proportional-hazards premium (see, [@Roletal99], page 82). $$\Pi \lbrack \rho ;F]=\int_{0}^{\infty }\left( 1-F(x)\right) ^{1/\rho }dx,
\label{Psi-ro}$$ with the concave distortion function $\psi (t)=t^{1/\rho }$ for every $\rho
\geq 1.$ Since we are concerned with with heavy-tailed losses with infinite second moment, then by following [@BrMeNeRi], we assume that $\gamma \in
(1/2,1)$ and $\rho \gamma \in \left( 0,1\right) ,$ thus we will work with $$1/2<\gamma <1/\rho . \label{condition gamabeta}$$ Suppose that we have an independent and identically distributed (iid) sample $X_{1},...,X_{n}$ of rv $X$ of size $n$ with a cdf $F$ satisfying condition ( \[first-condition\]) and let denote by $X_{1:n}\leq ...\leq X_{n:n}$ the corresponding order statistics. Also, let $1<k=k_{n}$ be the number of extreme observations used in the computation of the tail index. We assume that $k$ satisfies the conditions $$1<k<n,\text{ }k\rightarrow \infty \text{ and }k/n\rightarrow 0\text{ as } n\rightarrow \infty . \label{K}$$ [@NeMe09] proposed an alternative estimator of (\[Psi-ro\]) and establish its asymptotic normality by using the Weissman’s estimator of the high quantile $q_{t}=F^{\leftarrow }\left( 1-t\right) $ defined by $$\widehat{q}_{t}=\left( k/n\right) ^{\widehat{\gamma }^{H}}X_{n-k:n}t^{- \widehat{\gamma }^{H}},\text{ }t\downarrow 0,$$ where $F^{\leftarrow }$ denotes the generalized inverse of $F$ and $$\widehat{\gamma }^{H}=\widehat{\gamma }^{H}\left( k\right) :=\frac{1}{k} \sum\limits_{i=1}^{k}\log X_{n-i+1:n}-\log X_{n-k:n}, \label{Hill}$$ is the well-known Hill estimator [@Hill75] of the tail index $ \gamma .$ For a fixed aversion parameter $\rho ,$ their estimator is given by $$\widehat{\Pi }_{n}\left( \widehat{\gamma }^{H},k\right) :=\left( k/n\right)
^{1/\rho }\frac{X_{n-k:n}}{1-\widehat{\gamma }^{H}\rho }+\sum_{i=k+1}^{n} \left( \left( i/n\right) ^{1/\rho }-\left( \left( i-1\right) /n\right)
^{1/\rho }\right) X_{n-i+1:n}. \label{NeMe}$$ The Hill estimator is a pseudo-maximum likelihood estimator based on the exponential approximation of normalized log-spacings $Y_{j}=j\left( \log
X_{j:n}-\log X_{j+1:n}\right) $ for $j=1,...,k.$ In practice, the Hill estimator depends on the choice of the sample fraction $k$ and is inherently not very robust to large values $Y_{j},$ which makes the estimator proposed by [@NeMe09] sensitive. This constitutes a serious problem in terms of bias and root mean squared error (RMSE). To improve the quality of $\widehat{ \Pi }_{n}\left( \widehat{\gamma }^{H},k\right) $, instead of Hill’s one, we propose to estimate the tail index $\gamma $ by the so-called t-Hill estimator, proposed by [@Fab01], given by its harmonic mean $$\widehat{\gamma }=\widehat{\gamma }\left( k\right) :=\left( \frac{1}{k} \sum_{j=1}^{k}\frac{X_{n-k:n}}{X_{n-j+1:n}}\right) ^{-1}-1 \label{t-Hill}$$ known as score moment estimation (t-score or t-estimation method). The latter is more robust than the classical Hill estimator $\widehat{\gamma } ^{H}$ defined in (\[Hill\]) (see [@StFaS12] and the asymptotic normality is given in Theorem 2 of [@BeScSt14]). For other robust estimators for $\gamma $ we referred to [@PeWe01], [@JuSc04], [VaBeChHu07]{} and [@KiLe08]. The rest of the paper is organized as follows, in Section \[sec2\] we present a construction of a robust estimator of $\Pi \lbrack \psi ;F]$ in the case of heavy-tailed losses. In Section \[sec3\] we establish its asymptotic normality. In Section [sec4]{} we carry out a simulation study to illustrate empirical performance and robustness of the estimator. Concluding notes are given in Section [sec5]{}. Proofs are gathered in Section \[sec6\].
Throughout the paper, we use the standard notation $\overset{P}{ \rightarrow }$ for the convergence in probability and $\mathcal{N}\left( \mu
,\sigma \right) $ to denote a normal rv with mean $\mu $ and variance $ \sigma .$
**Defining the estimator\[sec2\]**
==================================
By using the generalized inverse $F^{\leftarrow }$ and for a fixed distortion function $\psi ,$ we may rewrite (\[eq1\]) into $$\Pi _{\psi }[X]:=-\int_{0}^{1}\psi \left( s\right) dF^{\leftarrow }\left(
1-s\right) . \label{DRMQ}$$ The empirical estimator of the risk premium $\Pi _{\psi }[X]$ is obtained by substituting $F^{\leftarrow }$ on the right-hand side of equation (\[DRMQ\] ) by its empirical counterpart $F_{n}^{\leftarrow }{\left( s\right) :=\inf
\{x\in \mathbb{R}:}$ ${F}_{n}{\left( x\right) \geq s\},}$ $0<s\leq 1,$ associated to the empirical cdf defined on the real line, defined by $ {\normalsize F_{n}\left( x\right) :=n}^{-1}\#\left\{ X_{i}\leq x,1\leq i\leq
n\right\} $ where $\#A$ denote the cardinality of a set $A.$ After straightforward computations, we obtain the formula $$\Pi _{n}[X]:=\int_{0}^{1}F_{n}^{\leftarrow }\left( 1-s\right) d\psi \left(
s\right)$$ which may be rewritten, in terms of $X_{1:n},...,X_{n:n},$ as an $L$ -statistic $$\Pi _{n}[X]=\sum_{i=1}^{n}c_{i,n}\left( \psi \right) X_{n-i+1:n},
\label{psin}$$ where $$c_{i,n}\left( \psi \right) \equiv \psi \left( i/n\right) -\psi \left( \left(
i-1\right) /n\right) . \label{ai}$$ The form (\[psin\]) is a linear combinations of the order statistics [see, @SW86 page 260]. The limit behavior was discussed by many authors: [@Chall67], [@Stigler74], [@Mason81], [@JoZi03] (see its Theorem 3.2 in the case that $X$ is not heavy-tailed) and in [BrMeNeRi]{} (in heavy-tailed case).
**Heavy-tailed losses case**
----------------------------
Let $X$ be a non-negative rv with cdf $F\in \mathcal{RV}_{\left(
-1/\gamma \right) }.$ The condition (\[first-condition\]) is equivalent to $$\lim_{t\rightarrow 0}\frac{F^{\leftarrow }\left( 1-tx\right) }{F^{\leftarrow
}\left( 1-t\right) }=x^{-\gamma },\text{ for every }x>0.
\label{First-Cond-Qua}$$ We say that the function $s\rightarrow F^{\leftarrow }\left( 1-s\right) $ satisfying condition (\[First-Cond-Qua\]) is regularly varying at zero with the index $(-\gamma )<0.$ The parameter $\gamma $ is called the tail index or extreme value index (EVI). A various tail index estimators have been suggested in the literature, based for instance of the conventional maximum likelihood method, moment estimation, ... (see, e.g. [@Hill75] , [@Pick75], [@Dekeretal89]. [@Csorgoetal85] and [@Drees95]). For the robustness and bias reduction (see, i.e. [@PenQui04] and [@StFaS12]). The regular-variation condition itself is not sufficient for establishing asymptotic distributions. To this end, we suppose that cdf $F$ satisfy the well-known by the second-order condition of regular variation with second-order parameter $\tau \leq 0$, that is: there exists a function $t\rightarrow a(t)$ with constant sign at infinity and converges to $0$ as $t\rightarrow \infty $ such that $$\underset{t\rightarrow \infty }{\lim }\dfrac{\overline{F}\left( tx\right) / \overline{F}\left( t\right) -x^{-1/\gamma }}{a\left( t\right) }=x^{-1/\gamma
}\dfrac{x^{\tau /\gamma }-1}{\gamma \tau }, \label{second-order}$$ for every $x>0.$ When $\tau =0,$ then the ratio $\dfrac{x^{\tau /\gamma }-1}{ \gamma \tau }$ should be interpreted as $\log x.$ In terms of the quantile function $F^{\leftarrow },$ condition (\[second-order\]) is equivalent to the following one $$\lim_{t\rightarrow 0}\frac{\dfrac{F^{\leftarrow }\left( 1-tx\right) }{ F^{\leftarrow }\left( 1-t\right) }-x^{-\gamma }}{A\left( t\right) } =x^{-\gamma }\frac{x^{\tau }-1}{\tau }, \label{second-cond}$$ for every $x>0,$ where $A\left( t\right) :=\gamma ^{2}a\left( F^{\leftarrow
}\left( 1-t\right) \right) ,$ (see @deHS96, [-@deHS96] or Theorem 3.2.9 in @deHF06, [-@deHF06 page 48]). The Weissman estimator [@Weis78] of high quantiles $F^{\leftarrow }$ is given by $$F_{n}^{\leftarrow \left( W\right) }(1-s):=(k/n)^{\widehat{\gamma } }X_{n-k:n}s^{-\widehat{\gamma }},\text{ }s\downarrow 0. \label{wies}$$ The formula (\[DRMQ\]) can be split into $$\Pi _{\psi }[X]=-\int_{0}^{k/n}\psi \left( s\right) dF^{\leftarrow
}(1-s)-\int_{k/n}^{1}\psi \left( s\right) dF^{\leftarrow }(1-s). \label{pi}$$ By using an integration by part to the second integral yields $$\begin{aligned}
\Pi _{\psi } &=&\psi \left( k/n\right) F^{\leftarrow
}(1-k/n)-\int_{0}^{k/n}\psi \left( s\right) dF^{\leftarrow
}(1-s)+\int_{k/n}^{1}F^{\leftarrow }(1-s)d\psi \left( s\right) \\
&:&=\Pi _{\psi }^{\left( 1\right) }+\Pi _{\psi }^{\left( 2\right) }+\Pi
_{\psi }^{\left( 3\right) }.\end{aligned}$$A simple estimator of $\Pi _{\psi }^{\left( 1\right) }$ is $$\Pi _{\psi ,n}^{\left( 1\right) }:=\psi \left( k/n\right) X_{n-k,n}.
\label{pi1}$$To estimate $\Pi _{\psi }^{\left( 2\right) }$, we note that $\widehat{\gamma
}$ is a consistent estimator for $\gamma $ [@StFaS12] and since $\rho <1/\gamma $, by substituting $F_{n}^{\leftarrow \left( W\right) }(1-s)$ given in (\[wies\]) instead of $F^{\leftarrow }(1-s)$ and integrating yield the following estimator$$\Pi _{\psi ,n}^{\left( 2\right) }:=\widehat{\gamma }\left( k/n\right) ^{ \widehat{\gamma }}X_{n-k:n}\int_{0}^{k/n}s^{-\widehat{\gamma }-1}\psi \left(
s\right) ds. \label{pi2}$$Finally, by plugging $F_{n}^{\leftarrow }$ instead of $F^{\leftarrow }$ on second integral of Equation (\[pi\]) we obtain the estimator $$\Pi _{\psi ,n}^{\left( 3\right) }:=\sum_{i=k+1}^{n}c_{i,n}\left( \psi
\right) X_{n-i+1:n}, \label{pi3}$$of $\Pi _{\psi }^{\left( 3\right) },$ where $F_{n}^{\leftarrow }{\normalsize \left( s\right) :=\inf \left\{ x\in \mathbb{R}:F_{n}\left( x\right) \geq
s\right\} ,\;0<s\leq 1,}$ denote the sample quantile function associated to the empirical cdf defined on the real line by ${\normalsize F_{n}\left(
x\right) :=n}^{-1}\sum\nolimits_{i=1}^{n}\mathbb{I}\left( X_{i}\leq x\right)
,$[ ]{}with $\mathbb{I}\left( \cdot \right) $ being theindicator function and the coefficients $c_{i,n}\left( \psi \right) $ are given in (\[ai\]). The final form of our estimator$$\widetilde{\Pi }_{\psi ,n}:=X_{n-k,n}\left( \psi \left( k/n\right) +\widehat{ \gamma }\left( k/n\right) ^{\widehat{\gamma }}\int_{0}^{k/n}s^{-\widehat{ \gamma }-1}\psi \left( s\right) ds\right) +\sum_{i=k+1}^{n}c_{i,n}\left(
\psi \right) X_{n-i+1:n}.$$To establish the asymptotic normality of our estimator and compared with the estimator proposed by @NeMe09 given in Equation (\[NeMe\]), we use the same function $\psi \left( t\right) =t^{1/\rho },$ for $\rho \geq 1$ used in [@NeMe09]. In this case our estimator have the following form$$\widetilde{\Pi }_{n}\left( \widehat{\gamma },k\right) :=\left( k/n\right)
^{1/\rho }\frac{X_{n-k,n}}{1-\widehat{\gamma }\rho }+\sum_{i=k+1}^{n}\left(
\left( i/n\right) ^{1/\rho }-\left( \left( i-1\right) /n\right) ^{1/\rho
}\right) X_{n-i+1:n}. \label{Br}$$
**Asymptotic distribution\[sec3\]**
===================================
We will begin to expose our results as asymptotic representations theorems in the lines of [@BeScSt14]. For that purpose, we need to describe the probability theory on which they hold. Indeed, we use the so-called Hungarian construction of [@CsCsHM86]. For this we define $\left\{ U_{n}(s),0\leq s\leq 1\right\} $, the uniform empirical distribution function and we consider the order statistics $U_{1:n}\leq ...\leq U_{n:n}$ pertaining to the independent standard uniform rv’s $U_{1},U_{2},...,$ we introduce the uniform empirical quantile function $\left\{ V_{n}(s),0\leq
s\leq 1\right\} $ based on the $n\geq 1$ first observations of $U_{1},U_{2},...,$ on $(0,1)$ such that$$V_{n}\left( s\right) =U_{i,n}\quad \text{for}\ \left( i-1\right) /n<s\leq
i/n,\text{ }i=1,...,n,\text{ and }V_{n}\left( 0\right) =U_{1,n}.$$Let $\beta _{n}\left( s\right) =\sqrt{n}\left( s-V_{n}\left( s\right)
\right) ,0\leq s\leq 1$, be the corresponding quantile empirical process, for $n\geq 1.$
We use the well-known by Gaussian approximation given in [CsCsHM86]{} Corollary 2.1. It says that: on the probability space $\left(
\Omega ,\mathcal{A},\mathbb{P}\right) ,$ there exists a sequence of Brownian bridges $\left\{ \mathbb{B}_{n}\left( s\right) ;\text{ }0\leq s\leq
1\right\} $ has the representation $$\left\{ \mathbb{B}_{n}\left( s\right) ;\text{ }0\leq s\leq 1\right\} \overset{D}{=}\left\{ W_{n}\left( s\right) -sW_{n}\left( 1\right) ;\text{ }0\leq
s\leq 1\right\} ,$$where $W_{n}$ is a standard Wiener process such that for every $0\leq \zeta
<1/2,$$$\sup_{1/n\leq s\leq 1-1/n}\frac{n^{\zeta }\left\vert \beta _{n}\left(
s\right) -\mathbb{B}_{n}\left( s\right) \right\vert }{\left( s\left(
1-s\right) \right) ^{1/2-\zeta }}=O_{\mathbb{P}}\left( n^{-\zeta }\right) .
\label{approxi}$$Our main result is the following
\[Theorem2\]Let $F$ be a df satisfying (\[second-cond\]) with $\gamma
>1/2$ and suppose that $F^{\leftarrow }\left( \cdot \right) $ is continuously differentiable on $[0,1).$ Let $k=k_{n}$ satisfying (\[K\]) such that $\sqrt{k}A\left( n/k\right) \rightarrow 0$ as $n\rightarrow \infty
.$ For any $1\leq \rho <1/\gamma ,$ we have $$\frac{\sqrt{n}\left( \widetilde{\Pi }_{\psi ,n}-\Pi _{\psi }\right) }{ (k/n)^{1/\rho -1/2}F^{\leftarrow }(1-k/n)}\overset{d}{\rightarrow }\mathcal{N }\left( 0,\sigma ^{2}\left( \gamma ,\rho \right) \right) ,\text{ as } n\rightarrow \infty ,$$ where $$\begin{aligned}
\sigma ^{2}\left( \gamma ,\rho \right) &=&\gamma ^{2}+\frac{\gamma ^{2}\rho
\left( \rho -2\rho \gamma ^{2}+2\gamma \right) }{\left( \gamma \rho
-1\right) ^{2}}+\frac{2\gamma ^{2}}{\left( \rho +\gamma \rho -1\right)
\left( \rho +2\gamma \rho -2\right) } \\
&&+\frac{2\gamma }{2\gamma -1}-\frac{2\gamma \rho \left( \rho \gamma
^{2}-\rho \gamma +1\right) }{\left( \gamma \rho -1\right) \left( \rho
+\gamma \rho -1\right) }.
\end{aligned}$$
**Simulation study**\[sec4\]
============================
**Performance and comparative study of** $\widetilde{\Pi } _{n}$ **and** $\widehat{\Pi }_{n}$
------------------------------------------------------------------------------------------------
In this simulation study we examine the performance of our estimator $\widetilde{\Pi }_{n}\left( \widehat{\gamma },k\right) $ given in (\[Br\]) and compare it with that of $\widehat{\Pi }_{n}\left( \widehat{ \gamma }^{H},k\right) $ given in (\[NeMe\]). Thus we follow the steps below.
**Step 1:** We generate $1000$ pseudorandom samples of size $n=100,200,500$ and $1000$ from Pareto cdf with $\gamma =0.6.$
**Step 2:** We estimate the tail index parameter by Hill and t-Hill estimators $\widehat{\gamma }^{H}(k_{1}^{\ast })$ and $\widehat{ \gamma }(k_{2}^{\ast }),$ respectively given in (\[Hill\]) and ([t-Hill]{}). We adopt the Reiss and Thomas algorithm (see @ReTo07, [-@ReTo07 page 137]), for choosing the optimal numbers of upper extremes $k_{1}$ and $k_{2}$. By this methodology, we define the optimal sample fraction of upper order statistics $k_{j}^{\ast }$ by$$k_{j}^{\ast }:=\arg \min_{k}\frac{1}{k}\sum_{i=1}^{k}i^{\theta }\left\vert
\widehat{\gamma }_{j}\left( i\right) -\text{median}\left\{ \widehat{\gamma }_{j}\left( 1\right) ,...,\widehat{\gamma }_{j}\left( k\right) \right\}
\right\vert ,j=1,2$$where $\widehat{\gamma }_{1}=\widehat{\gamma }^{H}$ and $\widehat{\gamma }_{2}=\widehat{\gamma }.$ On the light of our simulation study,** **we obtained reasonable results by choosing $\theta =0.3.$
**Step 3:** We fix the distortion parameter with respect to Condition (\[condition gamabeta\]) by $\rho =1.12,$ then we compute the bias and RMSE of the four estimators $\widehat{\gamma }^{H}(k_{1}^{\ast }),$ $\widehat{\gamma }(k_{2}^{\ast }),$ $\widetilde{\Pi }_{n}\left( \widehat{ \gamma },k_{1}^{\ast }\right) $ and $\widehat{\Pi} _{n}\left( \widehat{ \gamma }^{H},k_{2}^{\ast }\right) $. The results are summarized in Table [A1]{}. We see that when dealing with large samples our estimator performs better.
----------------- ----------------- -------------------- ------------------- --------------------------- ------------------- ---------------- -------------------- --------------------------- ------------------- -------------------
$k_{1}^{\ast }$ $k_{2}^{\ast }$
${\small n}$ bias RMSE bias RMSE bias RMSE bias RMSE
${\small 100}$ ${\small 10}$ ${\small -0.0733}$ ${\small 0.2511}$ $ {\small 0.3618}$ ${\small 0.5199}$ ${\small 17}$ ${\small -0.1641}$ $ {\small 0.2865}$ ${\small 0.4096}$ ${\small 0.7332}$
${\small 200}$ ${\small 23}$ ${\small -0.0571}$ ${\small 0.1821}$ $ {\small 0.3562}$ ${\small 0.5147}$ ${\small 34}$ ${\small -0.0993}$ $ {\small 0.2350}$ ${\small 0.3918}$ ${\small 0.7185}$
${\small 500}$ ${\small 62}$ ${\small -0.0299}$ ${\small 0.1142}$ $ {\small 0.3404}$ ${\small 0.4820}$ ${\small 86}$ ${\small -0.0301}$ $ {\small 0.0739}$ ${\small 0.3639}$ ${\small 0.6936}$
${\small 1000}$ ${\small 129}$ ${\small -0.0147}$ ${\small 0.0798}$ $ {\small 0.1966}$ ${\small 0.2687}$ ${\small 169}$ ${\small -0.0181}$ ${\small 0.0545}$ ${\small 0.2827}$ ${\small 0.5279}$
----------------- ----------------- -------------------- ------------------- --------------------------- ------------------- ---------------- -------------------- --------------------------- ------------------- -------------------
: $\widehat{\gamma }(k_{1}^{\ast }),$ $\widehat{\gamma } ^{H}(k_{2}^{\ast })$ $\widetilde{\Pi }_{n}\left( \widehat{\gamma},k_{1}^{\ast }\right) $ and $\widehat{\Pi}
_{n}\left( \widehat{\gamma},k_{2}^{\ast }\right)$ estimators based on 1000 samples of Pareto-distributed claim amounts with tail index 0.6 and distortion parameter $\rho=1.12$. The exact value of the premium is 2.0487.[]{data-label="A1"}
**Comparative robustness study**
--------------------------------
In this subsection we study the sensitivity to outliers of $\widetilde{\Pi }_{n}\left( \widehat{\gamma },k_{1}^{\ast }\right) $ and compare it with that of $\widehat{\Pi}_{n}\left( \widehat{\gamma }^{H},k_{2}^{\ast}\right).$ We consider an $\epsilon $-contaminated model known by mixture of Pareto distributions$$F_{\gamma _{1},\gamma _{2},\epsilon }\left( x\right) =1-\left( 1-\epsilon
\right) x^{-1/\gamma _{1}}+\epsilon x^{-1/\gamma _{2}},\text{ }
\label{mixture}$$where $\gamma _{1},\gamma _{2}>0$ and $0<\epsilon <0.5$ is the fraction of contamination. Note that for $\epsilon =0,$ $\widehat{\gamma }^{H}$ and $\widehat{\gamma}$ are asymptotically unbiased. Therefore, for $\epsilon >0,$ the effect of contamination becomes immediately apparent. If $\gamma_{1}<\gamma _{2}$ and $\epsilon >0,$ (\[mixture\]) corresponds to a Pareto distribution contaminated by a longer tailed distribution. For the implementation of mixtures models to the outliers study one refers, for instance, to [@BaLe95 page 43]. In this context, we proceed our study as follows.
First, we consider $\gamma _{1}=0.6,$ $\gamma _{2}=2$ to have the contaminated model and let $\rho =1.12.$ Then we consider four contamination scenarios according to $\epsilon=5\%,$ $10\%,$ $15\%,$ $25\%.$
For each value $\epsilon ,$ we generate $1000$ samples of size $n=100,$ $200$ and $1000$ from the model (\[mixture\]). Finally, we compare the $\widetilde{\Pi}_{n}\left(\widehat{\gamma},k_{1}^{\ast }\right)$ and $\widehat{\Pi}_{n}\left(\widehat{\gamma}^{H},k_{2}^{\ast}\right)$ estimators with this true value, by computing for each estimator, the appropriate bias and RMSE and summarize the results in Table \[A2\].
----------------- -------------------- ------------------- ------------------- -------------------------------- ---------------------------
${\small n}$ $\%$ contamination bias RMSE bias RMSE
${\small 100}$ ${\small 5}$ ${\small 0.4043}$ ${\small 0.6664}$ $ {\small -0.4286}$ ${\small 1.4727}$
${\small 10}$ ${\small 0.4389}$ ${\small 0.6862}$ ${\small -0.7291}$ ${\small 1.9123}$
${\small 15}$ ${\small 0.4598}$ ${\small 0.7464}$ ${\small -1.2786}$ ${\small 2.1247}$
${\small 25}$ ${\small 1.0578}$ ${\small 1.1305}$ ${\small -1.2103}$ ${\small 2.1828}$
$200$ ${\small 5}$ ${\small 0.3831}$ ${\small 0.5532}$ ${\small -0.4713}$ ${\small 1.5118}$
${\small 10}$ ${\small 0.3964}$ ${\small 0.5675}$ ${\small -1.2496}$ ${\small 2.1907}$
${\small 15}$ ${\small 0.4508}$ ${\small 0.6870}$ $-1.4355$ $ {\small 2.3107}$
${\small 25}$ ${\small 0.9470}$ ${\small 1.0197}$ ${\small -1.7366}$ ${\small 2.3274}$
${\small 1000}$ ${\small 5}$ ${\small 0.2124}$ ${\small 0.3211}$ $ {\small -0.3794}$ ${\small 2.1222}$
${\small 10}$ ${\small 0.2329}$ ${\small 0.3349}$ ${\small -1.0662}$ ${\small 2.3978}$
${\small 15}$ ${\small 0.2931}$ ${\small 0.3749}$ ${\small -1.2501}$ ${\small 2.0355}$
${\small 25}$ ${\small 0.8124}$ ${\small 0.9291}$ ${\small -1.5238}$ ${\small 2.3596}$
----------------- -------------------- ------------------- ------------------- -------------------------------- ---------------------------
: $\widetilde{\Pi }_{n}\left( \widehat{\gamma},k_{1}^{\ast }\right)
$ and $\widehat{\Pi}_{n}\left( \widehat{\gamma }^{H},k_{2}^{\ast}\right)$ are based on 1000 samples of mixture of Pareto distributions with tail index $0.6,$ $\epsilon=5\%, 10\%, 15\%, 25\%$ and distortion parameter $\rho=1.12$. The exact value of the premium is 2.0487.[]{data-label="A2"}
As expected, the estimator $\widehat{\Pi }_{n}\left( \widehat{ \gamma }^{H},k_{2}^{\ast }\right) $ as well as $\widetilde{\Pi }_{n}\left(
\widehat{\gamma },k_{1}^{\ast }\right) $ turn out to be more sensitive to this type of contaminations. For example, in $0\%$ contamination for $n=200$, the couple (bias, RMSE) for $\widehat{\Pi }_{n}\left( \widehat{\gamma }^{H},k_{2}^{\ast }\right) $ take the values $\left( 0.3918,0.7185\right) $, while for $15\%$ contamination the bias and the RMSE are given by the couple $\left( -1.4355,2.3107\right) $. We may conclude that the bias and RMSE of $\widehat{\Pi }_{n}\left( \widehat{\gamma }^{H},k_{2}^{\ast }\right) $ estimator are more sensitive (or note robust) to outliers. However, for $0\%$ contamination the (bias, RMSE) of $\widetilde{\Pi }_{n}\left( \widehat{ \gamma },k_{1}^{\ast }\right) $ is $\left( 0.3562,0.5147\right) ,$ while for $15\%$ contamination is $\left( 0.4508,0.6870\right) .$ Both the bias and the RMSE of $\widetilde{\Pi }_{n}\left( \widehat{\gamma },k_{1}^{\ast
}\right) $ estimation are note sensitive to outliers. Then we may conclude that is the better estimator.
**Concluding notes\[sec5\]**
============================
We showed that the new estimator of premium based on t-Hill estimator is more robust and performs better than the one based on Hill estimator proposed by [@NeMe09]. Our estimator $\widetilde{\Pi }_{n}\left( \widehat{\gamma },k\right) $ is based on Weissman’s estimation of high quantiles, so we would lead to improve our result to use one of several bias-reduced estimators have been proposed (see for example [@MaBe03]).
Proofs\[sec6\]
==============
To establish the asymptotic normality of $\widetilde{\Pi }_{\psi
,n}$ we need the asymptotic approximation of $\widehat{\gamma }$ with the same sequence of Brownian bridges as $\widetilde{\Pi }_{\psi ,n},$ for this reason we give the following results.
\[Cor\]Assume that the second order condition (\[second-cond\]) holds with $\gamma >1/2$ and let $k=k_{n}$ be an integer sequence satisfying ( [K]{}) and $\sqrt{k}A\left( n/k\right) \rightarrow 0.$ Then, there exists a sequence of Brownian bridges $\left\{ \mathbb{B}_{n}\left( s\right) ,\text{ } 0\leq s\leq 1\right\} $ such that $$\sqrt{k}\left( \widehat{\gamma }-\gamma \right) =\gamma \left( \gamma
+1\right) ^{2}\int_{0}^{1}s^{\gamma -1}\mathbb{B}_{n}\left( s\right)
ds+o_{p}\left( 1\right) ,$$ leading to $$\sqrt{k}\left( \widehat{\gamma }-\gamma \right) \overset{d}{\rightarrow } \mathcal{N}\left( 0,\frac{\gamma ^{2}\left( 1+\gamma \right) ^{2}}{\left(
1+2\gamma \right) }\right) ,\text{ as }n\rightarrow \infty ,$$
Our proofs are conducted in the probability space described in Section [sec3]{}. Then we are entitled to write $$S_{k}:=\frac{1}{k}\sum_{j=1}^{k}\frac{X_{n-k:n}}{X_{n-j+1:n}}$$ then $\widehat{\gamma }=S_{k}^{-1}-1$. The asymptotic normality of $\widehat{ \gamma }$ is established in [@BeScSt14] by given their Wiener process representation. Here we suppose that $\sqrt{k}A\left( n/k\right) \rightarrow
\lambda =0.$ So $$\sqrt{k}\left( S_{k}-\frac{1}{\gamma +1}\right) =\frac{\gamma }{\gamma +1} W_{n}\left( 1\right) -\gamma \int_{0}^{1}t^{\gamma -1}W_{n}\left( t\right)
dt+o_{p}\left( 1\right) .$$ Note that $$\mathbb{B}_{n}\left( s\right) \overset{d}{=}W_{n}\left( s\right)
-sW_{n}\left( 1\right) ,$$ it follows that $$\sqrt{k}\left( S_{k}-\frac{1}{\gamma +1}\right) =-\gamma
\int_{0}^{1}s^{\gamma -1}\mathbb{B}_{n}\left( s\right) dt+o_{p}\left(
1\right) .$$ Using the map $g\left( x\right) =1/x-1,$ since $g\left( 1/\left( \gamma
+1\right) \right) =\gamma $ and applying the delta method yields $$\sqrt{k}\left( \widehat{\gamma }-\gamma \right) =\left( \gamma +1\right)
^{2}\gamma \int_{0}^{1}s^{\gamma -1}\mathbb{B}_{n}\left( s\right)
ds+o_{p}\left( 1\right) .$$ It is clear that $\sqrt{k}\left( \widehat{\gamma }-\gamma \right) $ is a Gaussian rv with mean $0$ and variance $\frac{\gamma ^{2}\left( 1+\gamma
\right) ^{2}}{\left( 1+2\gamma \right) }.$ This completes the proof of Proposition \[Cor\].
**Proof of Theorem \[Theorem2\]**
---------------------------------
Making use of Proposition \[Cor\] and from [@NeMeMe07] we showe that under the assumptions of Theorem \[Theorem2\], there exists a sequence of Brownian bridges $\left\{ \mathbb{B}_{n}\left( s\right) ,0\leq
s\leq 1\right\} $ such that, for all large $n$$$\frac{\left( \Pi _{\psi ,n}^{\left( 1\right) }-\Pi _{\psi }^{\left( 1\right)
}\right) }{(k/n)^{1/\rho }F^{\leftarrow }\left( 1-k/n\right) }=-\gamma
\left( n/k\right) ^{1/2}\mathbb{B}_{n}\left( 1-k/n\right) +o_{p}\left(
1\right) .$$Let $\mathbb{U}$ be the left-continuous inverse of $1/(1-F).$ Note that $\mathbb{U}(t)$ is defined for $t>1.$ Let $Y_{1},Y_{2}....$ be independent and identically distributed rv’s with cdf $1-1/y,y>1,$ and let $Y_{1,n}\leq
Y_{2,n}\leq ...\leq Y_{n,n}$ be the associated order statistics. Then, for $\rho \geq 1,$ we may rewrite the statistic $\Pi _{\psi ,n}^{\left( 2\right)
} $ as$$\Pi _{\psi ,n}^{\left( 2\right) }=(k/n)^{1/\rho }\frac{\widehat{\gamma }\rho
}{1-\widehat{\gamma }\rho }X_{n-k:n}.$$ Then$$\Pi _{\psi ,n}^{\left( 2\right) }=(k/n)^{1/\rho }\frac{\widehat{\gamma }\rho
}{1-\widehat{\gamma }\rho }\mathbb{U}\left( Y_{n-k:n}\right) ,$$where $\widehat{\gamma }$ is the t-Hill estimator of $\gamma .$ So we have $$\frac{\sqrt{k}\left( \Pi _{\psi ,n}^{\left( 2\right) }-\Pi _{\psi }^{\left(
2\right) }\right) }{(k/n)^{1/\rho }\mathbb{U}\left( n/k\right) }:=\sum_{j=1}^{4}\Delta _{jn},$$where$$\begin{aligned}
\Delta _{1n}:= &&\sqrt{k}\frac{\widehat{\gamma }\rho }{1-\widehat{\gamma } \rho }\left( \frac{\mathbb{U}\left( Y_{n-k:n}\right) }{\mathbb{U}\left(
n/k\right) }-\left( \frac{Y_{n-k:n}}{n/k}\right) ^{\gamma }\right) , \\
\Delta _{2n}:= &&\sqrt{k}\frac{\widehat{\gamma }\rho }{1-\widehat{\gamma } \rho }\left( \left( \frac{Y_{n-k:n}}{n/k}\right) ^{\gamma }-1\right) , \\
\Delta _{3n}:= &&\sqrt{k}\left( \frac{\widehat{\gamma }\rho }{1-\widehat{ \gamma }\rho }-\frac{\gamma \rho }{1-\gamma \rho }\right)\end{aligned}$$and$$\Delta _{4n}:=\sqrt{k}(k/n)^{-1/\rho }\left( \frac{\frac{(k/n)^{1/\rho }\rho
}{\left( 1/\gamma -\rho \right) }\mathbb{U}\left( n/k\right) -\Pi _{\psi
}^{\left( 2\right) }}{\mathbb{U}\left( n/k\right) }\right) .$$As showed in [@NeMeMe07], we have : $\Delta _{1n}\rightarrow 0$ and $\Delta _{4n}\rightarrow 0$ as $n\rightarrow \infty .$
Next, we show that $\Delta _{2n}+\Delta _{3n}$ is asymptotically normal. Assume, without loss of generality, that the rv’s $(Y_{n})_{n\geq 1}$are defined on a probability space $(\Omega ,\mathcal{A},\mathbb{P})$ which carries the sequence $(U_{n})_{n\geq 1}$ in such a way that $Y_{n}=\left(
1-U_{n}\right) ^{-1}$ for $n=1,2,...$ and $Y_{i,n}=\left( 1-U_{i,n}\right)
^{-1},$ $i=1,...,n,$.$\ $Then, this allows us to write $Y_{n-i+1,n}=\left(
1-V_{n}\left( 1-s\right) \right) ^{-1},$ for $\dfrac{i-1}{n}<s\leq \dfrac{i}{ n},$ $i=1,...,n$. From [@NeMeMe07] we have For $\Delta _{2n}$ $$\Delta _{2n}=-\left( n/k\right) ^{1/2}\frac{\rho \gamma ^{2}}{1-\gamma \rho }\left( 1+o_{p}\left( 1\right) \right) \mathbb{B}_{n}\left( 1-k/n\right) .$$For $\Delta _{3n}$ and by using the map $h\left( \theta \right) =\rho
/\left( \frac{1}{\theta }-\rho \right) \ $and applying the delta method yields: $$\Delta _{3n}=\frac{\rho }{\left( \rho \gamma -1\right) ^{2}}\sqrt{k}\left(
\gamma -\widehat{\gamma }\right) .$$From Corollary \[Cor\] we get$$\Delta _{3n}=\frac{\gamma \rho \left( \gamma +1\right) ^{2}}{\left( \rho
\gamma -1\right) ^{2}}\int_{0}^{1}s^{\gamma -1}\mathbb{B}_{n}\left( s\right)
ds+o_{p}\left( 1\right) .$$Finally we have$$\frac{\sqrt{k}\left( \Pi _{\psi ,n}^{\left( 2\right) }-\Pi _{\psi }^{\left(
2\right) }\right) }{(k/n)^{1/\rho }F^{\leftarrow }\left( 1-k/n\right) }=\frac{\gamma \rho \left( \gamma +1\right) ^{2}}{\left( \rho \gamma -1\right)
^{2}}\int_{0}^{1}s^{\gamma -1}\mathbb{B}_{n}\left( s\right) ds-\left(
n/k\right) ^{1/2}\frac{\rho \gamma ^{2}}{1-\gamma \rho }\mathbb{B}_{n}\left(
1-k/n\right) +o_{p}\left( 1\right) .$$From [@NeMeMe07] we have$$\frac{\sqrt{k}\left( \Pi _{\psi ,n}^{\left( 3\right) }-\Pi _{\psi }^{\left(
3\right) }\right) }{(k/n)^{1/\rho }F^{\leftarrow }\left( 1-k/n\right) }=\frac{\int_{k/n}^{1}s^{1/\rho -1}\mathbb{B}_{n}\left( 1-s\right)
F^{\leftarrow \prime }\left( 1-s\right) ds}{\rho F^{\leftarrow }\left(
1-k/n\right) \left( k/n\right) ^{1/\rho -1/2}}+o_{p}\left( 1\right) .$$Then $$\frac{\sqrt{n}\left( \widetilde{\Pi }_{\psi ,n}-\Pi _{\psi }\right) }{ (k/n)^{1/\rho -1/2}F^{\leftarrow }(1-k/n)}=\Lambda \left( \gamma ,\rho
\right) +o_{p}\left( 1\right)$$where$$\Lambda \left( \gamma ,\rho \right) :=W_{n1}+W_{n2}+W_{n3}+o_{p}\left(
1\right)$$and$$\begin{aligned}
W_{n1} &:=&-\left( n/k\right) ^{1/2}\gamma \mathbb{B}_{n}\left( 1-k/n\right)
, \\
W_{n2} &:=&\frac{\gamma \rho \left( \gamma +1\right) ^{2}}{\left( \rho
\gamma -1\right) ^{2}}\int_{0}^{1}s^{\gamma -1}\mathbb{B}_{n}\left( s\right)
ds-\left( n/k\right) ^{1/2}\frac{\rho \gamma ^{2}}{1-\gamma \rho }\mathbb{B} _{n}\left( 1-k/n\right) , \\
W_{n3} &:=&\frac{\int_{k/n}^{1}s^{1/\rho -1}\mathbb{B}_{n}\left( 1-s\right)
F^{\leftarrow \prime }\left( 1-s\right) ds}{\rho F^{\leftarrow }\left(
1-k/n\right) \left( k/n\right) ^{1/\rho -1/2}}.\end{aligned}$$It is clear that $\Lambda \left( \gamma ,\rho \right) $ is a Gaussian rv with mean 0 and variance$$\begin{aligned}
E\left( \Lambda \left( \gamma ,\rho \right) \right) ^{2} &=&E\left(
W_{n1}^{2}\right) +E\left( W_{n2}^{2}\right) +E\left( W_{n3}^{2}\right)
+2E\left( W_{n1}W_{n2}\right) \\
&&+2E\left( W_{n1}W_{n3}\right) +2E\left( W_{n2}W_{n3}\right) .\end{aligned}$$An elementary calculation gives, we get $$\begin{aligned}
E\left( W_{n1}^{2}\right) &=&\gamma ^{2}+o\left( 1\right) , \\
E\left( W_{n2}^{2}\right) &=&\frac{\gamma ^{2}\rho ^{2}}{\left( 1-\gamma
\rho \right) ^{2}}+\frac{2\gamma }{2\gamma -1}+o\left( 1\right) , \\
E\left( W_{n3}^{2}\right) &=&\frac{2\gamma ^{2}}{\left( \rho +\gamma \rho
-1\right) \left( \rho +2\gamma \rho -2\right) }+o\left( 1\right) , \\
E\left( W_{n1}W_{n2}\right) &=&\frac{\rho \gamma ^{3}}{1-\gamma \rho } +o\left( 1\right) , \\
E\left( W_{n1}W_{n3}\right) &=&\frac{\gamma \rho }{\rho +\gamma \rho -1} +o\left( 1\right)\end{aligned}$$and$$E\left( W_{n2}W_{n3}\right) =-\frac{\gamma ^{3}\rho ^{2}}{\left( \gamma \rho
-1\right) \left( \rho +\gamma \rho -1\right) }+o\left( 1\right) .$$The proof of Theorem \[Theorem2\] is completed by combining all the preceding results.$\square $
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[^1]: [`brah.brahim@gmail.com`]{}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'An automorphism $\alpha$ of a group $G$ is normal if it fixes every normal subgroup of $G$ setwise. We give an algebraic description of normal automorphisms of relatively hyperbolic groups. In particular, we prove that for any relatively hyperbolic group $G$, $Inn(G)$ has finite index in the subgroup $Aut_n(G)$ of normal automorphisms. If, in addition, $G$ is non-elementary and has no non-trivial finite normal subgroups, then $Aut_n(G)=Inn(G)$. As an application, we show that $Out(G)$ is residually finite for every finitely generated residually finite group $G$ with more than one end.'
address:
- 'School of Mathematics, University of Southampton, Highfield, Southampton, SO17 1BJ, United Kingdom.'
- 'Department of Mathematics, Vanderbilt University, Nashville TN 37240, USA.'
author:
- 'A. Minasyan'
- 'D. Osin'
title: Normal automorphisms of relatively hyperbolic groups
---
[^1]
Introduction
============
Recall that an automorphism $\alpha \in Aut (G)$ of a group $G$ is said to be [*normal*]{} if $\alpha(N)=N$ for every normal subgroup $N$ of $G$. The subset of normal automorphisms, denoted by $Aut_n(G)$, is clearly a subgroup of $Aut (G)$. Obviously every inner automorphism is normal. Throughout this paper we denote by $Out_n(G)$ the quotient group $Aut_n(G)/Inn(G)$.
The study of normal automorphisms originates from the result of Lubotzky stating that $Out_n(G)$ is trivial for any non-abelian free group [@Lub]. Since then similar results have been proved for many other classes of groups. For example, $Out_n(G)=\{ 1\} $ for non-trivial free products [@Nesh], fundamental groups of closed surfaces of negative Euler characteristic [@BKZ], non-abelian free Burnside groups of large odd exponent [@Cher], non-abelian free solvable groups [@Rom], and free nilpotent group of class $c=2$ (for $c\ge 3$ this is not true) [@End]. On the other hand, every group can be realized as $Out (G)$ for a suitable simple group $G$ [@DGG]. Since every automorphism of a simple group is normal, every group appears as $Out_n(G)$ for some $G$. Furthermore, every countable group can be realized as $Out_n(G)$ for some finitely generated group $G$ [@CC; @Obr].
The main goal of this paper is to study normal automorphisms of relatively hyperbolic groups. The notion of a relatively hyperbolic group was originally suggested by Gromov [@Gro] and has been elaborated in many papers since then [@Bow; @DS; @F; @Hru; @RHG; @Yam]. The class of relatively hyperbolic groups includes (ordinary) hyperbolic groups, fundamental groups of finite-volume complete Riemannian manifolds of pinched negative curvature [@Bow; @F], groups acting freely on $\mathbb R^n$-trees [@Gui] (in particular, limit groups arising in the solutions of the Tarski problem [@KM; @Sel]), non-trivial free products and their small cancellation quotients [@RHG], groups acting geometrically on $CAT(0)$ spaces with isolated flats [@HK], and many other examples.
In this paper we neither assume relatively hyperbolic groups to be finitely generated nor the collection of peripheral subgroups to be finite. (The reader is referred to the next section for the precise definition.) However we do assume that all peripheral subgroups are proper to exclude the case of a group hyperbolic relative to itself. Further on, we will say that a group $G$ [*non-elementary*]{}, if it is not virtually cyclic.
In general $Out_n(G)$ is not necessarily trivial even for ordinary hyperbolic groups. Indeed, it is known (see [@Sah]) that certain finite groups $L$ possess non-inner automorphisms which map every element to its conjugate. One can therefore construct many hyperbolic groups $G$ with non-trivial $Out_n(G)$ by taking any hyperbolic group $H$ and considering the direct product $G=H\times L$. The first result of our paper shows that non-trivial finite normal subgroups are essentially the only sources of non-inner normal automorphisms.
More precisely, every relatively hyperbolic group $G$ contains a unique maximal finite normal subgroup (see Corollary \[KG\]). We denote it by $E(G)$. Further let $C(G)$ denote the centralizer of $E(G)$ in $G$.
\[main\] Suppose that $G$ is a non-elementary relatively hyperbolic group and $\alpha \in Aut_n(G)$. Then there exist an element $w\in G$ and a set map ${\varepsilon }\colon G\to E(G)$ such that ${\varepsilon }(C(G))=\{ 1\}$ and $\alpha (g)=wg{\varepsilon }(g)w^{-1}$ for every $g\in G$.
In fact, Theorem \[main\] is a particular case of a more general result about normal automorphisms of subgroups of relatively hyperbolic groups (see Theorem \[NormAutSubgr\]). The corollary below follows easily from Theorem \[main\] and the observation that $C(G)$ has finite index in $G$ being the centralizer of a finite normal subgroup.
\[cor1\] Suppose that $G$ is a relatively hyperbolic group. Then the following hold.
1. $Out_n(G)$ is finite.
2. If $G$ is non-cyclic and contains no non-trivial finite normal subgroups, then $Out_n(G)=\{ 1\} $.
This corollary generalizes the results about free groups [@Lub], free products [@Nesh], and surface groups [@BKZ] cited above. It also implies the result of Metaftsis and Sykiotis [@MS] stating that for every non-elementary finitely generated relatively hyperbolic group $G$, $Inn(G)$ has finite index in the group $Aut_c(G)$ of pointwise inner automorphisms of $G$. Recall that an automorphism of $G$ is [*pointwise inner*]{}, if it preserves conjugacy classes. Clearly $Aut_c(G)\le Aut_n(G)$. Thus finiteness of $Out_n(G)$ implies that of $Aut_c(G)/Inn(G)$. The converse is not true in general. For instance, if $G$ is free nilpotent of class at least $3$, we have $Aut_c(G) = Inn(G) $ while $\left| Out_n(G) \right| = \infty$ [@End].
It is also worth noting that our methods are quite different from those of [@MS]. Indeed we use the group-theoretic version of Dehn surgery introduced in [@GM1; @GM2; @CEP] and ‘component analysis’ developed in [@RHG; @CC], while Metaftsis and Sykiotis employed the Bestvina-Paulin approach [@Best; @Pau] based on ultralimits and group actions on $\mathbb R$-trees.
In order to prove Theorem \[main\], we introduce a new subclass of automorphisms of any given group, and investigate it in the case of relatively hyperbolic groups.
Let $G$ be a group. We say that an automorphism $\varphi \in Aut(G)$ is [*commensurating*]{} if for every $g\in G$ there exist $h \in G$ and $m,n \in {{\mathbb Z}}\setminus \{0\}$ such that $(\varphi (g))^n = h g^m h^{-1}$. In other words, $\varphi$ is commensurating if for each $g \in G$, $\varphi(g)$ is [*commensurable*]{} to $g$ in $G$ (see Definition \[def:commensurability\]).
It is clear that the subset $Aut_{comm}(G)$ of commensurating automorphisms of $G$ forms a subgroup of $Aut(G)$ and $Inn(G)\le Aut_c(G) \le Aut_{comm}(G)$.
In Section \[sec:comm-aut\] we study commensurating automorphisms of relatively hyperbolic groups and obtain a complete description of them:
\[cor:descr\_comm\_aut\] Let $G$ be a non-elementary relatively hyperbolic group and $\varphi \in Aut(G)$. The following conditions are equivalent:
1. $\varphi $ is commensurating;
2. there is a set map ${\varepsilon }: G \to E(G)$, whose restriction to $C(G)$ is a homomorphism, and an element $w \in G$ such that for every $g \in G$, $\varphi(g)=w \left( g {\varepsilon }(g) \right)w^{-1}$.
In particular, if $E(G)=\{ 1\}$, then every commensurating automorphism of $G$ is inner.
In Section \[sec:Dehn\_surgery\], using the algebraic version of Dehn filling, we show that each normal automorphism of a relatively hyperbolic group must be commensurating. After this, Theorem \[main\] follows quite quickly from the above description of commensurating automorphisms.
Our methods can also be used to prove residual finiteness of some outer automorphism groups. A well-known theorem of Baumslag states that the automorphism group of a finitely generated residually finite group is residually finite [@Bau]. In general, the analogous property does not hold for the group of outer automorphisms. Indeed, Bumagina and Wise showed that every finitely presented group is realized as $Out (G)$ for a suitable finitely generated residually finite group $G$ [@BW]. However we prove that Baumslag’s theorem does have an ‘outer’ analogue for groups with more than one end. We refer to [@Stall71] for the geometric definition of ends, and recall that the number of ends of a finitely generated group can be either $0$, $1$, $2$ or infinity.
\[infends\] Let $G$ be a finitely generated residually finite group with more than one end. Then $Out (G)$ is residually finite.
An infinite finitely generated group $G$ has two ends if and only if it is virtually cyclic; and $G$ has infinitely many ends if and only if it splits non-trivially as an amalgamated free product $A\ast _S B$ or an $HNN$-extension $A\ast_S$ over a finite group $S$ [@Stall71; @Stall68].
Note that the condition demanding residual finiteness of $G$ in Theorem \[infends\] cannot be removed. Indeed, if $H$ is any finitely generated group that has trivial center and is not residually finite, then the group $G=H * {{\mathbb Z}}$ has infinitely many ends and $H$ is embedded into $Out(G)$ ($H$ acts on itself by conjugation and trivially on ${{\mathbb Z}}$, which gives rise to an action of $H$ by outer automorphisms on the free product $H * {{\mathbb Z}}=G$). Thus $Out(G)$ is not residually finite.
The standard way of proving residual finiteness of $Out(G)$ is based on the following result of Grossman [@Gros]: if a group $G$ is finitely generated and conjugacy separable, then $Aut(G)/Aut_c(G)$ is residually finite. In particular, $Out (G)$ is residually finite whenever $G$ is finitely generated, conjugacy separable, and $Aut_c(G)=Inn(G)$. Recall that a group $G$ is said to be [*conjugacy separable*]{} if for any two non-conjugate elements $g,h\in G$ there exists a homomorphism $\varphi\colon G\to K$, where $K$ is finite, such that ${{\rm Lab }}(g)$ and $\varphi(h)$ are not conjugate in $K$.
This approach has been successfully used to prove residual finiteness of $Out(G)$, where $G$ is a free group of finite rank [@Gros], the fundamental group of a closed surface [@Gros; @AKT01], the fundamental group of a Seifert manifold with non-trivial boundary [@AKT03], etc. If $G$ is a finitely generated conjugacy separable non-elementary relatively hyperbolic group, the above mentioned result from [@MS] implies that every virtually torsion-free subgroup of $Out(G)$ is residually finite [@MS Theorem 1.1].
However there is no hope to use Grossman’s idea to prove Theorem \[infends\] since we only assume the group $G$ to be residually finite, which is much weaker than conjugacy separability. Indeed there are many examples of finitely generated residually finite groups that are not conjugacy separable (e.g., the group of unimodular matrices $GL(n,\mathbb{Z})$ for $n\ge 3$, see [@Rem]). To construct such an example with infinitely many ends, we can simply take $G=H\ast \mathbb Z$, where $H$ is finitely generated, residually finite, but not conjugacy separable. It is easy to show that $G$ will also be finitely generated, residually finite, but not conjugacy separable.
Our approach is different and is based on the following observation. Let $Aut_n^f(G)$ denote the group of automorphisms of $G$ that stabilize every normal subgroup of finite index (setwise). Then $Aut(G)/Aut_n^f(G)$ is residually finite for every finitely generated group $G$. The following result plays the crucial role in the proof of Theorem \[infends\]. It also seems to be of independent interest. Its proof essentially uses the fact that free products are hyperbolic relative to their free factors, which allows us to employ the techniques developed in the proof of Theorem \[main\].
\[fp\] Suppose that $G=A\ast B$, where $A,B$ are non-trivial residually finite groups. Then $Aut_n^f(G)=Inn(G)$.
[**Acknowledgments.**]{} We are grateful to A. Klyachko and V. Yedynak for useful discussions, and to the anonymous referee for his comments.
Preliminaries {#sec:prelim}
=============
[**Notation.**]{} Given a group $G$ generated by a subset $S\subseteq G$, we denote by ${\Gamma (G, S)}$ the Cayley graph of $G$ with respect to $S$ and by $|g|_S$ the word length of an element $g\in G$. If $p$ is a (combinatorial) path in ${\Gamma (G, S)}$, ${{\rm Lab }}(p)$ denotes its label, ${{\rm L}}(p)$ denotes its length, $p_-$ and $p_+$ denote its starting and ending vertex. The notation $p^{-1}$ will be used for the path in ${\Gamma (G, S)}$ obtained by traversing $p$ backwards. By saying that $o=p_1\dots p_k$ is a cycle in ${\Gamma (G, S)}$ we will mean that $o$ is obtained as a consecutive concatenation of paths $p_1,\dots p_k$ such that $(p_{i+1})_-=(p_i)_+$ for $i=1,\dots,k-1$ and $(p_k)_+=(p_1)_-$.
For a word $W$ written in the alphabet $S$, $\|W\|$ will denote its length. For two words $U$ and $V$ we shall write $U \equiv V$ to denote the letter-by-letter equality between them. The normal closure of a subset $K\subseteq G$ in a group $G$ (i.e., the minimal normal subgroup of $G$ containing $K$) is denoted by ${\left\langle\hspace{-.7mm}\left\langle }K{\right\rangle\hspace{-.7mm}\right\rangle }^G$, or simply by ${\left\langle\hspace{-.7mm}\left\langle }K{\right\rangle\hspace{-.7mm}\right\rangle }$ if omitting $G$ does not lead to a confusion. For any group elements $g$ and $t$, $g^t$ denotes $t^{-1}gt$. If $A\subseteq G$ then $A^t=\{a^t~|~a\in A\}$. For a subgroup $H\le G$, $N_G(H)$ denotes the normalizer of a $H$ in $G$. That is, $N_G(H)=\{g \in G~|~gHg^{-1}=H\}$. Similarly by $C_G(H)$ we denote the centralizer of $H$ in $G$, that is, $$C_G(H)=\{ g\in G~|~ gh=hg, \; \forall\, h\in H\} .$$ Finally for two subsets $A,B$ of $G$, their product is the subset $AB =\{ab~|~a \in A,b \in B\}$.
#### **Relatively hyperbolic groups.**
In this paper we use the notion of relative hyperbolicity which is sometimes called strong relative hyperbolicity and goes back to Gromov [@Gro]. There are many equivalent definitions of (strongly) relatively hyperbolic groups [@Bow; @DS; @F; @RHG]. We recall the isoperimetric characterization suggested in [@RHG], which is most suitable for our purposes. That relative hyperbolicity in the sense of [@Bow; @F; @Gro] implies relative hyperbolicity in the sense of Definition \[def:rel\_hyp\_gp\] stated below is essentially due to Rebbechi [@Reb]. (Indeed it was proved in [@Reb] under the additional technical condition that the groups under consideration are finitely presented.) In the full generality this implication and the converse one were proved in [@RHG].
Let $G$ be a group, ${\{ H_\lambda \} _{\lambda \in \Lambda } }$ – a collection of [*proper*]{} subgroups of $G$, $X$ – a subset of $G$. We say that $X$ is a [*relative generating set of $G$ with respect to ${\{ H_\lambda \} _{\lambda \in \Lambda } }$*]{} if $G$ is generated by $X$ together with the union of all $H_\lambda $. (In what follows we always assume $X$ to be symmetric.) In this situation the group $G$ can be regarded as a quotient group of the free product $$F=\left( \ast _{\lambda\in \Lambda } H_\lambda \right) \ast F(X),
\label{F}$$ where $F(X)$ is the free group with the basis $X$. If the kernel of the natural homomorphism $F\to G$ is the normal closure of a subset $\mathcal R$ in the group $F$, we say that $G$ has [*relative presentation*]{} $$\label{G}
\langle X,\; H_\lambda, \lambda\in \Lambda \; \mid \; \mathcal R
\rangle .$$ If $|X|<\infty $ and $|\mathcal R|<\infty $, the relative presentation (\[G\]) is said to be [*finite*]{} and the group $G$ is said to be [*finitely presented relative to the collection of subgroups ${\{ H_\lambda \} _{\lambda \in \Lambda } }$.*]{}
Set $$\label{H}
\mathcal H=\bigsqcup\limits_{\lambda\in \Lambda} (H_\lambda
\setminus \{ 1\} ) .$$ Given a word $W$ in the alphabet $X\cup \mathcal H$ such that $W$ represents $1$ in $G$, there exists an expression $$W\stackrel{F}{=} \prod\limits_{i=1}^k f_i^{-1}R_i^{\pm 1}f_i \label{prod}$$ with the equality in the group $F$, where $R_i\in \mathcal R$ and $f_i\in F $ for $i=1, \ldots , k$. The smallest possible number $k$ in a representation of the form (\[prod\]) is called the [*relative area*]{} of $W$ and is denoted by $Area^{rel}(W)$.
\[def:rel\_hyp\_gp\] A group $G$ is [*hyperbolic relative to a collection of proper subgroups*]{} ${\{ H_\lambda \} _{\lambda \in \Lambda } }$ if $G$ is finitely presented relative to ${\{ H_\lambda \} _{\lambda \in \Lambda } }$ and there is a constant $C>0$ such that for any word $W$ in $X\cup
\mathcal H$ representing the identity in $G$, we have $$\label{isop}
Area^{rel}
(W)\le C\| W\| .$$ The constant $C$ in (\[isop\]) is called an [*isoperimetric constant*]{} of the relative presentation (\[G\]) and ${\{ H_\lambda \} _{\lambda \in \Lambda } }$ is called the collection of [*peripheral (or parabolic) subgroups*]{} of $G$. In particular, $G$ is an ordinary (Gromov) [*hyperbolic group*]{} if $G$ is hyperbolic relative to the trivial subgroup. Later on by saying that a group $G$ is [*relatively hyperbolic*]{}, we will mean that there exists a collection of proper subgroups $\{H_\lambda \le G~|~\lambda \in \Lambda\}$ such that $G$ is hyperbolic relative to ${\{ H_\lambda \} _{\lambda \in \Lambda } }$.
This definition is independent of the choice of the finite generating set $X$ and the finite set $\mathcal R$ in (\[G\]) (see [@RHG]).
\[maln\] Let $G$ be a group hyperbolic relative to a collection of subgroups ${\{ H_\lambda \} _{\lambda \in \Lambda } }$. Then the following conditions hold.
1. For every $\lambda, \mu \in \Lambda $, $\lambda \ne \mu $, and every $g\in G$, we have $|H_\lambda \cap H_\mu^g |<\infty $.
2. For every $\lambda \in \Lambda $ and $g\in G\setminus H_\lambda $, we have $|H_\lambda \cap H_\lambda ^g|<\infty $.
#### **Components.**
Let $G$ be a group hyperbolic relative to a family of proper subgroups ${\{ H_\lambda \} _{\lambda \in \Lambda } }$. We recall some auxiliary terminology introduced in [@RHG], which plays an important role in our paper.
Let $q$ be a path in the Cayley graph ${\Gamma (G, X\cup \mathcal H)}$. A (non-trivial) subpath $p$ of $q$ is called an [*$H_\lambda $-component*]{} (or simply a [*component*]{}), if the label of $p$ is a word in the alphabet $H_\lambda\setminus \{ 1\} $ and $p$ is not contained in a longer subpath of $q$ with this property. Two $H_\lambda $-components $p_1, p_2$ of a path $q$ in ${\Gamma (G, X\cup \mathcal H)}$ are called [*connected*]{} if there exists a path $c$ in ${\Gamma (G, X\cup \mathcal H)}$ that connects some vertex of $p_1$ to some vertex of $p_2$, and ${{{\rm Lab }}(c)}$ is a word consisting of letters from $H_\lambda\setminus\{ 1\} $. In algebraic terms this means that all vertices of $p_1$ and $p_2$ belong to the same coset $gH_\lambda $ for a certain $g\in G$. Note that we can always assume that $c$ has length at most $1$, as every non-trivial element of $H_\lambda
\setminus\{ 1\} $ is included in the set of generators.
#### **Loxodromic elements and elementary subgroups.**
Recall that an element $g\in G$ is called [*parabolic*]{} if it is conjugate to an element of one of the subgroups $H_\lambda $, $\lambda \in \Lambda $. An element is said to be [*loxodromic*]{} if it is not parabolic and has infinite order. If $H$ is a subgroup of $G$, by $H^0 \subset H$ we will denote the set of all elements of $H$ that are loxodromic in $G$.
Recall also that a group is [*elementary*]{} if it contains a cyclic subgroup of finite index. The next result was obtained in [@ESBG]. The first part of the lemma is well known in the context of convergence groups [@Tuk]. In particular, it follows from [@Tuk] and [@Yam] in case $G$ is finitely generated. (The latter assumption is only essential for [@Yam].)
\[Eg\] Suppose a group $G$ is hyperbolic relative to a collection of subgroups ${\{ H_\lambda \} _{\lambda \in \Lambda } }$. Let $g$ be a loxodromic element of $G$. Then the following conditions hold.
1. There is a unique maximal elementary subgroup $E_G(g)\le G$ containing $g$.
2. $E_G(g)=\{ h\in G\mid \exists\, m\in \mathbb{N}~ \mbox{such that}~ h^{-1}g^mh=g^{\pm m}\} $.
3. The group $G$ is hyperbolic relative to the collection ${\{ H_\lambda \} _{\lambda \in \Lambda } }\cup \{ E_G(g)\} $.
For finitely generated relatively hyperbolic groups, a lemma similar to Lemma \[Eg\] (c) was also stated in [@Dah]. Namely it was claimed that if $G$ is a (finitely generated) relatively hyperbolic group and $Z$ is an infinite cyclic subgroup of $G$ such that $Z$ coincides with its normalizer, then $Z$ can be joined to the collection of peripheral subgroups of $G$ [@Dah Lemma 4.4]. We note that this is wrong even in case $G$ is an ordinary hyperbolic group.
The simplest counterexample is given by the group $$G=\langle z,c\mid c^3=1,\; zcz^{-1}=c^2\rangle$$ and the subgroup $Z=\langle z\rangle $. Obviously $G$ splits as $1\to C\to G\to \mathbb Z\to 1$, where $C=\langle c\rangle \cong \mathbb Z/3\mathbb Z$. In particular $G$ is hyperbolic (or, equivalently, hyperbolic relative to the trivial subgroup). It is straightforward to check that $Z$ coincides with its own normalizer in $G$. Indeed every element $g\in G$ has the form $z^kc^m$, where $k\in \mathbb Z$ and $m\in \{ 0,1,2\}$. If $m=1$, we have $$g^{-1}zg =(c^{-1}z^{-k})z(z^{k}c)=c^{-1}zc=c^{-1}(zcz^{-1})z=c^{-1}c^2z=cz\notin Z.$$ Similarly $g^{-1}zg\notin Z$ if $m=2$. On the other hand, $G$ is not hyperbolic relative to $Z$. Indeed $c^{-1}z^2c=z^2$ and hence $Z\cap c^{-1}Zc$ is infinite. This contradicts part (b) of Lemma \[maln\]. Similarly for every (finitely generated) group $H$, the free product $G\ast H$ is hyperbolic relative to $H$, and the subgroup $Z$ provides a counterexample. Note that $E_G(z)=E_{G\ast H}(z)=G$, so applying Lemma \[Eg\] (c) yields the correct result.
#### **Finite normal subgroups**
The following result was proved in [@AMO Lemma 3.3].
\[EH\] Let $H$ be a non-elementary subgroup of a relatively hyperbolic group $G$. Suppose that $H^0\ne \emptyset $. Then the subgroup $\displaystyle E_G(H)=\bigcap_{h \in
H^0} E_G(h)$ is the (unique) maximal finite subgroup of $G$ normalized by $H$.
\[KG\] Let $G$ be a relatively hyperbolic group. Then $G$ possesses a unique maximal finite normal subgroup $E(G)$.
If $G$ is finite then the statement is trivial. If $G$ contains an infinite normal cyclic subgroup $C$ of finite index, then denote by $K$ the union of all finite normal subgroups of $G$. It is easy to see that $K$ is a torsion normal subgroup of $G$ (because a product of two finite normal subgroups is itself a finite normal subgroup). Since $K \cap C=\{1\}$, $K$ injects into the finite quotient $G/C$, hence $K$ is finite.
Finally, if $G$ is non-elementary, then $G^0\ne \emptyset$ by [@ESBG Cor. 4.5] (if $G$ is finitely generated, this also follows from [@Tuk] and [@Yam]). It remains to apply Lemma \[EH\] to the case $G=H$.
Special elements in relatively hyperbolic groups
================================================
Let $G$ be a relatively hyperbolic group and let $H$ be a non-elementary subgroup of $G$ containing at least one loxodromic element.
We say that an element $h\in H$ is $H$-[*special*]{} if $h$ is loxodromic in $G$ and $E_G(h)=\langle h\rangle\times E_G(H)$. The set of all $H$-special elements will be denoted by $S_G(H)$.
Note that, by definition, any $g \in S_G(H)$ belongs to the centralizer ${C_H(E_G(H))}$. The result below was obtained in [@AMO Lemma 3.8].
\[SG\] If $G$ is a relatively hyperbolic group and $H \le G$ is a non-elementary subgroup such that $H^0\ne \emptyset$, then $S_G(H)$ is non-empty.
Special elements play a significant role in our approach to study automorphisms of relatively hyperbolic groups. They represent a useful tool that helps to deal with the technical problems which may arise when the group under consideration contains torsion. The main goal of this section is to prove the following important statement:
\[fi\] Suppose that $G$ is a relatively hyperbolic group and $H \le G$ is a non-elementary subgroup with $H^0\ne \emptyset$. Then $C_H(E_G(H))$ is generated by the set $S_G(H)$. In particular, $\langle S_G(H)\rangle $ has finite index in $H$.
Observe that the statement after ‘in particular’ follows from the fact that the centralizer of a finite subgroup of $G$, normalized by $H$, necessarily has finite index in $H$.
We begin with some auxiliary results. Let $G$ be a group hyperbolic relative to a family of proper subgroups ${\{ H_\lambda \} _{\lambda \in \Lambda } }$. If $G$ is infinite, it always contains a loxodromic element [@ESBG Corollary 4.5]. The next lemma provides us with a tool for constructing such elements. It was proved in [@ESBG Lemma 4.4].
\[ah\] Let $G$ be a group hyperbolic relative to a collection of subgroups ${\{ H_\lambda \} _{\lambda \in \Lambda } }$. For any $\lambda \in \Lambda $ and $a\in G\setminus H_\lambda $, there exists a finite subset $\mathcal F\subseteq H_\lambda $ such that if $h\in H_\lambda \setminus \mathcal F$, then $ah$ is loxodromic.
Suppose that $\Xi$ is a finite subset of $G$. Define ${\mathcal{W}}(\Xi)$ to be the set of all words $W$ over the alphabet $X \cup {\mathcal{H}}$ that have the following form: $$W \equiv x_0h_0x_1h_1 \dots x_l h_l x_{l+1},$$ where $ l \in {{\mathbb Z}}$, $l \ge -2$ (if $l=-2$ then $W$ is the empty word; if $l=-1$ then $W \equiv x_0$), $h_i$ and $x_i$ are considered as single letters and
- $x_i \in X \cup \{1\}$, $i=0,\dots,l+1$, and for each $i=0,\dots,l$, there exists $\lambda(i) \in \Lambda$ such that $h_i \in H_{\lambda(i)}$;
- if $\lambda(i)=\lambda(i+1)$ then $x_{i+1} \notin H_{\lambda(i)}$ for each $i=0,\dots,l-1$;
- $h_i \notin \Xi$, $i=0,\dots,l$.
The statement below was proved in [@CC Lemmas 6.3, 6.5].
\[lem:conseq-conn\] There is a finite subset $\Xi$ of $G$ such that the following holds. Suppose that $o=rqr'q'$ is a cycle in $\Gamma(G,X \cup{\mathcal{H}})$ with ${{\rm Lab }}(q),{{\rm Lab }}(q') \in {\mathcal{W}}(\Xi)$. Set $C=\max\{ {{\rm L}}(r),{{\rm L}}(r')\}$.
- If $l$ is the number of components of $q$, then at least $(l-6C)$ of components of $q$ are connected to components of $q'$; and two distinct components of $q$ cannot be connected to the same component of $q'$. Similarly for $q'$.
- For any $d\in {{\mathbb N}}$ there exists a constant $L=L(C,d) \in {{\mathbb N}}$ such that if ${{\rm L}}(q)\ge L$ then there are $d$ consecutive components $p_s,\dots,p_{s+d-1}$ of $q$ and $p'_{s'},\dots,p'_{s'+d-1}$ of $q'^{-1}$, so that $p_{s+i}$ is connected to $p'_{s'+i}$ for each $i=0,\dots,d-1$.
Proposition \[fi\] is an easy consequence of Lemma \[lem:spec-mod\] below. In the case when $G$ is an ordinary word hyperbolic group it was proved in [@paper3 Lemma 4.3].
\[lem:spec-mod\] Suppose that $g \in S_G(H)$ and $x \in C_H(E_G(H)) \setminus E_G(g)$. Then there exists $N_1 \in {{\mathbb N}}$ such that $g^n x \in S_G(H)$ for any $n \in {{\mathbb Z}}$ with $|n| \ge N_1$.
By part (3) of Lemma \[Eg\], $G$ is hyperbolic relative to the collection of subgroups ${\{ H_\lambda \} _{\lambda \in \Lambda } }\cup \{E_G(g)\}$. Denote $\mathcal{H}'=\left(\cup_{\lambda\in \Lambda} H_\lambda \cup E_G(g)\right)\setminus \{1\} \subset G$. After adding $x$ and $x^{-1}$ to the finite relative generating set of $G$, if necessary, we can assume that $x^{\pm 1} \in X$. Let $\mathcal F$ and $\Xi$ be the finite subsets of $G$ given by Lemmas \[ah\] and \[lem:conseq-conn\] respectively. Since $g$ has infinite order, there exists $N_1\in {{\mathbb N}}$ such that $g^n \notin \mathcal{F} \cup \Xi$ for any $n \in {{\mathbb Z}}$ with $|n| \ge N_1$.
Choose an arbitrary $n\in {{\mathbb Z}}$ such that $|n| \ge N_1$. By Lemma \[ah\], the element $g^nx=(xg^n)^x$ is loxodromic in $G$. Suppose that $y \in E_G(g^nx)$. By part (2) of Lemma \[Eg\], there are $m \in {{\mathbb N}}$ and $\epsilon \in \{-1,1\}$ such that $$\label{eq:m} y(g^nx)^my^{-1}=(g^nx)^{\epsilon m}.$$
Let $V$ be the letter from ${\mathcal{H}}'$ representing $g^n$ in $G$, let $W$ be the letter from $X$ representing $x$, and let $U$ be the shortest word over the alphabet $X \cup {\mathcal{H}}'$ representing $y$. Set $C=\|U\|$ and $d=1$. Now we apply Lemma \[lem:conseq-conn\].(b) to find the constant $L=L(C,d)$ from its claim. Evidently we can assume that the number $m$ from equation is larger than $L$.
Consider a cycle $o=rqr'q'$ in $\Gamma(G,X\cup{\mathcal{H}}')$ where ${{\rm Lab }}(r)\equiv U$, ${{\rm Lab }}(q) \equiv (VW)^m$, ${{\rm Lab }}(r')\equiv U^{-1}$, ${{\rm Lab }}(q') \equiv (VW)^{-\epsilon m}$. By construction, the cycle $o$ satisfies the assumptions of Lemma \[lem:conseq-conn\].(b), hence some components $p$ of $q$ and $p'$ of $q'^{-1}$ must be connected in $\Gamma(G,X\cup{\mathcal{H}}')$. That is, there is a path $s$ with $s_-=p_+$, $s_+=p'_+$ and $z={{\rm Lab }}(s)\in E_G(g)$ (see Figure \[pic:1\]). Note that ${{\rm Lab }}(p)
\equiv V$, ${{\rm Lab }}(p')\equiv V^{\epsilon}$.
Let $q_1$ be the subpath of $q$ starting at $r_+=q_-$ and ending at $p_+=s_-$; let $q_1'$ be the subpath of $q'$ starting at $s_+=p'_+$ and ending at $q'_+=r_-$. Considering the cycle $o_1=rq_1sq_1'$ in the case when $\epsilon=-1$ we get the following equality in $G$: $$(g^nx)^\xi y (g^{n}x)^\zeta= z^{-1} g^{-n} \in E_G(g^nx) \cap E_G(g)~\mbox{ for some } \xi,\zeta \in {{\mathbb Z}}.$$ Similarly, in the case when $\epsilon=1$, we get $$(g^nx)^\xi y (g^{n}x)^\zeta= g^n z^{-1} g^{-n} \in E_G(g^nx) \cap E_G(g)~\mbox{ for some } \xi,\zeta \in {{\mathbb Z}}.$$
Observe that by Lemma \[Eg\], the group $G$ is hyperbolic relatively to ${\{ H_\lambda \} _{\lambda \in \Lambda } }\cup \{E_G(g),E_G(g^nx)\}$, hence, by Lemma \[maln\], the intersection $E_G(g^nx) \cap E_G(g)$ is finite. Since $g$ is $H$-special, any finite subgroup of $E_G(g)$ is contained in $E_G(H)$. Therefore $E_G(g^nx) \cap E_G(g) \subset E_G(H)$. Thus, whatever $\epsilon \in \{-1,1\}$ is, we always have $(g^nx)^\xi y (g^{n}x)^\zeta= h \in E_G(H)$, implying that $y=(g^nx)^{-\xi-\zeta}h$ because $g,x \in C_H(E_G(H))$. By part (2) of Lemma \[Eg\], $\langle g^n x \rangle$ and $E_G(H)$ are both contained in $E_G(g^nx)$; consequently $E_G(g^nx)=\langle g^n x \rangle \times E_G(H)$.
By Lemma \[SG\] we can find an element $g \in S_G(H)$. Note that for any $x \in Z= E_G(H) \cap C_H(E_G(H))$, the element $gx$ is also $H$-special. Since $x=g^{-1}(gx)$, we have $Z \subset
\langle S_G(H) \rangle$. It is easy to see that $E_G(g)\cap
C_H(E_G(H)) = \langle g \rangle \times Z$, hence $E_G(g)\cap
C_H(E_G(H))\subset \langle S_G(H) \rangle$. Now, if $x \in
C_H(E_G(H))\setminus E_G(g)$, then by Lemma \[lem:spec-mod\], $g^nx \in S_G(H)$ for some $n \in {{\mathbb N}}$. Consequently, $x=g^{-n} (g^n
x) \in \langle S_G(H) \rangle$.
Technical lemmas
================
Our main goal here is to prove several auxiliary lemmas, which will be used in the next section to give an algebraic description of automorphisms preserving commensurability classes of elements in relatively hyperbolic groups. We begin with a definition.
\[def:commensurability\] Let $G$ be a group. Two elements $x,y \in G$, are said to be [*commensurable*]{} if there are $z \in G$, $m,n \in {{\mathbb Z}}\setminus \{0\}$ such that $y^n=zx^mz^{-1}$ in $G$. If the elements $x$ and $y$ are commensurable in $G$, we will write $x \stackrel{G}{\approx} y$; otherwise, we will write $x {\stackrel{G}{\not\approx}}y$.
\[rem:lox-comm\] Obviously any two elements of finite order are commensurable. Further, if $g$ and $h$ are commensurable elements of a relatively hyperbolic group $G$ and $g$ is loxodromic, then $h$ is loxodromic too. Indeed, evidently $h$ has infinite order. Suppose that $h$ is parabolic. Since $g {\stackrel{G}{\approx}}h$, there are $\lambda \in \Lambda$, $a \in G$ and $m \in {{\mathbb N}}$ such that $a^{-1} g^m a \in H_\lambda$. Since $g$ is loxodromic, $x=g^a \notin H_\lambda$ and the intersection $H^x \cap H$ contains an infinite order element $x^m$. The latter contradicts claim (2) of Lemma \[maln\].
Throughout the rest of this section, $G$ will denote a group hyperbolic relative to a collection of peripheral subgroups ${\{ H_\lambda \} _{\lambda \in \Lambda } }$, and $H \le G$ will denote a non-elementary subgroup with $H^0 \neq \emptyset$.
\[lem:non-comm\] Let $g \in G$ be a loxodromic element and $x\in G \setminus E_G(g)$. For any finite subset $Y$ of $G$ there is $N_2 \in {{\mathbb N}}$ such that $g^n x$ is loxodromic and is not commensurable with any $y \in Y$ whenever $|n| \ge N_2$.
In view of Lemma \[Eg\].(3), we can assume that $E_G(g)$ belongs to the family of peripheral subgroups ${\{ H_\lambda \} _{\lambda \in \Lambda } }$ of $G$ and each infinite order element $y \in Y$ is parabolic.
Now we can apply Lemma \[ah\], to find $N_2 \in {{\mathbb N}}$ such that for any $n \in {{\mathbb Z}}$ with $|n|\ge N_2$, the element $xg^n$ is loxodromic. Therefore, so is $h=g^nx=x^{-1}(xg^n)x$. Suppose that $h$ is commensurable with some $y \in Y$. Then $y$ must also be loxodromic (by Remark \[rem:lox-comm\]), which contradicts our assumption above.
\[lem:general\] Let $\{g_1,\dots,g_l\}$, $l\ge 2$, be a finite set of pairwise non-commensurable loxodromic elements in a relatively hyperbolic group $G$. For any finite subset $F \subset G$ there exists $N_3 \in {{\mathbb N}}$ such that for any permutation $\sigma$ of $\{1,2,\dots,l\}$ and arbitrary elements $h_i \in E_G(g_{\sigma(i)})$, $i=1,2,\dots,l$, of infinite order, the following hold.
- The element $g=g_1^{m_1}f_1g_2^{m_2}f_2\dots g_l^{m_l}f_l$ is loxodromic for any $f_i \in F$ and $m_i \in {{\mathbb Z}}$ with $|m_i|\ge N_3$, $i=1,2,\dots,l$.
- Suppose that $\left( g_1^{m_1}g_2^{m_2}\dots g_l^{m_l} \right)^\zeta$ is conjugate to $\left( h_1^{n_1}f_1 h_2^{n_2} f_2 \dots h_l^{n_l}f_l\right)^\eta$ in $G$, for some $f_i \in F$, $\zeta,\eta \in {{\mathbb N}}$, $m_i,n_i\in {{\mathbb Z}}$, $|m_i|\ge N_3$, $|n_i|\ge N_3$ for all $i=1,2,\dots,l$. Then $\zeta=\eta$, there is $k\in \{0,1,\dots,l-1\}$ such that $\sigma$ is a cyclic shift by $k$, that is $\sigma(i) \equiv i+k ~ ({\rm mod}~ l)$ for all $i \in \{1,2,\dots,l\}$, and $f_j \in E_G \left( g_{\sigma(j)} \right) E_G\left(g_{\sigma(j+1)} \right)$ when $j=1,2,\dots,l-1$, $f_l \in E_G\left( g_{\sigma(l)}\right) E_G \left( g_{\sigma(1)}\right)$.
By Lemma \[Eg\] and because $g_i {\stackrel{G}{\not\approx}}g_j$ when $i\neq j$, $G$ is hyperbolic relative to the extended collection of subgroups ${\{ H_\lambda \} _{\lambda \in \Lambda } }\cup \{E_G(g_i)\}_{i=1}^l$. Also, the finite relative generating set $X$ can be replaced by the bigger finite set $X'=X \cup F\cup F^{-1}$ retaining the relative hyperbolicity of $G$. Denote $\mathcal{H}'=\left(\cup_{\lambda\in \Lambda} H_\lambda \cup \cup_{i=1}^l E_G(g_i)\right)\setminus \{1\} \subset G$. Let $\Xi$ be the finite subset of $G$ given by Lemma \[lem:conseq-conn\].
Take any $i \in \{1,\dots,l\}$. By part (1) of Lemma \[Eg\], we have $|E_G(g_i): \langle g_i \rangle|<\infty$, hence any infinite order element $h \in E_G(g_i)$ belongs to the elementary subgroup $$E^+_G(g_i)=\{x \in G~|~\exists~m \in \mathbb{N}~\mbox{such that } x^{-1}g_i^mx=g_i^m\} \le E_G(g_i).$$ Clearly, the center of $E^+_G(G_i)$ has finite index in it, hence all finite order elements of $E_G^+(g_i)$ form the maximal torsion subgroup $T \lhd E_G^+(g_i)$. Let $\alpha: E_G^+(g_i) \to E_G^+(g_i)/T$ be the natural epimorphism. The image $\alpha(E_G^+(g_i))$ is infinite cyclic (because it is virtually cyclic and torsion-free), therefore there exists $K_i \in {{\mathbb N}}$ such that for every non-trivial element $y \in \alpha(E_G^+(g_i))$, one has $y^{n} \notin S_i$ whenever $|n|\ge K_i$, where $S_i=\alpha(E_G^+(g_i)\cap \Xi)$ is a finite subset of $\alpha(E_G^+(g_i))$. Set $N_3=\max\{K_i~|~i=1,\dots,l\}$. By construction, for every $i$ and each infinite order element $h \in E_G(g_i)$, we have $h^{n} \notin \Xi$ whenever $|n| \ge N_3$.
Choose any elements $f_i \in F$ and integers $m_i$ with $|m_i|\ge N_3$, $i=1,\dots,l$. Let $V_i$ and $W_i$ be the letters from ${\mathcal{H}}'$ and from $X'$ representing the elements $g_i^{m_i}$ and $f_i$, $i=1,\dots,l$, respectively.
Proving claim (i) by contradiction, suppose that the element $g$ is not loxodromic.
If $g$ has finite order $t \in {{\mathbb N}}$, then set $C=0$, $d=1$ and choose $L=L(C,d)$ according to Lemma \[lem:conseq-conn\].(b). In the Cayley graph $\Gamma(G,X'\cup{\mathcal{H}}')$ consider the cycle $o=rqr'q'$, where ${{\rm Lab }}(q) \equiv(V_1W_1V_2W_2\dots V_lW_l)^{Lt}$, and $r,r'$ and $q'$ are trivial paths consisting of single vertex $q_-=q_+=1$. Since ${{\rm L}}(q) \ge Lt \ge L$, it follows from Lemma \[lem:conseq-conn\].(b) that some component of $q$ must be connected to a component of $q'^{-1}$. But $q'^{-1}$ has no components at all. A contradiction.
Therefore $g$ must have infinite order and must be parabolic, i.e., $g=aha^{-1}$ for some $h \in {\mathcal{H}}'$ and $a\in G$. Let $C=|a|_{X'\cup {\mathcal{H}}'}$, $d=2$ and $L=L(C,d)$ be given by Lemma \[lem:conseq-conn\].(b). Since $h$ has infinite order (as a conjugate of $g$), there is $n \in {{\mathbb N}}$ such that $n \ge L$ and $h^n \notin \Xi$. Choose a shortest word $A$ over $X'\cup {\mathcal{H}}'$ representing $a$ in $G$, and let $U$ be the letter from ${\mathcal{H}}'$ corresponding to $h^n$. Consider a cycle $o=rqr'q'$ in $\Gamma(G,X'\cup{\mathcal{H}}')$ such that ${{\rm Lab }}(r)\equiv A$, $q_-=r_+$, ${{\rm Lab }}(q)\equiv (V_1W_1V_2W_2\dots V_lW_l)^{n}$, $r'_-=q_+$, ${{\rm Lab }}(r')\equiv A^{-1}$, $q'_-=r'_+$, ${{\rm Lab }}(q')\equiv U^{-1}$. Since ${{\rm L}}(r)={{\rm L}}(r')=C$, ${{\rm L}}(q)\ge n\ge L$, we can apply Lemma \[lem:conseq-conn\].(b) to $o$, claiming that two distinct components of $q$ must be connected to two distinct components of $q'^{-1}$. But $q'^{-1}$ has only one component by definition. This contradiction concludes the proof of claim (i).
To establish claim (ii), assume that $ b\left( g_1^{m_1}g_2^{m_2}\dots g_l^{m_l} \right)^\zeta b^{-1}=
\left( h_1^{n_1}f_1 h_2^{n_2} f_2 \dots h_l^{n_l}f_l\right)^\eta$ in $G$, for some infinite order elements $h_i \in E_G(g_{\sigma(i)})$, $b \in G$, $\zeta,\eta \in {{\mathbb N}}$, $m_i,n_i\in {{\mathbb Z}}$, $|m_i|\ge N_3$, $|n_i|\ge N_3$ for $i=1,2,\dots,l$. Then for every $\varkappa \in {{\mathbb N}}$ we have $$\label{eq:b-zeta} b\left( g_1^{m_1}g_2^{m_2}\dots g_l^{m_l} \right)^{\varkappa \zeta} b^{-1}=
\left( h_1^{n_1}f_1 h_2^{n_2} f_2 \dots h_l^{n_l}f_l\right)^{\varkappa \eta} .$$ Let $V_i$ and $W_i$ be as before. Choose a letter $U_i$ from ${\mathcal{H}}'$ corresponding to $h_{i}^{n_i}$, $i=1,\dots,l$, and a shortest word $B$ over $X'\cup {\mathcal{H}}'$ representing $b$ in $G$. Set $C=\|B\|$, $d=2l$ and let $L=L(C,d)\in {{\mathbb N}}$ be the constant given by Lemma \[lem:conseq-conn\].(b). Take $\varkappa \in {{\mathbb N}}$ so that $\varkappa\zeta l \ge L$ and $\varkappa l >6C$.
In the Cayley graph $\Gamma(G,X'\cup{\mathcal{H}}')$ equation gives rise to a cycle $o=rqr'q'$, in which ${{\rm Lab }}(r)\equiv B$, $q_-=r_+$, ${{\rm Lab }}(q)\equiv (V_1V_2\dots V_l)^{{\varkappa}\zeta}$, $r'_-=q_+$, ${{\rm Lab }}(r')\equiv B^{-1}$, $q'_-=r'_+$, ${{\rm Lab }}(q')\equiv \left( U_1 W_1 U_2 W_2\dots U_l W_l \right)^{-{\varkappa}\eta}$.
By construction, the paths $q$ and $q'$ have exactly $\varkappa \zeta l$ and $\varkappa\eta l$ components respectively. Suppose that $\zeta >\eta$. By Lemma \[lem:conseq-conn\].(a), at least $\varkappa\zeta l-6C > \varkappa l(\zeta-1)\ge \varkappa l\eta$ components of $q$ must be connected to components of $q'$, hence two distinct components of $q$ will have to be connected to the same component of $q'$, contradicting Lemma \[lem:conseq-conn\].(a). Hence $\zeta \le \eta$. A symmetric argument shows that $\eta \le \zeta$. Consequently $\zeta=\eta$.
Since ${{\rm L}}(q)=\varkappa \zeta l \ge {{\rm L}}$, we can apply Lemma \[lem:conseq-conn\].(b) to find $2l$ consecutive components of $q$ that are connected to $2l$ consecutive components of $q'^{-1}$. Therefore there are consecutive components $p_1,\dots ,p_{l+1}$ of $q$ and $p'_1,\dots, p'_{l+1}$ of $q'^{-1}$ such that $p_j$ is connected to $p'_j$ for each $j$, and ${{\rm Lab }}(p_i)\equiv V_i$ for $i=1,\dots,l$, ${{\rm Lab }}(p_{l+1})\equiv V_1$ (Figure \[pic:2\]). Therefore ${{\rm Lab }}(p_i') \in
E_G(g_i)$, $i=1,\dots,l$, ${{\rm Lab }}(p_{l+1}') \in E_G(g_1)$. From the form of ${{\rm Lab }}(q'^{-1})$ it follows that there is $k \in
\{0,1,\dots,l-1\}$ such that ${{\rm Lab }}(p_j')\equiv U_{j+k}$ for $j=1,\dots,l+1$ (indices at $U$ are taken modulo $l$). Thus $U_{j+k}= h_{j+k}^{n_{j+k}}\in E_G(g_j)$. On the other hand, $h_{j+k}^{n_{j+k}} \in E_G(g_{\sigma(j+k)})$ has infinite order. Hence the intersection $E_G(g_j) \cap E_G(g_{\sigma(j+k)})$ must be infinite, which yields (by Lemma \[maln\]) that $\sigma(j+k)=j$ for all $j$. Therefore $\sigma$ is a cyclic shift (by $l-k$) of $\{1,\dots,l\}$.
The subpath $w_i$ of $q'^{-1}$ between $(p'_i)_+$ and $(p'_{i+1})_-$ is labelled by $W_{\sigma^{-1}(i)}$. As we showed, the vertex $(p_i)_+=(p_{i+1})_-$ is connected to $(w_i)_-$ by a path $s_i$ with ${{\rm Lab }}(s_i) \in E_G(g_i)$, and to $(w_i)_+$ by a path $t_i$ with ${{\rm Lab }}(t_i) \in E_G(g_{i+1})$, $i =1,\dots,l$ (here we use the convention that $g_{l+1}=g_1$). Considering the cycle $t_i^{-1}s_i w_i$ we achieve the desired inclusion: $f_{\sigma^{-1}(i)}={{\rm Lab }}(w_i) \in E_G(g_i) E_G(g_{i+1})$, $i=1,\dots,l$. This concludes the proof.
\[lem:prod\_of\_three\] Suppose that $\varphi: H \to G$ is a homomorphism such that $\varphi(h) {\stackrel{G}{\approx}}h$ for all $h \in H^0$. Then for any $g_1,g_2,g_3 \in H^0$, satisfying $g_i \stackrel{G}{\not\approx} g_j$ for $i\neq j$, there exists $N_4 \in {{\mathbb N}}$ such that for arbitrary $n_1,n_2,n_3 \in {{\mathbb Z}}$, with $|n_i| \ge N_4$, $i=1,2,3$, and for $g =g_1^{n_1} g_2^{n_2} g_3^{n_3}$, one has $g \in H^0$ and $(\varphi(g))^\zeta= e g^\zeta e^{-1}$, for some $e \in G$ and $\zeta \in {{\mathbb N}}$.
According to the assumptions, there exist $x_i \in G$ and $\zeta_i,\eta_i \in {{\mathbb Z}}\setminus\{0\}$ such that $\left( \varphi(g_i) \right)^{\zeta_i}= x_i g_i^{\eta_i} x_i^{-1}$, $i=1,2,3$. Denote $h_i=x_i^{-1} \varphi(g_i) x_i$, $i=1,2,3$. Then $h^{\zeta_i}=g^{\eta_i}$ , hence $h_i \in E_G(g_i)$ (by part (2) of Lemma \[Eg\]) and $h_i$ has infinite order, $i=1,2,3$.
Set $f_1=x_1^{-1}x_2$, $f_2=x_2^{-1}x_3$ and $f_3=x_3^{-1}x_1$, and let $N_4 \in {{\mathbb N}}$ be the number $N_3$ from the claim of Lemma \[lem:general\] applied to the set of loxodromic elements $\{g_1,g_2,g_3\}$ and the finite set $F=\{f_1,f_2,f_3\}$. Take any $n_i \in {{\mathbb Z}}$ with $|n_i| \ge N_4$, $i=1,2,3$. By part (i) of Lemma \[lem:general\], $g =g_1^{n_1} g_2^{n_2} g_3^{n_3} \in H^0$. Hence there are $\zeta,\eta \in {{\mathbb Z}}\setminus \{0\}$ and $e \in G$ such that $e g^\zeta e^{-1}=(\varphi(g))^\eta $. Since $\varphi$ is a homomorphism, we get $$e(g_1^{n_1} g_2^{n_2} g_3^{n_3})^\zeta e^{-1}= (\varphi(g))^\eta =
(x_1h_1^{n_1}x_1^{-1} x_2 h_2^{n_2} x_2^{-1} x_3 h_3^{n_3} x_3^{-1})^\eta , ~\mbox{ hence }$$ $$\label{eq:three} (x_1^{-1}e)(g_1^{n_1} g_2^{n_2} g_3^{n_3})^\zeta (x_1^{-1}e)^{-1}=
(h_1^{n_1}f_1 h_2^{n_2} f_2h_3^{n_3} f_3)^\eta .$$
Without loss of generality we can assume that $\zeta >0$. Suppose that $\eta<0$. Then $(g_3^{-n_3} g_2^{-n_2} g_1^{-n_1})^\zeta$ is conjugate to $(h_1^{n_1}f_1 h_2^{n_2} f_2h_3^{n_3} f_3)^{-\eta} $ in $G$ and $-\eta>0$. Applying part (ii) of Lemma \[lem:general\] to this situation, we get a contradiction with the fact that the transposition $(1,3)$ is not a cyclic shift of $\{1,2,3\}$. Therefore, $\eta>0$ and we can apply part (ii) of Lemma \[lem:general\] to , achieving the required equality $\zeta=\eta$.
\[lem:first\_step\] Let $a,b \in G$ be non-commensurable loxodromic elements and let $y,z \in G$. There exists $N_5\in {{\mathbb N}}$ such that the following holds. Suppose that $a^{k'} y b^{l'} z\stackrel{G}{\approx} a^{k} b^{l}$ for some integers $k,l,k',l'$ with $|k|,|l|,|k'|,|l'|\ge N_5$. Then $y \in E_G(a) E_G(b)$ and $z \in E_G(b) E_G(a)$.
Choose $N_5 \in {{\mathbb N}}$ to be the number $N_3$ arising after an application of Lemma \[lem:general\] to $\{a,b\}$ and $F=\{y,z\}$. Choose any $k,l,k',l' \in {{\mathbb Z}}$ satisfying $|k|,|l|,|k'|,|l'|\ge N_5$.
Assume that there is $e \in G$, $\zeta \in {{\mathbb N}}$ and $\eta \in {{\mathbb Z}}\setminus \{0\}$ for which $e\left( a^{k} b^{l}\right)^\zeta e^{-1} = \left(a^{k'} y b^{l'} z\right)^\eta$. If $\eta>0$ then the statement immediately follows from part (ii) of Lemma \[lem:general\]. So, suppose that $\eta <0$. Then $-\eta >0$ and $\left( b^{-l}a^{-k}\right)^\zeta$ is conjugate to $\left(a^{k'} y b^{l'} z\right)^{-\eta}$ in $G$. Again, by part (ii) of Lemma \[lem:general\], $y \in E_G(a) E_G(b)$, $z \in E_G(b) E_G(a)$.
\[lem:spec-image\] Assume that $g \in S_G(H)$ and $\psi: H \to G$ is a homomorphism satisfying $\psi(g^n)=g^n z$ for some $n \in {{\mathbb N}}$ and $z \in E_G(H)$. Then there is $f \in E_G(H)$ such that $\psi(g)=gf$.
After replacing $n$ with $n'=n|E_G(H)|$, we can further assume that $z=1$, because $\psi(g^{n'})=g^{n'} z^{n'}=g^{n'}$.
Now, note that $\psi(g) g^n (\psi(g))^{-1}= \psi(g^n) =g^{n}$, hence $\psi(g) \in E_G(g)$ by part (2) of Lemma \[Eg\]. Since $g$ is $H$-special, there is $k \in {{\mathbb Z}}$ and $f \in E_G(H)$ such that $\psi(g)=g^k f$. Denote $l=|E_G(H)|$. Then $g^{ln}=\psi(g^{ln})=(g^k f)^{ln}=g^{lnk}f^{ln}=g^{lnk}$, implying that $k=1$, as required.
\[lem:lox\_non-comm\] Suppose that for an automorphism $\alpha \in Aut(H)$ there is $g \in H^0$ satisfying $g {\stackrel{G}{\not\approx}}\alpha(g)$. Then there exists an element $a \in H$ such that both $a$ and $\alpha(a)$ are loxodromic in $G$ and $a {\stackrel{G}{\not\approx}}\alpha(a)$.
If $\alpha(g) \in H^0$, there is nothing to prove. Thus, we can assume that $\alpha(g)$ is parabolic in $G$, i.e., there exists a peripheral subgroup $H_\lambda$ and elements $t\in G$, $h \in H_\lambda$ such that $\alpha(g) =h^t$. Denote $x=\alpha^{-1}(g) \in H$. If $x \in E_G(g)$, then $\langle g \rangle^x \cap \langle g \rangle$ is infinite (by Lemma \[Eg\].(b)), hence $\langle \alpha(g) \rangle^{\alpha(x)} \cap \langle \alpha(g) \rangle$ is infinite. Thus $H_\lambda^{(tgt^{-1})} \cap H_\lambda$ is infinite, which implies, by Lemma \[maln\], that $tgt^{-1} \in H_\lambda$, contradicting the loxodromicity of $g$.
Therefore $x \notin E_G(g)$. Since both $g$ and $\alpha(g)$ have infinite order and $y=tgt^{-1} \in G\setminus H_\lambda$, we can apply Lemmas \[lem:non-comm\] and \[ah\] to find $N \in {{\mathbb N}}$ such that for any integer $n \ge N$, the elements $g^n x$ and $h^n y$ are loxodromic in $G$. Note that $\alpha(g^n x)=(h^n y)^t$.
Suppose, first, that $$\label{eq:g^nx} g^n x {\stackrel{G}{\approx}}\alpha(g^n x)~\mbox{ for every } n \ge N.$$ By Lemma \[Eg\], $G$ is hyperbolic relative to ${\{ H_\lambda \} _{\lambda \in \Lambda } }\cup \{E_G(g)\}$. Without loss of generality, we can also assume that $x$ and $y$ belong to the finite relative generating set $X$ of $G$. Let $\Xi \subset G$ be the finite set from Lemma \[lem:conseq-conn\]. Evidently there is an integer $n \ge N$ such that $g^n,h^n \notin \Xi$. Our assumption implies that there is $b \in G$, $k,l \in {{\mathbb Z}}\setminus \{0\}$ such that $b (g^n x)^k b^{-1}=(h^n y)^l$. Choose a word $B$ in the alphabet $X \cup \mathcal{H}'$ representing $b$ in $G$, where $\mathcal{H}'=(\cup_{\lambda\in \Lambda} H_\lambda \cup E_G(g))\setminus\{1\}$, and let $W,Y \in X$, $U \in E_G(g)$, $V \in H_\lambda$ be the letters corresponding to $x,y,g^n,h^n$ respectively. Set $d=1$, $C=\|B\|$ and let $L=L(C,d)$ be the constant provided by part (b) of Lemma \[lem:conseq-conn\]. Without loss of generality we can assume that $|k|,|l| \ge L$.
Consider a cycle $o=rqr'q'$ in the Cayley graph $\Gamma(G,X\cup\mathcal{H}')$, where ${{\rm Lab }}(r) \equiv B$, $r_+=q_-$, ${{\rm Lab }}(q) \equiv (UW)^k$, $q_+=r'_-$, ${{\rm Lab }}(r') \equiv B^{-1}$, $q_-'=r'_+$ and ${{\rm Lab }}(q') \equiv (VY)^{-l}$. It is easy to see that $o$ satisfies all the conditions of Lemma \[lem:conseq-conn\], hence some component of $q$ must be connected to a component of $q'^{-1}$ in $\Gamma(G,X\cup\mathcal{H}')$. However, according to the construction, $q$ has only $E_G(g)$-components, and $q'^{-1}$ has only $H_\lambda$-components. Thus the assumption yields a contradiction. Hence, there exists $n \ge N$ such that for the element $a=g^nx$ we have $a \in H^0$, $\alpha(a) \in H^0$ and $a {\stackrel{G}{\not\approx}}\alpha(a)$.
Commensurating automorphisms of relatively hyperbolic groups {#sec:comm-aut}
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The purpose of this section is to study automorphisms of relatively hyperbolic groups preserving commensurability classes. Recall that $N_G(H)$ denotes the normalizer of a subgroup $H$ in a group $G$. Further, let $H$ be a non-elementary subgroup of a relatively hyperbolic group $G$ such that $H^0\ne \emptyset$. We denote by $\widehat H$ the product $H E_G(H)$. This is clearly a subgroup of $G$.
\[thm:comm-aut\] Let $G$ be a relatively hyperbolic group, let $H \le G$ be a non-elementary subgroup and let $\varphi \in Aut(H)$. Suppose that $H^0\ne \emptyset $ and $\varphi (h){\stackrel{G}{\approx}}h$ for every $h\in H^0$. Then there is a set map ${\varepsilon }: H \to E_G(H)$, whose restriction to ${C_H(E_G(H))}$ is a homomorphism, and an element $w \in N_G (\widehat H )$ such that for every $h \in H$, $\varphi(h)=w \left( h {\varepsilon }(h) \right)w^{-1}$.
Below is them main technical lemma of this section. It demonstrates how to construct the element $w$ and the restriction of the map ${\varepsilon }$ to ${C_H(E_G(H))}$ from the statement of Theorem \[thm:comm-aut\].
\[lem:comm-aut-centralizer\] Suppose that $G$ is a relatively hyperbolic group, $H \le G$ is a non-elementary subgroup and $\varphi \in Aut(H)$. Assume that $H^0\ne \emptyset $ and $\varphi (h){\stackrel{G}{\approx}}h$ for every $h\in H^0$. Then there is a homomorphism $\tilde {\varepsilon }: C_H \left( E_G(H) \right) \to E_G(H)$ and an element $w \in G$ such that for every $x \in C_H(E_G(H))$, $\varphi(x)=w \left( x \tilde {\varepsilon }(x) \right)w^{-1}$.
By Lemma \[SG\], $H$ contains an $H$-special element $g_1$. Since $H$ is non-elementary and ${C_H(E_G(H))}$ has finite index in it, ${C_H(E_G(H))}$ is also non-elementary. The subgroup $E_G(g_1)$ is elementary (by part (1) of Lemma \[Eg\]), thus there is an element $y \in {C_H(E_G(H))}\setminus E_G(g_1)$. By Lemma \[lem:non-comm\], there is $k_2 \in {{\mathbb N}}$ such that $g_2=g_1^{k_2}y \in {C_H(E_G(H))}$ is loxodromic and $g_2 {\stackrel{G}{\not\approx}}g_1$. Using the same lemma again we can find $k_3 \in {{\mathbb N}}$ such that $g_3=g_1^{k_3}y \in {C_H(E_G(H))}$ is loxodromic and $g_3 {\stackrel{G}{\not\approx}}g_i$, $i=1,2$. In particular, $E_G(g_2) \cap \langle g_3 \rangle = \{1\}$.
Choose $N_4 \in {{\mathbb N}}$ according to an application of Lemma \[lem:prod\_of\_three\] to $\varphi$, $g_1,g_2,g_3$, and let $n_3=N_4$. By Lemma \[lem:non-comm\], there is $n_2 \ge N_4$ such that $g_2^{n_2}g_3^{n_3} \in H^0$ is not commensurable with $g_1$ in $G$. Therefore $g_2^{n_2}g_3^{n_3}\in {C_H(E_G(H))}\setminus E_G(g_1)$, and by Lemma \[lem:spec-mod\] there is $N_1 \in {{\mathbb N}}$ such that the element $g_1^{n}g_2^{n_2}g_3^{n_3}$ is $H$-special for any $n\ge N_1$. Denote $n_1=\max\{N_1,N_4\}$ and apply Lemma \[lem:non-comm\] to find $m \in {{\mathbb N}}$ such that the elements $a=g_1^{n_1}g_2^{n_2}g_3^{n_3}$ and $b=g_1^{n_1+m}g_2^{n_2}g_3^{n_3}$ are not commensurable with each other in $G$. In view of Lemma \[lem:prod\_of\_three\] one can conclude that the elements $a,b \in {C_H(E_G(H))}$ are $H$-special and there exist $u,v \in G$, $\mu,\nu \in {{\mathbb N}}$ such that $\varphi(a^\mu)=u a^\mu u^{-1}$, $\varphi(b^\nu)=vb^\nu v^{-1}$.
Let $\chi: H \to G$ be the monomorphism, defined by $\chi(h)=u^{-1} \varphi(h) u$ for all $h \in H$. Then $\chi(a^\mu)=a^\mu$, $\chi(b^\nu)=(u^{-1}v)b^{\nu}(u^{-1}v)^{-1}$. Note that $\chi(h) {\stackrel{G}{\approx}}h$ for every $h \in H^0$. By part (i) of Lemma \[lem:general\], $a^{k\mu}b^{k\nu} \in H^0$ for every sufficiently large $k \in {{\mathbb N}}$. Therefore $$\label{eq:ab}
a^{k\mu} (u^{-1}v) b^{k\nu}(u^{-1}v)^{-1}=\chi(a^{k\mu}b^{k\nu}) {\stackrel{G}{\approx}}a^{k\mu} b^{k\nu}~\mbox{ for every sufficiently large } k \in {{\mathbb N}}.$$ Consequently, by Lemma \[lem:first\_step\], $u^{-1}v \in E_G(a)E_G(b)$, thus $u^{-1}v= a^s b^t f$ for some $s,t \in {{\mathbb Z}}$, $f \in E_G(H)$. Hence $\chi(b^\nu)=a^sb^{\nu} a^{-s}$ because $b \in {C_H(E_G(H))}$. Denote $w=ua^s \in G$ and let $\psi: H \to G$ be the monomorphism defined by the formula $\psi(h)=w^{-1} \varphi(h)w=a^{-s} \chi(h) a^s$ for all $h\in H$. By construction, we have $$\label{eq:psi} \psi(a^\mu)=a^\mu,~ \psi(b^\nu)=b^\nu ~\mbox{ and }~\psi(h) {\stackrel{G}{\approx}}h \mbox{ for each }h \in H^0.$$
Choose any element $g \in S_G(H)$. We will show that there is $f \in E_G(H)$ such that $\psi(g)=gf$.
If $g \in E_G(a)$ then there is $n \in {{\mathbb N}}$ such that $g^n \in \langle a^\mu \rangle$ because $|E_G(a):\langle a^\mu \rangle |<\infty$. Hence $\psi(g^n)=g^n$ and by Lemma \[lem:spec-image\], $\psi(g)=gf$ for some $f \in E_G(H)$.
Suppose, now, that $g \notin E_G(a)$. Since $g \in {C_H(E_G(H))}$ and $a$ is $H$-special, we can use Lemmas \[lem:spec-mod\] and \[lem:non-comm\] to find $l \in {{\mathbb N}}$ such that the element $d=a^{l\mu}g$ is $H$-special and is not commensurable with $a$ and $b$ in $G$. Arguing as in the beginning of the proof (using Lemmas \[lem:spec-mod\], \[lem:non-comm\] and \[lem:prod\_of\_three\]) we can find $m_1,m_2,m_3 \in {{\mathbb N}}$ such that $c=a^{m_1 \mu} b^{m_2 \nu} d^{m_3} \in S_G(H)$, $c {\stackrel{G}{\not\approx}}a$, $c {\stackrel{G}{\not\approx}}b$ and $\psi(c^\zeta)=e c^\zeta e^{-1}$ for some $\zeta \in {{\mathbb N}}$ and $e \in G$.
By part (i) of Lemma \[lem:general\], $a^{k\mu}c^{k\zeta} \in H^0$ for every sufficiently large $k \in {{\mathbb N}}$. Hence $a^{k\mu} e c^{k \zeta} e^{-1}=\psi\left( a^{k\mu} c^{k\zeta} \right) {\stackrel{G}{\approx}}a^{k\mu} c^{k\zeta}$ whenever $k$ is sufficiently large. Applying Lemma \[lem:first\_step\] we see that $e \in E_G(a)E_G(c)$. As before, this implies that $\psi(c^\zeta)= a^p c^\zeta a^{-p}$ for some $p \in {{\mathbb Z}}$.
Similarly, there is $q \in {{\mathbb Z}}$ such that $\psi(c^\zeta)= b^q c^\zeta b^{-q}$. Hence $(a^{-p}b^q) c^\zeta (a^{-p}b^q)^{-1}=c^\zeta$, yielding that $a^{-p}b^q \in E_G(c)$.
Suppose that $p \neq 0$ and $q\neq 0$. Then the element $a^{-p}b^q$ must have infinite order (otherwise we would have $a^{-p}b^q \in E_G(H)$ since $c$ is $H$-special, hence $b^q \in a^p E_G(H) \subset E_G(a)$ contradicting to $a {\stackrel{G}{\not\approx}}b$). This implies that $(a^{-p}b^q)^\alpha=c^\beta$ for some $\alpha\in {{\mathbb Z}}\setminus \{0\}$ and $\beta \in {{\mathbb N}}$. Recalling , we can apply Lemma \[lem:spec-image\] to find $f_1,f_2 \in E_G(H)$ such that $\psi(a)=af_1$ and $\psi(b)=bf_2$. Since $a,b \in {C_H(E_G(H))}$ we obtain $$\psi(c^\beta)=\psi \left((a^{-p}b^q)^\alpha\right)=\left(a^{-p}b^{q}\right)^\alpha f_3=c^\beta f_3~
\mbox{ for some } f_3 \in E_G(H).$$ Then for $\gamma=\beta\zeta |E_G(H)|$ we get $c^\gamma=\psi(c^\gamma)=a^p c^\gamma a^{-p}$, implying that $a^p \in E_G(c)$, which contradicts to $a {\stackrel{G}{\not\approx}}c$.
Therefore either $p=0$ or $q=0$, thus $\psi(c^\zeta)=c^\zeta$. By Lemma \[lem:spec-image\], there is $f_5 \in E_G(H)$ such that $\psi(c)=cf_5$. Since $c=a^{m_1 \mu} b^{m_2 \nu} d^{m_3}$, we can use to get $\psi(d^{m_3})=d^{m_3}f_5$. Applying Lemma \[lem:spec-image\] again, we find $f_6 \in E_G(H)$ such that $\psi(d)=df_6$. Finally, since $d=a^{l\mu}g$, in view of we achieve $\psi(g)=g f_6$, as needed.
To finish the proof, we observe that by Proposition \[fi\], ${C_H(E_G(H))}$ is generated by $S_G(H)$, therefore for each $x \in {C_H(E_G(H))}$ there is $\tilde {\varepsilon }(x) \in E_G(H)$ such that $\psi(x)=x \tilde {\varepsilon }(x)$. Since $\psi$ is a homomorphism, the map $\tilde {\varepsilon }:{C_H(E_G(H))}\to E_G(H)$ will be a homomorphism too. By construction, we have $\varphi(x)=w \psi(x) w^{-1}= w x \tilde {\varepsilon }(x) w^{-1}$.
Now we are ready to prove the main result of this section.
Let $w \in G$ and $\tilde {\varepsilon }:{C_H(E_G(H))}\to E_G(H)$ be as in the claim of Lemma \[lem:comm-aut-centralizer\]. Let $\psi:H \to G$ be the monomorphism that is defined according to the formula $\psi(h)=w^{-1} \varphi(h) w$ for all $h \in H$. Denote $l=|H:{C_H(E_G(H))}|$, $m=|E_G(H)|$ and $n=ml \in {{\mathbb N}}$.
Since ${C_H(E_G(H))}$ is a normal subgroup of $H$, for any $z \in H$ we have $z^l \in {C_H(E_G(H))}$ and $\psi(z^n)=z^n \tilde{\varepsilon }(z^l)^m=z^n$. Fix an arbitrary $h \in H$. For any $g \in H^0$ we see that $g^n,hg^nh^{-1} \in {C_H(E_G(H))}\cap H^0$, therefore $\psi(h) g^n \psi(h)^{-1}=\psi(hg^nh^{-1})=hg^nh^{-1}$, implying that $h^{-1}\psi(h) \in E_G(g)$. Thus, $h^{-1}\psi(h) \in \bigcap_{g\in H^0} E_G(g)=E_G(H)$. After defining ${\varepsilon }(h)=h^{-1}\psi(h)$ for each $h \in H$, one immediately sees that ${\varepsilon }:H \to E_G(H)$ is a map with the required properties. Obviously, the restriction of ${\varepsilon }$ to ${C_H(E_G(H))}$ coincides with $\tilde {\varepsilon }$.
It remains to prove that $w \in N_G(\widehat H)$. We will first show that $w \in N_G(E_G(H))$. Consider any element $f \in E_G(H)$. Since $\varphi$ is an automorphism of $H$, for any $g \in H^0$ there is $h \in H$ such that $\varphi(h)=g$. Then $h^n \in {C_H(E_G(H))}$ and $g^n=\varphi(h^n)=wh^nw^{-1}$ because ${\varepsilon }(h^n)=\tilde{\varepsilon }(h^l)^m=1$. Now we observe that $$wfw^{-1} g^n (wfw^{-1})^{-1}=w f h^n f^{-1} w^{-1}=wh^nw^{-1}=g^n.$$ Hence, $wfw^{-1} \in E_G(g)$ for every $g \in H^0$; consequently $w f w^{-1} \in E_G(H)$. The latter implies that $w E_G(H) w^{-1} \subseteq E_G(H)$ and since $E_G(H)$ is finite, we conclude that $w \in N_G(E_G(H))$.
Now, for any $h \in H$ we have $$whw^{-1}=w h {\varepsilon }(h)w^{-1} w {\varepsilon }(h)^{-1} w^{-1} = \varphi(h) \left( w {\varepsilon }(h) w^{-1} \right)^{-1} \in H E_G(H);$$ thus $w H w^{-1} \subseteq \widehat H$. Since $w^{-1} \varphi(h) w = h {\varepsilon }(h) \in H E_G(H)$ and $\varphi \in Aut(H)$, one gets $w^{-1} H w \subseteq \widehat H$. Therefore $w \widehat H w^{-1} \subseteq \widehat H w E_G(H) w^{-1}=\widehat H$, $w^{-1} \widehat H w \subseteq \widehat H w^{-1} E_G(H) w=\widehat H$, i.e., $w \in N_G(\widehat H)$.
We are now in a position to prove Corollary \[cor:descr\_comm\_aut\] mentioned in the Introduction. We establish it in a more general form:
\[cor:comm\_aut\_def\] Let $G$ be a non-elementary relatively hyperbolic group and $\varphi \in Aut(G)$. The following conditions are equivalent:
1. $\varphi $ is commensurating;
2. $\varphi (g){\stackrel{G}{\approx}}g$ for every loxodromic $g\in G$;
3. there is a set map ${\varepsilon }: G \to E(G)$, whose restriction to $C(G)$ is a homomorphism, and an element $w \in G$ such that for every $g \in G$, $\varphi(g)=w \left( g {\varepsilon }(g) \right)w^{-1}$.
In particular, if $E(G)=\{ 1\}$, then every commensurating automorphism of $G$ is inner.
\(a) implies (b) by definition, and (b) implies (c) by Theorem \[thm:comm-aut\]. It remains to show that (c) implies (a). Indeed, let $g$ be an arbitrary element of $G$, and let the automorphism $\varphi $ satisfy (c). If $g$ is of finite order, then so is $\varphi(g)$, and in this case evidently $\varphi (g){\stackrel{G}{\approx}}g$. Thus, we can suppose that $g$ has infinite order in $G$. By our assumptions, $\varphi (g)=w(g{\varepsilon }(g))w^{-1}$ for some $w\in G$ and ${\varepsilon }(g)\in E(G)$. Since $E(G)$ is finite and normal in $G$, $\langle g\rangle $ has finite index in the subgroup $\langle g\rangle E(G)$. Hence there exists a non-zero integer $k$ such that $(g{\varepsilon }(g))^k=g^l$ for some $l \in {{\mathbb Z}}$. And since the order of $g {\varepsilon }(g)=w^{-1} \varphi(g) w$ is infinite, we can conclude that $l\neq 0$. Therefore $\varphi(g)=wg{\varepsilon }(g)w^{-1}$ is commensurable with $g$ in $G$. Thus $\varphi $ in commensurating.
Recall that a result of Metaftsis and Sykiotis [@MS Lemma $2.2'$] states that for any relatively hyperbolic group $G$, one has $|Aut_c(G):Inn(G)|<\infty$, where $$Aut_c(G)=\{\alpha \in Aut(G)~|~\forall \, g\in G~\exists\, x=x(g) \in G~ \mbox{ such that } \alpha(g)=xgx^{-1}\}$$ is the group of all [*pointwise inner automorphisms*]{} of $G$. Theorem \[thm:comm-aut\] allows one to generalize their result to all non-elementary subgroups:
\[cor:conj\_aut\] Suppose that $H$ is a non-elementary subgroup of a relatively hyperbolic group $G$, with $H^0 \neq\emptyset$. Then $|Aut_c(H):Inn(H)|<\infty$. If, in addition, $E_G(H)=\{1\}$, then $Aut_c(H)=Inn(H)$.
By Theorem \[thm:comm-aut\], for any automorphism $\varphi \in Aut_c(H)$, there exist $w \in G$ and a map ${\varepsilon }:H \to E_G(H)$ such that $\varphi(h)=wh {\varepsilon }(h) w^{-1}$ for each $h \in H$. Take any element $h \in S_G(H)$. Then $h$ commutes with ${\varepsilon }(h) \in E_G(H)$, and, consequently, $(\varphi(h))^n=wh^nw^{-1}$ where $n=|E_G(H)|\in {{\mathbb N}}$.
Now, since $\varphi$ is a pointwise inner automorphism of $H$, there is $x \in H$ such that $\varphi(h)=xhx^{-1}$. Hence $x h^n x^{-1}=wh^nw^{-1}$, i.e., $w^{-1}x \in E_G(h)=\langle h \rangle \times E_G(H)$. Thus $w = f z$ for some $f \in H$ and $z \in E_G(H)$, and $w^{-1}x \in C_G(h)$ because $h$ is $H$-special. Consequently, we have $h=w^{-1}x h \left(w^{-1}x\right)^{-1}=h {\varepsilon }(h)$, which implies that ${\varepsilon }(h)=1$. Since the latter holds for any $h \in S_G(H)$, it follows from Proposition \[fi\] that ${\varepsilon }(C_H)=\{1\}$, where $C_H={C_H(E_G(H))}$.
Note that $|H:C_H|<\infty$, hence there are $h_1,\dots,h_l \in H$ such that $H=\bigsqcup_{i=1}^l C_H h_i$. For any $g \in H$ there are $a \in C_H$ and $i \in \{1,\dots, l\}$ such that $g=a h_i$. One has $$\begin{gathered}
\varphi(a) \varphi(h_i) = \varphi(g)=w g {\varepsilon }(g) w^{-1}=waw^{-1} w h_i {\varepsilon }(a h_i) w^{-1}= \\ \varphi(a)
\varphi(h_i) w ({\varepsilon }(h_i))^{-1}{\varepsilon }(a h_i) w^{-1}, \end{gathered}$$ hence ${\varepsilon }(g)={\varepsilon }(a h_i)={\varepsilon }(h_i)$, i.e., the map ${\varepsilon }$ is uniquely determined by the images of $h_1,\dots,h_l$. Thus, $\varphi(g)=f z (g {\varepsilon }(h_i)) z^{-1} f^{-1}$, implying that the automorphism $\varphi \in Aut_c(H)$, up to composition with an inner automorphism of $H$, is completely determined by the finite collection of elements $z,{\varepsilon }(h_1),\dots,{\varepsilon }(h_l) \in E_G(H)$, and since $E_G(H)$ is finite, we can conclude that $|Aut_c(H):Inn(H)|<\infty$.
Now, if $E_G(H)=\{1\}$ we obtain $w =f \in H$ and $\varphi(g)=wgw^{-1}$ for all $g \in H$, that is $\varphi \in Inn(H)$.
Group-theoretic Dehn surgery and normal automorphisms {#sec:Dehn_surgery}
=====================================================
In the context of relatively hyperbolic groups, the algebraic analogue of Dehn filling is defined as follows. Suppose that ${\{ H_\lambda \} _{\lambda \in \Lambda } }$ is a collection of (peripheral) subgroups of a group $G$. To each collection ${\mbox{\eufm N}}=\{ N_\lambda \} _{\lambda \in \Lambda }$, where $N_\lambda $ is a normal subgroup of $H_\lambda $, we associate the quotient-group $$G({\mbox{\eufm N}}) = G/{\left\langle\hspace{-.7mm}\left\langle }\mbox{$\bigcup_{\lambda \in \Lambda }$} N_\lambda
{\right\rangle\hspace{-.7mm}\right\rangle }^G .$$
\[def:periph\_fill\] Let $G$ and ${\{ H_\lambda \} _{\lambda \in \Lambda } }$ be as described above. We say that some assertion holds for [*most peripheral fillings of $G$*]{}, if there exists a finite subset $\mathcal F$ of non-trivial elements of $G$ such that the assertion holds for $G({\mbox{\eufm N}})$ for any collection ${\mbox{\eufm N}}=\{
N_\lambda \} _{\lambda \in \Lambda }$ of normal subgroups $N_\lambda
\lhd H_\lambda $ satisfying $N_\lambda \cap \mathcal F=\emptyset $ for all $\lambda \in \Lambda $.
The theorem below was proved in [@CEP]. In the particular case when $G$ is torsion-free, this theorem was independently proved in [@GM1; @GM2].
\[Fill\] Suppose that a group $G$ is hyperbolic relative to a collection of subgroups ${\{ H_\lambda \} _{\lambda \in \Lambda } }$. Then for most peripheral fillings of $G$, the following holds.
1. For each $\lambda\in \Lambda $, the natural map $H_\lambda\
/N_\lambda \to G({\mbox{\eufm N}})$ is injective.
2. The quotient-group $G({\mbox{\eufm N}})$ is hyperbolic relative to the collection $\{H_\lambda /N_\lambda \} _{\lambda \in \Lambda }$.
The following statement plays a key role in our paper.
\[conj\] Let $G$ be a relatively hyperbolic group, $H$ – a subgroup of $G$ and $\alpha \in Aut(H)$. Suppose that there exists a loxodromic element $g\in H$ such that $\alpha (g)$ is not conjugate to an element of $E_G(g)$ in $G$. Then $\alpha $ does not preserve some normal subgroup of $H$.
Suppose that $G$ is hyperbolic relatively to ${\{ H_\lambda \} _{\lambda \in \Lambda } }$. There are two cases to consider.
[**Case 1.**]{} Assume first that $\alpha (g) $ is loxodromic. Using Lemma \[Eg\] twice we obtain that $G$ is hyperbolic relatively to ${\{ H_\lambda \} _{\lambda \in \Lambda } }\cup \{ E_G(g), E_G(\alpha (g))\} $. Since $\langle g\rangle $ has finite index in $E_G(g)$, there is $m\ne 0$ such that $\langle
g^m\rangle $ (and each of its subgroups) is normal in $E_G(g)$. Let $\mathcal F$ be the finite set provided by Theorem \[Fill\] for the peripheral system ${\{ H_\lambda \} _{\lambda \in \Lambda } }\cup \{ E_G(g), E_G(\alpha(g))\} $. Taking $p$ to be a sufficiently large multiple of $m$, we can ensure the condition $\langle g^p\rangle \cap \mathcal F =\emptyset $. We now consider the filling of $G$ with respect to the collection of subgroups ${\mbox{\eufm N}}$ consisting of the trivial subgroups of $H_\lambda$’s, the trivial subgroup of $E_G(\alpha (g))$, and $\langle g^p\rangle\lhd E_G(g)$. By Theorem \[Fill\] elements $g$ and $\alpha (g)$ have orders $p$ and $\infty
$, respectively, in $Q=G/{\left\langle\hspace{-.7mm}\left\langle }g^p{\right\rangle\hspace{-.7mm}\right\rangle }^G$. Hence $\alpha $ does not induce an automorphism on the natural image of $H$ in $Q$, i.e., it does not preserve ${\left\langle\hspace{-.7mm}\left\langle }g^p{\right\rangle\hspace{-.7mm}\right\rangle }^G\cap H$.
[**Case 2.**]{} Now suppose that $\alpha (g) $ is parabolic, i.e., it is conjugate to an element of some peripheral subgroup $H_\lambda $. Again, by Lemma \[Eg\], $G$ is hyperbolic relatively to ${\{ H_\lambda \} _{\lambda \in \Lambda } }\cup \{
E_G(g)\} $. The rest of the proof is identical to that in Case 1. The only difference is that Theorem \[Fill\] is applied to the collection of subgroups ${\mbox{\eufm N}}$ consisting of trivial subgroups of $H_\lambda$’s and $\langle g^p\rangle\lhd E_G(g)$ for some $p>0$.
Theorem \[main\] is a particular case of the following result. (Recall that $\widehat H=H E_G(H)$.)
\[NormAutSubgr\] Let $G$ be a relatively hyperbolic group and let $H \le G$ be a non-elementary subgroup such that $H^0\ne \emptyset $. Then for any $\varphi\in Aut_n(H)$ there exists a map ${\varepsilon }: H \to E_G(H)$, whose restriction to ${C_H(E_G(H))}$ is trivial, and an element $w \in N_G (\widehat H
)$ such that for every $h \in H$, $\varphi(h)=w h {\varepsilon }(h)
w^{-1}$.
By Lemma \[conj\], $\varphi $ maps every loxodromic element $h\in H$ to a conjugate of an element of $E_G(h)$. As $\langle h\rangle $ has finite index in $E_G(h)$, every element of infinite order in $E_G(h)$ is commensurable with $h$ in $G$. In particular, $\varphi (h){\stackrel{G}{\approx}}h$ for every $h\in H^0$. Hence by Theorem \[thm:comm-aut\] there is a map ${\varepsilon }: H \to E_G(H)$, whose restriction to ${C_H(E_G(H))}$ is a homomorphism, and an element $w \in N_G (\widehat H )$ such that for every $h \in H$, $\varphi(h)=w h {\varepsilon }(h) w^{-1}$. It remains to show that ${\varepsilon }(h)=1$ for every $h\in {C_H(E_G(H))}$.
By Proposition \[fi\], it suffices to show that ${\varepsilon }(h)=1$ for all $h\in S_G(H)$. Suppose that $\varphi (h)=whrw^{-1}$ for some $r\in E_G(H)\setminus \{ 1\}$. Take any integer $p \equiv 1~ ({\rm mod }\; |r|)$, where $|r|$ denotes the (finite) order of $r$ in $G$. Note that $h$ commutes with $r$ as $h\in S_G(H)$. Thus $\varphi
(h^p)=wh^p rw^{-1}$. Since $\varphi$ should preserve ${\left\langle\hspace{-.7mm}\left\langle }h^p{\right\rangle\hspace{-.7mm}\right\rangle }^G
\cap H$, we obtain $h^p r\in {\left\langle\hspace{-.7mm}\left\langle }h^p{\right\rangle\hspace{-.7mm}\right\rangle }^G$. On the other hand, $h^pr\in E_G(h)$. By Lemma \[Eg\] we can join $E_G(h)$ to the collection of the peripheral subgroups. Without loss of generality we may assume that $p \gg 1$ so that the normal subgroup $N=\langle
h^p\rangle $ of $E_G(h)$ satisfies the requirement $N\cap \mathcal
F=\emptyset$ from Theorem \[Fill\] (and Definition \[def:periph\_fill\]). Then by the first part of Theorem \[Fill\] we have $h^pr \in {\left\langle\hspace{-.7mm}\left\langle }h^p{\right\rangle\hspace{-.7mm}\right\rangle }^G \cap E_G(h)
=\langle h^p\rangle $. Hence $r\in \langle h\rangle \cap E_G(H)=\{
1\}$, which contradicts $r\ne 1$.
\[outn\] Let $H$ be a non-elementary subgroup of a relatively hyperbolic group $G$ such that $H^0\ne \emptyset $. Then the following hold.
1. If $H$ has finite index in $N_G(HE_G(H))$, then $Out_n(H)$ is finite.
2. If $H$ does not normalize any non-trivial finite subgroup of $G$, and $H=N_G(H)$, then $Out _n(H)=\{ 1\}$.
The argument is similar to the one used to prove Corollary \[cor:conj\_aut\]. Observe that by Lemma \[EH\], $E_G(H)$ is a finite subgroup of $G$ normalized by $H$. Therefore $H$ acts on $E_G(H)$ by conjugation, and $C_H=C_H(E_G(H))$ has a finite index in $H$ as a kernel of this action. Let $h_1, \dots , h_l$ be elements of $H$ such that $H=\bigsqcup_{i=1}^l C_H h_i$. By Theorem \[NormAutSubgr\] we can argue as in the proof of Corollary \[cor:conj\_aut\] to conclude that every normal automorphism $\varphi$ of $H$ is uniquely determined by the images ${\varepsilon }(h_i)$ of $h_i$, $i=1, \dots,l$, and by the conjugating element $w \in N_G(\widehat H)$. As $E_G(H)$ is finite, for each $i$ there are only finitely many possibilities for ${\varepsilon }(h_i)$, and since $|N_G(\widehat H):H|<\infty$, we can deduce that $|Aut_n(H):Inn(H)|<\infty$.
Furthermore, if $H=N_G(H)$ and $H$ does not normalize any finite normal subgroup of $G$, we obtain $E_G(H)=\{1\}$, $N_G(\widehat H)=N_G(H)=H$, and ${C_H(E_G(H))}=H$. Hence $Aut_n(H)=Inn (H)$ by Theorem \[NormAutSubgr\]. This completes the proof.
The next lemma shows that Corollary \[cor1\] holds for elementary groups.
\[vc\] Let $G$ be a virtually cyclic group. Then $Out(G)$ is finite.
If $G$ is finite the claim is trivial, so assume that $G$ is infinite. Recall that every elementary group is ether finite-by-cyclic or finite-by-(infinite dihedral) (see, for example, [@FJ Lemma 2.5]). More precisely, as $G$ is infinite, the quotient $G/E(G)$ (where $E(G)$ is the maximal finite normal subgroup of $G$ given by Corollary \[KG\]) is either infinite cyclic or infinite dihedral. In both cases we have $$\label{eq:aut(g/eg)}\left|Aut(G/E(G)):Inn(G/E(G))\right|=2.$$
Every automorphism $\alpha \in Aut(G)$ induces an automorphism $\bar\alpha \in Aut(G/E(G))$. This gives rise to a homomorphism $\xi: Aut(G)\to Aut(G/E(G))$. If $\alpha \in \ker(\xi)$, then for every $x \in G$ there is $h=h(x) \in E(G)$ such that $\alpha(x)=x h$. By our assumptions, $G$ is generated by a finite set of elements $\{x_i~|~i=1,\dots,n\}$ and the automorphism $\alpha$ is uniquely determined by the images $\alpha(x_i)$, $i=1,\dots,n$. Since $|E(G)|<\infty$, for each $i$ there are only finitely many possibilities for $h(x_i)$. Therefore the kernel of $\xi$ is finite. Evidently $\xi (Inn(G))=Inn (G/E(G))$, and by we get $|Aut(G):(Inn(G) \ker (\xi))| \le 2$ yielding that $|Out(G)|=|Aut(G):Inn(G)|<\infty$.
Let us apply Theorem \[NormAutSubgr\] to the case $G=H$. Then $E_G(H)=E(G)$, ${C_H(E_G(H))}=C(G)$, $\widehat H=N_G(\widehat H)=G$, and the claim of Theorem \[main\] follows immediately.
First, suppose that $G$ is elementary. In this case the first part of the corollary follows from Lemma \[vc\]. To derive the second claim of the corollary, we observe that since $G$ is non-cyclic and does not have non-trivial finite normal subgroups, it must be infinite dihedral (this follows from the structure of an elementary group – see the proof of Lemma \[vc\]). Hence $G \cong {{\mathbb Z}}/2{{\mathbb Z}}* {{\mathbb Z}}/2{{\mathbb Z}}$ and, by Neshchadim’s theorem [@Nesh], $Out_n(G)=\{1\}$.
Thus we may assume that $G$ is non-elementary. In this case the corollary follows from Theorem \[main\] in the same way as Corollary \[outn\] from Theorem \[NormAutSubgr\]. Alternatively it follows immediately from Corollary \[outn\] applied to the case when $G=H$.
Free products and groups with infinitely many ends
==================================================
In order to prove Theorem \[fp\] we need two more statements below.
\[noncom\] Assume that $G$ is a relatively hyperbolic group and $g, h$ are two non-commensurable loxodromic elements. Then $g$ and $h$ are non-commensurable and loxodromic in most peripheral fillings of $G$.
Suppose that $G$ is hyperbolic relative to a collection of subgroups ${\{ H_\lambda \} _{\lambda \in \Lambda } }$. Applying Lemma \[Eg\] twice we obtain that $G$ is hyperbolic relative to the new collection ${\{ H_\lambda \} _{\lambda \in \Lambda } }\cup \{ E_1, E_2\}$, where $E_1=E_G(g)$, $E_2=E_G(h)$. Let $\mathcal F_1$ and $\mathcal F_2$ be the finite subsets provided by Theorem \[Fill\] for the collections of peripheral subgroups ${\{ H_\lambda \} _{\lambda \in \Lambda } }$ and ${\{ H_\lambda \} _{\lambda \in \Lambda } }\cup \{ E_1, E_2\}$, respectively. Set $\mathcal F=\mathcal F_1\cup \mathcal F_2$.
Consider any collection of subgroups $N_\lambda \lhd H_\lambda $ such that $N_\lambda \cap \mathcal F=\emptyset$, $\lambda \in \Lambda$. By Theorem \[Fill\], the filling of $G$ with respect to the collection of normal subgroups ${\mbox{\eufm N}}$, consisting of $N_\lambda \lhd H_\lambda $ for $\lambda \in \Lambda $ and the trivial subgroups of $E_1$, $E_2$, is hyperbolic relative to $\{ H_\lambda /N_\lambda \}_{\lambda \in \Lambda }\cup \{ E_1,
E_2\}$ as well as relative to $\{ H_\lambda /N_\lambda \}_{\lambda
\in \Lambda }$. (We keep the same notation for the isomorphic images of $E_1, E_2$ in $G({\mbox{\eufm N}})$ and the elements $g, h$.)
In particular, $E_1\cap E_2^t$ is finite for every $t\in
G({\mbox{\eufm N}})$. Clearly this implies that $g$ and $h$ are not commensurable in $G$. Similarly $g$ and $h$ are not conjugate to any elements of the subgroups $H_\lambda /N_\lambda $, $\lambda \in
\Lambda $, of $G({\mbox{\eufm N}})$. Thus $g$ and $h$ are loxodromic in $G({\mbox{\eufm N}})$ with respect to the peripheral collection $\{ H_\lambda /N_\lambda
\}_{\lambda \in \Lambda }$. As $\mathcal F$ is finite, $g$ and $h$ are non-commensurable and loxodromic in most peripheral fillings of $G$ (with respect to the peripheral structure ${\{ H_\lambda \} _{\lambda \in \Lambda } }$).
The proof of Theorem \[fp\] uses the following lemma, which is an immediate corollary of [@Yed Lemma 3]. (Recall that the [*Cartesian subgroup*]{} of a free product $A*B$ is, by definition, the kernel of the natural epimorphism $A*B \to A \times B$.)
\[Yed\] Let $G=A\ast B$, where $A$ and $B$ are finite groups. Let $u$, $v$ be non-commensurable elements of the Cartesian subgroup $C$ of $G$. Suppose that $u=a^k$, $v=b^l$ for some positive integers $k,l$, where $a$, $b$ are not proper powers. Assume also that $a^k$ (respectively, $b^l$) is the smallest non-zero power of $a$ (respectively, $b$) that belongs to $C$. Then there exists a finite quotient-group $Q$ of $G$ such that the images of $u$ and $v$ have different orders in $Q$.
Let $G$ be a non-trivial free product, i.e., $G=A\ast B$, where both $A$ and $B$ are non-trivial. Then $G$ is hyperbolic with respect to $\{A,B\}$ (the finite sets $X$ and $\mathcal{R}$, from the definition of relative hyperbolicity in Section \[sec:prelim\], can be taken to be empty; the isoperimetric constant $C$ for the corresponding relative presentation of $G$ will then be equal to zero). In what follows, we will fix this as a system of peripheral subgroups of $G$.
If $|A|=|B|=2$, the proof is an easy exercise. It also follows from the main result of [@Nesh], stating that every normal automorphism of a non-trivial free product is inner, and the observation that every non-trivial normal subgroup of the infinite dihedral group is of finite index. Thus we may assume that $G$ is non-elementary. Suppose that there exists an automorphism $\alpha \in Aut_n^f(G)\setminus Inn(G)$. Note that $E(G)=\{ 1\} $ because $G$, as a non-trivial free product, cannot contain non-trivial finite normal subgroups. Since $\alpha$ is not an inner automorphism of $G$, it follows from Corollary \[cor:descr\_comm\_aut\] that $\alpha $ is not commensurating. Therefore, by Corollary \[cor:comm\_aut\_def\] and Lemma \[lem:lox\_non-comm\] (applied to the case when $H=G$), there is a loxodromic element $g\in G$ such that $h=\alpha (g)$ is also loxodromic and is not commensurable with $g$. Further, by Lemma \[noncom\] there exist finite index normal subgroups $M\lhd A$ and $N\lhd B$ such that the natural images $\bar g$, $\bar h$ of $g$ and $h$, respectively, are not commensurable in $\overline{G}=A/M\ast B/N$. Without loss of generality we may assume that $\overline{G}$ is non-elementary.
Since $\overline{G}$ is a free product of two finite groups, it is residually finite. Therefore the kernel $K$ of the natural homomorphism $G\to \overline{G}$ is an intersection of finite index normal subgroups of $G$. As $\alpha \in Aut_n^f(G)$, $\alpha
$ stabilizes $K$. Hence $\alpha $ induces an automorphism $\bar\alpha $ of $\overline{G}$.
Let $\bar g=a^k$, where $k$ is a positive integer and $a$ is not a proper power. Clearly $b=\bar\alpha (a)$ is not a proper power as well and $b^k=\bar h$. Evidently $b^p=\bar\alpha (a^p)$ is not commensurable to $a^p$ for any non-zero integer $p$. Let $C$ denote the Cartesian subgroup of $\overline{G}$. Then $|\overline{G}:C|<\infty$, and replacing $\bar g$ with another positive power of $a$, if necessary, we may assume that $k>0$ and $\bar g=a^k$ is the smallest non-zero power of $a$ that belongs to $C$. Again, since $|\overline{G}:C|<\infty $, $\bar \alpha $ preserves $C$. In particular, $\bar h=b^k$ is the smallest power of $b$ that belongs to $C$.
By Lemma \[Yed\] there exists a finite index normal subgroup $K$ of $\overline{G}$ such that the images of $\bar g$ and $\bar h$ have different orders in $\overline{G}/K$. Therefore $\bar\alpha $ does not induce an automorphism on $\overline{G}/K$. Obviously this means that $\alpha $ does not preserve the full preimage of $K$ in $G$, which contradicts our assumption that $\alpha \in Aut_n^f(G)$.
The following lemma is well known and is easy to prove (see, for example, [@GL Lemma 5.4]).
\[outnormsub\] Suppose that $G$ is a finitely generated group and $N$ is a centerless normal subgroup of finite index in $G$. Then some finite index subgroup of $Out (G)$ is isomorphic to a quotient of a subgroup of $Out (N)$ by a finite normal subgroup. In particular, if $Out(N)$ is residually finite, then $Out (G)$ is residually finite.
The next observation is trivial.
\[finorb\] Suppose that a group $G$ acts on a set $\mathcal M$ faithfully with finite orbits. Then $G$ is residually finite.
Given $g\in G$, let $s\in \mathcal M$ be an element such that $g(s)\ne s$. Then the natural map from $G$ to the symmetric group on the orbit of $s$ provides us with a finite quotient of $G$, where the image of $g$ is non-trivial.
Since the outer automorphism group of any virtually cyclic group is finite (see Lemma \[vc\]), we can assume that $G$ has infinitely many ends.
By Stallings’s Theorem ([@Stall71; @Stall68]) there is a finite group $S$ such that $G$ splits as an amalgamated free product $A\ast_S B$ or an $HNN$-extension $A\ast _S$, where $(|A:S|-1)(|B:S|-1)\ge 2$ in the first case and $|A:S_i|\ge 2$, $i=1,2$, in the second case (where $S_1$ and $S_2$ are the two associated isomorphic copies of $S$ in $A$). Since $G$ is residually finite and $S$ is finite, there exists a finite index normal subgroup $N\lhd G$ such that $N\cap S=\{1\}$ if $G=A\ast_S B$, or $N \cap S_i = \{1\}$ for $i=1,2$, if $G=A\ast _S$. Note that the quotient of the Bass-Serre tree for $G$ modulo the action of $N$ is finite and the edge stabilizers in $N$ are trivial. The Bass-Serre structure theorem for groups acting on trees (see [@Serre]) yields a splitting of $N$ into a non-trivial free product. In particular, $N$ is centerless.
The group $Aut (N)$ naturally acts on the set $\mathcal M$ of finite index normal subgroups of $N$ and $Aut_n^f(N)$ is the kernel of this action. By Theorem \[fp\], $Aut_n^f(N)=Inn(N)$. Therefore, $Aut(N)/Aut_n^f(N)=Aut(N)/Inn(N)=Out(N)$ acts on $\mathcal M$ faithfully. Since $N$ is finitely generated, there are only finitely many subgroups of a given finite index in $N$, thus all orbits of the action of $Out(N)$ on $\mathcal M$ are finite. Hence $Out(N)$ is residually finite by Lemma \[finorb\]. The claim of the theorem is now a consequence of Lemma \[outnormsub\].
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[^1]: The first author was supported by the Swiss National Science Foundation grant PP002-116899. The second author was supported by the NSF grant DMS-0605093 and by the RFBR grant 05-01-00895.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'S.K. Randall'
- 'G. Fontaine'
- 'P. Brassard'
- 'V. Van Grootel'
bibliography:
- 'ms\_14780.bib'
date: 'Received date / Accepted date'
title: |
Mode identification from monochromatic amplitude and phase variations for the rapidly pulsating subdwarf B star\
EC 20338$-$1925[^1]
---
Introduction
============
Understanding the formation of subdwarf B (sdB) stars is one of the remaining challenges in stellar evolution theory today. As compact, low-mass, evolved objects located on the Extreme Horizontal Branch (EHB) it is generally accepted that they descend from the red giant branch, but loose too much of their hydrogen-rich envelope to ascend the asymptotic giant branch after core He-exhaustion [for a comprehensive recent review on subdwarf B stars see @heber2009]. However, the details surrounding the mass loss in particular are debatable, and a number of evolutionary scenarios have been modelled [e.g. @dorman1993; @han2002; @han2003; @yu2009]. These include both single star and binary formation channels such as common envelope evolution, stable and unstable Roche lobe overflow, and the merger of two Helium white dwarfs. Each of these channels should leave a distinct signature on the internal composition and the mass distribution of the resulting sdB star population. While the latter may be computed dynamically for the very rare case of an eclipsing binary, the former can be probed only on the basis of asteroseismology.
Over the last decade, the asteroseismological exploitation of subdwarf B stars has become increasingly successful. Full asteroseismological analyses leading to a precise estimate of the mass and hydrogen envelope thickness have now been presented for ten rapidly pulsating sdB stars [see, e.g. @charp2008; @val2008b; @randall2009 for the most recent results]. These objects, also known as EC 14026 or V361 Hya stars after the prototype [@kilkenny1997] exhibit low-order, low-degree $p$-mode oscillations on a typical time-scale of 100$-$200 s, thought to be excited by a classical $\kappa$-mechanism associated with a local overabundance of iron in the driving region [@charp1996; @charp1997]. While the asteroseismological results are beginning to indicate first evolutionary trends [see @charp2009 for a review], it is of course highly desirable to verify the accuracy of the claimed structural parameters using an independent means.
One of the most promising ways of doing this is to (partially) identify the modes of pulsation observationally. In the currently employed “forward method” in asteroseismology, the observed periodicities are fit to those computed for a grid of models, and an “optimal” model is identified using $\chi^2 $ minimisation. During the period fitting process, the observational frequencies are naturally associated with a theoretical mode of a given degree index $\ell$ and radial order $k$ (unless there is evidence for relatively fast rotation in the target, the computed period spectra are degenerate in azimuthal index $m$). Therefore, any independent knowledge of the modal indices can be used to reject an “optimal” model or, more usefully, be employed as an a priori constraint for the exploration of model parameter space. In certain cases, such constraints may be necessary to isolate just one “optimal” model from a number of very different possible solutions [@randall2009], in others they may simply confirm the original solution [@val2008b].
The degree index of a mode can be observationally inferred using several techniques. For the special case of a rapid rotator, the presence or absence of rotational splitting can give clues to the $\ell$ value of at least the higher amplitude modes [@charp2005b; @charp2008b]. Other, more universally applicable methods rely on the wavelength-dependent behaviour of the target flux during a pulsation cycle. Several recent projects have focussed on the interpretation of line profile variations observed in EC 14026 stars from high-resolution time-series spectroscopy [@telting2008; @maja2009; @telting2010], but these efforts were always hampered by the low S/N of the data. Indeed, the relative faintness of subdwarf B stars ($V\gtrsim$ 12) coupled with the short exposure times acceptable ($\lesssim$ 30 s) for these fast pulsators make this type of study a challenge even when granted access to the world’s largest telescopes, and none of the observations obtained to date were of sufficient quality to yield a unique identification of $\ell$. Meanwhile, numerous studies based on low-resolution time-series spectroscopy of EC 14026 stars [e.g. @jeffery2000; @otoole2005; @telting2006 to name just a few] have focussed mainly on measuring the radial velocity and equivalent width variations, and deducing the apparent changes in effective temperature and surface gravity over a pulsation cycle. Any attempts at mode identification were unfortunately inconclusive.
A more promising approach is to analyse the amplitude and phase variations of a pulsation mode as a function of wavelength. This has been done quite successfully on the basis of multi-colour photometry. While the S/N of the observations is still a major limitation, particularly for fainter targets and/or low-amplitude pulsations, unique identifications of the degree index $\ell$ have been possible in several cases [@jeffery2004; @randall2005; @charp2008b; @baran2008]. By far the nicest results to date were obtained for the brightest EC 14026 star Balloon 090100001 on the basis of UBV light curves of extraordinary quality obtained with the three-channel photometer LaPoune at the CFHT. Not only was the dominant pulsation clearly found to be a radial mode, but a further 8 frequencies could additionally be unambiguously identified as $\ell$=1, $\ell$=2 or $\ell$=4 modes [@charp2008b]. Beyond its intrinsic interest, this result proved beyond doubt the validity and accuracy of the theoretical colour-amplitudes calculated according to the method outlined in @randall2005.
The analysis presented here makes use of the same theoretical framework and numerical tools as our previous studies [@randall2005; @charp2008b], but it is based on time-series spectrophotometry rather than multi-colour photometry. To our knowledge this is the first time that this method of mode identification has been applied to a pulsating subdwarf B star. While the idea had been present in the back of our minds since the development of the theoretical tools, it was the very promising results presented by @vankerkwijk2000 and @clemens2000 for the ZZ Ceti white dwarf G29$-$38 that finally motivated us to apply for the necessary observing time. In that study, just under five hours worth of Keck LRIS time-series spectrophotometry were sufficient to measure the degree indices for six modes. Of course, G29$-$38 is an exceptionally bright ($V\sim$ 13) white dwarf with several relatively high amplitude ($A\gtrsim$ 1%) periodicities, and the monochromatic flux behaviour of $g$-mode pulsations in ZZ Ceti stars is somewhat different to that of $p$-modes in EC 14026 stars [cf. @randall2005; @brassard1995]. Also, the strong broadening of the Balmer lines in white dwarfs greatly facilitates the observational characterisation of the flux behaviour across the line profiles, as do the longer pulsation periods and correspondingly higher integration times acceptable. Thus it was with contained optimism that we embarked on the present study.
Given that the observations envisaged would be obtained with the Very Large Telescope (VLT) run by ESO on Cerro Paranal, Chile, we selected a target located in the southern hemisphere. EC 20338$-$1925 is a moderately bright ($V\sim$ 15.67) sdB star with no photometric indication of a cool companion [@kilkenny2006], and atmospheric parameters around $T_{\rm eff} \sim$ 35,500 K, $\log{g}\sim$ 5.75, and $\log{N(\rm He)/N(\rm H)}\sim - $ 1.71 [@ostensen2010]. It was discovered to belong to the class of rapid EC 14026 variables by @kilkenny2006, who detected five periodicities near 168, 151, 147, 141 and 135 s on the basis of white-light photometry gathered in 1998. The period spectrum was very clearly dominated by the 147-s oscillation, which exhibited a high amplitude of nearly 2.5%. This for an EC 14026 star unusually strong oscillation mode was the main reason we chose EC 20338$-$1925 over other, brighter target candidates. After the observing time had already been granted we learned that subsequent observations obtained from 1999 to 2007 had revealed quite striking amplitude variations for some of the modes. In particular, the 147-s oscillation had intermittently been measured at an amplitude as low as 0.3% [@Kilkenny2010], while some other frequencies appeared to have constant strengths. Such amplitude variations have been observed in a number of EC 14026 stars, however their origin is unclear, making EC 20338$-$1925 an enigmatic target for detailed study.
Observations and Data Reduction
===============================
Night Date (UT) Start time (UT) Length (h) spectra pairs
------- ------------ ----------------- ------------ ---------------
1 2009-08-22 23:57 04:48 654
3 2009-08-25 00:18 04:27 615
4 2009-08-25 23:48 04:52 656
: Time-series spectroscopy obtained for EC 20338$-$1925[]{data-label="obslog"}
The observations presented here were obtained using the HIT-MS mode of FORS2 mounted on Antu (UT1) at the VLT. This rarely used mode is ideal for fast time-series spectroscopy as it allows 42 observations per readout, achieved by exposing only a small part of the chip and successively shifting the exposed blocks of columns across the CCD while already integrating on the next spectrum. Moreover, up to two closely spaced targets can be observed, allowing for both a main target and a comparison star to be monitored simultaneously. For more detailed information on FORS2 and the HIT mode please see the User Manual, available online via the ESO webpages.
We were allocated four consecutive half-nights from 22-25 August 2009 for our proposed study of EC 20338$-$1925. Making use of the blue-efficient E2V CCD formerly installed on FORS1 together with the cross-disperser grism 600B we hoped to obtain spectra covering the 3400-6500 Å wavelength range. Unfortunately, we ran into severe technical problems at the beginning of the second night of observations, as one of the E2V chips of the mosaic died. Since in the HIT-MS mode each spectrum is projected along a block of columns spread across the two chips, this resulted in the loss of half the spectral wavelength range for the remainder of the observing run. To make matters worse, the mask we had cut before the beginning of the run placed our target on the dead chip, meaning that we could not resume our observations until a new mask had been cut the next day. Thus, despite good atmospheric conditions throughout the run we lost the second night completely, and obtained only half the wavelength range expected on the third and fourth nights. For a log of the observations please see Table \[obslog\].
Since the main aim of our study was to examine the monochromatic properties of the target flux as a function of time, we avoided slit losses by choosing the largest mask openings available in HIT-MS mode (5$\times$5), and essentially obtained spectrophotometry as the seeing was always well below the slit width. As a natural side effect, the spectral resolution of our data is not fixed but seeing-limited, and varies as a function of time. The typical seeing on the three nights we obtained data was $\sim$1.1at $\sim$4500 Å, which with the 600B cross-disperser corresponds to a wavelength resolution of $\sim$6.5Å. On the first night (when both chips were working) we covered the 3500-4950 Å (chip 2) and 5100-6600 Å (chip 1) ranges, and opted for the bluer 3700-5200 Å regime after chip 2 failed. That way, we included the Balmer lines from H$\alpha$ upwards on night 1, and from H$\beta$ upwards on nights 3 and 4 for our target star. We also monitored a F7V comparison star, HD 196286, picked purely for its location (7.5 from the target) and brightness ($V\sim$ 10.1, from Simbad). The spectral range covered for the comparison star is shifted $\sim$400 Å bluewards with respect to the target star due to its different location on the chip. We set the exposure time for each pair of spectra to 25 s, yielding a total integration time of 1025 s (for 41 measurements) on the CCD before each 36 s readout. This means that only 0.3% of the time-series is taken up by dead time in between exposures!
![Typical individual spectra obtained on 22 August 2009 for EC 20338$-$1915 (bottom) and the main sequence comparison star (top). The spectra have been arbitrarily normalised for visualisation purposes, and are not flux calibrated.[]{data-label="spectra"}](14780f1.ps){width="7.0cm"}
Given that the HIT-MS mode is not supported by the FORS pipeline, we wrote our own IRAF script to batch-process the data. This script essentially treats the 82 (41 each for the target and comparison star) spectra on each image as echelle orders, and writes the reduction products out as individual FITS files with a time-stamp corresponding to the middle of the exposure calculated for each pair of spectra. The steps performed are bias subtraction, bad pixel interpolation, cosmic ray cleaning, background fitting and subtraction, tracing and extraction of the spectra, and finally wavelength calibration. This last step was somewhat challenging, since we did not manage to take useful arc-lamp calibration data with the available wide-slit masks. In the end, the only way to obtain a meaningful wavelength scale was to use the spectral lines of the target and comparison star themselves, and compute the wavelength solutions for each night on the basis of the respective time-averaged combined spectra. Luckily, subdwarf B stars show a number of easily identifiable Balmer and Helium lines at short wavelengths, and we were thus able to obtain a reasonably accurate (within a few Å) wavelength calibration for at least the blue part of the target spectrum. This is sufficient for our purposes, since we do not attempt to detect radial velocity variations arising from the pulsations due to the low resolution of the spectra and the arbitrary wavelength variations induced by spatial shifts of the target across the wide slit. The red part of the target spectrum is largely devoid of features, and the wavelength solution obtained cannot be trusted for quantitative analysis. For the comparison star, we were able to identify several lines on both chips, and achieve a rough wavelength calibration. However, since these measurements are only used in wavelength-integrated form in what follows, this calibration is needed as nothing more than a guideline.
Figure \[spectra\] shows a typical individual spectrum for both the target and comparison star, taken from the first night where both chips were available. Note that the comparison star data were chopped at the short-wavelength end of the red chip as this part of the spectrum was saturated. The high quality is striking considering the relative faintness of the target [V$\sim$15.67 from @kilkenny2006] and the short exposure time used (25 s). In the central part of the blue target spectrum, we measure a signal-to-noise S/N $\sim$ 55, while it is lower in the red part at S/N $\sim$ 25. For the comparison star an accurate measurement of the S/N is complicated by the numerous metal lines, but we can roughly estimate S/N $\sim$ 90 for the central part of the blue spectrum and S/N $\geq$ 100 for the red part. It is apparent from looking at the relative depth of $H\alpha$ compared to the other Balmer lines in the target spectrum that the red part is a lot less useful than the blue part in terms of S/N of the lines, mostly due to the intrinsically low flux of sdB stars at longer wavelengths. Adding to this the fact that we were able to obtain only half a night’s worth of red data, and that an accurate wavelength calibration was not possible, they become unsuitable for a detailed interpretation. Therefore, the analyses presented in the following sections are based on the blue data only.
Data Analysis
=============
![Fourier transform computed on the basis of the integrated flux from all blue spectra of sufficient quality gathered during the observing run.[]{data-label="ft"}](14780f2.ps){width="7.0cm"}
Frequency Determination
-----------------------
Rank Period (s) Amp (%) A$_{Kilkenny}$(%)
------ ------------ --------- -------------------
1 146.940 2.30 0.32$-$2.40
2 168.435 0.45 0.33$-$0.46
3 126.382 0.38 0$-$0.33
4 140.588 0.20 0$-$0.24
: Periodicities extracted for EC 20338$-$1925. We list the period and amplitude obtained from our data set, as well as the ranges of amplitude measured by D. Kilkenny from 1998$-$2007. The uncertainty on our amplitudes is $\sim$0.05% from the least squares fit to the light curve.[]{data-label="freqs"}
![Measured amplitudes of the 4 periodicities listed in Table \[freqs\] over the last 12 years. Apart from the most recent data points, which were taken from the observations presented here, the measurements were kindly made available to us by D. Kilkenny. The errors were not provided, but should be similar to or smaller than those shown for our data. Zero amplitudes imply that the pulsation was not detected in a particular data set down to a threshold of $\sim$0.1%.[]{data-label="amps"}](14780f3.ps){width="6.5cm"}
The first step in the data analysis was to obtain pulsation frequencies for our data set, and at the same time assure the quality of the measurements retained for further analysis. In order to maximise the S/N we integrated the flux of each spectrum over the wavelength range of interest (3650$-$4950 Å) and produced broadband fluxes for the target as well as the comparison star. These were then used to compute normalised differential light curves additionally corrected for differential extinction by fitting a second-order polynomial to each night’s data. Individual outliers (usually associated with bad seeing) were removed from the light curves prior to Fourier analysis, and the corresponding spectra were discarded. While we initially included the red data, we found that this gave results of lower quality than using the blue part of the spectra alone, and thus decided to consider only the latter. In addition, the last $\sim$hour of data taken on the first night had to be disregarded because the stars had shifted towards the edge of the slit and some flux loss had occurred, making these measurements unsuitable for time-series flux analysis.
We show the Fourier transform (FT) of the combined light curve in Fig. \[ft\]. By pre-whitening we were able to detect four periodicities above the imposed 4$\sigma$ threshold for credible pulsations (0.2% of the mean brightness), listed in Table \[freqs\] together with their amplitudes. Three of these, including the strong dominant pulsation (146.9 s), correspond well to those measured in 1998 [@kilkenny2006], while the fourth (126.4 s) was previously only detected from a dataset taken in 2003 [@Kilkenny2010]. As briefly mentioned in the Introduction, the amplitudes of some of the periodicities observed for EC 20338$-$1925 (particularly the dominant 146.9-s peak) are highly variable on a time-scale of several years, whereas others (such as the 168.4-s pulsation) show a remarkably stable amplitude over the course of the last decade. This is visualised in Fig. \[amps\], where we have plotted the amplitudes measured by D. Kilkenny during 7 observing runs between 1998 and 2007 for the four oscillations detected from our data together with our more recent results (see also the last column of Table \[freqs\]). It is not clear whether these observed amplitude variations are due to real intrinsic amplitude modulations, or caused by the beating of very close unresolved modes. We hope to shed some light on this in the following sections.
Atmospheric analysis
--------------------
![Normalised averaged spectrum for EC 20338$-$1925 (thin line) overplotted with the best synthetic spectrum “with metals” (thick line). The vertical dashed lines indicate the positions of the SIII, OII and NII lines thought to be present in the observed spectrum. The atmospheric parameters derived from this fit are also given.[]{data-label="metalspec"}](14780f4.eps){width="6.5cm"}
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[{width="6.8cm"}]{} [{width="6.8cm"}]{}
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The second step in the data analysis was to determine the atmospheric parameters of EC 20338$-$1925, which are necessary for the modelling of the monochromatic flux behaviour (see below). For this purpose, we made a combined spectrum on the basis of 995 (blue) spectra spread over the three nights that have central wavelength FWHMs close to the average seeing during the observing run (between 1.0$-$1.15$\arcsec$). The result was a time-averaged spectrum with wavelength resolution 6.4$\pm$0.2 Å and high S/N of $\sim$400 at the central wavelength.
We fit this averaged spectrum with two grids of synthetic spectra computed from NLTE model atmospheres specially designed for subdwarf B stars using the public codes TLUSTY and SYNSPEC [@hubeny1995; @lanz1995]. The spectra in the first grid contain only H and He in the atmospheric composition (“no metals”), while those of the second set (“with metals”) have a chemical composition made up of H, He, C (0.1 solar), N (solar), O (0.1 solar), S (solar), Si (0.1 solar) and Fe (solar). This mixture was derived from the work of @blanchette2008, who determined the abundances of several astrophysically important elements for five representative subdwarf B stars on the basis of FUSE spectra in conjunction with NLTE model atmospheres. The abundance patterns were found to be very similar in all cases, therefore we deemed it appropriate to include the most abundant heavy elements in our model atmospheres at the level measured.
Fig. \[fitboth\] shows the fits to the Balmer and Helium lines obtained for both the “no metals” and “with metals” cases, and also gives the inferred values for the effective temperature, logarithmic surface gravity and logarithmic fractional Helium abundance. The uncertainty estimates given of course refer only to the formal fitting errors and most certainly underestimate the real errors. As is typically the case for EC 14026 stars with $T_{\rm eff}\sim$35,000 K, the resulting atmospheric parameters are only marginally influenced by the inclusion of metals in the model atmospheres, and are consistent with each other [@heber2000; @geier2007]. And indeed, the line profile fits shown in Fig. \[fitboth\] hardly change from one plot to the next.
There are nevertheless some indications that the solution incorporating metals is to be preferred. Despite its relatively low resolution, the averaged spectrum of EC 20338$-$1925 does hint at the presence of a few heavy elements, thanks to the high quality of the data. In particular, the normalised spectrum plotted in its entirety in Fig. \[metalspec\] contains lines corresponding to absorption from NII (4237.05 and 4241.79 Å), OII (4253.87 Å) and SIII (4284.97 Å). These are marked by vertical dotted lines in the plot, and are recovered quite nicely by the overplotted “with metals” synthetic spectrum. Another indicator for the presence of metals in EC 20338$-$1925 is the relatively poor fit to the He II line at 4686 Å (see Fig. \[fitboth\]). While the model incorporating metals fares better in the matching of this line than the synthetic spectrum without metals, it appears that even the “with metals” chemical composition does not include enough metals to accurately represent EC 20338$-$1925. Given the fact that the spectroscopic solution is not very sensitive to the metallicity of the model atmosphere for our target, this does not constitute a problem for the accuracy of the inferred atmospheric parameters. Therefore, we adopt the “with metals” values of $T_{\rm eff}$=34,153 K and $\log{g}$=5.966 in what follows.
Observed monochromatic amplitudes and phases
--------------------------------------------
![Fourier transforms obtained for 10 Å wavelength bins at the positions indicated in the left hand panel of Fig. \[ampphase\].[]{data-label="multifts"}](14780f6.ps){width="7.5cm"}
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[{width="7.5cm"}]{} [{width="7.5cm"}]{}
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The final, and arguably most interesting part of the data analysis was the computation of the observed monochromatic amplitudes and phases for the periodicities extracted. For this exercise we used only the blue target spectra also employed for the computation of the broadband Fourier transform shown in Fig. \[ft\], thereby ensuring a high data quality and the exact same sampling. The latter is important since we used the periods determined in Section 3.1 as input for the least-squares fitting procedure detailed below. All selected target spectra were chopped to the 3650$-$4950 Å range of interest and binned into common 3.25 Å wavelength units, which were chosen to adequately sample the average $\sim$6.5 Å resolution of the original data.
Light curves were then obtained for each of the 401 wavelength bins, and corrected for differential extinction using the comparison star data. In order to achieve the best possible S/N, we integrated the comparison star spectra from 3650 Å to their upper limit of $\sim$ 4700 Å and employed the resulting broadband fluxes for the calculation of the differential light curves. For each wavelength bin, and using the periods listed in Table \[freqs\] as fixed input parameters, the corresponding amplitudes and phases were determined from a least-squares fit to the light curve assuming a sinusoidal signal. While this did not involve the computation of Fourier spectra, we nevertheless calculated them for a small number of selected wavelength points as a sanity check. For illustration purposes we show some of these Fourier transforms in Fig. \[multifts\]. Note that here, the spectra were binned into units of 10 Å to somewhat decrease the noise level.
Fig. \[ampphase\] shows the monochromatic amplitudes and phases derived for the dominant mode $f_1$. For reference we also show the averaged spectrum of EC 20338$-$1925, and indicate the rest wavelengths of the Balmer and most prominent Helium lines. Looking at the amplitude spectrum we find that the amplitude of pulsation in the continuum drops quite rapidly between 3650 and 4050 Å and then stabilises somewhat. This was to be expected, since the same effect is observed from multi-colour photometry [e.g. @charp2008; @randall2005; @jeffery2004]. In the Balmer line cores, the pulsational amplitude is significantly higher than in the continuum, and again decreases with increasing wavelength of the line. Considering the Helium lines strong enough to leave a visible imprint on the amplitude spectrum, it appears that the neutral helium lines at 4026 and 4471 Å are associated with higher amplitudes compared to the continuum, while the pulsational signature is extremely weak in the He II line at 4686 Å. The observed phase on the other hand appears to be constant in the continuum, jumps occuring only in the line centers. This is consistent with the very small phase shifts observed on the basis of multi-colour photometry [@jeffery2004; @jeffery2005].
Modelling the observed monochromatic amplitude and phase variations
===================================================================
![Theoretical monochromatic amplitudes for modes with degree indices $\ell$=0-5 in the wavelength range of interest, normalised to 3650 Å. The computations were specifically carried out for the dominant mode in EC 20338$-$1925, and assume $T_{\rm eff}=$ 34,153 K, $\log{g}=$ 5.966, $M_{\ast}/M_{\odot}=$ 0.48, $\log{q(\rm H)}=-$4.3, and $P=$ 146.9 s. The wavelength resolution was artificially degraded to 6.5Å.[]{data-label="theoryamps"}](14780f8.eps){width="6.5cm"}
Computation of theoretical amplitudes and phases
------------------------------------------------
For the computation of the theoretical monochromatic amplitudes and phases we closely follow the approach developed by @randall2005 for the application to multi-colour photometry. We refer the interested reader to that paper for details on the underlying theory and the models, and only give a very brief overview of the methodology followed here.
Our theoretical amplitudes and phases rely primarily on monochromatic quantities derived from model atmospheres. The computation of these quantities involves not only the standard specific intensities, but also the corresponding derivatives with respect to effective temperature and surface gravity across the visible disk of the star. They are calculated on the basis of a grid of LTE model atmospheres characterised by a uniform composition specified by $\log{[N(\rm He)/N(\rm H)]}=-$2.0, i.e. without metals. Whereas it would be interesting to assess the effect that incorporating metals would have on this type of calculation in the future, the current grid is quite sufficient for the purposes of this study. In any case, the heavy element abundance pattern of EC 20338$-$1925 has not yet been characterised, and most likely deviates from the @blanchette2008 composition employed for the atmospheric analysis detailed in Section 3.2. We therefore compute the monochromatic atmospheric quantities using the existing H/He model grid, assuming an effective temperature $T_{\rm eff}$=34,153 K and a logarithmic surface gravity $\log{g}$=5.966 as derived above.
While the monochromatic atmospheric quantities thus computed are the main contributors to the relative first order perturbations of the emergent flux (from which the monochromatic amplitudes and phases directly follow), they neglect to take into account non-adiabatic effects. The quantities describing the departure from adiabacity in amplitude and phase must instead be derived from full non-adiabatic pulsation calculations on the basis of stellar models. For this purpose, we employ our well-known “second-generation” models [see, e.g. @charp1996; @charp2001] in conjunction with adiabatic and non-adiabatic pulsation codes based on finite-element techniques [@brassard1992; @fontaine1994]. These are exactly the same tools that have been used very successfully in the past for the asteroseismological analysis of EC 14026 stars. They require four structural input parameters: $T_{\rm eff}$, $\log{g}$, the total stellar mass $M_{\ast}$, and the fractional thickness of the hydrogen-rich envelope $\log{q(\rm H)}\equiv\log{M(\rm H)/M_{\ast}}$. In our computations we set the first two quantities to those inferred in Section 3.2, while for the other two we selected representative values of $M_{\ast}=$ 0.48 $M_{\odot}$ and $\log{q(\rm H)}=-$4.3. The two latter parameters were chosen to be typical of rapid sdB pulsators as found from asteroseismological studies, but do not significantly affect the results as long as they take on physically reasonable values.
The quantities describing the departure from adiabacity were first computed for the period spectrum of the representative EC 20338$-$1925 model, and subsequently determined for the dominant 146.9-s dominant mode using interpolation techniques. While for $p$-mode pulsators these quantities are quite sensitive to the period of the oscillation in question, they are luckily not dependent on the degree index of the mode, and can thus be inferred quite easily for any period in the excited frequency range predicted by the model. For reference, at the period of the main mode the non-adiabatic coefficients are $R=$ 0.810 and $\Psi_T=$ 2.761 rad compared to their adiabatic values of $R$ = 1 and $\Psi_T=\pi$ [for a definition of the parameters please see @randall2005]. With both the monochromatic atmospheric quantities and the non-adiabaticity coefficients available, we can now easily compute the monochromatic amplitudes and phases for EC 20338$-$1925.
![Theoretical monochromatic amplitudes for modes with $\ell=$ 0-5 overplotted on the observed amplitudes for the dominant $f_1$ mode in EC 20338$-$1925. The curves have all been normalised to unity at 3650 Å.[]{data-label="ampsf1"}](14780f9.eps){width="6.5cm"}
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[{width="7.5cm"}]{} [{width="7.5cm"}]{}
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Fig. \[theoryamps\] shows the relative amplitudes as expected for the 146.9-s mode of EC 20338$-$1925 for degree indices from $\ell$=0 to $\ell$=5 in the wavelength range of interest. The curves have been normalised to unity at 3650 Å and were degraded in wavelength resolution to the 6.5 Å typical of the observational data. Higher order modes are not illustrated as they are thought to have extremely low amplitudes in the continuum due to cancellation effects when integrating across the visible disk of the star, and should thus not be observable. As explained in @randall2005, the theoretical amplitudes and phases are, to first order, independent of the azimuthal index $m$, and can therefore be used for resolved or unresolved frequency components of multiplets split by slow rotation. The curves quite clearly show that the discrimination between low-order modes with $\ell\leq$ 2, and particularly between radial and dipole modes is challenging due to their very similar behaviour as a function of wavelength. This has always been the main difficulty also with the interpretation of multi-colour photometry and line-profile variability, and has limited many studies to the ambiguous identification of the main pulsations as low-order (i.e. $\ell$ = 0, 1 or 2) modes due to the insufficient S/N of the data [e.g. @jeffery2005; @tremblay2006; @maja2009]. Given that the amplitude dependence is greatest at short wavelengths, it is therefore imperative to observe as far in the blue as possible.
Comparing the curves displayed here for EC 20338$-$1925 to equivalent data for other EC 14026 stars [see in particular Fig. 2 of @randall2005], it turns out that we were quite lucky with our target in terms of discriminative power between the amplitudes of low-order modes. Indeed, as can be seen from Fig. 28a of @randall2005, the amplitude-wavelength behaviour of radial, dipole, and quadrupole modes diverges more strongly for stars with a higher surface gravity. By contrast, the $\ell$=3 mode shows a shallower amplitude gradient as a function of wavelength. EC 20338$-$1925 being one of the EC 14026 stars with the highest surface gravity measured is then definitely to our advantage, and increases the chances for the unique mode identification attempted below.
Fitting the observed monochromatic amplitudes and phases
--------------------------------------------------------
We now compare the theoretical monochromatic amplitudes computed in the previous section to the observations. Fig. \[ampsf1\] shows the curves displayed in Fig. \[theoryamps\] overplotted on the observed monochromatic amplitude of the dominant $f_1$ mode, which has also been normalised to unity at 3650 Å. From visual inspection alone it is obvious that the $\ell=$ 0 solution presents by far the best match to the observational data, followed by the curves for $\ell=$ 1 and, with a significant degradation $\ell=$ 2. The amplitudes for the higher degree modes match the data very poorly, and can be excluded from the outset. Computing the quality-of-fit $Q$ on the basis of $\chi^2$ fitting confirms this result, with $Q$=0.755 for $\ell=$ 0, $Q=$ 0.340 for $\ell=$ 1 and $Q=$ 0.056 for $\ell=$ 2. For the higher degree modes, $Q\ll$ 0.001, which implies that they can formally be excluded as possible solutions [@press1986]. The $Q$ value determined for the radial mode implies a very good fit, more than twice as good as that for the dipole mode. While the uncertainties on the observed amplitudes are too large to formally exclude the solutions for $\ell=$ 1 and $\ell=$ 2 as possible matches to the data, we believe that the far superior quality of the fit for $\ell=$ 0 leaves no doubt that the dominant pulsation observed in EC 20338$-$1925 is a radial mode.
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In order to illustrate the quality of the fit more clearly, we zoom in on the monochromatic amplitude plot for $\ell=$ 0 in the left panel of Fig. \[zoomampphase\]. Here, we also include the uncertainty on the observed amplitudes as obtained from the least-squares fits to the light curve for each wavelength bin (thin vertical line segments shifted downwards along the y-axis for clarity). It can be seen that, while the observed amplitude behaviour in the continuum is matched nearly perfectly by the theoretical curve, there are some discrepancies in the spectral lines. Most strikingly, the amplitude dip corresponding to the He II line at 4686 Å is predicted far weaker than observed, much like what we found for the model atmosphere fit to the time-averaged spectrum in Fig. \[fitboth\]. This is almost certainly due to the lack of metals in the model atmospheres used for the computation. A more puzzling feature is that the theoretical amplitudes in the Balmer line cores systematically deviate from those observed: while they are similar to, or slightly lower than the measurements for the higher Balmer lines (above H$\epsilon$), they increasingly overestimate the observed amplitudes in the lines at higher wavelengths (for H$\epsilon$, H$\delta$, and especially H$\gamma$ and H$\beta$). We are not sure whether this is due to some observational effect or caused by missing ingredients in our models (particularly concerning the chemical composition of the atmosphere). At the request of the referee we re-computed the $Q$-value of the fit taking into account only the continuum and the extended line wings (specifically, we masked out bands $\pm$6Åwide centered around H$\beta$, HeII 4686, H$\gamma$, H$\delta$, H$\epsilon$ and H8). As expected, the quality-of-fit is improved, with $Q$=0.999 ($\ell$=0), $Q$=0.503 ($\ell$=1), $Q$=0.223 ($\ell$=2) and $Q\ll$0.001 for higher $\ell$ values. This is simply associated with the fact that there are now less deviations between the predicted and observed amplitudes, and nicely illustrates the very good match achieved for a radial mode in the continuum.
While we are conviced that the dominant mode in EC 20338$-$1925 is radial from the amplitude data alone, we also analysed the corresponding phase data as a consistency check. Given the high noise level of the measurements, we assumed a degree of $\ell=$ 0 from the outset, and attempted to fit the theoretical phases to those observed. In order to be able to compute the monochromatic phases accurately, we extended our existing code for the computation of the pulsational flux perturbation by an optional radial velocity (RV) parameter. This parameter was not necessary for the theoretical quantities presented by @randall2005 because in that study we were interested only in wavelength-integrated phase measurements, which are dominated by the contributions from the continuum. As can be seen from Fig. \[ampphase\], these yield a relatively constant phase as a function of wavelength, which is consistent with the very small phase shifts predicted from one filter bandpass to the next in @randall2005. However, in a spectral line the moving matter produces significant distortions and phase delays over a pulsation cycle, and these need to be taken into account for monochromatic phase calculations. The distortions also affect the amplitudes in the spectral lines, but the effect is thought to be relatively small for the broader lines and was not included in the computations presented above for technical reasons. It remains to be investigated whether an incorporation of the Doppler shifts in the monochromatic amplitude computations could help improve the match between observations and theory in the Balmer lines.
The RV parameter incorporated into the phase calculations corresponds to the measurement of the semi-amplitude of the RV integrated over the disk, the same quantity that has been the subject of numerous time-series spectroscopy studies to date. Unfortunately, our data are simply not suitable for such an analysis due to the relatively low resolution, problems with the wavelength calibration (see Section 2), and spurious wavelength shifts caused by the star moving within the rather wide slit. We therefore have no observational constraint on the RV parameter, and instead treat it as a free parameter while fitting the observed monochromatic phases. That way, we were able to model the observed phases quite nicely, as can be seen in the right-hand panel of Fig. \[zoomampphase\]. Note that the y-axis now measures the phase in radians rather than in seconds as in Fig. \[ampphase\] for easier comparison to the computed values. This causes an apparent inversion of the curve due to a minus sign in the relation between phases specified in radians and those quoted in seconds. The radial velocity parameter used in the plot has a semi-amplitude of 11 km/s, which is quite similar to the values found for other high-amplitude EC 14026 pulsations from RV studies (e.g. $\sim$ 12 km/s for the dominant mode in PG 1325+101 from @telting2004, 19 km/s for the unusually strong pulsation in Balloon 09010001 from @telting2006, and 15 km/s for the main frequency in PG 1605+072 from @otoole2005). We believe that this consolidates our interpretation of the dominant pulsation as a radial mode. Nevertheless, it is clear that given the observational limitations on the S/N as well as the lack of an independent radial velocity estimate, the discriminative power for the degree index of the mode lies in the amplitude, rather than the phase information.
Considering the lower amplitude modes extracted from our data, the monochromatic quantities are a lot more noisy, making a unique mode identification impossible. Nevertheless, we repeated the exercise of overplotting the normalised theoretical amplitude curves on the measurements for $f_2$ and $f_3$ (the data for $f_4$ were too noisy due to the much lower amplitude). We used exactly the same theoretical values as for the dominant mode, because the small variation due to the difference in period is negligible when dealing with data this noisy. The results of the overplotting are shown in Fig. \[ampsf2f3\]. Unsurprisingly, the S/N of the observations is too low to distinguish between the low-order degree modes with $\ell=$ 0,1 and 2, however the higher degree modes present an inferior fit, and can quite safely be excluded. The corresponding monochromatic phases meanwhile are too noisy to gain any information whatsoever.
Conclusion
==========
In this study we attempted a mode identification of the main pulsation frequencies in the rapid sdB pulsator EC 20338$-$1925 on the basis of monochromatic amplitude and phase variations. We were able to obtain 3 half-nights of time-series spectrophotometry with the VLT instrument FORS2 in HIT-MS mode and detect four pulsations above the 4-$\sigma$ threshold. The S/N of the data was sufficient to extract high-quality monochromatic amplitudes and phases only for the dominant pulsation, noisier results being obtained for the remaining periodicities. Using full model atmosphere codes appropriate for the atmospheric parameters of our target and also incorporating non-adiabatic effects, we computed accurate theoretical amplitudes and phases for comparison to the observations. From the fits achieved to the observed amplitudes alone we determined the dominant pulsation to be a radial mode. This conclusion was found to be consistent with the monochromatic phase shifts observed under the assumption of a very reasonable radial velocity variation of 11 km/s. For two lower amplitude periodicities the data were found to be of sufficient quality to exclude higher degree modes with $\ell\geq$ 3.
To our knowledge, this is the first time that mode identification has been attempted for an EC 14026 star using monochromatic amplitudes. The fact that we were able to identify the main periodicity as a radial mode is therefore quite encouraging for future studies, particularly considering the relative faintness of our target. Moreover, the excellent match between theoretical predictions and the observations in the continuum confirms the basic validity of our approach and our models, although the discrepancies in the line cores indicate that there is still room for improvement, especially with respect to the chemical composition assumed in the model atmosphere calculations. Unfortunately, sdB stars are all chemically peculiar and contain heavy elements in varying abundances, implying that for a perfect match each target would first have to be submitted to a detailed abundance analysis and then analysed using a specially computed grid of model atmospheres. This is extremely time consuming both on the observational and the computational front, and not necessary for the practical application presented here, given that we have proved mode identification to be possible on the basis of the continuum behaviour alone.
It is quite striking that in the few cases where observational mode identification has been possible for rapidly pulsating subdwarf B stars, the dominant mode of pulsation was always found to be a radial mode (for Balloon 090100001 from @charp2008, KPD 2109+4401 from @randall2005, HS 2201+2610 from Silvotti et al. 2010, submitted, and now for EC 20338-1925). Of course, this cannot automatically be assumed to hold true for all the highest amplitude frequencies detected in EC 14026 stars, but it does seem reasonable to deduce that mode visibility across the disk of the star is a very important factor influencing the observed amplitude. Asteroseismological solutions assigning high-degree modes to the strongest frequencies observed can in all likelyhood be disregarded from the outset, as has occasionally been done in the past to distinguish between several possible “optimal models” [e.g. @randall2009]. This approach allows unambiguous model solutions to be found on the basis of fewer observed periods than if all observed frequencies are allowed to take on degree indices of $\ell$=0,1,2, and 4, and can therefore save valuable observing time.
One puzzling implication of our identification of the 146.9-s periodicity as a radial mode is that this frequency cannot then contain closely spaced components due to a rotationally split multiplet. Consequently, it is difficult to explain the strong amplitude variations observed over several years (see Fig. 3) in terms of the beating of unresolved frequencies. Given that the photometry data obtained by Dave Kilkenny have time baselines of several days, the two independent harmonic frequencies would have to be spaced less than $\sim$ 2 $\mu$Hz apart in order to not be resolved. Additionally, the strong amplitude variations observed can be produced only if the two pulsations have comparable amplitudes, effectively implying two very closely spaced modes with degree indices $\ell$ = 0 and/or 1. From our stellar models we know that the required proximity in frequency is impossible for modes of the same degree index, and extremely unlikely for the combination of $\ell=$ 0 and $\ell=$ 1 [see, e.g. @charp2000]. This is particularly true for a target with as high a surface gravity as EC 20338$-$1925, since the period density predicted is lower than for a more typical, less compact EC 14026 star. Therefore, we believe there is a high probability that the observed amplitude variations for the main mode are intrinsic to the star.
In conclusion, we have shown that the monochromatic amplitude and phase variations can be effectively used for mode identification in rapidly pulsating subdwarf B stars. The most critical factor is the quality of the observational data, as the S/N level attained limits the discriminative power especially between modes of low degree indices with $\ell\leq$ 2. Even though we were allocated observing time using a highly suitable instrument mounted on one of the world’s largest telescopes we were able to unambiguously identify the degree index for only the dominant, rather high amplitude mode. On the other hand, EC 20338$-$1925 is relatively faint, and there are several brighter targets available. Moreover, our observing run was affected by severe technical problems, due to which we lost a quarter of our allocated time completely, and did not obtain enough red data to constructively include it in our analysis. Therefore it seems likely that better results could be obtained in future similar studies.
It is not straightforward to objectively assess the relative merits of the different techniques currently in use for mode identification in subdwarf B stars, and opinions on this will vary. We believe that for the time being the exploitation of the pulsational amplitude’s signature on $\ell$ as a function of wavelength is the most promising method, and it is so far the only one to have yielded unambiguous determinations of the degree index $\ell$. Given that the work presented here constitutes the first study employing monochromatic rather than broadband amplitudes, it is too early to say which of these two closely related techniques is more efficient. One drawback of the amplitude-wavelength approach to mode identification is that medium-sized or large telescopes coupled with specialised instruments are needed in order to obtain rapid time-series data of sufficient quality. However, this is equally true for the study of line-profile variations, where the low S/N achievable even with the world’s largest telescopes has so far prevented unambiguous mode identification. Radial velocity surveys based on data from smaller telescopes have been successfully used to confirm periodicities previously detected from photometry, but we lack the theoretical framework necessary to use the information gained for mode identification. When supplementing simultaneous multi-colour photometry, radial velocity measurements can indeed improve the discriminative power of the former, as was shown by @baran2008 [@baran2010]. However, we would like to point out that for Balloon 09010001, the only case where a direct comparison of the two techniques is possible, four nights of standalone high S/N multi-colour photometry from a medium-size telescope [@charp2008] yielded an identification for significantly more modes than the combined results from 30 hours of simultaneous time-series spectroscopy obtained on a medium-size telescope and 120 hours of multi-colour photometry from a small telescope [@baran2008]. Therefore, we are confident that the exploitation of multi-wavelength amplitude information along the lines presented here makes the most efficient use of telescope time (at least until it becomes possible to obtain accurate monochromatic amplitudes and radial velocities with a single instrument), and will prove highly conducive to mode identification and, consequently, asteroseismology in the future.
S.K.R. would like to thank the Paranal Science Operations, Engineering and Software staff, in particular Kieran O’Brien and Pascal Robert, for working hard to get FORS2 ready for these observations. We are grateful to Heidi Korhonen for her input regarding the wavelength calibration, and to Dave Kilkenny for providing us with an unpublished list of pulsation frequencies and amplitudes. G.F. also wishes to acknowledge the contribution of the Canada Research Chair Program.
[^1]: Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile (proposal ID 083.D-0415).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
The description of spontaneous symmetry breaking that underlies the connection between classically ordered objects in the thermodynamic limit and their individual quantum mechanical building blocks is one of the cornerstones of modern condensed matter theory and has found applications in many different areas of physics. The theory of spontaneous symmetry breaking however, is inherently an [*equilibrium*]{} theory, which does not address the [*dynamics*]{} of quantum systems in the thermodynamic limit. Here, we will use the example of a particular antiferromagnetic model system to show that the presence of a so-called thin spectrum of collective excitations with vanishing energy –one of the well-known characteristic properties shared by all symmetry-breaking objects– can allow these objects to also spontaneously break time-translation symmetry in the thermodynamic limit. As a result, that limit is found to be able, not only to reduce quantum mechanical equilibrium averages to their classical counterparts, but also to turn individual-state quantum dynamics into classical physics. In the process, we find that the dynamical description of spontaneous symmetry breaking can also be used to shed some light on the possible origins of Born’s rule.
We conclude by describing an experiment on a condensate of exciton polaritons which could potentially be used to experimentally test the proposed mechanism.
author:
- Jasper van Wezel
title: 'Quantum Dynamics in the Thermodynamic Limit.'
---
1: Introduction
---------------
Combining many elementary particles into a single interacting system may result in collective behaviour that qualitatively differs from the properties allowed by the physical theory governing the individual building blocks. This realisation –immortalised by P.W. Anderson in his famous phrase ’More is Different’ [@Anderson72]– not only forms the basis of much of the research being done in condensed matter physics today, but has also found applications in areas ranging from string theory to cosmology. The theory of Spontaneous Symmetry Breaking which formalises these ideas first took shape over fifty years ago [@Landau37; @Goldstone62; @Anderson63:book; @Anderson52; @Anderson58; @Nambu60], and was completed in the context of quantum magnetism only two decades ago by the detailed description of the classical state as a combination of thin spectrum states, emerging as $N \to \infty$ because of the singular nature of the thermodynamic limit [@Lieb62; @Kaiser89; @Kaplan90]. The same description of the classical state emerging from the thin spectrum has since been shown to also directly apply to the cases of quantum crystals, antiferromagnets, Bose-Einstein condensates and superconductors [@vanWezel05; @vanWezel06; @vanWezel07; @vanWezel07:SC; @Birol07].
The connection between the quantum mechanical properties of microscopic particles and the classical behaviour of symmetry broken macroscopic objects has now again come to the forefront of modern science because of our technological capability to create ever larger and heavier quantum superpositions in the laboratory. Superconducting flux qubits harbour counterrotating streams of supercurrent consisting of up to $10^{11}$ Cooper pairs [@Wal00; @Chiorescu00; @Mooij:flux03], while Bose Einstein condensates of the order of $10^5$ Rubidium atoms can be routinely brought into superpositions of different momentum states [@Anderson95; @Davis95; @Stenger99; @Kozuma99]; Young’s double slit experiment has now been done using $C_{60}$ molecules instead of single photons or electrons [@Zeilinger99]; and an experiment has even been proposed to create a Schrödinger cat-like state of a mesoscopic mirror superposed over a macroscopically discernible distance [@Marshall03].
Almost all of these experiments employ the rigidity associated with a spontaneously broken symmetry to create and manipulate their ’macroscopic’ superpositions. Roughly speaking, the typical setup consists of a well defined, symmetry broken object in isolation (a superconductor, Bose Einstein condensate or crystal) which is brought into superposition by coupling it to a carefully selected quantum state. Although the theory of spontaneous symmetry breaking can be used to understand the stability and rigidity of macroscopic classical states such as superconductors or crystals, it says nothing about the quantum dynamics of such objects interacting with microscopic quantum states. The reason is that the standard description of spontaneous symmetry breaking is an inherently [*equilibrium*]{} description: it explains how macroscopic operators (such as the order parameter) can acquire finite expectation values and still be in stable equilibrium, but it does not say anything about the [*dynamics*]{} of these objects away from equilibrium.
A theoretical framework which does addresses the interaction of a macroscopic object with its microscopic quantum mechanical environment, is the study of decoherence [@Zurek81; @Joos85; @CaldeiraLeggett]. The basic idea of decoherence is that the entanglement of a certain quantum state with the many states of its environment can lead that state to behave effectively classically as long as the environmental states remain unobservable. This phenomenon has many practical implications, not in the least in the field of quantum information technology, where decoherence forms the main hurdle to be overcome in the race towards a working quantum computer. In the description of the interaction of a single macroscopic object with a single quantum state however, the theory of decoherence cannot be applied. The problem is that decoherence has to always refer to the properties of an [*ensemble average*]{}: after deciding which of the environmental degrees of freedom cannot be measured, one has to trace them out of the full density matrix describing the combined system of object and environment. Doing this (partial) trace is exactly equivalent to taking the quantum mechanical expectation value of the operators describing the unobserved states, and as such is only defined within an ensemble and cannot be used to say anything about the outcomes of [*single-shot*]{} experiments [@Adler; @Bassi]. In this paper, we will develop a description of dynamical spontaneous symmetry breaking that is meant to augment the earlier theories of equilibrium spontaneous symmetry breaking and decoherence in the areas where these theories do not apply. It will describe the quantum dynamics of individual experiments in which macroscopic and microscopic systems are allowed to interact. We will find that the presence of thin spectrum states in symmetry-broken objects allows these systems to also spontaneously break the unitarity of quantum mechanical time evolution. This result explains why truly macroscopic objects do not dynamically delocalise even if they are allowed to interact and entangle with an observable quantum mechanical environment. At the same time it also sheds light on what happens if the classical state is forced into a superposition state by an interaction with a carefully chosen quantum state.
In section 2 we start out with a short review of the equilibrium theory of spontaneous symmetry breaking. The role of the thin spectrum and the singular nature of the thermodynamic limit will be highlighted. In section 3 we then review the theory of decoherence and point out why it refers only to ensemble averages. We then turn to dynamic spontaneous symmetry breaking in section 4, using a model antiferromagnetic system as an example. It is argued there that the thin spectrum states and the thermodynamic limit can cooperate to allow the spontaneous breakdown of quantum mechanical unitarity. The resulting dynamics of a single quantum state in the thermodynamic limit is studied. We then continue in section 5 by describing the fate of a macroscopic object that is forced into a quantum superposition through the interaction with a microscopic quantum state. The results are again clarified using the example of the model antiferromagnet, and are shown to shed new light on the emergence of Born’s rule. Finally, in section 6, we describe a possible experimental test of the ideas of sections 4 and 5 using a condensate of exciton polaritons. We end in section 7 with a summary and conclusions.
2: Equilibrium Spontaneous Symmetry Breaking
--------------------------------------------
Classically, spontaneous symmetry breaking just corresponds to the evolution from a high symmetry metastable state into a ground state with lower symmetry. Quantum mechanically however, the situation becomes a bit more involved. First of all, there are in general nonzero tunneling matrix elements between different symmetry broken states, so that strictly speaking time evolution should cause any symmetry broken state to spread out and restore its symmetry. In practise though, this finite lifetime of a symmetry broken state can be easily shown to be long compared to the age of the universe for any realistic macroscopic system. Secondly, the symmetry broken states of a finite size system do not have to be ground states. In fact, they usually are not even eigenstates of the system. To establish how the system can end up in a state that is not an eigenstate of the underlying Hamiltonian, we will here use the specific example of the Lieb-Mattis model [@Lieb62; @Kaiser89; @Kaplan90; @vanWezel06; @vanWezel07]. This model is defined by the Hamiltonian: $$\begin{aligned}
H_{\text{LM}}
&= \frac{2 J}{N} {\bf S}_A \cdot {\bf S}_B \nonumber \\
&= \frac{J}{N} \left[ S^2 - S_A^2 - S_B^2 \right].
\label{Hlm}\end{aligned}$$ Here $N$ spin-$\frac{1}{2}$s are distributed over a bipartite lattice, with ${\bf S}_{A/B}$ the total spin of the $A/B$ sublattice and $S^z_{A/B}$ its $z$-projection. Each spin on the $A$ sublattice thus has an interaction with every spin on the $B$ sublattice and vice versa. The positive interaction strength $J$ is divided by $N$ to make the model extensive. $S$ is the total spin of the combined sublattices: $S=S_A+S_B$. The reason for considering specifically the Lieb-Mattis model with its infinitely long ranged interactions, is that it captures the relevant physics of a broad class of Heisenberg models with short ranged interactions. To say that a particular model for an antiferromagnet is invariant under SU(2) spin rotations is equivalent to stating that its Hamiltonian commutes with the total spin operator: $\left[H,S^2 \right]=0$. It is thus immediately obvious that total spin is a good quantum number for any isotropic antiferromagnet and that all their eigenstates can be labelled by such a total spin quantum number. For the description of the collective properties of the system as a whole (i.e. strictly infinite wavelength), the total spin is the only relevant part of the Hamiltonian. As far as the total spin is concerned, the Lieb-Mattis model coincides exactly with all other antiferromagnetic models. That is to say, if one takes [*any*]{} model for an antiferromagnet with short ranged interactions (such as for example the nearest neighbour Heisenberg model) and looks at the model in Fourier space, then the $k=0$ and $k=\pi$ modes together form [*exactly*]{} the Lieb Mattis-Hamiltonian [@vanWezel06; @vanWezel07]. At the same time, the finite wavelength, $k\neq 0,\pi$ modes are gapped and dispersionless in the Lieb-Mattis model due to the infinite ranged interactions, which makes it ideally suited for studying just the collective behaviour of antiferromagnets. The discussion of this model can also be easily adapted to describe spontaneous symmetry breaking in quantum crystals, superconductors and Bose-Einstein condensates [@vanWezel06; @vanWezel07; @vanWezel07:SC; @Birol07].
From the second expression in equation it is immediately clear that the ground state of the Lieb-Mattis system is a singlet state with zero total spin. This non-degenerate ground state is isotropic in spin space and thus fully respects the symmetry of its Hamiltonian. The heart of the workings of spontaneous symmetry breaking lies in the realisation that every many-particle Hamiltonian which possesses a continuous symmetry that is unbroken in its ground state (such as the Lieb-Mattis Hamiltonian), gives rise to a tower of low-energy states called the *thin spectrum* [@vanWezel05; @vanWezel07]. The states in this thin spectrum represent global (infinite wavelength) excitations that can be seen as the centre of mass properties of the collective system [@vanWezel07; @Birol07]. In the present model of equation the thin spectrum consists of total spin states, which only cost an energy of order $J/N$ to excite. These states thus become degenerate with the symmetric ground state in the thermodynamic limit. They are called the [*thin*]{} spectrum of the model because of the vanishing weight that these states have in the partition function. Excitations that change the size of the sublattice spins are separated from the ground state by an energy gap of size $J$, and can thus be ignored in the present (low energy) discussion. Without loss of generality we also set the $z$ projection of the total spin to be zero from here on.
The crucial observation is now that the strength of the field needed to give rise to a fully ordered ground state depends on the total number of particles, $N$, in the system. Because the energy separation between two consecutive thin spectrum states scales as $1/N$, the field strength necessary to explicitly break the symmetry decreases with system size. In the thermodynamic limit (where $N \to \infty$) all of the thin spectrum states collapse onto the ground state to form a degenerate continuum of states. Within this continuum even an *infinitesimally* small symmetry breaking field is enough stabilise a fully ordered, symmetry broken ground state. The system is thus said to spontaneously break its symmetry in that limit. To make this explicit in the present model, we add a symmetry breaking staggered magnetic field to the Hamiltonian: $$\begin{aligned}
H_{\text{LM}}=\frac{2 J}{N} {\bf S}_A \cdot {\bf S}_B - B \left( S^z_A - S^z_B \right).
\label{HlmSB}\end{aligned}$$ The staggered magnetisation only has non-zero matrix elements between consecutive thin spectrum levels [@vanWezel07]: $$\begin{aligned}
\left< S' \left| S^z_A - S^z_B \right| S \right> & = \delta_{S'+1,S} f_S + \delta_{S'-1,S} f_{S'} \nonumber \\
& \simeq \frac{N}{4} \left( \delta_{S'+1,S} + \delta_{S'-1,S} \right),
\label{Bmatrix}\end{aligned}$$ where $f_S \equiv \sqrt{\{ S^2 [ \left(S_A+S_B+1\right)^2 -S^2 ] \} /\{4S^2-1\}}$, and the approximation in the last line holds if $S_A=S_B=N/4$ and $1 \ll S \ll N$ [@Kaiser89]. The Schrödinger equation for the Lieb-Mattis model, $H_{\text{LM}} |n\rangle = E^n |n\rangle$, can be expanded in the total spin basis using $|n\rangle \equiv \sum_S u_S^n |S\rangle$. Upon taking the continuum limit it then reads $$\begin{aligned}
-\frac{1}{2}\frac{\partial^2}{\partial S^2} u_S^n + \frac{1}{2} \omega^2 S^2 u_S^n = \nu^n u_S^n,
\label{un}\end{aligned}$$ with $\omega=2/N \sqrt{J/B}$ and $\nu^n = 2 E^n / (B N) + 1$. This equation describes a harmonic oscillator and its eigenfunctions are given in terms of the well known Hermite polynomials. The expansion of these harmonic wavefuntions in the total spin basis brings to the fore the crucial role played by the thin spectrum in the mechanism of spontaneous symmetry breaking: because the total spin states all become degenerate in the limit $N \to \infty$, it then becomes arbitrarily easy to create the antiferromagnetic Néel state $|n=0\rangle = \sum_S u_S^0 |S\rangle$. Mathematically this translates into the non-commuting limits for the equilibrium expectation values of the order parameter [@vanWezel07] $$\begin{aligned}
\lim_{N \to \infty} \lim_{B \to 0} \left<\frac{S_A^z-S_B^z}{N/2}\right> &= 0 \nonumber \\
\lim_{B \to 0} \lim_{N \to \infty} \left<\frac{S_A^z-S_B^z}{N/2}\right> &= 1.
\label{limits}\end{aligned}$$ The same instability can also been seen by looking at the energy of the ground state in the presence of the symmetry breaking field. That energy is proportional to $-NB$ and thus an infinite amount of energy could be gained in the thermodynamic limit by aligning with an infinitesimally small symmetry breaking field.
An alternative, equivalent way of phrasing this singular property of the thermodynamic limit is to say that the limits of equation imply that even in the absence of $B$, quantum fluctuations of the order parameter which tend to disorder the symmetry broken state take an infinitely long time to have any measurable effect on a truly macroscopic system. Under equilibrium conditions, the system will thus be stable in a symmetry broken state that is not an eigenstate of its Hamiltonian.
Strictly speaking equation only allows truly infinite-size systems to spontaneously select a direction for their sublattice magnetisation. A large, but not infinitely large, system requires a finite symmetry breaking field to stabilise one of the symmetry broken states over the exact ground state. A true staggered magnetic field that points up on each site of the $A$ sublattice and down on the $B$ sublattice does not exist in nature. Because the strength of the required field becomes increasingly weaker as the size of the antiferromagnet grows, it can be argued however that [*any*]{} field which has a component that resembles a staggered magnetic field will be enough to stabilise the symmetry broken state in a large enough antiferromagnet. Such a weak staggered field could be provided in practise by magnetic impurities, local fields or even by a second antiferromagnet at an ever increasing distance from the first.
3: Decoherence
--------------
We have seen in the last section how spontaneous symmetry breaking enables a macroscopic collection of quantum mechanical particles to occur in an effectively classical symmetry broken state under equilibrium conditions. A different route from quantum mechanics to effectively classical behaviour is provided by the process of decoherence. Decoherence happens on all length scales (i.e. it does not require the object of interest to be macroscopic), and is a direct consequence of the inability of observers to monitor each and every degree of freedom of a typical quantum environment. At the heart, decoherence is the process in which a carefully prepared quantum state gets entangled with different states in its environment. Because the observer cannot measure all states of the environment, he can see only part of the final entangled state, and this partial state looks effectively classical. In this section we will use the Lieb-Mattis model as an example to highlight the different conceptual steps involved in the decoherence process.
Consider the Hamiltonian of equation . Its eigenstates can be written as $|m,S\rangle \equiv |S_A=S_B=N/4-m/2,S\rangle$ (where we have assumed $S^z=0$ and $S_A=S_B$ without loss of generality). The excitations $m$ represent magnons or spin waves while the excitations $S$ form the thin spectrum of this model. Because the thin spectrum excitations make only a vanishingly small contribution to the free energy of the Lieb-Mattis antiferromagnet if $N$ is large, they will be very hard to observe experimentally (for relatively small $N$ the thin spectrum states of molecular antiferromagnets can and have been experimentally observed [@Waldmann03; @Waldmann05]). For large systems we can thus regard the thin spectrum as a ’quantum environment’ for the magnon excitations. To study decoherence in this system we will first prepare a superposition state in the magnon sector, then we will let the magnon and the thin spectrum excitations interact and become entangled, and finally we will disregard the thin spectrum states and find that magnon states on their own have become an effectively classical mixture.
To prepare the initial magnon superposition, let us assume that we can access the exact ground state of the $N$-spin system and subsequently let it interact with a separate two-spin singlet state $\sqrt{1/2} \left[ |\uparrow \downarrow \rangle - |\downarrow \uparrow\rangle \right]$ through the instantaneous interaction defined by: $$\begin{aligned}
H = \left\{ \begin{array}{ll}
\frac{2 J}{N} {\bf S}_A \cdot {\bf S}_B + J {\bf S}_1 \cdot {\bf S}_2 & \text{for}~t<0 \\
\frac{2 J}{N+2} \left({\bf S}_A + {\bf S}_1 \right) \cdot \left( {\bf S}_B + {\bf S}_2 \right) & \text{for}~t>0.
\end{array} \right.
\label{Hint}\end{aligned}$$ Here ${\bf S}_{1/2}$ refer to the two initially separated spins, and the interaction is turned on at time $t=0$. In terms of the eigenstates of the Hamiltonian at positive times, the initial state can easily be shown to correspond to the state $\sqrt{1/2} \left[|m=0,S=0\rangle - |m=2,S=0\rangle \right]$ for large $N$ (where now $m$ and $S$ refer to the $N+2$-spin system). That is, for large $N$ the initial state of the two-spin system is encoded in the number and relative phase of the magnon excitations in the final state [@masterthesis].
Next, we would like to entangle the magnons with the thin spectrum so that the quantum information initially encoded in the magnon states is spread out over the environment. One way of achieving this is to instantaneously introduce a symmetry breaking field $B \left( S^z_A + S_1^z - S^z_B - S_2^z \right)$ into the Hamiltonian at some positive time $t_0$. After some straightforward algebra the state of our systems at times $\tau=t-t_0$ is then found to be $$\begin{aligned}
\hspace{-6pt} \left| \psi \right> = \sqrt{\frac{1}{2}} \sum_{n,S} u_S^n u_0^n \left[ e^{-\frac{i}{\hbar}E^n_0 \tau} \left| 0,S \right> - e^{-\frac{i}{\hbar}E^n_2 \tau} \left| 2,S \right> \right]
\label{state}\end{aligned}$$ where $u_S^n$ are the harmonic wavefunctions defined in equation and $E^n_m$ is the energy of the $n^{\text{\tiny{th}}}$ harmonic wavefunction in the presence of $m$ magnons. We can write this final entangled state in the form of a density matrix through the definition $\rho(\tau) = | \psi \rangle \langle \psi |$. Notice that all the quantum information encoded in the initial two-spin singlet state is still present in the final density matrix $\rho(\tau)$. Because purely quantum mechanical time evolution is always strictly unitary, time inversion symmetry is automatically preserved and there is always a way (at least in principle) to evolve the system back to its original state. If we now decide that the thin spectrum states are unobservable, and trace them out of our density matrix [@Neumann55], we end up with a reduced density matrix describing the dynamics of the magnons only. In doing so however, the time inversion symmetry is lost along with some of the quantum information. To be specific, the reduced density matrix $\rho_{\text{red}}$ will be given by: $$\begin{aligned}
\rho_{\text{red}}(\tau) &= \text{Tr}_{\text{thin}} \phantom{.} \rho(\tau) \nonumber \\
&= \sum_{S} \left<S \left| \right. \psi \right> \left< \psi \left| \right. S\right> \nonumber \\
&= \sum_{m,m'} \left|m\right> \left\{ \sum_S \psi^{\phantom{*}}_{m,S} \psi^{*}_{m',S} \right\} \left<m'\right|.
\label{reduced}\end{aligned}$$ In the last line we have written the entangled wavefunction as $|\psi\rangle=\sum_{m,S} \psi(m,S) |m\rangle|S\rangle$ to show explicitly that taking the partial trace over the thin spectrum states is equivalent to calculating the usual quantum mechanical ensemble-averaged expectation value with respect to these states.
To complete the analysis of our model interaction, we should explicitly calculate the reduced density matrix elements of equation . The diagonal elements of the resulting $2$x$2$ matrix are easily seen to be $1/2$. For the off-diagonal elements the calculation involves a summation over terms which differ only by the phase factor $e^{-\frac{i}{\hbar}(E^n_0-E^n_2) \tau}$. After some algebra one finds that these phases interfere destructively [@masterthesis], so that after a time $\tau_{\text{coh}}\sim \hbar / \sqrt{J B}$ the reduced density matrix becomes effectively diagonal. We thus find that the initial, pure density matrix loses its coherence and becomes a diagonal, mixed reduced density matrix within a time $\tau_{\text{coh}}$. Because for large enough $N$ the environmental states are unobservable this constitutes a ’for all practical purposes’ reduction from quantum to classical physics within the ensemble average. In any one single, individual experimental realisation of the above procedure however, one ends up with the full density matrix defined by equation , and one cannot use the expectation values of equation to conclude anything about that one specific experiment. In particular, in the classic Young’s double slit experiment, the observation that each single electron produces only a single dot on the photographic plate, can [*not*]{} be explained by invoking decoherence and averaging over the many degrees of freedom of the plate [@Tonomura89; @Adler; @Bassi].
Although the presence of the thin spectrum can lead to decoherence in real qubits [@vanWezel05], the interaction of the Lieb-Mattis antiferromagnet and the two-spin state considered in this section is of course a highly pathological example. In reality there will never be infinite ranged interactions, instantaneous changes to the Hamiltonian or full experimental control over the prepared states. Moreover, experiments typically involve finite temperatures and external environments that do not resemble the thin spectrum states of our model. However, the general idea of constructing a meaningful quantum superposition, letting it interact and entangle with its environment, and then looking only at the result averaged over the environmental degrees of freedom to find decoherence, remains essentially unaltered in more realistic situations [@CaldeiraLeggett]. In particular the conclusion that the the theory of decoherence is applicable only within the realm of ensemble averages remains intact throughout.
4: Dynamic Spontaneous Symmetry Breaking
----------------------------------------
As we have seen, both the theory of spontaneous symmetry breaking and the theory of decoherence have only a limited domain of applicability. Because macroscopic states typically have a lot of interaction with their environments, decoherence explains the reduction of pure macroscopic states to mixed states in situations where not all degrees of freedom can be explicitly monitored, but only in an (ensemble) averaged sense. Spontaneous symmetry breaking on the other hand can be used to demonstrate the stability of macroscopically ordered, classical states using the singular nature of the thermodynamic limit and the properties of the thin spectrum, but only under equilibrium conditions. The most general situation involving macroscopic objects –that of individual-state quantum dynamics in the thermodynamic limit– cannot be addressed within either of these frameworks.
In this section we will show that the presence of a thin spectrum in objects that can undergo spontaneous symmetry breaking also allows these objects to spontaneously break the (unitary) time translation symmetry of quantum mechanical time evolution. The resulting dynamical version of the process of spontaneous symmetry breaking naturally leads to the observed stability of macroscopic objects even in the presence of interactions with a quantum environment.
The approach to spontaneously breaking time translation symmetry is exactly analogous to the spontaneous breaking of more usual symmetries: we will introduce a vanishingly small non-unitary perturbation to the free Hamiltonian and demonstrate that this results in a qualitative change to the dynamics of a macroscopic object, even in the limit of taking the field strength to zero. The conclusion must thus be that the quantum dynamics of these macroscopic objects is infinitely sensitive to any non-unitary perturbation of the type considered. In other words: purely unitary quantum dynamics is unstable in the thermodynamic limit in the same way that the total spin singlet state of a macroscopic antiferromagnet is an unstable state under equilibrium conditions. As a result the unitary time translation symmetry of macroscopic quantum objects will be spontaneously broken and give rise instead to classical dynamics.
At this point one may wonder about the physical origin of the symmetry breaking field. As with the usual equilibrium symmetry breaking, large but finite sized systems will require a very small but nonetheless finite symmetry breaking field. Non-unitary fields however are strictly forbidden in quantum theory. The origin of a non-unitary symmetry breaking field must therefore lie outside of quantum mechanics. There are many possible candidates that could in principle insert a vanishingly small non-unitary correction into quantum mechanics. A notable example is the theory of general relativity, in which general covariance rather than unitarity is the guiding principle. Because of this, gravity has (in a different setting) been considered before as a possible non-unitary influence on mesoscopic systems [@Diosi89; @Penrose:96; @vanWezel:penrose]. In this paper we will not speculate about the possible origins of the non-unitary field, but merely recognise that there are non-unitary physical theories outside of the realm of quantum mechanics, and that only an infinitesimally small contribution from one of these sources would be enough to spontaneously break the unitarity of quantum dynamics in the thermodynamic limit.
We thus consider once again the Lieb-Mattis model for an antiferromagnet, but now in the presence of a non-unitary symmetry breaking field: $$\begin{aligned}
H=\frac{2 J}{N} {\bf S}_A \cdot {\bf S}_B + i b \left( S^z_A - S^z_B \right).
\label{Hdssb}\end{aligned}$$ The rationale of which specific form of non-unitary field is to be included in this equation is again exactly analogous to the case of equilibrium spontaneous symmetry breaking: one should in principle consider every conceivable field. The system will of course be stable with respect to the vast majority of them, but as long as there is one that has an effect in the limit in which its strength is sent to zero, the system will be unstable. In the equilibrium case considered before, we have seen that the symmetric singlet state is unstable with respect to a staggered magnetic field along the $z$-axis. We could have also considered other symmetry breaking fields, such as a uniform magnetic field along the $z$-axis. It is easy to show however that such a field would not lead to the non-commuting limits of equation . The Lieb-Mattis system is thus shown to be unstable under equilibrium conditions with respect to antiferromagnetic ordering, but not with respect to ferromagnetic ordering. The situation in the dynamical case is analogous: most fields have no effect on the quantum dynamics of the system if their strength is sent to zero; But as soon as there is one field that does influence the dynamics even if it is infinitesimally weak, the dynamics is found to be unstable. Notice also that in the equilibrium case, the antiferromagnet is in fact unstable towards staggered magnetic fields along any axis. In practise, the resulting orientation of the order parameter is therefore randomly chosen, just as in the case of classical symmetry breaking. In equation we have chosen a non-unitary version of the staggered magnetic field to break time translational symmetry, because to be able to have an effect in the thermodynamic limit, the symmetry breaking field must couple to the order parameter of the system. The orientation along the $z$-axis rather than any other axis is chosen for convenience only.
The time evolution operator $U(t) \equiv \exp (-iHt/\hbar)$ implied by equation has a non-unitary component, and thus no longer automatically conserves the total energy of the system (defined as $\langle H \rangle$ with $b \to 0$). This problem is automatically solved in the thermodynamic limit though. The staggered magnetisation only has non-zero matrix elements between consecutive states in the thin spectrum (see equation ). Since all thin spectrum states become degenerate with the ground state in the limit $N \to \infty$, the time evolution defined through $H$ cannot alter the total energy of the system in that limit. Other problems that are usually associated with non-unitary quantum dynamics (conservation of normalisability, commutativity, and so on) are likewise automatically solved in the limit of vanishing $b$ and large $N$.
To visualise the time evolution defined by $U(t)$, consider a general initial state $| \psi(t=0) \rangle = \sum_S \psi_S(t=0) |S\rangle$ (we again take $S_A$ and $S_B$ maximal and $S^z=0$). Using $| \psi(t) \rangle = U(t) | \psi(0) \rangle$ we then find the generalised (non-unitary) Schrödinger equation to be: $$\begin{aligned}
\hspace{-6pt} \dot{\psi}_S = \frac{-i}{\hbar}\frac{J}{N} S(S+1) \psi_S + \frac{b}{\hbar} \left( f_{S+1} \psi_{S+1} + f_S \psi_{S-1} \right)
\label{psidot}\end{aligned}$$ with $f_S$ the matrix elements defined in equation . This differential equation for the time evolution of a general initial wavefunction cannot easily be solved analytically (taking the limit in which $S$ becomes a continuous variable and $1 \ll S \ll N$, there is a solution in terms of Whittaker functions, but this explicit solution is not very enlightening for our present purposes). One can however integrate equation forward in time numerically, and we can study the effect of the unitarity breaking field by comparing the resulting time evolutions of different initial states. Two initial states of particular interest are the completely symmetric singlet state and the symmetry broken antiferromagnetic Néel state.
![(Color online) Left: The staggered magnetisation as a function of time. To make the plot the values $J=10$ and $b=1$ were used, and time was measured in units of $\hbar s$. The curves range from $N=20$ (rightmost curve) to $N=400$ (leftmost curve) and represent the evolution starting from the completely symmetric singlet state.\
Right: The dependence of the halftime on the parameters of the model. The top plot shows that $t_{1/2} \propto 1/b$, the middle plot that $t_{1/2} \propto 1/N$, and the bottom plot that $t_{1/2}$ is independent of $J$.[]{data-label="symplot"}](Fig1){width="\columnwidth"}
In the case of the symmetric initial state the time evolution of equation leads the unitarity breaking field to amplify the weight of states with a finite order parameter (i.e. its component in the wavefunction becomes a monotonously increasing exponential function), so that a fully ordered state is quickly formed. In figure \[symplot\] the time evolution of the order parameter is shown for different values of $b$, $J$ and $N$. It is immediately clear that the half-time associated with the reduction towards an ordered state must be proportional to $1/(Nb)$, so that the thermodynamic limit in this case is found to be a singular limit: if we let $b$ go to zero before sending $N$ to infinity, the symmetric singlet state remains an eigenstate of $H$ and under time evolution it can only pick up a total phase; if on the other hand even just an infinitesimally small field $b$ is present while the thermodynamic limit is taken, the time evolution governed by $H$ gives rise to an instantaneous reduction of the symmetric state to the fully ordered state with the order parameter pointing in the direction of $b$. Analogous to the equilibrium description, this non-commuting order of limits signals the sensitivity of the system to even infinitesimally small perturbations. In this case it is the unitary time translational symmetry of quantum dynamics itself that is spontaneously broken, and as a result the symmetric singlet state will be spontaneously and instantaneously reduced to an ordered Néel state.
![(Color online) Left: The staggered magnetisation along the $z$ axis as a function of time. To make the plot the values $J=10$ and $N=200$ were used, and time was measured in units of $\hbar s$. The curves range from $b=0.1$ (rightmost curve) to $b=2$ (leftmost curve) and represent the evolution starting from the state with full antiferromagnetic order along the $x$ axis.\
Right: The dependence of the halftime on the parameters of the model. The top plot shows that $t_{1/2} \propto \sqrt{1/b}$, the middle plot that $t_{1/2}$ is independent of $N$, and the bottom plot that $t_{1/2} \propto \sqrt{1/J}$.[]{data-label="xtozplot"}](Fig2){width="\columnwidth"}
Starting from a fully ordered initial state, the picture changes drastically. The state which has antiferromagnetic order aligned with the field $b$ to start with, will not be influenced at all. That state is just a stable state with respect to the generator of time evolution $U(t)$. The evolution of the initial state with full Néel order along the $x$ axis (at a $90$ degree angle with the field $b$) is shown in figure \[xtozplot\]. The effect of the presence of the unitarity breaking term is clearly to align the initial order parameter with the field $b$. The timescale on which this process takes place however is proportional to $\sqrt{1/(Jb)}$. This time is just the ergodic time of the Lieb-Mattis system and it becomes infinitely long in the thermodynamic limit with a vanishing symmetry breaking field. The difference between this ’turning time’ and the ’ordering time’ of the symmetric state considered before is due to the fact that for large objects any fully ordered state becomes exactly orthogonal to all differently ordered states, while the symmetric state always keeps a finite overlap with all of them [@Anderson:SolidState]. The lifetime of the symmetric state is therefore determined simply by the strength of the amplification due to the unitarity breaking field, while the turning time of a fully ordered initial state is set by the ergodic time of the system. Starting from the ordered state, the limit $N \to \infty$ is thus no longer singular: regardless of the size of $N$, the limit $b \to 0$ will reduce any dynamics to just the standard quantum mechanical time evolution. The dynamics of the ordered state, in other words, is stable with respect to the unitarity breaking field $b$.
Summarising, it has become clear that even an infinitesimally small unitarity breaking field is enough in the thermodynamic limit to instantaneously convert a symmetric initial state into a fully ordered state. Once such an ordered state has been formed however, it is stable with respect to any differently aligned unitarity breaking field. The former instability explains why the interaction with its environment cannot cause the wavefunction of a macroscopically ordered state to spread. After all, the more symmetric, spread-out wavepacket would be an unstable state, and it would spontaneously and instantaneously be brought back to the ordered state. At the same time the stability of the macroscopically ordered state itself ensures that such a state cannot spontaneously change the direction of its order parameter.
In the above analyses we have only considered symmetry breaking fields that are constant in time. Because of the dynamical nature of the spontaneous symmetry breaking process, it would actually be more natural to also include time dependent non-unitary fields. Since the strength of the field is taken to be infinitesimal, the time dependence of such a field must lie in its spatial orientation. As we have seen, ordered states are stable with respect to any orientation of the symmetry breaking field, and will thus also be stable with respect to a fluctuating field. The symmetric state on the other hand is sensitive to the direction of the field $b$: it is along this direction that the ordered state is formed. A fluctuating symmetry breaking field will thus cause the quantum dynamics of a symmetric state to amplify different orientations of the order parameter at different times. As a result both the direction and the size of the overall staggered magnetisation will undergo a random walk. As soon as the size of the magnetisation is large enough however, the dynamics again reduces to that of the ordered state, and the influence of the symmetry breaking field will no longer be felt. Because the symmetric state reacts infinitely fast to an infinitesimal perturbation in the thermodynamic limit, the whole process of undergoing a random walk and picking out an orientation for the order parameter will still be effectively instantaneous, and the earlier conclusions about the stability of quantum dynamics in the thermodynamic limit remain unaltered even in the presence of a fluctuating field.
5: Macroscopic Superpositions and Born’s Rule
---------------------------------------------
Having established that a macroscopically ordered state is stable and will not be driven into a quantum superposition of differently ordered states by its environment, the question arises what would happen to a macroscopic system that is forced into a superposition by some strong external force. Instead of a gentle and continuous spreading of the wavepacket (such as the one caused by the environment, which is subject to the instability discussed before), consider a quantum mechanical operation which quickly drives a macroscopic system into a superposition of ordered states with well separated orientations of their order parameters (the instantaneous coupling of the order parameter to a quantum superposition would in general do the trick). To be specific, consider the initial state $$\begin{aligned}
\left| \psi(0) \right> = \alpha \left| AFM \right>_x + \beta \left| AFM \right>_z.
\label{psi0}\end{aligned}$$ Here $|AFM\rangle_x$ signifies an antiferromagnetic Néel state ordered along the $x$ axis. The time evolution of the order parameter measured along the $z$ axis, starting from the initial state with $\alpha=\beta=\sqrt{1/2}$ is shown in figure \[theta\]. Here we again consider a constant symmetry breaking field $b$ and the time evolution defined by equation .
![(Color online) The evolution of the order parameter as a function of time (in units of $\hbar s$) for different constant orientations of the unitarity breaking field. Each set of three curves consists of different numbers of spins which are initially prepared in an equal-weight superposition of being ordered along the $z$ axis and along the $x$ axis. The angle $\theta$ between the unitarity breaking field and the $z$ axis $0.2~\pi$ for the upper set, $0.25~\pi$ in the middle and $0.3~\pi$ for the lowest set. The inset shows the fate of the order parameter in the thermodynamic limit, as a function of $\theta$.[]{data-label="theta"}](Fig3){width="0.8\columnwidth"}
The evolution of this initial state can be seen as a a combination of the two processes encountered before. First there is a fast reduction of the initial state to a single ordered state within a timescale $\propto 1/(Nb)$. The choice of which ordered state results from this fast initial evolution depends only on the chosen direction of the unitarity breaking field, and not on the weights of the different ordered states in $|\psi(0)\rangle$ (as can be seen in figure \[bornplot\]). After the fast reduction to a single ordered state, the slow process of rotating the order parameter towards alignment with the field $b$ takes over. This secondary process happens in a time which scales as $\propto \sqrt{1/(Jb)}$. In the limit that the number of particles goes to infinity before the unitarity breaking field is sent to zero, the result is thus a spontaneous, instantaneous reduction of the initial state to just a single one of the ordered states present in the original superposition.
The observation that the selection of the ordered state to be singled out by the spontaneous dynamics depends on the chosen (constant) orientation of $b$ signifies the fact that the initial state is unstable with respect to two different and competing perturbations: one for each orientation of the order parameter present in the initial superposition. The two possible stable final states are mutually exclusive since for any choice of unitarity breaking field, only one orientation of the staggered magnetisation results.
As mentioned before, the dynamical nature of the symmetry breaking process implies that we should really consider a time-dependent, fluctuating symmetry breaking field rather than only a constant field. In the presence of such a fluctuating field, it is clear that there must be a competition between the two instabilities of the initial state. In general, this gives rise to a statistical outcome of the reduction process (just like the instabilities of the singlet state gave rise to a statistical, random selection of the orientation of its order parameter under equilibrium conditions). The resulting dynamic process could be somewhat reminiscent of the evolutions considered in the GRW and CSL models for quantum state reduction [@Pearle89; @Ghirardi90], and consist of a random sequence of amplifying one or the other ordered state until one of them completely dominates.
![(Color online) The evolution of the order parameter as a function of time (in units of $\hbar s$) starting from the superposition state $\sqrt{ {1/5}} \left| AFM \right>_x + \sqrt{ {4/5}} \left| AFM \right>_z$. Each set of three curves represents the time evolution with three different values for $N$ in the presence of a single, constant orientation of the unitarity breaking field $b$. From top to bottom the angle between $b$ and the $z$ axis for the different sets is $0.1~\pi$, $0.2~\pi$, $0.25~\pi$, $0.3~\pi$ and $0.4~\pi$. The point at which the initial fast reduction process starts favouring the $x$ orientation over the $z$ orientation is seen to be at $0.25~\pi$.[]{data-label="bornplot"}](Fig4){width="0.8\columnwidth"}
It was shown recently by Zurek, using the concept of ENVariance [@Zurek03], that one can obtain conclusions about the statistics of the final results of a dynamic competition between instabilities such as the one considered here, without knowing the exact dynamics governing the competition process [@Zurek05]. It is shown in the appendix that Zurek’s proof is applicable here without the need for any assumptions regarding our system. Following his derivation one finds that the only possible result of the dynamic competition between different instabilities of the initial state of equation is the emergence of Born’s rule: the probability of a certain direction of the order parameter emerging from the process is given by the square of its weight in the initial wavefunction [@Born26]. Notice that this result is not an expectation value: it is valid even for the quantum dynamics of a single macroscopic object that is forced into a superposition state.
6: Experimental Predictions
---------------------------
The dynamic spontaneous symmetry breaking process described in the previous sections results in unaltered, purely unitary quantum dynamics for microscopic particles, but also gives rise to spontaneous and non-unitary effects in the thermodynamic limit. For truly macroscopic objects the non-unitarity will be effectively instantaneous, and the quantum dynamics of such objects correspondingly reduces to classical physics. Somewhere in between the micro and macro scales however, there must be a class of mesoscopic objects which are just sensitive enough to the presence of a small (but finite) time translation symmetry breaking field to undergo non-unitary dynamics on timescales that are measurable by human standards. The scale at which this happens should in fact be the same scale at which collections of interacting quantum particles become large enough to be meaningfully ascribed a (stable) orderparameter and considered classical, symmetry broken objects under equilibrium conditions. This prediction can in principle be exploited to experimentally test the ideas which are put forward in this paper.
The greatest obstacle in realising such an experimental test will be decoherence. Quite apart from the issue of its applicability to only ensemble averages, decoherence is of course a real physical phenomenon which severely complicates the observation of quantum effects in systems coupled to a reservoir. To observe the breakdown of unitary quantum dynamics, one will thus have to find a way to experimentally distinguish its effects from those of the usual environmental decoherence. The most obvious way of doing that is to look at single-shot experiments only. If the famous experiment of Zeilinger et al. [@Zeilinger99], interfering C$_{60}$ molecules one at a time, could be scaled up to truly macroscopic proportions, it would form the ideal testing ground for observing the transition from quantum to classical behaviour. The crossover scale could then be directly compared with the scale at which ordering and rigidity appear under equilibrium conditions, and this could be used to examine the role played by dynamic spontaneous symmetry breaking. Such macroscopic interference experiments however, seem to be very far from what can presently be experimentally realised.
We thus have to look for a different Schrödinger-cat like state of a mesoscopic system which is large enough to feel the effects of non-unitarity, but small enough to still have a measurably long reduction time. Creating such mesoscopic superpositions in the lab surely is not an easy task, but significant experimental progress towards its realisation is already being made in setups in for example quantum computation (superconducting flux qubits and Cooper pair boxes) or cold atom physics (Bose Einstein condensates in optical traps). Note however that the superposition must be a combination of states with different orientations of the order parameter itself. Superpositions of elementary excitations (such as magnons, phonons or supercurrents) which do not affect the order parameter, are not subject to the spontaneous reduction process described in the previous sections, even in a truly infinite system. Although distinguishing any non-unitary dynamics from the effects of decoherence is known to be very hard in most systems [@vanWezel:penrose], there is at least one experimental arena in which there seems to be, at least in principle, an opportunity for doing so: the Bose-Einstein condensation of exciton polaritons in semiconductor microcavities [@Kasprzak06; @Keeling07; @Wouters07].
Exciton polaritons are composite particles built partly from particle-hole pairs (excitons) and partly from photons. This unique combination of light and matter allows the particles to have strong interactions (due to their excitonic nature) while also being susceptible to direct experimental manipulation (due to their coupling to light). Although the short lifetime of the excitons implies that the condensate formed from polaritons in semiconductor microcavities is necessarily in a dynamical rather than a thermal equilibrium, it has been shown that the condensed phase shares many properties of the usual atomic Bose-Einstein condensate: it is a coherent state of spontaneously broken symmetry with an associated Goldstone mode [@Wouters07]. Recently it has been proposed that the dynamical nature of the polariton condensation can be used to explicitly break the U(1) phase symmetry present in a continuously, resonantly pumped experiment using an additional continuous probing laser [@Wouters07; @Amo07]. The coherence of the condensate can be independently tested by looking at the coherence and polarisation of the light emitted by recombining excitons [@Ciuti01]. If the pumping power is large enough to create a polariton condensate in a truly classical, symmetry broken state, then the condensate should retain its coherence even after the probing laser has been turned off. At lower power the condensate wavefunction will instead spread out over phase space and look symmetric again. Building on these results, the following experiment comes to mind. One can use the lack of number conservation in the condensate’s dynamical equilibrium [@Amo07], to create a superposition of different order parameters by subjecting the polaritons to a superposition of two different probing laser beams. The resulting macroscopic superposition is then expected to spontaneously collapse into just one ordered state for high enough pumping power due to dynamical spontaneous symmetry breaking, while lower pumping power (and the absence of symmetry breaking) should lead only to quantum beatings between the states of the initial superposition. If the transition from collapse behaviour to quantum beatings occurs at the same pumping power at which a single condensate has been seen to remain stable after turning off the probing laser, that would form a strong experimental indication of the involvement of dynamic spontaneous symmetry breaking.
7: Conclusions
--------------
In summary, we have shown here that macroscopic objects which spontaneously break a continuous symmetry under equilibrium conditions are also subject to a spontaneous breakdown of quantum mechanics’ unitary time translation symmetry. The coincidence of objects liable to dynamic spontaneous symmetry breaking with those liable to equilibrium spontaneous symmetry breaking is ensured by the crucial role played by the thin spectrum which is known to characterise the latter objects. Dynamic spontaneous symmetry breaking augments the well known theories of equilibrium spontaneous symmetry breaking and decoherence in the domains where these theories do not apply, and so leads to the symmetry broken state being not just the only stable ground state under equilibrium conditions, but also the only stable state dynamically. The quantum dynamics of any symmetric state, and more generally any superposition of differently ordered states, is almost infinitely sensitive to non-unitary perturbations in the thermodynamic limit, and such states must thus spontaneously and instantaneously be reduced to a state with only a single order parameter.
Applying this description of dynamic spontaneous symmetry breaking to the ordered states in our classical world, it becomes clear why these ordered classical states do not seem to be bothered by the interaction with their quantum environments: any buildup of quantum uncertainty is immediately reduced by the dynamical symmetry breaking process. Using the description instead to study the fate of a superposition of different classical states, one finds that only a single state can survive the spontaneous breakdown of quantum dynamics, and that the probability for finding any one particular outcome must be given by Born’s rule.
The predicted spontaneous breaking of unitary quantum time evolution can in principle be tested experimentally if one has a controlled way of constructing superpositions of differently ordered mesoscopic states. One type of system in which this may possibly be achieved is given by the polariton condensates in which the phase of the order parameter can be selected using the coherence of an incident laser beam.
Acknowledgements
----------------
I would like to gratefully acknowledge countless stimulating and insightful discussions with Jeroen van den Brink and Jan Zaanen.
Appendix A: Quantum Measurement
-------------------------------
In the main text we investigated the stability of a macroscopic state created by a quantum mechanical operation which quickly drives an ordered system into a superposition of differently ordered states with well separated orientations of their order parameters. One instance in which such a process is believed to occur is quantum measurement. By its very nature a quantum measurement is defined to be a process in which some property of a microscopic quantum state is translated into a specific pointer state of a macroscopic measurement machine [@Zurek81; @Joos85]. The different pointer states of such a machine must be easily distinguishable, classical states. In practise this always implies that they are symmetry broken states with different values or orientations for their order parameters. If we take these properties of the measurement machine at face value then it is clear that the measurement of a superposed quantum state must also lead to a superposition of pointer states in the measurement device because of the unitarity of quantum mechanical time evolution [@Bassi]. This simple observation already lead John von Neumann to postulate a collapse process which takes place after the usual quantum mechanical time evolution, and acts only on macroscopic superpositions [@Neumann55]. The explanation of why the collapse process exists, why it only acts on pointer states and not on microscopic states and why it gives rise to Born’s rule (dictating the probability of a certain outcome) is known as the quantum measurement problem. Many attempts have been made to either introduce a specific collapse process into quantum mechanics or to avoid the problem altogether by interpreting the mathematics of quantum mechanics in a different way. However, neither of these approaches has yet lead to a satisfactory resolution of all of the questions posed by the measurement problem.
Our analysis of the quantum dynamics of a superposition of differently ordered states in the thermodynamic limit suggests the following description of quantum measurement: a quantum measurement machine is any system with a well developed order parameter that can be coupled to a microscopic quantum system in such a way that the orientation of the order parameter after the coupling process has been completed, represents the property of the microscopic state that is to be measured. In general such a coupling should give rise to macroscopic superpositions of the order parameter, but the dynamical, spontaneous breakdown of quantum mechanics’ unitary time evolution ensures the spontaneous reduction of such superpositions into a state with just a single well defined order parameter. Because the macroscopic superposition state is subject to multiple competing instabilities, the outcome of the reduction process is probabilistic. The probability for obtaining a specific outcome is automatically guaranteed to agree with Born’s rule due to the properties of the process of dynamic spontaneous symmetry breaking (see also Appendix B).
Using dynamic spontaneous symmetry breaking, we have arrived at a clear-cut definition of what a measurement machine is; why it is subject to a collapse process; why this collapse does not influence microscopic quantum states; and we have recovered Born’s rule. The quantum measurement problem is thus reduced to the problem of identifying possible sources of non-unitary perturbations to the theory of quantum mechanics, which could drive the dynamic spontaneous symmetry breaking process. Regardless of its source, any non-unitary influence which can couple to a suitable order parameter will be amplified by the symmetry breaking process, and yield the expected macroscopic dynamics.
Appendix B: Detailed Derivation of Born’s Rule
----------------------------------------------
In this appendix we will give the detailed derivation of the emergence of Born’s rule from the dynamic spontaneous breaking of quantum mechanical time translation symmetry as applied to the case of the Lieb-Mattis antiferromagnet. There are three main requirements that need to be satisfied in order for the following derivation to be applicable. These requirements are: (1) The spontaneous evolution must yield a final state with only a single orientation of the order parameter, and the selection of the specific order parameter to be realised must be a probabilistic process; (2) The probability of obtaining a certain outcome may only depend on its weight in the initial superposition; (3) If the initial superposed state is entangled with some other, external quantum mechanical object with which the antiferromagnet has no further interaction, then the probability for finding a certain final orientation of the antiferromagnetic order parameter should not be affected by the precise state of the external quantum mechanical object.
To see that these requirements are all satisfied by the process of dynamic spontaneous symmetry breaking described before, consider the initial state $$\begin{aligned}
\left| \psi(0) \right> = \alpha \left| e1 \right> \otimes \left| AFM \right>_x + \beta \left| e2 \right> \otimes \left| AFM \right>_z,
\label{alphabeta}\end{aligned}$$ where $|\alpha|^2 + |\beta|^2=1$, $|AFM\rangle_x$ is the state with full antiferromagnetic order along the $x$ axis, and the states $|e1\rangle$ and $|e2\rangle$ are some external states which have no further interaction with the antiferromagnet whatsoever. The Hilbert space of the combined system of antiferromagnet and external states can be written as a product of the space of states of the antiferromagnet and the space of external states. Following the discussion of the quantum dynamics of a superposed macroscopic state in the main text, it is clear that the dynamics of the initial state $\left| \psi(0) \right>$ is unstable with respect to two orientations of the symmetry breaking field. Since the two instabilities of $\left| \psi(0) \right>$ must compete with each other, only one of the two available stable states can be realised, and the selection of which state is realised in the presence of a fluctuating symmetry breaking field is a probabilistic process, as stated in requirement one. Furthermore, since the competition between instabilities takes place on an infinitesimally short timescale, it cannot be influenced by the finite energy scale $J$. The [*fluctuating*]{} field $b$ is guaranteed by symmetry not to favour either one of the two possible final states. The only thing left to determine the probability of finding a certain final state is then the choice of the initial state itself: i.e. only the weights $\alpha$ and $\beta$ can determine the probability distribution of final states, in agreement with requirement two. That these weights in fact do influence the probability distribution is obvious from the fact that the initial states with $\alpha$ or $\beta$ equal to zero are stable states. The external states $|e1\rangle$ and $|e2\rangle$ cannot influence the spontaneous dynamics because all of the competition between the instabilities is governed by the unitarity breaking field $b$. This field acts only on the states of the antiferromagnet, and not on any other part of the Hilbert space (requirement three). The initial state $\left| \psi(0) \right>$ will thus be spontaneously and instantaneously reduced to either the state $\left| e1 \right> \otimes \left| AFM \right>_x$ or the state $\left| e2 \right> \otimes \left| AFM \right>_z$, while the probabilities $P_x(\psi)$ and $P_z(\psi)$ for finding either final state depend only on the values of $\alpha$ and $\beta$.
Building on these known properties of the final probabilities, let’s now follow Zurek’s arguments for obtaining the exact final probability distribution [@Zurek05]. First consider two different initial states: $$\begin{aligned}
\left| \psi \right> = \alpha \left| e1 \right> \otimes \left| AFM \right>_x + \beta \left| e2 \right> \otimes \left| AFM \right>_z \phantom{.} \nonumber \\
\left| \phi \right> = \alpha \left| e3 \right> \otimes \left| AFM \right>_x + \beta \left| e4 \right> \otimes \left| AFM \right>_z.
\label{statementA}\end{aligned}$$ Since the final probabilities can only depend on the weights of the classical states in the initial wavefunction (req. 2), it is immediately clear that $P_x(\psi)=P_x(\phi)$. This must hold independent of the external states $|e1\rangle$ through $|e4\rangle$ (req. 3), and thus it must also hold in the special case $|e1\rangle=e^{i \theta} |e3\rangle,~ |e2\rangle=|e4\rangle$, showing that the probability distribution cannot depend on the phases of the weights in the initial wavefunction.
Next, consider the initial states $$\begin{aligned}
\left| \psi \right> = \alpha \left| e1 \right> \otimes \left| AFM \right>_x + \beta \left| e2 \right> \otimes \left| AFM \right>_z \phantom{.} \nonumber \\
\left| \chi \right> = \alpha \left| e2 \right> \otimes \left| AFM \right>_z + \beta \left| e1 \right> \otimes \left| AFM \right>_x.
\label{statementB}\end{aligned}$$ Clearly, we must have $P_x(\psi)=P_z(\chi)$ for any choice of $\alpha$ and $\beta$. In the special case $|\alpha|=|\beta|$ we also know $P_z(\psi)=P_z(\chi)$, and thus we find that in that case $P_x(\psi)=P_z(\psi)$. In other words, if the sizes of the weights corresponding to two final states are equal, then so are the probabilities for finding these states. This statement can be trivially extended to yield the rule that a set of possible final states with equal weights in the initial wavefunction leads to equal probability for finding any one of the final states within that set. Continuing that line of thought, consider $$\begin{aligned}
\left| \psi \right> = \alpha \left| AFM \right>_i + \alpha \left| AFM \right>_j+ \alpha \left| AFM \right>_k + ...
\label{statementD}\end{aligned}$$ where $i$, $j$ and $k$ are different directions in real space. The combined probability $P_{i~\text{or}~j}(\psi)$ must then be equal to $P_i(\psi)+P_j(\psi)=2P_k(\psi)$, which follows directly from the additivity of probabilities and the mutual exclusivity of the three possible final states. That the final states are in fact mutually exclusive is guaranteed by requirement 1: in the thermodynamic limit $\left| AFM \right>_i$ and $\left| AFM \right>_j$ correspond to states with different directions of their order parameters, which can have no overlap and only one of which can be the result of the spontaneous dynamics. Extending this result, it is now clear that within a set of possible final states with equal weights in the initial wavefunction, a subset has a combined probability equal to the relative size of the subset times the total probability of the entire set.
Finally, consider the initial state $$\begin{aligned}
\hspace{-5pt} \left| \psi \right> = \sqrt{\frac{m}{N}} \left| e1 \right> \otimes \left| AFM \right>_x + \sqrt{\frac{n}{N}} \left| e2 \right> \otimes \left| AFM \right>_z.
\label{statementE}\end{aligned}$$ The probability $P_x(\psi)$ is independent of the external states (req. 3). We are therefore free to write $|e1\rangle$ and $|e2\rangle$ in a basis in which they are a sum of states with equal weights (such a basis can be shown to always exist [@Zurek05]):$$\begin{aligned}
\left| e1 \right> &= \sqrt{\frac{1}{m}} \left[ \left| E1_1 \right> + \left| E1_2 \right> + ... + \left| E1_m \right> \right] \phantom{.} \nonumber \\
\left| e2 \right> &= \sqrt{\frac{1}{n}} \left[ \left| E2_1 \right> + \left| E2_2 \right> + ... + \left| E2_n \right> \right] .
\label{statementE2}\end{aligned}$$ Reinserting these definitions into equation yields $$\begin{aligned}
\left| \psi \right> = \sqrt{\frac{1}{N}} \left[ \sum_{i=1}^m \left| E1_i \right> \otimes \left| AFM \right>_x + \right. \nonumber \\
\left. \sum_{j=1}^n \left| E2_j \right> \otimes \left| AFM \right>_z \right] .
\label{statementE3}\end{aligned}$$ In this expression all weights are equal, and using the previously found rules we must thus conclude that $P_x(\psi)=\frac{n}{m}P_z(\psi)$. In the case that the total probability for finding any outcome at all is one, this result precisely corresponds to Born’s rule: the probability for finding any specific final orientation of the order parameter is equal to the square of the weight of the corresponding state in the initial wavefunction [@Born26]. The extension of this result to include also weights which are square roots of non-rational numbers is trivial because the rational numbers are dense on the real line [@Zurek05].
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| {
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---
abstract: 'This paper presents a new generalized Mackey-Glass model with a non-linear harvesting term and mixed delays. The main purpose of this work is to study the existence and the exponential stability of the pseudo almost periodic solution for the considered model. By using fixed point theorem and under suitable Lyapunov functional, sufficient conditions are given to study the pseudo almost periodic solution for the considered model. Moreover, an illustrative example is given to demonstrate the effectiveness of the obtained results.'
author:
- Haifa Ben Fredj
- Farouk Chérif
date: 'Received: date / Accepted: date'
title: 'Positive pseudo almost periodic solutions to a class of hematopoiesis model: Oscillations and Dynamics'
---
[example.eps]{} gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore
Introduction {#intro}
============
In 1977, Mackey and Glass [@L] proposed the following non-linear differential equation with constant delay $$x'(t) = -\alpha x(t) + \dfrac{\beta x(t-\tau)}{\theta^n + x^n(t-\tau)},\quad 0 <n.$$ in order to describe the concentration of mature cells in the blood concentration. Here $\alpha$, $\beta$, $\tau$ and $\theta$ are positive constants, the unknown $x$ stands for the density of mature cells in blood circulation, $\alpha$ is the rate of lost cells from the circulation at time t, the flux
$f(x(t-\tau)):= \dfrac{\beta x(t-\tau)}{\theta^n+ x^n(t-\tau)}$
of cells in the circulation depends on $x(t - \tau)$ at the time $t -\tau$, where $\tau$ is the time delay between the production of immature cells in the bone marrow and their maturation.\
\
Since its introduction in the literature, the hematopoiesis model has gained a lot of attention and various extensions. Hence, under some additional conditions some authors [@K1; @K2; @M1; @N1] considered an extended version of eq.(1) and obtained the existence and attractivity of the unique positive periodic and almost periodic solutions of the following model $$x(t)= {\displaystyle -a(t) x(t) +\sum^{N}_{i=1} \dfrac{b_i(t)x^m(t-\tau_i(t))}{1 + x^n(t-\tau_i(t))}},\quad 0\leq m\leq 1, 0<n.$$
Recently, there have been extensive and valuable contributions dealing with oscillations of the hematopoiesis model with and without delays, see, e.g., [@K1; @B; @K2; @O; @H] and references therein.\
Also, the stability of various models was strongly investigated by many authors recently [@M; @A; @M1; @F; @B1; @N] and references therein.\
\
As we all know, in real-world applications equations with a harvesting term provide generally a more realistic and reasonable description for models of mathematical biology and in particular the population dynamics. Hence, the investigation of biological dynamics with harvesting is a meaningful subject in the exploitation of biological resources which is related to the optimal management of renewable resources [@K; @P].\
\
Besides, the study oscillations and dynamics systems of biological origin is an exciting topic. One can find a valuable results in this field [@H1; @H2; @B0; @M0; @M2] and references therein.\
\
Motivated by the discussion above the main subject of this paper is to study the existence and the global attractor of the unique and positive pseudo almost periodic solution for the generalized Mackey-Glass model with a nonlinear harvesting term and mixed delays. Roughly speaking, we shall consider the following hematopoiesis model $$\begin{aligned}
x'(t)=-a(t)x(t)+\sum^{N}_{i=1} \dfrac{b_i(t)x^m(t-\tau_i(t))}{1 + x^n(t-\tau_i(t))}- H(t,x(t-\sigma(t)), \quad 1<m \leq n, t\in {\mathbb{R}}.\end{aligned}$$ However, to the author’s best knowledge, there are no publications considering the pseudo and positive almost periodic solutions for Mackey-Glass model with harvesting term and $1<m\leq n$.\
\
The remainder of this paper is organized as follows: In Section 1, we will introduce some necessary notations, definitions and fundamental properties of the space PAP(${\mathbb{R}}$,${\mathbb{R}}^+$) which will be used in the paper. In Section 2, the model is given. In section 3, the existence of the unique positive pseudo almost periodic solution for the considered system is established. Section 4 is devoted to the stability of the pseudo almost periodic solution. In Section 5, based on suitable Lyapunov function and Dini derivative, we give some sufficient conditions to ensure that all solutions converge exponentially to the positive pseudo almost periodic solution of the equation (3). At last, an illustrative example is given. It should be mentioned that the main results of this paper are theorems 1, 2.
Preliminaries {#sec:1}
=============
In this section, we would like to recall some basic definitions and lemmas which are used in what follows. In this paper, $BC({\mathbb{R}}, {\mathbb{R}})$ denotes the set of bounded continued functions from ${\mathbb{R}}$ to ${\mathbb{R}}$. Note that $(BC({\mathbb{R}}, {\mathbb{R}}), \|.\|_{\infty})$ is a Banach space where the sup norm $$\|f\|_{\infty} :=\underset{t\in {\mathbb{R}}}{sup} |f(t)|.$$
Let u(.) $\in BC({\mathbb{R}}, {\mathbb{R}})$, u(.) is said to be almost periodic (a.p) on ${\mathbb{R}}$ if, for any $\epsilon > 0$, the set $$T(u, \epsilon) = \{\delta; |u(t + \delta) - u(t)| < \epsilon, \text{ for all }t \in {\mathbb{R}}\}$$ is relatively dense; that is, for any $\epsilon > 0$, it is possible to find a real number $l = l(\epsilon) > 0$; for any interval with length l($\epsilon$), there exists a number $\delta= \delta(\epsilon)$ in this interval such, that $$|u(t + \delta) - u(t)| <\epsilon, \text{ for all }t \in {\mathbb{R}}.$$
We denote by $AP({\mathbb{R}}, {\mathbb{R}})$ the set of the almost periodic functions from ${\mathbb{R}}$ to ${\mathbb{R}}$.
Let $u_i (.)$ , $1 \leq i \leq m$ denote almost periodic functions and $\epsilon > 0$ be an arbitrary real number. Then there exists a positive real number $L = L(\epsilon)> 0$ such that every interval of length L contains at least one common $\epsilon-almost$ period of the family of functions $ u_i (.)$, $1 \leq i \leq m.$
Besides, the concept of pseudo almost periodicity (p.a.p) was introduced by Zhang [@D] in the early nineties. It is a natural generalization of the classical almost periodicity. Precisely, define the class of functions $PAP_0({\mathbb{R}}, {\mathbb{R}})$ as follows $$\bigg\{f \in BC({\mathbb{R}}, {\mathbb{R}}); \underset{T\rightarrow+\infty}{lim} \dfrac{1}{2T} \int_{-T}^T |f(t)|dt = 0\bigg\} .$$ A function $f \in BC({\mathbb{R}}, {\mathbb{R}})$ is called pseudo almost periodic if it can be expressed as $f = h + \phi$, where $h \in AP({\mathbb{R}}, {\mathbb{R}})$ and $\phi \in PAP_0({\mathbb{R}}, {\mathbb{R}})$. The collection of such functions will be denoted by $PAP({\mathbb{R}}, {\mathbb{R}})$. The functions h and $\phi$ in the above definition are, respectively, called the almost periodic component and the ergodic perturbation of the pseudo almost periodic function $f$. The decomposition given in definition above is unique. It should be mentioned that pseudo almost periodic functions possess many interesting properties; we shall need only a few of them and for the proofs we shall refer to [@D; @D1; @M3].
Observe that (PAP(${\mathbb{R}}$, ${\mathbb{R}}$), $\|.\|_{\infty}$) is a Banach space and $AP({\mathbb{R}}, {\mathbb{R}})$ is a proper subspace of $PAP({\mathbb{R}}, {\mathbb{R}})$ since the function $\psi(t) = cos(2 t) +sin(2 \sqrt{5}t) +exp^{-t^2 |sin (t)|}$ is pseudo almost periodic function but not almost periodic.
[@D] If f,g $\in PAP({\mathbb{R}},{\mathbb{R}})$, then the following assertions hold:
1. f.g, f+g $\in PAP({\mathbb{R}}, {\mathbb{R}})$.
2. $\dfrac{f}{g} \in PAP({\mathbb{R}}, {\mathbb{R}})$, if $\underset{t\in {\mathbb{R}}}{inf}|g(t)|>0$.
[@D] Let $\Omega \subseteq {\mathbb{R}}$ and let K be any compact subset of $\Omega$. On define the class of functions\
$PAP_0(\Omega\times{\mathbb{R}}, {\mathbb{R}})$ as follows $$\bigg\{\psi\in C(\Omega\times {\mathbb{R}}; {\mathbb{R}}); \underset{T\rightarrow+\infty}{lim} \dfrac{1}{2T} \int_{-T}^T |\psi(s,t)|dt = 0\bigg\}$$ uniformly with respect to $s\in K$.
(Definition 2.12, [@E])
Let $\Omega \subseteq {\mathbb{R}}$. An continuous function f : ${\mathbb{R}}\times\Omega \longrightarrow {\mathbb{R}}$ is called pseudo almost periodic (p.a.p). in t uniformly with respect $x \in \Omega$ if the two following conditions are satisfied :\
i) $\forall x \in \Omega, f(., x) \in PAP({\mathbb{R}},{\mathbb{R}}),$\
ii) for all compact K of $\Omega$, $\forall \epsilon> 0, \exists \delta > 0, \forall t \in \mathbb{R}, \forall x_1, x_2 \in K$,
$|x_1 - x_2| \leq \delta \Rightarrow |f(t, x_1) - f(t, x_2)| \leq \epsilon$.
Denote by $PAP_U(\Omega\times {\mathbb{R}}; {\mathbb{R}})$ the set of all such functions.
The model {#sec:2}
=========
In order to generalize and improve the above models, let us consider the following Mackey-Glass model with a non-linear harvesting term and several concentrated delays $$\begin{aligned}
x'(t)=-a(t)x(t)+\sum^{N}_{i=1} \dfrac{b_i(t)x^m(t-\tau_i(t))}{1 + x^n(t-\tau_i(t))}- H(t,x(t-\sigma(t))\end{aligned}$$ where $t \in {\mathbb{R}}$ and
1. The function a : $\mathbb{R}\longrightarrow\mathbb{R^+}$ is pseudo almost periodic(p.a.p) and $\underset{t\in \mathbb{R}}{inf} a(t) >0.$
2. For all 1$\leq i \leq N$; the functions $\tau_i,\sigma$, b : $\mathbb{R}\longrightarrow\mathbb{R^+}$ are p.a.p.
3. The term H $\in PAP_U(\mathbb{R}\times\mathbb{R},{\mathbb{R}}^+$) satisfies the Lipschitz condition : $\exists L_H > 0$ such that $${\displaystyle \mid H(t,x)-H(t,y) \mid < L_H \mid x-y \mid,\quad \forall x,y,t \in \mathbb{R}}.$$
Throughout the rest of this paper, for every bounded function $f : {\mathbb{R}}\rightarrow {\mathbb{R}}$, we denote $$f^+ = \underset {t\in {\mathbb{R}}}{sup} f(t), f^- =\underset {t\in {\mathbb{R}}}{inf} f(t).$$ Pose $r =\underset{t \in \mathbb{R}}{sup} \bigg(\tau_i(t),\sigma(t) ; i=1,2...N\bigg).$ Denote by $BC ([-r, 0] , {\mathbb{R}}^+$) the set of bounded continuous functions from \[-r, 0\] to ${\mathbb{R}}^+$. If $x(.)$ is defined on $[-r + t_0, \sigma[$ with $t_0, \sigma \in {\mathbb{R}}$, then we define $x_t \in C([-r, 0] , {\mathbb{R}}$) where $x_t(\theta) = x(t + \theta)$ for all $\theta \in [-r, 0]$. Notice that we restrict our selves to ${\mathbb{R}}^+$-valued functions since only non-negative solutions of (4) are biologically meaningful. So, let us consider the following initial condition $$x_{t_0} = \phi,\quad \phi \in BC ([-r, 0] , {\mathbb{R}}^+) \text{ and }\phi (0) > 0.$$ We write $x_t (t_0, \phi)$ for a solution of the admissible initial value problem (4) and (5). Also, let $[t_0, \eta(\phi)[$ be the maximal right-interval of existence of $x_t(t_0, \phi)$.
Main results {#sec:3}
============
As pointed out in the introduction, we shall give here sufficient conditions which ensures existence and uniqueness of pseudo almost periodic solution of (4). In order to prove this result, we will state the following lemmas. For simplicity, we denote $x_t(t_0, \phi)$ by $x(t)$ for all $t\in {\mathbb{R}}$.
A positive solution $x(.)$ of model (4)-(5) is bounded on $[t_0, \eta(\phi)[$, and $\eta(\phi)=+\infty$.
We have for each $t \in [t_0, \eta(\phi)[$ the solution verifies $$x(t)= {\displaystyle e^{- \int^t_{t_0} a(u) du} \phi(0) +\int^t_{t_0} e^{-\int^t_{s} a(u) du}\bigg[\sum^{N}_{i=1} \dfrac{b_i(s)x^m(s-\tau_i(s))}{1 + x^n(s-\tau_i(s))}- H(t,x(s-\sigma(s))\bigg]ds}.$$ So, by $\underset{x\geq0}{sup}\dfrac{x^m}{1+x^n}\leq 1,\text{ }\forall 1<m\leq n$, we obtain $$\begin{array}{lll}
x(t)\leq {\displaystyle \phi(0) +\int^t_{t_0} e^{-a^-(t-s)}\sum^{N}_{i=1} b_i^+ds}&\leq {\displaystyle \phi(0) + \dfrac{1}{a^-}[1- e^{-a^-(t-t_0)}]\sum^{N}_{i=1} b_i^+}\\
&\leq \phi(0) + \dfrac{1}{a^-}\sum^{N}_{i=1} b_i^+< +\infty,
\end{array}$$ which proves that $x(.)$ is bounded. The second part of the conclusion is given by Thorem 2.3.1 in [@I], we have that $\eta(\phi)=+\infty$.
If $a^->\sum^N_{i=1}b_i^+,$ then each positive solution $x_t(t_0,\phi)$ of model (4)-(5) satisfies $$x(t) \underset{t \rightarrow +\infty}{\longrightarrow} 0.$$
We define the continuous function $$\begin{aligned}
G : [0,1]
&\longrightarrow& {\mathbb{R}}\\
y &\longmapsto& y-a^-+\sum^N_{i=1}b_i^+ e^{y t}.\end{aligned}$$ From the hypothesis, we obtain G(0)<0, then there exists $\lambda \in [0,1]$, where $$G(\lambda)<0. \qquad (C.1)$$ Let us consider the function $W(t)=x(t)e^{\lambda t}$. Calculating the left derivative $W(.)$ and by using the following inequality $$\dfrac{x^m}{1+x^n} \leq x ,\quad \forall 1<m\leq n.$$We obtain
$$\begin{aligned}
D^-W(t) &=& \lambda x(t)e^{\lambda t}+x'(t)e^{\lambda t}\\
\\
&=&\lambda x(t) e^{\lambda t}+e^{\lambda t}[-a(t)x(t)+\sum_{i=1}^{N} b_i(t) \frac{x^m(t-\tau_i(t))}{1+x^n(t-\tau_i(t))}-H(t,x(t-\sigma(t))]\\
\\ &\leq& e^{\lambda t}((\lambda- a^-)x(t) + \sum_{i=1}^{N} b^+_i x(t-\tau_i(t))).
\end{aligned}$$
Let us prove that $$\begin{aligned}
W(t) &=& x(t)e^{\lambda t}< e^{\lambda t_0} M = Q, \forall t\geq t_0.
\end{aligned}$$ Suppose that there exists $t_1>t_0$ such that $$\begin{aligned}
W(t_1) &=& Q, W(t)<Q, \text{ for all } t_0-r\leq t< t_1.
\end{aligned}$$ Then $$\begin{aligned}
0\leq D^-W(t_1)&\leq&(\lambda- a^-)x(t_1) e^{\lambda t_1}+ \sum_{i=1}^{N} b^+_i x(t_1-\tau_i(t_1))e^{\lambda t_1} \\
\\ &\leq& (\lambda- a^-) Q+ \sum_{i=1}^{N} b^+_i x(t_1-\tau_i(t_1))e^{\lambda t_1} e^{\lambda \tau_i} e^{-(\lambda \tau_i(t_1))}\\
\\ &=&(\lambda- a^-) Q+ \sum_{i=1}^{N} b^+_i x(t_1-\tau_i(t_1)) e^{\lambda (t_1-\tau_i(t_1))} e^{\lambda r}\\
\\&\leq& [\lambda- a^-+ \sum_{i=1}^{N} b^+_i e^{\lambda r}] Q\\
\\&<& 0 \qquad (\text{by \textbf{(C.1)}})
\end{aligned}$$
which is a contradiction. Consequently, $x(t) e^{\lambda t}< Q$. Then, $x(t)< e^{-\lambda t}Q \underset{t \rightarrow +\infty}{\longrightarrow} 0$.
Pose $f_{n,m}(u) =\dfrac{u^m}{1+u^n}$, one can get:\
: $$\left \{
\begin{array}{r c l}
f_{n,m}'(u) = \dfrac{u^{m-1}(m-(n-m)u^n)}{(1 + u^n)^2}> 0,\forall u \in \left[ 0,\sqrt[n]{\dfrac{m}{n-m} }\right] \qquad (C.3)\\
\\f_{n,m}'(u) = \dfrac{u^{m-1}(m-(n- m)u^n)}{(1 + u^n)^2}<0,\forall u \in \left] \sqrt[n]{\dfrac{m}{ n-m} },+\infty \right[ \qquad (C.4),
\end{array}
\right.$$\
and $m= n$: $$f_{n,m}'(u) = \dfrac{u^{m-1}m}{(1 + u^m)^2}> 0,\forall u \in [ 0,+\infty[.\qquad (C.5)$$
If $m<n$, one can choose $k \in \left]0, \sqrt[n]{\dfrac{m}{ n-m} }\right[ $ and combining with **(C.3)** and **(C.4)** there exists a constant\
$\overset{\backsim}{ k}>\sqrt[n]{\dfrac{m}{ n-m} } $ such that
$f_{n,m}(k)= f_{n,m}(\overset{\backsim}{k}).$ (C.6)
Moreover, $$\underset{u\geq0}{sup}\dfrac{u^m}{1+u^n}\leq 1, \quad \forall 1<m\leq n. \qquad (C.7)$$\
$$\text{Let }\mathcal{C}^0=\{\psi \in BC([-r,0] , \mathbb{R}^+); k\leq \psi \leq M\}.$$
A positive solution $ x(.)$ of the differential equation is permanent if there exists $t^*\geq 0$, A and B ; $B > A > 0$ such that $$A \leq x(t) \leq B \quad \text{ for }t \geq t^*.$$
Suppose that there exist a two positives constants M and k satisfying:
1. m<n: $$0<k < \sqrt[n]{\dfrac{m}{ n-m} } < M \leq \overset{\backsim}{k}\quad (\text {$\overset{\backsim}{k}$ was given by \textbf{(C.6)}})$$ $$0<k < M$$
2. ${\displaystyle -a^- M+\sum^N_{i=1}b_i^+ - H^-}<0$
3. ${\displaystyle -a^+ k+\sum^N_{i=1}b_i^- \dfrac{k^m}{1+k^n} -H^+}> 0$
and $\phi \in \mathcal{C}^0$, then the solution of (4)-(5) $ x(.)$ is permanent which $\eta(\phi)=+\infty$.
Actually, we prove that $x(.)$ is bounded in $[ t_0,\eta(\phi)[ .$\
$\bullet$ First, we claim that
$$x(t) < M, \forall t \in [ t_0,\eta(\phi)[ .\qquad (i)$$
Contrarily, there exists $t_1 \in ] t_0,\eta(\phi)[ $ such that: $$\left \{
\begin{array}{llll}
x(t)<M, \forall t \in \left[t_0-r,t_1\right[\\
x(t_1) = M
\end{array}
\right.$$
Calculating the right derivative of $x(.)$ and by $\textbf{(H2)}$ and **(C.7)**, we obtain $$\begin{array}{ll}
0 \leq x'(t_1)&=-a(t_1) x(t_1)+ \sum^{N}_{i=1} \dfrac{b_i(t_1)x^m(t_1-\tau_i(t_1))}{1 + x^n(t_1-\tau_i(t_1))}- H(t_1,x(t_1-\sigma(t_1))\\
\\&\leq -a(t_1) M +\sum^{N}_{i=1} b_i(t_1) -H^- \\
\\&< -a^- M +\sum^{N}_{i=1} b_i^+-H^-\\
\\& < 0,
\end{array}$$ which is a contradiction. So it implies that (i) holds.\
\
$\bullet $ Next, we prove that
$k < x(t) ,\forall t \in [ t_0,\eta(\phi)[ .$ (ii)
Otherwise, there exists $t_2 \in ] t_0,\eta(\phi)[ $ such that $$\left \{
\begin{array}{llll}
x(t)>k, \forall t \in \left[t_0-r,t_2\right[\\
x(t_2) = k
\end{array}
\right.$$
Calculating the right derivative of $x(.)$ and combining with $\textbf{(H3)}$, **(C.3)** and**(C.5)**, we obtain $$\begin{array}{ll}
0 \geq x'(t_2)&=-a(t_2) x(t_2)+ \sum^{N}_{i=1} \dfrac{b_i(t_2)x^m(t_2-\tau_i(t_2))}{1 + x^n(t_2-\tau_i(t_2))}- H(t_2,x(t_2-\sigma(t_2))\\
\\& \geq -a(t_2) k + \sum^{N}_{i=1} \dfrac{b_i(t_2) k^m}{1 +k^n}-H^+\\
\\& > -a^+ k + \sum^{N}_{i=1} \dfrac{b_i^- k^m}{1 +k^n}-H^+\\
\\& > 0,\\
\end{array}$$ which is a contradiction and consequnetly (ii) holds. From Thorem 2.3.1 in [@I], we have that $\eta(\phi)=+\infty$. The proof of Lemma 4.4 is now completed.
$$\text{Let }\mathcal{B}=\{\psi \in PAP(\mathbb{R} , \mathbb{R}); k\leq \psi \leq M\}.$$
$\mathcal{B}$ is a closed subset of $PAP({\mathbb{R}},{\mathbb{R}})$.
Let $(\psi_n)_{n \in {\mathbb{N}}} \subset \mathcal{B}$ such that $\psi_n \longrightarrow \psi$. $ \text{Let us prove that } \psi \in \mathcal{B}.$\
\
Clearly, $\psi \in PAP({\mathbb{R}},{\mathbb{R}})$ and we obtain that $$\begin{array}{lll} \psi_n \underset{n\longrightarrow +\infty}{\longrightarrow }\psi &\Leftrightarrow \forall \epsilon> 0, \exists n_0>0 \text{ such that } \mid \psi_n(t)-\psi(t)\mid \leq \epsilon, (\forall t \in {\mathbb{R}}, \forall n>n_0)\\&\Leftrightarrow \forall \epsilon> 0, \exists n_0>0 \text{ such that } -\epsilon \leq \psi_n(t)-\psi(t)\leq\epsilon,(\forall t \in {\mathbb{R}}, \forall n>n_0).\end{array}$$ Let t $\in {\mathbb{R}}$, we obtain then $$-\epsilon+ k \leq \psi(t)=[\psi(t)-\psi_n(t) ]+\psi_n(t) \leq \epsilon +M.$$ So, $\psi \in \mathcal{B}$.
If f $\in PAP_U({\mathbb{R}}\times {\mathbb{R}},{\mathbb{R}})$ and for each bounded subset B of ${\mathbb{R}}$, f is bounded on ${\mathbb{R}}\times B$, then the Nymetskii operator $$N_f : PAP({\mathbb{R}},{\mathbb{R}}) \longrightarrow PAP({\mathbb{R}}, {\mathbb{R}}) \text { with } N_f(u)=f(.,u(.))$$ is well defined.
[@G] Let f,g $\in AP({\mathbb{R}},{\mathbb{R}})$. If $g^->0$ then $F \in AP({\mathbb{R}},{\mathbb{R}})$ where $$F(t)={\displaystyle \int^t_{-\infty}e^{-\int^t_s g(u)du} f(s) ds}, \quad t\in{\mathbb{R}}.$$
[@D] Let F $\in PAP_U(\mathbb{R}\times {\mathbb{R}},{\mathbb{R}})$ verifies the Lipschitz condition: $\exists L_F >0$ such that $$| F(t,x)-F(t,y)| \leq L_F |x-y|,\quad \forall x,y\in {\mathbb{R}}\text{ and } t \in \mathbb{R}.$$
If h $\in PAP({\mathbb{R}},{\mathbb{R}})$, then the function $F(.,h(.)) \in PAP({\mathbb{R}},{\mathbb{R}})$.
If conditions $\textbf{(H1)-( H3)}$ and $$\textbf{[H4]}: {\displaystyle \underset{t\in \mathbb{R} }{sup}\bigg\{-a(t)+\sum^N_{i=1}b_i(t)\bigg[\dfrac{(n-m)}{4}+\dfrac{m}{(1+k^n)^2}\bigg]M^{m-1}+L\bigg\}}<0$$ are fulfilled, then the equation (4) has a unique p.a.p solution x(.) in the region $\mathcal{B}$, given by $${\displaystyle x(t)=\int^t_{-\infty} e^{-\int^t_s a(u)du}\sum^{N}_{i=1} \dfrac{b_i(s)x^m(s-\tau_i(s))}{1 + x^n(s-\tau_i(s))}- H(s,x(s-\sigma(s))ds}.$$
Step 1:\
Clearly $\mathcal{B}$ is a bounded set. Now, let $\psi \in \mathcal{B}$ and $f(t,z)=\psi(t-z)$, since the numerical application $\psi$ is continuous and the space PAP(${\mathbb{R}},{\mathbb{R}})$ is a translation invariant then the function $f \in PAP_U({\mathbb{R}}\times {\mathbb{R}}, {\mathbb{R}}^+)$. Furthermore $\psi$ is bounded, then $f$ is bounded on ${\mathbb{R}}\times B$. By the lemma 4, the Nymetskii operator $$\begin{aligned}
N_f : PAP({\mathbb{R}},{\mathbb{R}})&\longrightarrow & PAP({\mathbb{R}},{\mathbb{R}})
\\\tau_i &\longmapsto &f(.,\tau_i(.))\end{aligned}$$ is well defined for ${\displaystyle \tau_i \in PAP({\mathbb{R}},\mathbb{R})}$ such that $0 \leq i \leq N$. Consequently,$${\displaystyle \bigg[t\longmapsto\psi(t-\tau_i(t))\bigg]\in PAP({\mathbb{R}},\mathbb{R^+})}\text{ for all } i=1,...,N.$$
Since ${\displaystyle \underset{t\in {\mathbb{R}}}{inf} |1+\psi^n(t-\tau_i(t))|>0}$ and using properties 1, the p.a.p functions one has $${\displaystyle\bigg [ t \longmapsto \sum^{N}_{i=1} \dfrac{b_i(t)\psi^m(t-\tau_i(t))}{1 + \psi^n(t-\tau_i(t))} \bigg] \in PAP({\mathbb{R}},{\mathbb{R}})}.$$
Also, under the fact that the harvesting term verifies the Lipschitz condition, being the lemma 6, $${\displaystyle \bigg [G : t \longmapsto \sum^{N}_{i=1} \dfrac{b_i(t)\psi^m(t-\tau_i(t))}{1 + \psi^n(t-\tau_i(t))} -H(t,\psi(t-\sigma(t))\bigg] \in PAP({\mathbb{R}},\mathbb{R})}.$$
Step 2: Let us define the operator $\Gamma$ by $$\Gamma(\psi)(t)=\int^t_{-\infty} e^{-\int^t_s a(u)du} G(s) ds$$ We shall prove that $\Gamma$ maps $\mathcal{B}$ into itself. First, since the functions G(.) and a(.) are p.a.p one can write $$G=G_1+ G_2 \text{ and }a=a_1+ a_2,$$where $G_1,a_1\in AP({\mathbb{R}},{\mathbb{R}})$ and $G_2,a_2 \in PAP_0(\mathbb{R}, {\mathbb{R}})$. So, one can deduce $$\begin{array}{ll}
\Gamma(\psi)(t)&{\displaystyle=\int^t_{-\infty} e^{-\int^t_s a_1(u)du} G_1(s)ds+\int^t_{-\infty} \bigg[ e^{-\int^t_s a(u)du} G(s)-e^{-\int^t_s a_1(u)du}G_1(s)\bigg]ds}\\
\\&{\displaystyle=\int^t_{-\infty} e^{-\int^t_s a_1(u)du} G_1(s)ds+\int^t_{-\infty} e^{-\int^t_s a_1(u)du} G_1(s)\bigg[e^{-\int^t_s a_0(u)du}-1\bigg]ds}\\
\\&+ \int^t_{-\infty}e^{-\int^t_s a(u)du}G_2(s)ds\\
\\&{\displaystyle=I(t)+II(t)+III(t). }\end{array}$$\
By the lemma **5**, $I(.) \in AP({\mathbb{R}},{\mathbb{R}})$.\
\
Now, we show that II(.) is ergodic. One has $$\begin{array}{lll}
II(t)&={\displaystyle\int^t_{-\infty} e^{-\int^t_s a_1(u)du} G_1(s)\bigg[e^{-\int^t_s a_2(u)du}-1\bigg]ds}\\
\\&={\displaystyle \int^{+\infty}_0 e^{-\int^t_{t-v} a_1(u)du} G_1(t-v)\bigg[e^{-\int^t_{t-v} a_2(u)du}-1\bigg]dv}\\
\\&={\displaystyle \int^{v_0}_0 +\int^{+\infty}_{v_0} e^{-\int^{v}_0a_1(t-s)ds} G_1(t-v)\bigg[e^{-\int^{v}_0 a_2(t-s)ds}-1\bigg]dv}\\
\\&=II_1(t)+II_2(t).
\end{array}$$
Since $a^-_1\geq a^-$ and for large enough $v_0$, we obtain $$\begin{array}{lll}
|II_2(t)|&\leq {\displaystyle \int^{+\infty}_{v_0} e^{-\int^{v}_0 a_1(t-s)ds} |G_1(t-v)|\bigg[|e^{-\int^{v}_0 a_2(t-s)ds}|+1\bigg]dv}\\
\\&= {\displaystyle \int^{+\infty}_{v_0} \|G_1\|_{\infty}\bigg[e^{-\int^{v}_0 a(t-s)ds}+ e^{-\int^{v}_0 a_1(t-s)ds}\bigg]dv}\\
\\&\leq {\displaystyle \int^{+\infty}_{v_0} 2 \|G_1\|_{\infty}e^{- a^-v}dv <\dfrac{\epsilon}{2}}.
\end{array}$$ So, $II_2 \in PAP_0({\mathbb{R}},{\mathbb{R}})$.\
Thereafter, it has not yet been demonstrated that $II_1(.) \in PAP_0({\mathbb{R}},{\mathbb{R}})$.\
Firstly, we prove that the following function $$\mu(v,t)={\displaystyle \int^v_0 a_2(t-s)ds, \qquad (v\in [0,v_0], t\in {\mathbb{R}})}$$ is in $PAP_0([0,v_0] \times {\mathbb{R}}, {\mathbb{R}})$. Clearly $|\mu(v,t)|\leq {\displaystyle\int^v_0 |a_2(t-s)|ds}$, then it is obviously sufficient to prove that the function $${\displaystyle\int^v_0 |a_2(.-s)|ds \in PAP_0({\mathbb{R}},{\mathbb{R}}).}$$
We have $a_2 (.)\in PAP_0({\mathbb{R}},{\mathbb{R}})$, for $\epsilon >0$ there exists $T_0 >0$ such that $${\displaystyle \dfrac{1}{2T} \int^T_{-T} |a_2(t-s)|dt \leq \dfrac{\epsilon}{v}, \qquad (T\geq T_0, s\in [0,v])}.$$ Since $[0,v]$ is bounded, the Fubini’s theorem gives for $\epsilon >0$ there exists $T_0 >0$ such that $${\displaystyle \dfrac{1}{2T} \int^T_{-T} \int^v_0 |a_2(t-s)|ds dt\leq \epsilon, \qquad T\geq T_0}.$$ So, $${\displaystyle\int^v_0 |a_2(.-s)|ds \in PAP_0({\mathbb{R}},{\mathbb{R}})}$$ which implies the required result.\
Then, we obtain $$\begin{array}{ lll}
{\displaystyle\dfrac{1}{2T} \int^T_{-T}|II_1(t)|dt } & {\displaystyle\leq \dfrac{1}{2T} \int^T_{-T}\int^{v_0}_0 e^{-\int^{v}_0 a_1(t-s)ds} |G_1(t-v)||e^{-\int^{v}_0 a_2(t-s)ds}-1|dv dt }\\
\\& ={\displaystyle\dfrac{1}{2T} \int^T_{-T}\int^{v_0}_0 |G_1(t-v)||e^{-\int^{v}_0 a(t-s)ds}-e^{-\int^{v}_0 a_1(t-s)ds}|dv dt }.
\end{array}$$ By the mean value theorem, $\exists \eta \in ]0,1[$ such that $$\begin{aligned}
{\displaystyle\dfrac{1}{2T} \int^T_{-T}|II_1(t)|dt }\leq &&{\displaystyle \dfrac{1}{2T} \int^T_{-T}\int^{v_0}_0 |G_1(t-v)|e^{-[(1-\eta)\int^{v}_0 a(t-s)ds+ \eta \int^{v}_0 a_1(t-s)ds]}}\\
\\&&\times\bigg(\int^{v}_0|a_2(t-s)|ds\bigg) dv dt .\end{aligned}$$
Since the function $\mu \in PAP_U([0,x_0] \times {\mathbb{R}}, {\mathbb{R}})$ and in virtue of the Fubini’s theorem for $\epsilon>0$, $\exists T_1>0$ such that $$\begin{array}{lll}
{\displaystyle\dfrac{1}{2T} \int^T_{-T}|II_1(t)|dt }&{\displaystyle\leq \int^{v_0}_0 \|G_1\|_{\infty} \dfrac{\epsilon}{ 2\|G_1\|_{\infty} v_0}}=\dfrac{\epsilon}{2}, \qquad (\forall T\geq T_1).
\end{array}$$\
So, $II_1\in PAP_0({\mathbb{R}},{\mathbb{R}})$.\
Finally, we study the ergodicity of III(.). We have
$$\begin{array}{ll}
{\displaystyle \dfrac{1}{2T}\int^T_{-T} |III(t)|dt}& {\displaystyle \leq \dfrac{1}{2T} \int^T_{-T} \int ^t _{-\infty} e^{-(t-s) a^-} \mid G_2(s) \mid ds dt}\\
\\&\leq III_1(T) + III_2(T),
\end{array}$$ where $$III_1(T)={\displaystyle \dfrac{1}{2T} \int^T_{-T} \int ^t _{-T} e^{-(t-s) a^-} \mid G_2(s) \mid ds dt}$$ and $$III_2(T)={\displaystyle \dfrac{1}{2T} \int^{-T}_{-\infty} \int ^t _{-\infty} e^{-(t-s) a^-} \mid G_2(s) \mid ds dt}.$$
Next, we prove that $$\underset{T\rightarrow +\infty}{lim} III_1(T)=\underset{T\rightarrow +\infty}{lim}III_2(T)=0.$$ By the Fubini’s theorem, we obtain $$\begin{array}{ll}
III_1(T)&{\displaystyle = \int ^{+\infty}_0 e^{- a^- u} \dfrac{1}{2T} \int^T_{-T} \mid G_2(t-u) \mid dt du}\\
\\&{\displaystyle= \int ^{+\infty}_0 e^{- a^- u} \dfrac{1}{2T} \int^{T-u}_{-T-u} \mid G_2(t) \mid dt du}\\
\\&{\displaystyle \leq \int ^{+\infty}_0 e^{- a^- u} \dfrac{1}{2T} \int^{T+u}_{-(T+u)} \mid G_2(t) \mid dt du.}
\end{array}$$
Now, since $G_2 \in PAP_0(\mathbb{R},\mathbb{R})$, then the function $\Psi_T$ defined by $${\displaystyle \Psi_T(u)=\dfrac{T+u}{T} \dfrac{1}{2(T+u)} \int^{T+u}_{-(T+u)} \mid G_2(t) \mid dt}$$
is bounded and satisfy $\underset{T\longrightarrow +\infty}{lim} \Psi_T(u)=0$. From the Lebesgue dominated convergence theorem, we obtain $$\underset{T\rightarrow +\infty}{lim}III_1(T)=0.$$
On the other hand, notice that $\|G_2\|_\infty<0$ and by setting $\xi=t-s$ we obtain $$\begin{array}{ll}
III_2(T)&{\displaystyle \leq \dfrac{\|G_2\|_{\infty} }{2T} \int^T_{-T} \int ^{+\infty}_{2T} e^{- a^- \xi} d\xi dt}\\
\\&{\displaystyle =\dfrac{\|G_2\|_{\infty} }{a^-} e^{-2 a^- T}}\qquad \underset{T\longrightarrow +\infty }{\longrightarrow 0}.
\end{array}$$ which implies the required result.\
\
Step 3:\
Let $$\begin{aligned}
\gamma : [0,1]&\longrightarrow& \mathbb{R}\\
u&\longmapsto& {\displaystyle \underset{t\in \mathbb{R} }{sup}\bigg\{-a(t)+\bigg[\sum^N_{i=1}b_i(t) \bigg(\dfrac{(n-m)}{4}+\dfrac{m}{(1+k^n)^2}\bigg)M^{m-1}+L\bigg]e^u\bigg\}.}\end{aligned}$$ It is clear that $\gamma$ is continuous function on \[0,1\].\
From $\textbf{(H4)}$ : $\gamma$(0)<0, so $\exists \zeta \in [0,1]$ such that $$\gamma(\zeta)<0\qquad (C.8).$$
Next, we claim that $ \Gamma(\psi)(t) \in [k,M]$ for all $t\in {\mathbb{R}}.$\
For $\psi \in \mathcal{B}$, we have $$\begin{array}{lll}
\bullet \text{ }\Gamma(\psi)(t)&\leq{\displaystyle \int ^t_{-\infty} e^{-(t-s) a^-} \left[ \sum^{N}_{i=1} b_i^+ -H^-\right]ds} \qquad(\text{By \textbf{(C.1)}})\\
\\&\leq{\displaystyle \int ^t_{-\infty} e^{a^-(t-s)} a^- M ds} \qquad ({\text{By \textbf{(H2)}} )}\\
\\&=M.\\
\\ \bullet \text{ }\Gamma(\psi)(t)&\geq {\displaystyle \int^t_{-\infty} e^{-(t-s)a^+}\left[ \sum^{N}_{i=1} \dfrac{b_i^- k^m}{1+k^n} -H^+\right] ds} \quad (\text{By \textbf{(C.3)} and \textbf{(C.5)}})\\
\\ &\geq {\displaystyle \int ^t_{-\infty} e^{-(t-s)a^+} a^+ k ds} \qquad (\text{By \textbf{(H3)}} )\\
\\&=k.
\end{array}$$ Thus $\Gamma$ a self-mapping from $\mathcal{B}$ to $\mathcal{B}$.\
\
$\ast$ $\Gamma$ is a contraction. Indeed; Let $\varphi ,\psi \in \mathcal{B} $, we have $$\begin{aligned}
\Vert \Gamma(\varphi)-\Gamma(\psi)\Vert _\infty &=&\underset{t\in {\mathbb{R}}}{sup}|\Gamma(\phi)(t)-\Gamma(\psi)(t)|\\
\\ &\leq &{\displaystyle \underset{t\in \mathbb{R}}{sup} \int^t_{-\infty} e^{-\int^t_s a(u)du} \sum^{N}_{i=1} b_i(s) \bigg| \dfrac{\varphi^m(s-\tau_i(s))}{1 + \varphi^n(s-\tau_i(s))}-\dfrac{\psi^m(s-\tau_i(s))}{1 + \psi^n(s-\tau_i(s))}\bigg| }\\
\\&&+\bigg|H(s,\varphi(s-\sigma(s))-H(s,\psi(s-\sigma(s))\bigg|ds.\end{aligned}$$ By the mean value theorem, one can obtain $$\begin{array}{lll}
\bigg|\dfrac{x^m}{1+x^n}- \dfrac{y^m}{1+y^n}\bigg|&=|g'(\theta)| |x-y| \qquad \qquad\text{\qquad \text{\qquad }where $\bigg[g : t\in {\mathbb{R}}^+ \longrightarrow \dfrac{t^m}{1+t^n}\bigg]$}\\ \\
&=\bigg|\dfrac{\theta^{m-1+n}(m-n)+m\theta^{m-1}}{(1 + \theta^n)^2}\bigg| |x-y|\\
\\&\leq \bigg[\dfrac{\theta^{m-1}(n-m)}{4 } +\dfrac{m\theta^{m-1}}{(1+\theta^n)^2}\bigg]|x-y|,\\
\end{array}$$ where $x,y \in [k,M]$, $\theta$ lies between $x$ and $y$.\
Consequently, the following estimate hold $$\begin{array}{lll}
{\displaystyle \Vert \Gamma(\varphi)-\Gamma(\psi)\Vert_\infty}& \leq {\displaystyle \underset{t\in \mathbb{R}}{sup} \int^t_{-\infty} e^{-\int^t_s a(u)du} \bigg(\sum^{N}_{i=1} b_i(s) \bigg[ \dfrac{(n-m)M^{m-1}}{4}+\dfrac{m M^{m-1} }{(1+k^n)^2}\bigg]}\\
\\& \bigg| \varphi(s-\tau_i(s))-\psi(s-\tau_i(s))\bigg| +L \parallel \varphi-\psi \parallel_\infty\bigg) ds\\
\\& {\displaystyle \leq \underset{t\in \mathbb{R}}{sup} \int^t_{-\infty} e^{-\int^t_s a(u)du} \bigg(\sum^{N}_{i=1} b_i(s) \bigg[\dfrac{(n-m)}{4}+\dfrac{m}{(1+k^n)^2}\bigg]M^{m-1}+L\bigg)}\\
\\& {\displaystyle \times \parallel \varphi-\psi \parallel _\infty ds}\\
\\&\leq {\displaystyle \parallel \varphi-\psi \parallel _\infty \underset{t \in \mathbb{R}}{sup} \int ^t_{-\infty} e^{-\int^t_s a(u)du}a(s) e^{-\zeta}ds} \qquad (\text{By \textbf{(C.8)}})\\
\\& \leq {\displaystyle e^{-\zeta} \parallel \varphi-\psi \parallel_\infty},\end{array}$$
which proves that $\Gamma$ is a contracting operator on ${\displaystyle \mathcal{B}}$. By using fixed point theorem, we obtain that operator $\Gamma$ has a unique fixed point ${\displaystyle x^* (.)\in \mathcal{B}}$, which corresponds to unique p.a.p solution of the equation (4).
The stability of the pap solution {#sec:4}
=================================
[@I] Let $f : {\mathbb{R}}\longrightarrow {\mathbb{R}}$ be a continuous function, then the upper right derivative of $f$ is defined as $$D^+f(t)= \overline{\underset{h\rightarrow0^+}{lim}}\dfrac{f(t + h) - f(t)}{h}.$$
We say that a solution $x^*$ of Eq. (4) is a global attractor or globally asymptotically stable (GAS) if for any positive solution $x_t(t_0,\phi)$ $$\underset{t\rightarrow +\infty}{lim}|x^*(t)-x_t(t_0,\phi)|=0.$$
Under the assumptions **H(1)-H(4)**, the positive pseudo almost periodic solution $x^*(.)$ of the equation (4) is a global attractor.
Firstly, set $x_t(t_0,\phi)$ for $\phi \in \mathcal{C}^0$ by $x(t)$ for all $t \in {\mathbb{R}}$. Let $$y(.)=x(.)-x^*(.).$$Then, $$\begin{aligned}
y'(t) &=& -a(t)[x(t)-x^*(t)]+\sum^{N}_{i=1} b_i(t)\bigg[\dfrac{x^m(t-\tau_i(t))}{1 + x^n(t-\tau_i(t))}-\dfrac{x^*{^m}(t-\tau_i(t))}{1 + x^*{^n}(t-\tau_i(t))}\bigg] \\
\\ &&-\bigg [H(t,x(t-\sigma(t)))-H(t,x^*(t-\sigma(t)))\bigg].
\end{aligned}$$ Let us define a continuous function $\Delta :\mathbb{R^+}\longrightarrow \mathbb{R}$ by $$\Delta(u)= u-a^-+\bigg[L_{H}+\sum_{i=1}^{N} b^+_i\bigg(\dfrac{(n-m)}{4}+\dfrac{m}{(1+k^n)^2}\bigg)M^{m-1}\bigg]\exp(ut).$$
From $\textbf{(H4)}$, we have $\Delta (0)<0$, then there exists $\lambda\in \mathbb{R}^+$, such that$$\Delta(\lambda)<0 \qquad (C.11).$$
We consider the Lyapunov functional $V(t)=|y(t)|e^{\lambda t}$. Calculating the upper right derivative of $V(t)$, we obtain $$\begin{aligned}
D^+V(t) &\leq & \bigg[-a(t)|y(t)|+\sum_{i=1}^{N} b_i(t)\bigg|\frac{x^m(t-\tau_i(t))}{1+x^n(t-\tau_i(t))} -\frac{(x^*)^m(t-\tau_i(t))}{1+(x^*)^n(t-\tau_i(t))}\bigg|+ \bigg|H(t,x^*(t-\sigma(t)))\\
\\&&-H(t,x(t-\sigma(t)))\bigg|\bigg]e^{\lambda t}+\lambda|y(t)|e^{\lambda t}\\
\\&\leq & e^{\lambda t}\bigg((\lambda-a^-)|y(t)|+L_H|y(t-\sigma(t))|+\sum_{i=1}^{N} b^+_i \bigg|\frac{x^m(t-\tau_i(t))}{1+x^n(t-\tau_i(t))}-\frac{(x^*)^m(t-\tau_i(t))}{1+(x^*)^n(t-\tau_i(t))}\bigg|\bigg).
\end{aligned}$$ We claim that $$\begin{aligned}
V(t) &=& |y(t)|e^{\lambda t}< e^{\lambda t_0}(M+\max_{t<t_0}|x(t)-x^*(t)|)=M_1, \forall t\geq t_0.
\end{aligned}$$ Suppose that there exists $t_1>t_0$ such that $$\begin{aligned}
V(t_1) &=& M_1, V(t)<M_1, \forall \text{ } t_0-r\leq t< t_1.
\end{aligned}$$ Besides, $$\begin{aligned}
0\leq D^+V(t_1)&\leq& (\lambda-a^-)|y(t_1)|e^{\lambda t_1}+L_H|y(t_1-\sigma(t_1))|e^{\lambda (t_1-\sigma(t_1))}e^{\sigma(t_1) \lambda}\\
\\&&+\sum_{i=1}^{N} {b_i}^+ e^{\lambda t_1}\bigg[\frac{x^m(t_1-\tau_i(t_1))}{1+x^n(t_1-\tau_i(t_1))}-\frac{(x^*)^m(t_1-\tau_i(t_1))}{1+(x^*)^n(t_1-\tau_i(t_1))}\bigg].
\end{aligned}$$
On the other hand, for all $x,x^* \in {\mathbb{R}}^+$ we have
$$\bigg|\dfrac{x^m}{1+x^n}-\dfrac{(x^*)^m}{1+(x^*)^n}\bigg| \leq \bigg[\dfrac{\zeta^{m-1}(n-m)}{4 } +\dfrac{m\zeta^{m-1}}{(1+\zeta^n)^2}\bigg] |x-x^*|,$$
where $\zeta \in [k,M]$.\
So, we obtain $$\begin{aligned}
\bigg |\dfrac{x^m(t-\tau_i(t))}{1+x^n(t-\tau_i(t))}-\frac{(x^*)^m(t-\tau_i(t))}{1+(x^*)^n(t-\tau_i(t))}\bigg | \leq \bigg[\dfrac{M^{m-1}(n-m)}{4 } +\dfrac{M^{m-1}}{(1+k^n)^2}\bigg] |y(t-\tau_i(t))|.
\end{aligned}$$
Then, $$\begin{aligned}
0&\leq& D^+V(t_1)\\
\\&\leq& (\lambda-a^-)|y(t_1)|e^{\lambda t_1}+L_H|y(t_1-\sigma(t_1))|e^{\lambda (t_1-\sigma(t_1))}e^{\sigma(t_1)\lambda}\\
\\ &&+\sum_{i=1}^{N} \bar{b_i}e^{\lambda (t_1-\tau_i(t_1)) }e^{\tau_i(t_1) \lambda}|y(t_1-\tau_i(t_1))|\bigg[\dfrac{M^{m-1}(n-m)}{4 } +\dfrac{m M^{m-1}}{(1+k^n)^2}\bigg]\\
\\&\leq& \bigg(\lambda-a^-+L_H e^{r\lambda}+\sum_{i=1}^{N} b^+_i e^{r \lambda}\bigg[ \dfrac{(n-m)}{4}+\dfrac{m}{(1+k^n)^2}\bigg]M^{m-1} \bigg)M_1.
\end{aligned}$$
However, by **(C.11)** $$\begin{aligned}
\lambda-a^-+L_H e^{r\lambda}+\sum_{i=1}^{N} b^+_i e^{r \lambda}\bigg [ \dfrac{(n-m)}{4}+\dfrac{m}{(1+k^n)^2}\bigg]M^{m-1} <0,
\end{aligned}$$ which is contradicts the hypothesis. Consequently, $|y(t)|< M_1 e^{-\lambda t},\text{ } \forall t> t_0$.
An example {#sec:5}
==========
In this section, we present an example to check the validity of our theoretical results obtained in the previous sections.\
\
First, we construct a function $\omega(t)$. For $n = 1, 2,...$ and $0 \leq i < n$, $$a_n=\dfrac{n^3-n}{3}$$ and intervals $$I_n^i = [a_n + i, a_n + i + 1].$$ Choose a non-negative, continuous function g on \[0,1\] defined by $$g(s) =\dfrac{8}{\pi}\sqrt{s-s^2}.$$ Define the function $\omega$ on ${\mathbb{R}}$ by $$\omega(t)=
\left \{
\begin{array}{lll}
g[t-(a_n+i)], &\qquad t \in I_n^i,\\
0, &\qquad t\in {\mathbb{R}}^+ \ \cup \{I_n^i: n=1,2,...,0\leq i \leq n \}, \\
\omega(-t), &\qquad t<0.
\end{array}
\right.$$\
From , we know that $\omega \in PAP_0( {\mathbb{R}}, {\mathbb{R}})$ is ergodic. However, $\omega \notin C_0( {\mathbb{R}}, {\mathbb{R}})$.
Let us consider the case $n=m=2$ , $N=1$,\
\
$a(t)=0.38+ \dfrac{|sin(t)+sin (\pi t)|}{400} + \dfrac{ \pi\omega(t)}{800}$, $b_1(t) =1+\dfrac { sin^2( t) + sin^2(\sqrt{2}t)}{10}+ \dfrac{1}{100(1+t^2)},$\
\
$ \tau_1(t) = cos^2(t) + cos^2 (\sqrt{2}t) + 1 +e^{-t^2}, \text{}H(t,x)=0.01|sin(t)+cos(\sqrt{3}t)| \dfrac{|x|}{1+x^2},\text{ et }\sigma(t)=|sin(t)-sin(\pi t)|.$\
\
Clearly, $a^+=0.39, a^-=0.38, b^+=1.21, b^-=1, H^+=5*10^{-3}, H^-=0, r=4.$\
\
It is not difficult to see that H $\in PAP_U({\mathbb{R}}\times {\mathbb{R}},{\mathbb{R}}^+)$ and satisfies Lipschitz condition with $l=10^{-2}$.\
\
Let k=2, M=3.29. We obtain easily:
1. $0<2<3.29;$
2. ${\displaystyle -a^- M+\sum^N_{i=1}b_i^+ - H^-}= -0.04 < 0$;
3. ${\displaystyle -a^+ k+\sum^N_{i=1}b_i^- \dfrac{k^m}{1+k^n} -H^+}= 0.015> 0$;
4. ${\displaystyle \underset{t\in \mathbb{R} }{sup}\bigg\{-a(t)+\sum^N_{i=1}b_i(t)\bigg[\dfrac{(n-m)}{4}+\dfrac{m}{(1+k^n)^2}\bigg]M^{m-1}+L\bigg\}}$= -0.0515<0.
Then, all the conditions in Theorem [1]{} et [2]{} are satisfied, Therefore, there exists a unique pseudo almost periodic solution $x^*$ in $\mathcal{B}=\{\phi \in PAP({\mathbb{R}},{\mathbb{R}}); k\leq \phi (t) \leq M,\text{} \forall t\in {\mathbb{R}}\}$ which is global attractor.\
Notice that in vue of this above example, it follows that the condition of proposition 4.2 is necessary. Besides, The results are verified by the numerical simulations in fig(2).
Conclusion {#sec:6}
==========
In this paper, some new conditions were given ensuring the existence of the uniqueness positive pseudo almost periodic solution of the hematopoies model with mixed delays and with a non-linear harvesting term (which is more realistic).\
Also, the global attractivity of the unique pseudo almost periodic solution of the considered model is demonstrated by a new and suitable Lyapunov function.\
\
As the best of our knowledge, this is the first paper considering such solutions for this generalized model.\
Notice that our approach can be applied to the case of the almost automorphic and pseudo almost automorphic solutions of the considered model.
[100]{} Liu B., New results on the positive almost periodic solutions for a model of hematopoiesis, Nonlinear Anal. Real World Appl. 17(2014), 252-264. Zhang C.. Almost Periodic Type Functions and Ergodicity. Kluwer Academic/Science Press: Beijing, 2003. Zhang C., Pseudo almost periodic solutions of some diffe rential equations II, J. Math. Anal. Appl., 192(1995), pp. 543-561. Braverman E., Saker S.H., Permanence, oscillation and attractivity of the discrete hematopoiesis model with variable coefficients, Nonlinear Anal. Theory Methods Appl. 67 (2007) 2955–2965. Chérif F., Analysis of Global Asymptotic Stability and Pseudo Almost Periodic Solution of a Class of Chaotic Neural Networks, Mathematical Modelling and Analysis, 18:4, 489-504. Chérif F., Pseudo almost periodic solution of Nicholson’s blowflies model with mixed delays, Applied Mathematical Modelling 39 (2015) 5152–5163. Long F. and Yang M. Q. , “Positive periodic solutions of delayed Nicholsons blowies model with a linear harvesting term,” Electronic Journal of Qualitative Thory of Diffrential Equations, vol. 41, pp. 1–11, 2011. Liu G., Yan J., Zhang F. : Existence and global attractivity of unique positive periodic solution for a model of hematopoiesis. J. Math. Anal. Appl. 334, 157-171 (2007). Zhou H., Wang W., Zhou Z.F., Positive almost periodic solution for a model of hematopoiesis with infinite time delays and a nonlinear harvesting term, Abstr. Appl. Anal. 2013 (2013) 146729. Zhou H., Wang J., and Zhou Z., “Positive almost periodic solution for impulsive Nicholson’s blowfles model with multiple linear harvesting terms,” Mathematical Methods in the Applied Sciences, vol. 36, no. 4, pp. 456–461, 2013. Ding H.S., Liu Q.L., Nieto J.J.: Existence of positive almost periodic solutions to a class of hematopoiesis model. Appl. Math. Model. 40, 3289-3297 (2016). Ding H.S., N’Guérékata H.M., Nieto J.J., Weighted pseudo almost periodic solutions for a class of discrete hematopoiesis model, Rev. Mat. Complut. 26 (2013)427–443. Blot J. B., Blot J., N’Guérékata G. M., Pennequin D. On C(n)- Almost Periodic Solutions to Some Nonautonomous Differential Equations in Banach Spaces, annales societatis mathematycae polonae, Series I: commentationes mathematicae roczniki polskiego towarzystwa mathematycznego, Seria I: prace mathematyczne XLVI(2)(2006), 263-273. Meng J., Global exponential stability of positive pseudo-almost-periodic solutions for a model of hematopoiesis, Abstract and Applied Analysis, 2013, Art. ID 463076. Hale JK, Verduyn Lunel SM. Introduction to Functional Differential Equations. Springer-Verlag: New York, 1993. Alzabut J.O., Nieto J.J., Stamov G.Tr., Existence and exponential stability of positive almost periodic solutions for a model of hematopoiesis, Bound. Value Probl. 1193 (2009) 429–436. Mackey M.C, Glass L., Oscillation and chaos in physiological control system, Science 197 (1977) 287-289. Amdouni M., Chérif F., The pseudo almost periodic solutions of the new class of Lotka-Volterra recurrent neural networks with mixed delays,Chaos, Solitons and Fractals 113 (2018) 79–88. Cieutat P., Fatajou S. et N’Guérékata G.M., Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations. Annales mathématiques Blaise Pascal, tome 6(1999), p. 1-11. Rihani S., Amor K., Chérif F., Pseudo almost periodic solutions for a Lasota-Wazewska model, Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 62, pp. 1-17. Saker S.H., Oscillation and global attractivity in hematopoiesis model with periodic coefficients, Appl. Math. Comput. 142 (2003) 477–494. Diagana T., Pseudo Almost Periodic Functions in Banach Spaces, Nova Science, New York, 2007. Diagana T., Zhou H., Existence of positive almost periodic solutions to the hematopoiesis model. Applied Mathematics and Computation 274 (2016) 644–648. Chen X., Hui-Sheng, Ding, Positive Pseudo Almost Periodic Solutions for a Hemathopoies Model, Journal of Nonlinear Evolution Equations and Applications, Volume 2016, Number 2, pp. 25–36 (September 2016) Wang X., Zhang H., A new approach to the existence, nonexistence and uniqueness of positive almost periodic solution for a model of hematopoiesis, Nonlinear Anal. Real World Appl. 11 (1) (2010) 60–66. Wu X., Li J., Zhou H., A necessary and sufficient condition for the existence of positive periodic solutions of a model of hematopoiesis, Comput. Math. Appl. 54 (6) (2007) 840–849.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A unified treatment of mass varying dark matter coupled to cosmon-[*like*]{} dark energy is shown to result in [*effective*]{} generalized Chaplygin gas (GCG) scenarios. The mass varying mechanism is treated as a cosmon field inherent effect. Coupling dark matter with dark energy allows for reproducing the conditions for the present cosmic acceleration and for recovering the stability resulted from a positive squared speed of sound $c_{s}^{\2}$, as in the GCG scenario. The scalar field mediates the nontrivial coupling between the dark matter sector and the sector responsible for the accelerated expansion of the universe. The equation of state of perturbations is the same as that of the background cosmology so that all the effective results from the GCG paradigm are maintained. Our results suggest the mass varying mechanism, when obtained from an exactly soluble field theory, as the right responsible for the stability issue and for the cosmic acceleration of the universe.'
author:
- 'A. E. Bernardini'
date:
-
-
title: Mass varying dark matter in effective GCG scenarios
---
Introduction
============
The ultimate nature of the dark sector of the universe is the most relevant issue related with the negative pressure component required to understand why and how the universe is undergoing a period of accelerated expansion [@Zla98; @Wan99; @Ste99; @Bar99; @Ber00]. A natural and simplistic explanation for this is obtained in terms of a tiny positive cosmological constant introduced in the Einstein’s equation for the universe. Since the cosmological constant has a magnitude completely different from that predicted by theoretical arguments, and it is often confronted with conceptual problems, physicists have been compelled to consider other explanations for that [@Ame02; @Kam02; @Bil02; @Ber02; @Cal03; @Mot04; @Bro06A].
Motivated by the high energy physics, an alternative for obtaining a negative pressure equation of state considers that the dark energy can be attributed to the dynamics of a scalar field $\phi$ which realizes the present cosmic acceleration by evolving slowly down its potential $V\bb{\phi}$ [@Pee87; @Rat87]. These models assume that the vacuum energy can vary [@Bro33]. Following theoretical as well as phenomenological arguments, several possibilities have been proposed, such as $k$-essence [@Chi00; @Arm01], phantom energy [@Sch01; @Car03], cosmon fields [@Wet87], and also several types of modifications of gravity [@Def02; @Car04; @Ama06].
One of the most challenging proposals concerns mass varying particles [@Hun00; @Gu03; @Far04; @Bja08] coupled to the dark energy through a dynamical mass dependence on a light scalar field which drives the dark energy evolution in a kind of unified cosmological fluid. The idea in the well-known mass varying mechanism [@Far04; @Pec05; @Bro06A; @Bja08] is to introduce a coupling between a relic particle and the scalar field whose effective potential changes as a function of the relic particle density. This coupled fluid is either interpreted as dark energy plus neutrinos, or as dark energy plus dark matter [@Ber08A; @Ber08B]. Such theories can possess an adiabatic regime in which the scalar field always sits at the minimum of its effective potential, which is set by the local mass varying particle density. The relic particle mass is consequently generated from the vacuum expectation value of the scalar field and becomes linked to its dynamics by $m\bb{\phi}$.
A scenario which congregate dark energy and some kind of mass varying dark matter in a unified negative pressure fluid can explain the origin of the cosmic acceleration. In particular, any background cosmological fluid with an effective behaviour as that of the generalized Chaplygin gas (GCG) [@Kam02; @Bil02; @Ber02] naturally offers this possibility. The GCG is particularly relevant in respect with other cosmological models as it is shown to be consistent with the observational constraints from CMB [@Ber03], supernova [@Sup1; @Ber04; @Ber05], gravitational lensing surveys [@Ber03B], and gamma ray bursts [@Ber06B]. Moreover, it has been shown that the GCG model can be accommodated within the standard structure formation mechanism [@Kam02; @Ber02; @Ber04].
In the scope of finding a natural explanation for the cosmic acceleration and the corresponding adequation to stability conditions for a background cosmological fluid, our purpose is to demonstrate that the GCG just corresponds to an effective description of a coupled fluid composed by dark energy with equation of state given by $p\bb{\phi} = -\rho\bb{\phi}$ and a cold dark matter (CDM) with a dynamical mass driven by the scalar field $\phi$. Once one has consistently obtained the mass dependence on $\phi$, which is model dependent, it can be noticed that the cosmological evolution of the composed fluid is governed by the same dynamics prescribed by cosmon field equations. It suggests that the mass varying mechanism embedded into the cosmon-[*like*]{} dynamics reproduces the effective behaviour of the GCG. In addition, coupling dark matter with dark energy by means of a dynamical mass driven by such a scalar field allows for reproducing the conditions for the present cosmic acceleration and for recovering the stability prescribed by a positive squared speed of sound $c_{s}^{\2}$. At least implicitly, it leads to the conclusion that the dynamical mass behaviour is the main agent of the stability issue and of the cosmic acceleration of the universe.
At our approach, the dark matter is approximated by a degenerate fermion gas (DFG). In order to introduce the mass varying behaviour, we analyze the consequences of coupling it with and underlying dark energy scalar field driven by a cosmon-[*type*]{} equation of motion. We discuss all the relevant constraints on this in section II. In section III, we obtain the energy density and the equation of state for the unified fluid and compare them with the corresponding quantities for the GCG. In section IV, we discuss the stability issue and the accelerated expansion of the universe in the framework here proposed. The pertinent comparisons with a GCG scenario are evaluated. We draw our conclusions in section V by summarizing our findings and discussing their implications.
Mass varying mechanism for a DFG coupled to cosmon-[*like*]{} scalar fields
===========================================================================
To understand how the mass varying mechanism takes place for different particle species, it is convenient to describe the corresponding particle density, energy density and pressure as functionals of a statistical distribution. This counts the number of particles in a given region around a point of the phase space defined by the conjugate coordinates: momentum, [$\beta$]{}, and position, [$x$]{}. The statistical distribution can be defined by a function $f\bb{q}$ in terms of a comoving variable, $q = a\,|\mbox{\boldmath$\beta$}|$, where $a$ is the scale factor (cosmological radius) for the flat FRW universe, for which the metrics is given by $ds^{\2} = dt^{\2} - a^{\2}\bb{t}\delta_{\ii\j}dx^{\ii}dx^{\j}$. In the flat FRW scenario, the corresponding particle density, energy density and pressure are thus given by $$\begin{aligned}
n\bb{a} &=&\frac{1}{\pi^{\2}\,a^{\3}}
\int_{_{0}}^{^{\infty}}{\hspace{-0.3cm}dq\,q^{\2}\ \hspace{-0.1cm}f\bb{q}},\nonumber\\
\rho_m\bb{a, \phi} &=&\frac{1}{\pi^{\2}\,a^{\4}}
\int_{_{0}}^{^{\infty}}{\hspace{-0.3cm}dq\,q^{\2}\, \left(q^{\2}+ m^{\2}\bb{\phi}\,a^{\2}\right)^{\1/\2}\hspace{-0.1cm}f\bb{q}},\\
p_m\bb{a, \phi} &=&\frac{1}{3\pi^{\2}\,a^{\4}}\int_{_{0}}^{^{\infty}}{\hspace{-0.3cm}dq\,q^{\4}\, \left(q^{\2}+ m^{\2}\bb{\phi}\,a^{\2}\right)^{\mi\1/\2}\hspace{-0.1cm} f\bb{q}}.~~~~ \nonumber
\label{gcg01}\end{aligned}$$ where the last two can be depicted from the Einstein’s energy-momentum tensor [@Dod05]. For the case where $f\bb{q}$ is a Fermi-Dirac distribution function, it can be written as $$f\bb{q}= \left\{\exp{\left[(q - q_F)/T_{\0}\right]} + 1\right\}^{\mi\1}, \nonumber$$ where $T_{\0}$ is the relic particle background temperature at present. In the limit where $T_{\0}$ tends to $0$, it becomes a step function that yields an elementary integral for the above equations, with the upper limit equal to the Fermi momentum here written as $q_{F} = a \,\beta\bb{a}$. It results in the equations for a DFG [@ZelXX]. The equation of state can be expressed in terms of elementary functions of $\beta \equiv \beta\bb{a}$ and $m \equiv m\bb{\phi\bb{a}}$, $$\begin{aligned}
n\bb{a} &=& \frac{1}{3 \pi^{\2}} \beta^{\3},\nonumber\\
\rho_m\bb{a} &=& \frac{1}{8 \pi^{\2}}
\left[\beta(2 \beta^{\2} + m^{\2})\sqrt{\beta^{\2} + m^{\2}} -
\mbox{arc}\sinh{\left(\beta/m\right)}\right],\\
p_m\bb{a} &=& \frac{1}{8 \pi^{\2}}
\left[\beta (\frac{2}{3} \beta^{\2} - m^{\2})\sqrt{\beta^{\2} + m^{\2}} + \mbox{arc}\sinh{\left(\beta/m\right)}\right].\nonumber
\label{gcg01B}\end{aligned}$$ One can notice that the DFG approach is useful for parameterizing the transition between ultra-relativistic (UR) and non-relativistic (NR) thermodynamic regimes. It is not mandatory for connecting the mass varying scenario with the GCG scenario.
Simple mathematical manipulations allow one to easily demonstrate that $$n\bb{a} \frac{\partial \rho_m\bb{a}}{\partial n\bb{a}} = (\rho_m\bb{a} + p_m\bb{a}),
\label{gcg02B}$$ and $$m\bb{a} \frac{\partial \rho_m\bb{a}}{\partial m\bb{a}} = (\rho_m\bb{a} - 3 p_m\bb{a}),
\label{gcg02}$$
Noticing that the explicit dependence of $\rho_m$ on $a$ is intermediated by $\beta\bb{a}$ and $m\bb{a} \equiv m\bb{\phi\bb{a}}$, one can take the derivative of the energy density with respect to time in order to obtain $$\begin{aligned}
\dot{\rho}_m &=& \dot{\beta}\bb{a} \frac{\partial \rho_m\bb{a}}{\partial \beta\bb{a}} + \dot{m}\bb{a} \frac{\partial \rho_m\bb{a}}{\partial m\bb{a}}\nonumber\\
&=& \dot{n}\bb{a} \frac{\partial \rho_m\bb{a}}{\partial n \bb{a}} + \dot{m}\bb{a} \frac{\partial \rho_m\bb{a}}{\partial m\bb{a}}\nonumber\\
&=& - 3\frac{\dot{a}}{a} n\bb{a} \frac{\partial \rho_m\bb{a}}{\partial n \bb{a}} + \dot{\phi}\frac{\mbox{d} m}{\mbox{d} \phi}\frac{\partial \rho_m\bb{a}}{\partial m\bb{a}},
\label{gcg03BB}\end{aligned}$$ where the [*overdot*]{} denotes differentiation with respect to time ($^{\cdot}\, \equiv\, d/dt$). The substitution of Eqs. (\[gcg02B\]-\[gcg02\]) into the above equation results in the energy conservation equation given by $$\dot{\rho}_m + 3 H (\rho_m + p_m) - \dot{\phi}\frac{\mbox{d} m}{\mbox{d} \phi} (\rho_m - 3 p_m) = 0,
\label{gcg03}$$ where $H = \dot{a}/{a}$ is the expansion rate of the universe. If one performs the derivative with respect to $a$ directly from $\rho_m$ in the form given in Eq. (\[gcg01\]), the same result can be obtained. The coupling between relic particles and the scalar field as described by Eq. (\[gcg03\]) are effective just for NR fluids. Since the strength of the coupling is suppressed by the relativistic increase of pressure ($\rho\sim 3 p$), as long as particles become relativistic ($T\bb{a} = T_{\0}/a >> m\bb{\phi\bb{a}}$) the matter fluid and the scalar field fluid tend to decouple and evolve adiabatically. The mass varying mechanism expressed by Eq. (\[gcg03\]) translates the dependence of $m$ on $\phi$ into a dynamical behaviour. In particular, for a DFG, the consistent analytical transition between UR and NR regimes and their effects on coupling dark matter ($m$) and dark energy ($\phi$) are evident from Eq. (\[gcg03\]). The mass thus depends on the value of a slowly varying classical scalar field [@Wet94; @Bea08] which evolves like a [*cosmon*]{} field. The cosmon-[*type*]{} equation of motion for the scalar field $\phi$ is given by $$\ddot{\phi} + 3 H \dot{\phi} + \frac{\mbox{d} V\bb{\phi}}{\mbox{d} \phi} = Q\bb{\phi}.
\label{gcg04}$$ where, in the mass varying scenario, one identifies $Q\bb{\phi}$ as $ - (\mbox{d} m/\mbox{d} \phi)/(\partial \rho_m/\partial m)$. The corresponding equation for energy conservation can be written as $$\dot{\rho_{\phi}} + 3 H (\rho_{\phi} + p_{\phi}) + \dot{\phi}\frac{\mbox{d} m}{\mbox{d} \phi} \frac{\partial \rho_m}{\partial m} = 0.
\label{gcg05}$$ which, when added to Eq. (\[gcg03\]), results in the equation for a unified fluid $(\rho, p)$ with a dark energy component and a mass varying dark matter component, $$\dot{\rho} + 3 H (\rho + p) = 0,
\label{gcg06}$$ where $\rho = \rho_{\phi} + \rho_m$ and $p = p_{\phi} + p_m$.
As we shall notice in the following, this unified fluid corresponds to an effective description of the universe parameterized by a GCG equation of state.
Decoupling mass varying dark matter from the effective GCG
==========================================================
Irrespective of its origin, several studies yield convincing evidences that the GCG scenario is phenomenologically consistent with the accelerated expansion of the universe. This scenario is introduced by means of an exotic equation of state [@Ber02; @Kam02; @Ber03] given by $$p = - A_{\s} \left(\frac{\rho_{\0}}{\rho}\right)^{\al},
\label{gcg20}$$ which can be obtained from a generalized Born-Infeld action [@Ber02]. The constants $A_{\s}$ and $\alpha$ are positive and $0 < \alpha \leq 1$. Of course, $\alpha = 0$ corresponds to the $\Lambda$CDM model and we are assuming that the GCG model has an underlying scalar field, actually real [@Kam02; @Ber04] or complex [@Bil02; @Ber02]. The case $\alpha = 1$ corresponds to the equation of state of the Chaplygin gas scenario [@Kam02] and is already ruled out by data [@Ber03]. Notice that for $A_s =0$, GCG behaves always as matter whereas for $A_{\s} =1$, it behaves always as a cosmological constant. Hence to use it as a unified candidate for dark matter and dark energy one has to exclude these two possibilities so that $A_s$ must lie in the range $0 < A_{\s} < 1$.
Inserting the above equation of state into the unperturbed energy conservation Eq. (\[gcg06\]), one obtains through a straightforward integration [@Kam02; @Ber02] $$\rho = \rho_{\0} \left[A_{\s} + \frac{(1-A_{\s})}{a^{\3(\1+\alpha)}}\right]^{\1/(\1 \pl \al)},
\label{gcg21}$$ and $$p = - A_{\s} \rho_{\0} \left[A_{\s} + \frac{(1-A_{\s})}{a^{\3(\1+\alpha)}}\right]^{-\al/(\1 \pl \al)}.
\label{gcg22}$$
One of the most striking features of the GCG fluid is that its energy density interpolates between a dust dominated phase, $\rho \propto a^{-\3}$, in the past, and a de-Sitter phase, $\rho = -p$, at late times. This property makes the GCG model an interesting candidate for the unification of dark matter and dark energy. Indeed, it can be shown that the GCG model admits inhomogeneities and that, in particular, in the context of the Zeldovich approximation, these evolve in a qualitatively similar way as in the $\Lambda$CDM model [@Ber02]. Furthermore, this evolution is controlled by the model parameters, $\alpha$ and $A_{\s}$.
Assuming the canonical parametrization of $\rho$ and $p$ in terms of a scalar field $\phi$, $$\begin{aligned}
\rho &=& \frac{1}{2}\dot{\phi}^{\2} + V,\nonumber\\
p &=& \frac{1}{2}\dot{\phi}^{\2} - V,
\label{pap01}\end{aligned}$$ allows for obtaining the effective dependence of the scalar field $\phi$ on the scale factor, $a$, and explicit expressions for $\rho$, $p$ and $V$ in terms of $\phi$.
Following Ref. [@Ber04], one can obtain through Eq. (\[gcg05\]) the field dependence on $a$, $$\dot{\phi}^{\2}\bb{a} = \frac{\rho_{\0}(1 - A_{\s})}{a^{\3(\al\pl\1)}}
\left[A_{\s} + \frac{(1-A_{\s})}{a^{\3(\al\pl\1)}}\right]^{-\al/(\al \pl \1)},
\label{pap02}$$ and assuming a flat evolving universe described by the Friedmann equation $H^{\2} = \rho$ (with $H$ in units of $H_{\0}$ and $\rho$ in units of $\rho_{\mbox{\tiny Crit}} = 3 H^{\2}_{\0}/ 8 \pi G)$, one obtains $$\phi\bb{a} = - \frac{1}{3(\alpha + 1)}\ln{\left[\frac{\sqrt{1 - A_{\s}(1 - a^{\3(\al \pl \1)})} - \sqrt{1 - A_{\s}}}{\sqrt{1 - A_{\s}(1 - a^{\3(\al \pl \1)})} + \sqrt{1 - A_{\s}}}\right]},
\label{pap03}$$ where it is assumed that $$\phi_{\0} = \phi\bb{a_{\0} = 1} = - \frac{1}{3(\alpha + 1)}\ln{\left[\frac{1 - \sqrt{1 - A_{\s}}}{1 + \sqrt{1 - A_{\s}}}\right]}.
\label{pap04}$$ One then readily finds the scalar field potential, $$V\bb{\phi} = \frac{1}{2}A_{\s}^{\frac{\1}{\1 \pl \al}}\rho_{\0}\left\{
\left[\cosh{\left(3\bb{\alpha + 1} \phi/2\right)}\right]^{\frac{\2}{\al \pl \1}}
+
\left[\cosh{\left(3\bb{\alpha + 1} \phi/2\right)}\right]^{-\frac{\2\al}{\al \pl \1}}
\right\}.
\label{pap05}$$ If one supposes that energy density, $\rho$, may be decomposed into a mass varying CDM component, $\rho_{m}$, and a dark energy component, $\rho_{\phi}$, connected by the scalar field equations (\[gcg04\])-(\[gcg05\]), the equation of state (\[gcg20\]) is just assumed as an effective description of the cosmological background fluid of the universe. Since the CDM pressure, $p_m$, is null, the dark energy component of pressure, $p_{\phi}$, results in the GCG pressure, $p = p_{\phi}$. Assuming that dark energy obeys a de-Sitter phase equation of state, that is, $\rho_{\phi}\bb{\phi} = - p_{\phi}\bb{\phi}$, the dark energy density can be parameterized by a generic quintessence potential, $\rho_{\phi}\bb{\phi} = U\bb{\phi}$, since its kinetic component has to be null for a canonical formulation. It results in $U\bb{\phi} = - p_{\phi}\bb{\phi} = p$, where $p$ is the GCG pressure given by Eq. (\[gcg22\]). By substituting the result of Eq.(\[pap03\]) into the Eq.(\[gcg22\]), and observing that $H^{\2} = \rho$, with $\rho$ given by Eq.(\[gcg21\]), it is possible to rewrite the GCG pressure, $p$, in terms of $\phi$. It results in the following analytical expression for $U\bb{\phi}$, $$U\bb{\phi} = \rho_{\phi}\bb{\phi} = - p_{\phi}\bb{\phi} = \left[A_{s}\cosh{\left(\frac{3\bb{\alpha + 1}\phi}{2}\right)}\right]^{\frac{\2 \al}{1 \pl \al}},
\label{pap08}$$ which is consistent with the result for $V\bb{\phi} = (1/2)(\rho\bb{\phi} - p\bb{\phi})$ from Eq. (\[pap05\]). Since $\rho_{\phi}\bb{\phi} + p_{\phi}\bb{\phi} = 0$, the Eq. (\[gcg05\]) is thus reduced to $$\frac{\mbox{d} U\bb{\phi}}{\mbox{d}{\phi}} +
\frac{\partial \rho_{m}}{\partial m} \frac{\mbox{d} m\bb{\phi}}{\mbox{d}\phi} = 0,
\label{pap09}$$ and the problem is then reduced to finding a relation between the scalar potential $U\bb{\phi}$ and the variable mass $m\bb{\phi}$. From the above equation, the effective potential governing the evolution of the scalar field is naturally decomposed into a sum of two terms, one arising from the original quintessence potential $U\bb{\phi}$, and other from the dynamical mass $m\bb{\phi}$. For appropriate choices of potentials and coupling functions satisfying Eq. (\[pap09\]), the competition between these terms leads to a minimum of the effective potential. For [*quasi*]{}-static regimes, it is possible to adiabatically track the position of this minimum, in a kind of stationary condition. The timescale for $\phi$ to adjust itself to the dynamically modified minimum of the effective potential may be short compared to the timescale over which the background density is changing. In the adiabatic regime, the matter and scalar field are tightly coupled together and evolve as one effective fluid. At our approach, once we have assumed the dark energy equation of state as $p_{\phi} = - \rho_{\phi}$, the stationary condition is a natural issue that emerges without any additional constraint on cosmon-[*type*]{} equations. In the GCG cosmological scenario, the effective fluid description is valid for the background cosmology and for linear perturbations. The equation of state of perturbations is the same as that of the background cosmology where all the effective results of the GCG paradigm are maintained.
The Eq. (\[pap08\]) leads to $\rho + p = \rho_{m} + p_{m}$ which, in the CDM limit, gives $$\rho\bb{a} + p\bb{a} = m\bb{a} \, n\bb{a} + p_{m} (\equiv 0) = \frac{1}{3\pi^{\2}}\,m\bb{a}\, \beta^{\3}\bb{a}.
\label{pap10}$$ Since the dependence of $m$ on $a$ is exclusively intermediated by $\phi\bb{a}$, i. e. $m\bb{a} \equiv m\bb{\phi\bb{a}}$, from Eqs. (\[gcg21\]), (\[gcg22\]) and (\[pap03\]), after some mathematical manipulations, one obtains $$m\bb{\phi} = m_{\0} \left[\frac{\tanh{\left(3\bb{\alpha + 1}\frac{\phi}{2}\right)}}{\tanh{\left(3\bb{\alpha + 1}\frac{\phi_{\0}}{2}\right)}}\right]^{\frac{ \2 \al}{1 \pl \al}}
\label{pap11}$$ which is consistent with Eq. (\[pap09\]) once $n \propto a^{\mi\3}$. One can thus infer that the adequacy to the adiabatic regime is left to the mass varying mechanism which drives the cosmological evolution of the dark matter component.
To give the correct impression of the time evolution of the abovementioned dynamical quantities driven by $\phi$, in the Fig. \[Fpap-01\] we observe the behaviour of $m\bb{a}$ and $U\bb{a}$ in confront with $\phi\bb{a}$ and $V\bb{a}$ of the GCG. In the Fig. \[Fpap-02\] we verify how the energy density $\rho$ and the corresponding equations of state $\omega$ for the unified fluid, $\rho_{m} + \rho_{\phi}$, which imitates the GCG, deviates from the right GCG scenario. We assume that the mass varying dark matter behaves like a DFG in a relativistic regime (hot dark matter (HDM)) and in a non-relativistic regime (CDM). For mass varying CDM coupled with dark energy with $p_{\phi} = -\rho_{\phi}$, the effective GCC leads to similar predictions for $\omega$, independently of the scale parameter $a$. The same is not true for HDM which, in the DFG approach, when weakly coupled with dark energy, leads to the same behavior of the GCG just for late time values of $a$ ($a \sim 1$).
As one can observe, the mass varying mechanism allows for reconstituting the GCG scenario in terms of non-exotic primitive entities: CDM and dark energy. Furthermore, as we shall notice, the dynamical mass expressed by Eq. (\[pap11\]) overpasses the problematic issue of stability.
It is also important to emphasize that, as in the case of the Chaplygin gas, where $\alpha = 1$, the GCG model admits a d-brane connection as its Lagrangian density corresponds to the Born-Infeld action plus some soft logarithmic corrections. Space-time is shown to evolve from a phase that is initially dominated, in the absence of other degrees of freedom on the brane, by non-relativistic matter to a phase that is asymptotically De Sitter. The Chaplygin gas reproduces such a behaviour. In this context, the explicit dependence of the mass on a scalar field is relevant in suggesting that the mass varying mechanism, when obtained from an exactly soluble field theory, can be the right responsible for the cosmological dynamics.
Stability and accelerated expansion
===================================
Adiabatic instabilities in cosmological scenarios was predicted [@Afs05] in a context of a mass varying neutrino (MaVaN) model of dark energy. The dynamical dark energy, in this approach, is obtained by coupling a light scalar field to neutrinos but not to dark matter. Their consequent effects have been extensively discussed in the context of mass varying neutrinos, in which the light mass of the neutrino and the recent accelerative era are twinned together through a scalar field coupling. In the adiabatic regime, these models faces catastrophic instabilities on small scales characterized by a negative squared speed of sound for the effectively coupled fluid. Starting with a uniform fluid, such instabilities would give rise to exponential growth of small perturbations. The natural interpretation of this is that the Universe becomes inhomogeneous with neutrino overdensities subject to nonlinear fluctuations [@Mot08] which eventually collapses into compact localized regions.
In opposition, in the usual treatment where dark matter are just coupled to dark energy, cosmic expansion together with the gravitational drag due to CDM have a major impact on the stability of the cosmological background fluid. Usually, for a general fluid for which we know the equation of state, the dominant effect on the sound speed squared $c_{s}^{\2}$ arises from the dark sector component and not by the neutrino component.
For the models where the stationary condition (cf. Eq. (\[pap09\])) implies a cosmological constant type equation of state, $ p_{\phi} = - \rho_{\phi}$, one obtains $c_{s}^{\2} = -1$ from the very start of the analysis. The effective GCG is free from this inconsistency. The coupling of the dark energy component with dynamical dark matter is responsible for removing such inconsistency by setting $c_{s}^{\2} \simeq \frac{d p_{\phi}}{d\rho_{\phi}} > 0$. The exact behavior for dark energy plus mass varying dark matter fluid in correspondence with the GCG is exhibited in the Fig. \[Fpap-03\] for different GCG $\alpha$ parameters. A previous analysis of the stability conditions for the GCG in terms of the squared speed of sound was introduced in Ref. [@Ber04], from which positive $c_{s}^{\2}$ implies that $0 \leq \alpha \leq 1$. These results are consistent with the accelerated expansion of the universe ruled by the dynamical mass of Eq. (\[pap11\]) which sets positive values for $(1 + 3 (p_{\phi} + p_m)/(\rho_{\phi} + \rho_m))$, as we can notice in the Fig. \[Fpap-03\].
For CDM ($p << m$) the unified fluid reproduces the GCG scenario. For HDM ($p >> m$), in spite of not reproducing the GCG, the conditions for stability and cosmic acceleration are maintained. Fig. \[Fpap-03\] shows that the GCG can indeed be interpreted as the effective result for the coupling between mass varying dark matter and a kind of scalar field dark energy which is cosmologically driven by a $\Lambda$-type equation of state, $p_{\phi} = -\rho_{\phi}$.
Conclusions
===========
The dynamics of the cosmology of mass varying dark matter coupled with dark energy dynamically driven by cosmon-[*type*]{} equations was studied without introducing specific quintessence potentials, but assuming that the cosmological background unified fluid presents an effective behaviour similar to that of the GCG.
We have comprehensively analyzed the stability characterized by a positive squared speed of sound and the cosmic acceleration conditions for such a dark matter coupled to dark energy fluid, that exists whenever such theories enter an adiabatic regime in which the scalar field faithfully tracks the minimum of the effective potential, and the coupling strength is strong compared to gravitational strength. The matter and scalar field are tightly coupled together and evolve as one effective fluid. The effective potential governing the evolution of the scalar field is decomposed into a sum of two terms, one arising from the original scalar field potential $U\bb{\phi}$, and the other from the dynamical mass of the dark matter.
The mass varying behaviour of the dark matter component was determined from the assumption of a kind of $\Lambda$-type dark energy dynamics embedded in an effective cosmological scenario which reproduces the cosmological effects of the GCG. It is equivalent to decoupling mass varying dark matter from the effective GCG concomitantly with assuming the dark energy equation of state as $p_{\phi} = - \rho_{\phi}$. The adiabatic regime naturally occurs without any additional constraint on scalar field equations. The unified fluid description is valid for the background cosmology and for linear perturbations. The equation of state of perturbations is the same as that of the background cosmology where all the effective results of the GCG paradigm are maintained.
Unfortunately, we cannot provide a sharp criterion on the potential and on the mass varying dependence on the scalar field to discriminate between these two possibilities: the GCG scenario or an effective unified fluid imitating the GCG via scalar fields driven by cosmon-[*type*]{} equations. Many results for specific quintessence potentials that are found in the literature, when they reproduce stability and cosmic acceleration, are recovered, and speculative predictions for new scenarios featuring other mass dependencies on scalar fields can be made. It remains open if the present approach can lead to a natural solution of the cosmological constant problem. In the meanwhile we take the cosmon model analogy as an interesting phenomenological approach, through which we can reproduce the main characteristics of the GCG.
Given the fundamental nature of the underlying physics behind the Chaplygin gas and its generalizations, it appears that it contains some of the key ingredients in the description of the Universe dynamics at early as well as late times. Our results suggest that the mass varying mechanism, when eventually derived from an exactly soluble field theory, which is noway trivial, can be the effective agent for the stability issue and for the cosmic acceleration of the universe, once it can effectively reproduce the main characteristics of a GCG scenario. It also stimulates our subsequent investigation of the evolution of density perturbations, instabilities and the structure formation in such scenarios. To summarize, we expect that the future precise data can provide more strong evidence to judge whether the dark energy is the cosmological constant and whether dark energy and dark matter can be unified into one cosmological background effective component.
We would like to thank for the financial support from the Brazilian Agencies FAPESP (grant 08/50671-0) and CNPq (grant 300627/2007-6).
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| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Mohammadreza Doostmohammadian, Hamid R. Rabiee, Houman Zarrabi, and Usman Khan, [^1]'
bibliography:
- 'bibliography.bib'
---
[Doostmohammadian : Observational Equivalence in System Estimation: Contractions in Complex Networks]{}
networks have recently gained considerable attentions in control and estimation theory [@liu_pnas; @ruths2014control; @liu2016tutorial; @das2015distributed; @asilomar14; @Liu_nature; @barabasi2016social]. This interest stems from the challenge to understand and infer the fundamental aspects of system behavior. Such complex networks exist in nature for example in chemical reaction networks and biological networks [@liu_pnas] as in proteomics and gene networks. Other than these natural complex networks, synthetic large-scale networks are recently considered due to emergence of the so-called Internet-of-Things (IoT) and Cyber-Physical-Systems (CPS) [@das2015distributed; @asilomar14]. Interestingly the design of such man-made networks are significantly tied by control and estimation principles as they are genuinely constructed based on these principles [@liu2016tutorial]. Examples range from consensus networks [@scientia] and social networks [@barabasi2016social] to more technological networks including electric power grids, computer networks, the Internet, etc. Indeed, many networks are a formalism to describe phenomena and systems in real life[^2]. In these and other similar applications the research focus is to uncover the tie between the internal system/network dynamics and the controllability and estimation properties.
It is known that, the internal states of complex systems are to a great extent dependent on each other, which is due to interaction of different components on each other and therefore these complex systems are represented as networks [@liu_pnas]. This inter-dependence is such that by measuring and tracking certain variables of complex system one can infer sufficient information about the rest of the system for filtering and tracking purposes. This implies that measuring well-selected variables give an *observable* inference of complex system. The term observability is a measure defining whether the internal states of a system can be determined by knowledge of its measurements. The system is said to be observable if one can reconstruct the *complete* state of the complex system from the set of measured states also known as system outputs [@bay].
There are different methods to check for observability of dynamic systems, namely: (i) algebraic method based on Gramian test [@bay]; (ii) the symbolic method also known as Popov-Belevitch-Hautus (PBH) test [@hautus]; and, (iii) the structural observability method introduced by pioneering work of Lin [@lin]. The first two methods are based on numerical values of system parameters while the third method is irrespective of the parameters and only relies on the *structure* of the underlying system. Therefore, the third method has certain benefits over the two other methods as it is computationally efficient and only requires the structural information and sparsity pattern of the system instead of exact numerical values [@woude:03]. In other words, the structural method only relies on the knowledge of complex system as a graph/network and therefore is extensively studied in the literature [@lin; @liu_pnas; @ruths2014control; @liu2016tutorial; @asilomar14; @Liu_nature; @woude:03].
The system graph representation, referred to as *system graph*, is an abstract way of modeling complex systems and has recently applied widely in the literature to reduce the complexity of such systems [@liu_pnas; @asilomar14; @Liu_nature]. In the system graph, every graph node represents a state (a variable or a parameter) and every link (or edge) between a pair of nodes represents derivative functional connection relating the state variables [@liu_pnas; @Liu_nature; @woude:03]. In general, graph representation approach is more conventional in networked systems [@acc13_mesbahi; @jstsp14] where the network structure is embedded into the system structure. The structural observability of networks, or any system graph as a network, is also referred to as *network observability* [@liu_pnas; @Liu_nature] and is the adopted methodology in this paper.[^3]
The network observability, as an abstract observability model of the system is closely related to system graph properties.[^4] The main theorem on this topic is originally stated in [@lin] and further developed recently in [@liu_pnas; @Liu_nature; @asilomar11; @commault_recovery; @globalsip14; @boukhobza_recovery; @jstsp14][^5]. Existence of disjoint cycles and output connected paths in the graph is closely tied with its observability. In this direction, recently the concept of matching and dilations in graph [@Liu_nature], and Strongly Connected Components (SCC) [@asilomar11] are introduced to be related to network observability/controllability. Among these, the concept of contraction is the focus of this paper. An introductory description of contraction in the network is the set of nodes contracting (linking) to fewer number of nodes. In system estimation perspective, nodes in the same contraction are observationally equivalent, i.e. in case of losing observability of one (unmatched) node/state in the network/system another node in the same contraction can be measured to recover for the loss of observability. This is applicable in estimation of large scale systems such as power grid [@asilomar14] and internet based autonomous systems. For example when a sensor fails to measure a state –due to excessive noise, disturbance, or even external attacks– some necessary information of the system is lost and system/network cannot be tracked globally. To recover for this loss of information another sensor can be applied to measure equivalent state/node of the system/network. This is why the contractions play an important role in estimation. Indeed, one can apply a new sensor to measure an equivalent state in the same contraction and recover for the observability loss. In this regard, the size of contraction determines the possible number of equivalent options for observability recovery. Larger contractions imply more options among which one can choose the most efficient state measurement in terms of cost [@IJSS2017], reliability, etc. This is the main motivation to analyze the size and distribution of contractions as they play a major role in system observability recovery.
*Related Literature:* Structural observability of full-rank systems (having no contraction in system graph) is considered in [@liu_pnas; @asilomar11]. In these works, structural observability is shown to be closely related to network SCC classification. In [@Liu_nature] using cavity method the authors find considerable relation between average network degree and number of unmatched nodes. As one of their main results, they find that denser networks have less number of unmatched nodes and therefore it is less challenging to control and direct the network to the desired state. In [@globalsip14] the authors consider *distributed* estimation and formulate necessary and sufficient conditions for distributed structural observability. This work finds the connection between the structure of complex system and the structure of monitoring sensor network. In [@commault_recovery; @boukhobza_recovery], the authors classify sensors based on their essentiality for observability using combinatorial algorithms with application to sensor failure and diagnosis. Among these and other literature, what missing is on the concept of contraction and the relation between distribution of contractions and properties of the network (or system graph).
*Contribution:* In this paper, we study the properties of contractions in undirected networks/system-graphs as a key factor in estimation and observability. Adopting the structural/network observability method, the related question addressed here is that: how to find the equivalent state nodes in the network/system-graph to infer observationally equivalent information of the associated system? and we show that by finding contractions in the system-graph (or network), one can find the system states (or network nodes) equivalent in terms of observability and estimation. In this regard, the size of a contraction determines the potential number of *equivalent* sensing locations in networks as model of complex systems, which is discussed in Section \[sec\_cont\]. Further, a *polynomial order* algorithm is applied to find the contractions in the system graph. This algorithm is a modification of the algorithm for unmatched node detection given in [@murota]. Contractions are of particular interest in recovering sensor failure and loss of observability in tracking/filtering noise-corrupted global state of the system/network. Detailed discussion on application of contractions in system estimation and observability including example of observability recovery in Kalman filtering is provided in Section \[sec\_obsrv\]. Introducing the contraction set, the follow-up question is: how do the properties of these contraction sets change based on different characteristics of the underlying network? We investigate the effect of two factors on the size and distribution of contraction components: degree heterogeneity and clustering coefficient. First result of this paper is that the clustering coefficient as a network characteristic is related to average size and number of contractions. In particular, our results show that for Scale-Free networks, with power-law degree distribution, increase in clustering coefficient results in a decrease in average contraction size in the network. Further, we observe decrease in the number of contractions in high clustering coefficient Scale-Free networks. As the next contribution, we check the effect of degree heterogeneity in Small-World networks on contraction properties. Specifically, our results show that increase in randomness of link connectivity (tuning the $p$ factor) results in decrease in the average contraction size but increase in the number of contractions in the network. These results are addressed in Section \[sec\_rand\]. Further in Section \[sec\_real\], as a practical contribution, the contraction properties including the size distribution and prevalence are discussed for two real world networks: a Power-grid network and a Route-view network. Noting that the degree distribution of many real-world networks show power-law degree distribution, including the two example here, the results for these two practical examples corresponds with contraction properties of scale-free networks. More detailed discussion on these results and concluding remarks are stated in Section \[sec\_conc\]. It should be noted that in this paper the results are particularly stated for *undirected* networks/system graphs.
Notions on Graph Theory and Definition of Contraction {#sec_cont}
=====================================================
In this section we define the contraction sets in graphs by first introducing the relevant graph theoretic notions. Define a graph as $\mc{G}=(\mc{V},\mc{E})$, where $\mc{V}$ is the node set containing $n$ graph nodes, and $\mc{E}=\{(v_i,v_j)\}$ is the set of edges connecting the nodes. Define a path as a sequence of distinct nodes with every consecutive nodes as an edge in $\mc{E}$. Further, define a cycle as a path starting and ending at the same node. Define $\mc{N}(i)$ as the degree of node $i$. The adjacency matrix of the graph $A_G=\{a_{ij}\}$ is defined as $a_{ij}=1$, if $(v_j,v_i) \in \mc{E}$, otherwise $a_{ij}=0$. We further introduce the following graph-theoretic concepts to define contractions:
- *Bipartite graphs:* Define a bipartite graph, $\Gamma=(\mc{V}^+,\mc{V}^-,\mc{E}_\Gamma)$, such that its nodes are partitioned into two disjoint sets: $\mc{V}^+$ and $\mc{V}^-$, and all of its edges $\mc{E}_\Gamma$ start in $\mc{V}^+$ and end in $\mc{V}^-$. We construct a bipartite graph, $\Gamma$, from $\mc{G}$ with the edge set $\mc{E}_{\Gamma}$, defined as the collection of $(v_j^-,v_i^+)$, if $(v_j,v_i) \in \mc{E}$.[^6]
- *Matching:* A matching, $\underline{\mc{M}}$, on the system graph, $\mc{G}$, is defined as a subset of the edge set, $\mc{E}$, with no common end-nodes. In the bipartite graph, $\Gamma$, it is defined as a subset of edges where no two of them are incident on the same vertex in $\mc{V}^+$, i.e. all the edges in $\mc{M}$ are all disjoint. The number of edges, $|\underline{\mc{M}}|$, is the size of the matching. A matching, $\underline{\mc{M}}$, with maximum size is called maximum matching, denoted by $\mc{M}$, which is non-unique in general.
- *Matched/Unmatched nodes:* Let $\mc{M}$ be a maximum matching defined on the bipartite graph, $\Gamma$. Let $\partial \mc{M}^+$ and $\partial \mc{M}^-$ denote the nodes incident to $\mc{M}$ in $\mc{V}^+$ and $\mc{V}^-$ respectively. Denote by $\delta \mc{M}$ the set of unmatched nodes in $\mc{V}^+$ as $\delta \mc{M} = \mc{V}^+ \backslash \partial \mc{M}^+$. Note that maximum matching $\mc{M}$ is not unique in general.
- *Auxiliary graph*, denoted by $\Gamma^\mc{M}$, is a graph associated to a maximum matching, $\mc{M}$. It is constructed by reversing all the edges of maximum matching, $\mc{M}$, and keeping the direction of all other edges $\mc{E}_{\Gamma} \backslash \mc{M}$, in the bipartite graph, $\Gamma$. This graph is defined to localize the contractions in the system graph.
- *${\mc{M}}$-alternating path:* In the auxiliary graph, define an ${\mc{M}}$-alternating path as a sequence of edges starting from an unmatched node in $\delta \mc{M}$ and every second edge in $\mc{M}$, and denote it by $\mc{Q}_{\mc{M}}$. The name comes from the alternating edges between unmatched part, $\mc{E} \backslash \mc{M}$, and matched part, $\mc{M}$, of the auxiliary graph.
- *${\mc{M}}$-augmenting path:* In the auxiliary graph, define an ${\mc{M}}$-augmenting path, denoted by $\mc{P}_{\mc{M}}$, as an ${\mc{M}}$-alternating path with begin node and end node in $\delta \mc{M}$.
Having defined these preliminary notions on graph theory, the notion of a contraction set is defined as follows:
In the auxiliary graph representation of a network, $\Gamma^\mc{M} _A$, define a contraction set for every unmatched node $v_j \in \delta \mc{M}$, as the set of nodes containing all states in $\mc{V}^+$ reachable by ${\mc{M}}$-alternating paths starting from $v_j$. Denote this set by $\mc{C}_i$ and further define $\mc{C}$ as the set of all contractions, i.e. $\mc{C}=\{\mc{C}_1,...,\mc{C}_m\}$. Intuitively, in graph $\mc{G}$, a contraction set defines nodes that are connected (contracted) to less number of nodes.[^7]
![This figure shows a network contraction in the left, where the three contraction nodes are shown in red color. Figures in the right show three different maximum matching in bipartite representations of the same contraction. The red edges represent maximum matching and the red node represents the unmatched node.[]{data-label="fig_3nodecont"}](fig_3nodecont.pdf){width="3.5in"}
![This figure illustrates the procedure of finding contractions explained in the paper. Graph in the left shows one possible matching and the unmatched node in the bipartite representation of the graph in Fig \[fig\_3nodecont\]. The middle graph shows the auxiliary representation, where all matching edges are reversed. In the right graph an ${\mc{M}}$-alternating path is shown in black. Starting from the unmatched node, this path is used to find the contraction nodes (shown by dashed squares). Later in this paper we name these contraction nodes as observationally equivalent nodes.[]{data-label="fig_3nodecont2"}](fig_3nodecont2.pdf){width="2.7in"}
For better illustration of the above definitions a contraction of 3 nodes into 2 nodes is shown in Fig. \[fig\_3nodecont\]. The bipartite representation, the maximum matching, and the unmatched node are illustrated in the figure. We further illustrate the definition of auxiliary graph and ${\mc{M}}$-alternating path in Fig. \[fig\_3nodecont2\]. The algorithm to find the contraction sets in network is given in Algorithm \[alg\_cont\].
\[alg\_cont\] **Given:** System graph $\mc{G}_A$
Construct the bipartite graph $\Gamma=(\mc{V}^+,\mc{V}^-,\mc{E}_\Gamma)$ Find a matching $\underline{\mc{M}}$ as the set of edges with no common end nodes Construct the auxiliary graph $\Gamma^{\underline{\mc{M}}}_A$ by reversing the edges in matching $\underline{\mc{M}}$ Define $\partial {\underline{\mc{M}}}^+$ as the nodes in $\mc{V}^+$ incident to ${\underline{\mc{M}}}$ Define the set of unmatched nodes $\delta{\underline{\mc{M}}}$ as $\delta {\underline{\mc{M}}} = \mc{V}^+ \backslash \partial {\underline{\mc{M}}}^+$ Define the ${\mc{M}}$-alternating path, $\mc{Q}_{\underline{\mc{M}}}$, as a sequence of edges starting from an unmatched node in $\delta {\underline{\mc{M}}}$ and every second edge in ${\underline{\mc{M}}}$ Define an ${\mc{M}}$-augmenting path, $\mc{P}_{\underline{\mc{M}}}$, as an ${\mc{M}}$-alternating path with begin node and end node in $\delta {\underline{\mc{M}}}$
Construct the auxiliary graph $\Gamma^{\mc{M}}_A$ for the maximum matching ${\mc{M}}$ Define $\partial {\mc{M}}^+$ for the maximum matching ${\mc{M}}$ and define $\delta {\mc{M}} = \mc{V}^+ \backslash \partial \mc{M}^+$ Define the ${\mc{M}}$-alternating path, $\mc{Q}_{\mc{M}}$, for the maximum matching ${\mc{M}}$
**Return** $\mc{C}_i, i = \{1,...,m\}$
In Algorithm \[alg\_cont\], $\oplus$ is the XOR operator in set theory. As a result of this operator, each augmenting path increases the size of the matching till it reaches the maximum matching. The computational complexity of this algorithm is on the order of $\mc{O}(\sqrt{n} |\mc{E}|)$ or $\mc{O}(n^{\frac{5}{2}})$ in worst case. In general, given the system graph $\mc{G}_A$ there are other efficient algorithms to compute the maximum matching, $\mc{M}$, e.g., the maximum flow algorithm [@hopcraft]. The notions $\Gamma,\mc{M}$ can be obtained by the Dulmage-Mendelsohn (DM) decomposition [@dulmage58]. Other than DM decomposition, maximum matchings can be efficiently computed in $\mc{O}(\sqrt{n} |\mc{E}_A|)$ using the approach in [@maxmatching]. In the following, we state two main lemmas relating the maximum matchings and contractions.
\[lem\_unmatched\] Any choice of maximum matching, $\mc{M}$, includes one and only one unmatched node in every contraction $\mc{C}_i, i\in \{1,...,m\}$.
The detailed proof is provided in [@murota; @berge].
\[lem\_contr\] For two sets of maximum matching, $\mc{M}_1 \neq \mc{M}_2$, any unmatched node $v_i \in \delta \mc{M}_1$ can be reached along an alternating path from a node $v_j \in \delta \mc{M}_2$. This further implies that the set $\mc{C}$ is the same for any choice of maximum matching.
The proof is given in [@murota].
Application in Observability and System Estimation {#sec_obsrv}
==================================================
In this section, we first discuss the concept of structural observability in networks and then its application to system estimation. To further illustrate the results a network estimation example is provided.
Network Observability {#subsec_netobsrv}
---------------------
Observability of networks quantifies whether given measurements contain sufficient information to comprehensively reconstruct the states of all nodes in the network. For a network, or a system graph representing a complex system, the necessary and sufficient conditions for (structural) observability is given in the following theorem.
\[thm\_obsrv\] A network (or system graph) is structurally observable if and only if: (i) every node can reach to an output/measurement via a path of state nodes, and (ii) there exist a family of disjoint cycles and output-connected paths covering all nodes.
The proof is given in [@lin] and in [@rein_book] for the dual case of structural controllability. In Theorem \[thm\_obsrv\], condition (i) is known as *accessibility* and condition (ii) as the *S-rank* condition. Note that for connected undirected networks the accessibility is already satisfied. This is because, in a connected undirected network every node is reachable by every other node and therefore output connectivity of one node implies the reachability of all other nodes to that output.
\[thm\_unmatched\] In a connected undirected network with the set of unmatched nodes, $\delta \mc{M}$, observation/measurement of every unmatched node is necessary and sufficient for network observability.
The proof is given in [@Liu_nature] for the dual case of network controllability. Following the definition of contraction and results in previous section here we state the theorem on the concept of observational equivalence in contractions.
\[thm\_contr\] In a connected undirected network with the set of contractions $\mc{C}=\{\mc{C}_1,...,\mc{C}_m\}$, a measurement/observation of one state node in every contraction $\mc{C}_i, i\in \{1,...,m\}$ is necessary and sufficient for network observability.
From Theorem \[thm\_unmatched\] observation of every unmatched node is necessary and sufficient for network observability. Note that based on Lemma \[lem\_unmatched\] for every contraction $\mc{C}_i$, every node $v_j$ is an unmatched node for a choice of maximum matching $\mc{M}$. This implies that observing at least one node in every contraction is necessary for observability. Further, by measuring node $v_j$ in $\mc{C}_i$ from Lemma \[lem\_contr\] all other nodes in $\mc{C}_i \backslash v_j$ are matched for the choice of maximum matching $\mc{M}$ and therefore only one node is sufficient for network observability.
\[lem\_rankdeficiency\] Number of contractions in a network $\mc{G}$ equals the (structural) rank deficiency of its associated adjacency matrix, $A_G$. Indeed the rank deficiency of the adjacency matrix, $A_G$, equals the number of unmatched nodes in the network $\mc{G}$.
Indeed from Lemma \[lem\_unmatched\] the number of contractions equals the number of unmatched nodes in the network. Note that, by definition, unmatched nodes appear on acyclic part of network while the cyclic part is completely matched. It is known that the rank of the network adjacency matrix is structurally defined by the number of nodes included in a set of disjoint cycles [@godsil]. This implies that the rank deficiency can be structurally defined by the number of unmatched nodes, which are contained in the acyclic part of the network. The concept of contraction is closely related to the concept of *observational equivalence*. Let $C_i$ denote the measurement matrix of node/state $x^i$. Let $\mc{O}(A,C_i)$ represent the observability Grammian of the pair $A$ and $C_i$. The observational equivalence relation among two states/nodes $x^i$ and $x^j$, denoted by $x^i \sim x^j$, is defined as: $$\begin{aligned}
\mbox{rank}~\mc{O}(A, C_i) = \mbox{rank}~\mc{O}(A, C_j) = \mbox{rank}~\mc{O}\left(A, \left[
\begin{array}{c}
C_i\\
C_j
\end{array}
\right]
\right)\end{aligned}$$ Note that this follows the three properties of the equivalence relation, i.e. transitivity, reflexivity, and symmetry.[^8]
\[lem\_equiv\] The algebraic implication of observational equivalence relation among states in each contraction is defined as follows. For any two (or more) measurements of states $x^i$ and $x^j$ in the same contraction, the structural-rank recovery of system matrix $A$ is equal to $1$, i.e., $$\begin{aligned}
\label{eq_srank}
\mbox{S-rank}\left(
\left[
\begin{array}{c}
A \\
C_{i}
\end{array}
\right]
\right) &= &
\mbox{S-rank}\left(
\left[
\begin{array}{c}
A \\
C_{j}
\end{array}
\right]
\right)
\nonumber \\
&=& \mbox{S-rank}\left(
\left[
\begin{array}{c}
A \\
C_{i} \\
C_{j}
\end{array}
\right]
\right)
\nonumber \\
&=& \mbox{S-rank}(A)+1.
\end{aligned}$$ where S-rank implies the structural-rank[^9] of the matrix.
The proof directly follows the three properties of observational equivalence relation. One can easily check that the reflexivity, symmetry, and transitivity of equation directly follows.
\[cor\_equiv\] Theorem \[thm\_contr\] along with Lemma \[lem\_unmatched\], \[lem\_contr\], and \[lem\_equiv\] imply that all nodes in a contraction are *equivalent* in terms of observability. In other words, measurement of *any* node in each contraction, assuming that all other contractions have one observation, provides network observability. As a result of the equivalent observability relation, nodes in the same contraction recover loss of observability.
In other words, in the case of observation failure of a node, say node $v_i$, some information of the system is lost. In this case, observation of another node, say node $v_j$, sharing a contraction with node $v_i$ recovers the observability loss. In this regard, the *size of contraction* defines the possible number of equivalent sensing nodes for recovering observability loss.The implication of equivalent relation is further discussed in next subsection; we show how dynamic systems can be represented structurally as networks, where we can apply the above Theorems and Lemmas to find equivalent states in terms of system observability and estimation.
System Estimation
-----------------
Consider the system model to be a discrete-time linear dynamic system[^10]: $$\begin{aligned}
\label{sys1}
\mb{x}_{k+1} &=& A\mb{x}_k + \mb{v}_k,\end{aligned}$$ with $\mb{x}_k\in\mathbb{R}^n$ $$\begin{aligned}
\nonumber
\mb{x}_k =
\left(
\begin{array}{c}
x_k^1\\
\vdots\\
x_k^n
\end{array}
\right)\end{aligned}$$ as the state vector, $A=\{a_{ij}\}\in\mathbb{R}^{n\times n}$ as the system matrix, and $\mb{v}_k\sim\mathcal{N}(0,V)$ as the system noise. Assume the dynamical system to be monitored by measurement/observation model: $$\begin{aligned}
\label{sys2}
\mb{y}_k = C\mb{x}_k + \mb{r}_k,\end{aligned}$$ where $$\begin{aligned}
\nonumber
\mb{y}_k =
\left(
\begin{array}{c}
y_k^1\\
\vdots\\
y_k^m
\end{array}
\right),~~
C =
\left(
\begin{array}{c}
C_1\\
\vdots\\
C_m
\end{array}
\right),~~
\mb{r}_k =
\left(
\begin{array}{c}
r_k^1\\
\vdots\\
r_k^m
\end{array}
\right),\end{aligned}$$ Here, $\mb{r}_k\sim\mathcal{N}(0,R)$ is the observation noise with $R = \mbox{blockdiag}[R_1,\ldots,R_N]$, and $C$ is the measurement matrix.
In structured systems theory, the LTI system in Eqs. -, can be modeled as a *system graph*. In this scenario, every node is a system state and every edge represents the interaction of two states based on the system matrix, $A$. Denote the set of system states by $\mc{X}\triangleq\{x^1,\ldots,x^n\}$ and denote the set of system observations/measurements by $\mc{Y}\triangleq\{y^1,\ldots,y^m\}$. Then the system graph is defined by $\mc{G}_A=(\mc{X},\mc{E}_A)$ where the edge set, $\mc{E}_A$, is defined as $\mc{E}_A=\{(x^i,x^j)~|~a_{ji}\neq0\}$, to be interpreted as $x^i\rightarrow x^j$. One should note that, in this graph representation of system the structure of system graph only relies on free parametric entries of matrix $A$. In other words, the graph structure depends on each entry $a_{ij}$ being a free parameter and not on the exact numerical value of $a_{ij}$. Therefore, any Linear Structure Invariant (LSI) system with fixed structure and time-varying parameters can be modeled as a system graph.[^11] The motivation of applying graph representation of system is that one can check its characteristics by using equivalent graphical properties. The system characteristic of interest here is system observability, which plays a crucial role in system estimation and filtering. To illustrate more we consider the role of system graph observability in Kalman estimation as discussed next.
Let $\widehat{\mb{x}}_{k|k}$ be the Kalman estimator tracking system state $\mb{x}_k$ at time $k$ given all the measurements, $\mb{y}_k$, up to time $k$. The dynamics of this estimator is defined as follows [@kalman:61]: $$\begin{aligned}
\label{eq_kalman1}
\widehat{\mb{x}}_{k|k-1} &=& A\widehat{\mb{x}}_{k-1|k-1}
\\ \label{eq_kalman2}
\widehat{\mb{x}}_{k|k} &=&\widehat{\mb{x}}_{k|k-1} + K_k C^T (\mb{y}_k-C\widehat{\mb{x}}_{k|k-1})\end{aligned}$$ where the $K_k$ is the Kalman gain computed in a recursive procedure as proposed by Kalman [@kalman:61]. It can be shown that the error, $\widehat{\mb{e}}_{k|k} = \mb{x}_k - \widehat{\mb{x}}_{k|k}$, in this estimator is given by, $$\begin{aligned}
\label{ge}
\widehat{\mb{e}}_{k|k} = (A - K_kC^TCA)\widehat{\mb{e}}_{k-1|k-1} + \eta_k,\end{aligned}$$ where the vector $\eta_k$ collects the remaining terms (noise terms) that are independent of $\widehat{\mb{e}}_{k-1|k-1}$ and $\widehat{\mb{e}}_{k|k}$. It is known that the dynamics of Kalman error, $\widehat{\mb{e}}_{k|k}$ is stable if the measurements defined by matrix $C$ give an observable inference of system defined by $A$ [@kalman:61]. In other words, the Mean Squared Estimation Error (MSEE) reaches bounded stability over time if the pair $(A,C)$ is observable. Following the results of the graph-theoretic method in Section \[subsec\_netobsrv\], we consider two applications in the following.
(i) As the first application one can check the observability constraint using results of Theorem \[thm\_contr\]. For a system to be observable, according to Theorem \[thm\_contr\], one state node in each contraction set in the system graph has to be measured.
(ii) The other, and more important, application is in case of observability loss. Assume that one (or more) of the sensor measurements fail and therefore the system is not observable anymore. To recover for this loss of observability, one can assign measurements of equivalent states as stated in Corollary \[cor\_equiv\]. The set of equivalent states for observability and estimation can be determined by finding contractions in the system digraph representation using Algorithm \[alg\_cont\]. For example, loosing the measurement of state $x^i$ one can measure another state $x^j$ sharing a contraction with $x^i$ in $\mc{G}_A$ to recover for system observability.
These graph-theoretic applications are explained more in the following example. *Illustrative example:* Consider a system with $n=11$ states represented as a system graph in Fig. \[fig\_graph1\]-Top.
![(Top) This figure shows the example system graph with $3$ measurements of states $x^1,x^8,x^9$. The same-colored state nodes in red, green, and orange each represent states in the same contraction component and the blue states are not part of any contraction. (Bottom) The time evolution of the MSEE using the estimator in Eqs. - applied on the same system. The three measurements give an observable inference and therefore the MSEE is bounded steady-state stable.[]{data-label="fig_graph1"}](fig_graph1.pdf "fig:"){width="2.45in"} ![(Top) This figure shows the example system graph with $3$ measurements of states $x^1,x^8,x^9$. The same-colored state nodes in red, green, and orange each represent states in the same contraction component and the blue states are not part of any contraction. (Bottom) The time evolution of the MSEE using the estimator in Eqs. - applied on the same system. The three measurements give an observable inference and therefore the MSEE is bounded steady-state stable.[]{data-label="fig_graph1"}](fig_kalman_MSEE1.pdf "fig:"){width="2.45in"}
![(Top) This figure shows the same example system graph in Fig. \[fig\_graph1\] with $3$ new measurements of states $x^3,x^6,x^{11}$ sharing a contraction with the states measured in Fig. \[fig\_graph1\]. (Bottom) The time evolution of the MSEE of system estimation using the estimator in Eqs. -. As it can be seen, the equivalent measurements also provide an observable estimation with bounded steady-state MSEE.[]{data-label="fig_graph2"}](fig_graph2.pdf "fig:"){width="2.45in"} ![(Top) This figure shows the same example system graph in Fig. \[fig\_graph1\] with $3$ new measurements of states $x^3,x^6,x^{11}$ sharing a contraction with the states measured in Fig. \[fig\_graph1\]. (Bottom) The time evolution of the MSEE of system estimation using the estimator in Eqs. -. As it can be seen, the equivalent measurements also provide an observable estimation with bounded steady-state MSEE.[]{data-label="fig_graph2"}](fig_kalman_MSEE2.pdf "fig:"){width="2.45in"}
Assume this graph represents a dynamic system, where each node is system state and each link represents the dynamic interaction of two states (for more details on such representation of systems as networks see [@Liu_nature]). The associated system matrix elements in $A_G$ (i.e. the link weights) are chosen randomly. For sake of illustration and avoiding trivial solutions the elements in $A_G$ are such that the spectral radius of adjacency matrix is greater than $1$, $\rho (A_G)>1$, i.e. the system is unstable. To determine the necessary states for observability we find the set of contractions in the system graph using Algorithm \[alg\_cont\] as $\mc{C}_1=\{x^1,x^3,x^5\}$, $\mc{C}_2=\{x^6,x^8\}$, and $\mc{C}_3=\{x^9,x^{11}\}$.[^12] The system is tracked by $m=3$ measurement of three states each in one contraction set of the system graph. This satisfies the condition in Theorem \[thm\_contr\] for observability and thus leads to stable estimation. These measurements along with system parameters are used in estimator Eqs. - to estimate and track the global state $\mb{x}$ of the system over time iterations $k$. The Mean Squared Estimation Error (MSEE) over time is shown in Fig. \[fig\_graph1\]-Bottom, which is bounded. Note that if we loose the measurement of a state in a contraction, according to Theorem \[thm\_contr\] we loose the system observability. Without observability, we loose the stability of the MSEE and the estimation error goes unbounded. To recover for loss of observability, one can take a measurement of an equivalent state in the same contraction, as shown in Fig. \[fig\_graph2\]-Top. Indeed, measuring any state in the same contraction set is sufficient for observability and yields stable estimation as shown in Fig. \[fig\_graph2\]-Bottom. The key point is that number of possible state to recover for observability directly relates to the size of contraction sets. In this example, there are two options to recover for loss of observation of $x^1$, while there is only one replacement for observation of $x^8$ and replacement for observation of $x^9$.
Synthetic and Real Case Studies {#sec_sim}
===============================
In this section, we analyze the number and size of contraction sets in both real and random complex networks. Recall that the contraction size is of interest because it determines the number of equivalent nodes for observability recovery, and number of contractions determine the number of node measurements necessary and sufficient for observability. First, random networks as models of complex systems are reviewed and relation between features of these networks and size/number of contractions are discussed. Next, the distribution of contraction sets in some examples of real-world networks are analyzed. Here, we proceed by first reviewing the definitions of relevant network properties.
The local clustering coefficient of a node in a graph is defined as the fraction of pair of node neighbors that are linked together. On the other hand, the global clustering coefficient is defined as the fraction of the closed triplets (triangles) to the total number of the triplet paths in the graph [@newman2003structure]. Mathematically the clustering coefficient is defined as: $$\begin{aligned}
CC(i) = \frac{2.tr(i)}{\mc{N}(i)(\mc{N}(i)-1)}
\end{aligned}$$ where $tr(i)$ is the number of triangles that node $i$ forms with two of its neighbors. The global clustering coefficient of the network is defined as, $$\begin{aligned}
CC = \frac{3.tr}{trp}
\end{aligned}$$ where the $tr$ is the number of triangles and $trp$ is the number of connected triplets in the network. It is known that the clustering coefficient is a good measure of well-connectivity of the network and presence of strong community-structure in the network [@alstott2016cluster; @newman2003structure].
Degree heterogeneity is an intuitive concept related to the degree distribution of networks. Degree heterogeneity, as opposed to degree homogeneity, determines if the nodes in the network have various degrees (heterogeneous), or have similar degrees (homogeneous) to one another. Various measures of degree heterogeneity resembling the global differences in the node degrees are discussed in [@wu2010heterogeneity; @jacob2017measure]. The most well-known formula for degree heterogeneity is given by the variance of node degrees as follows [@jacob2017measure]: $$\begin{aligned}
VAR = \frac{1}{n}\sum_{i=1}^{n}(\mc{N}(i) - \bar{\mc{N}})^2
\end{aligned}$$ where $\bar{\mc{N}}$ is the average node degree, $$\begin{aligned}
\bar{\mc{N}} = \frac{1}{n}\sum_{i=1}^{n}\mc{N}(i)
\end{aligned}$$
More details on these definitions can be found in [@newman2003structure; @jacob2017measure].
Contraction sets in Random Networks {#sec_rand}
-----------------------------------
Random graphs are widely used to model complex systems facilitating analysis of different processes over networks, e.g., spreading processes or cascading failures [@newman2003structure; @barabasi2003scale; @barabasi_albert1999; @Holme2002clusteringScaleFree; @Toivonen2006social; @watts1998smallworld; @newman2002random]. The graphs are called random since the nodes in the graph are randomly connected with each other. We investigate two well-known models for random graphs. We particularly analyze the relation between number and size of contractions with clustering coefficient in Scale Free networks and with degree heterogeneity in Small-World Networks as discussed next.
*Scale-Free networks:* Many complex networks are modeled by this type of random network. It is known that degree distribution of such networks follows a power-law distribution [@barabasi2003scale], i.e. the portion of nodes having degree $d$, represented by $f(d)$, follows the following formula: $$\begin{aligned}
f(d) = d^{-\sigma},~2<\sigma <3\end{aligned}$$ In log-log scale, the distribution represents a linear function, hence it is named Scale-Free (SF) network. This implies that these graphs have a large number of low-degree nodes and few hubs with high connectivity. A well-known approach to build such networks is proposed by Barabasi and Albert [@barabasi_albert1999]. The Barabasi and Albert (BA) approach considers an initial graph of few number of nodes, called *initial seed* where recursively a new node with $m$ new links is added to the network. The probability that the new node makes a link to old nodes is proportional to the degree of old nodes, implying that the new node *preferably* links to high degree nodes, and is known as *preferential attachment*. In this method hubs with high degree are more likely to connect to the newly added nodes while the low degree nodes are unlikely to gain new links. These types of Scale-Free networks, e.g. BA model, are known to have low clustering coefficient. Therefore recently new random models of networks are proposed in the literature to account for high clustering of real networks [@Holme2002clusteringScaleFree; @Toivonen2006social]. These works propose to modify preferential attachment method such that the resulting networks, beside having power-law distribution, have high clustering. The network growth procedure is similar to the preferential attachment of BA model with some modification. Similar to BA model, they consider an initial seed. Then, recursively a new node connects to $m_r$ nodes in the network based on preferential attachment. But further, in each step the new node also makes $m_s$ links to randomly chosen neighbors of preferentially attached nodes in the network. This additional step is called *triad formation*. This increases the prevalence of triads (triangle cliques) in the network, and therefore results in high clustering coefficient. These random networks are called Clustered Scale Free (CSF) networks. For these networks we analyze the size and number of contractions. It should be noted that for simulation we consider $m=m_r+m_s$, i.e. number of links each new node makes in SF network equals number of links each new node makes in CSF network. This implies that the number of edges in both SF and CSF network are the same. This also implies that the average degree of the network is equal for both SF and CSF networks. This is important as all properties of both SF and CSF networks are similar except their clustering coefficient [@Holme2002clusteringScaleFree; @Toivonen2006social; @assenza2008enhancement]. Simulations are performed over $1000$ different realizations of 1000 node SF and CSF networks and the results are shown in Fig. \[fig\_SF\_CSF\].
![This figure shows the distribution of contraction size on SF and CSF networks with similar number of preferentially attached nodes. The simulation is performed over $1000$ realizations of $1000$-nodes networks.[]{data-label="fig_SF_CSF"}](fig_SF_CSF.pdf){width="3in"}
As it can be seen contractions are more prevalent in SF networks as compared to CSF networks and there are more contractions (and unmatched nodes) in SF networks as compared to CSF networks. Further, the contraction sets are in average larger in SF as compared to CSF networks. The results for $1000$ network realizations are summarized in Table \[tab\_SF\].
Type of network SF CSF
-------------------------------- ---------- ---------
Average Contraction size $18.89$ $6.72$
Average number of Contractions $156$ $109$
: Average size and number of contractions in $1000$ realizations of SF and CSF networks.
\[tab\_SF\]
This implies that by increasing the clustering coefficient as in CSF networks the number and size of contractions decreases.
*Small-World networks:* The idea behind this model is to imitate the graphical properties of real-world networks. One of the main structural feature of the real graphs is that they show high level of community structure while keeping small average distance (shortest path), which is known as the small-world phenomena. Such features are not present in typical random models, for example in Erdos-Renyi graphs. Therefore, Watts and Strogatz [@watts1998smallworld] proposed a new semi-random graph, named Small-World model. The Watts and Strogatz (WS) model starts with a $k$-regular network in which every node is connected to its $k$ nearest neighbors (in both sides). Randomly pick links in the $k$-regular network with uniform probability $p$ independent of each other. Then, choose the end node of this link and randomly rewire it to another node. The rewiring must be such that the new link is not a self-link or a link that already exist in the network. By increasing the rewiring probability $p$ one can generate random networks which are more random in terms of their degree heterogeneity and as $p \rightarrow 1$ the model reaches an Erdos-Renyi (ER) graph with Poisson degree distribution [@newman2002random]. On the other hand, small $p$ implies that network conserves its regularity and degrees of most nodes lie around the average degree $2k$. Such networks have adjustable degree heterogeneity by tuning $p$. Indeed, regular networks are the most degree homogeneous and next are small world networks with tunable degree heterogeneity by factor $p$. By increasing the factor $p$ the degree heterogeneity increases up to the point where $p=100\%$ and the graph models the ER network.
To relate the contraction size with degree heterogeneity,
$1000$ different realizations of 1000 node SW networks with $9$ different $p$ factors are considered and the simulation results are shown in Fig. \[fig\_SW\]. Note that for this simulation only rewiring probability $p$ is changing, therefore graph properties such as number of edges and average node-degree remains unchanged and the only property that changes is degree-heterogeneity [@dwivedi2017optimization]. The average size and number of contractions in $1000$ network realizations are also summarized in Table \[tab\_SW\].
![This figure shows the distribution of contraction size on different Small-World networks tuning the $p$ factor. The simulation is performed over $1000$ realizations of $1000$-nodes networks.[]{data-label="fig_SW"}](fig_SW.pdf){width="3in"}
$p$ factor of SW network $10\%$ $20\%$ $30\%$ $40\%$ $50\%$
-------------------------- --------- -------- -------- -------- ---------
avg Contraction size $13.71$ $7.62$ $5.62$ $4.68$ $4.14$
number of Contractions $25$ $47$ $68$ $86$ $101$
$p$ factor of SW network $60\%$ $70\%$ $80\%$ $90\%$ $100\%$
avg Contraction size $3.83$ $3.61$ $3.50$ $3.44$ $3.42$
number of Contractions $113$ $123$ $130$ $132$ $133$
: Average size and number of contractions in $1000$ realizations of SW networks with $10$ different $p$ factors.
\[tab\_SW\]
As it can be seen from the results by increasing the $p$ factor and degree heterogeneity, in average contractions are decreased in size but increased in number.
Contraction sets in Real Networks {#sec_real}
---------------------------------
*Power grid network:* As the first example, we consider the power grid network that represents the grid of the Western States of the United States of America. In this network a link is a power supply line and a node represents either a generator, a transformator or a substation. The network is originally addressed in [@watts1998smallworld] but the data is taken from [@UCI_power] where the description of state nodes can be found. It is known that such networks resemble the sparsity of system dynamic matrix where the states represent power flow, voltage, or phase angles [@asilomar14; @poor2012grid], and therefore the network can model a dynamic system type of Eq. , for more details see [@poor2012grid; @khan2013secure; @camsap11]. This network contains $6594$ interaction links connecting $4941$ state nodes. Applying the DM decomposition finds one possible set of unmatched nodes in the network. The network structure is shown in Fig. \[fig\_power\], including $575$ unmatched nodes represented in red color. Recall that From Theorem \[thm\_unmatched\], for observable estimation all the unmatched nodes must be observed by a sensor. These set of observable measurements gives one possible stable estimation of system state nodes over time.
![This figure shows the structure of Western-State Power grid network with $4941$ state nodes and $6594$ links; red nodes in the network represent unmatched nodes monitored by a sensor.[]{data-label="fig_power"}](fig_power.pdf){width="3in"}
The distribution of all $575$ contractions in this network are shown in Fig. \[fig\_power\_distribution\]. It should be noted that the clustering coefficient of this network is $10.3\%$ and the average contraction size is $4.98$.
![This figure shows the frequency of size of different contractions in power grid network.[]{data-label="fig_power_distribution"}](fig_power_distribution.pdf){width="2.5in"}
Applying the Contraction Detection Algorithm \[alg\_cont\] finds all the contraction sets in the network, where two examples are shown in Fig. \[fig\_power\_cont\]. These examples include contraction sets of size $3$ (green colored contraction) and $52$ (blue colored contraction). Recall that from Corollary \[cor\_equiv\] each contraction set associated with an unmatched node represents all possible options to recover for loss of measurement/observation. In this scenario, losing the observation of any node in the 3-nodes green contraction implies that there are measurements of only $2$ other nodes to recover for possible loss of observability, while for the blue colored contraction there are $51$ possible options to recover for the loss of observability.
![This figure includes two subnetworks of Power-grid network of Fig. \[fig\_power\]. Each subnetwork shows an example of nodes making a contraction, represented as blue and green colored nodes. These colored state nodes are equivalent in terms of network observability.[]{data-label="fig_power_cont"}](fig_power_cont1.pdf "fig:"){width="2in"} ![This figure includes two subnetworks of Power-grid network of Fig. \[fig\_power\]. Each subnetwork shows an example of nodes making a contraction, represented as blue and green colored nodes. These colored state nodes are equivalent in terms of network observability.[]{data-label="fig_power_cont"}](fig_power_cont2.pdf "fig:"){width="1in"}
*Route-view network:* This network represents the network of connected autonomous systems of Internet. Every node is an autonomous system and every link represents communication between two systems. The data is taken from [@Konect_Routeviews], but the original description of the network is given in [@leskovec2007graph]. As stated in [@leskovec2007graph] every node is indeed a subgraph of Internet-connected routers that exchanges traffic flow with its peer neighbors. The network contains $6474$ nodes connected with $13895$ links, and is represented in Fig. \[fig\_routeview\_unmatched\]. In this figure regular nodes are represented in black while $3568$ unmatched nodes are shown in red color.
![This figure shows the Route-views network representing internet-connected autonomous systems. The network contains $6474$ nodes connected with $13895$ links. Nodes in red color are unmatched nodes each monitored by a sensor.[]{data-label="fig_routeview_unmatched"}](fig_routeview_unmatched.pdf){width="3.3in"}
Applying the Contraction Detection Algorithm \[alg\_cont\] all $3568$ contractions in the network are found. The distribution of contraction sets is as shown in Fig. \[fig\_RouteView\_distribution\]. The average size of contractions in this network is $7.65$ and the clustering coefficient of the network is $0.959 \%$.
![This figure shows the frequency of size of different contractions in Route-view network.[]{data-label="fig_RouteView_distribution"}](fig_RouteView_distribution.pdf){width="2.5in"}
Two examples of contraction sets in the Route-view network are shown in Fig. \[fig\_Routeview\_cont\]; one includes set of $2$ contraction nodes (in green color) and the other one includes set of $7$ contraction nodes (in orange color). As mentioned before, each contraction set represents the state nodes giving equivalent information for network observability and estimation.
![This figure shows two subnetworks of Route-view network of Fig. \[fig\_routeview\_unmatched\]. In each subnetwork colored nodes in orange and green represent example of nodes making a contraction. In network observability, these colored nodes represent equivalent states.[]{data-label="fig_Routeview_cont"}](fig_Routeview_Cont2.pdf "fig:"){width="2in"} ![This figure shows two subnetworks of Route-view network of Fig. \[fig\_routeview\_unmatched\]. In each subnetwork colored nodes in orange and green represent example of nodes making a contraction. In network observability, these colored nodes represent equivalent states.[]{data-label="fig_Routeview_cont"}](fig_Routeview_Cont1.pdf "fig:"){width="1in"}
The results for these two networks are summarized in Table \[tab\_realnet\].
Name of network Power grid Route-view
------------------------ -------------- --------------
avg Contraction size $4.98$ $7.65$
Contractions/nodes $575/4941$ $3568/6474$
Clustering Coefficient $10.3\%$ $0.959\%$
: Characteristics of two examples of real networks including average contraction size, ratio of number of contractions to number of nodes, and clustering coefficient.
\[tab\_realnet\]
Discussion and Conclusions {#sec_conc}
==========================
Comparing the SF and CSF network, we observe a significant raise in average size of contractions in SF network. Noting that SF and CSF networks apply the same preferential attachment model and are similar in terms of most graph statistics including power-law degree distribution and logarithmically increasing average shortest-path length [@Holme2002clusteringScaleFree; @Toivonen2006social; @assenza2008enhancement], therefore, the only difference is low clustering coefficient as the key factor affecting the jump in average size of contractions in SF network. Similar statement holds for the average number of contractions in the network. Note that this number is decreased for clustered version of Scale-Free network while other network characteristics are unchanged. This implies that by increase in the clustering coefficient in average more contractions with larger size appear. This is also the case in real-world network examples stated in Section \[sec\_real\][^13]. For Power grid network with high clustering coefficient the ratio of number of contractions to the total number of nodes is lower than the Route-view network with low clustering coefficient. Similar statement holds for the average contraction size as the size of contractions are in average smaller in Power grid network. For observability and estimation of networks with power-law degree distribution (SF and CSF networks) these results imply that: (i) estimation/tracking of such networks with high clustering coefficient requires (in average) lower number of observations/measurements as there are less number of contractions, but (ii) in case of measurement/sensor failure there are less number of possible equivalent states for observability recovery as the average size of contractions are low. One application of these results is that one can tune the clustering coefficient of (synthetic) networks [@serrano2005tuning] to reduce the challenge for observability recovery and estimation.
The other result of this paper is that in Small World networks the average size of contractions is to a great extent related to the degree homogeneity. Increasing the heterogeneity in Small World networks, by increasing the rewiring probability $p$ [@watts1998smallworld], is one key factor on the decrease of average contraction size as mentioned in Table \[tab\_SW\]. Note that by only changing the rewiring probability in SW networks the number of links, average node degree, and the size of graphs are unchanged. On the other hand, by increasing the degree heterogeneity, while the other graph characteristics of in SW networks are unchanged, the average number of contractions is increased. In terms of observability and estimation of SW networks these results imply that: (i) estimation of networks with high level of degree heterogeneity requires more number of measurements/observations which is due to prevalence of contractions, and (ii) in case of sensor/observation failure there are less number of possible options to recover for the loss of observability as the contraction sets in average are smaller in degree heterogeneous SW networks. As an application of these results one may decrease the degree heterogeneity by tuning the $p$ factor in synthetic networks to reduce the number of necessary measurements for observability and further increase the contraction size providing more possible countermeasures for observability recovery.
Note that the above mentioned results are applicable for specific networks. In other words, we claim the results regarding the size/distribution of contractions and the clustering coefficient only for power-law degree distribution networks (SF-CSF networks). Further, the results on the relation of degree-heterogeneity and size/distribution of contractions are only stated for networks with Small-World property. For other kind of networks, for example Erdos-Renyi graphs, such results may not apply in general. Note that to make a justified claim about effect of clustering-coefficient/degree-heterogeneity we need to keep other graph properties (e.g. degree distribution, average degree, number of edges) unchanged so we can claim that the only effective property is clustering-coefficient/degree-heterogeneity. We cannot claim this for general graphs as they may differ in terms of, for example, degree distribution.
It should be noted that the algorithms to check the matching properties of the network, namely Hopcroft-Karp algorithm [@hopcraft] or the Dulmage-Mendelsohn decomposition [@dulmage58] are of $\mc{O}(n^{2.5})$ complexity. Particularly, the complexity of Algorithm \[alg\_cont\] is in polynomial order $\mc{O}(n^{2.5})$. Note that, polynomial time algorithms are suitable for large-scale system analysis as their running time is upper-bounded by a polynomial expression in system size. The polynomial order complexity of the algorithms motivates application in observability analysis of large-scale networks/systems similar to the real examples given in the previous section. It is worth mentioning that, the results in this paper can be extended to the dual case of large-scale network controllability.
As the final comment, it should be noted that this paper considers undirected networks and system graphs. The reason is that for directed networks *root SCCs* play important role in observability [@liu_pnas; @jstsp14]. Therefore, along with contractions, root SCCs are effective in observability recovery. In order to solely consider the role of contractions in observability recovery in this paper we focus on undirected networks. As the direction of future research, we plan to seek whether other graph properties such as network community structure and degree-degree correlation [@posfai2013correlation] are effective on the contraction analysis and observability properties.
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors would like to thank Professor Glenn Lawyer from Max-Planck-Institute for Informatics for providing us with real network data.
[Mohammadreza Doostmohammadian]{} received his B.Sc. and M.Sc. in Mechanical Engineering from Sharif University of Technology, Tehran, Iran, respectively in 2007 and 2010, where he worked on different applications of control systems and robotics. He received his PhD in Electrical and Computer Engineering from Tufts University, Medford, USA in 2015. During his PhD at Signal Processing and Robotic Network (SPARTN) lab he worked on control and signal processing over networks with particular application in social networks. From 2015 to 2017 he was a post-doctoral researcher at ICT Innovation Center for Advanced Information and Communication Technology (AICT), School of Computer Engineering, Sharif University of Technology, where he focused his research on control and estimation over networks and network epidemic. He is currently a researcher at Iran Telecommunication Research Center (ITRC), Tehran, Iran and a lecturer at Semnan University, Semnan, Iran. His general research interests include control and estimation, complex networks, and graph theory. He is a reviewer for IFAC and IEEE journals and conferences.
[Hamid R. Rabiee]{} received his B.S. and M.S. degrees (with great distinction) in electrical engineering from California State University, Long Beach, CA, in 1987 and 1989, respectively; the EEE degree in electrical and computer engineering from the University of Southern California, Los Angeles, CA, in 1993; and the Ph.D. degree in electrical and computer engineering from Purdue University, West Lafayette, IN, in 1996. From 1993 to 1996, he was a Member of Technical Staff at AT&T Bell Laboratories. From 1996 to 1999, he worked as a Senior Software Engineer at Intel Corporation. From 1996 to 2000, he was also an Adjunct Professor of electrical and computer engineering at Portland State University, Portland, OR; Oregon Graduate Institute, Beaverton, OR; and Oregon State University, Corvallis, OR. Since September 2000, he has been with the Department of Computer Engineering, Sharif University of Technology, Tehran, Iran, where he is currently a Professor of computer engineering and Director of Sharif University Advanced Information and Communication Technology Research Institute (AICT), Digital Media Laboratory (DML), and Mobile Value Added Services Laboratory (M-VASL). He was also the founder of AICT, Advanced Technologies Incubator (SATI), DML, and M-VASL. He has been the Initiator and Director of national- and international-level projects in the context of United Nation Open Source Network program and Iran National ICT Development Plan, and holds 3 patents. Prof. Rabiee has received numerous awards and honors for his industrial, scientific, and academic contributions. He is a Senior Member of IEEE.
[Houman Zarrabi]{} received his PhD from Concordia University in Montreal, Canada in 2011. Since then he has been involved in various industrial and research projects. His main expertise includes IoT, M2M, CPS, big data, embedded systems and VLSI. He is currently the national IoT program director and assistant professor at Iran Telecommunication Research Center (ITRC).
[Usman Khan]{} received his B.S. degree (with honors) in Electrical Engineering from University of Engineering and Technology, Lahore-Pakistan, in 2002, M.S. degree in Electrical and Computer Engineering (ECE) from the University of Wisconsin-Madison (UW-Madison) in 2004, and Ph.D. degree in ECE from Carnegie Mellon University in 2009. Currently, he is an Assistant Professor with the ECE Department at Tufts University. He received the NSF Career award in January 2014 and was elevated to IEEE Senior Member grade in March 2014. His research interests include statistical signal processing, networked control and estimation, and distributed linear/nonlinear iterative algorithms for efficient operation and planning of complex infrastructures.
Prof. Khan was a post-doctoral researcher in the Electrical and Systems Engineering department at the University of Pennsylvania from September 2009 to December 2010. He worked as a researcher in National Magnetic Resonance Facility at Madison (NMRFAM) from 2003 to 2005, as a research assistant in the Computer Science Dept. at UW-Madison from 2004 to 2005, and as an intern in AKAMAI Technologies in 2007. Dr. Khan is an associate member of the Sensor Array and Multichannel Technical Committee with the IEEE Signal Processing Society. He served on the Technical Program Committees of several IEEE conferences and has organized and chaired several IEEE workshops and sessions. His work was presented as Keynote speech at BiOS SPIE Photonics West-Nanoscale Imaging, Sensing, and Actuation for Biomedical Applications IX.
[^1]: Manuscript received April 5, 2017.
[^2]: In this paper a network describes the underlying dynamic system or phenomena. Therefore, throughout the paper the network and system are used interchangeably.
[^3]: See [@liu_pnas; @Liu_nature; @nonlin] for extension to nonlinear case.
[^4]: It should be noted that structural observability and graph theoretic method applied as a *tool* to solve network observability problem. See reference [@liu_pnas; @Liu_nature] for more information.
[^5]: Note that many of stated references deal with dual problem of network controllability. The graph properties and notions can be simply redefined for network observability.
[^6]: Note that, in general, edges in a bipartite graph have no direction. However in this paper, following the definition in [@murota], it is assumed that the edges have direction from $\mc{V}^+$ to $\mc{V}^-$. This kind of representation is later used in the definition of *Auxiliary graph*.
[^7]: It should be mentioned that the concept of contraction is dual of dilation defined in the network controllability problem [@Liu_nature]. In a dilation set, less number of nodes are dilated into more number of nodes. So that we don’t need to continually refer to the dual graph, we define a contraction that is the natural dual of dilation.
[^8]: Transitivity implies that if $x^i \sim x^j$ and $x^j \sim x^k$, then $x^i \sim x^k$. Reflexivity implies that every state/node is equivalent to itself, and symmetry implies that $x^i \sim x^j$, then $x^j \sim x^i$.
[^9]: Note that, the structural rank (or S-rank) is defined as the maximum rank of the system matrix, $A$, by changing its free parametric entries. In the system graph, $\mc{G}$, S-rank is the size of the *maximum matching*, $\mc{M}$, see [@shields; @van1999generic] for details.
[^10]: The results carry forward are also applicable to continuous-time systems.
[^11]: This is not a straighforward procedure as the edge weights vary over time while the structure is time-invariant. Note that here we only convey the idea behind LSI dynamics with fixed sparsity pattern on the adjacency matrix.
[^12]: Note that contraction $\mc{C}_1$ is similar to the contraction described in Fig. \[fig\_3nodecont\] and \[fig\_3nodecont2\].
[^13]: Note that it is known that most real-world networks including the two examples given in this paper follow a power-law degree distribution [@barabasi_albert1999].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The statics, stability and dynamical properties of dark-bright soliton pairs are investigated motivated by applications in a homogeneous system of two-component repulsively interacting Bose-Einstein condensate. One of the intra-species interaction coefficients is used as the relevant parameter controlling the deviation from the integrable Manakov limit. Two different families of stationary states are identified consisting of dark-bright solitons that are either antisymmetric (out-of-phase) or asymmetric (mass imbalanced) with respect to their bright soliton. Both of the above dark-bright configurations coexist at the integrable limit of equal intra- and inter-species repulsions and are degenerate in that limit. However, they are found to bifurcate from it in a transcritical bifurcation. The latter interchanges the stability properties of the bound dark-bright pairs rendering the antisymmetric states unstable and the asymmetric ones stable past the associated critical point (and vice versa before it). Finally, on the dynamical side, it is found that large kinetic energies and thus rapid soliton collisions are essentially unaffected by the intra-species variation, while cases involving near equilibrium states or breathing dynamics are significantly modified under such a variation.'
author:
- 'G. C. Katsimiga'
- 'P. G. Kevrekidis'
- 'B. Prinari'
- 'G. Biondini'
- 'P. Schmelcher'
title: 'Dark-bright soliton pairs: bifurcations and collisions'
---
Introduction
============
Multi-component Bose-Einstein condensates (BECs) and the nonlinear excitations that arise in them have been a focal research point over the past two decades since their experimental realization [@gpes1; @gpes2]. Among these excitations, dark-bright (DB) solitons constitute a fundamental example [@frantzkevre], whose experimental realization in a $^{87}$Rb mixture [@hamburg] has triggered a new era of investigations regarding the stability and interactions of these matter waves both with each other [@yan], as well as induced by the external traps [@ba; @hamner; @middelkamp; @alvarez].
Within mean-field theory the static and dynamical properties of such states are well described by a system of coupled Gross-Pitaevskii equations [@gpes1; @gpes2]. The latter is a variant of the so-called defocusing (repulsive) vector nonlinear Schr[ö]{}dinger equation [@nls; @siambook], to which it reduces in the absence of a confining potential. In this homogeneous setting, single DB solitons exist as exact analytical solutions when the repulsive interactions within (intra-) and between (inter-) the species are of equal strength; this is the integrable, so-called Manakov limit [@manakov]. In this setting, multiple solitonic states, both static and travelling ones, have been analytically derived by using the inverse scattering transform (IST) considering both trivial [@manakov], or more recently, non-trivial boundary conditions [@prinari; @biondini] allowing also for energy (and phase) exchanges between the bright soliton components. Additionally, the Hirota method has been used to explore different families of DB soliton solutions ranging from perfectly antisymmetric (out-of-phase) to fully asymmetric ones (mass imbalanced), with respect to their bright soliton counterpart [@shepkiv] (see also for a small sample among numerous additional studies the works [@vdbysk1; @ralak; @parkshin; @rajendran]).
As a matter of fact, given their versatility, BECs offer additional layers of tunability, enabling the controllable departure from this integrable Manakov limit. In particular, exploiting the tunablility of both the inter- and intra-species scattering lengths that can be achieved in current experimental settings with the aid of Feshbach resonances [@inouye; @roberts; @donley; @thalhammer; @chin], a new avenue opened towards exploring DB soliton interactions under parametric variations [@ef; @et; @ljpp; @carr; @ljpp1]. This allowed to address the robustness of these matter waves in BEC mixtures with genuinely different scattering lengths. However, typically in realistic settings all three coefficients of inter- and intra- component interactions are slightly different, and hence it is of particular interest to explore how things deviate from the integrable limit.
In the present work, our aim is to bring to bear the enhanced understanding of the integrable limit that exists through the recent works of [@prinari; @biondini] in order to controllably appreciate how statics, bifurcations and dynamics are affected upon deviations from this limit. More specifically, we will examine stationary states at the integrable limit and how they (and their respective stability properties) are modified upon deviation from integrability. In the process, we will uncover an unusual example of a transcritical bifurcation with symmetry involving bound DB soliton pairs of two kinds: antisymmetric and asymmetric ones. In the former, the bright components are out-of-phase whereas in the latter they are imbalanced in terms of their respective masses. We will then turn to dynamical states involving (from the integrable limit) solutions with different speeds. We will initialize such states in regimes close to and far from integrability to observe the implications of non-integrability on them. Our main conclusion there is that for states of high kinetic energy (where the latter dominates the DB interaction) implications of the non-integrability are rather limited. However, for states of proximal DBs with prolonged (or recurrent) interactions, non-integrability can have a significant impact in the outcome of their collisions, as we illustrate via suitable numerical computations.
The paper is organized as follows. In Sec. II we provide the setup of the multi-component system under consideration. In Sec. III the static properties of two-DB soliton solutions upon varying the intra-species interactions are exposed. Sec. IV is devoted to studying the dynamical properties of these matter waves, while Sec. V contains our conclusions and future perspectives.
Model setup
===========
As our prototypical playground, we consider the following one-dimensional (1D) system of coupled nonlinear Schr[ö]{}dinger equations: $$\begin{aligned}
i \partial_t \psi_d =& -&\frac{1}{2} \partial_{x}^2\psi_d
+(|\psi_d|^2 + g_{12}|\psi_b|^2 -\mu_d) \psi_d,\nonumber \\
\label{nls1}
\\
i \partial_t \psi_b =& -&\frac{1}{2} \partial_{x}^2\psi_b + (g_{12}|\psi_d|^2 + g_{22}|\psi_b|^2- \mu_b) \psi_b.\nonumber \\
\label{nls2}\end{aligned}$$ In the above equations, $\psi_d$ ($\psi_b$) is the wavefunction of the dark (bright) soliton component while $\mu_d$ ($\mu_b$) is the corresponding chemical potential. Furthermore, $g_{12}\equiv g_{12}/g_{11}$, and $g_{22}\equiv g_{22}/g_{11}$ denote the rescaled interaction coefficients which are left to arbitrarily vary, spanning both the miscible and the immiscible regime of interactions. Note that in this setting the miscibility of the two components occurs when $g_{12} \leqslant \sqrt{g_{22}}$, and refers to the absence of phase separation between the species [@aochui]. Additionally, Eqs. (\[nls1\])-(\[nls2\]) stem from the corresponding BEC system assuming a setting without a longitudinal trap, but only with a transverse trap of strength $\omega_{\perp}$. The coupling constants in 1D are $g_{jk}=2\hbar\omega_{\perp} a_{jk}$, where $a_{jk}$ denote the $s$-wave scattering lengths (with $a_{12}=a_{21}$) that account for collisions between atoms of the same ($j=k$) or different ($j \ne k$) species. The aforementioned dimensionless 1D system also assumes the measuring of densities [$|\psi_j|^2$]{}, length, time and energy in units of $2a_{11}$, $a_{\perp} = \sqrt{\hbar/\left(m \omega_{\perp}\right)}$, $\omega_{\perp}^{-1}$ and $\hbar\omega_{\perp}$, respectively. We note that in the following all our results are presented in dimensionless units.
From here, $\mu_d$ can be also scaled out via the transformations: $t \rightarrow
\mu_d t$, $x \rightarrow {\sqrt{\mu_d}}x$, $|\psi_{d,b}|^2 \rightarrow \mu_{d}^{-1} |u_{d,b}|^2$, and thus the system of equations (\[nls1\])-(\[nls2\]) acquires the following form $$\begin{aligned}
i \partial_t u_d + \frac{1}{2} \partial_{x}^2u_d -(|u_d|^2 + g_{12}|u_b|^2 -1) u_d &=& 0,
\label{deq11}
\\
i \partial_t u_b +\frac{1}{2} \partial_{x}^2u_b -(g_{12}|u_d|^2 + g_{22}|u_b|^2- \mu) u_b &=& 0,
\label{deq21}\end{aligned}$$ where $\mu\equiv\mu_b/\mu_d$ is the rescaled chemical potential. The above system of equations conserves the total energy $$\begin{aligned}
E &=& \frac{1}{2}\int_{-\infty}^{+\infty} dx \Big[ |\partial_{x} u_d|^2+|\partial_{x} u_b|^2+(|u_d|^2-1)^2 \Big. \nonumber \\
&+& \Big. g_{22}|u_b|^4 - 2\mu |u_b|^2 + 2g_{12} |u_d|^2 |u_b|^2 \Big],
\label{energy} \end{aligned}$$ as well as the total number of atoms $$\begin{aligned}
N&\equiv &N_d+N_b=\sum_{i=d,b}\int^{\infty}_{-\infty} dx|u_i|^2,
\label{ntot}\end{aligned}$$ with $N_d$, $N_b$, denoting the number of atoms in the first and second component of the system of Eqs. (\[deq11\])-(\[deq21\]) respectively. $N_d$ and $N_b$ are also individually conserved.
Stability analysis of bound antisymmetric and asymmetric dark-bright pairs
==========================================================================
By considering the time-independent version of the aforementioned system of Eqs. (\[deq11\])-(\[deq21\]), namely: $$\begin{aligned}
u_d&=& -\frac{1}{2} \partial_{x}^2u_d +(|u_d|^2 + g_{12}|u_b|^2) u_d ,
\label{tidd}
\\
\mu u_b &=&-\frac{1}{2} \partial_{x}^2u_b +(g_{12}|u_d|^2 + g_{22}|u_b|^2) u_b,
\label{tidb}\end{aligned}$$ bound states consisting of two DB solitons can be found in [@ljpp; @ljpp1] for out-of-phase or antisymmetric bright solitons for arbitrary nonlinear coefficients.
First, let us briefly recall what is known about the Manakov model. In such a case the system possesses exact two-DB soliton solutions that can be obtained by using either Hirota’s method studied in Ref. [@shepkiv] or the more recent exact expressions found in Ref. [@prinari] by using the inverse scattering transform (IST) but with non-trivial boundary conditions. In particular, the exact static two-DB solutions can be written in the following form: $$\begin{aligned}
u_d &=&\frac{(1-a)\cosh(\xi_1+\xi_2)-(1+a)\cosh(\xi_1-\xi_2)}{(1-a)\cosh(\xi_1+\xi_2)+(1+a)\cosh(\xi_1-\xi_2)}, \\
u_b &=&\frac{2(1-a^2) \sinh \xi_1}{(1-a)\cosh(\xi_1+\xi_2)+(1+a)\cosh(\xi_1-\xi_2)},\end{aligned}$$ where $\xi_1 = x - \delta_1$, $\xi_2 = a(x-\delta_2)$ and $\mu = 1- a^2/2$.
There are three free parameters here: $a$, $\delta_1$ and $\delta_2$. One of them is due to the translational invariance: Shifting $\delta_1 \rightarrow \delta_1 + \bar\delta$, $\delta_2 \rightarrow \delta_2 + \bar\delta$ only displaces the overall solution by $\bar \delta$, so fixing the overall location of the pair one can fix $\delta_1 = -\delta_2 =: \delta$. Then one is still left with a nontrivial two-parameter family, with parameters $a$ and $\delta$. Fixing the chemical potential ratio $\mu$ fixes $a$ and vice versa (note that there is no loss of generality in assuming $a>0$ since $a \rightarrow -a$ only flips the global sign of $u_d$). The parameter $a$ must satisfy $0 \leq a \leq 1$ and accordingly $1/2 \leq \mu
\leq 1$. The second parameter $\delta$ can also be taken positive since $\delta \rightarrow - \delta$, $x\rightarrow -x$ leaves $u_d$, $u_b$ invariant (up to a global sign), so changing the sign of $\delta$ just performs a reflection about $x=0$, exchanging the two DBs. For $\delta=0$, $u_d( x)=
u_d(-x)$, $u_b(x)=-u_b(-x)$, so the DB pair is antisymmetric. As $\delta$ becomes larger, the asymmetry increases towards a dark/DB state and the equilibrium distance between the solitons increases. In our simulations $\mu$ is fixed, so $a$ is fixed. But still, in the integrable limit there will be a one-parameter family of two-DB solutions of varying $\delta$, continuously ranging from perfectly antisymmetric to extremely asymmetric.
As we depart from integrability, such an explicit expression is no longer available. In light of that, the corresponding stationary states were numerically obtained by means of a Newton’s fixed point iteration method and using as an initial guess for identifying the numerically exact (up to a prescribed tolerance) two-DB spatial profile the following ansatz: $$\begin{aligned}
\!\!\!\!\!\!
u_d(x)&=&\tanh\left[D(x-x_0)\right] \times \tanh\left[D(x+x_0)\right],
\label{dark}
\\
\!\!\!\!\!\!
u_b(x)&=&\eta {\rm sech}\left[D(x-x_0)\right] +\eta {\rm sech}\left[D(x+x_0)\right] e^{i\Delta \theta}. \nonumber\\
\label{bright}\end{aligned}$$ In the above expressions $2x_0$ is the relative distance between the two DB solitons, $D$ denotes their common inverse width, while $\Delta \theta$ is their relative phase within the bright component, with $\Delta \theta=\pi$ ($\Delta \theta=0$) corresponding to out-of-phase (in-phase) bright solitons. Note also that the (background) amplitude of the dark soliton component is unity, while $\eta$ denotes the amplitude of the bright soliton counterpart. Here, we will solely focus on the out-of-phase or antisymmetric case (as the in-phase case does not produce a bound state pair in the homogeneous setting [@ljpp]). Upon varying $g_{12}$ typically within the interval $0.75\leqslant g_{12}\leqslant 1.5$, while keeping both $\mu=2/3$ and $g_{22}=0.95$ fixed, earlier numerical studies [@ljpp; @ljpp1] showcased that antisymmetric two-DB states exist as stable configurations only within a bounded interval of the inter-species repulsion coefficient $g_{12}$ limited by critical points both in the miscible and in the immiscible regime, associated with a supercritical and a subcritical pitchfork bifurcation respectively. Furthermore, new families of solutions consisting of mass imbalanced DB pairs, i.e. different amplitudes between the bright soliton constituents, were found to bifurcate through pitchfork bifurcations from the above obtained antisymmetric states.
However, by fixing $g_{22} \neq 1$ in these previous works it has not been possible to systematically approach the Manakov limit of $g_{12}=g_{22}=1$. To address this important special limit, and explore the effect of breaking the integrability, below we fix $g_{12}=1$ and perform a continuation in $g_{22}$, i.e., starting from $g_{22}=0.95$ (immiscible regime) up to $g_{22}=1$ and beyond, towards the miscible domain of interactions, considering the fate of both the antisymmetric and asymmetric bound states. Note also that for the numerical findings to be presented below the rescaled chemical potential is fixed to $\mu=0.7$ in the dimensionless units adopted herein. In Figs. \[fig1\] $(a_1)-(a_2)$ the linearization, or so-called Bogolyubov-de Gennes (BdG), spectrum of the antisymmetric bound pairs is shown as a function of the nonlinear coefficient $g_{22}$. This is obtained by expanding around an equilibrium configuration as follows:
$$\begin{aligned}
u_d &=& u_d^{(eq)} + \left(a(x) e^{-i \omega t} + b^{\star}(x) e^{i \omega^{\star} t}
\right),
\label{bdg1}
\\
u_b &=& u_b^{(eq)} + \left(c(x) e^{-i \omega t} + d^{\star}(x) e^{i \omega^{\star} t}
\right),
\label{bdg2}\end{aligned}$$
where “$\star$" stands for the complex conjugate. Then the system for the eigenfrequencies $\omega$ (or equivalently eigenvalues $\lambda=i \omega$) and eigenfunctions $(a,b,c,d)^T$ is numerically solved. If modes with purely real eigenvalues (imaginary eigenfrequencies) or complex eigenvalues (and thus eigenfrequencies) are identified, the configuration is characterized as dynamically unstable. Moreover, there is a class of modes that bears the potential to lead to instabilities. These are the modes with negative so-called energy or Krein signature [@krein]. The relevant quantity is defined as $$\begin{aligned}
K= \omega \int \Big(|a|^2 - |b|^2 + |c|^2 - |d|^2\Big) dx,
\label{krein}\end{aligned}$$ and can be directly evaluated on the basis of the eigenvector $(a,b,c,d)^T$ and eigenfrequency $\omega$. Both the real, $\rm{Re}(\omega)$, and the imaginary, $\rm{Im}(\omega)$, parts of the eigenfrequencies $\omega$ are depicted in Figs. \[fig1\] $(a_1)$ and $(a_2)$ respectively. Notice that in close contact with our previous findings [@ljpp; @ljpp1], two anomalous (namely, bearing negative Krein signature) modes appear in the linearization spectra and their trajectories are denoted with red squares \[see Fig. \[fig1\] $(a_1)$\]. Among these modes, the higher-lying one is found to be related to the out-of-phase vibration of the bound DB pair [@ljpp]. More importantly, the lower-lying anomalous mode is associated, through its eigenvector, with a symmetry breaking in the bright soliton component, resulting in mass imbalanced (with respect to their bright soliton counterpart) DB pairs that we will trace later on in the dynamics. In both cases the aforementioned findings can be identified by adding the corresponding eigenvector to the relevant antisymmetric DB solution. The background (continuous, in the limit of infinite domain) spectrum is also depicted in the same figure with blue circles. As it is observed, upon increasing $g_{22}$ towards the integrable limit the frequencies of both of the anomalous modes decrease. Following the lower-lying mode it becomes apparent that there exists a critical point $g_{22_{cr}}=1$ where this mode goes from linearly stable (for $g_{22}< g_{22_{cr}}$) to linearly unstable (for $g_{22}> g_{22_{cr}}$). Notice that exactly at the integrable limit this anomalous mode becomes neutrally stable, while past $g_{22_{cr}}=1$ it destabilizes as it is evident from the non-zero imaginary part presented in Fig. \[fig1\] $(a_2)$.
To verify the stability analysis results presented above, the spatio-temporal evolution of both the stable and unstable antisymmetric DB pairs is computed and shown in Figs. \[fig1\] $(b_1)-(c_3)$. Notice that the antisymmetric configuration is only slightly modified as $g_{22}$ is increased over the interval considered. Minor differences, mostly in the amplitude of the bound pairs upon increasing $g_{22}$, can be inferred by inspecting the overall decrease of the norm of the bright component (see Fig. 3 below). Here, Figs. \[fig1\] $(b_1)-(b_3)$ \[$(c_1)-(c_3)$\] correspond to the density of the dark \[bright\] soliton component. In particular, in all cases we use as an initial condition the numerically obtained stationary states at selected values of $g_{22}$, i.e., below, at and above the associated critical point, and we numerically evolve the system of Eqs. (\[deq11\])-(\[deq21\]) using a fourth order Runge-Kutta integrator. As anticipated from the aforementioned BdG outcome, for $g_{22}>g_{22_{cr}}=1$ the instability dynamically manifests itself via a dramatic mass redistribution between the bright soliton counterparts. The latter leads in turn to the splitting of the bound pair and results in asymmetric states with a dark and a DB soliton pair repelling one another and moving towards the boundaries (cf. also with the corresponding non-integrable cases in [@ljpp; @ljpp1]).
We now explore the same diagnostics for the asymmetric DB bound pairs that are degenerate with the antisymmetric ones in the integrable limit. These stationary asymmetric states are once again numerically identified and their stability outcome is summarized in Fig. \[fig2\]. As before, Fig. \[fig2\] $(a_1)$ depicts the real part, $\rm{Re}(\omega)$, of the eigenfrequncies $\omega$ as a function of $g_{22}$, while Fig. \[fig2\] $(a_2)$ shows the corresponding imaginary part, $\rm{Im}(\omega)$. In the real part of the spectrum the absence of the lower-lying anomalous mode is verified. Recall that the existence of this mode in the spectrum of the antisymmetric branch signalled the presence of the asymmetric branch of solutions \[see Fig. \[fig1\] $(a_1)$\]. In contrast to the antisymmetric states investigated above, this family of solutions is unstable for $g_{22}<1$ as is evident in that regime by a non-zero imaginary eigenfrequency illustrated in Fig. \[fig2\] $(a_2)$. On the other hand, the state remains spectrally stable for $g_{22}\geqslant 1$, and the formerly unstable mode, now becomes an anomalous one with a real eigenfrequency. It is important to note here, that the equilibrium distance is found to be larger for the asymmetric states when compared with the antisymmetric ones \[compare e.g. Fig. \[fig2\] $(b_2)$ with Fig. \[fig1\] $(b_2)$\]. The equilibrium distance also becomes larger for the asymmetric states upon increasing $g_{22}$, as can be deduced by more closely inspecting e.g. Figs. \[fig2\] $(b_2)$ and $(b_3)$. As before, our BdG results are confirmed via the long-time evolution of the stationary asymmetric states and are illustrated in Figs. \[fig2\] $(b_1)-(c_3)$. Once more, Figs. \[fig2\] $(b_1)-(b_3)$ depict the evolution of the density of the dark soliton component, while Figs. \[fig2\] $(c_1)-(c_3)$ illustrate the propagation of the density of the bright soliton counterpart. As it is expected, for values $g_{22_{cr}}<1$ instability sets in almost from the beginning of the dynamics, with the solitons featuring attraction, more pronounced in the dark soliton counterpart shown in Fig. \[fig2\] $(b_1)$, which results in a collision event at intermediate time scales. However, and as anticipated for $g_{22}=g_{12}=1$ shown in Figs. \[fig2\] $(b_2)$ and $(c_2)$ solitons remain intact throughout the propagation, a result that holds as such even upon considering parameters beyond the integrable limit and on the miscible side depicted in Figs. \[fig2\] $(b_3)$ and $(c_3)$.
![(Color online): $(a)$ Number of atoms, $N_b$, of the bright soliton components as a function of $g_{22}$ for both the antisymmetric and the asymmetric states. $(b)$ Transcritical bifurcation diagram obtained by measuring $\Delta N$ (see text) as a function of $g_{22}$. In all cases the stable (unstable) branches are denoted with solid (dashed) blue lines.[]{data-label="fig3a"}](Fig3_rep-eps-converted-to.pdf)
The above-observed differences between the antisymmetric and asymmetric branches of solutions can be further understood by inspecting the decrease in the number of atoms of the bright component, $N_b$, as a function of the nonlinear coefficient $g_{22}$ depicted for both cases in Fig. \[fig3a\] $(a)$. It is observed that as $g_{22}$ increases the bright norm decreases faster for the asymmetric states, while the two norms are exactly the same at the integrable point. Finally, the transcritical nature of the bifurcation diagram is illustrated in Fig. \[fig3a\] $(b)$. To obtain this bifurcation diagram we calculated the difference in the number of atoms in the bright part, $N_b$, for either the asymmetric or the antisymmetric branches of solutions defined as $\Delta N=N_b-N^{antisym.}_b$, upon varying the nonlinear coefficient $g_{22}$. Notice that the stability character of the antisymmetric and the asymmetric states is exchanged at the integrable point verifying the effectively transcritical nature of this bifurcation. It is worthwhile to comment a little on this bifurcation. Firstly, we point out that saddle-center and pitchfork examples are much more common than transcritical ones in our experience with Hamiltonian systems. In fact, the corresponding state where $\Delta N$ possesses the opposite sign (i.e., the parity symmetric variant of our DB-pair configuration) is also a solution. In that light, it can be thought of as transcritical bifurcation with symmetry. In fact, an even more crucial way in which the symmetry of the bifurcation can be appreciated is the [*neutrality*]{} discussed previously at the Manakov limit. The freedom in the variation of $\delta$ in that context represents a one-parameter family of solutions within which one can freely move and which represent different asymmetries in the bright component. A by-product of this invariance is the presence of a vanishing frequency eigenmode at the critical point of this bifurcation, i.e., at the transition point from stability to instability for the antisymmetric branch or vice-versa for the asymmetric one. However, it is important to point out that these features (neutrality, controllable asymmetry, and associated vanishing eigenfrequency) seem to disappear once we depart from the integrable limit, marking the absence of additional symmetry in the latter case.
Dark-bright soliton collisions
==============================
In what follows we consider collisions between two-DB states at the integrable limit of equal inter- and intra-species interactions, i.e. $g_{12}=g_{22}=1$, as well as deviating from it towards the miscible and the immiscible regime. In both cases we use as an initial ansatz ($t=0$) the exact solution for such two-DB states, namely [@prinari; @biondini]:
$$\begin{aligned}
q_1(x,0)&=&q_o \Bigg[1 + \frac{1}{D(x,0)} \Big[ \bar{\delta}_1 \frac{ (z_1 ^{\star})^2 } { q_o ^2 - z_1 z_2 }
\left( \delta_1 \frac{q_o ^2 - z_1 z_2 ^{\star} }{ z_1 (z_1 ^{\star} - z_1)} e^{-2 \nu_1 x}
- \delta_2 \frac{q_o ^2-|z_2|^2}{ z_2 (z_2-z_1 ^{\star}) } e^{ ix(z_2 - z_1 ^{\star}) } \right) \nonumber \\
&+& \frac{\bar{\delta}_2 (z_2 ^{\star})^2}{q_o^2- z_1 z_2} \left( \delta_2 \frac{q_o ^2-z_1^{\star} z_2}{z_2 (z_2^{\star}-z_2)}
e^{-2 \nu_2 x}
- \delta_1 \frac{ q_o ^2-|z_1|^2 }{ z_1 (z_1-z_2 ^{\star}) } e^{i x(z_1-z_2 ^{\star})} \right) \nonumber \\
&+& q_o ^4 |\delta_1 |^2 |\delta_2 | ^2
\frac{ \left( q_o ^2-|z_1|^2 \right) \left( q_o ^2-|z_2|^2 \right) |q_o ^2 - z_1^{\star} z_2|^2 |z_1-z_2|^4
\left( z_1^{\star} z_2 ^{\star} - z_1 z_2 \right)}{ 16 \nu_1 ^2 \nu_2 ^2 z_1 z_2 |q_o ^2 - z_1 z_2|^2 |z_1^{\star} - z_2|^4}
e^{-2x \left(\nu_1 + \nu_2 \right)} \Big] \Bigg],
\label{twodark} \\
\vspace{0.5cm}
q_2(x,0)&=& -\frac{q_o}{D(x,0)} \Bigg[\frac{\bar{\delta}_1 z_1 ^{\star} }{q_o^2} e^{-i z_1^{\star} x}
+ \frac{\bar{\delta}_2 z_2^{\star} }{q_o^2} e^{-iz_2 ^{\star} x}
+ \Big[ \frac{\bar{\delta}_1 \bar{\delta}_2 z_1^{\star} z_2^{\star}
\left(q_o^2- z_1^{\star} z_2^{\star}\right) \left(z_1^{\star} -z_2^{\star}\right)^2 } {q_o^2 \left(q_o^2-z_1 z_2\right)} \Big] \nonumber
\\
&\times & \Big[ \frac{\delta_1 z_1}{\left(z_1^{\star}-z_1\right)^2 \left(z_1-z_2^{\star}\right)^2} e^{-2 \nu_1 x-i z_2^{\star}x} +
\frac{\delta_2 z_2}{(z_2^{\star}-z_2)^2
(z_2-z_1^{\star})^2} e^{-2 \nu_2 x-i z_1^{\star} x} \Big]\Bigg].
\label{twobright} \end{aligned}$$
Here, $q_1(x,0)$ \[$q_2(x,0)$\] is the wavefunction for the two dark \[bright\] soliton solution of the first \[second\] component in the system of Eqs. (\[nls1\])-(\[nls2\]). $q_o$ is the amplitude of the background, $z_{j}=\kappa_j+i\nu_j$ correspond to the eigenvalues of the IST problem where $k_j=2\kappa_j$ is the soliton’s velocity, while $\delta_{j}$ are the so-called norming constants [@prinari; @biondini]. In all cases $j=1,2$ accounts for the first and the second component of the vector nonlinear Schr[ö]{}dinger system. Additionally, $\bar{\delta}_1=-\delta_1^{\star}( q_o^2(q_o^2-|z_1|^2) (q_o^2-z_1^{\star}z_2) )/( (z_1^{\star})^2 (q_o^2-z_1^{\star}z_2^{\star}) )$, and $\bar{\delta}_2=-\delta_2^{\star}( q_o^2(q_o^2-|z_2|^2) (q_o^2-z_2^{\star}z_1) )/( (z_2^{\star})^2 (q_o^2-z_1^{\star}z_2^{\star}) )$, are related to the complex conjugates of the aforementioned norming constants. The denominator of the above equations is given by
$$\begin{aligned}
D(x,0)=1&-&\frac{\bar{\delta}_1 (z_1^{\star})^2}{q_o^2-z_1 z_2} \left(\delta_1
\frac{q_o^2 - z_1 z_2^{\star} } {4 \nu_1 ^2} e^{-2 \nu_1 x}
- \delta_2 \frac{q_o^2-|z_2|^2 }{\left(z_2-z_1^{\star}\right)^2} e^{i x \left(z_2-z_1^{\star}\right)} \right) \nonumber \\
&+& \frac{\bar{\delta}_2 (z_2^{\star})^2}{q_o^2-z_1 z_2} \left(-\delta_2 \frac{q_o^2-z_2 z_1^{\star}}{4 \nu_2^2} e^{-2 \nu_2 x}
+ \delta_1 \frac{q_o^2 - |z_1|^2}{\left( z_1- z_2^{\star}\right)^2} e^{i x (z_1-z_2^{\star})} \right) \nonumber \\
&+& q_o^4 |\delta_1 |^2 |\delta_2 |^2
\frac{\left(q_o^2-|z_1|^2\right) \left(q_o^2-|z_2|^2\right) |q_o^2 - z_1 z_2^{\star}|^2 |z_1-z_2|^4}{16 \nu_1^2 \nu_2 ^2
|q_o^2-(z_1 z_2)|^2 |z_1^{\star}-z_2|^4} e^{-2 x \left(\nu_1 + \nu_2 \right)}.
\label{denominator}\end{aligned}$$
![(Color online): Collisions of DB states with zero initial velocities, $k_1=k_2=0$, and upon increasing $g_{22}$. The corresponding density profiles for $g_{22}=0.85$ are illustrated as insets in panels $(a_1)$ and $(b_1)$. Notice the breathing dynamics that the DB pairs undergo as the intra-species repulsion is increased. $(a_1)-(a_5)$ \[$(b_1)-(b_5)$\] Evolution of the density, $|q_1(x,t)|^2$ \[$|q_2(x,t)|^2$\], of the dark \[bright\] soliton component. Other parameters used are $\nu_1=1/2$, $\nu_2=1/3$, $\delta_1=1+i/5$, and $\delta_2=1-i/3$. All quantities shown are expressed in dimensionless units.[]{data-label="fig4"}](Fig5_rep-eps-converted-to.pdf)
In order to initialize the dynamics we first render the two-DB states of Eqs. (\[twodark\])-(\[twobright\]) well-separated. The latter can be achieved by parametrizing the norming constants $\delta_j$ ($j=1,2$) as: $\delta_j=(2 \nu_j/(q_o \sqrt{q_o^2-z_j^2})) \exp(x_j+i \phi_j)$ and varying the position offset $x_j$, and/or the phase $\phi_j$. Throughout this work, the amplitude of the background is fixed to $q_o=1$. In the case examples presented in Figs. \[fig3\] $(a_1)-(a_6)$, the DB pairs are located around $x_1\approx 0$ and $x_2\approx -10$ respectively. Furthermore, we fix the corresponding phases $\phi_1=\phi_2=\pi/4$, the velocity of the first DB pair $k_1=0$, and we vary $k_2$ within the interval \[0.35, 0.85\]. It is important to note that $k_2$ also significantly influences the asymmetry between the bright soliton counterparts. More specifically, for high speed solitons i.e., for $k_2>0.75$, the moving DB has a very weak bright component gradually becoming a single dark soliton impinging on a DB stationary wave. We have conducted numerous collisional simulations at and below as well as above the integrable limit (in terms of values of $g_{22}$). Our main finding in these cases where one of the solitons possesses a substantial speed is that generically the collisional phenomenology remains essentially similar \[see also Figs. \[fig3\] $(b_1)-(b_4)$, as well as Figs. \[fig3\] $(c_1)-(c_4)$\] to what is predicted by the analytical expressions of the integrable limit [@prinari].
This situation is in contrast to cases where the solitons have been initialized with vanishing speed, in which we have seen that the breaking of integrability has a maximal impact. A characteristic example of this kind is given in Figs. \[fig4\] $(a_1)-(b_5)$, where once more Figs. \[fig4\] $(a_1)-(a_5)$ \[$(b_1)-(b_5)$\] depict the breathing dynamics of the dark \[bright\] soliton constituent. In this case involving $k_1=k_2=0$, and the choice of the norming constants of $\delta_1=1+i/5$ and $\delta_2=1-i/3$ (for $\nu_1=1/2$ and $\nu_2=1/3$), we find that at the integrable limit the solution forms a beating state. This “fragile” beating is already seen to be significantly impacted by small deviations from integrability of the order of $5 \%$ as it is evident in Figs. \[fig4\] $(a_2), (b_2)$, and $(a_4),(b_4)$, which refer to deviations towards the miscible and the immiscible regime respectively. However, the phenomenology is dramatically affected for deviations of the order of $15 \%$ or more, whereby the former beating state gives way, upon already the first collision of the DB pair, to an indefinite separation between the two DBs. Remarkably, this deviation takes place both in the miscible, Figs. \[fig4\] $(a_1), (b_1)$, and in the immiscible regime, Figs. \[fig4\] $(a_5), (b_5)$. Under different values of the norming constants, the departure from the breathing state may be “decelerated”. A case example of this kind is depicted in Fig. \[fig5\]. In particular in this realization all parameters used are the same as in the aforementioned collisional scenario except for $\delta_1=5 + i/5$. Notice that for this choice of parameters the aforementioned deceleration against repulsion is more pronounced within the immiscible regime of interactions \[compare e.g. panels $(a_5)$ and $(b_5)$ here, with Figs. \[fig4\] $(a_5)$ and $(b_5)$\].
\[ht\] ![(Color online): Same as in Fig. \[fig4\] but for $\delta_1=5+i/5$, with all quantities shown expressed in dimensionless units.[]{data-label="fig5"}](Fig6_rep-eps-converted-to.pdf "fig:")
Furthermore, this change in the norming constant affects the bright soliton characteristics resulting in particular in slightly mass imbalanced bright soliton counterparts, illustrated as insets in Figs. \[fig5\] $(a_1)$ and $(b_1)$, when compared to the spatial profiles of the solitons shown as insets in Figs. \[fig4\] $(a_1)$ and $(b_1)$. On the other hand, when initializing the dynamics considering a genuinely stationary configuration, the results of the stability analysis of the previous section are confirmed. Namely, for a genuinely antisymmetric bright configuration, stability persists on the side of $g_{22}<1$, while a splitting (symmetry breaking) instability into a DB and a dark soliton arises for $g_{22}>1$, with the stability intervals being reversed for asymmetric initial conditions.
Conclusions and future challenges
=================================
In the present work we have investigated the stability and dynamics of matter-wave DB solitons in homogeneous binary BECs. We have done so by taking advantage in a systematic fashion of the understanding and knowledge imparted by the inverse scattering transform and the Hirota method within the integrable Manakov limit. In that limit, both antisymmetric and asymmetric DB pair waveforms are identified, but also solutions involving DB pairs interacting with different speeds can be explored via cumbersome, yet explicit analytically available formulae. We have benchmarked the results of our numerical simulations against these expressions and subsequently extended our analysis past the integrable limit to identify the nature of the deviations from the corresponding results. This has given rise to an array of interesting findings. In particular, in the case of stationary solutions we have identified an intriguing transcritical bifurcation with symmetry. Antisymmetric and asymmetric solutions in the bright soliton component of the DB pairs have been found, respectively to be, stable (unstable) for $g_{22}<1$ ($g_{22}>1$), exchanging their stability in the degenerate (invariant under symmetry breaking) case of the integrable limit. Moreover, as regards collisions, we have seen that those bearing significant kinetic energy were essentially unaffected by the breaking of integrability. On the other hand, the more delicate beating (or even stationary) states were drastically affected by the breaking of integrability, typically leading to the fission of the elements within the pair, possibly accompanied by a symmetry breaking between the bright components.
These findings suggest a multitude of interesting directions for future studies. A straight forward one would be to consider such multicomponent interactions in the presence of quantum fluctuations [@lgs]. In such a setting it has recently been shown that DB states decay into daughter DB ones, so it would be particularly interesting to explore how the collisional dynamics of the above beating states is altered by taking into account beyond mean-field effects. Yet another interesting aspect would be to extend our current considerations involving a higher number of species. In this spinor setting, solutions in the form of dark-dark-bright and dark-bright-bright solitons have been theoretically obtained [@BG], and also very recently experimentally observed [@PGK_Engels]. Thus a study of their static and dynamical properties will enhance our understanding of these soliton complexes. Furthermore, one could also explore multicomponent interactions as the ones considered herein but in higher dimensions. As is well-known there are no direct analogues of the Manakov model that are known to be integrable at present. However, it is nevertheless of interest to explore interactions of vortex-bright solitons in two dimensions [@pola] and of configurations such as vortex line-bright solitons or vortex-ring-bright solitons in three dimensions [@wenlong]. These possibilities are presently under consideration and will be reported in future publications.
Acknowledgements {#acknowledgements .unnumbered}
================
P.G.K. gratefully acknowledges the support of NSF-PHY-1602994, and the Alexander von Humboldt Foundation. B.P. gratefully acknowledges the support of NSF-DMS-1614601, and G.B. gratefully acknowledges the support of NSF-DMS-1614623, NSF-DMS-1615524. G.C.K and P.S. gratefully acknowledge fruitful discussions with J. Stockhofe.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We comment on the paper of S. Postnikov et al. in Phys. Rev. D 82, 024016 (2010) and give a modified formula that needs to be taken into account when calculating the tidal Love number of neutron stars in case a first order phase-transition occurs at non-zero pressure. We show that the error made when using the original formula tends to zero as $p \rightarrow 0$ and we estimate the maximum relative error to be $\sim 5\%$ if the density discontinuity is at larger densities.'
author:
- 'János Takátsy$^{1,2}$'
- 'Péter Kovács$^{1,2}$'
title: 'Comment on “Tidal Love numbers of neutron and self-bound quark stars”'
---
In Ref. [@postnikov2010] the authors investigated the qualitative differences between the tidal Love numbers of self-bound quark stars and neutron stars. In Eq. (14) they derived an expression for the extra term that should be subtracted from the logarithmic derivative $y(r)$ of the metric perturbation $H(r)$ in case there is a first-order phase transition in the equation of state (EoS). The authors applied this formula to quark stars where there is a core-crust phase transition at or below neutron-drip pressure. Since then multiple papers have included or applied this formula explicitly using EoSs with first-order phase transitions at non-negligible pressures ([*e.g.*]{} [@zhao2018; @han2019]). However, when the pressure $p_d$ corresponding to the density discontinuity is non-negligible compared to the central energy density of the neutron star, Eq. (14) of Ref. [@postnikov2010] should be modified as shown below. In this comment we derive the correct formula and estimate the error made when using the other formula instead.
It needs to be added, that although Ref. [@han2019] contains the uncorrected formula, the results presented in the paper were calculated using the correct relation, as it was reported by the authors and also verified by the authors of Ref. [@postnikov2010]. This also applies to more recent publications including the same authors [@han2019b; @chatziioannou2020]. Moreover, despite using the erroneous formula, the results of Ref. [@zhao2018] are also mainly unaffected by this error, since they only provide approximate analytic fits for the ratios of tidal deformabilities of the two components in binary neutron stars. Thus, uncertainties of a few percent are inherently contained in these fits, which encompass the errors of individual tidal deformabilities. The corrected fits – as it was claimed by the authors of Ref. [@postnikov2010] – are negligibly different from the reported fits in Ref. [@zhao2018].
The tidal $l=2$ tidal Love number can be expressed the following way: $$\begin{aligned}
k_2 &= \frac{8}{5} (1-2 \beta)^2 \beta^5 [2 \beta (y_R-1)-y_R+2]\nonumber\\
&\times \{2 \beta [4 (y_R+1) \beta^4+(6 y_R-4) \beta^3+(26-22 y_R) \beta^2\nonumber\\
&+3 (5 y_R-8) \beta-3 y_R+6]+3 (1-2 \beta)^2\nonumber\\
&\times[2 \beta (y_R-1)-y_R+2]\ln \left(1-2\beta\right)\}^{-1} ,
\label{eq:k2}\end{aligned}$$ where $\beta=M/R$ is the compactness parameter of the neutron star and $y_R=y(R)=[rH'(r)/H(r)]_{r=R}$ with $H(r)$ being a function related to the quadrupole metric perturbation (see [*e.g.*]{} [@damour2009]). $y_R$ is obtained by solving the following first-order differential equation: $$\begin{aligned}
ry'(r)&+y(r)^2+r^2 Q(r) \nonumber\\
&+ y(r)e^{\lambda(r)}\left[1+4\pi r^2(p(r)-\varepsilon(r))\right] = 0 ,
\label{eq:y}\end{aligned}$$ where $\varepsilon$ and $p$ are the energy density and pressure, respectively, and $$\begin{aligned}
Q(r)=4\pi e^{\lambda(r)}\left(5\varepsilon(r)+9p(r)+\frac{\varepsilon(r)+p(r)}{c_s^2(r)}\right) \nonumber\\
-6\frac{e^{\lambda(r)}}{r^2}-(\nu'(r))^2 .
\label{eq:Q}\end{aligned}$$ Here $c_s^2=\mathrm{d}p/\mathrm{d}\varepsilon$ is the sound speed squared, while $e^{\lambda(r)}$, $\nu(r)$ metric functions are given by $$\begin{aligned}
e^{\lambda(r)} &= \left[1-\frac{2m(r)}{r}\right]^{-1} \label{eq:tov_e} , \\
\nu'(r) &= \dfrac{2[m(r)+4\pi r^3 p(r)]}{r^2 - 2 m(r) r} \label{eq:tov_nu} ,\end{aligned}$$ with the line element for the unperturbed star defined as $$\mathrm{d}s^2 = e^{\nu(r)}\mathrm{d}t^2 - e^{\lambda(r)}\mathrm{d}r^2 - r^2(\mathrm{d}\vartheta^2 + \sin^2\vartheta \, \mathrm{d}\varphi^2),$$ and where $m(r)$ and $p(r)$ are calculated through the Tolman-Oppenheimer-Volkoff equations [@tolman1939; @oppenheimer1939]: $$\begin{aligned}
m'(r) &= 4\pi r^2 \varepsilon(r) , \label{eq:tov_m} \\
p'(r) &= - [\varepsilon(r)+p(r)]\dfrac{m(r)+4\pi r^3 p(r)}{r^2 - 2 m(r) r} .\label{eq:tov_p}\end{aligned}$$
In case there is a first-order phase transition in the EoS, there is a jump of $\Delta\varepsilon$ in the energy density at constant pressure, hence $c_s^2=0$ in that region and the term in Eq. (\[eq:Q\]) containing $1/c_s^2$ diverges. Expressing $1/c_s^2$ in the vicinity of the density discontinuity: $$\frac{1}{c_s^2} = \frac{\mathrm{d}\varepsilon}{\mathrm{d}p}\bigg|_{p\neq p_d} + \delta(p-p_d) \Delta \varepsilon .
\label{eq:cs2}$$ Changing the delta-function to a function in the radial position $r$, inserting Eq. (\[eq:cs2\]) into Eq. (\[eq:y\]) and integrating over an infinitesimal distance around $r_d$ one obtains: $$y(r_d^+) - y(r_d^-) = -4\pi r_d e^{\lambda(r_d)} [\varepsilon(r_d)+p(r_d)] \frac{\Delta \varepsilon}{|p'(r_d)|} .$$ Using Eq. (\[eq:tov\_p\]) we get: $$\begin{aligned}
y(r_d^+) - y(r_d^-) &= -\frac{4\pi r_d^3 \Delta \varepsilon}{m(r_d)+4\pi r_d^3 p(r_d)}\nonumber\\
&= -\frac{\Delta \varepsilon}{\tilde{\varepsilon}/3+p(r_d)} ,
\label{eq:ydisc}\end{aligned}$$ where $\tilde{\varepsilon}=m(r_d)/(4\pi r_d^3/3)$ is the average energy density of the inner ($r<r_d$) region. Eq. (\[eq:ydisc\]) shows that there is an extra $p(r_d)$ term in the denominator as compared to Eq. (14) of Ref. [@postnikov2010]. We see that if the phase transition is at very low densities compared to the central energy density then $p(r_d)/\tilde{\varepsilon}\rightarrow0$ [^1] and we get back the formula in Ref. [@postnikov2010].
![\[fig:css\]Illustration of the EoS in the constant-sound-speed construction [@alford2013; @han2019]. At $p_\mathrm{trans}$ a quark matter part with a constant sound speed of $c_\mathrm{QM}$ is attached to the nuclear matter EoS after an energy density jump of $\Delta\varepsilon$.](CSS_EoS){width="48.00000%"}
We investigated the difference caused by applying the two different formulas using a constant-sound-speed construction (see Fig. \[fig:css\]) [@alford2013; @han2019]: $$\varepsilon(p)=
\bigg\{\begin{array}{lr}
\varepsilon_\mathrm{NM}(p) &p<p_\mathrm{trans}\\
\varepsilon_\mathrm{NM}(p_\mathrm{trans}) + \Delta \varepsilon + c_\mathrm{QM}^{-2}(p-p_\mathrm{trans}) \quad &p>p_\mathrm{trans}
\end{array},$$ where we fixed $c_\mathrm{QM}^2 = 1$ as in Ref. [@han2019], while varying the values of $p_\mathrm{trans}$ (through $n_\mathrm{trans}\equiv n_\mathrm{NM}(p_\mathrm{trans})$) and $\Delta \varepsilon$. For the nuclear matter (NM) part we chose the SFHo [@steiner2013] and DD2 [@typel2010] as two representative EoSs. We varied the baryon number density at the phase transition $n_\mathrm{trans}$ between $n_0$ and $3.5 n_0$ with $n_0=0.16$ fm$^{-3}$ being the nuclear saturation density. The strength of the phase transition $\Delta\varepsilon/\varepsilon_\mathrm{trans}$ was varied between $0$ and $3$, where $\varepsilon_\mathrm{trans}\equiv\varepsilon_\mathrm{NM}(p_\mathrm{trans})$.
![\[fig:k2ex\]Tidal Love number – neutron star mass relations for the SFHo (orange line) and DD2 (blue line) EoSs, as well as for EoSs obtained from the constant-sound-speed construction. The different number pairs denote different values of $n_\mathrm{trans}/n_0$ and $\Delta\varepsilon/\varepsilon_\mathrm{trans}$, respectively. The tidal Love numbers calculated using Eq. (\[eq:ydisc\]) (solid lines) are reduced by a few percent compared to the ones calculated using Eq. (14) of Ref. [@postnikov2010] (dashed lines).](k2_examp){width="48.00000%"}
{width="48.00000%"} {width="48.00000%"}
In Fig. \[fig:k2ex\] we show some examples of tidal Love number – neutron star mass relations. For EoSs with first-order phase transitions, the Love numbers are reduced when using Eq. (\[eq:ydisc\]) (red and green solid lines) compared to using the formula in Ref. [@postnikov2010] (red and green dashed lines). The maximum relative difference in the tidal Love number as a function of the two parameters defining our constant-sound-speed EoSs is shown in Fig. \[fig:k2err\]. We see that the maximum relative difference reaches its maximum at $n_\mathrm{trans}/n_0\approx2.5$ and $\Delta\varepsilon/\varepsilon_\mathrm{trans}\approx1.5$ for the SFHo EoS, and at $n_\mathrm{trans}/n_0\approx2.0$ and $\Delta\varepsilon/\varepsilon_\mathrm{trans}\approx1.5$ for the DD2 EoS, however, it does not exceed $5\%$ for the whole parameter range. The relative difference also diminishes as we go to lower densities, as it is expected.
Acknowledgement {#acknowledgement .unnumbered}
===============
J. Takátsy and P. Kovács acknowledge support by the NRDI fund of Hungary, financed under the FK19 funding scheme, project no. FK 131982. P. Kovács also acknowledges support by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.
[99]{}
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[^1]: It is worth to note here that although $\tilde{\varepsilon}$ – the average energy density of the inner core – is not equal to the central energy density $\varepsilon_c$, it falls to the same order of magnitude ($\tilde{\varepsilon}/\varepsilon_c \gtrsim 0.25 - 0.5$ for $M>0.5$ $M_\odot$).
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'K. Gesicki, A. A. Zijlstra, A. Acker, S. K. Górny, K. Gozdziewski,'
- 'J. R. Walsh'
date: 'Received ; accepted '
title: 'Planetary nebulae with emission-line central stars'
---
Introduction
============
Some central stars (cores) of planetary nebulae (PNe) show broad, stellar emission lines similar to Wolf-Rayet stars. Although superficially similar, they differ from classical WR stars in their degenerate structure, much lower masses, a wider range of temperatures, and being limited almost exclusively to carbon-rich stars (there are very few counterparts to the massive WN stars). Similar to their massive WC-star counterparts, they are deficient in hydrogen. They form a class named \[WC\]-type which historically was further divided into early \[WCE\] and late \[WCL\] groups. A less numerous class of PNe central stars show narrower and weaker emission lines than the \[WR\]-type stars. These are named [*wels*]{} (weak emission-line stars), as defined in Tylenda et al. (1993).
The evolutionary status of the \[WR\]-type stars is still very uncertain, and it is unclear whether there is any evolutionary relation to the [*wels*]{}. Nebular data suggest an evolutionary sequence \[WC11\]$\rightarrow$\[WO2\] (Zijlstra et al. 1994, Acker et al. 1996, Peña et al. 2001), followed by PG1159 stars (Werner et al. 1992). Acker & Neiner (2003) propose a sequence: \[WC11\]$\rightarrow$\[WC4\]$\rightarrow$\[WO4\]$\rightarrow$\[WO1\]. Parthasarathy et al. (1998) have suggested that the [*wels*]{} are related to the PG1159 stars: \[WCL\]$\rightarrow$\[WCE\]$\rightarrow$[ *wels*]{}/PG1159$\rightarrow$PG1159. But this sequence is by no means proved; the precise location of the [*wels*]{} in relation to the \[WR\] stars is under discussion (Marcolino & de Araujo 2003). Peña et al. (2001) also argue against too closely identifying [*wels*]{} with PG1159 stars (see also Koesterke 2001). Not all PG1159 are hydrogen-poor (Dreizler et al. 1996), showing that not all PG1159 stars will have evolved from \[WR\] stars. Neither are all [*wels*]{} hydrogen-poor: a dual evolutionary sequence may be expected.
The lack of hydrogen is often taken to indicate that the \[WR\] stars have undergone a late thermal pulse (or helium flash), either during the post-AGB evolution (a so-called Late Thermal Pulse or LTP) or on the white dwarf cooling track (Very Late Thermal Pulse or VLTP: Herwig 2001, Hajduk et al. 2005). Following such a pulse, the star rejuvenates and retraces part of its earlier evolution. This predicts that nebulae around \[WR\] stars should be more evolved than around non-\[WR\] stars, but this is not confirmed by observations; the properties of the PNe around \[WR\] stars do not differ from those around other central stars (Górny 2001).
In this paper we discuss a much larger sample of \[WR\] stars and [*wels*]{} than has previously been available. We apply velocity-field analysis to locate the objects on derivatives of the HR diagram. In Sect.2 we describe the methods applied for analysis of PNe with specific attention towards automatic model fitting. Sect.3 presents the 33 newly analyzed PNe. In Sect.4 we discuss the nebular and stellar parameters for the full sample of 101 PNe. In Sect.5 we discuss possible evolutionary relations.
Crowther et al. (1998) have refined the WC and the WO schemes used for WR stars and define a unified classification for massive WR and low-mass \[WR\] stars. Acker & Neiner (2003) developed this scheme for a sample of 42 PN central stars, classified into \[WO 1-4\] and \[WC 4-11\] stars. This classification (used in the present paper) is based on the ionization level of the elements (depending on the wind-temperature of the PN core), showing essentially carbon lines for the coolest stars and oxygen lines for the hottest ones.
Methods
=======
We derive information on the nebulae using a combination of line ratios, diameters, and high resolution spectra. The diameters and line ratios are used to fit a photo-ionization model. The model is constrained to reproduce the intensity distribution of images, if any are available. Line profiles are obtained from the high resolution spectra, and are fitted using the emissivity distributions of the photoionization model, and assuming a velocity field. This is done using the Torun models (Gesicki et al. 1996, 2003). The velocity fields include separate contributions from expansion and turbulence. Turbulence always includes the instrumental broadening and the thermal broadening (calculated from the photoionization model), but some objects may require additional turbulence.
The models assume spherical symmetry. Some strongly bipolar objects cannot be fitted, because they show very irregular velocity profiles. But the majority of objects can be well reproduced with a spherical model. The model corrects for the size of the aperture used for the spectroscopy, and includes seeing effects.
The diameters and distances are adopted from the literature. The models find the density distribution, stellar temperature and the velocity field. The density as function of radius is usually assumed in the shape of a reverted parabola, however if images were available we compared the computed surface brightness and improved the run of the density. The stellar temperature assumes a black-body spectrum energy distribution – therefore we prefer to call it $T_{\rm b-b}$ instead of $T_{\rm eff}$. This assumption can be challenged, especially for the \[WR\] central stars where the comparison between the stellar temperature $T_*$ obtained from non-LTE modelling and $T_{2/3}$ which refers to the radius $R(\tau_{\rm Ross} = 2/3)$ shows differences. The differences are significant for hotter \[WO\] stars (Koesterke & Hamann 1997) while smaller for cooler \[WC\] objects (Leuenhagen et al. 1996). The general tendency is always the same: the $T_{2/3}$, $T_{\rm eff}$ or Zanstra temperatures are smaller than $T_*$. Despite this discrepancy we apply the $T_{\rm b-b}$ for further discussion since it has the advantage of providing a uniform measure of the temperature over a range of stellar classes.
The velocity distribution is assumed to vary arbitrarily and smoothly with radius. In Gesicki & Zijlstra (2000) we compared the Torun model analysis with more traditional methods of deriving the expansion velocities from the spectra. Since long time it was known that different observed lines resulted in different velocities (e.g. Weinberger 1989) indicating a velocity gradient. Our model has this advantage that it combines these data into a single velocity curve. Earlier it was not obvious but the Torun models have shown that even the shape of a single line can indicate a velocity gradient. Good velocity fields can be obtained if a larger number of lines are available. The resulting fields show detailed structure, with velocity peaks at the outer and sometimes at the inner radius of the nebula (Gesicki & Zijlstra 2003). If fewer lines are available, the velocity field is less constrained and simplifying assumptions need to be made. From the full velocity curve $V(r)$ we derive a single parameter characterizing the nebular expansion. We define the expansion velocity as a mass-weighted average over the nebula, $V_{\rm av}$. This parameter has been shown to be robust against the simplifying assumptions: it can be accurately determined even when the velocity field itself is uncertain (Gesicki et al 2003). This allows us to define a kinematic age to the nebula.
The genetic algorithm
---------------------
![The best fit luminosity and temperature for photoionization models for a single object (He2-113, PNG321.0+03.9) for 40 different PIKAIA runs. The larger the size of the circle the better the fit quality.[]{data-label="ltfits"}](figure1.ps){width="88mm"}
The Torun models were recently adopted to automate the search for a best-fit model, using an optimization routine. We applied the method widely known as the genetic algorithm. While it is still not a very popular optimization technique, it has proved to be very effective and robust in many problems, in particular of astrophysical origin. For a review, we refer to a paper by Charbonneau (1995) who is the author of the publicly available code PIKAIA for a genetic algorithm[^1], used for computations in this work.
The genetic algorithms have been invented as an optimization technique mimicking the processes of biological evolution (e.g. Koza 1992). They lead to the selection and adaptation of life forms to the conditions of the natural environment. In this sense the evolution can be thought of as a powerful optimization algorithm. The genetic optimization starts with a set (population) of randomly chosen individuals (parameters of the mathematical model encoded in the ‘genomes’). The whole parameter space of the problem creates the environment for these individuals. The likelihood of survival of a given individual is determined by its ‘fitness’ function $f$; usually, in the least squares minimization, $f=1/\sqrt{\chi^2}$. Then the fitness of a particular individual is a measure of goodness of fit to the studied data set. The population evolves through a number of generations. At every generation, the genetic algorithm evaluates $f$ resulting from each parameter set and processes information by applying to the genomes genetic operators like crossover, mutation and selection. The best fitted members of the population are used to produce a new generation and the process continues until some convergence criteria is reached. Thanks to keeping the memory of the best fitted individuals, the genetic algorithms are much more efficient than Monte Carlo based search techniques. They are robust and especially well suited for multi-dimensional problems, models possessing multiple local extrema and discontinuities. They are non-gradient methods and thus especially well suited for models dependent in a sophisticated way on their parameters. Details of practical application of the technique are given in an excellent introduction by Charbonneau (2002).
The genetic optimization replaced the ’trial and error’ search of parameters applied in our earlier publications (e.g. Gesicki & Zijlstra 2003, Gesicki et al. 2003). The algorithm appeared very effective in comparison with the earlier work. We note also that because of random initialization of the population the method is now free from initial biases. The search procedure again follows two steps. First PIKAIA searches for the values of the nebular mass and the stellar temperature and luminosity which optimize the fit to the observables, before attempting to find the velocity field which optimizes the fit to the line profiles. The following expression is optimized:
$$\chi^2 =
\frac{1}{N-p-1}\sum_{i=1}^N \left[ \frac{O(i) - C(i)}{\sigma(i)}\right]^2,$$
where $O(i)$ are $N$ observations ($i=1,\ldots,N$), $C(i)$ are model predictions dependent on $p$ parameters and $\sigma(i)$ are individual errors of the observations. In some cases considered in this work, $N$ is very small and the determination of uncertainties on the fitted is difficult (and without much meaning in terms of the formal, statistical approach). Therefore we did not perform a formal error analysis of the results. Nevertheless, in a well performed optimization search the error analysis is as important as the determination of the fit parameters.
To check the possible degeneracy of the solutions and to obtain an idea about the errors, we collected the values to which the search process converged in many independent runs with the same start, and projected them on chosen parameter planes. The distribution of these parameters gives insight on the significance of the minima, whether they can be well localized and fixed in the parameter space of the problem. In Fig.\[ltfits\] we present an example of fitting the photoionization model for a single PN (He2-113, PNG321.0+03.9). Results are plotted for 40 different runs of PIKAIA, using the same set of observables. A scatter in the obtained parameters is seen: few of the points are overlapping. The size of the symbols corresponds to the fit quality. For this PN the temperature is better constrained than the stellar luminosity. The situation for other PNe is similar. For higher temperatures the models seem to be better constrained. We estimated the accuracy in $T_{\rm b-b}$ as $\pm 2000 \rm K$ and in $\log L/L_{\odot}$ as $\pm 0.3$.
Similar test were performed for the velocity procedure. When fitting a simple velocity field (linear or parabola-like) the PIKAIA routine repeats very well the results. More complicated velocity fields result in different shapes for different runs; however the largest discrepancies appear in the least constrained parts of the velocity curves. Nevertheless the mass-averaged expansion velocity remains well determined and is similar in all runs. From the distribution of models, we estimate the accuracy of $V_{\rm av}$ as about $\pm 2{\rm km\,s^{-1}}$.
Observations
------------
![The observed and modelled lines $\ion{H}{i}$ 6563Å and \[$\ion{N}{ii}$\] 6583Å. The circles correspond to the observed profile, the line to the fitted model. The line fluxes are normalized to unity. The X-axis velocity scale given in the lowest boxes is the same for all plots. []{data-label="li_1"}](figure2.ps){width="88mm"}
![The observed and modelled lines $\ion{H}{i}$ and \[$\ion{N}{ii}$\]. Continuation of Fig.[\[li\_1\]]{}[]{data-label="li_2"}](figure3.ps){width="88mm"}
![The observed and modelled lines $\ion{H}{i}$ and \[$\ion{N}{ii}$\]. Continuation of Fig.[\[li\_2\]]{}[]{data-label="li_3"}](figure4.ps){width="88mm"}
A large number of PNe have been analyzed in previous papers using the Torun models. In most cases the velocity fields were determined from three lines (hydrogen H$\alpha$ 6563Å, \[\] 6583Å and \[\] 5007Å) or fewer. To this sample we add in the present paper 27 unpublished CAT observations, which cover only H$\alpha$ and \[\] 6583Å, supplemented with recent echelle NTT observations of 6 PNe with emission-line central stars. PNe with new data are listed in Table 1.
The ESO Coudé Auxiliary Telescope (CAT) was a subsidiary 1.4m telescope feeding into the Coudé Echelle Spectrograph (CES) located at the neighboring 3.6m telescope. The CAT observations of 27 objects were performed during 1993 and 1994. The long camera was used, giving a spectral resolution of 60000 (corresponding to 5kms$^{-1}$). The slit width was 2. The observations use a long slit, sampled with 2 pixels. However, the spectra analyzed here use only the central row of pixels. (As the nebulae studied here are compact and the CAT was not an imaging-quality telescope, no spatial information was expected.) The spectrum covers one order of the echelle, covering H$\alpha$ and the \[\] lines at 6548Å and 6853Åbut no other nebular lines.
Six objects with emission-line central stars were observed with the ESO New Technology Telescope (NTT) during June 2001. The echelle spectra were obtained using ESO Multi Mode Instrument (EMMI), with grating 14 and cross disperser 3, giving a resolution of 60000. The slit width was 1. The spectra covered the wavelength range 4300–8100Å. The spectra were summed over the slit length of 3. Exposure times were typically 120 seconds. Between 3 and 7 lines per object showed high enough S/N to be used in the velocity analysis. The details of the data reduction are given in Gesicki & Zijlstra (2003).
The profile fitting procedure requires specification of the center of the emission line with zero velocity. We did not perform the full radial velocity correction. Instead we assumed the zero position in the centre of symmetry for the almost symmetric lines and usually located the zero position in the centre between the two maxima of the asymmetric lines. In the second case the presence of two (or more) spectral lines helped to locate the zero position, the obviously hopeless cases were not analyzed.
The modelled planetary nebulae
==============================
[ l l r r r l l r r r r l l ]{} PNG & Name & $\log\,T_{\rm b-b}$ & $\log {L} $ & Dist. & R$_{\rm out}$ & M$_{\rm ion}$ & $V_{\rm av}$ & $V_{trb}$ & $M_{core}$ & $t_{dyn}$ & Central & Remarks\
& & \[K\] &\[L$_\odot$\] & \[kpc\] & \[pc\] & \[M$_{\odot}$\] & & \[M$_{\odot}$\] & \[kyrs\] & star &\
000.9-04.8 & M 3-23 & 5.18 & 3.0 & 4.0 & .11 & .10 & 24 & 13 & 0.61 & 5.1 & &\
002.6-03.4 & M 1-37 & 4.40 & 3.9 & 8.0 & .04 & .08 & 27 & 0 & 0.60 & 1.7 &\[WC11\]?&\
002.7-04.8 & M 1-42 & 4.92 & 3.0 & 4.0 & .08 & .16 & 13 & 0 & 0.60 & 5.4 & &\
003.6+03.1 & M 2-14 & 4.64 & 3.0 & 8.0 & .05 & .06 & 17 & 10 & 0.60 & 2.9 & wels & 6 lines\
004.6+06.0 & H 1-24 & 4.56 & 3.8 & 7.0 & .08 & .13 & 24 & 0 & 0.58 & 3.7 & wels &\
006.4+02.0 & M 1-31 & 4.76 & 3.8 & 8.0 & .06 & .29 & 19 & 0 & 0.61 & 3.2 & wels &\
008.2+06.8 & He 2-260 & 4.80 & 3.2 & 12.0 & .06 & .13 & 17 & 0 & 0.61 & 3.5 & &\
211.2-03.5 & M 1-6 & 4.60 & 3.5 & 4.0 & .03 & .05 & 24 & 0 & 0.62 & 1.4 & &\
217.4+02.0 & St 3-1 & 4.96 & 2.5 & 4.0 & .14 & .20 & 26 & 0 & 0.60 & 6.1 & &\
232.0+05.7 & SaSt 2-3 & 4.80 & 2.5 & 4.0 & .02 & .01 & 19 & 0 & 0.64 & 1.1 & &\
232.4-01.8 & M 1-13 & 5.04 & 2.9 & 4.0 & .10 & .19 & 14 & 0 & 0.60 & 6.5 & &\
253.9+05.7 & M 3-6 & 4.72 & 4.0 & 1.5 & .036& .03 & 18 & 0 & 0.61 & 2.0 & wels &\
258.1-00.3 & He 2-9 & 4.73 & 2.9 & 1.5 & .015& .01 & 25 & 0 & 0.65 & 0.7 & wels &\
261.6+03.0 & He 2-15 & 5.17 & 3.0 & 2.2 & .13 & .32 & 22 & 0 & 0.71 & 6.4 & &\
283.3+03.9 & He 2-50 & 5.12 & 2.9 & 5.1 & .14 & .32 & 17 & 0 & 0.59 & 8.1 & &\
292.4+04.1 & PB 8 & 4.76 & 3.6 & 5.0 & .06 & .18 & 22 & 9 & 0.61 & 2.9 &\[WC5-6\]&\
300.7-02.0 & He 2-86 & 4.83 & 3.4 & 3.0 & .03 & .05 & 14 & 11 & 0.62 & 2.0 &\[WC4\] &\
304.8+05.1 & He 2-88 & 4.72 & 2.8 & 4.0 & .03 & .03 & 23 & 0 & 0.62 & 1.4 & &\
309.0-04.2 & He 2-99 & 4.49 & 2.7 & 4.0 & .16 & .25 & 45 & 10 & 0.57 & 4.6 & \[WC9\] &\
316.1+08.4 & He 2-108 & 4.51 & 3.8 & 4.0 & .10 & .14 & 21 & 9 & 0.57 & 5.0 & wels? &\
321.0+03.9 & He 2-113 & 4.48 & 2.8 & 2.0 & .01 & .003& 18 & 15 & 0.64 & 0.6 & \[WC10\]&\
325.0+03.2 & He 2-129 & 4.87 & 3.8 & 7.0 & .03 & .08 & 18 & 13 & 0.63 & 1.7 & &\
325.8+04.5 & He 2-128 & 4.70 & 3.3 & 5.0 & .06 & .13 & 12 & 10 & 0.59 & 4.3 & &\
347.4+05.8 & H 1-2 & 4.96 & 3.9 & 7.0 & .02 & .06 & 13 & 10 & 0.64 & 1.4 & wels & 3 lines\
351.1+04.8 & M 1-19 & 4.72 & 3.2 & 11.0 & .10 & .20 & 26 & 0 & 0.61 & 4.3 & wels & 5 lines\
352.1+05.1 & M 2-8 & 5.11 & 3.2 & 7.0 & .06 & .16 & 14 & 14 & 0.61 & 3.9 &\[WO2-3\]& 7 lines\
355.9+03.6 & H 1-9 & 4.58 & 4.0 & 9.0 & .04 & .09 & 20 & 0 & 0.61 & 2.1 & &\
356.1+02.7 & Th 3-13 & 5.05 & 3.3 & 8.0 & .03 & .05 & 32 & 0 & 0.65 & 1.1 & wels & 3 lines\
356.5-02.3 & M 1-27 & 4.41 & 3.6 & 3.0 & .04 & .05 & 21 & 0 & 0.60 & 2.0 &\[WC11\]?&\
357.1+03.6 & M 3-7 & 4.72 & 3.2 & 4.0 & .05 & .07 & 23 & 0 & 0.61 & 2.4 & wels & 3 lines\
357.4-03.2 & M 2-16 & 4.95 & 3.3 & 7.0 & .09 & .25 & 21 & 14 & 0.61 & 4.5 & &\
357.4-03.5 & M 2-18 & 4.64 & 3.5 & 7.0 & .06 & .09 & 24 & 0 & 0.60 & 2.8 & &\
359.1-02.3 & M 3-16 & 4.57 & 3.8 & 7.0 & .08 & .14 & 30 & 0 & 0.59 & 3.1 & &\
\[lista\]
Figures \[li\_1\]–\[li\_3\] present the observed and modelled nebular emission lines $\ion{H}{i}$ 6563Å and \[$\ion{N}{ii}$\] 6583Å. Although for the indicated in the last column of the Table\[lista\] six PNe more lines were used for model fitting (\[\] 5007Å and in some cases \[\] 6302Å, \[\] 6732Å, \[\] 6311Å, 4686Å), we present only the two lines for a consistent presentation. We do not draw the model radial distributions of the velocity, density and surface brightness because our main results are the mass-averaged expansion velocities. The \[WR\]-subclass and [*wels*]{} classification is adopted from Acker & Neiner (2003) and for the lowest temperature \[WC11\] stars from Górny et al. (2004). The summary of nebular and central star parameters are listed in Table\[lista\]. The last column indicates whether more than two lines were used for modelling. A few objects are discussed in the following.
[*PNG002.6-03.4 (M 1-37), \[WC11\]“?”)*]{}. The density structure is adopted to agree approximately with the image of Sahai (2000). The brightest nebular structure is the inner ellipsoidal ring $1.5 \times
2.5\arcsec$ which is modelled by our $2\arcsec$ sphere. The multipolar lobes as well as the extended spherical halo are beyond our analysis. We obtained a lower temperature than Zhang & Kwok (1991) for this low excitation nebula.
[*PNG253.9+05.7 (M 3-6, wels)*]{}. The radio image (Zijlstra et al. 1989) reveals a complicated structure with four maxima. Our spherical model is density bounded, but the line ratios are not well fitted and suggest a mixture of ionization and density boundaries. A small central cavity is indicated. The best velocity field which reproduces the multicomponent structure of the \[\] line shows multiple maxima. Considering the complicated nebular image our model should be treated with caution, but because the spectral lines are almost symmetric the mass-averaged expansion velocity is believed to be reliable.
[*PNG321.0+03.9 (He 2-113, \[WC 10\])*]{}. The HST image (Sahai et al. 2000) reveals complex, highly aspherical structures which makes spherical modelling doubtful. Nevertheless the average velocity still is a useful parameter. In our observations this PN is unresolved and the emission lines are symmetric.
Derived parameters
==================
The newly analyzed data is combined with an earlier sample discussed in Gesicki et al. (2003). Together with the 33 new PNe, the full sample contains 101 objects (from 73 earlier objects we removed three extragalactic PNe, and two PNe are reanalyzed with the new spectra). The full sample contains 23 \[WR\], 21 [*wels*]{} and 57 non-emission-line central stars.
Several objects were reclassified based on the work of Acker & Neiner (2003) and Górny et al. (2004). The distinction between emission-line stars and non-emission-line stars depends on the depth of the available spectra. It is therefore possible that some of the non-emission-line stars would be classified as [*wels*]{} stars with deeper spectra. The effect of detectability of the stellar lines on the classification is discussed by Górny et al. (2004).
Temperatures {#temps}
------------
![Stellar photo-ionization temperatures versus \[WO\]-\[WC\] subclass. []{data-label="wct"}](figure5.ps){width="88mm"}
The subclass of the \[WR\] stars provides an indication of stellar temperature, but it can not be used to predict unique temperatures. This can for instance be seen for the earlier subclasses (hotter stars) from the data in Acker & Neiner (2003). De Araujo et al. (2002) show that deep spectra are needed for accurate association of a \[WR\] subclass. In this paper we use the photoionization (equivalent black-body) temperature $T_{\rm b-b}$, rather than the \[WR\] subclass. This parameter has the further advantage that it is also available for the non-\[WR\] stars. Fig.\[wct\] shows the relation between photoionization temperature and \[WR\] (\[WO\] and \[WC\]) subclass. There is a good relation overall, but the change in temperature is smaller for the later subclasses (8–11). One discrepant object is M2-43 (27.6+04.2) where the temperature is much higher than the subclass (\[WC 7-8\]) would suggest. The photo-ionization model fits the line ratios reasonably well, and the strong \[\] line with weak \[\] require a high temperature (Acker et al. 2002). Such high temperature was also obtained by non-LTE model analysis of Leuenhagen & Hamann (1998). This object shows a dual dust composition (Górny et al. 2001). The \[WC\] subclass should be checked.
Expansion velocities
--------------------
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For the whole sample of 101 PNe, the average (and the median) value of the mass-averaged expansion velocities is $22\, {\rm km\,s^{-1}}$ with standard error on the mean of $0.67 {\rm km\,s^{-1}}$. The average $V_{\rm av}$ for \[WR\] PNe is $25\pm 2\, {\rm km\,s^{-1}}$ while for [*wels*]{} it is $22\pm
1\, {\rm km\,s^{-1}}$. The 1-$\sigma$ difference in expansion velocity is much smaller than found in Peña et al. (2003a), who find an average expansion velocity for non-WR PNe of $21\,\rm km\,s^{-1}$ (virtually the same as ours) but for \[WR\] PNe $36\,\rm km\,s^{-1}$. The difference can be explained by their method of estimating the velocity: they use the half-width at half maximum of the nebular lines and this is affected by the turbulence in the velocity fields (Acker et al. 2002).
It is natural to explain the higher than average expansion velocity of \[WR\] PNe in terms of stronger stellar wind acting on the swept-up shell. Mellema & Lundqvist (2002) studied such interactions and confirmed the above tendency providing an example of \[WR\] PN expansion of $21\,\rm km\,s^{-1}$ versus $17\,\rm km\,s^{-1}$ for non-\[WR\] PN. This difference is smaller than obtained by Peña et al. (2003a), the values are rather comparable to ours.
Peña et al. (2003b) argue that the expansion velocities for evolved \[WR\] PNe are larger than for younger objects. In that paper the expansion velocity is defined as half-width at 1/10 of the maximum intensity, a measure introduced by Dopita et al. (1988). At this intensity level the emission lines are strongly broadened by a turbulent component common for \[WR\] PNe so it is no surprise that our values (corrected for the turbulence) are smaller. Peña et al. (2003b) arrive at their conclusion by plotting stellar temperatures versus expansion velocities. In our sample, no such effect is visible (Fig.\[velocities\], left panel), except that the three highest expansion velocities in the sample are from \[WR\] stars. The $V_{\rm exp}$ and $T_{\rm b-b}$ values are spread over the whole range. However, there is a trend for the \[WR\] stars to dominate towards the lower-right corner in the figure, i.e. lower values of $T_{\rm
b-b}/V_{\rm exp}$. This trend looks very similar to that from Fig.3 of Peña et al. (2003b). However their \[WR\] PNe clustering in the upper-right corner is not so similar.
Schönberner et al. (2005a) discuss their observations of 13 PNe and concluded that expansion speed increases with the $T_{\rm eff}$ of the central star, i.e. with the nebular age. Such a situation cannot be explained with simple models which assume an initial nebular density distribution described by a power law $\rho \propto r^{-\alpha}$ with a fixed index $\alpha$. Their conclusion is that the power-law index increases systematically with $T_{\rm eff}$ (see their Fig.12). We plot their model sequences for four values of the power law index in the left panel of Fig.\[velocities\]. The indexes are from left to right: 2.00, 2.5, 3.0 and 3.25. Our data for 101 PNe overlap with the lines (the velocities are not defined exactly in the same way). We do not confirm the Schönberner et al. (2005a) conclusion, but the range of power law indices which they derive provides a good fit to the spread seen in our sample. The trend in Fig.\[velocities\] that \[WR\] stars dominate the distribution towards lower values of $T_{\rm b-b}/V_{\rm exp}$, in combination with the slope of the model tracks, suggests that some \[WR\] PNe may have steeper initial density distributions than non-emission-line PNe.
Dynamical ages
--------------
The expansion velocity and radius of the nebula can be used to derive the age of the nebula. One can use the outermost radius and the maximum expansion velocity, but this is affected by the acceleration caused by the overpressure in the ionized region. Instead we apply the mass-averaged velocity, and use for the radius 0.8 of the outer radius, roughly corresponding to the mass-averaged radius. To account for the acceleration of the nebula we average between the current velocity and that of the original outflow velocities on the Asymptotic Giant Branch. The procedure is described in Gesicki et al. (2003). We made a simplifying assumption about the AGB outflow velocity setting it equal to $10\,\rm km\,s^{-1}$ for all objects.
The dynamical ages can directly be compared between the different groups of stars. However, they may not be equal to the true ages of the nebulae, due to the various assumptions made. This problem was discussed together with non-LTE analysis of some central stars of PNe. Rauch et al. (1994) obtained for the object K1-27 that the dynamical age was more than one order of magnitude smaller than the evolutionary age. In another paper Rauch et al. (1999) analyzed four more PNe and obtained dynamical ages a few times larger than evolutionary ages. The nebular and stellar ages were compared on a much bigger sample by McCarthy et al. (1990). They obtained that generally the empirical ages were larger than the evolutionary ones however their method for estimating dynamical ages was recently strongly criticized by Schönberner et al. (2005b). The contemporary situation is far from being conclusive.
In the recent article we found a nice support for our method for estimating nebular ages. Schönberner et al. (2005b) show that both radii and velocities increase gradually with time what justifies our averaging between original and current velocity. For ionization bounded models (dominant in our sample) the ages derived from ions \[\] and \[\] diverge in opposite directions from true ages what supports our multi-line analysis where such effects compensate. For density bounded PNe the situation is even better since both lines result in the same age equal to the evolutionary one.
The dynamical age in combination with the stellar temperature gives the rate at which the central star is increasing in temperature. This rate is very sensitive to the core mass of the star, with more massive stars evolving much faster. To assign a core mass, we interpolate between published tracks (see e.g Górny et al. 1997, Frankowski 2003). For a given dynamical age and stellar temperature we obtain the core mass (and luminosity). The dynamical ages and core masses are listed in Table\[lista\]. As for the ages, the mass determinations are internally consistent between different objects, but may show systematic offsets with respect to the true masses. As a check of the consistency of the procedure the luminosities can be compared. In Gesicki et al. (2003) we obtained that the luminosities from photoionization modelling are in most cases smaller than those interpolated from evolutionary tracks. However this can be interpreted as leaking of the nebulae: the PN does not intercept and transform all stellar ionizing flux. This points to asymmetry or clumping.
Turbulence
----------
For a number of nebulae, satisfactory fits to the velocity fields could not be obtained. In these cases we used enhanced turbulence, above what is expected from thermal broadening and instrumental resolution. Indicative of turbulence is that lines of all ions show Gaussian shapes of comparable width: thermal broadening gives much wider lines for the lightest element (hydrogen). Comparing the H$\alpha$ with metal lines is here crucial because differences in thermal broadening between different metal ions are small. Therefore we searched for turbulent solutions only for those PNe with H$\alpha$ and at least one metal line available. Turbulence was indicated for 30 of the 101 nebulae in the full sample.
Gesicki & Acker (1996) and Acker et al. (2002) find that \[WR\]-type PNe are characterized by strong turbulent motions while non-emission-line PNe are not. This general tendency is confirmed in the present analysis, but there are some exceptions.
Among the [*wels*]{} we find both turbulent and non-turbulent objects. Our full sample contains 21 [*wels*]{} of which 5 show turbulence and 16 do not. Of these 21 objects, 10 come from the new observations described here (3 turbulent and 7 non-turbulent).
We also find well-determined, turbulent solutions for 4 PNe not classified as \[WR\] nor [*wels*]{} (M2-16, M3-23, He2-128, He2-129). Not all \[WR\]-type PNe in the new sample show turbulent motions: M1-27 and M1-37 have good solutions, without requiring turbulence. The two non-turbulent \[WR\] PNe are both classified by Górny et al. (2004) as ’\[WC11\]?’. They are the coolest objects in our sample. However, the strong correlation between emission-line stars and turbulence remains: the fraction showing turbulence for non-emission-line stars, [*wels*]{} and \[WR\] stars is 7%, 24% and 91%, respectively.
Uncertainties in the \[WR\] identifications may affect these fractions. The turbulent non-WR PNe could have emission-line central stars. This is unlikely for M2-16, a well-studied, highly metal-rich PN (Cuisinier et al. 2000). He2-128 and He2-129 were studied spectroscopically by Dopita &Hua (1997): the non-WR identification cannot be attributed to a lack of observations, but faint stellar lines may have been missed. Some objects may have too few nebular lines to be able to detect turbulence: a velocity gradient broadens a single line profile similar to turbulence – we only accepted turbulent solutions where they fitted clearly better the line shapes than non-turbulent ones. Deeper spectra would help here.
The onset of the turbulence and the \[WR\] phenomenon may not be completely simultaneous. However, we find no evidence that the turbulence increases for more evolved objects: Fig.\[velocities\] (right panel) shows no relation between turbulence and stellar temperature.
In trying to understand why the presence of turbulence in [*wels*]{} PNe is so mixed, let us consider a possibility that a PN can switch between turbulent and non-turbulent regimes during its lifetime. The transition time for this should be comparable to the shock travel time across the nebula. Assuming a radius of 0.1pc and isothermal sound speed of $10\,{\rm km\,s^{-1}}$ we obtain a time of $10^4$ years. Perinotto et al. (2004) stated that the shock front velocity can be 3–4 times faster than the isothermal sound speed: this reduces the time to a few thousands years. This is comparable to the ages of our PNe. In our sample we have a couple of (compact) objects with an age of about 1000 years which have already developed turbulence. Acker et al. (2002) proposed that turbulence in PNe is triggered or enhanced by \[WR\] stellar wind inhomogeneities. The nebulae surrounding very active \[WR\] cores are turbulent. The less active [*wels*]{} with much weaker winds, if they could change their wind characteristics, they would develop turbulence, or exhibit decaying turbulence, over the lifetime of the PN. Thus, the mixed occurrence of turbulence in [*wels*]{} is compatible with a transitory status, not unlikely a \[WR\]-star progenitor, provided the life time of the phase does not exceed $\sim 10^3\,$yr.
IRAS colours
------------
{width="12cm"}
Planetary nebulae are strong infrared emitters, due to their heated dust. The dust colours follow the evolution: as the nebula expands, the dust cools and the dust colours redden. Zijlstra (2001) has shown that \[WR\] stars are stronger infrared emitters than other PNe, with a tendency for bluer colours. The brightest infrared PNe are the IR-\[WR\] stars, a subgroup of the cooler \[WC\] stars.
Fig.\[iras\] shows the colour-colour diagram for the present sample. Different symbols distinguish the \[WR\] stars (filled circles), the [*wels*]{} (open circles) and the non-emission-line stars (pluses). In addition, overplotted star symbols indicate turbulent nebulae. We excluded the confused source M2-6 where the IRAS colours appear to be those of a nearby AGB star. The boxes are source classification regions of Zijlstra et al. (2001). Planetary nebulae fall in box II. The cooler \[WC\] stars are found mainly towards the left in the diagram, in three cases with colours like those of (young) post-AGB stars. (The three objects are BD +303639, M4-18 and He2-142.) The [*wels*]{} are shifted to redder colours, and their distribution is similar to those of normal PNe. Note that the cooler non-emission-line stars in our sample have 60/25-micron colours similar to the cool \[WC\] PNe, but their 25/12-micron colours are very different.
We constructed dust models following the nebular expansion occurring as the star heats up. We used a $r^{-2}$ wind, with the model and recipe of Siebenmorgen et al. (1994). The model calculates the IRAS fluxes at any particular time. The results are shown in Fig.\[iras\], for a carbon rich (long dashes) and a oxygen rich (short dashes) nebular dust. The evolutionary progression from cool stars with bluer IRAS colours to hot stars with (more evolved) redder IRAS colours is visible for the \[WR\] stars , but not for the non-emission-line stars and the (less populated) [*wels*]{} group (Zijlstra 2001).
Detection statistics show the differences between the groups. Out of the 101 stars, 60 have three good IRAS detections. The detection rates are 28 out of 57 non-emission-line stars, 12 out of 21 [*wels*]{} and 20 out of 23 \[WR\] stars: respectively 49%, 57% and 87%. This shows that the \[WR\] stars are on average much stronger IRAS emitters. If we only consider 25 and 60 micron, the detection rates become 89%, 86% and 100%, i.e. the difference is rather reduced. The numbers indicate that \[WR\] stars are much brighter at 12-micron, but less so at the other bands.
The [*wels*]{} have the same detection rate as other non-\[WR\] stars. However, all 5 [*wels*]{} with turbulence have three-band detections; their colours are similar to the \[WO\] stars. All [*wels*]{} without 12-micron detection are non-turbulent objects.
Evolutionary parameters
=======================
Temperature distributions
-------------------------
![Temperature distribution of \[WR\], [*wels*]{} and non-emission-line stars, in logarithmic temperature bins. The bottom panel, discussed further in the text, compares the observed data with the histogram obtained from H-burning evolutionary models of Blöcker (1995).[]{data-label="temp_lin"}](figure8.ps){width="88mm"}
The temperature distribution of the three groups of stars, defined as in section \[temps\], is shown in Fig.\[temp\_lin\]. The \[WR\] stars show some clustering at low and high temperatures with a gap in between (see upper panel). This gap has been noted before long time (see e.g. Górny 2001). The [*wels*]{} stars, in contrast, peak precisely in this gap. In contrast, the non-emission-line stars show a peak in their distribution at high temperature (see central panel).
Summing the two groups of emission-line stars (central panel), we find a flat distribution (in logarithmic bins!). Summing all three groups together (bottom panel, drawn line) shows an increasing number of objects with $\log
T$.
We compare the temperature distribution of the various PNe samples with that predicted by the evolutionary tracks of Bloöcker (1995). We interpolate the tracks to constant temperature steps, in order to calculate the predicted number of stars per temperature range (proportional to the time spent in the range). The model tracks have to first order a fairly constant temperature increase, $dT/dt$. In logarithmic bins, this means that the number of stars per bin increases with temperature. To be consistent with our model analysis we only use the the horizontal part of the evolutionary track, before the knee in the H-R diagram. Fig.\[temp\_lin\] (bottom panel, dotted line) shows the predicted distribution for hydrogen burning $0.605 {\rm M}_{\odot}$ track of Blöcker (1995). The predicted number of objects (per logarithmic temperature bin) gradually increases towards higher temperatures, as expected from the increasing width of the bins and from slowing the evolution.
When compared to the observational data we see that no single group of PNe produces a histogram similar to the models. The solid line in the bottom panel of Fig.\[temp\_lin\] shows that the sum of all classes produce a histogram most similar to the theoretically expected distribution. Thus, the full sample is representative for a uniformly sampled Blöcker track. The flat distribution of the emission-line stars implies that this group is biassed towards lower temperatures, relative to standard evolution. The [*wels*]{} as a separate group show a strong temperature preference and cannot represent a full evolutionary track.
It is possible that our observations misrepresent the oldest PNe with the hottest cores. Our sample is mainly restricted to the spherical PNe with an ionization boundary while for old PNe the ionization front breaks through and may cause fragmentation of a nebula making it not suitable for our analysis. We also cannot exclude a systematic error of our $T_{\rm b-b}$ values which may shift the maximum of the observed distribution, and this error may also depend on temperature.
Dynamical ages of the PNe
-------------------------
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Evolutionary tracks should connect objects along sequences of increasing dynamical age and increasing stellar temperature, until the stars reach the knee in the HR diagram, after which the stellar temperature slowly decreases. Helium-burning tracks generally also follows this sequence, but the stars evolve a few times slower (factor of 3 according to Blöcker 1995); the details depend on the precise timing of the thermal pulse preceding the switch to helium burning.
Fig.\[evo\] plots the dynamical age versus the temperature for the stars in our sample. The interpolated (hydrogen-burning) evolutionary tracks are overplotted with a solid line. As in Fig.\[iras\] different symbols distinguish types of stars and indicate turbulence.
The figure shows that some regions in this age-temperature plane are dominated by certain types of objects. The \[WR\] stars are mainly found in the narrow region between the 0.605 and 0.625M$_\odot$ tracks, with a few further objects at low temperatures and a range of ages. The [*wels*]{} stars are spread mostly out at intermediate temperatures. The non-emission-line stars tend to avoid the low temperature region, but cover a region in the top-right corner which the other stars tend to avoid.
The statistics of our full sample of 101 PNe shows that the average dynamical age of non-emission-line central stars and of [*wels*]{} is 3200 years while for \[WR\]-stars it is 2800 years, with standard error on the mean of about 300 years in both cases. This difference is not significant and the result may be contaminated by a selection effect because it is easier to detect a \[WR\] star at lower temperatures (see Górny et al. 2004).
Non-emission-line objects
-------------------------
In Fig.\[evo\] the non-emission line objects are mixed with the emission-line ones except of the upper right area. This is the place where the cooling parts of evolutionary tracks of massive stars cross with heating parts of evolutionary tracks of low mass stars. The hot ($10^5$K), non-emission-line stars with old nebulae could be high-mass stars located on the cooling track. The alternative interpretation is that they have lower-mass central stars. In either case, the absence of emission-line stars in this region is obvious. Either a minimum mass is required for a hot \[WO\] star of about 0.61M$_\odot$, or the stellar wind ceases when the star reaches the cooling track. The latter is plausible since the luminosity drops quickly by a factor of 100 at this time. This fact is well known for hot star winds (see e.g. Owocki 1994, Owocki & Gayley 1995). Koesterke & Werner (1999) show that PG1159 stars have much weaker winds than [*wels*]{} stars, and suggest that the wind mass-loss rate decreases rapidly when the star enters the cooling track.
Nebular evolution
-----------------
Schönberner et al. (2005a) conclude that during the evolution the PN material always enters the ’champagne phase’ where the ionization front breaks through and the PN becomes density bounded. This disagrees with our result that most of the 101 PNe are ionization bounded despite their age and core $T_{\rm eff}$. In our sample the presence of the ionization boundary was deduced from the strengths of the low-ionization low-excitation lines. A possible explanation of this disagreement can be found in Gesicki et al. (2003) where we compared the model luminosities with those obtained from evolutionary tracks and suggested that ’PNe are leaking’. This indicates that PNe are a mixture of regions transparent and opaque for ionizing radiation. The spherical Schönberner’s and Torun models cannot investigate asymmetric or clumpy nebulae. However there is another explanation - the ionization boundary is not so exceptional. The evolutionary sequence No.4 of Perinotto et al. (2004), with high AGB mass loss rate and low AGB wind velocity, never becomes optically thin and can serve here as a hint. It seems to be promising to follow this idea and to search for other hydrodynamical models that remain ionization bounded for a significant part of PN evolution. In Gesicki & Zijlstra (2003) and Zijlstra et al. (2005 in preparation) we presented models supported by HST images showing very small central cavities of the nebulae. This finding supports the above idea of a low AGB wind velocity, at least at the end of AGB and for some PNe. Finding an appropriate hydrodynamic model might constrain the still unknown AGB mass loss history.
The hydrogen-poor sequences
===========================
The \[WR\] stars
----------------
A global nebular quantity, turbulence, is predominant among the \[WR\] stars. Mellema (2001) links the metals-enriched \[WR\] winds with long lasting momentum-driven phase which drives the turbulence. This confirms that they form a distinct group, and it can be interpreted as evidence for a separate different evolutionary channel. Fig.\[iras\] and Fig.\[evo\] show that there is a marked uniformity among the majority of the \[WR\] stars. Fig.\[evo\] suggests this is related to the core mass of the star, where stars of a relatively narrow range in core mass account for the majority of the \[WR\] stars. Caution should be taken when quoting precise masses: the location in the age-temperature diagram depends on modeling assumptions as discussed above.
The \[WR\] stars are in general hydrogen-poor, and are considered helium-burning central stars. These stars will show a slower temperature increase than do hydrogen-burning tracks of the same mass, although the detailed evolution is complicated. The distribution of \[WR\] stars in Fig.\[evo\] is not inconsistent with such an inclined track, but strong conclusions are not warranted.
There are three \[WC\]-PNe located near the 0.565M$_{\odot}$ track: from the left M4-18, He2-99 and NGC40. All are old nebulae with cool stars. The first is determined as old because of small expansion velocity, the other two PNe are large ($\approx$0.15pc) as expected for old objects. Their three central stars were analyzed by Leuenhagen et al. (1996). All have long been suspected to be born-again PNe, where the central stars have suffered a Very Late Thermal Pulse (VLTP), leading to a rejuvenated star inside an old nebula (e.g. Hajduk et al. 2005). Helium-burning stars are not expected to experience another VLTP. The stars along the VLTP sequence therefore represent a separate evolutionary channel from the majority of the \[WR\] stars. The difference may largely be one of timing, whether the last thermal pulse takes place along the horizontal sequence (LTP), or on the cooling track (VLTP). The only non-emission-line star in this region is the halo object PNG359.2-33.5, which could well be a genuine low-mass object (McCarthy et al. 1991). The [*wels*]{} in this region is He 2-108 (its classification is uncertain).
There is a group of 7 \[WC\] stars which combine very compact nebulae (low ages) with very cool central stars. This group has temperatures in a narrow range around $\log T_{\rm b-b} = 4.7$, well separated from hotter \[WR\] objects (see Fig.\[wct\] and Fig.\[evo\]). The IR-\[WC\] stars and the mixed-chemistry objects are found here (Zijlstra 2001). It is not clear that these belong to either of the two groups discussed above. The mixed chemistry has been associated with binary evolution (Zijlstra et al. 2001, de Marco et al. 2004), and in one case (CPD$-$568032 — Cohen et al. 1999, de Marco et al. 2002), there are strong indications for binarity. The precise role of a binary companion is not clear, but it may involve retaining old ejecta in a disk, triggering accretion and possibly providing a source of ionization. They could have similar origins to the R Cor Bor stars (see the review of Clayton 1996), although the very different circumstellar environments suggest the relation is not an evolutionary one.
This group of cool, compact objects could in principle have descendants among the hot \[WO\] stars, which have more evolved nebulae. However, the mixed chemistry (i.e. evidences for both a neutral oxygen-rich and an inner carbon-rich regions) and unique IRAS colours argues against this. But there are no obvious descendants among the non-emission-line stars either, leaving the evolution of this group unexplained. If the stars are accreting, the extended, cool photosphere could hide a much hotter underlying star: this would explain the lack of intermediate-temperature related stars, but still leaves the question of descendants open.
The gap in \[WR\] temperature distribution
------------------------------------------
The lack of PNe cores of \[WC 5–7\] type is well known (Crowther et al. 1998, Górny 2001). This feature can also be seen in our temperature distributions presented in Fig.\[wct\] and Fig.\[evo\]. One of the two \[WR\] star filling this gap is PB8 (PNG292.4+04.1) of \[WC5-6\] type (Acker & Neiner 2003). However by Parthasarathy et al. (1998) it was classified as [*wels*]{} and Mendez (1991) found it as H-rich. In Fig.\[iras\] this object is positioned among [*wels*]{} stars. Therefore we propose that PB8 is a [*wels*]{} and no longer fills the \[WR\] temperature gap. Another object, the only known \[WC 6\] type - M1-25 (PNG004.9+04.9) fits the centre of the gap. The Torun models resulted in a stellar temperature $\log T_{\rm b-b} = 4.6$ (Gesicki & Acker 1996). However much higher temperatures have been published for the M1-25 central star, reaching $\log T_{\rm b-b} = 4.9$ (Samland et al. 1992) based on the helium ionization. In Fig.\[iras\] this object is placed amongst other regular \[WR\] objects. Therefore we propose that M1-25 is a \[WC6\] object (we don’t question the WR classification) but its temperature may be above the discussed gap.
The uncertain status of the interlopers strengthens the case that the gap in the temperature distribution of \[WR\] stars is real and quite prominent: the region $4.55 < \log T_{\rm b-b} < 4.8$ appears devoid of \[WR\] stars. This supports the proposed separate evolutionary sequences of cool and hot \[WR\] stars. The same conclusion was reached by Crowther et al. (1998) who applied a unified classification scheme to massive WR and low mass \[WR\] stars and realized that the mentioned above gap is not visible for WR objects.
The wels
--------
The [*wels*]{} appear to be a mixed zoo. We tentatively identify three groups. First, a group which shows similarities in distribution in Fig.\[evo\] to the \[WR\] stars. These include one object among the VLTP sequence, and approximately 5 objects which form a low-temperature extension to the hot \[WO\] stars and may be their progenitors. Second, around 6 [*wels*]{} are found along the high-mass track ($>0.625\,\rm M_\odot$). These we suggest are objects where the high stellar luminosity drives a stellar wind–they may be H-rich and not related to the \[WR\] stars. Finally, a group along a low-mass track ($M\approx 0.6\,\rm M_\odot$). These could represent failed \[WR\] stars, where the luminosity is insufficient to drive a full WR-type wind.
The [*wels*]{} show on average a shift in location in Fig.\[velocities\]: they fall in a region consistent with a lower power index than do \[WR\] stars. The nebulae around \[WR\] stars tend to be more centrally condensed than those of [*wels*]{} stars. This points to a different mass-loss history and is consistent with the suggestion that some fraction, possibly the majority, of [*wels*]{} are not evolutionarily related to the \[WR\] stars.
The small group which may be \[WO\] progenitors suggests a sequence [*wels*]{}–\[WO\], reflecting increasing temperature. This could reflect a bi-stability jump, where the strength of a stellar wind suddenly changes when a change in ionization equilibrium makes different lines act in the absorption of radiation. A similar situation is known among early-type supergiants (e.g. Vink et al. 1999, Tinkler & Lamers 2002). But the latter authors find that among PNe cores, the wind suddenly [*decreases*]{} by a factor of up to 100, precisely at the temperature where we find the [*wels*]{}-\[WO\] transition. They attribute this to a CNO bi-stability jump. Their analysis used stars classified as O(H) only, and could be biassed if hotter stars with high $\dot M$ appear as \[WO\] stars.
For the other stars, the driving factor differentiating [*wels*]{} and \[WR\] stars appears to be luminosity,
The PG1159 stars
----------------
The hot PG1159 stars are generally considered the most natural descendants of \[WR\] stars (Werner 2001). They are not a uniform group: Werner et al. (1996) refer to them as the ’PG1159 zoo’. There is also a group of so-called ’\[WC\]-PG1159 transition objects’ which are found between \[WCL\] and \[WCE\] groups in the $\log g-\log T_{\rm eff}$ plane (Werner 2001). Because our results suggest separate sequences for cooler \[WC\] and hotter \[WO\], one could envisage that cool \[WC\] stars evolve towards white dwarfs via a \[WC\]-PG1159 phase, whilst hot \[WO\] stars do so via a PG1159 phase. However, this remains speculative.
It has been reported that the central star of the Longmore4 nebula changed from a PG1159 type to a \[WC 2-3\], for a time of a few months (Werner et al. 1992). Its absorption spectrum transformed into an emission spectrum, with changes proceeding on a timescale of days. It was interpreted as strongly enhanced mass loss, and may have been triggered by an accretion event. As yet it is the sole documented example of such impressive variability among PN cores, but even so it shows that the picture is a complicated one, where the chaotic picture of [*wels*]{} could be interpreted in terms of stellar wind activity.
Conclusions
===========
We have deduced for 33 PNe the velocity field. Since this was done in most cases by considering two nebular lines we regard as robust the mass averaged value only. For six PNe the analysis was based on three or more lines and [*is*]{} therefore more reliable. The discussion of details of velocity field like e.g. turbulence should be therefore treated with caution while the analysis based on the mass averaged velocity (ages, masses) should be reliable. We combined this sample with earlier published data and discussed the full sample of 101 PNe.
We found no evidence for an increase in nebular expansion velocity with time. This correlation was checked against $T_{\rm b-b}$ which should correspond to stellar age and on the nebular dynamical age i.e. on two independent characteristics. The \[WR\] stars show a tendency to low ratios of $T_{\rm
b-b}/V_{\rm exp}$. Comparison with the Schönberner et al. (2005a) models suggests that these objects have more centrally condensed nebulae, indicative of increasing mass-loss rate with time on the AGB. Emission-line stars are not found on the cooling track.
We find a correlation between IRAS 12-micron excess and the \[WR\] stars. This excess is not shown by the [*wels*]{}. The cause of 12-micron excess is typically a population of small grains. These may form in the wind, or form in the collision between the stellar wind and the nebula. Without exception, the [*wels*]{} do not show the 12-micron excess.
We confirm the stong correlation between turbulence and \[WR\] stars. However, the correlation is not one to one: there are a few non-emission-line stars with turbulence, and a few \[WR\] stars without. The [*wels*]{} appear intermediate in occurrence of turbulence.
The distribution in the age-temperature plane suggests that there are several groups of \[WR\] stars and [*wels*]{}. One group shows evolved nebulae with cool stars and likely originates from a Very Late Thermal Pulse. The hot \[WO\] stars show a narrow distribution corresponding to a small range in core mass. Some [*wels*]{} fall on the same sequence but at lower temperature–they may be the \[WO\] progenitors. Other [*wels*]{} are distributed on higher and on lower mass tracks, which lack \[WR\] stars. Finally, a group of cool \[WC\] stars with IR-bright compact nebulae have no counterpart among any other group of PNe.
The relation between the \[WR\] stars and the [*wels*]{} is therefore a complicated one, with neither group being uniform. For the majority of the [*wels*]{}, there is evidence against them being evolutionarily related to the \[WR\] stars. There is a noticeable apparent difference in core mass and dust properties and nebular turbulence. A small group (5 objects) of [*wels*]{} have characteristics consistent with \[WO\] star progenitors. Apart from this [*wels*]{}–\[WO\] sequence, the [*wels*]{} contain stars at both the high and low luminosity end of the distribution. The former are likely stars where the strong radiation field drives the wind, but are not necessarily H-poor. The latter may be H-poor, failed \[WR\] stars where the luminosity is too low to maintain a strong \[WR\] wind. For the \[WO\] progenitors, the determining factor appears to be stellar temperature: the particular ionization balance may reduce the efficiency of the radiation-driven wind.
The determined discontinuity in temperatures of \[WR\]-type stars is in favor of the suggested separate evolutionary channels of cool and hot \[WR\] objects.
This project was financially supported by the “Polish State Committee for Scientific Research” through the grant No. 2.P03D.002.025, by the CNRS through the LEA Astro-PF programme, and by a NATO collaborative program grant No. PST.CLG.979726. ESO provided support via its scientific visitor program. The NTT observations were associated with observing program 67.D-0527.
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[^1]: http://www.hao.ucar.edu/Public/models/pikaia/pikaia.html
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A new polynomial analogue of the Rogers–Ramanujan identities is proven. Here the product-side of the Rogers–Ramanujan identities is replaced by a partial theta sum and the sum-side by a weighted sum over Schur polynomials.'
address: 'Department of Mathematics and Statistics, The University of Melbourne, Vic 3010, Australia'
author:
- 'S. Ole Warnaar'
title: 'Partial-sum analogues of the Rogers–Ramanujan identities'
---
[^1]
Introduction
============
The famous Rogers–Ramanujan identities are given by [@Rogers94] $$\label{RR1}
1+\sum_{n=1}^{\infty} \frac{q^{n^2}}{(1-q)(1-q^2)\cdots(1-q^n)}=
\prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+1})(1-q^{5j+4})}$$ and $$\label{RR2}
1+\sum_{n=1}^{\infty} \frac{q^{n(n+1)}}{(1-q)(1-q^2)\cdots(1-q^n)}
=\prod_{j=0}^{\infty}\frac{1}{(1-q^{5j+2})(1-q^{5j+3})}$$ for $|q|<1$. In one of his two proofs of these identities, Schur [@Schur17] introduced two sequences of polynomials $(e_n)_{n\geq 2}$ and $(d_n)_{n\geq 2}$, where $e_n$ ($d_n$) is the generating function of partitions with difference between parts at least $2$ (and no part equal to $1$), and largest part at most $n-2$. The partitions $\{\emptyset,(1),(2),(3),(4),(3,1),(4,1),(4,2)\}$, for example, contribute to $e_6=1+q+q^2+q^3+2q^4+q^5+q^6$ and the partitions $\{\emptyset,(2),(3),(4),(4,2)\}$ contribute to $d_6=1+q^2+q^3+q^4+q^6$.
By standard combinatorial arguments, see e.g., [@MacMahon16; @GR90], it follows that $e_{\infty}:=\lim_{n\to\infty} e_n
=\text{LHS}\eqref{RR1}$ and $d_{\infty}:=\lim_{n\to\infty} d_n=\text{LHS}\eqref{RR2}$. Schur proved the Rogers–Ramanujan identities by showing that these limits also hold when $\text{LHS}$ is replaced by $\text{RHS}$. This he achieved by showing that both $e_n$ and $d_n$ satisfy the recurrence $$\label{xrec}
x_{n+2}=x_{n+1}+q^n x_n,$$ and by solving this recurrence subject to the initial conditions $d_1=0$, $e_1=e_2=d_2=1$ (consistent with the combinatorial definition of $e_n$ and $d_n$ for $n\geq 2$). Specifically, Schur’s solution to reads
\[edbos\] $$\begin{aligned}
e_n&=\sum_{j=-\infty}^{\infty}(-1)^j q^{j(5j+1)/2}
{\genfrac{[}{]}{0pt}{}{n-1}{\lfloor{(n-5j-1)/2}\rfloor}} \\
d_n&=\sum_{j=-\infty}^{\infty}(-1)^j q^{j(5j+3)/2}
{\genfrac{[}{]}{0pt}{}{n-1}{\lfloor{(n-5j-2)/2}\rfloor}}\end{aligned}$$
for $n\geq 1$ and $\lfloor x \rfloor$ denoting the integer part of $x$. Here the $q$-binomial coefficients are given by ${{\textstyle\genfrac{[}{]}{0pt}{}{n}{m}}}=(q;q)_n/(q;q)_m(q;q)_{n-m}$ for $0\leq m\leq n$ and zero otherwise, where $(a;q)_n=\prod_{j=0}^{n-1}(1-aq^j)$.
Employing the notation $(a_1,\dots,a_k;q)_n=
(a_1;q)_n\cdots(a_k;q)_n$ and recalling to the Jacobi triple product identity [@GR90 Eq. (II.28)] $$\label{tpi}
\sum_{n=-\infty}^{\infty}(-1)^n a^n q^{\binom{n}{2}}=(a,q/a,q;q)_{\infty}$$ it is now easy to obtain the desired limits; $$\begin{aligned}
e_{\infty}&=\frac{1}{(q;q)_{\infty}}
\sum_{j=-\infty}^{\infty}(-1)^j q^{j(5j+1)/2}=\frac{1}{(q,q^4;q^5)_{\infty}}
=\text{RHS}\eqref{RR1} \\
d_{\infty}&=\frac{1}{(q;q)_{\infty}}
\sum_{j=-\infty}^{\infty}(-1)^j q^{j(5j+3)/2}=\frac{1}{(q^2,q^3;q^5)_{\infty}}
=\text{RHS}\eqref{RR2}.\end{aligned}$$
Representations for the Schur polynomials similar to the left sides of the Rogers–Ramanujan identities are also known [@MacMahon16 §286 and §289], $$\label{edfer}
e_n=\sum_{r=0}^{\infty}q^{r^2}{\genfrac{[}{]}{0pt}{}{n-r-1}{r}} \quad\text{and}\quad
d_n=\sum_{r=0}^{\infty}q^{r(r+1)}{\genfrac{[}{]}{0pt}{}{n-r-2}{r}}.$$ Equating this with yields the following polynomial analogue of the Rogers–Ramanujan identities [@Andrews70]: $$\label{RRpoly}
\sum_{r=0}^{\infty}q^{r(r+a)}{\genfrac{[}{]}{0pt}{}{n-r-a}{r}}=
\sum_{j=-\infty}^{\infty}(-1)^j q^{j(5j+2a+1)/2}
{\genfrac{[}{]}{0pt}{}{n}{\lfloor{(n-5j-a)/2}\rfloor}}$$ for $n\geq 0$ and $a\in\{0,1\}$.
Recently there has been renewed interest in the Schur polynomials [@GIS99; @AKP00; @IPS00; @BP01; @W01] sparked by the following nice generalization of the Rogers–Ramanujan identities due to Garrett, Ismail and Stanton [@GIS99 Eq. (3.5)] $$\label{GIS}
\sum_{r=0}^{\infty}\frac{q^{r(r+m)}}{(q;q)_r}=
\frac{(-1)^m q^{-\binom{m}{2}}d_m}{(q,q^4;q^5)_{\infty}}-
\frac{(-1)^m q^{-\binom{m}{2}}e_m}{(q^2,q^3;q^5)_{\infty}},$$ where $m$ is a nonnegative integer and $e_0=0$ and $d_0=1$ consistent with .
In this paper we show that may be used to prove new polynomial analogues of the Rogers–Ramanujan identities involving the Schur polynomials. These polynomial identities are fundamentally different from in that the product-side is replaced by a partial theta series.
\[thm\] For $k\in\{0,1\}$ and $n\geq 0$ there holds $$\begin{aligned}
\notag
\sum_{j=-n-k}^n&(-1)^j q^{j(5j+1)/2} \\
&=\sum_{r=0}^n e_{2r+k+2}(-1)^{n-r}
q^{(n-r)(5n+3r+4k+5)/2}\frac{(q;q)_{n+r+k}}{(q;q)_{n-r}}
\label{RRP1}\\
\intertext{and}
\notag
\sum_{j=-n-k}^n&(-1)^j q^{j(5j+3)/2} \\
&=\sum_{r=0}^n d_{2r+k+2}(-1)^{n-r}
q^{(n-r)(5n+3r+4k+5)/2}\frac{(q;q)_{n+r+k}}{(q;q)_{n-r}}.
\label{RRP2}\end{aligned}$$
Partial theta-sum identities of this type were first discovered by Shanks [@Shanks51].
When $n$ tends to infinity (for $|q|<1$) only the term with $r=n$ contributes to the sums on the right. Hence the first identity of the theorem implies $$\sum_{j=-\infty}^{\infty}(-1)^j q^{j(5j+1)/2}=
(q;q)_{\infty}e_{\infty}=(q;q)_{\infty}
\sum_{n=0}^{\infty} \frac{q^{n^2}}{(q;q)_n},$$ which is transformed into by the triple product identity. Likewise, arises as the the large $n$ limit of the second identity of the theorem.
Polynomial analogues of the Rogers–Ramanujan strikingly similar to those of Theorem \[thm\] have previously been discovered by Andrews [@Andrews83]. For $n\geq 0$ let $K_n(x)$ denote the Szegö polynomial [@Szego26] $$K_n(x)=\sum_{r=0}^n x^r q^{r(r+1)}{\genfrac{[}{]}{0pt}{}{n}{r}}.$$ Then Andrews posed in the problems section of SIAM Review [@Andrews83] the problem of showing that $$\begin{gathered}
\label{AJ}
\sum_{j=-n-k}^n(-1)^j q^{j(5j+2k+1)/2} \\
=\sum_{r=0}^n K_r(q^{2n-2r+k-1})(-1)^{n-r}
q^{(n-r)(5n-3r+4k+1)/2}\frac{(q;q)_{n+k}}{(q;q)_{n-r}}\end{gathered}$$ for $k=\{0,1\}$ and $n\geq 0$. Note here that the left side of coincides with the left side of () when $k=0$ ($k=1$).
The remainder of this paper is divided in two parts with section \[secp\] containing a proof and section \[secd\] a discussion of Theorem \[thm\]. In the first part of this discussion we examine two simple proofs of found by Jordan and Andrews and indicate our failure in generalizing these to a proof Theorem \[thm\]. The second part of our discussion focuses on some of the combinatorial aspects of Theorem \[thm\].
Proof of Theorem \[thm\] {#secp}
========================
A more general identity
-----------------------
Key to the proof of Theorem \[thm\] is the following proposition.
\[prop2\] For $k\in\{0,1\}$ and $|a|,|q|<1$ there holds $$\begin{gathered}
\label{key}
\sum_{n=0}^{\infty}\frac{a^{2n} q^{n(n+k)}}{(q;q)_n}=
\frac{(a;q)_{\infty}^2}{(q;q)_{\infty}^3}\sum_{j=1}^{\infty}
(q^{2j-k},q^{5+k-2j},q^5;q^5)_{\infty} \\
\times \sum_{r=0}^{\infty}\frac{(-1)^{j+r+1}
q^{\binom{j+r}{2}}(1-q^{2r+k+1})(aq^{-r};q)_r}{(a;q)_{r+k+1}}.\end{gathered}$$
It is perhaps not immediately clear that the sums on the right converge, but inspection of the potentially problematic terms shows that for $k\in\{0,1\}$ and $j\geq 1$, $$\begin{gathered}
O\Bigl(q^{\binom{j+r}{2}}(q^{5+k-2j};q^5)_{\infty}(aq^{-r};q)_r\Bigr) \\
=\begin{cases}
q^{(j-1)(j+10r+4k+6)/10} & \text{$j\equiv 1,k+4 \pmod{5}$} \\
q^{(j-2)(j+10r+4k+7)/10+r+1} & \text{$j\equiv 2,k+3 \pmod{5}$,}
\end{cases}\end{gathered}$$ which shows that both sums on the right converge and that their order is irrelevant.
Before proving Proposition \[prop2\] we will show how it implies Theorem \[thm\]. Starting point is the observation that $$\begin{gathered}
\label{inffin}
\sum_{j=1}^{\infty}(-1)^j q^{\binom{j+r}{2}}
\frac{(q^{2j-k},q^{5+k-2j},q^5;q^5)_{\infty}}{(q;q)_{\infty}} \\
=\sum_{i=1}^2 \frac{(-1)^{i+r}q^{-\binom{2r+k+2}{2}}}
{(q^{i+2k},q^{5-i-2k};q^5)_{\infty}}
\sum_{j=-r-k}^r (-1)^j q^{j(5j+2i+4k-5)/2}.\end{gathered}$$ To prove this we use that for $f_j$ such that $f_j=0$ if $j\equiv 3k \pmod{5}$ there holds $$\sum_{j=1}^{\infty}f_j=\sum_{i=1}^2\sum_{j=0}^{\infty}
(f_{5j+i}+f_{5j+5-i+k}).$$ This, together with the simple to verify identities $$\begin{gathered}
(q^{m+5n},q^{5-5n-m};q^5)_{\infty}=
(q^m,q^{5-m};q^5)_{\infty}(-1)^n q^{-nm-5\binom{n}{2}} \\[2mm]
\frac{(q^{2i-k},q^{5+k-2i},q^5;q^5)_{\infty}}{(q;q)_{\infty}}=
\frac{1}{(q^{i+2k},q^{5-i-2k};q^5)_{\infty}}, \;
i,k+1\in\{1,2\} \\[2mm]
\binom{i+r}{2}-(r+1)(5r+2i+4k)/2=-\binom{2r+k+2}{2}, \;
i,k+1\in\{1,2\} \\[2mm]
\sum_{i=1}^2 (-1)^i=0\end{gathered}$$ and the Jacobi triple product identity , yields $$\begin{aligned}
&\text{LHS}\eqref{inffin} \\
&\; = \sum_{i=1}^2 \frac{(-1)^i q^{\binom{i+r}{2}}}
{(q^{i+2k},q^{5-i-2k};q^5)_{\infty}}
\biggl(\sum_{j=-\infty}^{-2r-k-2}\!+\sum_{j=0}^{\infty}\,\biggr)
(-1)^j q^{j(5j+10r+2i+4k+5)/2} \\
&\; =\sum_{i=1}^2 \frac{(-1)^{i+r} q^{\binom{i+r}{2}-(r+1)(5r+2i+4k)/2}}
{(q^{i+2k},q^{5-i-2k};q^5)_{\infty}} \\
&\qquad \times\biggl[\,\sum_{j=-r-k}^r(-1)^j q^{j(5j+2i+4k-5)/2}
-(q^{i+2k},q^{5-i-2k},q^5;q^5)_{\infty}\biggr] \\
&\; =\text{RHS}\eqref{inffin}.\end{aligned}$$ After substituting in we obtain $$\begin{gathered}
\sum_{n=0}^{\infty}\frac{a^{2n} q^{n(n+k)}}{(q;q)_n} \\
=\frac{(a;q)_{\infty}^2}{(q;q)_{\infty}^2}\sum_{r=0}^{\infty} \sum_{i=1}^2
\frac{(-1)^{i+1}q^{-\binom{2r+k+2}{2}}(1-q^{2r+k+1})(aq^{-r};q)_r}
{(q^{i+2k},q^{5-i-2k};q^5)_{\infty}(a;q)_{r+k+1}} \\
\times \sum_{j=-r-k}^r(-1)^j q^{j(5j+2i+4k-5)/2}.\end{gathered}$$ Here the reader is warned that the order of the sums over $r$ and $i$ must be strictly adhered to. Indeed, our earlier considerations about convergence and the fact that is true, guarantee the not so obvious fact that after summing over $i$ the sum over $r$ converges.
Our next step removes any further convergence issues as we now specialize $a=q^{m+1}$ with $m$ a nonnegative integer. The sum over $r$ then terminates, yielding $$\begin{gathered}
\label{am}
\sum_{n=0}^{\infty}\frac{q^{n(n+2m+k+2)}}{(q;q)_n}
=\sum_{i=1}^2 \frac{(-1)^{i+1}}{(q^{i+2k},q^{5-i-2k};q^5)_{\infty}}\\
\times \sum_{r=0}^m
\frac{q^{-\binom{2r+k+2}{2}}(1-q^{2r+k+1})}{(q;q)_{m-r}(q;q)_{m+r+k+1}}
\sum_{j=-r-k}^r(-1)^j q^{j(5j+2i+4k-5)/2}.\end{gathered}$$ Rewriting the left-hand side using the Garrett–Ismail–Stanton identity gives $$\frac{d_{2m+k+2}}{(q,q^4;q^5)_{\infty}}-
\frac{e_{2m+k+2}}{(q^2,q^3;q^5)_{\infty}}
=(-1)^k q^{\binom{2m+k+2}{2}}\text{RHS}\eqref{am}.$$ Multiplying both sides by $(q;q)_{2m+k+1}$ this is of the form $$\frac{P(q)}{(q^2,q^3;q^5)_{\infty}}=\frac{Q(q)}{(q,q^4;q^5)_{\infty}}$$ with $P(q)$ and $Q(q)$ polynomials. An identity of this type can only be true if $P(q)=Q(q)=0$, and we infer $$\begin{aligned}
e_{2m+k+2}&=q^{\binom{2m+k+2}{2}}
\sum_{r=0}^m \frac{q^{-\binom{2r+k+2}{2}}(1-q^{2r+k+1})}
{(q;q)_{m-r}(q;q)_{m+r+k+1}}\sum_{j=-r-k}^r (-1)^j q^{j(5j+1)/2} \\
d_{2m+k+2}&=q^{\binom{2m+k+2}{2}}
\sum_{r=0}^m \frac{q^{-\binom{2r+k+2}{2}}(1-q^{2r+k+1})}
{(q;q)_{m-r}(q;q)_{m+r+k+1}}\sum_{j=-r-k}^r (-1)^j q^{j(5j+3)/2}\end{aligned}$$ for $m\geq 0$. All that remains is to invert these new representations of the Schur polynomials. This is easily done recalling the Bailey transform [@Andrews84], which states that if $$\label{ba}
\beta_n=\sum_{r=0}^n\frac{\alpha_r}{(q;q)_{n-r}(aq;q)_{n+r}}$$ then $$\label{ab}
\alpha_n=(1-aq^{2n})\sum_{r=0}^n\frac{(-1)^{n-r}q^{\binom{n-r}{2}}
(aq;q)_{n+r-1}}{(q;q)_{n-r}}\,\beta_r.$$ For later reference we remark that a pair of sequences $(\alpha,\beta)$ that satisfies (or, equivalently, ) is called a Bailey pair relative to $a$.
Since our expressions for the Schur polynomials take the form with $a=q^{k+1}$, we may invoke to find the identity claimed in Theorem \[thm\].
Proof of Proposition \[prop2\]
------------------------------
Our proof relies on the following lemma.
\[lem1\] For $k\in\{0,1\}$ and $M$ and $n$ integers there holds $$\begin{gathered}
\frac{q^{n(n+2)}}{(q;q)_{\infty}^3}
\sum_{j=1}^{\infty}\sum_{r=0}^{\infty}\sum_{l=0}^M (-1)^{M+j+r+1}
q^{\binom{j+r}{2}+\binom{M-l}{2}+\binom{l}{2}+l(r+k+1)} \\
\times q^{-r(M-l)-n(2j-k)}
\frac{(1-q^{(2j-k)(2n+1)})(1-q^{2r+k+1})}
{(1-q^{2n+1})(q;q)_{M-l}(q;q)_l} \\[2mm]
=\begin{cases}\displaystyle
\frac{q^{m(m+k)}}{(q;q)_{m-n}(q;q)_{m+n+1}} & \text{$M=2m$} \\[4mm]
0 &\text{$M=2m+1$.}
\end{cases}\end{gathered}$$
Here our earlier definition of $(a;q)_n$ is extended to all integers $n$ by $(a;q)_n=(a;q)_{\infty}/(aq^n;q)_{\infty}$. Note in particular that $1/(q;q)_n=0$ for $n<0$.
Given the triple sum on the left, Lemma \[lem1\] perhaps appears complicated and not readily applicable. However, in view of it is in fact quite useful, and if we multiply boths sides by $\alpha_n$ and then sum $n$ over the nonnegative integers we get $$\begin{gathered}
\frac{q^{n(n+2)}}{(q;q)_{\infty}^3}
\sum_{n=0}^{\infty}\sum_{j=1}^{\infty}\sum_{r=0}^{\infty}
\sum_{l=0}^M \alpha_n (-1)^{M+j+r+1}
q^{\binom{j+r}{2}+\binom{M-l}{2}+\binom{l}{2}+l(r+k+1)} \\
\times q^{-r(M-l)-n(2j-k)}
\frac{(1-q)(1-q^{(2j-k)(2n+1)})(1-q^{2r+k+1})}
{(1-q^{2n+1})(q;q)_{M-l}(q;q)_l} \\[2mm]
=\begin{cases}
q^{m(m+k)}\beta_m & \text{$M=2m$} \\[2mm]
0 &\text{$M=2m+1$,}
\end{cases}\end{gathered}$$ where $(\alpha,\beta)$ is a Bailey pair relative to $q$.
Next we multiply both sides by $a^M$ and sum over $M$. If on the left we interchange the sums over $M$ and $l$, shift $M\to M+l$ and then sum over $l$ and $M$ using Euler’s $q$-exponential sum [@GR90 Eq. (II.2)] $$\sum_{n=0}^{\infty}\frac{(-1)^n a^n q^{\binom{n}{2}}}{(q;q)_n}=(a;q)_{\infty}$$ this yields $$\begin{gathered}
\frac{(a;q)_{\infty}^2}{(q;q)_{\infty}^3}
\sum_{n=0}^{\infty}\sum_{j=1}^{\infty}\alpha_n q^{n(n-2j+k+2)}
\frac{(1-q)(1-q^{(2j-k)(2n+1)})}{(1-q^{2n+1})} \\
\times \sum_{r=0}^{\infty} \frac{(-1)^{j+r+1}
q^{\binom{j+r}{2}}(1-q^{2r+k+1})(aq^{-r};q)_r}{(a;q)_{r+k+1}}
=\sum_{n=0}^{\infty} a^{2n} q^{n(n+k)}\beta_n.\end{gathered}$$ We have nearly arrived at . All that is needed is the following Bailey pair relative to $q$ due to Rogers [@Rogers17]: $$\alpha_n=(-1)^n q^{n(3n+1)/2}\frac{(1-q^{2n+1})}{(1-q)}
\quad\text{and}\quad \beta_n=\frac{1}{(q;q)_n}.$$ Substituting this, interchanging the sum over $n$ and $j$ (with the above choice for $\alpha_n$ this may indeed be done) and using the triple product identity gives .
Replacing $M$ by $2m+i$ where $i\in\{0,1\}$, and shifting $l\to l+m+i$ leads to $$\begin{gathered}
\label{lquad}
\frac{q^{n(n+2)}}{(q;q)_{\infty}^3}
\sum_{j=1}^{\infty}\sum_{r=0}^{\infty}\sum_{l=-m-i}^m (-1)^{j+r+1}
q^{\binom{j+r}{2}+ir-n(2j-k)+l(l+2r+k+i+1)} \\
\times \frac{(1-q^{(2j-k)(2n+1)})(1-q^{2r+k+1})}
{(1-q^{2n+1})(q;q)_{m-l}(q;q)_{m+l+i}}
=\frac{\delta_{i,0}}{(q;q)_{m-n}(q;q)_{m+n+1}}.\end{gathered}$$ By the $q$-Chu–Vandermonde summation [@GR90 Eq. (II.6)] $$\label{qCV}
\sum_{j=0}^n \frac{(a,q^{-n};q)_j \,q^j}{(q,c;q)_j}=
\frac{(c/a;q)_n}{(c;q)_n}\,a^n$$ this follows from the simpler to prove identity $$\begin{gathered}
\label{llin}
\frac{q^{2n(n+1)}}{(q;q)_{\infty}^3}
\sum_{j=1}^{\infty}\sum_{r=0}^{\infty}\sum_{l=-m-i}^m (-1)^{j+r+1}
q^{\binom{j+r}{2}+ir-n(2j-k)+l(2r+k+1)} \\
\times \frac{(1-q^{(2j-k)(2n+1)})(1-q^{2r+k+1})}
{(1-q^{2n+1})(q;q)_{m-l}(q;q)_{m+l+i}}
=\frac{\delta_{i,0}q^{m-n}}{(q;q)_{m-n}(q;q)_{m+n+1}}.\end{gathered}$$ Indeed, if we multiply both sides of by $q^{m(m+i)}/(q;q)_{M-m}$, the resulting identity can be summed over $m$ by the $c=0$ instance of (after first replacing $m\to M-m$). On the right we of course only need to do this sum when $i=0$. Replacing $M$ by $m$ then gives . Those familiar with the concept of a Bailey chain [@Andrews84] will have recognized that the reduction of to corresponds to a simplifying (i.e., backwards) iteration along a Bailey chain relative to $q^i$.
Since is of the form with $a=q^i$ we can use to invert. Hence $$\begin{gathered}
\label{inverted}
\frac{q^{2n(n+1)}}{(q;q)_{\infty}^3}
\sum_{j=1}^{\infty}\sum_{r=0}^{\infty}(-1)^{j+r+1}
q^{\binom{j+r}{2}+ir-n(2j-k)-(m+i)(2r+k+1)} \\
\times \bigl(1-q^{(2n+1)(2j-k)}\bigr)\bigl(1-q^{2r+k+1}\bigr)
\bigl(1+q^{(2m+i)(2r+k+1)}\bigr) \\
=\delta_{i,0}(1-q^{2m})(1-q^{2n+1})
\sum_{r=0}^m\frac{(-1)^{m-r}q^{\binom{m-r}{2}+r-n}(q;q)_{r+m-1}}
{(q;q)_{m-r}(q;q)_{r-n}(q;q)_{r+n+1}},\end{gathered}$$ with the convention that $(1-q^{2m})(q^m;q)_{r+m-1}=2$ for $m=r=0$ in accordance with $(1-q^{2m})(q;q)_{m-1}=(1+q^m)(q;q)_m$. The sum over $r$ on the right may be carried out by the $q$-Chu–Vandermonde sum , leading to $$\begin{gathered}
\text{RHS}\eqref{inverted} \\ =
\delta_{i,0}\frac{(-1)^{m+n}(1-q^{2m})(1-q^{2\max\{n,-n-1\}+1})
(q;q)_{m+\max\{n,-n-1\}-1}}{(q;q)_{m-n}(q;q)_{m+n+1}
(q^2;q)_{\max\{n,-n-1\}-m}},\end{gathered}$$ which is nonzero for $n=\pm m$ and $n=\pm m-1$ only. If we also multiply by $q^{i(k+1)/2}$ and note that on the right this may again be dropped, we obtain $$\begin{gathered}
\label{del}
\frac{q^{2n(n+1)}}{(q;q)_{\infty}^3}
\sum_{j=1}^{\infty}\sum_{r=0}^{\infty}(-1)^{j+r+1}
q^{\binom{j+r}{2}-n(2j-k)-(2m+i)(2r+k+1)/2} \\
\times \bigl(1-q^{(2n+1)(2j-k)}\bigr)\bigl(1-q^{2r+k+1}\bigr)
\bigl(1+q^{(2m+i)(2r+k+1)}\bigr) \\[1mm]
=\delta_{i,0}\bigl(\delta_{m,n}+\delta_{-m,n}-
\delta_{m-1,n}-\delta_{-m-1,n}\bigr).\end{gathered}$$ Since both sides are invariant under the substitution $m\to -m-i$ this must hold for all $m,n\in{\mathbb{Z}}$ and $i,k\in\{0,1\}$.
Next we observe that is a consequence of the stronger result $$\begin{gathered}
\label{symm}
\frac{q^{\binom{n}{2}+(n-m)/2}}{(q;q)_{\infty}^3}
\sum_{j=1}^{\infty}\sum_{r=0}^{\infty}(-1)^{j+r+n}
q^{\binom{j+r}{2}-(n-1)(2j-k)/2-(m-1)(2r+k)/2} \\
\times \bigl(1-q^{n(2j-k)}\bigr)\bigl(1-q^{m(2r+k+1)}\bigr)
=\delta_{m,n}-\delta_{-m,n},\end{gathered}$$ for $m,n\in{\mathbb{Z}}$ and $k\in\{0,1\}$. If we denote the above two identities by $\eqref{del}|_{m,n}$ and $\eqref{symm}|_{m,n}$ and note that $\delta_{2m+i\pm 1,2n+1}=
\delta_{i,0}\delta_{m-(1\mp 1)/2,n}$, then $\eqref{del}|_{m,n}=
\eqref{symm}|_{2m+i+1,2n+1}-\eqref{symm}|_{2m+i-1,2n+1}$.
Before proving let us point out that without loss of generality we may fix $k=0$. For, if we take with $k=1$, replace $r\leftrightarrow j-1$, and multiply the result by $(-1)^{m-n}q^{(m^2-n^2)/2}$ we find $\eqref{symm}|_{m,n;k=1}=\eqref{symm}|_{n,m;k=0}$. Equation for $k=0$ is a linear combination of yet another identity, given by $$\begin{gathered}
\label{symm2}
\frac{q^{\binom{n}{2}}}{(q;q)_{\infty}^3}
\sum_{j=0}^{\infty}\sum_{r=0}^{\infty}(-1)^{j+r+n}
q^{\binom{j+r}{2}-(n-1)j-(m-1)r} \\
+\frac{q^{\binom{n}{2}}}{(q;q)_{\infty}^3}
\sum_{j=1}^{\infty}\sum_{r=0}^{\infty}(-1)^{j+r+n}
q^{\binom{j+r}{2}-(n-1)j-(m-1)r+2nj+m(2r+1)}
=\delta_{m,n}.\end{gathered}$$ Here it should be noted that the first sum over $j$ now includes the term $j=0$. It is easily seen that this extra term is cancelled out in the following linear combination, and that $\eqref{symm2}|_{m,n}-q^{-n}\eqref{symm2}|_{m,-n}=
q^{(m-n)/2}\eqref{symm}|_{m,n;k=0}$.
After this string of reductive steps we are finally in a position to carry out a proof. Replacing $m$ by $m+n$ in and changing the summation variable $j\to n-j$ ($j\to j-n)$ in the first (second) double sum gives $$\begin{gathered}
\sum_{j=-\infty}^n\sum_{r=0}^{\infty}(-1)^{j+r}
q^{\binom{j-r}{2}-mr}
+\sum_{j=n+1}^{\infty}\sum_{r=0}^{\infty}(-1)^{j+r}
q^{\binom{j+r+1}{2}+m(r+1)} \\
=(q;q)_{\infty}^3 \delta_{m,0}.\end{gathered}$$ In the second term on the left we rewrite the sum over $r$ using $$\label{zero1}
\sum_{r=0}^{\infty}(-1)^r q^{\binom{r+1}{2}+a(r+1)}
=\sum_{r=0}^{\infty}(-1)^r q^{\binom{r+1}{2}-ar}$$ as follows from $$\label{zero2}
\sum_{r=-\infty}^{\infty}(-1)^r q^{\binom{r+1}{2}-ar}=0.$$ (To prove replace $r\to 2a-1-r$.) As a result we are left with $$\label{fin}
\sum_{j=-\infty}^{\infty}\sum_{r=0}^{\infty}(-1)^{j+r}
q^{\binom{j-r}{2}-mr}=(q;q)_{\infty}^3\delta_{m,0}.$$ Using on the sum over $r$ and negating $j$ yields $\eqref{fin}|_m=q^m \eqref{fin}|_{-m}$, so that we may assume $m\leq 0$ when proving . If $m<0$ the order of the sums may be interchanged. By this completes the proof. If $m=0$ we need $$\sum_{r=0}^{2j-1}(-1)^r q^{\binom{j-r}{2}}=0$$ for $j\geq 0$ (to prove this replace $r\to 2j-1-r$), and Jacobi’s identity [@Jacobi29 §66, (5.)] $$\sum_{i=0}^{\infty}(-1)^i (2i+1) q^{\binom{i+1}{2}}=(q;q)_{\infty}^3.$$ Equipped with these the rest is easy; $$\begin{aligned}
\sum_{j=-\infty}^{\infty}\sum_{r=0}^{\infty}(-1)^{j+r}q^{\binom{j-r}{2}}
&=\Bigl\{\sum_{j=0}^{\infty}\sum_{r=2j}^{\infty}+
\sum_{j=-\infty}^{-1}\sum_{r=0}^{\infty}\Bigr\}
(-1)^{j+r} q^{\binom{j-r}{2}} \\
&=\Bigl\{\sum_{j=0}^{\infty}
+\sum_{j=1}^{\infty}\Bigr\} \sum_{r=j}^{\infty}
(-1)^r q^{\binom{r+1}{2}}\\
&=\sum_{r=0}^{\infty}(-1)^r (2r+1) q^{\binom{r+1}{2}}
=(q;q)_{\infty}^3. \qquad \qed\end{aligned}$$
Discussion {#secd}
==========
Eqs and versus
----------------
The proof of Theorem \[thm\] as given in the previous section is very lengthy and complicated, and, as a result, not very illuminating. Here we briefly discuss the proofs of as found by Jordan and Andrews as we hold some hope that at least one of these may be generalized to also prove and .
Perhaps simplest is Jordan’s proof [@Jordan84]. Denoting the right side of by $f_{n;k}$ and the summand on the right of by $f_{n,r;k}$, it is not difficult to show that the functional equation $$(1-x q^{n+2})K_{n+1}(x)=K_n(x)-x^2 q^{n+4}K_n(x q^2)$$ satisfied by the Szegö polynomials implies the recurrence $$\label{frec}
\sum_{r=m+1}^n (f_{n,r;k}-f_{n-1,r-1;k})=
-\frac{1-q^{m-n}}{1-q^{n+k}}f_{n,m;k}.$$ By the $m=0$ instance hereof it is found that $$\begin{aligned}
f_{n;k}-f_{n-1;k}&=f_{n,0;k}+\sum_{r=1}^n(f_{n,r;k}-f_{n-1,r-1;k}) \\
&=q^{-n}\frac{1-q^{2n+k}}{1-q^{n+k}}f_{n,0;k} \\[2mm]
&=(-1)^n q^{n(5n+2k+1)/2}+(-1)^{n+k}q^{(n+k)(5(n+k)-2k-1)/2},\end{aligned}$$ from which follows by induction. Unfortunately, at present we have been unable to find an analogue of for the summands on the right of Theorem \[thm\].
Andrews’ proof of relies on the following multiple series generalization of Watson’s $q$-Whipple transform [@Andrews75 Thm. 4; $k=3$]: $$\begin{gathered}
\label{k3}
{_{10}}W_9(a;b,c,d,e,f,g,q^{-n};q,a^3 q^{n+3}/bcdefg;q,q) \\
=\frac{(aq,aq/fg;q)_n}{(aq/f,aq/g;q)_n}
\sum_{j=0}^n \sum_{k=0}^{n-j}
\frac{(aq/bc,d,e;q)_j}{(q,aq/b,aq/c;q)_j}
\frac{(aq/de;q)_k}{(q;q)_k} \\ \times
\frac{(f,g,q^{-n};q)_{j+k}}{(aq/d,aq/e,fgq^{-n}/a)_{j+k}}
\Bigl(\frac{aq^2}{de}\Bigr)^j q^k.\end{gathered}$$ Here and in the following we employ standard notation for basic hypergeometric series, see e.g., [@GR90]. Taking $b=aq^{n+1}$ and letting $c,d,e,f,g$ tend to infinity yields with $k=0$ if $a=1$ ($k=1$ if $a=q$).
Now if we apply Sears’ $_{4}\phi_3$ transformation [@GR90 Eq. (III.15)] to Watson’s Watson’s $q$-Whipple transform [@GR90 Eq. (III.18)] we readily obtain $$\begin{gathered}
\label{WS}
{_8W_7}(a;b,c,d,e,q^{-n};q,a^2 q^{n+2}/bcde;q,q) \\
=\frac{(aq,b,a^2 q^2/bcde;q)_n}{(aq/c,aq/d,aq/e;q)_n}
{_4\phi_3}\Bigl[\genfrac{}{}{0pt}{}
{aq/bc,aq/bd,aq/be,q^{-n}}{aq/b,a^2q^2/bcde,q^{1-n}/b};q,q\Bigr].\end{gathered}$$ Taking $b=aq^{n+1}$ and letting $c,d,e,f,g$ this simplifies to $$\label{WSlim}
\sum_{j=0}^n \frac{1-aq^{2j}}{1-a}
\frac{(a;q)_j(-1)^j q^{3\binom{j}{2}}(aq)^j}
{(q;q)_j}=(aq;q)_{2n}\sum_{j=0}^n \frac{q^j}{(q,q^{-2n}/a;q)_j}.$$ Chosing $a=q^k$ for $k=0,1$ and making the variable change $j\to n-r$ on the right we obtain the following polynomial analogue of Euler’s identity $$\label{Eulerpol}
\sum_{j=-n-k}^n(-1)^j q^{j(3j+1)/2}=\sum_{r=0}^n (-1)^{n-r}
q^{(n-r)(3n+r+2k+3)/2}\frac{(q;q)_{n+r+k}}{(q;q)_{n-r}}.$$ (Incidentally, this identity is very similar and can easily be transformed into a polynomial version of Euler’s identity due to Shanks [@Shanks51].) Given the similarity between and the identities and , and given Andrews’ proof of by means of it seems very natural to ask for a proof of Theorem \[thm\] by means of a multiple series generalization of the transformation . If we take with $k=0$ and with $k=1$ and replace $r\to n-j$ in the sums on the right we find that the resulting identities are the $a=1$ and $a=q$ instances of $$\begin{gathered}
\sum_{j=0}^n \frac{1-aq^{2j}}{1-a}
\frac{(a;q)_j(-1)^j q^{5\binom{j}{2}}(aq)^{2j}}{(q;q)_j} \\
=(aq;q)_{2n}\sum_{j=0}^n\sum_{k=0}^{n-j}
\frac{(-1)^k a^{j+k} q^{j^2-\binom{2j+k}{2}+(2n+1)(j+k)}
(q^{-2n-1};q)_{2j+2k}}
{(q,q^{-2n}/a;q)_j(q;q)_k(q^{-2n-1};q)_{2j+k}}\end{gathered}$$ This is to be compared with . Despite numerous attempts we failed to extend this to a multiple series transformation similar to and generalizing . Of course one can try to prove the above by equating coefficients of $a^m$, but the resulting identity $$\begin{gathered}
\sum_{j=0}^n \frac{(-1)^j q^{\binom{j}{2}+j(4j-2m+1)}}{(q;q)_j}
\biggl({\genfrac{[}{]}{0pt}{}{j}{m-2j}}+q^{4j-m+1}{\genfrac{[}{]}{0pt}{}{j}{m-2j-1}}\biggr) \\
=\sum_{r=0}^n\sum_{s=0}^{n-r}
\frac{(-1)^{r+s}q^{\binom{r+s}{2}+r(4n-2m+4)+s(s+r-m+1)}}{(q;q)_r}
\\ \times
{\genfrac{[}{]}{0pt}{}{2n+1-r}{m-2r-s}}{\genfrac{[}{]}{0pt}{}{2n-2r-s+1}{s}}\end{gathered}$$ for $n\geq 0$ and $0\leq m\leq 3n+1$ is not particularly simple (and would only prove half of Theorem \[thm\]).
Some combinatorics related to Theorem \[thm\]
---------------------------------------------
In order to discuss some of the combinatorics of Theorem \[thm\] we need to review several standard results from partition theory [@Andrews76].
Let $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_r)$ be a partition, defined as a weakly decreasing sequence of positive integers $\lambda_j$ (the parts of $\lambda$). The weight $|\lambda|$ of $\lambda$ is given by the sum of its parts. We say that $\lambda$ is a partition of $l$ if $|\lambda|=l$. The Ferrers graph of $\lambda$ is the graph obtained by drawing $r$ left-aligned rows of dots with the $j$th row containing $\lambda_j$ dots. The conjugate $\lambda'$ of $\lambda$ is obtained by transposing its Ferrers graph. The number $d(\lambda)$ is the number of rows in the maximal square of dots of the Ferrers graph of $\lambda$. An alternative way to represent a partition $\lambda$ is as a two-rowed matrix of $d(\lambda)=d$ columns $\bigl(\begin{smallmatrix} t_1 t_2 \dots t_d \\ b_1 b_2 \dots b_d
\end{smallmatrix}\bigr)$, where $t_j=\lambda_j-j$ and $b_j=\lambda_j'-j$, so that, in particular, $t_j>t_{j+1}$ and $b_j>b_{j+1}$. Conversely, any such matrix (also called Frobenius symbol) corresponds to the unique partition $\lambda$ by $\lambda_j=t_j+b_j+1$. We will in the following identify the standard and Frobenius notations for partitions. Note that $|\lambda|=d+\sum_{j=1}^d(t_j+b_j)$. The rank of a partition $\lambda$ is defined as its largest part minus its number of parts, i.e., as $\lambda_1-\lambda'_1=t_1-b_1$. More generally, the $i$th successive rank of $\lambda$ is given by $t_i-b_i$, and $r(\lambda)=(t_1-b_1,t_2-b_2,\dots,t_d-b_d)$ denotes the sequence of successive ranks of $\lambda$. For example, if $\lambda=(7,7,5,3,3,1,1,1)$, then $|\lambda|=28$, $\lambda'=(8,5,5,3,3,2,2)$, $d(\lambda)=3$, $\lambda=\bigl(\begin{smallmatrix} 6 5 2 \\ 7 3 2 \end{smallmatrix}\bigr)$, $\lambda'=\bigl(\begin{smallmatrix} 7 3 2 \\ 6 5 2 \end{smallmatrix}\bigr)$, and $r(\lambda)=(-1,2,0)$.
Now let $b_2(l,n)$ denote the set of all partitions of $l$, with largest part at most $n-2$ and difference between parts at least $2$, and let $B_2(l,n)$ be its cardinality. Then $e_n=\sum_{l\geq 0}B_2(l,n) q^l$. Given a partition $\lambda\in b_2(l,n)$ with exactly $r$ parts, one can form a new partition $\mu$ as follows [@Andrews76 §9.3]: $\mu=\bigl(\begin{smallmatrix} s_1,\dots,s_r \\ c_1,\dots,c_r
\end{smallmatrix}\bigr)$, where $s_j=\lfloor \lambda_j/2 \rfloor$ and $c_j=\lfloor (\lambda_j-1)/2 \rfloor$. Because of the difference-$2$ condition one indeed has $s_j>s_{j+1}$ and $c_j>c_{j+1}$. Since (for $n\in{\mathbb{Z}}$) $\lfloor n/2 \rfloor +\lfloor (n-1)/2 \rfloor=n-1$ one finds that $|\mu|=r+\sum_{j=1}^r (s_j+c_j)=|\lambda|=l$. Furthermore, the restriction that $\lambda_j-\lambda_{j+1}\geq 2$ translates into the fact that the successive ranks of $\mu$ must take the values $0$ and $1$ only. Finally the restriction that $\lambda_1\leq n-2$ implies that $s_1+c_1+1\leq n-2$. Since $s_1-c_1\in\{0,1\}$ this is equivalent to requiring that $\mu_1\leq\lfloor n/2\rfloor$ and $\mu'_1\leq\lfloor (n-1)/2\rfloor$. If we denote the set of all partitions of $l$ with successive ranks in $\{0,1\}$, largest part not exceeding $\lfloor n/2 \rfloor$ and number of parts not exceeding $\lfloor (n-1)/2\rfloor$ by $q_2(l,n)$ (with cardinality $Q_2(l,n)$) then clearly each partition $\mu\in q_2(l,n)$ can also be mapped back onto a partition in $b_2(l,n)$. Specifically, if $\mu\in q_2(l,n)$ has Frobenius symbol $\bigl(\begin{smallmatrix} s_1,\dots,s_r \\ c_1,\dots,c_r
\end{smallmatrix}\bigr)$, then $\lambda=(s_1+c_1+1,\dots,s_r+c_r+1)\in b_2(l,n)$, since $\lambda_j-\lambda_{j+1}=s_j+c_j-s_{j+1}-c_{j+1}\geq 2$ and $\lambda_1=s_1+c_1+1=\mu_1+\mu'_1-1\leq n-2$. Hence $Q_2(l,n)=B_2(l,n)$ and $e_n=\sum_{l\geq 0}Q_2(l,n) q^l$. For example, $\cup_{l\geq 0}q_2(l,n)=
\{\emptyset,(1),(2),(2,1),(3,1),(2,2),(3,2),(3,3)\}$ so that $e_6=1+q+q^2+q^3+2q^4+q^5+q^6$.
The above discussion can be repeated for the Schur polynomial $d_n$ and we define $b_1(l,n)$ as the subset of $b_2(l,n)$ obtained by removing all partitions which have a part equal to $1$. Hence $d_n=\sum_{l\geq 0}B_1(l,n)q^l$. If we also define $q_1(m,n)$ as the set of partitions with successive ranks in $\{1,2\}$, largest part not exceeding $\lfloor (n+1)/2 \rfloor$ and number of parts not exceeding $\lfloor (n-2)/2\rfloor$, then it is not hard to show that $Q_1(l,n)=B_1(l,n)$ so that $d_n=\sum_{l\geq 0} Q_1(l,n) q^l$. For example, $\cup_{l\geq 0} q_1(l,n)=\{\emptyset,(2),(3),(3,1),(3,3)\}$ so that $d_6=1+q^2+q^3+q^4+q^6$.
So far, we have given a combinatorial interpretation of the Schur polynomials $e_n$ and $d_n$ in terms of partitions with restrictions on their size and successive ranks. Next we will discuss the combinatorial interpretation of the partial theta sum $\sum_{j=-n+k}^n (-1)^j q^{j(5j-2i+5)/2}$ in terms of successive ranks.
First we recall some further known properties of $Q_i(l,n)$ [@Andrews72; @Andrews76]. Let $\lambda$ be a partition and $r(\lambda)$ its sequence of successive ranks. The length of the largest subsequence $r'$ of $r(\lambda)$ such that the odd (even) elements of $r'$ are at least $4-i$ and the even (odd) elements of $r'$ are at most $1-i$, is called the $(2,i)$-positive ($(2,i)$-negative) oscillation of $\lambda$. The number of partitions of $l$ that have at most $b$ parts, largest part not exceeding $a$ and $(2,i)$-positive ($(2,i)$-negative) oscillation at least $j$ is denoted by $p_i(a,b;j;l)$ ($m_i(a,b;j;l)$). By inclusion-exclusion arguments it then follows that $$Q_i(l,n)=\sum_{j=0}^{\infty}(-1)^j p_i(\bar{a},\bar{b};j,l)
+\sum_{j=1}^{\infty}(-1)^j m_i(\bar{a},\bar{b};j,l),$$ with $\bar{a}=\bar{a}(n,i)=\lfloor (n-i+2)/2\rfloor$ and $\bar{b}=\bar{b}(n,i)=\lfloor (n+i-3)/2\rfloor$. Furthermore, $$q^{j(5j-2i+5)/2}{\genfrac{[}{]}{0pt}{}{n-1}{\lfloor \frac{n+i-5j-3}{2}\rfloor}}=
\begin{cases}
\sum_{l=0}^{\infty} p_i(\bar{a},\bar{b};-j,l)q^l
&\text{$j\leq 0$, $j$ even} \\[1mm]
\sum_{l=0}^{\infty} m_i(\bar{a},\bar{b};-j,l)q^l
&\text{$j\leq 0$, $j$ odd} \\[1mm]
\sum_{l=0}^{\infty} p_i(\bar{a},\bar{b};j,l)q^l
&\text{$j\geq 0$, $j$ odd} \\[1mm]
\sum_{l=0}^{\infty} m_i(\bar{a},\bar{b};j,l)q^l
&\text{$j\geq 0$, $j$ even,}
\end{cases}$$ from which immediately follows. But now it is also clear what our partial theta sums represent. If we denote by $\lambda^{\pm}_{i,j}$ the (unique) partition of minimal weight that has a positive/negative $(2,i)$-oscillation $j$ and by $M_i$ the set of all such minimal partitions, i.e., $M_i=\{\lambda^{\sigma}_{i,j}\}_{j\geq 0;\sigma\in\{0,1\}}$, then (for $k\in\{0,1\}$ and $i\in\{1,2\}$) $$\sum_{j=-n-k}^n (-1)^j q^{j(5j-2i+5)/2}=
\sum_{\substack{\lambda\in M_i \\
\lambda_1\leq \lfloor (5n+2ki-2i+5)/2\rfloor \\
\lambda'_1\leq \lfloor (5n+2ki)/2\rfloor}}
(-1)^{d(\lambda)} q^{|\lambda|}.$$ One can in fact easily find the partition $\lambda^{\pm}_{i,j}$. For example, using the Frobenius notation it follows immediately that for $j$ even $$\lambda^{+}_{i,j}=
\begin{pmatrix} 5j/2-1,&5j/2-5,&\dots,& 9,& 5, &4, & 0 \\
5j/2+i-5,&5j/2+i-6,& \dots,& i+5,& i+4,& i,& i-1\end{pmatrix}.$$ When converted into standard notation this gives $$\lambda^{+}_{i,j}=
(5j/2,(5j/2-3)^2,\dots,(j+6)^2,(j+3)^2,j^i,(j-2)^3,\dots,4^3,2^3)$$ where $p^f$ stands for $f$ parts of size $p$. Calculating the weight of this partition gives $$|\lambda^{+}_{i,j}|=
5j/2+2\sum_{k=1}^{j/2-1}(j+3k)+ij+3\sum_{k=1}^{j/2-1}(2k)=j(5j+2i-5)/2,
\quad \text{$j$ even}$$ as it should. Similarly one can use the Frobenius notation to find $$\lambda^{-}_{i,j}=
((5j-3)/2)^2,\dots,(j+6)^2,(j+3)^2,j^i,(j-2)^3,\dots,3^3,1^3),$$ $(\lambda^{+}_{i,j})'=\lambda^{-}_{5-i,j}$ both for $j$ odd, and $(\lambda^{-}_{i,j})'=\lambda^{+}_{5-i,j}$ for $j$ even, and thus $|\lambda^{\pm}_{i,j}|=j(5j\mp 2i \pm 5)/2$ for odd $j$ and $|\lambda^{-}_{i,j}|=j(5j-2i+5)/2$ for even $j$.
Summarizing, we have the following remarkable situation. The Schur polynomials, which are the generating functions of certain size and successive rank restricted partitions, can be expressed as an alternating sum over the generating functions of partitions with certain restrictions on their $(2,i)$-oscillations. This well-known fact [@Andrews72; @Andrews76] provides a combinatorial explanation of Schur’s result . But now we see that according to the Theorem \[thm\] there is another side to the coin; the alternating sum over the generating function of a very special subset of partitions with certain restrictions on their $(2,i)$-oscillations can in its turn be expressed as a weighted sum over Schur polynomials. However, by no means is this an example of a trivial (or nontrivial but known) inversion result. Indeed, naively one might think that if we substitute in Theorem \[thm\] to get $$\begin{gathered}
\sum_{j=-n-k}^n(-1)^j q^{j(5j-2i+5)/2}=
\sum_{j=-\infty}^{\infty}(-1)^j q^{j(5j-2i+5)/2} \\
\times \sum_{r=0}^n (-1)^{n-r}
q^{(n-r)(5n+3r+4k+5)/2}\frac{(q;q)_{n+r+k}}{(q;q)_{n-r}}
{{\textstyle\genfrac{[}{]}{0pt}{}{2r+k+1}{\lfloor{(2r+i+k-5j-1)/2}\rfloor}}},\end{gathered}$$ that this is just a consequence of the second line being $\chi(-n-k\leq j\leq n)$ with $\chi(\text{true})=1$ and $\chi(\text{false})=0$. However, it is readily checked that this is only correct when $n=k=0$. It thus seems an extremely challenging problem to find a combinatorial proof of Theorem \[thm\], especially since our analytic proof provides so little insight as to why this theorem is true.
To conclude we remark that the previous discussion has a representation theoretic counterpart. As is well-known, $$\frac{1}{(q;q)_{\infty}}
\sum_{j=-\infty}^{\infty}(-1)^j q^{j(5j-2i+5)/2}$$ is the (normalized) character of the $c=-22/5$ Virasoro algebra corresponding to the highest weight vector $v_{h_i}$ of weight $h_i=(1-i)/5$. According to the Feigin–Fuchs construction [@FF84] the above character can be constructed from the Verma module $V(c,h_i)$ by eliminating submodules generated by singular or null vectors. Because of the embedded structure of these submodules this leads to an inclusion-exclusion type of sum. Specifically, the character corresponding to the submodule $V(c,h_i')$ with singular vector of weight $h_i'$ is given by $q^{h_i'-h_i}/(q;q)_{\infty}$, with the set of weights of singular vectors (including $v_{h_i}$) given by $h_i'=h_i+j(j-2i+5)/2$ for $j\in{\mathbb{Z}}$. Therefore, if we denote by $V_s(c,h_i;N)$ the set comprising of the $N$ singular vectors of $V(c,h_i)$ of smallest weight, and if we denote by $d(v)+1$ the number of (sub)modules $V(c,h_i')$ that contain the singular vector $v$ (so that $d(v)=0$ iff $v=v_{h_i}$), then $$\sum_{j=-n-k}^n (-1)^j q^{j(5j-2i+5)/2}=
\sum_{v\in V_s(c,h_i;2n+k+1)} (-1)^{d(v)} q^{|v|-h_i},$$ where $|v|$ is the weight of $v$, $c=-22/5$ and $h_i=(1-i)/5$. Again it is a challenge to explain Theorem \[thm\] from the above representation theoretic point of view.
Note added {#note-added .unnumbered}
----------
Robin Chapman has informed me that he has found a combinatorial proof of Theorem \[thm\] using Schur’s involution.
[99]{}
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[^1]: Work supported by the Australian Research Council
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Global hypothesis tests are a useful tool in the context of, e.g, clinical trials, genetic studies or meta analyses, when researchers are not interested in testing individual hypotheses, but in testing whether none of the hypotheses is false. There are several possibilities how to test the global null hypothesis when the individual null hypotheses are independent. If it is assumed that many of the individual null hypotheses are false, combinations tests have been recommended to maximise power. If, however, it is assumed that only one or a few null hypotheses are false, global tests based on individual test statistics are more powerful (e.g., Bonferroni or Simes test). However, usually there is no a-priori knowledge on the number of false individual null hypotheses. We therefore propose an omnibus test based on the combination of p-values. We show that this test yields an impressive overall performance. The proposed method is implemented in the R-package omnibus.\
Keywords: Multiple testing, global null hypothesis
author:
- Andreas Futschik
- Thomas Taus
- Sonja Zehetmayer
bibliography:
- 'bibliography.bib'
title: An omnibus test for the global null hypothesis
---
Introduction
============
When testing multiple hypotheses, the global null hypothesis is often of specific interest. It states that none of the individual null hypotheses is false. In some applications, rejecting the global null can be a goal in itself, whereas in other situation such a test may occur as part of a more sophisticated multiple test procedure. Think for instance of the closure test principle, where the global null needs to be rejected before looking at specific tests. Also, in an ANOVA, the global null is usually tested before testing for pairwise differences.
In meta analysis, rejecting the global null implies an effect at least under some circumstances. Another application is experimental evolution, where several replicate populations of micro- or higher organisms are maintained under controlled laboratory conditions and their response to selection pressures is studied. Further applications where such a test is of interest in its own merit are testing for overall genomic differences in gene expression, and signal detection [see @ING].
Several approaches to test the global null hypothesis are known. If we assume alternative scenarios where all or most null hypotheses do not hold, combination tests , that sum up two or more independent transformed p-values to a single test statistic, have been recommended to maximize power. If, however, it is assumed that the null hypothesis holds in most cases, global tests based on individual test statistics are more powerful [e.g., Bonferroni @Simes86]. If a larger number of hypotheses is tested, and the alternative hypothesis holds sufficiently often, goodness of fit tests for a uniform distribution of p-values could also be used. They test however for any type of deviation from uniformity, and do not focus specifically on too small p-values. Under more specific models, such as the comparison of several normal means, more specialised tests such as a Tukeys multiple range test, or an ANOVA are further options.
Higher criticism, and checking for [*overall significance*]{} are alternative terms used instead of global testing. Originating from biblical science, the term [*higher criticism*]{} was first used by [@Tukey] in a statistical context. Making the point that a certain number of falsely rejected null hypotheses can be expected when testing several null hypotheses at level $\alpha$, he then proposed a [*second level significance test*]{} to check for [overall significance.]{} Later Donoho and Jin provided an asymptotic analysis of this and related tests, when the number of hypotheses tends to infinity [@Don-Jin1; @Don-Jin2] . Their results show that there are situations where there is sufficient power to detect deviations from the global null hypothesis, but no chance to reliably identify in which cases the alternative holds.
Our focus is on a general situation where independent p-values are available from several hypothesis tests that are assumed to be uniformly distributed under the null hypothesis. As there is often no a priori knowledge on the number of false individual null hypotheses, we propose a test that enjoys good power properties, both if few and many null hypotheses are false. Our test is based on cumulative sums of the (possibly transformed) sorted p-values.
In comparison to other available methods, our simulations show that this test yields an excellent overall behavior. It typically performs better than combination tests, if the alternative holds in only a few cases. If the alternative holds in most cases, it performs better than the Bonferroni and Simes test. The performance relative to methods that combine evidence across all p-values tends to be even better under those one-sided testing scenarios, where parameters are in the interior of the null hypothesis for some of the tests. For these tests, the corresponding p-values will be stochastically larger than uniformly distributed ones, reducing in particular the power of combination tests.
We also present real data applications in the context of meta analysis and experimental evolution.
Testing the global null hypothesis based on p-values
====================================================
Consider a multiple testing procedure with $m$ null hypotheses $H_{0i}$, $i=1,\dots,m$. We assume that $m$, possibly different, hypotheses tests are carried out leading to stochastically independent p-values $p_1,\dots,p_m$. Our focus is on testing the global null $$H_0=\bigcap_{i=1}^m H_{0i},$$ i.e, that none of the null hypotheses is false. We assume that the p-values are either uniformly distributed $$p_i\sim U_{[0,1]}$$ under the global null hypothesis $H_{0}$, or that the p-values are stochastically larger than uniformly distributed ones. In other words, we assume that $P(p_i\le x)\le x$ for $0\le x\le 1.$
Some tests for the global null hypothesis use a combined endpoint, summing up the evidence across all available p-values to a single test statistic (Fisher combination function, Stouffer test). Alternatively other approaches focus on those individual test statistics that lead to extreme p-values, such as in the Bonferroni and Simes tests. As combination tests aggregate evidence across all hypotheses, these tests are particularly powerful when there are (small) effects in many considered null hypotheses. When there are only a few (large) effects, global tests based on individual test statistics are more powerful. Other approaches are goodness of fit tests or higher criticism.
Omnibus test
------------
#### General outline
Starting with independent p-values $p_1,\dots,p_m$, we denote the sorted p-values by $$p_{(1)} \leq \dots \leq p_{(m)},$$ and transform them with a monotonously decreasing function $h(\cdot)$ so that small p-values lead to large scores. Possible choices for $h(\cdot)$ will be discussed below. Next we obtain the L-statistics $S_i=\sum_{j=1}^i h(p_{(j)})$, $i=1,\dots,m$. Each of these partial sums could in principle be chosen as a test statistic for the global test and the best choice in terms of power for a specific scenario will depend both on $(m_0,m_1)$ and the respective effect sizes. Since these quantities are unknown, we propose to select the most unusual test statistic out of $S_i$, $1\le i\le m$. If the scores $S_i$ were approximately normally distributed, we could standardize them to figure out how unusual they are. Here however, the distribution of the $S_i$ with small index will be closer to an extreme value distribution, Therefore, we transform the sums using the distribution function $G_i$ of $S_i$ under the global null hypothesis, and take $$T^{\ast}=\max_{1\le i\le m} G_i(S_i).$$ as our test statistic. Although the cumulative sums $S_i$ may be viewed as L-statistic, and conditions that ensure the asymptotic normality of L-statistics $m\rightarrow\infty$ are known (see e.g.[@Stigler69], these conditions are not satisfied for some of the $S_i$, and furthermore the number of hypotheses is small to moderate. We therefore estimate the distribution of $T^\ast$ by simulating uniformly distributed p-values under the global null. Notice however that for some underlying distributions of $h(p_j)$, such as uniform, exponential or (skewed) normal, exact distributions are available for $S_i$ ([@Crocetta], [@Nagaraja2006] ).
Later on, we will consider four transformations $h(p)$ in more detail:
- $h(p)=1-p$ (omnibus p)
- $h(p)=-\log p$ (omnibus log p)
- $h(p)=\Phi^{-1} (1-p)$ (omnibus z)
- $h(p)=p^{-\alpha}$ with $\alpha=0.5$ (omnibus power).
Notice that for small enough $p$, we have that $$1-p
\le \Phi^{-1} (1-p)\approx \sqrt{2\log(1/p)}
\le \log (1/p)\le p^{-\alpha}.$$ Thus different choices of $h(.)$ assign different relative weights to small p-values.
Alternative test statistics
---------------------------
We briefly explain the most popular approaches that use p-values for testing the global null hypothesis.
#### Fisher combination test
@Fisher32 proposed the combined test statistic given by $T=-\sum_{i=1}^m 2 \log p_i$. Under the assumption of independent uniformly distributed p-values, the null distribution is $T \sim \chi^2_{2m}$.
#### Stouffer’s z
Based on z-values $Z_i=z_{1-p_i}$, where $z_{1-p_i}$ denotes the $1-p_i$ quantile of the standard normal distribution, the combined test statistic is given by $Z=\sum_{i=1}^m Z_i/\sqrt m$. Assuming again independent uniformly distributed p-values under the global null, it can be easily seen that $Z \sim N(0,1)$ [@Stouffer].
#### Bonferroni test
The Bonferroni test rejects the global null hypothesis, if the minimum p-value falls below $\alpha/m$, i.e. $\min_i p_i \leq \alpha/m$ [see, e.g., @Dickhaus]. The Bonferroni test controls the family-wise error rate at level $\alpha$ in the strong sense. The test makes no assumption on the dependence structure of endpoints. For independent test statistics, $\alpha/m$ may be replaced by the slightly more liberal upper bound $1-(1-\alpha)^{1/m}.$
#### Simes test
An improvement of the Bonferroni test in terms of power was proposed by [@Simes86]. For the $m$ hypotheses $H_{0i}$, $i=1,\dots,m$ with p-values $p_i$, the Simes test rejects the global null hypothesis if for some $k=1,\dots,m$, $p_{(k)} \leq \alpha k /m$. In the last decades the Simes test has become very popular for testing individual hypotheses controlling the False Discovery Rate.
#### Higher criticism
Based on an idea by [@Tukey], [@Don-Jin1; @Don-Jin2] introduced the higher criticism HC to test the global null hypothesis of no effect for independent hypotheses. It is defined by $$HC^*_m=\max_{1 \leq i \leq \alpha_0m} \left\{ \sqrt m \frac{i/m-p_{(i)}}{\sqrt{p_{(i)}(1-p_{(i)})}} \right\}.$$ $\alpha_0$ is a tuning parameter often set to 1/2, and has been studied in particular for large scale testing problems.
#### Goodness of fit tests
For our global test problem of independent p-values and under a point null hypothesis, the p-values $p_i$, $i=1,\dots,m$, usually follow a uniform distribution $U(0,1)$. Thus any goodness of fit test for uniformity, such as the Kolmogorov-Smirnov (KS), the Chi-square and the Cramer-von Mises tests also provide tests for the global null hypothesis. The KS test, for example, would use the maximum distance between the empirical distribution function of the observed p-values and the uniform distribution function, $D_n=sup_{0\le x\le 1} |F_n(x)-x|$ as test statistic. A disadvantage of goodness of fit tests in our context is that they test not only for smaller than expected p-values but against any deviation from uniformity. As also confirmed by our simulations, these tests therefore provide lower power compared to more specialized tests in our situation (data not shown).
Results
=======
We start our simulation study by comparing the power of our test when different transformations $h(\cdot)$ are used. It will turn out that $h(p)=-\log p$ leads to a particularly good overall behavior across different scenarios, and we thus focus on this transformation when comparing our approach with alternative tests for the global null, such as the Bonferroni and the Simes procedure, as well as Fisher’s and Stouffer’s combination test. Although typically used for a large number of hypotheses, we will also consider higher criticism as a competing method (with the tuning parameter $\alpha_0=0.5$). As the asymptotic approximations do not necessarily hold for small numbers of hypotheses, we simulate critical values under the null model for this test.
We simulate different scenarios by varying both the total number $m$ of hypotheses, and the number $m_1$ of instances where the alternative holds. We assume independence between the p-values, which was a condition in our derivation of the omnibus test.
Although our test is based on p-values that may arise in a multitude of settings, we want to specify effect sizes and alternative distributions in an intuitive way, and therefore compute our p-values from normally distributed data with known variance $\sigma^2=1$ and equal sample sizes $n$. More specifically, we consider the one-sample z-test for one-sided hypotheses $$H_{0i}: \mu_i=0 \quad \mbox{versus}\quad H_{1i}:\, \mu_i>0, \; i=1, \dots, m,$$ for the mean of the observations.
In the simulations, we first assume that all alternatives have the same mean effect $\Delta/\sigma$ and for the true null hypotheses $\Delta=0$. Later on we also consider the following setups:
(i)
: Negative effect sizes that are in the interior of the null hypotheses: We assume that under the true null hypothesis, the data have a negative effect size of $-\Delta/\sigma$ and under the alternative hypotheses a positive effect size of $\Delta/\sigma$.
(ii)
: Different effect sizes of alternative hypotheses: We assume randomly chosen exponentially distributed effect sizes with a rate parameter of $3 \sqrt m_1$.
(iii)
: Different effect sizes of alternative hypotheses and different effect sizes in the interior of the null hypotheses: We assume randomly chosen exponentially distributed effect sizes with a rate parameter of $3 \sqrt m_1$ or $-3 \sqrt m_1$, respectively.
All computations were performed using the statistical language R [@R], the Fisher and the Stouffer combination test were calculated using the function combine.test in the survcomp package [@survcomp]. For each scenario at least 10000 simulation runs were performed.
For all following simulation results, the methods control the Type I error of $5\%$ if the global null hypothesis is true (data not shown).
Influence of the chosen transformation on the omnibus method
------------------------------------------------------------
Fig. 1 shows power curves for the omnibus test using the four proposed transformations. We consider $m =10$, $m_1 \in \{1,3,5,10\}$, and $\Delta/\sigma=0.3/\sqrt m_1$. These variants show similar power values for a lot of scenarios. Nevertheless the performance of the power (“power”) and identity transforms (“p”) seems to be somewhat less satisfactory. In particular the power transform performs considerably worse when the alternative is true in several instances, while giving only slightly better results in the case of only one true alternative. The [*z*]{} and [*log p*]{} transforms both show a good overall behavior. The [*log p*]{} transform performs slightly better for small $m_1$ (i.e. a few larger effects), whereas the omnibus [*z*]{} method turns out to be slightly better if $m_1$ is large (i.e. several smaller effects). In section \[powcomp\] we provide a between methods comparison of the worst case power across all possible choices of $m_1$ with constant cumulative effect sizes. According to Table \[tab1\], the omnibus [*log p*]{} transform slightly outperforms the $z$ transform. Thus we will use the [*log p*]{} transformation with our omnibus test subsequently.
![Power values for omnibus log p, power, z, and p are given for increasing $n$, $m=10$, $m_1 \in \{1,3,5,10\}$, $\Delta/\sigma=0.3/\sqrt m_1$.[]{data-label="fig:01"}](Powercurve_Transfxeffektm10.pdf){width="90.00000%"}
Power comparison between different testing methods \[powcomp\]
---------------------------------------------------------------
Figure \[fig:02\] shows power curves for omnibus log p, Bonferroni, Simes, Fisher, Stouffer test, and HC for $m =10$, $m_1 \in \{1,3,5,10\}$, and $\Delta/\sigma=0.3/ \sqrt m_1$. It can be seen that the omnibus method is among the top methods concerning power for all scenarios (black solid curves).
The Bonferroni and Simes methods give the best power results in the case of only one false null hypothesis, $m_1=1$, however, the difference to the omnibus log p variant of our test is only marginal. For increasing $m_1$ the power of the Bonferroni and Simes methods is inferior compared to all other methods. As expected the Simes test outperforms the Bonferroni procedure (or is equal), though, for the considered scenarios the improvement in power is only small.
The Fisher combination test is slightly superior in scenarios with large $m_1$ in comparison to the omnibus tests, however, it has low power for small $m_1$. E.g, for scenarios with $m_1=1$ the omnibus test has nearly 20 percentage points higher power than the Fisher test. The Stouffer test only shows competitive power values for high number of false null hypotheses for the considered scenarios. In contrast the HC method for $\alpha_0=0.5$ has similar power values as Bonferroni and Simes for $m_1=1$, for increasing $m_1$ the omnibus log p, Fisher, and Stouffer test are clearly more powerful.
![Power values for increasing $n$, $m=10$, $m_1 \in \{1,3,5,10\}$, $\Delta/\sigma=0.3 / \sqrt m_1$ for omnibus log p, Bonferroni, Simes, Fisher, and Stouffer test, HC test)[]{data-label="fig:02"}](Powercurve8_Fig1m10.pdf){width="90.00000%"}
#### Worst case behavior
We assess also the overall behavior of the statistical tests we considered by looking at the minimax power across scenarios that involve all possible numbers $m_1$ of true alternative hypothesis. We define the minimax power as then the lowest power across all these scenarios. With $m_1$ alternatives, the individual effect size was chosen $\Delta/\sigma=\gamma/\sqrt m_1$. This leads to a constant cumulative effect size of $\frac{\sqrt{m_1}\gamma}{\sqrt{m_1\sigma^2}}=\gamma/\sigma^2$. This constant cumulative effect size would lead to equal power for any value of $m_1$ with a likelihood ratio test in the simplified scenario assuming $m_1$ null hypotheses that are either all true or false. In the theoretical case that $m_1$ and the position of the $m_1$ hypotheses is known, an optimal test could be obtained this way that leads to constant non-centrality parameters for all values of $m_1$. The below table uses $\gamma=0.3$, leading to intermediate power values. As can be seen, the considered omnibus tests outperform the other tests with respect to the worst case behavior, with the omnibus [*log p*]{} test performing best.
--------------- --------- --------- --------- --------- --------- --------- -- -- -- --
$n=100$ $n=200$ $n=100$ $n=200$ $n=100$ $n=200$
Omnibus log p 0.63 0.92 0.50 0.84 0.23 0.50
Omnibus z 0.62 0.92 0.49 0.83 0.23 0.49
Omnibus p 0.59 0.90 0.46 0.82 0.22 0.48
Bonferroni 0.41 0.68 0.28 0.47 0.11 0.18
Simes 0.44 0.73 0.30 0.52 0.12 0.19
Fisher 0.49 0.83 0.35 0.67 0.15 0.28
Stouffer 0.24 0.39 0.16 0.25 0.09 0.11
HC half 0.53 0.86 0.40 0.73 0.14 0.30
--------------- --------- --------- --------- --------- --------- --------- -- -- -- --
: \[tab1\] Minimax power. Worst case power values for $m_1$ from 1 to $m$ (minimum over all simulation scenarios) for $n=\{100,200\}$, $m=\{10,20,1000\}$, $\Delta/\sigma=0.3/\sqrt m_1$.
#### Behavior for small numbers $m_1$ of true alternatives
To further compare the power of our omnibus [*log p*]{} test and Fisher’s test, we performed simulations when $m_1$ is small, either in absolute terms or compared to $m$. More specifically we considered $m_1=1$, $m_1=5$, as well as $m_1=m/10.$ We assigned the same fixed effect sizes $\Delta/\sigma \in \{0.25, 0.5\}$ to each alternative hypothesis. Figure 3 shows the power curves of the omnibus test (black curves) and the Fisher combination test (grey curves) for $n \in \{20,40\}$, and increasing $m$. The omnibus test provides a higher power in most scenarios. Only in the situation of small effect sizes ($\Delta=0.25$), the Fisher combination test behaves better under some circumstances. This occurs in particular when $m_1=5$, and $m$ fairly small, implying a fairly large proportion $m_1/m$ of alternatives. Notice however, that the difference in power is small in these cases compared to the excess power of the omnibus test for larger effect sizes.
![Omnibus log p (black line) and the Fisher combination test (grey line) for $n \in \{20,40\}$, $\Delta \in \{0.25, 0.5\}$, increasing $m$ and $m_1=m/10$, $m_1=1$, or $m_1=5$, respectively. []{data-label="fig:03"}](Powercurve10_Fig2.pdf){width="90.00000%"}
Distributed/negative effect sizes
---------------------------------
In Fig. 4 (first row) we show simulation results for distributed effect sizes with a mean effect $\Delta$ distributed according to an exponential distribution with a rate parameter of $3 \sqrt m_1 $ for $m_1 \in \{1,3,5\}$, $m=10$. Generally, the power values are much lower than for equal mean effect sizes. Still, the omnibus log p method has maximum power in nearly all scenarios, only for $m_1=10$ the Fisher combination test is more powerful.
![ Power values are given for increasing $n$, $m=10$, $m_1 \in \{1,3,10\}$ for omnibus log p, Bonferroni, Fisher, and HC. The first row shows results for distributed effect sizes of alternative hypotheses according to an exponential distribution with rate parameter $3 \sqrt m_1 $. The second row shows results for $\Delta/\sigma=-0.3/ \sqrt m_1$ under the null hypothesis and $\Delta/\sigma = 0.3/ \sqrt m_1$ under the alternative. The third row shows results for distributed effect sizes according to an exponential distribution under the alternative as well as under the null hypothesis. []{data-label="fig:04"}](Powercurve_all.pdf){width="100.00000%"}
Fig. 4 (second row) shows results for negative effect sizes under the null hypothesis, leading to p-values that are stochastically larger than uniform. A comparison with Figure 2 reveals that this does not much influence the power of the omnibus test ($\Delta=0.3 \sqrt m_1$ for $m_1 \in \{1,3,5\}$, $m=10$), but it reduces the power of the Fisher combination test a lot for small $m_1$. The same is true for the Stouffer test (not shown), as it also uses the sum over all (transformed) p-values. The power difference between the omnibus test and the Fisher combination test reaches more than 70 percentage points for, e.g., $m=10$, $m_1=1$, $\Delta=0.3$. The power of the Bonferroni test and of higher criticism changes even less compared to the omnibus test when parameters are in the interior of the null hypothesis.
If both alternative and null hypotheses have effect sizes distributed according to an exponential distribution (with a rate parameter of $3\sqrt m_1$ for alternative hypotheses and $-3\sqrt m_1$ for null hypotheses), the relative behavior of the methods (Fig. 4, third row) is qualitatively similar to that implied in the second row. As observed in the first row however, the power clearly decreases for all methods with randomly distributed effect sizes.
P-values from discrete data
---------------------------
The assumption of uniformly distributed p-values under the null hypothesis is not always satisfied. Besides the possibility of parameter values in the interior of the null hypothesis, also discrete models lead to p-values that are not uniformly distributed on the interval $[0,1]$. To also cover the case of discrete data, we performed a simulation study under a two sample binomial model. For the first group, the simulated data were $B(n,p_0)$ distributed, for the second group, again generated from $B(n,p_0)$ under the null hypothesis and from $B(n,p_1)$ under the alternative. Here, $n$ denotes the per-group sample size. A $\chi^2$-test with one degree of freedom was performed and a corresponding p-value was calculated. If both groups showed only successes or only failures, the p-value was set to $p=1$.
We first checked whether the type I error is still controlled under our discrete model. For this purpose we considered sample sizes $n$ between 10 and 100, as well as allele frequencies $p_0$ in $[0.05,\,0.5]$ under the null hypothesis. Although for small $n$ the distribution of the test statistic is not well approximated by the chi-square distribution, we nevertheless used the standard p-values produced by the R function [*chisq.test*]{}. Our simulations showed no violations of the type I error probability of $\alpha=0.05$. This is since the chi-square test tends to become conservative (and the p-values stochastically larger than uniform) for small $n$. In other testing situations where this is not the case, type error control may however be an issue.
Figure 5 provides the power obtained when using our omnibus test on several scenarios. The left plot shows the power values as a function of $n$ from 10 to 100 for omnibus log p for $m=10$, $m_1\in \{1,3,5,10\}$. The plot in the middle shows the power values as a function of $p_0$ with constant $n=50$ and $p_1=p_0+0.2$ increasing in the same amount as $p_0$. For the right plot, $p_0=0.4$, $n=50$ and $p_1$ is increasing from 0.4 to 0.9.
![Power values for discrete data simulation for omnibus log p for $m_1=\{1,3,5,10\}$, $m=10$. The left chart shows results for increasing $n$ and constant $p_0=0.4$, $p_1=0.6$; the chart in the middle, constant $n=50$, increasing $p_0$ and $p_1=p_0+0.2$; The right chart $n=50$, $p_0=0.4$ and increasing $p_1$.[]{data-label="fig:05"}](Powercurve_chisq1.pdf){width="100.00000%"}
Examples
========
Meta Analysis
-------------
In meta analysis the evidence from several studies on a topic is combined. There are several examples in the literature, showing that the efficacy of a treatment can vary among studies. Reasons for such a variation can among other factors be differences in the underlying study populations, or environmental factors. If effect size estimates are available for all considered studies, a random effect meta analysis is often carried out. Global tests, such as the Fisher and the Stouffer test are a popular alternative option that does not require effect size estimates.
As an illustration, we applied our omnibus test to a data set from a meta analysis provided by the R-package metafor [@metafor]. We chose the data set dat.fine1993 where results from 17 studies are presented which compare post operative radiation therapy with or without adjuvant chemotherapy in patients with malignant gliomas [@Fine]. For each study the data set specifies the number of patients in the experimental group (receiving radiotherapy plus adjuvant chemotherapy) as well as the number of patients in the control group (receiving radiotherapy alone). In addition the number of survivors after 6, 12, 18, and 24 months follow-up within each group is given. One of the 17 studies recorded survival only at 12 and 24 months. For illustration purposes we performed a separate meta analysis for each time point and calculated a $\chi^2$-test (or Fisher’s exact test, where appropriate) for each study. The resulting p-values were then applied to test the global null hypothesis using the following methods: Bonferroni, Simes, Fisher, Stouffer, higher critisism, and omnibus log p.
Table \[tab2a\] shows the resulting p-values for the global tests. Note that the table does not display the results for Stouffer’s method which in all cases results in a p-value close to 1 and will not be discussed further. As in the simulation study, the omnibus method is among the top methods for all time points except for the 12 months data, were the p-value of the Fisher combination test is approximately one third smaller than the p-values of the omnibus method. For the 6 months data, however, the advantage of the omnibus method as well as Bonferroni and Simes methods (all p-values between 0.12 and 0.13) over the Fisher test (p-value: 0.51) is considerable. The largest p-value across all scenarios turns out to be smallest for the omnibus test.
----------- --------- ------------ ------- -------- ------- -- --
omnibus
$log p$ Bonferroni Simes Fisher HC
6 months 0.119 0.118 0.118 0.509 0.500
12 months 0.257 0.406 0.235 0.178 0.355
18 months 0.116 0.279 0.279 0.094 0.152
24 months 0.013 0.006 0.006 0.019 0.033
----------- --------- ------------ ------- -------- ------- -- --
: \[tab2a\] Meta analysis example I. Global tests have been applied to a meta analysis comparing post operative radiation therapy with or without adjuvant chemotherapy in patients with malignant gliomas. The p-values of the methods are shown when testing the global null hypothesis at different time points.
We next analysed the data examples from the R-package metap [@metap]. We used five of the eight different data examples, ignoring three that involve only hypothetical data. For each of these data sets a vector of p-values of lengths ranging from 9 to 34 is provided in the package. For instance the data taken from the meta analysis by [@Sutton] involves 34 randomized clinical trials where cholesterol lowering interventions were compared between treatment and control groups. The actual treatments were mostly drugs and diets. For each study, a test was performed to analyze, if the effect sizes (log Odds Ratio) are smaller than 0 (one-sided test), and p-values were calculated based on the normal distribution ([@Sutton], Table 14.3). For details on the other data sets we refer to the original publications, for references see the documentation of the metap package. Note that for some studies p-values were derived from independent subgroup analyses.
Table 3 compares again different tests of the global null in terms of their p-values. Three of the methods (Simes, Fisher, log p) lead to significant p-values at level $\alpha=0.05$ for four of the five data sets. The omnibus log p method however, is the only test that also provides four significant results at level $\alpha=0.01$.
---------- ---------- ------------ ----------- ---------- ---------- ------- -- -- --
omnibus
$log p$ Bonferroni Simes Fisher Stouffer HC
Sutton 0.24 0.13 0.13 0.79 1 0.57
mourning 0.007 0.07 0.04 0.017 0.11 0.013
naep $<$0.001 $<$0.001 $<$0.001 $<$0.001 $<$0.001 0.056
teach 0.0007 0.019 0.019 0.0014 0.0077 0.24
validity $<$0.001 $<$0.001 $<$0.0001 $<$0.001 $<$0.001 0.025
---------- ---------- ------------ ----------- ---------- ---------- ------- -- -- --
: \[tab2\] Meta analysis example II. The table states the p-values obtained from several global null hypothesis tests. The underlying data have been taken from the examples provided with the R-package [*metap.*]{}
Experimental Evolution
----------------------
With the development of large scale inexpensive sequencing technologies, experiments became popular that aim to elucidate biological adaptation at the molecular level of DNA and RNA. In such experiments, organisms are often exposed to stress factors for several generations, and their genetic adaptation is studied. With microorganisms, such stress factors can for instance result from antibiotics, with the adaptation being resistance. With higher organisms examples of stress factors are temperature, or toxic substances. While evolution in nature usually takes place only once under comparable circumstances, experimental evolution can be done with replicate populations. Among other things, replication permits to investigate the reproducibility of adaptation, a key topic in evolutionary genetics. The statistical challenge is to identify genomic positions (called loci) involved in adaptation. There is a large number of candidate loci, for which adaptation has to be distinguished from random temporal allele frequency changes due to genetic drift, as well as sampling and sequencing noise.
Furthermore, recent research suggests that replicate populations often do not show a consistent behavior, with signals of adaptation showing up partially at different loci. Two biological explanations for this finding is that beneficial alleles may be lost due to drift, and that the same adaptation at a phenotypic level can often be achieved in multiple ways at the genomic level.
When testing for significant allele frequency changes, a test like our omnibus test is therefore desirable, as it enjoys good power also when signals of adaptation are not consistent across replicates. We illustrate the application of our omnibus $log p$ test to data from an experiment on [*Drosophila*]{} described in @Griffin.
![Manhattan plots of negative logarithm of p-values from a genome wide scan of five replicate populations. The p-values have been corrected for multiple testing. Data are taken from @Griffin. []{data-label="figG01"}](exp-ev1.jpg){width="80.00000%"}
![Plot of combined evidence across replicates. Manhattan plots of the negative logarithm of the p-values obtained with our (log-p) omnibus test. The p-values have been corrected for multiple testing. Data are taken from @Griffin. []{data-label="figG02"}](exp-ev2.jpg){width="80.00000%"}
Discussion
==========
In this manuscript we introduced new non-parametric omnibus tests for testing the global null hypothesis. Our proposed approach enjoys very good power properties, no matter in how many cases the alternative holds. In our comparison with alternative approaches, it is not always the best method, but we did not find scenarios, where the omnibus test performs considerably worse than the best alternative method for a given setup (as it is the case, e.g., for Bonferroni and Simes for large number of alternative hypotheses or Fisher and Stouffer for small number of alternative hypotheses).
For our test, we compute successive cumulative sums of the suitably transformed sorted individual p-values. The most unusual cumulative sum is then obtained by computing the p-value of each sum under the global null hypothesis. The smallest p-value is then used as test statistic.
We considered different transformations of the initial p-values $p_i$, in particular $1-p_i$, $-\log(p_i)$, $\Phi^{-1}(1-p_i)$, and $p_i^{-1/2}$. Our results showed only small differences in power between the transformations. However, the log p transfrom seems to lead to a particularly good trade off in power across many scenarios.
As expected the Simes test outperforms the Bonferroni procedure (or is equal) in the simulation study, though, for the considered scenarios the improvement in power is not remarkable.
All our simulations are based on one-sided tests, but the methods also work for the two-sided testing scenario. For two-sided tests however, it is also possible to reject the global null hypothesis even when the individual hypotheses show clear effects in differing directions.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study the voter model, under node and link update, and the related invasion process on a single strongly connected component of a directed network. We implement an analytical treatment in the thermodynamic limit using the heterogeneous mean field assumption. From the dynamical rules at the microscopic level, we find the equations for the evolution of the relative densities of nodes in a given state on heterogeneous networks with arbitrary degree distribution and degree-degree correlations. We prove that conserved quantities as weighted linear superpositions of spin states exist for all three processes and, for uncorrelated directed networks, we derive their specific expressions. We also discuss the time evolution of the relative densities that decay exponentially to a homogeneous stationary value given by the conserved quantity. The conservation laws obtained in the thermodynamic limit for a system that does not order in that limit determine the probabilities of reaching the absorbing state for a finite system. The contribution of each degree class to the conserved quantity is determined by a local property. Depending on the dynamics, the highest contribution is associated to influential nodes reaching a large number of outgoing neighbors, not too influenceable ones with a low number of incoming connections, or both at the same time.'
author:
- 'M. Ángeles Serrano'
- Konstantin Klemm
- Federico Vazquez
- 'Víctor M. Eguíluz'
- Maxi San Miguel
title: 'Conservation laws for voter-like models on directed networks'
---
Introduction
============
Conservation laws are intimately related to symmetries in the systems they hold for. They play an important role in the characterization and classification of different nonequilibrium processes of ordering dynamics. For example, in Kinetic Ising models one distinguishes between Glauber (spin flip) and Kawasaki (spin exchange) dynamics. Kawasaki dynamics fulfils a microscopic conservation law, such that the total magnetization is conserved in each individual dynamical step of a stochastic realization. This conservation law does not hold for Glauber As a consequence, the Glauber and Kawasaki dynamics give rise to different scaling laws for domain growth in coarsening processes [@Gunton:1983], and they define different nonequilibrium universality classes.
In other types of nonequilibrium lattice models non-microscopic conservation laws are known to hold. They are statistical conservation laws in which the conserved quantity is an ensemble average defined over different realizations of the stochastic dynamics for the same distribution of initial conditions. Examples of such conservation laws occur for the voter model [@Clifford:1973; @Holley:1975] or the invasion process [@Castellano:2005]. In particular, the role of the conservation law of the magnetization and of the $Z_2$ symmetry (±1 states) in the voter dynamics universality class has been studied in detail in the critical dimension d = 2 of regular lattices [@Dornic:2001]. The voter model is a paradigmatic model of consensus dynamics in the social context [@SanMiguel:2005; @Castellano:2008] or, in the biological context, of competition of plant species in ecological communities [@Chave:2001]. In general, any Markov chain with at least two absorbing states reachable from all other configurations has a conserved quantity when averaged over the ensemble. Such a quantity determines the probability to eventually reach a particular absorbing configuration in a finite system. In some cases, this conservation law is of rather trivial nature as in the zero temperature Ising Glauber dynamics where the magnetization sign is conserved. The voter model, the zero temperature Ising Glauber dynamics, and other related models of language evolution [@Castello:2006] or population dynamics [@Tilman:1997], belong to the class of models with two absorbing states while epidemic spreading dynamics, like the contact process [@Boguna:2008a] or the Susceptible-Infected-Susceptible model [@Romualdo:2001], usually have a single absorbing state with no conservation law.
While some of these questions have been studied for spin lattice models for a long time, conservation laws for dynamical processes on complex networks [@Albert:2002; @Dorogovtsev:2003; @Newman:2003; @BarratBook:2008] still remain a challenge. This issue has been considered for the voter model [@Clifford:1973; @Holley:1975] or the invasion process [@Castellano:2005] on undirected uncorrelated networks [@Wu:2004; @Suchecki:2005b; @Sood:2005; @VazquezF:2008]. The link-update dynamics for the voter model has been found to conserve the global magnetization [@Suchecki:2005a], while the node update dynamics [@Suchecki:2005a] and the invasion process [@Sood:2005] preserve a weighted global magnetization where the contribution of each spin is calibrated by some function of the degree of the corresponding node in the undirected network. Such ensemble average conservation laws characterize processes with two absorbing states accessible to the dynamics, that compete to maintain an active state in the thermodynamic limit. In finite networks, the conserved quantities give the probabilities of reaching the uniform states and so act as a bridge that enables some probabilistic predictive power of the final dynamical state based on information about the initial conditions. In addition, different finite size dynamical scaling properties can be related to different conservation laws [@Suchecki:2005a].
Much less has been done exploring dynamical processes on directed networks, with the exception of the Ising model [@Sanchez:2002] and Boolean dynamics mainly applied to biological problems [@Kauffman:1969]. However, interactions between pairs of elements are asymmetric in different systems including some social networks [@Newman:2001c], where social ties are perceived or implemented differently by the two individuals forming the connected pairs. Directed network representations rather than undirected ones become more informative and adjusted to reality. In general, directed networks present characteristic large-scale connectivity structures, the so-called bow-tie architecture formed by a strongly connected component as a core structure and peripheral in- and out-components [@Broder:2000]. This organization, coupled to the initial condition of the dynamics running on top, have an impact both on the evolution of the processes and the final possible states of the systems [@Lieberman:2005; @MinPark:2006; @Jiang:2008]. In the voter model, leaf nodes in the in-component never change their state thus sending an invariable signal that can potentially propagate to the rest of the components of the system. This is closely related to phenomena such as the presence of zealots [@Mobilia:2003; @Mobilia:2007] in undirected networks. Both input or output directional large-scale components and zealotry imply at the end an external forcing on the dynamical processes that prevents reaching one of the absorbing states even for a finite network. This is clearly illustrated by the evolution of dynamical processes running on networks at the transition from a pure strongly connected component to a complete bow-tie structure. In an isolated and strongly connected component, the voter dynamics keeps an active dynamical state in the thermodynamic limit, but it leads to a consensus (absorbing state) in a finite network as it happens on undirected networks. Thus, the appearance of an input component in the large-scale structure of the network prevents the system from reaching an absorbing state for random initial conditions [@MinPark:2006].
In this paper, we focus on dynamics of coupled two-state spin variables and consider conserved quantities that are weighted sums of the spin values. Specifically, we investigate the form of the conservation law for the voter model — under node and link update — and the invasion process in directed networks with arbitrary degree distribution and degree-degree correlations. The directionality of the interactions is therefore encoded in the topology. We restrict to a single strongly connected component so that the absorbing state can be reached in a finite system, what seems realistic for a number of densely connected real networks like the world trade web [@Serrano:2007b]. In Sec. II, we present a detailed study of the node update version of the voter model and implement an analytical treatment using the heterogeneous mean field assumption in the thermodynamic limit. From the dynamical rules at the microscopic level, we find the equations for the evolution of the relative densities of nodes in one of the two possible states on heterogeneous networks with arbitrary degree distribution and degree-degree correlations. In this case, we prove that a conserved quantity as a weighted linear superposition of spin states exists. In Sec. III, we discuss the node-update voter model in uncorrelated directed networks to derive analytical expression for the conservation law and we also discuss the exponential decay of the relative densities to their homogeneous stationary value, which is basically a function of the conserved quantity. We show how the conserved quantity determines the probability of reaching one of the two states in a finite network. In Sec. IV and V, we present the results of applying the same methodology to the voter model with link update and the invasion process, respectively. We conclude in Sec. VI with a summary of results and open questions for future research.
The voter model on strongly connected components
================================================
In the voter model under node update (VM), each node of a network can exist in one of two possible states, $1$ or $0$ [^1]. In a single dynamical event, a randomly selected node copies the state of one of its neighbors, also selected at random. The link update dynamics of the Voter model selects instead a link [@Suchecki:2005a]. Time is increased by $1/N$, so that the physical time is incremented by 1 after $N$ of such events. On undirected networks, the node-update voter model conserves the ensemble average of a weighted magnetization, where the contribution of each spin is multiplied by the degree of the corresponding node.
As defined above, the interactions in the voter dynamics are instantaneously asymmetric since the updates always go in the same direction once the original node is chosen independently of the undirectionality of the substrate. Hence, the discussion of the voter model on directed networks comes out as a natural one, where the directionality of the interaction is decoupled from the dynamics and encoded in the structure of the substrate. The straightforward generalization of the voter model on directed networks under node update consists of selecting a node at random, and then assigning to it the state of one of its incoming neighbors, also chosen at random. We will discuss this dynamics next in this section and Sec. III, and the voter model with link update will be discussed later in Sec. IV.
Directed networks
-----------------
The topological structure of directed networks is more complex than the one of undirected graphs. In purely directed networks, without bidirectional links, the edges are differentiated into incoming and outgoing, so that each vertex has two coexisting degrees $k_\textrm{in}$ and $k_\textrm{out}$, with total degree $k=k_\textrm{in}+k_\textrm{out}$. Hence, the degree distribution for a directed network is a joint degree distribution $P(k_\textrm{in},k_\textrm{out})\equiv P({\bf k})$ of in- and out-degrees that in general may be correlated. We consider degree correlations $P_\textrm{in}({\bf k'} | {\bf k})$ and $P_\textrm{out}({\bf k'} | {\bf k})$, which respectively measure the probability to reach a vertex of degree ${\bf k'}$ leaving from a vertex of degree ${\bf k}$ using an incoming or outgoing edge of the source vertex, and are related through the following degree detailed balance condition [@Boguna:2005] $$k_\textrm{out} P({\bf k}) P_\textrm{out}({\bf k'} | {\bf k})=k'_\textrm{in} P({\bf k'}) P_\textrm{in}({\bf k} | {\bf
k'}).
\label{detailed_dir}$$ This ensures that the network is closed and $\left< k_\textrm{in} \right>=\left< k_\textrm{out} \right>$. Apart from the prescribed degrees and two point correlations, networks are maximally random.
At the macroscopic scale, the giant weakly connected component, [*i.e.*]{}, the set of nodes that can communicate to each other when considering the links as undirected [@Molloy:1995; @Molloy:1998; @Havlin:2000; @Callaway:2000; @Newman:2001b], becomes internally structured in three giant connected components, as well as other secondary structures such as tubes or tendrils, forming a bow-tie architecture [@Broder:2000]. The main component is the strongly connected component (SCC), a central core formed by the set of vertices that can be reached from each other following a directed path. The other two main components are peripheral components, the in component (IN) formed by all vertices from which the SCC is reachable by a directed path but that cannot be reached from there, and the out component (OUT) formed by all vertices that are reachable from the SCC by a directed path but cannot reach the SCC themselves. Percolation theory for purely directed networks was first developed for uncorrelated networks [@Callaway:2000; @Newman:2001b; @Dorogovtsev:2001; @Dorogovtsev:2001b; @Serrano:2007c], and directed random networks with arbitrary two point degree correlations and bidirectional edges [@Boguna:2005].
We restrict to networks forming a strongly connected component without peripheral components that would act on the SCC as sources of external forcing. We will see that within the strongly connected component, conservation laws preserve weighted magnetizations, where the weights are dictated by the directed degrees.
From microscopic dynamics to the drift equation under the heterogeneous mean field assumption
---------------------------------------------------------------------------------------------
To study the time evolution of the system, we consider the drift part of the Langevin equation for the density of nodes in one of the states from a microscopic description of the evolution of single nodes’ states applying a heterogeneous mean field approach. To our knowledge, this methodology, which allows us to deal with dynamical processes on complex networks, was first presented in Refs. [@Catanzaro:2005b; @Catanzaro:2008] and recently used to study the contact process [@Boguna:2008a]. In [@VazquezF:2008b; @VazquezF:2008], a homogeneous mean field pair approximation was instead developed.
We focus on the microscopic state of nodes at some time $t$. Let $s_u(t), u=1,...,N$, be a stochastic binary variable defined for each of the $N$ nodes in the network which describes its state, $0$ or $1$. The vector ${\bf s}(t) \equiv \{s_u(t)\},
u=1,\ldots,N$, completely defines the dynamical state of the system at time $t$. Two more independent binary stochastic variables $\mu(dt)$ and $\xi_u$ are defined in order to model the transitions between states of single nodes in an iteration. After a time interval $dt$, the variable $\mu(dt)$ for a given node $u$ takes the value $1$ or $0$ if $u$ was chosen or not, respectively. In case node $u$ was selected, then $\xi_u$ assumes the value 1 \[0\] if $u$ copies a neighbor with state 1 \[0\]. We assume that the occurrence of events in the voter dynamics follows an independent Poisson process for each node, with constant rate $\lambda$ for all of them, which corresponds to a Montecarlo step. In the remainder we be set to $\lambda=1$ without loss of generality. Thus, $\mu(dt)$ and $\xi_u$ have probability distributions $$P(\mu(dt))= dt \delta_{\mu(dt),1}+(1- dt) \delta_{\mu(dt),0},
\label{mu}$$ $$P(\xi_u) = \Phi_u/k_{u,\textrm{in}} \delta_{\xi_u,1} +
(1-\Phi_u/k_{u,\textrm{in}})\delta_{\xi_u,0},
\label{xi}$$ where $k_{u,\textrm{in}}$ is the incoming degree of node $u$, and we have defined $\Phi_u(t)=\sum_v a_{vu}s_v(t)$. The adjacency matrix $\{a_{vu}\}$ encodes the topological properties of the directed network. Element $a_{vu}$ has value one if there is a directed link from $v$ to $u$ and zero otherwise, so that $\Phi_u(t)$ stands for the number of state-one incoming neighbors of node $u$ at time $t$. The matrix $\{a_{vu}\}$ is symmetric for undirected networks but for directed ones it is in general asymmetric.
In terms of the above variables, the dynamical state $s_u(t)$ of node $u$ after an increment of time $dt$ is $$s_u(t+dt)=\mu(dt)\xi_u + (1-\mu(dt))s_u(t). \label{micro1node}$$ This equation, together with Eqs. (\[mu\]) and (\[xi\]), give the complete description of the evolution of the system, making the formalism general and applicable to any network structure.
Although exact, this microscopic description is unmanageable. In order to reduce the degrees of freedom, we apply a heterogeneous mean-field hypothesis [@Romualdo:2001] so that nodes with the same degree ${\bf k}$ are assumed to be statistically independent and equivalent and can be aggregated in the same degree class $\Upsilon({\bf k})\equiv \Upsilon(k_\textrm{in},k_\textrm{out})$. Properties are then defined for each degree class, that will be characterized by the relative density $m_{\bf k}(t)$, the ratio between the number of state-one nodes within class $\Upsilon({\bf
k})$ and its number of nodes $N_{\bf k}$, $$m_{\bf k}(t)=\frac{\sum_{u\epsilon\Upsilon({\bf k})}s_u(t)}{N_{\bf k}}.
\label{defrho}$$ In the thermodynamic limit, the relative densities $m_{\bf k}(t)$ can be considered as continuous variables. Their time evolution can be described by a Langevin equation [@Gardiner:2004] with drift and diffusion coefficients that are respectively given by the first and second infinitesimal moments of the stochastic variables $m_{\bf k}(t)$. Those moments can be derived from the microscopic equation Eq. (\[micro1node\]) along with the definition in Eq. (\[defrho\]). In the thermodynamic limit, it is possible to prove that the diffusion term has a dependence $1/\sqrt{N_k}$ on the system size as for undirected networks [@VazquezF:2008], so that the drift term $A_k$ will dominate. It is given by the average value over all possible configurations of $m_{\bf k}(t+dt)$ conditioned to the state of the system at time $t$, $$\left< m_{\bf k}(t+dt)\right>_{m_{\bf k}(t)}=m_{\bf k}(t)+A_k(t)dt.$$ From the microscopic dynamics $$\left< s_u(t+dt) \right>_{{\bf s}(t)}=s_u(t) -dt \left[
s_u(t)-\Phi_u(t)/k_{u,\textrm{in}} \right],
\label{averagemicro}$$ and summing this equation for all nodes in the degree class ${\bf
k}$ and dividing by the number of nodes $N_{\bf k}$, we arrive to $$\left< m_{\bf k}(t+dt)\right>_{m_{\bf k}(t)}=m_{\bf k}(t) - dt
\left[m_{\bf k}(t)-\frac{1}{N_{\bf k}}\frac{1}{k_\textrm{in}
}\sum_{u\epsilon\Upsilon({\bf k})} \Phi_u(t) \right], \label{averagemeso}$$ and from here to $$A_k(t)=-m_{\bf k}(t)+\frac{1}{N_{\bf k}}\frac{1}{k_\textrm{in}
}\sum_{u\epsilon\Upsilon({\bf k})} \Phi_u(t).
\label{driftermV}$$ The adjacency matrix contained in $\Phi_u(t)$ can be coarse-grained as well, so that a differential equation for the relative densities can eventually be written. This coarse-graining restricts the validity of the equations to random complex networks (and not lattices), since we assume all nodes in the same degree class to be statistically independent. With these assumptions, $$\begin{aligned}
\sum_{u\epsilon\Upsilon({\bf k})} \Phi_u(t)&=&\sum_{\bf
k'}\sum_{v\epsilon\Upsilon({\bf k'})}\sum_{u\epsilon\Upsilon({\bf k})}
a_{vu}s_v(t)\nonumber \\ &=& \sum_{\bf k'}\bar{a}_{\bf k'k}N_{\bf k} N_{\bf
k'}m_{\bf k'}(t).\end{aligned}$$ At this point, we restrict to directed networks organized at the large scale into a SCC without IN and OUT. This allows us to write $$\bar{a}_{\bf k'k}=\frac{E_{\bf k'k}}{N_{\bf k} N_{\bf k'}}=\frac{k'_\textrm{out}
P_\textrm{out}({\bf k}|{\bf k'})}{N_{\bf k}}=\frac{k_\textrm{in}P_\textrm{in}({\bf k'}|{\bf k})}{N_{\bf k'}},$$ where $E_{\bf k'k}$ is the asymmetric matrix of the number of connections from the class of vertices of degree ${\bf k'}$ to the class of vertices of degree ${\bf k}$, and where we have made use of the detailed balance condition Eq. (\[detailed\_dir\]).
Inserting these results into Eq. (\[driftermV\]), we arrive to the equation for the evolution of the relative density in the degree class ${\bf k}$ of a purely directed correlated network (disregarding diffusion terms), $$\frac{d m_{\bf k}(t)}{dt}=-m_{\bf k}(t)+ \sum_{\bf k'} P_\textrm{in}({\bf k'}|{\bf k})
m_{\bf k'}(t).
\label{driftV}$$ Let us recall that this result is valid for the ensemble of networks defined by the degree distribution $P({\bf k})$ and the degree correlations $P_\textrm{in}({\bf k'} | {\bf k})$ and $P_\textrm{out}({\bf k'} | {\bf k})$, but otherwise maximally random. Notice that big enough networks present good statistical quality at the level of degree classes and are also well described by this equation. Finally, in the thermodynamic limit, the Langevin equation loses its noise term because of the dependence on the system size and reduces to Eq. (\[driftV\]), so that $m_{\bf k}(t)$ becomes a deterministic variable. Nevertheless, since the process is linear, Eq. (\[driftV\]) is always valid even for finite systems understanding that in this case the variables are averages over realizations of the process with the same distribution of initial conditions.
Conserved quantity on directed networks with degree-degree correlations
-----------------------------------------------------------------------
For correlated networks, $m_{\bf k}(t)=\sum_{\bf k'}
P_\textrm{in}({\bf k'}|{\bf k}) m_{\bf k'}(t)$ in the stationary state and hence all relative densities are entangled through topological correlations. This equation corresponds indeed to an eigenvector problem, since $\{m_{\bf k}(t)\}$ can be thought as the eigenvector of the matrix $\{P_\textrm{in}({\bf k'}|{\bf k})\}$ with eigenvalue one.
We prove next that, within the heterogeneous mean field approach and for the correlated directed networks we are considering, there is a conserved quantity given as a linear superposition of the form $\omega=\sum_{\bf k} \varphi_{\bf k}
m_{\bf k}(t)$. From Eq. (\[driftV\]), its evolution is given by $$\frac{d \omega}{dt}=-\omega+ \sum_{\bf k}\sum_{\bf k'} \varphi_{\bf k}
P_\textrm{in}({\bf k'}|{\bf k}) m_{\bf k'}(t),$$ and imposing that $d\omega/dt=0$, we obtain $$\sum_{\bf k}\varphi_{\bf k}m_{\bf k}(t)=\sum_{\bf k}\sum_{\bf k'} P_\textrm{in}({\bf
k}|{\bf k'}) \varphi_{\bf k'}m_{\bf k}(t). \label{cq}$$ For each density $$\varphi_{\bf k}=\sum_{\bf k'} P_\textrm{in}({\bf k}|{\bf k'}) \varphi_{\bf k'}.
\label{varphiV}$$ This is an eigenvector equation that has a solution if the matrix $\{P_\textrm{in}({\bf k}|{\bf k'})\}$ has an eigenvalue equal to one with $\{\varphi_{\bf k}\}$ the corresponding eigenvector. One can prove that this eigenvector with eigenvalue one exists by summing both sides of the previous equation over ${\bf k}$. Using the normalization of the conditional probability $\sum_{\bf k}
P_\textrm{in}({\bf k}|{\bf k'})=1$, one eventually arrives to a trivial identity [^2]. The fact that the coefficients $\varphi_{\bf k}$ that modulate the contributions of the different $m_{\bf k}$ to the conserved weighted magnetization correspond to the entries of the eigenvector of a certain characteristic matrix with eigenvalue one also applies to other similar dynamical processes, such as the link dynamics and the invasion process, as we will show.
This proves that a conserved quantity of the form of a linear functional exists but, in general, it is not possible to derive its value without further specifying the form of the degree-degree correlations in the network.
Voter model on uncorrelated SCCs
================================
When two-point correlations are absent, the transition probabilities become independent of the degree of the source vertex. In this situation, $$P_\textrm{out}({\bf k'} | {\bf k})=\frac{k'_\textrm{in} P({\bf k'})}{\langle k_\textrm{in} \rangle} \mbox{
, } P_\textrm{in}({\bf k'} | {\bf k})=\frac{k'_\textrm{out} P({\bf k'})}{\langle k_\textrm{in} \rangle},
\label{transitioninout_uncorrelated}$$ and using these expressions, Eq. (\[driftV\]) becomes $$\frac{d m_{\bf k}(t)}{dt}=-m_{\bf k}(t)+ \omega_\textrm{out},
\label{driftVunc}$$ where we have defined $$\omega_\textrm{out}=\frac{1}{\left< k_\textrm{in} \right>}\sum_{\bf k} k_\textrm{out}P({\bf k}) m_{\bf k}(t).
\label{wo}$$ Therefore, in the stationary state $m_{\bf k}=\omega_\textrm{out} \mbox{ } \forall
{\bf k}$ and $\omega_\textrm{out}$ is a conserved quantity in uncorrelated networks, which immediately follows from Eq. (\[driftVunc\]). In general, it is not preserved in strongly connected components of directed networks with degree-degree correlations. This is in contrast to undirected networks, where the conserved quantity $\omega=\left( \sum_{k} kP(k)m_{k}(t) \right) / \langle k
\rangle$ is preserved even in the correlated case and indeed for any structure [@Suchecki:2005a]. Going back to the uncorrelated case, notice that the out-degree is the quantity that weights the contribution of the nodes to the conserved quantity. From a local perspective, what seems therefore important in the VM is to be able to influence a large number of partners
In uncorrelated networks, the convergence of the state-one relative densities to their stationary value can be easily computed. From Eq. (\[driftVunc\]), taking into account that $\omega_\textrm{out}$ is a conserved quantity and for a given initial condition $m_{\bf k}(0)$, it is straightforward to arrive to the solution $$m_{\bf k}(t)=\omega_\textrm{out} + \left( m_{\bf
k}(0)- \omega_\textrm{out} \right) e^{-t},
\label{decayV}$$ where we have substituted $\left <k_\textrm{in} \right>$ by $\left
<k_\textrm{out} \right>$. Thus, all the densities decay exponentially fast to the stationary value $m_{\bf k}^{st} = \omega_\textrm{out}$ and the relaxation time is for all of them equal and independent of the degrees.
In the thermodynamic limit, the partially ordered stationary state is stable, while finite-size fluctuations eventually bring the system to one of the two possible unanimity states. The probability $P_1$ that the system ends up with all nodes in state one ($m_{\bf k}=1, \forall {\bf k}$) is given by the initial condition, that fixes the value of the conserved quantity at the beginning of the process. To see this, one takes into account that $\omega_\textrm{out}$ is an ensemble average conserved quantity of the form in Eq. (\[wo\]), from which $$\omega_\textrm{out}=P_1.$$ This is in agreement with the fact that, in general, the Markov property of a stochastic process, if present, trivially ensures that the exit probability is a conserved quantity corresponding to a time-translation invariance. If the process has one absorbing state, the exit probability has a constant value one but, if the process has two or more absorbing barriers, the probability of reaching one of those is not trivial any more.
It is also interesting to investigate what happens to the quantity $\upsilon_{i}(t)=\left( \sum_{\bf k} k_\textrm{in}P({\bf k})
m_{\bf k}(t)\right) / \langle k_\textrm{in} \rangle$, which involves in-degree instead of out-degree. In the uncorrelated case, and disregarding fluctuations, $\upsilon_{i}(t)=(\upsilon_{i}(0)-\omega_\textrm{out})e^{-t}+\omega_\textrm{out}$, that is, in general $\upsilon_{i}$ decays exponentially fast to $\omega_\textrm{out}$. The quantity $\upsilon_{i}(0)$ depends on the initial condition. If this is homogeneous over degree classes, then $\upsilon_{i}(0)=\omega_\textrm{out}$ and $\upsilon_{i}(t)$ remains constant.
In order to check the convergence of the sate one relative densities to the conserved quantity, we have run numerical simulations of the voter model dynamics on a random uncorrelated network of size $N=10^5$, scale-free in-degree distribution with exponent $2.5$ and exponential out-degree distribution. To obtain an initial state that is inhomogeneous in the densities $m_{\bf k}$, we have chosen an initial configuration in which half of the nodes with the lowest out-degree have state zero, and the other half have state one. In this way, initial densities $m_{\bf
k}(0)$ in classes with $k_\textrm{out}$ lower than $4$ were small or zero, while densities in classes with $k_\textrm{out}$ larger than $4$ were one.
In Fig. \[Fig1\], we plot the average of the conserved quantity $\omega_\textrm{out}$ and the densities for classes ${\bf
k}=(k_\textrm{in},k_\textrm{out})=(2,1)$, $(4,3)$ and $(3,9)$ vs time, over $100$ independent realizations starting from the same initial condition as mentioned above. As predicted by the theory, we observe that $\left< \omega_\textrm{out} \right>$ stays constant over time, whereas the three densities converge to the average of the stationary value $m_{\bf k}^{st}$, in a time of order $10$. We note that, apart from finite size fluctuations, the convergence of the densities to $m_{\bf k}^{st}$ happens for every realization. This can be seen in Fig. \[Fig2\], where we show the evolution of $m_{(2,1)}$ and $m_{(3,9)}$ vs $\omega_\textrm{out}$ in a single run. After a short transient, the densities and the conserved quantity start to evolve in a coupled manner (except from small deviations around the $m_{\bf k} = \omega_\textrm{out}$ line), they fluctuate from $0$ to $1$ until they reach the homogeneous zero-state. We also observe that fluctuations in $m_{(3,9)}$ are larger than in $m_{(2,1)}$, given that degree distribution make the number of nodes in class $(2,1)$ larger than in class $(3,9)$.
![Time evolution of the conserved quantity $\omega_\textrm{out}$ (circles) and the densities of state-one nodes $m_{\bf k}$ in degree classes ${\bf k}=(k_\textrm{in},k_\textrm{out})=(2,1)$ (squares), $(4,3)$ (diamonds) and $(3,9)$ (triangles), for the voter model dynamics. Curves correspond to averages over $100$ realizations on a single random uncorrelated network with $N=10^5$ nodes, scale-free in-degree distribution with exponent $2.5$ and exponential out-degree distribution. While $\left< \omega_\textrm{out} \right>$ remains roughly constant over time, the densities quickly decay to the stationary value $\left< \omega_\textrm{out} \right>$. The inset shows that the ratio between the densities and the conserved quantity is close to one during the entire evolution.[]{data-label="Fig1"}](Fig1.eps){width="3.4in"}
![Densities of state-one nodes $m_{(2,1)}$ and $m_{(3,9)}$ vs $\omega_\textrm{out}$ in a single realization of the voter model dynamics on the same network of Fig. \[Fig1\]. The trajectories of classes $(2,1)$ and $(3,9)$ start at the positions $(0.8,0)$ and $(0.8,1.0)$ respectively, then they quickly hit and move along the diagonal $m_{\bf k}=\omega_\textrm{out}$, until they reach the zero-state consensus point $m_{(2,1)}=m_{(3,9)}=0$.[]{data-label="Fig2"}](Fig2.eps){width="3.0in"}
Voter model with link update
============================
The same assumptions and procedures apply to the link-update voter model and the invasion process. The link update (LU) dynamics selects first a directed connection, so that the node at the tail will always transmit its state to the neighbor at the head.
The microscopic dynamics of the link-update voter model is described by $$s_u(t+dt)=\mu_u(dt)\xi_u + (1-\mu_u(dt))s_u(t), \label{micro1nodeLU}$$ where as for the voter dynamics $\xi_u$ is given by Eq. (\[xi\]) and the binary variable $\mu_u(dt)$ for the selection of a link has a probability distribution $$P(\mu_u(dt)) = k_{u,\textrm{in}} dt \delta_{\mu_u(dt),1} + (1-k_{u,\textrm{in}}
dt)\delta_{\mu_u(dt),0}.
\label{muLU}$$ A factor $\lambda/(N\left<k_\textrm{in}\right>)$ has been reabsorbed in the definition of $dt$. Proceeding as for the voter model (we skip the details), we arrive to the equation for the evolution of the relative densities $m_{\bf k}$ for the different degree classes, $$\frac{d m_{\bf k}(t)}{dt}=-k_\textrm{in}m_{\bf k}(t)+ k_\textrm{in}\sum_{\bf k'} P_\textrm{in}({\bf k'}|{\bf
k}) m_{\bf k'}(t).
\label{driftL}$$ Regarding the stationary state, the same result as for the voter model is found. The state-one relative densities behave again as $m_{\bf k}(t)= \sum_{\bf k'} P_\textrm{in}({\bf k'}|{\bf k}) m_{\bf
k'}(t)$, so that all the relative densities are entangled through topological correlations. We can once again prove, within the heterogeneous mean field approach and for correlated strongly connected components, that a conserved quantity of the form $\omega=\sum_{\bf k} \varphi_{\bf k} m_{\bf k}(t)$ exists and is defined by the eigenvector problem $$\tilde{\varphi}_{\bf k}=\sum_{\bf k'} P_\textrm{in}({\bf k}|{\bf k'})
\tilde{\varphi}_{\bf k'},
\label{varphiV2}$$ where now $\tilde{\varphi}_{\bf k}=k_\textrm{in} \varphi_{\bf k}$. In general, it is not possible to derive these coefficients without further specifying the form of degree-degree correlations in the network.
When two-point correlations are absent, $$\frac{d m_{\bf k}(t)}{dt}=-k_\textrm{in}m_{\bf k}(t)+k_\textrm{in}\omega_\textrm{out}(t).
\label{driftLunc}$$ In the stationary state, $m_{\bf k}=\omega_\textrm{out}(t) \mbox{ }
\forall {\bf k}$, but $\omega_\textrm{out}(t)$ is not a conserved quantity for the link update process as it was for the voter model. Instead, the conserved quantity is $$\begin{aligned}
\omega_\textrm{oi}&=&\left<\frac{k_\textrm{out}}{k_\textrm{in}} m_{\bf
k}(t)\right>/\left<\frac{k_\textrm{out}}{k_\textrm{in}} \right> \nonumber \\
&=&\sum_{\bf
k}\frac{k_\textrm{out}}{k_\textrm{in}}P({\bf k})m_{\bf k}(t)/\langle
\frac{k_\textrm{out}}{k_\textrm{in}}\rangle,
\label{consLU}\end{aligned}$$ which follows from Eq. (\[driftLunc\]). Compare this expression with that for the total magnetization in uncorrelated undirected networks $w=\omega=\left( \sum_{k}P(k)m_{k}(t) \right) / \langle k\rangle$ which corresponds to the conserved quantity for those structures [@Suchecki:2005a]. The dependence of the conserved weighted magnetization on the ratio between out- and in-degree for directed networks highlights the fact that in LU it is important to have both a high out-degree to be influential and at the same time to have a low in-degree not to be too influenceable. Notice that the ratio of the directed degrees is well defined since we are assuming that the network is organized at the macroscopic scale into a SCC without peripheral components all nodes having at least one incoming and one outgoing link. Finally, in finite systems the probability of the state-one absorbing state is given by the conserved quantity, $\omega_\textrm{oi}=P_1$, and so fixed by the initial condition.
The derivation of how the state-one relative densities converge to their stationary value in uncorrelated networks is more intricate than for the voter model, but we can make use of a quasi-stationary approximation [@Gardiner:2004] in order to solve Eq. (\[driftLunc\]), exploiting the fact that $\omega_\textrm{oi}$ is the conserved quantity. In the stationary state $\omega_\textrm{out}=\omega_\textrm{oi}$, and we approximate the equation by $$\frac{d m_{\bf k}(t)}{dt}=-k_\textrm{in} m_{\bf k}(t)+k_\textrm{in} \omega_\textrm{oi}.$$ For a given initial condition $m_{\bf k}(0)$, the solution is $$m_{\bf k}(t)=\omega_\textrm{oi} (m_{\bf
k}(0)-\omega_\textrm{oi})e^{-k_\textrm{in}t}.
$$ As in the voter model, all the densities decay exponentially fast to the stationary value $\omega_\textrm{oi}$, but in contrast not all the densities decay with the same velocity, which depends on the in-degree. Higher in-degree classes have smaller relaxation times and decay faster than lower ones, but the transient is always faster as compared to the VM.
[l|cccc]{} & $\omega_\textrm{corr}$ & $\omega_\textrm{unc}$ & $m_{\bf k}^{st}$ & $m_{\bf k}$\
\
VM & $\exists$ & $\omega_\textrm{out}=\frac{1}{\langle k_\textrm{out} \rangle}\sum_{\bf k}k_\textrm{out}P({\bf k})m_{\bf
k}(t)$ & $\omega_\textrm{out}$ & $m_{\bf k}^{st}+(m_{\bf k}(0)-m_{\bf
k}^{st})e^{-t}$\
\
LU & $\exists$ & $\omega_\textrm{oi}=\frac{1}{\langle k_\textrm{out}/k_\textrm{in} \rangle}\sum_{\bf k}k_\textrm{out}/k_\textrm{in}P({\bf k})m_{\bf
k}(t)$ & $\omega_\textrm{oi}$ & $m_{\bf k}^{st}+(m_{\bf k}(0)-m_{\bf
k}^{st})e^{-k_\textrm{in}t}$\
\
IP & $\exists$ & $\omega_\textrm{in}=\frac{1}{\langle 1/k_\textrm{in} \rangle}\sum_{\bf k}
\frac{1}{k_\textrm{in}}P({\bf k})m_{\bf k}(t)$ & $\omega_\textrm{in}$ & $m_{\bf k}^{st}+(m_{\bf
k}(0)-m_{\bf k}^{st})e^{-\frac{k_\textrm{in}}{\left<k_\textrm{in} \right>}t}$\
\[-0.2 cm\]\
\[table1\]
Invasion process
================
The invasion process (IP) picks nodes at random that export their state to a randomly chosen outgoing neighbor. A certain node $u$ will update its state in a passive form only when one of its incoming neighbors $v$ is selected as the first node in one iteration of the dynamics and then $v$ chooses $u$ among all its outgoing neighbors to transmit it its state. In this situation, it is more convenient to work with the probability of node $u$ undergoing a state update with final state 1, $\xi_u^{(1)}$, and the probability of node $u$ undergoing a state update with final state 0, $\xi_u^{(0)}$. The probability distributions of these dichotomic stochastic variables are $$\begin{aligned}
P(\xi_u^{(1)}) &=& \Phi_u^1 dt \delta_{\xi_u^{(1)},1} +
(1-\Phi_u^1 dt)\delta_{\xi_u^{(1)},0}, \\ P(\xi_u^{(0)}) &=&
\Phi_u^0 dt \delta_{\xi_u^{(0)},1} + (1-\Phi_u^0
dt)\delta_{\xi_u^{(0)},0},
\label{xiIP}\end{aligned}$$ with $$\begin{aligned}
\Phi_u^1(t) &=& \sum_v a_{vu}s_v(t)/k_{v,\textrm{out}},\\
\Phi_u^0(t)&=& \sum_v
a_{vu}(1-s_v(t))/k_{v,\textrm{out}}\end{aligned}$$ and the parameter $\lambda$ of the Poisson process for the happening of events reabsorbed in $dt$. Using these expressions, the dynamics is described at the microscopic scale by $$\begin{aligned}
s_u(t+dt)&=&\xi_u^{(1)}(dt)(1-\xi_u^{(0)}(dt)) \nonumber \\ &+&
(1-\xi_u^{(1)}(dt))(1-\xi_u^{(0)}(dt))s_u(t). \label{micro1nodeIP}\end{aligned}$$
Following the same methodology as for the voter model, the drift equations for the relative densities in the different degree classes read $$\frac{d m_{\bf k}(t)}{dt}=k_\textrm{in}\sum_{\bf k'}\frac{1}{k'_\textrm{out}} P_\textrm{in}({\bf k'}|{\bf k})
(m_{\bf k'}(t)-m_{\bf k}(t)).
\label{driftIP}$$ The existence of a conserved quantity $\omega=\sum_{\bf k}
\varphi_{\bf k} m_{\bf k}(t)$ in the correlated case is governed by the eigenvalue problem $$\tilde{\varphi}_{\bf k}=\sum_{\bf k'} \frac{P_\textrm{in}({\bf k}|{\bf
k'})/k_\textrm{out}}{\sum_{\bf k''}P_\textrm{in}({\bf k''}|{\bf k'})/k_\textrm{out}''}\tilde{\varphi}_{\bf k'},$$ where $\tilde{\varphi}_{\bf k}=\varphi_{\bf k} k_\textrm{in}
\sum_{\bf k''}P_\textrm{in}({\bf k''}|{\bf k})/k_\textrm{out}''$. Summing both sides of this equation over ${\bf k}$, one arrives once more to a trivial identity and so a conserved quantity exists in general on networks with degree-degree correlations. As we see next, we can be more specific on uncorrelated networks, for which Eq. (\[driftIP\]) reduces to $$\frac{d m_{\bf k}(t)}{dt}=\frac{k_\textrm{in}}{\left<k_\textrm{in}\right>}(m(t)-m_{\bf k}(t)),
\label{driftIPunc}$$ where $m(t)=\sum_{\bf k} P({\bf k}) m_{\bf k}(t)$ is the total density of state-one nodes in the network.
In the stationary state, $m_{\bf k}(t)= m(t) \mbox{ } \forall {\bf k}$, but here $m(t)$ is not a conserved quantity for the IP in uncorrelated directed networks. Instead, the conserved quantity is $$\begin{aligned}
\omega_\textrm{in}(t)&=&\left< \frac{m_{\bf k}(t)}{k_\textrm{in}} \right> /
\left< \frac{1}{k_\textrm{in}} \right> \nonumber \\
&=&\sum_{\bf k}\frac{1}{k_\textrm{in}}P({\bf
k})m_{\bf k}(t)/\left<\frac{1}{k_\textrm{in}} \right>.\end{aligned}$$ In finite systems, the probability of the state-one absorbing state is given by this conserved quantity, $\omega_\textrm{in}=P_1$, and is therefore fixed by the initial condition. The dependence of the weights on the inverse of the in degree implies that those nodes with low in-degree, so less influenceable, have the highest contribution and control the process. This dependence on the in degree is analogous to the dependence on the degree of the conserved quantity $w=\omega=\left( \sum_{k}1/k P(k)m_{k}(t) \right) / \langle k\rangle$ in uncorrelated undirected networks [@Sood:2005].
After a transient, $m(t)$ reaches the value $\omega_\textrm{in}$, so that the stationary values of the relative densities are $m_{\bf k}(t)=\omega_\textrm{in} \mbox{
} \forall {\bf k}$. This result tells us that all the densities become independent of ${\bf k}$ and reach the same stationary value, as in the previous processes.
The derivation of how the state-one relative densities converge to their stationary value in uncorrelated networks is more intricate than for the voter model, but like for the link update we can make use of a quasi-stationary approximation [@Gardiner:2004] in order to solve Eq. (\[driftIPunc\]). Substituting into Eq. (\[driftIPunc\]) that in the stationary state $m(t)=\omega_\textrm{in}$, $$\frac{d m_{\bf
k}(t)}{dt}=\frac{k_\textrm{in}}{\left<k_\textrm{in}\right>}\left(\omega_\textrm{in}-m_{\bf
k}(t)\right).$$ For a given initial condition $m_{\bf k}(0)$, the solution is $$m_{\bf k}(t)=\omega_\textrm{in} + \left(m_{\bf
k}(0)- \omega_\textrm{in} \right)e^{-\frac{k_\textrm{in}}{\left<k_\textrm{in}\right>}t}.
$$ All the densities decay exponentially to the stationary value $\omega_\textrm{in}$. Higher in-degree classes decay faster than lower ones with a relaxation time that is proportional to the inverse of the in-degree, as is the case for LU. Due to the average degree in the relaxation time, however, transients are generally slower in the IP than in the LU. When compared with the VM, the IP dynamics exhibits a slower transient for degree classes with in-degree below average while those with in-degree above the average converge faster to the stationary state.
Conclusions
===========
We have introduced an analytical formalism from microscopic dynamics to show that three different nonequilibrium dynamical models with two-absorbing states running on strongly connected components of directed networks with heterogeneous degrees and degree-degree correlations have associated ensemble average conservation laws. These conservation laws have been fully determined when degree-degree correlations are absent. The existence of ensemble average conservation laws is a general characteristic of Markov processes with two or more absorbing states.
The constraints imposed on the dynamics by the conservation laws lead to interesting and nontrivial behavior. From a practical point of view, they are related to the stationary values and the characteristic relaxation times of the relative densities of nodes in state one in each degree class and, in finite systems, gives the probabilities of reaching the two possible absorbing states. In this sense, the conservation laws obtained in the thermodynamic limit for a system that does not order in that limit (i.e. does not reach the absorbing state) determine the probabilities of reaching each absorbing state for a finite system. The contribution of each node to he conserved global weighted magnetization is always a specific function of the directed degrees. In the case of the VM, the out-degree is the weight that controls the importance of the node as a measure of its influence, while in the IP it is the inverse of the in-degree, and in the LU it is the ratio between out and in-degree. In all cases, the conserved quantities are determined by local properties that encode the importance of each node in the network. Depending on the dynamics, what seems important from a local perspective is to be influential reaching a large number of neighbors, or not to be too influenceable, with a low number of incoming connections, or both at the same time.
From a broad perspective, these studies help in the understanding of how the rich structure of real systems affects the dynamical processes that run on top. However, many questions still remain unsolved. In which specific way do degree correlations alter the results for uncorrelated networks? How is the diffusive fluctuations regime in SCCs of finite directed networks? Is the finite size scaling of consensus times the same as in undirected networks? On the other hand, it seems realistic to restrict to SCCs for a number of densely connected systems, like for instance the world trade web [@Serrano:2007b], but in sparse directed networks the whole structure of core and peripheral components should be taken into account. Numerical simulations in some specific model networks [@MinPark:2006] show that the appearance of an input component seems to prevent the system, even if finite, from reaching an absorbing state for specific initial conditions. How does the complete structure of a directed network couples to the initial conditions of the dynamics to induce the presence of zealots and how do they affect in quantitative terms the behavior of the whole system still needs further research.
During the final completion of this work, we became aware of a recent preprint [@Masuda:2008] discussing the fixation probabilities of mutants for Voter-like dynamics on directed networks. Since there exists a direct relation between fixation probabilities of mutants and exit probabilities, and so conserved quantities, some of the results derived in that paper –without reference to conservation laws- concerning the dependence on the directed degrees are in correspondence to some of our results on uncorrelated strongly connected components.
We thank Marián Boguñá for helpful discussions. We acknowledge financial support from MCIN (Spain) and FEDER through project FISICOS (FIS2007-60327); M. A. S. acknowledges support by DGES grant No. FIS2007-66485-C02-01, K. K. acknowledges financial support from the Volkswagen Stiftung.
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[^1]: We us this values $s=1,0$ in order to simplify computations instead of the usual spin notation $\sigma=\pm 1$. There is a direct mapping between both schemes $\sigma=2s-1$, and therefore for all the properties defined as a function of the states. For instance, the total magnetization $m$ in the $\{\pm 1\}$ scheme is related to the total magnetization $m'$ in the $\{0,1\}$ scheme through $m=2m'-1$.
[^2]: For a wider validity range, the same can be proved at the microscopic level from equation Eq. (\[micro1node\]), which is exact for any graph, with the only assumption that the adjacency matrix represents a SCC. One has to assume $\omega=\sum_{u} c_u s_u$, but the procedure is the same.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
Benjamin D. Haeffele bhaeffele@jhu.edu\
René Vidal rvidal@jhu.edu\
Department of Biomedial Engineering\
Johns Hopkins University\
Baltimore, MD 21218, USA
bibliography:
- 'posfactor.bib'
- '../biblio/sparse.bib'
- '../biblio/vidal.bib'
title: 'Global Optimality in Tensor Factorization, Deep Learning, and Beyond'
---
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Motivated by recent results in random matrix theory we will study the distributions arising from products of complex Gaussian random matrices and truncations of Haar distributed unitary matrices. We introduce an appropriately general class of measures and characterize them by their moments essentially given by specific Jacobi polynomials with varying parameters. Solving this moment problem requires a study of the Riemann surfaces associated to a class of algebraic equations. The connection to random matrix theory is then established using methods from free probability.'
author:
- 'Wolfgang Gawronski[^1], Thorsten Neuschel, Dries Stivigny [^2]'
title: Jacobi polynomial moments and products of random matrices
---
[**Keywords:** Moment problem; Jacobi polynomials; Raney distributions; Random matrices; Distribution of eigenvalues; Free probability theory; Free multiplicative convolution]{}
[**Mathematics Subject Classification 2010:** 30E05 ; 15B52 , 30F10 , 46L54]{}
Introduction
============
Products of random matrices are subject to research for many years now. It dates back to the 1960’s when Furstenberg and Kesten [@Furstenberg_Kesten] studied products of random matrices letting the number of factors grow to infinity while keeping the dimension fixed. This work was very influential and had applications to, for example, Schödinger operator theory [@Bougerol_Lacroix]. A more recent development is the study of the distribution of the eigenvalues and (squared) singular values of products consisting of a fixed number of factors as the dimensions grow to infinity. Different approaches have been found, e.g. free probability theory, to obtain the so-called limiting global eigenvalue or (squared) singular value distribution [@Alexeev_Gotze; @Burda_Janik_Waclaw; @Burda_1; @Burda_extended; @Dupic; @Gotze_Tikhomirov; @ORourke_Soshnikov; @Penson_Z]. In particular, the product of independent standard complex Gaussian matrices (these matrices are called Ginibre random matrices) has attracted interest with applications in, e.g., so-called multiple-input and multiple-output (MIMO) communication networks [@Akemann_Kieburg_Wei] (see [@Tulino_Verdu] for a more general introduction). In this context, it is also of interest to consider products involving Ginibre matrices and other random matrices. In [@Forrester] this was done for the product of Ginibre matrices and inverses of Ginibre matrices and in [@Kuijlaars_Stivigny] for the product of Ginibre matrices and truncations of unitary matrices (for applications, see, e.g., the introduction of [@Akemann_Nagao]).
Let $r,s \in \mathbb{N} = \{0, 1, 2, \ldots\}$ with $s < r$ and let $T_1, \ldots, T_s$ be $s$ independent truncations of Haar distributed unitary matrices (such a matrix $T_j$ can be considered to be the upper left block of a Haar distributed unitary matrix). Moreover, let $G_{s+1}, \ldots, G_r$ be $r-s$ independent Ginibre random matrices. The motivation of this paper is to characterize the limiting distributions of the squared singular values of the product $$Y_{r,s} := G_r \ldots G_{s+1} T_s \ldots T_1.$$ This is equivalent with studying the limiting eigenvalue distribution of the Wishart-type matrix $Y_{r,s}^{\ast}Y_{r,s}$. The case $s = 0$, where we only have Ginibre random matrices, has been studied in [@Penson_Z; @Neuschel]. The limiting distribution was shown to be characterized by its moments $$FC_r(n) := \frac{1}{rn+1} {rn+n \choose n}, \qquad n \in \mathbb{N},$$ for fixed $r$. These numbers are called Fuss-Catalan numbers of order $r$ and historically arose in the context of combinatorial problems [@Knuth]. We will denote the corresponding distributions by $FC_r$. In case $s=1$, it turns out that the limiting distribution of the squared singular values of $Y_{r,1}$ coincides with a specific Raney distribution (Theorem \[thm: connection\_rmt\], Remark \[remark: lim\_distr\_case\_s\_1\] and also [@MNPZ; @Neuschel_Stivigny]). These distributions are a generalization of the Fuss-Catalan distributions and are defined by their moments, the so-called Raney numbers $$R_{\alpha, \beta}(n) := \frac{\beta}{n\alpha + \beta} {n\alpha + \beta \choose n}, \qquad n \in \mathbb{N},$$ for given $\alpha, \beta \in \mathbb{N}$ such that $\alpha > 1$ and $0 \leq \beta \leq \alpha$. These numbers have a combinatorial interpretation as well, see [@Forrester_Liu] for an overview. We will denote the corresponding distributions by $R_{\alpha, \beta}$. One can easily see that $R_{r+1, 1}(n) = FC_r(n)$ and so in the cases $s=0$ and $s=1$ the above mentioned limiting distributions of squared singular values are contained in the class of Raney distributions. However, for $s > 1$ the limiting distribution turns out not to belong to this class anymore. The main goal of this paper is to introduce and characterize an appropriately general class of measures that contains all these limiting distributions. In the language of free probability theory, this means we want to characterize a class of measures containing all multiplicative free convolutions (see Section 3.1) of the form $$FC_{r-s} \boxtimes R_{1, \frac{1}{2}}^{\boxtimes s}.$$
With this in mind, in Section \[sec: jac\_moments\] we introduce a sequence, depending on $a > 0$ and $r,s \in \mathbb{N}$ such that $s < r$, of positive numbers $$\label{eq: def_mu_0}
J_{r,s,a}(0) := a$$ and for $n \in \mathbb{N}$ and $n \geq 1$ $$\label{eq: def_mu_n}
J_{r,s,a}(n) := \frac{a}{n} \left(\frac{a^r}{(1+a)^s}\right)^n P_{n-1}^{(\alpha_{n-1}, \beta_{n-1})}\left(\frac{1-a}{1+a}\right),$$ where $P_n^{(\alpha_n, \beta_n)}(x)$ are the Jacobi polynomials with varying parameters $\alpha_n = rn + r + 1$ and $\beta_n = -(r+1-s)n - (r+2-s)$ as defined in [@Szego_Orth_Pol]. We will prove in Section 2 that these numbers indeed form a (Hausdorff) moment sequence of a compactly supported measure $J_{r,s,a}$. More precisely, we prove the following theorem.
Let $r,s \in \mathbb{N}$ such that $s < r$ and let $a$ be a positive real number. Then there exists a unique measure $J_{r,s,a}$ on $[0, x^{\ast}]$ with total mass $a$ such that the moments are given by the numbers and .
The right endpoint $x^{\ast}$ of the support of $J_{r,s,a}$ is defined below in . The proof of this result heavily relies on a study of the Riemann surface associated to the algebraic equation $$w^{r+1} - x(w-a)(w+1)^s = 0$$ which is done in Proposition \[prop: study\_alg\_eq\].
In Section \[sec: appl\_rmt\] we establish the connection with random matrix theory and we prove in Theorem \[thm: connection\_rmt\] that for $s < r$ we have $$J_{r,s,1} = FC_{r-s} \boxtimes R_{1, \frac{1}{2}}^{\boxtimes s}.$$ In particular, we can identify $J_{r,s,1}$ in the case $s = 0$ with the Fuss-Catalan distribution $FC_r$ and in the case $s = 1$ with the Raney distribution $R_{\frac{r+1}{2}, \frac{1}{2}}$.
Finally, we want to emphasize the remarkable fact that the combination of Theorem \[thm: measure\_mu\] and Theorem \[thm: connection\_rmt\] establishes a further connection between random matrix theory and the theory of classical orthogonal polynomials.
Jacobi polynomial moments {#sec: jac_moments}
=========================
We start by showing that $J_{r,s,a}(n)$ is a positive-valued sequence.
\[prop: positivity\_mu\_n\] Let $a > 0$ be a positive real number and $r,s$ positive integers such that $s \leq r$. Then $J_{r,s,a}(n) > 0$ for all $n \in \mathbb{N}$ where $J_{r,s,a}(n)$ is given by .
To prove this, we need the following lemma.
\[lemma: mu\_n\_id\_derivative\] Let $a > 0$ be a positive real number and $r,s$ positive integers such that $s \leq r$. Then $$\label{eq: mu_n_id_derivative}
J_{r,s,a}(n) = \frac{1}{n!}\frac{d^{n-1}}{dz^{n-1}} \left.\left(\frac{z^{n(r+1)}}{(1+z)^{ns}}\right)\right|_{z = a}$$ for all $n \in \mathbb{N}$ and $n \geq 1$.
This follows from Leibniz’ rule. Indeed, we know that $$\begin{aligned}
\frac{d^{n-1}}{dz^{n-1}} \left(\frac{z^{n(r+1)}}{(1+z)^{ns}}\right) &= \sum_{k = 0}^{n-1} {n-1 \choose k} \frac{d^{n-1-k}}{dz^{n-1-k}}\left(z^{n(r+1)}\right) \frac{d^k}{dz^k}\left((1+z)^{-ns}\right) \\
&= \sum_{k = 0}^{n-1} {n-1 \choose k} (nr + k + 2)_{n-1-k} z^{nr +k + 1} (-ns - k +1)_k (1+z)^{-ns-k} \\
&= \frac{z^{nr + 1}}{(1+z)^{ns}} P_{n-1}\left(\frac{z}{1+z}\right)\end{aligned}$$ where $(a)_k$ denotes the Pochhammer symbol and $$P_n(x) := \sum_{k = 0}^n {n \choose k} ((n+1)r + k + 2)_{n-k} (-(n+1)s - k +1)_k x^k.$$ Using the representation (see, e.g., [@Szego_Orth_Pol], p.62) $$P_n^{(\alpha_n, \beta_n)}(z)=\frac{1}{n!} \sum_{k=0}^n\binom{n}{k} (n+\alpha+\beta+1)_k (\alpha+k+1)_{n-k} \left(\frac{z-1}{2}\right)^k,$$ it is now straightforward to check that $$P_n(x) = n! P_n^{(\alpha_n, \beta_n)}(1-2x)$$ with $\alpha_n = rn + r + 1$ and $\beta_n = -(r+1 - s)n - (r+2 - s)$.
This is clearly true for $n = 0$, so let $n \in \mathbb{N}$ and $n \geq 1$. We claim that $$\frac{d^{k}}{dz^{k}} \left.\left(\frac{z^{n(r+1)}}{(1+z)^{ns}}\right)\right|_{z = x} > 0, \quad k = 0, \ldots, n, \quad x > 0$$ from which the statement then immediately follows by using Lemma \[lemma: mu\_n\_id\_derivative\]. A simple argument using Leibniz’ rule shows that it suffices to prove this claim for $r = s$. We start with the identity $$\frac{1}{(1+z)^{nr}} = \frac{1}{\Gamma(rn)} \int_{0}^{\infty} e^{-(1+z)t} t^{rn - 1} dt, \quad z>0.$$ Hence, after the substitution $y = tz$, we obtain $$\begin{aligned}
\frac{d^{k}}{dz^{k}} \left(\frac{z^{n(r+1)}}{(1+z)^{nr}}\right) &= \frac{d^k}{dz^k} \left(\frac{z^n}{\Gamma(rn)} \int_0^{\infty} e^{-y(1 + \frac{1}{z})} y^{rn-1} dy\right) \\
&= \frac{1}{\Gamma(rn)} \int_0^{\infty} \frac{d^{k}}{dz^{k}} \left(z^n e^{-\frac{y}{z}}\right) e^{-y} y^{rn-1} dy \\
&= \frac{1}{\Gamma(rn)} \int_0^{\infty} \frac{d^{k}}{du^{k}} \left.\left(u^n e^{-\frac{1}{u}}\right)\right|_{u = \frac{z}{y}} y^{n-k} e^{-y} y^{rn-1} dy.\end{aligned}$$ Using [@Szego_Orth_Pol Ex. 73 p. 388] and the sum representation for Laguerre polynomials (see e.g. [@Szego_Orth_Pol Formula 5.1.6]) $$L_k^{\alpha}(x) = \sum_{j = 0}^k (-1)^j {k + \alpha \choose k-j} \frac{x^j}{j!},$$ this can be rewritten as $$\frac{k!}{\Gamma(rn)} \int_0^{\infty} e^{-y(1 + \frac{1}{z})} y^{rn-1} z^{n-k} \left(\sum_{j = 0}^k {k-n-1 \choose k-j} \frac{(-1)^{k-j}}{j!} \left(\frac{y}{z}\right)^j\right) dy.$$ Since the sign of ${k-n-1 \choose k-j}$ is given by $(-1)^{k-j}$ the claim follows.
We are now ready to state our main result of this section. First, we define $$\label{eq: w_ast}
w^{\ast} := \frac{a(r + 1 - s) - r + \sqrt{(a(r + 1 - s) - r)^2 + 4a(r+1)(r-s)}}{2(r-s)}$$ and $$\label{eq: x_ast}
x^{\ast} := \frac{r+1}{s+1} \frac{(w^{\ast})^r}{(w^{\ast} + 1)^{s-1}\left(w^{\ast} - \frac{as - 1}{s+1}\right)}.$$ These quantities are derived in Proposition \[prop: study\_alg\_eq\].
\[thm: measure\_mu\] Let $r,s \in \mathbb{N}$ such that $s < r$ and let $a$ be a positive real number. Then there exists a unique measure $J_{r,s,a}$ on $[0, x^{\ast}]$ with total mass $a$ such that the moments are given by the numbers and .
First, notice that $x^{\ast} > 0$ for all $a > 0$ and $s < r$. Indeed, one can easily check that $w^{\ast} > a$ and thus, since $\frac{as-1}{s+1} < a$, we have that $x^{\ast} > 0$. Consider now the algebraic equation $$\label{eq: algebraic_eq_w}
w^{r+1} - x(w - a)(w+1)^s = 0.$$ As we will show in Proposition \[prop: study\_alg\_eq\] this algebraic equation has a unique solution $w(x)$ which is analytic at infinity such that $w(x) \to a$ as $x \to \infty$. Furthermore, this solution has an analytic continuation to $\mathbb{C}\setminus[0, x^{\ast}]$. Since $w(x)$ is a solution of we know that $$w(x) = a + \frac{1}{x} \frac{w(x)^{r+1}}{(w(x)+1)^s}$$ and now applying the Lagrange-Bürmann theorem gives us that $$w(x) = a + \sum_{n = 1}^{\infty} \frac{1}{n!} \frac{d^{n-1}}{dz^{n-1}} \left.\left(\frac{z^{n(r+1)}}{(1+z)^{ns}}\right)\right|_{z = a} x^{-n}$$ in a neighbourhood of infinity. Because of Lemma \[lemma: mu\_n\_id\_derivative\] this can be rewritten as $$\label{eq: w_moments_series}
w(x) = \sum_{n = 0}^{\infty} J_{r,s,a}(n) x^{-n}.$$ We define now $$\label{eq: def_density}
\rho(x) := \frac{1}{2\pi i} \left\lbrace\frac{w_{-}(x)}{x} - \frac{w_+(x)}{x}\right\rbrace, \qquad x \in (0, x^{\ast})$$ where $w_-(x)$, resp. $w_+(x)$, denotes the limiting value of $w(z)$ as $z$ approaches $x$ with $\text{Im}(z) < 0$, resp. $\text{Im}(z) > 0$. First of all, we notice that $w_-(x) = \overline{w_+(x)}$ so that $\rho(x)$ is a real-valued function. Furthermore, we claim that $\rho(x)$ is an integrable, everywhere positive function such that $$\int_0^{x^{\ast}} x^n \rho(x) dx = J_{r,s,a}(n) \qquad \text{for all } n \in \mathbb{N}$$ and thus is the density of a (unique) measure on $[0, x^{\ast}]$ with the numbers $J_{r,s,a}(n)$ as moments.
By standard arguments and using $w(0) = 0$ one can see that $$\int_0^{x^{\ast}} x^n \rho(x) dx = \oint_{K} z^n \frac{w(z)}{z} dz = \oint_{K} z^{n-1} w(z) dz$$ with $K$ a positively oriented, closed contour encircling the cut $[0, x^{\ast}]$. Since $z^{n-1} w(z)$ has no singularities in $\mathbb{C}\setminus [0, x^{\ast}]$, we can can compute the residue at infinity and use to obtain that $$\oint_{K} z^{n-1} w(z) dz = J_{r,s,a}(n)$$ and hence $$\int_0^{x^{\ast}} x^n \rho(x) dx = J_{r,s,a}(n).$$ Finally, to prove that $\rho(x) > 0$ for all $x \in (0, x^{\ast})$, it now suffices to show that $w_-(x) \neq w_+(x)$ if $x \in (0, x^{\ast})$. Indeed, if $w_-(x) \neq w_+(x)$, then $\rho(x) \neq 0$ and thus, because of the continuity, $\rho(x)$ is either everywhere positive or everywhere negative. Since $$\int_0^{x^{\ast}} \rho(x) dx = J_{r,s,a}(0) = a$$ we obtain that $\rho(x)$ must be positive-valued.
So let $x \in (0, x^{\ast})$ and let $K_x$ be a positively oriented circle around the origin with radius $x$ starting at the point $x$. Then we have $$1 = \frac{1}{2\pi i} \oint_{K_x} \frac{1}{z} dz = \frac{1}{2\pi i} \oint_{K_x} \frac{1}{f(w(z))} dz$$ where we used the fact that $$z = \frac{w(z)^{r+1}}{(w(z) - a)(w(z)+1)^s} =: f(w(z)).$$ Making $u = w(z)$ the new variable of integration and observing that $1 = f'(w(z)) w'(z)$ and $$f'(w) = \left(\frac{r+1}{w} - \frac{1}{w-a} - \frac{s}{w+1}\right)f(w)$$ we get that $$1 = \frac{1}{2\pi i} \oint_{w(K_x)} \left(\frac{r+1}{u} - \frac{1}{u-a} - \frac{s}{u+1}\right) du.$$ Here $w(K_x)$ is a contour starting at $w_+(x)$ and ending at $w_-(x)$. Assume now that $w_-(x) = w_+(x)$, i.e. $w_-(x) = w_+(x)$ is a real number. Because cannot have nonnegative solutions for $w$ if $x \in (0, x^{\ast})$ we see that $w(K_x)$ will be a closed contour starting in a point on the negative axis, going to the complex plane, crossing the real axis exactly one more time between the origin and $a$ and returning to its starting point. The orientation can be positive or negative, and the contour can encircle $-1$ but not $a$. Thus we get the following possibilities $$\frac{1}{2\pi i} \oint_{w(K_x)} \left(\frac{r+1}{u} - \frac{1}{u-a} - \frac{s}{u+1}\right) du = \begin{cases}r+1 \\ r+1-s \\ -(r+1) \\ -(r+1-s)\end{cases} \neq 1.$$ We see that in all cases we get a contradiction and thus $w_-(x) \neq w_+(x)$. Hence we can conclude that indeed defines a density.
We end this section with the following proposition, which was needed in our proof of Theorem \[thm: measure\_mu\].
\[prop: study\_alg\_eq\] Let $r, s$ be positive integers such that $s < r$ and let $a > 0$. Then the equation $$\label{eq: algebraic_eq_w_2}
w^{r+1} - x(w - a)(w+1)^s = 0$$ defines an algebraic function $w(x)$ which has an analytic branch at infinity with $w(x) \to a$, as $x \to \infty$. Moreover, this branch admits an analytic continuation to $\mathbb{C} \setminus [0, x^{\ast}]$ where $x^{\ast}$ is given by .
The existence of a solution which is analytic at infinity can be seen by rewriting as $$w = a + \frac{1}{x} \frac{w^{r+1}}{(w+1)^s},$$ which permits one to apply Lagrange-Bürmann’s theorem. We confine this solution to the first sheet of the Riemann surface associated to and we denote it by $w_1(x)$. It remains to prove that this solution has an analytic continuation to $\mathbb{C} \setminus [0, x^{\ast}]$. To this end, we want to find all the branch points and thus we have to solve the following system of equations in the variables $x$ and $w$ $$\label{eq: solve_system}
\begin{cases}w^{r+1} - x(w-a)(w+1)^s = 0 \\(r+1)w^r - x(w+1)^{s-1}((s+1)w - (as-1)) = 0\end{cases}.$$ One can now immediately see that $x = 0$ is a branch point with $w = 0$. This is a multiple branch point connecting all the $r+1$ sheets of the associated Riemann surface. Moreover, in the case $a = \frac{1}{s}$, the second equation of gives us that $w = 0$ and thus $x = 0$. So from now on we assume that $a \neq \frac{1}{s}$. Then the second equation can be rewritten as $$x = \frac{r+1}{s+1} \frac{w^r}{(w+1)^{s-1} \left(w - \frac{as-1}{s+1}\right)}.$$ Substituting this in the first equation gives us $$w^{r+1} - \frac{r+1}{s+1} \frac{w^r}{(w+1)^{s-1} \left(w - \frac{as-1}{s+1}\right)} (w-a)(w+1)^s = 0.$$ Assuming that $w \neq 0$, this can be simplified to $$w((s+1)w - (as-1)) - (r+1)(w-a)(w+1) = 0.$$ One can now easily check that the two solutions of this quadratic equation are given by $w = w^{\ast}$ where $w^{\star}$ is defined in and by $w = \tilde{w}$ with $$\label{eq: w_tilde}
\tilde{w} := \frac{a(r + 1 - s) - r - \sqrt{(a(r + 1 - s) - r)^2 + 4a(r+1)(r-s)}}{2(r-s)}.$$ Hence, we can conclude that the only possible branch points are at the real points $x = 0$, $x = x^{\ast}$, $x = \tilde{x}$ with $$\label{eq: x_tilde}
\tilde{x} := \frac{r+1}{s+1} \frac{(\tilde{w})^r}{(\tilde{w} + 1)^{s-1}\left(\tilde{w} - \frac{as - 1}{s+1}\right)}$$ and at infinity.
To conclude that on the first sheet $w(x)$ only has branch points at $x = 0$ and at $x = x^{\ast}$, it now suffices to show that there is no branch point at $x = \tilde{x}$ on the first sheet. Indeed, due to the analyticity of $w_1(x)$ there cannot be a branch point at infinity on this sheet. From equation it can be observed that $w_1(x)$ admits an analytic continuation starting at infinity travelling along the negative real axis up to the origin. In the same manner, $w_1(x)$ can be analytically continued starting at infinity travelling along the positive real axis up to $x = x^{\ast}$. Moreover, we have $w_1(x) > 0$ on $\mathbb{R}\setminus [0, x^{\ast}]$. As all branch points are real, we can conclude that $w_1(x)$ admits an analytic continuation onto $\mathbb{C}\setminus [0, x^{\ast}]$. Taking into account that $\tilde{x} \notin [0, x^{\ast}]$ and $w(\tilde{x}) = \tilde{w} < 0$, this shows that there can be no further branch point on the first sheet which completes the proof.
Using the observations made in the proof of Proposition \[prop: study\_alg\_eq\] we can now describe the geometry of the Riemann surface associated to the algebraic equation . We illustrate this in Figure \[fig: riemann\_surface\] for the case $r = 5, s=3$.
(0,0)–(1,1)–(4,1)–(3,0)–cycle; (8,0)–(9,1)–(12,1)–(11,0)–cycle; (2, 0.5)–(2.8,0.5); (8.5,0.5)–(10,0.5); (10,0.5)–(10,-3.5); (8.5,0.5)–(8.5,-3.5); (2,0.5)–(2,-1.5); (8.5,0.5)–(5.5,-1.5); (2.8,0.5)–(2.8,-1.5); (10,0.5)–(7,-1.5); (2, -1.5)–(2.8,-1.5); (5.5,-1.5)–(7,-1.5); (0,-2)–(1,-1)–(4,-1)–(3,-2)–cycle; (5,-2)–(6,-1)–(9,-1)–(8,-2)–cycle; (0.5,-1.5)–(2,-1.5); (7,-1.5)–(8,-1.5); (0.5,-1.5)–(0.5,-3.5); (7,-1.5)–(2,-3.5); (2,-1.5)–(2,-3.5); (8,-1.5)–(3,-3.5); (0.5,-3.5)–(2,-3.5); (2,-3.5)–(3,-3.5); (8.5,-3.5)–(5.5,-1.5); (10,-3.5)–(7,-1.5); (8.5,-3.5)–(10,-3.5); (0,-4)–(1,-3)–(4,-3)–(3,-4)–cycle; (8,-4)–(9,-3)–(12,-3)–(11,-4)–cycle;
at (2.8,0.5) [$x^{\ast}$]{}; (2.8,0.5) circle (2pt); at (2,0.5) [$0$]{}; (2,0.5) circle (2pt); at (2.8,-1.5) [$x^{\ast}$]{}; (2.8,-1.5) circle (2pt); (2,-1.5) circle (2pt); (2,-3.5) circle (2pt); at (3,-3.5) [$\tilde{x}$]{}; (3,-3.5) circle (2pt); at (8,-1.5) [$\tilde{x}$]{}; (8,-1.5) circle (2pt); (7,-1.5) circle (2pt); (10,0.5) circle (2pt); (10,-3.5) circle (2pt);
Application to random matrix theory {#sec: appl_rmt}
===================================
In this section we will show how the measures obtained in Theorem \[thm: measure\_mu\] arise naturally in random matrix theory and free probability. We start with a small introduction in free probability theory which we need to state our second theorem. For more details, we refer the reader to [@Voiculescu_DN], [@Anderson_GZ], [@Speicher] or [@Akemann_Handbook].
Free probability and random matrices
------------------------------------
Given a (compactly supported) probability measure $\mu$ on $\mathbb{R}$ such that $\int_{\mathbb{R}}x d\mu(x) \neq 0$ we define its $S$-transform as follows. Let $G_{\mu}$ denote the Stieltjes transform of the measure $\mu$, i.e. $$\label{eq: def_stieltjes}
G_{\mu}(z) := \int_{\mathbb{R}} \frac{1}{z-x} d\mu(x), \qquad z \in \mathbb{C}\setminus\text{supp}(\mu)$$ and define $$\label{eq: def_psi_mu}
\psi_{\mu}(z) := \frac{1}{z} G_{\mu}\left(\frac{1}{z}\right) - 1.$$ Let $\chi_{\mu}$ be the unique function, analytic in a neighbourhood of zero, satisfying $$\label{eq: def_chi_mu}
\chi_{\mu}(\psi_{\mu}(z)) = z.$$ Then the $S$-transform, denoted by $S_{\mu}$, is defined as $$\label{eq: def_s_tr}
S_{\mu}(z) := \frac{z+1}{z} \chi_{\mu}(z).$$ Given two (compactly supported) probability measures $\mu$ and $\nu$ with non-vanishing first moments, the free multiplicative convolution, denoted by $\mu \boxtimes \nu$, is the unique (compactly supported) probability measure that satisfies the identity $$S_{\mu \boxtimes \nu}(z) = S_{\mu}(z) S_{\nu}(z).$$ Notice that this identity shows us that the free multiplicative convolution is commutative, i.e. $\mu \boxtimes \nu = \nu \boxtimes \mu$. The $S$-transform is an important tool in free probability theory to compute the distribution of, for instance, the product of free random variables.
By $\mu_A$ we denote the empirical eigenvalue distribution of an $n \times n$ random matrix $A$, i.e. $$\mu_A = \frac{1}{n} \sum_{i = 1}^n \delta_{\lambda_i(A)}$$ with $\lambda_i(A)$ the $n$ random eigenvalues of $A$.
With this in mind, we now have the following result ([@Couillet_Debbah], Theorem 4.7, p. 82, see also [@Voiculescu] and [@Akemann_Handbook]).
\[thm: distr\_prod\_matrices\] Let $\{A_n\}$ and $\{B_n\}$ be two sequences of random matrices of size $n \times n$ such that $A_n > 0$, i.e. all eigenvalues are positive (with probability 1), and such that $A_n$ and $B_n$ are asymptotically free almost surely for all $n$. Moreover, suppose that there exist two compactly supported probability measures $\mu_1$ and $\mu_2$ such that $$\mu_{A_n} \stackrel{w}{\longrightarrow} \mu_1 \qquad a.s. \qquad \text{and} \qquad \mu_{B_n} \stackrel{w}{\longrightarrow} \mu_2 \qquad a.s.,$$ as $n \to \infty$, and where $\mu_{A_n}$ resp. $\mu_{B_n}$ denote the empirical eigenvalue distribution of $A_n$ resp. $B_n$. Then $$\mu_{A_nB_n} \stackrel{w}{\longrightarrow} \mu_1 \boxtimes \mu_2 \qquad a.s.,$$ as $n \to \infty$.
Here, by $$\mu_{A_n} \stackrel{w}{\longrightarrow} \mu_1 \qquad a.s.,$$ as $n \to \infty$, we mean that $$\int_{\mathbb{R}} f(t) d\mu_1(t) = \lim_{n \to \infty} \frac{1}{n} \sum_{i = 1}^n f(\lambda_i(A_n))$$ holds with probability $1$ for each bounded, continuous function $f(t)$. We say that $\mu_{A_n}$ converges weakly, almost surely to $\mu_1$.
Basically, Theorem \[thm: distr\_prod\_matrices\] holds for all sequences $\{A_n\}$ and $\{B_n\}$ for which $A_n$ and $B_n$ are independent and for which at least the distribution of $A_n$ or $B_n$ is invariant under left and right multiplication by Haar distributed unitary matrices [@Couillet_Debbah; @Akemann_Handbook; @Voiculescu].
Product of Ginibre and truncated unitary matrices
-------------------------------------------------
Let $U$ be a Haar distributed unitary random matrix of size $l \times l$ and let $T$ be the $m \times n$ upper left block of $U$ such that $l \geq m+n$. We call $T$ a truncated unitary (random) matrix of size $m \times n$ and the distribution is proportional to (see, e.g., [@Fyodorov_Sommers Eq. (69)]) $$\det(I - T^{\ast}T)^{l - m-n} \chi_{T^{\ast}T \leq I}(T) dT$$ where $$\chi_{T^{\ast}T \leq I}(T) := \begin{cases}1 & \mbox{ if } I-T^{\ast}T \mbox{ is positive-definite} \\ 0 & \mbox{else}\end{cases}.$$
A complex Ginibre matrix $G$ of size $m \times n$ has independent entries whose real and imaginary parts are independent and have a standard normal distribution with fixed variance. The probability distribution of $G$ is proportional to $$e^{- \operatorname{Tr}G^{\ast} G} dG.$$
We now take $s$ independent truncated unitary matrices $T_j$ of size $(n + \nu_j) \times (n + \nu_{j-1})$, with $\nu_j \geq 0$ and $\nu_0 = 0$, coming from an $l_j \times l_j$ unitary matrix. Furthermore, we take $r-s$ independent Ginibre random matrices $G_j$ of size $(n + \nu_j) \times (n + \nu_{j-1})$ for $j = s+1, \ldots, r$ and we define the product of independent matrices $$\label{eq: def_prod_yrs}
Y_{r,s} := G_r \ldots G_{s+1}T_s \ldots T_1.$$ We then have the following theorem:
\[thm: connection\_rmt\] Let $T_j$ and $G_j$ be as described above. Furthermore, suppose that we have that $l_j - 2n\geq 0$ and $\nu_i$ remain fixed for all $j = 1, \ldots, s$ and all $i = 0, \ldots, r$, as $n \to \infty$. Then we have $$\mu_{Y_n} \stackrel{w}{\longrightarrow} J_{r,s,1} \qquad a.s.,$$ as $n \to \infty$, and where $J_{r,s,1}$ is described in Theorem \[thm: measure\_mu\] and $Y_n$ is defined as the rescaled Wishart-type product $$Y_n := \frac{1}{n^{r-s}}Y_{r,s}^{\ast}Y_{r,s}.$$
We start with the observation that $Y_n$ has the same non-zero eigenvalues (counted with multiplicity) as $$Z_n := \frac{1}{n^{r-s}} (G_r \ldots G_{s+1})^{\ast}(G_r \ldots G_{s+1})(T_s \ldots T_1)(T_s \ldots T_1)^{\ast}$$ Moreover, the difference in the number of eigenvalues equal to zero is $\nu_{s}$ and thus the limiting eigenvalue distributions of $Y_n$ and $Z_n$ have to be equal, if they exist. It is known that the empirical eigenvalue distribution of $\frac{1}{n^{r-s}}(G_r \ldots G_{s+1})^{\ast}(G_r \ldots G_{s+1})$ converges weakly, almost surely to the Fuss-Catalan distribution $FC_{r-s}$, as $n \to \infty$ (see, e.g., [@Penson_Z]), which can be written as the Raney distribution $R_{r-s+1,1}$. Moreover, the distribution of each $G_j$ is invariant under left and right multiplication of Haar distributed unitary matrices and thus the same holds true for $$\frac{1}{n^{r-s}}(G_r \ldots G_{s+1})^{\ast}(G_r \ldots G_{s+1}).$$ In order to apply Theorem \[thm: distr\_prod\_matrices\] we now have to determine the limiting eigenvalue distribution of $\tilde{T}_n := (T_s \ldots T_1)(T_s \ldots T_1)^{\ast}$.
By the same arguments as before, we know that $(T_s \ldots T_1)(T_s \ldots T_1)^{\ast}$ has the same non-zero eigenvalues (counted with multiplicity) as $(T_{s-1} \ldots T_1)(T_{s-1} \ldots T_1)^{\ast}T_s^{\ast}T_s$ and the difference in the number of eigenvalues equal to zero is $|\nu_s - \nu_{s-1}|$. It is known that the empirical eigenvalue distribution of $T_s^{\ast}T_s$ converges weakly, almost surely to the arcsine measure on $(0, 1)$ if $l_s - 2n$ is fixed, as $n \to \infty$. By comparing the moments, one can see that this is the Raney distribution $R_{1, \frac{1}{2}}$. Moreover, one can check that the distribution of $T_s$ is also invariant under left and right multiplication of Haar distributed unitary matrices and so is the distribution of $T_s^{\ast}T_s$. As before, to apply Theorem \[thm: distr\_prod\_matrices\] we now have to determine the limiting eigenvalue distribution of $(T_{s-1} \ldots T_1)(T_{s-1} \ldots T_1)^{\ast}$. Repeating this argument $s-1$ times we can conclude that $$\mu_{\tilde{T}_n} \stackrel{w}{\longrightarrow} R_{1, \frac{1}{2}}^{\boxtimes s} \qquad a.s.,$$ as $n \to \infty$, and with $\tilde{T}_n := (T_s \ldots T_1)(T_s \ldots T_1)^{\ast}$.
An application of Theorem \[thm: distr\_prod\_matrices\] gives us that $$\mu_{Z_n} \stackrel{w}{\longrightarrow} \kappa \qquad a.s.,$$ as $n \to \infty$, where $$\label{eq: def_kappa}
\kappa := R_{r-s+1, 1} \boxtimes R_{1, \frac{1}{2}}^{\boxtimes s}.$$
It remains to show that $\kappa = J_{r,s,1}$. Using the result from Mlotkowski [@Mlotkowski Proposition 4.3], we know that $$S_{R_{r-s+1}, 1}(z) = \frac{1}{(1+z)^{r-s}}, \qquad S_{R_{1, \frac{1}{2}}}(z) = \frac{z+2}{z+1}$$ and hence $$S_{\kappa}(z) = \frac{(z+2)^s}{(z+1)^r}.$$ Using this can be rewritten as $$\chi_{\kappa}(z) = \frac{z(z+2)^s}{(z+1)^{r+1}}.$$ Thus, if we replace $z$ by $\psi_{\kappa}(z)$ and use we obtain $$z = \frac{\psi_{\kappa}(z) (\psi_{\kappa}(z) + 2)^s}{(\psi_{\kappa}(z) + 1)^{r+1}}.$$ Finally applying identity and replacing $z$ by $1/z$ we arrive at $$\frac{1}{z} = \frac{(zG_{\kappa}(z) - 1)(z G_{\kappa}(z) + 1)^s}{(z G_{\kappa}z)^{r+1}}$$ and from this we can conclude that $w(x) = x G_{\kappa}(x)$ satisfies the algebraic equation $$w(x)^{r+1} - x (w(x) - 1)(w(x) + 1)^s = 0.$$ This is equation with $a = 1$ and thus, because of Theorem \[thm: measure\_mu\], we obtain $$\label{eq: kappa_equal_mu1}
\kappa = J_{r,s,1}.$$
\[remark: lim\_distr\_case\_s\_1\] Theorem \[thm: connection\_rmt\] in combination with equations and gives us that $$J_{r,0, 1} = FC_r, \qquad J_{r,1,1} = R_{\frac{r+1}{2}, \frac{1}{2}}.$$ This can be seen immediately by using $$R_{r, 1} \boxtimes R_{1, \frac{1}{2}} = R_{\frac{r+1}{2}, \frac{1}{2}},$$ which is a special case of the identity stated in [@Mlotkowski Proposition 4.3]. This means in particular that $$J_{r, 0, 1}(n) = FC_r(n), \qquad J_{r, 1, 1}(n) = R_{\frac{r+1}{2}, \frac{1}{2}}(n)$$ for every $n \in \mathbb{N}$. It is interesting to remark that for these distributions explicit and elementary forms of the densities can be found by the method of parametrization (see, e.g., [@Forrester_Liu; @MNPZ; @Neuschel; @Neuschel_Stivigny]).
The statements of Theorem \[thm: measure\_mu\] and Theorem \[thm: connection\_rmt\] are restricted to the case $s < r$ because of several technical issues that arise in the case $r = s$. However, it is natural and interesting to ask whether our results can be extended to this case.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Prof. Arno Kuijlaars for many valuable discussions. The last two authors are supported by KU Leuven Research Grant OT/12/073 and the Belgian Interuniversity Attraction Pole P07/18.
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[^1]: Department of Mathematics, University of Trier, 54286 Trier, Germany. E-mail: Gawron@uni-trier.de
[^2]: Department of Mathematics, KU Leuven, Celestijnenlaan 200B box 2400, BE-3001 Leuven, Belgium. E-mail: Thorsten.Neuschel@wis.kuleuven.be, Dries.Stivigny@wis.kuleuven.be
| {
"pile_set_name": "ArXiv"
} |
---
address:
- |
A. Connes: Collège de France\
3, rue d’Ulm\
Paris, F-75005 France
- |
M. Marcolli: Max–Planck Institut für Mathematik\
Vivatsgasse 7\
Bonn, D-53111 Germany
author:
- Alain Connes
- Matilde Marcolli
title: Quantum Fields and Motives
---
Renormalization: particle physics and Hopf algebras
===================================================
The main idea of renormalization is to correct the original Lagrangian of a quantum field theory by an infinite series of counterterms, labelled by the Feynman graphs that encode the combinatorics of the perturbative expansion of the theory. These counterterms have the effect of cancelling the ultraviolet divergences. Thus, in the procedure of perturbative renormalization, one introduces a counterterm $C(\G)$ in the initial Lagrangian for every divergent one particle irreducible (1PI) Feynman diagram $\Gamma$. In the case of a [*renormalizable*]{} theory, all the necessary counterterms $C(\G)$ can be obtained by modifying the numerical parameters that appear in the original Lagrangian. It is possible to modify these parameters and replace them by (divergent) series, since they are not observable, unlike actual physical quantities that have to be finite. One of the fundamental difficulties with any renormalization procedure is a systematic treatment of nested and overlapping divergences in multiloop diagrams.
Dimensional regularization and minimal subtraction {#dimensional-regularization-and-minimal-subtraction .unnumbered}
--------------------------------------------------
One of the most effective renormalization techniques in quantum field theory is dimensional regularization (DimReg). It is widely used in perturbative calculations. It is based on an analytic continuation of Feynman diagrams to complex dimension $d\in \C$, in a neighborhood of the integral dimension $D$ at which UV divergences occur. For the complex dimension $d\to D$, the analytically continued integrals become singular and the expression admits a Laurent series expansion. Thus, within the framework of dimensional regularization, one can implement a renormalization by minimal subtraction, where the singular part of the Laurent series in $z=d-D$ is subtracted at each order in the loop expansion. This renormalization method (DimReg plus minimal subtraction) was developed by ‘t Hooft and Veltman [@tHV], who applied it to one-loop calculations in scalar electrodynamics, discussed the problem of overlapping divergences, the Ward identities, the case of theories with fermions, and anomalies. The method has since been applied widely to perturbative calculations and it quickly became the standard regularization and renormalization method for nonabelian gauge theories and the standard model.
Hopf algebras and the combinatorics of renormalization {#hopf-algebras-and-the-combinatorics-of-renormalization .unnumbered}
------------------------------------------------------
The modern viewpoint on combinatorics, which unfolded in the 70s around the systematic and rigorous restructuring of its foundations advocated by Giancarlo Rota, showed how algebraic structures such as coalgebras, bialgebras, and Hopf algebras govern elaborate combinatorial phenomena ([[*cf.*]{} ]{}[@JoRo], [@Ro], [@Sch]). The reason why such algebraic structures are naturally present lies in the fact that combinatorial objects tend to admit decomposition laws that reduce them to simpler pieces. Such laws are the source of coproduct rules. This principle was illustrated by many examples of incidence Hopf algebras arising from classes of graphs and matroids. The typical situation is families of finite graphs, closed under disjoint union and taking vertex induced subgraphs. These admit a coproduct of the form $$\label{coprodcomb}
\Delta(\Gamma)= \sum_{W\subseteq V} \gamma_W \otimes
\gamma_{V-W},$$ where $V$ is the set of vertices of $\Gamma$ and $\gamma_W$ is the induced subgraph on a set of vertices $W\subseteq V$. Results from Hopf algebras in combinatorics were used, for instance, to study graph coloring problems.
During 1960s and 1970s Quantum Field Theory underwent a season of extraordinary theoretical development. The detailed knowledge theoretical physicists gained on the subject not only made it into something of an art, but refined it into a highly sophisticated instrument, capable of producing theoretical predictions that, to this day, match experiments with unprecedented precision. Renormalization plays a central role in the quantum theory of fields, in as it provides a consistent scheme for extracting from divergent expressions finite values that can be matched to physically observed quantities. Various renormalization schemes can be implemented (though here we will be concerned only with the “dimensional regularization and minimal subtraction” scheme described above). A renormalization scheme produces an extremely elaborate combinatorial recipe that accounts for structuring of subgraphs in a hierarchy of subdivergences and counterterms. Perturbative renormalization hence appears as one of the most elaborate combinatorial recipes imposed on us by nature.
Conceptually, the crucial issue in the combinatorics of perturbative renormalization is a scheme that accounts for subdivergences. This is achieved by a [*forest formula*]{}, which assigns to a graph $\Gamma$ a formal expression where the subdivergences have been dealt with through inductively defined counterterms. Subtraction of the corresponding counterterm from this formal expression finally yields the renormalized value for $\Gamma$. The definition of such formal expressions, as we discuss more in detail below, is related to decomposing a graph by extracting all possible divergent subgraphs $\gamma\subset \Gamma$ and considering corresponding graphs $\Gamma/\gamma$ obtained by collapsing $\gamma\subset \Gamma$ to a single vertex. Such decomposition is more complicated than those derived from incidence relations on graphs in many combinatorial problems, as it is adapted to the specific divergences of the physical theory and has to take into account other data like the distribution of external momenta. Still, one can see a suggestive analogy between the type of decomposition expressed by coproducts and the decomposition $$\label{forest}
\Delta(\Gamma)= \sum_{\gamma \subseteq \Gamma} \gamma \otimes \Gamma/\gamma$$ in a sum over divergent subgraphs, which underlies the combinatorics of the forest formulae. It was the seminal work of Dirk Kreimer [@DK1] in 1997 that paved the way to a conceptual mathematical formulation of perturbative renormalization, precisely by encoding the complicated combinatorics of forest formulae via a coproduct and identifying the Hopf algebra that governed the renormalization procedure.
The extraction of a renormalized value from divergent Feynman integrals was related in [@DK1] and [@CK3] to the antipode in the Hopf algebra. However, the precise formula for the renormalized value given by the BPHZ procedure ([@BP], [@Hepp], [@Zi]) requires a further operation that twists the antipode, which, in this formulation, is not given directly in terms of the Hopf algebra structure. The main conceptual breakthrough in the understanding of the renormalization procedure, that fully reconciles it with the Hopf algebra structure, was then obtained in a later stage of development of the Connes–Kreimer theory of perturbative renormalization, [@CK1], [@CK2], where the BPHZ recursive formulae (see , , below) are described in terms of the Birkhoff factorization of loops. We shall return to this point in Section \[SectBirk\].
Given the state of affairs in combinatorics and in quantum field theory around the late ’70s, it may seem surprising that the pursuit of a conceptual mathematical interpretation of the procedure of perturbative renormalization had to wait, as it did, until the late 1990s. One should keep in mind though that, during the 1970s, mathematicians and physicists were maximally apart. The tendency among physicists was to shift the emphasis heavily towards deriving efficient computational recipes at the expense of conceptual understanding, the latter being often dismissed as a mere exercise of pedantry. This position, though justifiable in developing a theoretical apparatus that could be continuously tested against experiments, had the effect of alienating mathematicians. While quantum mechanics stimulated and in turn benefited from a lot of advancements in modern mathematics (operator algebras, functional analysis), mathematicians shied away from quantum field theory, which they perceived as ill founded, riddled with inevitable divergences, and governed by obscure hands-on recipes. In more recent times, mathematicians and theoretical physicists found a renewed harmony of language, but this happened mostly in the context of string theory. This, however, bypasses many of the crucial problems posed by quantum field theory, by proposing a large restructuring of the foundations of high energy physics, which at present still awaits experimental confirmation. Thus, in particular, the new developments left pretty much untouched the problem of a conceptual understanding of the foundations of quantum field theory. Of course, there were at various times attempts to axiomatize quantum field theory in a way that would be palatable for mathematicians (algebraic and constructive quantum field theory, for instance). Such attempts unfortunately fell short of incorporating the full complexity of quantum field theory, especially with respect to the issue of perturbative renormalization. On the other hand, at present perturbative quantum field theory still remains the most accurate instrument for theoretical predictions in elementary particle physics and this impressive agreement between theory and nature calls for the best possible conceptual understanding of its foundational principles.
Bogoliubov–Parasiuk preparation {#bogoliubovparasiuk-preparation .unnumbered}
-------------------------------
The Bogoliubov–Parasiuk preparation, or BPHZ method (for Bogoliubov–Parasiuk–Hepp–Zimmermann, [@BP], [@Hepp], [@Zi]) accounts for the presence of subdivergences, simultaneously taking care of the problem of the appearance of non-local terms and the organization of subdivergences via an inductive procedure.
The BP preparation of a graph $\Gamma$, whose divergent integral we denote by $U(\Gamma)$, is given by the formal expression $$\label{BPprep}
\overline{R}( \G) = U(\G) + \sum_{\g \sbs \G} C(\g) U( \G / \g),$$ where the sum is over divergent subgraphs. The $C(\g)$ are inductively defined counterterms, obtained (in the minimal subtraction scheme) by taking the pole part (here denoted by $T$) of the Laurent expansion in $z=d-D$ of a divergent expression, $$\label{counter}
C(\G) = -T(\overline{R}( \G)) = -T\left(U(\G) + \sum_{\g \sbs \G}
C(\g) U( \G /
\g)\right).$$ The renormalized value of $\Gamma$ is then given by the formula $$\label{Rgamma}
R(\G) = \overline{R}( \G) +C(\G) =U(\G) +C(\G) + \sum_{\g \sbs \G}
C(\g) U( \G /
\g).$$
Before continuing with the physics, we need to introduce some algebraic notions that will be useful in the rest of the paper.
Hopf algebras and affine group schemes {#hopf-algebras-and-affine-group-schemes .unnumbered}
--------------------------------------
While affine schemes are the geometric manifestation of commutative algebras, affine group schemes are the geometric counterpart of commutative Hopf algebras. The theory of affine group schemes is developed in SGA 3 [@SGA3].
Consider a commutative Hopf algebra $\Hc$ over a field $k$, which we assume here of characteristic zero. Thus, $\Hc$ is a commutative algebra with unit over $k$, endowed with a (not necessarily co-commutative) coproduct $\Delta: \Hc \to
\Hc\otimes_k \Hc$, a counit $\ve: \Hc \to k$, which are $k$-algebra morphisms and an antipode $S: \Hc \to \Hc$ which is a $k$-algebra antihomomorphism. These satisfy the “co-rules” $$\label{corules}
\begin{array}{ll}
(\Delta \otimes id)\Delta = (id\otimes \Delta)\Delta & : \Hc \to
\Hc\otimes_k \Hc \otimes_k \Hc , \\[2mm]
(id\otimes \ve)\Delta =id = (\ve \otimes id)\Delta & : \Hc \to \Hc , \\[2mm]
m (id \otimes S)\Delta = m (S\otimes id) \Delta = 1\,\ve & : \Hc \to \Hc,
\end{array}$$ where we used $m$ to denote multiplication in $\Hc$.
One then lets $G=\,{\rm Spec}\,\Hc$ be the set of prime ideals of the commutative $k$-algebra $\Hc$, with the Zariski topology. The Zariski topology is too coarse to fully recover the “algebra of coordinates” $\Hc$ from the topological space $\Sp(\Hc)$, but one recovers it through the data of the structure sheaf, [[*i.e.*]{} ]{}by considering global sections of the “sheaf of functions” on $\Sp(\Hc)$.
Since $\Hc$ is a commutative $k$-algebra, $G=\Sp(\Hc)$ is an affine scheme over $k$, while the additional structure given by the co-rules endow $G=\Sp(\Hc)$ with a product operation, a unit, and an inverse.
More precisely, one can view such $G$ as a functor that associates to any unital commutative algebra $A$ over $k$ a group $G(A)$, whose elements are the $k$-algebra homomorphisms $$\phi \,: \Hc \to A\,,\quad \phi(x\,y)= \phi(x) \phi(y) \qq
x,y\in \Hc\,, \quad\phi(1)=1\,.$$ The product in $G(A)$ is given as the dual of the coproduct, by $$\label{dualprod}
\phi_1\,\star\,\phi_2\,\,(x)=\,\langle \phi_1\otimes
\phi_2\,,\;\Delta(x)\rangle\,.$$ The inverse and the unit of $G(A)$ are determined by the antipode and the co-unit of $\Hc$. The co-rules imply that these operations define a group structure on $G(A)$. The resulting covariant functor $$A \,\rightarrow G(A)$$ from commutative algebras to groups is representable (in fact by $\Hc$). The functor $G$ obtained in this way is called an [*affine group scheme*]{}. Conversely, any covariant representable functor from the category of commutative algebras over $k$ to groups, is an affine group scheme $G$, represented by a commutative Hopf algebra, uniquely determined up to canonical isomorphism.
Some simple examples of affine group schemes:
- The multiplicative group $G={\mathbb{G}}_m$ is the affine group scheme obtained from the Hopf algebra $\Hc=k[t,t^{-1}]$ with coproduct $\Delta(t)=t\otimes
t$.
- The additive group $G={\mathbb{G}}_a$ corresponds to the Hopf algebra $\Hc=k[t]$ with coproduct $\Delta(t)=t\otimes 1 + 1 \otimes t$.
- The affine group scheme $G=\GL_n$ corresponds to the Hopf algebra $$\Hc=k[x_{i,j},t]_{i,j=1,\ldots,n} / \det(x_{i,j})t-1,$$ with coproduct $\Delta(x_{i,j})= \sum_k x_{i,k}\otimes x_{k,j}$.
The latter example is quite general. In fact, if $\Hc$ is finitely generated as an algebra over $k$, then the corresponding affine group scheme $G$ is a linear algebraic group over $k$, and can be embedded as a Zariski closed subset in some $\GL_n$. Moreover, in the more general case, one can find a collection $\Hc_i\subset \Hc$ of finitely generated algebras over $k$ such that $\Delta(\Hc_i)\subset \Hc_i\otimes \Hc_i$, $S(\Hc_i)\subset \Hc_i$, for all $i$, and such that, for all $i,j$ there exists a $k$ with $\Hc_i \cup \Hc_j \subset \Hc_k$, and $\Hc=\cup_i \Hc_i$. In this case, one obtains linear algebraic groups $G_i=\Sp(\Hc_i)$ such that $$\label{Gprojlim}
G=\varprojlim_i G_i.$$ Thus, in general, an affine group scheme is a projective limit of linear algebraic groups. If the $G_i$ are unipotent, then $G$ is a pro-unipotent affine group scheme.
The Lie algebra ${{\mathfrak{g}}}(k)=\Lie\, G(k)$ is given by the set of linear maps $ L\,:\Hc \to k$ satisfying $$\label{Liescheme}
L(X\,Y)=\, L(X)\,\ve(Y) +\, \ve(X)\, L(Y)\,,\quad \forall X\,,Y
\in \Hc\,,$$ where $\ve$ is the co-unit of $\Hc$, playing the role of the unit in the dual algebra. Equivalently, ${{\mathfrak{g}}}=\hbox{Lie} \ G$ is a covariant functor $$\label{LieGA}
A \,\rightarrow {{\mathfrak{g}}}(A)\,,$$ from commutative $k$-algebras to Lie algebras, where ${{\mathfrak{g}}}(A)$ is the Lie algebra of linear maps $ L\,:\Hc \to A$ satisfying .
Hopf algebra of Feynman graphs and diffeographisms {#hopf-algebra-of-feynman-graphs-and-diffeographisms .unnumbered}
--------------------------------------------------
The Kreimer Hopf algebra of [@DK1] is based on rooted trees, which organize the hierarchy of subdivergences in a given graph. The Hopf algebra depends on the particular physical theory $\cT$ through the use of trees whose vertices are decorated by the divergence free Feynman graphs of the theory ([[*cf.*]{} ]{}[@DK1] [@CK3]). In the work of Connes–Kreimer [@CK1] this Hopf algebra was refined to a Hopf algebra $\Hc(\sT)$, also dependent on the physical theory $\sT$ by construction, which is directly defined in terms of Feynman graphs.
The CK Hopf algebra is the free commutative algebra over $k=\C$ generated by one particle irreducible (1PI) graphs $\Gamma(p_1,\ldots,p_n)$, where $\Gamma$ is not a tree. A graph $\Gamma$ is 1PI if it cannot be disconnected by the removal of a single edge. Here one considers graphs endowed with external momenta $(p_1,\ldots,p_n)$. To account for this external structure one considers distributions $\sigma\in C_c^{-\infty}(E_\Gamma)$ for $$E_\Gamma=\,\left\{ (p_i)_{i=1 , \ldots , N} \ ; \
\sum \, p_i = 0 \right\},$$ and the symmetric algebra $\Hc=\,S(C_c^{-\infty}(\cup E_\Gamma))$, with $\cup E_\Gamma$ the disjoint union.
The coproduct is given by a formula that reflects the BP preparation , namely, it is given on generators by the expression $$\label{CKcoprod}
\Delta(\G) = \G \ot 1 + 1 \ot \G + \sum_{\g \sbs \G} \g_{(i)} \ot \G /
\g_{(i)}.$$ Here the sum is over divergent subgraphs $\gamma \sbs \G$ and $\G
/\g$ denotes the graph obtained by contracting $\g$ to a single vertex. In the notation $\gamma_{(i)}$ accounts for the fact that one has to specify how to assign the external structure to $\gamma$, depending on the type of the corresponding vertex in $\G /
\g_{(i)}$, [[*cf.*]{} ]{}[@CK1].
Up to passing to the Hopf subalgebra constructed on 1PI graphs with fixed external structure, one can reduce to a Hopf algebra $\Hc(\sT)$ that is finite dimensional in each degree, where the degree is defined on 1PI graphs by the loop number. There is an affine group scheme associated to this Hopf algebra $\Hc(\sT)$. This is called the group of [*diffeographisms*]{} $G={\rm
Difg}(\sT)$ of the physical theory. It is a pro-unipotent affine group scheme.
The reason for the terminology lies in the fact that ${\rm
Difg}(\sT)$ has a close relation to the group of formal diffeomorphisms of the complexified coupling constants of the theory. In the simplest case this group is the group ${\rm
Diff}(\C)$ of formal diffeomorphisms of the complex line tangent to the identity. The latter corresponds to the Hopf algebra $\Hc_{\text{diff}}$ whose generators $a_n$ are obtained by writing formal diffeomorphisms as $ \varphi(x) = x + \sum_{n\geq 2}
a_n(\varphi)\, x^n $, and with coproduct $\lgl \D a_n \, , \,
\vp_1 \ot \vp_2 \rgl = a_n (\vp_2 \circ \vp_1)$. A Hopf algebra homomorphism is obtained by writing the effective coupling constant as a formal power series $g_{{\rm eff}}(g) = g +
\sum_{n\geq 2} \alpha_n \, g^n$, where all the coefficients $\alpha_n$ are finite linear combinations of products of graphs, $\alpha_{n} \in \Hc$, for all $n\geq 1$ and mapping $a_n \mapsto \alpha_n$, [[*cf.*]{} ]{}[@CK1].
Birkhoff factorization and renormalization {#SectBirk}
==========================================
Suppose given a complex Lie group $G(\C)$ and a smooth simple curve $C\sbs {{\mathbb P}}^1 (\C)$, with $C^\pm$ the two complementary regions, with $\infty\in C^-$. For a given loop $\g : C \to
G(\C)$, the problem of Birkhoff factorization asks whether there exist holomorphic maps $\g_{\pm} : C_{\pm} \to G(\C)$, such that $$\label{Birk}
\g \, (z) = \g_- (z)^{-1} \, \g_+ (z) \qquad z \in C .$$
This procedure of factorization of Lie group valued loops became well known in algebraic geometry because of its use in the Grothendieck–Birkhoff decomposition [@Gro1] of holomorphic vector bundles on the sphere ${{\mathbb P}}^1(\C)$. In this case, the Lie group is $\GL_n(\C)$ and a weaker form of holds, whereby loops factor as $$\label{BirkL}
\gamma(z) = \gamma_-(z)^{-1}\, \lambda(z)\, \gamma_+(z),$$ where $ \lambda(z)$ is a diagonal matrix with entries $(z^{k_1},z^{k_2},\cdots,z^{k_n})$. The Grothendieck–Birkhoff decomposition hence states that a holomorphic vector bundle on ${{\mathbb P}}^1(\C)$ can be described as $E = L^{k_1} \op \ldots \op L^{k_n}$, where the line bundles $L^{k_i}$ have Chern class $c_1 \, (L^{k_i})
=k_i$. This corresponds to the Birkhoff decomposition when $c_1 \, (L^{k_i}) = 0$.
From a more analytic viewpoint ([[*cf.*]{} ]{}[[*e.g.*]{} ]{}[@Boj]), the Birkhoff factorizations or can be viewed as a (homogeneous) [*transmission problem*]{}, which can be formulated in terms of systems of singular integral equations, with various regularity assumptions. Such transmission problems can be recast in the context of the theory of Fredholm pairs, obtained by considering the spaces of boundary values, on a simple closed curve $C$, of sections of holomorphic vector bundles on ${{\mathbb P}}^1(\C)$.
BPHZ as a Birkhoff factorization {#bphz-as-a-birkhoff-factorization .unnumbered}
--------------------------------
One of the key results of the Connes–Kreimer theory of perturbative renormalization [@CK1] [@CK2] is a reformulation of the BPHZ procedure as a Birkhoff factorization in the pro-unipotent Lie group $G(\C)$ associated to the affine group scheme $G={\rm Difg}(\sT)$.
Unlike the case of $\GL_n$, where the Birkhoff decomposition only holds when $k_i=0$, in the case of interest for renormalization one always has a factorization . This follows from a result of Connes–Kreimer, which we recall in Proposition \[BirkHopf\] below. For the general case where $G$ is the pro-unipotent affine group scheme of a Hopf algebra that is graded in positive degree and connected, the result shows that a factorization of the form always exists. The result, in fact, provides an explicit recursive formula, in Hopf algebra terms, which determines both terms in the factorization.
In this setup, the Lie group $G(\C)$ is the set of complex points of an affine group scheme $G$, whose commutative Hopf algebra $\cH$ is graded in positive degrees $\cH=\cup_k \cH_k$ and connected ([[*i.e.*]{} ]{}the only elements of degree $0$ in $\cH$ are the scalars).
We let $K=\C(\{ z \})$ be the field of Laurent series convergent in some neighborhood of the origin ([[*i.e.*]{} ]{}germs of meromorphic functions at the origin) and ${{\mathcal O}}=\C\{ z \}$ be the ring of convergent power series, and we let $\Qc=\,z^{-1}\,\C([z^{-1}])$, with $\tilde\Qc=\C([z^{-1}])$ the corresponding unital ring. Then a loop $\gamma\,:C\to G$, for $C$ an infinitesimal circle around the origin, is equivalently described by a homomorphism $\phi: \cH
\to K$, [[*i.e.*]{} ]{}by a point in $G(K)$. Because the group structure on $G$ corresponds to the co-rules of the Hopf algebra $\cH$, the product of loops $\gamma(z)=\gamma_1(z)\,\gamma_2(z)$, for $z\in
C$, corresponds to $\phi= \phi_1\star\phi_2$ (dual to the coproduct in $\cH$) and the inverse $ z\mapsto \gamma(z)^{-1}$ to the antipode $\phi \circ S$.
For $z=0 \in C^+$, the condition that the loop $\gamma$ extends to a holomorphic function $\gamma\,:P_1 (\Cb)\backslash\{ 0 \}\to G$ is equivalent to the condition that the homomorphism $\phi$ lies in $G(\tilde\Qc)=\{
\phi\,,\phi({\mathcal H})\subset\;\tilde\Qc \}$, while the condition that $\gamma(0)$ is finite translates in the condition that $\phi$ belongs to $G({{\mathcal O}})=\{ \phi\,, \phi({\mathcal
H})\subset\;{{\mathcal O}}\}$. The normalization condition $\gamma(\infty)=1$ translates algebraically into the condition $\ve_-\circ \phi=\,\ve$, where $\ve_-$ is the augmentation in the ring $\tilde\Qc$ and $\ve$ is the augmentation (co-unit) of $\Hc$. This dictionary shows how interpreting affine group schemes as functors of unital commutative algebras to groups provides a very convenient language in which to reformulate the problem of Birkhoff factorization.
\[BirkHopf\] [*([@CK1])*]{} Let $\cH$ be a Hopf algebra that is graded in positive degree and connected, and $G$ the corresponding affine group scheme. Then any loop $\gamma: C \to G(\C)$ admits a Birkhoff factorization . An explicit recursive formula for the factorization is given, in terms of the corresponding homomorphism $\phi: \cH \to \C(\{ z \})$, by the expressions $$\label{Hbirkhoff1}
\phi_-(X)=-T\left(\phi(X)+\sum\phi_-(X^\prime)
\phi(X^{\prime\prime}) \right)$$ and $$\label{Hbirkhoff2}
\phi_+(X)=\phi(X)+\phi_-(X)+\sum\phi_-(X^\prime)
\phi(X^{\prime\prime}),$$ where $T$ is the projection along ${{\mathcal O}}$ to the augmentation ideal of $\tilde\Qc$ (taking the pole part), and $X'$ and $X''$ denote the terms of lower degree in the coproduct $\Delta(X)= X \otimes 1 + 1 \otimes X + \sum X^\prime
\otimes X^{\prime\prime}$, for $X \in{\mathcal H}$.
Applied to the Hopf algebra $\cH(\sT)$ of Feynman graphs, with $G={\rm
Difg}(\sT)$, the formulae and yield the counterterms and the renormalized values in the BPHZ renormalization procedure.
Mass parameter, counterterms, and the renormalization group {#MassSect .unnumbered}
-----------------------------------------------------------
In DimReg, when analytically continuing the Feynman graphs to complex dimension, in order to preserve the dimensionality of the integrand in physical units, one needs to replace the momentum space integration $d^{D-z}k$ by $\mu^z d^{D-z} k$, where $\mu$ is a mass parameter, so that the resulting quantity has the correct dimensionality of (mass)$^D$. This introduces a dependence on the parameter $\mu$ in the loop $\gamma_\mu(z)$ describing the unrenormalized theory. The behavior of a renormalizable theory under rescaling of the mass parameter $\mu \mapsto e^t\mu$, for $t\in \R$, was analyzed in [@tH].
An important result, which will play a crucial role in our geometric formulation in Section \[GalSect\], is that [*the counterterms do not depend on the mass parameter*]{} $\mu$ ([[*cf.*]{} ]{}[@Collins] §5.8 and §7.1). This result translates in terms of the Birkhoff factorization to the condition that the negative part $\gamma_{\mu^-}(z)$ of the factorization $\g_{\mu} (z) = \g_{\mu^-}
(z)^{-1} \, \g_{\mu^+} (z)$ satisfies $$\frac{\partial}{\partial \mu} \, \g_{\mu^-} (z) = 0 \, . \label{gammamu-}$$
The effect of scaling the mass parameter on the loop $\gamma_\mu(z)$ is instead described by the action of the 1-parameter group of automorphisms generated by the grading by loop number. Namely, if $\theta_t$ denotes the 1-parameter group with infinitesimal generator $\frac{d}{dt}\theta_t|_{t=0}=Y$, where $Y$ is the grading by loop number, we have $$\label{thetagamma}
\gamma_{e^t\mu}(z) =\theta_{tz}(\gamma_\mu(z)) , \ \ \ \ \forall t\in
\R,$$ and for all $z$ in an infinitesimal punctured neighborhood $\Delta^*$ of the origin $z=d-D=0$.
A well known but unpublished result of ‘t Hooft shows that the counterterms in a renormalizable quantum field theory can be reconstructed from the beta function of the theory. In the context of the Connes–Kreimer theory of perturbative renormalization, this can be seen in the following way.
The beta function here is lifted from the space of the coupling constants of the theory to the group of Diffeographisms, namely, it can be regarded as an element in the Lie algebra $\Lie G$ satisfying $$\label{beta}
\b = Y \, {\rm Res} \, \g ,$$ where $Y$ is the grading by loop number, and the residue of $\gamma$ is given by $$\label{resgamma}
{\rm Res}_{z = 0}^{} \g = - \left( \frac{\partial}{\partial u} \,
\g_- \left( \frac{1}{u} \right) \right)_{u=0}.$$ The beta function is the infinitesimal generator $\b= \frac{d}{dt}
{\bf rg}_t
|_{t=0}$ of the renormalization group $$\label{rengroup}
{\bf rg}_t ={\rm lim}_{z \ra 0} \:\g_- (z) \, \t_{t z} (\g_-
(z)^{-1}).$$ Correspondingly, the renormalized value, that is, the finite value $\g_{\mu}^+(0)$ of the Birkhoff decomposition satisfies the equation $$\label{renaction}
\mu \frac{\partial}{\partial \mu}
\,\g^+_{\mu}(0)=\, \b\,\g^+_{\mu}(0)\,.$$
A strong form of the ‘t Hooft relations, deriving the counterterms from the beta function, is given by the following result.
\[thooftrel\] [*([@CK2])*]{} The negative part of the Birkhoff factorization $\gamma_-(z)$ satisfies $$\label{gammaminussum}
\gamma_-(z)^{-1} = 1 + {\displaystyle
\sum_{n=1}^{\ify}} \ \frac{d_n}{z^n},$$ where the coefficients $d_n$ are given by iterated integrals $$\label{dncond}
d_n = \int_{s_1 \geq s_2 \geq \cdots \geq s_n \geq 0} \t_{-s_1}
(\b) \, \t_{-s_2} (\b) \ldots \t_{-s_n} (\b) \, \, ds_1 \cdots
ds_n \, .$$
The result can be formulated ([[*cf.*]{} ]{}[@CK2]) as a scattering formula $$\label{scattering}
\g_- (z) = \lim_{t \ra \ify}
e^{-t \left( \frac{\b}{z} + Z_0 \right)} \, e^{t Z_0},$$ where $Z_0$ is the additional generator of the Lie algebra of $G
\rtimes_\theta {\mathbb{G}}_a$, satisfying $$[Z_0 , X] = Y(X) \qquad \forall \, X \in \Lie\, G \, . \label{LieGstar}$$
This form of the ‘t Hooft relations and the explicit formula in terms of iterated integrals are the starting point for our formulation of perturbative renormalization in terms of the Riemann–Hilbert correspondence and for the relation to motivic Galois theory.
Before continuing with a more detailed account of these topics, we give an introductory tour of some ideas underlying the theory of motives and the Riemann–Hilbert correspondence, that we will need in order to introduce the main result of [@cmln].
The yoga of motives: cohomologies as avatars {#MotSect}
============================================
There are several possible cohomology theories that can be applied to algebraic varieties. Over a field $k$ of characteristic zero one has de Rham cohomology $H_{dR}^\cdot(X)={\mathbb{H}}^\cdot(X,\Omega_X^\cdot)$, defined in terms of sheaves of differential forms, and Betti cohomology $H^\cdot_B(X,\Q)$, which is a version of singular homology for $\sigma
X(\C)$, for an embedding $\sigma: k\hookrightarrow \C$. These are related by the periods isomorphism $$H^i_{dR}(X,k)\otimes_\sigma \C \cong H^i_B (X,\Q)\otimes_\Q \C.$$ Over a perfect field of positive characteristic there is also crystalline cohomology, while in all characteristics one can consider étale cohomology given by finite dimensional $\Q_\ell$-vector spaces $H^i_{et}(\bar X, \Q_\ell)$, where $\bar X$ is obtained by extension of scalars to an algebraic closure $\bar k$, and $\ell\neq
{\rm char}\, k$. In the smooth projective case, these have the expected properties of Poincaré duality, Künneth isomorphisms, etc. Moreover, étale cohomology provides interesting $\ell$-adic representations of $\Gal(\bar k/k)$. There are comparison isomorphisms $$H^i_B(X,\Q)\otimes_\Q \Q_\ell \cong H^i_{et} (\bar X,\Q_\ell).$$
The natural question is then what type of information, such as maps or operations on one cohomology, can be transferred to the other ones. This gave rise to the idea, proposed by Grothendieck, of the existence of a “universal cohomology theory” with realization functors to all the known cohomology theories for algebraic varieties. He called this the theory of [*motives*]{}.
A metaphor [@Gro3] justifying the terminology is provided by music scores, some of which (such as Bach’s “Art of the fugue") are not written for any particular instrument. They are just the motive, which in turn can be realized on different musical instruments. Another powerful metaphor is provided by the notion of avatar in Hindu philosophy, which expresses the idea of a single entity manifesting itself in manifold incarnations (the ten avatars of Vishnu).
We will present here only a very short overview of some ideas and results about motives, following [@De3], [@Man2], [@Se1], and [@Blo], [@dg], [@Gon], [@Le]. We start first by recalling some general algebraic formalism we will need in the following.
Tannakian categories {#tannakian-categories .unnumbered}
--------------------
The basis for a Galois theory of motives lies in a suitable categorical formalism. This was first proposed by Grothendieck, who used the term Galois–Poincaré categories (or rigid tensor categories), and was then developed by Saavedra [@Saa], who introduced the now currently adopted terminology of Tannakian categories, and by Deligne–Milne [@DeMi] ([[*cf.*]{} ]{}also the more recent [@De2]).
It is well known that there are many deep analogies between the theory of coverings of topological spaces and Galois theory. The analogy starts with the observation that, in cases where the covering spaces are defined by algebraic equations, the Galois symmetries of the equation actually correspond to deck transformations of the covering space.
Grothendieck brought this initial simple analogy to far reaching consequences. He developed a common formalism where fundamental groups (of a space, a scheme, or much more generally a topos) and Galois groups both fit naturally. The idea is that, in this very general setting, the group always arises as the group of automorphisms of a fiber functor on a suitable “category of coverings”. The theory of the (pro-finite) fundamental groups is based on the existence of a fiber functor from a certain category $\sC$ of finite étale covers of a connected scheme $S$, with values in finite sets. Then such functor $\omega$ yields an equivalence of categories between $\sC$ and $G$-sets for $G=\Aut(\omega)$ a pro-finite group. This yields a profinite completion of the fundamental group. For $S=\Spec(K)$, it gives Galois theory, thus effectively bringing fundamental groups and Galois groups within the same general formalism.
This is the fundamental idea that guided the development of a motivic Galois theory. The latter appeared as a “linear” version of the general formalism described above, where the fiber functor is a faithful and exact tensor functor with values in vector spaces (instead of finite sets), and the Galois group is the affine group scheme $G=\Aut^\otimes(\omega)$.
More precisely, an abelian category is a category to which the tools of homological algebra apply, that is, a category where the sets of morphisms are abelian groups, there are products and coproducts, kernels and cokernels always exist and satisfy the same basic rules as in the category of modules over a ring. A tensor category over a field $k$ of characteristic zero is a $k$-linear abelian category $\T$ endowed with a tensor functor $\otimes: \T\times \T \to \T$ satisfying associativity and commutativity laws defined by functorial isomorphisms, and with a unit object. Moreover, for each object $X$, there exists a dual $X^\vee$ and maps $ev: X\otimes X^\vee \to 1$ and $\delta: 1 \to X
\otimes X^\vee$, such that the composites $(ev\otimes 1) \circ
(1\otimes \delta)$ and $(1\otimes ev) \circ (\delta \otimes 1)$ are the identity, with an identification $k \simeq \End(1)$.
A [*Tannakian category*]{} $\T$ over $k$ is a tensor category endowed with a fiber functor, namely a functor $\omega$ to finite dimensional vector spaces ${\rm Vect}_K$, for $K$ an extension of $k$, satisfying $\omega(X)\otimes \omega(Y)\simeq \omega(X\otimes
Y)$ compatibly with associativity commutativity and unit. (A more general formulation can be given with values in locally free sheaves over a scheme, see [@De2]). A [*neutral*]{} Tannakian category $\T$ has a ${\rm Vect_k}$-valued fiber functor $\omega$. In this case, the main result is that the fiber functor $\omega$ induces an equivalence of categories between $\T$ and the category ${\rm Rep}_G$ of finite dimensional linear representations of a uniquely determined affine group scheme $G=\Aut^\otimes(\omega)$, given by the automorphisms of the fiber functor.
A $k$-linear abelian category $\T$ is semi-simple if there exists $A\subset Ob(\T)$ such that all objects $X$ in $A$ are simple (namely $\Hom(X,X)\simeq k$), with $\Hom(X,Y)=0$ for $X\neq Y$ in $A$, and such that every object of $\T$ is isomorphic to a direct sum of objects in A. The affine group scheme $G$ of a neutral Tannakian category is pro-reductive if and only if the category is semi-simple.
As an example, one can consider the category of finite dimensional complex linear representations of a group. It is not hard to see what is in this case the structure of neutral Tannakian category, with fiber functor the forgetful functor to complex vector spaces. The affine group scheme determined by this neutral Tannakian category is called the “algebraic hull" of the group. In the case of the group $\Z$, the algebraic hull is an extension of $\hat\Z$, with the corresponding commutative Hopf algebra given by $\Hc=\C
[e(q),t]$, for $q\in \C/\Z$, with the relations $e(q_1+q_2)=e(q_1)e(q_2)$ and the coproduct $\Delta(e(q))=e(q)\otimes e(q)$ and $\Delta(t)=t\otimes 1 + 1
\otimes t$.
The non-neutral case where $\omega$ takes values in ${\rm Vect}_K$ for some extension of $k$, or the more general case of locally free sheaves over a scheme, can also be identified with a category of representations, but now the group $G$ is replaced by a groupoid (Grothendieck’s Galois–Poincaré groupoid). This corresponds to the fact that, even in the original case of fundamental groups of topological spaces, it is more natural to work with the notion of fundamental groupoid, rather than with the base point dependent fundamental group. For our purposes, however, it will be sufficient to work with the more restrictive notion of neutral Tannakian category.
Gauge groups and categories {#gauge-groups-and-categories .unnumbered}
---------------------------
In [@De2], §7, Deligne gives a characterization of Tannakian categories, over a field $k$ of characteristic zero, as tensor categories where the dimensions are positive integers. The dimension of $X\in \T$ is defined in this context as $\Tr(1_X)$, where $\Tr(f) =ev\circ \delta\,(f)$.
This characterization is very close to results developed via different techniques by Doplicher and Roberts in the context of algebraic quantum field theory, [@DR]. Their motivation was to derive the existence of a global compact gauge group, given the local observables of the theory. The group is obtained from a monoidal $C^*$-category where the objects are endomorphisms of certain unital $C^*$-algebras and the arrows are intertwining operators between these endomorphisms. They obtain a characterization of those monoidal $C^*$-categories that are equivalent to the category of finite dimensional continuous unitary representations of a compact group, unique up to isomorphism. Though the context and the techniques employed in the proof are different, the result has a flavor similar to the relation between Tannakian categories and affine group schemes. In their proof, a characterization analogous to the one of [@De2], §7 of the integer dimensions also plays an important role.
Pure and mixed motives {#pure-and-mixed-motives .unnumbered}
----------------------
The first constructions of a category of motives proposed by Grothendieck covers the case of smooth projective varieties. The corresponding motives form a $\Q$-linear abelian category $\cM_{pure}(k)$ of [*pure motives*]{}. There is a contravariant functor assigning a motive to a variety $$\label{hX}
X \mapsto h(X)=\oplus_i h^i(X).$$ If $h^j=0$, for all $j\neq i$, the motive is [*pure of weight $i$*]{}. This way a pure motive can be thought of as a “direct summand of an algebraic variety”. The morphisms $\Hom(X,Y)$ in the category of motives are given by [*correspondences*]{}, namely algebraic cycles in the product $X\times Y$ of codimension equal to the dimension of $X$, modulo a suitable equivalence relation. Different choices of the notion of equivalence for algebraic cycles produce variants of the theory, ranging from the coarsest numerical equivalence to the finest rational equivalence (Chow groups). The objects of the category also include kernels of projectors, namely of idempotents in $\Hom(X,Y)$. Thus, for $p=p^2\in \End(X)$ and $q=q^2\in \End(Y)$, one takes $\Hom((X,p),(Y,q))=q \Hom(X,Y) p$.
One also adds to the objects the Tate motive $\Q(1)$, which is the inverse of $h^2({{\mathbb P}}^1)$. This is a pure motive of weight $-2$. The category is endowed with a tensor product $\otimes$ and a unit $\Q(0)=h(pt)$. The Tate objects $\Q(n)$ satisfy the rule $\Q(n+m)\cong \Q(n)\otimes \Q(m)$.
Grothendieck formulated a set of [*standard conjectures*]{} about pure motives, which are at present still unproven. Assuming the standard conjectures, the category of pure motives is a neutral Tannakian category, with fiber functors given by Betti cohomology (characteristic zero case). Thus, the category of pure motives is equivalent to the category of representations $Rep_G$ of an affine group scheme $G$. This group is called the [*motivic Galois group*]{}. The category of pure motives is conjecturally semi-simple, hence for pure motives $G$ is pro-reductive.
When one considers certain subcategories of the category of motives, one obtains a corresponding Galois group, which is a quotient of the original $G$. For instance, if the subcategory is generated by a single $X$, one obtains a quotient $G_X$, whose identity component is the Mumford–Tate group of $X$. The subcategory of pure Tate motives, generated by $\Q(1)$ has as motivic Galois group the multiplicative group ${\mathbb{G}}_m$.
Some of the first unconditional results about motives were obtained in [@Man2]. In general, a serious technical obstacle in the development of the theory of motives, which accounts for the fact that, decades after its conception, the theory is still largely depending on conjectures, is the fact that not enough is known about algebraic cycles. The situation gets even more complicated when one wishes to consider more general algebraic varieties, which need not be smooth projective. This leads to the notion of [*mixed motives*]{} with $\M_{pure}(k)\subset
\M_{mix}(k)$.
Over a field of characteristic zero (where one has resolution of singularities), one can always write such $X$ as a disjoint union of $X_i - D_i$, where the $X_i$ are smooth projective and the $D_i$ are lower dimensional. Thus, one can assign to $X$ a virtual object in a suitable Grothendieck group of algebraic varieties; however, if one wants a theory that satisfies the main requirements of a category of motives, including the fact of providing a universal cohomology theory (via the Ext functors), the construction of such a category of mixed motives remains a difficult task.
The main properties for a category of mixed motives are that it should be a $\Q$-linear tensor category containing the Tate objects $\Q(n)$ with the usual properties, endowed with a functor $X\mapsto h(X)$ that assigns motives to algebraic varieties, with properties like Künneth isomorphisms. Moreover, the Ext functors in this category of mixed motives define a “motivic cohomology” $$\label{motH}
E^{i,j}_2 = \Ext^i (\Q(0),h^j(X)\otimes \Q(n)) \Rightarrow H^{i+j}_{mot}(X,\Q(n)).$$ One expects also this motivic cohomology to come endowed with Chern classes from algebraic $K$-theory. In fact, if one uses the decomposition $K_n(X)\otimes \Q = \oplus_j K_n(X)^{(j)}$, where the Adams operation $\Psi_k$ acts on $K_n(X)^{(j)}$ as $k^j$, then one expects isomorphisms given by Chern classes $$ch^j: K_n(X)^{(j)} \stackrel{\simeq}{\to}
H_{mot}^{2j-n}(X,\Q(j)).$$ Such motivic cohomology will be universal with respect to all cohomology theories for algebraic varieties satisfying certain natural properties (Bloch–Ogus axioms). Namely, for any such cohomology $H^*(\cdot ,\Gamma(*))$ there will be a natural transformation $H^*_{mot}(\cdot, \Z(*)) \to H^*(\cdot,\Gamma(*))$, compatible with the above isomorphisms. Mixed motives have increasing weight filtrations preserved by the realizations to cohomology theories. More generally, instead of working over a field $k$, one can consider a category $\M_{mix}(S)$ of motives (or “motivic sheaves”) over a scheme $S$. In this case, the functors above are natural in $S$ and to a map of schemes $f: S_1 \to S_2$ there correspond functors $f^*$, $f_*$, $f^{!}$, $f_{!}$, behaving like the corresponding functors of sheaves.
The motivic Galois group for mixed motives will then be an extension of the pro-reductive motivic Galois group of pure motives by a pro-unipotent group. The pro-unipotent property reflects the presence of the weight filtration on mixed motives.
Though, at present, there is not yet a general construction of such a category of mixed motives $\M_{mix}(S)$, there are constructions of a triangulated tensor category $\cD\M(S)$, which has the right properties to be the bounded derived category of the category of mixed motives. The constructions of $\cD\M(S)$ due to Levine [@Le] and Voevodsky [@Vo] are known to be equivalent. In general, given a construction of a triangulated tensor category, one can extract from it an abelian category by considering the [*heart of a $t$-structure*]{}. A caveat with this procedure is that it is not always the case that the given triangulated tensor category is in fact the bounded derived category of the heart of a $t$-structure. The available constructions, in any case, are obtained via this general procedure of $t$-structures developed in [@BBD], which can be summarized as follows. A triangulated category ${\mathcal D}$ is an additive category with an automorphism $T$ and a family of distinguished triangles $X \to Y \to Z \to T(X)$, satisfying suitable axioms (which we do not recall here). We use the notation ${\mathcal D}^{\geq n} = {\mathcal D}^{\geq 0}[-n]$ and ${\mathcal D}^{\leq n} = {\mathcal D}^{\leq 0}[-n]$, with $X[n]=T^n(X)$ and $f[n]=T^n(f)$. A $t$-structure consists of two full subcategories ${\mathcal D}^{\leq 0}$ and ${\mathcal D}^{\geq 0}$ with the properties: ${\mathcal D}^{\leq -1} \subset {\mathcal D}^{\leq 0}$ and ${\mathcal D}^{\geq 1} \subset {\mathcal D}^{\geq 0}$; for all $X\in {\mathcal D}^{\leq 0}$ and all $Y\in {\mathcal D}^{\geq 1}$ one has $\Hom_{\mathcal D} (X,Y)=0$; for all $Y\in {\mathcal D}$ there exists a distinguished triangle as above with $X\in {\mathcal D}^{\leq 0}$ and $Z\in {\mathcal D}^{\geq 1}$. The heart of the t-structure is the full subcategory ${\mathcal D}^0= {\mathcal D}^{\leq 0}\cap
{\mathcal D}^{\geq 0}$. It is an abelian category. This type of construction may be familiar to physicists in the context of mirror symmetry, where continuous families of hearts of $t$-structures play a role in [@Dou].
For our purposes, we will be mostly interested in the full subcategory of Tate motives. The triangulated category of [*mixed Tate motives*]{} $\sD\M\sT(S)$ is then defined as the full triangulated subcategory of $\sD\M(S)$ generated by the Tate objects. It is possible to define on it a $t$-structure whose heart gives a category of mixed Tate motives $\M\sT_{mix}(S)$, provided the Beilinson–Soulé vanishing conjecture holds, namely when $$\label{BSconj}
\Hom^j(\Q(0),\Q(n))=0, \ \ \ \text{ for } n>0, j\leq 0.$$ where $\Hom^j(M,N)=\Hom(M,N[j])$. The conjecture is known to hold in the case of a number field, where one has $$\label{ExtK}
\Ext^1(\Q(0),\Q(n))= K_{2n-1}(k)\otimes \Q$$ and $\Ext^2(\Q(0),\Q(n))=0$. Thus, in this case it is possible to extract from the triangulated tensor category a Tannakian category $\M\sT_{mix}(k)$ of mixed Tate motives, with fiber functor $\omega$ to $\Z$-graded $\Q$-vector spaces, $M
\mapsto \omega(M)=\oplus_n \omega_n(M)$ with $$\label{omegaGr}
\omega_n(M)=\Hom(\Q(n),\Gr_{-2n}^w(M)),$$ where $\Gr_{-2n}^w(M)=W_{-2n}(M)/W_{-2(n+1)}(M)$ is the graded structure associated to the finite increasing weight filtration $W$.
The motivic Galois group of the category $\M\sT_{mix}(k)$ is then an extension $G=U\rtimes {\mathbb{G}}_m$, where the reductive piece is ${\mathbb{G}}_m$ as in the case of pure Tate motives, while $U$ is pro-unipotent. By the results of Goncharov (see [@Gon], [@dg]), it is known that the pro-unipotent affine group scheme $U$ corresponds to a graded Lie algebra ${\rm Lie}\, (U)$ that is free with one generator in each odd degree $n\leq -3$.
A similar construction is possible in the case of the category $\M\sT_{mix}(S)$, where the scheme $S$ is the set of $V$-integers ${{\mathcal O}}_V$ of a number field $k$, for $V$ a set of finite places of $k$. In this case, objects of $\M\sT_{mix}({{\mathcal O}}_V)$ are mixed Tate motives over $k$ that are unramified at each finite place $v\notin V$. For $\M\sT_{mix}({{\mathcal O}}_V)$ we have $$\label{ExtOS}
\Ext^1(\Q(0),\Q(n))=\left\{ \begin{array}{ll} K_{2n-1}(k)\otimes
\Q & n\geq 2 \\[2mm] {{\mathcal O}}_V^*\otimes \Q & n=1 \\[2mm] 0 & n\leq 0.
\end{array} \right.$$ and $\Ext^2(\Q(0),\Q(n))=0$. In fact, the difference between the Ext in $\M\sT_{mix}({{\mathcal O}}_V)$ of and the Ext in $\M\sT_{mix}(k)$ of is the $\Ext^1(\Q(0),\Q(1))$ which is finite dimensional in and infinite dimensional in . The category $\M\sT_{mix}({{\mathcal O}}_V)$ is also a neutral Tannakian category, and the fiber functor determines an equivalence of categories between $\M\sT_{mix}({{\mathcal O}}_V)$ and finite dimensional linear representations of an affine group scheme of the form $U \rtimes {\mathbb{G}}_m$ with $U$ pro-unipotent. The Lie algebra $\Lie(U)$ is freely generated by a set of homogeneous generators in degree $n$ identified with a basis of the dual of $\Ext^1(\Q(0),\Q(n))$ ([[*cf.*]{} ]{}Prop. 2.3 of [@dg]). There is however no [*canonical*]{} identification between $\Lie(U)$ and the free Lie algebra generated by the graded vector space $\oplus
\Ext^1(\Q(0),\Q(n))^\vee$.
We mention the following case, which will be the one most relevant in the context of perturbative renormalization.
\[SNmotives\] [*([@dg], [@Gon])*]{} Consider the scheme $S_N={{\mathcal O}}[1/N]$ for $k=\Q(\zeta_N)$ the cyclotomic field of level $N$ and ${{\mathcal O}}$ its ring of integers. For $N=3$ or $4$, the motivic Galois group of the category $\M\sT_{mix}(S_N)$ is of the form $U \rtimes {\mathbb{G}}_m$, where the Lie algebra $\Lie(U)$ is (noncanonically) isomorphic to the free Lie algebra with one generator $e_n$ in each degree $n\leq -1$.
Hilbert’s XXI problem and the Riemann–Hilbert correspondence
============================================================
Consider an algebraic linear ordinary differential equation, in the form of a system of rank $n$ $$\label{ODE}
\frac{d}{dz} f(z) + A(z) f(z) =0$$ on some open set $U=
\bP^1(\C)\smallsetminus \{ a_1,\ldots a_r \}$, where $A(z)$ is an $n\times n$ matrix of rational functions on $U$. In particular, this includes the case of a linear scalar $n$th order differential equation. The space ${\mathcal S}$ of germs of holomorphic solutions of at a point $z_0\in U$ is an $n$-dimensional complex vector space. Moreover, given any element $\ell \in
\pi_1(U,z_0)$, analytic continuation along a loop representing the homotopy class $\ell$ defines a linear automorphism of ${\mathcal S}$, which only depends on the homotopy class $\ell$. This defines the [*monodromy representation*]{} $\rho:
\pi_1(U,z_0) \to {\rm Aut}({\mathcal S})$ of the differential system . A slightly different formulation requires not the [*Fuchsian condition*]{} ($A(z)$ has simple poles) but the weaker [*regular singular condition*]{} for . The regularity condition at a singular point $a_i \in \bP^1(\C)\smallsetminus U$ is a growth condition on the solutions, namely all solutions in any strict angular sector centered at $a_i$ have at most polynomial growth in $1/|z-a_i|$. The system is regular singular if every $a_i \in \bP^1(\C)\smallsetminus U$ is a regular singular point. The Hilbert 21st problem (or Riemann–Hilbert problem) asks whether any finite dimensional complex linear representation of $\pi_1(U,z_0)$ is the monodromy representation of a differential system with regular (or Fuchsian) singularities at the points of $\bP^1(\C)\smallsetminus U$. A solution to the Hilbert 21st problem in the regular singular case is given by Plemelj’s theorem ([[*cf.*]{} ]{}[@AnBol] §3). The argument first produces a system with the assigned monodromy on $U$, where in principle an analytic solution has no constraint on the behavior at the singularities. Then, one restricts to a [*local problem*]{} in small punctured disks $\Delta^*$ around each of the singularities, for which a system exists with the prescribed behavior of solutions at the origin. The global trivialization of the holomorphic bundle on $U$ determined by the monodromy datum yields the patching of these local problems that produces a global solution with the correct growth condition at the singularities.
From problem to correspondence {#from-problem-to-correspondence .unnumbered}
------------------------------
A modern revival of interest in Fuchsian differential equation, with a new algebraic viewpoint that slowly transformed the original Riemann–Hilbert problem into the broad landscape of the Riemann–Hilbert correspondence, was pioneered in the early 1960s by the influential paper of Yuri Manin [@Man] on Fuchsian modules. This new perspective influenced the work of Deligne [@De1] in 1970, who solved the Riemann–Hilbert problem for regular singular equations on an arbitrary smooth projective variety. In this viewpoint, if $X$ is a smooth projective variety and $U$ is a Zariski open set, with $X\smallsetminus U$ a union of divisors with normal crossing, the data of an algebraic differential system are given by a pair $(M,\nabla)$ of a locally free coherent sheaf on $U$ with a connection $\nabla : M \to
M\otimes \Omega^1_{U/\C}$, while the regular singular condition says that there exists an algebraic extension $(\bar M,
\bar\nabla)$ of the data $(M,\nabla)$ to $X$, where the extended connection $\bar\nabla : \bar M \to \bar M \otimes
\Omega^1_{X/\C}(\log D)$ has log singularities at the divisor $D$. The reconstruction argument for algebraic linear differential systems with regular singularities in terms of their monodromy representation consists then of first producing an analytic solution $(M,\nabla)$ on $U$ with the prescribed monodromy and then restricting to a local problem in punctured polydisks $\Delta^*$ around the singularities, to obtain a local extension of the form $H(z) \prod_j z_j^{B_j}$, where $H\in
\GL_n({{\mathcal O}}_{\Delta^*})$ and the $B_j$ are commuting matrices that give the local monodromy representation $\exp(2\pi i B_j)$ of $\pi_1(\Delta^*)$. An important point of the argument is to show that these local extensions can be patched together. The patching problem does not arise when $\dim U=1$, since in that case the divisor $D$ consists of isolated points. The construction is then completed by showing that the global analytic extension $(\bar M,
\bar \nabla)$ obtained this way on $X$ is equivalent to an algebraic extension.
Starting with the early 1980s, with the work of Mebkhout [@Me1] [@Me2] and of Kashiwara [@Ka1] [@Ka2], and with the development of the theory of perverse sheaves by Beilinson, Bernstein, Deligne, and Gabber [@BBD], the Riemann–Hilbert correspondence was recast in terms of an equivalence of derived categories between regular holonomic $\Dc_X$-module and perverse sheaves. A reason for introducing the language of $\Dc$-modules ([[*cf.*]{} ]{}[[*e.g.*]{} ]{}[@GeMa] §8 or [@LM] for an overview) is that this captures more information in a differential system $(M,\nabla)$, than what was possible with the previous formulations. For instance, the data $(M,\nabla)$ fit into a de Rham type complex. Also, one may want to consider different classes of solutions (smooth, holomorphic, distributional, etc). This type of extra information is taken care of by the formalism of $\Dc$-modules. Namely, a differential equation determines a module $\Mc$ over $\Dc_X$ (differential operators on $X$ with holomorphic coefficients), with solutions to the equation given by $\Hom_{\Dc_X}(\Mc,\Oc_X)$. One can alter the type of solutions by replacing $\Oc_X$ by another module $\Nc$ over $\Dc_X$, and account for the extra structure in the data $(M,\nabla)$ by considering the de Rham complex $\Mc \otimes_{\Oc_X} \Omega_X$. The condition of regular singularities can be extended to modules $\Mc$ subject to another ‘growth’ condition, related to the module structure, compatibly with a natural filtration of $\Dc_X$ (holonomic $\Dc$-modules). Then the equivalence of categories extends to an equivalence of derived categories, between regular holonomic $\Dc_X$-module and perverse sheaves.
With the regular singular hypothesis replaced by the stronger Fuchsian condition, as in Hilbert’s original formulation, counterexamples to the Riemann–Hilbert problem were later found by Bolibruch [@Bol], in the simplest case of $X={{\mathbb P}}^1(\C)$. On the other hand, one can instead relax the regular singular condition and look at classes of differential systems with irregular singularities. It is immediately clear that finite dimensional complex linear representations of the fundamental group no longer suffice to distinguish equations that can have very different analytic behavior at the singularities and equal monodromy. One can see this in a simple example, where all equations of the form $\frac{d}{dz} f(z) + \frac{1}{z^2} P\left( \frac{1}{z} \right)f(z)
=0$ have trivial monodromy, for any polynomial $P$, but they all have inequivalent behavior at the singularity $z=0$.
Thus, one needs a refinement of the fundamental group, whose finite dimensional linear representations are equivalent to (a given class of) irregular differential systems. There are different approaches to the irregular case. Since we are directly interested in the case relevant to perturbative renormalization, we might as well restrict our attention to the one dimensional setting, namely where $\dim U=1$ and $X$ is a compact Riemann surface. In fact, in our case $X={{\mathbb P}}^1(\C)$ will be sufficient, as we will be interested only in the local problem in a punctured disk $\Delta^*$. As we discuss in Section \[GalSect\] below, in physical terms $\Delta^*$ represents the space of complexified dimensions around a given integer dimension $D$ at which the Feynman integrals of the specified theory $\sT$ are divergent.
In this context, the theory that best fits our needs for the application to renormalization was developed by Martinet and Ramis [@MR], where instead of the usual fundamental group one considers representations of a [*wild fundamental group*]{}, which arises from the asymptotic theory of divergent series and differential Galois theory.
Differential Galois theory and the wild fundamental group {#DiffGalSect .unnumbered}
---------------------------------------------------------
We consider a local version of the irregular Riemann–Hilbert correspondence, in a small punctured disk $\Delta^*$ in the complex plane around a singularity $z=0$. We work in the context of differential Galois theory ([[*cf.*]{} ]{}[@vPS], [@vP]). In this setting, one works over a differential field $(K,\delta)$, such that the field of constants $k=\Ker(\delta)$ is an algebraically closed field of characteristic zero. One considers differential systems of the form $\delta f = Af$, for some $A\in {\rm End}(n, K)$.
For $k=\C$, at the formal level we are then working over the differential field of formal complex Laurent series $K=\C((z))=\C[[z]][z^{-1}]$, with differentiation $\delta=z\frac{d}{dz}$, while at the non-formal level one considers the subfield $K=\C(\{z\})$ of convergent Laurent series.
Given a differential system $\delta f = Af$, its [*Picard–Vessiot ring*]{} is a $K$-algebra with a differentiation extending $\delta$. As a differential algebra it is simple and is generated over $K$ by the entries and the inverse determinant of a fundamental matrix for the equation $\delta f = Af$. The [*differential Galois group*]{} of the differential system is given by the automorphisms of the Picard–Vessiot ring commuting with $\delta$.
The formalism of Tannakian categories, that we discussed in Section \[MotSect\] in the context of motives, reappears in the present context and allows for a description of the differential Galois group that fits in the same general picture we recalled regarding motivic Galois groups.
In fact, if we consider the set of all possible such differential systems (differential modules over $K$), these form a neutral Tannakian category, which can therefore be identified with the category of finite dimensional linear representations of a unique affine group scheme over the field $k$.
Similarly to what we discussed in the case of motivic Galois groups, any subcategory $\T$ that inherits the structure of a neutral Tannakian category in turn corresponds to an affine group scheme $G_\T$. This is the universal differential Galois group of the class of differential systems that form the category $\T$. It can be realized as the automorphisms group of the universal Picard–Vessiot ring $R_\T$. The latter is generated over $K$ by the entries and inverse determinants of the fundamental matrices of all the differential systems considered in the category $\T$.
There is therefore a clear analogy between the induced motivic Galois groups of certain subcategories of, say, the category of mixed Tate motives that we discussed in Section \[MotSect\], and the differential Galois group of certain classes of differential systems defining subcategories of the neutral Tannakian category of irregular differential systems over a differential field $K$. Our main result of [@cmln], which we discuss in Section \[GalSect\] below, shows that the theory of perturbative renormalization (in the DimReg and minimal subtraction scheme) identifies a class of differential systems (dictated by physical assumptions), whose differential Galois group is the motivic Galois group of Proposition \[SNmotives\].
The regular–singular case can be seen in this context as follows. The subcategory of regular–singular differential modules over $K=\C((z))$ is a neutral Tannakian category equivalent to $Rep_G$, where the affine group scheme $G$ is the algebraic hull $\bar\Z$ of $\Z$, generated by the formal monodromy $\gamma$. The latter is the automorphism of the universal Picard–Vessiot ring acting by $\gamma \, Z^a = \exp(2\pi i a)\, Z^a$ and $\gamma\, L = L +
2\pi i$ on the generators $\{ Z^a \}_{a\in \C}$ and $L$, which correspond, in angular sectors, to the powers $z^a$ and the function $\log(z)$ ([[*cf.*]{} ]{}[@vPS] §III, [@vP]).
When one allows for an arbitrary degree of irregularity for the differential systems $\delta f =Af$, the universal Picard–Vessiot ring of the formal theory $K=\C((z))$ is generated by elements $\{ Z^a
\}_{a\in \C}$ and $L$ as before, and by additional elements $\{ E(q)
\}_{q\in \Ec}$, where $\Ec= \cup_{\nu\in \N^\times} \Ec_\nu$, for $\Ec_\nu=
z^{-1/\nu} \C [ z^{-1/\nu} ]$. These additional generators correspond, in local sectors, to functions of the form $\exp(\int q\, \frac{dz}{z})$ and satisfy relations $E(q_1+q_2)=E(q_1)E(q_2)$ and $\delta E(q)=q E(q)$.
Correspondingly, the universal differential Galois group $\cG$ is described by an extension ${\mathcal T}\rtimes \bar\Z$, where ${\mathcal T}=\Hom(\Ec, \C^*)$ is the [*Ramis exponential torus*]{}. The algebraic hull $\bar\Z$ generated by the formal monodromy $\gamma$ acts as an automorphism of the universal Picard–Vessiot ring by the same action as above on the $Z^a$ and on $L$, and by $\gamma\, E(q) = E(\gamma q)$ on the additional generators, where the action on $\Ec$ is given by the action $ \gamma : q( z^{-1/\nu}) \mapsto q(
\exp( -2\pi i/\nu) \, z^{-1/\nu})$ of $\Z/\nu\Z$ on $\Ec_\nu$. The exponential torus acts by automorphisms of the universal Picard–Vessiot ring $\tau\,Z^a=Z^a$, $\tau\, L=L$ and $\tau\, E(q)=\tau(q) E(q)$, and the formal monodromy acts on the exponential torus by $(\gamma\tau)(q)=\tau(\gamma q)$.
Thus, at the formal level, the local irregular Riemann–Hilbert correspondence establishes an equivalence of categories between the differential modules over $K=\C((z))$ and finite dimensional linear representations of $G={\mathcal T}\rtimes \bar\Z$. Ramis’ wild fundamental group [@MR] further extends this irregular Riemann–Hilbert correspondence to the non-formal setting. In general, when passing to the non-formal level over convergent Laurent series $K=\C(\{z\})$, the universal differential Galois group acquires additional generators, which depend upon resummation of divergent series and are related to the Stokes phenomenon. However, there are specific classes of differential systems (subcategories of differential modules over $K$), for which the differential Galois group is the same over $\C((z))$ and over $\C(\{z\})$ ([[*cf.*]{} ]{}[[*e.g.*]{} ]{}Proposition 3.40 of [@vPS]). In such cases, the wild fundamental group consists only of the exponential torus and the formal monodromy. This is, in fact, the case in the class of differential systems we obtain from the theory of perturbative renormalization, hence we do not need to discuss here the more complicated case where Stokes phenomena are present, and we simply refer the interested reader to [@MR], [@vPS], and [@vP].
Cartier’s dream of a cosmic Galois group {#GalSect}
========================================
In the section “I have a dream” of [@Cart], Pierre Cartier formulated the hypothesis of the existence of a “cosmic Galois group”, closely related to the Grothendieck–Teichmüller group [@Gro2], underlying the Connes–Kreimer theory of perturbative renormalization, that would relate quantum field theory to the theory of motives and multiple zeta values.
We present in this section the main result of [@cmln], which realizes Cartier’s suggestion, by reformulating the Connes–Kreimer theory of perturbative renormalization in the form of a suitable Riemann–Hilbert correspondence.
Equisingular connections {#equisingular-connections .unnumbered}
------------------------
The first step, in order to pass to this type of geometric formulation, is to identify the loops $\gamma_\mu(z)=\gamma_{\mu,-}(z)^{-1}
\gamma_{\mu,+}(z)$ with solutions of suitable differential equations. The idea of reformulating a Birkhoff factorization problem in terms of a class of differential equations is familiar to the analytic approach to the Riemann–Hilbert problem ([[*cf.*]{} ]{}[@Boj]). In our setting, the key that allows us to pass from the Birkhoff factorization to an appropriate class of differential systems is provided by the ‘t Hooft relations in the form of Proposition \[thooftrel\] and the scattering formula , reformulated more explicitly in terms of iterated integrals.
Here the main tool is the [*time ordered exponential*]{}, formulated mathematically in terms of Chen’s iterated integrals [@Chen1], [@Chen2], also known (in the operator algebra context) as Araki’s expansional [@Araki].
We consider a commutative Hopf algebra $\cH$ that is graded in positive degree and connected, with $G$ the corresponding affine group scheme and ${{\mathfrak{g}}}=\Lie G$. We assume that $\Hc$ is, in each degree, a finite dimensional vector space. Given a ${{\mathfrak{g}}}(\C)$-valued smooth function $\a(t)$ where $t\in[a,b]\subset \R$ is a real parameter, the expansional is defined by the expression $$\label{expansional}
{\bf {\rm T}e^{\int_a^b\,\a(t)\,dt}}=\,1+\, \sum_1^\infty \int_{a\leq
s_1\leq \cdots\leq
s_n\leq b} \,\a(s_1)\cdots\,\a(s_n) \prod ds_j \,,$$ where the products are in the dual algebra $\Hc^*$ and $1\in \Hc^*$ is the unit given by the augmentation $\ve$. When paired with any element $x\in \Hc$, reduces to a finite sum, which defines an element in $G(\C)$.
The fact that, when pairing with elements in $\Hc$ one reduces to an algebraic (polynomial) case plays an important role. In particular, it is related to the fact that, for the class of differential systems we consider, the differential Galois group remains the same in the formal and in the non-formal case, and we need not take into account the possible presence of Stokes’ phenomena.
We are particularly interested in the following property of the expansional: is the value $g(b)$ at $b$ of the unique solution $g(t)\in G(\C)$ with value $g(a)=1$ of the differential equation $$\label{diffexp}
dg(t)=\,g(t)\,\a(t)\,dt\,.$$
More generally, if $(K,\delta)$ is a differential field with $K\supset
\C$ and if $g\in
G(K)$ and $g'=\delta(g)$ is the linear map $\Hc\to K$ defined as $g'(x)=\, \delta(g(x))$ for $x\in \Hc$, then the logarithmic derivative $D(g)$ is defined as the linear map $\Hc\to K$ of the form $D(g)=\,g^{-1}\star\, g'$, with the product dual to the coproduct of $\cH$. It satisfies $$\langle D(g),x\,y\rangle=\,\langle D(g),x\rangle\,\ve(y)
+\,\ve(x)\,\langle D(g),y\rangle \qq x,y \in \Hc\,,$$ hence it gives an element in the Lie algebra $D(g) \in {{\mathfrak{g}}}(K)$. We will work here with the field of convergent Laurent series $K=\C(\{z
\})$.
If we consider over $\Delta^*$ a differential system of the form $$\label{Dfomega}
Df =\omega,$$ where $\omega$ is a flat ${{\mathfrak{g}}}(K)$-valued connection, then the condition of trivial monodromy $$\label{mono}
{\bf {\rm T}e^{\int_0^1\,\g^*\omega}} =1,$$ for $\gamma\in \pi_1(\Delta^*,z_0)$, ensures the existence of a solution. In the expansional form this is given by $$\label{solution}
g(z) =\;{\bf {\rm T}e^{\int_{z_0}^z\,\omega}}\,,$$ independently of the path in $\Delta^*$ from $z_0$ to $z$.
The notion of equivalence relation that we consider for differential systems of the form is the following: two connections $\omega$ and $\omega'$ are equivalent iff they are related by a gauge transformation $h \in G({{\mathcal O}})$, with ${{\mathcal O}}\subset {K}$ the subalgebra of regular functions, $$\label{gaugetransf}
\omega' = Dh + h^{-1} \omega\, h.$$ The behavior of solutions at the singularity is the same for all equivalent connections. When we regard the solutions as $G(\C)$-valued loops, the equivalence of the connections translates to the fact that the loops have the same negative part of the Birkhoff decomposition.
So far we have not taken into account the fact that, in the case of perturbative renormalization, the loop $\gamma_\mu(z)$ that corresponds to the unrenormalized theory depends on the mass parameter $\mu$, as discussed above in Section \[MassSect\]. Because of the presence of this parameter, the geometric reformulation in terms of a class of differential systems takes place, in fact, not just on the 1-dimensional (infinitesimal) punctured disk $\Delta^*$ representing the complexified dimensions of DimReg, but on a principal ${\mathbb{G}}_m(\C)=\C^*$-bundle over $\Delta^*$.
As we discuss below, the fact that the loop $\gamma_\mu(z)$ satisfies the properties and will make it possible to treat this case, which lives naturally over a 2-dimensional space, by applying the same techniques described in Section \[DiffGalSect\] for the irregular Riemann–Hilbert correspondence over the 1-dimensional domain $\Delta^*$.
The conditions and determine a class of differential systems associated to perturbative renormalization. This is given by equivalence classes of flat [*equisingular*]{} $G(\C)$-connections, where $G={\rm Difg}(\sT)$.
Let $\pi\,:B\to \Delta$ be a principal ${\mathbb{G}}_m(\C)=\C^*$-bundle, identified with $\Delta \times \C^*$ by the non-canonical choice of a section $\sigma\,: \Delta \to B$, $\sigma(0)= y_0$. Physically, the latter corresponds to a choice of the Planck constant. Let $P=B\times G(\C)$ be the trival principal $G(\C)$-bundle, and $B^*$ and $P^*$ the restrictions to the punctured disk $\Delta^*$.
We say that the connection $\omega$ on $P^*$ is [*equisingular*]{} if it is ${\mathbb{G}}_m$-invariant and its restrictions to sections of the principal bundle $B$ that agree at $0\in \Delta$ are mutually equivalent, in the sense that they are related by a gauge transformation by a $G(\C)$-valued ${\mathbb{G}}_m$-invariant map $h$ regular in $B$.
The notion of [*equisingularity*]{} is introduced as a geometric reformulation of the properties and . In fact, the property that, when approaching the singular fiber, the type of singularity does not depend on the section along which one restricts the connection but only on the value of the section at $0\in \Delta$ corresponds to the fact that the counterterms are independent of the mass scale, as in .
Thus, we have identified a class of differential systems associated to a physical theory $\sT$, namely the equivalence classes of flat equisingular $G(\C)$-valued connections on $P$, where $G={\rm
Difg}(\sT)$. We can then proceed to investigate the Riemann–Hilbert correspondence underlying this class of differential systems.
The first step consists of writing solutions of and in expansional form through the following result, which we can view as a stronger version of the ‘t Hooft relations.
\[expans\] Let $\g_\mu(z)$ be a family of $G(\C)$-valued loops satisfying the properties and . Then there exists a unique $\beta \in {\rm Lie}\,G(\C)$ and a loop $\g_{\rm reg}(z)$ regular at $z=0$, such that $$\label{solexp}
\g_\mu(z) =\,{\bf {\rm T}e^{-\frac{1}{z}\,
\int^{-z \log\mu}_\infty\,\t_{-t}(\beta)\,dt}}\;
\t_{z\log\mu}(\g_{\rm reg}(z))\,.$$ Conversely, for any $\beta$ and regular loop $\g_{\rm reg}(z)$ the expression gives a solution to and . The Birkhoff decomposition of $\g_\mu(z)$ is of the form $$\label{gammaplusminus}
\begin{array}{ll} \g_{\mu^+}(z)= & {\bf {\rm T}e^{-\frac{1}{z}
\,\int_0^{-z \log\mu}\,\t_{-t}(\beta)\,dt}}\;
\t_{z\log\mu}(\g_{\rm reg}(z))\,, \\[3mm]
\g_-(z) = & \,{\bf {\rm
T}e^{-\frac{1}{z}\,\int_0^\infty\,\t_{-t}(\beta)\,dt}}\,.
\end{array}$$
Using the equivalent geometric formulation in terms of flat equisingular connections, one then obtains the following result.
\[connbeta\] Let $\omega$ be a flat equisingular $G(\C)$-connection. There exists a unique element $\beta \in \Lie\, G(\C)$, such that $\omega$ is equivalent to the flat equisingular connection $D\g$ for $$\label{solexpm}
\g(z,v) =\,{\bf {\rm T}e^{-\frac{1}{z}\,
\int^{v}_0\,u^Y(\beta)\,\frac{du}{u}}}\;
\in G(\C)\; \,,$$ with the integral performed on the straight path $u=t v$, $t\in[0,1]$.
Here a crucial point is the fact that the monodromies with respect to the two generators of $\pi_1(B^*)$ vanish for flat equisingular connections. As we will see in the next section, this fact will be reflected in the form of the affine group scheme associated to the category of equivalence classes of flat equisingular connections (the differential Galois group), which will only contain the part corresponding to the Ramis exponential torus and no contribution from the monodromy. The correspondence of Proposition \[connbeta\] is independent of the choice of the trivialising section $\sigma$ of $B$.
The Riemann–Hilbert correspondence {#the-riemannhilbert-correspondence .unnumbered}
----------------------------------
So far we have been working with an assigned quantum field theory $\sT$ and the corresponding affine group scheme $G={\rm
Difg}(\sT)$. We now pass to considering a universal setting, which encompasses all theories. This is achieved by considering, instead of flat equisingular $G(\C)$-connections, the category of equivalence classes of all [*flat equisingular bundles*]{}. For a specific physical theory, the corresponding category of equivalence classes of flat equisingular $G(\C)$-connections can be recovered from this more general setting by considering the subcategory of those flat equisingular bundles that are finite dimensional linear representations of $G^*=G\rtimes {\mathbb{G}}_m$. This is analogous to what happens when one specializes motivic Galois groups to sucategories of motives, or differential Galois groups to subcategories of differential systems. We describe now in detail the universal setting, with the corresponding group of symmetries and the way it specializes to a given physical theory.
The category of equivalence classes of flat equisingular bundles has as objects $\Theta=(E,\nabla)$ pairs of a finite dimensional $\Z$-graded vector space $E$ and an equisingular flat $W$-connection $\nabla$. To define the latter, we consider the vector bundle $\tilde{E}=B\times E$ with the action of ${\mathbb{G}}_m$ given by the grading and with the weight filtration defined by $W^{-n}(E)=\oplus_{m\geq n} E_m$. A $W$-connection is a connection on the restriction of $\tilde E$ to $B^*$, which is compatible with the weight filtration and induces the trivial connection on the associated graded. The connection $\nabla$ in the data above is a flat $W$-connection that satisfies the equisingular condition, that is, it is ${\mathbb{G}}_m$-invariant and the restrictions to sections $\sigma$ of $B$ with $\sigma(0)=y_0$ are all $W$-equivalent on $B$, where the equivalence relation is realized by an isomorphism of the vector bundles over $B$, compatible with the filtration and identity on the associated graded, that conjugates the connections. We consider the data $\Theta=(E,\nabla)$ as $W$-equivalence classes.
As usual, it is a bit more delicate to define morphisms than objects.
For a linear map $T: E\to E'$, consider the $W$-connections $\nabla_j$, $j=1,2$, on $\tilde{E'}\oplus \tilde{E}$ of the form $$\label{nablaj}
\nabla_1 = \left( \begin{matrix}\nabla' &0 \cr 0 &\nabla \cr
\end{matrix} \right) \ \ \ \text{ and } \ \ \
\nabla_2 = \left( \begin{matrix}\nabla' &T\,\nabla-\,\nabla'\,T
\cr 0 &\nabla \cr \end{matrix} \right),$$ where $\nabla_2$ is the conjugate of $\nabla_1$ by the unipotent matrix $$\left( \begin{matrix}1 &T \cr 0 &1 \cr \end{matrix}\right)\,.$$ Morphisms $T\in{\rm Hom}(\Theta, \Theta')$ in the category of equisingular flat bundles are linear maps $T: E\to E'$ compatible with the grading and such that the connections $\nabla_j$ of are $W$-equivalent on $B$. The condition is independent of the choice of representatives for the connections $\nabla$ and $\nabla'$.
The category $\Ec$ of equisingular flat bundles is a tensor category over $k=\C$, with a fiber functor $\omega: \Ec \to
Vect_\C$ given by $$\label{omegaE}
\omega: \Theta=(E,\nabla) \mapsto E.$$ In fact, one can refine the construction and work over the field $k=\Q$, since the universal singular frame (see below), in which one expresses the connections, has rational coefficients. In this case, the fiber functor $\omega: \Ec_\Q \to
Vect_\Q$ is of the form $\omega = \oplus \,\omega_n$, with $$\omega_n(\Theta)=\,{\rm Hom}(\Q(n), \Gr_{-n}^W(\Theta))\,,$$ where $\Q(n)$ denotes the object in $\Ec_\Q$ given by the class of the pair of the trivial bundle over $B$ with fiber a one-dimensional $\Q$-vector space placed in degree $n$ and the trivial connection.
Let ${{\mathcal}{F}}(1,2,3,\cdots)_{\bullet}$ be the free graded Lie algebra generated by one element $e_{-n}$ in each degree $n\in \Z_{>0}$, and let $$\label{hopfu}
\Hc_u=\; {{\mathcal}{U}}({{\mathcal}{F}}(1,2,3,\cdots)_{\bullet})^\vee$$ be the commutative Hopf algebra obtained by considering the graded dual of the enveloping algebra ${{\mathcal}{U}}({{\mathcal}{F}})$. We can then identify explicitly the affine group scheme associated to the neutral Tannakian category of flat equisingular bundles as follows ([[*cf.*]{} ]{}[@cmln] [@CM2]).
\[EU\] The category $\Ec$ of flat equisingular bundles is a neutral Tannakian category, with fiber functor . It is equivalent to the category ${\rm
Rep}_{U^*}$ of finite dimensional linear representations of the affine group scheme $U^*=U\rtimes {\mathbb{G}}_m$, where $U$ is the pro-unipotent affine group scheme associated to the Hopf algebra $\Hc_u$ of .
The affine group scheme $U^*$ is a motivic Galois group. In fact, by results of Goncharov and Deligne ([@dg], [@Gon], see Proposition \[SNmotives\] above), we have the following identification of the “cosmic Galois group” $U^*$.
\[motgal\] There is a (non-canonical) isomorphism $$\label{MotU}
U^* \cong G_{\cM_T}({{\mathcal{O}}}) \,.$$ of the affine group scheme $U^*$ with the motivic Galois group $G_{\cM_T}({{\mathcal{O}}})$ of the scheme $S_N$ of $N$-cyclotomic integers, for $N=3$ or $N=4$.
The fact that we only have a noncanonical identification suggests that there should be an explicit identification dictated by the form of the iterated integrals that give the expansionals defining the equisingular connections as in Proposition \[expans\]. This should be related to Kontsevich’s formula for multiple zeta values as iterated integrals generalized by Goncharov to multiple polylogarithms ${\rm Li}_{\,k_1,\ldots,k_m}(z_1,z_2,\ldots,z_m)$, in terms of the expansional ${\bf {\rm
T}e^{\int_0^1\,\a(z)\,dz}}$, with the connection $$\alpha(z)dz = \sum_{a\in \mu_m\cup \{0\}} \, \frac{dz}{z-a} \,\, e_a.$$
Notice, moreover, that the group $U^*$, as the differential Galois group in the formal theory of equisingular connections, corresponds to the Ramis exponential torus. In fact, we have no contribution from the monodromy, a fact on which the proof of Proposition \[connbeta\] depends essentially, and we also do not have Stokes phenomena, hence, as far as the differential Galois group is concerned, we can equally work in the formal or in the non-formal setting.
The renormalization group as a Galois group {#the-renormalization-group-as-a-galois-group .unnumbered}
-------------------------------------------
The formulation of Theorem \[EU\] is universal with respect to physical theories. When we consider a particular choice of a renormalizable theory $\sT$, we restrict the category of equisingular flat bundles to the full subcategory of finite dimensional linear representations of $G^*=G\rtimes {\mathbb{G}}_m$, for $G={\rm Difg}(\sT)$. In this case, the Riemann–Hilbert correspondence specializes to a morphism of differential Galois groups, as follows.
\[rhoG\] Let $G$ be a positively graded pro-unipotent affine group scheme. Then there exists a canonical bijection between equivalence classes of flat equisingular $G(\C)$-connections and graded representations $\rho \, : U\,\to G$, of the affine group scheme $U$ in $G$. Compatibility with the grading implies that $\rho$ extends to a homomorphism $\rho^* \, : U^*\,\to G^*$, which is the identity on ${\mathbb{G}}_m$.
This is a reformulation of the result of Proposition \[connbeta\]. In fact, more explicitly, the representation $\rho$ of Proposition \[rhoG\] is obtained as follows. We can write an element $\beta$ in $\Lie\,G$ as an infinite formal sum $$\label{betasum}
\beta=\; \sum_1^\infty\;\beta_n\,,$$ where, for each $n$, $\beta_n$ is homogeneous of degree $n$ for the grading, [[*i.e.*]{} ]{}$Y(\beta_n)=n \beta_n$. Thus, assigning $\beta$ with the action of the grading is the same as giving a collection of homogeneous elements $\beta_n$, with no restriction besides $Y(\beta_n)=n \beta_n$. In particular, there is no condition on their Lie brackets, hence assigning such data is equivalent to giving a homomorphism from the affine group scheme $U$ to $G$, by assigning, at the Lie algebra level, the generator $e_{-n}$ to the component $\beta_n$ of $\beta$.
In particular, the result above means that we can realize the renormalization group as a Galois group. In fact, recall that, for an assigned theory $\sT$, the corresponding $\beta$ that determines the counterterms $\gamma_{-}(z)$ is the infinitesimal generator of the renormalization group . The representation $\rho: U^* \to G^*$ then determines a lifting of the renormalization group ${\bf rg}$ to a canonical 1-parameter subgroup of $U^*$, obtained by considering the element $$\label{esum}
e=\; \sum_1^\infty\;e_{-n}\,,$$ in the Lie algebra $\Lie\,U$. As $U$ is a pro-unipotent affine group scheme, $e$ defines a morphism of affine group schemes $$\label{rgU}
{\bf{rg}}\; :\,{\mathbb{G}}_a \,\to \,U\,,$$ from the additive group ${\mathbb{G}}_a$ to $U$.
Thus, the rest of the affine group scheme $U$ can be throught of as further symmetries that refine the action of the renormalization group on a given physical theory. More precisely, restricting the attention to a generator $e_{-n}$ of the Lie algebra of $U$ corresponds to considering the flow generated by the degree $n$ component of the $\beta$ function with respect to the grading by loop number. Thus, from a physical point of view the Galois group $U$ accounts for a decomposition of the action of the renormalization group in terms of a family of flows restricted to the $n$-loops theory.
Universal singular frame {#universal-singular-frame .unnumbered}
------------------------
The element $e\in \Lie\,U$ defined in determines a “universal singular frame” given by $$\label{univ}
\g_U(z,v) =\,{\bf {\rm T}e^{-\frac{1}{z}\,
\int^{v}_0\,u^Y(e)\,\frac{du}{u}}}\;
\in U\;
\,.$$ This is obtained by applying Proposition \[connbeta\] to the affine group scheme $U$. This can be expressed explicitly in terms of iterated integrals in the form $$\label{framecoeff}
\g_U(z,v) =\,\sum_{n \geq
0}\,\sum_{k_j>0}\,\frac{e_{-k_1}e_{-k_2}\cdots e_{-k_n}}
{k_1\,(k_1+k_2)\cdots (k_1+k_2+\cdots +k_n)}\,v^{\sum
k_j}\,z^{-n},$$ with $e_{-n}$ the generators of $\Lie \,U$. This expansion has rational coefficients. The coefficients are the same as those occurring in in the local index formula of Connes–Moscovici [@cmindex], where the renormalization group idea is used in the case of higher poles in the dimension spectrum.
The Birkhoff factorization in $U$, applied to the universal singular frame, yields universal counterterms that maps under the representation $\rho: U \to {\rm Difg}(\sT)$ to the counterterms of a specific theory $\sT$.
Renormalization and geometry
============================
Quantum mechanics allows for two equivalent formulations of physics at the macroscopic scale, based on coordinate and momentum space, dual to one another by Fourier transform, while gravity, relativistically formulated in terms of the geometry of space-time, appears to privilege coordinates over momenta.
In the quantum theory of fields, at the perturbative level, Feynman integrals are computed in momentum space, using the dimensional regularization scheme. A nice historical and motivational perspective on how this came to be the general “accepted paradigm” in the context of renormalizable perturbation theory can be found in Veltman’s paper [@Vel]. As Veltman suggests, one can assume perturbative field theory as the starting point, defined in terms of Feynman diagrams using dimensional regularization (he refers to this as the “dimensional formulation”). This is very much the approach followed by the Connes–Kreimer theory and by our present work, where such physical data, taken as the given starting point, are reformulated in a more satisfactory conceptual perspective.
It is also possible to follow a different approach and to consider the problem of perturbative renormalization in coordinate space, working geometrically in terms of Fulton–MacPherson compactifications. A mathematical theory of perturbative renormalization under this point of view was developed recently by Kontsevich [@Kon]. It has the advantage of introducing directly geometric objects like algebraic varieties, hence a natural setting for an explicit action of motivic Galois symmetries ([[*cf.*]{} ]{}also [@Kon1]).
As stressed by Veltman [@Vel], space and time do not occur at all in the dimensional formulation, as coordinate space exists solely as Fourier transform of momentum space, which ceases to be defined when momentum space is continued to complex dimension. Notions associated to coordinate space, such as length and time measurements, must be recovered through the gravitational field, with graviton-fermion interactions determined by gauge invariance and Ward identities. Thus, a viewpoint that favors momentum rather than coordinate space is necessarily closer to noncommutative geometry than to classical algebraic geometry. In noncommutative geometry the metric properties of space are assigned not by a local coordinates description of the metric tensor, but through a “dual viewpoint”, spectrally, in terms of the Dirac operator, hence they continue to make sense on spaces that no longer exist classically. This appears to be a promising approach to reconcile space (no longer defined classically) with the dimensional formulation.
It is important to stress, in this respect, that the formulation of Riemannian spin geometry in the setting of noncommutative geometry, in fact, has already built in the possibility of considering a geometric space at dimensions that are complex numbers rather than integers. This is seen through the dimension spectrum, which is the set of points in the complex plane at which a space manifests itself with a nontrivial geometry. There are examples where the dimension spectrum contains points off the real lines ([[*e.g.*]{} ]{}the case of Cantor sets), but here one is rather looking for something like a deformation of the geometry in a small neighborhood of a point of the dimension spectrum, which would reflect dimensional regularization. The possibility of recasting the dimensional formulation in the setting of noncommutative geometry may prove very useful in the problem of extending at a fully quantum level the geometric interpretation of the standard model of elementary particle physics provided by noncommutative geometry ([@CoSM], [@ChCo]).
An important related question, which may be a starting point for such broader program, is to understand the precise relation between the universal singular frame and the local index formula, which in turn may cast some new light on the issue of the relation of the theory of perturbative renormalization illustrated here and noncommutative geometry. Since the local index formula of Connes–Moscovici is closely related to chiral anomalies, a direct comparison with the local index formula will involve a well known problem associated to dimensional regularization in the chiral case, namely the technical issue of how to extend the definition of the product $$\label{gamma5}
\gamma_5= i \gamma^0\gamma^1\gamma^2\gamma^3,$$ of the $\gamma$ matrices, which integer dimension $D=4$ satisfies the Clifford relations $\{ \gamma^\mu,\gamma^\nu \} = 2
g^{\mu\nu}\, I$, with $\Tr(I)=4$, and anticommutativity $\{
\gamma_5, \gamma^\mu \} =0$. The $\gamma^5$ problem, however, is not considered a serious obstacle to the application of dimensional regularization, as there are good methods to address it ([[*cf.*]{} ]{}[@MSR] for a recent discussion of this issue). For instance, the $\gamma^5$ problem is addressed by the Breitenlohner–Maison approach, in which one does not give an explicit expression for the gamma matrices in complex dimension, but just defines them (and the $\gamma_5$ given by ) through their formal properties. In [@DK1], Kreimer described another approach to the problem, in which $\gamma_5$ still anticommutes with $\gamma^\mu$ but the trace is no longer cyclic, an approach that is expected to be equivalent to the one of Breitenlohner–Maison ([[*cf.*]{} ]{}[@DK1], §5).
Finally, we would like to end on a more speculative tone, by mentioning a very different source for the idea of the existence of a deformation of geometry to non-integral complex dimensions. In arithmetic geometry, the Beilinson conjectures relate the values of the first non-vanishing derivative at integer points of the motivic $L$-functions of algebraic varieties to periods, namely numbers obtained by integration of algebraic differential forms on algebraic varieties ([[*cf.*]{} ]{}[[*e.g.*]{} ]{}[@KoZa]). This process of considering the expansion in a neighborhood of an integer point is reminiscent of the procedure of Dim-Reg, where one considers the Feynman integrals in an infinitesimal neighborhood of the integer dimension $D$. Based on this analogy, it becomes extremely suggestive to imagine that the Beilinson conjectures may be related to a “dimensional regularization of algebraic varieties and periods”, and that there may be a geometric interpretation even for the values at non-integer points, in terms of some (noncommutative) geometry in complex dimension.
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---
abstract: |
The cyclic shift graph of a monoid is the graph whose vertices are the elements of the monoid and whose edges connect elements that are cyclic shift related. The Patience Sorting algorithm admits two generalizations to words, from which two kinds of monoids arise, the ${{\smash{\mathrm{rps}}}}$ monoid and the ${{\smash{\mathrm{lps}}}}$ (also known as Bell) monoid. Like other monoids arising from combinatorial objects such as the plactic and the sylvester, the connected components of the cyclic shift graph of the ${{\smash{\mathrm{rps}}}}$ monoid consists of elements that have the same number of each of its composing symbols. In this paper, with the aid of the computational tool SageMath, we study the diameter of the connected components from the cyclic shift graph of the ${{\smash{\mathrm{rps}}}}$ monoid.
Within the theory of monoids, the cyclic shift relation, among other relations, generalizes the relation of conjugacy for groups. We examine several of these relations for both the ${{\smash{\mathrm{rps}}}}$ and the ${{\smash{\mathrm{lps}}}}$ monoids.
address:
- 'Centro de Matemática e Aplicações, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal'
- 'Centro de Matemática e Aplicações and Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal'
- 'Departamento de Matemática and CEMAT-CIÊNCIAS, Faculdade de Ciências, Universidade de Lisboa, Lisboa 1749-016, Portugal.'
author:
- 'Alan J. Cain'
- António Malheiro
- 'Fábio M. Silva'
bibliography:
- '\\jobname.bib'
title: Conjugacy in Patience Sorting monoids
---
Introduction {#sec:introduction}
============
Patience Sorting has its origins in the works of Mallows [@Mallows62; @10.2307/2028347] and can be regarded as an insertion algorithm on standard words over a totally ordered alphabet ${\mathcal{A}}_n=\{1<2<\dots<n\}$, that is, words over ${\mathcal{A}}_n$ containing exactly one occurrence of each of the symbols from ${\mathcal{A}}_n$. As noticed by Burstein and Lankham [@BL2005], this algorithm can be viewed as a non-recursive version of Schensted’s insertion algorithm. This perspective suggests that a construction similar to the plactic monoid must also hold for this case. The plactic monoid can be constructed as the quotient of the free monoid over ${\mathcal{A}}$ (the infinite totally ordered alphabet of natural numbers), ${\mathcal{A}}^*$, by the congruence which relates words of ${\mathcal{A}}^*$ inserting to the same (semistandard) Young tableaux under Schensted’s insertion algorithm.
According to Aldous and Diaconis [@MR1694204] we can consider two generalizations of Patience Sorting to words, which we will call the right Patience Sorting insertion and the left Patience Sorting insertion ( and insertion, respectively, for short). Considering the alphabet ${\mathcal{A}}$, these generalizations lead to two distinct monoids, the monoid, denoted by ${{\smash{\mathrm{rps}}}}$, and the monoid (also known in the literature as the Bell monoid [@Maxime07]), denoted by ${{\smash{\mathrm{lps}}}}$, which are, respectively, the monoids given by the quotient of ${\mathcal{A}}^*$ by the congruence which relates words having the same insertion under the and insertion.
In a monoid $M$, two elements $u$ and $v$, are said to be related by a cyclic shift, denoted $u{\sim_{{\mathrm{p}}}}v$, if there exists $x,y\in M$ such that $u=xy$ and $v=yx$. In their seminal work concerning the plactic monoid [@MR646486], Lascoux and Schützenberger proved that any two elements in the plactic monoid, ${{\smash{\mathrm{plac}}}}$, having the same evaluation (that is, elements that contain the same number of each generating symbol) can be obtained one from the other by applying a finite sequence of cyclic shift relations. The same characterization is known to hold for other plactic-like monoids, such as the hypoplactic monoid [@1709.03974], the Chinese monoid [@MR1847182], the sylvester monoid [@MR2081336; @MR2142078], and the taiga monoid [@1709.03974]. In Section \[subsection:conjugacy\] we show that an analogous result holds for the monoids (of finite rank) and for the monoid of rank $1$, ${{\smash{\mathrm{lps}}}}_1$. Note that all these monoids are multihomogeneous, that is, they are defined by presentations where the two side of each defining relation contains the same number of each generator. Thus, the evaluation of an element of the monoid corresponds to the evaluation of some (and hence any) word that represents it.
The previous results can be rewritten in another form by considering what we will call as cyclic shift graph of a monoid $M$, denoted ${{\smash{\mathrm{K}}}}(M)$, which is the undirected graph whose vertices are the elements of $M$ and whose edges connect elements that differ by a cyclic shift. So, if $M={{\smash{\mathrm{plac}}}}$ or $M={{\smash{\mathrm{rps}}}}$, or their finite analogues, then the results mentioned in the previous paragraph can be restated as saying that the connected components of ${{\smash{\mathrm{K}}}}(M)$ consist of the elements of $M$ which have the same evaluation. Thus, it follows that the connected components of ${{\smash{\mathrm{K}}}}(M)$ are finite. With the aid of the computational tool SageMath we studied the diameter of the connected components of the cyclic shift graph ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$. In SageMath we wrote a program based on the insertion algorithm, which given a word of ${\mathcal{A}}^*$, outputs the connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ containing the element of ${{\smash{\mathrm{rps}}}}$ that corresponds to the evaluation of the inserted word .
Aiming to parallel the result obtained by Choffrut and Merca[ş]{} [@Choffrut2013], and refined by Cain and Malheiro [@1709.03974], concerning the maximal diameter of connected components of the cyclic shift graph of the plactic monoid of finite rank, we used the tools available in the SageMath library to construct tables containing the number of vertices and the diameter of connected components from ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}}_n)$. The experimental results obtained from these calculations lead us to establish some conjectures regarding diameters of specific connected components. In Section \[subsection:cyclic\_shift\], we show that some of these conjectures are in fact true. In particular we prove that the maximum diameter of a connected components of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}}_n)$, for $n\geq 3$, lies between $n-1$ and $2n-4$. We also draw some conclusions for the diameter of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}}_n)$ for particular elements of ${{\smash{\mathrm{rps}}}}_n$.
The cyclic shift relation previously defined generalizes the usual conjugacy relation for groups. That is, when considering groups, the cyclic shift relation is just the usual conjugacy relation. Since for monoids this relation is, in general, not transitive, it is natural to consider the transitive closure of ${\sim_{{\mathrm{p}}}}$, which we will henceforth denote by ${{\sim}_{{\mathrm{p}}}^*}$. (Note that ${{\sim}_{{\mathrm{p}}}^*}$-classes correspond to connected components of the cyclic shift graph.) We consider two other notions of conjugacy (see [@Araujo201493; @araujo2015four] for other conjugacy notions, their properties, and relations among them). The relation ${\sim_{{\mathrm{l}}}}$ on $M$, proposed by Lallement in [@MR530552], which can be defined as follows: given $u,v\in M$ $$u{\sim_{{\mathrm{l}}}}v\ \Leftrightarrow\ \exists g\in M\ ug=gv.$$ (There is a dual notion ${\sim_{{\mathrm{r}}}}$ relating elements for which $gu=vg$, instead.) As this relation is reflexive and transitive but, in general, not symmetric, in [@MR742135], Otto considered the equivalence relation ${\sim_{{\mathrm{o}}}}$ given by the intersection of ${\sim_{{\mathrm{l}}}}$ and ${\sim_{{\mathrm{r}}}}$.
All the mentioned relations are equal in the group case, and in any monoid, ${\sim_{{\mathrm{p}}}}\ \subseteq\ {{\sim}_{{\mathrm{p}}}^*}\ \subseteq\ {\sim_{{\mathrm{o}}}}\
\subseteq\ {\sim_{{\mathrm{l}}}}$ (cf. [@Araujo201493]). Denoting by ${\sim_{{\mathrm{ev}}}}$ the binary relation that pairs elements with the same evaluation, it is easy to see that for multihomogeneous monoids ${\sim_{{\mathrm{l}}}}\
\subseteq\ {\sim_{{\mathrm{ev}}}}$ (cf. [@cm_conjugacy Lemma 3.2]), and thus for all the above multihomogeneous monoids (plactic, hypoplactic, chinese, sylvester, taiga and ) we have ${{{\sim}_{{\mathrm{p}}}^*}} = {{\sim_{{\mathrm{o}}}}} = {{\sim_{{\mathrm{l}}}}} = {{\sim_{{\mathrm{ev}}}}}$. This property, is not a general property of multihomogeneous monoids, as it is known that in the stalactic monoid connected components of the cyclic shift graph are properly contained in ${\sim_{{\mathrm{ev}}}}$ [@1709.03974 Proposition 7.2]. In this paper we show that a similar situation occurs for monoids of rank greater than 1, since we will prove that ${{\sim_{{\mathrm{l}}}}} \subsetneq {{\sim_{{\mathrm{ev}}}}}$ in these cases.
Preliminaries and notation
==========================
In this section we introduce the fundamental notions that we will use along the paper. For more details regarding these concepts check for instance [@1706.06884], [@MR1905123], and [@howie1995fundamentals].
Words and presentations {#alphabetswords}
-----------------------
In this paper, we denote by ${\mathcal{A}}$ the infinite totally ordered alphabet $\{1<2<\dots \}$, that is, the set of natural numbers with the usual order viewed as an alphabet. For any $n\in\mathbb{N}$, the resriction of ${\mathcal{A}}$ to the first $n$ natural numbers is denoted by ${\mathcal{A}}_n$.
In general, if $\Sigma$ is an alphabet, then $\Sigma^+$ denotes the *free semigroup* over $\Sigma$, that is, the set of non-empty words over $\Sigma$, and if $\varepsilon$ denotes the empty word, then the *free monoid* over $\Sigma$ is $\Sigma^*=
\Sigma^+\cup \{\varepsilon\}$.
Next, we define several concepts that are directly related with the notion of word. Let $w\in{\mathcal{A}}^*$. Then:
- a word $u\in {\mathcal{A}}^*$, is said to be a *factor* of $w$ if there exist words $v_1,v_2\in {\mathcal{A}}^*$, such that $w=v_1uv_2$;
- for any symbol $a$ in ${\mathcal{A}}$, the number of occurrences of $a$ in $w$, is denoted by ${\left|w\right|}_{a}$;
- the *content of* $w$, is the set ${{\smash{\mathrm{cont}}}}(w)=\left\{{a}\in
{\mathcal{A}}: {\left|w\right|}_{a}\geq 1\right\}$;
- the *evaluation of* $w$, denoted by ${{{\mathrm{ev}}}\parens[]{w}}$, is the sequence of non-negative integers whose $a$-th term is ${\left|w\right|}_{a}$, for any $a\in {\mathcal{A}}$;
- the word is said to be *standard* if each symbol from ${\mathcal{A}}_n$, for a given $n$, occurs exactly once.
A *monoid presentation* is a pair $(\Sigma,
\mathcal{R})$, where $\Sigma$ is an alphabet and $\mathcal{R}\subseteq
\Sigma^*\times \Sigma^*$. We say that a monoid $M$ is *defined by a presentation* $(\Sigma,\mathcal{R})$ if $M\simeq \Sigma^*/\mathcal{R}^\#$, where $\mathcal{R}^\#$ is the smallest congruence containing $\mathcal{R}$ (see [@howie1995fundamentals Proposition 1.5.9] for a combinatorial description of the smallest congruence containing a relation).
A presentation is *multihomogeneous* if, for every relation $(w,w') \in\mathcal{R}$, we have ${{{\mathrm{ev}}}\parens[]{w}}={{{\mathrm{ev}}}\parens[]{w'}}$, in other words, if $w$ and $w'$ contain the same number of each of its composing symbols. Then, a monoid is multihomogeneous if there exists a multihomogeneous presentation defining the monoid.
PS. *tableaux* and insertion
-----------------------------
In this subsection we recall the basic concepts regarding patience sorting *tableaux*, and the insertion on such *tableaux*.
A *composition diagram* is a finite collection of boxes arranged in bottom-justified columns, where no order on the length of the columns is imposed. Let $\Sigma$ be a totally ordered alphabet. Then, an (resp. ) *tableau over* $\Sigma$ is a composition diagram with entries from $\Sigma$, so that the sequence of entries of the boxes in each column is strictly (resp., weakly) decreasing from top to bottom, and the sequence of entries of the boxes in the bottom row is weakly (resp., strictly) increasing from left to right. So, if $$\label{exmp2}
\ytableausetup
{aligntableaux=center, boxsize=1.25em}
R=\begin{ytableau}
\none &\none & 4 \\
4 & 5 & 3 \\
1 & 1 & 2
\end{ytableau}\ \text{ and }\ S=\begin{ytableau}
\none & 5 \\
4 & 4 \\
1 & 3 \\
1 & 2
\end{ytableau},$$ then $R$ is an *tableau*, and $S$ is an *tableau* both over ${\mathcal{A}}_n$, for $n \geq 5$. Henceforth, we shall often refer to an tableau or to an tableau simply as a PS. tableau, not distinguishing the cases whenever they can be dealt in a similar way.
The left and right Patience Sorting monoids can be given as the quotient of the free monoid ${\mathcal{A}}^*$ over the congruence which relates words that yield the same PS. tableau under a certain algorithm [@1706.06884 § 3.6]. This algorithm is presented in the following paragraph and merges in one the Algorithms 3.1 and 3.2 of [@1706.06884]. (Observe that we will use the notation ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}()$, ${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}()$ instead of, respectively, $\mathfrak{R}_\ell()$, $\mathfrak{R}_r()$ used in [@1706.06884].)
\[alg:PSinsertion\]
*Input:* A word $w$ over a totally ordered alphabet $\Sigma$.
*Output:* An *tableau* ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(w)$ (resp., *tableau* ${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(w)$).
*Method:*
1. If $w=\varepsilon$, output an empty *tableau* $\emptyset$. Otherwise:
2. $w=w_1\cdots w_n$, with $w_1,\ldots,w_n\in \Sigma$. Setting $$\begin{aligned}
{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(w_1)=\ytableausetup
{boxsize=1.1em, aligntableaux=center}\begin{ytableau}
w_1 \end{ytableau} ={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(w_1),
\end{aligned}$$ then, for each remaining symbol $w_j$ with $1<j\leq n$, denoting by $r_1\leq \dots\leq r_k$ (resp., $r_1< \dots< r_k$) the symbols in the bottom row of the *tableau* ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(w_1\cdots w_{j-1})$ (resp., ${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(w_1\cdots w_{j-1})$), proceed as follows:
- if $r_k\leq w_j$ (resp., $r_k < w_j$), insert $w_j$ in a new column to the right of $r_k$ in ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(w_1\cdots w_{j-1})$ (resp., ${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(
w_1\cdots w_{j-1})$);
- otherwise, if $m=\min\left\{i\in\{1,\ldots,
k\}:w_j< r_i\right\}$, (resp. $m=\min\left\{i\in\{1,\ldots,
k\}:w_j\leq r_i\right\}$) construct a new empty box on top of the column of ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(w_1\cdots w_{j-1})$ (resp. ${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(w_1\cdots w_{j-1})$) containing $r_m$. Then bump all the symbols of the column containing $r_m$ to the box above and insert $w_j$ in the box which has been cleared and previously contained the symbol $r_m$.
Output the resulting *tableau*.
Observe that the insertion of a given word $w=w_1\cdots w_n$ under Algorithm \[alg:PSinsertion\] is obtained through the insertion of each of its symbols, from left to right in the previously obtained tableaux (starting with the empty tableaux $\emptyset$). For instance, if $R$ is the tableau from Example \[exmp2\], and $u=4511432
\in{\mathcal{A}}^*_5$, then ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u)=R$ (see Figure \[figure:extended\_insertion\]). The reader can check that ${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(u)=S$.
$$\begin{aligned}
&\emptyset \xleftarrow[4]{\ \ \ }\
\ytableausetup
{mathmode, boxsize=1.3em, aligntableaux=center}
\begin{ytableau}
4
\end{ytableau}\
\xleftarrow[5]{\ \ \ }\
\begin{ytableau}
4 & 5
\end{ytableau}\
\xleftarrow[1]{\ \ \ }\
\begin{ytableau}
4 & \none\\
1 & 5
\end{ytableau}\
\xleftarrow[1]{\ \ \ }\
\begin{ytableau}
4 & 5\\
1 & 1
\end{ytableau}\\
&\xleftarrow[4]{\ \ \ }\
\begin{ytableau}
4 & 5 & \none \\
1 & 1 & 4
\end{ytableau}\
\xleftarrow[3]{\ \ \ }\
\begin{ytableau}
4 & 5 & 4 \\
1 & 1 & 3
\end{ytableau}
\xleftarrow[2]{\ \ \ }\
\begin{ytableau}
\none & \none & 4\\
4 & 5 & 3\\
1 & 1 & 2
\end{ytableau}={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u ).\
\end{aligned}$$
The Patience Sorting monoids
----------------------------
For each $\mathrm{x}\in \{{{\mathrm{l}}},{{\mathrm{r}}}\}$, we define a binary relation ${\equiv_{{\smash{\mathrm{xps}}}}}$ in ${\mathcal{A}}^*$ in the following way: given $u,v\in{\mathcal{A}}^*$, $$u{\equiv_{{\smash{\mathrm{xps}}}}}v\quad \textrm{iff}\quad {{\smash{\mathrm{P}}}_{\smash{\mathrm{xps}}}}(u)={{\smash{\mathrm{P}}}_{\smash{\mathrm{xps}}}}(v).$$ This relation is a congruence [@1706.06884 Proposition 3.21], and the quotient of ${\mathcal{A}}^*$ by ${\equiv_{{\smash{\mathrm{lps}}}}}$ is the so-called monoid, denoted ${{\smash{\mathrm{lps}}}}$, and the quotient of ${\mathcal{A}}^*$ by ${\equiv_{{\smash{\mathrm{rps}}}}}$ is the monoid which is denoted by ${{\smash{\mathrm{rps}}}}$. The rank-$n$ analogues of these monoids, denoted by ${{\smash{\mathrm{lps}}}}_n$ and ${{\smash{\mathrm{rps}}}}_n$, are obtained by restricting the alphabet and the relation to the set ${\mathcal{A}}_n^*$. Note that each equivalence class of these monoids is represented by a unique tableau, and hence we will identify elements of the monoid with their tableaux representation.
Words yielding the same PS. tableau (and hence in the same ${\equiv_{{\smash{\mathrm{xps}}}}}$-class) have necessarily the same content, and even the same evaluation. Thus, we can refer to the content and evaluation of an element of the monoid, and similarly to the content and evaluation of a tableau. Also, we shall refer to an element of ${{\smash{\mathrm{xps}}}}_n$ (or to its tableau representative) as *standard* if one (and hence any) of its words in the ${\equiv_{{\smash{\mathrm{xps}}}}}$-class has one occurrence of each of the symbols from ${\mathcal{A}}_n$.
As shown in [@1706.06884 § 3.6 & § 3.7], the left and right Patience Sorting monoids are defined by the multihomogeneous presentations $({\mathcal{A}}^*,{\mathcal{R}}_{{\smash{\mathrm{lps}}}})$ and $({\mathcal{A}}^*,{\mathcal{R}}_{{\smash{\mathrm{rps}}}})$, where $$\begin{aligned}
{\mathcal{R}}_{{\smash{\mathrm{lps}}}}&=\{\,(yux,yxu): m\in \mathbb{N},\, x,y,u_1,\ldots , u_m\in
{\mathcal{A}}, \\
&\qquad u=u_m\cdots u_1,\, x<y\leq
u_1< \cdots < u_m\,\}\end{aligned}$$ and $$\begin{aligned}
{\mathcal{R}}_{{\smash{\mathrm{rps}}}}&=\{\,(yux,yxu): m\in \mathbb{N},\, x,y,u_1,\ldots , u_m\in
{\mathcal{A}}, \\
&\qquad u=u_m\cdots u_1,\, x\leq y<
u_1\leq \cdots \leq u_m\,\}.\end{aligned}$$ Hence, the left and right Patience Sorting monoids, and their finite rank analogues, are multihomogeneous monoids.
We have seen how to obtain a PS. tableau from a word in ${\mathcal{A}}^*$. Now, we explain how to pass from PS. tableaux to words representing such diagrams. Given $\textrm{x}\in \{{{\mathrm{l}}},{{\mathrm{r}}}\}$ and an $x$PS. tableau $P$, the *column reading of* $P$ is the word obtained from reading the entries of the $x$PS. tableau $P$, column by column, from the leftmost to the rightmost, starting on the top of each column and ending on its bottom. For example, the column reading of the tableau $R$ in Example \[exmp2\] is $41\, 51\, 432$, while the column reading of the tableau $S$ is $411\,
5432$.
Combinatorics of cyclic shifts {#Subsection4.1}
==============================
\[subsection:cyclic\_shift\]
As noted in the introduction, the *cyclic shift graph of* a monoid $M$, ${{\smash{\mathrm{K}}}}(M)$, is the undirected graph with vertex set $M$, whose edges connect vertices that differ by a single cyclic shift. Since, ${{\smash{\mathrm{rps}}}}$ is a multihomogeneous monoid, we have ${{{\sim}_{{\mathrm{p}}}^*}}\subseteq
{{\sim_{{\mathrm{ev}}}}}$, and thus each connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ cannot contain elements with different evaluations and therefore they have finitely many vertices.
Our goal in this subsection is to study the diameter of the connected components from ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}}_n)$, which as we will show are bounded by a value that depends on the rank $n$. Note that in [@1709.03974 Example 3.1], the authors provide a finitely presented multihomogeneous monoid for which the connected components of the cyclic shift graph have unbounded diameter. Therefore, these are not particular cases of a more general result that holds for all multihomogeneous monoids.
The experimental results within this subsection were obtained with the aid of SageMath [@cocalc]. This computational tool allowed us to write a program for which: given an element of ${{\smash{\mathrm{rps}}}}_n$, provides the connected component from the cyclic shift graph of ${{\smash{\mathrm{rps}}}}_n$ containing that element.
The program starts by creating a vertex for each word from ${\mathcal{A}}^*_n$ that has the same evaluation as the given element from ${{\smash{\mathrm{rps}}}}_n$. Afterwards, it adds edges between the words that are cyclic shift related. Finally, by merging the vertices whose $x$PS. insertion is the same into a single vertex, it constructs the connected component of the cyclic shift graph of ${{\smash{\mathrm{rps}}}}_n$, ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}}_n)$, containing the given element from ${{\smash{\mathrm{rps}}}}_n$.
For instance in Figure \[fig:connected\_component\] we show the connected component of the cyclic shift graph of ${{\smash{\mathrm{rps}}}}_4$ containing the element ${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(1234)$ that can be seen to have diameter $4$.
(0.5,-3.5) – (9.5,6); (1)
[1]{} & [2]{} & [3]{} & [4]{}
; (2) \[right of=1\]
[2]{} & &\
[1]{} & [3]{} & [4]{}
; (3) \[above of=2\]
[3]{} & [4]{}\
[1]{} & [2]{}
; (4) \[below of=2\]
[4]{} & &\
[1]{} & [2]{} & [3]{}
; (5) \[right of=2\]
[3]{} &\
[2]{} &\
[1]{} & [4]{}
; (6) \[above of=5\]
[4]{} &\
[2]{} &\
[1]{} & [3]{}
; (7) \[above of=6\]
& [4]{} &\
[1]{} & [2]{} & [3]{}
; (8) \[below of=5\]
& [3]{} &\
[1]{} & [2]{} & [4]{}
; (9) \[right of=5\]
[2]{} & [4]{}\
[1]{} & [3]{}
; (10) \[above of=9\]
[4]{}\
[3]{}\
[2]{}\
[1]{}
; (11) \[above of=10\]
& [4]{}\
& [3]{}\
[1]{} & [2]{}
; (12) \[below of=9\]
[4]{} & [3]{}\
[1]{} & [2]{}
; (13) \[right of=9\]
& & [4]{}\
[1]{} & [2]{} & [3]{}
; (14) \[above of=13\]
[4]{} &\
[3]{} &\
[1]{} & [2]{}
; (15) \[below of=13\]
[3]{} & &\
[1]{} & [2]{} & [4]{}
;
\(1) edge node (2) (1) edge node (3) (1) edge node (4) (2) edge node (3) (2) edge node (4) (2) edge node (5) (2) edge node (6) (2) edge node (7) (2) edge node (8) (3) edge \[bend right=37pt\] node (4) (3) edge node (6) (3) edge node (7) (5) edge node (6) (5) edge node (8) (5) edge node (9) (5) edge node (10) (5) edge node (11) (5) edge node (12) (6) edge node (7) (6) edge \[bend right=30pt\] node (8) (8) edge node (9) (8) edge node (12) (9) edge node (10) (9) edge \[bend right=27pt\] node (11) (9) edge node (12) (9) edge node (13) (9) edge node (14) (9) edge node (15) (10) edge node (11) (13) edge node (14) (13) edge node (15) (14) edge \[bend left=38pt\] node (15) ;
The results of computer experimentation on the diameter of connected compontents is shown in Tables \[tab:std\_cyclic\_shift\_graph\] and \[tab:cyclic\_shift\_graph\]. In Table \[tab:std\_cyclic\_shift\_graph\] we present the diameter and number of vertices in the connected component of the cyclic shift graph of standard elements of lengths $1$ up to $9$, whereas in Table \[tab:cyclic\_shift\_graph\] the same information is presented but for some (non-standard) words of given fixed evaluations.
--- ------- ---- -------- --
1 1 0 $n-1$
2 2 1 $n-1$
3 5 2 $2n-4$
4 15 4 $2n-4$
5 52 6 $2n-4$
6 203 8 $2n-4$
7 877 10 $2n-4$
8 4140 12 $2n-4$
9 21147 14 $2n-4$
--- ------- ---- -------- --
: Examples of diameter and number of vertices in the connected component of the cyclic shift graph ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ for given evaluations of standard elements.
\[tab:std\_cyclic\_shift\_graph\]
The results in Table \[tab:std\_cyclic\_shift\_graph\] suggest the following:
\[conj:diameter\_std\] The diameter of a connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ containing a standard element of length $n\geq 3$ is $2n-4$.
Note that the connected components of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ and ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{lps}}}})$ coincide when restricted to standard elements.
The data gathered in both Table \[tab:std\_cyclic\_shift\_graph\] and Table \[tab:cyclic\_shift\_graph\] leads us to propose the following:
\[conj:diameter\_nonstd\] The diameter of a connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ containing an element with $n\geq 3$ symbols, with possible multiple appearences of each symbol, lies between $n-1$ and $2n-4$.
Evaluation
------------------- ------- ---- ------------
(5) 1 0 $n-1$
(5,3) 4 1 $n-1$
(4,1,4) 20 2 $n-1=2n-4$
(3,3,1,2) 75 3 $n-1=2n-5$
(1,2,4,2) 287 4 $n=2n-4$
(1,3,2,1,2) 656 5 $n=2n-5$
(2,1,1,2,3) 554 4 $n-1=2n-6$
(1,2,1,2,2) 711 6 $n+1=2n-4$
(1,1,1,3,1,2) 2409 7 $n+1=2n-5$
(1,1,2,2,1,2) 2840 6 $n=2n-6$
(1,2,1,1,2,2) 2373 8 $n+2=2n-4$
(1,1,1,1,2,1,2) 6499 9 $n+2=2n-5$
(1,1,1,2,1,1,2) 6078 8 $n+1=2n-6$
(1,1,1,1,1,2,2) 6768 10 $n+3=2n-4$
(1,1,1,1,1,2,1,1) 11695 11 $n+3=2n-5$
(1,1,1,1,2,1,1,1) 11224 10 $n+2=2n-6$
(1,1,1,1,1,1,2,1) 12002 12 $n+4=2n-4$
: Examples of diameter and number of vertices in the connected component of the cyclic shift graph ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ for given evaluations of non-standard elements.
\[tab:cyclic\_shift\_graph\]
One of the first results that was possible to obtain from the data was
\[prop:diameter12\] All elements of ${{\smash{\mathrm{rps}}}}$ containing two symbols, with the same evaluation, form a connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$. Furthermore, the component has diameter $1$.
As already noticed each connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ cannot contain elements with different evaluations. Let $u$ and $v$ be two elements of ${{\smash{\mathrm{rps}}}}$ with the same evaluation such that ${\left|{{\smash{\mathrm{cont}}}}(w)\right|}= 2$. Suppose without loss of generality that ${{\smash{\mathrm{cont}}}}(w)=\{1,2\}$. Then, these elements are of the form ${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^i1^j2^k)$, for some $i, k\in \mathbb{N}_0$ and $i+k,j\in\mathbb{N}$. So, $u={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^i1^j2^k)$ and $v={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^l1^n2^m)$ with $j=n$ and $i+k=l+m$. Therefore, $v={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^l1^j2^m)$ and we consider the following cases:
If $i\geq l$, then $k+i-l=m$. Setting $x={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{i-l})$ and $y={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^l1^j2^k)$, we have $$\begin{aligned}
&u={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{i-l}2^l1^j2^k)={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{i-l}) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^l1^j2^k)=x y\,
\text{ and}\\
&v={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^l1^j2^k2^{i-l})={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^l1^j2^k) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{i-l})=y x.
\end{aligned}$$
Otherwise, if $i<l$, then $m+l-i=k$. Setting $x={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{i}1^{j}2^{m})$ and $y={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{l-i})$, we get $$\begin{aligned}
&u={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{i}1^{j}2^{m}2^{l-i})={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{i}1^{j}2^{m})
{{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{l-i})=x y\, \text{ and}\\
&v={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{l-i}2^{i}1^{j}2^{m})={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{l-i})
{{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(2^{i}1^{j}2^{m})=y x.
\end{aligned}$$ In both cases, $u{\sim_{{\mathrm{p}}}}v$. Therefore, the diameter of the connected component from ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ containing such elements is 1. The result follows.
In the following lemma we provide an upper bound for the diameter of the connected components from ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ of elements whose content is greater or equal to $3$, thus answering the upper bound part of Conjecture \[conj:diameter\_nonstd\].
By observing several connected components obtained with the program constructed with SageMath, we concluded that for any element $w\in {{\smash{\mathrm{rps}}}}$, with ${{\smash{\mathrm{cont}}}}(w)=\{1,\ldots,
n\}$ and $n\geq 3$, the element $${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left((n-1)^{{{\left|w\right|}}_{n-1}}
(n-2)^{{{\left|w\right|}}_{n-2}}\cdots 3^{{\left|w\right|}_3}
2^{{\left|w\right|}_2}1^{{\left|w\right|}_1}\ n^{{{\left|w\right|}}_{n}}\right)$$ plays a key role in the connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ which contains $w$. For instance, in Figure \[fig:connected\_component\], we see that the element $${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(3214)={\color{Grey}\begin{ytableau}
{\color{black}3} & \none\\
{\color{black}2} & \none\\
{\color{black}1} & {\color{black}4}
\end{ytableau}}$$ is in the center of the connected component. Using this insight we were able to prove the following result:
\[prop:diameter\_upper\_bound\] All elements of ${{\smash{\mathrm{rps}}}}$ containing $n\geq 3$ symbols, with the same evaluation, form a connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$. Furthermore, the component has diameter at most $2n-4$.
Let $w$ be an element of ${{\smash{\mathrm{rps}}}}$ with ${\left|{{\smash{\mathrm{cont}}}}(w)\right|}=n\geq 3$. Suppose without loss of generality that ${{\smash{\mathrm{cont}}}}(w)=\{1,\ldots, n\}$. Since each connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}}_n)$ cannot contain elements with different evaluations, to prove this result, it suffices to check that from $w$, by applying at most $n-2$ cyclic shift relations we can always obtain the element $$w'={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left((n-1)^{{\left|w\right|}_{n-1}} (n-2)^{
{\left|w\right|}_{n-2}}\cdots 2^{{\left|w\right|}_2}1^{{\left|w\right|}_1}
n^{{\left|w\right|}_{n}}\right)$$ of ${{\smash{\mathrm{rps}}}}$.
We will construct a path in ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}}_n)$ from $w$ to $w'$ of length at most $n-2$. We aim to find a sequence $w_0,w_1,\ldots,w_{n-2}$ of elements of ${{\smash{\mathrm{rps}}}}_n$ such that $w=w_0$, $w'=w_{n-2}$, and $w_i{\sim_{{\mathrm{p}}}}w_{i+1}$, for $i=0,\ldots, n-3$. The construction is inductive. First note that all the symbols $1$ occur in the bottom of the first column of $w$. If $w$ has only one column, then $w$ has column reading $n^{{\left|w\right|}_{n}}(n-1)^{{\left|w\right|}_{n-1}}
(n-2)^{{\left|w\right|}_{n-2}}\cdots 2^{{\left|w\right|}_2}1^{{\left|w\right|}_1}$ and applying one cyclic shift we get the intended result. Suppose $w$ has at least two columns. Let $k$ (necessarily $k\geq 2$) be the bottom symbol of the second column of $w$. Observe that any symbol $j$ less than $k$ must lie in the first column of $w$. Set $w=w_0=\dots =w_{k-1}$. We calculate the element $w_k$ from $w$ in the following way. Consider the column reading $ukv$, of $w$, for $u,v\in {\mathcal{A}}_n^*$, where $u$ is the prefix just up to before the first occurrence of a symbol $k$ occurring in the second column. Fix $w_k={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(kv) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(u)$. Note that $w={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(u) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(kv)$ and so $w{\sim_{{\mathrm{p}}}}w_k$.
Then, the first column of $w_k$ has column reading $k^{{\left|w\right|}_k}\dots 2^{{\left|w\right|}_2}1^{{\left|w\right|}_1}$, because all symbols in $v$ are greater or equal to $k$, and symbols in $u$ that are strictly less than $k$ appear in decreasing order. For $i\in\{k, \ldots, n-2\}$, let $w_{i}={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(u\right) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left((i+1)v\right)$ where $u$ is the prefix of the column reading of $w_{i}$ up to just before the first occurrence of a symbol $i+1$ (in the second column) and let $w_{i+1}={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left((i+1)v\right) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(u\right)$. Using this process we ensure that the first column of $w_{i+1}$ is precisely $${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left((i+1)^{{\left|w\right|}_{i+1}} i^{{\left|w\right|}_i}
\cdots 2^{{\left|w\right|}_2} 1^{{\left|w\right|}_1}\right).$$ The result follows by induction.
Regarding the lower bound of Conjecture \[conj:diameter\_nonstd\], we are only able to establish it for standard elements of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$. To prove such a result, we will use the notion of cocharge sequence for standard words over ${\mathcal{A}}$ and follow an approach similar to the one used in the case of the plactic monoid in [@1709.03974]. Note that it will be sufficient to prove the result for standard words over the alphabet ${\mathcal{A}}_n$.
For any standard word $w$ over ${\mathcal{A}}_n$, the *cocharge labels* from the symbols of $w$ are calculated as follows:
- draw a circle, place a point $*$ somewhere on its circumference, and, starting from $*$, write $w$ anticlockwise around the circle;
- let the cocharge label of the symbol $1$ be $0$;
- iteratively, suppose the cocharge label of the symbol $a$ from $w$ is $k$, then proceed clockwise from the symbol $a$ to the symbol $a+1$ and:
- if the symbol $a+1$ of $w$ is reached without passing the point $*$, then the cocharge label of $a+1$ is $k$;
- otherwise, if the symbol $a+1$ is reached after passing the point $*$, then the cocharge label of $a+1$ is $k+1$.
The *cocharge sequence* of a standard word $w$, ${{\smash{\mathrm{cochseq}}}}(w)$, is the sequence of the cocharge labels from the symbols of $w$, whose $a$-th term is the cocharge label from the symbol $a$ of $w$. So, it follows from the definition that if $w$ is a standard word over ${\mathcal{A}}_n$, then ${{\smash{\mathrm{cochseq}}}}(w)$ is a sequence of length $n$.
For example, the labelling of the standard word $w=4572631$ over ${\mathcal{A}}_7$, proceeds in the following way
(0,0) circle\[radius=5mm\];
(4:-4mm) arc\[radius=4mm,start angle=-170,end angle=-10\];
(-90:2mm);
(-18:13.5mm) arc\[radius=13mm,start angle=-10,end angle=-170\];
(-165:18mm) arc\[radius=18mm,start angle=-170,end angle=-10\];
i/in [0/\*,9/4,8/5,7/7,6/2,5/6,4/3,3/1]{} [ at ($ (90-\i*30:7.8mm) $) [$\ilabel$]{}; ]{}; i/in [9/2,8/2,5/2,7/3,3/0,6/1,4/1]{} [ at ($ (90-\i*30:11.2mm) $) [$\ilabel$]{}; ]{};
and thus ${{\smash{\mathrm{cochseq}}}}(w)=(0,1,1,2,2,2,3)$.
From the definition it also follows that the cocharge sequence is a weakly increasing sequence which starts at $0$ and such that each of the remaining terms is either equal to the previous term or greater by $1$.
\[standardinvariant\] For standard words $u,v$ over ${\mathcal{A}}_n$, if $u{\equiv_{{\smash{\mathrm{rps}}}}}v$, then ${{\smash{\mathrm{cochseq}}}}(u)={{\smash{\mathrm{cochseq}}}}(v)$.
It is enough to show that any two standard words over ${\mathcal{A}}_n$ such that one is obtained from the other by applying a relation from ${\mathcal{R}}_{{{\smash{\mathrm{rps}}}}_n}$ have the same cocharge sequence.
So, there is a factor $yu_i\cdots u_1x$ of one of the standard words, with $x,y,u_1,\ldots,u_i\in{\mathcal{A}}_n$ and $x<y< u_1<\cdots <u_i$, that is changed to the factor $yxu_i\cdots u_1$ of the other. Given any symbol $a\in {\mathcal{A}}_n\setminus\{1\}$, when applying such relation, the relative position between the symbols $a$ and $a-1$ is not changed. That is, if $a-1$ occurs to the right (resp. left) of $a$ in one of the standard words, then $a-1$ also occurs to the right (resp. left) of $a$ in the other. Thus, equal symbols of these standard words have the same cocharge label and therefore the cocharge sequence of these words is the same.
Given a standard element $u$ of ${{\smash{\mathrm{rps}}}}_n$ on $n$ generators, let ${{\smash{\mathrm{cochseq}}}}(u)$ be ${{\smash{\mathrm{cochseq}}}}(w)$ for any word $w\in {\mathcal{A}}^*_n$ such that ${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(w)=u$. Using the previous lemma we conclude that ${{\smash{\mathrm{cochseq}}}}(u)$ is well-defined.
\[prop:diameter\_lower\_bound\] The diameter of a connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}}_n)$, with $n\geq 2$, containing a standard element is at least $n-1$.
The case $n=2$ follows from Lemma \[prop:diameter12\]. Suppose $n\geq 3$.
From [@1709.03974 Lemma 2.2], we deduce that any two standard elements of ${{\smash{\mathrm{rps}}}}$ that differ by a single cyclic shift, have cocharge sequences whose corresponding terms differ by at most $1$.
The standard elements $u={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(1\cdot 2\cdots
n\right)$ and $v={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(n(n-1)\cdots 1\right)$ of ${{\smash{\mathrm{rps}}}}_n$ are in the same connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ by Lemma \[prop:diameter\_upper\_bound\]. Now, notice that ${{\smash{\mathrm{cochseq}}}}(u)=(0,0,\ldots,0)$ and that ${{\smash{\mathrm{cochseq}}}}(v)=(0,1,\ldots,n-1)$. Since the last term of both sequences differs by $n-1$, the standard elements $u$ and $v$ are at distance of at least $n-1$.
For instance, in Figure \[fig:connected\_component\], the distance between the elements $${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(1234)={\color{Grey}\begin{ytableau}
{\color{black}1} & {\color{black}2} & {\color{black}3} & {\color{black}4}
\end{ytableau}}\ \text{ and }\ {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(4321)={\color{Grey}\begin{ytableau}
{\color{black}4} \\
{\color{black}3} \\
{\color{black}2} \\
{\color{black}1}
\end{ytableau}}$$ in that connected component is precisely $3$, which is in accordance with the previous result.
Since for standard words the cyclic shif graphs ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{lps}}}})$ and ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ coincide, the previous result also give us a lower bound for connected components of standard words of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{lps}}}})$.
Combining Lemmata \[prop:diameter12\], \[prop:diameter\_upper\_bound\] and \[prop:diameter\_lower\_bound\] we get
1. Connected components of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ coincide with ${\sim_{{\mathrm{ev}}}}$-classes of ${{\smash{\mathrm{rps}}}}$.
2. The maximum diameter of a connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}}_n)$ is $n-1$, for $n=1,2$, and lies between $n-1$ and $2n-4$, for $n\geq 3$.
Other observations from computer experimental results lead us to conclude that the number of vertices in a given connected component is equal to the number of vertices in the connected component that has one more symbol $1$. This makes sense since the elements of the new connected component will be the elements of the former with an additional symbol 1 in the bottom of the first column.
Also, it seems that in a standard component, the addition of a new symbol $1$ leads to a connected component whose diameter can possibly decrease by 2 when compared with the original. In fact, we were able to establish the following result:
Let $w$ be an element of ${{\smash{\mathrm{rps}}}}$, with $n\geq 4$ symbols, such that the minimum symbol of $w$ has at least two occurences, and the second smallest symbol only occurs once. Then the diameter of the connected component of ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ containing $w$ is at most $2n-6$.
Without lost of generality, suppose that ${{\smash{\mathrm{cont}}}}(w)=\{1,\dots ,n\}$, with $n\geq 4$. The proof strategy is similar to the proof of Lemma \[prop:diameter\_upper\_bound\]. We aim to construct a path in ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}})$ from $w$ to $$w'={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(1^{{\left|w\right|}_1}
(n-1)^{{\left|w\right|}_{n-1}}(n-2)^{{\left|w\right|}_{n-2}}\cdots 3^{{\left|w\right|}_3}2
n^{{\left|w\right|}_n}\right)$$ by applying at most $n-3$ cyclic shifts relations.
For an element $w$ of ${{\smash{\mathrm{rps}}}}$, under the given assumptions, we will distinguish particular readings of its tableau representation. For simplicity, we call these readings *delayed column readings*. Note that the symbol $1$ occurs more than once, and that all symbols $1$ appear on the bottom of the first column of such tableaux. If we proceed as in the column reading, but we read the symbol on the bottom of the first column (necessarily a symbol $1$) latter on, we obtain a delayed column reading. Following Algorithm \[alg:PSinsertion\], it is clear that all these words corresponding to delayed column readings also insert to the same element. For example, the element $S$ of has column reading $411\, 5432$ and has delayed column readings, $4151432$, $4154132$, $4154312$, and $4154321$.
If the tableau representation of $w$ has only one column, then it has the form $${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(
n^{{\left|w\right|}_n}(n-1)^{{\left|w\right|}_{n-1}}(n-2)^{{\left|w\right|}_{n-2}}\cdots
3^{{\left|w\right|}_3}2
1^{{\left|w\right|}_1}\right)$$ which is cyclic shift related to $${{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(
(n-1)^{{\left|w\right|}_{n-1}}(n-2)^{{\left|w\right|}_{n-2}}\cdots
3^{{\left|w\right|}_3}2
1^{{\left|w\right|}_1}n^{{\left|w\right|}_n}\right)$$ which in turn has delayed column reading $$(n-1)^{{\left|w\right|}_{n-1}}(n-2)^{{\left|w\right|}_{n-2}}\cdots
3^{{\left|w\right|}_3}2
1^{{\left|w\right|}_1-1}n^{{\left|w\right|}_n}1.$$ By applying a cyclic shift we get the intended form since $$w'= {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(
1(n-1)^{{\left|w\right|}_{n-1}}(n-2)^{{\left|w\right|}_{n-2}}\cdots
3^{{\left|w\right|}_3}2
1^{{\left|w\right|}_1-1}n^{{\left|w\right|}_n}\right).$$
Otherwise, suppose first that the bottom symbol of the second column is $2$. Note that the symbol $3$ can appear in the first three columns of $w$, and if it appears in the third column, then its bottom symbol is a $3$. Consider the delayed column reading of $w$, $u13v$, where $u$ is the prefix up to before the first occurence of a symbol $3$ in the rightmost column where a symbol $3$ appears (necessarily on the first three columns). So, either $u$ or $v$ has the unique symbol $2$, and if $u$ or $v$ has the symbol $2$ then all symbols $3$ appear to its left. Let $w_3={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(13v) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(u)$, and so $w{\sim_{{\mathrm{p}}}}w_3$, since $w={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(u) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(13v)$. The first column of $w_3$ has column reading $1^{{\left|w\right|}_1}$ and the second column $3^{{\left|w\right|}_3}2$.
Now suppose the bottom symbol of the second column is $k>2$. Consider the delayed column reading of $w$, $u1kv$, where $u$ is the prefix up to before the first occurence of a symbol $k$ in the second column. Note that all symbols in $v$ are greater or equal to $k$, and symbols in $u$ that are strictly less than $k$ appear in decreasing order (from left to right). Let $w_k={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(1kv) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}(u)$, and so $w{\sim_{{\mathrm{p}}}}w_k$. The first column of $w_k$ has column reading $1^{{\left|w\right|}_1}$ and the second column $k^{{\left|w\right|}_{k}}\cdots
3^{{\left|w\right|}_3}2 $.
We will construct a path in ${{\smash{\mathrm{K}}}}({{\smash{\mathrm{rps}}}}_n)$ from $w_k$ to $w'$ of length at most $n-4$, by considering a sequence $w_k,\ldots,w_{n-1}$ of elements of ${{\smash{\mathrm{rps}}}}_n$, with $k\geq 3$, such that $w'=w_{n-1}$, and $w_i{\sim_{{\mathrm{p}}}}w_{i+1}$, for $i=k,\ldots, n-1$. For $i\in\{k, \ldots, n-2\}$, let $w_{i}={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(u\right) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(1(i+1)v\right)$ where $u$ is the prefix of the delayed column reading $u1(i+1)v $ of $w_{i}$ up to just before the first occurrence of a symbol $i+1$ (on the third column) and let $w_{i+1}={{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(1(i+1)v\right) {{\smash{\mathrm{P}}}_{\smash{\mathrm{rps}}}}\left(u\right)$. Note that all symbols in $v$ are greater or equal to $i+1$, and all symbols in $u$ that are strictly less than $i+1$ appear in decreasing order (from left to right). Thus the two first columns of $w_{i+1}$ have column readings $1^{{\left|w\right|}_1}$ and $(i+1)^{{\left|w\right|}_{i+1}} i^{{\left|w\right|}_i}\dots
3^{{\left|w\right|}_3}2$, respectively. The result follows by induction.
Conjugacy in the and monoids {#subsection:conjugacy}
==============================
Restating the results of Section \[subsection:cyclic\_shift\] in terms of the conjugacy relation ${\sim_{{\mathrm{p}}}}$ we have shown that in ${{\smash{\mathrm{rps}}}}_n$ we have ${{\sim_{{\mathrm{p}}}}}={{\sim_{{\mathrm{ev}}}}}$, for $n\in\{1,2\}$; and that ${{\sim_{{\mathrm{p}}}}}\subsetneq {{{\sim}_{{\mathrm{p}}}^*}}={{\sim_{{\mathrm{ev}}}}}$, for $n>2$. Thus, ${{{\sim}_{{\mathrm{p}}}^*}}={{\sim_{{\mathrm{ev}}}}}$ in the (infinite rank) right Patience Sorting monoid. In all cases, we deduce that any of the conjugacy relations ${{{\sim}_{{\mathrm{p}}}^*}}$, ${{\sim_{{\mathrm{o}}}}}$, and ${{\sim_{{\mathrm{l}}}}}$ coincides with ${{\sim_{{\mathrm{ev}}}}}$.
The case proves to be distinct from the case. In ${{\smash{\mathrm{lps}}}}_1$, it is immediate that ${{\sim_{{\mathrm{p}}}}}={{\sim_{{\mathrm{ev}}}}}$, but for $n\geq 2$, we will see that ${{\sim_{{\mathrm{p}}}}}\subsetneq
{{{\sim}_{{\mathrm{p}}}^*}}$ and ${{\sim_{{\mathrm{l}}}}}\subsetneq {{\sim_{{\mathrm{ev}}}}}$, in ${{\smash{\mathrm{lps}}}}_n$, and thus in ${{\smash{\mathrm{lps}}}}$. Whether the inclusion ${{{\sim}_{{\mathrm{p}}}^*}}\subseteq {{\sim_{{\mathrm{l}}}}}$ is strict or, in fact an equality, is left as an open question.
For any $n\geq 2$, in ${{\smash{\mathrm{lps}}}}_n$ we have ${{\sim_{{\mathrm{p}}}}}\subsetneq {{{\sim}_{{\mathrm{p}}}^*}}$.
From Lemma \[prop:diameter\_lower\_bound\], we deduce that ${{\sim_{{\mathrm{p}}}}}
\subsetneq {{{\sim}_{{\mathrm{p}}}^*}}$, for ${{\smash{\mathrm{lps}}}}_n$ with $n\geq 3$.
Regarding the ${{\smash{\mathrm{lps}}}}_2$ case, consider the elements ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21121)$ and ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21112)$ of ${{\smash{\mathrm{lps}}}}_2$. We have that $$\begin{aligned}
{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21121)&={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(211) {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21){\sim_{{\mathrm{p}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21) {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(211)={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21211)\\
&={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(22111)={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(2) {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(2111){\sim_{{\mathrm{p}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(2111) {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(2)\\
&={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21112),
\end{aligned}$$ and so ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21121)\ {{\sim}_{{\mathrm{p}}}^*}\ {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21112)$ in ${{\smash{\mathrm{lps}}}}_2$. It is easy to check that ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21121){\nsim_{{\mathrm{p}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21112)$ in ${{\smash{\mathrm{lps}}}}_2$. Indeed, notice that the unique words $u$ and $v$ of ${\mathcal{A}}^*_2$ such that ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u)={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21121)$ and ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v)={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21112)$ are precisely, $21121$ and $21112$, respectively. Moreover, if ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21121)={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(st)$, for words $s,t\in{\mathcal{A}}_2^*$, then ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(ts)\neq {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21112)$.
Resuming, we have a pair of elements of ${{\smash{\mathrm{lps}}}}_2$ which belong to ${{\sim}_{{\mathrm{p}}}^*}$ but not to ${\sim_{{\mathrm{p}}}}$.
In order to prove that ${{\sim_{{\mathrm{l}}}}}\subsetneq {{\sim_{{\mathrm{ev}}}}}$, in ${{\smash{\mathrm{lps}}}}_n$, we first prove two auxiliary results.
For any $k,n\in \mathbb{N}$ and $u,v\in {{\smash{\mathrm{lps}}}}_k$, if $n\geq k$, then: $$u{\sim_{{\mathrm{l}}}}v \text{ in } {{\smash{\mathrm{lps}}}}_n\ \Leftrightarrow\ u{\sim_{{\mathrm{l}}}}v \text{ in }
{{\smash{\mathrm{lps}}}}_k.$$
Let $u,v\in {{\smash{\mathrm{lps}}}}_k$ and $n\geq k$. Suppose that $u{\sim_{{\mathrm{l}}}}v$ in ${{\smash{\mathrm{lps}}}}_n$. Note that $u$ and $v$ have the same evaluation. There exists $g\in {{\smash{\mathrm{lps}}}}_n$ such that $u g= g v$. If $g$ is the identity then the result holds trivially. Assume that the tableau representation of $g$ has $j$ columns.
Since $u g= g v$, then $u^2g=u u g
= u g v= g v v=g v^2$. Using the same reasoning, it follows that for any $i\geq 1$, $u^i g=g v^i$. Note that if $a$ is the minimum symbol occuring in $u$, then $u^i$ has bottom row beginning (from left to right) with (at least) $i$ symbols $a$.
Suppose $g$ has a symbol greater than $k$. As ${{\smash{\mathrm{cont}}}}(u)\subseteq {\mathcal{A}}_k$, the symbols from ${g}$ that are greater or equal than $k$ have to be inserted in the tableau representation of $u^j$ to the right of the first $j$ columns. Now, in the tableau representation of $gv^i$, the symbols from $g$ are inserted into the first $j$ columns. This is a contradiction, since $u^ig=gv^i$. So all symbols from $g$ are less or equal than $k$, that is, $g\in{{\smash{\mathrm{lps}}}}_k$.
The converse direction of the lemma is obvious from the definition of ${\sim_{{\mathrm{l}}}}$.
Let $C_2=\left\{{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(1),{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21)\right\}$. As proved in [@1706.06884 Proposition 4.1], the submonoid of ${{\smash{\mathrm{lps}}}}_2$ generated by $C_2$, denoted $\langle C_2\rangle$, is free. Observe that the elements of $\langle C_2\rangle$ are precisely the elements of ${{\smash{\mathrm{lps}}}}_2$ whose tableau representation has bottom row filled with symbols $1$.
\[lem:left\_conjugacy\_Bell2\] For any $u,v\in \langle C_2\rangle$ and $n\geq 2$, $$u{\sim_{{\mathrm{l}}}}v \text{ in } {{\smash{\mathrm{lps}}}}_n\ \Leftrightarrow\ u{\sim_{{\mathrm{l}}}}v
\text{ in } \langle C_2\rangle.$$
Let $u,v\in \langle C_2\rangle$, $n\geq 2$ and suppose that $u{\sim_{{\mathrm{l}}}}v$ in ${{\smash{\mathrm{lps}}}}_n$. Suppose that $u\in\langle {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21)\rangle$. Since $u{\sim_{{\mathrm{ev}}}}v$, then also $v\in
\langle{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21)\rangle$, and thus $u=v$. Therefore the result holds.
Suppose now that $u\notin \langle {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21)\rangle$. Then at least one of the columns of the tableau representation of $u$ has height one and is filled with the symbol $1$. Note that the tableau representation of $v$ has the same number of columns of heigth two, and the same number of columns of heigth one (and each such box is filled with the symbol $1$).
Let $g\in {{\smash{\mathrm{lps}}}}_n$ be such that $u g= g v$. By the previous lemma we can assume $g\in{{\smash{\mathrm{lps}}}}_2$. If $g$ is the identity then the result holds trivially. Suppose that the tableau representation of $g$ has at least one column with height one filled with the symbol $2$. Attending to Algorithm \[alg:PSinsertion\] and since the bottom row of $u$ is filled with the symbol $1$, $ug$ is represented by a tableau that is composed by the columns of $u$ followed by the columns of $g$.
Now, the tableau representation of $gv$ has at least one less column. Indeed, consider the column reading of the tableau representation of $v$, which is a word from $\{1,21\}^*$, where at least one single symbol $1$ is used, that is, it does not belong to $\{21\}^*$. Applying Algorithm \[alg:PSinsertion\] we will first insert symbols from $g$, and get the tableau representation of $g$, followed by the insertion of the column reading from $v$. Now, each time a word $21$ is inserted we obtain a new column, but the first time a single symbol $1$ is inserted it will take place in the leftmost column of height one filled with the symbol $2$, becoming a column of heigth two and column reading $21$. Thus, the tableau representation of $gv$ cannot have the same number of columns as the tableau representation of $ug$. This is a contradiction. Therefore, the tableau representation of $g$ has bottom row filled with the symbol $1$, and hence $g\in\langle C_2\rangle$.
Since the converse direction is immediate, the result follows.
\[prop511\] For the monoid of rank $n$, with $n\geq 2$, we have $${\sim_{{\mathrm{l}}}}\ \subsetneq\
{\sim_{{\mathrm{ev}}}}.$$
In the free monoid of rank $2$ the relation ${{\sim}_{{\mathrm{p}}}^*}$ is equal to ${\sim_{{\mathrm{l}}}}$ [@lentin1967combinatorial Theorem 3], and it is properly contained in ${\sim_{{\mathrm{ev}}}}$ (For example, in ${\mathcal{A}}_2^*$, there are words with the same evaluation $2121$ and $2112$, for which $2121{\nsim_{{\mathrm{l}}}}2112$).
Consider the embedding $\eta:{\mathcal{A}}_2^*\to{{\smash{\mathrm{lps}}}}_n$ given by $1\mapsto
{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(1)$ and $2\mapsto {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(21)$. This map yields an isomorphism between ${\mathcal{A}}_2^*$ and the free submonoid of ${{\smash{\mathrm{lps}}}}_n$, $\langle C_2\rangle$. Using the example of the first paragraph and the isomorphism, we conclude that the elements ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(211211)$ and ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(211121)$ of $\langle C_2\rangle$ that have the same evaluation, satisfy ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(211211) {\nsim_{{\mathrm{l}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(211121)$ in $\langle
C_2\rangle$. By Lemma \[lem:left\_conjugacy\_Bell2\] we get ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(211211)
{\nsim_{{\mathrm{l}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(211121)$ in ${{\smash{\mathrm{lps}}}}_n$. The result follows.
Regarding the relation between ${{\sim}_{{\mathrm{p}}}^*}$ and ${\sim_{{\mathrm{l}}}}$ in the monoids of rank greater or equal than $3$ we leave the following:
In any multihomogeneous monoid the inclusion ${{\sim}_{{\mathrm{p}}}^*}\ \subseteq\ {\sim_{{\mathrm{l}}}}$ holds. For the monoid of rank $n$, ${{\smash{\mathrm{lps}}}}_n$, with $n\geq 3$, is the inclusion strict, or does the equality hold?
Considering this problem we were able to prove the following result:
Let $u,v$ be elements of ${{\smash{\mathrm{lps}}}}_n$ with exactly two symbols (with possible multiple occurrences) and $n\geq 2$. In ${{\smash{\mathrm{lps}}}}_n$, the following holds $$u\ {{\sim}_{{\mathrm{p}}}^*}\ v\ \Leftrightarrow\ u\ {\sim_{{\mathrm{l}}}}\ v.$$
Without lost of generality, assume that $u,v\in {{\smash{\mathrm{lps}}}}_2$ and that $u{\sim_{{\mathrm{l}}}}v$ in ${{\smash{\mathrm{lps}}}}_n$. Hence $u {\sim_{{\mathrm{ev}}}}v$ and thus for $a\in{\mathcal{A}}_2$, the number of symbols $a$ in $u$ and $v$ is the same.
As $u,v\in {{\smash{\mathrm{lps}}}}_2$, $u={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u'u'')$ and $v={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v'v'')$ where ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u'),{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v')
\in\langle C_2\rangle$, and ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''),{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v'')\in \langle{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(2)\rangle$. Note that $u{\sim_{{\mathrm{p}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u')$ and $v{\sim_{{\mathrm{p}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v')$ in ${{\smash{\mathrm{lps}}}}_n$.
We consider two cases. If ${\left|u'u''\right|}_2\geq {\left|u'u''\right|}_1$, then ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u')={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}\left((21)^i2^j\right)$ and ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v')
={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}\left((21)^k2^l\right)$ for some $i,j,k,l\in\mathbb{N}_0$. As ${\left|u''u'\right|}_a={\left|v''v'\right|}_a$ for all $a\in{\mathcal{A}}_2$, we deduce that $i=k$ and $i+j=k+l$, and thus it follows that $j=l$. So, we conclude that ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u')={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v')$. Therefore $u{\sim_{{\mathrm{p}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u')={{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v'){\sim_{{\mathrm{p}}}}v$ and thus $u\ {{\sim}_{{\mathrm{p}}}^*}\ v$ in ${{\smash{\mathrm{lps}}}}_n$.
Now suppose that ${|u'u''|}_1>{|u'u''|}_2$. In this case ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u'),
{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v')\in \langle C_2\rangle$. As in ${{\smash{\mathrm{lps}}}}_n$ ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u') {\sim_{{\mathrm{p}}}}u$, $u{\sim_{{\mathrm{l}}}}v$, ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v'){\sim_{{\mathrm{p}}}}v$ and ${{\sim_{{\mathrm{p}}}}}\subseteq
{{\sim_{{\mathrm{l}}}}}$, it follows that ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u'){\sim_{{\mathrm{l}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v')$ in ${{\smash{\mathrm{lps}}}}_n$, by the transitivity of ${\sim_{{\mathrm{l}}}}$. Hence, by Lemma \[lem:left\_conjugacy\_Bell2\], ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u'){\sim_{{\mathrm{l}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v')$ in the free monoid $\langle C_2\rangle$. In a free monoid we have ${{{\sim}_{{\mathrm{p}}}^*}} ={{\sim_{{\mathrm{l}}}}}$ [@lentin1967combinatorial Theorem 3]. Therefore ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u')\ {{\sim}_{{\mathrm{p}}}^*}\ {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v')$ in $\langle C_2\rangle$. So, ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u')\ {{\sim}_{{\mathrm{p}}}^*}\ {{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v')$ in ${{\smash{\mathrm{lps}}}}_n$. Combining this with fact that $u{\sim_{{\mathrm{p}}}}{{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(u''u')$ and ${{\smash{\mathrm{P}}}_{\smash{\mathrm{lps}}}}(v''v'){\sim_{{\mathrm{p}}}}v$ in ${{\smash{\mathrm{lps}}}}_n$, it follows that $u\ {{\sim}_{{\mathrm{p}}}^*}\ v$ in ${{\smash{\mathrm{lps}}}}_n$.
In both cases $u\ {{\sim}_{{\mathrm{p}}}^*}\ v$ in ${{\smash{\mathrm{lps}}}}_n$ and the result follows.
| {
"pile_set_name": "ArXiv"
} |
---
address:
- 'Pontifícia Universidade Católica do Rio de Janeiro (PUC–Rio)'
- 'Institut de Mathématiques de Bordeaux, Université Bordeaux 1'
author:
- Jairo Bochi
- Nicolas Gourmelon
title: Note on the dimension of certain algebraic sets of matrices
---
Preamble
========
In this short note we prove a lemma about the dimension of certain algebraic sets of matrices. This result is needed in our paper [@BG_control]. The result presented here has also applications in other situations and so it should appear as part of a larger work [@BG_transitive].
Statement of the result {#s.statement}
=======================
If $A \in {\mathrm{Mat}}_{n \times m}({\mathbb{C}})$, let $\operatorname{col}A \subset {\mathbb{C}}^n$ denote the column space of $A$. A set $X \subset {\mathrm{Mat}}_{n \times m}({\mathbb{C}})$ is called *column-invariant* if $$\left.
\begin{array}{c}
A \in X \\
B \in {\mathrm{Mat}}_{n \times m}({\mathbb{C}})\\
\operatorname{col}A = \operatorname{col}B
\end{array}
\right\} \ \Rightarrow \
B \in X.$$ So a column-invariant set $X$ is characterized by its set of column spaces. We enlarge the latter set by including also subspaces, thus defining: $$\label{e.bracket_notation}
\ldbrack X \rdbrack := \big\{ E \text{ subspace of } {\mathbb{C}}^n ; \; E \subset \operatorname{col}A \text{ for some } A \in X \big\}.$$ Then we have:
\[t.main\] Let $X \subset {\mathrm{Mat}}_{n \times m}({\mathbb{C}})$ be a nonempty algebraically closed, column-invariant set. Suppose $E$ is a vector subspace of ${\mathbb{C}}^n$ that does not belong to $\ldbrack X \rdbrack$. Then $$\operatorname{codim}X \ge m + 1 - \dim E \, .$$
It is obvious that the algebraicity hypothesis is indispensable.
follows without difficulty from intersection theory of the grassmannians (“Schubert calculus”). We tried to make the exposition the least technical as possible, to make it accessible to non-experts (like ourselves).
A particular case {#s.particular}
=================
Define $$\label{e.R_k}
R_k := \big \{ A \in {\mathrm{Mat}}_{n \times m}({\mathbb{C}}) ; \; \operatorname{rank}A \le k \big\} \, .$$ We recall (see [@Harris Prop. 12.2]) that this is an irreducible algebraically closed set of codimension $$\label{e.cod_R_k}
\operatorname{codim}R_k = (m-k)(n-k) \qquad \text{if } 0 \le k \le \min(m,n).$$
If $E = {\mathbb{C}}^n$ then the hypothesis ${\mathbb{C}}^n \not\in \ldbrack X \rdbrack$ means that $X \subset R_{n-1}$. We can assume that $n-1 \le m$, otherwise the conclusion of the is vacuous. Thus $\operatorname{codim}X \ge \operatorname{codim}R_{n-1} = m + 1 - n$, as we wanted to show.
It does not seem likely that the general \[t.main\] can be reduced to .
Reduction to a property of grassmannians {#s.reduction}
========================================
We will show that to prove \[t.main\] it is sufficient to prove a dimension estimate (\[t.schubert\] below) for certain subvarieties of a grassmaniann.
Grassmannians
-------------
Given integers $n > k \ge 1$, the *grassmanniann* $G_k({\mathbb{C}}^n)$ is the set of the vector subspaces of ${\mathbb{C}}^{n}$ of dimension $k$.
The grassmannian can be interpreted a subvariety of a higher dimensional complex projective space as follows. The *Plücker embedding* is the map $G_k({\mathbb{C}}^n) \to P(\bigwedge^k C^n)$ defined as follows: for each $V \in G_k({\mathbb{C}}^n)$, take a basis $\{v_1, \dots, v_k\}$ and map $V$ to $[v_1 \wedge \cdots v_k]$. This is clearly an one-to-one map. It can be shown (see e.g.[@Harris p. 61ff]) that the image is an algebraically closed subset of $P(\bigwedge^k C^n)$. Its dimension is $$\label{e.dim_G}
\dim G_k({\mathbb{C}}^n) = k(n-k).$$
If $E \subset {\mathbb{C}}^n$ is a vector space with $\dim E = e \le k$ then we consider the following subset of $G_k({\mathbb{C}}^n)$: $$\label{e.special schubert}
S_k(E) := \big\{V \in G_k({\mathbb{C}}^n) ; \; V \supset E \big\}.$$ (This is a Schubert variety of a special type, as we will see later.) Since any $V \in S_k(E)$ can be written as $E \oplus W$ for some $V \subset W^\perp$, we see that $S_k(E)$ is homeomorphic to $G_{k-e}({\mathbb{C}}^{n-e})$.
We will show that an algebraic set that avoids $S_k(E)$ cannot be too large:
\[t.schubert\] Fix integers $1 \le e \le k < n$. Suppose that $Y$ is an algebraically closed subset of $G_k({\mathbb{C}}^n)$ that is disjoint from $S_k(E)$, for some $e$-dimensional subspace $E \subset {\mathbb{C}}^n$. Then $\operatorname{codim}Y \ge k + 1 - e$.
Proof of \[t.main\] assuming \[t.schubert\] {#ss.reduction}
-------------------------------------------
Assuming \[t.schubert\] for the while, let us see how it yields \[t.main\].
Recalling notation , define the quasiprojective variety $$\hat{R}_k := R_k {\smallsetminus}R_{k-1} \, .$$ We define a map $\pi_k \colon \hat{R}_k \to G_k({\mathbb{C}}^n)$ by $A \mapsto \operatorname{col}A$.
\[l.projection\] If $X$ is an algebraically closed column-invariant subset of $\hat{R}_k$ then $Y = \pi_k(X)$ is algebraically closed subset of $G_k({\mathbb{C}}^n)$, and the codimension of $Y$ inside $G_k({\mathbb{C}}^n)$ is the same as the codimension of $X$ inside $\hat{R}_k$.
First, let us see that $\pi_k \colon \hat{R}_k \to G_k({\mathbb{C}}^n)$ is a regular map. We identify $G_k({\mathbb{C}}^n)$ with the image of the Plücker embedding. In a Zariski neighborhood of each matrix $A \in \hat{R}_k$, the map $\pi_k$ can be defined as $A \mapsto [a_{j_1} \wedge \dots \wedge a_{j_k}]$ for some $j_1 < \dots < j_k$, where $a_j$ is the $j^\text{th}$ column of $A$. This shows regularity.
Next, let us see that $Y = \pi_k (X)$ is closed with respect to the classical (not Zariski) topology. Consider the subset $K$ of $X$ formed by the matrices $A \in \hat{R}_k$ whose first $k$ columns form an orthonormal set, and whose $m-k$ remaining columns are zero. Then $K$ is compact (in the classical sense), and thus so is $\pi_k(K)$. But column-invariance of $X$ implies that $\pi_k(K) = Y$, so $Y$ is closed (in the classical sense).
It follows (see e.g. [@Harris p.39]) from regularity of $\pi_k$ is regular that the set $Y$ is constructible, i.e., it can be written as $$Y = \bigcup_{i=1}^{p} Z_i {\smallsetminus}W_i \, ,$$ where $Z_i \varsupsetneq W_i$ are algebraically closed subsets of $G_k({\mathbb{C}}^n)$. We can assume that each $Z_i$ is irreducible. It follows from [@Mumford Thrm. 2.33] that $\overline{Z_i {\smallsetminus}W_i} = Z_i$, where the bar denotes closure in the classical sense. In particular, $Y = \overline{Y} = \bigcup_{i=1}^{p} Z_i$, showing that $Y$ is algebraically closed.
We are left to show the equality between codimensions. Since the codimension of an algebraically closed set equals the minimum of the codimensions of its components, we can assume that $X$ is irreducible.
By column-invariance of $X$, for each $y\in Y$, the whole fiber $\pi^{-1}(y)$ is contained in $X$. All those fibers have the same dimension $\mu = km$. By [@Harris Thrm. 11.12], $\dim X = \dim Y + km$. By and , we have $\dim \hat{R}_k - \dim G_k = km$, so the claim about codimensions follows.
Let $X \subset {\mathrm{Mat}}_{n \times m}({\mathbb{C}})$ be a nonempty algebraically closed, column-invariant set. Suppose $E$ is a vector subspace of ${\mathbb{C}}^n$ that does not belong to $\ldbrack X \rdbrack$. Let $e = \dim E$. We can assume $e > 0$ (otherwise the result is vacuously true), and $e<n$ (because the $e=n$ case was already considered in \[s.particular\]).
Notice that $X \subset R_{n-1}$. Let $$X_k := X \cap \hat{R}_k \quad \text{and} \quad
Y_k := \pi_k(X_k) , \quad \text{for } 0 \le k \le \min(m,n-1).$$ For every $k$ with $e \le k < n$, the set $Y_k$ is disjoint from the set $S_k(E)$ defined by . In view of \[l.projection\] and \[t.schubert\], we have $$\operatorname{codim}_{\hat{R}_k} X_k = \operatorname{codim}Y_k \ge k + 1 - e \, .$$ So the codimension of $X_k$ as a subset of ${\mathrm{Mat}}_{n\times m}({\mathbb{C}})$ is $$\begin{aligned}
\operatorname{codim}X_k &= \operatorname{codim}\hat{R}_k + \operatorname{codim}_{\hat{R}_k} X_k \\
&\ge (m-k)(n-k) + k + 1 - e =: f(k) \, .\end{aligned}$$ One checks that the function $f(k)$ is decreasing on the interval $0 \le k \le \min(m,n-1)$. Therefore: $$\begin{gathered}
\operatorname{codim}X
= \min_{0 \le k \le \min(m,n-1)} \operatorname{codim}X_k
\ge \min_{0 \le k \le \min(m,n-1)} f(k) \\
= f(\min(m,n-1))
= m + 1 - e,\end{gathered}$$ as claimed. This proves \[t.main\] modulo \[t.schubert\].
The proof of \[t.schubert\] will be given in \[s.end\], after we explain the necessary tools in \[s.schubert,s.intersection\].
Schubert calculus {#s.schubert}
=================
Here we will outline some facts about the intersection of Schubert varieties. The readable expositions [@Blasiak; @Vakil] contain more information.
A (complete) flag in ${\mathbb{C}}^{n}$ is a sequence of subspaces $F_0 \subset F_1 \subset \cdots \subset F_{n}$ with $\dim F_j = j$. We denote $F_\bullet = \{F_i\}$.
Given $V \in G_k ({\mathbb{C}}^n)$, its *rank table* (with respect to the flag $F_\bullet$) is the data $\dim (V \cap F_j)$, $j=0,\dots,n$. The *jumping numbers* are the indexes $j \in \{1,\dots,n\}$ such that $\dim (V \cap F_j) - \dim (V \cap F_{j-1})$ is positive (and thus equal to $1$). Of course, if one knows the jumping numbers, one know the rank table and vice-versa. Let us define a third way to encode this information: Consider a rectangle of height $m$ and width $n-m$, divided in $1 \times 1$ squares. We form a path of square edges: Start in the northeast corner of the rectangle. In the $j^\text{th}$ step ($1 \le j \le n$), if $j$ is a jumping number then we move one unit in the south direction, otherwise we move one unit in the west direction. Since there are exactly $k$ jumping numbers, the path ends at the southwest corner of the rectangle. The *Young diagram* of $V$ with respect to the flag $F_\bullet$ is the set of squares in the rectangle that lie northwest of the path. We denote a Young diagram by $\lambda = (\lambda_1, \lambda_2, \dots, \lambda_k)$, where $\lambda_i$ is the number of squares in the $i^\text{th}$ row (from north to south). Its *area* $\lambda_1+\cdots+\lambda_k$ is denoted by $|\lambda|$.
\[ex.Young\] Here is a possible rank table with $k=5$, $n=12$; the jumping numbers are underlined:
---------------------- --- --- --- --- --- --- --- --- --- --- ---- --- ----
$j = $ 0 1 2 4 5 7 10 12
$\dim (W \cap F_j)=$ 0 0 0 1 1 1 2 2 3 4 4 5 5
---------------------- --- --- --- --- --- --- --- --- --- --- ---- --- ----
The associated path in the rectangle is:
and so the Young diagram is $$\lambda=\tiny{\yng(5,3,2,2,1)} = (5,3,2,2,1).$$
In general, we have:
- $\lambda = (\lambda_1, \dots, \lambda_k)$ is a possible Young diagram if and only if $n-k \ge \lambda_1 \ge \dots \ge \lambda_k \ge 0$.
- If $j_1 < \dots < j_k$ are the jumping numbers then $\lambda_i = n-k-j_i+i$.
The set of $V \in G_k({\mathbb{C}}^n)$ that have a given Young diagram $\lambda$ is called a *Schubert cell*, denoted by $\Omega(\lambda)$ or $\Omega(\lambda,F_\bullet)$. Each Schubert cell is a topological disk of real codimension $2|\lambda|$. The Schubert cells (for a fixed flag) give a CW decomposition of the space $G_k({\mathbb{C}}^n)$. The closure of $\Omega(\lambda)$ (in either classical or Zariski topologies) is the set of $V \in G_k({\mathbb{C}}^n)$ such that $\dim (V \cap F_{j_i}) \ge i$ for each $i=1,\ldots,n$ (where $j_1 < \dots < j_k$ are the jumping numbers associated to $\lambda$). These sets are closed irreducible varieties, called *Schubert varieties*. (See e.g. [@Fulton §9.4].)
\[ex.special schubert\] If $E \subset {\mathbb{C}}^n$ is a subspace with $\dim E = e \le k$ then the set $S_k(E)$ defined by is a Schubert variety $\bar\Omega(\lambda,F_\bullet)$, where $F_\bullet$ is any flag with $F_e = E$ and $$\label{e.special young}
\lambda = \big( \underbrace{n-k,\dots,n-k}_{e \text{ times}}, \underbrace{0,\dots,0}_{k-e \text{ times}} \big) =
\raisebox{-4\unitlength}{
\begin{picture}(12,8)
\thinlines
\put(0,0){\grid(12,8)(1,1)}
\multiput(0,5)(0,1){3}{\multiput(0,0)(1,0){12}{{\drawline(.5,0)(1,.5)\drawline(0,0)(1,1)\drawline(0,.5)(.5,1)}}}
\end{picture}
}$$
Let $A^*(k,n)$ denote the set of formal linear combinations with integer coefficients of Young diagrams in the $k \times (n-k)$ rectangle. This is by definition an abelian group.
There is a second ${\smallsmile}$ called the *cup product* that makes $A^*(k,n)$ a commutative ring, and is characterized by the following propertis:
If $\lambda$ and $\mu$ are Young diagrams with respective areas $r$ and $s$ then their cup product is of the form: $$\lambda {\smallsmile}\mu = \nu_1 + \cdots + \nu_N \, .$$ where $\nu_1$, …, $\nu_N$ are Young diagrams with area $r+s$ (possibly with repetitions, possibly $N=0$). Moreover, there are flags $F_\bullet$, $G_\bullet$, $H^{(i)}_\bullet$ such that the manifolds $\bar\Omega(\lambda,F_\bullet)$ and $\bar\Omega(\mu,G_\bullet)$ are transverse and their intersection is $\bigcup \bar\Omega(\nu_i,H^{(i)}_\bullet)$.
Working in $A^*(2,4)$, let us compute the products of the Young diagrams $\lambda = {\tiny \yng(2)}$ and $\mu={\tiny \yng(1,1)}$. Fix a flag $F_\bullet$. Then $\bar\Omega(\lambda, F_\bullet)$ is the set of $W \in G_2({\mathbb{C}}^4)$ that contain $F_1$, and $\bar\Omega(\mu, F_\bullet)$ is the set of $W \in G_2({\mathbb{C}}^4)$ that are contained in $F_3$. Take another flag $G_\bullet$ which is in general position with respect to $F_\bullet$, that is $F_i \cap G_{4-i} = \{0\}$. Then:
- The set $\bar\Omega(\lambda, F_\bullet) \cap \bar\Omega(\lambda, G_\bullet)$ contains a single element, namely $F_1 \oplus G_1$, and thus equals $\bar\Omega((2,2),H_\bullet) = \{H_2\}$ for an appropriate flag $H_\bullet$. This shows that $\lambda {\smallsmile}\lambda = {\tiny \yng(2,2)}$.
- The space $F_3 \cap G_3$ is $2$-dimensional and thus is the single element of $\bar\Omega(\mu, F_\bullet) \cap \bar\Omega(\mu, G_\bullet)$. So $\mu {\smallsmile}\mu = {\tiny \yng(2,2)}$.
- The set $\bar\Omega(\lambda, F_\bullet) \cap \bar\Omega(\mu, G_\bullet)$ is empty, thus $\lambda {\smallsmile}\mu = 0$.
However, if we work in $A^*(4,8)$ then it can be shown that: $${\tiny \yng(2)} {\smallsmile}{\tiny \yng(2)} = {\tiny \yng(2,2)} + {\tiny \yng(4)} + {\tiny \yng(3,1)}, \quad
{\tiny \yng(1,1)} {\smallsmile}{\tiny \yng(1,1)} = {\tiny \yng(2,2)} + {\tiny \yng(2,1,1)} + {\tiny \yng(1,1,1,1)}, \quad
{\tiny \yng(2)} {\smallsmile}{\tiny \yng(1,1)} = {\tiny \yng(3,1)} + {\tiny \yng(2,1,1)}.$$ If we drop the terms that do not fit in a $2 \times 2$ rectangle, we reobtain the results for $G_2({\mathbb{C}}^4)$.
The general computation of the product $\lambda {\smallsmile}\mu$ is not simple and can be done in various ways – see e.g. [@Vakil; @Fulton].[^1] For our purposes, however, it will be sufficient to know when the product is zero or not. The answer is provided by the following simple [^2]:
\[l.overlap\] Let $\lambda$ and $\mu$ be Young diagrams in the $k \times (n-k)$ rectangle. The following two conditions are equivalent:
1. \[i.nonzero\] $\lambda {\smallsmile}\mu \neq 0$.
2. \[i.nonoverlap\] If one draws inside the $k \times (n-k)$ rectangle the Young diagrams of $\lambda$ and $\mu$, being the later rotated by $180^{\circ}$ and put in the southeast corner, then the two figures do not overlap (see \[f.nooverlap\]). Equivalently, $\lambda_i + \mu_{k+1-i} \le n-k$ for every $i=1, \ldots, n$.
(7,5)
(0,0)[(7,5)(1,1)]{}
(0,0)[[(.75,0)(1,.25)(.5,0)(1,.5)(.25,0)(1,.75)(0,0)(1,1)(0,.25)(.75,1)(0,.5)(.5,1)(0,.75)(.25,1)]{}]{} (0,1)[[(.75,0)(1,.25)(.5,0)(1,.5)(.25,0)(1,.75)(0,0)(1,1)(0,.25)(.75,1)(0,.5)(.5,1)(0,.75)(.25,1)]{}]{} (1,1)[[(.75,0)(1,.25)(.5,0)(1,.5)(.25,0)(1,.75)(0,0)(1,1)(0,.25)(.75,1)(0,.5)(.5,1)(0,.75)(.25,1)]{}]{} (0,2)[[(.75,0)(1,.25)(.5,0)(1,.5)(.25,0)(1,.75)(0,0)(1,1)(0,.25)(.75,1)(0,.5)(.5,1)(0,.75)(.25,1)]{}]{} (1,2)[[(.75,0)(1,.25)(.5,0)(1,.5)(.25,0)(1,.75)(0,0)(1,1)(0,.25)(.75,1)(0,.5)(.5,1)(0,.75)(.25,1)]{}]{} (0,3)[[(.75,0)(1,.25)(.5,0)(1,.5)(.25,0)(1,.75)(0,0)(1,1)(0,.25)(.75,1)(0,.5)(.5,1)(0,.75)(.25,1)]{}]{} (1,3)[[(.75,0)(1,.25)(.5,0)(1,.5)(.25,0)(1,.75)(0,0)(1,1)(0,.25)(.75,1)(0,.5)(.5,1)(0,.75)(.25,1)]{}]{} (2,3)[[(.75,0)(1,.25)(.5,0)(1,.5)(.25,0)(1,.75)(0,0)(1,1)(0,.25)(.75,1)(0,.5)(.5,1)(0,.75)(.25,1)]{}]{} (0,4)[[(.75,0)(1,.25)(.5,0)(1,.5)(.25,0)(1,.75)(0,0)(1,1)(0,.25)(.75,1)(0,.5)(.5,1)(0,.75)(.25,1)]{}]{} (1,4)[[(.75,0)(1,.25)(.5,0)(1,.5)(.25,0)(1,.75)(0,0)(1,1)(0,.25)(.75,1)(0,.5)(.5,1)(0,.75)(.25,1)]{}]{} (2,4)[[(.75,0)(1,.25)(.5,0)(1,.5)(.25,0)(1,.75)(0,0)(1,1)(0,.25)(.75,1)(0,.5)(.5,1)(0,.75)(.25,1)]{}]{} (3,4)[[(.75,0)(1,.25)(.5,0)(1,.5)(.25,0)(1,.75)(0,0)(1,1)(0,.25)(.75,1)(0,.5)(.5,1)(0,.75)(.25,1)]{}]{} (4,4)[[(.75,0)(1,.25)(.5,0)(1,.5)(.25,0)(1,.75)(0,0)(1,1)(0,.25)(.75,1)(0,.5)(.5,1)(0,.75)(.25,1)]{}]{}
(2,0)[[(.25,0)(.25,.25)(.75,.25)(.75,.5)(.25,.5)(.25,.75)(.75,.75)(.75,1)(0,0)[(1,1)(1,.25)]{}]{}]{} (3,0)[[(.25,0)(.25,.25)(.75,.25)(.75,.5)(.25,.5)(.25,.75)(.75,.75)(.75,1)(0,0)[(1,1)(1,.25)]{}]{}]{} (4,0)[[(.25,0)(.25,.25)(.75,.25)(.75,.5)(.25,.5)(.25,.75)(.75,.75)(.75,1)(0,0)[(1,1)(1,.25)]{}]{}]{} (5,0)[[(.25,0)(.25,.25)(.75,.25)(.75,.5)(.25,.5)(.25,.75)(.75,.75)(.75,1)(0,0)[(1,1)(1,.25)]{}]{}]{} (6,0)[[(.25,0)(.25,.25)(.75,.25)(.75,.5)(.25,.5)(.25,.75)(.75,.75)(.75,1)(0,0)[(1,1)(1,.25)]{}]{}]{} (2,1)[[(.25,0)(.25,.25)(.75,.25)(.75,.5)(.25,.5)(.25,.75)(.75,.75)(.75,1)(0,0)[(1,1)(1,.25)]{}]{}]{} (3,1)[[(.25,0)(.25,.25)(.75,.25)(.75,.5)(.25,.5)(.25,.75)(.75,.75)(.75,1)(0,0)[(1,1)(1,.25)]{}]{}]{} (4,1)[[(.25,0)(.25,.25)(.75,.25)(.75,.5)(.25,.5)(.25,.75)(.75,.75)(.75,1)(0,0)[(1,1)(1,.25)]{}]{}]{} (5,1)[[(.25,0)(.25,.25)(.75,.25)(.75,.5)(.25,.5)(.25,.75)(.75,.75)(.75,1)(0,0)[(1,1)(1,.25)]{}]{}]{} (6,1)[[(.25,0)(.25,.25)(.75,.25)(.75,.5)(.25,.5)(.25,.75)(.75,.75)(.75,1)(0,0)[(1,1)(1,.25)]{}]{}]{} (3,2)[[(.25,0)(.25,.25)(.75,.25)(.75,.5)(.25,.5)(.25,.75)(.75,.75)(.75,1)(0,0)[(1,1)(1,.25)]{}]{}]{} (4,2)[[(.25,0)(.25,.25)(.75,.25)(.75,.5)(.25,.5)(.25,.75)(.75,.75)(.75,1)(0,0)[(1,1)(1,.25)]{}]{}]{} (5,2)[[(.25,0)(.25,.25)(.75,.25)(.75,.5)(.25,.5)(.25,.75)(.75,.75)(.75,1)(0,0)[(1,1)(1,.25)]{}]{}]{} (6,2)[[(.25,0)(.25,.25)(.75,.25)(.75,.5)(.25,.5)(.25,.75)(.75,.75)(.75,1)(0,0)[(1,1)(1,.25)]{}]{}]{} (5,3)[[(.25,0)(.25,.25)(.75,.25)(.75,.5)(.25,.5)(.25,.75)(.75,.75)(.75,1)(0,0)[(1,1)(1,.25)]{}]{}]{} (6,3)[[(.25,0)(.25,.25)(.75,.25)(.75,.5)(.25,.5)(.25,.75)(.75,.75)(.75,1)(0,0)[(1,1)(1,.25)]{}]{}]{}
Intersection of subvarieties of the grassmannian {#s.intersection}
================================================
Next we explain how the Schubert calculus sketched above can be used to obtain information about intersection of general subvarieties of the Grassmannian, by means of cohomology and Poincaré duality. Our primary source is [@Fulton Appendix B]; also, [@Hutchings] is a very readable account about the geometric interpretation of the cup product in cohomology.
Any topological space $X$ has singular homology groups $H_i X$ and cohomology groups $H^i X$ (here taken always with integer coefficients). With the cup product $H^i X \times H^j X \to H^{i+j} X$, the cohomology $H^* X = \bigoplus H^i X$ has a ring structure.
If $X$ is a real compact oriented manifold of dimension $d$ then the homology group $H_d X$ is canonically isomorphic to ${\mathbb{Z}}$, with a generator $[X]$ called the *fundamental class* of $X.$ In addition, there is *Poincaré duality isomorphism* $H^i X \to H_{d-i} X$, which is given by $\alpha \mapsto \alpha \smallfrown [X]$ (taking the cap product with the fundamental class). Let us denote by $\omega \mapsto \omega^*$ the inverse isomorphism.
Next suppose $Y$ and $Z$ are compact oriented submanifolds of $X$, of codimensions $i$ and $j$ respectively. Also suppose that $Y$ and $Z$ have transverse intersection $Y \cap Z$, which therefore is either empty or a compact submanifold of codimension $i + j$, which is oriented in a canonical way. The images of the fundamental classes of $Y$, $Z$, and $Y\cap Z$ under the inclusions into $X$ define homology classes that we denote (with a slight abuse of notation) by $[Y] \in H_{d-i} X$, $[Z] \in H_{d-j} X$, $[Y\cap Z] \in H_{d-i-j} X$. Then their Poincaré duals $[Y]^*\in H^i X$, $[Z]^* \in H^j X$, and $[Y \cap Z]^* \in H^{i+j} X$ are related by: $$[Y]^* {\smallsmile}[Z]^* = [Y \cap Z]^* \, .$$ That is, *cup product is Poincaré dual to intersection.*
Now consider the case where $X$ is a projective nonsingular (i.e., smooth) complex variety, and $Y$ and $Z$ are irreducible subvarieties of $X$. Obviously, the fundamental class $[X]$ makes sense, because $X$ is a compact manifold with a canonical orientation induced from the complex structure. A deeper fact (see [@Fulton Appendix B]) is that fundamental classes $[Y]$ and $[Z]$ can also be canonically associated to the (possibly singular) subvarieties $Y$ and $Z$, and the Poincaré duality between cup product and intersection works in this situation. More precisely, suppose that $Y$ and $Z$ are transverse in the algebraic sense: $Y \cap Z$ is a union of subvarieties $W_1$, …, $W_\ell$ whose codimensions are the sum of the codimensions of $Y$ and $Z$, and for each $i=1,\dots,\ell$, the tangent spaces $T_w Y$ and $T_w Z$ are transverse for all $w$ in a Zariski-open subset of $W_i$. Then each $W_i$ has its canonical fundamental class, and the following duality formula holds: $$[Y]^* {\smallsmile}[Z]^* = [W_1]^* + \cdots + [W_\ell]^* \, .$$
In our application of this machinery, $X$ will be the grassmannian $G_k({\mathbb{C}}^n)$. In this case:
- The fundamental classes of the Schubert varieties $[\bar\Omega(\lambda, F_\bullet)]$ do not depend on the flag $F_\bullet$.
- Let $\sigma_\lambda$ denote the Poincaré dual of $[\bar\Omega(\lambda, F_\bullet)]$. Then $H^{2r} G_k({\mathbb{C}}^n)$ is a free abelian group and the elements $\sigma_\lambda$ with $|\lambda| = r$ form a set of generators. (The cohomology groups of odd codimension are zero.)
- The cup product on cohomology agrees with the “cup” product of Young diagrams explained in the previous section.
End of the proof {#s.end}
================
We are now able to give to prove \[t.schubert\].[^3]
Let $1 \le e \le k < n$. Let $E \subset {\mathbb{C}}^n$ be a subspace of dimension $e$, and consider the set $S_k(E)$ defined by . Recall from \[ex.special schubert\] that this is the Schubert variety for the Young diagram $\lambda$ given by .
Now consider a (nonempty) subvariety $Y \subset G_k({\mathbb{C}}^n)$ that is disjoint from $S_k(E)$. We want to give a lower bound for the codimension $c$ of $Y$. We can of course assume that $Y$ is irreducible.
Let $[Y]^*$ be the dual of fundamental class of $Y$. This is a nonzero element of $H^{2c} G_k({\mathbb{C}}^n)$. It can be expressed as $\sum n_i \sigma_{\mu_i}$, where $\mu_i$ are Young diagrams with area $|\mu_i|=c$, and $n_i$ are nonzero integers. In fact we have $n_i>0$, because of the canonical orientations induced by complex structure. Since the intersection between $S_k(E)$ and $Y$ is empty (and in particular transverse), Poincaré duality gives $[S_k(E)]^* {\smallsmile}[Y]^* = 0$. Therefore we have $\sigma_\lambda {\smallsmile}\sigma_{\mu_i} = 0$ for each $i$.
By \[l.overlap\], if we draw the Young diagram of $\mu_i$ rotated by $180^{\circ}$ and put in the southeast corner of the $k \times (n-k)$ rectangle, then it overlaps the Young diagram $\lambda$ pictured in . This is only possible if $c \ge k-e+1$; indeed the Young diagram $\mu$ with least area such that $\lambda {\smallsmile}\mu \neq 0$ is $$\mu = \big( \underbrace{1,\dots,1}_{k-e+1 \text{ times}}, \underbrace{0,\dots,0}_{e-1 \text{ times}} \big),$$ for which the overlapping picture becomes:
(12,8) (0,0)[(12,8)(1,1)]{} (0,5)(0,1)[3]{}[(0,0)(1,0)[12]{}[[(.5,0)(1,.5)(0,0)(1,1)(0,.5)(.5,1)]{}]{}]{} (11,0)(0,1)[6]{}[[(.25,0)(.25,.25)(.75,.25)(.75,.5)(.25,.5)(.25,.75)(.75,.75)(.75,1)(0,0)[(1,1)(1,.25)]{}]{}]{} (11,5)[(1,1)(1,1)]{}
This concludes the proof of \[t.schubert\].
As explained in \[ss.reduction\], \[t.main\] follows.
[BG2]{}
<span style="font-variant:small-caps;">Blasiak, J.</span> Cohomology of the complex Grassmannian. [www-personal.umich.edu/ $\sim$jblasiak/grassmannian.pdf](http://www-personal.umich.edu/~jblasiak/grassmannian.pdf)
<span style="font-variant:small-caps;">Bochi, J.; Gourmelon, N.</span> Universal regular control for generic semilinear systems. [Preprint [arXiv:[1201.1672]{}](http://arxiv.org/abs/1201.1672)]{}
. Transitivity of spaces of matrices. In preparation.
<span style="font-variant:small-caps;">Fulton, W.</span> *Young tableaux. With applications to representation theory and geometry.* Cambridge Univ. Press, 1997.
<span style="font-variant:small-caps;">Harris, J.</span> *Algebraic geometry: a first course.* Springer, 1992.
<span style="font-variant:small-caps;">Hutchings, M.</span> *Cup product and intersection.* Course notes. [math.berkeley.edu/ $\sim$hutching/teach/215b-2011/cup.pdf](http://math.berkeley.edu/~hutching/teach/215b-2011/cup.pdf)
<span style="font-variant:small-caps;">Mumford, D.</span> *Algebraic geometry. I. Complex projective varieties.* Springer, 1976.
<span style="font-variant:small-caps;">Vakil, R.</span> A geometric Littlewood-Richardson rule. *Ann. of Math. *164 (2006), no. 2, 371–421.
[^1]: Here is an online calculator: [young.sp2mi.univ-poitiers.fr/ cgi-bin/ form-prep/ marc/ LiE[\_]{}form.act?action=LRR](http://young.sp2mi.univ-poitiers.fr/cgi-bin/form-prep/marc/LiE_form.act?action=LRR)
[^2]: In [@Vakil] condition \[i.nonoverlap\] of the is expressed as “the white checkers are happy”.
[^3]: Probably the result could also be proved using the Chow ring, but we feel more comfortable with cohomology.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'An overview of the experimental results on high-[$p_T$]{} light hadron production and open charm production is presented. Data on particle production in elementary collisions are compared to next-to-leading order perturbative QCD calculations. Particle production in Au+Au collisions is then compared to this baseline.'
address: 'Lawrence Berkeley National Laboratory, Berkeley, California 94720'
author:
- M van Leeuwen
title: 'Overview of Hard processes at RHIC: high-[$p_T$]{} light hadron and charm production'
---
Introduction
============
The goal of research in high-energy heavy-ion collisions is to study the properties of strongly interacting matter at extreme energy density, including the possible phase transition to a colour-deconfined state: the Quark Gluon Plasma (QGP). The matter produced in these collisions can be probed using hadrons produced in partonic processes with a large momentum transfer (‘hard scatterings’). These processes take place early in the collision and are only sensitive to short distance scales. In the absence of nuclear effects the hard production yields in nucleus-nucleus collisions are therefore expected to scale as if the collision were an independent superposition of nucleon-nucleon collisions. Measurements at SPS have shown that dilepton production in the Drell-Yan process indeed follows this expectation [@Abreu:1997ji].
Among the first measurements at RHIC was the measurement of light hadron production at high transverse momentum [$p_T$]{} in Au+Au collisions, which shows a suppression with respect to the scaled p+p results. Since these first observations, measurements in d+Au collisions have confirmed that the observed suppression is a final state effect. Recently there has also been an increased activity to verify that high-[$p_T$]{} light-hadron production in proton-proton collisions can be understood in terms of perturbative QCD (pQCD) calculations, as expected for hard processes. Some of the relevant results will be reviewed in the next section.
While for light hadron production much of the groundwork has been done and analyses are clearly moving towards more advanced observables like identified hadron spectra and correlation measurements, first results on open charm production are becoming available. Like high-[$p_T$]{} light hadrons, open charm is expected to be dominantly produced in hard processes and can therefore serve as a calibrated probe of the medium. Due to their large mass, however, charm quarks and hadrons are expected to be affected differently by the medium than high-[$p_T$]{} light hadrons [@molnar_sqm04].
In the second part of this paper an overview will be presented of the existing results on open charm production at RHIC. The present results are based on run-2 and run-3 data. First results from the large statistics Au+Au data sample from run-4 are to be expected soon. These will greatly improve the precision and [$p_T$]{}-coverage of the open charm measurements in Au+Au collisions at RHIC.
High-[$p_T$]{} light hadron production
======================================
High-[$p_T$]{} hadron production at is the most readily accessible observable for hard processes at RHIC. At sufficiently high [$p_T$]{}, all hadrons are expected to be produced in jet fragmentation. The non-perturbative dynamics of jet-fragmentation can be characterised by a universal fragmentation function $D(z)$ which is parametrised using data from $e^+e^-$ collisions at different energies [@Kniehl:2000fe; @Kretzer:2000yf]. With these fragmentation functions and the parton densities from deep inelastic scattering experiments, the expected cross sections for high-[$p_T$]{} hadron production can be calculated in perturbative QCD (pQCD).
Neutral pions and charged hadrons in p+p collisions
---------------------------------------------------
shows [$p_T$]{}-spectra of [$\pi^0$]{} measured by PHENIX [@Adler:2003pb] (left panel) and charged hadrons from STAR [@Adams:2003kv] and BRAHMS [@Arsene:2004ux] (right panel) in p+p collisions at $\sqrt{s}=200$ GeV. The data are compared to a next-to-leading order (NLO) pQCD calculation [@Jager:2002xm]. The uncertainty in the theoretical calculation is estimated by varying the renormalisation and factorisation scales $\mu_R$ and $\mu_F$ to half and twice the nominal value of $\mu_R=\mu_F={\ensuremath{p_T}}$. These scale variations change the calculated cross-sections by about 20% for ${\ensuremath{p_T}}>5$ GeV. To illustrate the uncertainty in the fragmentation functions, the calculation was performed with two different sets of fragmentation functions, from Kniehl, Kramer and Potter (KKP) [@Kniehl:2000fe] and from Kretzer [@Kretzer:2000yf]. Both sets were independently determined from similar selections of $e^+e^-$ data using slightly different assumptions about relations between the fragmentation functions for different partons. This turns out to be the dominant source of uncertainty for the [$\pi^0$]{} spectrum: variations of the order of 50% are seen, mainly due to uncertainties in the gluon fragmentation function. Note, however, that these are partly normalisation uncertainties and do not change the shape of the spectra very much. The measurements have an overall normalisation uncertainty of about 10%, which is not indicated in the figures. The systematic offset between the STAR and BRAHMS results in can probably be attributed to this normalisation uncertainty.
Both for neutral pions and charged hadrons, data and theory agree over more than 5 orders of magnitude, which gives confidence that hadron production at high [$p_T$]{} ($>3$ GeV) is indeed governed by hard point-like processes.
Strange hadron production in p+p collisions
-------------------------------------------
The above comparisons can be extended to the strange hadrons $K^0_S$ and $\Lambda$. While the kaon fragmentation function is relatively well-constrained by the data from $e^+e^-$ collisions, data on $\Lambda$ production are scarce [@deFlorian:1997zj].
A first comparison of $K^0_S$ and $\Lambda$ spectra in $\sqrt{s}=200$ GeV p+p collisions to NLO calculations as presented at this conference is shown in [@heinz_sqm04]. The agreement between the measured $K^0_S$ spectrum and the expectation from NLO pQCD is reasonable, although the shape of the calculated spectrum is slightly more concave than the measured one. This difference is mainly apparent at relatively low [$p_T$]{} (1-2 GeV), where soft production processes may still contribute significantly.
For the $\Lambda$ on the other hand, the agreement between the data and the NLO calculations is far from satisfactory. This might be indicative of the breakdown of the massless formalism and the factorization ansatz for particles with mass that is significant compared to ${\ensuremath{p_T}}$ [@Kretzer:2004ie]. Before drawing this conclusion, however, the uncertainties in the $\Lambda$ fragmentation functions should be better quantified.
Suppression in Au+Au collisions
-------------------------------
To compare measured the particle spectra in Au+Au collisions to the expected $N_{coll}$ scaling from p+p, the nuclear modification factor $$R_{AA}= \frac{\left.dN/d{\ensuremath{p_T}}\right|_{Au+Au}}{N_{coll}
\left.dN/d{\ensuremath{p_T}}\right|_{p+p}}$$ is generally used. In these ratios are shown for peripheral and central Au+Au collisions at $\sqrt{s}=200$ GeV, both for [$\pi^0$]{} from PHENIX (left panel) [@Adler:2003qi] and charged hadrons from STAR (right panel) [@Adams:2003kv]. The suppression ratio $R_{AA}$ in peripheral collisions is close to unity, while for central collisions a suppression of up to a factor 5 is observed. This suppression was one of the first indications of a strong final state modification of particle production in Au+Au collisions that is now generally ascribed to energy loss of the fragmenting parton in the hot and dense medium.
Open charm production in d+Au and Au+Au collisions
==================================================
While light hadron production is only expected to be calculable in perturbative QCD at higher [$p_T$]{}, the charm quark mass ($m_c\approx 1.35$ GeV) is large enough to expect pQCD calculations to be valid for all [$p_T$]{}. Final state effects on charm quarks and hadrons are expected to be smaller than for the light hadrons due to the large charm mass [@molnar_sqm04].
Charmed meson spectra in d+Au collisions
----------------------------------------
None of the RHIC experiments currently has a vertex detector with sufficient resolution to reconstruct secondary vertices of charm decays. Even without secondary vertex reconstruction, STAR has been able to statistically reconstruct decays to charged hadrons in d+Au collisions at $\sqrt{s}=200$ GeV using an invariant mass method [@Tai:2004bf]. Different charmed mesons ($D^0$, $D^{\pm}$, and $D^{*\pm}$) can be reconstructed in different [$p_T$]{} ranges, thus providing a charm measurement up to ${\ensuremath{p_T}}{}=11$ GeV, as shown in the left hand panel of . The $D^\pm$ (triangles) and $D^{*\pm}$ (squares) spectra have been scaled to match the $D^0$ spectra (circles). Note that the $D^0$ has been measured at low [$p_T$]{} and at high [$p_T$]{} using different decay modes and thus provides a normalisation for the whole [$p_T$]{} range.
Also shown in (left) are NLO pQCD calculations of [*charm quark spectra*]{} [@Vogt:2001nh]. To illustrate the sensitivity of those calculations to the charm quark mass $m_c$ and the choice of renormalisation scale, curves are drawn for $m_c=1.2$ GeV and $m_c=1.5$ GeV and with two choices for the renormalisation and factorisation scales: $\mu_R=\mu_F={\ensuremath{m_T}}$ and $\mu_R=\mu_F=2{\ensuremath{m_T}}$. Note that the shape of the spectra at low [$p_T$]{} is most sensitive to both the charm quark mass and the choice of scales.
For a detailed comparison of the data to theory, the calculated charm quark spectrum should be convoluted with the charm fragmentation function. This would lead to a softening of the spectrum and a reduction of the yield, both of which may in principle depend on the meson species. Given the limited [$p_T$]{} range of the spectra for the separate species and the relatively large uncertainties in their fragmentation functions, we have chosen to compare the shape of the combined meson spectrum directly to the charm quark spectra. For this purpose, the calculated spectra were scaled up by a factor of 4 to approximately match the data. The shapes of the calculated charm quark spectra and the measured meson spectra are surprisingly similar, leaving little room for softening due to fragmentation.
All in all, it is far from clear that the present data can be matched with a NLO pQCD calculation. Before drawing any conclusions about charm production in elementary collisions at RHIC, though, we should wait until the present data are finalised. Although it is expected that $N_{coll}$ scaling is valid for charm production in d+Au, it would be good to confirm this by similar measurements in p+p. There are also some open questions for theory. For example, a matched next-to-leading logarithm calculation is needed to describe beauty production at the Tevatron [@Cacciari:2003uh]. In addition, a new way of extracting fragmentation functions, by fitting the Mellin moments instead of a direct $z$-space fit is found to lead to an effectively harder fragmentation function [@Cacciari:2002pa]. Similar considerations may also affect calculations of charm production at RHIC.
Total charm cross section
-------------------------
The right-hand panel of shows the measured energy dependence of the total charm quark cross section [@Adams:2004fc; @Kelly:2004qw], compared to NLO pQCD calculations [@Vogt:2001nh]. Estimating the total charm quark cross section from experimental data involves substantial extrapolations to the unmeasured regions of momentum space and corrections to include unmeasured charmed hadron species. These corrections lead to sizeable systematic uncertainties on the data, as can be seen from the figure. The uncertainties on the NLO pQCD calculations due to higher order corrections and the choice of the charm mass are also significant. It is therefore preferable to directly compare data and calculations in the measured regions (see also [@Frixione:2004md]).
Note also that it seems that the charm cross section per nucleon-nucleon collision as measured by STAR in d+Au collisions from a combination of the electron measurements and the invariant mass method is somewhat higher than expected from the trend observed by other experiments and the pQCD calculations. The deviations are within the present uncertainties on the measurements.
Centrality dependence of charm production in Au+Au collisions
-------------------------------------------------------------
A first indication of the centrality dependence of charm production in Au+Au collisions can be taken from electron spectra. After subtraction of the contributions from light hadrons (mainly through photon conversions, but also from Dalitz decays of [$\pi^0$]{}, $\eta$, $\eta'$, $\rho$, $\omega$ and $\phi$) the electrons from heavy flavour decays remain (‘non-photonic’ electrons). In the left hand panel of , the electron spectra from heavy flavour decays as measured by PHENIX in centrality-selected Au+Au collisions [@Adler:2004ta] are shown. The lines show reference spectra obtained from a fit to the measured spectrum in p+p, scaled by the number of collisions. At each centrality, the spectra agree with the expected $N_{coll}$ scaling from p+p, albeit within large errors.
In the right-hand panel of , the yields of non-photonic electrons with $0.8<{\ensuremath{p_T}}<4.0$ GeV per nucleon-nucleon collision are shown as a function of centrality. There is no indication of a suppression as seen for light hadrons (see ). One should keep in mind, however, that electrons with ${\ensuremath{p_T}}>0.8$ GeV have contributions from semi-leptonic charm decays at all [$p_T$]{}. The presented results are therefore not very sensitive to a possible suppression of charm production at moderate or high [$p_T$]{} ($>2$ GeV).
Charm flow
----------
A measurement of the elliptic flow $v_2$ of charmed mesons is an independent way of assessing the sensitivity of charmed mesons to final state interactions. Here again, we have to rely on measurements of decay electrons for the time being. In the elliptic flow of non-photonic electrons is shown [@laue_sqm04]. Both STAR and PHENIX observe non-zero electron flow, which is a strong indication that charmed mesons flow. This is an intriguing possibility, because it would show decisively that charm production is sensitive to the dense hadronic or even partonic environment in the collision. At the moment the statistical and systematic errors are still large, precluding a precise quantitative extraction of flow values. The situation is expected to dramatically improve with the larger Au+Au data samples which were recorded this year.
Summary and outlook
===================
A comparison of neutral pion and charged hadron [$p_T$]{}-spectra measured in p+p collisions at $\sqrt{s}=200$ GeV at RHIC to NLO pQCD calculations shows that high-[$p_T$]{} light hadron production is well described by perturbative QCD, albeit within relatively large uncertainties, mainly from the fragmentation functions. This gives confidence that high-[$p_T$]{} particle production is governed by hard, point-like processes, for which the cross section in Au+Au collisions is expected to scale with the number of nucleon-nucleon collisions. A suppression of light hadrons by approximately a factor of 5 compared to the expected scaling is observed in central Au+Au collisions, due to final state interactions of the fragmenting quarks and/or the produced hadrons.
For strange hadrons, $K_S^0$ and $\Lambda$, the agreement between data and NLO pQCD calculations is not as good, or even unsatisfactory (for $\Lambda$). In those cases, however, the data do not extend to very high [$p_T$]{} and the fragmentation functions are not as well known as for the light hadrons. This needs more investigation before conclusions can be drawn about the applicability of pQCD.
The same is true for open charm production in d+Au collisions, where shape of the measured $D$ meson spectra is similar to the calculated charm quark spectra, leaving little room for softening due to fragmentation.
First results on electron production from PHENIX indicate that there is no or very little suppression of charm production in Au+Au collisions. Measurements of electron flow, on the other hand, indicate significant flow of the charmed mesons, which can only be due to significant final state interactions.
In the near future, a measurement of the [$p_T$]{}-dependence of nuclear modification factors for non-photonic electrons or maybe even open charm can be expected from the large statistics Au+Au data samples collected in run-4 at RHIC. This, together with a more accurate measurement of charm flow, will map out the interactions of charm quarks and hadrons with the medium and, through comparison with the light hadron results, may eventually shed more light on the nature of these interactions.
{#section .unnumbered}
[88]{} Abreu M C [*et al.*]{} (NA50 Collaboration) [*Phys. Lett.*]{} B [**410**]{} 327 Molnar D [*these proceedings*]{}
Kniehl B A, Kramer G and Potter B [*Nucl. Phys.*]{} B [**582**]{} 514 \[arXiv:hep-ph/0010289\] Kretzer S [*Phys. Rev.*]{} D [**62**]{} 054001 \[arXiv:hep-ph/0003177\] Adler S S [*et al.*]{} (PHENIX Collaboration) [*Phys. Rev. Lett.*]{} [**91**]{} 241803 \[arXiv:hep-ex/0304038\] Adams J [*et al.*]{} (STAR Collaboration) [*Phys. Rev. Lett.*]{} [**91**]{} 172302 \[arXiv:nucl-ex/0305015\] Arsene I [*et al.*]{} (BRAHMS Collaboration) [*Preprint*]{} arXiv:nucl-ex/0403005 Jager B, Schafer A, Stratmann M and Vogelsang W [*Phys. Rev.*]{} D [**67**]{} 054005 \[arXiv:hep-ph/0211007\] and private communication de Florian D, Stratmann M and Vogelsang W [*Phys. Rev.*]{} D [**57**]{} 5811 \[arXiv:hep-ph/9711387\] Heinz M [*et al.*]{} (STAR collaboration) [*these proceedings*]{}
Kretzer S [*Preprint*]{} arXiv:hep-ph/0410219 Adler S S[*et al.*]{} (PHENIX Collaboration) [*Phys. Rev. Lett.*]{} [**91**]{} 072301 \[arXiv:nucl-ex/0304022\] Tai A (STAR Collaboration) [*J. Phys.*]{} G [**30**]{} S809 \[arXiv:nucl-ex/0404029\] Vogt R (Hard Probe Collaboration) [*Int. J. Mod. Phys.*]{} E [**12**]{} 211 \[arXiv:hep-ph/0111271\] and private communication Cacciari M, Frixione S, Mangano M L, Nason P and Ridolfi G JHEP [**0407**]{} 033 \[arXiv:hep-ph/0312132\] Cacciari M and Nason P [*Phys. Rev. Lett.*]{} [**89**]{} 122003 \[arXiv:hep-ph/0204025\] Adams J[*et al.*]{} (STAR Collaboration) [*Preprint*]{} arXiv:nucl-ex/0407006 Kelly S (PHENIX Collaboration) [*J. Phys.*]{} G [**30**]{} S1189 \[arXiv:nucl-ex/0403057\] Frixione S [*Preprint*]{} arXiv:hep-ph/0408317 Adler S S [*et al.*]{} (PHENIX Collaboration) [*Preprint*]{} arXiv:nucl-ex/0409028. Laue F [*et al.*]{} (STAR collaboration) [*these proceedings*]{}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: '[It is shown that Bogoliubov quasi-averages select the pure or ergodic states in the ergodic decomposition of the thermal (Gibbs) state. Our examples include quantum spin systems and many-body boson systems. As a consequence, we elucidate the problem of equivalence between Bose-Einstein condensation and the quasi-average spontaneous symmetry breaking (SSB) discussed in [@LSYng], [@LSYng1] for continuous boson systems. The multi-mode extended van den Berg-Lewis-Pulé condensation of type III [@vdBLP], [@BZ] demonstrates that the only physically reliable quantities are those that defined by Bogoliubov quasi-averages. ]{}'
author:
- |
Walter F. Wreszinski\
Instituto de Fisica USP\
Rua do Matão, s.n., Travessa R 187\
05508-090 São Paulo, Brazil\
`wreszins@gmail.com`\
and\
Valentin A. Zagrebnov\
Aix-Marseille Université, CNRS, Centrale Marseille, I2M\
Institut de Mathématiques de Marseille - UMR 7373\
CMI - Technopôle Château-Gombert\
13453 Marseille, France\
`valentin.zagrebnov@univ-amu.fr`\
title: 'On ergodic states, spontaneous symmetry breaking and the Bogoliubov quasi-averages'
---
Introduction and summary {#sec:Intr-Summ}
========================
The concept of Spontaneous Symmetry Breaking (SSB) is a central one in quantum physics, both in statistical mechanics and quantum field theory and particle physics. In this paper we restrict ourselves to continuous SSB since the breaking of discrete symmetries has been extensively studied and it has quite different properties, in particular regarding the Goldstone-Mermin-Wagner theorem for both zero $T=0$ and non-zero $T>0$ temperatures, see e.g. [@SimonSM] and references given there. The definition of SSB is also well-known since the middle sixties and is well expounded in Ruelle’s book [@Ru], Ch.6.5.2., and references given there, as well as [@BR87], Ch.4.3.4, and, from the point of view of local quantum theory in [@Haag], Ch.III.3.2. Roughly speaking, one starts from a state (ground or thermal), assumed to be invariant under a symmetry group $G$, but which has a nontrivial decomposition into extremal states, which may be physically interpreted as pure thermodynamic phases. The latter, however, do not exhibit invariance under $G$, but only under a proper subgroup $H$ of $G$.
There are basically two ways of constructing extremal states: (1) by a choice of boundary conditions (b.c) for Hamiltonians $H_{\Lambda}$ in finite regions; (2) by replacing $H_{\Lambda}
\rightarrow H_{\Lambda} + \lambda B_{\Lambda}$, where $B_{\Lambda}$ is a suitable extensive operator and $\lambda$ a real parameter, taking first $\Lambda \nearrow \mathbf{Z}^{d}$ or $\Lambda \nearrow \mathbf{R}^{d}$, and then $\lambda \to +0$ (or $\lambda \to -0$). Here one assumes that the states considered are locally normal or locally finite, see e.g. [@Sewell1] and references there. Method (2) is known as Bogoliubov’s *quasi-averages* method [@Bog07]-[@Bog70].
Note that the method (1) is not of general applicability to, e.g., continuous many-body systems or quantum field theory. It is thus of particular interest to show that the Bogoliubov quasi-average “trick” may be shown to constitute a method, whose applicability is universal, explaining, at the same time, its physical meaning. This is one of the main purpose of our paper.
An important element of discussion is a general connection between SSB and Off-Diagonal Long-Range Order (ODLRO), that was studying in papers by Fannes, Pulè and Verbeure [@FPV1] (see also [@PVZ]), by Lieb, Seiringer and Yngvason ([@LSYng], [@LSYng1]), and by Sütö [@Suto1]. The central role played by ODLRO in the theory of phase transitions in quantum spin systems was scrutinised by Dyson, Lieb and Simon [@DLS], see also the review by Nachtergaele [@Ntg]. For its importance in the theories of superconductivity and superfluidity, we refer to the books by Sewell [@Sewell] and Verbeure [@Ver], as well as to review [@SeW], Sec.3.
As a consequence of our results, a general question posed by Lieb, Seiringer and Yngvason [@LSYng] concerning the equivalence between Bose-Einstein condensation $\rm{(BEC)}_{qa}$ and Gauge Symmetry Breaking $\rm{(GSB)}_{qa}$ both defined via the one-mode Bogoliubov quasi-average is elucidated for any *type* of generalised BEC *à la* van den Berg-Lewis-Pulè [@vdBLP] and [@BZ].
[Setup: continuous SSB, ODLRO and examples]{} {#sec:Setup}
=============================================
To warm up we start by some indispensable basic notations and definitions, see e.g., [@BR87], [@BR97], [@Sewell1] and [@Wrepp] Let $\mathcal{A}$ be a unital (i.e. $\mathds{1}\in {\cal A}$) quasi-local $C^*$-algebra of observables. Recall that positive linear functionals $\omega$ over ${\cal A}$ are called **states** if they are normalised: $\|\omega\| =1$. Note that these functionals are automatically continuous and bounded: $\|\omega\| = \omega(\mathds{1})$. The state $\omega$ is called faithful if $\omega(A^* A) = 0$ implies $A = 0$. To construct states and dynamics of quantum (boson) systems the $C^*$-setting is too restrictive and one has to use the $W^*$-setting ([@BR87], [@BR97]). One defines an abstract $W^*$-algebra $\mathfrak{M}$ as a unital $C^*$-algebra that possesses (as a Banach space) a predual $\mathfrak{M}_*$, i.e., $\mathfrak{M} = (\mathfrak{M}_*)^*$. Every abstract $W^*$-algebra is $\ast$-isomorphic to a *concrete* $W^*$-algebra $\mathcal{B}(\mathcal{H})$ of bounded operators on a Hilbert space $\mathcal{H}$. Now we can introduce *normal* states on $W^*$-algebra as those that any $\omega$ on the corresponding concrete $W^*$-algebra is defined by a positive trace-class operator $\rho\in \mathcal{C}_{1}(\mathcal{H})$ with trace-norm $\|\rho\|_{1} = 1$ such that $$\omega (A) = {\rm{Tr}}_{{\cal H}} (\rho \ A) \ , \ {\rm{for \ all}} \ A \in \mathcal{B}(\mathcal{H}) \ .$$
For a finite system in the Hilbert space ${\cal H}_{\Lambda}$, the Gibbs (thermal) state is normal $$\label{2.1}
\omega_{\beta,\mu,\Lambda}(A) = {\rm{Tr}}_{{\cal H}_{\Lambda}} (\rho_{\Lambda} A) \ , \
{\rm{for \ all}} \ A \in \mathcal{B}(\mathcal{H}_{\Lambda}) \ .$$ and defined by the trace-class density matrix $$\label{2.2}
\rho_{\Lambda} = \frac{\exp(-\beta(H_{\Lambda}-\mu N_{\Lambda}))}{\Xi_{\Lambda}(\mu,\beta)} \ .$$ Here $\Lambda$ is a finite domain in $\mathbf{Z}^{d}$ for quantum spin systems, or in $\mathbf{R}^{d}$ for continuous many-body systems and $\Xi_{\Lambda}$ is the grand-canonical partition function $$\label{2.3}
\Xi_{\Lambda}(\mu,\beta) := {\rm{Tr}}_{{\cal H}_{\Lambda}} \exp(-\beta(H_{\Lambda}-\mu N_{\Lambda})) \ ,$$ with $\beta = 1/k_{B}T$ the inverse temperature, $\mu$ the chemical potential for continuous quantum system. For a boson continuous quantum system the Hilbert space ${\cal H}_{\Lambda}$ coincides with the symmetric Fock space $\mathfrak{F}_{symm}(L^{2}(\Lambda))$, and ${\cal C}^{2}_{\Lambda} = \otimes_{i=1}^{N} {\cal C}_{i}^{2}$ for quantum spin systems, with $N=V=|\Lambda|$, the number of points in $\Lambda$. The thermodynamic limit in both cases will be denoted by $V \to \infty$. Operator $A$ in (\[2.1\]) is an element of a local algebra ${\mathfrak{M}}_{\Lambda}= {\cal B}({\cal H}_{\Lambda})$ of bounded operators on ${\cal H}_{\Lambda}$. By $H_{\Lambda}$ we denote the Hamiltonian of the system in a finite domain $\Lambda$, and by $N_{\Lambda}$ the corresponding number operator. If $\Omega_{\Lambda} \in {\cal H}_{\Lambda}$ is the ground-state vector of operator $H_{\Lambda} -\mu N_{\Lambda}$, then the ground state ($T=0$) is defined by $$\label{2.4}
\omega_{\infty,\mu,\Lambda}(A) := (\Omega_{\Lambda}, A \Omega_{\Lambda}) \mbox{ for } A \in {\cal A}_{\Lambda} \ .$$
By $\omega_{\beta,\mu}$ and $\omega_{\infty,\mu}$ we denote thermal and ground states for the infinite-volume (thermodynamic) limit of the finite-volume states (\[2.1\]) and (\[2.3\]), in the sense that $$\label{2.5-6}
\omega_{\beta,\mu}(A) = \lim_{V \to \infty} \omega_{\beta,\mu,\Lambda}(A) \ \ {\rm{and}} \ \
\omega_{\infty,\mu}(A) = \lim_{V \to \infty} \omega_{\infty,\mu,\Lambda}(A) \ , \
A \in \bigcup_{\Lambda \subset (\mathbb{R}^d \, {\rm{or}}\, \mathbb{Z}^d)}{\cal A}_{\Lambda} \ .$$ Now we recall that a $C^*$-dynamics on a $C^*$-algebra ${\cal A}$ is a strongly continuous one-parameter group of $\ast$-automorphisms: $\mathbf{R}\ni t \mapsto \tau_{t}$ of $C^*$. Then a $C^*$-*dynamical system* is a corresponding pair $({\cal A}, \tau_{t})$. Note that the strong continuity of $\{\tau_{t}\}_{t\in \mathbb{R}}$ on ${\cal A}$ means that the map $t \mapsto \tau_{t}(A)$ is norm-continuous for any $A\in {\cal A}$. Therefore, $C^*$-*dynamical systems* are completely characterised by the corresponding densely defined and closed in ${\cal A}$ infinitesimal generators.
It is also well-known that the $C^*$-[dynamical systems]{} are too restrictive for boson systems, that forces to use the $W^*$-setting. Let $\mathfrak{M}$ be a von Neumann algebra ($W^*$-algebra) and let $\mathbf{R}\ni t \mapsto \tau_{t}$ be a one-parameter group of weak\*-continuous $\ast$-automorphisms ($W^*$-dynamics) of $\mathfrak{M}$. Then the pair $(\mathfrak{M}, \tau_{t})$ is called a $W^{\ast}$-*dynamical system*. The continuity condition on the group $\{\tau_{t}\}_{t\in \mathbb{R}}$ means that the weak\*-densely defined and closed in $\mathfrak{M}$ infinitesimal generator corresponding to the $W^*$-dynamics can be defined in the weak\*-topology similar to the $C^*$-setting.
We comment that it is this $W^*$-setting, which is appropriate for representations of the Canonical Commutation Relations (CCR) and description of boson systems by the Weyl algebra [@BR97], [@PiMe]. We shall also use it for quantum spin systems, and, therefore, throughout the whole paper.
\[QDS\] Consider a $W^{\ast}$-[dynamical system]{} $(\mathfrak{M}, \tau_{t})$. A state on $\mathfrak{M}$ is called $\tau$-invariant if $\omega \circ \tau_{t} = \omega$ for all $t\in \mathbb{R}$. If in addition this state is normal, we refer to the triplet $(\mathfrak{M}, \tau_{t}, \omega)$ as to a Quantum Dynamical System (QDS) generated by $(\mathfrak{M}, \tau_{t})$.
Recall that GNS representation $\pi_{\omega}$ of the QDS, which is induced by the invariant state $\omega$, is denoted by the triplet $({\cal H}_{\omega}, \pi_{\omega}, \Omega_{\omega})$. Here, $\Omega_{\omega}$ is a cyclic vector for $\pi_{\omega}(\mathfrak{M})$ in the Hilbert space ${\cal H}_{\omega}$. The unicity of the GNS representation implies that there exists a unique one-parameter group $t \mapsto U_{\omega}(t)$ of unitary operators on ${\cal H}_{\omega}$ such that $$\label{GNS}
\pi_{\omega}(\tau_{t}(A)) = U_{\omega}(t)\pi_{\omega}(A)U_{\omega}^{*}(t) \ ,
\ U_{\omega}(t) \Omega_{\omega} = \Omega_{\omega} \ ,$$ for any $t\in \mathbb{R}$ and $A \in \mathfrak{M}$.
Since we assumed $\omega$ to be normal, the group $\{U_{\omega}(t)\}_{t\in \mathbb{R}}$ is strongly continuous and there exists (by the Stone theorem) a unique self-adjoint generator $H_{\omega}$ of this unitary group such that $$\label{GNS-H}
\pi_{\omega}(\tau_{t}(A)) = e^{i t H_{\omega}}\pi_{\omega}(A)e^{- i t H_{\omega}} \ , \
H_{\omega} \Omega_{\omega} = 0 \ .$$
We note that GNS construction applied directly to a $C^*$-[dynamical system]{} with invariant state $\omega$ defines a normal extension of this state to an enveloping von Neumann algebra. Therefore, it maps the $C^*$-[dynamical system]{} into a $W^*$-[dynamical system]{} with a normal invariant state. Hence, instead of QDS one can start with GNS representation of the $C^*$-algebra ${\cal A}$.
[In the context of infinite boson system we suppose also that the time-invariant $\omega$ is such that restriction to ${\cal A}_{\Lambda}$ (or $\mathfrak{M}_{\Lambda}$) is given by $\omega_{\beta,\mu,\Lambda}$ (\[2.1\]),(\[2.2\]), (\[2.3\])]{}
Now let $G$ be a group and $\{\tau_{g}\}_{g \in G}$ be the associated group of $\ast$-automorphisms in ${\cal A}$. Suppose that $\tau_{g}$ leaves $\omega$ invariant: $$\label{2.9}
\omega(\tau_{g}(A)) = \omega(A), \ \forall A \in {\cal A}, \ \forall g \in G \ .$$ Then one can find on the GNS Hilbert space ${\cal H}_{\omega}$ a unique group of unitary operators $\{U_{g}\}_{g \in G}$ such that $$\label{2.10}
\pi_{\omega}(\tau_{g}(A))= U_{g} \pi_{\omega} U_{g}^{\ast} \ \mbox{ with } \ U_{g}\Omega_{\omega}=\Omega_{\omega} \ .$$ It is easy to show (see, e.g., [@Wrepp]) that the natural candidate for $U_{g}$, given by $$\label{2.11}
U_{g} \pi_{\omega}(A) \Omega_{\omega}= \pi_{\omega}(\tau_{g}(A))\Omega_{\omega} \ ,$$ indeed fulfills these requirements. The $G$-invariant states forms a convex and compact in the weak\*-topology set, that we denote by $E_{{\cal A}}^{G}$.
The same properties are evidently shared by the set $E_{{\cal A}}$ of all states on ${\cal A}$. An **extremal invariant** or **ergodic** state is a state $\omega \in E_{{\cal A}}^{G}$, which cannot be written as a proper convex combination of two distinct states $\omega_{1},\omega_{2} \in E_{{\cal A}}^{G}$: $$\label{2.12}
\omega \ne \lambda \omega_{1} + (1-\lambda) \omega_{2} \mbox{ with } 0<\lambda<1 \ \mbox{ unless } \
\omega_{1}=\omega_{2}=\omega \ .$$
[There exists an alternative characterization:]{} we say that a state $\omega_{1}$ **majorizes** another state $\omega_{2}$ if $\omega_{1}-\omega_{2}$ is a positive linear functional on ${\cal A}$, i.e., $(\omega_{1}-\omega_{2})(A^* A) \ge 0 \ \ \forall A \in {\cal A}$. Clearly, if a state is a convex combination of two others, it majorizes both, and a state $\omega$ is said to be **pure** if the only positive linear functionals majorized by $\omega$ are of the form $\lambda \omega$, with $0\le \lambda \le 1$. By [@BR87], Theorem 2.3.15, we are allowed to use the terms pure and extremal interchangeably. When (\[2.12\]) does not hold, it is natural to consider $\omega$ as a **mixture** of two **pure phases** $\omega_{1}$ and $\omega_{2}$, with proportions $\lambda$ and $(1-\lambda)$, respectively.
Thermal states $\omega_{\beta,\mu}$ satisfy the equilibrium (KMS) condition ([@BR97],[@Hug]) and will be called KMS or thermal equilibrium states, or, for short, thermal states. The **commutant** $\pi_{\omega}({\cal A})^{'}$ of $ \pi_{\omega}({\cal A})$ is defined as $\pi_{\omega}({\cal A})^{'} =
\{B \in {\cal B}({\cal H}_{\omega}): \ [A,B]=0 \ \ \forall A \in \pi_{\omega}({\cal A})\}$. the strong closure of $\pi_{\omega}({\cal A})$, called the von neumann algebra generated by $ \pi_{\omega}({\cal A})$, which also equals $ \pi_{\omega}({\cal A})^{''}$ by von Neumann’s theorem [@BR87], is called a **factor** if its **center** $$\label{2.13}
Z_{\omega} = \pi_{\omega}({\cal A})^{'} \cap \pi_{\omega}({\cal A})^{''} \ ,$$ is a multiple of the identity operator $$\label{2.14}
Z_{\omega} = \{ \mathbb{C} \ \mathds{1}\} \ .$$ The corresponding representation is called **factor** or **primary**, and the extension of $\omega$ to $\pi_{\omega}({\cal A})^{''}$ is called a factor or primary state. Consider the central decomposition of a KMS state $\omega_{\beta}$ [@BR87] (we omit the $\mu$ for brevity): $$\label{2.15}
\omega_{\beta}(A) = \int_{E_{\cal A}^{G}} d\mu(\omega_{\beta}^{'}) \omega_{\beta}^{'}(A)\ ,$$ which, for a KMS state is identical to the extremal or ergodic decomposition, see Theorem 4.2.10 of [@BR87]. The states $\omega_{\beta}^{'}$ in (\[2.15\]) are extremal or factor states, and the decomposition is along the center $ Z_{\omega_{\beta}}$ which is of the form (\[2.14\]). In the examples we shall treat, $Z_{\omega_{\beta}}$ coincides with the so-called **algebra at infinity** ([@BR87], Example 4.2.11). Let $\omega$ be a spatially ($\mathbf{Z}^{d}$ - or $\mathbf{R}^{d}$) - translation invariant state (we shall no longer distinguish these two possibilities explicitly): $$\label{2.16}
\omega(\tau_{{x}}(A)) = \omega(A) \ \ \forall A \in {\cal A} , \ \forall {x} \ ,$$ where $\tau_{{x}}$ denotes the group of automorphisms of ${\cal A}$ corresponding to translations. Let us define $$\label{2.17}
\eta(A) := s-\lim_{V \to \infty} \eta_{\Lambda}(A) \ ,$$ where $$\label{2.18}
\eta_{\Lambda}(A) = \frac{1}{V}\int_{\Lambda} d{x} \pi_{\omega}(\tau_{{x}}(A)) \ ,$$ again not distinguishing the lattice from the continuous case, in the former one has a sum instead of the integral in (\[2.18\]). The existence of (\[2.17\]) is well-known, see [@BR97], or Proposition 6.7 in [@MWB]. Then by construction $\eta(A) \in Z_{\omega}$. If $\omega$ is an extremal (=factor=primary), which is also ergodic for space translations, then (\[2.14\]) holds and therefore $$\label{2.19}
\eta(A) = \omega(A) \ \mathds{1} \ .$$ Hence, the states occurring in the extremal or ergodic decomposition of a KMS state correspond to ”freezing” the observables at infinity to their expectation values. In correspondence with (\[2.17\]), we extend (\[2.5-6\]) to space averages by $$\label{eqn3.2.20}
\omega_{\beta,\mu}(\prod_{i=1}^{m} \eta(A_{i}) B) = \\
\lim_{V \to \infty} \omega_{\beta,\mu,\Lambda}(\prod_{i=1}^{m}\eta_{\Lambda}(A_{i}) B)
=\lim_{V \to \infty} \omega_{\beta,\mu}(\prod_{i=1}^{m} \eta_{\Lambda})(A_{i}) B)\ .$$ Here we assumed that $\omega_{\beta,\mu,\Lambda} $ is space translation-invariant, which may be achieved by imposing the periodic b.c. on $\Lambda$.
Let, now, $G$ be a group, $\{\tau_{g}\}$ denote the corresponding group of $\ast$-automorphisms of ${\cal A}$, and assume that $\tau_{g} \circ \tau_{{x}}= \tau_{{x}} \circ \tau_{g}$ for all $g \in G$ and ${x}$, i.e., $G$ commutes with space translations. We assume henceforth that all states are space translation-invariant, (\[2.16\]), and thus all states in decomposition (\[2.15\]) are also invariant under space translations.
\[SSB\] We say that the state $\omega$ undergoes a (*conventional*) Spontaneous Symmetry Breaking (SSB) of the group $G$ if:\
(i) $\omega$ is $G$-invariant, i.e., (\[2.9\])-(\[2.11\]) hold;\
(ii) $\omega$ has a nontrivial decomposition (\[2.15\]) into ergodic states $\omega_{\beta}^{'}$, which means that at least two such distinct states occur in representation (\[2.15\]), and $$\label{2.21}
\omega_{\beta}^{'}(\tau_{g}(A)
\ne \omega_{\beta}^{'}(A) \ ,$$ for some $g \in G$, and for some $A \in {\cal A}$.
As previously remarked, we shall use the W\* description: ${\cal A}$ shall henceforth be identified with the von neumann algebra $\pi_{\omega}({\cal A})^{''}$ corresponding to a given state $\omega$; thus, $\eta(A)$ is a special element of ${\cal A}$.
Note that if there is no nontrivial decomposition, then there exists only one equilibrium state, which is then automatically $G$-invariant.
The physical interpretation of condition (ii) in Definition \[SSB\] is well-known (see [@Sewell1], or [@Ru]). By (\[2.19\]), for an ergodic state $\omega^{'}$ in (\[2.15\]), $$\lim_{V \to \infty} \frac{1}{V} \, \int_{\Lambda}d{x}\ \omega^{'}(\tau_{{x}}(A) B)
= \omega^{'}(A) \, \omega^{'}(B) \ \ \ \forall A,B \in {\cal A} \ .
\label{eqn2.2.22}$$ By (\[2.14\]) and (\[eqn2.2.22\]), the spatial averages do not fluctuate in an ergodic state $\omega^{'}$: $$\lim_{V \to \infty} \omega^{'}\left\{\left(\frac{1}{V} \,\int_{\Lambda}d{x} \ \tau_{{x}}(A)\right)^{2}-
\left(\frac{1}{V} \, \int_{\Lambda} d{x}\ \omega^{'}(\tau_{{x}}(A))\right)^{2}\right\} = 0 \ .
\label{eqn2.2.23}$$ This is a characteristic property of a **pure** thermodynamic phase, in which average values, such as the density, do not fluctuate (in contrast to a mixture).
How does this relate to SSB ? The part (ii) of Definition \[SSB\] implies that $\tau_{g}$ cannot be implemented by a group of unitary operators in ${\cal H}_{\omega}$ in the form (\[2.11\]), in particular suitable generators of the unitary group do not exist. A natural alternative to (\[eqn2.2.22\]) is to replace it (see [@SwiecaJ], [@Wrepp]) by $$\label{2.24}
\lim_{R \to \infty, \delta \to 0} \omega_{\beta}^{'} ([Q_{R,\delta},A]) \ne 0 \mbox{ for some } A \in {\cal A}_{L}
\ .$$ Here ${\cal A}_{L}$ is the dense subalgebra of local observables, and $Q_{R,\delta}$ is a smooth approximation to the charge in space and time, i.e., $$\label{2.25}
Q_{R,\delta} := \int d{x} dt f_{R}({x}) f_{d}(t) j^{0}({x},t) \ ,$$ with $\lim_{|{x}| \to \infty} f_{R}({x})= 1$, $ f_{\delta}$ tends to delta-function as $\delta \to 0$, and $ j^{0}({x},t)$ is the ”charge density”. In statistical mechanics one may ignore time-smoothing, and choose $f_{R}$ as characteristic function of a region $\Lambda$. The limit (\[2.24\]) exists as a consequence of locality [@SwiecaJ]. For quantum statistical mechanics one uses the the property of “causality” $[{\cal A}_{\Lambda}, {\cal A}_{\Lambda^{'}}]= 0$ if $\Lambda \cap \Lambda^{'} = \emptyset$. To illustrate the ideas presented above we recall a standard example of quantum spin systems corresponding to the simplest Heisenberg ferromagnet $$\label{2.26}
H_{\Lambda} = - \sum_{{x},{y} \in \Lambda; \|{x}-{y}\|=1} {\sigma}_{{x}} \cdot
{\sigma}_{{y}} \ ,$$ where $\sigma_{{x}}^{i},i=1,2,3$ are the Pauli matrices at ${x}$, on the Hilbert space ${\cal H}_{\Lambda} = \otimes_{{x} \in \Lambda} {\cal C}_{{x}}^{2}$. Assuming that $H_{\Lambda}$ in (\[2.26\]) is defined with periodic b.c., so that the momentum is also well-defined, the Gibbs state $\omega_{\beta,\Lambda}$ in (\[2.1\]) (with $\mu = 0$) is invariant under the rotation group $G= SO(3)$. Hence, $\omega_{\beta}$ satisfies (\[2.9\]) with $G=SO(3)$, and, moreover, (\[2.16\]) also holds by translation invariance of $\omega_{\beta,\Lambda}$.
The “charge” (\[2.25\]) coincides with magnetisation $$\label{2.27}
{Q}_{\Lambda} = {M}_{\Lambda} = \sum_{{x} \in \Lambda} {\sigma}_{{x}} \ .$$ In an ergodic state the spatial average (\[2.17\]), (\[2.18\]) of the observable ${\sigma}$, $$\label{2.28}
\eta({\sigma}) = s-\lim_{V \to \infty} (\eta_{\Lambda} =\frac{1}{V} \sum_{{x} \in \Lambda} {\sigma}_{{x}}) \ ,$$ is equal by (\[2.19\]) to $$\label{2.29}
\eta({\sigma}) = \lambda \, {n} \ ,$$ where ${n}$ is a fixed unit vector and coefficient $\lambda = \lambda(\beta,\mu)$. Note that a rotation $g=R \in G$ acts on ${n}$, by (\[2.28\]),(\[2.29\]), in the form $$\label{2.30}
\tau_{g}(\eta({\sigma})) = \lambda \, R \, {n} \ .$$ Since (\[2.15\]) is a central decomposition, for the Gibbs state we may write it in the form $$\label{2.31}
\omega_{\beta}(A) = \int d\mu_{{n}} \omega_{\beta,{n}}(A) \ ,$$ where $\mu$ is the normalized measure on the sphere $S_{2}$ and each $\omega_{\beta,{n}}$ is ergodic. Further, $$\omega_{\beta,{n}}(\tau_{g}(A)) = \omega_{\beta,{n}}(\tau_{g}(\eta(A)))
= \omega_{\beta,R {n}}(\eta(A)) \ .
\label{eqn2.2.32}$$
Now we recall the concept of *Off-Diagonal Long Range Orde*r (ODLRO), which is relevant to our discussion of SSB in the Heisenberg ferromagnet: in this definition $\omega_{\beta}$ is assumed to be an equilibrium state of the Heisenberg ferromagnet (2.25): for a discussion of other examples, see remark 2.5.
\[ODLRO\] For a given $\beta$ the state $\omega_{\beta}$ is said to exhibit ODLRO if $$\label{2.33}
\lim_{V \to \infty} \omega_{\beta} (\eta_{\Lambda}({\sigma})^{2}) > 0 \ .$$
Since $\eta_{\Lambda}$ is given by (\[2.18\]), the both $\sigma$ and $\eta_{\Lambda}(\sigma)$ are three-component vectors, and $(\eta_{\Lambda}(\sigma))^{2} = \|\eta_{\Lambda}(\sigma)\|^2$. Hence, the space-averaged magnetization: $\eta$ (\[2.28\]), **fluctuates** in the state $\omega_{\beta}$. The following well-known proposition relates ODLRO and SSB for the Heisenberg ferromagnet:
\[prop:2.1\] If $\omega_{\beta}$ exhibits conventional ODLRO, it undergoes the SSB defined by (\[2.21\]), with $A$ defined by (2.27). Conversely, if (2.20) holds for some $\omega_{\beta,n}$ in the decomposition (2.30), with $A$ given by (2.27), then (2.32) holds.
[If $\omega_{\beta}$ exhibits ODLRO, it follows from (2.21) and (2.28) that $\lambda \ne 0$ in (2.28), and thus the ergodic decomposition (2.30) is nontrivial. hence, SSB holds, with $A$ in (2.20) given by $\eta(A)$, defined by (2.27). The converse statement is a direct consequence of the ergodicity of$\omega_{\beta,n}$, and the fact that (2.20) implies that $\lambda \ne 0$.]{}
\[rem:2.1\] The ergodic states are not invariant under $G=SO(3)$ but rather under the isotropy (stationary) subgroup $H_{{n_{0}}}$ of $G$, and $S_{d-1}$ may be identified as the harmonic space $G/H$.
\[rem:2.2\] The connection between ODLRO and the existence of several equilibrium states for quantum spin systems was first pointed out by Dyson, Lieb and Simon in their seminal paper [@DLS], see also the review by Nachtergaele [@Ntg] and references given there. By [@DLS], both the spin one-half XY model for $\beta \ge \beta_{c}^{1}$, and the Heisenberg antiferromagnet for suitable spin and $\beta \ge \beta_{c}^{2}$, with $\beta_{c}^{1},\beta_{c}^{2}$ explicitly given in [@DLS], display ODLRO in the sense of definition 2.3, but with different $\eta_{\Lambda}$ in (2.27) (for the antiferromagnet the sum over $x \in \Lambda $ being replaced by a sum over $\Lambda \cap A$, where $\mathbf{Z}^{d} = A \cup B$, $A$ and $B$ being disjoint sublattices. We expect that proposition 2.1 is applicable to the above mentioned cases, yielding SSB (of the rotation group ($SO(2)$ in the XY case) according to Definition \[SSB\], but a choice of $\eta_{\Lambda}$ for a general Heisenberg hamiltonian, with arbitrary spin, is not known, as well as what the correct order parameters are, and how the set of pure phases should be parametrized and constructed (We thank B. Nachtergaele for this last remark). We therefore restrict ourselves to the the ferromagnet as our quantum spin example.
\[rem:2.3\] By (\[2.29\]) we have different values for the “charge density” $\eta({\sigma})$ labelled by ${n} \in S_{2}$. By a well-known result (see, e.g., [@MWB], Corollary 6.3), the GNS representations $\pi_{\omega_{{n}}}$ associated to the corresponding states $\omega_{{n}}$ in the (central) decomposition (\[2.31\]) are not unitary equivalent (they are, more precisely, disjoint, see Definition 6.6 in [@MWB]), and the GNS Hilbert space splits into a direct integral of disjoint “sectors” ${\cal H}_{{n}}$ (see e.g. [@BR87]).
We note that in this respect the case of boson systems is more complicated than spin lattice systems. It becomes clear even on the level of the perfect Bose-gas.
To see this, consider the Perfect Bose-gas (PBG) in a three-dimensional anisotropic parallelepiped $\Lambda:= V^{\alpha_1}\times V^{\alpha_2}\times V^{\alpha_3}$, with *periodic boundary condition* (p.b.c.) and $\alpha_1 \geq \alpha_2 \geq \alpha_3$, $\alpha_1 + \alpha_2 + \alpha_3 = 1$, i.e. the volume $|\Lambda| = V$. In the boson Fock space $\mathcal{F}:= \mathcal{F}_{boson}(\mathcal{L}^2 (\Lambda))$ the Hamiltonian of this system for the grand-canonical ensemble with chemical potential $\mu < 0$ is defined by : $$\begin{aligned}
\label{G-C-PBG}
H^{0}_{\Lambda} (\mu) \, = T_{\Lambda} - \mu \, N_{\Lambda} =
\sum_{k \in \Lambda^{*}} (\varepsilon_{k} - \mu)\, b^{*}_{k} b_{k} \ .\end{aligned}$$ Here one-particle kinetic-energy operator spectrum $\{\varepsilon_{k} = k^2\}_{k \in \Lambda^{*}}$, where the dual to $\Lambda$ set is : $$\label{dual-Lambda}
\Lambda ^{\ast }= \{k_{j}= \frac{2\pi }{V^{{\alpha_{j}}}}n_{j} : n_{j}
\in \mathbb{Z} \}_{j=1}^{d=3}
\ \ \ {\rm{then}} \ \ \ \varepsilon _{k}= \sum_{j=1}^{d} {k_{j}^2} \ .$$ We denote by $b_{k}:=b(\phi_k^\Lambda)$ and $b^{*}_{k}= b^{*}(\phi_k^\Lambda)$ the boson annihilation and creation operators in the Fock space $\mathcal{F}$. They are indexed by the ortho-normal basis $\{\phi_k^\Lambda(x) = e^{i k x}/\sqrt{V}\}_{k \in \Lambda^{*}} \subset \mathcal{L}^2 (\Lambda)$ generated by the eigenfunctions of the self-adjoint one-particle kinetic-energy operator $(- \Delta)_{p.b.c.}$ in $\mathcal{L}^2 (\Lambda)$. Formally these operators satisfy the Canonical Commutation Relations (CCR): $[b_{k},b^{*}_{k'}]=\delta_{k,k'}$. Then $N_k = b^{*}_{k} b_{k}$ is occupation-number operator of the one-particle state $\phi_k^\Lambda$ and $N_{\Lambda} = \sum_{k \in \Lambda^{*}} N_k$ denotes the total-number operator in $\Lambda$.
If we denote by $\omega_{\beta,\mu,\Lambda}^{0}(\cdot)$ the grand-canonical Gibbs state of the PBG generated by (\[G-C-PBG\]), then the problem of existence of conventional Bose-Einstein condensation is related to solution of the equation $$\label{BEC-eq}
\rho = \frac{1}{V} \sum_{k \in \Lambda^{*}} \omega_{\beta,\mu,\Lambda}^{0}(N_k) =
\frac{1}{V} \sum_{k\in \Lambda ^{\ast}}\frac{1}{e^{\beta \left(\varepsilon_{k}-\mu \right)}-1} \ ,$$ for a given total particle density $\rho$ in $\Lambda$. Note that by (\[dual-Lambda\]) the thermodynamic limit $\Lambda \uparrow \mathbb{R}^3$ in the right-hand side of (\[BEC-eq\]) $$\label{I}
\mathcal{I}(\beta,\mu) = \lim_{\Lambda} \frac{1}{V} \sum_{k \in \Lambda^{*}} \omega_{\beta,\mu,\Lambda}^{0}(N_k)
= \frac{1}{(2\pi)^3}\int_{\mathbb{R}^3} d^3 k \ \frac{1}{e^{\beta \left(\varepsilon_{k}-\mu \right)}-1} \ ,$$ exists for any $\mu <0$. It reaches its (finite) maximal value $\mathcal{I}(\beta,\mu =0) = \rho_c(\beta)$, which is called the critical particle density for a given temperature.
The existence of finite $\rho_c(\beta)$ triggers (via *saturation mechanism*) a non-zero BEC $\rho_0(\beta) := \rho - \rho_c(\beta)$, when the total particle density $\rho > \rho_c(\beta)$.
Note that for $\alpha_1 < 1/2$, the whole condensate is sitting in the one-particle ground state mode $k=0$: $$\begin{aligned}
&&{\rho_0} (\beta)= {\rho} - \rho _{c}(\beta) = \lim_{\Lambda} \frac{1}{V} \omega_{\beta,\mu,\Lambda}^{0}(N_0)
= \lim_{\Lambda} \frac{1}{V} \ \left\{e^{-\beta \, {\mu_{\Lambda}(\beta,\rho\geq
\rho _{c}(\beta))} }-1\right\}^{-1}\\
&&{\mu_{\Lambda}(\beta,\rho\geq \rho _{c}(\beta))}= {- \, \frac{1}{V}} \ \frac{1}
{\beta(\rho -\rho _{c}(\beta))} + {o}({1}/{V}) \ ,\end{aligned}$$ where $\mu_{\Lambda}(\beta,\rho)$ is a unique solution of equation (\[BEC-eq\]).
This is a well-known *conventional* (or the *type* I [@vdBLP]) condensation. In particular, in this case it make sense the ODLRO for the Bose-field $$\label{b-field}
b(x) = \sum_{k \in \Lambda^{*}} b_{k} \phi_{k}^{\Lambda}(x) \ .$$ Indeed, by Definition \[ODLRO\] one gets for the spacial average of (\[b-field\]) $$\label{PBG-ODLRO}
\lim_{\Lambda} \omega_{\beta,\mu,\Lambda}^{0}(\frac{1}{V}\int_{\Lambda}dx b^*(x)\ \frac{1}{V}\int_{\Lambda}dx b(x))=
\lim_{\Lambda} \omega_{\beta,\mu,\Lambda}^{0}(\frac{b^{*}_{0} b_{0}}{V}) = \rho_{0} (\beta) \ ,$$ i.e. the ODLRO coincides with the condensate density [@Ver].
For $\alpha_1 = 1/2$ (the Casimir box [@ZBru]) one observes the infinitely-many levels macroscopic occupation called the *type* II condensation.
On the other hand, when $\alpha_1 > 1/2$ (van den Berg-Lewis-Pulé boxe [@vdBLP]) one obtains $$\label{BEC=0}
\lim_{\Lambda} \omega_{\beta,\mu,\Lambda}^{0}(\frac{b^{*}_{k} b_{k}}{V}) =
\lim_{\Lambda}\frac{1}{V}\left\{e^{\beta(\varepsilon_{k}-{\mu_{\Lambda}(\beta,\rho)})}-
1 \right\}^{-1} = 0 \ , \ \forall k \in \Lambda^{*} \ ,$$ i.e., there is no macroscopic occupation of any mode for any value of particle density $\rho$. But a generalised BEC (gBEC of type III) does exist in the following sense: $$\label{gBEC}
\rho -\rho_{c}(\beta)= \lim_{\eta \rightarrow +0}\lim_{\Lambda }
\frac{1}{V}\sum_{\left\{ k\in \Lambda^{\ast }, \left\| k\right\|
\leq \eta \right\}}\left\{e^{\beta(\varepsilon_{k}- {\mu_{\Lambda}(\beta,\rho)})}- 1
\right\}^{-1} , \ {\rm{for}} \ \ \rho > \rho_{c}(\beta) \ .$$ Note that (\[PBG-ODLRO\]) and (\[BEC=0\]) imply triviality of the ODLRO, whereas the condensation in the sense (\[gBEC\]) is nontrivial.
We comment that this unusual condensation is not exclusively due to the special geometry $\alpha_1 > 1/2$. In fact the same phenomenon of the gBEC (*type* III) [@BZ] happens due to interaction in the model with Hamiltonian [@ZBru]: $$\label{Int-TypeIII}
H_{\Lambda }= {\sum_{k\in \Lambda^{*}} }\varepsilon_{k}b_{k}^{*}b_{k}+
\frac{a}{2V}{\sum_{k\in\Lambda^{*}}} b_{k}^{*}b_{k}^{*}b_{k}b_{k}\ , \ \text{ } a>0 \ .$$
These examples show that connection between BEC, ODLRO, and SSB is a subtle matter. This motivates and bolsters a relevance of the Bogoliubov *quasi-average method* [@Bog07]-[@Bog70], that we discuss in the next two sections.
[Selection of pure states by the Bogoliubov quasi-averages: spin systems]{} {#sec:QA-spin}
===========================================================================
Considering further the simple example of spin system (\[2.26\]) for the sake of argument, at least two methods of selecting pure states may be suggested: (1) by taking in (\[2.1\]), (\[2.2\]) $H_{\Lambda}$ with special boundary conditions (b.c.), i.e., upon imposing on the boundary $\partial \Lambda$ of $\Lambda$ $$\label{3.1}
|{n})_{{x}} \mbox{ such that } {\sigma}_{{x}} |{n})_{{x}}= |{n})_{{x}}$$ The above choice leads, presumably, to the limiting states $\omega_{\beta,{n}}$ in (\[2.31\]); (2) by replacing in (\[2.1\]), (\[2.2\]) $H_{\Lambda}$ by the *quasi-*Hamiltonian $$\label{3.2}
H_{\Lambda,{B}} := H_{\Lambda} + H_{\Lambda}^{{B}} \ ,$$ with the *symmetry-breaking* vector field ${B} \, n$ directed along the unit vector $n$: $$\label{3.3}
H_{\Lambda}^{{B}} = - {B} \, n \cdot \sum_{{x} \in \Lambda} {\sigma}_{{x}} \ , \ B > 0 \ .$$ We take $B \to +0$ after the thermodynamic limit $V \to \infty$. This method, which is known as the [Bogoliubov *quasi-averages*]{} ([@Bog07]-[@Bog70], [@ZBru] ), is currently employed as a trick, i.e., without explicit connection to ergodic states. The quantity $\sum_{{x} \in \Lambda} {\sigma}_{{x}}$ (the magnetization) in the symmetry-breaking field is known as the **order parameter**. As spelled out in (\[3.3\]), it is appropriate to the Heisenberg ferromagnet (\[2.26\]) and for the XY model, but not for the antiferromagnet, in which case the order parameter should be replaced by the sub-lattice magnetization $\sum_{{x} \in \Lambda \cap A} {\sigma}_{{x}}$, where $\mathbf{Z}^{d}= A \cup B$, $A,B$ denoting two disjoint sublattices.
If we consider first $0 < \beta < \infty$, $G=SO(3)$ and $H_{\Lambda}$ the Hamiltonian (\[2.26\]) (or its antiferromagnetic or XY analog), with free or periodic b.c., then $H_{\Lambda}$ is G-invariant, and thus $\omega_{\beta,\Lambda}$, defined by (\[2.1\]),(\[2.2\]), is also G-invariant. Taking, now, $H_{\Lambda}$ with the b.c. (\[3.1\]), **both** $H_{\Lambda}$ and $\omega_{\beta,\Lambda}$ are **not** G-invariant. Consider, now, $\beta = \infty$, i.e., theground state, with $H_{\Lambda}$ given by (\[2.26\]), defined with free or periodic b.c.. Again, $H_{\Lambda}$ is invariant under $G$, and we may regard a ground state $$\label{3.5}
\omega_{\infty,\Lambda} = (\Omega_{\Lambda}, \cdot \Omega_{\Lambda})) \ ,$$ with $$\label{3.6}
|\Omega_{\Lambda} = \otimes_{{x} \in \Lambda} |{n})_{{x}} \ .$$ Then, clearly, $\omega_{\infty,\Lambda}$ as well as its infinite volume counterpart is **not** G-invariant. Note that (\[3.5\]) leads, however, presumably to the ergodic states $\omega_{\infty,{n}}$ in the decomposition (\[2.31\]), when taking the weak\* limit as $\Lambda \nearrow \mathbf{Z}^{3}$.
If we take, however, the weak\* limit, as $\beta \to \infty$ along a subsequence, of $\omega_{\beta}$, it may be conjectured that the $G$-invariant ground state $$\omega_{\infty} := \int d\mu_{{n}} \omega_{\infty,{n}} \ ,$$ is obtained. The limits $V \to \infty$ and $\beta \to \infty$ are not expected to commute, and we believe, in consonance with the third principle of thermodynamics [@WreA], that it is more adequate, both physically and mathematically, to regard the states $\omega_{\beta}$ for $0 < \beta < \infty$ as fundamental, with ground states defined as their (weak\*) limit as $\beta \to \infty$ (along a subsequence or subnet). In this sense, the assertion found in most textbooks, see also [@LSYng] beginning of Section 2, that SSB occurs when the Hamiltonian is invariant, but not the state, is not correct, or, at least, not precise. Note, however, that, in the textbooks, “state” is understood as the ground state or the vacuum state, but not as the thermal state, for which the equivalence between the invariance of the Hamiltonian and the state is essentially obvious.
If one uses the method of Bogoliubov quasi-averages, such difficulties do not appear, because $\omega_{\beta,{n}}$ is thereby directly connected to $\omega_{\infty,{n}}$ for each ${n}$. Moreover, as we motivated at the end of Section \[sec:Setup\] by the example of gBEC, the quasi-average method is even indispensable for quantum continuous Bose-systems. An example of its use appears in the next Section \[sec:QA-Boson\]. See also the conclusion.
Note that for quantum continuous many-body systems or relativistic quantum field theory imposition of boundary conditions is very questionable, or even not feasible.
The proof of (2) for the ferromagnet follows [@LSYng], but using Bloch coherent states, instead of Glauber coherent states, in the manner of Lieb’s classic work on the classical limit of quantum spin systems [@Lieb1]. It will not be spelled out here, because the next section will be devoted to a similar proof in the case $G=U(1)$ and Boson systems, but we note the result:
\[3.1\] The ergodic states $\omega_{\beta,{n}}$ in the decomposition (\[2.31\]) may be obtained by the Bogoliubov quasi-average method: $$\label{3.7}
\omega_{\beta, {n}} = \lim_{B \to +0} \lim _{V \to \infty} \omega_{\beta,\Lambda,{n}}$$ where $$\label{3.8}
\omega_{\beta,\Lambda,{n}}(A) \equiv \frac{{\rm{Tr}}_{{\cal H}_{\Lambda}}
(\exp(-\beta H_{\Lambda,{B}})A)}{{\rm{Tr}}_{{\cal H}_{\Lambda}} \exp(-\beta H_{\Lambda,{B}})} \ ,$$ with $A \in {\cal B}({\cal H}_{\Lambda})$, and $H_{\Lambda,{B}}$ is defined by (\[3.2\]), (\[3.3\]) for the ferromagnet (\[2.26\]). The limit (\[3.7\]) is taken along a (double) subsequence of the variables $(B,V)$. For $A$ of the form (2.27), the actual double limit in (3.6) exists, and, if ODLRO holds in the form (2.32), SSB in the form of definition 2.2 holds for the states (3.6).
[The identification of $\omega_{\beta,n}$ in (3.6) with those occurring in the decomposition (2.30) is possible by the unicity of the ergodic decomposition, in view of the fact that the spin algebra is asymptotically abelian for the space translations, see [@BR87], pp 380,381. Since for KMS states the ergodic decomposition coincides with the central decomposition, the extension of the states to elements of the center is also unique.]{}
[Continuous boson systems: quasi-averages, condensates, and pure states]{} {#sec:QA-Boson}
==========================================================================
We now study the states of Boson systems, and, for that matter, assume, together with Verbeure ([@Ver], Ch.4.3.2) that they are analytic in the sense of [@BR97], Ch.5.2.3. We start, with [@LSYng], with the Hamiltonian for Bosons in a cubic box $\Lambda$ of side $L$ and volume $V=L^{3}$, $$\label{4.1}
H_{\Lambda,\mu} = H_{0,\Lambda,\mu} + V_{\Lambda} \ ,$$ where $$\label{4.2}
V_{\Lambda} = \frac{1}{V} \sum_{{k},{p},{q}}\nu({p})b_{{k}+{p}}^{*}b_{{q}-{p}}^{*}b_{{k}}b_{{q}} \ ,$$ with periodic b.c., $\hbar=2m=1$, and $k,p,q \in \Lambda^*$. Here $\Lambda^*$ is dual (with respect to Fourier transformation) set corresponding to $\Lambda$. Here $\nu$ is the Fourier transform of the two-body potential $v({x})$, with bound $$\label{4.3}
|\nu({k})| \le \phi < \infty \ ,$$ and $$\label{4.4}
H_{0,\Lambda,\mu} = \sum_{{k}} {k}^{2} b_{{k}}^{*}b_{{k}} -\mu N_{\Lambda} \ ,$$ $$\label{4.5}
N_{\Lambda} = \sum_{{k}} b_{{k}}^{*}b_{{k}} \ ,$$ with $[b_{{k}},b_{{l}}^{*}]=\delta_{{k},{l}}$ the second quantized annihilation and creation operators. The quasi-Hamiltonian corresponding to (3.2) is taken to be $$\label{4.6}
H_{\Lambda,\mu,\lambda} = H_{\Lambda,\mu} + H_{\Lambda}^{\lambda} \ ,$$ with the symmetry-breaking field analogous to (3.3) given by $$\label{4.7}
H_{\Lambda}^{\lambda} = \sqrt{V}(\bar{\lambda}_{\phi} b_{{0}}+\lambda_{\phi} b_{{0}}^{*}) \ .$$ Above, $$\label{4.8}
\lambda_{\phi} = \lambda \exp(i\phi) \ \mbox{ with } \lambda \geq 0 \, , \,
\mbox{ where } \, {\rm{arg}}(\lambda) = \phi \in [0,2\pi) \ .$$ We take initially $\lambda \geq 0$ and consider first the perfect Bose-gas to define $$\label{4.9.1}
H_{0,\Lambda, \mu, \lambda} = H_{0,\Lambda,\mu} + H_{\Lambda}^{\lambda} \ .$$ We may write $$H_{0,\Lambda,\mu,\lambda}= H_{{0}}+H_{{k}\ne{0}} \ ,$$ where $H_{{0}} = -\mu \ b_{{0}}^{*}b_{{0}}+\sqrt{V}(\bar{\lambda}_{\phi} b_{{0}}+
\lambda_{\phi} b_{{0}}^{*})$. The grand partition function $\Xi_{\Lambda}$ splits into a product over the zero mode and the remaining modes. We introduce the canonical shift transformation $$\label{4.9.2}
\widehat{b}_{{0}} := b_{{0}}+\frac{\lambda_{\phi} \sqrt{V}}{\mu} \ ,$$ without altering the nonzero modes, and assume henceforth $\mu < 0$. We thus obtain for the grand partition function $\Xi_{\Lambda}$, $$\label{4.9.3}
\Xi_{\Lambda}(\beta,\mu,\lambda) = (1-\exp(\beta \mu))^{-1}\exp(-\frac{\beta |\lambda|^{2}V}{\mu})\
\Xi^{\prime}_{\Lambda} \ ,$$ where $$\label{4.9.4}
\Xi^{\prime}_{\Lambda} := \prod_{{k} \ne {0}} (1-\exp(-\beta(\epsilon_{{k}}-\mu)))^{-1} \ ,$$ with $\epsilon_{{k}}={k}^{2}$. Recall that the grand-canonical state for the perfect Bose-gas is $$\label{4.9.5}
\omega^{0}_{\beta,\mu,\Lambda,\lambda}(\cdot):= {\frac{1}{\Xi_{\Lambda}}} \
{\rm{Tr}}[e^{-\beta H_{0,\Lambda,\mu,\lambda}} \ (\cdot )] \ ,$$ see Section \[sec:Setup\]. Then it follows from (\[4.9.3\])-(\[4.9.5\]) that the mean density ${\rho}$ equals to $${\rho}=\omega_{\beta,\mu,\Lambda,\lambda}(\frac{N_{\Lambda}}{V})= \frac{1}{V(\exp(-\beta \mu)-1)}
+ \frac{|\lambda|^{2}}{\mu^{2}} + \\
\frac{1}{V} \sum_{{k} \ne {0}} \frac{1}{\exp(\beta(\epsilon_{{k}}-\mu))-1} \ .
\label{4.9.6}$$
Equation (\[4.9.6\]) is the starting point of our analysis. Let $$\label{4.10}
{\rho_{c}}(\beta) \equiv \int \frac{d{k}}{2\pi^{3}}(\exp(\beta \epsilon_{{k}})-1)^{-1} \ .$$
\[4.1\] Let $0 < \beta <\infty$ be fixed. Then, for each $$\label{4.11}
{\rho_{c}} < {\rho} < \infty \ ,$$ and for each $\lambda >0$, $V <\infty$, there exists a unique solution of (\[4.9.6\]) of the form $$\mu(V,|\lambda|,{\rho}) = -\frac{|\lambda|}{\sqrt{{\rho}-{\rho_{c}}(\beta)}}\\
+ \alpha(|\lambda|,V) \ ,
\label{4.12.1}$$ with $$\label{4.12.2}
\alpha(|\lambda|,V) \ge 0 \ \ \forall \ |\lambda|, V \ ,$$ and such that $$\label{4.13}
\lim_{|\lambda| \to 0} \lim_{V \to \infty} \frac{\alpha(|\lambda|,V)}{|\lambda|} = 0 \ .$$
\[4.2\] [We skip the proof of this lemma, but we note that besides the cube $\Lambda$, it is also true for the case of three-dimensional anisotropic parallelepiped $\Lambda:= V^{\alpha_1}\times V^{\alpha_2}\times V^{\alpha_3}$, with *periodic boundary condition* (p.b.c.) and $\alpha_1 \geq \alpha_2 \geq \alpha_3$, $\alpha_1 + \alpha_2 + \alpha_3 = 1$, i.e. the volume $|\Lambda| = V$.]{}
We have now that $$\lim_{\lambda \to +0} \lim_{V \to \infty}\omega^{0}_{\beta,\mu,\Lambda,\lambda}(\eta_{\Lambda}(b_{{0}}^{*}))=
\lim_{\lambda \to +0} \lim_{V \to \infty} \frac{\partial}{\partial \lambda_{\phi}}
p_{\beta,\mu,\Lambda,\lambda_{\phi}}
\ ,
\label{4.14.1}$$ where $\eta$ is defined as in (\[2.17\]),(\[2.18\]). Above we denote by $$\label{4.14.2}
p_{\beta,\mu,\Lambda,\lambda}=\frac{1}{\beta V}\ln \Xi_{\Lambda}(\beta,\mu,\lambda) \ ,$$ the pressure. By (\[4.9.6\]),(\[4.14.2\]) and the fact that the second term in (\[4.9.6\]) equals $(\lambda_{\phi}\bar{\lambda}_{\phi})/{\mu^{2}}$, we obtain $$\label{4.14.3}
\frac{\partial}{\partial \lambda_{\phi}} p_{\beta,\mu,\Lambda,\lambda_{\phi}}=
-\frac{\bar{\lambda}_{\phi}}{\mu} \ .$$ By (\[4.14.3\]), (\[4.12.1\]) and (\[4.14.1\]), $$\lim_{\lambda \to +0} \lim_{V \to \infty}\omega^{0}_{\beta,\mu,\Lambda,\lambda}(\eta_{\Lambda}(b_{{0}}^{*}))
= \sqrt{\rho_{{0}}} \exp(i\phi) \ ,
\label{4.15.1}$$ where, for the perfect Bose-gas, $$\rho_{{0}} = {\rho}-{\rho_{c}}(\beta) \ .$$ We see therefore that the phase in (\[4.14.1\]) remains in (\[4.15.1\]) even after the limit $\lambda \to +0$. Define the states $$\label{PBG-LimSt-phi}
\omega^{0}_{\beta,\mu,\phi} := \lim_{\lambda \to +0}
\lim_{V \to \infty}\omega^{0}_{\beta,\mu,\Lambda,\lambda_{\phi}} \ ,$$ where the double limit along a subnet exists by weak\* compactness [@Hug], [@BR87].
For this and the forthcoming definitions, we are referring to the full interacting Bose gas (\[4.1\])-(\[4.3\]), with $\omega$ replaced by $\omega^{0}$. The corresponding definitions for the general case of the quantities $\omega_{\beta,\mu,\Lambda,\lambda}$, $\omega_{\beta,\mu,\lambda,\phi}$ and $\omega_{\beta, \mu, \phi}$ are the obvious analogues of (\[4.9.5\]) and (\[PBG-LimSt-phi\]), with $H_{0,\Lambda,\mu,\lambda}$ replaced by $H_{\lambda,\mu,\lambda}$.
We say (cf Section \[sec:Setup\]) that the interacting Bose-gas undergoes the *zero-mode* **Bose-Einstein condensation** (BEC) (and/or ODLRO) if $$\label{4.16}
\lim_{V \to \infty} \omega_{\beta,\mu,\Lambda}(\frac{b_{{0}}^{*}b_{{0}}}{V}) = \rho_{{0}}>0 \ .$$
We define the group of *gauge* transformations $\{\tau_{\lambda}|\lambda \in [0,2\pi)\}$ by the operations $$\tau_{\lambda}(b^{*}(f)) = \exp(i\lambda)b^{*}(f)\\
\tau_{\lambda}(b(f)) = \exp(-i\lambda)b(f) \ ,
\label{4.17}$$ where $b^{*}(f), b(f)$ are the creation and annihilation operators smeared over test-functions $f$ from the Schwartz space. This group is isomorphic to the group $U(1)$.
Note that (\[4.15.1\]), (\[PBG-LimSt-phi\]) show that the states $\omega_{\beta,\mu,\phi}$ are not gauge invariant. Assuming that they are the ergodic states in the ergodic decomposition of $\omega_{\beta,\mu}$, which we shall prove next, in greater generality, for the interacting system, it follows that BEC is equivalent to SSB for the free Bose gas. It is illuminating to see, however, in the free case, a different explicit mechanism for the appearance of the phase, which is connected with (\[4.12.1\]) of Lemma \[4.1\], i.e., that the chemical potential remains proportional to $|\lambda|$ even after the thermodynamic limit (together with (\[4.14.3\])). This property persists for the *interacting* system, see below.
\[4.3\]
Note that these results are independent of the anisotropy, i.e. of whether the condensation for $\lambda =0$ is in single mode ($k=0$) or it is extended as the gBEC-type III, Section \[sec:Setup\]. This means that the Bogoliubov quasi-average method solves the question about equivalence between $\rm{(BEC)}_{qa}$, $\rm{(SSB)}_{qa}$ and $\rm{(ODLRO)}_{qa}$ if they are defined via *one-mode* quasi-average.
To this aim we re-consider the prefect Bose-gas (\[G-C-PBG\]) with symmetry breaking sources (\[4.7\]) in a single mode $q \in \Lambda^{*}$: $$\begin{aligned}
\label{freeQE}
H^{0}_{\Lambda} (\mu; \eta) \, := \, H^{0}_{\Lambda}(\mu) \, + \, \sqrt{V} \ \big( \overline{\eta} \ b_{{q}} +
\eta \ b^{*}_{{q}} \big) \ , \ \mu < 0.\end{aligned}$$ Then for a fixed density ${\rho}$, the the grand-canonical condensate equation (\[BEC-eq\]) for (\[freeQE\]) takes the following form: $$\begin{aligned}
\label{perfect-gas-with-source-density-equation-finite-volume}
&&{\rho} = \rho_{\Lambda}(\beta, \mu, \eta) \, := \, \frac{1}{V} \sum_{k \in \Lambda^{*}_{l}}
\omega_{\beta,\mu,\Lambda,\eta}^{0}(b^{*}_{k}b_{k}) = \\
&&\frac{1}{V} (e^{\beta(\varepsilon_{{q}} - \mu)}-1)^{-1} \, + \, \frac{1}{V}
\sum_{k\in \Lambda^{*}\setminus{q}} \frac{1}{e^{\beta(\varepsilon_{k} - \mu)}-1} \, + \,
\frac{\vert \eta \vert\, ^{2}}
{(\varepsilon_{{q}} - \mu)\, ^{2}} \ . \nonumber\end{aligned}$$
According the quasi-average method, to investigate a possible condensation, one must first take the thermodynamic limit in the right-hand side of (\[perfect-gas-with-source-density-equation-finite-volume\]), and then switch off the symmetry breaking source: $\eta \rightarrow 0$. Recall that the critical density, which defines the threshold of boson saturation is equal to $\rho_c(\beta) = \mathcal{I}(\beta,\mu=0)$ (\[I\]), where $\mathcal{I}(\beta,\mu)=\lim_{\Lambda} \rho_{\Lambda}(\beta, \mu , \eta = 0)$.
Since $\mu < 0$, we have to distinguish two cases:\
(i) Let ${q}\in \Lambda^{*}$ be such that $\lim_{\Lambda} \varepsilon_{{q}} > 0$, we obtain from (\[perfect-gas-with-source-density-equation-finite-volume\]) the condensate equation $$\begin{aligned}
{\rho} \, = \, \lim_{\eta \rightarrow 0}
\lim_{\Lambda} \rho_{\Lambda}(\beta, \mu, \eta) \, = \, \mathcal{I}(\beta, \mu) \ ,\end{aligned}$$ i.e. the quasi-average coincides with the average. Hence, we return to the analysis of the condensate equation (\[perfect-gas-with-source-density-equation-finite-volume\]) for $\eta =0$. This leads to finite-volume solutions $\mu_{\Lambda}(\beta,\rho)$ and consequently to all possible types of condensation as a function of anisotropy $\alpha_1$, see Section \[sec:Setup\] for details.\
(ii) On the other hand, if ${q}\in \Lambda^{*}$ is such that $\lim_{\Lambda} \varepsilon_{{q}} = 0$, then thermodynamic limit in the right-hand side of the condensate equation (\[perfect-gas-with-source-density-equation-finite-volume\]) yields: $$\begin{aligned}
\label{perfect-gas-with-source-density-equation-infinite-volume}
{\rho} = \lim_{\Lambda} \rho_{\Lambda}(\beta, \mu, \eta)
\, = \, \mathcal{I}(\beta, \mu) + \frac{\vert \eta \vert\, ^{2}}{\mu\, ^{2}} \ .\end{aligned}$$
Now, if ${\rho} \leq \rho_{c}(\beta)$, then the limit of solution of (\[perfect-gas-with-source-density-equation-infinite-volume\]): $\lim_{\eta \rightarrow 0}{\mu}(\beta, {\rho}, \eta) = {\mu}_{0} (\beta, {\rho}) <0$, where ${\mu}(\beta,{\rho}, \eta)= \lim_{\Lambda}{\mu}_{\Lambda} (\beta,{\rho}, \eta)<0 $ is thermodynamic limit of the finite-volume solution of condensate equation (\[perfect-gas-with-source-density-equation-finite-volume\]). Therefore, there is no condensation in any mode.
But if ${\rho} > \rho_{c}(\beta)$, then $\lim_{\eta \rightarrow 0}{\mu}(\beta, {\rho},\eta) =0$ and the density of condensate is $$\label{BEC-qa}
\rho_{0}(\beta) = {\rho} - \rho_{c}(\beta) =
\lim_{\eta \rightarrow 0}\frac{\vert \eta \vert\, ^{2}}{\mu(\beta, {\rho},\eta)\, ^{2}} \ .$$ Note that expectation of the particle density in the $q$-mode (see (\[perfect-gas-with-source-density-equation-finite-volume\])) is $$\omega_{\beta,\mu,\Lambda,\eta}^{0}({b^{*}_{q}b_{q}}/{V}) = \frac{1}{V} (e^{\beta(\varepsilon_{{q}} - \mu)}-1)^{-1}
+ \frac{\vert \eta \vert\, ^{2}} {(\varepsilon_{{q}} - \mu)\, ^{2}} \ .$$ Then by (\[BEC-qa\]) the corresponding Bogoliubov quasi-average for ${b^{*}_{q}b_{q}}/{V}$ is equal to $$\begin{aligned}
\label{Bog-qa}
&&{\rho} - \rho_{c}(\beta)=\lim_{\eta \rightarrow 0}\lim_{\Lambda}\omega_{\beta,{\mu}_{\Lambda}
(\beta,{\rho}, \eta),\Lambda,\eta}^{0}({b^{*}_{q}b_{q}}/{V}) = \\
&&\lim_{\eta \rightarrow 0}\lim_{\Lambda}\frac{1}{V} (e^{\beta(\varepsilon_{{q}} - {\mu}_{\Lambda}
(\beta,{\rho}, \eta))}-1)^{-1} +
\frac{\vert \eta \vert\, ^{2}} {(\varepsilon_{{q}} - {\mu}_{\Lambda} (\beta,{\rho}, \eta))\, ^{2}} \ , \nonumber\end{aligned}$$ where ${\mu}_{\Lambda} (\beta,{\rho}, \eta)<0$ is a unique solution of the condensate equation (\[perfect-gas-with-source-density-equation-finite-volume\]) for ${\rho} > \rho_{c}(\beta) $.
Note that by virtue of (\[BEC-qa\]) one has ${\mu}(\beta,{\rho}, \eta \neq 0)<0$. Hence, for any $k \neq q$ such that $\lim_{\Lambda} \varepsilon_{{k}} = 0$ we get $$\label{zero-non-zero-modes}
\lim_{\eta \rightarrow 0}\lim_{\Lambda}\omega_{\beta,{\mu}_{\Lambda}
(\beta,{\rho}, \eta),\Lambda,\eta}^{0}({b^{*}_{k}b_{k}}/{V}) =
\lim_{\eta \rightarrow 0}\lim_{\Lambda} \frac{1}{V}
\frac{1}{e^{\beta(\varepsilon_{{k}}- {\mu}_{\Lambda} (\beta,{\rho}, \eta)))}-1} = 0 \ ,$$ i.e., for any $\alpha_1$ the quasi-average condensation $\rm{(BEC)}_{qa}$ occurs only in one mode (type I), whereas for $\alpha_1 >1/2$ the BEC is of the type III, see Section \[sec:Setup\].
Similarly, diagonalisation (\[4.9.2\]) and (\[BEC-qa\]) allow to apply the quasi-average method to calculate a nonvanishing for ${\rho} > \rho_{c}(\beta)$ gauge-symmetry breaking $\rm{(SSB)}_{qa}$: $$\label{GSB-qa}
\lim_{\eta \rightarrow 0}\lim_{\Lambda}\omega_{\beta,{\mu}_{\Lambda}
(\beta,{\rho}, \eta),\Lambda,\eta}^{0}({b_{q}}/\sqrt{V}) =
\lim_{\eta \rightarrow 0}\frac{\eta}{\mu(\beta,{\rho}, \eta)} =
e^{i \, {\rm{arg}}(\eta)} \, \sqrt{{\rho} - \rho_{c}(\beta)} \ ,$$ along $\{\eta = |\eta| e^{i \, {\rm{arg}}(\eta)} \wedge |\eta|\rightarrow 0\}$. Then by inspection of (\[Bog-qa\]) and (\[GSB-qa\]) we find that $\rm{(SSB)}_{qa}$ and $\rm{(BEC)}_{qa}$ are equivalent: $$\begin{aligned}
\label{Bog=GSB-qa}
&&\lim_{\eta \rightarrow 0}\lim_{\Lambda} \ \omega_{\beta,{\mu}_{\Lambda}
(\beta,{\rho}, \eta),\Lambda,\eta}^{0}({b^{*}_{q}}/\sqrt{V}) \ \omega_{\beta,{\mu}_{\Lambda}
(\beta,{\rho}, \eta),\Lambda,\eta}^{0}({b_{q}}/\sqrt{V}) = \\
&& = \lim_{\eta \rightarrow 0}\lim_{\Lambda} \ \omega_{\beta,{\mu}_{\Lambda}
(\beta,{\rho}, \eta),\Lambda,\eta}^{0}({b^{*}_{q}b_{q}}/{V}) = {\rho} - \rho_{c}(\beta) \ . \nonumber\end{aligned}$$
Note that by (\[PBG-ODLRO\]) the $\rm{(SSB)}_{qa}$ and $\rm{(BEC)}_{qa}$ are in turn equivalent to $\rm{(ODLRO)}_{qa}$, whereas for the conventional BEC on gets $$\lim_{\Lambda} \ \omega_{\beta,{\mu}_{\Lambda}
(\beta,{\rho}, \eta =0),\Lambda,\eta= 0}^{0}({b^{*}_{q}b_{q}}/{V}) =
\lim_{\Lambda} \ \omega_{\beta,{\mu}_{\Lambda}
(\beta,{\rho}, 0),\Lambda, 0}^{0}({b^{*}_{q}}/\sqrt{V}) \ \omega_{\beta,{\mu}_{\Lambda}
(\beta,{\rho}, 0),\Lambda, 0}^{0}({b_{q}}/\sqrt{V})= 0 \ ,$$ for any $\rho$ and $q\in \Lambda^{*}$ as soon as $\alpha_1 > 1/2$.
We now consider the [[interacting case (4.1)-(4.5)]{}]{}. The famous **Bogoliubov approximation** of replacing $\eta_{\Lambda}(b),\eta_{\Lambda}(b^{*})$ by $c$-numbers [@ZBru], [@Za14] will be instrumental. It was proved by Ginibre [@Gin], Lieb, Seiringer and Yngvason ([@LSYng1], [@LSYng]) and Sütö [@Suto1], but we shall rely on the method of [@LSYng], which uses the Berezin-Lieb inequality [@Lieb1].
Let $z$ be a complex number , $|z\rangle = \exp(-|z|^{2}/2 +z b_{{0}}^{*})|0\rangle$ the Glauber coherent vector in ${\cal F}_{{0}}$ and, as in [@LSYng], let $(H_{\Lambda,\mu,\lambda})^{'}(z)$ be the *lower symbol* of $H_{\Lambda,\mu,\lambda}$. Then $$\exp(\beta V p_{\beta,\Lambda,\mu,\lambda}^{'})=\Xi_{\Lambda}(\beta,\mu,\lambda)^{'} =
\int d^{2}z {\rm{Tr}}_{{\cal H}^{'}} \exp(-\beta (H_{\Lambda,\mu,\lambda})^{'}(z)) \ ,
\label{eqn2.4.18}$$ where ${\cal H}^{'}= {\cal F}_{{k} \ne {0}}$, with obvious notations for the Fock spaces associated to the zero mode and the remaining modes. Consider the weight $${\cal W}_{\mu,\Lambda, \lambda}(z) := \Xi_{\Lambda}(\beta,\mu,\lambda)^{-1}\\
{\rm{Tr}}_{{\cal H}^{'}}\langle z| \exp(-\beta H_{\Lambda,\mu,\lambda})|z \rangle \ .
\label{eqn2.4.19}$$ For almost all $\lambda >0$ it was proved in [@LSYng] that the density of distribution ${\cal W}_{\mu,\Lambda, \lambda} (\zeta \sqrt{V})$ converges, as $V \to \infty$, to a $\delta$ function at the point $\zeta_{max}(\lambda)=\lim_{V \to \infty} {z_{max}(\lambda)}/{\sqrt{V}}$, where $ z_{max}(\lambda)$ maximizes the partition function ${\rm{Tr}}_{{\cal H}^{'}} \exp(-\beta (H_{\Lambda,\mu,\lambda})^{'}(z))$. Although [@LSYng] took $\phi=0$ in (\[4.8\]), their results in the general case (\[4.8\]) may be obtained by the trivial substitution $b_{{0}}\to b_{{0}}\exp(-i\phi)$, $b_{{0}}^{*} \to b_{{0}}^{*} \exp(i\phi)$ coming from (\[4.6\]). Note that their expression (34) in [@LSYng] may be thus re-written as $$\begin{aligned}
&& \lim_{V \to \infty} \omega_{\beta,\mu,\Lambda,\lambda}(\eta_{\Lambda}(b_{{0}}^{*}\exp(i\phi))=
\lim_{V \to \infty} \omega_{\beta,\mu,\Lambda,\lambda}(\eta_{\Lambda}(b_{{0}}\exp(-i\phi)) \nonumber \\
&& = \zeta_{max}(\lambda)=\frac{\partial p(\mu,\lambda)}{\partial \lambda} \ , \label{eqn2.4.20}\end{aligned}$$ and consequently $$\lim_{V \to \infty} \omega_{\beta,\mu,\Lambda,\lambda}(\eta_{\Lambda}(b_{{0}}^{*})\eta_{\Lambda}(b_{{0}}))
= |\zeta_{max}(\lambda)|^{2} \ .
\label{eqn2.4.21}$$ Here above, $$\label{4.22}
p(\beta,\mu,\lambda) = \lim_{V \to \infty} p_{\beta,\mu,\Lambda,\lambda} \ ,$$ is the pressure in the thermodynamic limit. Equality (\[eqn2.4.20\]) follows from the convexity of $p_{\beta,\mu,\Lambda,\lambda}$ in $\lambda$ by the Griffiths lemma [@Gri66]. As it is shown in [@LSYng] the pressure $p(\beta,\mu,\lambda)$ is equal to $$\label{4.23}
p(\beta,\mu,\lambda)^{'} = \lim_{V \to \infty} p_{\beta,\mu,\Lambda,\lambda}^{'} \ .$$ As well as it is also equal to the pressure $p(\beta,\mu,\lambda)^{''}$, which is the thermodynamic limit of the pressure associated to the *upper symbol* of $H_{\Lambda,\mu,\lambda}$.
It is crucial in the proof of [@LSYng] that all of these three pressures coincide with $p_{max}(\beta,\mu,\lambda)$, which is the pressure associated to ${\rm{max}}_{z} {\rm{Tr}}_{{\cal H}^{'}} \exp(-\beta (H_{\Lambda,\mu,\lambda})^{'}(z))$.
\[theo:4.1\] Consider the system of interacting Bosons (\[4.1\])-(\[4.8\]). If the system displays ODLRO in the sense of (\[4.16\]), the limit $\omega_{\beta,\mu,\phi}:=
\lim_{\lambda \to +0} \lim_{V \to \infty}\omega_{\beta,\mu,\Lambda,\lambda_{\phi}} $, on the set $\{\eta(b_{{0}}^{*})^{m}\eta(b_{{0}})^{n}\}_{m,n=0,1}$ exists and satisfies $$\label{4.24.1}
\omega_{\beta,\mu,\phi} (\eta(b_{{0}}^{*})) = \sqrt{\rho_{0}} \exp(i\phi) \ ,$$ $$\label{4.24.2}
\omega_{\beta,\mu,\phi} (\eta(b_{{0}})) = \sqrt{\rho_{0}} \exp(-i\phi) \ ,$$ together with $$\label{eqn3.4.24.3}
\omega_{\beta,\mu,\phi} (\eta(b_{{0}}^{*})\eta(b_{{0}}) = \omega_{\beta,\mu}((\eta(b_{{0}}^{*})\eta(b_{{0}}))
= \rho_{{0}} \ \ \forall \phi \in [0,2\pi) \ ,$$ and $$\label{4.24.4}
\omega_{\beta,\mu} = \frac{1}{2\pi} \int_{0}^{2\pi} d\phi \ \omega_{\beta,\mu,\phi} \ .$$ On the Weyl algebra the limit defining $\omega_{\beta,\mu,\phi}, \ \phi \in [0,2\pi)$ exists along a net in the $(\lambda,V)$ variables, and defines ergodic states coinciding with those states that explicitly constructed in Theorem \[theo:A.1\]. Conversely, if SSB occurs in the special sense that (4.41) and (4.42) hold, with $\rho_{0} \ne 0$, then ODLRO in the sense of (4.25) takes place.
We need only prove the direct statement, because the converse follows by applying the Schwarz inequality to the states $\omega_{\beta,\mu,\phi}$, together with the forthcoming (\[eqn3.4.27\]).
We thus prove ODLRO $\Rightarrow$ SSB. We first assume that some state $ \omega_{\beta,\mu,\phi_{0}},\phi_{0}
\in [0,2\pi)$ satisfies ODLRO. Then by (\[eqn2.4.21\]), $$\label{eqn2.4.25.1}
\lim_{\lambda \to +0} \lim_{V \to \infty} \omega_{\beta,\mu,\Lambda,\lambda}(\eta(b_{{0}}^{*})\eta(b_{{0}}))
= \lim_{\lambda \to +0} |\zeta_{max}(\lambda)|^{2} =: \rho_{{0}} > 0 \ .$$ The above limit exists by the convexity of $p(\mu,\lambda)$ in $\lambda$ and (\[4.14.1\]) by virtue of (\[eqn2.4.25.1\]), $$\label{4.25.2}
\lim_{\lambda \to +0} \frac{\partial p(\mu,\lambda)}{\partial \lambda} \ne 0 \ .$$ At the same time, (\[eqn2.4.20\]) shows that all states $\omega_{\beta,\mu,\phi}$ satisfy (\[eqn2.4.25.1\]). Thus, SSB is broken in the states $\omega_{\beta,\mu,\phi},\phi \in [0,2\pi)$. We now prove that the original assumption (\[4.16\]) implies that all states $\omega_{\beta,\mu,\phi},\phi \in [0,2\pi)$ exhibit ODLRO.
Gauge invariance of $\omega_{\beta,\mu,\Lambda}$ (or equivalently $H_{\Lambda,\mu}$) yields, by (\[4.7\]), (\[4.17\]), $$\omega_{\beta,\mu,\Lambda,\lambda}(\eta(b_{{0}}^{*})\eta(b_{{0}}))
=\omega_{\beta,\mu,\Lambda,-\lambda}(\eta(b_{{0}}^{*})\eta(b_{{0}})) \ .
\label{eqn2.4.26.1}$$ Again by (\[4.7\]), (\[4.12.1\]) and gauge invariance of $H_{\Lambda,\mu}$, $$\lim_{\lambda \to -0} \frac{\partial p(\mu,\lambda)}{\partial \lambda}=
-\lim_{\lambda \to +0} \frac{\partial p(\mu,\lambda)}{\partial \lambda} \ ,$$ and, since by convexity the derivative ${\partial p(\mu,\lambda)}/{\partial \lambda}$ is monotone increasing, we find $$\lim_{\lambda \to +0} \frac{\partial p(\mu,\lambda)}{\partial \lambda}
= \lim_{\lambda \to +0} \zeta_{max}(\lambda) = \sqrt{\rho_{0}} \ ,
\label{eqn2.4.26.2}$$ $$\lim_{\lambda \to -0} \frac{\partial p(\mu,\lambda)}{\partial \lambda}
= -\lim_{\lambda \to +0} \zeta_{max}(\lambda)= -\sqrt{\rho_{0}} \ .
\label{eqn2.4.26.3}$$ Again by (\[eqn2.4.26.1\]), $$\lim_{\lambda \to -0}\lim_{V \to \infty} \omega_{\beta,\mu,\Lambda,\lambda}(\eta(b_{{0}}^{*})\eta(b_{{0}}))
= \lim_{\lambda \to +0}\lim_{V \to \infty} \omega_{\beta,\mu,\Lambda,\lambda}(\eta(b_{{0}}^{*})\eta(b_{{0}})) \ .
\label{eqn2.4.26.4}$$ By [@LSYng], the weight ${\cal W}_{\mu,\lambda}$ is, for $\lambda=0$, supported on a disc with radius equal to the right-derivative (\[4.25.2\]). Convexity of the pressure as a function of $\lambda$ implies $$\begin{aligned}
\frac{\partial p(\mu,\lambda_{0}^{-})}{\partial \lambda_{0}^{-}} \le \lim_{\lambda \to -0}\frac{\partial
p(\mu,\lambda)}{\partial \lambda}
\le \lim_{\lambda \to +0}\frac{\partial p(\mu,\lambda)}{\partial \lambda} \le \frac{\partial p(\mu,
\lambda_{0}^{+})}{\partial \lambda_{0}^{+}} \ ,\end{aligned}$$ for any $\lambda_{0}^{-}<0<\lambda_{0}^{+}$. Therefore, by the Griffiths lemma (see e.g. [@Gri66], [@LSYng]) one gets $$\lim_{\lambda \to -0}\lim_{V \to \infty} \omega_{\beta,\mu,\Lambda,\lambda}(\eta(b_{{0}}^{*})\eta(b_{{0}}))
\le \lim_{V \to \infty} \omega_{\beta,\mu,\Lambda}(\frac{b_{{0}}^{*}b_{{0}}}{V})
\le \lim_{\lambda \to +0}\lim_{V \to \infty} \omega_{\beta,\mu,\Lambda,\lambda}(\eta(b_{{0}}^{*})\eta(b_{{0}})) \ .
\label{eqn3.4.27}$$ Then (\[eqn2.4.26.4\]) and (\[eqn3.4.27\]) yield $$\lim_{V \to \infty} \omega_{\beta,\mu,\Lambda}(\frac{b_{{0}}^{*}b_{{0}}}{V})=\\
\lim_{\lambda \to +0}\lim_{V \to \infty} \omega_{\beta,\mu,\Lambda,\lambda}(\eta(b_{{0}}^{*})\eta(b_{{0}})) \ \
\forall \phi \in [0,2\pi) \ .
\label{eqn3.4.28}$$ This proves that all $\omega_{\beta,\mu,\phi},\phi \in [0,2\pi)$ satisfy ODLRO, as asserted.
By (\[eqn2.4.20\]) and (\[eqn2.4.26.2\]) one gets (\[4.24.1\]) and (\[4.24.2\]). Then (\[4.24.4\]) is a consequence of the gauge-invariance of $\omega_{\beta,\mu}$. Ergodicity of the states $\omega_{\beta,\mu,\phi},\phi \in [0,2\pi$ follows from (\[eqn3.4.28\]) and (\[4.24.1\]), (\[4.24.2\]).
An equivalent construction is possible using the Weyl algebra instead of the polynomial algebra, see [@Ver], pg. 56 and references given there for theorem 6.1 and similarly we could have proceeded so here. The limit along a subnet in the $(\lambda,V)$ variables exists by weak\* compactness, and, by asymptotically abelianness of the Weyl algebra for space translations (see, e.g., [@BR97], Example 5.2.19), the ergodic decomposition (\[4.24.4\]), which is also a central decomposition, is unique. Thus, the $\omega_{\beta,\mu,\phi},\phi \in [0,2\pi)$ coincide with the states constructed in Theorem 6.1.
\[4.1\] [Our Remark \[4.3\] and Theorem \[theo:4.1\] elucidate a problem discussed in [@LSYng]. In this paper the authors defined a generalised Gauge Symmetry Breaking via quasi-average $\rm{(GSB)}_{qa} \ $, i.e. by $\lim_{\lambda \to +0} \lim_{V \to \infty} \omega_{\beta,\mu,\Lambda,\lambda}(\eta_{\Lambda}(b_{{0}})) \ne 0$. (If it involves other than gauge group, we denote this by $\rm{(SSB)}_{qa}$.) Similarly they modified definition of the one-mode condensation denoted by $\rm{(BEC)}_{qa}$ (\[eqn2.4.25.1\]), and established the equivalence: $\rm{(GSB)}_{qa} \Leftrightarrow \rm{(BEC)}_{qa}$. They asked whether $\rm{(BEC)}_{qa} \Leftrightarrow \rm{BEC}$ ? We show that $\rm{(GSB)}_{qa}$ coincides with GSB (Definition 2.2), and that BEC is indeed equivalent to $\rm{(BEC)}_{qa}$. ]{}
\[4.2\] The states $\omega_{\beta,\mu,\phi}$ in Theorem \[theo:4.1\] have the property ii) of Theorem \[theo:A.1\], i.e., if $\phi_{1} \ne \phi_{2}$, then $\omega_{\beta,\mu,\phi_{1}} \ne \omega_{\beta,\mu,\phi_{2}}$. By a theorem of Kadison [@Kadison], two factor states are either disjoint or quasi-equivalent (see Remark 3.1 and references given there), and thus the states $\omega_{\beta,\mu,\phi}$ for different $\phi$ are mutually disjoint. This fact has a simple explanation: only for a finite system is the Bogoliubov transformation (\[4.9.2\]) (which also applies to the interacting system), which connects different $\phi$, unitary: for infinite systems one has to make an infinite change of an extensive observable $b_{{0}} \sqrt{V}$, and mutually disjoint sectors result. This phenomenon also occurs with regard to the magnetization in quantum spin systems, in correspondence to (\[2.31\]) and it is in this sense that the word ”degeneracy” must be understood (compare with the discussion in [@Bog70]).
Concluding remarks
==================
In this paper, we reexamined the issue of ODLRO versus SSB by the method of Bogoliubov quasi-averages, commonly regarded as a *symmetry-breaking trick*. We showed that it represents a general method of construction of extremal, pure or ergodic states, both for quantum spin systems (Proposition \[3.1\]) and many-body Boson systems (Theorem \[theo:4.1\]). The breaking of gauge symmetry in the latter has some analogy with the breaking of gauge and $\gamma_{5}$ invariance in the Schwinger model (quantum electrodynamics of massless electrons in two dimensions) ([@LSwi]), in which the vacuum state decomposes in a manner similar to (\[4.24.4\]). We believe, and argued so in Section \[sec:QA-spin\], that the quasi-average method is the only universally applicable method, in particular to relativistic quantum field theory, to which the imposition of classical boundary conditions is bound to be inconsistent with the general principles of local quantum theory, as in the case of the Casimir effect [@KNW].
A general necessary feature for the applicability of the Bogoliubov method is the existence of an order parameter. In the two examples treated, the Heisenberg ferromagnet (Section \[sec:Setup\], see also remark 2.5 concerning order parameters for quantum spin systems in the general case) and many-body Boson systems (Section \[sec:QA-Boson\]), the respective symmetry-breaking fields (\[3.3\]) and (\[4.7\]) are qualitatively different. Note that (\[3.3\]) commutes with $H_{\Lambda}$ and the corresponding order parameter, the magnetization, is physically measurable. Whereas (\[4.7\]) does not commute with $H_{\Lambda}- \mu N_{\Lambda}$ (even in the free-gas case!), and the order parameter involves a **phase** by (\[4.24.1\]),(\[4.24.2\]), which, at first glance, is not physically measurable. It has been observed, however, in the interference of two condensates of different phases [@ATMDKK] , [@BZ], in the case of trapped gases. In the latter case, Condensation takes place at ${k} \ne {0}$, and the version of Theorem \[theo:4.1\] due to Pulè et al [@PVZ] is the relevant one. Finally, in the quantum spin case there is a residual symmetry (Remark \[rem:2.1\], but none, of course, in the Boson case. These remarks exemplify the rather wide diversity of types of the *Bogoliubov quasi-avearge*, which make its conjectured universal applicability further plausible, see e.g. random boson systems [@JaZ10].
As remarked by Swieca [@SwiecaJ], it is the fluctuations occurring all over space which do not allow to take the “charge” (e.g. (\[2.27\])) in the limit $V \to \infty$ as a well-defined operator (this would, in particular, contradict (\[2.24\])), even if a meaning has been given to the density - as in (\[2.25\]) - see also Remark \[rem:2.3\]. The additional input we offer is that the fluctuation of the charge density (or of a related operator) is precisely a very nontrivial condition of ODLRO ((\[2.33\]) or (\[4.16\]) respectively).
As a final question, the treatment of the free Bose gas suggests that the chemical potential $\mu(\lambda) < 0$ for $\lambda \ne 0$ even after the thermodynamic limit also for interacting systems. It should be interesting to look at Bose gases with repulsive interactions [@BraRo] from the point of view of quasi-average: $\rm{(SSB)}_{qa}$, using the symmetry breaking term (\[4.7\]).
Appendix A
==========
In this Appendix we reproduce, for the reader’s convenience, the statement of the basic theorem of Fannes, Pulè and Verbeure [@FPV1], see also [@PVZ] for the extension to nonzero momentum, and Verbeure’s book [@Ver]. Unfortunately, neither [@FPV1] nor [@PVZ] show that the states $\omega_{\beta,\mu,\phi},\phi \in [0,2\pi)$ in the theorem below are ergodic. The simple, but instructive proof of this fact was given by Verbeure in his book [@Ver].
\[theo:A.1\] Let $\omega_{\beta,\mu}$ be an analytic, gauge-invariant equilibrium state. If $\omega_{\beta,\mu}$ exhibits ODLRO (\[4.16\]), then there exist ergodic states $ \omega_{\beta,\mu,\phi},\phi \in [0,2\pi)$, not gauge invariant, satisfying (i) $\forall \theta,\phi \in [0,2\pi)$ such that $\theta \ne \phi$, $\omega_{\beta,\mu,\phi} \ne
\omega_{\beta,\mu,\theta}$; (ii) the state $\omega_{\beta,\mu}$ has the decomposition $$\omega_{\beta,\mu} = \frac{1}{2\pi} \int_{0}^{2\pi} d\phi\omega_{\beta,\mu,\phi} \ .$$ (iii) For each polynomial $Q$ in the operators $\eta(b_{{0}})$,$\eta(b_{{0}}^{*})$, and for each $\phi \in [0,2\pi)$, $$\begin{aligned}
\omega_{\beta,\mu,\phi}(Q(\eta(b_{{0}}^{*}),\eta(b_{{0}})X)
= \omega_{\beta,\mu,\phi}(Q(\sqrt{\rho_{0}} \exp(-i\phi),\sqrt{\rho_{0}} \exp(i \phi)X)\ \
\forall X \in {\cal A} \ .\end{aligned}$$
We remark, with Verbeure [@Ver], that the proof of Theorem A.1 is **constructive**. One essential ingredient is the separating character (or faithfulness) of the state $\omega_{\beta,\mu}$, i.e., $\omega_{\beta,\mu}(A) = 0$ implies $A=0$. This property, which depends on the extension of $\omega_{\beta,\mu}$ to the von-Neumann algebra $\pi_{\omega}({\cal A})^{''}$ (see [@BR97], [@Hug]) is true for thermal states, but is not true for ground states, even without this extension: in fact, a ground state (or vacuum) is non-faithful on ${\cal A}$ (see proposition 3 of [@Wrep]). We see, therefore, that thermal states and ground states might differ with regard to the ergodic decomposition (ii). Compare also with our discussion in the Concluding remarks.
**Acknowledgements**
Some of the issues dealt with in this paper originate in the open problem posed in Sec.3 of [@SeW] and at the of [@JaZ10]. One of us (W.F.W.) would like to thank G. L. Sewell for sharing with him his views on ODLRO along several years. He would also like to thank the organisers of the Satellite conference “Operator Algebras and Quantum Physics” of the XVIII conference of the IAMP (Santiago de Chile) in SãoPaulo, July 17th-23rd 2015, for the opportunity to present a talk in which some of the ideas of the present paper were discussed. We are thankful to Bruno Nachtergaele for very useful remarks, suggestions, and corrections, which greatly improved and clarified the paper.
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| {
"pile_set_name": "ArXiv"
} |
---
author:
- Miao Li
- Da Liu
- Jun Meng
- Tower Wang
- Lanjun Zhou
title: Replaying neutrino bremsstrahlung with general dispersion relations
---
Introduction {#sec:intr}
============
The OPERA experiment stirred the physics community recently with its astonishing result that neutrinos in this experiment travel apparently faster than light at a high confidence level [@OPERA:2011zb]. If not attributed to systematic errors in the measurement, this result would imply the violation of special relativity. Subsequently, it has inspired a lot of speculation.[^1]
However, as quickly claimed in [@Cohen:2011hx; @Bi:2011nd], several high-energy processes disfavor the superluminal interpretation of the OPERA data. An outstanding example is the bremsstrahlung-like process $$\label{bremsstrahlung}
\nu_{\mu}\rightarrow\nu_{\mu}+e^{+}+e^{-},$$ where electron-positron pairs are radiated and hence neutrinos lose their energy efficiently. To explicitly demonstrate this point, the authors of [@Cohen:2011hx] assumed a special dispersion relation roughly of the form $E=v_{\nu}p$, where $v_{\nu}$ is a constant greater than light velocity. A similar assumption was also taken in [@Bi:2011nd].
On the other hand, a dispersion relation of the form $E=v_{\nu}p$ is too oversimplified to accommodate more observational data of neutrino velocity, as summarized and analyzed in [@Li:2011ue]. Therefore, it is urgent to extend the calculation of [@Cohen:2011hx; @Bi:2011nd] to general dispersion relations. The present paper is devoted to such a calculation. Our results would help to rule out more phenomenological models and hunt for viable models.
In the absence of Lorentz invariance, the calculation is complicated even for the special dispersion relation $E=v_{\nu}p$, so an effective “mass” was assigned to neutrinos and a “rest frame” was employed in [@Cohen:2011hx]. Our calculation does not resort to such a nontrivial reference frame. Or more explicitly, one may interpret our work as a direct calculation in the laboratory frame. Applied to the above dispersion relation, our result provides a crosscheck for the results in [@Cohen:2011hx; @Bi:2011nd].
The paper is organized as follows. Section \[sec:assum\] collects the basic assumptions and conventions of notation in this work. Kinematically there is a threshold energy for the process . We discuss the dependence of this threshold on dispersion relations in section \[sec:thres\]. Section \[sec:width\] is the main part of our paper, where we calculate the “decay width” of superluminal neutrino via . Details and techniques are presented clearly. To check and apply our general results in sections \[sec:thres\] and \[sec:width\], we work out some specific examples in section \[sec:exam\] with given dispersion relations. The results are consistent with [@Cohen:2011hx] quantitatively and [@Mohanty:2011rm] qualitatively. Section \[sec:con\] concludes this paper.
Basic assumptions and notation conventions {#sec:assum}
==========================================
Throughout the paper, we assume
1. The ordinary conservation law for energy and momentum is intact. In other words, the time and space translations are exact symmetries in the working frame. A case study for violating this assumption can be found in [@AmelinoCamelia:2011bz].
2. The space is Euclidean and isotropic. Thereby we can work in a spherical coordinate system and define the magnitude of momentum as $p=|\vec{p}|$.
3. In the relevant energy range, the dispersion relation of electron and positron is well characterized by $E^2=p^2+m_e^2$. This assumption is in accordance with experiments to date.
4. The dispersion relation of neutrino is either $E=E(p)$ or $p=p(E)$. Here $E(p)$ and $p(E)$ are arbitrary devisable functions. They could be non-monotonic and may involve some parameters such as mass, etc.
In section \[sec:thres\], when deriving the threshold energy of process , we make one more assumption:
- The neutrino’s dispersion relation reduces to $E^2=p^2+m_{\nu}^2$ at very low energies, typically lower than the threshold energy by a factor $\mathcal{O}(10^{-4})$. See details in section \[sec:thres\].
In section \[sec:width\], this assumption is replaced by two other assumptions:
- The squared amplitude is the same as that in standard model. We will comment on possible loopholes of this assumption in section \[sec:con\]. But to alleviate the complexity of our calculation, we should make such an assumption at the moment.
- The masses of electron and positron are neglected. This assumption seems to be reasonable, because when we study the decay width of high-energy neutrino which is practically around or above $\GeV$ scale, the phase space should be dominated by high-energy particles in principle.
Let us clarify the conventions of notation by writing in the form $$\nu(p)\rightarrow\nu(p')+e^{+}(k')+e^{-}(k),$$ where we have specified the notation of momentum for each particle. One must be careful of the notations of momenta. Taking $p$ for instance, sometimes it stands for the four-vector, but sometimes it stands for the magnitude of three-vector $\vec{p}$. In our equations, the four-vectors appear usually together with a dot, indicating the inner product with Lorentz signature $\diag(+,-,-,-)$. For example, $p\cdot p=E_{p}^2-|\vec{p}|^2$ but $p^2=|\vec{p}|^2$. So there are no confusions if one is careful.
The velocity of light is a constant, so we set it to $1$ throughout this paper.
Energy threshold {#sec:thres}
================
The assumptions made in the previous section simplify the derivation of threshold energy for considerably.[^2] From the energy conservation relation $$\label{econs}
E_{p}=E_{p'}+E_{k'}+E_{k},$$ we observe that to minimize $E_{p}$, one should lower $E_{p'}$, $E_{k'}$ and $E_{k}$ as much as possible. This can be achieved by deleting all of the transverse momentum components, leading to the reduced conservation law of momentum $$\label{pcons}
p=p'+k'+k,$$ where $p=|\vec{p}|$, $p=|\vec{p'}|$ and so on are magnitude of momenta as our conventions.
Substituting and into the dispersion relation of $e^{+}(k')$ and $e^{-}(k)$, we have $$\label{Epfun}
(E_{p}-E_{p'})^2-2E_{k}(E_{p}-E_{p'})-(p-p')^2+2k(p-p')=0.$$ In light of dispersion relations of $\nu(p)$, $\nu(p')$ and $e^{-}(k)$, this equation may be understood as an implicit function of $E_{p}$, $E_{p'}$ and $E_{k}$. Then we can extremize $E_{p}$ with respect to $E_{p'}$ and $E_{k}$, obtaining $$\begin{aligned}
\label{extrem}
(E_{p}-E_{k})-(p-k)\frac{E_{p'}}{p'}&=&0,\nonumber\\
(E_{p}-E_{p'})-(p-p')\frac{E_{k}}{k}&=&0,\end{aligned}$$ where conditions $\partial E_{p}/\partial E_{p'}=\partial
E_{p}/\partial E_{k}=0$ and dispersion relations $dp'/dE_{p'}=E_{p'}/p'$, $dk/dE_{k}=E_{k}/k$ are used.
From eqs. and , it is not hard to get $$2E_{k}=E_{p}-E_{p'},\qquad2k=p-p',\qquad\frac{E_{p}}{p}=\frac{E_{p'}}{p'}$$ and subsequently $$\label{midstep}
E_{p}^2-2E_{p}E_{p'}+m_{\nu}^2=p^2-2pp'+4m_e^2,\qquad E_{p'}=\frac{m_{\nu}}{\sqrt{1-\frac{p^2}{E_{p}^2}}}.$$ As a result, we find the threshold of is given by $$\label{threshold}
\left(E_{p}^2-p^2\right)_{\th}=\left(2m_{e}+m_{\nu}\right)^2.$$ This result is in accordance with [@Villante:2011pk]. To the leading order, it is also compatible with [@Cohen:2011hx] where neutrino’s dispersion relation is $E_{p}/p=v_{\nu}$. This will be shown in subsection \[subsec:cg\]. As a concrete example, in subsection \[subsec:lw\] we will utilize to get the decay threshold for a toy model of mass-dependent Lorentz violation [@Li:2011ue].
Note that when writing down and , we have assumed the dispersion relation $E_{p'}^2=p'^2+m_{\nu}^2$ of $\nu(p')$ at low energies. But the dispersion relation of $\nu(p)$ is irrelevant throughout the derivation. From eqs. and we can see $(E_{p'}/E_{p})_{\th}=m_{\nu}/(2m_{e}+m_{\nu})\sim\mathcal{O}(10^{-4})$. That means we have assumed the dispersion relation $E_{p'}^2=p'^2+m_{\nu}^2$ for neutrinos at an energy lower than the threshold energy by $\mathcal{O}(10^{-4})$. This assumption is natural and consistent with the observational data of neutrino velocity summarized in [@Li:2011ue]. Replacing this assumption with other dispersion relations, one may also restart from and follow our method to derive the threshold energy.[^3]
Decay width {#sec:width}
===========
We are interested in the following process: $\nu(p)\rightarrow\nu(p')+e^{+}(k')+e^{-}(k)$. Considering the neutral current of this process, we have[^4] $$\sum_{spin}\mathcal{M}\mathcal{M}^{*}=128G_{F}^{2}[(p\cdot k')(k\cdot p')(-\frac{1}{2}+\sin^{2}\theta)^{2}+(p\cdot k)(k'\cdot p')\sin^{4}\theta)].$$ To calculate the decay width, we will integrate over $p'$, $k$ and $k'$, hence we can make use of the symmetry between $k$ and $k'$ to write the squared amplitude as $$\label{amplitude}
\sum_{spin}\mathcal{M}\mathcal{M}^{*}=128G_{F}^{2}(p\cdot k')(k\cdot p')\left[\left(-\frac{1}{2}+\sin^{2}\theta_W\right)^{2}+\sin^{4}\theta_W\right].$$ The decay width is formally given by $$\begin{aligned}
\label{Gamma1}
\Gamma&=&\frac{1}{2E_{p}}\int\frac{d^{3}\vec{p'}}{(2\pi)^{3}2E_{p'}}\int\frac{d^{3}\vec{k}}{(2\pi)^{3}2E_{k}}\int\frac{d^{3}\vec{k'}}{(2\pi)^{3}2E_{k'}}\frac{1}{2}|\mathcal{M}|^{2}(2\pi)^{4}\delta^{4}(p-p'-k'-k)\\
&=&\frac{8G_{F}^{2}}{(2\pi)^{5}E_{p}}\left(\frac{1}{4}-\sin^{2}\theta_W+2\sin^{4}\theta_W\right)\int\frac{d^{3}\vec{p'}}{E_{p'}}\int\frac{d^{3}\vec{k}}{E_{k}}\int\frac{d^{3}\vec{k'}}{2E_{k'}}(p\cdot k')(k\cdot p')\delta^{4}(p-p'-k'-k).\nonumber\end{aligned}$$ Here $\theta_W$ is the Weinberg angle.
In the main part of this section, we will focus on the calculation of integral by temporarily forgetting the overall coefficient. Because we are interested in high-energy neutrino decay, the masses of electron and positron will be neglected in our calculation. Then the integral in can be briefly rewritten as $$\begin{aligned}
\Gamma_{\ab}&=&\int\frac{d^{3}\vec{p'}}{E_{p'}}\int\frac{d^{3}\vec{k}}{E_{k}}\int\frac{d^{3}\vec{k'}}{2E_{k'}}(p\cdot k')(k\cdot p')\delta^{4}(p-p'-k'-k)\nonumber\\
&=&\int\frac{d^{3}\vec{p'}}{E_{p'}}\int\frac{d^{3}\vec{k}}{E_{k}}\int d^{4}k'\delta\left(k'\cdot k'\right)|_{k'^{0}>0}(p\cdot k')(k\cdot p')\delta^{4}\left(p-p'-k'-k\right)\\
&=&\int\frac{d^{3}\vec{p'}}{E_{p'}}\int\frac{d^{3}\vec{k}}{E_{k}}\left[p\cdot(p-p'-k)\right]\left(k\cdot p'\right)\delta\left((p-p'-k)\cdot(p-p'-k)\right)|_{E_{p}-E_{p'}-k>0}.\nonumber\end{aligned}$$
Making use of relations $k\cdot k=k'\cdot k'=0$ and $$k\cdot p'=k\cdot (p-k-k')=k\cdot p-\frac{(k+k')\cdot(k+k')}{2}=k\cdot p-\frac{(p-p')\cdot(p-p')}{2},$$ we obtain $$\begin{aligned}
\Gamma_{\ab}&=&\int\frac{d^{3}\vec{p'}}{E_{p'}}\int\frac{d^{3}\vec{k}}{k}\left[p\cdot(p-p')-p\cdot k\right]\left[k\cdot p-\frac{(p-p')\cdot(p-p')}{2}\right]\nonumber\\
&&\times\delta\left((p-p')\cdot(p-p')-2k\cdot(p-p')\right)\biggr|_{E_{p}-E_{p'}-k>0}\nonumber\\
%&=&\int\frac{d^{3}\vec{p'}}{E_{p'}}\int\frac{d^{3}\vec{k}}{k}\left[p\cdot(p-p')-E_{p}E_{k}+\vec{p}\cdot\vec{k}\right]\left[E_{p}E_{k}-\vec{p}\cdot\vec{k}-\frac{(p-p')\cdot(p-p')}{2}\right]\nonumber\\
%&&\times\delta\left((p-p')\cdot(p-p')-2k\cdot(p-p')\right)\biggr|_{E_{p}-E_{p'}-k>0}\nonumber\\
&=&\int\frac{d^{3}\vec{p'}}{E_{p'}}\int kdk\sin\theta_{1}d\theta_{1} d\varphi_{1}\left[p\cdot(p-p')-E_{p}E_{k}+\vec{p}\cdot\vec{k}\right]\nonumber\\
&&\times\left[E_{p}E_{k}-\vec{p}\cdot\vec{k}-\frac{(p-p')\cdot(p-p')}{2}\right]\nonumber\\
&&\times\delta\left((p-p')\cdot(p-p')-2E_{k}(E_{p}-E_{p'})+2\vec{k}\cdot(\vec{p}-\vec{p'})\right)\biggr|_{E_{p}-E_{p'}-k>0}.\end{aligned}$$ Here $\theta_{1}$ is defined as the angle between $\vec{k}$ and $\vec{p}-\vec{p'}$. We will define $\theta_{2}$ as the angle between $\vec{p}$ and $\vec{p'}$. The relative directions and angles between the relevant momentum vectors are depicted in figure \[fig-coord\].
![(color online). The relative directions of vectors $\vec{p}$, $\vec{p'}$, $\vec{p}-\vec{p'}$ and $\vec{k}$ in three spatial dimensions. The zenith angle $\theta_{1}$ is defined as the angle between $\vec{k}$ and $\vec{p}-\vec{p'}$, and $\theta_{2}$ as the angle between $\vec{p}$ between $\vec{p'}$. As shown in the picture, axes $\vec{\mathbf{e}}_{z}$ and $\vec{\mathbf{e}}_{z'}$ coincide with directions of $\vec{p}$ and $\vec{p}-\vec{p'}$ respectively. Projecting $\vec{k}$ on the plane perpendicular to $\vec{\mathbf{e}}_{z'}$, we can define one azimuth angle $\varphi_1$. Similarly, the other azimuth angle $\varphi_2$ can be defined by projection of $\vec{p'}$ on the plane perpendicular to $\vec{\mathbf{e}}_{z}$.[]{data-label="fig-coord"}](coordinates.eps){width="35.00000%"}
It is convenient to express the inner product of $\vec{p}$ and $\vec{k}$ in terms of the new coordinates, $$\label{pk}
\vec{p}\cdot\vec{k}=pk\frac{\cos\theta_{1}(p-p'\cos\theta_{2})+p'\sin\theta_{1}\cos\varphi_{1}\sin\theta_{2}}{|\vec{p}-\vec{p'}|}.$$ In terms of the new coordinates, the integration takes the form $$\begin{aligned}
\Gamma_{\ab}&=&\int\frac{d^{3}\vec{p'}}{E_{p'}}\int kdk\sin\theta_{1}d\theta_{1}d\varphi_{1}\biggl[p\cdot(p-p')-E_{p}E_{k}\nonumber\\
&&+pk\frac{\cos\theta_{1}(p-p'\cos\theta_{2})+p'\sin\theta_{1}\cos\varphi_{1}\sin\theta_{2}}{|\vec{p}-\vec{p'}|}\biggr]\nonumber\\
&&\times\left[E_{p}E_{k}-pk\frac{\cos\theta_{1}(p-p'\cos\theta_{2})+p'\sin\theta_{1}\cos\varphi_{1}\sin\theta_{2}}{|\vec{p}-\vec{p'}|}-\frac{(p-p')\cdot(p-p')}{2}\right]\nonumber\\
&&\times\delta\left((p-p')\cdot(p-p')-2E_{k}(E_{p}-E_{p'})+2k|\vec{p}-\vec{p'}|\cos\theta_{1}\right)\biggr|_{E_{p}-E_{p'}-k>0}\nonumber\\
&=&\int\frac{d^{3}\vec{p'}}{E_{p'}}\int kdkd\varphi_{1}\int^{1}_{-1}dx\biggl[p\cdot(p-p')-E_{p}E_{k}\nonumber\\
&&+pk\frac{x(p-p'\cos\theta_{2})+p'\sqrt{1-x^{2}}\cos\varphi_{1}\sin\theta_{2}}{|\vec{p}-\vec{p'}|}\biggr]\nonumber\\
&&\times\left[E_{p}E_{k}-pk\frac{x(p-p'\cos\theta_{2})+p'\sqrt{1-x^{2}}\cos\varphi_{1}\sin\theta_{2}}{|\vec{p}-\vec{p'}|}-\frac{(p-p')\cdot(p-p')}{2}\right]\nonumber\\
&&\times\delta\left((p-p')\cdot(p-p')-2E_{k}(E_{p}-E_{p'})+2k|\vec{p}-\vec{p'}|x\right)\biggr|_{E_{p}-E_{p'}-k>0}.\end{aligned}$$
With the newly introduced variable $x=\cos\theta_{1}\in[-1,1]$, we note that the function $$f(x)=(p-p')\cdot(p-p')-2k(E_{p}-E_{p'})+2k|\vec{p}-\vec{p'}|x$$ has a single root $$x_{0}=\frac{2k(E_{p}-E_{p'})-(p-p')\cdot(p-p')}{2k|\vec{p}-\vec{p'}|}$$ and its first-order derivative $$f'(x)=2k|\vec{p}-\vec{p'}|.$$ Therefore we can integrate $x$ out of the delta function and quickly obtain $$\begin{aligned}
\label{Gamma2}
\Gamma_{\ab}&=&\int\frac{d^{3}\vec{p'}}{E_{p'}}\int kdkd\varphi_{1}\biggl[p\cdot(p-p')-kE_{p}\nonumber\\
&&+kp\frac{x_{0}(p-p'\cos\theta_{2})+p'\sqrt{1-x_{0}^{2}}\cos\varphi_{1}\sin\theta_{2}}{|\vec{p}-\vec{p'}|}\biggr]\nonumber\\
&&\times[E_{p}k-pk\frac{x_{0}(p-p'\cos\theta_{2})+p'\sqrt{1-x_{0}^{2}}\cos\varphi_{1}\sin\theta_{2}}{|\vec{p}-\vec{p'}|}-\frac{(p-p')\cdot(p-p')}{2}]\nonumber\\
&&\times\frac{1}{2k|\vec{p}-\vec{p'}|}\biggr|_{E_{p}-E_{p'}-k>0}\nonumber\\
%&=&\int\frac{d^{3}\vec{p'}}{2E_{p'}|\vec{p}-\vec{p'}|}\int dkd\varphi_{1}\biggl[p\cdot(p-p')-kE_{p}+p\frac{kx_{0}(p-p'\cos\theta_{2})}{|\vec{p}-\vec{p'}|}\nonumber\\
%&&+\frac{pp'\sqrt{k^2-k^2x_{0}^{2}}\cos\varphi_{1}\sin\theta_{2}}{|\vec{p}-\vec{p'}|}\biggr]\biggl[E_{p}k-p\frac{kx_{0}(p-p'\cos\theta_{2})}{|\vec{p}-\vec{p'}|}\nonumber\\
%&&-\frac{pp'\sqrt{k^2-k^2x_{0}^{2}}\cos\varphi_{1}\sin\theta_{2}}{|\vec{p}-\vec{p'}|}-\frac{(p-p')\cdot(p-p')}{2}\biggr]\biggr|_{E_{p}-E_{p'}-k>0}\nonumber\\
&=&\int\frac{d^{3}\vec{p'}}{2E_{p'}|\vec{p}-\vec{p'}|}\int dk\biggl\{2\pi\left[p\cdot(p-p')-kE_{p}+p\frac{kx_{0}(p-p'\cos\theta_{2})}{|\vec{p}-\vec{p'}|}\right]\biggl[E_{p}k\nonumber\\
&&-p\frac{kx_{0}(p-p'\cos\theta_{2})}{|\vec{p}-\vec{p'}|}-\frac{(p-p')\cdot(p-p')}{2}\biggr]-\pi\frac{(pp')^{2}(k^2-k^2x_{0}^{2})\sin^{2}\theta_{2}}{|\vec{p}-\vec{p'}|^{2}}\biggr\}\biggr|_{E_{p}-E_{p'}-k>0}\end{aligned}$$
One may check that the integrand of is independent of $\varphi_{2}$. Its dependence on $k$ is quite simple. Its domain of integration is determined by $-1<x_{0}<1$, $E_{p}-E_{p'}-k>0$, or namely $$\begin{aligned}
\frac{(p-p')\cdot(p-p')}{2(E_{p}-E_{p'})+2|\vec{p}-\vec{p'}|}<k&<&\frac{(p-p')\cdot(p-p')}{2(E_{p}-E_{p'})-2|\vec{p}-\vec{p'}|},\nonumber\\
E_{p}-E_{p'}&>&|\vec{p}-\vec{p'}|.\end{aligned}$$ So we can integrate over variables $k$ and $\varphi_{2}$ straightforwardly. The calculation is a little tedious, yielding $$\begin{aligned}
\label{Gamma4}
\Gamma&=&\frac{8G_{F}^{2}}{(2\pi)^{5}E_{p}}\left(\frac{1}{4}-\sin^{2}\theta_W+2\sin^{4}\theta_W\right)\pi^{2}\int\int\frac{p'^2d p'dy}{6E_{p'}}\Bigl[3E_{p}E_{p'}^{3}\nonumber\\
&&+(-6 E_{p}^2+2p^{2}-3pp'y)E_{p'}^{2}+(3E_{p}^{3}-3E_{p}p^{2}-3E_{p}p'^{2}+8E_{p}pp'y)E_{p'}\nonumber\\
&&+3pyp'^{3}+(2E_{p}^{2}-2p^{2}-4p^{2}y^{2})p'^{2}+(3p^{3}y-3pE_{p}^{2}y)p'\Bigr].\end{aligned}$$ The domain of integration is $$\begin{aligned}
p^2+p'^2-2pp'y&<&(E_{p}-E_{p'})^{2},\label{domain1}\\
-1<y&<&1.\label{domain2}\end{aligned}$$ In the above, we employed notation $y=\cos\theta_2$ and turned on the overall factor in front of the integral as given by .
The expression for decay width is one of our main results. In section \[sec:exam\], we will apply it to several examples with explicit dispersion relations. Before proceed, let us make some comments on .
First, we would like to show that the decay width is positive-definite, as one should have anticipated from its definition . For this purpose, it is enough to focus on the integrand inside the square brackets, which can be transformed to $$\begin{aligned}
[\cdots]&=&3(E_{p}E_{p'}-pp'y)\left[(E_{p}-E_{p'})^2-(p^{2}+p'^{2}-2pp'y)\right]+2(E_{p}p'-E_{p'}p)^2\nonumber\\
&&+2pp'(1-y)\left[2E_{p}E_{p'}-pp'(1+y)\right].\end{aligned}$$ Taking account of inequalities and , this expression is nonnegative if $E_{p}E_{p'}\geq pp'$. This condition is well-satisfied if $E_{p}-p\geq0$, $E_{p'}-p'\geq0$ at lower energy and $dE_{p}/dp\geq1$, $dE_{p'}/dp'\geq1$ in the energy region of superluminal neutrino.
Second, we note that puts a lower limit of integration $y>[p^{2}+p'^{2}-(E_{p}-E_{p'})^2]/(2pp')$. In most situations, we have $E_{p}-E_{p'}\leq p+p'$, then this limit is more stringent than $y>-1$, and hence the domain of integration is simply $[p^{2}+p'^{2}-(E_{p}-E_{p'})^2]/(2pp')<y<1$, leading to the reduced decay width $$\begin{aligned}
\Gamma_{E_{p}-E_{p'}\leq p+p'}&=&\frac{8G_{F}^{2}}{(2\pi)^{5}E_{p}}\left(\frac{1}{4}-\sin^{2}\theta_W+2\sin^{4}\theta_W\right)\pi^{2}\int\frac{p'^2dp'}{6E_{p'}}\frac{1}{24pp'}\Bigl\{5E_{p}^6\nonumber\\
&&-3\left(15E_{p'}^2+5p^2-3p'^2\right)E_{p}^4+8E_{p'}\left(10E_{p'}^2-6p'^2+9pp'\right)E_{p}^3\nonumber\\
&&-3E_{p}^2\left[15E_{p'}^4-6\left(3p^2-8p'p+3p'^2\right)E_{p'}^2-(p-p')^2\left(5p^2+10p'p-3p'^2\right)\right]\nonumber\\
&&-24E_{p}E_{p'}p\left[(2p-3p')E_{p'}^2+3(p-p')^2p'\right]+5E_{p'}^6+3E_{p'}^4\left(3p^2-5p'^2\right)\nonumber\\
&&-3E_{p'}^2(p-p')^2\left(3p^2-10p'p-5p'^2\right)-5(p-p')^4\left(p^2+4p'p+p'^2\right)\Bigr\}.\end{aligned}$$
Examples {#sec:exam}
========
In sections above, we have derived the kinematical threshold and “decay width” of the bremsstrahlung-like process for general dispersion relations of neutrino. This was done under the assumptions made in section \[sec:assum\]. To check our main results and , we will apply them to muon decay process in subsection \[subsec:muon\] and to Cohen-Glashow model in subsection \[subsec:cg\]. As further applications, we will use them to study some other models in subsections \[subsec:lw\], \[subsec:step\], \[subsec:hl\].
Muon decay {#subsec:muon}
----------
Process can be regarded as a three-body decay process by weak interaction. It is analogous to muon decay in standard model. So the basic test of our result is comparison with $\mu(p)\rightarrow\nu_{\mu}(p')+\bar{\nu}_{e}(k')+e(k)$ by neglecting particle masses in the final states and replacing neutrino with muon in the initial state. Setting $E_{p'}=p'$, $E_{p}^2=p^2+m_{\mu}^2$, we work out directly $$\Gamma=\frac{G_{F}^{2}m_{\mu}^6}{192\pi^3E_{p}}\left(\frac{1}{4}-\sin^{2}\theta_W+2\sin^{4}\theta_W\right).$$ It is different from the muon decay width by a factor $(1/4-\sin^{2}\theta_W+2\sin^{4}\theta_W)$. This factor should be replaced by $1$ if we incorporate the charged current. So our result exactly passes the muon decay test.
Cohen-Glashow model {#subsec:cg}
-------------------
The second test is to recover the result in [@Cohen:2011hx]. For this purpose, we set $E_{p}^2=p^2(1+\delta)$, $E_{p'}^2=p'^2(1+\delta)$ in and get $$\label{cg}
\Gamma=\frac{4}{7}\times\frac{G_{F}^{2}E_{p}^5\delta^3}{192\pi^3}\left(\frac{1}{4}-\sin^{2}\theta_W+2\sin^{4}\theta_W\right).$$ The decay width of [@Cohen:2011hx] can be numerically reproduced[^5] by taking $\sin^{2}\theta_W\simeq1/4$.
Another main result of this paper is the threshold . Applying this threshold condition to the model of [@Cohen:2011hx], we find $$E_{\th}=\frac{2m_{e}+m_{\nu}}{\sqrt{1-\frac{1}{1+\delta}}}\simeq\frac{2m_{e}}{\sqrt{\delta}},$$ the same as the threshold in [@Cohen:2011hx].
Mass-dependent Lorentz violation {#subsec:lw}
--------------------------------
In ref. [@Li:2011ue], two of the authors proposed the mass-dependent Lorentz violation scenario to explain the observed neutrino velocity as a function of energy. In this scenario, the mass-energy relation of neutrino has the form $$\label{m-E}
1-v^2=\lambda-f(\lambda),\quad\lambda=m^2/E^2$$ where $f(\lambda)$ is a model-dependent function. The new function $f(\lambda)$ is useful phenomenologically, because we can get its information directly from experiments which constrain neutrino velocity as a function of energy. With the definition of group velocity $v=dE/dp$, relation can be taken as a differential equation and integrated into dispersion relation $$p=\int_{m}^{E}\frac{d\tilde{E}}{\sqrt{1-\frac{m^2}{\tilde{E}^2}+f\left(\frac{m^2}{\tilde{E}^2}\right)}}.$$ A concrete toy model of mass-dependent Lorentz violation was devised in [@Li:2011ue], well fitting observational data of neutrino velocity. For the toy model and parameters given in [@Li:2011ue], we combine this relation with eq. , and numerically get the threshold energy at about $0.5\GeV$. This is illustrated in figure \[fig:lw\]. This threshold is higher than that of the Cohen-Glashow model [@Cohen:2011hx].
![(color online). Numerical solution of the threshold for a toy model of mass-dependent Lorentz violation. The solid blue line depicts function $2m_e(1-p^2/E_{p}^2)^{-1/2}$. It crosses the dashed black line at the threshold energy, as highlighted by a black dot in the right graph. The electron/positron mass is set to $0.5\MeV$. The toy model and other parameter values are chosen to be the same as in [@Li:2011ue].[]{data-label="fig:lw"}](Ethlog.eps "fig:"){width="45.00000%"} ![(color online). Numerical solution of the threshold for a toy model of mass-dependent Lorentz violation. The solid blue line depicts function $2m_e(1-p^2/E_{p}^2)^{-1/2}$. It crosses the dashed black line at the threshold energy, as highlighted by a black dot in the right graph. The electron/positron mass is set to $0.5\MeV$. The toy model and other parameter values are chosen to be the same as in [@Li:2011ue].[]{data-label="fig:lw"}](Ethlin.eps "fig:"){width="45.00000%"}
Because the dispersion relation takes a very complicated form, it is difficult to work out the decay width in this toy model, even numerically. As an alternative, we will deal with a simplified model in subsection \[subsec:step\].
Velocity of step form {#subsec:step}
---------------------
In the toy model of ref. [@Li:2011ue], the dependence of velocity on energy looks like a delta function, and well explains the observational data of neutrino velocity. But it is very difficult to work out the decay width for that model. As an alternative, let us study a model in which the neutrino’s velocity depends on energy in a step form $$\label{step}
dp/dE=1+\left(\frac{1}{\sqrt{1+\delta}}-1\right)H(E-E_{c}).$$ Here $H(x)$ is a Heaviside unit step function and $E_{c}$ is an critical energy.
Fixing $\delta=2\times10^{-5}$, we numerically computed the decay width and got the results in figure \[fig:step\]. From the figure we can see the value of decay width becomes closer and closer to as $E$ increases. The figure also tells us that the decay width gets larger if neutrino is faster than light in a wider energy range. Reversing the logic, the decay width can be suppressed by narrowing the energy range in which the neutrino is faster than light. Perhaps this is realizable torturously if the dependence of neutrino velocity on energy takes a comb-like form.
![(color online). The dependence of decay width on neutrino energy $E_{p}$ in model . We have fixed $\delta=2\times10^{-5}$, $E_{c}=5\GeV$ for the blue points and $E_{c}=17\GeV$ for the red points.[]{data-label="fig:step"}](step.eps){width="45.00000%"}
Horava-Lifshitz model {#subsec:hl}
---------------------
Motivated by Horava-Lifshitz theories, ref. [@Mohanty:2011rm] has studied the dispersion relation for neutrinos of the form $$\label{hl0}
E^2=p^2+m^2+\eta'p^2+\frac{\eta p^4}{M^2}.$$ Here the mass of neutrino $m$ is negligible. When the $\eta'$ correction dominates, this model reduces to the Cohen-Glashow model.
When the $\eta$ correction dominates, dispersion relation becomes $$\label{hl2}
E^2=p^2+\frac{\eta p^4}{M^2}$$ with energy-dependent velocity. For this form of dispersion relation, we can calculate the decay width with . To the leading order of $\eta$, it is
$$\label{hl}
\Gamma=\frac{1665}{2002}\times\left(\frac{E_{p}^2}{M^2}\right)^3\times\frac{G_{F}^{2}E_{p}^5\eta^3}{192\pi^3}\left(\frac{1}{4}-\sin^{2}\theta_W+2\sin^{4}\theta_W\right).$$
Remembering that the neutrino velocity is $v^2-1\simeq3\eta
p^2/M^2$, it is convenient to take the notation $\delta=3\eta
E_{p}^2/M^2$ and rewrite decay width as $$\label{hl}
\Gamma=\frac{185}{6006}\times\frac{G_{F}^{2}E_{p}^5\delta^3}{192\pi^3}\left(\frac{1}{4}-\sin^{2}\theta_W+2\sin^{4}\theta_W\right).$$ At energies relevant to the OPERA experiment, such a decay width is one or two orders smaller than that of the Cohen-Glashow model. This ratio of suppression is consistent with the results of ref. [@Mohanty:2011rm]. A naive application of to yields $\eta p_{\th}^4/M^2=(2m_{e}+m_{\nu})^2$, which gives the threshold energy in leading order $$E_{\th}\simeq\frac{\sqrt{2m_{e}M}}{\eta^{1/4}}.$$ It is unsafe to take $E_{\th}\simeq2m_{e}\sqrt{3/\delta}$, because here $\delta$ is energy-dependent.
Conclusion {#sec:con}
==========
In this paper, under the assumptions enumerated in section \[sec:assum\], we studied the kinematic threshold and decay width of superluminal neutrinos for the bremsstrahlung-like process . This was done for general dispersion relations of neutrino, without resorting to any nontrivial frame such as the effective “rest frame”. The main results are represented by eqs. and . Our results confirmed and generalized the previous results in [@Cohen:2011hx; @Bi:2011nd].
Before concluding this paper, we would like to make some relevant remarks on the assumption of squared amplitude, which leaves a loose end for future investigation. As has been emphasized in section \[sec:assum\], when calculating the decay width, we assumed that the squared amplitude is the same as that in standard model. Strictly speaking, this is not always a consistent assumption when general dispersion relations are involved. In general, dispersion relations will enter into both kinematics and dynamics of particle physics. However, without an assumption on the squared amplitude, we cannot do any calculation about decay width. At the same time, since the deviation of neutrino dispersion relation is not too far from special relativity, we expect that the deformed amplitude should not deviate significantly from the standard model. Therefore, our assumption is not only necessary but also natural to some extent. We thus expect our result provides a good estimation of decay width in order of magnitude. Of course, further investigation is required to confirm this expectation and improve the present situation. See ref. [@Bezrukov:2011qn] for a recent progress along this direction.
Another interesting project is employing our general results to rule out more phenomenological models and hunt for viable models, as shown by some examples in section \[sec:exam\]. We feel this project will be challenging but rewarding, given the importance of special relativity in modern physics.
[99]{} OPERA, arXiv:1109.4897 \[hep-ex\]. C. Pfeifer and M. N. R. Wohlfarth, arXiv:1109.6005 \[gr-qc\]. J. Alexandre, J. Ellis and N. E. Mavromatos, Phys. Lett. B [**706**]{}, 456 (2012) \[arXiv:1109.6296 \[hep-ph\]\]. P. Wang, H. Wu and H. Yang, arXiv:1109.6930 \[hep-ph\]. J. Franklin, arXiv:1110.0234 \[physics.gen-ph\]. N. D. H. Dass, arXiv:1110.0351 \[hep-ph\]. W. Winter, Phys. Rev. D [**85**]{}, 017301 (2012) \[arXiv:1110.0424 \[hep-ph\]\]. P. Wang, H. Wu and H. Yang, arXiv:1110.0449 \[hep-ph\]. T. Li and D. V. Nanopoulos, arXiv:1110.0451 \[hep-ph\]. I. Y. Aref’eva and I. V. Volovich, arXiv:1110.0456 \[hep-ph\]. E. N. Saridakis, arXiv:1110.0697 \[gr-qc\]. S. Nojiri and S. D. Odintsov, Eur. Phys. J. C [**71**]{}, 1801 (2011) \[arXiv:1110.0889 \[hep-ph\]\]. C. Y. Zhu, H. Fan and S. P. Ding, arXiv:1110.1943 \[hep-ph\]. T. Li and D. V. Nanopoulos, arXiv:1110.3451 \[hep-ph\]. N. Qin and B. Q. Ma, arXiv:1110.4443 \[hep-ph\]. J. Bramante, arXiv:1110.4871 \[hep-ph\]. L. A. Zhao and X. Zhang, arXiv:1110.6577 \[hep-ph\]. Z. Chang, X. Li and S. Wang, arXiv:1110.6673 \[hep-ph\]. M. Matone, arXiv:1111.0270 \[hep-ph\]. J. Evslin, arXiv:1111.0733 \[hep-ph\]. A. G. Cohen and S. L. Glashow, Phys. Rev. Lett. [**107**]{}, 181803 (2011) \[arXiv:1109.6562 \[hep-ph\]\]. X. J. Bi, P. F. Yin, Z. H. Yu and Q. Yuan, Phys. Rev. Lett. [**107**]{}, 241802 (2011) \[arXiv:1109.6667 \[hep-ph\]\]. M. Li and T. Wang, arXiv:1109.5924 \[hep-ph\]. G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman and L. Smolin, arXiv:1110.0521 \[hep-ph\]. F. L. Villante and F. Vissani, arXiv:1110.4591 \[hep-ph\]. S. Mohanty and S. Rao, arXiv:1111.2725 \[hep-ph\]. F. Bezrukov and H. M. Lee, arXiv:1112.1299 \[hep-ph\].
[^1]: As a partial list, see [@Pfeifer:2011ve; @Alexandre:2011bu; @Wang:2011sz; @Franklin:2011ws; @Dass:2011yj; @Winter:2011zf; @Wang:2011zk; @Li:2011zm; @Aref'eva:2011zp; @Saridakis:2011eq; @Nojiri:2011ju; @Zhu:2011fx; @Li:2011rt; @Qin:2011md; @Bramante:2011uu; @Zhao:2011sb; @Chang:2011td; @Matone:2011fn; @Evslin:2011vq] and references therein for speculation on this issue from various aspects.
[^2]: When our work was in preparation, ref. [@Villante:2011pk] appeared. The topics in this section and [@Villante:2011pk] overlap partly. For the completeness of our paper, we keep this section in its own form. Ref. [@Mohanty:2011rm], starting form different assumptions, also concerns a partly overlapped subject of this paper. It appeared more recently when we were polishing our work.
[^3]: This method is valid if $E_{p}$ has local minima as an implicit function of $E_{p'}$ and $E_{k}$. The situation will be more complicated if the configuration of does not have a local minimum.
[^4]: In the first version of our manuscript, the squared amplitude is incomplete. Here we corrected this error and included the last term in . The difference does not affect most of our calculations. It only modifies an overall factor in the decay width. We are grateful to Zhaohuan Yu, Fedor Bezrukov and Evslin Jarah for communications on this point.
[^5]: This corrects the wrong claim in the first version of our manuscript, because the modification of squared amplitude changes the overall factor in the decay width.
| {
"pile_set_name": "ArXiv"
} |
---
bibliography:
- 'Refs.bib'
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=15.5pt
[**Vacuum Decay in CFT**]{}5 pt [**and the Riemann-Hilbert problem**]{}5 pt
[Guilherme L. Pimentel$^{\bigstar}$, Alexander M. Polyakov$^{\blacklozenge}$ and Grigory M. Tarnopolsky$^{\blacklozenge}$]{}
*$^\bigstar$ Department of Applied Mathematics and Theoretical Physics, Cambridge University* *Cambridge, CB3 0WA, UK*
*$^\blacklozenge$ Joseph Henry Laboratories, Princeton University* *Princeton, NJ 08544, USA*
------------------------------------------------------------------------
[**Abstract**]{}\
We study vacuum stability in $1+1$ dimensional Conformal Field Theories with external background fields. We show that the vacuum decay rate is given by a non-local two-form. This two-form is a boundary term that must be added to the effective in/out Lagrangian. The two-form is expressed in terms of a Riemann-Hilbert decomposition for background gauge fields, and its novel “functional” version in the gravitational case.
------------------------------------------------------------------------
Introduction
============
In this article we discuss vacuum decay in $1+1$ dimensional Conformal Field Theories with external fixed background fields. As an example, we consider a theory of massless fermions in $1+1$ dimensions coupled to Abelian, non-Abelian or gravitational background fields. The computation of the vacuum decay rate involves evaluating the effective action, which is given by the logarithm of the determinant of the quantum fields in the fixed background. The pioneer example, due to Schwinger [@Schwinger:1951nm], is of fermions in a constant background electric field. The example we study in our paper is interesting, as we can find formulas for vacuum decay in generic field profiles (which satisfy a few technical assumptions that we state below). Some exact results for generic field profiles were also obtained in [@Tomaras:2000ag; @Tomaras:2001vs], in $1+1$ dimensional QED.
Let us briefly review a case with no particle production. Consider free massless fermions interacting with a fixed non-Abelian gauge field background. The effective action is obtained by the Gaussian integration over the fermion fields, and is given by a one loop determinant. If the field profile satisfies a “good" behavior, that we specify later, the effective action is real and is expressed [@Polyakov:1984et] in terms of the Wess-Zumino-Novikov-Witten (WZNW) action [@Wess:1971yu; @Novikov:1982ei; @Witten:1983tw]. In this case particles are not created, since the vacuum decay rate is nonzero only when the effective action has an imaginary part.
Our goal is to determine the effective action for background fields that do lead to particle production. In this case, we have to discuss the in/out effective action which has an imaginary part, reflecting vacuum decay. The imaginary piece in the effective action is determined by a careful treatment of the Feynman $i\varepsilon$ prescription in a massless theory.
Our main result is that the effective action is modified by the inclusion of extra boundary terms, which are complex, and whose imaginary part gives the vacuum decay rate. The boundary term is a two-form which appears to be novel. To compute the boundary terms we need a certain Riemann-Hilbert decomposition. While the Abelian and non-Abelian decompositions are standard Riemann-Hilbert problems, the gravitational case has not been considered before. The vacuum decay rate for Abelian background fields is given by the same formula of dissipative quantum mechanics obtained by Caldeira and Leggett [@Caldeira:1981rx; @Caldeira:1982uj]. Our results generalize their formulas for non-Abelian and gravitational backgrounds.
The rest of the paper is organized as follows. In section \[Absec\] we compute the effective action and the new boundary term for an Abelian gauge field and discuss the general logic of the computation, which helps in the more complicated cases. In section \[nonAbsec\] we find the effective action and the new boundary term for the non-Abelian gauge field. Finally, in section \[gravsec\] we find the effective action and the new boundary term in the case of the gravitational field. In appendix \[bdyapp\], we discuss an alternative method of computation of the boundary terms. In appendix \[app:gauge-gravity\], we review the gauge-gravity duality between the non-Abelian and gravitational cases [@Alekseev:1988ce; @Bershadsky:1989mf; @Polyakov:1989dm]. Finally, in appendix \[CaldLeg\], we show the first perturbative correction to the Caldeira-Leggett formula coming from non-Abelian and gravitational backgrounds.
Vacuum decay in an Abelian background {#Absec}
=====================================
To set the stage, let us look at the Abelian case first. The Lagrangian is $$\begin{aligned}
\mathcal{L} = \bar{\psi} \gamma^{\mu} (i\partial_{\mu}+ A_{\mu})\psi\,=\bar{\psi}_-(i\partial_+ +A_+)\psi_- + \bar{\psi}_+ (i\partial_- +A_-)\psi_+ ,\end{aligned}$$ where the metric is $\eta^{\mu\nu}=(1,-1)$, and we introduced light cone coordinates $x^{\pm}=(x^{0}\pm x^{1})/\sqrt{2}$. From the Lagrangian it is clear that the left movers $\psi_{+}$ and right movers $\psi_{-}$ are sourced by $A_-$ and $A_+$ fields, respectively. Therefore, the determinant will split into a right-moving piece, a left-moving piece, and a contact term that ensures gauge invariance [@Polyakov:1984et][^1] $$\begin{aligned}
\label{AbCompForm}
S(A_+,A_-) = \log \det (\gamma^{\mu}(i\partial_{\mu}+A_{\mu}))\,=W_+(A_+)+W_-(A_-)-2\int d^{2}x A_+A_-\, .\end{aligned}$$ The contact term comes from short distance cutoff regulators; it is not related to particle production. In the case of strong fields which lead to particle production, $W_{+}$ and $W_{-}$ have imaginary parts. The vacuum decay rate factorizes and is given by $$\begin{aligned}
|_{\textrm{out}}\langle 0 |0\rangle_{\textrm{in}}|^{2} = e^{-2 \textrm{Im}\, S(A)}=e^{-2\textrm{Im}W_+}e^{-2\textrm{Im}W_-}\,.\end{aligned}$$ Let us compute the contribution to the effective action coming from $A_+$. We will treat $x^{+}$ as a time coordinate, while in the $x^{-}$ direction we assume that $A_+(x^{+},x^{-})\to0$ as $x^-\to\pm\infty$. An easy calculation of the diagram
{width="6cm"}
leads to ($d^{2}p=dp_{+}dp_{-}$) $$\begin{aligned}
\label{eamom}
W_+(A_{+}) =\int \frac{d^{2}p}{(2\pi)^{2}}\, \frac{p_{-}}{p_{+}+i\varepsilon\, \textrm{sgn}\,p_{-}} A_{+}(p)A_{+}(-p)\,.\end{aligned}$$ As is well known, this result is exact and higher order corrections in $A_{+}$ are zero. The “$i\varepsilon$" prescription follows from the Feynman rule $\frac{1}{p^{2}} \Rightarrow \frac{1}{p^{2}+i\varepsilon}=\frac{1}{p_{-}}\big(\frac{1}{p_{+}+i\varepsilon\, \textrm{sgn}\,p_{-}}\big)$. The term in parenthesis is the Feynman Green’s function. We see that[^2] $$\begin{aligned}
\textrm{Im} \,W_+(A_{+}) =-\int \frac{d^{2}p}{4\pi} |p_{-}| \delta(p_{+})A_{+}(p)A_{+}(-p) =-\frac{1}{4\pi}\int dp_{-}|p_{-}| A_{+}(0,p_{-})A_{+}(0,-p_{-})\,. \label{EffInPspace}\end{aligned}$$ The condition of vacuum stability ($\textrm{Im}\,W_+=0$) is thus $\int_{-\infty}^{+\infty} A_{+}(y^{+},x^{-})dy^{+}=0$. It is useful to rewrite the formula (\[EffInPspace\]) in position space. If we denote $$\begin{aligned}
\omega(x^{-})\equiv \int_{-\infty}^{+\infty}A_{+}(y^{+},x^{-})dy^{+}\,, \label{abomega}\end{aligned}$$ from (\[EffInPspace\]) we obtain[^3] $$\begin{aligned}
\textrm{Im}\, W_+(A_{+}) = \frac{1}{4\pi}\int_{-\infty}^{+\infty} dx^{-}dy^{-} \frac{(\omega(x^{-})-\omega(y^{-}))^{2}}{(x^{-}-y^{-})^{2}}\,. \label{imw1}\end{aligned}$$ We recognize this formula as the friction term in Caldeira-Leggett’s dissipative quantum mechanics [@Caldeira:1981rx; @Caldeira:1982uj]. Below we will find the non-Abelian and gravitational generalizations of this action.
It is instructive to rewrite (\[imw1\]) in a slightly different form. Let us introduce two complex functions $\omega_{\textrm{up}}(x^{-})$ and $\omega_{\textrm{down}}(x^{-})$, which are analytic in the upper and lower half-planes, respectively. They are related to $\omega(x^{-})$ as $$\begin{aligned}
\label{RHScalar}
\omega_{\textrm{up}}(x^{-})-\omega_{\textrm{down}}(x^{-})=\omega(x^{-})\end{aligned}$$ for real $x^-$. This decomposition of the function $\omega(x^-)$ is called the [*scalar Riemann-Hilbert problem*]{} and the explicit solution in this case is given by $$\begin{aligned}
\omega_{\textrm{up}/\textrm{down}}(x^{-}) = \frac{1}{2\pi i}\int_{-\infty}^{+\infty} \frac{\omega(y^{-})dy^{-}}{y^{-}-x^{-}\mp i \varepsilon}\,. \label{omupdown}\end{aligned}$$ In terms of $\omega_{\textrm {up/down}}$, the imaginary part of the effective action can be written as $$\begin{aligned}
\label{EffAcRH}
\boxed{\textrm{Im}\, W_+(A_{+}) = \textrm{Im}\int_{-\infty}^{+\infty}dx^{-} \left(\omega_{\textrm{down}}\partial_- \omega_{\textrm{up}} \right)}\,.\end{aligned}$$ The generalization of the formula (\[EffAcRH\]) for the strong non-Abelian and gravitational cases is the main goal of this paper.
There is yet another way of obtaining (\[EffAcRH\]), which will be useful below. We can parametrize $A_{+}$ as $$\begin{aligned}
A_+(x^+,x^-)=\partial_+\phi(x^+,x^-)\,\end{aligned}$$ and we notice that the “Wilson line" $\phi(x^+,x^-)$ has residual gauge invariance $\phi \to \phi + u(x^-)$. We say that $\phi =\phi_R(x^+,x^-)$ is in [*retarded*]{} gauge if it obeys the boundary condition $\phi_{R}(-\infty,x^-)\to0$ and therefore $$\begin{aligned}
\phi_{R}(x^+,x^-)= \int_{-\infty}^{x^+}A_+(y^{+},x^-)dy^{+} \,.\end{aligned}$$ We see that $\phi_R$ is manifestly real and causal, as $\phi_R(x^+,x^-)$ only depends on $A_+(y^+,x^-)$ for $y^+<x^+$; moreover, $\phi_R(+\infty,x^-)=\omega(x^-)$, so the imaginary part of the effective action (\[EffAcRH\]) is written in terms of the boundary value of $\phi_R$ and the whole $W_{+}(A_{+})$ reads $$\begin{aligned}
W_{+}(A_{+}) = \int d^{2}x \, \partial_{+}\phi_{R}\partial_{-}\phi_{R}+\int_{-\infty}^{+\infty}dx^{-} \left(\, \omega_{\textrm{down}}\partial_- \omega_{\textrm{up}} \right)\,. \label{abeffret}\end{aligned}$$ We can use the solution of the Riemann-Hilbert problem (\[RHScalar\]) and the residual gauge invariance of $\phi$ to define a [*spectral*]{} (or Feynman) gauge, namely $$\begin{aligned}
\label{spec gauge}
\phi_S(x^+,x^-)\equiv\phi_R(x^+,x^-)-\omega_{\textrm{down}}(x^-)\to\begin{cases} \omega_{\textrm{up}}(x^-), &x^+\to+\infty\\ -\omega_{\textrm{down}}(x^-), &x^+\to-\infty\end{cases}.\end{aligned}$$ In the spectral gauge the effective action (\[abeffret\]) reads $$\begin{aligned}
W_{+}(A_{+}) = \int d^{2}x \,\, \partial_{+}\phi_{S}\partial_{-}\phi_{S}\,, \label{abeffspec}\end{aligned}$$ and has the form of the usual result [@Schwinger:1962tp]. In our case, the difference is that the function $\phi_{S}$ is complex valued and (\[abeffspec\]) contains both real and imaginary parts of the effective action! The conclusion is that in the [*spectral*]{} gauge, we do not require boundary terms in the effective action, whereas in the [*retarded*]{} gauge, we have boundary terms, which are complex and account for the vacuum decay.
The logic is summarized as follows. If we use the [*spectral*]{} gauge, then the expressions for the effective actions are well known [@Schwinger:1962tp; @Polyakov:1984et; @Polyakov:1987zb], as the boundary terms evaluate to zero. Then passing from the [*spectral*]{} gauge to [*retarded*]{} gauge we determine the functional form of the boundary terms. In the retarded gauge, the boundary term contains the imaginary part of the effective action. In Appendix \[bdyapp\] we discuss an alternative method to compute the full effective action, by exploiting (chiral or trace) anomaly equations.
Vacuum decay in a non-Abelian background {#nonAbsec}
========================================
In the non-Abelian case the general form of the effective action reads $$\begin{aligned}
\label{nAbCompForm}
S(A_+,A_-)= \log \det (\gamma^{\mu}(i\partial_{\mu}+A_{\mu}))=W_+(A_+)+W_-(A_-)+2\int\, d^{2}x\, \Tr (A_+A_-)\,\end{aligned}$$ and imaginary terms responsible for the particle production are present only in $W_{+}$ and $W_{-}$. We concentrate again on $W_{+}(A_{+})$, which is formally given by the following sum of Feynman diagrams
{width="9.5cm"}
If we parametrize $A_{+}=g^{-1}\partial_{+}g$ we get $W_{+}(g)$. If $g(x^{+}\to \pm \infty,x^{-})=\mathbbm{1}$ then $W_{+}(g)$ is the WZNW action [@Polyakov:1983tt] $$\begin{aligned}
\label{wznwact}
W_{\rm WZNW}(g)\equiv {1 \over 2} \int \dd^{2}x\, \Tr (\partial^{\mu}g^{-1}\partial_{\mu}g) - {1 \over 3}\int \dd^{2}x dt \,\varepsilon^{\mu\nu\lambda} \Tr (g^{-1}\partial_{\mu}{g} g^{-1}\partial_{\nu}g g^{-1}\partial_{\lambda}g)\,,\end{aligned}$$ where in the last Wess-Zumino (WZ) term we introduced the extra $t$-dependence: $g(x^{+},x^{-},t)$ such that $g(x^{+},x^{-},0)=\mathbbm{1}$ and $g(x^{+},x^{-},1)= g(x^{+},x^{-})$; and $\mu,\nu,\lambda = (\pm,0)$, and $\varepsilon^{0-+}=1$, where zero corresponds to the $t$ coordinate.
From the Abelian case, we expect that vacuum decay occurs for $A_{+}$ with $g^{-1}(-\infty,x^{-})$ $\cdot g(+\infty,x^{-})\neq \mathbbm{1}$, or, in different notation: $$\begin{aligned}
\Omega(x^{-}) \equiv P \exp \int_{-\infty}^{\infty} A_{+}(y^{+},x^{-}) dy^{+}\neq \mathbbm{1},\end{aligned}$$ where “$P \exp$" is the path-ordered exponential. In this case the effective action is not given by (\[wznwact\]); it must include new boundary terms. Indeed, looking at the variation of the WZ term $$\begin{aligned}
S_{\textrm{WZ}} \equiv \int d^{2}x dt \, \varepsilon^{\mu\nu\lambda} \textrm{Tr}(a_{\mu}a_{\nu}a_{\lambda})\,, \end{aligned}$$ where $a_{\mu}\equiv g^{-1} \partial_{\mu}g$, with $\delta a_{\mu}= \nabla_{\mu}\varepsilon$, we obtain $$\begin{aligned}
\delta S_{\textrm{WZ}} &= \int d^{2}x dt \, \varepsilon^{\mu\nu\lambda} \textrm{Tr}(a_{\mu}a_{\nu}\nabla_{\lambda}\varepsilon) \sim \int d^{2}x dt \, \varepsilon^{\mu\nu\lambda} \nabla_{\lambda}\textrm{Tr}\big( (\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu})\varepsilon\big) \notag\\
&=\int d^{2}x dt \,\varepsilon^{ij}\,\partial_{0} \textrm{Tr}(\partial_{i}a_{j} \varepsilon) - \int d^{2}x dt \, \partial_{+} \textrm{Tr} \big((\partial_{-}a_{0}-\partial_{0}a_{-})\varepsilon\big) \notag\\
&= \int d^{2}x \, \varepsilon^{ij} \textrm{Tr} (\partial_{i}a_{j}\varepsilon) -\int dx^{-} dt\, \left.\textrm{Tr}\big((\partial_{-}a_{0}-\partial_{0}a_{-})\varepsilon\big)\right|^{x^{+}=+\infty}_{x^{+}=-\infty}\,. \label{varWZ}\end{aligned}$$ The first term here is standard while the time-boundary term explicitly violates $t$-symmetry. In other words, $S_{\textrm{WZ}}$ is dependent on the $t$-parametrization. This unphysical dependence on the extrapolation disappears when the right boundary terms are added to WZNW action.
Notice that the matrix $g$ has a gauge symmetry $$\begin{aligned}
g(x^{+},x^{-})\to u(x^{-})g(x^{+},x^{-}) \,,\end{aligned}$$ where $u(x^{-})$ is an arbitrary complex matrix. The retarded gauge is defined by $$\begin{aligned}
g_R(x^{+} \to -\infty, x^{-})= \mathbbm{1}\ \Rightarrow g_{R}(x^{+},x^{-}) = P \exp \int_{-\infty}^{x^{+}} dy^{+} A_{+}(y^{+},x^{-})\, .\end{aligned}$$ Like in the Abelian case, we see that $\Omega(x^-)=g_R(+\infty,x^-)$.
Proceeding by analogy, we should look for complex valued matrices $\Omega_{\textrm{down}}(x^{-})$ and $\Omega_{\textrm{up}}(x^{-})$ that are a solution to the [*matrix Riemann-Hilbert problem*]{} $$\begin{aligned}
\label{matriemhilb}
\Omega_{\textrm{down}}(x^{-}) \Omega_{\textrm{up}}(x^{-})=\Omega(x^{-}) \,, \end{aligned}$$ for real values of $x^-$. We assume that $\Omega^{-1}_{\textrm{up}}(x^{-})$ and $\Omega^{-1}_{\textrm{down}}(x^{-})$ are also analytic in the upper and lower half-planes, respectively. Unfortunately, the matrix Riemann-Hilbert problem does not have an explicit general solution.[^4]
As we see from (\[varWZ\]) the retarded gauge choice requires extra terms in the WZ term in order to cancel the unacceptable boundary contributions. However, we can use our gauge freedom in choosing $g$ to eliminate the boundary terms. Let us introduce the spectral (or Feynman) gauge: $$\begin{aligned}
g_{S}(x^{+},x^{-}) \equiv \Omega^{-1}_{\textrm{down}}(x^{-})\ g_{R}(x^{+},x^{-}) \to\begin{cases} \Omega_{\textrm{up}}(x^-),&x^+\to+\infty\\ \Omega^{-1}_{\textrm{down}}(x^-),&x^+\to-\infty \end{cases}. \label{feyngauge}\end{aligned}$$ It follows from here that $g_{S}(x^{+},x^{-})$ at $x^{+}\to \pm \infty$ is analytic in the lower/upper half-planes and thus all boundary terms vanish after $x^{-}$ integration. By analogy with the Abelian case, we come to the conclusion that in the spectral gauge there are no boundary terms! The effective action is just the standard WZNW action (\[wznwact\]), which is complex valued, as $g_S$ is complex $$\begin{aligned}
\label{easg}
W_{+}(A_{+})= W_{\rm WZNW}(g_{S})\,.\end{aligned}$$ A more physical justification of the absence of boundary terms in the spectral gauge is discussed in the Appendix \[bdyapp\].
From (\[easg\]), we now determine the boundary term that must be present in the effective action written in an arbitrary gauge. For example, in going from spectral to retarded gauge, we do not change $A_{+}=g^{-1}\partial_{+}g$, therefore the effective actions must be the same, \[releffa\] W\_[+]{}(A\_[+]{})=W\_[WZNW]{}(\^[-1]{}\_ g\_[R]{})=W\_[WZNW]{}(g\_[R]{})+W\_B(\_[up]{},\_[down]{}).
In order to proceed we use exterior calculus to derive a composition formula for the WZ term. Let us introduce two $1$-forms $a$ and $b$, with $a= g^{-1} d g, b = dh h^{-1}\,$. $a$ and $b$ satisfy the equations $da = -a\wedge a$, $d b= b\wedge b\,$. Consider the $1$-form $c = (gh)^{-1}d(gh) = h^{-1}(a+b)h$. Then we have $$\begin{aligned}
\Tr (c\wedge c \wedge c)&=\Tr(a \wedge a\wedge a)+\Tr(b\wedge b\wedge b) -3d (\Tr\, a \wedge b)\,. \label{rel}\end{aligned}$$ Now we apply (\[rel\]) with $g_S=\Omega_{\rm down}^{-1}g_R$. From the quadratic term in the WZNW action we obtain[^5] \[chh\] [1 2]{} \^[2]{}x (\^g\_[S]{}\^[-1]{}\_g\_[S]{}) =\^[2]{}x (\^g\_[R]{}\^[-1]{}\_g\_[R]{})+\^[2]{}x (\_-\_[down]{} \_[down]{}\^[-1]{}\_[+]{}g\_[R]{} g\_[R]{}\^[-1]{}), and using (\[releffa\]) and (\[rel\]) for the WZ term in (\[wznwact\]) we find $$\begin{aligned}
\label{wzm1}
W_{\textrm{WZ}}(g_{S})=W_{\textrm{WZ}}(g_{R})- 3\int_{(x^{+},x^{-},t)} d\,\big( \Tr (\Omega_{\rm down} d \Omega^{-1}_{\rm down} \wedge dg_{R}g_{R}^{-1} )\big)\,. \end{aligned}$$ Notice that the Penrose diagram for our space-time with the embedding dimension is a pyramid; we call it the Penrose-Nefertiti diagram (see figure \[Nefertiti\]).
![Penrose-Nefertiti diagram. The usual Penrose diagram of $1+1$ dimensional Minkowski spacetime is the base of a pyramid. The embedding coordinate $t$ runs from the apex ($t=0$) to the base ($t=1$). The new boundary terms in the effective action are supported at the $t$ - $x$ faces of the pyramid.[]{data-label="Nefertiti"}](PenroseDiag.pdf){width="3.9cm"}
The first term in (\[wzm1\]) is real (we assume $A_+$ is real), while the boundary term, which has support at the faces of the pyramid, is complex valued. Using Stokes’ theorem in (\[wzm1\]), we obtain $$\begin{aligned}
W_{\textrm{WZ}}(g_{S})=W_{\textrm{WZ}}(g_{R})- 3\int d^{2}x \Tr (\Omega_{\rm down} \partial_{-} \Omega^{-1}_{\rm down} \partial_{+}g_{R}g_{R}^{-1} )+ 3\int\limits_{(x^{-},t)} \Tr (\Omega_{\rm down}^{-1} d \Omega_{\rm down} \wedge \Omega_{\rm up} d\Omega_{\rm up}^{-1})\,. \label{wzbound}\end{aligned}$$ Using (\[releffa\]), (\[chh\]) and (\[wzbound\]) we finally obtain $$\begin{aligned}
\boxed{W_{B}(\Omega_{\rm up},\Omega_{\rm down})= \int\limits_{(x^{-},t)} \Tr (\Omega_{\rm down}^{-1} d \Omega_{\rm down} \wedge \Omega_{\rm up}d\Omega_{\rm up}^{-1})\,}\,\,. \label{nonAbBound}\end{aligned}$$ The formula (\[nonAbBound\]) is one of the main results of our paper.[^6] The boundary term is complex valued, and although not manifestly imaginary, contains the imaginary part of the effective action[^7].
We emphasize that this boundary term is the non-Abelian generalization of the Caldeira-Leggett dissipative term, and is given by a two-form[^8]. We also notice that our two-form is a Minkowski space counterpart of the Atiyah-Patodi-Singer $\eta$-invariant [@Atiyah:1975jf; @Atiyah:1976jg; @Atiyah:1980jh], which appears in Euclidean manifolds with boundary. We present the leading order, non-Abelian correction to the Caldeira-Leggett formula in Appendix \[CaldLeg\].
Vacuum decay in the gravitational field {#gravsec}
=======================================
Now we consider a theory of fermions coupled to a fixed gravitational field. It is convenient to parametrize the metric in the light cone coordinates, \[mlcg\] ds\^2=h\_[+-]{}(x\^+,x\^-)dx\^+dx\^-+h\_[++]{}(x\^+,x\^-)dx\^+dx\^++h\_[–]{}(x\^+,x\^-)dx\^-dx\^- . We assume that the background fields are asymptotically flat, i.e. $h_{++}(x^{+},x^{-})\to0$, $h_{--}(x^{+},x^{-})\to0$ and $h_{+-}(x^+,x^-)\to 1$ as $x^\pm\to\pm\infty$. The Lagrangian is[^9] \[lgra\] [L]{} = \_-(\_+-h\_[++]{}\_-)\_-+\_+(\_- -h\_[–]{}\_+)\_+. Like in the non-Abelian case, the effective action is $$\begin{aligned}
S(h_{++},h_{--},h_{+-})=W_{+}(h_{++})+W_{-}(h_{--})+L(h_{++},h_{--},h_{+-})\,,\end{aligned}$$ where the last term $L$ is a local and real term and appears due to the UV regulator. We concentrate on the calculation of the contribution from left-moving fermions, $W_+(h_{++})$. For gravity we use the same logic as in the non-Abelian case. We parametrize the metric tensor $h_{++}(x^{+},x^{-})$ using the function $f(x^{+},x^{-})$ defined by the equation $$\begin{aligned}
(\partial_{+}-h_{++}\partial_{-})f=0\,, \label{defoff}\end{aligned}$$ which is a gravitational analog of the Wilson line. Lines of constant $f$ correspond to the characteristics of light-like, right-moving geodesics in the background spacetime. Notice that there is an ambiguity in $f$, namely $$\begin{aligned}
f(x^{+},x^{-})\; \Rightarrow \; u(f(x^{+},x^{-}))\,, \label{gaugesymgrav}\end{aligned}$$ where $u(x^{-})$ is an arbitrary invertible complex function of one variable. The [*retarded*]{} gauge is defined by \[ordgbc\] f\_[R]{}(x\^[+]{}-,x\^[-]{})=x\^-f\_[R]{}(x\^[+]{},x\^[-]{}) P ( \_[-]{}\^[x\^[+]{}]{} dy\^[+]{}h\_[++]{}(y\^[+]{},x\^[-]{})\_[-]{}) x\^[-]{}. As in the case of gauge fields, we need to add a suitable boundary term to the effective action [@Polyakov:1987zb] $$\begin{aligned}
W_{\textrm{gWZ}}(f)\equiv \int \dd^{2}x \left({\partial_{-}^{2}f \partial_{+}\partial_{-}f \over (\partial_{-}f)^{2}}-{(\partial_{-}^{2}f)^{2}\partial_{+}f \over (\partial_{-}f)^{3}}\right)\,, \label{la}\end{aligned}$$ where gWZ stands for “gravitational Wess-Zumino" and we omit an overall normalization factor, which is $-1/48\pi$ in our case. Alternatively, we can use the gauge symmetry (\[gaugesymgrav\]) to eliminate the boundary term. Let us introduce the [*spectral*]{} gauge by \[spgg\] f\_[S]{}(x\^[+]{},x\^[-]{})\^[-1]{}\_[down]{}(f\_[R]{}(x\^[+]{},x\^[-]{}))=
\_[up]{}(x\^[-]{}), x\^[+]{}+, \^[-1]{}\_[down]{}(x\^[-]{}), x\^[+]{}-,
where $\Gamma_{\rm up}(x^{-})$ and $\Gamma_{\rm down}(x^{-})$ are analytic functions in the upper and lower $x^-$ half-plane. We also assume that the inverse functions $\Gamma_{\rm up}^{-1}(x^{-})$ and $\Gamma^{-1}_{\rm down}(x^{-})$ are analytic in the upper and lower $x^-$ half-plane respectively. In this case, to determine $\Gamma_{\rm up, down}$, we need to solve a [*“functional” Riemann-Hilbert problem*]{} [^10], \[frhp\] \_[down]{}(\_[up]{}(x\^[-]{})) =(x\^[-]{}) , where $$\begin{aligned}
\Gamma(x^{-}) \equiv P \exp \Big( \int_{-\infty}^{+\infty} dy^{+}h_{++}(y^{+},x^{-})\partial_{-}\Big) x^{-} = f_{R}(+\infty,x^{-})\,.\label{gammagrav}\end{aligned}$$ To our knowledge, the Riemann-Hilbert problem (\[frhp\]) has not been considered in the mathematics literature before. We also notice that (\[frhp\]) doesn’t have an explicit solution. [^11]
By similar arguments as in the previous sections, the effective action is $$\begin{aligned}
W_{+}(h_{++})=W_{\textrm{gWZ}}(f_{S})\,, \label{effgravspect}\end{aligned}$$ where $W_{\textrm{gWZ}}$ is given by (\[la\]). See also Appendix \[bdyapp\] for a different derivation of (\[effgravspect\]). This effective action is complex valued. In retarded and spectral gauges the metric $h_{++}(x^{+},x^{-})$ is the same, therefore we have the equality \[effrelg\] W\_[+]{}(h\_[++]{}) = W\_[gWZ]{}(\_[down]{}\^[-1]{}(f\_[R]{})) =W\_[gWZ]{}(f\_[R]{})+W\_[B]{}(\_[up]{},\_[down]{}). Using (\[la\]), (\[effrelg\]) we get \[gbac\] W\_[B]{}(\_[up]{},\_[down]{})=d\^[2]{}x \_[-]{}f\_[R]{} [(\_[down]{}\^[-1]{})” (\_[down]{}\^[-1]{})’]{}\_[-]{}([\_[+]{}f\_[R]{} \_[-]{}f\_[R]{}]{}). Now, introducing new variables $y^- = f_{R}(x^{+},x^{-})$, $y^{+}=x^{+}$ one can get[^12] $$\begin{aligned}
W_{B}(\Gamma_{\rm up},\Gamma_{\rm down})=\int dy^{-} {\partial \over \partial y^{-}} \log \big((\Gamma^{-1}_{\rm down})'(y^{-})\big) \log \big(\Gamma'(\Gamma^{-1}(y^{-}))\big). \label{gbaccc}\end{aligned}$$ Finally introducing a coordinate $s = \Gamma^{-1}(y^{-})$ and using that $(\Gamma^{-1}_{\rm down})'(\Gamma(s)) = 1/\Gamma'_{\rm down}(\Gamma_{\rm up}(s))$ and $\Gamma'(s)= \Gamma'_{\rm down}(\Gamma_{\rm up}(s))\Gamma'_{\rm up}(s)$ we obtain[^13] $$\begin{aligned}
\boxed{W_{B}(\Gamma_{\rm up},\Gamma_{\rm down})=\int \dd s \log \big( \Gamma'_{\rm down}(\Gamma_{\rm up}(s))\big){\partial \over \partial s} \log \big( \Gamma'_{\rm up}(s)\big)\,.} \label{gbacccc}\end{aligned}$$ Therefore the effective action in the retarded gauge is \[gravefa\] W\_[+]{}(h\_[++]{}) = \^[2]{}x ([\_[-]{}\^[2]{}f\_[R]{} \_[+ -]{}f\_[R]{} (\_[-]{}f\_[R]{})\^[2]{}]{}-[(\_[-]{}\^[2]{}f\_[R]{})\^[2]{}\_[+]{}f\_[R]{} (\_[-]{}f\_[R]{})\^[3]{}]{}) + s ( ’\_[down]{}(\_[up]{}(s)))[s]{} ( ’\_[up]{}(s)). The bulk term is manifestly real, while the boundary term is complex, and, in particular, contains the imaginary piece of the effective action.
In appendix \[app:gauge-gravity\], we review a connection between gauge theory and gravity in two dimensions [@Alekseev:1988ce; @Bershadsky:1989mf; @Polyakov:1989dm], and phrase in terms of a matrix Riemann-Hilbert problem, in the hope that this simple connection might be useful in finding explicit solutions of the functional Riemann-Hilbert problem.
Conclusions {#sec:conclusions}
===========
We conclude with a few open questions that we find interesting:
- We considered fixed background fields. One can also integrate over these backgrounds, in a similar fashion as in perturbative string theory. Does the boundary term play any role in that case?
- Is the functional Riemann-Hilbert problem solvable for some set of functions? Perhaps there are relevant gravitational backgrounds for which one could compute the vacuum decay rate explicitly.
- It would be quite interesting to classify the backgrounds that, although curved, keep the vacuum stable.
- The boundary terms in the non-Abelian and gravitational cases are complex valued. We tried, but could not find a compact expression for the imaginary part of the effective action. In particular, this supposed expression for the imaginary part, in the non-Abelian case, should be manifestly $t$-independent.
- If we use retarded Wilson lines, then we can write causal equations of motion for the background fields which include quantum friction. Can the quantum friction screen certain backgrounds once we solve for them dynamically?
- In the gravitational case we assume that $f_{R}(x^{+},x^{-})$ is invertible for any $x^{+}$. An interesting variation of this property is the case when $f_{R}(x^{+}=+\infty,x^{-})$ is not invertible. This loss of information may be related to the backgrounds with horizons, which have intrinsic entropy.
In summary, we are still scratching the surface in terms of potential applications of these new results.
### Acknowledgements {#acknowledgements .unnumbered}
We thank T. Banks, D. Baumann, J. Cardy, G. Dunne, R. Flauger, A. Kisil, I. Klebanov, C. Mafra, N. Nekrasov, H. Osborn, M. Rangamani, H. Reall, S. Shenker, H. Verlinde and A. Zhiboedov for helpful discussions. We also thank D. Baumann for comments on a draft. G.L.P. thanks the Aspen Center for Physics (supported in part by NSF Grant PHY10-66293) and the University of Amsterdam for their hospitality. He also thanks the KITP for hospitality during the program ‘Quantum Gravity Foundations: UV to IR’. Research at the KITP is supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. G.L.P. acknowledges support from a Starting Grant of the European Research Council (ERC STG grant 279617). The work of A.M.P. and G.M.T. was supported in part by the US NSF under Grant No. PHY-1314198.
Boundary conditions on induced currents and alternative derivation of the boundary actions {#bdyapp}
==========================================================================================
In this appendix we derive the effective action for non-Abelian and gravitational cases using the anomaly equations. We start with the non-Abelian case. We define $J_{\mu}\equiv\delta W/\delta A_{\mu}$; then the anomaly equations read [@Polyakov:1983tt][^14] $$\begin{aligned}
\begin{cases}
\partial_{\mu}J^{\mu}+[A_{\mu},J^{\mu}]=0\,, \\
\varepsilon^{\mu\nu}(\partial_{\mu}J_{\nu}+[A_{\mu},J_{\nu}])= \varepsilon^{\mu\nu} F_{\mu\nu}\,.
\end{cases}\end{aligned}$$ Working in the light-cone cone coordinates $x^{\pm}$ and choosing the axial gauge $A_{-}=0$, we get $\partial_{-}(A_{+}-J_{+})=0$ and ($\varepsilon^{-+}=1$) $$\begin{aligned}
\partial_{-}A_{+}+\partial_{+}J_{-} -[J_{-},A_{+}]=0. \label{anomeq1}\end{aligned}$$ Parametrizing $A_{+}=g^{-1}\partial_{+}g$ one can find that the general solution of (\[anomeq1\]) is $$\begin{aligned}
J_{-}=-g^{-1}\partial_{-}g-g^{-1}j_{-}g\,,\end{aligned}$$ where $j_{-}=j_{-}(x^{-})$ is, at this stage, an arbitrary complex matrix function, which depends only on $x^{-}$, and has to be fixed by additional physical arguments. On the other hand the variation of the effective action is $$\begin{aligned}
\delta W(A_{+}) = \int d^{2}x \, \Tr (J_{-}\delta A_{+})\,. \label{varofw}\end{aligned}$$ As we will see below, it is exactly the term $g^{-1}j_{-}g$ in the current $J_{-}$ which is responsible for the imaginary part of the effective action.
In order to fix $j_{-}(x^{-})$ we use the “analyticity" argument. Namely we say that the induced current ${}_{\rm out}\langle J_-(x^+,x^-) \rangle_{\rm in}$ must satisfy the analytical (spectral) boundary conditions[^15]: \[bc\] \_[out]{}J\_-(x\^+,x\^-) \_[in]{}
J\_[up]{}(x\^-),& x\^++,\
J\_[down]{}(x\^-),& x\^+ -,
where $J_{\rm up}(x^{-})$ and $J_{\rm down}(x^{-})$ are complex matrix functions analytic in the upper and lower $x^-$ half-planes correspondingly[^16].
Now we return to determining $j_-$ in the expression for the induced current. Working in the retarded gauge $g_{R}(x^{+},x^{-}) \equiv P \exp \int_{-\infty}^{x^{+}} dy^{+} A_{+}(y^{+},x^{-})$ and using (\[bc\]) one finds $$\begin{aligned}
j_{-R}(x^-)=-\partial_{-} \Omega_{\rm down} \Omega_{\rm down}^{-1}\,,\end{aligned}$$ where $\Omega_{\rm down}$ and $\Omega_{\rm up}$ are matrices analtyic in the lower and upper half-planes, and solve the matrix Riemann-Hilbert problem $$\begin{aligned}
\Omega_{\textrm{down}}(x^{-}) \Omega_{\textrm{up}}(x^{-})= P \exp \int_{-\infty}^{+\infty} dy^{+} A_{+}(y^{+},x^{-})\,.\end{aligned}$$ Correspondingly we find $J_{\rm up}(x^-)=-\Omega_{\rm up}^{-1}\partial_{-}\Omega_{\rm up}$ and $J_{\rm down}(x^-)=\partial_{-}\Omega_{\rm down}\Omega_{\rm down}^{-1}$.
Notice that in the spectral gauge (\[feyngauge\]) we have $j_{-S}(x^{-})=0$. From this it follows that, in the spectral gauge, the effective action is the WZNW action (\[easg\]), evaluated at $g_S$, and there are no boundary terms. Now, as we determined the current $$\begin{aligned}
J_{-}=-g_{R}^{-1}\partial_{-}g_{R}-g_{R}^{-1}j_{-R}g_{R}\,,\end{aligned}$$ one can check that the variation of $W_+(A_{+})$ (see (\[wznwact\]) and (\[nonAbBound\])) indeed equals to (\[varofw\]).
In the gravitational case everything is similar to the non-Abelian case. In the light-cone coordinates and the axial gauge $h_{--}=0$, the anomaly equation reads [@Polyakov:1987zb][^17] $$\begin{aligned}
(\partial_{+} -h_{++}\partial_{-}-2(\partial_{-}h_{++})) T_{--} = -2 \partial_{-}^{3}h_{++}\,. \label{gravaneq}\end{aligned}$$ Parametrizing $h_{++}$ by $f(x^{+},x^{-})$, with $(\partial_{+}-h_{++}\partial_{-})f=0$, the general solution of the equation (\[gravaneq\]) is $$\begin{aligned}
T_{--}(x^{+},x^{-}) =-2\mathcal{D}_{-}f+ (\partial_{-}f)^{2} t_{-}(f)\,,\end{aligned}$$ where we define the Schwarzian $$\begin{aligned}
\mathcal{D}_{-}f \equiv \frac{\partial_{-}^{3}f}{\partial_{-}f}- \frac{3}{2}\frac{(\partial_{-}^{2}f)^{2}}{(\partial_{-}f)^{2}}\end{aligned}$$ and $t_{-}(f)$ is at this stage is an arbitrary complex function, which has to be fixed by additional physical arguments[^18]. So analogously to the non-Abelian case we say that the induced current ${}_{\rm out}\langle T_{--}(x^+,x^-) \rangle_{\rm in}$ must satisfy the analytical (spectral) boundary conditions: \[bcgrav\] \_[out]{}T\_[–]{}(x\^+,x\^-) \_[in]{}
T\_[up]{}(x\^-),& x\^++,\
T\_[down]{}(x\^-),& x\^+ -,
where $T_{\rm up}(x^{-})$ and $T_{\rm down}(x^{-})$ are some complex functions analytic in the upper and lower $x^-$ half-planes correspondingly. Again, working in the retarded gauge, defined by the condition $f_{R}(x^{+}\to -\infty,x^{-})=x^{-}$ we find that[^19] $$\begin{aligned}
t_{-R}(f)=-2 \mathcal{D}_{f}\Gamma^{-1}_{\rm down}(f)\,,\end{aligned}$$ where $\Gamma_{\rm up}(x^{-})$ and $\Gamma_{\rm down}(x^{-})$ are invertible, analytic functions in the upper and lower $x^-$ half-plane, and they are solutions of the functional Riemann-Hilbert problem ($\Gamma(x^{-})\equiv f_{R}( +\infty,x^{-})$) $$\begin{aligned}
\Gamma_{\rm down}(\Gamma_{\rm up}(x^{-})) = \Gamma(x^{-})\,.\end{aligned}$$ We have $T_{\rm up}(x^-)=-2\mathcal{D}_{-}\Gamma_{\rm up}$ and $T_{\rm down}(x^-)=-2\mathcal{D}_{-}\Gamma_{\rm down}^{-1}$ and we again notice that $t_{-S}(f)=0$ in the spectral gauge $f_{S}$, defined in (\[spgg\]), which leads to the formula (\[effgravspect\]).
Having the expression for the current $T_{--}= -2\mathcal{D}_{-}\Gamma_{\rm down}^{-1}(f_{R})$, we can check that the variation of (\[gravefa\]) is indeed equal to $$\begin{aligned}
\delta W(h_{++}) = \int d^{2}x\, T_{--}\delta h_{++}\,.\end{aligned}$$
Gauge-gravity duality in two dimensions {#app:gauge-gravity}
=======================================
In this appendix, we review the duality between 2-dimensional gravity and $ \textrm{SL}(2,\mathbb{C})$ gauge theory [@Alekseev:1988ce; @Bershadsky:1989mf; @Polyakov:1989dm]. We find it useful, as the functional Riemann-Hilbert problem can be related to $\textrm{SL}(2,\mathbb{C})$ matrix Riemann-Hilbert problem.[^20]
The main idea is to consider the gauge theory on a nontrivial background, and study one particular component of the gauge field. The gauge field has three flavor indices and two spacetime indices, $A^a_\mu$, $a=+,0,-$ (a new occurrence of $\pm$, unrelated to the others in the paper) and $\mu=+,-$. Now, instead of fixing the axial gauge $A^a_-=0$, we partially fix the gauge by setting A\^+\_-T\_[–]{}, A\^-\_-=1, A\^0\_-=0. It turns out that the remaining gauge freedom on the component $A^+_-$ acts as the Virasoro generators on a stress tensor $T_{--}$. Thus, there is a beautiful duality between a component of a gauge field and the stress tensor of a certain gravitational theory. To complete the duality, one notices that the anomaly equations for the gauge field $A$ are equivalent to the anomaly equations for a metric $g_{++}$, if we identify the induced current in the gauge theory with the metric in the gravitational theory, $J^-_+=g_{++}$.
In terms of the action functionals, for the $ \textrm{SL}(2,\mathbb{C})$ non-Abelian gauge theory one can establish a relation $$\begin{aligned}
W_{\rm WZ}(h)= W_{\rm gWZ}(g_{++})\,,\end{aligned}$$ where the Wess-Zumino action and gravitational Wess-Zumino actions are given by the formulas $$\begin{aligned}
W_{\rm gWZ}(g_{++}) &= \frac{1}{4}\int d^{2}x \left({\partial_{-}^{2}f \partial_{+}\partial_{-}f \over (\partial_{-}f)^{2}}-{(\partial_{-}^{2}f)^{2}\partial_{+}f \over (\partial_{-}f)^{3}}\right)\,, \notag\\
W_{\rm WZ}(h)&= \frac{1}{2} \int_{0}^{1}dt d^{2}x\, \Tr (h^{-1}\dot{h} [h^{-1}\partial_{-}h, h^{-1}\partial_{+}h])\,,\end{aligned}$$ and the $ \textrm{SL}(2,\mathbb{C})$ matrix $h(x^{+},x^{-},t)$ and the metric $g_{++}(x^{+},x^{-})$ are related as follows: $$\begin{aligned}
&A_-=h^{-1}\partial_{-}h = \left(\begin{array}{cc}
0 & T_{--} \\
1 & 0 \\
\end{array}\right),\quad J^-_+ =(-h^{-1}\partial_{+}h)_{21}=g_{++},\quad (\partial_{+}-g_{++}\partial_{-})f=0\,,\notag\\
&\qquad\partial_{+}T_{--}-g_{++}\partial_{-}T_{--}-2(\partial_{-}g_{++})T_{--}=-\frac{1}{2}\partial_{-}^{3}g_{++}\,.\end{aligned}$$ One can prove these relations using a nice parametrization for the matrix $h$: $$\begin{aligned}
h= \left(\begin{array}{cc}
a & \partial_{-}a \\
b & \partial_{-}b \\
\end{array}\right),\quad \textrm{with} \quad a\partial_{-}b-b\partial_{-}a=1\,.\end{aligned}$$ In this parametrization one has $g_{++}= a\partial_{+}b-b\partial_{+}a$ and $T_{--}=\partial_{-}^{2}a/a=\partial_{-}^{2}b/b$ and $f= \mathcal{F}(a/b)$, where $\mathcal{F}$ is an arbitrary invertible function. Thus, in terms of $h$, we can find the characteristic function $f$. It is inetersting to understand whether this makes a connection between functional and matrix Riemann-Hilbert problems.
Non-Abelian and gravitational corrections to Caldeira-Leggett formula {#CaldLeg}
=====================================================================
In the case of a weak non-Abelian field profile, we may try to solve the matrix Riemann-Hilbert problem perturbatively $$\begin{aligned}
\Omega_{\rm down}(x^{-})\Omega_{\rm up}(x^{-}) = \Omega(x^{-})\,,\end{aligned}$$ where $\Omega(x^{-}) \equiv P \exp \int_{-\infty}^{\infty} A_{+}(y^{+},x^{-}) dy^{+}$ and we assume the following perturbative decomposition for $\Omega_{\rm down}$ and $\Omega_{\rm up}$: $$\begin{aligned}
&\Omega_{\rm up}=\mathbbm{1}+\Omega_{\rm up}^{(1)}+\Omega_{\rm up}^{(2)}+\dots, \quad \Omega_{\rm down}=\mathbbm{1}-\Omega_{\rm down}^{(1)}-\Omega_{\rm down}^{(2)}+\dots\,.\end{aligned}$$ Expanding $\Omega(x^{-})$ to first order we get $$\begin{aligned}
\Omega^{(1)}_{\rm up}(x^{-})-\Omega^{(1)}_{\rm down}(x^{-})= \omega(x^{-})\,,\end{aligned}$$ where $\omega(x^{-})\equiv\int_{-\infty}^{\infty} A_{+}(y^{+},x^{-}) dy^{+}$, thus $$\begin{aligned}
\Omega_{\rm up}^{(1)} = \omega_{\rm up}(x^{-}), \quad \Omega_{\rm down}^{(1)} =\omega_{\rm down}(x^{-})\,, \label{g1pert}\end{aligned}$$ where $ \omega_{\rm up/down}(x^{-})$ are given in (\[omupdown\]). At second order we have $$\begin{aligned}
\Omega_{\rm up}^{(2)}-\Omega_{\rm down}^{(2)} =\int_{-\infty}^{+\infty}dy_{1}^{+}\int_{-\infty}^{y_{1}^{+}}dy_{2}^{+}\, \Tr (A_{+}(y_{1}^{+},x^{-})A_{+}(y_{2}^{+},x^{-}))+\omega_{\rm down}\omega_{\rm up}\,,\end{aligned}$$ where we used (\[g1pert\]), and so we have just a scalar Riemann-Hilbert problem, which we and can solve explicitly. Now plugging this perturbative decomposition in the $2$-form (\[nonAbBound\]) we obtain $$\begin{aligned}
\textrm{Im}\,W_{B}(\Omega_{\rm up},\Omega_{\rm down}) &= \,\textrm{Im} \int dx^{-} \, \Tr \Big(\omega_{\rm down} \partial_{-}\omega_{\rm up} +\notag\\&+ \Omega_{\rm down}^{(2)}\partial_{-}\omega_{\rm up}+\omega_{\rm down}\partial_{-}\Omega_{\rm up}^{(2)}-\frac{1}{2}\big(\omega_{\rm up}\partial_{-}\omega_{\rm down}^{2}+\omega_{\rm down} \partial_{-}\omega_{\rm up}^{2}\big)+\dots\Big)\,,\end{aligned}$$ where the term in the first line is the standard Caldeira-Leggett formula, and the terms in the second line are the first perturbative corrections to it, cubic in $A_+$. Notice that, perturbatively, it is clear that the imaginary part of $W_B$ does not depend on the $t$-interpolation.
Now, in analogy with the non-Abelian case, we can solve the functional Riemann-Hilbert problem perturbatively. This assumes that the gravitational field is weak. It is convenient to write $\Gamma_{\rm up/down}(x^{-})=x^{-}\pm \gamma_{\rm up/down}(x^{-})$ and $\Gamma(x^{-})=x^{-}+\gamma(x^{-})$, then for (\[frhp\]) we have $$\begin{aligned}
\gamma_{\rm up}(x^{-})- \gamma_{\rm down}(x^{-}+\gamma_{\rm up}(x^{-}))=\gamma(x^{-})\,.\end{aligned}$$ Then writing a perturbative decomposition for $\gamma_{\rm up/down}$ $$\begin{aligned}
\gamma_{\rm up}= \gamma_{\rm up}^{(1)}+ \gamma_{\rm up}^{(2)}+\dots, \quad
\gamma_{\rm down}= \gamma_{\rm down}^{(1)}+ \gamma_{\rm down}^{(2)}+\dots\end{aligned}$$ we find at the first and the second order $$\begin{aligned}
&\gamma_{\rm up}^{(1)}(x^{-})-\gamma_{\rm down}^{(1)}(x^{-})= \gamma(x^{-})\,,\notag\\
&\gamma_{\rm up}^{(2)}(x^{-})-\gamma_{\rm down}^{(2)}(x^{-}) =\gamma_{\rm up}^{(1)}(x^{-})\partial_{-}\gamma_{\rm down}^{(1)}(x^{-})\,.\end{aligned}$$ So we see that step by step we just need to solve the scalar Riemann-Hilbert problem, which has the explicit solution (\[omupdown\]). Thus, the boundary action (\[gbacccc\]) reads $$\begin{aligned}
W_{B} = \int dx^{-}\Big(\gamma_{\rm down}^{(1)}\partial_{-}^{3}\gamma_{\rm up}^{(1)}-\big((\partial_{-}\gamma_{\rm up}^{(1)})^{2}\partial_{-}^{2}\gamma_{\rm down}^{(1)}+(\partial_{-}\gamma_{\rm down}^{(1)})^{2}\partial_{-}^{2}\gamma_{\rm up}^{(1)}-(\partial_{-}^{2}\gamma_{\rm down}^{(1)})^{2}\gamma_{\rm up}^{(1)}\big)+\dots\Big)\,.\end{aligned}$$ We also checked this result using Feynman diagrams.
[^1]: In this section we write the effective action up to an unimportant overall factor $-\frac{e^{2}}{4\pi}$. In other words, we set $e^{2}=-4\pi$. The charge $e$ can be restored by the substitution $A_{\mu}\to eA_{\mu}$.
[^2]: We use $1/(p_{+}+i\varepsilon \textrm{sgn}\, p_{-})= \mathcal{P}(1/p_{+})-i \pi\, \textrm{sgn}\,( p_{-}) \delta(p_{+}) $.
[^3]: The fact that the effective action depends on $\omega(x^{-})$ demonstrates that the gauge symmetry in our system is restricted by the condition that gauge transformations for $A_{+}$ and $A_-$ must be trivial at the boundary of spacetime.
[^4]: For a review on the subject and the cases where an explicit solution is available, see [@gohberg2003overview]. Notice that the right and left decompositions are inequivalent, namely, we could look for $\Omega(x^-)=\widetilde{\Omega}_{\textrm{up}}(x^{-}) \widetilde{\Omega}_{\textrm{down}}(x^{-})$, but in terms of these matrices we do not obtain spectral boundary conditions in a simple way. In general $\widetilde{\Omega}_{\textrm{up}/\textrm{down}}(x^{-})\ne{\Omega}_{\textrm{up}/\textrm{down}}(x^{-})$. We thank A. Kisil for discussions on the matrix Riemann-Hilbert problem.
[^5]: In light-cone coordinates $\Tr(\partial^{\mu}g^{-1}\partial_{\mu}g) =\Tr(\partial_{-}g^{-1}\partial_{+}g)+\Tr(\partial_{+}g^{-1}\partial_{-}g)=2\Tr(\partial_{+}g^{-1}\partial_{-}g)$.
[^6]: The effective action for arbitrary $g$ is $W_{+}(A_{+}) = W_{\rm WZNW}(g^{-1}(-\infty,x^{-})g)+W_{B}(\Omega_{\rm up},\Omega_{\rm down})$.
[^7]: Notice that, although the boundary term does depend on the $t$-interpolation, its imaginary part does not! One can see this by looking at the variation of (\[nonAbBound\]), $
\delta W_{B}=\frac{1}{2} \int dx^{-} \Tr ( \Omega_{\rm down}^{-1}\delta \Omega_{\rm down}\Omega_{\rm up}\partial_{-}\Omega_{\rm up}^{-1}+ \Omega_{\rm down}^{-1}\partial_{-}\Omega_{\rm down} \delta \Omega_{\rm up}\Omega_{\rm up}^{-1})+\frac{1}{2}\int dt dx^{-}\Tr([\Omega^{-1}\partial_{0}\Omega,\Omega^{-1}\partial_{-}\Omega]\Omega^{-1}\delta \Omega) \, .$ We see that the last term is $t$-dependent but explicitly real ($\Omega$ is a real matrix), whereas the first term is $t$-independent and complex. We also notice that the $t$-dependent term cancels with the $t$-dependent term in the variation of the WZ term (\[varWZ\]) in the effective action. Thus the variation of the effective action is also $t$-independent.
[^8]: A similar two-form was found in Euclidean manifolds with a boundary, in [@Baumgartl:2004iy; @Baumgartl:2006xb]. We thank N. Nekrasov for bringing these papers to our attention.
[^9]: For simplicity, we consider Majorana fermions. As in [@Knizhnik:1988ak], we perform a field redefinition to write the Lagrangian in the form (\[lgra\]). In the previous sections, we considered Dirac fermions, as these can carry electric and color charge.
[^10]: In an analogous fashion to the matrix Riemann-Hilbert problem, we can have right and left decompositions of the function $f$. Namely, we can consider functions $\tilde{\Gamma}_{\rm up/down}$ such that $ \tilde{\Gamma}_{\rm up}(\tilde{\Gamma}_{\rm down}(x^{-}))=\Gamma(x^{-}) $ in the real line. In general, $\tilde{\Gamma}_{\rm up/down}\ne \Gamma_{\rm up/down}$.
[^11]: Finding a physically relevant explicit solution to (\[frhp\]) seems to be hard. On the other hand one can find solutions in terms of meromorphic funtions. For example, $\Gamma_{\rm down}(x)= \frac{\epsilon}{1-x_{3}^{2}}\frac{(x-a)^{2}}{(x-ix_{1})(x-i x_{2})}$, $\Gamma_{\rm up}(x)= \frac{ax-b}{x+ix_{3}}$ and $\Gamma(x)=\frac{\epsilon}{1+x^{2}}$ is a solution to (\[frhp\]), where $a=\frac{i}{2}(x_{1}+x_{2}+(x_{1}-x_{2})x_{3})$, $b= \frac{1}{2}(x_{1}-x_{2}+(x_{1}+x_{2})x_{3})$, $x_{1},x_{2},x_{3}>0$ and $\epsilon$ is an arbitrary real parameter.
[^12]: To arrive at this formula we need two steps. At step 1 we define the inverse function $f_{R}^{-1}(\cdot ,\cdot)$ by $
f_{R}^{-1}(x^{+},f_{R}(x^{+},x^{-}))=x^{-}$ and notice that $\int d^{2}x \partial_{-}f_{R}=\int d^{2}y$ and $\partial_{-}=(\partial_{-}f_{R})\partial/\partial y^{-}=(\partial f_{R}^{-1}/\partial y^{-})^{-1}\partial/\partial y^{-}$ and $\partial_{+}f_{R}/\partial_{-}f_{R}=-\partial f_{R}^{-1}/\partial y^{+}$. At step 2 we integrate over $y^{+}$ and use that $\partial f_{R}^{-1}/\partial y^{-}(+\infty,y^{-}) =1/\Gamma'(\Gamma^{-1}(y^{-}))$ and $ \partial f_{R}^{-1}/\partial y^{-}(-\infty,y^{-}) =1$. It was crucial here to assume that $f_{R}(x^{+},x^{-})$ is invertible for all $x^{+}$.
[^13]: The expression (\[gbaccc\]) is very similar to formula (5.23) in [@Chung:1993rf]. The reason for the similarity of the results is puzzling to us and is an interesting open question. We thank H. Verlinde for pointing this to us.
[^14]: To restore the unimportant overall factor in front of the effective action one needs to replace $\varepsilon^{\mu\nu} F_{\mu\nu} \to \frac{\varepsilon^{\mu\nu}}{2\pi} F_{\mu\nu}$.
[^15]: Although $J_{-}(x^{+},x^{-})$ is a hermitian operator, the matrix element ${}_{\rm out}\langle J_-(x^+,x^-) \rangle_{\rm in}$ can be complex valued, as we are not computing an expectation value of the current for a given state, but rather evaluating a transition amplitude between states without particles in the past and without particles in the future.
[^16]: We can justify (\[bc\]) as follows. First, we checked (\[bc\]) diagramatically in perturbation theory, to third order in the background field. The other general argument invokes consideration of the correlation function $
{}_{\rm out}\langle0| \bar\psi_{+} (y^{+},y^{-}) \psi_{+}(x^{+},x^{-})|0\rangle_{\rm in} \, ,
$ where $x^{+}\to -\infty$. In this limit $\psi_{+}(x^{+},x^{-})$ is a free field and we have $
\psi_+(-\infty, x^{-}) = \sum_{p>0}(a_{p} e^{ipx^{-}}+a_{p}^{\dag}e^{-ipx^{-}})\, .
$ As $a_{p}|0\rangle_{\rm in}=0$ we see that only $e^{-ipx^{-}}$ modes survive. These modes define an analytic function in $x^-$ in the lower half plane because $e^{-ipx^{-}}$ decays when $p>0$ and $\textrm{Im} \,x^{-}<0$. This argument can be applied for any operator $O(\psi_{+})$, to show that a correlation function $\langle ...O(-\infty,x^{-})\rangle$ is analytic in $x^{-}$ in the lower half-plane.
[^17]: To restore the overall factor in front of the effective action one needs to replace $ -2 \partial_{-}^{3}h_{++} \to \frac{1}{24\pi}\partial_{-}^{3}h_{++}$.
[^18]: The logic is very similar to that of the paper [@Callan:1992rs], where the term $t_{-}(f)$ in the stress-energy tensor is fixed by choosing a particular state.
[^19]: It is convenient here to use the composition formula for the Schwarzian: $\mathcal{D}_{x}g(f)=\mathcal{D}_{x}f+ (\partial_{x}f)^{2}\mathcal{D}_{f}g\,. $
[^20]: We need to extend the gauge group to be complex valued, as we are interested in both real and imaginary parts of the action. Originally the duality was found using $\textrm{SL}(2,{\mathbb R})$ gauge group. The only new subtleties arise in treating integrations by parts, but, as long as we use the spectral gauge condition, the formulas are similar to the ones in the literature.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We study contributions to the nucleon strange quark vector current form factors from intermediate states containing $K^{*}$ mesons. We show how these contributions may be comparable in magnitude to those made by $K$ mesons, using methods complementary to those employed in quark model studies. We also analyze the degree of theoretical uncertainty associated with $K^{*}$ contributions.\
PACS numbers: 14.20.Dh, 12.40.-y\
author:
- |
L.L. Barz$^{1,2}$, H. Forkel$^{3,4}$, H.-W. Hammer$^{5,6}$, F.S. Navarra$^1$, M. Nielsen$^1$, and M.J. Ramsey-Musolf$^{6,7}$[^1]\
\[0.5cm\] [*$^1$Instituto de Física, Universidade de São Paulo*]{}\
[*C.P. 66318, 05315-970 São Paulo, SP, Brazil*]{}\
[*$^2$Faculdade de Engenharia de Joinville, Universidade Estadual de Santa Catarina*]{}\
[*82223-100 Joinville, SC, Brazil*]{}\
[*$^3$European Centre for Theoretical Studies in Nuclear Physics and Related Areas*]{},\
[*Villa Tambosi, Strada delle Tabarelle 286, I-38050 Villazzano, Italy*]{}\
[*$^4$Institut f[ü]{}r Theoretische Physik, Universit[ä]{}t Heidelberg*]{},\
[*Philosophenweg 19, D-69120 Heidelberg, Germany*]{}\
[*$^5$ TRIUMF, 4004 Wesbrook Mall, Vancouver, B.C., Canada V6T 2A3*]{}\
[*$^6$ Institute for Nuclear Theory, University of Washington, Seattle, WA 98195, USA*]{}\
[*$^7$ Department of Physics, University of Connecticut, Storrs, CT 08629 USA*]{}
title: '**$K^*$ Mesons and Nucleon Strangeness** '
---
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Introduction {#intro}
============
The role played by virtual $q\bar{q}$ pairs in the low-energy structure of hadrons remains one of the outstanding questions for hadron structure physics. Despite the evidence for important $q\bar{q}$ sea effects obtained with deep inelastic scattering, the experimental manifestations of explicit sea-quark effects at low energies are minimal. Partial explanations for this absence have been given using a non-relativistic quark model framework by the authors of Ref. [@Gei90], who noted that in the adiabatic approximation, virtual $q\bar{q}$ pairs renormalize the string tension and, therefore, do not have any discernable impact on the low-lying spectrum of hadronic states. Similarly, virtual $q\bar{q}$ effects – in the guise of virtual mesonic loops – which could conceivably lead to large $\rho-\omega$ and $\phi-\omega$ mixing were shown to cancel at second order in strong couplings when a sum is performed over a tower of virtual hadronic states [@Gei91]. The latter result provides insight into the applicability of the OZI Rule to $V$-$V'$ mixing despite the naïve scale of $q\bar{q}$ effects expected at one-loop order.
Nevertheless, several mysteries involving $q\bar{q}$ pairs remain to be solved. Of particular interest are those involving nucleon matrix elements of strange quark operators, $\bra{N}\bar{s}\Gamma s\ket{N}$. The latter explicitly probe properties of the $q\bar{q}$ sea at low energies, since the nucleon contains no valence strange quarks. Moreover, the mass scale associated with $s\bar{s}$ pairs – $m_s\sim\Lambda_{QCD}$ – implies that such pairs live for sufficiently long times and propagate over sufficiently large distances to produce observable effects when probed explicitly. In this respect, $s\bar{s}$ pairs stand in contrast with, [*e.g.*]{}, $c\bar{c}$ pairs, whose effects one expects to be suppressed by powers of $\Lambda_{QCD}/m_c\sim 0.1$ [@Kap88].
Some support for these simple-minded expectations is provided by determinations of $\bra{N}\bar{s} s\ket{N}$ from the $\pi N$ sigma" term [@Che76] and of $\bra{N}\bar{s}\gamma_\mu\gamma_5 s\ket{N}$ from polarized deep inelastic scattering [@EMC] and neutrino-nucleus quasi-elastic scattering [@Ahr87]. The former suggests that roughly 15% of the nucleon mass is generated by $s\bar{s}$ pairs, while the latter implies that strange quarks contribute about 30% of the total quark contribution to the nucleon spin[^2]. Measurements of $\bra{N}\bar{s}\gamma_\mu s\ket{N}$, which would provide information about the strange quark contribution to the nucleon magnetic moment and rms radius are presently underway at MIT-Bates [@MIT], Mainz [@Mai], and the Jefferson Laboratory [@TJN]. The first results for the strangeness magnetic form factor have been reported in Ref. [@MIT]. One expects this set of $\bra{N}\bar{s}\Gamma
s\ket{N}$ determinations to provide a clearer picture of the $q\bar{q}$ sea than obtained from existing spectroscopic data alone.
Despite over a decade of theoretical efforts to study nucleon strangeness, the theoretical understanding of s-quark matrix elements remains in its infancy. In the case of $\bra{N}\bar{s}\gamma_\mu s\ket{N}$, a plethora of predictions have been reported in the literature [@Lat; @Mod; @Jaf89; @Had; @Gei97; @Mrm97a; @Mus97b; @Mrm97c; @HWH97]. While a few lattice results have been obtained by different groups[@Lat], they are not entirely consistent with each other nor with the recent first results for the strange magnetic moment" obtained by the SAMPLE collaboration[@MIT]. The remaining predictions – based generally on QCD-inspired nucleon models [@Mod; @Gei97] or low-energy truncations of QCD in a hadronic basis [@Jaf89; @Had; @Mrm97a] – display a broad range in magnitude and sign. Recently, it has been shown why such truncations – either in the strong coupling constant ($g$) expansion (loop order) [@Mus97b; @Mrm97c] or hadronic excitation energy ($\Delta E$) [@Gei97] – are untrustworthy and may produce misleading results. The implication of these studies is that the intuitively appealing picture of a kaon cloud around the nucleon does not suffice to describe $s\bar{s}$ fluctuations in the nucleon. It appears that one must include both the full set of virtual hadronic intermediate states [@Gei97] as well as the full set of higher-order rescattering effects for a given state [@Mus97b; @Mrm97c] in order to obtain a physically realistic prediction. In principle, corrections to the leading order truncations in $\Delta E$ and $g$ could be accounted for by the appropriate low-energy constants in chiral perturbation theory (CHPT); however, chiral symmetry does not afford a determination of the low-energy constant relevant to nucleon vector current strangeness [@Mrm97a]. Hence, one must understand in some detail the short-distance strong interaction mechanisms responsible for the low-energy structure of the strange quark sea.
In the present study, we amplify on the themes of Refs. [@Gei97; @Mrm97a; @Mus97b; @Mrm97c] by studying the $K^{\ast}$ contribution to $\bra{N}\bar{s}\gamma_\mu s\ket{N}$. Our objective is two-fold: (i) to illustrate, using an alternative framework to that of Ref. [@Gei97], how inclusion of higher-lying intermediate states may alter conclusions obtained when only the lightest OZI-allowed" fluctuation is included, and (ii) to demonstrate the theoretical uncertainty associated with computing higher-lying contributions. For these purposes, we restrict ourselves to second order in the strong meson-baryon coupling, $g$, when treating hadronic amplitudes $N\to
YK^*$ [*etc.*]{}, fully cognizant of the shortcomings such a truncation entails. In fact, the kind of analysis of higher-order effects reported in Ref. [@Mrm97c] for the $K\bar{K}$ intermediate state does not appear feasible at present for higher lying states. Consequently, some form of model-dependent truncation is necessary when treating these states, and we do not, therefore, pretend to make any reliable numerical predictions. Rather, we use the ${\cal O}(g^2)$ (one-loop) truncation to illustrate the two main points stated above. In this respect, our study is similar in spirit to that of Ref. [@HWH97], where a comparison at one-loop order was made to show that contributions from intermediate states containing no valence strangeness ($3\pi$) and those containing valence s-quarks ($K\bar{K}$) may be comparable in magnitude.
In order to estimate the degree of theoretical uncertainty one has in the numerical prediction for the $K^{\ast}$ contribution, we use two approaches to carry out the calculation: (a) an explicit one-loop calculation, where form factors are included at hadronic vertices and the intermediate state $\bar{s}\gamma_\mu s$ matrix elements are taken to be point-like, and (b) a computation using dispersion relations, in which the $N\bar{N}\to KK,\ KK^{\ast},\ K^{\ast}K^{\ast}$ amplitudes are computed in the Born approximation but form factors are included at the $\bar{s}\gamma_\mu s$ insertions. These computations are outlined, respectively, in Sections II and III. In Section IV, we discuss the results of the calculations and compare with the conclusions drawn in Ref. [@Gei97].
One-loop calculation {#ext}
====================
The first kaon cloud" estimates of $\bra{N}\bar{s}\gamma_\mu s\ket{N}$ were obtained from the amplitudes associated with the diagrams of Fig. 1, where only the contributions for $B=B'=\Lambda, \Sigma$ and $M=M'=K$ were included [@Had]. Here we consider the next heaviest contributions by including the octet of spin-one mesons as well as the pseudoscalars, and compute the following amplitudes where, in each case, $B=B'=\Lambda$ or $\Sigma$: (1a) for $M=K^{\ast}$; (1b) for $M=M'=K^{\ast}$; (1b) for $M=K$, $M'=K^{\ast}$; (1c) for $M=K^{\ast}$. As we discuss below, the diagrams (1c) are required for consistency with the Ward-Takahashi identities.
The resulting contributions to the strange-quark vector current matrix element are embodied in the Dirac and Pauli form factors defined via
N(p\^) | |s\_s| N(p) = [|U]{}(p\^) U(p) , \[ma\] where $U(p)$ denotes the nucleon spinor. Recall that $F_1^{(s)}(0)=0$, due to the zero strangeness charge of the nucleon. The leading nonvanishing moments of the corresponding Sachs form factors G\_E\^[(s)]{}(q\^2)= F\_1\^[(s)]{}(q\^2)+ [q\^24m\_N\^2]{}F\_2\^[(s)]{} (q\^2),\
G\_M\^[(s)]{}(q\^2)= F\_1\^[(s)]{}(q\^2)+F\_2\^[(s)]{}(q\^2) are the strangeness radius r\_s\^2 \_S = .6[ddq\^2]{} G\_E\^[(s)]{}(q\^2)|\_[q\^2=0]{} , \[rs\] and the strangeness magnetic moment \_s=G\_M\^[(s)]{}(0)=F\_2\^[(s)]{}(0) . \[mus\] For future reference, we note that the Sachs radius $\langle r_s^2 \rangle_S$ is related to the corresponding Dirac radius as \[sachsdirac\] r\^2\_s \_S= r\^2\_s \_ + \_s.
In order to extend the $K - \Lambda$ loop framework to include $K^*$-meson contributions, we start from the meson baryon effective lagrangians $$\begin{aligned}
\label{1aa}
{\cal L}_{MB} & = &-ig_{ps} \bar{B} \gamma_5 B K\ \ \ , \\
{\cal L}_{VB} & = & -g_v(\bar{B} \gamma_\alpha B V^\alpha +
{\kappa\over 2m_N}\bar{B}
\sigma_{\alpha\beta} B \partial^\alpha V^\beta)\; ,
\label{la}\end{aligned}$$ where $B$, $K$, and $V^\alpha$ are the baryon, kaon, and $K^*$ vector-meson fields respectively, $m_N=939\MeV$ is the nucleon mass and $\kappa$ is the ratio of tensor to vector coupling, $\kappa=g_t/g_v=3.26$, with $g_v/\sqrt{4\pi}=
-1.588$ [@hol89]. The strength of the pseudoscalar coupling is $g_{ps}/\sqrt{4\pi} = -3.944$ [@hol89].
In order to account in some way for the finite extent of the hadrons appearing in the loops of Fig. 1, we include form factors at the hadronic vertices. For simplicity, we adopt a monopole form F(k\^2)=[m\^2-\^2k\^2-\^2]{} . \[ff\] Although there is no rigorous justification for this choice, form factors of this type for the $KN\Lambda$ and $K^* N \Lambda$ vertices are used in the Bonn potential. Their cut-off parameters are determined from hyperon-nucleon scattering data [@hol89]: $\Lambda_{K^*}=2.2$ (2.1), $\Lambda_K=1.2 (1.4) \GeV$ with masses $m_K= 495$ MeV and $m_{K^*}=895\MeV$[@PDG]. The numbers in parenthesis denote values obtained in an alternate model for the baryon-baryon interaction.The momentum of the $K^*$ is $k$. These form factors render all the following loop integrals finite and reproduce the on-shell values of the mesonic couplings (since $F(m^2)=1$) .
In the presence of electroweak fields the non-local meson-baryon interaction of Eqs. (\[1aa\]-\[ff\]) gives rise to vertex currents. In order to maintain gauge invariance we introduce the photon field by minimal substitution of the momentum variable in the form factors[^3]. This procedure generates the nonlocal seagull vertex [@ohta; @wan96; @Mrm97a] i\_\^[(s)]{}(k,q)=ig\_vQ\_[K\^\*]{} (q2k)\_ , \[seag\] where the upper/lower signs correspond to an incoming/outgoing vector meson (with index $\alpha$), $Q_{K^*}=-1$ is the $K^*$ strangeness charge, and $q$ is the photon momentum.
Due to the derivative in eq. (\[la\]), the minimal substitution also generates an additional seagull vertex (even in the absence of meson-nucleon form factors) i\_\^[(v)]{}(k)= F((qk)\^2) \_ , \[ver\] where the sign convention is the same as above.
The diagonal matrix elements of $\bar{s}\gamma_\mu s$ for strange mesons and baryons is straightforwardly determined by current conservation and the net strangeness charge of each hadron. The structure of the s-quark current spin-flip transition from $K$ to $K^*$ is K\^\*\_a(k\_1,)|[s]{}\_s|K\_b(k\_2)= [(q\^2)m\_[K\^\*]{}]{}\_k\_1\^k\_2\^\^[\*]{}\_[ab]{} , where $a$ and $b$ are isospin indices, $\varepsilon^\beta$ is the polarization vector of the $K^*$, and $k_1, k_2$ are the meson momenta. In a loop calculation, $\ffkks(q^2)$ is taken to be a constant equal to its value at the photon point. In order to estimate this constant, we follow Ref. [@gm] and assume $\ffkks(q^2)$ to be dominated at low-$q^2$ by the lightest $I^G(J^{PC})=0^-(1^{--})$ vector mesons[^4]: \[omegaphi\] [(q\^2)m\_[K\^\*]{}]{}=-\_[V=,]{} [G\_[K\^\*VK]{}S\_Vq\^2-m\_V\^2]{} , where $G_{K^*VK}$ are the couplings of the vector meson $V$ to $K$ and $K^*$. $S_V$ determines the strength of the strange-current conversion into $V$: 0|[s]{}\_s|V=S\_V \_=[m\_V\^2f\_V]{}[f\_Vf\_V\^[(s)]{}]{}\_ .
>From the known isoscalar electromagnetic couplings $f_{\omega,\phi}$ one can delineate the corresponding strange-current couplings with the help of a simple quark counting prescription based on flavor symmetry [@Jaf89]: \[fvrel\] [f\_f\_\^[(s)]{}]{}=- , = - , Here $\epsilon=0.053$ [@jain] is the mixing angle between the pure $\overline{u}u+ \overline{d}d$ and $\overline{s}s$ states and the physical vector mesons $\omega$ and $\phi$, and $\theta_0$ is the “magic angle” defined by $\sin^2\theta_0
=1/3$. From the above we find $f_\omega / f_\omega^{(s)}
= -0.21$ and $f_\phi/ f_\phi^{(s)} = -3.11$. Combined with the strong couplings $G_{K^*\phi K} = -8.94\GeV^{-1}$ and $G_{K^*\omega K} = 6.84 \GeV^{-1}$ estimated in Ref. [@gm] we finally obtain (0)=1.84.
After these preparations[^5], we can evaluate the $K^*$ loop contributions to the nucleon’s strangeness radius and magnetic moment. Explicit expressions for the loop amplitudes are given in Appendix A. The results for the different diagrams are listed in Table I. The implications of these results are discussed in Section IV.
Dispersion relation calculation {#dispa}
===============================
An alternative approach to computing virtual hadronic contributions to strange quark form factors is the use of dispersion relations (DR’s). In principle, DR’s provide a method for including information beyond second order in $g$, both via the strong amplitudes $N\to YK^{*}\to N$ and through the form factors $F_n^{(s)}$ describing the intermediate state matrix elements $\bra{Y}\bar{s}
\gamma_\mu s\ket{Y}$, $\bra{K^{*}}\bar{s}\gamma_\mu s\ket{K}$, $\ldots$. The one-loop calculation of Section II is equivalent to the use of a DR in which the strong amplitudes $N\to YK^{*}\to N$ are computed in the Born approximation and the form factors assumed to be point-like: $F_n^{(s)}(q^2)=F_n^{(s)}(0)=
{\hbox{const}}$[^6].
The inclusion of rescattering and resonance effects in the $N\to YK^{*}\to N$ amplitude would require the existence of sufficient data for $KN\to NK\pi,\ldots$ or $N\bar{N}\to KK\pi, KK\pi\pi$ [*etc.*]{} to permit analytic continuation of these amplitudes to the unphysical regime as needed for the dispersion relation. Although such a program is feasible to some degree for the $K\bar{K}$ intermediate state [@Mrm97c], it does not appear practical at present for the case of higher mass strange mesons of interest here. Consequently, we include the amplitude for $N\to YK^{*}\to N$ at the level of the Born approximation. In the case of the $F_n^{(s)}(q^2)$, however, it is possible to introduce some structure beyond the point-like approximation, albeit in a model-dependent way. Our strategy for doing so is discussed below.
First, we review the formalism for treating strangeness form factors with DR’s. We write an unsubtracted dispersion relation for the Pauli form factor $F^{(s)}_2$ and subtract the one for the Dirac form factor $F^{(s)}_1$ once at $t=0$ (where $F^{(s)}_1$ vanishes, see above): $$\begin{aligned}
\label{disp1}
F^{(s)}_1(t) &=& \frac{t}{\pi}\int\limits_{t_0}^\infty dt'
\frac{\hbox{Im}\ F^{(s)}_1(t')}{t'(t' -t)}\ \ \ , \\
F^{(s)}_2(t) &=& \frac{1}{\pi}\int\limits_{t_0}^\infty dt'
\frac{\hbox{Im}\ F^{(s)}_2(t')}{t' -t} \ \ \ ,
\nonumber\end{aligned}$$ where $t\equiv q^2$. The cut along the real $t$-axis starts at the threshold $t_0$ of a given multi-particle intermediate state, as [*e.g.*]{} $t_0 =4\mks$ for the $K\bar{K}$ state. From Eqs. (\[disp1\]) one expects that contributions from the lightest intermediate states will mainly determine the behavior of the form factors at $t=0$. The imaginary part of the form factors is readily obtained by means of a spectral decomposition. Since the matrix elements $\bra{N(p)}\bar{s}\gamma_\mu s \ket{N(p')}$ and $\bra{N(p);\bar{N}(\pbar)}\bar{s}\gamma_\mu s\ket{0}$ are simply related by crossing symmetry, we write the spectral decomposition for the latter one as [@Mrm97c], $$\begin{aligned}
\label{spec_t}
& &{\hbox{Im}}\ \bra{N(p);\bar{N}(\pbar)}\bar{s}\gamma_\mu s\ket{0} =
{\hbox{Im}}\ \bar{U}(p)\left[F_1^{(s)}(t)\gamma_\mu + i{\sigma_{\mu\nu}(p+
\pbar)^\nu \over 2\mn} F_2^{(s)}(t)\right]V(\pbar)\\
& &\qquad\rightarrow
{\pi\over\sqrt{Z}}(2\pi)^{3/2}{\cal N}\sum_{n}
\bra{N(p)}\bar{J_N}(0)\ket{n}\bra{n}\bar{s}\gamma_\mu s\ket{0} V(\pbar)
\delta^4(p+\pbar-p_n)\, ,
\nonumber\end{aligned}$$ where ${\cal N}$ is a spinor normalization factor, $Z$ is the nucleon’s wave function renormalization constant, and $J_N(x)$ is a nucleon source. Nonzero contributions arise only from physical states $\ket{n}$ with the same quantum numbers as the current $\bar{s}\gamma_\mu s$, [*i.e.*]{} $I^G(J^{PC})=0^-(1^{--})$ and zero baryon number. These asymptotic states $\ket{n}$ in the above sum do not explicitly contain resonances. Resonance contributions arise via the matrix elements $\bra{N(p)}\bar{J_N}(0)\ket{n}$ and $\bra{n}\bar{s}\gamma_\mu s\ket{0}$. In the vector meson dominance approximation, one assumes the product of the two matrix elements in Eq. (\[spec\_t\]) to be strongly peaked near vector meson masses. This approximation has been used in several pole analyses of the strange vector form factors [@Jaf89].
The lightest contributing intermediate states are purely mesonic: $3\pi$, $5\pi$, $7\pi$, $K\bar{K}$, $K\bar{K}\pi$, $9\pi$, $K\bar{K}\pi\pi$, $\ldots$, in order. Intermediate baryon states $N\bar{N}$, $\Lambda\bar{\Lambda},\ldots$ appear with significantly higher thresholds, $t_0$. In the present study, we restrict ourselves to the strange states and consider corrections to the $K\bar{K}$ state. The first such corrections (in order of threshold) are those involving the $K\bar{K}\pi$ and $K\bar{K}\pi\pi$ intermediate states. In the previous section, these states were included using the narrow resonance approximation: $K\bar{K}\pi
\to K^{*}\bar{K}$ and $K\bar{K}\pi\pi\to K^{*}\bar{K}^{*}$. In order to make contact with the loop results of Section II as well as with the calculation of Ref. [@Gei97] where in effect the same approximation was made, we adopt the narrow resonance approximation here. We also include the $\Lambda\bar{\Lambda}$ and $\Sigma\bar{\Sigma}$ intermediate states, even though they are not among the lightest in the series, in order to compare the DR results with those of the loop and quark model calculations, which contain these states.
As noted earlier, we also include the strong amplitudes $\bra{N}\bar{J}_N(0)\ket{n}$ at the level of the Born approximation. For the matrix elements $\bra{n}\bar{s}\gamma_\mu s\ket{0}$, parameterized by form factors $F_n^{(s)}(t)$, we go beyond the point-like approximation, F\_n\^[(s)]{}(t)F\_n\^[(s)]{}(0) F\_n\^0 , of the one-loop and quark model calculations by allowing for some structure in the form factors. For the mesonic intermediate states, we make a simple vector meson dominance (VMD) [*ansatz*]{}. This [*ansatz*]{} is well justified for the $K\bar{K}$ state, following from $e^+e^-\to K\bar{K}$ cross section data [@Del81] and simple flavor rotation arguments [@Jaf89]. The $e^+ e^-\to K\bar{K}$ data indicates a strong peak in the vicinity of the $\phi$ resonance, with a subsequent rapid fall-off as $q^2$ (time-like) increases away from $m_\phi^2$. Inclusion of a VMD-type form factor peaked near the $\phi$-resonance significantly affects the $K\bar{K}$ component of the spectral functions \[Eqs. (\[spec\_t\])\] and the resulting contribution to the strangeness moments as compared with the use of a point-like form factor.
In the case of the $KK\pi\sim
KK^{*}$ and $KK\pi\pi\sim K^{*}K^{*}$ states, we take the $\fns(t)$ to be dominated by either the $\phi(1020)$ or the $\phi'(1680)$. Following Ref. [@Mus97b] we write $$\label{fk_vdm}
| \fns(t)_{VDM}| =F_n^0
\left\{ {(\xi^2)^2+M^2\Gamma^2\over [(\xi^2-t)^2
+M^2\Gamma^2]}\right\}^{1/2}\,, \label{phidom}$$ where $M=m_{\phi}=1020$ MeV or $m_{\phi'}=1680\pm 20$ MeV, $\Gamma= \Gamma_{\phi}=4.43\pm 0.05$ MeV or $\Gamma_{\phi'}= 150\pm 50$ MeV are the total widths of the $\phi$ or $\phi'$ [@PDG], and $\xi^2\equiv M^2-\Gamma^2 /4$. As we note below, we need only the magnitude of the form factor in the present calculation, as the $n\to N\bar{N}$ amplitudes are real in the Born approximation. Because the states $KK\pi\sim
KK^{*}$ and $KK\pi\pi\sim K^{*}K^{*}$ contribute to the DR of Eq. (\[disp1\]) for $t_0> m_\phi$, we expect higher mass vector mesons to play a significant role in the $\fns(t)$ in the region of integration.
The case for $\phi'$ dominance is most convincing for the $K\bar{K}\pi$ intermediate state. Data for $\sigma(e^+e^-\to K_S^0 K^\pm \pi^\mp)$ in the range $1.4 \leq \sqrt{s}
\leq 2.18$ GeV display a pronounced peak near $\sqrt{s}=1.680$ GeV [@Man82]. Furthermore, Dalitz plot analyses imply that the final state is dominated by a $K^{\ast}K\leftrightarrow KK\pi$ resonance. The OZI rule implies that the $\phi'$ is nearly a pure $s\bar{s}$ state, while SU(3) relations and data for $\sigma(e^+e^-\to \rho\pi;\sqrt{s}\approx 1.65)$ constrain the $\omega'-\phi'$ mixing angle to deviate by less than $10^\circ$ from ideal mixing [@Buo82]. While the tails of the $\rho(770)$, $\omega(780)$, and $\phi(1020)$ affect details of the peak structure, the dominant effect is that of the $\phi'$ [@Buo82]. In the absence of any other structure in $\sigma(e^+e^-\to KK\pi)$ in the region $t>t_0$, we conclude that $\ffkks(t)$ should also be dominated by the $\phi'(1680)$. Indeed, the $\omega\phi$ model of Eq. (\[omegaphi\]), which is credible for low-$t$, is inconsistent with annihilation data for $t> t_0$. Using it in this region would generate an artificial suppression of the $KK\pi$ spectral function.
With these considerations in mind, it is straightforward to determine the normalization $F_{KK^*}^0$ appearing in Eq. (\[fk\_vdm\]). Following the notation of Ref. [@gm], we obtain \[fkks\_zero\] F\_[KK\^\*]{}\^0= G\_[KK\^\*’]{} m\_[K\^\*]{}/f\_[’]{}\^[(s)]{} , where $1/f_{\phi'}^{(s)}\approx -3/f_{\phi'}$, and $G_{KK^*\phi'}$ is the strong $\phi'\to KK^*$ coupling. The latter may be obtained from $\Gamma(\phi'\to KK^*)$ which, for a single final charge state is [@gm] (’KK\^\*) = [|G\_[KK\^\*’]{}|\^212]{} |k\_F|\^3 , where $k_F=463$ MeV is the $K$ or $K^*$ CM momentum. Assuming $\Gamma(\phi'\to {\hbox{all}})$ is dominated by $\Gamma(\phi'\to KK^*)$ [@PDG], we obtain $|G_{KK^*\phi'}|\approx 3.8$ GeV$^{-1}$.
Similarly, the $\phi'$ electronic width determines $f_{\phi'}$: (’e\^+e\^-) = [43]{} \^2 [M\_[’]{}f\_[’]{}\^2]{} . Analyses of $e^+e^-$ data yield $\Gamma(\phi'\to e^+e^-)=0.7$ keV [@Buo82; @Bis91], from which we obtain $f_{\phi'}\approx 23$. The factor of $-3$ appearing in the relation between $f_{\phi'}$ and $f_{\phi'}^{(s)}$ assumes ideal mixing \[see Eq. (\[fvrel\])\]. Allowing for a small deviation $|\epsilon|< 10^\circ$ does not change our results appreciably, especially since the $\omega'$ is not observed to decay to $KK\pi$.
Substituting these results into Eq. (\[fkks\_zero\]) yields $F_{KK^*}^0=\ffkks(0)=0.43$, to be compared with the value $\ffkks(0)=1.84$ used in loop calculation. We emphasize the latter value results from assuming only the $\omega$ and $\phi$ contribute to $\ffkks(t)$, whereas the former is obtained when [*only*]{} the $\phi'$ is included. Depending on the relative phase of the $(\rho\omega\phi)$ and $(\rho\omega\phi)'$ contributions in $e^+e^-\to KK\pi$, the $\phi'$ will either increase or decrease the point-like value (1.84) for this form factor by about 25% . At the $KK^*$ threshold, the $\phi'$ contribution to $\ffkks$ is about half as large as that from the $\phi$, but becomes nearly five times larger in the vicinity of $t=m_{\phi'}^2$. For purposes of estimating the $t$-dependence of $\ffkks$ in the region $t>t_0$, then, inclusion of only the $\phi'$ appears to be a reasonable approximation[^7]. We note in passing that our estimate of the $\phi'$ contribution carries an uncertainty of 25% or more, as the experimental values for $\Gamma(\phi'\to KK\pi,
e^+e^-)$ carry experimental errors of $\geq 25\% $ [@Cle94].
The implications of $e^+e^-$ data for $F_{K^*}^{(s)}(t)$ are less clear. To our knowledge, there exists no annihilation data giving $K^*K^*$ branching ratios. In $e^+e^-\to KK\pi\pi$ ($1.4\leq
\sqrt{s}\leq 2.18$), for example, the $K\pi$ invariant mass distribution is consistent with production of only one $K^*$ per event [@Cor82]. Consequently, the data cannot be used to infer a $K^*$ EM or strangeness form factor for $t>t_0$, and we must rely on a model. Given the evidence for $\phi'$ dominance of $\ffkks(t)$ and for $\phi$ dominance of $F_K^{(s)}(t)$ as well as the absence of experimental observation of any $0^-(1^{--})$ $s\bar{s}$ mesons with mass $\geq 2 m_{K^*}$, it is natural to assume that the $t$-dependence of $F_{K^*}^{(s)}(t)$ is governed by the tails of the known $s\bar{s}$ vector mesons. For simplicity, we include only one $s\bar{s}$ resonance – either the $\phi$ or the $\phi'$ – using the form of Eq. (\[fk\_vdm\]). The normalization $F_{K^*}^0=|Q_{K^*}|$. In the DR results displayed in Table I, we quote a range of values, the limits of which correspond to using either the $\phi$ or $\phi'$. A more realistic parameterization of $F_{K^*}^{(s)}(t)$ is likely to include some linear combination of $\phi$ and $\phi'$ poles, as well as small contributions from the $\omega$ and $\omega'$. Existing information does not permit us to determine this linear combination. Consequently, we use the ranges appearing in Table I to estimate the uncertainty in the $K^*K^*$ contribution associated with lack of knowledge of the $K^*$ strangeness form factor.
For the intermediate hyperon form factors, we are aware of no electromagnetic data to provide guidance for the choice of $F_n^{(s)}(t)$. We therefore work in analogy with the proton EM form factors, since both $F_B^{(s)}(t)$ ($B=\Lambda,\ \Sigma$) and $F_\sst{PROTON}^\sst{EM}(t)$ involve matrix elements of vector currents having unit conserved charge in the states of interest. Consequently, we adopt the standard dipole form factor for the Dirac strangeness form factors of the intermediate hyperons. Since the corresponding strange magnetic couplings are unknown, we omit magnetic form factors altogether. Because the resulting contributions to the strangeness moments are generally small compared to the mesonic contributions, we do not expect the uncertainty associated with $F_B^{(s)}(t)$ to be problematic.
Under these assumptions, our calculation proceeds as follows. The spectral functions entering Eqs. (\[disp1\]) have the general form \[genform\] F(t) = = |A\_[J=1]{}\^n(t)| |(t)| (1+\_n) , where $A_{J=1}^n$ is the appropriate combination of $J=1$ partial waves for the process $n\to N\bar{N}$ and $\gamma_n$ is a correction arising from the difference in phases between the amplitude $A_{J=1}^n$ and $\fns$ [@Mus97b]. This correction can vary between $-2$ and 0 and depends on $t$. At present, we are unable to determine $\gamma_n$ for the intermediate states considered here, and set $\gamma_n=0$ to obtain an upper bound.
To compute the $A_{J=1}^n(t)$ in Born approximation, we calculate the imaginary parts of the diagrams (a) and (b) in Fig. 1 assuming point-like strangeness form factors, $\fns(t)\equiv 1$. We neglect the hyperon-nucleon mass difference and take $m_\sst{Y}=\mn$. The seagull diagrams do not have an imaginary part, so we obtain no contributions from diagrams 1c. Furthermore, from Eq. (\[spec\_t\]) the individual contributions are manifestly gauge invariant in this approach. We calculate the imaginary parts of the corresponding diagrams with cutting rules [@cutru] and insert them into the dispersion relations Eqs. (\[disp1\]). To obtain the imaginary parts it is convenient to consider the crossed $t$-channel matrix element $\bra{N(p);\bar{N}(\pbar)}\bar{s} \gamma_\mu s\ket{0}$. The generic form of such a diagram is shown in Fig. 2. The different choices for the internal lines I, II, and III are shown in Table II. The equivalent of the previous kaon loop result is recovered if the internal lines are chosen as in case 1 and 2. In the following, we outline our calculation for the cases 3 - 5. In case 3 and 4, both kaons have been replaced by $K^*$ vector mesons, while one kaon and one $K^*$ contribute in case 5.
We choose to work in the center-of-momentum (CM) frame of the nucleon-antinucleon pair, where $q=(\omega,\vec{0})$. The loop diagrams lead to a physical reaction for $t \geq 4 \mns$, which is the minimal energy required for the creation of a $\bar{N}N$-pair, and we have $p'=(\omega/2,\vec{p'})$ and $p=(\omega/2,-\vec{p'})$ with $p_t=|\vec{p'}|=\sqrt{t/4-\mns}$. We define the contribution of a particular Feynman diagram with vertex function $\Gamma_\mu$ as $$\label{vert}
{\cal M}^{(i)}_\mu= -i\, \bar{u}(p') \Gamma^{(i)}_\mu v(p)\,.$$ These vertex functions are then multiplied by the strangeness form factor $|F^{(s)}(t)_{VDM}|$ from above as indicated by Eq. (\[genform\]). Our choice for the momenta of the internal lines is indicated in Fig. 2.
For the cases 3 - 5 we obtain the vertex functions shown in Appendix B. The imaginary part of $\Gamma_\mu^{(i)}$ is always finite; hence, the divergencies of the $d^4k$ integrals are without consequences. The vertex functions $\Gamma^\mu$ have branch cuts on the real axis for $ t \geq (m_I+m_{II})^2$. Their real part is continuous, such that the discontinuity associated with the cut is reflected only in the imaginary part. In the CM-frame of the nucleon and antinucleon, we have to calculate $$\label{discon}
{\hbox{Im}}\,\Gamma^\mu = \frac{1}{2\,i}\Delta\Gamma^\mu =
\frac{1}{2\,i}\lim_{\delta \to 0}
\left(\Gamma^\mu(\omega+i\delta)-\Gamma^\mu(\omega-
i\delta)\right) \,.$$ In particular, we obtain the discontinuity $\Delta\Gamma^\mu$ using the Cutkosky rules [@cutru] by cutting the lines I and II, i.e. by replacing the propagators of these lines by $\delta$ functions, $$\label{cuma}
\frac{1}{p^2 - m^2 + i\varepsilon} \longrightarrow
-2\,\pi\,i\,\theta(p_0)\, \delta(p^2 - m^2)\; .$$ As a consequence, the discontinuity arises when the particles I and II in Fig. 2 are on-shell. Due to the delta functions, the $d^4 k$ integration covers only a finite part of the $k$ space, leading to a finite value of the integral. Next we write $d^4k$ as $dk_0\, k^2 dk \,d\Omega_k$ and use the delta functions to carry out the $dk_0$ and $dk$ integrations. Moreover, the $d\Omega_k$ integration involves only $x$, the cosine of the angle between $\vec{k}$ and $\vec{p'}$. The denominator of the remaining propagator acquires the structure $z + x$, where $z$ depends on the particles internal to the loop. 3&:& z = =-(1+ )\
[Case]{} 4 &:& z=\
& &q\_t =\
5&:& z =\
& & q\_t = Finally, ${\hbox{Im}}\,\Gamma_\mu$ can be expressed through Legendre functions of the second kind, and, using the relation $$\label{f_zer}
{\hbox{Im}} \,\Gamma_\mu = \gamma_\mu {\hbox{Im}}\,F_1 +i
\frac{\sigma_{\mu\nu}}{2m} q^\nu {\hbox{Im}}\, F_2 \; ,$$ the contributions to the imaginary parts of the Dirac and Pauli form factors for $t \geq 4\,\mns$, respectively, can be identified. The emerging spectral functions are valid for $t \geq 4\,\mns$. The dispersion integrals, however, start at $t_0=(m_I + m_{II})^2$, with $m_I$ and $m_{II}$ the masses of the loop particles I and II, respectively. Consequently, the imaginary parts of the diagrams with two internal meson lines have to be analytically continued into the unphysical region $(m_I + m_{II})^2 \leq t < 4\,\mns$, by replacing the momentum $p_t = \sqrt{t/4 -\mns}$ by $i\,p_{-} = i\sqrt{\mns -t/4}$. Similarly, the variables $z$ become complex ($z \to i \xi$), and the Legendre functions of the second kind must be analytically continued as well.
Inserting now the imaginary parts and their analytical continuations in the unphysical region into the dispersion relations of Eq. (\[disp1\]), we obtain the $KK^*$ and $K^*K^*$ contributions to the strangeness form factors of the nucleon. In particular, the dispersion relations for the $K^*$ loop contributions to the strangeness radius and magnetic moment read $$\begin{aligned}
\label{rhosi}
\langle r^2_s \rangle_D &=&{6\over\pi}\int_{t_0}^\infty dt {{\hbox{Im}}\,
\FOS(t)\over t^2} \\
\label{musi}
\mu^{(s)}&=& {1\over\pi} \int_{t_0}^\infty dt {{\hbox{Im}}\, \FTS(t)\over t}\,,\end{aligned}$$ where $\langle r^2_s\rangle_D$ is related to the Sachs radius via Eq. (\[sachsdirac\]). For most of the intermediate states considered here, the dispersion integrals in Eqs. (\[rhosi\], \[musi\]) converge when a non-pointlike form for the $F_n^{(s)}(t)$ is employed. However, the tensor $K^*NB$ ($B=\Lambda, \Sigma$) coupling renders the $K^{*}K^*$ divergent even when the VDM form factor is included. To regulate this integral, we note that the unitarity of the S-matrix implies that the $N\bar{N}\to K^{*}K^{*}$ amplitude is bounded in magnitude for scattering in the physical region, $t> 4m_N^2$. The Born approximation for this amplitude does not respect this boundedness property, signalling the importance of higher-order rescattering corrections [@Mus97b]. At present, since we wish only to obtain an estimate for the $K^{*}$ contributions, we replace the $A_{J=1}^n(t>4 m_N^2)$ by its value at the physical threshold, $A_{J=1}^n(t=4 m_N^2)$. We make the same replacement in the integrals for the $KK^{*}$ intermediate state. This procedure leads to a crude upper bound on the contribution to the integrals from the integration region $t> 4 m_N^2$.
The results of the DR estimates of the various contributions are quoted in Table I. The DR results for the $K\bar{K}$ contribution given in Table I were obtained using the rigorous unitarity bound. We stress that the $K^*$ results give rough upper bounds on the various contributions, not only because of the boundedness of the strong amplitudes but also because the phase difference correction, $\gamma_n$, is not known. We also do not compute the total contributions from the various states, as we cannot presently determine their relative phases. Only in the one-loop calculation of the previous section are the relative phases fixed by the model.
Discussion and Conclusions {#disc}
===========================
The results shown in Table I illustrate the two primary conclusions of our analysis: (i) contributions from higher mass intermediate states to the strangeness moments are not necessarily small compared with those from the lightest “OZI allowed" state $K\bar{K}$ ; (ii) estimating these higher mass contributions can entail a significant degree of theoretical uncertainty.
In the one-loop model, the $K^*$ contributions can be as much as an order of magnitude larger than those from the kaon loop. The origin of this result can be traced to two factors: the tensor coupling of the $N\Lambda K^*$ vertex is much larger than the $N
\Lambda K$ coupling, and the cut-off of the Bonn form factor involving the $K^*$ is about twice as large as that involving the kaon ($\Lambda_K = 1.2$ GeV). In the case of the former, omitting the tensor coupling reduces the contribution to the strangeness radius by a factor of five to ten and yields a near exact cancellation between the $KK$, $KK^{*}$, and $K^{*}K^{*}$ contributions. In the case of the strange magnetic moment, the large $K^{*}K^{*}$ and $KK^{*}$ contributions drop by two orders of magnitude when $\kappa$ is set to zero.
The effect of the larger cut-off is particularly emphasized in graphs which contain derivative ([*i.e.*]{} tensor) couplings of the $K^*$. These couplings bring in additional powers of the loop momentum $k$ and the corresponding loop integrals therefore receive larger contributions from $k$ of the order of the cut-off. However, the importance of loop momenta above $\sim$2 GeV points to weaknesses of the one loop approximation. As we discuss in more detail below, the large $K^*K\Lambda$ and $K^*K^*\Lambda$ contributions (1b) appear to result from un-physical, un-realistically large values of the integrand for large loop momenta. Physically realistic contributions from these intermediate states are likely to be much smaller.
In fact, the DR contributions from the $KK^*$ and $K^*K^*$ states are significantly smaller in magnitude than those generated in the loop model, though they are still comparable to, or larger than, the $K\bar{K}$ contribution. The reduction in the magnitude of these contributions from the loop model estimate reflects two factors: the boundedness of the $n\to N\bar{N}$ scattering amplitude in the physical region and the presence of more realistic, non-pointlike $\fns(t)$. Although we have only implemented the boundedness crudely for the $KK^{*}$ and $K^{*}K^{*}$ states, the requirement that the partial waves are bounded in the physical region ($t>4\mns$) is a rigorous one, following from the unitarity of the S-matrix. Since a one-loop calculation is equivalent to a DR in which the $\fns(t)$ are taken to be pointlike and the $A_{J=1}^n$ computed in the Born approximation, the one-loop results do not respect the boundedness requirement. The presence of hadronic form factors \[Eqs. (\[ff\])\] does not remedy this violation since they preserve the on-shell form for the $n\to N\bar{N}$ amplitudes.
In the $K\bar{K}$ case, the unitarity violation of the one-loop calculation was shown to be a serious one [@Mus97b]. For the intermediate states containing a $K^*$, this violation appears to be all the more serious, as a comparison of the DR and loop results suggests. The tensor coupling of the $K^*$ to baryons weights the $K^*K\to N\bar{N}$ and $K^*K^*\to N\bar{N}$ amplitudes more strongly in the physical region, relative to the un-physical region ($t_0\leq t\leq 4\mns$), than in the $K\bar{K}\to N\bar{N}$ case. Consequently, the physical region contributes a substantial fraction of the entries (1b) for the $K^*K$ and $K^*K^*$ states (80% of the total in the $K^*K^*$ case) – even after the imposition of a crude bound on the $A_{J=1}^n$ and inclusion of non-pointlike $F_K^{(s)}(t)$. Had we not imposed even our rough bound, the $K^*K^*$ contribution to $\langle r_s^2\rangle_D$, for example, would have been five times larger. We conclude that the large contributions to the strangeness moments resulting from the one-loop model are not physically realistic.
We emphasize that the DR calculation given here – though containing more physical information than the one-loop model – remains incomplete. A rigorous unitarity bound for the $K^*K$ and $K^*K^*$ amplitudes remains to be implemented, as has been done in the $K\bar{K}$ case. More importantly, the impact of higher order (in $g$) rescattering corrections and possible resonance effects in the $A_{J=1}^n(t_0\leq t\leq 4\mns)$ must also be estimated. In the $K\bar{K}$ case, these effects significantly enhance the $\langle r_s^2\rangle$ contribution over the entry $KK\Lambda$ (1b) in Table I [@Mrm97c]. This enhancement arises primarily from a near threshold $\phi(1020)$-resonance in the $K\bar{K}\to N\bar{N}$ amplitude. Similarly, we expect inclusion of $K^*K$ and $K^*K^*$ rescattering and $\phi'$ resonance effects in the $A_{J=1}^n$ to modify the $K^*K$ and $K^*K^*$ entries in Table I. Unfortunately, sufficient $KK\pi\to N\bar{N}$ (or $KN\to
K\pi N$) and $KK\pi\pi\to N\bar{N}$ ($KN\to KN\pi\pi$ [*etc.*]{}) data do not presently exist to afford a model-independent determination of these effects.
Given that higher mass contributions to the strangeness moments need not be small compared to that from the $K\bar{K}$, it is desireable to reduce the theoretical uncertainty in the former as much as possible. The $K^*K^*\Lambda$ (1b) entry hints at the level of this uncertainty. Our reasonable range" for this contribution allows for about a factor of four to seven variation, which follows from the choice of different, but reasonable, $K^*$ strangeness form factors. Based on our previous study of the $K\bar{K}$ contribution, as well as the behavior of the scattering amplitudes in the physical region, we may reasonably expect a similar level of uncertainty associated with the presently unknown rescattering and resonance effects in the $A_{J=1}^{K^*K,\ K^*K^*}$.
To summarize, we have estimated $K^*K$ and $K^*K^*$ contributions to the nucleon strangeness moments, using two approaches which complement the quark model calculation of Ref. [@Gei97]. Our results confirm the conclusions reached in that work that higher mass hadronic states can be as important as the $K\bar{K}$ state and that a calculation of the strangeness moments based on a truncation in $\Delta E$ is not reliable. Similarly, we illustrate the significant theoretical ambiguities involved in estimating these higher mass contributions – particularly those associated with effects going beyond ${\cal O}(g^2)$ and with the intermediate state strangeness form factors. In this study, we have taken the first steps toward including the latter in a realistic way. We find that inclusion of physically reasonable parameterizations of the $\fns(t)$ can appreciably affect the $K^*K$ and $K^*K^*$ contributions. Even here, however, our efforts are limited by a lack of existing EM data. In the case of higher-order and resonance effects in the strong amplitudes, it should be evident that simple models which do not account for them can produce physically unrealistic estimates of the higher mass intermediate state contributions. Clearly, more sophisticated approaches are needed in order to understand how $s\bar{s}$ pairs live as virtual hadronic states.
We would like to thank D. Drechsel and N. Isgur for useful discussions. This work has been supported in part by FAPESP and CNPq. M.N. would like to thank the Institute for Nuclear Theory at the University of Washington for its hospitality and H.F. acknowledges an HCM grant from the European Union and a DFG habilitation fellowship. MJR-M has been supported in part under U.S. Department of Energy contracts \# DE-FG06-90ER40561 and \# DE-AC05-84ER40150 and under a National Science Foundation Young Investigator Award. HWH has been supported by the Deutsche Forschungsgemeinschaft (SFB 201) and the German Academic Exchange Service (Doktorandenstipendium HSP III/ AUFE).
Vertex Functions: Loops
=======================
In the following appendices we list the explicit expressions for the one-loop diagrams considered in Section \[ext\].
They are numbered as in the figures: (1a) for $M=K^{\ast}$ and $B=B'=\Lambda,\Sigma$; (1b) for $M=M'=K^{\ast}$ and $B=B'=\Lambda,\Sigma$; (1b) for $M=K$, $M'=K^{\ast}$, and $B=B'=\Lambda,\Sigma$; (1c) for $M=K^{\ast}$ and $B=B'=\Lambda,\Sigma$.
$$\begin{aligned}
\Gamma^{(1a)}_\mu(p^\prime,p)& =& ig^2_v Q_B \int \frac{d^4k}
{(2\pi)^4} (F(k^2))^2 D^{\alpha\beta}(k)\left(\gamma_\alpha
+i{\kappa\over2m_N}
\sigma_{\alpha\nu}k^\nu\right)
S(p^\prime-k) \gamma_\mu\times
\nonumber\\*[7.2pt]
&&S(p-k) \left(\gamma_\beta-i{\kappa\over2m_N}\sigma_{\beta\gamma}k^\gamma
\right) \; ,
\label{1a}\end{aligned}$$
$$\begin{aligned}
\Gamma^{(1b)}_\mu(p^\prime,p)& =&- ig^2_v Q_{K^*} \int
\frac{d^4k}{(2\pi)^4} F((k+q)^2)F(k^2) D^{\alpha\lambda}(k+q)
D^{\sigma\beta}(k)\left(\gamma_\alpha + \right.
\nonumber\\*[7.2pt]
&+&\left.i{\kappa\over2m_N}
\sigma_{\alpha\nu}(k+q)^\nu\right)
[(2k+q)_\mu \, g_{\sigma\lambda}-(k+q)_\sigma g_{\lambda\mu}-k_\lambda
g_{\sigma\mu}]\times
\nonumber\\*[7.2pt]
&&S(p-k)\left(\gamma_\beta-i{\kappa\over2m_N}\sigma_{\beta\gamma}
k^\gamma\right) \; ,\;{\mbox{for $M=M^\prime=K^*$}}
\nonumber\\*[7.2pt]
& =&-{g_vg_{ps}F_{K^*K}^{(s)}(0)\over m_{K^*}}
\epsilon_{\mu\nu\lambda\alpha}\int \frac{d^4k}{(2\pi)^4}\left\{F((k+q)^2)F_K
(k^2)D^{\alpha\beta}(k+q)\times\right.
\nonumber\\*[7.2pt]
&&\Delta(k^2)(k+q)^\nu k^\lambda \left(\gamma_\beta +i{\kappa\over2m_N}
\sigma_{\beta\delta}(k+q)^\delta\right)S(p-k)\gamma_5+
\nonumber\\*[7.2pt]
&+&F(k^2)F_K((k+q)^2)D^{\alpha\beta}(k)\Delta((k+q)^2)k^\nu
(k+q)^\lambda \gamma_5 \times
\nonumber\\*[7.2pt]
&&\left.S(p-k)\left(\gamma_\beta -i{\kappa\over2m_N}\sigma_{\beta\delta}
k^\delta\right)\right\}\; ,\;{\mbox{for $M=K\;,M^\prime=K^*$}}
\label{1b}\end{aligned}$$
$$\begin{aligned}
\Gamma^{(1c)}_\mu(p^\prime,p)& =& g^2_v Q_{K^*} \int \frac{d^4k}
{(2\pi)^4} F(k^2) D^{\alpha\beta}(k) \left\{i
\left[\frac{ (q+2k)_\mu}{ (q+k)^2-k^2}
\left(F(k^2)\, - F((k+q)^2)\right) \times \right.\right.
\nonumber\\*[7.2pt]
& & \left(\gamma_\alpha +i{\kappa\over2m_N}\sigma_{\alpha\nu}k^\nu\right)
S(p-k)\left(\gamma_\beta-i{\kappa\over2m_N}\sigma_{\beta\gamma}k^\gamma\right)
- \frac{ (q-2k)_\mu}{ (q-k)^2-k^2} (F(k^2)+
\nonumber\\*[7.2pt]
&&\left.-F((k-q)^2))\left(\gamma_\alpha +i{\kappa\over2m_N}\sigma_{\alpha\nu}
k^\nu\right)
S(p^\prime-k)\left(\gamma_\beta-i{\kappa\over2m_N}\sigma_{\beta\gamma}k^\gamma
\right) \right] \; +
\nonumber\\*[7.2pt]
&+&\;{\kappa\over2m_N}\left[F((k+q)^2)\sigma_{\alpha\mu}
S(p-k)\left(\gamma_\beta-i{\kappa\over2m_N}\sigma_{\beta\gamma}k^\gamma
\right)\right. +
\nonumber\\*[7.2pt]
&-&\left.\left.F((k-q)^2)\left(\gamma_\alpha +i{\kappa\over2m_N}
\sigma_{\alpha\nu}k^\nu
\right)S(p^\prime-k)\sigma_{\beta\mu}\right]\right\} \; ,
\label{1c}\end{aligned}$$
In the above equations we define $p^\prime=p+q$ and use the notation $D_{\alpha\beta}(k)=(-g_{\alpha\beta} +
k_\alpha k_\beta/m_{K^*}^2)(k^2-m_{K^*}^2+i\epsilon)^{-1}$ for the $K^*$ propagator, $\Delta(k^2)=(k^2-m_K^2+i\epsilon)^{-1}$ for the kaon propagator, $S(p-k) = (p\kern-.5em\slash- k\kern-
.5em\slash-m_B+ i\epsilon)^{-1}$ for the hyperon, $B$, propagator with mass $m_\Lambda=1116\MeV$, $m_\Sigma=1193\MeV$ and strangeness charge $Q_B =1$.
Vertex Functions: Dispersion Calculation
========================================
Here, we display the vertex functions for the dispersion relation calculation of Section III. We require the product of propagator denominators and $|F^{(s)}(t)_{VDM}|$ for the cases 3-5. This product is abbreviated by \[denab\] [D]{}\_3 &=&{\[(k-q/2)\^2 --i\]\[(k+q/2)\^2 -- i\] .\
& & .\[(p’-k-q/2)\^2 -m\_\^2-i\] }\^[-1]{}|F\^[(s)]{}(t)\_[VDM]{}|,for case 3 and accordingly for cases 4 and 5. The vertex functions are labelled as in section III (Table I). We obtain:
- Case 3 ($K^*K^*B$ 1a) : $$\begin{aligned}
\Gamma_\mu^{(3)} &=& -iQ_B g_v^2\int\frac{d^4 k}{(2\pi)^4}\,
(\gamma_\alpha +\frac{i \kappa }{2\mn}\sigma_{\alpha\nu}
(p'-k-q/2)^\nu)\\ &{\hphantom{-}}&({/ \!\!\! k}+{/ \!\!\! q}/2+\mn)
\gamma_\mu({/ \!\!\! k}-{/ \!\!\! q}/2+\mn) \nonumber\\
&{\hphantom{-}}&(\gamma_{\alpha'}-\frac{i \kappa}{2\mn}
\sigma_{\alpha'\nu'}(p'-k-q/2)^{\nu'}) \nonumber \\ &{\hphantom{-}}&
(g^{\alpha\alpha'}-(p'-k-q/2)^\alpha
(p'-k-q/2)^{\alpha'}/m_\sst{K^\ast}^2)\, {\cal D}_3\; \nonumber\end{aligned}$$
- Case 4 ($K^*K^*B$ 1b) : $$\begin{aligned}
\Gamma_\mu^{(4)} &=& -iQ_{K^*} g_v^2 \int\frac{d^4 k}{(2\pi)^4}\,
(\gamma_{\beta'}+\frac{i \kappa}{2\mn}\sigma_{\beta'\nu}
(k+q/2)^\nu)\\
&{\hphantom{-}}& (g^{\beta'\beta}-(k+q/2)^{\beta'}(k+q/2)^{\beta}/
m_\sst{K^*}^2) \nonumber \\ &{\hphantom{-}}&
(g^{\alpha\alpha'}-(k-q/2)^\alpha (k-q/2)^{\alpha'}/m_\sst{K^*}^2)
\,({/ \!\!\! p'}-{/ \!\!\! k}-{/ \!\!\! q}/2+\mn)
\nonumber \\ &{\hphantom{-}}&(2k_\mu g_{\beta\alpha}
-g_{\beta\mu}(k+q/2)_\alpha-g_{\alpha\mu}(k-q/2)_\beta)
\nonumber\\
&{\hphantom{-}}&(\gamma_{\alpha'}-\frac{i \kappa}{2\mn}\sigma_{\alpha'\nu'}
(k-q/2)^{\nu'}){\cal D}_4\; \nonumber\end{aligned}$$
- Case 5 ($KK^*B$ 1b) : $$\begin{aligned}
\Gamma_\mu^{(5)} &=&-2 g_{ps}g_v \frac{F_{K^*K}^{(s)}(0)}{m_\sst{K^*}}
\int\frac{d^4 k}{(2\pi)^4}\,
(\gamma_{\beta'}+\frac{i\kappa}{2\mn}\sigma_{\beta'\nu}
(k+q/2)^\nu)\\
& & (g^{\beta'\beta}-(k+q/2)^{\beta'}(k+q/2)^{\beta}/m_\sst{K^*}^2)
\nonumber \\ & &\epsilon_{\sigma\beta\rho\mu}
(k+q/2)^\sigma q^\rho ({/ \!\!\! p'}-{/ \!\!\! k}-{/ \!\!\! q}/2+\mn) \gamma_5
{\cal D}_5\; \nonumber\end{aligned}$$
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Contribution $\langle r_s^2 \rangle_D \; (\mbox{fm}^2) \; \; $ loop $|\langle r_s^2 \rangle_D |\; (\mbox{fm}^2) \; \; $ DR $\mu_s \; $ loop $|\mu_s| \; $ DR
--------------- ---------------------------------------------------------- ------------------------------------------------------------- ------------------ -------------------
$KKB$ 1a 0.006 $0.001$ -0.107 $0.023$
$KKB$ 1b -0.009 $0.036$ -0.078 $0.143$
$KKB$ 1c -0.004 0 -0.069 0
$KKB$ tot $-0.007$ $-0.24$
$K^*K^*B$ 1a $ 0.075$ 0.001 $-2.283$ 0.053
$K^*K^*B$ 1b $-0.038$ $0.003\to 0.012$ $-2.343$ $0.059\to 0.408$
$K^*K^*B$ 1c $-0.007$ $0$ 0.499 $0$
$K^*K^*B$ tot $0.030$ $-4.127$
$KK^*B$ 1b 0.078 0.035 1.015 0.425
total $0.101$ $-3.352$
: \[kstartab4\] Intermediate state contributions to the strange magnetic moment $\mu_s$ and the electric strangeness radius $\langle
r_s^2 \rangle_D $. The contributions are labelled according to the diagrams in Fig. 1 and the intermediate state particles.
Case I II III
------ -------------------- -------------------- --------------------------------
1 $\qquad K\qquad$ $\qquad K\qquad$ $\qquad\Lambda,\;\Sigma\qquad$
2 $\Lambda,\;\Sigma$ $\Lambda,\;\Sigma$ $K$
3 $\Lambda,\;\Sigma$ $\Lambda,\;\Sigma$ $K^*$
4 $K^*$ $K^*$ $\Lambda,\;\Sigma$
5 $K$ $K^*$ $\Lambda,\;\Sigma$
: \[kstartab1\] Particles assigned to the internal lines in the loop diagram of Fig. 2.
[^1]: National Science Foundation Young Investigator
[^2]: Theoretical uncertainties associated with SU(3) breaking qualify the conclusions drawn from deep inelastic scattering experiments, however.
[^3]: As noted in [@ohta; @Mrm97a] and elsewhere this procedure is not unique since the Ward-Takahashi identity does not restrict the transverse part of the vertex.
[^4]: The validity of this assumption is discussed in more detail in the following section.
[^5]: Note also that the small SU(3) values for the $\Sigma K N$ couplings [@hol89] lead to a strong suppression of the contributions from $\Sigma K$ intermediate states [@Had]. This argument does not affect, however, the $\Sigma^* K$ and $\Sigma^* K^*$ contributions.
[^6]: The equivalence holds only when the hadronic form factors of Eq. (\[ff\]) are set to unity.
[^7]: A more sophisticated treatment, including the tails of the $\phi$ and $\omega$, would – as in the purely EM case – affect the shape of the form factor near the $\phi'$ peak and the resultant $KK^*$ spectral function.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'M.V. Ignatev[^1]'
title: |
Subregular characters of the unitriangular group\
over a finite field
---
Let $k$ be a field and $n$ be a natural number. By $G_n(k)=\mathrm{UT}(n, k)$ we denote the group of all unipotent lower-triangular $n\times n$-matrices with coefficients from $k$; this group is called a *unitriangular* group. By $\mathfrak{g}_n(k)=\mathfrak{ut}(n, k)$ we denote its Lie algebra over $k$; this Lie algebra consists of all nilpotent lower-triangular matrices with coefficients in $k$.
If $k=\mathbb{F}_q$ is a finite field, then $G_n(q)=G_n(k)$ is a finite group; so, there are finitely many classes of equivalency of irreducible complex representations of this group. A description of all irreducible characters (or of some series of them) is a classical problem of representation theory. The orbit method of A.A. Kirillov [@Kirillov1], [@Kirillov2] allows to reduce the similar problem of description of unitary irreducible representations of Lie groups to the problem of classification of coadjoint orbits; this method is also valid for $G_n(q)$ (see. [@Kazhdan]), but complete classification of coadjoint orbits for an arbitrary $n$ is unknown.
A description of orbits of the principal series (i.e., orbits of maximum dimension) of Lie groups was presented in the pioneering work on the orbit method [@Kirillov2]; it’s also valid over a finite field [@Kirillov3]. In C. Andre’s works (see [@Andre1], [@Andre2]) so-called *basic characters* are described (in particular, an exact formula for characters of the principal series is found).
The problem of description of orbits, representations and characters of sub-maximum dimension is a natural generalisation of these results; such orbits, representations and characters are called *subregular*. They play an important role in algebraic geometry and $K$-theory (see, for example, [@Lusztig]). Subregular orbits are described in [@IgnatevPanov]. The main goal of this paper is to give an exact formula for the corresponding characters. This formula shows that subregular characters (as characters of the principal series) can be described in terms of coefficients of minors of the characteristic matrix.
The paper is organized as follows. In section \[paragr\_osn\_ser\] we collect some basic facts about the group $G_n$ and discuss Andre’s formulae for characters of the principal series (Theorem \[char\_reg\]). In section \[paragr\_sreg\_formulir\] we formulate the Main Theorem including exact formulas for subregular characters (Theorem \[theo\_sreg\]). More precisely, for an arbitrary subregular character we find the elements such that the value of this character on the conjugacy classes of these elements is non-zero and compute this value. Section \[class\_sopr\] is devoted to the description of these conjugacy classes (Theorem \[theo\_klass\_sopr\]). In section \[paragr\_semidirect\] we recall some facts about semi-direct decomposition of the group $G_n$, wich are needed for the proof of the main Theorem, which is given in section \[paragr\_proof\]. Finally, section \[utochn\_obob\] includes some remarks about discussed problems.
\[paragr\_osn\_ser\]
Let $k=\mathbb{F}_q$ and $\mathrm{char}\,k=p$, i.e., $q=p^r$, where $p$ is a prime number. Throughout the paper, we suppose $p\geqslant
n$. According to [@Kazhdan], under these assumptions the orbit method is valid: there is a one-to one correspondence between the set of all irreducible complex characters of our group and the set of all coadjoint orbits, i.e, $G_n(q)$-orbits in the dual space $\mathfrak{g}_n^*(q)$. Moreover, to each orbit $\Omega\subset\mathfrak{g}_n(q)$ the character $$\chi_{\Omega}(\mathrm{exp}\,a)
=q^{-\frac{1}{2}\dim\,\Omega}\cdot\sum_{f\in\Omega}
\theta(f(a)),\;a\in\mathfrak{g}_n(q),\label{tupaya_formula_char}$$ is assigned (here $\theta\colon\mathbb{F}_q\to\mathbb{C}$ is a non-trivial character of the additive group of the field $\mathbb{F}_q$). On the other hand, there are no exact formulas that allow to find the value of the character of a given orbit on a given element of the unitriangular group.
**Remark ..** [There exists a form $\langle A, B\rangle=\mathrm{tr}(AB)$ which is non-degenerate on $\mathfrak{gl}_n(k)$. This allows to identify $\mathfrak{ut}^*(n,
q)$ with the space of all nilpotent upper-triangular matrices by the formula $f(x)=\langle f, x\rangle=\sum_{i, j}\xi_{ji}x_{ij}$, where $x=(x_{ij})\in\mathfrak{g}_n(q),\,f=(\xi_{ij})\in\mathfrak{g}_n^*(q)$. Then the coadjoint action $$K(g)\colon\mathfrak{g}^*_n\to\mathfrak{g}^*_n\colon(K(g)f)(x)=f(\mathrm{Ad}_
{g^{-1}}x),\,g\in G_n,\,f\in\mathfrak{g}_n^*,x\in\mathfrak{g}_n$$ is given by $$K(g)\colon
x\mapsto(gxg^{-1})_{high},\,x\in\mathfrak{g}_n^*,g\in\mathfrak{g}_n,$$ where $(a)_{high}$ denotes the following matrix: its elements below the main diagonal coincide with the corresponding elements of the matrix $a$ and its elements on the main diagonal and below are equal to zero.]{}
Representations of maximum dimension play an important role in this theory. These representations (and their orbits and characters) form the so-called *principal series*; orbits of maximum dimension are called *regular*. We’ll use the following
**Notation ..** [Let $n\in\mathbb{N}$. Set $n_0=[n/2]$, $n_1=[(n-1)/2]$. Note that $n=n_0+n_1+1$.]{}
**Notation ..** [Let $g=(y_{ij})\in\mathrm{Mat}(n, k)$. By $\Delta_{i_1,\ldots,i_k}^{j_1,\ldots,j_k}(g)$ we denote the minor of the matrix $g$ with the rows $i_1,\ldots,i_k$ and the columns $j_1,\ldots,j_k$ in the given order (for a given $1{\leqslant}k{\leqslant}n,1{\leqslant}i_1,\ldots,i_k,j_1,\ldots,j_k{\leqslant}n$). In particular, for an arbitrary $1\leqslant d\leqslant n_0$ we set $$\Delta_d(g)=
\Delta_{n-d+1,n-d+2,\ldots,n}^{1,2\ldots,d}(g).$$ We also denote $\Delta^X(g)=\Delta_{\sigma(i_1,\ldots,i_k)}^{\tau(j_1,\ldots,j_k)}(g)$, where $X$ is the set of the pairs $X=\{(i_1,j_1),\ldots,(i_k,j_k)\}$ and $\sigma, \tau$ are permutations such that $\sigma(i_1)<\ldots<\sigma(i_k)$ and $\tau(j_1)<\ldots<\tau(j_k)$.]{}
A complete description of orbits of the principal series is well-known [@Kirillov3]:
**Theorem ..** [Let $k=\mathbb{F}_q$ be a finite field. An arbitrary regular orbit has the following defining equations: $$\Delta_d(^ta)=\beta_d,\;a\in\mathfrak{g}_n^*(q),\;1\leqslant d\leqslant n_0$$ (here $^ta$ is the transpose of the matrix $a$, $\beta_d$ are arbitrary scalars from $\mathbb{F}_q$, $\beta_1,\ldots,\beta_{n_0-1}\in\mathbb{F}_q^*$ and $\beta_{n_0}\neq
0$ for odd $n$).]{} $\square$
\[theo\_reg\]
**Corollary ..** [For an arbitrary orbits of the principal series there exists the unique *canonical form of a regular orbit*, i.e., the matrix of the form $$f=\begin{pmatrix}
0&\ldots&0&\xi_{1, n}\\
0&\ldots&\xi_{2, n-1}&0\\
\vdots&\ddots&\vdots&\vdots\\
0&\ldots&0&0
\end{pmatrix},$$ where $\xi_{1, n}=\beta_1,\xi_{d, n-d+1}=
\dfrac{\beta_d}{\beta_{d-1}}, d=2,\ldots,n_0$, that consists in this orbit.]{} $\square$
\[can\_reg\]
Moreover, dimension of a representation (and an orbit) of the principal series is known [@Lehrer]:
**Theorem ..** [Let $\mu(n)=(n-2)+(n-4)+\ldots$ and $T_{\Omega}$ be the representation that corresponds to an orbit $\Omega$ of the principal series. Then $\dim\Omega=2\mu(n),\;\dim T_{\Omega}=q^{\mu(n)}$.]{} $\square$
Note that there are exactly $q^{2\mu(n)}$ points on an orbit $\Omega$ of dimension $2\mu(n)$ [@Kirillov4].
In this section we recall the formula for characters of the principal series, i.e., characters of the form $\chi=\chi_f$, where $f\in\Omega_f\subset{\mathfrak{g}}_n^*$ is the canonical form on a regular orbit. More precisely, let $\chi$ be an irreducible character of the group $G_n$. By $\mathrm{Supp}\mathop\chi$ denote its *support* (i.e., the set $\{g\in G_n\mid\chi(g)\neq0\}$); obviously, the support is a union of certain conjugacy classes and the value of the character on an arbitrary conjugacy class is constant. So, it’s enough to describe the support explicitly and compute the value of the character on an arbitrary conjugacy class containing in the support.
We’ll use the following notations from Andre’s paper [@Andre2]:
**Notation ..** [We denote by $\Phi(n)$ the set of all pairs $\{(i, j)\mid
1{\leqslant}j<i{\leqslant}n\}$ (we call them *roots*, because to each $(i,j)\in\Phi(n)$ the root vector $e_{ji}$, i.e., the matrix with 1 in the $(j,i)$-th entry and zeroes elsewhere, is assigned). Let $D\subset\Phi(n)$ be a subset that contains at most one element from each row and at most one element from each column; then this subset is called *basic*. If a basic subset consists from the roots of the form $(n-j+1,j)$ then it’s called *regular*.]{}
**Example ..**
Here we draw one of regular subsets: $D=\{(6, 1), ((4, 3))\}
\subset\Phi(6)$. The entries $(i, j)\in D$ are marked by the symbol $\otimes$:
$\mymatrix{
{\phantom{\otimes}}& {\phantom{\otimes}}&{\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}\\
\Top{2pt}\Rt{2pt} {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}\\
{\phantom{\otimes}}& \Top{2pt}\Rt{2pt} {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}\\
{\phantom{\otimes}}& {\phantom{\otimes}}& \Top{2pt}\Rt{2pt} \otimes& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}\\
{\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& \Top{2pt}\Rt{2pt} {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}\\
\otimes & {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& \Top{2pt}\Rt{2pt} {\phantom{\otimes}}& {\phantom{\otimes}}\\
}$
**Notation ..** [For an arbitrary basis subset $D\subset\Phi(n)$ and arbitrary map $\varphi\colon D\to{\mathbb{F}}_q^*$ consider the element of the group $G_n(q)$ of the form $$x_D(\varphi)=1_n+\sum_{(n-j+1, j)\in
D}\varphi(n-j+1, j)e_{n-j+1, j},$$ where $1_n$ is the identity $n\times n$-matrix (if $D=\varnothing$ then $x_D(\varphi)=1_n$). By $\EuScript K_D(\varphi)$ denote the conjugacy class of this element and by $\EuScript{K}_{\mathrm{reg}}$ denote the (disjoint) union of $\EuScript K_D(\varphi)$ such that $D$ is a regular subset and $\varphi\colon D\to{\mathbb{F}}_q^*$ is a map.]{}
**Definition ..** [Let $D$ be a subset of $\Phi(n)$. A root $(i,j)\in\Phi(n)$ is called $D$-*regular*, if $(i,k)\notin D$ and $(k,j)\notin D$ for any $i>k>j$. By $R(D)$ denote the set of all $D$-regular roots.]{}
**Example ..**
Let $n=6$; $D=\{(3, 2), (6, 4)\}\subset\Phi(6)$ is a basic subset. Here we mark the roots $(i, j)\notin R(D)$:
$\mymatrix{
{\phantom{\otimes}}& {\phantom{\otimes}}&{\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}\\
\Top{2pt}\Rt{2pt} {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}\\
\gray {\phantom{\otimes}}& \Top{2pt}\Rt{2pt} \otimes& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}\\
{\phantom{\otimes}}& \gray {\phantom{\otimes}}& \Top{2pt}\Rt{2pt} {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}\\
\gray {\phantom{\otimes}}& \gray {\phantom{\otimes}}& \gray {\phantom{\otimes}}& \Top{2pt}\Rt{2pt} \otimes& {\phantom{\otimes}}& {\phantom{\otimes}}\\
{\phantom{\otimes}}& \gray {\phantom{\otimes}}& {\phantom{\otimes}}& \gray {\phantom{\otimes}}& \Top{2pt}\Rt{2pt} {\phantom{\otimes}}& {\phantom{\otimes}}\\
}$
Now, set $\Phi_{{\mathrm{reg}}}=\{(i,j)\in\Phi(n)\mid i>n-j+1\}$, $m_D=|R(D)\cap\Phi_{{\mathrm{reg}}}|$ for an arbitrary regular subset $D\subset\Phi(n)$. Fix also any non-trivial character $\theta\colon{\mathbb{F}}_q\to\mathbb{C}$ of the additive group of the ground field.
**Theorem ..**
Let $\Omega=\Omega_f\subset{\mathfrak{g}}_n^*(q)$ be a regular orbit, let $f=(\xi_{ij})$ be the canonical form of this orbit and $\chi=\chi_f$ be the corresponding character. Then
1. ${\mathrm{Supp}\mathop{\chi}}=\EuScript{K}_{\mathrm{reg}}$.
2. $\chi(g)=q^{m_D}\cdot\theta_f(e_D(\varphi))$ for any $g\in\EuScript
K_D(\varphi)\subset\EuScript{K}_{\mathrm{reg}}$, where $e_D(\varphi)=x_D(\varphi)-1_n\in{\mathfrak{g}}_n(q)$ and $\theta_f\colon{\mathfrak{g}}_n(q)\to\mathbb{C}$ is given by the formula $$\theta_f(x)=\theta(f(x))=\prod_{(i,j)\in\Phi(n)}\theta(\xi_{ji}x_{ij}),
\quad x=(x_{ij})\in{\mathfrak{g}}_n(q).$$
**Proof**. [This is a special case of [@Andre2 Theorem 5.1].]{} $\square$
\[char\_reg\]
In Andre’s work [@Andre2] one can find explicit description of $\EuScript K_D(\varphi)$ for any basic subset $D\subset\Phi(n)$. More precisely, $g\in\EuScript K_D(\varphi)\subset G_n$ if and only if $$\Delta^{R_D(i, j)}(g)=\Delta^{R_D(i,
j)}(x_D(\varphi))\label{reg_uravn}$$ for all $(i,j)\in R(D)$. Here $R_D(i, j)=\{(i, j)\}\cup\{(k, l)\in
D\mid l> j\text{ and }k< i\}$.
In particular, if $m=\max_{(i, j)\in D}j$ then $$y_{ij}=0\text{ if }j>m\text{ or }i<n-m+1.\label{reg_zero}$$ One can see that characters of the principal series and regular orbits can be described in terms of minors of matrices from $G_n$; by definition, this is also true for all Andre’s basic characters. In the next section we’ll see that this is *not* true for subregular characters in general (see [@Andre2]).
\[paragr\_sreg\_formulir\] It follows from Theorem \[theo\_reg\] that a coadjoint orbit “in general position” has maximum dimension $2\mu(n)$. More precisely, the set of points such that their orbits are regular is a dense open subset of ${\mathfrak{g}}_n^*(q)$ in Zariski topology. Denote this subset by $\mathcal O_{{\mathrm{reg}}}$, it’s given by the set of inequalities $\Delta_d\neq0$, $1{\leqslant}d{\leqslant}n_1$ (thats’s why we use the term “regular orbits”). Recall that $n=n_0+n_1+1$, where $n_0=[n/2]$, $n_1=[(n-1)/2]$.
On the other hand, for an arbitrary $1{\leqslant}d{\leqslant}n_1$ one can consider the hyper-surface $\mathcal O_d\subset{\mathfrak{g}}_n^*$ given by $\Delta_d=0$. This hyper-surface splits into the union of certain orbits.
**Definition ..** [An orbit $\Omega\in\mathfrak{g}_n^*$ (and the corresponding representation $T_{\Omega}$ and character $\chi_{\Omega}$) is called *subregular*, if $\dim\Omega=2\mu(n)-2$ (resp., $\dim T_{\Omega}=q^{\mu(n)-1}$), i.e., it has dimension $=$ dimension of a regular orbit $-\;2$ (in the other words, this orbit has sub-maximum dimension, because any orbit is even-dimensional [@Kirillov4]).]{}
Each subregular orbits contains in the unique $\mathcal O_d$ and has maximum dimension among all orbits containing in $\mathcal O_d$. Hence, we say that a subregular orbit is $d$-*subregular* if it contains in $\Omega\subset O_d$. In [@IgnatevPanov] defining equations of subregular orbits are found, but we will not use them in the sequel; we only list here elements of ${\mathfrak{g}}^*_n$ such that their orbits are subregular.
**Definition ..**
An element $f=(\xi_{ij})\in{\mathfrak{g}}^*_n$ is called *a canonical form of a subregular orbit* (of the first, second or third type resp.) if
1. There are a number $1{\leqslant}d<n_1$ and $\beta_1,\ldots,\beta_{d-1},\beta',\beta'',\beta_{d+1}$, $\ldots$, $\beta_{n_0-1}\in k^*$, $\beta_{n_0},\beta\in k$, where $\beta_{n_0}\neq0$ for odd $n$, such that $\xi_{j,n-j+1}=\beta_j$ for any $1{\leqslant}j{\leqslant}d-1$ and $d+2{\leqslant}j{\leqslant}n_0$, $\xi_{d,n-d}=\beta',\xi_{d+1,n-d+1}=\beta'',\xi_{n-d,n-d+1}=\beta$, and $\xi_{ij}=0$ for all other roots.
2. $n$ is odd and there are $\beta_1,\ldots,\beta_{n_1-1}\in k^*,\beta',\beta''
\in k$ such that $\xi_{j,n-j+1}=\beta_j$ for any $1{\leqslant}j{\leqslant}n_1-1$, $\xi_{n_1,n_0+1}=\beta',\xi_{n_1+1,n_0+2}=\beta''$,and $\xi_{ij}=0$ for all other roots.
3. $n$ is even and there are $\beta_1,\ldots,\beta_{n_1-1},\beta\in k^*,
\beta',\beta''\in k$, such that $\xi_{j,n-j+1}=\beta_j$ for any $1{\leqslant}j{\leqslant}n_1-1$ and either $\xi_{n_1,n_0+1}=\beta,\xi_{n_1+1,n_0+2}=\beta',\xi_{n_1+2,n_0+2}=\beta''$ (and all other $\xi_{ij}=0$), or $\xi_{n_1,n_0}=\beta',\xi_{n_1+1,n_0+2}=\beta$ (and all other $\xi_{ij}=0$).
\[defi\_sreg\_orbits\]
**Example ..**
Here we draw some canonical forms of subregular orbits for $n=8$, $d=1$ and $d=2$. The symbol $\otimes$ marks the roots $(i,
j)$ such that $\xi_{ji}\neq0$:
$\mymatrix{
{\phantom{\otimes}}& {\phantom{\otimes}}&{\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}&{\phantom{\otimes}}&{\phantom{\otimes}}\\
\Top{2pt}\Rt{2pt} \otimes & {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}&{\phantom{\otimes}}&{\phantom{\otimes}}\\
{\phantom{\otimes}}& \Top{2pt}\Rt{2pt} {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}&{\phantom{\otimes}}&{\phantom{\otimes}}\\
{\phantom{\otimes}}& {\phantom{\otimes}}& \Top{2pt}\Rt{2pt} {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}&{\phantom{\otimes}}&{\phantom{\otimes}}\\
{\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& \Top{2pt}\Rt{2pt} \otimes& {\phantom{\otimes}}& {\phantom{\otimes}}&{\phantom{\otimes}}&{\phantom{\otimes}}\\
{\phantom{\otimes}}& {\phantom{\otimes}}& \otimes& {\phantom{\otimes}}& \Top{2pt}\Rt{2pt} {\phantom{\otimes}}& {\phantom{\otimes}}&{\phantom{\otimes}}&{\phantom{\otimes}}\\
\otimes & {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& \Top{2pt}\Rt{2pt} {\phantom{\otimes}}&{\phantom{\otimes}}&{\phantom{\otimes}}\\
{\phantom{\otimes}}& \otimes& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}&\Top{2pt}\Rt{2pt} \otimes &{\phantom{\otimes}}\\
}$$\mymatrix{
{\phantom{\otimes}}& {\phantom{\otimes}}&{\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}&{\phantom{\otimes}}&{\phantom{\otimes}}\\
\Top{2pt}\Rt{2pt} {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}&{\phantom{\otimes}}&{\phantom{\otimes}}\\
{\phantom{\otimes}}& \Top{2pt}\Rt{2pt} \otimes& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}&{\phantom{\otimes}}&{\phantom{\otimes}}\\
{\phantom{\otimes}}& {\phantom{\otimes}}& \Top{2pt}\Rt{2pt} {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}&{\phantom{\otimes}}&{\phantom{\otimes}}\\
{\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& \Top{2pt}\Rt{2pt} \otimes& {\phantom{\otimes}}& {\phantom{\otimes}}&{\phantom{\otimes}}&{\phantom{\otimes}}\\
{\phantom{\otimes}}& \otimes& {\phantom{\otimes}}& {\phantom{\otimes}}& \Top{2pt}\Rt{2pt} {\phantom{\otimes}}& {\phantom{\otimes}}&{\phantom{\otimes}}&{\phantom{\otimes}}\\
{\phantom{\otimes}}& {\phantom{\otimes}}& \otimes& {\phantom{\otimes}}& {\phantom{\otimes}}& \Top{2pt}\Rt{2pt} \otimes &{\phantom{\otimes}}&{\phantom{\otimes}}\\
\otimes & {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}&\Top{2pt}\Rt{2pt} {\phantom{\otimes}}&{\phantom{\otimes}}\\
}$
**Theorem ..** [The orbit of a canonical form is subregular. More over, for each subregular orbit there exists the unique canonical form that contains in this orbit.]{}
**Proof**. [ See [@IgnatevPanov] (there the case $\mathrm{char}\,k=0$ is considered, but the proof is still valid in our case $k={\mathbb{F}}_q$ if ${\mathrm{char}\,}k=p{\geqslant}n$).]{} $\square$
This allows to define $d$-subregular forms and subregular orbits of the first, second or third type by the obvious way. Clearly, if $1{\leqslant}d<n_1$, then $d$-subregular orbit is of the first type, and $n_1$-subregular orbits are of the second type, if $n$ is odd, and the third one, if $n$ is even.
So, the main goal is to find exact formulas for all characters of the form $\chi_f$, where $f$ is a canonical form of a subregular orbit. As for characters of the principal series, we’ll describe the support of a given subregular character and compute the value of this characters on an arbitrary conjugacy class contained in the support (we’ll firstly consider the case $1{\leqslant}d<n_1$ and then (in section \[utochn\_obob\]) the case of $n_1$-subregular orbits).
**Notation ..** [For an arbitrary $1{\leqslant}d< n_1$ by $D_0(d)$ and $D_1(d)$ denote one of the following sets resp.: $$\begin{split}
D_0(d):\quad&\varnothing,\{(n-d+1, d)\},\{(n-d,d)\},
\{(n-d+1,d+1)\},\\
&\{(n-d,d), (n-d+1,d+1)\},\\
D_1(d):\quad&\{(d+1,d), (n-d+1,n-d)\},\\
&\{(d+1,d), (n-d+1,n-d), (n-d,d), (n-d+1,d+1)\}.
\end{split}$$]{}
**Definition ..** [A subset $D\subset\Phi(n)$ is called $d$-*subregular* ($1{\leqslant}d<n_1$) if it has the form $D=D'\cup
D_i(d)$, where either $i=0$ or $i=1$ and $D'$ is a regular subset that doesn’t consist the roots $(n-d+1,d)$ and $(n-d+1, d+1)$. Note that subregular subsets consisting $D_1(d)$ are not basic in general.]{}
[\[defi\_sreg\_subset\]]{}
**Remark ..** [Denote ${\mathrm{Supp}\mathop{f}}=\{(i,
j\in\Phi(n))\mid f(e_{ij})\neq0\}$. One can see that $D\subset{\mathrm{Supp}\mathop{f}}\cup(n-d+1,n-d)\cup(n-d+1,d)$, if $f$ is a canonical form of a subregular orbit.]{}
**Example ..**
Here we draw one of $1$-subregular subsets for $n=6$:
$D=\{(2, 1), (5, 1), (6, 2), (4, 3), (6, 5)\}\subset\Phi(6)$.
$\mymatrix{
{\phantom{\otimes}}& {\phantom{\otimes}}&{\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}\\
\Top{2pt}\Rt{2pt} \otimes & {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}\\
{\phantom{\otimes}}& \Top{2pt}\Rt{2pt} {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}\\
{\phantom{\otimes}}& {\phantom{\otimes}}& \Top{2pt}\Rt{2pt} \otimes& {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}\\
\otimes & {\phantom{\otimes}}& {\phantom{\otimes}}& \Top{2pt}\Rt{2pt} {\phantom{\otimes}}& {\phantom{\otimes}}& {\phantom{\otimes}}\\
{\phantom{\otimes}}& \otimes& {\phantom{\otimes}}& {\phantom{\otimes}}& \Top{2pt}\Rt{2pt} \otimes& {\phantom{\otimes}}\\
}$
Define $R(D)$, $x_D(\varphi)$, $e_D(\varphi)$, $\EuScript
K_D(\varphi)$, $\theta$ and $\theta_f$, as for regular orbits. For a subregular orbit $\Omega_f\subset\mathcal O_d$ denote $$\EuScript K_f=\bigcup_{(D,\varphi)}\EuScript K_D(\varphi),$$ where the union is over all $d$-subregular subsets $D$ and over maps $\varphi\colon D\to{\mathbb{F}}_q^*$ such that $$\xi_{d,n-d}\cdot\varphi(d+1,d)=\xi_{d+1,n-d+1}
\cdot\varphi(n-d+1,n-d),\text{where
$f=(\xi_{ij})$}.\label{dop_usl}$$
For a $d$-subregular subset $D$ containing $D_1(d)$ we denote $$\Phi_d=\{(i,j)\in\Phi(n)\mid i>n-j+1, j\notin\{d, n-d\},i\notin
\{n-d+1, n-d\}\}.$$ Finally, for a $d$-subregular subset $D$ let $$m_D=\begin{cases}|R(D)\cap\Phi_{{\mathrm{reg}}}|-1,&\text{if }D\supset
D_0(d),\\
|R(D)\cap\Phi_d|+n-2d-1,&\text{if }D\supset D_1(d).
\end{cases}$$
**Theorem ..**
Let $1{\leqslant}d<n_1$, let $\Omega_f\subset\mathcal
O_d\subset{\mathfrak{g}}_n^*(q)$ be a subregular orbit, and $\chi=\chi_f$ be the corresponding character. Then
1. ${\mathrm{Supp}\mathop{\chi}}=\EuScript K_f$.
2. $\chi(g)=q^{m_D}\cdot\theta_f(e_D(\varphi))$ for an arbitrary $g\in\EuScript K_D(\varphi)\subset\EuScript K_f$.
**Sketch of the proof**.
In Lemma \[lemm\_proofs\_inv\] we present certain equations the element $x_D(\varphi)$ satisfies, and show that the ideal $J$ of $k[G_n]$ generated by these equations is invariant under the adjoint action. In Lemma \[lemm\_proofs\_prim\] we prove that $J$ is a prime ideal. In Lemma \[lemm\_centr\] we find the stabilizer $\mathcal
C=\mathrm{Stab}_{\,G_n}(x_D(\varphi))$ and prove that $$\dim\EuScript K_D(\varphi)={\mathrm{codim}\mathop}{\mathcal C}=\dim V(J)$$ (here $V(J)=\{g\in G_n\mid F(g)=0\text{ for all }F\in J\}$, as usual). This shows that $\EuScript K_D(\varphi)=V(J)$, i.e., our equations are exactly the defining equations for the conjugacy class of the element $x_D(\varphi)$ (Theorem \[theo\_klass\_sopr\]).
On the other hand, in Lemmas \[lemm\_O\_1\_neq\] — \[lemm\_for\_all\_d\] we prove that if $\chi(g)\neq0$, then $g$ satisfies these equations (and the additional equation (\[dop\_usl\])), and compute the value $\chi(g)$ in this case. This concludes the proof.
$\square$
\[theo\_sreg\]
**Remark ..** [If $D\supset D_1(d)$ is $d$-subregular, ${\varphi}, {\widetilde}{\varphi}$ are maps from $D$ to ${\mathbb{F}}_q^*$, and $$\begin{split}&{\varphi}(d+1, d)\cdot{\varphi}(n-d+1, d+1)+
{\varphi}(n-d, d)\cdot{\varphi}(n-d+1,n-d)=\\
&={\widetilde}{\varphi}(d+1, d)\cdot{\widetilde}{\varphi}(n-d+1, d+1)+{\widetilde}{\varphi}(n-d,
d)\cdot{\widetilde}{\varphi}(n-d+1, n-d),
\end{split}$$ then ${\EuScript{K}}_D({\varphi})={\EuScript{K}}_D({\widetilde}{\varphi})$ (see section \[class\_sopr\]). But one can see (using the results of section \[class\_sopr\]) that in this case $\theta_f(e_D({\varphi}))=\theta_f(e_D({\widetilde}{\varphi}))$. Since $m_D$ is independent of ${\varphi}$, the character value in the formulation of the Theorem is well-defined.]{}
It’s easy to prove a variant of this Theorem for the case of $n_1$-subregular orbits by some modifications of definitions and formulations (see section \[utochn\_obob\]).
\[class\_sopr\] Here we present an explicit description of the conjugacy class ${\EuScript{K}}_D(\varphi)$ of an element $x_D(\varphi)$. Fix an arbitrary $1{\leqslant}d<n_1$ and a subregular subset $D\supset D_1(d)$ (in this section and in the two next sections we consider the case $D(d)=D_1(d)$; the case $D(d)=D_0(d)$ is similar, see section \[utochn\_obob\]).
**Notation ..**
It’s convenient to split a $d$-subregular subset into the union $$D=D^-\sqcup D_1(d)\sqcup D^+,$$ where $D^-=\{(i,j)\in D\mid j<d\}$, $D^+=\{(i,j)\in D\mid d<j<n-d\}$.
Recall that $D'=D\setminus D_1(d)=D^-\sqcup D^+$ is a regular subset (see Definition \[defi\_sreg\_subset\]). Let $D''=D\setminus\{(n-d,
d), (n-d+1, d+1)\})$ (it’s a basic subset of $\Phi(n)$, and $D'\subset D''\subset D$).
It’s impossible to describe $\EuScript K_D(\varphi)$ in terms of minors of matrices $g=(y_{ij})\in G_n$. Let $(i, j)\in D^+$ and $m=\max_{(i, j)\in D^+}j$. Consider the following polynomials: $$\alpha_{ij}=\sum_{l=n-m+1}^{n-d}y_{n-d+1,l}y_{l,j},\quad \beta_{ij}=
\sum_{l=d+1}^{m}y_{i,l}y_{l,d},\quad
\gamma=\sum_{l=d+1}^{n-d}y_{n-d+1,l}y_{l,d}$$ (more conceptual description of these polynomials is given in section \[utochn\_obob\]). For simplicity, we’ll write $\Delta_{ij}$ instead of $\Delta^{R_{D''}(i,j)}(g)$, and $y_{\beta},
y_{\alpha}$ instead of $y_{d+1,d},y_{n-d+1,n-d}$ resp. ($R_{D''}(i,j)$ are defined similar to $R_D(i, j)$ in (\[reg\_uravn\])).
Let us now start with proving Theorem \[theo\_sreg\]. Let $c_0\in
k$, $c_{\alpha}, c_{\beta},c_{ij}\in k^*$, where $(i,j)\in D'$, are arbitrary scalars. Consider the ideal $J$ of $k[G_n]$ generated by the elements $$\begin{split}
&y_{\alpha}-c_{\alpha}, y_{\beta}-c_{\beta},\gamma-c_0,\\
&\alpha_{ij},\beta_{ij},\quad(i,j)\in D^+,\\
&\Delta_{ij}-c_{ij},\quad(i,j)\in D',\\
&\Delta_{ij},\quad(i,j)\in R(D'')\setminus D'.
\end{split}\label{sreg_uravn}$$
**Lemma ..** [$J$ is $G_n$-invariant, i.e., if $g\in V(J)$, then $xgx^{-1}\in V(J)$ for all $x\in G_n$.]{}
**Proof**.
Since the set $D''$ is basic, all $\Delta_{ij}$ invariant [@Andre2 Lemma 2.1].
Let $g=(y_{ij})\in V(J)$. Denote $x_{rs}(\lambda)=1+\lambda e_{rs}$. Since every element $x\in G_n$ can be written as a product $x=x_{r_1s_1}(\lambda_1)\ldots x_{r_ms_m}(\lambda_m)$ for a certain $m$ and $(r_i,s_i)\in\Phi(n)$, $\lambda_i\in k$, it’s enough to prove that if $x=x_{rs}(\lambda)$, then $xgx^{-1}\in V(J)$. But if $x$ has this form, then $$(xgx^{-1})_{ij}=\begin{cases} y_{ij},&\text{if
}i\neq r\text{ and
}j\neq s,\\
y_{ij},&\text{if }i=r\text{ and }j{\geqslant}s\text{, or }j=s\text{ and }i{\leqslant}r,\\
y_{rj}+\lambda y_{sj},&\text{if }i=r,j<s,\\
y_{is}-\lambda y_{ir},&\text{if }j=s,i>r.
\end{cases}$$ Hence, $y_{\alpha},y_{\beta}$ are invariant obviously, and the proof of invariance of other elements is by direct enumeration of possible values of $r$ and $s$.
For example, consider the polynomial $\alpha_{ij}$. If $s{\leqslant}j$ or $r> n-d+1$, then all coordinate functions involved in this polynomial are invariant themselves; this is also true, if $r=n-d+1$, $s{\leqslant}n-m+1$. If $r=n-d+1,s> n-m+1$, then $$\begin{split}
\alpha_{ij}(xgx^{-1})&=\sum_{l=n-m+1}^{s-1}(y_{n-d+1,l}+\lambda
y_{s,l})y_{l,j}+
\sum_{l=s}^{n-d}y_{n-d+1,l}y_{l,j}=\\
&=\alpha_{ij}(g)+\lambda\cdot\sum_{l=n-m+1}^{s-1}
y_{s,l}y_{l,j}=\alpha_{ij}(g),
\end{split}$$ because $y_{s,l}=0$ for all $l{\geqslant}n-m+1{\geqslant}n-n_0+1>m$ (it’s a particular case of (\[reg\_zero\])).
If $r<n-d+1,s= j$, then $\alpha_{ij}$ is invariant for the same reasons. Finally, if $r<n-d+1,s> j$, then $$\begin{split} \alpha_{ij}(xgx^{-1})&= \sum_{l\neq
r,s}y_{n-d+1,l}y_{l,j}+(y_{r,j}+\lambda
y_{s,j})y_{n-d+1,r}+y_{s, j}(y_{n-d+1,s}-\lambda y_{n-d+1,r})=\\
&=\alpha_{ij}(g)+\lambda\cdot(y_{s,j}y_{n-d+1,r}-y_{s,
j}y_{n-d+1,r})=\alpha_{ij}(g).
\end{split}$$ Invariance of $\beta_{ij}$ and $\gamma$ can be proved similarly.
$\square$
\[lemm\_proofs\_inv\]
This means that $V(J)$ is a union of cojugacy classes. It’s easy to see that $x_D(\varphi)\in V(J)$, if $$\begin{split}
&c_{ij}=\Delta_{ij}(x_D(\varphi)),\\
&c_0=\gamma(x_D(\varphi)),\\
&c_{\alpha}=\varphi(n-d+1,n-d),\\
&c_{\beta}=\varphi(d+1,d).\end{split}\label{const_x_D}$$ Thus, ${\EuScript{K}}_D(\varphi)\subset V(J)$ for this values of constants.
For any root $\xi=(i,j)\in\Phi(n)$, we define its *level* as the number $u(\xi)=i-j$. Then the formula $$\xi=(i_1,j_1)<\eta=(i_2,j_2)\Leftrightarrow
\text{ either }u(\xi)<u(\eta)\text{ or }u(\xi)=u(\eta),j_1<j_2,$$ defines a complete order on the set of all roots. For each root $\xi=(i,j)\in\Phi(n)$ let $I_{\xi}$ be the ideal in $k[G_n]$ generated by all $y_{\eta},\eta<\xi$, and $\xi_0=(i_0,j_0)$ be the maximal root in $D''$, which is less than $\xi$ (if exists).
**Lemma ..** [The ideal $J$ is a prime ideal of $\bar k[y_{ij}]$ (here $\bar k$ is the algebraic closure of $k$).]{}
**Proof**.
Consider the following transformation of coordinates: $$\begin{split}
&{\widetilde}y_{n-d+1,n-d}=y_{\alpha}-c_{\alpha},\quad {\widetilde}y_{d+1,d}=
{\widetilde}y_{\beta}-c_{\beta},\quad {\widetilde}y_{n-d+1,d+1}=\gamma-c_0,\\
&{\widetilde}y_{n-d+1,i}=\alpha_{ij},\quad
{\widetilde}y_{i,d+1}=\beta_{ij},\quad(i,j)\in D^+,\\
&{\widetilde}y_{ij}=\Delta_{ij}-c_{ij},\quad(i,j)\in D',\\
&{\widetilde}y_{ij}=\Delta_{ij},\quad(i,j)\in R(D'')\setminus D'.
\end{split}\label{zamena}$$ (it’s easy to see that $J=\langle {\widetilde}y_\xi\rangle_{\xi\in B}$, where $B\subset\Phi(n)$ denotes the set of all roots $(i,j)$ from the left-side hand of (\[zamena\])). Note that for any $\xi\in B$ we have $${\widetilde}y_{\xi}\equiv y_{\xi}^0\cdot
y_{\xi}+a_{\xi}\pmod{I_{\xi}},\label{treug}$$ where $y_{\xi}^0$ is an invertible element of $\bar k[y_{ij}]/J$, and $a_{\xi}\in k$ is a certain scalar. Indeed, this is evident for ${\widetilde}y_{n-d+1,n-d}$, ${\widetilde}y_{d+1,d}$. For other roots we have: $$\begin{split}
&{\widetilde}y_{n-d+1,d+1}=\gamma-c_0=y_{n-d+1,d+1}y_{d+1,d}+\ldots,\\
&{\widetilde}y_{n-d+1,i}=\alpha_{ij}= y_{n-d+1,i}y_{i,j}+\ldots,\quad(i,j)\in
D^+,\\
&{\widetilde}y_{i,d+1}=\beta_{ij}=y_{i,d+1}y_{d+1,d}+\ldots,\quad(i,j)\in D^+,\\
&{\widetilde}y_{ij}=\Delta_{ij}-c_{ij}=y_{ij}\cdot\Delta_{i_0,j_0}+\ldots,\quad(i,j)\in D',\\
&{\widetilde}y_{ij}=\Delta_{ij}=y_{ij}\cdot\Delta_{i_0,j_0}+\ldots,\quad(i,j)\in
R(D'')\setminus D',
\end{split}$$ where for any ${\widetilde}y_{\xi}$, $\xi\in B$, dots denote elements, equal to zero modulo $I_{\xi}$, and scalars (we assume that $\Delta_{i_0,j_0}=1$, if the root $\xi_0$ does not exist for a given $\xi\in\Phi(n)$). But one can easily obtain the following equatities modulo $J$: $$\begin{split} &y_{d+1,d}\equiv
c_{\beta}\neq0,\quad y_{n-m+1,m}\equiv c_{n-m+1,m}\neq0,\\
& y_{i,j}\equiv c_{ij}/c_{i_0,j_0}\neq0,\quad(i,j)\in D^+,j<m,\\
&\Delta_{i_0,j_0}\equiv c_{i_0,j_0}\neq0,\quad(i_0,j_0)\in D''
\end{split}$$ (recall that $m=\max_{(i,j)\in D^+}j$). This concludes the proof of (\[treug\]).
Hence, in $\bar k[y_{ij}]/J$ all $y_{\xi}$ are polynomial in ${\widetilde}y_{\xi}$ (for each $\xi\in B$). Consequently, $$\bar k[y_{ij}]/J=\left.\bar k[y_{\xi}]_{\xi\in\Phi(n)}
\right/\langle{\widetilde}y_{\xi}\rangle_{\xi\in B} \cong\left.\bar
k[\{y_{\xi}\}_{\xi\notin B}\cup\{{\widetilde}y_{\xi}\}_{\xi\in
B}]\right./\langle{\widetilde}y_{\xi}\rangle_{\xi\in B}\cong \bar k[{\widetilde}y_{\xi}]_{\xi\notin B}.$$ In particular, $\bar k[y_{ij}]/J$ is a domain, so $J$ is a prime ideal.
$\square$
\[lemm\_proofs\_prim\]
This Lemma shows that $V(J)$ is an irreducible subvariety in $G_n(\bar k)$, because $\bar k[G_n]\cong\bar k[y_{ij}]$.
**Lemma ..** [Let $\mathcal C=\mathrm{Stab}_{\,G_n}(x_D(\varphi))=\{g\in G_n\mid
gx_D(\varphi)=x_D(\varphi)g\}$ be the stabilizer (the centralizer) of $x_D(\varphi)$. Then $\dim \mathcal C={\mathrm{codim}\mathop}V(J)$, where $J$ is generated by the elements (\[sreg\_uravn\]) with the scalars given by (\[const\_x\_D\]).]{}
**Proof**.
Firstly, we’ll present the defining equations of the centralizer $\mathcal C$. For any $\xi=(i,j)\in\Phi(n)$, let $\Phi_{\xi}$ be the union of all roots form the $i$-th column and the $j$-th row. We put $$\begin{split}
&\Phi_{\alpha}=\{(n-d+1,i)\mid(i,j)\in
D^+\},\quad{\widetilde}\Phi_{\alpha}=\{(n-d+1,j)\mid j<d\},\\
&\Phi_{\beta}=\{(i,d+1)\mid(i,j)\in
D^+\},\quad{\widetilde}\Phi_{\beta}=\{(i,d)\mid i>n-d+1\},\\
&\Phi_{\gamma}=\{(n-d,d)\},\quad\Phi_{\delta}=(\cup_{\xi\in D}\Phi_{\xi})\cap (R(D)\setminus D),\\
&A=\Phi_{\alpha}\sqcup{\widetilde}\Phi_{\alpha}\sqcup
\Phi_{\beta}\sqcup{\widetilde}\Phi_{\beta}\sqcup\Phi_{\gamma}\sqcup\Phi_{\delta}\subset\Phi(n)
\end{split}$$ (these sets are really disjoint). It’s easy to check that $\mathcal
C$ is given by the following equations: $$\begin{split}
&(\alpha)\quad y_{n-d+1,i}a_{ij}=a_{n-d+1,n-d}y_{n-d,j},\quad(i,j)\in D^+,\\
&(\beta)\quad y_{i,d+1}a_{d+1,d}=a_{ij}y_{jd},\quad(i,j)\in D^+,\\
&({\widetilde}\alpha)\quad a_{n-d+1,n-d}y_{n-d,j}+a_{n-d+1,d+1}y_{d+1,j}=0,
\quad j<d,\\
&({\widetilde}\beta)\quad y_{i,d+1}a_{d+1,d}+y_{i,n-d}a_{n-d,d}=0,\quad i>n-d+1,\\
&(\gamma)\quad y_{n-d+1,n-d}a_{n-d,d}+y_{n-d+1,d+1}a_{d+1,d}=
a_{n-d+1,n-d}y_{n-d,d}+a_{n-d+1,d+1}y_{d+1,d},\\
&(\delta)\quad y_{ij}=0,\quad(i,j)\in\Phi_{\delta},\\
\end{split}\label{centralizer}$$ (here we write $a_{ij}=\varphi(i,j)\in k^*$ for simplicity).
So, $\mathcal C$ is defined by the equations (\[centralizer\]), which are labeled by roots from $A$, and $V(J)$ is defined by the equations (\[sreg\_uravn\]), which are labeled by roots from $B$. Consider the maps $\sigma_A\colon A\to\Phi(n)$ and $\sigma_B\colon
B\to\Phi(n)$, given by the formulas
$$\begin{split}
&\sigma_A(\xi)=\begin{cases}(n-i+1,j),&
\xi=(i,j),(n-i+1,i)\in D'',j\neq d+1,i<n-j+1,\\
(i,n-j+1),&
\xi=(i,j),(n-j+1,j)\in D'',i\neq n-d,i>n-j+1,\\
(i,d),&
\xi=(i,d+1)\in\Phi_{\delta},i\neq n-d,\\
(n-d+1,j),&
\xi=(n-d,j)\in\Phi_{\delta},j\neq d+1,\\
(n-d+1,d),&
\xi=(n-d,d+1)\in\Phi_{\delta},\\
\xi,&\text{elsewhere},
\end{cases}\\
&\\
&\sigma_B(\xi)=\begin{cases} (n-d,j),&
\xi=(n-d+1,i),(i,j)\in D^+,\\
\xi,&\text{elsewhere}.
\end{cases}
\end{split}$$
They define the map $\sigma\colon A\sqcup B\to\Phi(n)$ (here $A\sqcup B$ is the disjoint union of $A$ and $B$) by the rule $\sigma\mathop{\mid}_A=\sigma_A$, $\sigma\mathop{\mid}_B=\sigma_B$; it’s easy to see that this map is a bijection. Since $\dim{\EuScript{K}}_D(\varphi)={\mathrm{codim}\mathop}\mathcal C=|A|$, ${\mathrm{codim}\mathop}V(J)=|B|$, this concludes the proof.
$\square$
\[lemm\_centr\]
**Theorem ..** [The defining ideal $J$ of the conjugacy call ${\EuScript{K}}_D(\varphi)\subset G_n(k)$ of an element $x_D(\varphi)$ is generated by the elements (\[sreg\_uravn\]) with the scalars given by (\[const\_x\_D\]).]{}
**Proof**. [Since conjugacy classes of $G_n(\bar k)$ are Zariski-closed [@Steinberg Proposition 2. 5], previous Lemmas follow that ${\EuScript{K}}_D(\varphi)=V(J)$ over $\bar k$. Hence, the sets of their $k$-points also coincide.]{} $\square$
\[theo\_klass\_sopr\]
\[paragr\_semidirect\] The next goal is to describe the support of a subregular character $\chi=\chi_f$ explicitly (we’ll see that the support coincides with ${\EuScript{K}}_f$) and to compute the value of this character on a counjugacy class contained in the support. We’ll use induction on dimension of the group and the Mackey’s method of semi-direct decomposition (see, f.e., [@Lehrer]). In this section we collect some basic facts which are needed for the sequel.
Let $G$ be a finite group, $A, B$ be its subgroups and $G=A\rtimes
B$ be their semi-direct product (i.e., $G=AB$ and $A\triangleleft
G$).
**Definition ..** [Let $G=A\rtimes B$ be a finite group and $A$ be abelian. For a given irreducible character $\psi$ of the group $A$, the subset $B^{\psi}=\{b\in B\mid\psi\circ\tau_b=\psi\}$ of the group $B$ is said to be the *centralizer* of this character (here $\tau_b\colon A\to A\colon a\mapsto bab^{-1}$).]{}
The following Theorem satisfies [@Lehrer]:
**Theorem ..** [Let $G=A\rtimes
B$ be a finite group and $A$ be abelian. Then every irreducible representation $\tau$ of the group $G$ has the form $\tau={\mathrm{Ind}_{A\rtimes B^{\chi}}^{G}{\psi\otimes\widetilde\tau}}$, where $\psi$ is a certain irreducible character of the group $A$, and ${\widetilde}\tau$ is a certain irreducible representation of the centralizer $B^{\psi}$. Hence, every irreducible character $\chi$ of the group $G$ has the form $\chi={\mathrm{Ind}_{A\rtimes
B^{\psi}}^{G}{\psi\widetilde\chi}}$, where $\widetilde\chi$, $\psi$ are certain irreducible characters of the groups $A$, $B_{\psi}$ resp. On the other hand, any character ${\mathrm{Ind}_{A\rtimes
B^{\psi}}^{G}{\psi\widetilde\chi}}$ is an irreducible character of $G$.]{} $\square$
\[semi\_direct\]
We denote $$\begin{split}&P_n=\{g=(y_{ij})\in
G_n\mid
y_{ij}=0\text{ for }j\neq1\},\\
&G_{n-1}\cong\{g=(y_{ij})\in G_n\mid y_{ij}=0\text{ for
}j=1\}\hookrightarrow G_n.\\
\end{split}$$
Then $G_n=P_n\rtimes G_{n-1}$, moreover, the group $P_n\cong{\mathbb{F}}_q^{n-1}$ is abelian; hence, all conditions of Theorem \[semi\_direct\] are satisfied.
We fix a non-trivial additive character $\theta\colon{\mathbb{F}}_q\to\mathbb{C}$. Any irreducible character of the group $P_n$ has the form $$p=(p_{ij})\in P_n\mapsto\theta(s_2 p_{21})\cdot\ldots\cdot\theta(s_n
p_{n1}),$$ where $s_i\in{\mathbb{F}}_q$, $2{\leqslant}i{\leqslant}n$, are arbitrary scalars. We’ll consider the case $\theta_n(p)=\theta(s_np_{n1})$ and the case $\theta_{n-1}(p)=\theta(s_{n-1}p_{n-1,1})$. One can see that their centralizers in the subgroup $G_{n-1}$ are $$\begin{split}
&G_{n-1}^{\theta_n}=\{g=(y_{ij})\in G_{n-1}\subset G_n\mid
y_{nj}=0\text{ for }1{\leqslant}j{\leqslant}n-1\},\\
&G_{n-1}^{\theta_{n-1}}=\{g=(y_{ij})\in G_{n-1}\subset G_n\mid
y_{n-1,j}=0\text{ for }1{\leqslant}j{\leqslant}n-2\}.
\end{split}$$ Note that $G_{n-1}^{\theta_n}\cong G_{n-2}$, and $G_{n-1}^{\theta_{n-1}}\cong{\widetilde}G_{n-2}=G_{n-2}\times{\mathbb{F}}_q$ (we’ll consider only these embeddings of these subgroups into $G_n$).
For a linear function $f\in{\mathfrak{g}}_n^*$, let $\pi(f)\in{\mathfrak{g}}_{n-2}^*$ and ${\widetilde}\pi(f)$ denote its restrictions to ${\mathfrak{g}}_{n-2}$ and ${\widetilde}{\mathfrak{g}}_{n-2}$ respectively (here ${\widetilde}{\mathfrak{g}}_{n-2}=\mathop{\mathrm{Lie}}{\widetilde}G_{n-2}$ and embeddings of subalgebras into ${\mathfrak{g}}_n$ correspond to embeddings of subgroups into $G_n$). Due to the Mackey’s method, if $1{\leqslant}d<n_1$ and $f=(\xi_{ij})\in\mathcal O_d$ is the canonical form of a $d$-subregular orbit, then $$\begin{split}
&T_f={\mathrm{Ind}_{P_n\rtimes G_{n-2}}^{G_n}{\theta_n\otimes
T_{\pi(f)}}},\text{ where }s_n=\xi_{1n},\quad\text{
if }d>1,\\
&T_f={\mathrm{Ind}_{P_n\rtimes{\widetilde}G_{n-2}}^{G_n}{\theta_{n-1}\otimes
T_{{\widetilde}\pi(f)}}},\text{ where }s_{n-1}=\xi_{1,n-1},\quad\text{
if }d=1,\\
\end{split}\label{ind_reps}$$ where $T_f$ (resp. $T_{\pi(f)}$ and $T_{{\widetilde}\pi(f)}$) denotes the representation of the group $G_n$ (resp. $G_{n-2}$ and ${\widetilde}G_{n-2}$), corresponding to the orbit $\Omega_f\subset{\mathfrak{g}}_n^*$ (resp. $\Omega_{\pi(f)}\subset{\mathfrak{g}}_{n-2}^*$ and $\Omega_{{\widetilde}\pi(f)}\subset{\widetilde}{\mathfrak{g}}_{n-2}^*$). All terms in the last equality are well-defined, because the orbit method is valid for the group ${\widetilde}G_{n-2}$. So, the problem can be reduced to the study of representations of dimension less than the first one has, and we can use an inductive argument.
Finally, certain coset decompositions of the group $G_n$ are needed for construction of induced representations. It’s easy to check that $$\begin{split}
& H_n=\{h=(t_{ij})\in G_{n-1}\mid t_{ij}=0\text{ for }i\neq n\},\\
& {\widetilde}H_n=\{h=(t_{ij})\in G_{n-1}\mid t_{ij}=0\text{ for }i\neq
n-1\}.
\end{split}$$ are complete systems of representatives of $G_n/(P_n\rtimes G_{n-2})$ and $G_n/(P_n\rtimes{\widetilde}G_{n-2})$ resp.
**Remark ..** [Since ${\widetilde}G_{n-2}\cong G_{n-2}\times{\mathbb{F}}_q$, we have a complete description of the representation $T_{{\widetilde}\pi(f)}$. Precisely, we note that $G_{n-2}$ is isomorphic to the subgroup $G_{n-2}\times0$ of ${\widetilde}G_{n-2}$ (this induces the embedding of Lie algebras ${\mathfrak{g}}_{n-2}\hookrightarrow{\widetilde}{\mathfrak{g}}_{n-2}$); let $\psi$ be the projection ${\widetilde}G_{n-2}\to G_{n-2}=G_{n-2}\times0$ and $g=(y_{ij})\in{\widetilde}G_{n-2}$. Then the character of the representation $T_{{\widetilde}\pi(f)}$ has the form $g\mapsto\chi(\psi(g))\cdot\theta(\xi_{n-1,n}y_{n,n-1})$, where $\chi$ is the character of the principal series of $G_{n-2}$, corresponding to the orbit $({\widetilde}\pi(f))\mathbin{\mid}_{{\mathfrak{g}}_{n-2}}$.]{}
\[nota\_w\_pi\]
**Remark ..** [There is another (“symmetric”) decomposition $G_n=P_n'\rtimes G_{n-1}'$, where $$\begin{split}
&P_n'=\{g=(y_{ij})\in G_n\mid y_{ij}=0\text{ for }j\neq1\},\\
&G_{n-1}'=\{g=(y_{ij})\in G_n\mid y_{ij}=0\text{ for }j=1\}.
\end{split}$$ Reflecting other subgroups and subsets in the anti-diagonal, we get ${\widetilde}G_{n-2}'$, $H_n'$ and ${\widetilde}H_n'$ ($G_{n-2}$, embedded into $G_n$ as above, is invariant under this reflection, i.e, $G_{n-2}'=G_{n-2}$). Then the irreducible character $\theta_{n-1}'\colon P_n'\to\mathbb{C}$ has the form $P_n'\ni
p=(p_{ij})\mapsto\theta(\xi_{2,n}p_{n,2}+\xi_{n-1,n}p_{n,n-1})$.]{}
\[nota\_symm\_rtimes\] Now we are able to conclude the proof of Theorem \[theo\_sreg\].
\[paragr\_proof\] Let $1{\leqslant}d<n_1$, $f\in\mathcal O_d$ be a canonical form of a $d$-subregular orbit and $\chi=\chi_f$ be the corresponding character (as above). We’ll prove in this section that its support coincides with ${\EuScript{K}}_f$ and compute its value on an arbitrary conjugacy class ${\EuScript{K}}_D(\varphi)\subset{\EuScript{K}}_f$. We’ve proved in section \[class\_sopr\] that the subvariety ${\EuScript{K}}_D(\varphi)$ of $G_n$ is defined by the equations (\[sreg\_uravn\]) with the scalars given by (\[const\_x\_D\]); in the other words, $V(J)$ is the defining ideal of ${\EuScript{K}}_D(\varphi)$ in $k[G_n]$.
The following proof is by induction on $n$; the base can be checked directly (f.e., using (\[tupaya\_formula\_char\])). We’ll assume that if $g=(y_{ij})\in G_n$, then $y_{n-d+1,n-d}\neq0$ (this means that $D\supset D_1(d)$). Firstly, we have to consider the case $d=1$. For convenience, we denote $$\begin{split}
&\Phi_{i_0}=\{(i,j)\in\Phi(n)\mid i=i_0\},\quad
{\widetilde}\Phi_{i_0}=\cup_{i{\geqslant}i_0}\Phi_i,\\
&\Phi^{j_0}=\{(i,j)\in\Phi(n)\mid j=j_0\},\quad
{\widetilde}\Phi^{j_0}=\cup_{j{\leqslant}j_0}\Phi^j.\\
\end{split}$$
**Lemma ..** [Let $f\in\mathcal O_1$ be a canonical form of a $1$-subregular orbit, and $g\in G_n$. If $\chi(g)\neq0$, then $g\in{\EuScript{K}}_D(\varphi)$ for a certain $x_D(\varphi)$ satisfying (\[dop\_usl\]).]{}
**Proof**.
Recall that $$T_f={\mathrm{Ind}_{P_n\rtimes{\widetilde}G_{n-2}}^{G_n}{\theta_{n-1}\otimes
T_{{\widetilde}\pi(f)}}}$$ (see (\[ind\_reps\])). Since $G_n=P_n\rtimes
G_{n-1}$, an arbitrary element $g\in G_n$ can be uniquely represented as $g=pg'$, $p\in P_n$, $g'\in G_{n-1}$ (in fact, $p$ and $g'$ are given by replacing the corresponding entries of $g$ by zeroes). Hence, for a given $g\in G_n$, the element $p(g)\in P_n$ is well-defined. By ${\widetilde}\pi(g)$ we denote the element of ${\widetilde}G_{n-2}$, which is given by replacing the corresponding entries of $g$ by zeroes. We have $$\chi(g)=\chi_f(g)={\mathrm{Ind}_{P_n\rtimes{\widetilde}G_{n-2}}^{G_n}{\theta_{n-1}(p(g))\cdot\chi_{{\widetilde}\pi(f)}({\widetilde}\pi(g))}}=\sum_{h\in{\widetilde}H_n}\theta_{n-1}(p(h^{-1}gh))\cdot\chi_{{\widetilde}\pi(f)}({\widetilde}\pi(h^{-1}gh))
\label{first_sreg_char_formula}$$ (here the summation is over all $h\in{\widetilde}H_n$ such that $h^{-1}gh\in
P_n\rtimes{\widetilde}G_{n-2}$).
Note that for any $h=(t_{ij})\in{\widetilde}H_n$, $$\begin{split}&(h^{-1}gh)_{n,j}=y_{n,j}+y_{n,n-1}t_{n-1,j},
\quad\text{if $2{\leqslant}j{\leqslant}n-2$},\\
&(h^{-1}gh)_{n-1,j}=y_{n-1,j}-\sum_{i=j+1}^{n-2}t_{n-1,i}y_{i,1},\quad\text{if
$1{\leqslant}j{\leqslant}n-3$},\\
&(h^{-1}gh)_{ij}=y_{ij}\quad\text{for all other $1{\leqslant}j<i{\leqslant}n$}.
\end{split}{\label{hgh}}$$ If $\chi_{{\widetilde}\pi(f)}({\widetilde}\pi(g))\neq0$, then it follows from the equations of regular orbits (Theorem \[theo\_reg\]), formulas for characters of the principal series (Theorem \[char\_reg\]) and remark \[nota\_w\_pi\] that $$\chi_{{\widetilde}\pi(f)}({\widetilde}\pi(g))=q^s\cdot\chi_{\pi^2(f)}(\pi^2(g))\cdot\theta(\xi_{2,n}
\cdot(-1)^{|X_1|-1}\cdot\frac{\Delta^{X_1}(g)}{\Delta^{X_2}(g)})
\cdot\theta(\xi_{n-1,n}y_{n,n-1}). \label{second_sreg_char_formula}$$ Here $X_1=D\setminus(\Phi^1\cup\Phi^{n-1})$, $X_2=X_1\setminus\Phi_n$, $s=|(R(D^+)\cap\Phi_n)\setminus({\widetilde}\Phi^2\cup\Phi^{n-1})|$ and $\pi^2$ denotes the maps $G_n\to G_{n-4}$ and ${\mathfrak{g}}_n^*\to{\mathfrak{g}}_{n-4}^*$ (we assume that $G_{n-4}$ is embedded into $G_n$ like $G_{n-4}\subset G_{n-2}\subset G_n$). In particular, $\chi_{\pi^2(f)}$ is a character of the principal series of the group $G_{n-4}$.
On the other hand, \[hgh\]) shows that $\chi_{\pi^2(f)}(\pi^2(g))=\chi_{\pi^2(f)}(\pi^2(h^{-1}gh))$ for all $h\in{\widetilde}H_n$. Substituting this to (\[first\_sreg\_char\_formula\]) and using (\[second\_sreg\_char\_formula\]), we obtain $$\begin{split}
&\chi(g)=q^s\cdot\chi_{\pi^2(f)}(\pi^2(g))\times\\
&\times\sum_{h\in{\widetilde}H_n}\theta_{n-1}(p(h^{-1}gh))
\cdot\theta(\xi_{2,n}
\cdot(-1)^{|X_1|-1}\cdot\frac{\Delta^{X_1}(h^{-1}gh)}
{\Delta^{X_2}(h^{-1}gh)}+\xi_{n-1,n}y_{n,n-1}).
\end{split}
\label{formula_char_sreg}$$
Furthermore, $$\begin{split}&\theta_{n-1}(p(h^{-1}gh))=\theta(\xi_{1,n-1}\cdot(y_{n-1,1}
-\sum_{i=2}^{n-2}t_{n-1,i}y_{i,1})),\\
&\frac{\Delta^{X_1}(h^{-1}gh)}{\Delta^{X_2}(h^{-1}gh)}=
\frac{\Delta^{X_1}(g)+y_{n,n-1}\cdot\Delta^{X_1}(g_t)}{\Delta^{X_2}(g)},
\end{split}
\label{delta_formula}$$ where $g_t$ denotes the matrix given by replacing $y_{nj}$ by $t_{nj}$ for $2{\leqslant}j{\leqslant}n-2$ in the matrix $g$ (the last equality can be checked directly).
Suppose now that $\chi(g)\neq0$. Thus $\chi_{\pi^2(f)}(\pi^2(g))\neq0$ (see (\[formula\_char\_sreg\])), some conditions from the equations (\[sreg\_uravn\]), defining ${\EuScript{K}}_D({\varphi})$, are satisfied automatically. Precisely, we have to check only the following equalities: $$\begin{split}
&\xi_{1,n-1}y_{2,1}=\xi_{2,n}y_{n,n-1},
\quad\text{(condition (\ref{dop_usl}))}\\
&\alpha_{ij}=\beta_{ij}=0,\quad(i, j)\in D^+,\\
&\Delta_{ij}=0,\quad(i, j)\in (R(D'')\setminus
D')\cap({\widetilde}\Phi^2\cup{\widetilde}\Phi_{n-1}).
\end{split}$$ But $(n,n-1),(2,1)\in D''$, so $R(D'')\cap(\Phi^1\cup\Phi_n)=\varnothing$, hence, the last set of equalities in fact has the form $\Delta_{n-1,j}=0$, $\Delta_{i,2}=0$.
Consider the equalities $\Delta_{n-1,j}=0$ firstly. If $j>m=\max_{(i,j)\in D}j$, then they have the form $\Delta_{n-1,j}=y_{n-1,j}=0$, and other equalities (with $j{\leqslant}m$) are exactly Kronecker-Capelli conditions of compatibility of the system of linear in $t_{n-1,j}$ equations $$y_{n-1,j}-\sum_{i=j+1}^{n-2}t_{n-1,i}y_{i,1}=0,\quad 2{\leqslant}j{\leqslant}m.
\label{uravn_Kron}$$ In both cases equalities under consideration are satisfied, because they express the condition $h^{-1}gh\in P_n\rtimes{\widetilde}G_{n-2}$ (see (\[hgh\])).
On the other hand, it follows from this condition and (\[hgh\]) that $t_{n-1,j}=-\frac{y_{n,j}}{y_{n,n-1}}$ for all $m<j{\leqslant}n-2$. Substituting this to (\[uravn\_Kron\]), we obtain that $\alpha_{ij}=0$ for any $(i,j)\in D^+$ (all other equalities of this form are satisfied because of compatibility of the system (\[uravn\_Kron\])). The equalities $\beta_{ij}=0$, $(i,j)\in D^+$, and the other equalities $\Delta_{i,2}=0$ can be obtained by using symmetric semi-direct decomposition of $G_n$ (see remark \[nota\_symm\_rtimes\]).
Finally, we’ll prove that (\[dop\_usl\]) is satisfied. Note that $\sum_{c\in{\mathbb{F}}_q}\theta(ct)=0$ for all $t\in{\mathbb{F}}_q^*$. Hence, the coefficients of $t_{n-1,j}$ in the character formula, that vary independently, have to be zero. For example, the coefficient of $t_{n-1,2}$ equals zero. Since $\Delta^{X_1}(g^t)=
(-1)^{|X_1|-2}\cdot t_{n-1,2}\cdot\Delta^{X_2}(g)+\ldots$ (members without $t_{n-1,2}$), the coefficient of $t_{n-1,2}$ in the character formula is equal to $$-\xi_{1,n-1}y_{21}+\xi_{2,n}\cdot(-1)^{|X_1|}\cdot
y_{n,n-1}\cdot\dfrac{(-1)^{|X_1|-2}\cdot\Delta^{X_2}(g)}
{\Delta^{X_2}(g)}=-\xi_{1,n-1}y_{21}+\xi_{2,n}y_{n,n-1},$$ so, $-\xi_{1,n-1}y_{21}+\xi_{2,n}y_{n,n-1}=0$.
$\square$
\[lemm\_O\_1\_neq\]
**Lemma ..** [Let $f\in\mathcal O_1$ be a canonical form of a $1$-subregular orbit, $g\in{\EuScript{K}}_D(\varphi)\subset {\EuScript{K}}_f$. Then $\chi(g)=q^{m_D}\cdot\theta_f(e_D(\varphi))$.]{}
**Proof**.
We’ll compute the value of the character under the assumption that this value isn’t equal to zero. Substituting (\[delta\_formula\]) to (\[formula\_char\_sreg\]), we have: $$\begin{split}
&\chi(g)=q^s\cdot\chi_{\pi^2(f)}(\pi^2(g))\cdot\sum_{h\in{\widetilde}H_n}\theta(\xi_{1,n-1}\cdot(y_{n-1,1}
-\sum_{i=2}^{n-2}t_{n-1,i}y_{i,1})+\\
&+\xi_{2,n} \cdot(-1)^{|X_1|}\cdot\frac{\Delta^{X_1}(g)+
y_{n,n-1}\cdot\Delta^{X_1}(g_t)}{\Delta^{X_2}(g)}+\xi_{n-1,n}y_{n,n-1}),
\end{split}$$ if $\chi_{{\widetilde}\pi(f)}({\widetilde}\pi(g))\neq0$. If $\chi(g)\neq0$, then the following four groups of conditions are satisfied.
1\. The first condition $\chi_{\pi^2(f)}(\pi^2(g))\neq0$ is satisfied automatically, because $g\in{\EuScript{K}}_D(\varphi)$. Indeed, it’s evident, that $\pi^2(D)\in\Phi(n-4)$ is a regular subset and $\pi^2(g)\in{\EuScript{K}}_{\pi^2(D)}(\varphi\mid_{\pi^2(D)})$. But $\chi_{\pi^2(f)}(\pi^2(g))$ is a character of the principal series of $G_{n-4}$, so, Theorem \[char\_reg\] shows that $$\chi_{\pi^2(f)}(\pi^2(g))=q^{m_{\pi^2(D)}}\cdot\theta_{\pi^2(f)}(\pi^2(e_D(\phi)))$$ (the map $\pi^2\colon{\mathfrak{g}}_n\to{\mathfrak{g}}_{n-4}$ is defined by the obvious way).
2\. The second group of conditions provides the inequality $\chi_{{\widetilde}\pi(f)}({\widetilde}\pi(g))\neq0$. Since $\chi_{\pi^2(f)}(\pi^2(g))$ $\neq0$, it’s enough to prove that $\Delta_{ij}(h^{-1}gh)=0$ for $(i,j)\in (R(D'')\setminus D)\cap(\Phi^2\cup\Phi_n)$. More precisely, these conditions have the form $$\begin{split}
&\Delta_{i,2}(g)=0,\quad (i,2)\in R(D)''\setminus D\\
&\Delta_{nj}(h^{-1}gh)=\Delta_{nj}(g)+y_{n,n-1}
\cdot\Delta_{nj}(g_t)=0,\quad (n,j)\in R(D)''\setminus D.\\
\end{split}$$
The first set of equalilies is contained in the Kronecker-Capelli conditions of compatibility of the system (\[uravn\_Kron\]). Since $t_{n-1,j}$, $3{\leqslant}j{\leqslant}m$, contain only in the second set of equalities, it follows that $t_{n-1,j}$, $3{\leqslant}j{\leqslant}m$, $(n-1,j)\notin R(D'')$, may be arbitrary, and other $t_{n-1,j}$ are uniquely determined by the formulas $$t_{n-1,j}=(-1)^{|X_1|-3}\cdot\frac{-\Delta_{n,j}(g)-\sum_{\substack{r>j\\(n-1,r)\notin
R(D'')}}\pm
t_{n-1,r}\cdot\Delta^{Y_r(j)}(g)}{y_{n,n-1}\cdot\Delta^{Y(j)}},
\label{small_t}$$ where $Y(j)=D^+\setminus{\widetilde}\Phi^j$, $Y_r(j)=(D^+\setminus(\Phi^r\cup{\widetilde}\Phi^{j-1}))\cup\{(n-r+1,j)\}$, and signs of $t_{n-1,r}$ alternates.
3\. The third condition says that we must consider $h\in{\widetilde}H_n$, such that $h^{-1}gh\in P_n\rtimes{\widetilde}G_{n-2}$. This condition is always satisfied, if $g\in{\EuScript{K}}_D(\varphi)$ (see the proof of Lemma \[lemm\_O\_1\_neq\]).
4\. Finally, the fourth condition says that the coefficients of $t_{n-1,j}$ in the character formula, that vary independently, are equal to zero. These are $t_{n-1,2}$ and $t_{n-1,j},3{\leqslant}j{\leqslant}m$, $(n-1,j)\notin R(D'')$. The end of the proof of Lemma \[lemm\_O\_1\_neq\] shows that the condition (\[dop\_usl\]) is satisfied if and only if the coefficient of $t_{n-1,2}$ equals zero; hence, we must prove only that the coefficients of all $t_{n-1,j},3{\leqslant}j{\leqslant}m$, $(n-1,j)\notin R(D'')$, are equal to zero. Simplifying (\[small\_t\]) under all our assumptions, one can check that this is equivalent to the set of equalities $\beta_{ij}=0$, $(i,j)\in D^+$, which are satisfied, because $g\in{\EuScript{K}}_D(\varphi)$.
So, the character formula contains only the following expression: $$\begin{split}
\label{coeff_1}
&\xi_{n-1,n}y_{n,n-1}+\xi_{1,n-1}\cdot(y_{n-1,1}-\sum_{i=m+1}^{n-2}y_{i,1}\cdot(-
\dfrac{y_{n-1,i}}{y_{n,n-1}}))+\xi_{2,n}\cdot\ldots=\\
&=\xi_{n-1,n}y_{n,n-1}+\xi_{1,n-1}\cdot\dfrac{\sum_{i=n-m+1}^{n-1}y_{n,i}y_{i,1}}
{y_{n,n-1}}+\xi_{1,n-1}\cdot\dfrac{\sum_{i=m+1}^{n-m}y_{n,i}y_{i,1}}
{y_{n,n-1}}+\xi_{2,n}\cdot\ldots,\\
\end{split}$$ where the numerator of every summand of the group marked by dots contains the only one of the elements $y_{i,1}$, $2{\leqslant}i{\leqslant}n-m$, and these elements aren’t contained in other summands. On the other hand, the symmetric decomposition of the group $G_n$ (see remark \[nota\_symm\_rtimes\]) give the following expression in the character formula $$\begin{split}
\label{coeff_2}
&\xi_{n-1,n}y_{n,n-1}+\xi_{2,n}\cdot\dfrac{\sum_{j=2}^{n-m}y_{n,j}y_{j,1}}
{y_{21}}+\xi_{1,n-1}\cdot\ldots=\\
&=\xi_{n-1,n}y_{n,n-1}+\xi_{2,n}\cdot\dfrac{\sum_{j=2}^{m}y_{n,j}y_{j,1}}
{y_{21}}+\xi_{2,n}\cdot\dfrac{\sum_{j=m+1}^{n-m}y_{n,j}y_{j,1}}
{y_{21}}+\xi_{1,n-1}\cdot\ldots,\end{split}$$ which has to coincide with (here the numerator of every summand of the group marked by dots contains the only one of the elements $y_{n,j}$, $n-m+1{\leqslant}j{\leqslant}n-1$, and these elements aren’t contained in other summands). Since $\xi_{1,n-1}y_{21}=\xi_{2,n}y_{n,n-1}$, we have that, in fact, and have the common form $$\xi_{1,n-1}\cdot\dfrac{\sum_{i=n-m+1}^{n-1}y_{n,i}y_{i,1}}
{y_{n,n-1}}+\xi_{2,n}\cdot\dfrac{\sum_{j=2}^{n-m}y_{n,j}y_{j,1}}
{y_{21}}+\xi_{n-1,n}y_{n,n-1}
=\xi_{1,n-1}\cdot\dfrac{\gamma(g)}{y_{n,n-1}}+\xi_{n-1,n}y_{n,n-1}.$$
We also have $$\label{q_deg}
\sum_{h\in{\widetilde}H_n}\theta(\xi_{1,n-1}\cdot\dfrac{\gamma(g)}{y_{n,n-1}}+\xi_{n-1,n}y_{n,n-1})=
q^{s_1}\cdot\theta(\xi_{1,n-1}\cdot\dfrac{\gamma(g)}{y_{n,n-1}}+\xi_{n-1,n}y_{n,n-1}),$$ where $s_1=|R(D)\cap\Phi_{n-1}\cap{\widetilde}\Phi^m|$. Indeed, $t_{n-1,j}$, $2{\leqslant}j{\leqslant}m$, $(n-1, j)\in R(D)$, vary independently, and if we fix them, then all other $t_{n-1,j}$ are uniquely determined.
Note that $$\label{coeff_raven_eDphi}
\begin{split}
&\xi_{1,n-1}\cdot\dfrac{\gamma(g)}{y_{n,n-1}}=
\xi_{1,n-1}\cdot\dfrac{\gamma(x_D(\varphi))}{y_{n,n-1}}=
\xi_{1,n-1}\cdot\dfrac{a_{n-d+1,n-d}a_{n-d,1}+
a_{n-d+1,2}a_{2,1}}{a_{n,n-1}}=\\
&=\xi_{1,n-1}a_{n-1,1}+ \dfrac{\xi_{1,n-1}}{a_{n,n-1}}\cdot
a_{n-d+1,2}a_{2,1}= \xi_{1,n-1}a_{n-1,1}+
\dfrac{\xi_{2,n-d+1}}{a_{2,1}}\cdot a_{n-d+1,2}a_{2,1}=\\
&=\xi_{1,n-1}a_{n-1,1}+\xi_{2,n-d+1}a_{n-d+1,2}
\end{split}$$ (here $a_{ij}=\varphi(i, j)$, as above).
Finally, substitute (\[q\_deg\]) and (\[coeff\_raven\_eDphi\]) to (\[formula\_char\_sreg\]): $$\begin{split}
&\chi_{n,f}(g)=q^s\cdot\chi_{\pi^2(f)}(\pi^2(g))\cdot\sum_{h\in{\widetilde}H_n}\theta(\xi_{1,n-1}\cdot\dfrac{\gamma(g)}{y_{n,n-1}}+\xi_{n-1,n}y_{n,n-1})=\\
&=q^{s+s_1}\cdot\chi_{\pi^2(f)}(\pi^2(g))\cdot\theta(\xi_{n-1,1}a_{n,n-1}+
\xi_{1,n-1}a_{n-1,1}+\xi_{2,n-d+1}a_{n-d+1,2})=q^{m_D}\cdot\theta_f(e_D(\varphi)),
\end{split}$$ because $m_{\pi^2(D)}+s+s_1=m_D$. The proof is complete.
$\square$
**Lemma ..** [Let $1<d<n_1$, $f\in\mathcal O_d$ be a canonical form of a $d$-subregular, $\chi=\chi_f$ be the corresponding character, and $g\in G_n$. Then $\chi(g)\neq0$ if and only if $g\in{\EuScript{K}}_D(\varphi)\subset{\EuScript{K}}_f$; in this case $\chi(g)=q^{m_D}\cdot\theta_f(e_D(\varphi))$.]{}
**Proof**.
Suppose that the conditions of the Lemma are satisfied. According to (\[ind\_reps\]), $$T_f={\mathrm{Ind}_{P_n\rtimes G_{n-2}}^{G_n}{\theta_n\otimes T_{\pi(f)}}},$$ hence, $$\chi(g)=\sum_{h\in
H_n}\theta_n(p(h^{-1}gh))\cdot\chi_{\pi(f)}(\pi(h^{-1}gh))$$ (here the summation is over $h\in H_n$ such that $h^{-1}gh\in
P_n\rtimes G_{n-2}$). Since $\pi(f)$ is a canonical form of a subregular orbit of $G_{n-2}$, we may assume that Theorem \[theo\_sreg\] is valid for $\chi_{\pi(f)}$ (inductive assumption).
Using the same arguments, as in two previous Lemmas (Kronecker-Capelli criterion of compatibility of systems of linear equations, conditions of the form $h^{-1}gh\in P_n\rtimes G_{n-2}$, and vanishing the coefficients of independent variables in the character formula), one can obtain the required result.
$\square$
\[lemm\_for\_all\_d\]
\[utochn\_obob\] Above we considere $d$-subregular subsets $D$ with $D(d)=D_1(d)$ and assume that $d<n_1$. Here we’ll consider all other cases.
First, if $D\supset D_0(d)$, then the subset $D$ is basic and the conjugacy class ${\EuScript{K}}_D(\varphi)$ is given by the equations (\[reg\_uravn\]) (see [@Andre2]).
Second, we should consider subregular orbits of the second and the third types (these are $n_1$-subregular orbits for even and odd $n$ respectively). Note that if $f=(\xi_{ij})\in\mathcal O_{n_1}$ is a canonical form of a subregular orbit of the second type, then the set $D=\{(i,j)\in\Phi(n)\mid\xi_{ji}\neq0)\}$ is basic itself, so in this case the subregular character is a basic character (in the sense of Andre); a complete description of such characters is given in [@Andre2]. This is also true for subregular orbits of the third type, if $\xi_{n_1,n_0+1}=0$ (see definition \[defi\_sreg\_orbits\]).
If ${\mathcal{O}}_f$ is a subregular orbit of the third type and $\xi_{n_1,n_0+1}\neq0$, then all statements are the same as in the case of orbits of the first type (except that the ideal $J$ doesn’t contain the polynomial $\gamma-c_0$). All proofs are quite similar to proofs in case of orbits of the first type, so we don’t reply them.
Finally, we should explain the genesis of polynomials $\alpha_{ij}$, $\beta_{ij}$ and $\gamma$ from our description of subregular characters of the first type with $D\supset D_1(d)$: (these polynomials are *not* minors!). Consider the following *characteristic matrix* $$M(t)=\begin{pmatrix}
1&0&\ldots&0&0\\
t\cdot y_{21}&1&\ldots&0&0\\
\vdots&\vdots&\ddots&\vdots&\vdots\\
t\cdot y_{n-1, 1}&t\cdot y_{n-1, 2}&\ldots&1&0\\
t\cdot y_{n, 1}&t\cdot y_{n, 2}&\ldots&t\cdot y_{n, n-1}&1
\end{pmatrix}.$$
A minor of the matrix $M(t)$ is polynomial on $t$, and its coefficient are polynomial on $y_{ij}$ (hence, we can consider them as function on $G_n$ or as elements of ${\mathbb{F}}_q[{\mathfrak{g}}_n]$).
**Notation ..** [By $M^{i_1,\ldots,i_k}_{j_1,\ldots,j_k}(t^d, g)$ we denote the value of the coefficient of $t^d$ in the minor of the matrix $M(t)$ with rows $1{\leqslant}i_1,\ldots,i_k{\leqslant}n$ and columns $1{\leqslant}j_1,\ldots,j_k{\leqslant}n$ on an element $g=(y_{ij})\in G_n$. We also denote $M^X(d,g)=M^{\sigma(i_1,\ldots,i_k)}_{\tau(j_1,\ldots,j_k)}(t^d,
g)$, where $X$ is the set of pairs of the form $X=\{(i_1,j_1),\ldots,(i_k,j_k)\}$, and $\sigma, \tau$ are permutations such that $\sigma(i_1)<\ldots<\sigma(i_k)$ and $\tau(j_1)<\ldots<\tau(j_k)$.]{}
For example, for any $1{\leqslant}d{\leqslant}n_0$ we have $M^X(d,
g)=\Delta_d(g)$, where $X=\{(n, 1), (n-1, 2),\ldots,(n_1+2,n_0)\}$; hence, all minors considered above are also coefficients of minors of characteristic matrix. On the other hand, if $(i,j)\in D^+$ (where $D\supset D_1(d)$ is a $d$-subregular subset and $1{\leqslant}d<n_1$), $m=\max_{(i,j)\in D}j$, then $$\begin{split}
&\alpha_{ij}(g)=\pm M^{X^{\alpha}_{ij}}(2,g), \quad
X^{\alpha}_{ij}=\{n-d+1,j\}\cup\bigcup_{d<i{\leqslant}m}\{(i,i)\},\\
&\beta_{ij}(g)=\pm M^{X^{\beta}_{ij}}(2,g), \quad
X^{\beta}_{ij}=\{i,d\}\cup\bigcup_{d<j{\leqslant}m}\{(j,j)\},\\
&\gamma(g)=\pm M^{X^{\gamma}}(2,g), \quad
X^{\gamma}=\{n-d+1,d\}\cup\bigcup_{d<i<n-d+1}\{(i,i)\}\\
\end{split}$$ (the choice of signs depends on the cardinality of $D^+$).
We see that subregular characters can be described in terms of coefficients of minors of the characteristic matrix. In fact, subregular orbits, all orbits for $n{\leqslant}7$ [@IgnatevPanov] and all irreducible characters for $n{\leqslant}5$ [@Ignatev] can be described in these terms too. So, we may conjecture that all orbits and characters of the unitriangular group of an arbitrary dimension can be describes in terms of coefficients of minors of the characteristic matrix.
The author expresses his sincere gratitude to professor A.N. Panov for suggesting the ideas that gave rise to this work.
[XXXX]{} Andre C.A.M. Basic characters of the unitriangular group. J. Algebra, v. **175**, 1995, p. 287-319. Andre C.A.M. The basic character table of the unitriangular group. J. Algebra, v. **241**, 2001, p. 437-471. Ignatev M.V. Characters of the unitriangular group over a finite field. Proceedings of the XIII International Conference for Undergraguate and Graduate Students and Young Scientists “Lomonosov”. Vol. IV. Moscow, Moscow University Press, 2006, p. 65-66. Ignatev M.V., Panov A.N. Coadjoint orbits of the group $\mathrm{UT}(7, K)$. arXiv: math.RT/0603649. Kazhdan D. Proof of Springer’s hypothesis. Israel J. Math., v. **28**, 1977, p. 272-286. Kirillov A.A. Lectures on the orbit method. Novosibirsk, Nauchnaya kniga IDMI, 2002. Kirillov A.A. Unitary representations of nilpotent Lie groups. Uspekhi Mat. Nauk, v. **17**, 1962, p. 57-110. Kirillov A.A. The orbit method and finite groups. Moscow, MCCME, MC IUM, 1998. Kirillov A.A. Variations on the triangular theme. Amer. Math. Soc. Transl, v. **169**, 1995, p. 43-73. Lehrer G.I. Discrete series and the unipotent subgroup. Composito Math., v. **28**, fasc. 1, 1974, p. 9-19. Lusztig G. Subregular nilpotent elements and bases in $K$-theory. Canad, J. Math., v. **51**(**6**), 1999, p.1194-1225. Steinberg R. Conjugasy classes in algebraic groups. Lecture Notes in Mathematics, v. **366**. New York and Berlin, Springer, 1974.
Department of Algebra and Geometry, Samara State University, ul. Akad. Pavlova, 1, Samara 443011, Russia
E-mail address: mihail\_ignatev@mail.ru
[^1]: This research was supported by Samara regional grant for students and young scientists 23E2.1D
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we introduce two discrete curvature flows, which are called $\alpha$-flows on two and three dimensional triangulated manifolds. For triangulated surface $M$, we introduce a new normalization of combinatorial Ricci flow (first introduced by Bennett Chow and Feng Luo [@CL1]), aiming at evolving $\alpha$ order discrete Gauss curvature to a constant. When $\alpha\chi(M)\leq0$, we prove that the convergence of the flow is equivalent to the existence of constant $\alpha$-curvature metric. We further get a necessary and sufficient combinatorial-topological-metric condition, which is a generalization of Thurston’s combinatorial-topological condition, for the existence of constant $\alpha$-curvature metric. For triangulated 3-manifolds, we generalize the combinatorial Yamabe functional and combinatorial Yamabe problem introduced by the authors in [@GX2; @GX4] to $\alpha$-order. We also study the $\alpha$-order flow carefully, aiming at evolving $\alpha$ order combinatorial scalar curvature, which is a generalization of Cooper and Rivin’s combinatorial scalar curvature, to a constant.'
author:
- 'Huabin Ge, Xu Xu'
title: '**$\alpha$-curvatures and $\alpha$-flows on low dimensional triangulated manifolds**'
---
Introduction {#Introduction}
============
Calculations in local coordinate charts are indispensable for the study of smooth manifolds. However, an entirely different way came from Regge [@Re]. The basic procedure is to triangulate a manifold to simplexes, and finally obtain the geometrical and topological information of the manifold by studying the geometry in a simplicial complex and the combinatorial structure in the triangulation. On triangulated manifolds, the most general discrete metric seems to be piecewise linear metric which is defined on all edges. Discrete curvatures are determined by discrete metrics. Roughly speaking, discrete metrics are edge lengthes while discrete curvatures are angles between sub-simplexes and their combinatorics. Besides the general piecewise linear metrics, there are discrete metrics defined on all vertices. This type of metrics may be considered as a discrete conformal class. In his work on constructing hyperbolic metrics on 3-manifolds, Thurston [@T1] introduced circle packing metric on a triangulated surface with prescribed intersection angles. For triangulated 3-manifolds, Cooper and Rivin introduced a sphere packing metric. We shall study circle (sphere) packing metrics and the corresponding discrete curvatures in the following.
Suppose $M$ is a closed surface with triangulation $\mathcal{T}=\{V,E,F\}$, where $V,E,F$ represent the sets of vertices, edges and faces respectively. $\Phi: E\rightarrow [0,\frac{\pi}{2}]$ is a function evaluating each edge $i\sim j$ a weight $\Phi_{ij}$. The triple $(M, \mathcal{T}, \Phi)$ will be referred as a weighted triangulation of $M$ in the following. All the vertices are supposed to be ordered one by one, marked by $1, \cdots, N$, where $N=|V|$ is the number of vertices. Throughout this paper, all functions $f: V\rightarrow \mathds{R}$ will be regarded as column vectors in $\mathds{R}^N$ and $f_i$ is the value of $f$ at $i$. And we use $C(V)$ to denote the sets of functions defined in this way.
Each map $r:V\rightarrow (0,+\infty)$ is called a circle packing metric. Given $(M, \mathcal{T}, \Phi)$, we can attach each edge $\{ij\}$ a length $l_{ij}=\sqrt{r_i^2+r_j^2+2r_ir_j\cos \Phi_{ij}}$. It is proved [@T1] that the lengths $\{l_{ij}, l_{jk}, l_{ik}\}$ satisfies the triangle inequality for each face $\{ijk\}\in F$, which ensures that the face $\{ijk\}$ could be realized as a Euclidean triangle with lengths $\{l_{ij}, l_{jk}, l_{ik}\}$. Suppose $\theta_i^{jk}$ is the inner angle of triangle $\{ijk\}$ at the vertex $i$, the classical discrete Gauss curvature at the vertex $i$ is defined as $$\label{Def-Gauss curv}
K_i=2\pi-\sum_{\{ijk\}\in F}\theta_i^{jk},$$ where the sum is taken over all the triangles with $i$ as one of its vertices. Then $$\label{Gauss-Bonnet without weight}
\sum_{i=1}^NK_i=2\pi \chi(M),$$ which may be considered as a discrete version of Gauss-Bonnet formula. Denote $K_{av}=2\pi \chi(M)/N$. Notice that constant $K$-curvature metric, i.e. a metric $r$ with $K_i=K_{av}$ for all $i\in V$, does not always exist. In fact, for any proper subset $I\subset V$, let $F_I$ be the subcomplex whose vertices are in $I$ and let $Lk(I)$ be the set of pairs $(e, v)$ of an edge $e$ and a vertex $v$ such the following three conditioins: (1) the end points of $e$ are not in $I$; (2) $v$ is in $I$; (3) $e$ and $v$ form a triangle. Thurston [@T1] proved that the existence of constant $K$-curvature metric is equivalent the following combinatorial-topological conditions $$\label{condition-Thurston}
2\pi\chi(M)\frac{|I|}{|V|} >-\sum_{(e,v)\in Lk(I)}(\pi-\Phi(e))+2\pi\chi(F_I), \;\;\forall I: \phi\subsetneqq I\subsetneqq V.$$ Moreover, the constant $K$-curvature metric is unique, if exists, up to the scaling of $r$.
Chow and Luo [@CL1] first established an intrinsic connection between Thurston’s circle packing and surface Ricci flow. They introduced a combinatorial Ricci flow $$\label{Def-ChowLuo's flow}
\frac{dr_i}{dt}=-K_ir_i$$ and its normalization $$\label{Def-ChowLuo's normalized flow}
\frac{dr_i}{dt}=(K_{av}-K_i)r_i.$$ Then they proved that flow (\[Def-ChowLuo’s normalized flow\]) converges iff. the constant $K$-curvature metric exists, iff. Thurston’s combinatorial and topological conditions (\[condition-Thurston\]) are satisfied.
Inspired by Chow and Luo’s work, the first author introduced a combinatorial Calabi flow $$\frac{dr_i}{dt}=\Delta K_ir_i$$ in [@Ge] and proved similar results. The authors also studied the corresponding problem in hyperbolic background geometry using combinatorial Calabi flow [@GX1].
The paper is organized as follows. In Section 2, we introduce the $\alpha$th order Yamabe flow for surfaces and use it to study the corresponding constant and prescribing curvature problem. In Section 3, we study the constant $\alpha$-curvature problem for triangulated 3-manifolds. And Section 4 is devoted to some useful lemmas used in the paper.
$\alpha$ order combinatorial flow in two dimension {#section-2d}
==================================================
As pointed by the authors in [@GX4], there are two disadvantages of the classical definition of $K_i$. For one thing, classical discrete Gauss curvature does not perform so perfectly in that it is scaling invariant, i.e. if $\tilde{r}_i=\lambda r_i$ for some positive constant $\lambda$, then $\tilde{K}_i=K_i$, which is different from the transformation of scalar curvature $R_{\lambda g}=\lambda ^{-1}R_g$ in smooth case. For another, classical discrete Gauss curvature can’t be used directly to approximate smooth Gauss curvature since it always tends to zero as the triangulation becomes finer and finer. To amend this flaw, we suggested a new definition of discrete Gauss curvature by dividing an “area element" $A_i$.
We observed that the easiest form of “area element" may be $A_i=\pi r_i^{\alpha}$, where $\alpha$ is any real number. For convenience, we omitted the coefficient $\pi$ in the following definition.
\[def-R-curvature\] Given a weighted triangulated surface $(M, \mathcal{T}, \Phi)$ with circle packing metric $r: V\rightarrow (0,+\infty)$, the discrete Gauss curvature of order $\alpha$ (“$\alpha$-curvature" for short) at the vertex $i$ is defined to be $$R_{\alpha,i}=\frac{K_i}{r_i^{\alpha}},$$ where $K_i$ is the classical discrete Gauss curvature defined as the angle deficit at $i$ by (\[Def-Gauss curv\]).
For the classical discrete Gauss curvature $K$, [@T1; @MR; @CL1] provide a complete description of the space of admissible $K$-curvature $\{K=K(r)|r\in \mathds{R}^N_{>0}\}$, see also Lemma \[Lemma-admissible-curvature space\] in this paper. Similarly, it’s interesting to know how to describe the space of admissible $\alpha$-curvatures.
For the special case $\alpha=2$, $A_i=\pi r_i^2$ is just the area of the disk packed at $i$. This case is especially interesting. It was shown [@GX4] that $r_i^2$ is a suitable analogue of smooth Riemannian metric tensor $g$. Furthermore, the $2$-curvature is very similar to the smooth Gauss curvature. On one hand, it has similar scaling law; On the other hand, it can be used to approximate the smooth Gauss curvature.
One may ask if there are circle packing metrics whose corresponding $\alpha$-curvatures are constants or have prescribing curvatures. Curvature flow method gives an efficient approach to study these questions. The smooth 2-dimensional Yamabe flow, which is the same as Ricci flow, is $\partial g/\partial t=(r-R)g$. It is Chow and Luo’s insight to consider $r_i$ as an analogue of smooth Riemannian metric tensor $g$, $K_i$ as an analogue of smooth Riemannian curvature $K=R/2$ and $K_{av}$ as an analogue of smooth average curvature $r$. Thus the classical combinatorial Ricci flow (\[Def-ChowLuo’s normalized flow\]) comes into being. Inspired by their work and the idea of taking $r_i^2$ as a suitable analogue of smooth Riemannian metric tensor $g$ and $K_i/r_i^2$ as an appropriate discrete version of Gaussian curvature $K$, we [@GX4] defined a combinatorial Ricci flow as $$\label{Def-new alpha-Ricci flow}
\frac{dg_i}{dt}=(R_{av}-R_i)g_i,$$ where $g_i=r_i^2$ and $R_{av}=2\pi\chi(M)/\|r\|^2$. We further proved that, for surfaces with non-positive Euler characteristic, the flow (\[Def-new Ricci flow\]) converges iff. there exists a constant $2$-curvature metric. For $\alpha$-curvature, we proved similar results for surface $M$ with $\alpha \chi(M)\leq0$ by introducing a modified flow $$\label{Def-new Ricci flow}
\frac{dr_i}{dt}=(R_{\alpha,av}-R_{\alpha,i})r_i,$$ where $R_{\alpha,av}=2\pi\chi(M)/\|r\|^{\alpha}_{\alpha}$. However, there are no similar results for surface $M$ with $\alpha\chi(M)>0$. Even worse, we don’t know whether the solution of flow (\[Def-new alpha-Ricci flow\]) exists for all time $t\in(-\infty,+\infty)$. For this reason, we introduce another flow here, the solution of which always exists for all time $t\in \mathds{R}$.
Set $u_i=\ln r_i$.
Given $(M, \mathcal{T}, \Phi)$ with circle packing metric $r$, denote $s_{\alpha}=2\pi\chi(M)/\|r\|^{\alpha}_{\alpha}$. The $\alpha$th order combinatorial Ricci (Yamabe) flow (“$\alpha$-flow" for short) is $$\label{Def-Yamabe flow-2d}
\frac{du_i}{dt}=s_{\alpha}r_i^{\alpha}-K_i.$$
Chow and Luo’s normalized Ricci flow (\[Def-ChowLuo’s normalized flow\]) is in fact the $0$-flow defined above with $\alpha=0$. Hence $\alpha$-flow (\[Def-Yamabe flow-2d\]) may be considered as a different normalization of Chow and Luo’s flow (\[Def-ChowLuo’s flow\]).
Formally, $\alpha$-flow seems plainer and simpler than flow (\[Def-new Ricci flow\]). Furthermore, we have
\[Prop-longtime-existence\] The solution to discrete Yamabe flow (\[Def-Yamabe flow-2d\]) exists for all time $t\in \mathds{R}$.
Notice that $|s_{\alpha}r_i^{\alpha}|\leq2\pi|\chi(M)|$, $(2-d)\pi\leq K_i\leq 2\pi$, where $d=\max\limits_{1\leq i\leq N} d_i$ and $d_i$ is the degree of vertex $i$. Hence $|s_{\alpha}r_i^{\alpha}-K_i|$ is uniformly bounded by a constant depending only on the topology and the triangulation. Then by the extension theorem for solution in ODE theory, the solution to (\[Def-Yamabe flow-2d\]) exists for all $t\in(-\infty,+\infty)$.\
\[Prop-negative-gradient-flow\] $\alpha$-flow (\[Def-Yamabe flow-2d\]) is a negative gradient flow. Moreover, $\prod_{i=1}^N r_i(t)$ preserves to be a constant along $\alpha$-flow.
It is remarkable that Chow and Luo [@CL1] introduced a functional $$F(u)=\int_{u_0}^u\sum\nolimits_{i=1}^N\left(K_i-K_{av}\right)du_i,$$ which is called discrete Ricci potential, to prove the convergence of their Ricci flow (\[Def-ChowLuo’s normalized flow\]). With a slight modification, we define the $\alpha$-order discrete Ricci potential (“$\alpha$-potential" for short) as $$F(u)=\int_{u_0}^u\sum\nolimits_{i=1}^N\left(K_i-s_{\alpha}r_i^{\alpha}\right)du_i,$$ where $u_0\in\mathds{R}^N$ is arbitrary selected. The $\alpha$-potential is well defined since $\sum(K_i-s_{\alpha}r_i^{\alpha})du_i$ is a closed differential form. Thus $\alpha$-flow is in fact $\dot{u}=-\nabla_u F$.
Furthermore, $(\sum u_i)'=\sum (s_{\alpha}r_i^{\alpha}-K_i)=0$ implies that $\sum u_i(t)$ and hence $\prod r_i(t)$ are invariant along the $\alpha$-flow.\
For this reason, we always assume the initial circle packing metric $r(0)$ belongs to the hypersurface $\prod_{i=1}^N r_i=1$ in the following.
\[Prop-converge imply CCCP-metric\] If the solution to $\alpha$-flow (\[Def-Yamabe flow-2d\]) converges, then there exists at least a constant $\alpha$-curvature metric.
Notice that constant $\alpha$-curvature metrics are exactly the critical points of ODE system (\[Def-Yamabe flow-2d\]). If the flow converges, it converges to its critical points. This implies the existence of constant $\alpha$-curvature metric.\
Discrete Laplace operators are closely related to discrete curvature flows. Denote $$L=\frac{\partial(K_1, \cdots, K_N)}{\partial (u_1,\cdots,u_N)}$$ as the Jacobian matrix of the curvature map. It’s very interesting that $L: C(V)\rightarrow C(V)$ with $f\mapsto Lf$ can be considered as the prototype of a type of discrete Laplace operator [@CL1; @G3; @Ge; @GX4], which comes from the dual structure of circle patterns. Chow and Luo proved that $L$ is positive semi-definite everywhere in $\mathds{R}_{>0}^N$. Moreover, $rank(L)=N-1$ and the kernel of $L$ is $t(1,\cdots,1)^T$, $t\in \mathds{R}$. Other properties of $L$ is exhibited in Lemma \[Lemma-property of L\]. The authors once defined [@GX4] a two dimensional $\alpha$-order combinatorial Laplacian as $$\label{Def-alpha Laplacian-2d}
\Delta_\alpha f_i=-\frac{1}{r_i^\alpha}\sum_{j=1}^N \frac{\partial K_i}{\partial u_j} f_j=-\frac{1}{r_i^\alpha}\sum_{j\thicksim i} \frac{\partial K_i}{\partial u_j}(f_j-f_i).$$ Recall that we have considered each $f\in C(V)$ as a column vector, hence the two dimensional $\alpha$-Laplacian (\[Def-alpha Laplacian-2d\]) can be written in a matrix form, $$\Delta_{\alpha}=-\Sigma^{-\alpha}L$$ with $\Delta_{\alpha} f=-\Sigma^{-\alpha}Lf$ for each $f\in C(V)$, where $\Sigma=diag\big\{r_1,\cdots,r_N\big\}$. It’s interesting that we can consider $r_i^\alpha$ as an analogy of $d\mu$, where $d\mu$ is the area element in smooth case. We can define a $\alpha$ order inner product $\langle\cdot, \cdot\rangle_{\alpha}$ on $(M, \mathcal{T}, \Phi)$ with circle packing metric $r$ by $$\label{inner product}
\langle f, h \rangle_{\alpha}=\sum_{i=1}^{N}f_ih_ir_i^{\alpha}=h^T\Sigma^{\alpha}f$$ for real functions $f, h\in C(V)$. Then the $\alpha$-Laplacian $\Delta_{\alpha}: C(V)\rightarrow C(V)$ is self-adjoint since $$\langle \Delta_{\alpha}f, h\rangle_{\alpha}=\langle f, \Delta_{\alpha}h\rangle_{\alpha}$$ for any $f, h\in C(V)$.
Denote the first positive eigenvalue of $-\Delta_{\alpha}$ as $\lambda_1(-\Delta_{\alpha})$. The following theorem shows that the first positive eigenvalue of $\alpha$ order combinatorial Laplace operator is closely related to the behavior of discrete $\alpha$-flow.
\[Thm-convergence-nonpositive Euler number\] Assuming $\lambda_1(-\Delta_{\alpha})>\alpha s_{\alpha}=\alpha\frac{2\pi\chi(M)}{\|r\|_{\alpha}^{\alpha}}$ at all $r\in \mathds{R}_{>0}^N$. Then the $\alpha$-flow (\[Def-Yamabe flow-2d\]) converges iff. there exists at least a constant $\alpha$-curvature metric on $(M, \mathcal{T}, \Phi)$.
We give a direct and self-contained proof here. We just need to prove the “if" part. Set $\Lambda_{\alpha}=\Sigma^{-\frac{\alpha}{2}}L\Sigma^{-\frac{\alpha}{2}}$, then $$\Delta_{\alpha}=-\Sigma^{-\frac{\alpha}{2}}\Lambda_{\alpha}\Sigma^{\frac{\alpha}{2}},$$ which implies that $\lambda_1(-\Delta_{\alpha})=\lambda_1(\Lambda_{\alpha})$. Assuming $r^*$ is a constant $\alpha$-curvature metric. Scaling $r^*$ to any $tr^*$ with $t>0$, the corresponding $\alpha$-curvature is still a constant. Hence we may suppose $r^*$ belongs to the hypersurface $\prod r_i=1$. Denote $u^*$ as the $u$-coordinate of $r^*$, and $\alpha$-potential as $$\label{Def-alpha-potential}
F(u)=\int_{u^*}^u\sum_{i=1}^N\left(K_i-s_{\alpha}r_i^{\alpha}\right)du_i.$$ Denote $r^{\alpha}=(r_1^{\alpha},\cdots,r_N^{\alpha})^T$, then it’s easy to calculate $$\label{Hession of F}
\begin{aligned}
Hess_uF
=L-\alpha s_{\alpha}
\left(\Sigma^{\alpha}-\frac{r^{\alpha}(r^{\alpha})^T}{\|r\|_{\alpha}^{\alpha}}\right)=\Sigma^{\frac{\alpha}{2}}
\left(\Lambda_{\alpha}-\alpha s_{\alpha}\left(I-\frac{r^{\frac{\alpha}{2}}(r^{\frac{\alpha}{2}})^T}{\|r\|^{\alpha}_{\alpha}}\right)\right)
\Sigma^{\frac{\alpha}{2}}.
\end{aligned}$$ Choose an orthonormal matrix $P$ such that $P^T\Lambda_{\alpha} P=diag\{0,\lambda_1(\Lambda_{\alpha}),\cdots,\lambda_{N-1}(\Lambda_{\alpha})\}$. Suppose $P=(e_0,e_1,\cdots,e_{N-1})$, where $e_i$ is the $(i+1)$-column of $P$. Then $\Lambda_{\alpha} e_0=0$ and $\Lambda_{\alpha} e_i=\lambda_i e_i,\,1\leq i\leq N-1$, which implies $e_0=r^\frac{\alpha}{2}/\|r^\frac{\alpha}{2}\|$ and $r^\frac{\alpha}{2}\perp e_i,\,1\leq i\leq N-1$. Hence $\big(I-\frac{r^{\frac{\alpha}{2}}(r^{\frac{\alpha}{2}})^T}{\|r\|^{\alpha}_{\alpha}}\big)e_0=0$ and $\big(I-\frac{r^{\frac{\alpha}{2}}(r^{\frac{\alpha}{2}})^T}{\|r\|^{\alpha}_{\alpha}}\big)e_i=e_i$, $1\leq i\leq N-1$, which implies $$Hess_uF=\Sigma^{\frac{\alpha}{2}}P diag\big\{0,\lambda_1(\Lambda_{\alpha})-\alpha s_{\alpha},\cdots,\lambda_{N-1}(\Lambda_{\alpha})-\alpha s_{\alpha}\big\}\Sigma^{\frac{\alpha}{2}}P^T.$$ If $\lambda_1(\Lambda_{\alpha})>\alpha s_{\alpha}=\alpha\frac{2\pi\chi(M)}{\|r\|_{\alpha}^{\alpha}}$, then $HessF\geq0$, $rank(F)=N-1$. Using Lemma \[Lemma-positive-definite\], Lemma \[Lemma-proper\] and Lemma \[Lemma-injective\], we know that $F$ is proper on $\mathscr{U}$, where $\mathscr{U}\triangleq\{u\in \mathds{R}^N|\sum_i u_i=0\}$. Moreover, $\lim\limits_{u\in\mathscr{U},\,u\rightarrow \infty}F(u)=+\infty$ and $u^*$ is the unique zero point of $\nabla F$ which is also the unique minimum point of $F$ on $\mathscr{U}$. Let $\varphi(t)=F(u(t))$, then $\varphi'(t)=-\sum_i\left(K_i-s_{\alpha}r_i^{\alpha}\right)^2=-\|\nabla F\|^2\leq0$, which implies that $u(t)$ lies in a compact region of $\mathscr{U}$. We can further get $\varphi''(t)=2(K-s_{\alpha}r^{\alpha})^THessF(K-s_{\alpha}r^{\alpha})\geq0$. The fact that $\varphi'\leq0$, $\varphi''\geq0$ and $\varphi$ is bounded below implies $\varphi'(+\infty)=0$. Combining $u^*$ is the unique zero point of $\nabla F$ and $\{u(t)\}\subset\subset\mathscr{U}$, we know $u(t)\rightarrow u^*$ exponentially fast.\
If $\alpha\chi(M)\leq 0$, we always have $\lambda_1(-\Delta_{\alpha})>0\geq \alpha s_{\alpha}=\alpha\frac{2\pi\chi(M)}{||r||^{\alpha}_{\alpha}}$, thus we have
\[Thm-convergence-negative Euler\] Suppose $(M, \mathcal{T}, \Phi)$ is a weighted triangulated surface with $\alpha\chi(M)\leq 0$. Then the solution to the $\alpha$-flow (\[Def-Yamabe flow-2d\]) converges if and only if there exists a constant $\alpha$-curvature metric $r^*$. Furthermore, if the solution converges, it converges exponentially fast to the metric of constant curvature.
When $\alpha=0$, above theorem is obtained in [@CL1].
Next we want to study the combinatorial and topological conditions for the existence of constant $\alpha$-curvature metric. For any proper subset $I\subset V$, denote $$\mathscr{Y}_I\triangleq\{x\in \mathds{R}^N |\sum_{i\in I}x_i >-\sum_{(e,v)\in Lk(I)}(\pi-\Phi(e))+2\pi\chi(F_I)\}$$ and $\mathscr{K}_{GB}\triangleq\{x\in \mathds{R}^N|\sum_{i=1}^Nx_i=2\pi\chi(X)\}$. The authors once proved
\[Proposition-topo-com-condition-ofGeXu\] ([@GX4], Theorem 2.34) Given a weighted triangulated surface $(M, \mathcal{T}, \Phi)$. Consider Thurston’s circle packing metric and the $\alpha$-curvature. When $\alpha>0$ and $\chi(M)<0$, the existence of constant $\alpha$-curvature metric is equivalent to $\mathscr{K}_{GB} \cap (\mathop{\cap}_{\phi\neq I \subsetneqq V} \mathscr{Y}_I )\cap\mathds{R}^N_{<0}\neq\phi$; When $\alpha<0$ and $\chi(M)>0$, the existence of constant $\alpha$-curvature metric is equivalent to $\mathscr{K}_{GB} \cap (\mathop{\cap}_{\phi\neq I \subsetneqq V} \mathscr{Y}_I )\cap\mathds{R}^N_{>0}\neq\phi$; When $\chi(M)=0$, the existence of constant $\alpha$-curvature metric is equivalent to $\mathscr{K}_{GB} \cap (\mathop{\cap}_{\phi\neq I \subsetneqq V} \mathscr{Y}_I )\cap \{0\}\neq\phi$.
Similar to Thurston’s condition (\[condition-Thurston\]), conditions in Proposition (\[Proposition-topo-com-condition-ofGeXu\]) also show that the combinatorial structure of the triangulation and the topology of surface, which have no relation with circle packing metrics, contains $\alpha$-curvature information surprisingly. Using Proposition \[Proposition-topo-com-condition-ofGeXu\], we can derive a combinatorial-topological-metric condition which contains Thurston’s condition (\[condition-Thurston\]) as a special case.
\[Thm-combtopo-condition-CCCPmetric\] Suppose $(M, \mathcal{T}, \Phi)$ is a weighted triangulated surface with $\alpha\chi(M)\leq 0$. Then there exists a constant $\alpha$-curvature metric iff. there exists a circle packing metric $r^*$ such that for any nonempty proper subset $I$ of vertices $V$, $$\label{combtopo-condition of GX}
2\pi\chi(M)\frac{\sum_{i\in I}r_i^{*\alpha}}{\|r^*\|^{\alpha}_{\alpha}}>-\sum_{(e,v)\in Lk(I)}(\pi-\Phi(e))+2\pi\chi(F_I).$$
**Proof.** By Lemma \[Lemma-admissible-curvature space\], for any circle packing metric $r$ and any proper subset $I\subset V$, the classical combinatorial Gauss curvature $K$ satisfies $$\label{K-satisfy}
\sum\limits_{i\in I}K_i(r)>-\sum_{(e,v)\in Lk(I)}(\pi-\Phi(e))+2\pi\chi(F_I).$$ If there exists a constant $\alpha$-curvature metric $r^*$, substituting $K^*=K(r^*)=(K_1^*,\cdots,K_N^*)$ into (\[K-satisfy\]), where $K_i^*=\frac{2\pi\chi(M)}{||r^*||^{\alpha}_{\alpha}}r_i^{*\alpha}$, we get (\[combtopo-condition of GX\]). On the other hand, it’s easy to see,
$\bullet$ when $\alpha>0$ and $\chi(M)<0$, (\[combtopo-condition of GX\]) implies that $\mathscr{K}_{GB} \cap (\mathop{\cap}_{\phi\neq I \subsetneqq V} \mathscr{Y}_I )\cap\mathds{R}^N_{<0}\neq\phi$;
$\bullet$ when $\alpha<0$ and $\chi(M)>0$, (\[combtopo-condition of GX\]) implies that $\mathscr{K}_{GB} \cap (\mathop{\cap}_{\phi\neq I \subsetneqq V} \mathscr{Y}_I )\cap\mathds{R}^N_{>0}\neq\phi$;
$\bullet$ when $\chi(M)=0$, (\[combtopo-condition of GX\]) implies that $0\in\mathscr{K}_{GB} \cap (\mathop{\cap} _{\phi\neq I \subsetneqq V} \mathscr{Y}_I )$.
Proposition \[Proposition-topo-com-condition-ofGeXu\] shows that conditions in above three cases all implies the existence of constant $\alpha$-curvature metrics. For $\alpha=0$ case, (\[combtopo-condition of GX\]) is in fact Thurston’s condition (\[condition-Thurston\]). Thus we finish the proof.\
The proof of Proposition \[Proposition-topo-com-condition-ofGeXu\] and hence Theorem \[Thm-combtopo-condition-CCCPmetric\] deeply rely on deriving a discrete maximum principle for the flow (\[Def-new Ricci flow\]). We want to know if there are more direct proofs without using discrete maximum principle for the flow (\[Def-new Ricci flow\]).
As to the uniqueness of constant $\alpha$-curvature metric, we restate Theorem 2.33 [@GX4] here for completeness.
\[Thm-uniqueness-CCCPmetric\] Suppose $(M, \mathcal{T}, \Phi)$ is a weighted triangulated surface with $\alpha\chi(M)\leq 0$, then the constant $\alpha$-curvature metric is unique if it exists. Specificly, if $\alpha\chi(M)=0$, then there exists at most one constant $\alpha$-curvature metric up to scaling. If $\alpha\chi(M)<0$, then for any $c^*$, there exists at most one metric with $\alpha$-curvature $R_{\alpha,i}\equiv c^*$.
When $\alpha\chi(M)>0$, such as $\alpha=2$ and $M$ is a sphere, Example 2 in [@GX4] shows that the conclusions in Theorem \[Thm-convergence-negative Euler\] are not true. For the tetrahedron triangulation of the sphere, if the initial metric $r(0)$ is close enough to $r^*=(1,\cdots,1)^T$, then the solution to flow (\[Def-Yamabe flow-2d\]) converges to $r^*$ when $t\rightarrow -\infty$. However, the limit behavior of $r(t)$ depends on the selection of initial metric $r(0)$. Indeed, there exists $r(0)$ such that the solution $r(t)$ diverges to $\infty$ either $t$ tends to $+\infty$ or $-\infty$. Example 3 in [@GX4] shows that the constant $\alpha$-curvature metric is not unique. For the existence of constant $\alpha$-curvature, we have
\[Thm-uniqueness-CCCPmetric\] Suppose $(M, \mathcal{T}, \Phi)$ is a weighted triangulated surface with $\alpha\chi(M)>0$.\
(1) There exists a constant $\alpha$-curvature metric.\
(2) There exists a circle packing metric $r^*$ such that for any nonempty proper subset $I$ of vertices $V$, $$\label{comb-topo-cond-for positive case}
2\pi\chi(M)\frac{\sum_{i\in I}r_i^{*\alpha}}{\|r^*\|^{\alpha}_{\alpha}}>-\sum_{(e,v)\in Lk(I)}(\pi-\Phi(e))+2\pi\chi(F_I).$$ (3) When $\alpha>0$ and $\chi(M)>0$, then $\mathscr{K}_{GB} \cap (\mathop{\cap}_{\phi\neq I \subsetneqq V} \mathscr{Y}_I )\cap\mathds{R}^N_{>0}\neq\phi$; When $\alpha<0$ and $\chi(M)<0$, then $\mathscr{K}_{GB} \cap (\mathop{\cap}_{\phi\neq I \subsetneqq V} \mathscr{Y}_I )\cap\mathds{R}^N_{<0}\neq\phi$.\
Then (1) implies (2) which implies (3).
**Proof.** For (1) $\Rightarrow$ (2), using formula (\[K-satisfy\]). For (2) $\Rightarrow$ (3), it’s obviously.\
\[Def-modified-alpha-flow\] Suppose $(M, \mathcal{T}, \Phi)$ is a weighted triangulated surface with circle packing metric $r$, $\overline{R}\in C(V)$ is a function defined on $M$. The modified $\alpha$-flow with respect to $\overline{R}$ is defined to be $$\label{Equation-modified-alpha-flow}
\frac{du_i}{dt}=\overline{R}_ir_i^{\alpha}-K_i.$$
$\overline{R}$ is called admissible if there is a circle packing metric $\overline{r}$ with curvature $\overline{R}$. The modified $\alpha$-flow can be used to study prescribing curvature problem. On one hand, if the solution to the modified $\alpha$-flow (\[Equation-modified-alpha-flow\]) converges, then $\overline{R}$ is admissible. On the other hand, we have
\[Thm-prescribing curvature\] Suppose $(M, \mathcal{T}, \Phi)$ is a weighted triangulated surface and $\overline{R}\in C(V)$ is a function defined on $M$. If $\alpha\overline{R}_i\leq 0$ for all $i$, but not identically zero, and $\overline{R}$ is admissible by a metric $\overline{r}$. Then $\overline{r}$ is the unique metric in $\mathds{R}^N_{>0}$ such that it’s $\alpha$-curvature is $\overline{R}$. Moreover, the solution to the modified flow (\[Equation-modified-alpha-flow\]) converges exponentially fast to $\overline{r}$.
**Proof.** The first part is obviously, and $\overline{R}$ is admissible by metric $r(+\infty)$. For the second part, for given function $\overline{R}\in C(V)$, we can introduce the following modified $\alpha$-potential $$\label{definition of modified Ricci potential}
\overline{F}(u)=\int_{u_0}^u\sum_{i=1}^N\left(K_i-\overline{R}_ir_i^{\alpha}\right)du_i.$$ It is easy to check that the modified $\alpha$-potential $\overline{F}$ is well-defined. Furthermore, by direct calculation, we have $$\begin{aligned}
Hess_u\overline{F}=L-\Sigma^{\frac{\alpha}{2}}
\left(
\begin{array}{ccc}
\alpha\overline{R}_1 & & \\
& \ddots & \\
& & \alpha\overline{R}_N \\
\end{array}
\right)\Sigma^{\frac{\alpha}{2}}.
\end{aligned}$$ It is easy to check that, if $\alpha\overline{R}_i\leq 0$ for $i=1, \cdots, N$ and not identically zero, $Hess_u\overline{F}$ is positive definite. By Lemma \[Lemma-injective\], $\nabla_u \overline{F}=(K_1-\overline{R}_1r_1^{\alpha},\cdots,K_N-\overline{R}_Nr_N^{\alpha})^T$ is an injective map from $u\in \mathds{R}^N$ to $\mathds{R}^N$. Hence $\overline{r}$ is the unique zero point of $\nabla_u \overline{F}$. This fact implies that $\overline{r}$ is the unique metric in $\mathds{R}^N_{>0}$ such that it’s curvature is $\overline{R}$. By Lemma \[Lemma-proper\], we know that $\overline{F}$ is proper and $\lim\limits_{u\rightarrow\infty}\overline{F}(u)=+\infty$. Furthermore, $\frac{d}{dt}F(u(t))=-\sum_i(K_i-\overline{R}_ir_i^{\alpha})^2\leq 0$ implies that the solution of (\[Equation-modified-alpha-flow\]) lies in a compact region. The following of the proof is the same as that of Theorem \[Thm-convergence-nonpositive Euler number\], so we omit it here.\
\[prescribing problem for R=0\] The second part of Theorem \[Thm-prescribing curvature\] implies that $\alpha\chi(M)<0$. If $\alpha\overline{R}_i=0$ for all $i$, then the corresponding prescribing curvature problem is already solved in Theorem \[Thm-convergence-negative Euler\]. In this case, the metric $\overline{r}$ is not unique. However, it’s unique up to scaling. This is slightly different from Theorem \[Thm-prescribing curvature\].
In the following of this section, we consider more general “area element" $A_i$. It’s interesting to define the “$A$-curvature" as $R_i=K_i/A_i$, where $A_i>0$ is a function of circle packing metric $r\in \mathds{R}_{>0}^N$. We can consider the following discrete flow $$\label{Def-Aflow}
u'_i(t)=\frac{2\pi\chi(M)}{\sum A_i}A_i-K_i,$$ which is called “$A$-flow" for short. This generalized $A$-flow can be used to evolve a metric to a metric with constant $A$-curvature, i.e. a metric $r$ satisfying $K_i=sA_i$ for all $i\in V$, where $s=\frac{2\pi\chi(M)}{\sum A_i}$. It’s easy to see the solution to $A$-flow always exists for all time $t\in (-\infty,+\infty)$. So this flow seems simpler than other flows such as $\dot{u}_i=s-R_i$, which is an $A$-generalization of flow (\[Def-new Ricci flow\]). Furthermore, if the solution $r(t)$ to $A$-flow converges to $r(+\infty)\in\mathds{R}_{>0}^N$, then $r(+\infty)$ has constant $A$-curvature. It’s very interesting that we can select $A_i$ as the real area instead of the area of disk packed at $i$, hence $$\label{Equation-sum-A=area-of-M}
\sum_{i=1}^NA_i=Area(M, \mathcal{T}, \Phi, r)$$ is necessary, where $Area(M, \mathcal{T}, \Phi, r)$ is the total real area of a weighted triangulated surface $(M, \mathcal{T}, \Phi)$ with a fixed circle packing metric $r$. Then it’s easy to see the following three selections of $A_i$ all satisfy (\[Equation-sum-A=area-of-M\]).
Select $A_i=\sum\limits_{\triangle ijk \in F}Area(\triangle ijk)/3$, where the sum is taken over all the triangles with $i$ as one of its vertices.
Consider the dual structure determined by Thurston’s circle patterns. For any face $\{ijk\}\in F$, denote $C_i,C_j,C_k$ as the closed disks centered at $i$, $j$ and $k$ so that their radii are $r_i$, $r_j$ and $r_k$. They both intersect with each other. Let $\mathcal{L}_i$, $\mathcal{L}_j$, $\mathcal{L}_k$ be the geodesic lines passing through the pairs of the intersection points of $\{C_k,C_j\}$, $\{C_k,C_i\}$, $\{C_i,C_j\}$. These three lines $\mathcal{L}_i$, $\mathcal{L}_j$, $\mathcal{L}_k$ must intersect in a common point $O_{ijk}$. Connect $O_{ijk}$ and $O_{ijl}$ whenever triangles $\{ijk\}$ and $\{ijl\}$ share a common edge $\{ij\}\in E$. Thus we get a dual graph. For more details see [@CL1; @GX4; @G3]. Select $A_i=Area(D_i)$, where $D_i$ is the dual $2$-cell of $i$.
Inspired by [@MY], we can select $A_i=Area(V_i)$, where $V_i$ is the Voronoi dual $2$-cell of $i$ in the Delaunay triangulation.
Under the assumption (\[Equation-sum-A=area-of-M\]), the constant $A$-curvature is $\frac{2\pi\chi(M)}{Area(M, \mathcal{T}, \Phi, r)}$. If $(M, \mathcal{T}, \Phi, r)$ approximates a smooth Riemannian surface $(M, g)$, then $Area(M, \mathcal{T}, \Phi, r)$ approximates $Area(M, g)$. Hence the constant $A$-curvature $\frac{2\pi\chi(M)}{Area(M, \mathcal{T}, \Phi, r)}$ approximates the smooth average curvature $\frac{2\pi\chi(M)}{Area(M, g)}$=$\frac{\int_M Kdvol}{\int_M dvol}$. This fact inspires us to consider the following problem.
Fix a smooth Riemannian surface $(M, g)$. Suppose $(M_n, \mathcal{T}_n, \Phi_n, r_n)$ is a sequence of weighted triangulation of $M$ with initial circle packing metric $r_n$. $M_n$ is different with $(M, g)$ as metric space, although they are topologically equal. For each $n$, one can evolve $A$-flow (\[Def-Aflow\]) and get a solution $r_n(t)$, $t\in [0,+\infty)$. Meanwhile, one can evolve $(M, g)$ by smooth Ricci flow and derive a solution $g(t)$, $t\in [0,+\infty)$. Assuming the initial Gromov-Hausdorff distance between $(M, g)$ and $(M_n, \mathcal{T}_n, \Phi_n, r_n)$ tends to zero, then $r_n(t)\rightarrow g(t)$ as $n\rightarrow+\infty$.
$\alpha$ order combinatorial flow in three dimension {#3-dimensional combinatorial Yamabe problem}
====================================================
Suppose $M$ is a 3-dimensional compact manifold with a triangulation $\mathcal{T}=\{V,E,F,T\}$, where the symbols $V,E,F,T$ represent the sets of vertices, edges, faces and tetrahedrons respectively. A sphere packing metric is a map $r:V\rightarrow (0,+\infty)$ such that the length between vertices $i$ and $j$ is $l_{ij}=r_{i}+r_{j}$ for each edge $\{i,j\}\in E$, and the lengths $l_{ij},l_{ik},l_{il},l_{jk},l_{jl},l_{kl}$ determines a Euclidean tetrahedron for each tetrahedron $\{i,j,k,l\}\in T$. Glickenstein pointed out [@G1] that a tetrahedron $\{i,j,k,l\}\in T$ generated by four positive radii $r_{i},r_{j},r_{k},r_{l}$ can be realized as a Euclidean tetrahedron if and only if $$\label{nondegeneracy condition}
Q_{ijkl}=\left(\frac{1}{r_{i}}+\frac{1}{r_{j}}+\frac{1}{r_{k}}+\frac{1}{r_{l}}\right)^2-
2\left(\frac{1}{r_{i}^2}+\frac{1}{r_{j}^2}+\frac{1}{r_{k}^2}+\frac{1}{r_{l}^2}\right)>0.$$ Thus the space of admissible Euclidean sphere packing metrics is $$\mathfrak{M}_{\mathcal{T}}=\left\{\;r\in\mathds{R}^N_{>0}\;\big|\;Q_{ijkl}>0, \;\forall \{i,j,k,l\}\in T\;\right\}.$$ Cooper and Rivin [@CR] called the tetrahedrons generated in this way conformal and proved that a tetrahedron is conformal if and only if there exists a unique sphere tangent to all of the edges of the tetrahedron. Moreover, the point of tangency with the edge $\{i,j\}$ is of distance $r_i$ to $v_i$. They further proved that $\mathfrak{M}_{\mathcal{T}}$ is a simply connected open subset of $\mathds{R}^N_{>0}$, but not convex.
For a triangulated 3-manifold $(M, \mathcal{T})$ with sphere packing metric $r$, there is also the notion of combinatorial scalar curvature. Cooper and Rivin [@CR] defined combinatorial scalar curvature $K_{i}$ at a vertex $i$ as angle deficit of solid angles $$\label{Def-CR curvature}
K_{i}= 4\pi-\sum_{\{i,j,k,l\}\in T}\alpha_{ijkl},$$ where $\alpha_{ijkl}$ is the solid angle at the vertex $i$ of the Euclidean tetrahedron $\{i,j,k,l\}\in T$ and the sum is taken over all tetrahedrons with $i$ as one of its vertices. For this curvature, Glickenstein [@G1] first defined a combinatorial Yamabe flow $$\label{Flow-Glickenstein}
\frac{dr_i}{dt}=-K_ir_i$$ and give some very interesting and inspiring results.
Similar to the two dimensional case, Cooper and Rivin’s definition of combinatorial scalar curvature $K_i$ is scaling invariant, which is not so satisfactory. The authors [@GX4] once defined a new combinatorial scalar curvatures as $R_i=K_i/r_i^2$ on 3-dimensional triangulated manifold $(M, \mathcal{T})$ with sphere packing metric $r$. Consider $r_i^2$ as the analogue of the smooth Riemannian metric. If $\widetilde{r}_i^2=c r_i^2$ for some positive constant $c$, we have $\widetilde{R}_i=c^{-1}R_i$. This is similar to the transformation of scalar curvature in smooth case under scaling. For this type of combinatorial scalar curvature, the authors defined a combinatorial Yamabe functional $$Q(r)=\frac{\mathcal{S}}{V^{\frac{1}{3}}}=\frac{\sum_{i=1}^NK_ir_i}{(\sum_{i=1}^Nr_i^3)^{1/3}}, \ \ r\in\mathfrak{M}_{\mathcal{T}},$$ and proposed to study the corresponding constant curvature problem which is called combinatorial Yamabe problem. For this, the authors defined a new discrete Yamabe flow $$\frac{dg_i}{dt}=-R_ig_i,$$ with normalization $$\label{normalized comb Yamabe flow}
\frac{dg_i}{dt}=(R_{av}-R_i)g_i,$$ where $g_i=r_i^2$ and $R_{av}=\frac{\mathcal{S}}{\sum_{i=1}^Nr_i^3}$ is the average of the combinatorial scalar curvature.
Constant $R$-curvature metric means that $R_i\equiv$constant, which implies $K=R_{av}r^2$. The authors [@GX2] once defined the so called discrete quasi-Einstein metric satisfying $K=\lambda r$, which is similar to constant $R$-curvature metric. Motivated by these phenomena, we can generalize these properties to $\alpha$ order combinatorial scalar curvature (“$\alpha$-curvature" for short).
\[Def-alpha-curvature-3d\] For a triangulated 3-manifold $(M, \mathcal{T})$ with sphere packing metric $r$, the $\alpha$-curvature at the vertex $i$ is defined as $$\label{alpha-curvature 3d}
R_{\alpha,i}=\frac{K_i}{r_i^{\alpha}}$$ for any $\alpha\in\mathds{R}$, where $K_i$ is given by (\[Def-CR curvature\]).
The study of smooth Einstein-Hilbert functional has a long history. For piecewise linear metric case, Regge [@Re] first give a discretization of this functional. For sphere packing metric case, the Einstein-Hilbert-Regge functional $\mathcal{S}=\sum_{i=1}^N K_i r_i$ was introduced by Cooper and Rivin in [@CR].
\[Def-alpha-normalize-Regge-functional\] Suppose $(M, \mathcal{T})$ is a triangulated 3-manifold with a fixed triangulation $\mathcal{T}$. For any $\alpha\in \mathds{R}$, $\alpha\neq-1$, the $\alpha$ order combinatorial Yamabe functional (“$\alpha$-functional" for short) is defined as $$\label{Def-3d-Yamabe-functional}
Q_{\alpha}(r)=\frac{\mathcal{S}}{\|r\|_{\alpha+1}}=\frac{\sum_{i=1}^NK_ir_i}{\big(\sum_{i=1}^Nr_i^{\alpha+1}\big)^{\frac{1}{\alpha+1}}}, \ \ r\in\mathfrak{M}_{\mathcal{T}}.$$ The $\alpha$ order combinatorial Yamabe invariant with respect to $\mathcal{T}$ is defined as $$Y_{M,\mathcal{T}}=\inf_{r\in\mathfrak{M}_{\mathcal{T}}} Q_{\alpha}(r),$$ while the $\alpha$ order combinatorial Yamabe constant of $M$ is defined as $Y_{M}=\sup\limits_{\mathcal{T}}\inf\limits_{r\in\mathfrak{M}_{\mathcal{T}}} Q_{\alpha}(r).$
When $\alpha\geq0$, then $|Q_{\alpha}(r)|\leq\|K\|_{1+\frac{1}{\alpha}}$. When $-1<\alpha<0$, then $|Q_{\alpha}(r)|\leq \sum_i|K_i|$. Hence the $\alpha$ order combinatorial Yamabe invariant $Y_{M,\mathcal{T}}$ is well defined when $\alpha>-1$. For $\alpha\geq0$ case, $Y_{M,\mathcal{T}}$ attains the minimum value $-\|K\|_{1+\frac{1}{\alpha}}$ at a metric $r^*$ if and only if $r^*$ is a constant $\alpha$-curvature metric with $s_{\alpha}^*\leq0$. As noted in [@GX4], the admissible sphere packing metric space $\mathfrak{M}_{\mathcal{T}}$ for a given triangulated manifold $(M,\mathcal{T})$ may be considered as the combinatorial conformal class for $(M,\mathcal{T})$, which is an analogue of the conformal class $[g_0]$ of a Riemannian manifold $(M, g_0)$. Denote $s_{\alpha}=\mathcal{S}/\|r\|_{\alpha+1}^{\alpha+1}$. Then we have $$\label{gradient of Q-alpha}
\nabla_{r}Q_{\alpha}=\frac{K-s_{\alpha}r_i^{\alpha}}{\|r\|_{\alpha+1}}.$$ Hence $r$ is a constant $\alpha$-curvature metric iff. it is a critical point of $\alpha$-functional $Q_{\alpha}(r)$. We raise the following discrete $\alpha$-Yamabe problem on 3-dimensional triangulated manifold.\
\
**The Combinatorial $\alpha$-Yamabe Problem.** Given a 3-dimensional manifold $M$ with triangulation $\mathcal{T}$, find a sphere packing metric with constant combinatorial $\alpha$-curvature in the combinatorial conformal class $\mathfrak{M}_{\mathcal{T}}$.\
It’s easy to see, if $r$ is a constant $\alpha$-metric with $K=\lambda r^{\alpha}$, then $\lambda=s_{\alpha}=\mathcal{S}/\|r\|_{\alpha+1}^{\alpha+1}$. To study the combinatorial $\alpha$-Yamabe problem, we introduce 3-dimensional $\alpha$-Yamabe flow.
Given a triangulated 3-manifold $(M, \mathcal{T})$ with sphere packing metric $r$. For any $\alpha\in\mathds{R}$, the $\alpha$ order combinatorial Yamabe flow (“$\alpha$-flow" for short) is $$\label{Def-alpha-Yamabe flow-3d}
\frac{dr_i}{dt}=s_{\alpha}r_i^{\alpha}-K_i.$$
The flow $\dot{r}=\lambda r-K$ ($\lambda=s_1$) introduced by the authors in [@GX2] is in fact the $\alpha$-flow defined above with $\alpha=1$. Hence $\alpha$-flow (\[Def-alpha-Yamabe flow-3d\]) may be considered as a different normalization of the flow $\dot{r}=\lambda r-K$.
Along the 3-dimensional $\alpha$-flow, $\|r(t)\|_2^2=\sum r_i^2(t)$ is invariant. Hence we always assume $r(0)\in\mathbb{S}^{N-1}$ in the following. It’s easy to see that, if the solution of (\[Def-alpha-Yamabe flow-3d\]) converges to a metric $r(\infty)$, then $r(\infty)$ is a metric with constant $\alpha$-curvature. It’s interesting that almost all 3-dimensional results in [@GX2; @GX4] are still true for $\alpha$-flow (\[Def-alpha-Yamabe flow-3d\]). We just state some of them here.
\[Thm-3d-compact-exist-const-alpha-metric\] If the solution of (\[Def-alpha-Yamabe flow-3d\]) lies in a compact region in $\mathfrak{M}_{\mathcal{T}}\cap \mathbb{S}^{N-1}$, then there exists at least one sphere packing metric with constant $\alpha$-curvature on $(M, \mathcal{T})$.\
Given a triangulated 3-manifold $(M, \mathcal{T})$. For any $\alpha\in \mathds{R}$, the $\alpha$ order combinatorial Laplacian (“$\alpha$-Laplacian" for short) $\Delta_{\alpha}:C(V)\rightarrow C(V)$ is defined as $$\label{Def-alpha-Laplacian}
\Delta_{\alpha} f_i=\frac{1}{r_i^{\alpha}}\sum_{j\sim i}(-\frac{\partial K_i}{\partial r_j}r_j)(f_j-f_i)$$ for $f\in C(V)$.
This definition of $\alpha$-Laplacian is a generalization of $\alpha=2$ case, which was carefully studied by the authors in [@GX4]. Similar to the two dimensional $\alpha$-Laplacian, the three dimensional $\alpha$-Laplacian (\[Def-alpha-Laplacian\]) can also be written in a matrix form, $$\Delta_{\alpha}=-\Sigma^{-\alpha}\Lambda \Sigma$$ with $\Delta_{\alpha} f=-\Sigma^{-\alpha}\Lambda \Sigma f$ for each $f\in C(V)$, where $\Sigma=diag\big\{r_1,\cdots,r_N\big\}$ and $$\Lambda=Hess_r\mathcal{S}=\frac{\partial(K_{1},\cdots,K_{N})}{\partial(r_{1},\cdots,r_{N})}.$$ It was proved [@CR; @Ri; @G1; @G2] that $\Lambda$ is positive semi-definite with rank $N-1$ and the kernel of $\Lambda$ is the linear space spanned by the vector $r$ (see Lemma \[Lemma-3d-Lambda matrix\]).\
Set $\Gamma_i(r)=s_{\alpha}r_i^{\alpha}-K_i$, $1\leq i\leq N$. Then the $\alpha$-flow (\[Def-alpha-Yamabe flow-3d\]) can be written as $\dot{r}=\Gamma(r)$, which is an autonomy ODE system. Differentiate $\Gamma$, we get $$D\Gamma(r)=-\Lambda+\alpha s_{\alpha}\left(diag\left\{r_1^{\alpha-1},\cdots,r_N^{\alpha-1}\right\}-\frac{r^\alpha (r^\alpha)^T}{\|r\|_{\alpha+1}^{\alpha+1}}\right)-\frac{r^{\alpha}\left(K-s_{\alpha}r^{\alpha}\right)^T}{\|r\|_{\alpha+1}^{\alpha+1}}.$$ If $r^*\in\mathfrak{M}_{\mathcal{T}}$ is a sphere packing metric with constant $\alpha$-curvature, then $$\label{Diff-gamma at-r^*}
D\Gamma|_{r^*}=\left(-\Lambda+\alpha s_{\alpha}\left(diag\left\{r_1^{\alpha-1},\cdots,r_N^{\alpha-1}\right\}-\frac{r^\alpha (r^\alpha)^T}{\|r\|_{\alpha+1}^{\alpha+1}}\right)\right)_{r^*}.$$
\[Proposition-3d-semi-definite\] Given $(M^3, \mathcal{T})$, suppose $r^*$ is a constant $\alpha$-curvature metric. If the first positive eigenvalue of $-\Delta_{\alpha}$ at $r^*$ satisfies $$\label{3d-lamda1>alfa*s}
\lambda_1(-\Delta_{\alpha})>\alpha s_{\alpha}^*=\frac{\alpha\mathcal{S^*}}{\|r^*\|_{\alpha+1}^{\alpha+1}}$$ then $-D\Gamma|_{r^*}$ is positive semi-definite with $rank$ $N-1$ and kernel $\{tr^*|t\in\mathds{R}\}$.
**Proof.** Denote $\widetilde{\Lambda}=\Sigma^{\frac{1-\alpha}{2}}\Lambda \Sigma^{\frac{1-\alpha}{2}}$. Then $$-\Delta_{\alpha}=\Sigma^{-\alpha}\Lambda \Sigma=\Sigma^{-\frac{1+\alpha}{2}}\widetilde{\Lambda}\Sigma^{-\frac{1+\alpha}{2}},$$ which implies that $$\lambda_1(-\Delta_{\alpha})=\lambda_1(\widetilde{\Lambda}).$$ Choose a matrix $P\in O(N)$ such that $P^T\widetilde{\Lambda}P=diag\{0,\lambda_1(\widetilde{\Lambda}),\cdots,\lambda_{N-1}(\widetilde{\Lambda})\}$. Suppose $P=(e_0,e_1,\cdots,e_{N-1})$, where $e_i$ is the $(i+1)$-column of $P$. Then $\widetilde{\Lambda} e_0=0$ and $\widetilde{\Lambda} e_i=\lambda_i e_i,\,1\leq i\leq N-1$, which implies $e_0=r^\frac{\alpha+1}{2}/\|r^\frac{\alpha+1}{2}\|$ and $r^\frac{\alpha+1}{2}\perp e_i,\,1\leq i\leq N-1$. Hence $\left(I-\frac{r^{\frac{\alpha+1}{2}}(r^{\frac{\alpha+1}{2}})^T}{\|r\|^{\alpha+1}_{\alpha+1}}\right)e_0=0$ and $\left(I-\frac{r^{\frac{\alpha+1}{2}}(r^{\frac{\alpha+1}{2}})^T}{\|r\|^{\alpha+1}_{\alpha+1}}\right)e_i=e_i$, $1\leq i\leq N-1$. Furthermore, $$-D\Gamma|_{r^*}=\Sigma^{\frac{\alpha-1}{2}}P diag\left\{0,\lambda_1(\widetilde{\Lambda})-\alpha s_{\alpha}^*,\cdots,\lambda_{N-1}(\widetilde{\Lambda})-\alpha s_{\alpha}^*\right\}\Sigma^{\frac{\alpha-1}{2}}P^T.$$ Hence the conclusion is derived.\
\[Thm-3d-isolat-const-alpha-metric\] The constant $\alpha$-curvature metrics satisfying $\lambda_1(-\Delta_{\alpha})>\alpha s_{\alpha}$ are isolated in $\mathfrak{M}_{\mathcal{T}}\cap \mathbb{S}^{N-1}$. Specifically, the constant $\alpha$-curvature metrics with $\alpha s_{\alpha}\leq0$ are isolated.
**Proof.** Consider the map $\Gamma:\mathfrak{M}_{\mathcal{T}}\rightarrow \mathds{R}^{N}$, $r\mapsto \Gamma(r)$. It is easy to see that the zero point of $\Gamma$ corresponds to the metric with constant $\alpha$-curvature. By (\[Diff-gamma at-r\^\*\]) and Proposition \[Proposition-3d-semi-definite\], if the conditions in this theorem are satisfied, then $D\Gamma|_{r^*}$, the Jacobian of $\Gamma$ at the constant $\alpha$-curvature metric $r^*$, is negative semi-definite with $rank$ $N-1$ and kernel $\{tr^*|t\in\mathds{R}\}$. Notice that the kernel is the normal of $\mathbb{S}^{N-1}$. Restricted to $\mathbb{S}^{N-1}$, $D\Gamma$ is negative definite and then nondegenerate, which implies the conclusions.\
Proposition \[Proposition-3d-semi-definite\] shows that constant $\alpha$-curvature metric $r^*$ with $\lambda_1(-\Delta_{\alpha}^*)>\alpha s_{\alpha}^*$ is a asymptotically stable point of the $\alpha$-flow. Hence we have
\[Thm-3d-convergence of CYF under existence\] Given a triangulated manifold $(M^3, \mathcal{T})$. Suppose $r^*\in\mathbb{S}^{N-1}$ is a constant $\alpha$-curvature metric satisfying $\lambda_1(-\Delta_{\alpha}^*)>\alpha s_{\alpha}^*$, or more specifically, $r^*\in\mathbb{S}^{N-1}$ is a constant $\alpha$-curvature metric with $\alpha s_{\alpha}^*\leq0$. If $\|r(0)-r^*\|$ is small enough, then the solution of $\alpha$-flow (\[Def-alpha-Yamabe flow-3d\]) exists for all time and converges to $r^*$.\
Let’s look at what happens when $\alpha=0$. The $0$-curvature is just Cooper and Rivin’s scalar curvature. By Theorem \[Thm-3d-isolat-const-alpha-metric\] and Theorem \[Thm-3d-convergence of CYF under existence\], constant curvature metrics $r^*$ are always isolated in $\mathbb{S}^{N-1}$ and are always local attractors of the ODE system (\[Def-alpha-Yamabe flow-3d\]).
$\alpha$-flow (\[Def-alpha-Yamabe flow-3d\]) is not a negative gradient flow, since $\partial (s_{\alpha}r_i^{\alpha})/\partial r_j\neq \partial (s_{\alpha}r_j^{\alpha})/\partial r_i$. We could get similar results by considering the negative gradient flow of $\alpha$-functional (\[Def-3d-Yamabe-functional\]), i.e. $$\label{3d-gradient-flow}
\dot{r}=-\nabla_rQ_{\alpha}=\frac{1}{\|r\|_{\alpha+1}}\left(s_{\alpha}r_i^{\alpha}-K\right).$$ For this flow, Theorem \[Thm-3d-compact-exist-const-alpha-metric\] and Theorem \[Thm-3d-convergence of CYF under existence\] are still true.
Some useful lemmas {#usful lemma}
==================
For reader’s convenience, we list some lemmas in this section which are used in the proof of our main results. Some of them are proved in detail while some of them we just give related references.
\[Lemma-positive-definite\] Given a function $F\in C^2(\mathds{R}^N)$. Assuming $Hess(F)\geq0$, $rank(Hess(F))=N-1$, $\nabla F$ has at least one zero point and $F(u+t(1,\cdots,1)^T)=F(u)$ for any $u\in\mathds{R}^N$ and $t\in\mathds{R}$. Denote $\mathscr{U}=\{u\in\mathds{R}^N|\sum_iu_i=0\}$. Then the Hessian of $F\big|_{\mathscr{U}}$ (considered as a function of $N-1$ variables) is positive definite.
**Proof.** Chow and Luo [@CL1] stated similar results for $\alpha$-potential $F$ with $\alpha=0$. Since this lemma is very important, we give a direct and rigorous proof here for completeness. Set $\gamma=(1,\cdots,1)^T/\sqrt{N}$. Choose $A\in O(N)$, such that $A\gamma=(0,\cdots,0,1)^T$, meanwhile, $A$ transforms $\mathscr{U}$ to $\{\zeta\in\mathds{R}^N|\zeta_N=0\}$. Define $g(\zeta_1,\cdots,\zeta_{N-1})\triangleq f(A^T(\zeta_1,\cdots,\zeta_{N-1},0)^T)$, we can finish the proof by showing that $Hess(g)$ is positive definite. Partition $A$ into two blocks, $A=\left[S^{N\times(N-1)}, \gamma\right]$. Write $L=Hess(F)$ for short, then $Hess(g)=S^TLS$. Notice that, $F(u+t\gamma)=F(u)$ implies $L\gamma=0$. Therefore we have $$A^TLA=
\begin{bmatrix}
S^T \\ \gamma^T
\end{bmatrix}
L\big[S,\gamma\big]=
\begin{bmatrix}
S^TLS & S^TL\gamma \\
\gamma^T LS & \gamma^T L\gamma
\end{bmatrix}
=\begin{bmatrix}
S^TLS& 0\,\,\\
0 & 0\,\,
\end{bmatrix},$$ which implies that $S^TLS$ is positive semi-definite. Furthermore, $rank(S^TLS)=N-1$. Hence $Hess(g)=S^TLS$ is positive definite, which implies that $F\big|_{\mathscr{U}}$ is strictly convex.\
\[Lemma-proper\] Given a function $F\in C^2(\mathds{R}^N)$ with $Hess(F)>0$. Assuming $\nabla F$ has at least one zero point, then $\lim\limits_{\|x\|\rightarrow \infty} F(x)=+\infty$. Moreover, $F$ is proper.
**Proof.** Without loss of generality, assume $\nabla F(0)=0$. First, we consider the case $N=1$. So we have $F: ~\mathds{R}\to \mathds{R}$ with $F''>0$ and $F'(0)=0$. Since $F''>0$ and $F'(0)=0$, then $F'(1)=a>0$ and $F'(-1)=-b<0$. Again, by $F''>0$, $F'(x)>F'(1)=a>0$ for $x>1$, and $F'(x)<F'(-1)=-b$. Integrating this gives $$\begin{aligned}
&F(x)\ge a(x-1)+F(1)~~~\mbox{ for $x>1$}\\
&F(x)\ge b(-1-x)+F(-1)~~~\mbox{ for $x<-1$}.
\end{aligned}$$ That means $F(x)\ge \min\{a,b\}|x|+C$, where $C=\min\{-a+F(1),-b+F(-1)\}$. Obviously, $\lim\limits_{|x|\to\infty}F(x)=+\infty$ and $F$ is proper.
For $N>1$, for any $\omega\in\mathbb{S}^{N-1}$, we consider the ray $\{t\omega\}_{0<t<\infty}\subset \mathds{R}^N$. Let $f_\omega(t)=F(t\omega )$, then $f_\omega'(0)=0$ and $f_\omega''(t)=Hess_F(\omega,\omega)>0$, then as in the dimension $N=1$ case, there exists constant $a_\omega=f_\omega'(1)=\nabla F(\omega)\cdot\omega~>0$ and such that $$\begin{aligned}
F(t\omega)=f_\omega(t)\ge a_\omega t-a_\omega+F(\omega),~~~\mbox{ for $t>1$}.
\end{aligned}$$ By the arguments above, we have proved $\omega\cdot \nabla F(\omega)>0$ for any $\omega\in \mathbb{S}^{N-1}$, by compactness of $\mathbb{S}^{N-1}$ and $F$ is $C^2$, $$\begin{aligned}
A:=\inf_{\omega\in \mathbb{S}^{N-1}}\omega\cdot \nabla F(\omega)>0.
\end{aligned}$$ Let $B:=\min\limits_{\omega\in \mathbb{S}^{N-1}}F(\omega)$, then we have $$\begin{aligned}
F(x)\ge A\|x\|-A+B,~~~\mbox{ for $\|x\|\ge 1$}.
\end{aligned}$$ It implies $\lim\limits_{\|x\|\to\infty}F(x)=+\infty$ and $F$ is proper.\
([@CL1])\[Lemma-injective\] Suppose $\Omega\subset \mathds{R}^N$ is convex, the function $h:\Omega\rightarrow \mathds{R}$ is strictly convex, then the map $\nabla h:\Omega\rightarrow \mathds{R}^N$ is injective.
\[Lemma-property of L\] ([@CL1], Proposition 3.9.) Suppose $(M, \mathcal{T}, \Phi)$ is a weighted triangulated surface. $r=(r_1,\cdots,r_N)^T$ is circle packing metric, while $K=(K_1,\cdots,K_N)^T$ is classical discrete Gauss curvature. Then $L=\frac{\partial(K_1, \cdots, K_N)}{\partial (u_1,\cdots,u_N)}$ is positive semi-definite with rank $N$-$1$ and kernel $\{t(1,\cdots,1)^T| t\in \mathds{R}\}$. Moreover, $\frac{\partial K_i}{\partial u_i}>0$, $\frac{\partial K_i}{\partial u_j}<0$ for $i\sim j$ and $\frac{\partial K_i}{\partial u_j}=0$ for others.
\[Lemma-admissible-curvature space\] ([@T1; @MR; @CL1]) Given a weighted triangulated surface $(M, \mathcal{T}, \Phi)$. Consider Thurston’s circle packing metric $r$ and classical discrete Gauss curvature $K$. For any proper subset $I\subset V$, denote $\mathscr{Y}_I\triangleq\{x\in \mathds{R}^N |\sum_{i\in I}x_i >-\sum_{(e,v)\in Lk(I)}(\pi-\Phi(e))+2\pi\chi(F_I)\}$, $\mathscr{K}_{GB}\triangleq\{x\in \mathds{R}^N|\sum_{i=1}^Nx_i=2\pi\chi(X)\}$. Then the space of all admissible $K$-curvature is $\{K=K(r)|r\in \mathds{R}^N_{>0}\}=\mathscr{K}_{GB} \cap (\mathop{\cap} _{\phi\neq I \subsetneqq V} \mathscr{Y}_I )$.
\[Lemma-3d-Lambda matrix\] ([@CR; @Ri; @G1; @G2]) Suppose $(M, \mathcal{T})$ is a triangulated 3-manifold with sphere packing metric $r$, $\mathcal{S}=\sum K_ir_i$ is the Einstein-Hilbert-Regge functional. Then we have $$\nabla_r\mathcal{S}=K.$$ If we set $$\Lambda=Hess_r\mathcal{S}=
\frac{\partial(K_{1},\cdots,K_{N})}{\partial(r_{1},\cdots,r_{N})}=
\left(
\begin{array}{ccccc}
{\frac{\partial K_1}{\partial r_1}}& \cdot & \cdot & \cdot & {\frac{\partial K_1}{\partial r_N}} \\
\cdot & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot \\
{\frac{\partial K_N}{\partial r_1}}& \cdot & \cdot & \cdot & {\frac{\partial K_N}{\partial r_N}}
\end{array}
\right),$$ then $\Lambda$ is positive semi-definite with rank $N-1$ and the kernel of $\Lambda$ is the linear space spanned by the vector $r$.
**Acknowledgements**\
The authors would like to thank Dr. Wenshuai Jiang, Liangming Shen for many helpful conversations. The first author would also like to give special thanks to Dr. Yurong Yu for her supports and encouragements during the work. The research of the second author is partially supported by National Natural Science Foundation of China under grant no. 11301402 and 11301399. He would also like to thank Professor Guofang Wang for the invitation to the Institute of Mathematics of the University of Freiburg and for his encouragement and many useful conversations during the work.
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(Huabin Ge) Department of Mathematics, Beijing Jiaotong University, Beijing 100044, P.R. China
E-mail: hbge@bjtu.edu.cn\
(Xu Xu) School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P.R. China
E-mail: xuxu2@whu.edu.cn\
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Let $R$ be a commutative Noetherian ring with non-zero identity and ${\mathfrak{a}}$ an ideal of $R$. Let $M$ be a finite $R$–module of of finite projective dimension and $N$ an arbitrary finite $R$–module. We characterize the membership of the generalized local cohomology modules $\operatorname{H}^{i}_{{\mathfrak{a}}}(M,N)$ in certain Serre subcategories of the category of modules from upper bounds. We define and study the properties of a generalization of cohomological dimension of generalized local cohomology modules. Let $\mathcal S$ be a Serre subcategory of the category of $R$–modules and $n \geqslant \operatorname{pd}M$ be an integer such that $\operatorname{H}^{i}_{{\mathfrak{a}}}(M,N)$ belongs to $\mathcal S$ for all $i> n$. If ${\mathfrak{b}}$ is an ideal of $R$ such that $\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N/{{\mathfrak{b}}}N)$ belongs to $\mathcal S$, It is also shown that the module $\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N)/{{\mathfrak{b}}}\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N)$ belongs to $\mathcal S$.'
address: ' Department of Mathematic, Faculty of Science, Arak University, Arak, 38156-8-8349, Iran.'
author:
- 'M. Aghapournahr'
title: Upper bounds for finiteness of generalized local cohomology modules
---
Introduction
============
Throughout this paper $R$ is a commutative noetherian ring. Let ${\mathfrak{a}}$ be an ideal of $R$, $M$ be a finite $R$–module of of finite projective dimension and $N$ an arbitrary finite $R$–module. The notion of generalized local cohomology was introduced by J. Herzog [@He]. The $i$–th generalized local cohomology modules of $M$ and $N$ with respectt to ${\mathfrak{a}}$ is defined by
$\operatorname{H}^{i}_{{\mathfrak{a}}}(M,N) \cong \underset{n}\varinjlim
\operatorname{Ext}^{i}_{R}(M/{{\mathfrak{a}}}^{n}M,N).$
It is clear that $\operatorname{H}^{i}_{{\mathfrak{a}}}(R,N)$ is just the the ordinary local cohomology module $\operatorname{H}^{i}_{{\mathfrak{a}}}(N)$. This concept was studied in the articles [@S], [@BZ] and [@Ya].
For ordinary local cohomology module there is the important concept [*[cohomological dimension]{}*]{} of an $R$–module $N$ with respect to an ideal ${\mathfrak{a}}$ of $R$. It is denoted by
$\operatorname{cd}_{{\mathfrak{a}}}(N)= \sup\{i\geqslant 0 | \operatorname{H}^{i}_{{\mathfrak{a}}}(N)\neq 0\}$
This notion has been studied by several authors; see, for example [@F], [@Ha], [@O], [@HL] and [@DNT].
Hartshorn [@Ha] has defined the notion $\operatorname{q}_{{\mathfrak{a}}}(R)$ as the greatest integer $i$ such that $\operatorname{H}^{i}_{{\mathfrak{a}}}(R)$ is not Artinian. Dibaei and yassemi [@DY] extended this notion to arbitrary finite $R$–modules as
$\operatorname{q}_{{\mathfrak{a}}}(N)=\sup \{i\geqslant 0 | \operatorname{H}^{i}_{{\mathfrak{a}}}(N)~~\text{is not Artinian}\}$
Recall that a subclass of the class of all modules is called Serre class, if it is closed under taking submodules, quotients and extensions. Examples are given by the class of finite modules, Artinian modules and etc. In [@AM Theorem 3.1 and 3.3] the Author and Melkersson characterized the membership of ordinary local cohomology modules in certain Serre class of the class of modules from upper bounds they also introduced [*Serre cohomological dimension of a module with respect to an ideal*]{} [@AM Definition 3.5] as
${\operatorname{cd}}_{({\mathfrak{a}},\mathcal S)}(N)=\sup\{n\geq 0|\operatorname{H}^{i}_{{\mathfrak{a}}}(N)
\text{ is not in } \mathcal S \}$.
see also [@AT Definition 3.4]. Note that when $\mathcal S=\{0\}$ then ${\operatorname{cd}}_{({\mathfrak{a}},\mathcal S)}(N)={\operatorname{cd}}_{{\mathfrak{a}}}(N)$ and when $\mathcal S$ is the class of Artinian modules, then ${\operatorname{cd}}_{({\mathfrak{a}},\mathcal S)}(N)={\operatorname{q}}_{{\mathfrak{a}}}(N)$.
Amjadi and Naghipour in [@AN] (resp. Asgharzadeh, Divaani-Aazar and Tousi in [@DNT]) extended ${\operatorname{cd}}_{{\mathfrak{a}}}(N)$ (resp. ${\operatorname{q}}_{{\mathfrak{a}}}(N)$) to generalized local cohomology modules as
$\operatorname{cd}_{{\mathfrak{a}}}(M,N)=\sup \{i\geqslant 0 | \operatorname{H}^{i}_{{\mathfrak{a}}}(M,N)\neq 0\}$
$(\text{resp}.~\operatorname{q}_{{\mathfrak{a}}}(M,N)=\sup \{i\geqslant 0 | \operatorname{H}^{i}_{{\mathfrak{a}}}(M,N)~~\text{is not Artinian}\}).$
They also proved basic results about related notions. Also there are some other attempts to study generalized local cohomology modules from upper bounds, see [@CT Corollary 2.7] and [@CH Theorem 5.1, Lemma 5.2 and Corollary 5.3 ].
Our objective in this paper is to characterize the membership of generalized local cohomology modules in certain Serre class of the category of $R$–modules from upper bounds. We will do it in section 2. Our main results in this section are theorems \[2-1\], \[T:casen+1\] and \[P:loc/aloc\]. In section 3, we will define and study the [*Serre cohomological dimension of two modules with respect to an ideal*]{}. Our definition and results in this paper improve and generalize all of the above mentioned one. For unexplained terminology we refer to [@BSh] and [@BH].
main results
============
The following theorem characterize the membership of generalized local cohomology modules to a certain Serre class from upper bounds.
\[2-1\] Let $\mathcal S$ be a Serre subcategory of the category of $R$–modules. Let ${\mathfrak{a}}$ an ideal of $R$, $M$ be a finite $R$–module of finite projective dimension and $N$ an arbitrary finite $R$–module. Let $n \geqslant\operatorname{pd}M$ be a non-negative integer. Then the following statements are equivalent:
- $\operatorname{H}^{i}_{{\mathfrak{a}}}(M,N)$ is in $\mathcal S$ for all $i> n$.
- $\operatorname{H}^{i}_{{\mathfrak{a}}}(M,L)$ is in $\mathcal S$ for all $i> n$ and for every finite $R$–module $L$ such that $\operatorname{Supp}_R(L)\subset\operatorname{Supp}_R(N)$.
- $\operatorname{H}^{i}_{{\mathfrak{a}}}(M,R/{\mathfrak{p}})$ is in $\mathcal S$ for all ${\mathfrak{p}}\in\operatorname{Supp}_R(N)$ and all $i> n$.
- $\operatorname{H}^{i}_{{\mathfrak{a}}}(M,R/{\mathfrak{p}})$ is in $\mathcal S$ for all ${\mathfrak{p}}\in\operatorname{Min}\operatorname{Ass}_R(N)$ and all $i> n$.
We use descending induction on $n$. So we may assume that all conditions are equivalent when $n$ is replaced by $n+1$ using [@Ya Theorem 2.5].
(i)$\Rightarrow$(iii). We want to show that $\operatorname{H}^{n+1}_{{\mathfrak{a}}}(M,R/{\mathfrak{p}})$ is in $\mathcal S$ for each ${\mathfrak{p}}\in\operatorname{Supp}_R(N)$. Suppose the contrary and let ${\mathfrak{p}}\in\operatorname{Supp}_R(N)$ be maximal of those ${\mathfrak{p}}\in\operatorname{Supp}_R(N)$ such that $\operatorname{H}^{n+1}_{{\mathfrak{a}}}(M,R/{\mathfrak{p}})$ is not in $\mathcal S$. Since ${\mathfrak{p}}\in\operatorname{Supp}_R(N)$, there is by [@Bo Chap.(ii), § 4, $n^o$ 4, Proposition 20] a nonzero map $f:M{\longrightarrow}R/{\mathfrak{p}}$. Let ${\mathfrak{b}}\supsetneqq {\mathfrak{p}}$ be the ideal of $R$ such that $\operatorname{Im}f=
{\mathfrak{b}}/{\mathfrak{p}}$. The exact sequence $0\rightarrow \operatorname{Ker}{f}\rightarrow
M\rightarrow \operatorname{Im}{f}\rightarrow 0$, yields the exact sequence $$\operatorname{H}^{n+1}_{{\mathfrak{a}}}(M,N){\longrightarrow}\operatorname{H}^{n+1}_{{\mathfrak{a}}}(M,\operatorname{Im}f){\longrightarrow}\operatorname{H}^{n+2}_{{\mathfrak{a}}}(M,\operatorname{Ker}f).$$ Since $\operatorname{Supp}_R(\operatorname{Ker}f)\subset\operatorname{Supp}_R(N)$, by induction $\operatorname{H}^{n+2}_{{\mathfrak{a}}}(M,\operatorname{Ker}f)$ belongs to $\mathcal S$. It follows that $\operatorname{H}^{n+1}_{{\mathfrak{a}}}(M,\operatorname{Im}f)$ belongs to $\mathcal S$. There is a filtration $$0=N_t\subset N_{t-1}\subset N_{t-2}\subset \dots\subset N_0=R/{\mathfrak{b}}$$ of submodules of $R/{\mathfrak{b}}$, such that for each $0\leqslant i\leqslant
t$, $N_{i-1}/N_i\cong R/{{\mathfrak{q}}}_i$ where ${{\mathfrak{q}}}_i\in \operatorname{V}({\mathfrak{b}})$. Then by the maximality of ${\mathfrak{p}}$, $\operatorname{H}^{n+1}_{{\mathfrak{a}}}(M,R/{{\mathfrak{q}}}_i)$ is in $\mathcal S$. Use the exact sequences $0\rightarrow N_{i}\rightarrow
N_{i-1}\rightarrow R/{{\mathfrak{q}}}_i\rightarrow 0$, to conclude that $\operatorname{H}^{n+1}_{{\mathfrak{a}}}(M,R/{\mathfrak{b}})$ is in $\mathcal S$. Next the exact sequence $0\rightarrow \operatorname{Im}f\rightarrow R/{\mathfrak{p}}\rightarrow
R/{{\mathfrak{b}}}\rightarrow 0$, yields the exact sequence $$\operatorname{H}^{n+1}_{{\mathfrak{a}}}(M,\operatorname{Im}f){\longrightarrow}\operatorname{H}^{n+1}_{{\mathfrak{a}}}(M,R/{\mathfrak{p}}){\longrightarrow}\operatorname{H}^{n+1}_{{\mathfrak{a}}}(M,R/{\mathfrak{b}}).$$ It follows that $\operatorname{H}^{n+1}_{{\mathfrak{a}}}(M,R/{\mathfrak{p}})$ is in $\mathcal S$ which is a contradiction.
(iii)$\Rightarrow$(ii). Use a filtration for $N$ as above.
(iv)$\Rightarrow$(iii). Let ${\mathfrak{p}}\in\operatorname{Supp}_R(N)$. Then ${\mathfrak{p}}\supset
{\mathfrak{q}}$ for some ${\mathfrak{q}}\in\operatorname{Min}\operatorname{Ass}_R(N)$. Hence ${\mathfrak{p}}\in\operatorname{Supp}_R(R/{\mathfrak{q}})$. Applying (i)$\Rightarrow$ (iii), it follows that $\operatorname{H}^{i}_{{\mathfrak{a}}}(M,R/{\mathfrak{p}})$ is in $\mathcal S$ for all $i> n$.
\[C:supN=supM\] Let $\mathcal S$ be a Serre subcategory of the category of $R$–modules. Let ${\mathfrak{a}}$ an ideal of $R$ and $M$ be a finite $R$–module of finite projective dimension. Let $n \geqslant\operatorname{pd}M$ be a non-negetive integer. If $L$ and $N$ are finite $R$–modules such that $\operatorname{Supp}_R(L)=\operatorname{Supp}_R(N)$, then $\operatorname{H}^{i}_{{\mathfrak{a}}}(M,L)$ is in $\mathcal S$ for all $i> n$ if and only if $\operatorname{H}^{i}_{{\mathfrak{a}}}(M,N)$ is in $\mathcal S$ for all $i> n$.
\[2-4\] [(]{}see [@AM Definition 2.1] and [@ATV Definition 3.1][)]{} Let $\mathcal{M}$ be a Serre subcategory of the category of $R$–modules. We say that $\mathcal{M}$ is a [*Melkersson subcategory with respect to the ideal ${\mathfrak{a}}$*]{} if for any ${\mathfrak{a}}$–torsion $R$–module $X$, $0:_{X}{\mathfrak{a}}$ is in $\mathcal{M}$ implies that $X$ is in $\mathcal{M}$. $\mathcal{M}$ is called [*Melkersson subcategory*]{} when it is a Melkersson subcategory with respect to all ideals of $R$.
When $\mathcal M$ is Melkersson subcategory of the category of $R$–modules, we are able to weaken the condition $(iii)$ in \[2-1\] to require that $\operatorname{H}^{i}_{{\mathfrak{a}}}(M,R/{\mathfrak{p}})$ is in $\mathcal M$ for all ${\mathfrak{p}}\in\operatorname{Supp}_R(N)$, just for $i=n+1$.
\[T:casen+1\] Let $\mathcal M$ is Melkersson subcategory of the category of $R$–modules $R$–module. Let ${\mathfrak{a}}$ an ideal of $R$ and $M$ be a finite $R$–module of finite projective dimension. Let $n \geqslant\operatorname{pd}M$ be a non-negetive integer. Then for each finite $R$–module $N$ the conditions in theorem \[2-1\] are equivalent to:
1. $\operatorname{H}^{n+1}_{{\mathfrak{a}}}(M,R/{\mathfrak{p}})$ is in $\mathcal M$ for all ${\mathfrak{p}}\in\operatorname{Supp}_R(N)$.
(v)$\Rightarrow$(iv). We prove by induction on $i\geq n+2$ that $\operatorname{H}^{i}_{{\mathfrak{a}}}(M,R/{{\mathfrak{p}}})$ is in $\mathcal M$ for all ${\mathfrak{p}}\in\operatorname{Supp}_R(N)$. It is enough to treat the case $i=n+2$. Suppose that $\operatorname{H}^{n+2}_{{\mathfrak{a}}}(M,R/{{\mathfrak{p}}})$ is not in $\mathcal M$ for some ${\mathfrak{p}}\in \operatorname{Supp}_R(M)$. It follows that ${\mathfrak{a}}\not\subset {\mathfrak{p}}$, since otherwise $\operatorname{H}_{{\mathfrak{a}}}^{n+2}(M,R/{{\mathfrak{p}}})=0$, because $n+2>0$. Take $x\in{\mathfrak{a}}\setminus {\mathfrak{p}}$ and put $L=R/({{\mathfrak{p}}}+{x}R)$. Then $\operatorname{Supp}_R(L)\subset\operatorname{Supp}_R(N)$. We have a finite filtration $$0=L_t\subset L_{t-1}\subset L_{t-2}\subset \dots\subset L_0=L$$ such that $L_{i-1}/L_{i}\cong R/{{\mathfrak{p}}}_i$ for each $1 \leq i\leq t$ where ${{\mathfrak{p}}}_i\in \operatorname{Supp}_R(N)$. Using the exact sequence $$\operatorname{H}^{n+1}_{{\mathfrak{a}}}(M,L_i){\longrightarrow}\operatorname{H}^{n+1}_{{\mathfrak{a}}}(M,L_{i-1}){\longrightarrow}\operatorname{H}^{n+1}_{{\mathfrak{a}}}(M,R/{{\mathfrak{p}}}_i)$$ for each $1 \leq i\leq t$, shows that $\operatorname{H}^{n+1}_{{\mathfrak{a}}}(M,L)$ is in $\mathcal M$. Consider the exact sequence $0\rightarrow
R/{\mathfrak{p}}\overset x \rightarrow R/{\mathfrak{p}}\rightarrow L\rightarrow 0,$ which induces the following exact sequence $$\operatorname{H}^{n+1}_{{\mathfrak{a}}}(M,L){\longrightarrow}\operatorname{H}^{n+2}_{{\mathfrak{a}}}(M,R/{{\mathfrak{p}}}) \overset x{\longrightarrow}\operatorname{H}^{n+2}_{{\mathfrak{a}}}(M,R/{{\mathfrak{p}}}).$$ This shows that $0:_{\operatorname{H}^{n+2}_{{\mathfrak{a}}}(M,R/{{\mathfrak{p}}})}{x}$ is in $\mathcal
M$. Since $\operatorname{H}^{n+2}_{{\mathfrak{a}}}(M,R/{{\mathfrak{p}}})$ is ${\mathfrak{a}}$–torsion, by [@AM Lemma 2.3] $\operatorname{H}^{n+2}_{{\mathfrak{a}}}(M,R/{{\mathfrak{p}}})$ is in $\mathcal M$, which is a contradiction.
\[R:appl\] In theorems \[2-1\] and \[T:casen+1\] we may specialize $\mathcal S$ to any of the Melkersson subcategories, given in [@AM Example 2.4] to obtain characterizations of artinianness, vanishing, finiteness of the support etc.
Cho and Tang gave some parts of them in [@CT Theorem 2.5, 2.6 and Corrolary 2.7] for the case of artinianness in local rings. In the case of vanishing, i.e., when $\mathcal S$ merely consists of zero modules and finiteness of support (in local case) some parts of them were studied in [@AN Theorem B] and [@CH Theorem 5.1 part (a)]. These authors used Gruson’s theorem, [@V Theorem 4.1], while we just used the maximal condition in a noetherian ring.
\[P:loc/aloc\] Let $\mathcal S$ be a Serre subcategory of the category of $R$–modules and $M$ be a finite $R$–module of finite projective dimension. Let $N$ be a finite $R$–module, ${\mathfrak{a}}$ be an ideal of $R$ and $n \geqslant \operatorname{pd}M$ be an integer such that $\operatorname{H}^{i}_{{\mathfrak{a}}}(M,N)$ belongs to $\mathcal S$ for all $i> n$. If ${\mathfrak{b}}$ is an ideal of $R$ such that $\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N/{{\mathfrak{b}}}N)$ belongs to $\mathcal S$, then the module $\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N)/{{\mathfrak{b}}}\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N)$ belongs to $\mathcal S$.
Suppose $\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N)/{{\mathfrak{b}}}\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N)$ is not in $\mathcal S$. Let $L$ be a maximal submodule of $N$ such that $\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N/L)\otimes_R{R/{\mathfrak{b}}}$ is not in $\mathcal S$. Let $T\supset L$ be such that $\operatorname{\Gamma}_{{\mathfrak{b}}}(N/L)=T/L$. Since $\operatorname{Supp}_R(T/L)\subset{\operatorname{V}({\mathfrak{b}})\cap \operatorname{Supp}_R(N)}$,$\operatorname{H}^{i}_{{\mathfrak{a}}}(M,T/L)$ belongs to $\mathcal S$ for all $i\geq n$ by \[2-1\].
From the exact sequence $0 \rightarrow T/L\rightarrow N/L\rightarrow
N/T\rightarrow 0$, we get the exact sequence
$\operatorname{H}^{n}_{{\mathfrak{a}}}(M,T/L){\longrightarrow}\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N/L)\overset{f}{\longrightarrow}\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N/T){\longrightarrow}\operatorname{H}^{n+1}_{{\mathfrak{a}}}(M,T/L)$.
$\operatorname{Tor}^R_{i}(R/{\mathfrak{b}},\operatorname{Ker}f)$ and $\operatorname{Tor}^R_{i}(R/{\mathfrak{b}},\operatorname{Coker}f)$ are in $\mathcal S$ for all $i$, because $\operatorname{Ker}f$ and $\operatorname{Coker}f$ are in $\mathcal S$. It follows from [@Mel Lemma 3.1], that $\operatorname{Ker}(f\otimes{R/{\mathfrak{b}}})$ and $\operatorname{Coker}(f\otimes{R/{\mathfrak{b}}})$ are in $\mathcal
S$. Since $\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N/L)\otimes_R{R/{\mathfrak{b}}}$ is not in $\mathcal
S$, the module $\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N/T)\otimes_R{R/{\mathfrak{b}}}$ can not be in $\mathcal S$. By the maximality of $L$, we get $T=L$. We have shown that $\operatorname{\Gamma}_{{\mathfrak{b}}}(N/L)=0$ and therefore we can take $x\in {\mathfrak{b}}$ such that the sequence $0\rightarrow N/L\overset{x}\rightarrow N/L\rightarrow N/(L+{x}N)\rightarrow 0$ is exact. Thus we get the exact sequence
$\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N/L)\overset{x}\rightarrow \operatorname{H}^{n}_{{\mathfrak{a}}}(M,N/L)\rightarrow
\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N/L+{x}N)\rightarrow \operatorname{H}^{n+1}_{{\mathfrak{a}}}(M,N/L).$
This yields the exact sequence
$0\rightarrow
\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N/L)/{x}\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N/L)\rightarrow
\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N/L+{x}N)\rightarrow C\rightarrow 0$,
where $C\subset \operatorname{H}^{n+1}_{{\mathfrak{a}}}(M,N/L)$ and thus $C$ is in $\mathcal
S$.
Note that $x\in {\mathfrak{b}}$. Hence we get the exact sequence
$\operatorname{Tor}^R_{1}(R/{\mathfrak{b}},C){\longrightarrow}\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N/L)\otimes_R{R/{\mathfrak{b}}}{\longrightarrow}\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N/(L+{x}N))\otimes_R{R/{\mathfrak{b}}}$
However $L\subsetneqq{(L+{x}N)}$ and therefore $\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N/(L+{x}N))\otimes_R{R/{\mathfrak{b}}}$ belongs to $\mathcal
S$ by the maximality of $L$. Consequently $$\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N/L)\otimes_R{R/{\mathfrak{b}}}$$ is in $\mathcal S$ which is a contradiction.
\[2-0\] Let $M$ and $N$ be two finite $R$–modules such that $\operatorname{Supp}_R(M)
\cap \operatorname{Supp}_R(N)\subseteq \operatorname{V}({\mathfrak{a}})$. Then $\operatorname{H}^{i}_{{\mathfrak{a}}}(M, N)\cong
\operatorname{Ext}^i_{R}(M, N)$ for all $i\geq 0$.
There is a minimal injective resolution $E^\bullet$ of $N$ such that $\operatorname{Supp}_R(E^i)\subseteq \operatorname{Supp}_R(N)$ for all $i\geq 0$. Since $\operatorname{Supp}_R(\operatorname{Hom}_{R}(M, E^i))\subseteq \operatorname{Supp}_R(M) \cap
\operatorname{Supp}_R(N)\subseteq \operatorname{V}({\mathfrak{a}})$, $\operatorname{Hom}_{R}(M, E^i)$ is ${\mathfrak{a}}$–torsion. Therefore, for all $i\geq 0$, $$\begin{array}{llll}
\operatorname{H}^i_{\mathfrak{a}}(M, N)\!\!&= \ \ \operatorname{H}^{i}(\operatorname{\Gamma}_{{\mathfrak{a}}}(\operatorname{Hom}_{R}(M,
E^\bullet)))\\&= \ \ \operatorname{H}^{i}(\operatorname{Hom}_{R}(M, E^\bullet))\\&= \ \
\operatorname{Ext}^i_{R}(M, N),
\end{array}$$ as we desired.
Asgharzadeh, Divaani-Aazar and Tousi, in [@ADT Theorem 3.3 (i)] proved the following corollary when $\mathcal S$ is the category of Artinian $R$–modules with an strong assumption that $N$ has finite Krull dimension. This condition is very near to local case, while it is a simple conclusion of Theorem \[P:loc/aloc\] without that strong assumption.
\[T:H/bH\] Let $\mathcal S$ be a Serre subcategory of the category of $R$–modules and $M$ be a finite $R$–module of finite projective dimension. Let $N$ be a finite $R$–module, ${\mathfrak{a}}$ be an ideal of $R$ and $n>\operatorname{pd}M$ be an integer such that $\operatorname{H}^{i}_{{\mathfrak{a}}}(M,N)$ belongs to $\mathcal S$ for all $i>n$ , then $\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N)/{{\mathfrak{a}}}\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N)$ belongs to $\mathcal
S$.
Note that $\operatorname{H}^{i}_{{\mathfrak{a}}}(M,N/{{\mathfrak{a}}}N)\cong \operatorname{Ext}^{i}_{R}(M,N/{{\mathfrak{a}}}N)$ for all $i\geqslant 0$ by lemma \[2-0\], so $\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N/{{\mathfrak{a}}}N)=0$, now the proof is complete by theorem \[P:loc/aloc\] .
\[T:H/aH\] Let $\mathcal S$ be a Serre subcategory of the category of $R$–modules and $M$ be a finite $R$–module of finite projective dimension. Let $N$ be a finite $R$–module, ${\mathfrak{a}}$ be an ideal of $R$ and $n>\operatorname{pd}M$ be an integer such that $\operatorname{H}^{i}_{{\mathfrak{a}}}(M,N)$ belongs to $\mathcal S$ for all $i>n$ , then $\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N)$ is not finitely generated. In particular $\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N)\neq 0$.
Suppose $\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N)$ is finitely generated. Then there exist an integer $t$ such that ${{\mathfrak{a}}}^t\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N)=0$ but $\operatorname{H}^{n}_{{{\mathfrak{a}}}^t}(M,N)\cong \operatorname{H}^{n}_{{\mathfrak{a}}}(M,N)$ for all $i>0$. So $\operatorname{H}^{n}_{{\mathfrak{a}}}(M,N)\cong
\operatorname{H}^{n}_{{{\mathfrak{a}}}^t}(M,N)/{{\mathfrak{a}}}^t\operatorname{H}^{n}_{{{\mathfrak{a}}}^t}(M,N)$, is in $\mathcal S$, which is a contradiction.
Cohomological dimension with respect to Serre class
===================================================
In the following we introduce the last integer such that the generalized local cohomology modules belong to a Serre class.
\[D:t\] Let $\mathcal S$ be a Serre subcategory of the category of $R$–modules. Let ${\mathfrak{a}}$ be an ideal of $R$ and $M,N$ two $R$–modules. We define
${\operatorname{cd}}_{({\mathfrak{a}},\mathcal S)}(M,N)=\sup\{n\geq 0|\operatorname{H}^{i}_{{\mathfrak{a}}}(M,N)
\text{ is not in } \mathcal S \}$.
with the usual convention that the suprimum of the empty set of integers is interpreted as $-\infty$.
For example when $\mathcal S$={0}, then ${\operatorname{cd}}_{({\mathfrak{a}},\mathcal
S)}(M,N)= \operatorname{cd}_{{\mathfrak{a}}}(M,N)$ and when $\mathcal S$ is the class of Artinian modules, then ${\operatorname{cd}}_{({\mathfrak{a}},\mathcal S)}(M,N)=\operatorname{q}_{{\mathfrak{a}}}(M,N)$ as in [@AN] and [@ADT].
In the following we study the main properties of this invariant.
\[P:main\] Let $\mathcal S$ be a Serre subcategory of the category of $R$–modules. Let ${\mathfrak{a}}$ be an ideal of $R$ and $M$ a finite $R$–module of finite projective dimension. The following statements hold.
1. Let $\mathcal S_1,\mathcal S_2$ be two Serre subcategories of the category of $R$–modules such that $\mathcal S_1\subset \mathcal S_2$. Then ${\operatorname{cd}}_{({\mathfrak{a}},\mathcal
S_2)}(M,N)\leq{\operatorname{cd}}_{({\mathfrak{a}},\mathcal S_1)}(M,N)$ for every finite $R$–module $N$. In particular ${\operatorname{cd}}_{({\mathfrak{a}},\mathcal
S)}(M,N)\leq\operatorname{cd}_{{\mathfrak{a}}}(M,N)$ for each Serre subcategory $\mathcal S$ of the category of $R$–modules.
2. If $L$ and $N$ are finite $R$–modules s.t. $\operatorname{Supp}_R(L)\subset\operatorname{Supp}_R(N)$, then ${\operatorname{cd}}_{({\mathfrak{a}},\mathcal S)}(M,L)\leq{\operatorname{cd}}_{({\mathfrak{a}},\mathcal S)}(M,N)$ and equality holds if $$\operatorname{Supp}_R(L)= \operatorname{Supp}_R(N).$$
3. Let $0\rightarrow N^{\prime}\rightarrow N
\rightarrow N^{\prime\prime}\rightarrow 0$ be an exact sequence of finite $R$–modules. Then $${\operatorname{cd}}_{({\mathfrak{a}},\mathcal S)}(M,N)=
\max \{{\operatorname{cd}}_{({\mathfrak{a}},\mathcal S)}(M,N^{\prime}),
{\operatorname{cd}}_{({\mathfrak{a}},\mathcal S)}(M,N^{\prime\prime})\}.$$
4. ${\operatorname{cd}}_{({\mathfrak{a}},\mathcal S)}(M,R)=\sup\{{\operatorname{cd}}_{({\mathfrak{a}},\mathcal S)}(M,N)|N
\text{ is a finite } R\text{--module } \}$.
5. ${\operatorname{cd}}_{({\mathfrak{a}},\mathcal S)}(M,N)=
\sup\{{\operatorname{cd}}_{({\mathfrak{a}},\mathcal S)}(M,R/{\mathfrak{p}})|{\mathfrak{p}}\in\operatorname{Supp}_R(N) \}$.
6. ${\operatorname{cd}}_{({\mathfrak{a}},\mathcal S)}(M,N)=
\sup\{{\operatorname{cd}}_{({\mathfrak{a}},\mathcal S)}(M,R/{\mathfrak{p}})|{\mathfrak{p}}\in\operatorname{Min}\operatorname{Ass}_R(N) \}$.
If $\mathcal M$ is Melkersson subcategory, then the following statements hold:
1. ${\operatorname{cd}}_{({\mathfrak{a}},\mathcal M)}(M,N)=
\min\{r\geq 0|\operatorname{H}^{r+1}_{{\mathfrak{a}}}(M,R/{\mathfrak{p}})\in\mathcal M
\text{ for all }{\mathfrak{p}}\in\operatorname{Supp}_R(N) \}$.
2. For each integer $i$ with $1+\operatorname{pd}M\leq i\leq{\operatorname{cd}}_{({\mathfrak{a}},\mathcal M)}(M,N)+\operatorname{pd}M$, there exists ${\mathfrak{p}}\in \operatorname{Supp}_R(N)$ with $\operatorname{H}^{i}_{{\mathfrak{a}}}(M,R/{\mathfrak{p}})$ not in $\mathcal M$.
3. ${\operatorname{cd}}_{({\mathfrak{a}},\mathcal M)}(M,R)=
\min\{r\geq 0|\operatorname{H}^{r+1}_{{\mathfrak{a}}}(M,R/{\mathfrak{p}})\in \mathcal M
\text{ for all }{\mathfrak{p}}\in\operatorname{Spec}(R)\}$.
4. ${\operatorname{cd}}_{({\mathfrak{a}},\mathcal M)}(M,R)=
\min\{r\geq 0|\operatorname{H}^{r+1}_{{\mathfrak{a}}}(M,N)\in \mathcal M
\text{ for all finite }$$R\text{--modules } N \}$.
\(a) By definition.
\(b) Follows from \[2-1\].
\(c) The inequality “$\geq$”, holds by (b) and we get the opposite inequality from the following exact sequence $$\dots{\longrightarrow}\operatorname{H}^{i}_{{\mathfrak{a}}}(M,N^{\prime}){\longrightarrow}\operatorname{H}^{i}_{{\mathfrak{a}}}(M,N) {\longrightarrow}\operatorname{H}^{i}_{{\mathfrak{a}}}(M,N^{\prime\prime}){\longrightarrow}\dots$$
The assertions (d), (e) and (f) follow from theorem \[2-1\] $(i)\Leftrightarrow{(ii)}$, $(i)\Leftrightarrow{(ii)}$ and ${(i)}\Leftrightarrow{(iv)}$, respectively.
(g), (h), (i) and (j) follow from \[T:casen+1\].
[9]{}
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| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Daniel Barlet[^1].'
date: '28/09/08 ; révisée le 24/09/09.'
title: 'Un théorème à la “Thom-Sebastiani” pour les intégrale-fibres.'
---
Abstract {#abstract .unnumbered}
========
The aim of this article is to prove a Thom-Sebastiani theorem for the asymptotics of the fiber-integrals. This means that we describe the asymptotics of the fiber-integrals of the function $f \oplus g : (x,y) \to f(x) + g(y)$ on $(\mathbb{C}^p\times \mathbb{C}^q, (0,0))$ in term of the asymptotics of the fiber-integrals of the holomorphic germs $f : (\mathbb{C}^p,0) \to (\mathbb{C},0)$ and $g : (\mathbb{C}^q,0) \to (\mathbb{C},0)$. This reduces to compute the asymptotics of a convolution $\Phi*\Psi$ from the asymptotics of $\Phi$ and $\Psi$ modulo smooth terms.\
To obtain a precise result, giving the non vanishing of expected singular terms in the asymptotic expansions of the fiber-integrals associated to $f\oplus g$, we have to compute the constants coming from the convolution process. We show that they are given by rational fractions of Gamma factors. This enable us to show that these constants do not vanish.\
Résumé. {#résumé. .unnumbered}
=======
L’objet de cet article est de démontré un théorème “à la Thom-Sebastiani” pour les développements asymptotiques des intégrale-fibres des fonctions du type $f \oplus g : (x,y) \to f(x) + g(y)$ sur $(\mathbb{C}^p\times \mathbb{C}^q, (0,0))$ en terme des développements asymptotiques des intégrale-fibres associées aux germes holomorphes $f : (\mathbb{C}^p,0) \to (\mathbb{C},0)$ et $g : (\mathbb{C}^q,0) \to (\mathbb{C},0)$. Ceci se ramène à calculer les développements asymptotiques d’une convolution $\Phi*\Psi$ à partir des développements asymptotiques de $\Phi$ et $\Psi$ modulo les termes non singuliers.\
Pour obtenir un résultat précis donnant la non nullité des termes singuliers attendus dans les développements asymptotiques des intégrale-fibres associées à $f\oplus g$, nous devons calculer les constantes qui apparaissent dans la convolution. Nous montrons qu’elles sont données par des fractions rationnelles de facteurs Gamma, ce qui nous permet de montrer qu’elles sont non nulles.\
AMS Classification (2000) : 32-S-25, 32-S-40, 32-S-50.
Key words : Asymptotic expansions, fiber-integrals, Thom-Sebastiani theorem.\
Mots clefs : Développements asymptotiques, intégrale-fibres, théorème de Thom-Sebastiani.
Titre abrégé : Thom-Sebastiani ...
Titre anglais : A Thom-Sebastiani theorem for fiber-integrals.
Introduction.
=============
Le but du present article est de montrer directement un théorème à la “Thom-Sebastiani” (voir \[S.-T. 71\]) pour les développements asymptotiques des intégrale-fibres d’une fonction de la forme $(x,y) \to f(x) + g(y)$ où $f$ et $g$ sont deux fonctions holomorphes au voisinage de l’origine dans $\mathbb{C}^p$ et $\mathbb{C}^q$ respectivement.\
Il résulte du théorème général de développement asymptotique des intégrale-fibres (voir \[B. 82\]), qu’une intégrale-fibre, c’est-à-dire une fonction de la forme $$s \to F_{\varphi}(s) = \int_{f=s} \quad \frac{\varphi}{df\wedge d\bar f}$$ où $f : U \to \mathbb{C}$ est une fonction holomorphe non constante sur une variété complexe $U$ de dimension $n+1$ et $\varphi$ une $(n+1,n+1)-$forme $\mathscr{C}^{\infty}$ à support compact dans $U$, peut s’écrire, au voisinage de $s = 0$, $$F_{\varphi}(s) = \sum_{\alpha \in A, j\in [0,\mu(\alpha)]} \ \theta_{\alpha,j}(s).\vert s\vert^{2\alpha}.(Log\vert s\vert)^j \ + \ \xi(s)$$ où les fonctions $\theta_{j,\alpha} $ et $\xi$ sont $\mathscr{C}^{\infty}$ au voisinage de $s = 0$, où $A \subset ]-1,+\infty[ \cap \mathbb{Q}$ est un ensemble fini et où $\mu : A \to \mathbb{N}$ est une application à valeurs dans $[0,n]$.\
Nous dirons qu’une fonction qui admet une telle écriture au voisinage de l’origine admet un [développement asymptotique standard uniforme]{}[^2] de type $(A,\mu)$ en $s = 0$.\
En fait, le résultat que nous présentons consiste essentiellement à montrer la stabilité par convolution de la classe des fonctions admettant à l’origine de $\mathbb{C}$ des “développements asymptotiques standards uniformes”, en précisant les exposants et les degrés des termes logarithmiques de la convolée à partir des informations correspondantes pour les fonctions initiales.\
Donnons une définition qui facilitera l’énoncé de notre résultat.
Pour deux types donnés $(A,\mu)$ et $(B,\nu)$ nous définirons le type $(A*B, \mu*\nu)$ de la fa[ç]{}on suivante : $$\begin{aligned}
& A*B = \{ \alpha+\beta+1, \alpha \in A, \beta \in B \}.\\
& (\mu*\nu)(\alpha+\beta+1) = \mu(\alpha) + \nu(\beta) \quad {\rm quand} \quad \alpha, \beta \ {\rm et} \ \alpha+\beta+1 \ {\rm ne \ sont\ pas \ dans} \ \mathbb{N}.\\
& (\mu*\nu)(\alpha+\beta+1) = \mu(\alpha) + \nu(\beta) + 1 \quad {\rm quand} \quad \alpha+\beta+1 \in \mathbb{N}, \alpha \ {\rm et} \ \beta \ {\rm non \ dans} \ \mathbb{N}.\\
& (\mu*\nu)(\alpha+\beta+1) = \mu(\alpha) + \nu(\beta) - 1\quad {\rm quand} \quad \alpha \ {\rm ou} \ \beta \ {\rm (ou \ les \ deux) \ sont \ dans} \ \mathbb{N}.\end{aligned}$$
Précisément nous montrons le premier théorème suivant :
\[Th. Seb.\] Soient $f$ et $g$ deux fonctions holomorphes au voisinage de l’origine dans $\mathbb{C}^{p}$ et $\mathbb{C}^{q}$ respectivement. Supposons que les intégrale-fibres de $f$ et $g$ admettent des développements asymptotiques standards uniformes de type respectifs $(A,\mu)$ et $(B,\nu)$. Alors les intégrale-fibres de la fonction $f \oplus g : (x,y) \to f(x) + g(y)$ admettent des développements asymptotiques standards uniformes de type $(A*B,\mu*\nu)$.
Mais au delà du résultat qualitatif du théorème \[Th. Seb.\] qui permet de prévoir quel terme peut apparaître dans le développement asymptotique standard uniforme de la convolution de deux fonctions admettant de tels développements asymptotiques, il est important de savoir si les termes “candidats” donnés dans ce théorème donnent effectivement des termes non nuls pour la convolée. Ce problème est assez délicat, et en présence de plusieurs termes dans les développements asymptotiques des fonctions initiales, des phénomènes de compensations peuvent se produire et faire disparaître un terme attendu. Aussi dans la seconde partie de cet article étudions-nous en détail le cas de la convolution de deux fonctions admettant un seul terme singulier non nul dans leurs développements asymptotiques respectifs[^3]. Nous montrons dans ce cas que le terme attendu pour la convolution est effectivement non nul, c’est à dire que nous prouvons le théorème précis énoncé ci-dessous.\
Il n’est pas difficile de se convaincre que pour ce faire on doit calculer précisement les constantes qui apparaissent lors de l’opération de convolution, ce qui amène à faire toute une série de calculs qui sont assez fastidieux et pas aussi simples qu’on pourrait le penser à priori.\
Mais, heureusement, les constantes trouvées s’expriment dans tous les cas comme des “fractions rationnelles de facteurs Gammas”. On peut donc montrer la non nullité des constantes qui interviennent et donc répondre complètement à la question posée dans le cas considéré.\
Un corollaire combinant la proposition \[DA uni\] ci-dessous et le théorème précis \[Precis\] montre alors que, quitte à bien choisir la forme test et à admettre un décalage des exposants entiers, décalage borné ne dépendant que de $f,g$ et des compacts considérés, le terme attendu associé à des termes apparaissant effectivement pour $f$ et $g$, apparaît effectivement pour la fonction $f \oplus g$.\
On notera que le calcul explicite des constantes permettra à l’utilisateur potentiel qui souhaite aller au dela de l’énoncé de notre “théorème précis” et de son corollaire de déterminer dans le cas général, c’est à dire pour un produit de formes test données arbitraires, si un phénomène de compensation se produit dans le cas de la fonction convolée des deux intégrale-fibres données. Le cas où la somme $r + r'$ n’admet qu’une seule écriture avec $r \in A_f, r' \in B_g$ est probablement simple à élucider.
Avant de donner l’énoncé du théorème précis, il est important de disposer du résultat suivant qui permet effectivement de satisfaire l’hypothèse de ce théorème.
\[DA uni\] Soit $\tilde{f} : (\mathbb{C}^{n+1}, 0) \to (\mathbb{C}, 0)$ un germe de fonction holomorphe, et soit $ f : X \to D$ un représentant de Milnor de $\tilde{f}$. Soit $K \subset X$ un compact ; il existe un entier $\kappa$ ne dépendant que de $f$ et de $K$ vérifiant la propriété suivante:\
soit $\varphi \in \mathscr{C}^{\infty}_K(X,\mathbb{C})^{n,n}$ une $(n,n)-$forme test telle que la fonction $$s \to \int_{f = s} \ \varphi$$ admette le terme $c.\vert s\vert^{2r}.s^m.\bar{s}^{m'}.(Log\vert s\vert )^j $ dans son développement asymptotique en $s= 0$, avec $c \in \mathbb{C}^*$, $r \in \mathbb{Q} \cap [0,1[, j \in [0,n]$ et $(m,m') \in \mathbb{N}^2$, où nous supposerons l’entier $j$ maximal pour $r,m,m'$ donnés. Alors pour tout $N \geq \kappa + 1 + (m + m')/2 $ il existe $\varphi_N \in \mathscr{C}^{\infty}_K(X,\mathbb{C})^{n,n}$ vérifiant $$\int_{f = s} \ \varphi_N = \vert s\vert^{2r}.s^{m+\kappa}.\bar{s}^{m'+\kappa}.P_j(Log\vert s\vert ) + \mathcal{O}(\vert s\vert^{2N}) ,$$ où $P_j$ est un polynôme unitaire de degré $j$.
\[Precis\] Soient $\tilde{f}$ et $\tilde{g}$ deux germes de fonctions holomorphes au voisinage de l’origine dans $\mathbb{C}^{p}$ et $\mathbb{C}^{q}$ respectivement, et notons $f : X \to D$ et $g : Y \to D'$ des représentants de Milnor de ces germes. Soient $\varphi \in \mathscr{C}^{\infty}_c(X)^{p,p}$ et $\psi \in \mathscr{C}^{\infty}_c(Y)^{q,q}$ des formes test vérifiant les propriétés suivantes :
- Il existe des rationnels $r,r' \in ]-1,0]$ des entiers $m,n,m',n',j,j'$ et un entier $N > m+n+m'+n' +1$ tels que l’on ait $$\begin{aligned}
& \int_{f=s} \ \frac{\varphi}{df\wedge d\bar f} = \vert s \vert^{2r}.s^m.\bar s^n.P_j(Log\vert s \vert) + \mathcal{O}(\vert s\vert^{2N}) \\
& \int_{g=s} \ \frac{\varphi}{dg\wedge d\bar g} = \vert s \vert^{2r'}.s^{m'}.\bar s^{n'}.Q_{j'}(Log\vert s \vert) + \mathcal{O}(\vert s\vert^{2N})\end{aligned}$$
où $P_j$ et $Q_{j'}$ sont des polynômes [**unitaires**]{} de degrés respectifs $j$ et $j'$. On supposera que pour $r = 0$ (resp $r' = 0$) on a $j \geq 1$ (resp. $j' \geq 1$).\
Alors on a, modulo un polynôme $\chi_N$ de degré (total) $\leq 2N-1$ l’égalité $$\begin{aligned}
& \int_{f(x)+g(y)= s} \frac{\varphi(x)\wedge \psi(y)}{d(f(x)+g(y))\wedge d(\overline{f(x)+g(y)})} = \\
& \\
& \qquad \qquad \qquad \qquad c.\vert s\vert^{2(r+r'+1)}.s^{m+m'}.\bar s^{n+n'}.R(Log\vert s\vert) + \chi_N(s,\bar s) + \mathcal{O}(\vert s\vert^{2N}) \end{aligned}$$ où $c$ est un nombre complexe [**non nul**]{} et où $R$ est un polynôme [**unitaire**]{} dont le degré est déterminé de la fa[ç]{}on suivante :
- si $r , r' $ et $ r +r'+1$ sont non nuls le degré de $R$ est $j+j'$.
- si $r$ et $r'$ sont non nuls mais $r+r'+1 = 0$ le degré de $R$ est $j+j'+1$.
- si $r$ ou $r'$ est nul (ou les deux) le degré de $R$ est $j+j'-1$.
####
Si on a $r = 0 $ et $j = 0$ alors l’intégrale fibre de $\varphi$ ne présente qu’un terme non singulier (modulo $\mathcal{O}(\vert s\vert^{2N})$) et donc la convolution des deux intégrale-fibres aura au moins la même régularité que l’intégrale fibre de $\varphi$. On ne peut donc pas espérer de terme singulier dans ce cas. $\hfill \square$
On notera que la valeur précise de la constante $c$ qui dépend de $r,r',m,m',n,n'$ est donnée dans la section 3.
La démonstration du théorème précis est l’objet de la section 3 et découle des corollaires \[Cas 1 complet\], \[Cas 2 complet\] et \[Calcul 3\].
Combiné avec la proposition \[DA uni\], ce théorème donne immédiatement le corollaire suivant qui précise “presque complètement” le module des développements asymptotiques de la fonction $ (x,y) \mapsto f(x) + g(y)$ à partir de ceux de $f$ et $g$.
Dans la situation du théorème précédent, fixons des compacts $K \subset X$ et $L \subset Y$ ; il existe un entier $\kappa$, ne dépendant que de $f,g, K, L$, tel que, si les intégrale-fibres des formes $\varphi \in \mathscr{C}^{\infty}_K(X)^{p,p}$ et $\psi\in \mathscr{C}^{\infty}_L(Y)^{q,q}$ admettent respectivement dans leurs développements asymptotiques les termes $$c.\vert s\vert^{2r}.s^m.\bar{s}^{n}.(Log\vert s\vert )^j \quad {\rm resp.} \quad c'.\vert s\vert^{2r'}.s^{m'}.\bar{s}^{n'}.(Log\vert s\vert )^{j'}$$ avec $c.c' \not= 0$ les entiers $j$ et $j'$ étant maximaux pour $r,m,n$ (resp. $r', m', n'$) donnés, alors il existe une forme test $\theta $ dans $\mathscr{C}^{\infty}_c(X\times Y)^{p+q,p+q}$ telle que le développement asymptotique de l’intégrale-fibre $$\int_{f(x)+g(y)= s}\quad \frac{\theta(x,y)}{d(f(x)+g(y))\wedge d(\overline{f(x)+g(y)})}$$ contienne le terme $$c.c'.\vert s\vert^{2(r+r'+1)}.s^{m+m'+\kappa}.\bar s^{n+n'+\kappa}.R(Log\vert s\vert)$$ où $R$ est un polynôme [**unitaire**]{} dont le degré est déterminé de la fa[ç]{}on suivante :
- si $r , r' $ et $ r +r'+1$ sont non nuls le degré de $R$ est $j+j'$.
- si $r$ et $r'$ sont non nuls mais $r+r'+1 = 0$ le degré de $R$ est $j+j'+1$.
- si $r$ ou $r'$ est nul (ou les deux) le degré de $R$ est $j+j'-1$.
####
Comme on l’a déjà fait remarquer plus haut, des compensations éventuelles entre divers termes des développements asymptotiques ne permettent pas, en général, d’assurer que la forme test $\varphi \wedge \psi$ aura dans le développement asymptotique de son intégrale-fibre le terme attendu correspondant aux termes donnés dans les développements asymptotiques des intégrale-fibres de $\varphi$ et $\psi$.\
Autrement dit, la forme test $\theta$ sera, en général, différente de $\varphi \wedge \psi$, et le décalge par l’entier $\kappa$ sera nécessaire.
Une conséquence simple de ces résultats est le corollaire suivant, sur le polynôme de Bernstein à l’origine (voir par exemple \[K. 76\]) d’une fonction de la forme $(x,y) \mapsto f(x) + g(y)$, qui peut également se déduire du théorème de Thom-Sebastiani topologique (voir \[Sak.73\]) et des résultats de Malgrange-Kashiwara (voir \[M.83\] ou bien \[Bj.93\] ch. 6) sur le lien entre la monodromie du complexe des cycles evanescents et les racines du polynôme de Bernstein.\
Mais, bien sûr, les résultats ci-dessus sont bien plus précis puisque, non seulement il sont au niveau des termes singuliers des développements asymptotiques, mais aussi puisqu’ils contrôlent les décalages entiers dans les puissances de $s$ et $\bar s$.
\[Bern\] Soient $f : (\mathbb{C}^p,0) \to \mathbb{C},0)$ et $g : (\mathbb{C}^q,0) \to (\mathbb{C},0)$ deux germes non constants de fonctions holomorphes. Alors toute racine du polynôme de Bernstein à l’origine du germe $f \oplus g : (x,y) \to f(x) + g(y)$ est modulo $\mathbb{Z}$ somme d’une racine du polynôme de Bernstein de $f$ en $0$ et d’une racine du polynôme de Bernstein de $g$ en $0$.\
Réciproquement, si $\alpha$ et $\beta$ sont des racines respectivement des polynômes de Bernstein de $f$ et $g$ à l’origine, il existe une racine $\gamma$ du du polynôme de Bernstein de $ f\oplus g : (x,y) \to f(x) + g(y)$ qui est congrue modulo $\mathbb{Z}$ à $\alpha + \beta$.
Le théorème de convolution.
===========================
Utilisation du théorème de Fubini.
----------------------------------
Commen[ç]{}ons par montrer comment le théorème de Fubini permet de ramener notre problème à la détermination du développement asymptotique à l’origine d’une convolution.
\[T-S\] Soient $f : X \to D$ et $g : Y \to D$ des représentants de Milnor de deux germes de fonctions holomorphes à l’origine de $\mathbb{C}^p$ et $\mathbb{C}^q$ respectivement. Pour $\rho$ (resp. $\sigma$ ) une fonction $\mathscr{C}^{\infty}$ à support compact dans $X$ (resp. dans $Y$) notons $\Phi_{f,\rho}$ (resp. $\Psi_{g,\sigma}$ ) l’intégrale fibre $$\begin{aligned}
& \Phi_{f,\rho}(s) : = \int_{f(x) = s} \ \rho(x).\frac{dx\wedge d\bar x}{df\wedge d\bar f} \quad {\rm ( et \ respectivement) } \\
& \qquad\\
& \Psi_{g,\sigma}(t) : = \int_{g(y) = t} \ \sigma(y).\frac{dy\wedge d\bar y}{dg\wedge d\bar g}
\end{aligned}$$ Alors on a l’égalité, pour $s\in \mathbb{C}$ : $$\int_{f(x) + g(y) = s} \ \rho(x).\sigma(y).\frac{dx\wedge d\bar x\wedge dy\wedge d\bar y}{d(f+g)\wedge d\overline{(f + g)}} = \int_{\mathbb{C}} \ \Phi_{f,\rho}(s-u).\Psi_{g,\sigma}(u).du\wedge d\bar u .$$
####
Commen[ç]{}ons par remarquer que, d’après le Nullstellensatz, il existe un entier $N$ en une $(p-1,p-1)-$forme $\omega$ de classe $ \mathscr{C}^{\infty}$ au voisinage de $Supp(\rho)$ dans $\mathbb{C}^p$ vérifiant $$\omega\wedge df\wedge d\bar f = \vert f(x)\vert^{2N}.dx\wedge d\bar x .$$ On a alors, au voisinage de $Supp(\rho)\times Supp(\sigma)$, l’égalité $$\omega\wedge dy\wedge d\bar y\wedge d(f+g)\wedge d\overline{(f+g)} = \vert f(x)\vert^{2N}.dx\wedge d\bar x\wedge dy\wedge d\bar y .$$ On peut alors écrire, toujours au voisinage de $Supp(\rho)\times Supp(\sigma)$, sur la fibre $\{f(x) + g(y) = s \}$ : $$\frac{dx\wedge d\bar x\wedge dy\wedge d\bar y}{d(f+g)\wedge d\overline{(f+g)}} = \frac{dx\wedge d\bar x}{df\wedge d\bar f}\wedge dy\wedge d\bar y .$$ Alors le Théorème de Fubini donne $$\begin{aligned}
& \int_{f(x) + g(y) = s} \ \rho(x).\sigma(y).\frac{dx\wedge d\bar x\wedge dy\wedge d\bar y}{d(f+g)\wedge d\overline{(f + g)}} \quad = \\
&\quad \\
& \int_{y \in \mathbb{C}^q} \sigma(y).dy\wedge d\bar y.\Big( \int_{f(x)= s-g(y)} \rho(x). \frac{dx\wedge d\bar x}{df\wedge d\bar f}\, \Big) \quad = \\
& \quad \\
& \int_{u \in \mathbb{C}} \Phi_{f,\rho}(s-u).\Big( \int_{g(y) = u} \sigma(y).\frac{dy\wedge d\bar y}{dg\wedge d\bar g}\, \Big).du\wedge d\bar u \quad = \\
& \quad \\
& \int_{u \in \mathbb{C}} \Phi_{f,\rho}(s-u).\Psi_{g,\sigma}(u).du\wedge d\bar u \qquad \qquad \qquad \qquad \qquad \blacksquare
\end{aligned}$$
Le théorème de convolution.
---------------------------
Montrons maintenant notre théorème de développement asymptotique pour une convolution.
\[T-S\] Considérons deux nombres réels $\alpha$ et $\beta$ strictement plus grands que $-1$ et soient $\theta$ et $\eta$ deux fonctions dans $\mathscr{C}^{\infty}_c(\mathbb{C})$, et posons, pour $s \in \mathbb{C}^*$, $$F_{\alpha,\beta,j,k}(s) : = \frac{1}{2i\pi}.\int_{\mathbb{C}} \theta(s-u).\eta(u).\vert s-u\vert^{2\alpha}.(Log\vert s-u\vert^2)^j.\vert u\vert^{2\beta}.(Log\vert u\vert^2)^k.du\wedge d\bar u .$$ Alors il existe des fonctions $(\zeta_l)_{l \in [0,L]}$ et $\xi$ dans $\mathscr{C}^{\infty}_c(\mathbb{C})$ telles que l’on ait $$F_{\alpha,\beta,j,k}(s) = \sum_{l=0}^L \ \zeta_l(s).\vert s\vert^{2(\alpha+\beta+1)}.(Log\vert s\vert^2)^l + \xi(s)$$ où l’on a
- $L = j+k $ si $\alpha, \beta ,\alpha+\beta+1 \not\in \mathbb{N}$,
- $L = j+k+1$ si $\alpha, \beta \not\in \mathbb{N}$ et si $\alpha+\beta+1 \in \mathbb{N}$,
- $L = j+k-1$ si $\alpha$ ou $\beta$ ou les deux sont entiers.
De plus, on a $L = -1$ dès que $\alpha \in \mathbb{N}$ et $j = 0$ ou, symétriquement, dès que $\beta \in \mathbb{N}$ et $k = 0$.
####
Nous la ferons en plusieurs étapes.
####
Remarquons déjà que les conditions $\alpha > -1$ et $\beta > -1$ assurent que, pour $s \not= 0$ la fonction intégrée est localement intégrable. De plus le support compact de la fonction $\eta$ montre l’intégrabilité globale. La fonction est donc bien définie sur $\mathbb{C}^*$.\
Le changement de variable $v = s - u$ dans l’intégrale définissant la fonction $F_{\alpha,\beta,j,k}$ montre que les triplets $(\theta, \alpha,j)$ et $(\eta, \beta,k)$ jouent des rôles symétriques.\
Par ailleurs il est clair que pour $\alpha \in \mathbb{N}$ et $j=0$ on a une convolution entre une fonction $\mathscr{C}^{\infty}_c$ est une fonction intégrable à support compact . On obtient donc une fonction $\mathscr{C}^{\infty}_c$, et de même pour $\beta \in \mathbb{N}$ et $k = 0$.
####
Fixons $\varepsilon > 0 $. L’intégrale $$\frac{1}{2i\pi}.\int_{\vert u \vert \geq \varepsilon} \ \theta(s-u).\eta(u).\vert s-u\vert^{2\alpha}.(Log\vert s-u\vert^2)^j.\vert u\vert^{2\beta}.(Log\vert u\vert^2)^k.du\wedge d\bar u$$ donne manifestement une fonction $\mathscr{C}^{\infty}$ sur l’ouvert $ \{ \vert s \vert < \varepsilon/2\} $, puisque maintenant la fonction intégrée est $\mathscr{C}^{\infty}$ en $s$ et majorée par une fonction intégrable fixe ainsi que chacune de ses dérivée partielles en $s, \bar s$.\
Pour prouver l’existence du développement asymptotique à l’origine de la fonction $F_{\alpha,\beta,j,k}$ il suffit donc de le faire pour la fonction obtenue en intégrant seulement sur le disque $\{ \vert u \vert \leq \varepsilon \}$.
####
Montrons maintenant que si la fonction $\theta$ est plate à l’ordre $N \geq 1$ à l’origine, alors la fonction $F_{\alpha,\beta,j,k}$ est de classe $\mathscr{C}^{N-1}$ au voisinage de l’origine. Comme notre hypothèse permet d’écrire $$\theta(v) = \sum_{i=0}^{N+1} \ v^i.\bar v^{N-i+1}.\theta_i(v)$$ où les fonctions $\theta_i$ sont dans $\mathscr{C}^{\infty}_c(\mathbb{C})$, il s’agit de voir que, pour chaque $i \in [0,N+1]$, la fonction $s \mapsto G_i(s)$ définie par l’intégrale $$\int_{\vert u\vert \leq \varepsilon} \ (s-u)^i.\overline{(s-u)}^{N-i+1}.\theta_i(s-u).\eta(u).\vert s-u\vert^{2\alpha}.(Log\vert s-u\vert^2)^j.\vert u\vert^{2\beta}.(Log\vert u\vert^2)^k.du\wedge d\bar u$$ est de classe $\mathscr{C}^{N-1}$ au voisinage de l’origine. Montrons ceci par récurrence sur $N \geq 1$. Pour $N = 1$ il s’agit de voir que les fonctions $G_0, G_1$ et $G_2$ sont continues au voisinage de l’origine. Ceci résulte immédiatement du fait que la fonction $$(s-u)^i.\overline{(s-u)}^{2-i}.\theta_j(s-u).\vert s-u\vert^{2\alpha}.(Log\vert s-u\vert^2)^j$$ est continue en $(s,u)$ pour $i = 0,1,2$ puisque $\alpha > -1$, et de l’intégrabilité de la fonction $u \mapsto \eta(u).\vert u\vert^{2\beta}.(Log\vert u\vert^2)^k$.\
Supposons l’assertion montrée pour $N-1 \geq 1$ et montrons-là pour $N$.\
On a, d’après le théorème de dérivation de Lebesgue, $$\begin{aligned}
& \frac{\partial G_i}{\partial s}(s) = (i+\alpha).\int_{\vert u\vert \leq \varepsilon} (s-u)^{i-1}.\overline{(s-u)}^{N-i+1}.\theta_i(s-u).\eta(u).R_{\alpha,\beta,j.k}(s,u).du\wedge d\bar u + \\
& \int_{\vert u\vert \leq \varepsilon} \ (s-u)^i.\overline{(s-u)}^{N-i+1}.\frac{\partial \theta_i}{\partial v}(s-u).\eta(u).R_{\alpha,\beta,j,k}(s,u).du\wedge d\bar u + \\
& j.\int_{\vert u\vert \leq \varepsilon} (s-u)^{i-1}.\overline{(s-u)}^{N-i+1}.\theta_i(s-u).\eta(u).R_{\alpha,\beta,j-1,k}(s,u).du\wedge d\bar u \end{aligned}$$ où l’on a posé, pour simplifier l’écriture, $$\begin{aligned}
& R_{\alpha,\beta,j,k}(s,u) = \vert s-u\vert^{2\alpha}.(Log\vert s-u\vert^2)^j.\vert u\vert^{2\beta}.(Log\vert u\vert^2)^k \end{aligned}$$
On constate alors que, comme les fonctions $$v^{i-1}.\bar v^{N-i+1}.\theta_i(v) \quad {\rm et} \quad v^i.\bar v^{N-i+1}.\frac{\partial \theta_i}{\partial v}(v)$$ sont plates à l’ordre $N-1$ à l’origine, l’hypothèse de récurrence donne que $ \frac{\partial G_i}{\partial s}$ est de classe $\mathscr{C}^{N-2}$ au voisinage de l’origine. Comme il en est de même pour $\frac{\partial G_i}{\partial \bar s}$, par un calcul analogue à ce que l’on vient de voir, ceci achève la preuve de notre assertion.
####
Etudions maintenant les fonctions suivantes : $$G_{\alpha,\beta,j,k}^{p,q}(s) : = \int_{\vert u\vert \leq 1} u^p.\bar u^q.\vert s-u\vert^{2\alpha}.(Log\vert s-u\vert)^j.\vert u\vert^{2\beta}(Log\vert u\vert)^k.du\wedge d\bar u$$ où $j,k,p, q$ sont des entiers et où $\alpha$ et $\beta$ sont des réels strictement plus grands que $-1$. On va établir la proposition suivante :
La fonction $G_{\alpha,\beta,j,k}^{p,q}$ est de la forme suivante:
- pour $p \geq q$ : $$\sum_{l=0}^{L} c_l.\vert s\vert^{2(\alpha+\beta+1)}.s^p.\bar s^{q}.(Log\vert s\vert)^l \ + \ s^{p-q}.\Phi(\vert s\vert^2)$$
- pour $p \leq q $ : $$\sum_{l=0}^{L} c_l.\vert s\vert^{2(\alpha+\beta+1)}.s^p.\bar s^{q}.(Log\vert s\vert)^l \ \ + \ \bar s^{q-p}.\Phi(\vert s\vert^2)$$
où $\Phi$ est une fonction analytique au voisinage de l’origine qui dépend de $\alpha,\beta,j,k,p,q$, où les constantes $c_l$ dépendent également de $\alpha,\beta,j,k,p,q$, et où l’on a
- $L = j + k$ quand $\alpha, \beta, \alpha + \beta + 1 \not\in \mathbb{N}$,
- $L = j + k + 1$ quand $\alpha + \beta + 1 \in \mathbb{N}$ et $\alpha,\beta \not\in \mathbb{N}$,
- $ L = j + k - 1$ quand $\alpha$ ou $\beta$ ou les deux sont entiers.
On notera que ceci implique que la fonction $G_{\alpha,\beta,j,k}^{p,q}$ est $\mathscr{C}^{\infty}$ en dehors de l’origine et admet, quand $s \to 0$ un développement asymptotique avec des termes “singuliers”[^4] éventuels bien précis, et que ce développement asymptotique est indéfiniment dérivable, c’est-à-dire que toute dérivée partielle de $G_{\alpha,\beta,j,k}^{p,q}$ admet quand $s \to 0$ le développement asymptotique obtenu en effectuant formellement les mêmes dérivées partielles sur le développement asymptotique trouvé que celles effectuées sur $G_{\alpha,\beta,j,k}^{p,q}$.
####
Posons $u = s.t $ pour $s \not= 0$. Alors $G_{\alpha,\beta,j,k}^{p,q}(s)\big/ s^p.\bar s^q.\vert s\vert^{2(\alpha+\beta+1)}$ est donné par l’intégrale : $$\int_{\vert t\vert \leq 1/\vert s\vert} \ t^p.\bar t^q.\vert 1-t\vert^{2\alpha}.\vert t\vert^{2\beta}.(Log\vert s.(1-t)\vert)^j.(Log\vert s.t\vert)^k.dt\wedge d\bar t .$$ Commen[ç]{}ons par supposer que $\alpha+\beta+1 \not\in \mathbb{N}$.\
Comme l’intégrale pour $\vert t\vert \leq 3$ est un polynôme en $Log\vert s\vert$ de degré au plus $j+k$ dont les coefficients dépendent de fa[ç]{}on $\mathscr{C}^{\infty}$ de $(\alpha,\beta) \in ]-1,+\infty[^2$, nous pouvons nous contenter d’étudier la fonction $ \Gamma_{\alpha,\beta,j,k}^{p,q}(s) \big/ s^p.\bar s^q.\vert s\vert^{2(\alpha+\beta+1)}$ donnée par l’intégrale : $$\int_{3 \leq \vert t\vert \leq 1/\vert s\vert} \ t^p.\bar t^q.\vert 1-t\vert^{2\alpha}.\vert t\vert^{2\beta}.(Log\vert s.(1-t)\vert)^j.(Log\vert s.t\vert)^k.dt\wedge d\bar t .$$ Ecrivons $$\begin{aligned}
& ( Log\vert s.t\vert)^k = (Log\vert s\vert + Log\vert t\vert)^k \quad\quad {\rm et}\\
& (Log\vert s.(1-t)\vert)^j = (Log\vert s\vert + Log\vert t\vert + Log\vert 1-1/t\vert)^j \end{aligned}$$ et développons notre intégrale par la formule du binôme. On obtient ainsi un polynôme de degré $\leq j+k$ en $Log\vert s\vert$ et le coefficient de $(Log\vert s\vert)^{j+k-h}$ est, à une combinaison linéaire à coefficients constants de fonctions de la forme $$\int_{3 \leq \vert t\vert \leq 1/\vert s\vert} \ t^p.\bar t^q.\vert 1-t\vert^{2\alpha}.\vert t\vert^{2\beta}.(Log\vert 1-1/t\vert)^{h_1}.(Log\vert t\vert)^{h_2}.dt\wedge d\bar t$$ où $h_1 + h_2 = h$.
En coordonnées polaires $t = \rho.e^{i\varphi}$ l’intégrale précédente donne, à une constante non nulle près, $$\int_3^{1/\vert s\vert} \rho^{p+q+2(\alpha+\beta+1)}.(Log\, \rho)^{h_2}.\frac{d\rho}{\rho}.\int_0^{2\pi} \vert 1- \frac{e^{-i\varphi}}{\rho}\vert^{2\alpha}.(Log\vert 1- \frac{e^{-i\varphi}}{\rho}\vert)^{h_1}.e^{i(p-q)\varphi}.d\varphi .$$
Comme on a $\rho \geq 3$ on a un développement en série qui est normalement convergeant $$\vert 1- \frac{e^{-i\varphi}}{\rho}\vert^{2\alpha}.(Log\vert 1- \frac{e^{-i\varphi}}{\rho}\vert)^{h_1} = \sum_{\nu = 0}^{\infty} \gamma_{\alpha,\nu}^{h_1}(cos \varphi).\rho^{-\nu}$$ ce qui donne que $\Gamma_{\alpha,\beta,j,k}^{p,q}(s) \big{/} s^p.\bar s^q.\vert s\vert^{2(\alpha+\beta+1)} $ est combinaison linéaire à coefficients constants quand $h_2 $ décrit $[0,j+k]$ de termes tels que $$\begin{aligned}
& \sum_{\nu = 0}^{\infty} \ C_{\alpha,\nu}^{h_1}. \int_3^{1/\vert s\vert} \rho^{p+q+2(\alpha+\beta+1)-\nu}.(Log \,\rho)^{h_2}.\frac{d\rho}{\rho} \tag{*}\end{aligned}$$ où $C_{\alpha,\nu}^{h_1} : = \int_0^{2\pi} \ \gamma_{\alpha,\nu}^{h_1}(cos \varphi).e^{i.(p-q).\varphi}.d\varphi $.\
On notera que le polynôme en $cos(\varphi), \gamma_{\alpha,\nu}^{h_1}(cos(\varphi))$, est une fonction $\mathscr{C}^{\infty}$ de\
$\alpha \in ]-1,+\infty[$, et donc que $C_{\alpha,\nu}^{h_1}$ dépend également de fa[ç]{}on $\mathscr{C}^{\infty}$ de $\alpha \in ]-1,+\infty[$.
Remarquons maintenant que la fonction $$H_{\alpha,h_1}^{p,q}(x) = \int_0^{2\pi} \vert 1- x.e^{-i\varphi}\vert^{2\alpha}.(Log\vert 1- x.e^{-i\varphi}\vert)^{h_1} .e^{i(p-q)\varphi}.d\varphi$$ dont le développement en série à l’origine est donné par $$H_{\alpha,h_1}^{p,q}(x) = \sum_{\nu = 0}^{\infty} \ C_{\alpha,\nu}^{h_1}.x^{\nu}$$ est $(-1)^{p+q}-$paire. Ceci se voit facilement en changeant $x$ en $-x$ et $\varphi$ en $\varphi + \pi$.\
Ceci montre que $C_{\alpha,\nu}^{h_1}$ est non nul seulement quand les entiers $p + q$ et $\nu$ ont même parité.\
On en déduit que, sous notre hypothèse $\alpha+\beta + 1 \not\in \mathbb{N}$, l’exposant\
$ 2(\alpha+\beta+1) + p + q - \nu = 0 $ n’apparaît pas dans le développement $(^*)$.
On a donc, pour $\alpha + \beta +1 \not\in \mathbb{N}$ :
$$\Gamma_{\alpha,\beta,j,k}^{p,q}(s) = s^p.\bar s^q.\vert s\vert^{2(\alpha+\beta+1)}.P_{j,k}(Log\vert s\vert) + (\frac{s}{\vert s\vert})^p.(\frac{\bar s}{\vert s\vert})^q.\Phi_{j,k}(\vert s\vert) \tag{**}$$
où $P_{j,k}$ est un polynôme de degré $\leq j+k$, dont les coefficients dépendent de $(\alpha,\beta)$ de fa[ç]{}on $\mathscr{C}^{\infty}$ ainsi que de $p$ et $q$, et où $\Phi_{j,k}$ (qui dépend également de $\alpha,\beta,p,q$) est analytique au voisinage de l’origine.\
On remarquera que le calcul explicite des coefficients du développement en série de la fonction $\Phi_{j,k}$ montre qu’elle dépend de fa[ç]{}on $\mathscr{C}^{\infty}$ de $(\alpha,\beta) \in (]-1, +\infty[)^2$, $\alpha+ \beta+1 \not\in \mathbb{N}$.
Montrons maintenant par récurrence sur $j+k$ que la fonction $$s \mapsto (\frac{s}{\vert s\vert})^p.(\frac{\bar s}{\vert s\vert})^q.\Phi_{j,k}(\vert s\vert)$$ est $\mathscr{C}^{\infty}$ en $s$ au voisinage de l’origine.\
Pour $\lambda \in \mathbb{C}^*$, en posant $u = \lambda.v$ dans l’intégrale qui définit la fonction $G_{\alpha,\beta,0,0}^{p,q}$, on obtient : $$G_{\alpha,\beta,0,0}^{p,q}(\lambda.s) = \lambda^p.\bar \lambda^q.\vert \lambda\vert^{2(\alpha+\beta+1)}.\int_{\vert v\vert \leq 1/\vert \lambda\vert} v^p.\bar v^q.\vert s-v\vert^{2\alpha}.\vert v\vert^{2\beta}.dv\wedge d\bar v$$ ce qui donne, puisque la fonction $$s \mapsto \int_{1 \leq \vert v\vert \leq 1/\vert \lambda\vert} v^p.\bar v^q.\vert s-v\vert^{2\alpha}.\vert v\vert^{2\beta}.dv\wedge d\bar v$$ est $\mathscr{C}^{\infty}$ au voisinage de $s = 0$, $$G_{\alpha,\beta,0,0}^{p,q}(\lambda.s) - \lambda^p.\bar \lambda^q.\vert \lambda\vert^{2(\alpha+\beta+1)}.G_{\alpha,\beta,0,0}^{p,q}(s) \in \mathscr{C}^{\infty}.$$ On aura donc, pour tout $\lambda \in \mathbb{C}^*$ fixé, $$\frac{1}{\vert \lambda\vert^{p+q}}.\lambda^p.\bar\lambda^q. (\frac{s}{\vert s\vert})^p.(\frac{\bar s}{\vert s\vert})^q.\big{[}\Phi_{0,0}(\vert \lambda.s\vert) - \vert \lambda\vert^{2(\alpha+\beta+1)+p +q}.\Phi_{0,0}(\vert s\vert)\big{]} \in \mathbb{C}[[s,\bar s]]. \tag{$^{***}$}$$ Écrivons le développement à l’origine de la fonction analytique $$\Phi_{0,0}(x) = \sum_{m =0}^{\infty} \ c_m.x^m .$$ La relation $(^{***})$ donne alors que $$\frac{1}{\vert \lambda\vert^{p+q}}.\lambda^p.\bar\lambda^q. (\frac{s}{\vert s\vert})^p.(\frac{\bar s}{\vert s\vert})^q.\Big[\sum_{m=0}^{\infty} \ c_m.(\vert \lambda\vert^m - \vert \lambda\vert^{2(\alpha+\beta+1)+p +q}).\vert s\vert^m \Big] \in \mathbb{C}[[s,\bar s]].$$ Comme on a supposé $(\alpha +\beta + 1) \not\in \mathbb{N}$, et que $\Phi_{0,0}$ est $(-1)^{p+q}-$paire, on en déduit que pour chaque $m \in \mathbb{N}$ tel que $c_m \not= 0$, on a $$(\frac{s}{\vert s\vert})^p.(\frac{\bar s}{\vert s\vert})^q.\vert s\vert^m \in \mathbb{C}[[s,\bar s]]$$ ce qui prouve notre assertion pour $j+k = 0$.
Supposons l’assertion démontrée pour $j+k \leq n$ et montrons-là pour $j+k = n+1$. Supposons, par exemple $j \geq 1$. Alors on a, grâce à l’hypothèse de récurrence $$G_{\alpha,\beta,j-1,k}^{p,q}(s) - s^p.\bar s^q.\vert s\vert^{2(\alpha+\beta+1)}.P(Log\vert s\vert) \in \mathscr{C}^{\infty}.$$ Comme on a $$\frac{\partial G_{\alpha,\beta,j-1.k}^{p,q}}{\partial \alpha} = G_{\alpha,\beta,j,k}^{p,q}$$ et que la dépendance en $(\alpha,\beta)\in (]-1,+\infty[)^2$ des calculs précédents est $\mathscr{C}^{\infty}$, on obtient en dérivant en $\alpha$ la relation $(^{**})$ pour le couple $(j-1,k)$ $$G_{\alpha,\beta,j,k}^{p,q}(s) - s^p.\bar s^q.\vert s\vert^{2(\alpha+\beta+1)}.\big[P_{j-1,k}(Log\vert s\vert).Log\vert s\vert^2 + \frac{\partial P_{j-1,k}}{\partial \alpha}(Log\vert s\vert)\big] \in \mathscr{C}^{\infty}$$ ce qui prouve notre assertion pour le couple $(j,k)$ en vertu de l’unicité du développement asymptotique.
####
Si $\alpha $ ou $\beta$ est entier, mais toujours en supposant que $\alpha + \beta +1 \not\in \mathbb{N}$, on constate que pour $j + k = 0$ on n’a pas de terme singulier pour la fonction $G^{p,q}_{\alpha,\beta,0,0}$. La récurrence sur $j+k$ montre que l’on peut prendre $L = j+k-1$ dans ce cas. $\hfill \square$
Dans le cas où $\alpha + \beta +1 = m \in \mathbb{N}$ la fonction $\Phi_{0,0}$ sera la somme d’une fonction analytique et d’un terme logarithmique du type $ \delta.\vert s\vert^{2m +p+q}.Log \vert s\vert^2 $, qui sera obtenu pour $\nu = 2m + p+ q $. On peut alors mettre le terme initial $c.\vert s\vert^{2m}.s^p.\bar s^q$ dans les termes non singuliers, sortir le terme logarithmique et conclure de fa[ç]{}on analogue au cas précédent pour la fonction $G_{\alpha,\beta,0,0}^{p,q}$. Le cas où $(j,k)\in \mathbb{N}^2$ est arbitraire s’en déduit alors par dérivation $j-$fois en $\alpha$ et $k-$fois en $\beta$.
Il nous reste à prouver que pour $(\alpha,\beta) \in \mathbb{N}^2$ on a en fait un polynôme en $Log\vert s\vert$ de degré $\leq j+k-1$. Il suffit en fait de montrer ce résultat pour $j+k = 1$ car le cas général s’en déduit immédiatement par dérivation en $\alpha$ et $\beta$.\
Il s’agit donc de montrer que pour $(\alpha,\beta) \in \mathbb{N}^2$ et, par exemple[^5] $j=0,k = 1$ la fonction $G_{\alpha,\beta,0,1}^{p,q}$ est $\mathscr{C}^{\infty}$. Mais ceci est évident puisque la fonction $u \to \vert u\vert^{2\alpha}$ est $\mathscr{C}^{\infty}$ dans ce cas. $\hfill \blacksquare$
####
D’après la seconde étape, on peut se contenter d’integrer sur le disque $\{ \vert u\vert \leq 1\}$ pour prouver l’assertion. Fixons un entier $N$. D’après la troisième étape, on peut alors remplacer dans l’intégrale les fonctions $\theta$ et $\eta$ par des polynômes de degrés $N +1$ pour établir l’existence et la forme du développement asymptotique d’ordre $N$ puisque l’erreur commise sera de classe $\mathscr{C}^N$ au voisinage de l’origine.\
En développant le polynôme en $s-u$ on est alors ramené à montrer l’existence d’un développement asymptotique du type désiré pour les fonctions du type $$s^{p'}.\bar s^{q'}.G_{\alpha,\beta,j,k}^{p,q}(s) .$$ Ceci est donné par la proposition de l’étape 4.\
On remarquera que l’on a prouvé en même temps que les fonctions $F_{\alpha,\beta,j,k}$ sont $\mathscr{C}^{\infty}$ en dehors de l’origine, puisque c’est le cas pour les fonctions $G_{\alpha,\beta,j,k}^{p,q}$. Ceci achève la preuve du théorème \[T-S\]. $\hfill \blacksquare$
Preuves de la proposition \[DA uni\] et du corollaire \[Bern\].
---------------------------------------------------------------
Commen[ç]{}ons par démontrer la proposition \[DA uni\].
####
Soit donc $K$ un compact de $X$ contenant le support de $\varphi$ et notons $$\mathcal{M}_K : = \{ DA\big[ \int_{f=s} \ \psi ] ,\ \psi \in \mathscr{C}^{\infty}_K(X,\mathbb{C})^{n,n} \}$$ où $DA(F)$ désigne le développement asymptotique à l’origine de la fonction $F$. Alors $ \mathcal{M}_K$ est un $\mathbb{C}[[s,\bar s]]-$module de type fini d’après \[B. 82\], qui est contenu dans $$\vert \Xi\vert^2_{R,n} : = \sum_{(r,j) \in R \times [0,n]} \quad \mathbb{C}[[s,\bar s]].\vert s\vert^{2r}.(Log\vert s\vert )^j ,$$ où $R \subset \mathbb{Q} \cap [0,1[$ est un sous-ensemble fini.\
Notons $\tilde{\mathcal{M}}_K$ le saturé de $\mathcal{M}_K$ par les opérateurs $s\frac{\partial}{\partial s}$ et $\bar s\frac{\partial}{\partial \bar s}$. Comme $\vert \Xi\vert^2_{R,n}$ est stable par ces deux opérateurs, la noethérianité de $\mathbb{C}[[s,\bar s]]$ implique que $\tilde{\mathcal{M}}_K$ est également de type fini sur $\mathbb{C}[[s,\bar s]]$.\
Mais d’après \[B.-S. 74\][^6], on a $s^{n+1}.\frac{\partial}{\partial s} \mathcal{M}_K \subset \mathcal{M}_K$ ainsi que $\bar s^{n+1}.\frac{\partial}{\partial \bar s} \mathcal{M}_K \subset \mathcal{M}_K$. Ceci montre que l’on a l’inclusion $$\tilde{\mathcal{M}}_K \subset \mathcal{M}_K[s^{-1}, \bar s^{-1}] .$$ La finitude de $ \tilde{\mathcal{M}}_K$ sur $\mathbb{C}[[s,\bar s]]$ donne alors l’existence d’un entier $\kappa$, ne dépendant que de $f$ et du compact $K$, tel que l’on ait $$\vert s\vert^{2\kappa}. \tilde{\mathcal{M}}_K \subset \mathcal{M}_K .$$ Il nous suffit donc de prouver la proposition pour $\tilde{\mathcal{M}}_K$ en prenant $\kappa = 0$ dans ce cas.\
Comme $\tilde{\mathcal{M}}_K$ est stable par $s\frac{\partial}{\partial s}$ et $\bar s\frac{\partial}{\partial \bar s}$, l’utilisation itérée des opérateurs $$s\frac{\partial}{\partial s} - (r_1+m_1) \quad {\rm et} \quad \bar s\frac{\partial}{\partial \bar s} - (r_1+m'_1)$$ permet de supprimer dans le développement asymptotique de $\varphi$ tous les termes qui ne sont pas des $\mathcal{O}(\vert s\vert^{2N})$ (il n’y en a qu’un nombre fini) et qui ne sont pas de la forme $\vert s\vert^{2r}.s^m.\bar s^{m'}.P(Log\vert s\vert)$ où $P$ est un polynôme de degré égal à $j$ (rappelons que l’on a supposé que $j$ est maximal pour $(r,m,m')$ donné). Ceci permet alors de conclure. $\hfill \blacksquare$
####
Si la fonction $f$ est contenue dans son idéal jacobien au voisinage du compact $K$, on a $\tilde{\mathcal{M}}_K = \mathcal{M}_K$ et l’on peut prendre $\kappa = 0$ dans la proposition. On a donc le résultat optimal dans ce cas.
En utilisant de manière anticipée les résultats de la section suivante, nous allons démontrer le le corollaire \[Bern\].
####
La première partie du corollaire est immédiate puisque l’on sait que modulo $\mathbb{Z}$ les racines du polynôme de Bernstein de $f$ à l’origine correspondent aux exposants qui apparaissent effectivement dans les\
développements asymptotiques des intégrales fibres au voisinage de l’origine ; voir par exemple \[Bj. 93\] chapitre 6.5.\
La réciproque est conséquence de la proposition \[DA uni\] qui permet d’appliquer le théorème précis \[Precis\] et ainsi d’éviter les phénomènes de compensation entre différents termes des développements asymptotiques de $f$ et $g$. $\hfill \blacksquare$
Calcul des constantes :\
démonstration du théorème précis.
=================================
Préliminaires.
--------------
####
Pour $a, b, \cdots$ des nombres complexes, nous noterons $\alpha, \beta, \cdots$ leurs parties réelles respectives.\
Notons pour $p,q$ entiers positifs, $$\begin{aligned}
& U_{p,q} : = \{ (a,b)\in \mathbb{C}^2 \ / \ \alpha + p/2 > -1 \quad {\rm et} \quad \beta + q/2 > -1 \} \\
& U^0_{p,q} : = U_{p,q} \cap \{(a,b)\in \mathbb{C}^2 \ / \ \alpha + \beta +1+ \frac{p+q}{2} < 0 \}\\
& V_{p,q} : = U_{p,q} \setminus \{(a,b)\in \mathbb{C}^2 \ / \ a+b+1 \in \mathbb{Z} \}.
\end{aligned}$$ On remarquera que pour $(a,b) \in U^0_{p,q}$, on a $a+b+1 \not\in \mathbb{N}$ puisque les inégalités imposées impliquent $-1-\frac{p+q}{2} < \alpha+\beta+1 < -\frac{p+q}{2}$. $\hfill \square$
Nous poserons, pour $ x \in \mathbb{R}, \vert x\vert < 1, m \in \mathbb{Z}$ et $a \in \mathbb{C}$ tel que $\alpha > -1$ $$\frac{1}{2\pi}\int_0^{2\pi} \vert 1 - x.e^{-i\theta}\vert^{2a}.e^{im\theta}.d\theta = \sum_{r=0}^{+\infty} \gamma_{a,m}^r.x^{r} . \tag{@}$$ On remarquera que $ \gamma_{a,m}^r = 0 $ si $r \not= m $ modulo 2, puisque l’intégrale considérée est invariante par le changement $x \to - x$ et $\theta \to \theta + \pi$.\
La conjuguaison complexe montre que l’on a $$\overline{\gamma_{\bar a,-m}^r} = \gamma_{a,m}^r$$ il nous suffira donc de considérer le cas où $m$ est dans $\mathbb{N}$.\
On remarquera également que les coefficients $ \gamma_{a,m}^r $ dépendent holomorphiquement de $a$ sur l’ouvert $\alpha > -1$, d’après la formule de Cauchy.
La fonction holomorphe définie sur l’ouvert $U_q^0 : = U^0_{0,q}$ de $\mathbb{C}^2$ en posant $$G_{q}(a,b) : = \frac{i}{4\pi} \int_{\mathbb{C}} \vert 1-t\vert^{2a}.t^q.\vert t\vert^{2b}.dt\wedge d\bar t$$ est égale à $$\frac{1}{2}.\frac{\Gamma(a+1)\Gamma(b+q+1)\Gamma(-a-b -1)}{\Gamma(-a)\Gamma(-b)\Gamma(a+b+q+2)}. \tag{*}$$
####
Commen[ç]{}ons par remarquer que la convergence absolue de l’intégrale définissant $G_{q}$ est assurée en $t = 0$ par la condition $2\beta + q +1 > -1$, en $t = 1$ par la condition $\alpha > -1$ et en $\vert t\vert = + \infty $ par la condition $ 2(\alpha + \beta) + q + 1 < -1$. Donc la fonction $G_{q}$ est bien holomorphe sur l’ouvert $U^0_q$.\
Pour prouver la formule $(^*)$ il suffit de le faire quand $a = \alpha$ et $b = \beta$. Ce que nous supposerons donc dans la suite.\
En coordonnées polaires $ t = \rho.e^{i\theta}$ on aura, pour $(\alpha,\beta) \in U^0_q$ : $$\begin{aligned}
& G_{q}(\alpha,\beta) = \int_0^{+\infty} (1+\rho^2)^{\alpha}.\rho^{2(\beta+q/2+1)}.\frac{d\rho}{\rho}.\Big(\frac{1}{2\pi}\int_0^{2\pi} \big(1 -\frac{2\rho.cos\,\theta}{1+\rho^2}\big)^{\alpha}.e^{iq\theta}.d\theta \Big) \tag{**}
\end{aligned}$$ Mais on a $$\begin{aligned}
& \frac{1}{2\pi}\int_0^{2\pi} \big(1 -\frac{2\rho.cos\,\theta}{1+\rho^2}\big)^{\alpha}.e^{iq\theta}.d\theta = \\
& \qquad \sum_{k=0}^{\infty} \frac{\Gamma(\alpha+1)}{\Gamma(\alpha-k+1).k!}.\frac{(-1)^k.2^k.\rho^k}{(1+\rho^2)^k}.\frac{1}{2\pi}\int_0^{2\pi} cos^k\theta.e^{iq\theta}.d\theta .
\end{aligned}$$ Et, comme on a $$\begin{aligned}
& \frac{1}{2\pi}\int_0^{2\pi} cos^k\theta.e^{iq\theta}.d\theta \
= \frac{1}{2^k}.C_k^j \quad \quad {\rm si } \quad k = q + 2j , \ {\rm avec} \quad j \in [0,k], \ {\rm et} \\
& \qquad \qquad \qquad \qquad \quad \ = 0 \quad {\rm sinon, \ on \ obtient}\end{aligned}$$
$$\begin{aligned}
& \frac{1}{2\pi}\int_0^{2\pi} \big(1 -\frac{2\rho.cos\,\theta}{1+\rho^2}\big)^{\alpha}.e^{iq\theta}.d\theta = \\
& \qquad \qquad (-1)^q.\sum_{j=0}^{+\infty} C_{q+2j}^j\times\frac{\Gamma(\alpha+1)}{\Gamma(\alpha-q -2j +1).(q+2j)!}\times\frac{\rho^{q+2j}}{(1+\rho^2)^{q+2j}}.
\end{aligned}$$
En reportant dans $(^{**})$ cela donne $$\begin{aligned}
& G_{q}(\alpha,\beta) = \\
& (-1)^q.\sum_{j=0}^{+\infty} C_{q+2j}^j\times \frac{\Gamma(\alpha+1)}{\Gamma(\alpha-q -2j +1).(q+2j)!}\times\int_0^{+\infty} (1+\rho^2)^{\alpha -q-2j}.\rho^{2(\beta+q+j+1)}.\frac{d\rho}{\rho}.
\end{aligned}$$ Utilisons alors la formule classique $$\int_0^{+\infty} (1+x^2)^{-u}.x^v.dx = \frac{1}{2}.\frac{\Gamma(\frac{v+1}{2}).\Gamma(u -\frac{v+1}{2})}{\Gamma(u)}$$ avec $ u = q + 2j - \alpha$ et $1+v = 2(\beta+q+j+1) $. On obtient $$\begin{aligned}
& G_{q}(\alpha,\beta) = \frac{(-1)^q}{2}.\Gamma(\alpha+1).\sum_{j=0}^{+\infty} \ \frac{\Gamma(\beta+q+j+1).\Gamma(-\alpha-\beta+j-1).(q+2j)!}{\Gamma(\alpha-q-2j+1).[(q+2j)!]\Gamma(q+2j-\alpha).j! (q+j)!}.
\end{aligned}$$ Après simplification par $(q+2j)!$ et utilisation de la formule des compléments qui donne $$\begin{aligned}
& \Gamma(\alpha-q-2j+1).\Gamma(q+2j-\alpha) = \frac{\pi}{\sin \pi(q+2j-\alpha)} \\
& \qquad = (-1)^{q+1} \frac{\pi}{\sin\pi\alpha} = (-1)^{q}.\Gamma(1+\alpha).\Gamma(-\alpha)
\end{aligned}$$ on arrive à $$\begin{aligned}
& G_{q}(\alpha,\beta) = \frac{1}{2.\Gamma(-\alpha)}\times \sum_{j=0}^{+\infty} \ \frac{\Gamma(\beta+q+j+1).\Gamma(-\alpha-\beta+j-1)}{\Gamma(q+j+1).j!}
\end{aligned}$$ Utilisons maintenant la formule (voir Erdelyi, Magnus ... p. 10) $$\sum_{j=0}^{+\infty} \ \frac{\Gamma(j+x).\Gamma(j+y)}{\Gamma(j+z).j!} = \frac{\Gamma(x).\Gamma(y).\Gamma(z-x-y)}{\Gamma(z-x).\Gamma(z-y)} .$$ On obtient avec $x = \beta+q+1, y = -\alpha-\beta-1$ et $z = q+1$ $$G_{q}(\alpha,\beta) \ = \frac{1}{2} . \frac{\Gamma(\beta+q+1).\Gamma(-\alpha-\beta-1).\Gamma(\alpha+1)}{\Gamma(-\alpha).\Gamma(-\beta).\Gamma(\alpha+\beta+q+2)}$$ c’est à dire la formule annoncée.$\hfill \blacksquare$
####
1) Le changement $t \to 1/t$ transforme $G_{q}(\alpha,\beta)$ en $\overline{G_{q}(\alpha,-(\alpha+\beta+q+2))}$.\
On constate que ceci est bien compatible avec la formule obtenue puisque si\
$\alpha' = \alpha, \ \beta' = -(\alpha+\beta+q+2)$ on a $$\begin{aligned}
& \beta' + q + 1 = -\alpha-\beta-1, \quad -\alpha'-\beta'-1 = \beta+q+1,\\
& \alpha'+\beta'+q+2 = -\beta \quad \quad etc...
\end{aligned}$$
2) On notera également que l’ouvert $U^0_{0,q}$ est stable par cette involution, puisque\
$$\beta' + q/2 +1= -\alpha-\beta -q/2- 1 > 0 \quad {\rm ainsi \ que}$$ $$\alpha'+\beta'+q/2 +1 = -\beta -q/2 -1 < 0$$ pour $(\alpha,\beta) \in U^0_{0,q}$.
3) On remarquera que pour $(\alpha,\beta) \in U^0_{0,q} \cap \mathbb{R}^2$ le nombre $G_{q}(\alpha,\beta)$ est réel. On en déduit que l’on a, pour tout $(\alpha,\beta) \in U^0_{0,q} \cap \mathbb{R}^2$ $$G_{\bar q}(\alpha,\beta) : = \frac{i}{4\pi} \int\int_{\mathbb{C}} \vert 1-t\vert^{2\alpha}.\bar{t}^q.\vert t\vert^{2\beta}.dt\wedge d\bar t = G_{q}(\alpha,\beta).$$ On a donc, pour $(a,b) \in U^0_{0,q}$ l’égalité $ G_{\bar q}(a,b) = \overline{G_{q}(\bar a,\bar b)}$. $\hfill \square $
Considérons maintenant la même intégrale que dans le lemme précédent mais en demandant seulement à $(a,b)$ de vérifier les conditions de convergence en $t = 0$ et $t =1$ c’est à dire d’être dans l’ouvert $U_{0,q}$ défini par les inégalités $\alpha > -1$ et $\beta + q/2 > -1$. L’intégrale diverge alors à l’infini, mais le lemme suivant montre que sa “partie finie”, c’est à dire le terme constant dans le développement asymptotique à l’infini de cette intégrale donne le prolongement méromorphe de la fonction $G_q$ à l’ouvert $U_{0,q}$.
\[Calcul de la constante\] Soit $q$ un entier positif ou nul. Pour $(a,b) \in V_{0,q}$ on a, pour $N$ entier assez grand, $$\begin{aligned}
& \tag{***} \lim_{s \to 0} \Big[ \frac{i}{4\pi}\int_{\vert t \vert \leq 3} \vert 1-t\vert^{2a}.t^{q}.\vert t\vert^{2b}.dt\wedge d\bar t \ + \\
& \qquad \int_3^{1/\vert s\vert} \rho^{2(a+b+1)+q}.\frac{d\rho}{\rho}.\big[\frac{1}{2\pi}\int_0^{2\pi} \vert1-\frac{e^{-i\theta}}{\rho}\vert^{2a}.e^{iq\theta}.d\theta - \sum_0^N \gamma_{a,q}^r.\rho^{-r}\big]\Big] \ = \\
& \frac{1}{2}. \frac{\Gamma(a+1).\Gamma(b+q+1).\Gamma(-a-b-1)}{\Gamma(-a).\Gamma(-b).\Gamma(a+b+q+2)} + \sum_{r=0}^N \frac{\gamma_{a,q}^r}{2(a+b+1)+q-r}.3^{2(a+b+1)+q-r}
\end{aligned}$$
####
Remarquons déjà que pour $N$ fixé, la différence $$\big[\frac{1}{2\pi}\int_0^{2\pi} \vert1-\frac{e^{-i\theta}}{\rho}\vert^{2a}.e^{iq\theta}.d\theta - \sum_0^N \gamma_{a,q}^r.\rho^{-r}\big]$$ est un $O(\rho^{-N-1})$ et donc que pour $2(\alpha+\beta+1) + q-N-1 < 0$ l’intégrale de $3$ à $+\infty$ converge et la limite cherchée est simplement la valeur de cette intégrale.\
Il s’agit donc simplement de montrer que le prolongement analytique de la fonction holomorphe $G_{q}$ définie sur l’ouvert $U^0_{0,q}$ est bien donné par cette intégrale. On concluera alors grâce au lemme précédent.\
Mais pour $(a,b) \in U^0_{0,q}$, en coupant l’intégrale définissant $G_q$ pour $\vert t\vert \leq 3$ et pour $\vert t \vert \geq 3$ et en utilisant le développement $(@)$ à l’ordre $N$ dans cette dernière, on obtient bien l’égalité de la limite du membre de gauche de $(^{***})$ avec $G_q$ sur l’ouvert $U^0_{0,q}$, puisque $\vert s\vert^{-[2(a+b+1)+q-r]}$ tend vers $0$ quand $s \to 0$, étant donné que l’on a $\alpha+ \beta+q/2 + 1 < 0$ et $r\geq 0$. $\hfill \blacksquare$
On remarquera que ceci montre que le prolongement analytique de la fonction $G_{q}$ a des pôles simples (au plus) sur les droites $$a+b+1 = \frac{r - q}{2} \quad {\rm pour} \quad r \in \mathbb{N}, \quad r = q \quad modulo \ 2.$$ Le fait que ces points soient réellement des pôles simples quand $a+b+1 \in \mathbb{N}$ est montré à posteriori par la formule que l’on a établie. Ceci permet de calculer les coefficients $\gamma_{a,q}^r $. On obtient, pour $a \not\in \mathbb{N}$, en utilisant la formule des compléments (on notera que $a \not\in -\mathbb{N}$ sous nos hypothèses, puisque l’on a $\alpha > -1$ et aussi que $\frac{r+q}{2} - a \not\in -\mathbb{N}$, puisque $\frac{r+q}{2} - \alpha = \beta+q+1 > 0$) $$\begin{aligned}
& \gamma_{a,q}^r = \frac{\Gamma(a+1).\Gamma(\frac{r+q}{2}-a).(-1)^{\frac{r-q}{2}}}{\Gamma(-a).\Gamma(a+1-\frac{r-q}{2}).\Gamma(\frac{r+q}{2}+1).(\frac{r-q}{2})!} \\
& \qquad = (-1)^r.\frac{\Gamma(a+1)^2}{\Gamma(a+1-\frac{r-q}{2}).\Gamma(a+1-\frac{r+q}{2}).(\frac{r-q}{2})! (\frac{r+q}{2})!}\quad . \tag{@@}
\end{aligned}$$ Pour $a \in \mathbb{N}$ le calcul direct donne, pour $q \leq r \leq r+q \leq 2a$ $$\gamma_{a,q}^r = (-1)^r.C_a^{(r+q)/2}.C_a^{(r-q)/2},$$ et $0$ sinon, ce qui co[ï]{}ncide bien avec $(@@)$.
La situation précédente où $a+b+1 \in \mathbb{N}$ avec $a$ non entier est examinée dans le lemme suivant.
\[Calcul de la constante bis\] Soit $q$ un entier positif ou nul. Pour $(a,b) \in U_{0,q}, a+b+1 \in \mathbb{N}$, $a$ [**non entier**]{}, posons $r_0 = 2(a+b+1) + q $. Alors on a, pour $N \geq r_0$ entier $$\begin{aligned}
& \lim_{s \to 0} \Big[ \frac{i}{4\pi}\int_{\vert t \vert \leq 3} \vert 1-t\vert^{2a}.t^{q}.\vert t\vert^{2b}.dt\wedge d\bar t \ + \\
& \qquad \int_3^{1/\vert s\vert} \rho^{2(a+b+1)+q+j}.\frac{d\rho}{\rho}.\big[\frac{1}{2\pi}\int_0^{2\pi} \vert1-\frac{e^{-i.\theta}}{\rho}\vert^{2a}.e^{iq\theta}.d\theta - \sum_0^N \gamma_{a,q}^r.\rho^{-r}\big]\Big] \ + \\
&\qquad \qquad - \sum_{r=0, r \not= r_0}^N \frac{\gamma_{a,q}^r}{2(a+b+1)+q-r}.3^{2(a+b+1)+q-r} + \gamma_{a,q}^{r_0}.Log\, 3
\end{aligned}$$ est égale à $$\frac{(-1)^{r_0}}{2}\frac{\big[\Gamma'(1) + \sum_{j=1}^{(r_0 -q)/2} \frac{1}{j}\big].\Gamma(a+1)^2}{\Gamma((a+1-\frac{r_0+q}{2}).\Gamma((a+1-\frac{r_0-q}{2}).(\frac{r_0+q}{2}!).(\frac{r_0-q}{2}!)} \quad .$$
####
La preuve est la même que pour le lemme précédent sauf que le terme pour $r = r_0$ donne un terme logarithmique dans ce cas. On doit donc évaluer la valeur en $2(a+b+1)+q = r_0$ de la somme $$\frac{1}{2}\, \frac{\Gamma(a+1).\Gamma(b+q+1).\Gamma(-a-b-1)}{\Gamma(-a).\Gamma(-b).\Gamma(a+b+q+2)} \ + \ \frac{1}{2}\,\frac{\gamma_{a,q}^{r_0}}{a+b+1+(q-r_0)/2}.$$
Compte tenu de la formule $(@@)$, du fait que l’on a, pour $k \in \mathbb{N}$, $$\Gamma(z-k) = \frac{(-1)^k}{k!}.[\frac{1}{z} + \Gamma'(1) + \sum_{j=1}^k \frac{1}{j} + o(z)]$$ quand $ z \to 0$, un calcul simple montre que la valeur de cette limite est bien celle annoncée. $\hfill \blacksquare$
On notera que cette limite n’est jamais nulle puisque l’on suppose que $a$ n’est pas un entier
\[Calcul des constantes\] Soient $0 \leq p \leq q$ deux entiers. Posons pour $(a,b)\in U^0_{p,q}$ $$\begin{aligned}
& F_{p,q}(a,b) : = \frac{i}{\pi} \int\int_{\mathbb{C}} \vert 1-t\vert^{2a}.(1-t)^p.\vert t\vert^{2b}.t^q.dt\wedge d\bar t \\
& F_{p,\bar q}(a,b) : = \frac{i}{\pi} \int\int_{\mathbb{C}} \vert 1-t\vert^{2a}.(1-t)^p.\vert t\vert^{2b}.\bar t^q.dt\wedge d\bar t
\end{aligned}$$ On a alors $$F_{p,q}(a,b) = \frac{\Gamma(a+p+1).\Gamma(b+q+1).\Gamma(-a-b-1)}{\Gamma(a+b+p+q+2).\Gamma(-a).\Gamma(-b)}$$ $$F_{p,\bar q}(a,b) = (-1)^p .\frac{\Gamma(a+p+1).\Gamma(b+q+1).\Gamma(-a-b-p-1)}{\Gamma(a+b+q+2).\Gamma(-a).\Gamma(-b)} \quad .$$
####
i) On vérifie facilement les égalités “évidentes à priori” sur $U^0_{p,q} \cap \mathbb{R}^2$ $$F_{p,0} = F_{p,\bar 0} \quad {\rm et} \quad F_{0,q} = F_{0,\bar q} .$$
ii) Une conséquence simple du lemme précédent est le prolongement méromorphe des fonctions $F_{p,q}$ et $F_{p,\bar q}$ à l’ouvert $U_{p,q}$ avec des pôles au plus simples sur les droites $a + b+ 1 \in \mathbb{N}$.
####
La formule suivante sera la clef de cette preuve.
\[Formule\] Soit $p \in \mathbb{N}$. Pour $x,y$ dans $\mathbb{C}\setminus \{-\mathbb{N}\}$ on a $$A_p(x,y) : = \sum_{j=0}^p (-1)^j.C_p^j.\frac{\Gamma(x+j)}{\Gamma(x+y+j)} = \frac{\Gamma(x).\Gamma(y+p)}{\Gamma(x+y+p).\Gamma(y)}\quad .$$
####
Montrons cette égalité par récurrence sur $p \geq 0$. Le cas $p = 0$ est trivial. Supposons la formule démontrée pour $p$ et montrons-là pour $p+1$. En utilisant l’égalité $C_{p+1}^j = C_p^j + C_p^{j-1} $ on obtient $$\begin{aligned}
& A_{p+1}(x,y) = A_p(x,y) - A_p(x+1,y) = \frac{\Gamma(x).\Gamma(y+p)}{\Gamma(x+y+p).\Gamma(y)} - \frac{\Gamma(x+1).\Gamma(y+p)}{\Gamma(x+y+p+1).\Gamma(y)}\\
& \quad \quad = \frac{\Gamma(x).\Gamma(y+p)}{\Gamma(x+y+p+1).\Gamma(y)}[x+y+p - x] \\
& \quad \quad = \frac{\Gamma(x).\Gamma(y+p+1)}{\Gamma(x+y+p+1).\Gamma(y)}\quad . \qquad \qquad \qquad\qquad \qquad \qquad \qquad \qquad \qquad\qquad \blacksquare
\end{aligned}$$
####
La formule du binôme donne $$\begin{aligned}
& F_{p,q}(a,b) = \sum_{j=0}^p (-1)^j.C_p^j.G_{q+j}(a,b) \\
& \qquad \qquad = \frac{\Gamma(a+1).\Gamma(-a-b-1)}{\Gamma(-a).\Gamma(-b)}\times \Big(\sum_{j=0}^p (-1)^j.C_p^j.\frac{\Gamma(b+q+j+1)}{\Gamma(a+b+q+j+2)} \Big)
\end{aligned}$$ et en utilisant le lemme ci-dessus avec $x = b+q+1, y= a+1 $ on obtient la formule annoncée.
La formule du binôme donne $$\begin{aligned}
& F_{p,\bar q}(a,b) = \sum_{j=0}^p (-1)^j.C_p^j.G_{q-j}(a,b+j) \\
& \qquad \qquad = \frac{\Gamma(a+1).\Gamma(b+q+1)}{\Gamma(-a).\Gamma(a+b+q+2)}\times\Big(\sum_{j=0}^p (-1)^j.C_p^j.\frac{\Gamma(-a-b-j-1)}{\Gamma(-b-j)} \Big)
\end{aligned}$$ La formule des compléments donne alors la formule annoncée. $\hfill \blacksquare$
####
Pour $a+b+1 \not\in \mathbb{N}$, les nombres complexes $F_{p,q}(a,b)$ et $F_{p,\bar q}(a,b)$ sont non nuls. $\hfill \square$
Le premier cas.
---------------
####
Soient $0 \leq p \leq q$ deux entiers et soit $D$ le disque unité du plan complexe. Pour $(a,b) \in U_{p,q}$ et $s \in D^* : = D \setminus \{0\}$ posons $$\begin{aligned}
& F_{p,q}(a,b)[s] : = \frac{i}{4\pi}\int_{\vert u \vert \leq 1} \vert s-u\vert^{2a}.(s-u)^p.\vert u \vert^{2b}.u^q.du \wedge d\bar u \\
& F_{p,\bar q}(a,b)[s] : = \frac{i}{4\pi}\int_{\vert u \vert \leq 1} \vert s-u\vert^{2a}.(s-u)^p.\vert u \vert^{2b}.\bar u^q.du \wedge d\bar u
\end{aligned}$$
\[Cas 1\] On suppose que $(a,b) \in V_{p,q}$. Alors il existe des fonctions $\Phi_{p,q}$ et $\Phi_{p,\bar q}$ qui sont $\mathscr{C}^{\infty}$ sur $V_{p,q} \times D$, holomorphes sur $V_{p,q}$ pour $s \in D$ fixé, telles que l’on ait sur $V_{p,q}\times D^*$ $$\begin{aligned}
& F_{p,q}(a,b)[s] = F_{p,q}(a,b).s^{p+q}.\vert s\vert^{2(a+b+1)} + s^{p+q}.\Phi_{p,q}(a,b,s) \\
& F_{p,\bar q}(a,b)[s] = F_{p,\bar q}(a,b).s^p.\bar s^q.\vert s\vert^{2(a+b+1)} + \bar s^{q-p}.\Phi_{p,\bar q}(a,b,s)
\end{aligned}$$
####
Remarquons déjà que le fait que ces fonctions soient $\mathscr{C}^{\infty}$ sur l’ouvert $U^0_{p,q}\times D^*$, holomorphes sur $U^0_{p,q}$ pour $s \in D^*$ fixé, est conséquence immédiate des définitions. Nous allons commencer par traiter le cas de $F_{p,q}$. Pour $s \in D^*$ effectuons le changement de variable $u = s.t$. On obtient $$\begin{aligned}
& F_{p,q}(a,b)[s] = s^{p+q}.\vert s\vert^{2(a+b+1)}.\frac{i}{4\pi}\int_{\vert t \vert \leq 1/\vert s\vert} \vert 1-t\vert^{2a}.(1-t)^p.\vert t\vert^{2b}.t^q.dt\wedge d\bar t \\
& \qquad = s^{p+q}.\vert s\vert^{2(a+b+1)}. \sum_{j=0}^p (-1)^j.C_p^j.\frac{i}{4\pi}\int_{\vert t \vert \leq 1/\vert s\vert} \vert 1-t\vert^{2a}.t^{q+j}.\vert t\vert^{2b}.dt\wedge d\bar t .
\end{aligned}$$ Supposons $\vert s \vert < 1/3$ et posons $$C^0_{p,q}(a,b): = \frac{i}{4\pi}\int_{\vert t \vert \leq 3} \vert 1-t\vert^{2a}.(1-t)^p.\vert t\vert^{2b}.t^q.dt\wedge d\bar t .$$ L’intégrale pour $3 \leq \vert t \vert \leq 1/\vert s\vert $ correspondant au terme $j \in [0,p]$ donne en coordonnées polaires $$I_j = \int_3^{1/\vert s\vert} \rho^{2(a+b)+q+j+1}.d\rho \frac{1}{2\pi}\int_0^{2\pi} \vert 1-\frac{e^{-i\theta}}{\rho}\vert^{2a}.e^{i(q+j)\theta}.d\theta. \tag{A}$$ Utilisons le développement $(@)$, en se souvenant que l’on a $r = q+j$ modulo 2 : $$\begin{aligned}
& I_j = \sum_{r=0}^{+\infty} \gamma_{a,q+j}^r.\int_0^{1/\vert s\vert} \rho^{2(a+b+(q+j-r)/2))+1}.d\rho \\
& \quad = \sum_{r=0}^{+\infty} \frac{\gamma_{a,q+j}^r}{2(a+b+(q+j-r)/2)+1)}.\Big[\vert s\vert ^{-2(a+b+(q+j-r)/2)+1)} - 3^{-2(a+b+(q+j-r)/2)+1)}\Big] \\
& \quad = C^1_{j,q}(a,b) + \vert s\vert^{-2(a+b+1)}.\Psi_j(a,b,\vert s\vert^2) . \tag{B}
\end{aligned}$$ la fonction $\Psi_j$ étant une série de Laurent en $\vert s\vert^2$ dont les coefficients sont holomorphes en $(a,b) $ pourvu que $a+b \not\in \mathbb{Z}$. On notera que la puissance maximale négative en $\vert s\vert^2$ dans $\Psi_j$ est $(j+q)/2 \leq (p+q)/2$.\
On obtient alors $$F_{p,q}(a,b)[s] = C_{p,q}(a,b).s^{p+q}.\vert s\vert^{2(a+b+1)} + \sum_{j=0}^p (-1)^j.C_p^j.s^{p+q}.\Psi_j(a,b,\vert s\vert^2)$$ où nous avons posé $$C_{p,q}(a,b) = C^0_{p,q}(a,b) + \sum_{j=0}^p (-1)^j.C_p^j.C^1_{j,q}(a,b) .$$ Ceci établit l’assertion annoncée pour la fonction $F_{p,q}$ sauf qu’il nous reste à montrer les deux points suivants :
1. La fonction $\Phi_{p,q}(a,b,s) : = \sum_{j=0}^p (-1)^j.C_p^j.s^{p+q}.\Psi_j(a,b,\vert s\vert^2)$ est bien $\mathscr{C}^{\infty}$ en $s=0$, c’est à dire ne présente pas de termes non nul de la forme $s^{p+q}/\vert s\vert^{2k}$ avec $k\geq 1$.
2. Montrer que la constante $C_{p,q}(a,b)$ est bien égale à $F_{p,q}(a,b)$.
Pour établir le premier point considérons un nombre complexe $\lambda \in \mathbb{C}^*$ et calculons la différence $F_{p,q}(a,b)[\lambda.s] - \lambda^{p+q}.\vert \lambda\vert^{2(a+b+1)}.F_{p,q}(a,b)[s] $. Le changement de variable $v = \lambda.u$ montre que cette différence est donnée, pour $\vert \lambda \vert < 1$ par l’intégrale $$\frac{i}{4\pi}\int_{1 \leq \vert v\vert \leq 1/\vert\lambda\vert} \vert s-v\vert^{2a}.(s-v)^p.\vert v\vert^{2b}.v^q.dv\wedge d\bar v .$$ Cette différence est donc $\mathscr{C}^{\infty}$ sur $\mathbb{C}^2\times D$, holomorphe sur $\mathbb{C}^2$ à $s \in D$ fixé. Mais la présence d’un terme non nul de la forme $c.s^{p+q}/\vert s\vert^{2k}$ avec $k \geq 1$ produirait dans le développement à l’origine de la différence précédente le terme $$c.\Big[ \lambda^{p+q}.(\vert \lambda\vert^{-2k} - \vert \lambda\vert^{2(a+b+1)})\Big].s^{p+q}/\vert s\vert^{2k}$$ nécessairement non nul puisque $\alpha+\beta+1 > -1$, ce qui contredirait l’aspect $\mathscr{C}^{\infty}$ en $s = 0$ de cette différence.
L’identification de la constante s’obtient facilement en utilisant les lemmes \[Calcul de la constante\] et \[Calcul des constantes\].\
La preuve pour la fonction $F_{p,\bar q}$ est tout à fait analogue.\
On notera que le terme $s^p.\bar s^q/\vert s \vert^{2k}$ est $\mathscr{C}^{\infty}$ pour $k \in [0,p]$, puisque l’on a supposé $0 \leq p \leq q$, ce qui explique le facteur $\bar s^{q-p}$ dans ce cas. $\hfill \blacksquare$
####
Pour $a$ ou $b$ dans $\mathbb{N}$ on a $F_{p,q}(a,b) =0 $ ainsi que $F_{p,\bar q}(a,b) = 0$ ce qui montre que la fonction considérée est $\mathscr{C}^{\infty}$ en $s = 0$. Ceci est évident à priori puisque l’on convole une fonction $\mathscr{C}^{\infty}$ avec une fonction localement intégrable à support compact.
\[Cas 1 complet\] Dans la situation de la proposition précédente, soient $j$ et $k$ deux entiers, et définissons pour $(a,b) \in V_{p,q}$ les fonctions $$\begin{aligned}
& F_{p,q}^{j,k}(a,b)[s] : = \frac{i}{4\pi}\int_{\vert u \vert \leq 1} \vert s-u\vert^{2a}.(s-u)^p.(Log\vert s-u\vert^2)^j.\vert u \vert^{2b}.u^q.(Log \vert u\vert^2)^k.du \wedge d\bar u \\
& F_{p,\bar q}^{j,k}(a,b)[s] : = \frac{i}{4\pi}\int_{\vert u \vert \leq 1} \vert s-u\vert^{2a}.(s-u)^p.(Log\vert s-u\vert^2)^j.\vert u \vert^{2b}.\bar u^q.(Log \vert u\vert^2)^k.du \wedge d\bar u
\end{aligned}$$ Alors on a $$\begin{aligned}
& F^{j,k}_{p,q}(a,b)[s] = P^{j,k}_{p,q}(a,b)[Log\vert s\vert^2].s^{p+q}.\vert s\vert^{2(a+b+1)} + s^{p+q}.\Phi_{p,q}(a,b,s) \\
& F_{p,\bar q}^{j,k}(a,b)[s] = P^{j,k}_{p,\bar q}(a,b)[Log\vert s\vert^2].s^p.\bar s^q.\vert s\vert^{2(a+b+1)} + \bar s^{q-p}.\Phi_{p,\bar q}(a,b,s)
\end{aligned}$$ où $P^{j,k}_{p,q}$ et $P^{j,k}_{p,\bar q}$ sont des polynômes de degré $j+k$ dont les coefficients dépendent holomorphiquement de $(a,b)$ et dont les coefficients dominants sont donnés par $ F_{p,q}(a,b) $ et $ F_{p,\bar q}(a,b)$ respectivement, et où les fonctions $\Phi_{p,q}^{j,k}$ et $\Phi_{p,\bar q}^{j,k}$ sont obtenues via l’opération $\frac{\partial^{j+k}}{\partial^ja \partial^kb}$ sur les fonctions $\Phi_{p,q}$ et $\Phi_{p.\bar q}$ de la proposition précédente.
####
Il suffit d’appliquer l’opérateur différentiel $\frac{\partial^{j+k}}{\partial^ja \partial^kb}$ dans l’assertion de la proposition précédente. $\hfill \blacksquare$
####
Dans le cas où $a$ (ou bien $b$) est dans $\mathbb{N}$, avec $(a,b) \in U_{p,q}$, le terme de degré $k+j$, c’est à dire le coefficient de\
$s^{p+q}.\vert s\vert^{2(a+b+1)}.\big[Log\vert s\vert^2\big]^{j+k}$ (resp. de $s^p.\bar s^q.\vert s\vert^{2(a+b+1)}.\big[Log\vert s\vert^2\big]^{j+k}$) est nul.\
Pour avoir le terme singulier dominant non nul on doit donc calculer le coefficient de $s^{p+q}.\vert s\vert^{2(a+b+1)}.\big[Log\vert s\vert^2\big]^{j+k-1}$ (resp. $s^p.\bar s^q.\vert s\vert^{2(a+b+1)}.\big[Log\vert s\vert^2\big]^{j+k-1}$). Un calcul simple donne que ce coefficient vaut $$\begin{aligned}
& \Big[ \frac{\partial F_{p,q}}{\partial a} + \frac{\partial F_{p,q}}{\partial b}\Big](a,b) \quad {\rm (resp.} \qquad \Big[ \frac{\partial F_{p,\bar q}}{\partial a} + \frac{\partial F_{p,\bar q}}{\partial b}\Big](a,b)).
\end{aligned}$$ Comme on a $\Gamma(z-k) = \frac{(-1)^k}{k!}\frac{1}{z} + holomorphe(z)$ pour $k \in \mathbb{N}$ et $z$ voisin de $0$, on constate que pour obtenir le coefficient cherché il suffit de remplacer (si par exemple c’est $a$ qui est dans $\mathbb{N}$) le facteur $1/\Gamma(-a)$ dans l’expression de $F_{p,q}$ par le nombre $(-1)^{a+1}.a!$ (resp. dans l’expression de $F_{p,\bar q}$ ) . Ceci montre que ces coefficients sont non nuls (car $a$ et $b$ ne peuvent être simultanément dans $\mathbb{N}$ puisque $a + b+ 1 \not\in \mathbb{N}$ par hypothèse).\
Donc dans ces cas les polynômes en $Log\vert s\vert^2 $ $P^{j,k}_{p,q}(a,b)$ et $P^{j,k}_{p,\bar q}(a,b)$ sont de degré exactement $j+k-1$ . $\hfill \square$
Le second cas.
--------------
\[Cas 2\] On suppose maintenant que $(a,b) \in U_{p,q}$ mais que $a+b+1 $ est un entier. Alors on a $$\begin{aligned}
& F_{p,q}(a,b)[s] = \big[\tilde{F}_{p,q}(a,b).Log\vert s\vert^2 + c_{p,q}(a,b)\big]. s^{p+q}.\vert s\vert^{2(a+b+1)} \ + \ s^{p+q}.\Psi_{p,q}(a,b,s) \\
& F_{p,\bar q}(a,b)[s] = \big[\tilde{F}_{p,\bar q}(a,b).Log\vert s\vert^2 + c_{p,\bar q}(a,b)\big].s^p.\bar s^q.\vert s\vert^{2(a+b+1)} + \bar s^{q-p}.\Psi_{p,\bar q}(a,b,s)
\end{aligned}$$ où les coefficients $\tilde{F}_{p,q}(a,b)$ et $\tilde{F}_{p,\bar q}(a,b)$ sont donnés par les formules suivantes $$\begin{aligned}
& \tilde{F}_{p,q}(a,b) = \frac{(-1)^{a+b}}{\Gamma(-a).\Gamma(-b)}.\frac{\Gamma(a+p+1).\Gamma(b+q+1)}{\Gamma(a+b+2).\Gamma(a+b+p+q+2)} \\
& \tilde{F}_{p,\bar q}(a,b) = \frac{(-1)^{a+b}}{\Gamma(-a).\Gamma(-b)}.\frac{\Gamma(a+p+1).\Gamma(b+q+1)}{\Gamma(a+b+q+2).\Gamma(a+b+p+2)}
\end{aligned}$$ où les fonctions $\Psi_{p,q}$ et $\Psi_{p,\bar q}$ sont $\mathscr{C}^{\infty}$ sur $U_{p,q} \times D$, holomorphes sur $U_{p,q}$ pour $s \in D$ fixé, et les fonctions $c_{p,q}$ et $c_{p,\bar q}$ sont holomorphes sur $U_{p,q}$.
####
Quand on suppose de plus que $a$ et $b$ ne sont pas entiers, les nombres $\tilde{F}_{p,q}(a,b) $ et $\tilde{F}_{p,\bar q}(a,b)$ ne sont pas nuls. $\hfill \square$
####
Elle est analogue au cas de la proposition \[Cas 1\] sauf qu’il faut prendre en compte l’apparition du logarithme puisque le fait que $a+b+1$ soit entier oblige à rencontrer dans la somme l’intégrale $\int_3^{1/\vert s\vert} \frac{d\rho}{\rho}$. Posons $ r_j = q+j+ 2(a+b+1) $, et reprenons le calcul de l’intégrale $I_j$ (voir (A) dans la preuve de la proposition \[Cas 1\]). Le terme en $s^{p+q}.\vert s\vert^{2(a+b+1)} .Log\vert s\vert$ aura pour coefficient $- \gamma_{a,q+j}^{r_j}$. On obtiendra ainsi que $$\begin{aligned}
& \tilde{F}_{p,q}(a,b) = - \sum_{j=0}^p \ (-1)^j.C_p^j.\gamma_{a,q+j}^{r_j}\\
& \quad \quad = (-1)^{a+b}\frac{\Gamma(a+1)}{\Gamma(-a).\Gamma(-b).\Gamma(a+b+2)}.\sum_{j=0}^p (-1)^j.C_p^j.\frac{\Gamma(b+q+j+1)}{\Gamma(a+b+q+j+2)} \\
& \quad \quad = \frac{(-1)^{a+b}}{\Gamma(-a).\Gamma(-b)}\frac{\Gamma(a+p+1).\Gamma(b+q+1)}{\Gamma(a+b+2).\Gamma(a+b+p+q+2)}
\end{aligned}$$ d’après le lemme \[Formule\].\
Le calcule analogue pour le coefficient $ \tilde{F}_{p,\bar q}(a,b) $ donne, $$\begin{aligned}
& \tilde{F}_{p,\bar q}(a,b) = - \sum_{j=0}^p \ (-1)^j.C_p^j.\gamma_{a,j-q}^{r_j}\\
& \quad \quad = (-1)^{a+b+q}\frac{\Gamma(a+1)}{\Gamma(-a).\Gamma(-b-q).\Gamma(a+b+q+2)}.\sum_{j=0}^p (-1)^j.C_p^j.\frac{\Gamma(b+j+1)}{\Gamma(a+b+j+2)} \\
& \quad \quad = \frac{(-1)^{a+b+q}}{\Gamma(-a).\Gamma(-b-q)}\frac{\Gamma(a+p+1).\Gamma(b+1)}{\Gamma(a+b+q+2).\Gamma(a+b+p+2)}\\
& \quad \quad = \frac{(-1)^{a+b}}{\Gamma(-a).\Gamma(-b)}\frac{\Gamma(a+p+1).\Gamma(b+q+1)}{\Gamma(a+b+q+2).\Gamma(a+b+p+2)}
\end{aligned}$$ d’après la formule des compléments. $\hfill \blacksquare$
\[Cas 2 complet\] Dans la situation de la proposition précédente, soient $j$ et $k$ deux entiers, et définissons pour $(a,b) \in U_{p,q}$ vérifiant $a+b+1 \in \mathbb{N}$, les fonctions $$\begin{aligned}
& F_{p,q}^{j,k}(a,b)[s] : = \frac{i}{4\pi}\int_{\vert u \vert \leq 1} \vert s-u\vert^{2a}.(s-u)^p.(Log\vert s-u\vert^2)^j.\vert u \vert^{2b}.u^q.(Log \vert u\vert^2)^k.du \wedge d\bar u \\
& F_{p,\bar q}^{j,k}(a,b)[s] : = \frac{i}{4\pi}\int_{\vert u \vert \leq 1} \vert s-u\vert^{2a}.(s-u)^p.(Log\vert s-u\vert^2)^j.\vert u \vert^{2b}.\bar u^q.(Log \vert u\vert^2)^k.du \wedge d\bar u
\end{aligned}$$ Alors on a $$\begin{aligned}
& F^{j,k}_{p,q}(a,b)[s] = P^{j,k}_{p,q}(a,b)[Log\vert s\vert^2].s^{p+q}.\vert s\vert^{2(a+b+1)} + s^{p+q}.\Phi_{p,q}(a,b,s) \\
& F_{p,\bar q}^{j,k}(a,b)[s] = P^{j,k}_{p,\bar q}(a,b)[Log\vert s\vert^2].s^p.\bar s^q.\vert s\vert^{2(a+b+1)} + \bar s^{q-p}.\Phi_{p,\bar q}(a,b,s)
\end{aligned}$$ où $P^{j,k}_{p,q}$ et $P^{j,k}_{p,\bar q}$ sont des polynômes de degré $j+k+1$ dont les coefficients dépendent holomorphiquement de $(a,b)$ et dont les coefficients dominants sont donnés par $ \tilde{F}_{p,q}(a,b) $ et $ \tilde{F}_{p,\bar q}(a,b)$ respectivement, et où les fonctions $\Phi_{p,q}^{j,k}$ et $\Phi_{p,\bar q}^{j,k}$ sont obtenues via l’opération $\frac{\partial^{j+k}}{\partial^ja \partial^kb}$ sur les fonctions $\Phi_{p,q}$ et $\Phi_{p.\bar q}$ de la proposition précédente.
####
Il suffit à nouveau d’appliquer l’opérateur différentiel $\frac{\partial^{j+k}}{\partial^ja \partial^kb}$ dans l’assertion de la proposition précédente. $\hfill \blacksquare$
Le dernier cas à traiter est celui où $a$ et $b$ sont entiers. Ceci ne peut s’obtenir comme précédemment par dérivation du cas où les logarithmes n’apparaissent pas. Il faut donc traiter directement l’analogue des corollaires \[Cas 1 complet\] et \[Cas 2 complet\].
Le dernier cas.
---------------
\[Cas 3 complet\] Donnons-nous deux entiers\
$0 \leq p \leq q$. Supposons maintenant que $a$ et $b$ sont dans $\mathbb{Z}$ et que l’on a $a+p/2 > -1$ et $b+ q/2 > - 1$. Pour $(j,k) \in (\mathbb{N}^*)^2$ et $s \in D^*$ posons alors $$\begin{aligned}
& F^{j,k}_{p,q}(a,b)[s] : = \frac{i}{4\pi} \int_{\vert u \vert \leq 1} \vert s-u\vert^{2a}.(s-u)^p.(Log\vert s-u\vert^2)^j.\vert u\vert^{2b}.u^q.(Log\vert u\vert^2)^k.du\wedge d\bar u \\
& F^{j,k}_{p,\bar q}(a,b)[s] : = \frac{i}{4\pi} \int_{\vert u \vert \leq 1} \vert s-u\vert^{2a}.(s-u)^p.(Log\vert s-u\vert^2)^j.\vert u\vert^{2b}.\bar u^q.(Log\vert u\vert^2)^k.du\wedge d\bar u
\end{aligned}$$ Alors on a $$\begin{aligned}
& F^{j,k}_{p,q}(a,b)[s] = \check{P}^{j,k}_{p,q}(a,b)[Log\vert s\vert^2].s^{p+q}.\vert s\vert^{2(a+b+1)} + s^{p+q}.\Phi_{p,q}(a,b)[\vert s\vert^2] \\
& F^{j,k}_{p,\bar q}(a,b)[s] = \check{P}^{j,k}_{p,\bar q}(a,b)[Log\vert s\vert^2].s^p.\bar s^q.\vert s\vert^{2(a+b+1)} + \bar s^{q-p}.\Phi_{p,\bar q}(a,b)[\vert s\vert^2] \tag{C}
\end{aligned}$$ où $\check{P}^{j,k}_{p,q}$ et $\check{P}^{j,k}_{p,\bar q}$ sont des polynômes de degré exactement $j+k-1$ et où les fonctions $\Phi_{p,q}^{j,k}(a,b)$ et $\Phi_{p,\bar q}^{j,k}(a,b)$ sont analytiques réelles.
####
Commen[ç]{}ons par rappeler que pour $j =0$ ou $k = 0$ (cas exclus de l’énoncé ci-dessus) les fonctions considérées sont $\mathscr{C}^{\infty}$ comme convolées d’une fonction $\mathscr{C}^{\infty}$ et d’une fonction localement intégrable à support compact.\
Le changement de variable $ u = t.s$, pour $s \not= 0$ fixé donne $$\begin{aligned}
& F^{j,k}_{p,q}(a,b)[s] = s^{p+q}.\vert s\vert^{2(a+b+1)}.I(s)
\end{aligned}$$ où nous avons posé $$\begin{aligned}
& I(s) : = \int_{\vert t \vert \leq 1/\vert s\vert} \vert 1-t\vert^{2a}.(1-t)^p.(Log\vert 1-t\vert^2 + Log\vert s\vert^2)^j.\vert t\vert^{2b}.t^q.(Log\vert s.t\vert^2)^k.dt\wedge d\bar t
\end{aligned}$$ ainsi qu’une expression analogue pour $ F^{j,k}_{p,\bar q}(a,b)[s] $. Des calculs analogues à ceux déjà détaillés plus haut montrent facilement que l’on a des expressions du type $({\rm C})$, mais avec des polynômes en $Log\vert s\vert^2$ à priori de degrés inférieurs ou égaux à $k+j+1$.\
Nous allons démontrer l’assertion sur le degré de ces polynômes par récurrence sur $j+k = n \geq 2$. Pour $n = 2$ on a nécessairement $j = k = 1$ et il s’agit de montrer que les polynômes $\check{P}^{1,1}_{p,q}(a,b)$ et $\check{P}^{1,1}_{p,q}(a,b)$ sont de degrés exactement égal à $1$. Le pas de récurrence qui va suivre montrera qu’ils sont de degrés au plus égal à $1$. Il nous suffit donc de montrer que le coefficient de $Log\vert s\vert^2$ est non nul.\
Ceci résulte du calcul de la constante $\gamma_{1,1}$ qui est fait au paragraphe 3.5.
Supposons démontré que pour $n \geq 2$ le degré des polynômes $\check{P}^{j,k}_{p,q}(a,b)$ et $\check{P}^{j,k}_{p,\bar q}(a,b)$ est au plus égal à $j+k-1 $ et montrons ceci pour un couple $(j,k) \in (\mathbb{N}^*)^2$ vérifiant $j+k = n+1$. Soit $\lambda \in \mathbb{C}^*$ et calculons la différence $$F^{j,k}_{p,q}(a,b)[\lambda.s] - \lambda^{p+q}.\vert \lambda\vert^{2(a+b+1)}.F^{j,k}_{p,q}(a,b)[s]$$ en utilisant le changement de variable $u = \lambda.v$. On obtient $ \frac{i}{4\pi}.\lambda^{p+q}.\vert \lambda\vert^{2(a+b+1)}$ multiplié par $$\begin{aligned}
&\int_{1\leq \vert v\vert \leq 1/\vert \lambda\vert} \vert s - v\vert^{2a}.(s-v)^p.(Log\vert \lambda\vert + Log\vert s-v\vert)^j.\vert v\vert^{2b}.v^q.(Log\vert \lambda\vert + Log\vert v\vert)^k.dv\wedge d\bar v \\
&\qquad \qquad \qquad \qquad + \int_{\vert v\vert \leq 1} \vert s-v\vert^{2a}.(s-v)^p.\vert v\vert^{2b}.v^q.Z.dv\wedge d\bar v
\\
&{\rm avec} \quad Z : = \Big[(Log\vert \lambda\vert + Log\vert s-v\vert)^j.(Log\vert \lambda\vert + Log\vert v\vert)^k - ( Log\vert s-v\vert)^j.( Log\vert v\vert)^k\Big]
\end{aligned}$$ On constate que la première intégrale est $\mathscr{C}^{\infty}$ sur $D$, et que la seconde est une combinaison linéaire des fonctions $F^{j',k'}_{p,q}(a,b)$ avec $j'+k' \leq n $. On en déduit facilement notre assertion.\
De plus, si $\gamma(j,k)$ désigne le coefficient de $(Log\vert s\vert)^{j+k-1}$ dans $ \check{P}^{j,k}_{p,q}(a,b)$ le calcul ci-dessus donne facilement la relation $$\gamma(j,k) = (j+k)! \gamma(1,1) \quad \forall j, k \geq 1$$ ce qui montre que si $\gamma(1,1)$ est non nul, il en est de même pour tous les\
$\gamma(j,k), \forall (j,k) \in (\mathbb{N}^*)^2 $. Nous allons montrer au paragraphe suivant que la constante $\gamma(1,1) $ (qui dépend de $a,b,p,q$) est non nulle, ce qui achèvera la preuve. $\hfill \blacksquare$
Le calcul de $\gamma(1,1)$.
----------------------------
\[Calcul 1\] Pour $0 \leq x < 1 $ et $p \in \mathbb{Z}$ on a $$\begin{aligned}
& C_p(x) : = \frac{1}{2\pi}\int_0^{2\pi} Log\vert 1- x.e^{i\theta}\vert^2.e^{ip\theta}.d\theta = \frac{x^{\vert p\vert}}{\vert p\vert} \quad {\rm pour} \quad p \not= 0 \\
& C_0(x) : = \frac{1}{2\pi}\int_0^{2\pi} Log\vert 1- x.e^{i\theta}\vert^2.d\theta = 0.
\end{aligned}$$ Pour $x > 1$ on a $C_p(x) = C_p(1/x)$ pour $p \not= 0$ et $C_0(x) = Log\, x^2 $.
####
Posons pour $x \in ]0,1[$ $$A_p : = \frac{1}{2i\pi}\int_{\vert z\vert = x} Log(1-z).z^p \frac{dz}{z} .$$ Alors on a $C_p(x) = x^{-p}.A_p - x^p.\bar A_{-p}$. Comme la formule de Cauchy donne $$\begin{aligned}
& A_p = 0 \quad {\rm pour} \quad p \geq 0 \quad {\rm et} \\
& A_p = -\frac{1}{p} \quad {\rm pour} \quad p < 0
\end{aligned}$$ on conclut facilement.$\hfill \blacksquare$
\[Calcul 2\] Pour $p,q$ deux entiers naturels, posons pour $s \in D$ $$F_{p,q}(s) : = \frac{i}{4\pi}\int_{\vert u \vert \leq 1} \ u^p.\bar u^q.Log\vert s-u\vert.Log\vert u\vert.du\wedge d\bar u .$$ Alors le coefficient du terme en $s^p.\bar s^q.Log\vert s\vert^2 $ dans le développement asymptotique en $s = 0$ de $F_{p,q}$ vaut $- \frac{1}{4(p+1)(q+1)}$.
On remarquera que le terme en $s^p.\bar s^q.Log\vert s\vert^2 $ est le seul terme non $\mathscr{C}^{\infty}$ dans le développement de cette fonction à l’origine.
####
Commen[ç]{}ons par le cas $p \not= q$. On a, en posant $u = s.t$ pour $s \not= 0$ $$\begin{aligned}
& F_{p,q}(s) = s^{p+1}.\bar s^{q+1}\frac{i}{4\pi}\int_{\vert t\vert \leq 1/\vert s\vert} t^p.\bar t^q.Z.dt\wedge d\bar t \\
& {\rm avec} \quad Z : = (Log\vert s\vert + Log\vert 1-t\vert).(Log\vert s\vert + Log\vert t\vert)
\end{aligned}$$ Cela donne les trois termes suivants $$\begin{aligned}
& A : = s^{p+1}.\bar s^{q+1}.(Log\vert s\vert)^2.I_1 \\
& B : = s^{p+1}.\bar s^{q+1}.(Log\vert s\vert).I_2 \\
& C : = s^{p+1}.\bar s^{q+1}.I_3
\end{aligned}$$ où les intégrales $I_1, I_2$ et $I_3$ vont être examinées ci-dessous.\
Remarquons déjà que le développement asymptotique de $A$ ne donnera jamais de contribution au terme qui nous intéresse.\
Pour $B$ nous cherchons le terme constant dans le développement asymptotique de $$I_2(s) : = \frac{i}{4\pi}\int_{\vert t\vert \leq 1/\vert s\vert} t^p.\bar t^q.(Log\vert 1-t\vert + Log\vert t\vert).dt\wedge d\bar t \quad.$$ Cherchons déjà le terme constant dans le développement de l’intégrale $$\begin{aligned}
& I'_2(s) : = \frac{i}{4\pi}\int_{1 \leq \vert t\vert \leq 1/\vert s\vert} t^p.\bar t^q.(Log\vert 1-t\vert + Log\vert t\vert).dt\wedge d\bar t \\
& \quad = \frac{1}{2}.\int_1^{1/\vert s\vert} \rho^{p+q+1}.C_{p-q}(\rho).d\rho \ + \ \int_1^{1/\vert s\vert} \rho^{p+q+1} Log\, \rho .d\rho \quad .
\end{aligned}$$ Comme le lemme précédent donne $C_{p-q}(\rho) = \frac{\rho^{-\vert p-q\vert}}{\vert p-q\vert}$ puisque l’on suppose $p \not= q$, on obtient facilement que le terme constant du développement de $I'_2(s)$ vaut $$-\frac{1}{2}.\frac{1}{p+q+2-\vert p-q\vert} \ + \ \frac{1}{(p+q+2)^2} .$$ Il nous reste encore à évaluer la constante $$I_2" : = \frac{i}{4\pi}\int_{\vert t\vert \leq 1} t^p.\bar t^q.(Log\vert 1-t\vert + Log\vert t\vert).dt\wedge d\bar t$$ ce qui est simple à l’aide du lemme précédent : il donne $$\begin{aligned}
& I_2" = \frac{1}{2}.\int_0^1 \rho^{p+q+1}.C_{p-q}(\rho).d\rho \ + \ \int_0^1 \rho^{p+q+1} Log\, \rho .d\rho \\
& \quad = \frac{1}{2.\vert p-q\vert}.\Big[\frac{\rho^{p+q+2+\vert p-q\vert}}{p+q+2+\vert p-q\vert}\Big]_0^1 \ - \ \frac{1}{(p+q+2)^2}\\
& \quad = \frac{1}{2.\vert p-q\vert}.\frac{1}{p+q+2+\vert p-q\vert} - \frac{1}{(p+q+2)^2}
\end{aligned}$$ On trouve finalement, comme contribution de $I_2$ la constante $$\begin{aligned}
& -\frac{1}{2\vert p-q\vert}.\Big[\frac{1}{p+q+2-\vert p-q\vert} \ - \frac{1}{p+q+2+\vert p-q\vert} \Big]\\
& \quad = -\frac{1}{4(p+1)(q+1)}
\end{aligned}$$
Cherchons la contribution de ${\rm C}$ c’est à dire le terme en $Log \vert s\vert$ dans le développement asymptotique de
$$\begin{aligned}
& I_3(s) : = \frac{i}{4\pi}\int_{\vert t\vert \leq 1/\vert s\vert} t^p.\bar t^q.(Log\vert 1-t\vert).(Log\vert t\vert).dt\wedge d\bar t
\end{aligned}$$
Comme le terme constant ne nous intéresse pas, on peut se contenter de regarder $$\begin{aligned}
& I'_3(s) : = \frac{i}{4\pi}\int_{1 \leq \vert t\vert \leq 1/\vert s\vert} t^p.\bar t^q.(Log\vert 1-t\vert).(Log\vert t\vert).dt\wedge d\bar t \\
& \quad = \int_1^{1/\vert s\vert} \rho^{p+q+1}.C_{p-q}(\rho).Log\,\rho.d\rho \\
& \quad = \frac{1}{\vert p-q\vert}.\int_1^{1/\vert s\vert} \rho^{p+q+1-\vert p-q\vert}.Log\,\rho.d\rho
\end{aligned}$$ et il n’y a pas de terme en $Log \vert s\vert$ dans le développement asymptotique de $I_3(s)$, puisque $p+q+2-\vert p-q\vert \geq 2$.\
Le cas $p = q$ est analogue en utilisant le calcul de $C_0(x)$ dans le lemme précédent, en prenant garde au cas $x > 1$. $\hfill \blacksquare$
\[Calcul 3\] Soient $a,b,p,q$ des entiers naturels. Pour $s \in D^*$, posons $$\begin{aligned}
& F_{p,q}(a,b)[s] : = \frac{1}{4i\pi}\int_{\vert u \vert \leq 1} \vert s-u\vert^{2a}.(s-u)^p.Log\vert s-u\vert.\vert u\vert^{2b}.u^q.Log\vert u\vert.du\wedge d\bar u \\
& F_{p,\bar q}(a,b)[s] : = \frac{1}{4i\pi}\int_{\vert u \vert \leq 1} \vert s-u\vert^{2a}.(s-u)^p.Log\vert s-u\vert.\vert u\vert^{2b}.\bar u^q.Log\vert u\vert.du\wedge d\bar u
\end{aligned}$$ Le coefficient du terme en $\vert s\vert^{2(a+b+1)}.s^{p+q}.Log\vert s\vert $ dans le développement asymptotique de $ F_{p,q}(a,b)$ en $s = 0$ est égal à $$\begin{aligned}
& -\frac{1}{4}\frac{\Gamma(a+1).\Gamma(b+1)}{\Gamma(a+b+2)}.\frac{\Gamma(a+p+1).\Gamma(b+q+1)}{\Gamma(a+b+p+q+2)} \quad .
\end{aligned}$$ Le coefficient du terme en $\vert s\vert^{2(a+b+1)}.s^p.\bar s^q.Log\vert s\vert $ dans le développement asymptotique de $ F_{p,\bar q}(a,b)$ en $s = 0$ est égal à $$-\frac{1}{4}\frac{\Gamma(a+1).\Gamma(b+q+1)}{\Gamma(a+b+q+2)}.\frac{\Gamma(a+p+1).\Gamma(b+1)}{\Gamma(a+b+p+2)} \quad .$$
On remarquera à nouveau que, pour chacune de ces fonctions, le terme considéré est le seul terme non $\mathscr{C}^{\infty}$ du développement asymptotique.
####
La formule du binôme et la proposition précédente donne que le coefficient cherché vaut, pour la fonction $ F_{p,q}(a,b)$, $$\begin{aligned}
& \frac{-1}{4}. \sum_{j=0}^{a+p}\sum_{k=0}^a (-1)^{j+k}.C_{a+p}^j.C_a^k \frac{1}{(b+q+j+1)(b+k+1) } \quad {resp.} \\
& \frac{-1}{4}. \sum_{j=0}^{a+p}\sum_{k=0}^a (-1)^{j+k}.C_{a+p}^j.C_a^k\frac{1}{(b+j+1)(b+q+k+1) }
\end{aligned}$$ En utilisant la formule $$\sum_{k=0}^m (-1)^k.C_m^k\frac{1}{n+k} = \frac{\Gamma(m+1).\Gamma(n)}{\Gamma(m+n+1)}$$ on obtient facilement le résultat annoncé. L’autre cas est analogue. $\hfill \blacksquare$
####
Le corollaire précédent montre, en particulier, que ces coefficients ne sont jamais nuls.
Références.
===========
- [\[B.82\]]{} Barlet, D. *Développements asymptotiques des fonctions obtenues par intégration sur les fibres*, Inv. Math. vol. 68 (1982), p. 129-174.
- [\[B.86\]]{} Barlet, D. *Calcul de la forme hermitienne canonique pour $X^a + Y^b + Z^c $*, in Sem. P. Lelong, Lecture Notes, vol. 1198 Springer Verlag (1986), p. 35-46.
- [\[Bj.93\]]{} Bjork,J.-E. [*Analytic D-Modules and Applications*]{}, Kluwer Academic Publishers, Dordrecht/Boston/London 1993.
- [\[B.-S.74\]]{} Brian[ç]{}on, J. et Skoda, H. [*Sur la clôture intégrale d’un idéal de germes de fonctions holomorphes en un point de $\mathbb{C}^n$*]{}, C.R.Acad.Sci. Paris série A, 278 (1974), p.949-951.
- [\[K.76\]]{} Kashiwara,M. [*b-function and holonomic systems*]{}, Inv. Math. 38, (1976), p.33-53.
- [\[M.74\]]{} Malgrange, B. [*Intégrale asymptotique et monodromie.*]{} Ann. Scient. Ec. Norm. Sup. , t.7 (1974), p.405-430.
- [\[M.83\]]{} Malgrange, B. [*Polynôme de Bernstein-Sato et cohomologie évanescente*]{}, Astérisque 101-102 (1983), p.243-267.
- [\[Sak.73\]]{} Sakamoto, K. [*Milnor fibering and their characteristic maps*]{}, Proc. Intern. Conf. on manifolds and Related Topics in Topology, Tokyo 1973.
- [\[S.-T.71\]]{} Sebastiani, M. and Thom, R. [*Un résultat sur la monodromie*]{}, Inv. Math. 13 (1971), p. 90-96.
Barlet Daniel, Institut Elie Cartan UMR 7502\
Nancy-Université, CNRS, INRIA et Institut Universitaire de France,\
BP 239 - F - 54506 Vandoeuvre-lès-Nancy Cedex.France.\
e-mail : barlet@iecn.u-nancy.fr.
[^1]: Barlet Daniel, Institut Elie Cartan UMR 7502 Nancy-Université, CNRS, INRIA et Institut Universitaire de France, BP 239 - F - 54506 Vandoeuvre-lès-Nancy Cedex.France.e-mail : barlet@iecn.u-nancy.fr
[^2]: nous considérons ici des fonctions “uniformes” contrairement aux fonctions holomorphes “multiformes” que l’on obtient dans les intégrales “à la Malgrange”, c’est à dire en intégrant des formes holomorphes sur des familles horizontales de cycles (voir \[M. 74\]).
[^3]: en fait on regroupe ensemble les différentes puissances de $Log\vert s\vert$ correspondant aux mêmes exposants pour $s$ et $\bar s$.
[^4]: c’est-à-dire qui ne sont pas dans $\mathbb{C}[[s,\bar s]]$.
[^5]: rappelons que l’on a symétrie entre $j$ et $k$.
[^6]: en combinant l’inclusion $f^{n+1}.\Omega^{n+1}_X \subset df\wedge \Omega^n_X$ avec la formule de dérivation $$\frac{\partial}{\partial s} \int_{f=s} \varphi = \int_{f=s} \frac{d'\varphi}{df} .$$
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'It is conjectured that every closed manifold admitting an Anosov diffeomorphism is, up to homeomorphism, finitely covered by a nilmanifold. Motivated by this conjecture, an important problem is to determine which nilmanifolds admit an Anosov diffeomorphism. The main theorem of this article gives a general method for constructing Anosov diffeomorphisms on nilmanifolds. As a consequence, we give new examples which were overlooked in a corollary of the classification of low-dimensional nilmanifolds with Anosov diffeomorphisms and a correction to this statement is proven. This method also answers some open questions about the existence of Anosov diffeomorphisms which are minimal in some sense.'
author:
- |
Jonas Deré[^1]\
KU Leuven Kulak, E. Sabbelaan 53, BE-8500 Kortrijk, Belgium
bibliography:
- 'G:/algebra/ref.bib'
title: '**A new method for constructing Anosov Lie algebras**'
---
A diffeomorphism $f: M \to M$ on a closed manifold is called Anosov if the tangent bundle splits continuously into two $df$-invariant vector bundles $E^s$ and $E^u$, such that $df$ is contracting on $E^s$ and expanding on $E^u$. The standard examples of Anosov diffeomorphisms are induced by unimodular hyperbolic automorphisms of ${\mathbb{R}}^n$ on the $n$-torus ${\mathbb{T}}^n$, considered as quotient space $\faktor{{\mathbb{R}}^n}{{\mathbb{Z}}^n}$. The first non-toral example was given by S. Smale in his paper [@smal67-1] where he also raised the question of classifying all closed manifolds admitting an Anosov diffeomorphism.
It is conjectured that every Anosov diffeomorphism is topologically conjugate to an affine infra-nilmanifold automorphism. A proof of this conjecture would thus imply that every closed manifold admitting an Anosov diffeomorphism is homeomorphic to an infra-nilmanifold admitting an Anosov diffeomorphism. More details about the definition and the conjecture can be found in [@deki11-1; @smal67-1]. In recent years, there has been quite some research about classifying infra-nilmanifolds supporting an Anosov diffeomorphism as well as constructing new examples with specific properties.
The conjecture motivates the study of infra-nilmanifolds supporting an Anosov diffeomorphism and in this paper we give a general method for constructing examples in the case of nilmanifolds. The existence of an Anosov diffeomorphism on a nilmanifold is equivalent to the existence of an Anosov automorphism on the corresponding nilpotent Lie algebra, i.e. a hyperbolic and integer-like automorphism (see Section \[intro\] for more details). One way to construct Anosov automorphisms on Lie algebras is to start from an automorphism of a free nilpotent Lie algebra and take a quotient by an ideal which is invariant under the automorphism, see for example the papers [@dm05-1; @dd03-2; @frie81-1; @payn09-1]. Another type of construction, as was used in [@laur03-1; @lw08-1; @lw09-1; @mw07-1], starts from a nilpotent Lie algebra ${\mathfrak{n}}^E$ over some field extension $E$ of ${\mathbb{Q}}$ with a hyperbolic automorphism and then gives a rational form of the Lie algebra which is invariant under the automorphism. The hard step in the latter case is to construct an explicit basis of ${\mathfrak{n}}^E$ such that the structure constants are rational and the matrix representation of the automorphism has rational entries. Note that in these papers, it is checked for each example separately that the given set of vectors is in fact a basis, that the structure constants do lie in ${\mathbb{Q}}$ and that the matrix of $f$ with respect to this basis has entries in ${\mathbb{Q}}$. The computations needed for these steps are rather cumbersome and somewhat time-consuming and these can be avoided by the main theorem of this paper.
The main theorem of this article generalizes this second method and states that given a hyperbolic and integer-like automorphism of a Lie algebra ${\mathfrak{n}}^E$ which behaves ‘nicely’ under the action of $\operatorname{Gal}(E,{\mathbb{Q}})$ (see Section \[secGC\] for the exact statement), there always exists a rational form of ${\mathfrak{n}}^E$ which is an Anosov Lie algebra. Not only does this technique significantly shorten the construction of the examples in [@laur03-1; @lw08-1; @lw09-1; @mw07-1], it also allows us to give new interesting examples of Anosov diffeomorphisms on nilmanifolds. For example, we can give a positive answer to some existence questions stated in [@laur03-1; @lw09-1].
The construction from the main theorem is also important for the classification of infra-nilmanifolds supporting an Anosov diffeomorphism. The only general classification is given by Porteous in [@port72-1] for flat manifolds and this was generalized to infra-nilmanifolds modeled on free nilpotent Lie groups in [@dd13-1], based on the work of [@dv09-1; @dv11-1]. For nilmanifolds, there is a classification of Anosov automorphisms on Lie algebras up to dimension $8$ in [@lw09-1]. One of the consequences of the main theorem is the construction of some new examples which were overlooked in this classification and we formulate a correction to this statement in Section \[verb\].
This article is built up as follows. In the first section, we give the main definitions about Anosov diffeomorphisms on nilmanifolds and state the correspondence between Anosov diffeomorphisms and hyperbolic, integer-like automorphisms of rational Lie algebras. The second part discusses the exact statement and proof of the main theorem. The last part describes the consequences of the main theorem, including the new examples answering the questions mentioned above.
Anosov diffeomorphisms on nilmanifolds {#intro}
======================================
We start with recalling the definitions and main results about Anosov diffeomorphisms on nilmanifolds. We also introduce the signature of an Anosov diffeomorphism and the type of a nilpotent Lie algebra, including some existence questions about Anosov automorphisms we will answer in this paper.
Let $N$ be a connected, simply connected, nilpotent Lie group with a lattice $\Gamma$, i.e. a discrete subgroup $\Gamma \subset N$ with compact quotient $\Gamma \backslash N$. The quotient space $\Gamma \backslash N$ is a closed manifold with fundamental group $\Gamma$ and is called a nilmanifold. The group $\Gamma$ is nilpotent, finitely generated and torsion-free and every group satisfying those three properties occurs as the fundamental group of a nilmanifold (see e.g. [@deki96-1]). Every Lie group automorphism $\alpha: N \to N$ induces a Lie algebra automorphism on the Lie algebra of $N$ and the eigenvalues of $\alpha$ are defined as the eigenvalues of this Lie algebra automorphism. If $\alpha \in \operatorname{Aut}(N)$ is a hyperbolic automorphism, meaning that it has no eigenvalues of absolute value $1$, and if $\alpha(\Gamma) = \Gamma$, then $\alpha$ induces an Anosov diffeomorphism on the nilmanifold $\Gamma \backslash N$ and the induced map is called a hyperbolic nilmanifold automorphism. For more details and the general definitions of infra-nilmanifolds and affine infra-nilmanifold automorphisms we refer to the paper [@deki11-1].
Let $\Gamma \backslash N$ be a nilmanifold and denote by $\Gamma^{\mathbb{Q}}$ the radicable hull of $\Gamma$ (also called rational Mal’cev completion). Corresponding to this radicable hull $\Gamma^{\mathbb{Q}}$, there is also a rational Lie algebra ${\mathfrak{n}}^{\mathbb{Q}}$ and every finite dimensional rational nilpotent Lie algebra can be found in this way (see e.g. [@sega83-1]). Two lattices $\Gamma_1$ and $\Gamma_2$ of $N$ are called commensurable if there exist finite index subgroups $\Gamma_1^\prime \le \Gamma_1$ and $\Gamma_2^\prime \le \Gamma_2$ such that $\Gamma_1^\prime$ and $\Gamma_2^\prime$ are isomorphic. This is equivalent to the property that the Lie algebras of the radicable hulls $\Gamma_1^{\mathbb{Q}}$ and $\Gamma_2^{\mathbb{Q}}$ are isomorphic.
From [@deki99-1 Corollary 3.5.] and [@mann74-1] it follows that a nilmanifold $\Gamma \backslash N$ admits an Anosov diffeomorphism if and only if there exists a hyperbolic and integer-like automorphism of ${\mathfrak{n}}^{\mathbb{Q}}$. Recall that a matrix is called integer-like if its characteristic polynomial has coefficients in ${\mathbb{Z}}$ and its determinant has absolute value $1$. Since the eigenvalues and characteristic polynomial of a matrix are invariant under conjugation, the properties hyperbolic and integer-like are invariant under change of basis. An automorphism is hyperbolic respectively integer-like if the matrix representation of the automorphism for some (and thus for every) basis has the same property. This motivates the following definition:
An automorphism $\alpha \in \operatorname{Aut}({\mathfrak{n}}^E)$ of a Lie algebra ${\mathfrak{n}}^E$ over some field $E \subseteq {\mathbb{C}}$ is called *Anosov* if it is hyperbolic and integer-like. A rational Lie algebra ${\mathfrak{n}}^{\mathbb{Q}}$ with an Anosov automorphism is called *Anosov*.
The superscripts $^{\mathbb{Q}}$ and $^E$ indicate over which field we are working. Thus a classification of all nilmanifolds admitting an Anosov diffeomorphism is equivalent to a classification of all Anosov Lie algebras. A theorem by Jacobson (see [@jaco55-1]) shows that every Anosov Lie algebra is necessarily nilpotent.
Theorem \[main\] will give us a very general way of constructing Anosov Lie algebras. By using the low-dimensional classification of Anosov Lie algebras, we can describe the isomorphism class for some of those Lie algebras in Section \[verb\], but it will be hard to do the same in general. On the other hand, there will be some properties of the Lie algebra or the Anosov automorphisms which follow immediately from the construction, e.g. the signature of the Anosov automorphism and the type of the Lie algebra, properties which we introduce below.
The definition of the signature of an Anosov diffeomorphism $f: M \to M$ makes use of the $df$-invariant splitting $TM = E^s \oplus E^u$:
The signature $\operatorname{sgn}(f)$ of an Anosov diffeomorphism $f: M \to M$ is defined as the set $\{\dim_{\mathbb{R}}E^s, \dim_{\mathbb{R}}E^u\}$.
We define the signature as a set rather than as an ordered pair since the order of the elements does not play a role. Indeed, the inverse of an Anosov diffeomorphism is again Anosov with the bundles $E^s$ and $E^u$ interchanged. For Anosov automorphisms $f: {\mathfrak{n}}^{\mathbb{Q}}\to {\mathfrak{n}}^{\mathbb{Q}}$ on a rational Lie algebra ${\mathfrak{n}}^{\mathbb{Q}}$, the signature is defined as the set $\{p,q\}$ where $p$ is the number of eigenvalues with absolute value $<1$ and $q$ the number of eigenvalues of absolute value $> 1$. Note that the proof of [@deki99-1 Corollary 3.5.] implies that there exists an Anosov diffeomorphism $f$ with signature $\operatorname{sgn}(f)$ if and only if there exists an Anosov automorphisms with signature $\operatorname{sgn}(f)$ on the corresponding Lie algebra.
The following question is stated in [@lw09-1]:
\[q1\] Does there exist an Anosov automorphism on a non-abelian Lie algebra with signature $\{2,k\}$ for some $k \in {\mathbb{N}}$?
In Section \[sign\] it is shown that $\min(\operatorname{sgn}(f)) \geq c$ where $c$ is the nilpotency class of the Lie algebra. So a more general question is the existence of Anosov automorphisms which attain this lower bound for the signature. The main theorem gives us a positive answer in Section \[sign\].
As a consequence of a low-dimensional classification of Anosov automorphisms, the following is stated as a corollary in [@lw09-1]:
\[corlaur\] Let $\Gamma \backslash N$ be a nilmanifold of dimension $\leq 8$ which admits an Anosov diffeomorphism. Then $\Gamma \backslash N$ is a torus or the dimension is $6$ or $8$ and the signature is $\{3,3\}$ or $\{4,4\}$ respectively.
Examples illustrating that some cases were overlooked in this corollary are given in Section \[verb\] with signature $\{2,4\}$ and $\{3,5\}$. A complete list of Anosov Lie algebras admitting such signatures is given and thus this section forms a correction to the result of [@lw09-1].
If ${\mathfrak{n}}^E$ is a Lie algebra over a field $E$, then its lower central series $\gamma_i({\mathfrak{n}}^E)$ is recursively defined by $\gamma_1({\mathfrak{n}}^E) = {\mathfrak{n}}^E$ and $\gamma_i({\mathfrak{n}}^E) = [{\mathfrak{n}}^E, \gamma_{i-1}({\mathfrak{n}}^E) ]$ for all $i \geq 2$. The Lie algebra ${\mathfrak{n}}^E$ is said to be $c$-step nilpotent (or to have nilpotency class $c$) if $\gamma_{c+1}({\mathfrak{n}}^E) = 0$ and $\gamma_{c}({\mathfrak{n}}^E) \neq 0$. The type of a Lie algebra gives us information about the quotients of the lower central series:
The type of a nilpotent Lie algebra ${\mathfrak{n}}^E$ of nilpotency class $c$ is defined as the $c$-tuple $(n_1,\ldots,n_c)$, where $n_i = \dim_E \faktor{\gamma_{i}({\mathfrak{n}}^E)}{\gamma_{i+1}({\mathfrak{n}}^E)}$.
If ${\mathfrak{n}}^{\mathbb{Q}}$ is an Anosov Lie algebra of type $(3, n_2, \ldots,n_c)$, then [@payn09-1 Theorem 1.3.] states that $3 \mid n_i$ for every $i \in \{2, \ldots, c\}$. Combining this result with [@lw08-1 Proposition 2.3.] we get that every Anosov Lie algebra ${\mathfrak{n}}^{\mathbb{Q}}$ of type $(n_1,\ldots,n_c)$ satisfies one of the following:
(i) ${\mathfrak{n}}^{\mathbb{Q}}$ is abelian or
(ii) $n_1 \geq 4$ and $n_i \geq 2$ for all $i \in \{2, \ldots, c\}$ or
(iii) $n_1 = n_2 = 3$ and $3 \mid n_i $ for all $i \in \{3, \ldots, c\}$.
In [@laur03-1] it was proved that the lower bound of (ii) occurs for every $c$ and the main theorem shows that also the lower bound of (iii) is attained for every $c$, answering the following question of [@lw08-1]:
\[q2\] Does there exist a $c$-step Lie algebra of type $(3, \ldots, 3)$ of nilpotency class $c \geq 3$ which is Anosov?
An Anosov Lie algebra which attains one of the lower bounds (ii) or (iii) for the type will be called of minimal type. In fact, in Section \[sign\] we will show that Question \[q2\] has a positive answer even if we replace $3$ by any integer $n > 2$.
To end this first section, we give a short summary of the known results about nilmanifolds supporting an Anosov diffeomorphism. In a few cases, a complete classification is given, for example in low-dimensional cases (see [@lw08-1]) and in the case of free nilpotent Lie algebras (see [@dani03-1; @dd13-1]). In [@dm05-1; @dd03-2; @frie81-1; @payn09-1], some new examples of Anosov diffeomorphisms on nilmanifolds were constructed by taking quotients of free nilpotent Lie algebras. This method is based on the work of L. Auslander and J. Scheuneman in [@as70-1]. All other examples are explicitly constructed and the main result of this article is to generalize this construction in Theorem \[main\]. This theorem avoids all the computations and allows us to construct more complicated examples of Anosov automorphisms. It seems plausible that all questions about the existence of Anosov automorphisms of specific signature on a Lie algebra of specific type can be answered in this way. The disadvantage of this method is that in general it is hard to describe the rational Lie algebra in terms of a basis and relations on this basis.
Construction of Anosov Lie algebras {#con}
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The examples of Anosov Lie algebras in [@laur03-1; @lw09-1; @mw07-1] are constructed from a nilpotent Lie algebra over some field $E$ given by its decomposition into eigenspaces of an Anosov automorphism. An explicit basis is then constructed which is ‘symmetric’ under the action of the Galois group of $E$. Instead of working with a basis, we rather use this ‘symmetric’ property of the basis as the definition of a rational form of the Lie algebra. We will make this statement precise in Section \[secGC\]. For automorphisms, a similar ‘symmetric’ property can be defined such that the automorphism induces an automorphism on the rational form. In this way, no explicit calculations are needed and Theorem \[main\] allows us to easily show the existence of several new Anosov Lie algebras. This method further generalizes the work of [@payn09-1] which starts from polynomials instead of field extensions.
First let us fix some notations for this section. Let $E$ be a subfield of ${\mathbb{C}}$ such that $E$ has finite degree over ${\mathbb{Q}}$. If $n$ is the extension degree of $E \supseteq {\mathbb{Q}}$, then $E$ is called a Galois extension of ${\mathbb{Q}}$ if the group $$\operatorname{Gal}(E,{\mathbb{Q}}) = \{ \sigma: E \to E \mid \sigma \text{ is a field automorphism} \}$$ has order $n$. We will always assume that our field extensions are Galois.
A vector space $V$ over $E$ will be denoted by $V^E$ to indicate over which field we are working, as we already did above in Section \[intro\]. All the vector spaces and Lie algebras we consider are finite dimensional. If $F \supseteq E$ is a field extension, then we can consider the vector space $F \otimes_E V^E$ which we will denote as $V^F$. If $G$ is a finite group and $\rho: G \to \operatorname{GL}(V^E)$ is a representation, then there also exists a representation $\rho^F: G \to \operatorname{GL}(V^F)$ by extending the scalars, i.e. by considering $\operatorname{GL}(V^E)$ as a subgroup of $\operatorname{GL}(V^F)$. The same notations will be used for Lie algebras, e.g. a Lie algebra over the rationals ${\mathbb{Q}}$ will be denoted by ${\mathfrak{n}}^{\mathbb{Q}}$. We say that two representations $\rho_1: G \to \operatorname{GL}(V^E)$ and $\rho_2: G \to \operatorname{GL}(W^E)$ are $E$-equivalent if there exists an isomorphism $\varphi: V^E \to W^E$ such that $\rho_2(g) \circ \varphi = \varphi \circ \rho_1(g)$ for all $g \in G$.
A rational subspace $W^{\mathbb{Q}}\subseteq V^E$ is called a rational form if some (and hence every) basis of $W^{\mathbb{Q}}$ over ${\mathbb{Q}}$ is also a basis of $V^E$ over $E$. If ${\mathfrak{n}}^E$ is a Lie algebra over $E$, we call a rational subalgebra ${\mathfrak{m}}^{\mathbb{Q}}\subseteq {\mathfrak{n}}^E$ a rational form if it is a rational form seen as subspace of ${\mathfrak{n}}^E$ as vector space.
Construction of a rational form {#secGC}
-------------------------------
Instead of focusing on the basis of the rational form of a vector space, we focus on the defining property of the rational form as being ‘symmetric’ under the action of the Galois group. For this we first introduce the action of the Galois group on a vector space and define for each representation a rational subspace.
Consider the natural right action of $\operatorname{Gal}(E,{\mathbb{Q}})$ on the field $E$, given by $$\forall \sigma \in \operatorname{Gal}(E,{\mathbb{Q}}), \forall x \in E: x^\sigma = \sigma^{-1}(x).$$ By defining the action component-wise, there is also a natural right action on the vector space $E^m$. Note that the relations $$\label{relatie}
\begin{aligned}[c]
\left(\sigma(\lambda) x\right)^\sigma = \lambda x^\sigma
\end{aligned}
\text{ and }
\begin{aligned}[c]
\left(x + y\right)^\sigma = x^\sigma + y^\sigma
\end{aligned}$$ hold for all $x,y \in E^m$, $\sigma \in \operatorname{Gal}(E,{\mathbb{Q}})$ and $\lambda \in E$. Let $\rho: \operatorname{Gal}(E,{\mathbb{Q}}) \to \operatorname{GL}_m(E)$ be a representation, then it follows immediately that the subset defined as $$\begin{aligned}
V^{\mathbb{Q}}_\rho = \{ v \in E^m \mid \forall \sigma \in \operatorname{Gal}(E,{\mathbb{Q}}), \rho_\sigma(v) = v^\sigma \} \label{def}\end{aligned}$$ is a rational subspace of $E^m$. This subspace is already close to being a rational form of $E^m$ in the sense of the following lemma:
\[ratform\] If a set of vectors $v_1, \ldots, v_k$ of $V^{\mathbb{Q}}_\rho$ is linearly independent over ${\mathbb{Q}}$, then this set is also linearly independent over $E$ as vectors of $E^m$.
Assume that the lemma does not hold and take vectors $v_1, \ldots, v_k$ of $V^{\mathbb{Q}}_\rho$ with $k$ minimal which are linearly independent over ${\mathbb{Q}}$ but contradict the statement. This means that there exists $x_i \in E$ such that $$\begin{aligned}
\label{linind} \sum_{i=1}^k x_i v_i = 0.\end{aligned}$$ From the minimality of $k$ it follows that $x_1 \neq 0$ and thus by multiplying this equation by $x_1^{-1}$ we can assume that $x_1 = 1$. Since $k$ is minimal, it follows that the $x_i$ are the unique elements of $E$ such that $x_1 = 1$ and equation (\[linind\]) is true.
If we apply the map $\rho_\sigma$ to the equation, we get that $$0 = \rho_\sigma\left(\sum_{i=1}^k x_i v_i \right) = \sum_{i=1}^k x_i \rho_\sigma(v_i) = \sum_{i=1}^k x_i v_i^\sigma = \left(\sum_{i=1}^k \sigma(x_i) v_i\right)^\sigma$$ because of (\[relatie\]). We also have that $$\sum_{i=1}^k \sigma(x_i) v_i = 0.$$ Minimality of $k$ and the fact that $\sigma(x_1) = \sigma(1) = 1$ imply that $\sigma(x_i) = x_i$ for all $i$. Because this statement holds for all $\sigma \in \operatorname{Gal}(E,{\mathbb{Q}})$, we conclude that the coefficients $x_i$ lie in ${\mathbb{Q}}$ and thus we get a contradiction since $v_i$ was a set of linearly independent vectors over ${\mathbb{Q}}$.
The lemma shows that the rational subspace $V^{\mathbb{Q}}_\rho$ is a rational form of $E^m$ if its dimension is maximal. This motivates the following definition:
A representation $\rho: \operatorname{Gal}(E,{\mathbb{Q}}) \to \operatorname{GL}_m(E)$ is called *Galois compatible* (abbreviated as GC) if and only if $\dim_{\mathbb{Q}}(V^{\mathbb{Q}}_\rho) = m$. Equivalently, the representation $\rho
$ is GC if and only if $E \otimes V^{\mathbb{Q}}_\rho = E^m$ or if $V^{\mathbb{Q}}_\rho $ is a rational form of $E^m$.
The trivial representation is the easiest example of a GC representation, since $V^{\mathbb{Q}}_\rho = {\mathbb{Q}}^m$. A simple computation shows that also the regular representation is GC:
\[reg\] *Let $\rho$ be the regular representation of $\operatorname{Gal}(E,{\mathbb{Q}})$, i.e. take the vector space over $E$ spanned by the basis $\{v_\sigma \mid \sigma \in \operatorname{Gal}(E,{\mathbb{Q}})\}$ and the representation $\rho$ induced by the relations $\rho_\tau(v_\sigma) = v_{\tau\sigma}$ for all $\tau, \sigma \in \operatorname{Gal}(E,{\mathbb{Q}})$. Every element $v$ of the rational vector space $V^{\mathbb{Q}}_\rho$ is given by $$v = \sum_{\sigma \in \operatorname{Gal}(E,{\mathbb{Q}})} \sigma(x) v_\sigma$$ for some $x \in E$. This is a rational vector space of dimension $[E:{\mathbb{Q}}]$, which is also the order of the group $\operatorname{Gal}(E,{\mathbb{Q}})$ and thus the dimension of the regular representation $\rho$. This shows that $\rho$ is indeed GC.*
In a similar way it is easy to check that every representation which is given by permutation matrices is GC. The goal of the remaining part of this section is to show that every rational representation is GC:
\[GC\] Let $E$ be a Galois extension of ${\mathbb{Q}}$. Every representation $\operatorname{Gal}(E,{\mathbb{Q}}) \to \operatorname{GL}_m({\mathbb{Q}})$ is Galois compatible.
To prove this statement we will first show that it holds for irreducible representations and then apply the following lemma:
\[som\] Let $\rho_i: \operatorname{Gal}(E,{\mathbb{Q}}) \to \operatorname{GL}_{n_i}(E)$ with $i \in \{1,2\}$ be representations, then the following are equivalent:
1. $\rho_1$ and $\rho_2$ are Galois compatible.
2. $\rho_1 \oplus \rho_2: \operatorname{Gal}(E,{\mathbb{Q}}) \to \operatorname{GL}_{n_1+n_2}(E)$ is Galois compatible.
Let $V_{\rho_1}^{\mathbb{Q}}$ and $V_{\rho_2}^{\mathbb{Q}}$ be the rational subspaces corresponding to $\rho_1$ and $\rho_2$, then by defition it follows that $$V_{\rho_1 \oplus \rho_2}^{\mathbb{Q}}= V_{\rho_1}^{\mathbb{Q}}\oplus V_{\rho_2}^{\mathbb{Q}}.$$ The statement of the lemma then easily follows by using Lemma \[ratform\].
The proof of the proposition is now immediate by using the previous lemma:
If two representation $\rho_1, \rho_2: \operatorname{Gal}(E,{\mathbb{Q}}) \to \operatorname{GL}_m({\mathbb{Q}})$ are ${\mathbb{Q}}$-equivalent, then $\rho_1$ is GC if and only if $\rho_2$ is GC. So from Lemma \[som\] it follows that it is sufficient to prove the proposition for the ${\mathbb{Q}}$-irreducible representations of $\operatorname{Gal}(E,{\mathbb{Q}})$. Since every ${\mathbb{Q}}$-irreducible representation is a subrepresentation of the regular representation (see e.g. [@isaa76-1 Corollary 9.5.]), the statement follows from the example above.
Construction of automorphisms on rational forms of Lie algebras {#cons}
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In the previous subsection we constructed rational forms of vector spaces $E^m$ from representations of the Galois group $\operatorname{Gal}(E,{\mathbb{Q}})$. In this subsection we use this technique to construct Lie algebras and automorphisms on them, allowing us to construct Anosov Lie algebras as well.
Let ${\mathfrak{n}}^{\mathbb{Q}}$ be a rational Lie algebra and $E$ a Galois extension of ${\mathbb{Q}}$. By taking a basis for ${\mathfrak{n}}^{\mathbb{Q}}\subseteq {\mathfrak{n}}^E$, the vector space ${\mathfrak{n}}^E$ is isomorphic to $E^m$ with $m = \dim_{\mathbb{Q}}({\mathfrak{n}}^{\mathbb{Q}})$ and thus the right action of $\operatorname{Gal}(E,{\mathbb{Q}})$ is defined on ${\mathfrak{n}}^E$. It is an exercise to check that this action does not depend on the choice of basis for ${\mathfrak{n}}^{\mathbb{Q}}$. The right action of $\operatorname{Gal}(E,{\mathbb{Q}})$ satisfies the property $$[X,Y]^\sigma = [ X^\sigma,Y^\sigma]$$ for all $\sigma \in \operatorname{Gal}(E,{\mathbb{Q}})$, $X,Y \in {\mathfrak{n}}^E$. For every representation $$\rho: \operatorname{Gal}(E,{\mathbb{Q}})\to \operatorname{Aut}({\mathfrak{n}}^{\mathbb{Q}}) \le \operatorname{GL}({\mathfrak{n}}^{\mathbb{Q}}),$$ with $\operatorname{GL}({\mathfrak{n}}^{\mathbb{Q}})$ the isomorphisms of ${\mathfrak{n}}^{\mathbb{Q}}$ as vector space, it follows from the definition that the rational subspace $V^{\mathbb{Q}}_\rho$ forms a subalgebra of ${\mathfrak{n}}^E$ and is thus a rational form of ${\mathfrak{n}}^E$ by Proposition \[GC\]. We will denote this subspace by ${\mathfrak{m}}^{\mathbb{Q}}_\rho$ to emphasize the fact that it is a subalgebra.
We now answer the question when a given automorphism $f: {\mathfrak{n}}^E \to {\mathfrak{n}}^E$ induces an automorphism on ${\mathfrak{m}}^{\mathbb{Q}}_\rho$. Note that $\operatorname{Gal}(E,{\mathbb{Q}})$ also acts on the right on $\operatorname{Aut}({\mathfrak{n}}^E)$, by defining for all $f \in \operatorname{Aut}({\mathfrak{n}}^E)$ the action as $$f^\sigma(v) = \left( f(v^{\sigma^{-1}}) \right)^\sigma.$$ By fixing a basis for ${\mathfrak{n}}^{\mathbb{Q}}$ and thus also for ${\mathfrak{n}}^E$, the matrix representation of $f^\sigma$ is given by applying $\sigma^{-1}$ to every entry of the matrix representation of $f$. Also every representation $\rho: \operatorname{Gal}(E,{\mathbb{Q}}) \to \operatorname{Aut}({\mathfrak{n}}^E)$ induces a left action on $\operatorname{Aut}({\mathfrak{n}}^E)$ by conjugation. The automorphisms where this left action corresponds with the right action are exactly those that induce an automorphism on ${\mathfrak{m}}^{\mathbb{Q}}_\rho$:
\[auto\] Let ${\mathfrak{n}}^{\mathbb{Q}}$ be a rational Lie algebra and $\rho: G \to \operatorname{Aut}({\mathfrak{n}}^{\mathbb{Q}})$ a representation. An element $f \in \operatorname{Aut}({\mathfrak{n}}^E)$ induces an automorphism on ${\mathfrak{m}}^{\mathbb{Q}}_\rho$ if and only if $f^\sigma = \rho_\sigma f \rho_{\sigma^{-1}}$ for all $\sigma \in \operatorname{Gal}(E,{\mathbb{Q}})$.
First assume that $f$ satisfies the condition, i.e. that $f^\sigma = \rho_\sigma f \rho_{\sigma^{-1}}$ for $\sigma \in \operatorname{Gal}(E,{\mathbb{Q}})$. For every $v \in {\mathfrak{m}}^{\mathbb{Q}}_\rho$ and $\sigma \in \operatorname{Gal}(E,{\mathbb{Q}})$, we have that $$\begin{aligned}
\big(f(v)\big)^\sigma = f^\sigma (v^\sigma) = \rho_\sigma f \rho_{\sigma^{-1}} (\rho_\sigma(v)) = \rho_\sigma(f(v))\end{aligned}$$ and thus $f(v) \in {\mathfrak{m}}^{\mathbb{Q}}_\rho$ because $\sigma$ was taken arbitrary. Since ${\mathfrak{m}}^{\mathbb{Q}}_\rho$ is a rational form, the restriction of $f$ to ${\mathfrak{m}}^{\mathbb{Q}}_\rho$ is invertible and $f$ induces an automorphism of ${\mathfrak{m}}^{\mathbb{Q}}_\rho$.
For the other direction, fix $\sigma \in \operatorname{Gal}(E,{\mathbb{Q}})$ and fix a basis $\{v_1, \ldots v_m\} \subseteq {\mathfrak{m}}^{\mathbb{Q}}_\rho$ for the Lie algebra ${\mathfrak{n}}^E$. Since $f$ induces an automorphism on ${\mathfrak{m}}^{\mathbb{Q}}_\rho$ we know that $f(v_i) \in {\mathfrak{m}}^{\mathbb{Q}}_\rho$ and thus that $\big(f(v_i)\big)^\sigma = \rho_\sigma(f(v_i))$. Since the vectors $\rho_{\sigma}(v_i)$ also form a basis for the Lie algebra ${\mathfrak{n}}^E$, it suffices to prove the relation $f^\sigma = \rho_\sigma f \rho_{\sigma^{-1}}$ on this basis. A computation shows that $$\begin{aligned}
\rho_\sigma f \rho_{\sigma^{-1}} (\rho_\sigma(v_i)) = \rho_\sigma (f(v_i)) = \big(f(v_i)\big)^\sigma = f^\sigma(v_i^\sigma) = f^\sigma(\rho_\sigma(v_i)) \end{aligned}$$ and thus the relation holds.
The main theorem of this article is the combination of Proposition \[GC\] and Lemma \[auto\]:
\[main\] Let ${\mathfrak{n}}^{\mathbb{Q}}$ be a rational Lie algebra and $\rho: \operatorname{Gal}(E,{\mathbb{Q}}) \to \operatorname{Aut}({\mathfrak{n}}^{\mathbb{Q}})$ a representation. Suppose there exists a Lie algebra automorphism $f: {\mathfrak{n}}^E \to {\mathfrak{n}}^E$ such that $\rho_\sigma f \rho_{\sigma^{-1}} = f^\sigma$ for all $\sigma \in \operatorname{Gal}(E,{\mathbb{Q}})$. Then there also exists a rational form ${\mathfrak{m}}^{\mathbb{Q}}\subseteq {\mathfrak{n}}^E$ such that $f$ induces an automorphism of ${\mathfrak{m}}^{\mathbb{Q}}$.
If all eigenvalues of $f$ are algebraic units of absolute value different from $1$, then ${\mathfrak{m}}^{\mathbb{Q}}$ is Anosov.
The only statement left to show is the last one. We claim that if $f$ is an automorphism of a rational Lie algebra with only algebraic units as eigenvalues, then $f$ is integer-like. Note that the coefficients of the characteristic polynomial of $f$ are formed by taking sums and products of the eigenvalues and thus all these coefficients are algebraic integers. Since these coefficients also lie in ${\mathbb{Q}}$, they are integers. The determinant of $f$ is equal to the product of all eigenvalues and therefore is an algebraic unit. Since the only algebraic units in ${\mathbb{Q}}$ are $\pm 1$, the claim now follows and this ends the proof.
Note that the type of the Lie algebra ${\mathfrak{m}}^{\mathbb{Q}}$ of the theorem is equal to the type of ${\mathfrak{n}}^{\mathbb{Q}}$ and is thus completely determined. The signature of the automorphism $f$ also does not change by restricting it to a rational form. But in general the Lie algebras ${\mathfrak{n}}^{\mathbb{Q}}$ and ${\mathfrak{m}}^{\mathbb{Q}}$ will not be isomorphic, so this theorem does not allow us to show that a specific Lie algebra is Anosov. In Section \[verb\] we determine the isomorphism class of the Lie algebra in a low-dimensional case by using the classification of low-dimensional Anosov algebras given in [@lw09-1].
In the special case where the Lie algebra ${\mathfrak{n}}^{\mathbb{Q}}$ is given by the eigenspaces of an automorphism, the theorem becomes:
\[main2\] Let ${\mathfrak{n}}^{\mathbb{Q}}$ be a rational Lie algebra and assume there exists a decomposition of ${\mathfrak{n}}^{\mathbb{Q}}$ into subspaces $${\mathfrak{n}}^{\mathbb{Q}}= \bigoplus_{\lambda \in E} V_\lambda$$ such that $[V_\lambda,V_\mu] \subseteq V_{\lambda \mu}$. Let $\rho:\operatorname{Gal}(E,{\mathbb{Q}}) \to \operatorname{Aut}({\mathfrak{n}}^{\mathbb{Q}})$ be a representation such that $\rho_\sigma(V_\lambda)= V_{\sigma(\lambda)}$ for all $\sigma \in \operatorname{Gal}(E,{\mathbb{Q}})$. Then the linear map $f: {\mathfrak{n}}^E \to {\mathfrak{n}}^E$ given by $f(X) = \lambda X$ for all $X \in V_\lambda$ induces an automorphism on some rational form ${\mathfrak{m}}^{\mathbb{Q}}\subseteq {\mathfrak{n}}^E$.
If every $\lambda$ is an algebraic unit of absolute value different from $1$, then ${\mathfrak{m}}^{\mathbb{Q}}$ is Anosov.
We will make use of Theorem \[main\] to prove this. The condition on the Lie bracket implies that $f$ is a Lie algebra automorphism and the last condition is identical to the last condition of Theorem \[main\]. So it is left to show that $$\rho_\sigma f \rho_{\sigma^{-1}} = f^\sigma$$ for all $\sigma \in \operatorname{Gal}(E,{\mathbb{Q}})$. It suffices to prove this relation for vectors $X \in V_\lambda$ for all possible $\lambda$. For such a vector, we have that $$\begin{aligned}
\rho_\sigma f \rho_{\sigma^{-1}} \left( X \right) = \rho_\sigma \Big(f \big( \rho_{\sigma^{-1}}\left( X \right) \big) \Big)= \sigma^{-1}(\lambda) \rho_\sigma \big( \rho_{\sigma^{-1}} \left( X \right) \big) = \sigma^{-1}\left( \lambda \right) X\end{aligned}$$ since $\rho_{\sigma^{-1}}(X) \in V_{\sigma^{-1}(\lambda)}$ and $$\begin{aligned}
f^\sigma \left(X \right) = \left( f(X) \right)^\sigma = \left( \lambda X \right)^\sigma = \sigma^{-1}(\lambda) X\end{aligned}$$ since $X^{\sigma^{-1}} = X$ and thus equality holds.
\[oned\] Corollary \[main2\] is the version of the main theorem we will use most of the times. In many examples, the spaces $V_\lambda$ will be one-dimensional and thus given by basis vectors $X_\lambda$ for $\lambda \in E$. In these examples, the Lie bracket is of the form $$[X_\lambda,X_\mu] = \pm X_{\lambda\mu}$$ and the representation $\rho$ is given by $\rho_\sigma(X_\lambda) = \pm X_{\sigma(\lambda)}$ for all $\lambda \in E, \sigma \in \operatorname{Gal}(E,{\mathbb{Q}})$. To use the corollary, the only thing left to check is that $\rho$ is indeed a representation, since all other conditions are straightforward.
Consequences of Theorem \[main\] {#conseq}
================================
The main application of the theorem and corollary above lies in constructing Anosov Lie algebras of specific types and with Anosov automorphisms of specific signatures. We present here three different consequences of the main theorem, going from simplifying and correcting existing results to constructing new examples of minimal signature and minimal type.
Simplification of existing results
----------------------------------
By using our main theorem, all the examples of Anosov Lie algebras in [@laur03-1; @lw08-1; @lw09-1; @mw07-1] are now straightforward to construct. For instance, the examples in [@lw09-1; @mw07-1] follow from Remark \[oned\] since they start from a basis for the one-dimensional eigenspaces of a hyperbolic automorphism. As another example of the simplification we demonstrate how [@lw08-1 Theorem 3.2.] follows after a few lines.
Recall that a Lie algebra ${\mathfrak{n}}^E$ over a field $E$ is called graded if there exist subspaces ${\mathfrak{n}}_i^E \subseteq {\mathfrak{n}}^E$ for $i \in \{1, \ldots, k\}$ such that $$\begin{aligned}
&{\mathfrak{n}}^E = {\mathfrak{n}}_1^E \oplus {\mathfrak{n}}_2^E \oplus \ldots \oplus {\mathfrak{n}}_k^E \hspace{1mm} \text{ and} \\
&[{\mathfrak{n}}_i^E, {\mathfrak{n}}_j^E] \subseteq {\mathfrak{n}}_{i+j}^E.\end{aligned}$$ If ${\mathfrak{n}}^E$ is graded, then there exists an automorphism $f_\lambda: {\mathfrak{n}}^E \to {\mathfrak{n}}^E$ for every $\lambda \in E, \lambda \neq 0$, which is defined by $$f_\lambda (X) = \lambda^i X \hspace{2mm} \forall X \in {\mathfrak{n}}_i.$$ The proof of [@lw08-1 Theorem 3.2.] now follows immediately from the main theorem:
\[laur\] Let ${\mathfrak{n}}^{\mathbb{Q}}$ be a graded Lie algebra and consider the direct sum $$\tilde{{\mathfrak{n}}}^{\mathbb{Q}}= \underbrace{{\mathfrak{n}}^{\mathbb{Q}}\oplus \ldots \oplus {\mathfrak{n}}^{\mathbb{Q}}}_{m\ {\rm times}}$$ with $m \geq 2$. Then there exists a rational form of $\tilde{{\mathfrak{n}}}^{\mathbb{R}}$ which is Anosov.
Let $E$ be any real Galois extension of ${\mathbb{Q}}$ of degree $m$ with $\operatorname{Gal}(E,{\mathbb{Q}}) = \{ \sigma_1, \ldots, \sigma_m\}$. Every $\sigma \in \operatorname{Gal}(E,{\mathbb{Q}})$ induces a permutation $\pi \in S_m$ via $$\sigma \sigma_i = \sigma_{\pi(i)} \hspace{1mm} \text{ for all } \hspace{1mm} i \in \{1, \ldots, m\},$$ and thus also an automorphism $\rho_\sigma \in \operatorname{Aut}(\tilde{{\mathfrak{n}}}^{\mathbb{Q}})$ by permuting the components of $\tilde{{\mathfrak{n}}}^{\mathbb{Q}}$ according to $\pi$. Note that $\rho$ is a representation $\operatorname{Gal}(E,{\mathbb{Q}}) \to \operatorname{Aut}(\tilde{{\mathfrak{n}}}^{\mathbb{Q}})$.
Take an unit Pisot number $\lambda$ in $E$, which is possible since $m \geq 2$ (see Appendix for more details) and take the grading $ {\mathfrak{n}}^{\mathbb{Q}}= {\mathfrak{n}}_1^{\mathbb{Q}}\oplus \dots \oplus {\mathfrak{n}}_k^{\mathbb{Q}}$ for ${\mathfrak{n}}^{\mathbb{Q}}$. By writing the subspace ${\mathfrak{n}}^{\mathbb{Q}}_i$ of the $j$-th component of $\tilde{{\mathfrak{n}}}^{\mathbb{Q}}$ as $V_{\sigma_i(\lambda^j)}$, it is immediate from the construction of $\rho$ that all conditions of Corollary \[main2\] are satisfied. Since every $\sigma_i(\lambda^j)$ is an algebraic unit of absolute value different from $1$, it follows that $\tilde{{\mathfrak{n}}}^E$ (and therefore also $\tilde{{\mathfrak{n}}}^{\mathbb{R}}$) has a rational form which is Anosov.
Correction to a result of [@lw09-1] {#verb}
-----------------------------------
In [@lw09-1], a classification of all Anosov Lie algebras up to dimension $8$ is given. As one of the consequences, as we recalled in (False) Claim \[corlaur\], it is stated in [@lw09-1 Corollary 4.3.] that every Anosov diffeomorphism on a nilmanifold of dimension $\leq 8$ which is not a torus, has signature $\{3,3\}$ or $\{4,4\}$. There is not really a proof of this statement given in [@lw09-1] and in fact, by using Theorem \[main\] we can give new examples which were overlooked by the author.
First we recall some notions of [@lw09-1] about the Pfaffian form of a Lie algebra. Let ${\mathfrak{n}}^E$ be a Lie algebra over any field $E$ and take $\langle \hspace{1mm},\hspace{1mm} \rangle$ an inner product on ${\mathfrak{n}}^E$, i.e. a non-degenerate symmetric bilinear form. For every $Z \in {\mathfrak{n}}^E$, there exists a linear map $J_Z: {\mathfrak{n}}^E \to {\mathfrak{n}}^E$ defined by $$\langle J_Z X, Y \rangle = \langle [X,Y], Z \rangle \hspace{3mm} \forall X,Y \in {\mathfrak{n}}^E.$$ Note that $J_Z$ is skew-symmetric with respect to the inner product, meaning that $$\langle J_Z X, Y \rangle = - \langle X, J_Z Y \rangle$$ for all $X,Y \in {\mathfrak{n}}^E$. It is easy to check that an isomorphism $\alpha: {\mathfrak{n}}^E \to {\mathfrak{n}}^E$ is an automorphism of ${\mathfrak{n}}^E$ if and only if $$\begin{aligned}
\label{aut} \alpha^T J_Z \alpha = J_{\alpha^T (Z)}\end{aligned}$$ for all $Z \in {\mathfrak{n}}^E$, where $\alpha^T: {\mathfrak{n}}^E \to {\mathfrak{n}}^E$ is the adjoint map of $\alpha$, defined by $\langle X, \alpha^T( Y) \rangle = \langle \alpha (X), Y \rangle$.
Now assume that ${\mathfrak{n}}^E$ is a $2$-step nilpotent Lie algebra of type $(2 m,k)$ and take $V^E \subseteq {\mathfrak{n}}^E$ such that $V^E \oplus \gamma_2({\mathfrak{n}}^E) = {\mathfrak{n}}^E$ as a vector space. Take an inner product satisfying $\langle V^E , \gamma_2({\mathfrak{n}}^E)\rangle = 0$ and such that there exists an orthonormal basis for $V^E$ with respect to the inner product. Denote the vector space (in fact it is a Lie algebra) of all skew-symmetric endomorphisms of $V^E$ by ${\mathfrak{so}}(V^E)$. The construction above then induces a linear map $$\begin{aligned}
J : \gamma_2({\mathfrak{n}}^E) &\to {\mathfrak{so}}(V^E)\\ Z &\mapsto J_Z |_{V^E}.\end{aligned}$$ After taking an orthonormal basis for $V^E$, we can identify ${\mathfrak{so}}(V^E)$ with the skew-symmetric matrices. Recall that the Pfaffian on $V^E$ is the unique polynomial function $$\operatorname{Pf}: {\mathfrak{so}}(V^E) \to E$$ such that $\operatorname{Pf}(A)^2 = \det(A)$ for all $A \in {\mathfrak{so}}(V^E)$ and $\operatorname{Pf}(S) = 1$ for some fixed $S \in {\mathfrak{so}}(V^E)$ with $\det(S) = 1$ (where this last condition is needed to fix the sign). The Pfaffian obviously satisfies the relation $\operatorname{Pf}(A^T B A) = \det(A) \operatorname{Pf}(B)$ for all endomorphisms $A: V^E \to V^E$. The composition $$h = \operatorname{Pf}\circ J: \gamma_2({\mathfrak{n}}^E) \to E$$ is called the Pfaffian form of the 2-step nilpotent Lie algebra ${\mathfrak{n}}^E$. By taking another vector space $V^E$ or changing the inner product on ${\mathfrak{n}}^E$ or the basis of $V^E$, the Pfaffian form changes to a polynomial $$\begin{aligned}
\gamma_2({\mathfrak{n}}^E) &\to E \\ Z &\mapsto e h(\beta(Z))\end{aligned}$$ for some $e \in E \setminus \{ 0 \}$ and some isomorphism $\beta: \gamma_2({\mathfrak{n}}^E) \to \gamma_2({\mathfrak{n}}^E)$. Thus the Pfaffian form is uniquely determined by ${\mathfrak{n}}^E$ up to projective equivalence (see [@laur08-1 Proposition 2.4.] for the exact definition and proof of this statement). The polynomial $h$ is homogeneous of degree $m$ in $k$ variables and with coefficients in $E$. An automorphism of the Pfaffian form is an isomorphism $\beta: \gamma_2({\mathfrak{n}}^E) \to \gamma_2({\mathfrak{n}}^E)$ such that $h\circ \beta =h$.
A binary quadratic form over ${\mathbb{Q}}$ is a homogeneous polynomial of degree 2 in 2 variables, i.e. polynomials that can be written as $$h(X,Y) = a X^2 + b XY + c Y^2$$ with $a,b,c \in {\mathbb{Q}}$. The Pfaffian form of a rational Lie algebra of type $(4,2)$ is such a polynomial. The discriminant $\Delta(h)$ of $h$ is defined as $$\Delta(h) = b^2 - 4 ac.$$ In [@gss82-1] it is shown that two rational Lie algebras of type $(4,2)$ with Pfaffian forms $h_1$ and $h_2$ are isomorphic if and only if there exists $q \in {\mathbb{Q}}\setminus \{ 0 \}$ such that $\Delta(h_1) = q^2 \Delta(h_2)$. Given any $k \in {\mathbb{Z}}$, we can consider the Lie algebra ${\mathfrak{n}}^{\mathbb{Q}}_k$ given by the basis $X_1, X_2, X_3, X_4, Z_1, Z_2$ and relations $$\begin{aligned}[c]
[X_1,X_3] &= Z_1 \\
[X_2,X_3] &= k Z_2
\end{aligned}
\hspace{2cm}
\begin{aligned}[c]
[X_1,X_4] &= Z_2 \\
[X_2,X_4] &= Z_1.
\end{aligned}$$ The Lie algebra ${\mathfrak{n}}^{\mathbb{Q}}_k$ has Pfaffian form equal to $h(X,Y) = X^2 - k Y^2$ with discriminant $4k$. So every rational Lie algebra of type $(4,2)$ is isomorphic to ${\mathfrak{n}}^{\mathbb{Q}}_k$ for some $k \in {\mathbb{Z}}$ and ${\mathfrak{n}}^{\mathbb{Q}}_k$ is isomorphic to ${\mathfrak{n}}^{\mathbb{Q}}_{k^\prime}$ if and only if there exists a natural number $q > 0$ such that $k = q^2 k^\prime$.
The automorphisms $\beta \in \operatorname{SL}(2,{\mathbb{Z}})$ of a binary quadratic form $h(X,Y) = a X^2 + b XY + c Y^2$ are completely known and described e.g. in [@bv07-1 Theorem 2.5.5.]. Given a solution $x,y \in {\mathbb{Z}}$ of the Pell equation $$x^2 - \Delta(h) y^2 = 4,$$ the matrix $$U(x,y) = \begin{pmatrix} \frac{ x - yb}{2} & -cy \\ ay & \frac{x + yb}{2} \end{pmatrix} \in \operatorname{SL}(2,{\mathbb{Z}})$$ is an automorphism of the quadratic form $h$. The map $U$ is a bijection between the solutions of the Pell equation and the automorphisms of $h$ which lie in $\operatorname{SL}(2,{\mathbb{Z}})$. The eigenvalues of $U(x,y)$ are equal to $\frac{x \pm \sqrt{\Delta(h)} y}{2}$, so the field in which these eigenvalues lie gives us information about the discriminant of the form $h$.
Let ${\mathfrak{n}}^{\mathbb{Q}}$ be a rational Lie algebra of type $(4,2)$ and $\alpha \in \operatorname{Aut}({\mathfrak{n}}^{\mathbb{Q}})$ an Anosov automorphism. By squaring $\alpha$ if necessary, we can also assume that $\det(\alpha) = 1$. From the equation $\alpha^T J_Z \alpha = J_{\alpha^T (Z)}$ (see (\[aut\])) and by applying the Pfaffian, we get that $h(\alpha^T (Z)) = h(Z)$ for all $Z \in V^{\mathbb{Q}}$. So $\alpha$ induces a hyperbolic and integer-like automorphism $\beta = \alpha^T |_{\gamma_2({\mathfrak{n}}^{\mathbb{Q}})}$ of the Pfaffian form $h$. The eigenvalues of $\beta$ (and thus also of $\alpha |_{\gamma_2({\mathfrak{n}}^{\mathbb{Q}})}$) lie in the field ${\mathbb{Q}}(\sqrt{\Delta(h)})$ and thus if we know these eigenvalues, we can determine the discriminant of the Pfaffian form of ${\mathfrak{n}}^{\mathbb{Q}}$ up to a square and therefore also the isomorphism class of ${\mathfrak{n}}^{\mathbb{Q}}$. This also implies that every Anosov Lie algebra of type $(4,2)$ is isomorphic to a Lie algebra ${\mathfrak{n}}^{\mathbb{Q}}_k$ with $k$ a square free natural number $>1$ (since all other values of $k$ imply that the eigenvalues have absolute value $1$).
We now have all the tools to construct new Anosov automorphisms on the Anosov Lie algebras ${\mathfrak{n}}^{\mathbb{Q}}_k$:
\[count\] Let $k$ be a natural number with $k > 1$ and $k$ square free. Let ${\mathfrak{n}}^{\mathbb{Q}}_k$ be the Lie algebra with basis $X_1, X_2, X_3, X_4, Z_1, Z_2$ and relations $$\begin{aligned}[c]
[X_1,X_3] &= Z_1 \\
[X_2,X_3] &= k Z_2
\end{aligned}
\hspace{2cm}
\begin{aligned}[c]
[X_1,X_4] &= Z_2 \\
[X_2,X_4] &= Z_1.
\end{aligned}$$ Then there exists an Anosov automorphism $f$ on ${\mathfrak{n}}^{\mathbb{Q}}_k$ with $\operatorname{sgn}(f) = \{2,4\}$.
Fix the $k$ of the theorem and let $l$ be a different natural number with $l$ square free and $l>1$. Take $E = {\mathbb{Q}}(\sqrt{k},\sqrt{l})$, then $E$ is Galois over ${\mathbb{Q}}$ with $\operatorname{Gal}(E,{\mathbb{Q}}) = {\mathbb{Z}}_2 \oplus {\mathbb{Z}}_2$. Let $\tau \in \operatorname{Gal}(E,{\mathbb{Q}})$ be the unique element with $\tau(\sqrt{k}) = \sqrt{k}, \tau(\sqrt{l}) = -\sqrt{l}$ and take another $\sigma \in \operatorname{Gal}(E,{\mathbb{Q}})$ such that $\operatorname{Gal}(E,{\mathbb{Q}}) = \{ 1, \sigma, \tau, \sigma \tau\}$. Take $\lambda_1 \in E$ an unit Pisot number as introduced in the Appendix. Write the Galois conjugates of $\lambda_1$ as $\lambda_1, \tau(\lambda_1) = \lambda_2, \sigma(\lambda_1) = \lambda_3, \sigma \tau(\lambda_1) = \lambda_4$
Consider the Lie algebra ${\mathfrak{n}}^{\mathbb{Q}}$ with basis $X_{\lambda_1},X_{\lambda_2},X_{\lambda_3}, X_{\lambda_4}, Y_{\lambda_1 \lambda_2}, Y_{\lambda_3 \lambda_4}$ and Lie bracket given by $$\begin{aligned}
[X_{\lambda_1},X_{\lambda_2}] &= Y_{\lambda_1 \lambda_2} \\
[X_{\lambda_3}, X_{\lambda_4}] &= Y_{\lambda_3 \lambda_4}\end{aligned}$$ and all other brackets zero (so ${\mathfrak{n}}^{\mathbb{Q}}$ is isomorphic to the direct sum of two copies of the Heisenberg algebra of dimension $3$). Each of these basis vectors spans a $1$-dimensional subspace indexed by the same algebraic unit, corresponding to the decomposition in Corollary \[main2\]. Consider the representation $\rho: \operatorname{Gal}(E,{\mathbb{Q}}) \to \operatorname{Aut}({\mathfrak{n}}^{\mathbb{Q}})$ induced by $\rho_\sigma(X_\lambda) = X_{\sigma(\lambda)}$ and $\rho_\tau(X_\lambda) = X_{\tau(\lambda)}$. A small computation shows that this is indeed a representation. By using Corollary \[main2\] we then get a rational form ${\mathfrak{m}}^{\mathbb{Q}}$ of ${\mathfrak{n}}^E$ with Anosov automorphism $f: {\mathfrak{m}}^{\mathbb{Q}}\to {\mathfrak{m}}^{\mathbb{Q}}$.
Note that $f$ has only two eigenvalues $>1$, namely $\lambda_1$ and $\lambda_1\lambda_2$ and thus $\operatorname{sgn}(f) = \{2,4\}$. Since $\tau(\lambda_1 \lambda_2) = \lambda_1 \lambda
_2$ and ${\mathbb{Q}}(\sqrt{k})$ is the unique subfield of $E$ fixed by $\tau$, the eigenvalue $\lambda_1 \lambda_2$ is in ${\mathbb{Q}}(\sqrt{k})$, showing that the Pfaffian form $h$ of ${\mathfrak{m}}^{\mathbb{Q}}$ satisfies $\Delta(h) = q^2 k$ for some $q \in {\mathbb{Q}}$. This shows that ${\mathfrak{m}}^{\mathbb{Q}}$ is isomorphic to ${\mathfrak{n}}^{\mathbb{Q}}_k$ and thus ${\mathfrak{n}}^{\mathbb{Q}}_k$ also has an Anosov automorphism of signature $\{2,4\}$.
It is also possible to start the proof from a Galois extension ${\mathbb{Q}}\subseteq E$ with $\operatorname{Gal}(E,{\mathbb{Q}}) = {\mathbb{Z}}_4$ as we show in the example below. By computing a basis for ${\mathfrak{m}}^{\mathbb{Q}}$ we give an explicit example which was overlooked in (False) Claim \[corlaur\]:
Start from the polynomial $$p(X) = X^4 -4X^3 - 4X^2+X+1,$$ which has $4$ distinct real roots, say $\lambda_1 > \lambda_2 > \lambda_3 > \lambda_4$. These roots satisfy $\lambda_1 > 1$ and $\vert \lambda_i \vert < 1$ for $i \in \{ 2, 3, 4 \}$, showing that $\lambda_1$ is an unit Pisot number. The Galois group of the field $E = {\mathbb{Q}}(\lambda_1)$ is isomorphic to ${\mathbb{Z}}_4$, which can be checked e.g. with GAP, see [@gap14-1]. A generator $\sigma \in \operatorname{Gal}(E,{\mathbb{Q}})$ is given by $$\sigma(\lambda_1) = \lambda_2, \hspace{1mm} \sigma(\lambda_2) = \lambda_3, \hspace{1mm} \sigma(\lambda_3) = \lambda_4, \hspace{1mm} \sigma(\lambda_4) = \lambda_1.$$ Just as in the proof of Proposition \[count\], consider the Lie algebra ${\mathfrak{n}}^{\mathbb{Q}}$ with basis $$X_{\lambda_1},X_{\lambda_2},X_{\lambda_3}, X_{\lambda_4}, Y_{\lambda_1 \lambda_3}, Y_{\lambda_2 \lambda_4}$$ and Lie bracket given by $$\begin{aligned}
[X_{\lambda_1},X_{\lambda_3}] &= Y_{\lambda_1 \lambda_3} \\
[X_{\lambda_2}, X_{\lambda_4}] &= Y_{\lambda_2 \lambda_4}\end{aligned}$$ Consider the representation $\rho: \operatorname{Gal}(E,{\mathbb{Q}}) \to \operatorname{Aut}({\mathfrak{n}}^{\mathbb{Q}})$ induced by $\rho_\sigma(X_{\lambda_i}) = X_{\sigma(\lambda_i)}$. The main theorem guarantees us the existence of a rational form ${\mathfrak{m}}^{\mathbb{Q}}$ of ${\mathfrak{n}}^E$ which is Anosov, but we now compute this Lie algebra explicitly by giving a basis for ${\mathfrak{m}}^{\mathbb{Q}}$.
Consider the basis $U_1, U_2, U_3, U_4, V_1, V_2$ given by $$\begin{aligned}
U_i &=& \sum_{j=1}^4 \lambda_j ^{i-1} X_{\lambda_j } \\
V_i &=& \left( \lambda_3^{i} - \lambda_1^{i} \right) Y_{\lambda_1 \lambda_3} + \left(\lambda_4^{i} - \lambda_2^{i}\right) Y_{\lambda_2 \lambda_4}.\end{aligned}$$ To simplify the computations, we will use the notations $\lambda_0 = \lambda_4$ and $\lambda_5=\lambda_1$. The basis vectors $U_i$ satisfy $$U_i^\sigma = \sum_{j=1}^4 \left(\lambda_j^{i-1}\right)^\sigma X_{\lambda_{j}} = \sum_{j=1}^4 \sigma^{-1}\left(\lambda_j^{i-1}\right) X_{\lambda_{j}} = \sum_{j=1}^4 \lambda_{j-1}^{i-1} X_{\lambda_{j}}$$ and $$\rho_{\sigma}(U_i) = \sum_{j=1}^4 \lambda_j ^{i-1} \rho_\sigma\left(X_{\lambda_j }\right) = \sum_{j=1}^4 \lambda_j^{i-1} X_{\lambda_{j+1}} = \sum_{j=2}^5 \lambda_{j-1} ^{i-1} X_{\lambda_{j}}.$$ We conclude that $\rho_{\sigma}(U_i) = U_i^\sigma$ and similarly this equation also holds for the vectors $V_i$. This shows that the basis vectors $U_1, U_2, U_3, U_4, V_1, V_2$ satisfy the defining relation of the rational form given in equation (\[def\]) of Section \[secGC\] and thus they indeed span the rational form ${\mathfrak{m}}^{\mathbb{Q}}$ of Theorem \[main\]. The induced Anosov automorphism on ${\mathfrak{m}}^{\mathbb{Q}}$ guaranteed by Theorem \[main\] is given by the matrix $$\begin{aligned}
\begin{pmatrix}
0 & 0 & 0 & -1 & 0 & 0\\
1 & 0 & 0 & -1 & 0 & 0\\
0 & 1 & 0 & 4 & 0 & 0\\
0 & 0 & 1 & 4 & 0 & 0\\
0 & 0 & 0 & 0 & -\frac{1}{2} & -\frac{1}{2} \\[0.3em]
0 & 0 & 0 & 0 & -\frac{1}{2} & -\frac{5}{2}
\end{pmatrix}\end{aligned}$$ in the basis $\hspace{0.5 mm} U_1, U_2, U_3, U_4, V_1, V_2$. By using the matrix representation of this Anosov automorphism, one can compute the Lie bracket: $$\begin{aligned}[c]
\left[U_1, U_2 \right] &= V_1 \\
\left[U_2, U_3 \right] &= -\frac{1}{2} V_1 - \frac{1}{2} V_2 \\
\left[U_3, U_4 \right] &= \frac{1}{2} V_1 + \frac{3}{2} V_2
\end{aligned}
\hspace{2cm}
\begin{aligned}[c]
\left[U_1, U_3 \right] &= V_2 \\
\left[U_2, U_4 \right] &= -\frac{1}{2} V_1 - \frac{5}{2} V_2 \\
\left[U_1, U_4 \right] &= \frac{3}{2} V_1 + \frac{9}{2} V_2.
\end{aligned}$$ The discriminant of the Pfaffian form of ${\mathfrak{m}}^{\mathbb{Q}}$ is $\frac{5}{4}$, so ${\mathfrak{m}}^{\mathbb{Q}}$ is isomorphic to ${\mathfrak{n}}_5^{\mathbb{Q}}$. The characteristic polynomial of the Anosov automorphism restricted to $\gamma_2({\mathfrak{m}}^{\mathbb{Q}})$ is equal to $X^2+3X+1$ which has ${\mathbb{Q}}(\sqrt{5})$ as splitting field.
Of course, Proposition \[count\] also gives us examples of Anosov automorphisms $f: {\mathfrak{n}}^{\mathbb{Q}}_k \oplus {\mathbb{Q}}^2 \to {\mathfrak{n}}^{\mathbb{Q}}_k \oplus {\mathbb{Q}}^2$ with $\operatorname{sgn}(f) = \{3,5\}$. But by using the notion of Scheuneman duality (see [@sche67-1]) for $2$-step nilpotent Lie algebras, it is possible to give another class of Anosov automorphisms overlooked in (False) Claim \[corlaur\]. First we recall some details about this method as described in [@laur08-1].
Let $V^{\mathbb{Q}}$ be any vector space with an inner product and consider the standard inner product $B$ on ${\mathfrak{so}}(V^{\mathbb{Q}})$, given by $$B(Z_1,Z_2) = \operatorname{Tr}(Z_1^T Z_2) = - \operatorname{Tr}(Z_1 Z_2).$$ For every subspace $W^{\mathbb{Q}}$ of ${\mathfrak{so}}(V^{\mathbb{Q}})$, there exists a rational Lie algebra ${\mathfrak{n}}^{\mathbb{Q}}= V^{\mathbb{Q}}\oplus W^{\mathbb{Q}}$ with Lie bracket $[\hspace{1mm},\hspace{1mm}]: V^{\mathbb{Q}}\times V^{\mathbb{Q}}\to W^{\mathbb{Q}}$ defined by $$B([X,Y], Z) = \langle Z(X),Y \rangle$$ for all $Z \in {\mathfrak{so}}(V^{\mathbb{Q}})$, $X,Y \in V^{\mathbb{Q}}$. This is a $2$-step nilpotent Lie algebra where the map $J: \gamma_2(V^{\mathbb{Q}}\oplus W^{\mathbb{Q}}) = W^{\mathbb{Q}}\to {\mathfrak{so}}(V^{\mathbb{Q}})$ is the inclusion. If we take the vector space $W^{\mathbb{Q}}= {\mathfrak{so}}(V^{\mathbb{Q}})$, then the result is the free $2$-step nilpotent Lie algebra on $V^{\mathbb{Q}}$. An isomorphism $\alpha: V^{\mathbb{Q}}\to V^{\mathbb{Q}}$ induces an automorphism $\bar{\alpha}: {\mathfrak{n}}^{\mathbb{Q}}\to {\mathfrak{n}}^{\mathbb{Q}}$ if and only if $\alpha^T Z \alpha \in W^{\mathbb{Q}}$ for all $Z \in W^{\mathbb{Q}}$. Let ${\mathfrak{n}}^{\mathbb{Q}}$ be a Lie algebra of type $(m,k)$ and consider the map $J: \gamma_2({\mathfrak{n}}^{\mathbb{Q}}) \to {\mathfrak{so}}(V^{\mathbb{Q}})$ as introduced above. Denote the image of $J$ as $W^{\mathbb{Q}}$, then it follows by definition that ${\mathfrak{n}}^{\mathbb{Q}}$ is isomorphic to the Lie algebra $V^{\mathbb{Q}}\oplus W^{\mathbb{Q}}$ of the previous paragraph. The dual of ${\mathfrak{n}}^{\mathbb{Q}}$ is then the Lie algebra $\tilde{n}^{\mathbb{Q}}= V^{\mathbb{Q}}\oplus \tilde{W}^{\mathbb{Q}}$ with Lie bracket as in the previous paragraph, where $\tilde{W}^{\mathbb{Q}}$ is the orthogonal complement of $W^{\mathbb{Q}}$ in ${\mathfrak{so}}(V^{\mathbb{Q}})$ relative to the inner product $B$ given above. The dual of the Lie algebra ${\mathfrak{n}}^{\mathbb{Q}}_k$ is denoted by ${\mathfrak{h}}_k^{\mathbb{Q}}$. If $\alpha \in \operatorname{GL}(V^{\mathbb{Q}})$ induces an automorphism $\bar{\alpha}$ of ${\mathfrak{n}}^{\mathbb{Q}}$, then $\alpha^T$ induces an automorphism on $\tilde{{\mathfrak{n}}}^{\mathbb{Q}}$ since $\alpha \tilde{W}^{\mathbb{Q}}\alpha^T = \tilde{W}^{\mathbb{Q}}$ and this map is called the dual automorphism of $\bar{\alpha}$. The combined eigenvalues of $\bar{\alpha}$ and its dual on $\gamma_2({\mathfrak{n}}^{\mathbb{Q}})$ and $\gamma_2(\tilde{{\mathfrak{n}}}^{\mathbb{Q}})$ are equal to the eigenvalues of the map that $\alpha$ induces on $\gamma_2(V^{\mathbb{Q}}\oplus {\mathfrak{so}}(V^{\mathbb{Q}}))$ where $V^{\mathbb{Q}}\oplus {\mathfrak{so}}(V^{\mathbb{Q}})$ is the free $2$-step nilpotent Lie algebra on $V^{\mathbb{Q}}$.
The dual Lie algebra of ${\mathfrak{n}}^{\mathbb{Q}}_k$ is of type $(4,4)$ and denoted as ${\mathfrak{h}}^{\mathbb{Q}}_k$. The Lie algebra ${\mathfrak{h}}^{\mathbb{Q}}_k$ can also be described as the one with basis $X_1, X_2, X_3, X_4, Z_1, Z_2, Z_3, Z_4$ and relations $$\begin{aligned}[c]
[X_1,X_2] &= Z_1 \\
[X_1,X_3] &= Z_2 \\
[X_1,X_4] &= kZ_3
\end{aligned}
\hspace{2cm}
\begin{aligned}[c]
[X_2,X_3] &= -Z_3 \\
[X_2,X_4] &= -Z_2 \\
[X_3,X_4] &= Z_4,
\end{aligned}$$ see for example [@laur08-1]. From the Scheuneman duality, the following proposition is immediate:
For every $k \in {\mathbb{N}}$ with $k > 1$ and $k$ not a square, there exists an Anosov automorphism on ${\mathfrak{h}}^{\mathbb{Q}}_k$ with $\operatorname{sgn}(f) = \{3,5\}$. On ${\mathfrak{h}}^{\mathbb{Q}}_1$ every Anosov automorphism has signature $\{4,4\}$.
Note that the dual of an Anosov automorphism with $\operatorname{sgn}(f)= \{2,4\}$ is also Anosov with signature $\{3,5\}$ and vice versa. So the first part follows from the fact that ${\mathfrak{h}}^{\mathbb{Q}}_k$ is the dual of ${\mathfrak{n}}^{\mathbb{Q}}_k$. Also the second statement follows since the Lie algebra ${\mathfrak{n}}^{\mathbb{Q}}_1$ is not Anosov.
For all other Lie algebras, (False) Claim \[corlaur\] is correct (and the arguments to prove it are the same as the ones used to prove the classification of Anosov Lie algebras up to dimension $8$). Also, the Lie algebra ${\mathfrak{h}}^{\mathbb{Q}}_k$ does not admit an Anosov automorphism of signature $(2,6)$, for example by using the same number theoretical arguments as in [@lw09-1]. So the combined results above determine completely for which Anosov Lie algebras (False) Claim \[corlaur\] is indeed false.
Note that the examples of Proposition \[count\] also answer Question \[q1\] about non-abelian examples of signature $\{2,q\}$ for some $q \in {\mathbb{N}}_0$. We give a more general approach to this question in the next section.
Anosov automorphisms of minimal signature and minimal type {#sign}
----------------------------------------------------------
In this subsection, we show how the main theorem can be used to construct Anosov automorphisms of minimal signature and Anosov Lie algebras of minimal type. These examples answer Questions \[q1\] and \[q2\] which we already mentioned in the introduction.
First we recall some basic properties of unit Pisot numbers. If $E$ is a real Galois extension of ${\mathbb{Q}}$ of degree $n$, then we call an algebraic integer a Pisot number if $ \lambda > 1$ and for all $1 \neq \sigma \in \operatorname{Gal}(E,{\mathbb{Q}})$, it holds that $\vert \sigma(\lambda) \vert <1$. An unit Pisot number is then an algebraic unit which is also a Pisot number. We say that an algebraic unit $\lambda$ with Galois conjugates $\lambda_1= \lambda, \ldots, \lambda_n$ satisfies the full rank condition if for all $d_1, \ldots, d_n$ integers with $$\prod_{j=1}^n \lambda_j^{d_j} = \pm 1,$$ it must hold that $d_1 = d_2 = \ldots = d_n$. From [@payn09-1 Proposition 3.6.] it follows that every unit Pisot number satisfies the full rank condition.
Let ${\mathfrak{n}}^{\mathbb{Q}}$ be an Anosov Lie algebra of nilpotency class $c$ and $f: {\mathfrak{n}}^{\mathbb{Q}}\to {\mathfrak{n}}^{\mathbb{Q}}$ a hyperbolic integer-like automorphism with signature $\{p,q\}$. The characteristic polynomial $h(X)$ of $f$ has integer coefficients and constant term $\pm 1$. This implies that if $g(X)$ is a rational polynomial which divides $h(X)$, then it must have at least one root of absolute value strictly smaller than $1$. We know that $f$ induces an isomorphism on each quotient $\faktor{\gamma_{i-1}({\mathfrak{n}}^{\mathbb{Q}})}{ \gamma_i({\mathfrak{n}}^{\mathbb{Q}})}$ and thus the polynomial $h(X)$ has at least $c$ irreducible factors. Therefore $f$ has at least $c$ eigenvalues of absolute value strictly smaller than $1$ and thus $p \geq c$. By considering $f^{-1}$ as well, we get that $q \geq c$ and this shows that $\min (\operatorname{sgn}(f)) \geq c$. We say that an Anosov automorphism $f: {\mathfrak{n}}^{\mathbb{Q}}\to {\mathfrak{n}}^{\mathbb{Q}}$ has minimal signature if equality holds, i.e. if $\min(\operatorname{sgn}(f)) = c$.
Question \[q1\] asks if there exists Anosov automorphisms of minimal signature for $c= 2$ and already in Section \[verb\] we gave a positive answer to this question as a consequence of the main theorem. So the existence of Anosov automorphisms of minimal signature is a generalization of Question \[q1\] and with Theorem \[main\] we can also give a positive answer to the generalized question:
\[csig\] For every $c$, there exists an Anosov automorphism $f: {\mathfrak{n}}^{\mathbb{Q}}\to {\mathfrak{n}}^{\mathbb{Q}}$ on a Lie algebra of nilpotency class $c$ such that $f$ is of minimal signature.
Let $E$ be a real Galois extension of ${\mathbb{Q}}$ with $\operatorname{Gal}(E,{\mathbb{Q}}) \cong {\mathbb{Z}}_{2n}$ for some $n > 1$ and $\sigma$ a generator of $\operatorname{Gal}(E,{\mathbb{Q}})$. Take $\lambda$ an unit Pisot number in $E$ with the extra condition that $\vert \lambda \sigma^n(\lambda^2) \vert < 1$ (see the Appendix for more details). Since $n > 1$, we have that $ \vert \lambda \sigma^n(\lambda) \vert > \vert \prod_{i=1}^{2n} \sigma^i(\lambda) \vert = 1$. Consider the collection of algebraic integers $$\mu_{i,j} = \sigma^i(\lambda^{j-1}) \sigma^{i+n}(\lambda)$$ for all $i \in \{1, \ldots, 2n\}$ and all $j \in \{1,\ldots, c\}$. The Galois conjugates $\sigma^i(\lambda)$ of $\lambda$ are the $\mu_{i,j}$ with $j = 1$. Note that the definition implies that $\mu_{i+n,2} = \mu_{i,2}$ and all other $\mu_{i,j}$ are distinct because of the full rank condition.
Every $\mu_{i,j}$ with $i \notin \{n,2n\}$ has absolute value $<1$ since $\lambda$ is a Pisot number. The algebraic unit $\mu_{n,3} = \lambda \sigma^n(\lambda^2)$ satisfies $\vert \mu_{n,3} \vert < 1$ because of our choice of $\lambda$ and therefore also all $\mu_{n,j} = \sigma^n\left(\lambda^{j-3}\right) \mu_{n,3} $ with $j \geq 3$ have absolute value $<1$. Thus it follows that of all $\mu_{i,j}$, only $$\mu_{n,1}= \lambda,\mu_{n,2} = \mu_{2n,2} = \lambda \sigma^n(\lambda), \mu_{2n,3}= \lambda^2 \sigma^n(\lambda), \ldots, \mu_{2n,c} = \lambda^{c-1} \sigma^n(\lambda)$$ have absolute value $>1$, so in total there are $c$ of the $\mu_{i,j}$ with $\vert \mu_{i,j} \vert > 1$.
Now consider the Lie algebra ${\mathfrak{n}}^{\mathbb{Q}}$ with basis $X_{\mu_{ij}}$ for all values of $i$ and $j$, where we write the $\mu_{i,1}$ as the conjugates of $\lambda$. The Lie bracket on ${\mathfrak{n}}^{\mathbb{Q}}$ is given by on the one hand $$\begin{aligned}
[ X_{\sigma^i(\lambda)}, X_{\sigma^{i+n}(\lambda)}] = X_{\mu_{i,2}}\end{aligned}$$ for all $i \in \{1, \ldots, n\}$ and on the other hand by $$\begin{aligned}
[ X_{\sigma^i(\lambda)}, X_{\mu_{i,j}}] = X_{\mu_{i,j+1}}\end{aligned}$$ for all $i \in \{1,\ldots, 2n\}, \hspace{1mm} j \in \{2, \ldots, c\}$ (and all other brackets are $0$). It is easy to check that these relations define a Lie algebra (i.e. that the Jacobi identity holds) and that the $1$-dimensional subspaces spanned by each basis vector satisfy the conditions of Corollary \[main2\]. The map $\rho_\sigma$ given by $\rho_\sigma(X_{\mu_{i,j}}) = - X_{\sigma(\mu_{i,j})}$ for $i \in \{ n, 2n \}$ and $j \geq 2$ and $\rho_\sigma(X_{\mu_{i,j}}) = X_{\sigma(\mu_{i,j})}$ for all other $(i,j)$ defines a representation $\rho: \operatorname{Gal}(E,{\mathbb{Q}}) \to \operatorname{Aut}({\mathfrak{n}}^{\mathbb{Q}})$. The minus sign in the first case comes from the relation $\mu_{i+n,2} = \mu_{i,2}$. The conditions of Corollary \[main2\] are satisfied for $\rho$ and thus this gives us a rational form ${\mathfrak{m}}^{\mathbb{Q}}$ with Anosov automorphism $f$. There are only $c$ eigenvalues of $f$ with absolute value $>1$, so $f$ is of minimal signature.
The type of the example constructed in this theorem is equal to $$\underbrace{(2n,n,2n, 2n,\ldots, 2n)}_{c\ {\rm components}}$$ for all $n \geq 2$. This is not the only possibility for Anosov automorphisms of minimal signature since one can construct examples on Lie algebras of type $$\underbrace{(2n,n,2n, n, 2n,\ldots)}_{c\ {\rm components}},$$ where the induced eigenvalues on $\faktor{\gamma_{2j}({\mathfrak{n}}^{\mathbb{Q}})}{ \gamma_{2j+1}({\mathfrak{n}}^{\mathbb{Q}})}$ are of the form $\left( \sigma^{i}\left(\lambda\right) \sigma^{n+i}\left(\lambda\right)\right)^j$. The construction of such examples is similar as in Theorem \[csig\]. We conjecture that these are the only possibilities:
Let $f: {\mathfrak{n}}^{\mathbb{Q}}\to {\mathfrak{n}}^{\mathbb{Q}}$ be an Anosov automorphism of minimal signature, then the type of ${\mathfrak{n}}^{\mathbb{Q}}$ is one of the following:
(i) $(2n,n,2n, 2n,\ldots, 2n)$ or
(ii) $(2n,n,2n, n, 2n,\ldots)$,
where $n > 1$.
The methods of this paper are useful to prove or disprove this conjecture.
In a similar way, the main theorem also gives a positive answer to Question \[q2\]. We state this theorem in a more general setting:
\[last\] For every $c \in {\mathbb{N}}_0$ and $n \in {\mathbb{N}}$ with $n > 2$, there exists an Anosov Lie algebra of type $(n,\ldots,n)$ and nilpotency class $c$.
Let $E \supseteq {\mathbb{Q}}$ be a real Galois extension with cyclic Galois group of order $n$ and let $\sigma \in \operatorname{Gal}(E,{\mathbb{Q}})$ be a generator. Take $\lambda_1 \in E$ an unit Pisot number and consider the Galois conjugates $\lambda_1,\lambda_2 = \sigma(\lambda_1),\ldots, \lambda_n = \sigma^{n-1}(\lambda_1)$. Define the algebraic units $$\mu_{i,j} = \lambda_i^{j-1} \sigma(\lambda_i)$$ for all $i \in \{1,\ldots, n\}$ and $j \in \{1,\ldots, c\}$, where the algebraic units $\lambda_i$ occur as the $\mu_{i,j}$ with $j=1$. Let ${\mathfrak{n}}^{\mathbb{Q}}$ be the Lie algebra with basis $X_{\mu_{i,j}}$ for all $i\in \{1,\ldots,n\}$ and $j \in \{1,\ldots, c\}$, with Lie bracket given by $$\begin{aligned}
[X_{\lambda_i},X_{\mu_{i,j}}] &= X_{\mu_{i,j+1}}\end{aligned}$$ for all $i\in \{1,\ldots,n\}$ and $j \in \{1, \ldots, c\}$ and all other brackets $0$. It is easy to see that the Jacobi identity holds and thus that ${\mathfrak{n}}^{\mathbb{Q}}$ is indeed a Lie algebra.
The linear map $h: {\mathfrak{n}}^{\mathbb{Q}}\to {\mathfrak{n}}^{\mathbb{Q}}$ defined by $h(X_{\mu_{i,j}}) = X_{\sigma(\mu_{i,j})}$ is an automorphism of this Lie algebra of order $n$. So the map $\rho: \sigma \mapsto h$ defines a representation $\rho: \operatorname{Gal}(E,{\mathbb{Q}}) \to \operatorname{Aut}({\mathfrak{n}}^{\mathbb{Q}})$. This Lie algebra and the representation $\rho$ satisfy the conditions of Corollary \[main2\] (where the spaces $V_\lambda$ are one-dimensional and spanned by the basis vectors) and thus there exists a rational form of ${\mathfrak{n}}^E$ which is Anosov. The type of this rational form is equal to the type of ${\mathfrak{n}}^{\mathbb{Q}}$.
The case where $n = 3$ gives an answer to Question \[q2\]. It is an open question to determine all possibilities for the types $(n_1,\ldots, n_c)$ of Anosov Lie algebras with $n_1 = n_2 = 3$. To solve this problem, a careful study of the conjugates of algebraic units of degree $3$ is needed.
Appendix {#app .unnumbered}
========
An important ingredient we used during this article is the existence of unit Pisot numbers in a real Galois extension $E$ of ${\mathbb{Q}}$. In this appendix we extend some results of [@dd13-1] about $c$-hyperbolic units to unit Pisot numbers with extra conditions on them.
If $E$ is a real Galois extension of ${\mathbb{Q}}$ of degree $n$, then we call an algebraic integer a Pisot number if $ \lambda > 1$ and for all $1 \neq \sigma \in \operatorname{Gal}(E,{\mathbb{Q}})$, it holds that $\vert \sigma(\lambda) \vert <1$. An unit Pisot number is then an algebraic unit which is also a Pisot number. Denote by $U_E$ the algebraic units of $E$ and by $\sigma_1, \ldots, \sigma_n$ all elements of the Galois group $\operatorname{Gal}(E,{\mathbb{Q}})$ with $\sigma_1 = 1$. From Dirichlet’s Unit Theorem we know that the map $$\begin{aligned}
l: U_E \to {\mathbb{R}}^{n}: \lambda \mapsto \Big( \log \big( \vert \sigma_1\left(\lambda\right)\vert\big) , \ldots, \log\big( \vert \sigma_n \left(\lambda \right) \vert \big) \Big)\end{aligned}$$ maps $U_E$ onto a cocompact lattice of the subspace $V \subseteq {\mathbb{R}}^n$, where $V$ is given by the equation $x_1 + \ldots + x_n = 0$. The unit Pisot numbers are mapped to the open subset $O \subseteq V$ given by the equations $x_1 > 0$ and $x_i < 0$ for all $i \geq 2$. So for the existence of unit Pisot numbers, one has to show that $O \cap l(U_E) \neq \emptyset$. The following lemma asserts that this is indeed the case:
\[simpel\] Let $L \subseteq {\mathbb{R}}^n$ be a cocompact lattice and $O \subseteq {\mathbb{R}}^n$ a nonempty open subset such that for all $v_1, v_2 \in O$ also $v_1 + v_2 \in O$. Then $O \cap L \neq \emptyset$.
Since $O$ is open and $L \otimes {\mathbb{Q}}$ is dense, there exists $x \in O \cap L \otimes {\mathbb{Q}}$. By taking $n x = x + \ldots + x$ for some $n \in {\mathbb{N}}_0$, we find $x \in L \cap O$.
So this lemma implies that there exist unit Pisot numbers in every real Galois extension $E \neq {\mathbb{Q}}$. The lemma also implies that every open nonempty subset of $V$ which is invariant under addition gives rise to possible algebraic units. For example, there also exists unit Pisot numbers with an extra condition on them:
\[existence\] Let $E$ be a real Galois extension of ${\mathbb{Q}}$ and fix some $\sigma \in \operatorname{Gal}(E,{\mathbb{Q}})$ with $\sigma \neq 1$. Then there always exists an unit Pisot number $\lambda \in E$ such that $ \vert \sigma(\lambda^2) \lambda \vert < 1$.
Assume that $\sigma_2$ of the map $l$ given above is equal to $\sigma$. Let $O \subseteq V$ be the open nonempty subset of $V$ given by $x_1 > 0, x_i < 0$ for all $i \geq 2$ and $x_1 + 2 x_2 <0$, then there exists $x \in O \cap l(U_E)$ because of the previous lemma. Any element of the preimage of $x$ will satisfy the conditions of the proposition.
**Acknowledgments**
I would like to thank my advisor Karel Dekimpe for his useful comments on a first version of this article and the referee for his/her remarks which have improved this paper.
[^1]: The author was supported by a Ph.D. fellowship of the Research Foundation – Flanders (FWO). Research supported by the research Fund of the KU Leuven
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'World-class human players have been outperformed in a number of complex two person games (Go, Chess, Checkers) by Deep Reinforcement Learning systems. However, owing to tractability considerations minimax regret of a learning system cannot be evaluated in such games. In this paper we consider simple games (Noughts-and-Crosses and Hexapawn) in which minimax regret can be efficiently evaluated. We use these games to compare Cumulative Minimax Regret for variants of both standard and deep reinforcement learning against two variants of a new Meta-Interpretive Learning system called *MIGO*. In our experiments all tested variants of both normal and deep reinforcement learning have worse performance (higher cumulative minimax regret) than both variants of *MIGO* on Noughts-and-Crosses and Hexapawn. Additionally, *MIGO*’s learned rules are relatively easy to comprehend, and are demonstrated to achieve significant transfer learning in both directions between Noughts-and-Crosses and Hexapawn.'
author:
- Céline Hocquette$^1$
- |
Stephen H. Muggleton$^1$ $^1$Department of Computing, Imperial College London, London, UK\
{celine.hocquette16, s.muggleton}@imperial.ac.uk
bibliography:
- 'biblio.bib'
title: 'Can Meta-Interpretive Learning outperform Deep Reinforcement Learning of Evaluable Game strategies?'
---
=1
Introduction
============
Deep Reinforcement Learning systems have been demonstrated capable of mastering two-player games such as Go [@GO], outperforming the strongest human players. However, these systems 1) generally require a very large training set to converge toward a good strategy, 2) are not easily interpretable as they provide limited explanation about how decisions are made and 3) do not provide transferability of the learned strategies to other games.
We demonstrate in this work how machine learning strategies as logic programs can overcome these limitations. For example, an applicable strategy for playing Noughts-and-Crosses is to create double attacks when possible. An example of this is shown in Figure \[fig:intro\]. Player O executes a move from board A to board B which creates the two threats represented in green, and results in a forced win for O. The rules in Figure \[fig:intro\] describe such a strategy. A,B and C are variables representing state descriptions which encode the board together with the active player. The rules state that a move by the active player from A to B is a winning move if the opponent cannot immediately win and the opponent cannot make a move to prevent an immediate win by the active player. These rules provide an understandable strategy for winning in two moves. Moreover, these rules are transferable to more complex games as they are generally true for describing double attacks.
We introduce in this article a new logical system called *MIGO* (*Meta-Interpretive Game Ordinator*)[^1] designed for learning two player game optimal strategies of the form presented in Figure \[fig:intro\]. It benefits from a strong inductive bias which provides the capability to learn efficiently from a few examples of games played. Learned hypotheses are provided in a symbolic form, which allows their interpretation. Moreover, learned strategies are generally true for all two-player games, which provides straightforward transferability to more complex games.
*MIGO* uses Meta-Interpretive Learning (MIL), a recently developed Inductive Logic Programming (ILP) framework that supports predicate invention, the learning of recursive programs [@[MIL1]; @[MIL]] and Abstraction [@[Abstraction]]. *MIGO* additionally supports Dependent Learning [@[Onsshot]]. The learning operates in a staged fashion: simple definition are first learned and added to the background knowledge [@[Onsshot]], allowing them to be reused during further learning tasks, and thus to build up more and more complex definitions. For instance, *MIGO* would first learn a simple definition of *win\_1/1* for winning in one move. Next, a predicate *win\_2/1* describing the action of winning in two moves can be built from *win\_1/1* as shown in Figure \[fig:intro\].
To evaluate performance we consider two evaluable games (Noughts and Crosses and Hexapawn). Our results demonstrate that substantially lower Cumulative Minimax Regret can be achieved by *MIGO* compared to variants of reinforcement learning.
Our contributions are the introduction of a system for learning optimal two-player-game strategies (Section 3) and the description of its implementation (Section 4). We demonstrate experimentally it converges faster than reinforcement learning systems and that learned strategies are transferable to more complex games (Section 5).
Related Work
============
Learning game strategies
------------------------
Various early approaches to game strategies [@shapnib; @Quinlan] used the decision tree learner ID3 to classify minimax depth-of-win for positions in chess end games. These approaches used a set of carefully selected board attributes as features. Conversely, *MIGO* is provided with a set of three relational primitives ([*move/2*]{}, [*won/1*]{}, [*drawn/1*]{}) representing the minimal information a human would expect to know before playing a two person game. An ILP approach learned optimal chess endgame strategies at depth 0 or 1 [@Chess]. Examples are board positions taken from a database. Conversely, MIGO learns from game play.
Reinforcement Learning
----------------------
Reinforcement Learning considers the task of identifying an optimal agent policy to maximise the cumulative reward perceived by an agent. MENACE (Matchbox Educable Noughts And Crosses Engine) [@[MENACE]] was the world’s earliest reinforcement learning system and was specifically designed to learn to play Noughts-and-Crosses. An early manual version of MENACE used a stack of matchboxes, one for each accessible board position. Each box contained coloured beads representing possible moves. Moves were selected by randomly drawing a bead from the current box. After having completed a game, MENACE’s punishment or reward consisted of subtracting or adding beads according to the outcome of the game. This modified the probability of the selected move being played in the position [@Brooks]. HER (Hexapawn Educational Robot) [@Hexapawn] is a similar system for the game of Hexapawn.
More generally, Q-learning [@QLearning] addresses the problem of learning an optimal policy from delayed rewards and by trial and error. The learned policy takes the form of Q-values for each actions available from a state. A guarantee of asymptotic convergence to optimal behaviour has been proved [@QLearning2].
Deep Q-learning [@DQL] is an extension that uses a deep convolutional neural network to approximate the different Q-values for each actions given a particular state. It provides better scalability which has been demonstrated through a diverse range of tasks from the Atari 2600 games. However, this framework generally requires the execution of many games to converge. Moreover, the learned strategy is implicitly encoded into the Q-value parameters, which do not provide interpretability. In [@Murray], a hybrid neural-symbolic system is described which address some of these drawbacks. A neural back-end transforms images into a symbolic representation and generates features. A symbolic front end performs action selection. Conversely, *MIGO* is based upon a purely symbolic approach and the number of primitives considered is reduced.
Relational Reinforcement Learning
---------------------------------
Relational reinforcement learning [@RRL] is a reinforcement learning framework where states, actions and policies are represented relationally. It benefits from background knowledge and declarative bias. It learns a Q-function using a relational regression tree algorithm. Conversely, the learning framework *MIGO* is not based upon the identification of Q-values but aims at deriving hypotheses describing an optimal strategy. Relational reinforcement learning also provides the ability to carry over the policies learned in simple domains to more complex situations. However, most systems aim at learning single agent policy and, in contrast to *MIGO*, are not designed to learn to play two person games.
Theoretical Framework
=====================
Credit Assignment
-----------------
One can evaluate the success of a game by looking at its outcome. However, a problem arises for assigning the reward to the various moves performed. Reinforcement learning systems usually tackle this so-called Credit Assignment Problem by adjusting parameter values associated with the moves responsible for the reward observed. We introduce theorems for identifying moves that are necessarily positive examples for the task of winning and drawing.
We assume the learner $P_{1}$ plays against an opponent $P_{2}$ that follows an optimal strategy and that the game starts from a randomly chosen initial board $B$. We consider the following ordering over the different outcomes for $P_{1}$ and demonstrate the lemma below: $$\begin{aligned}
won \succ drawn \succ loss\end{aligned}$$
\[lemma\] The expected outcome of $P_{1}$ can only decrease during a game.
$P_{2}$ plays optimally and therefore any move of $P_{2}$ maintains or lowers the expected outcome. Therefore $P_{1}$ cannot increase its outcome.
We demonstrate the Theorems below given these assumptions and Lemma \[lemma\]:
\[theorem1\] If the outcome is won for $P_{1}$, then every move of $P_{1}$ is a positive example for the task of winning.
Suppose there exists a move of $P_{1}$ from the board $B_{1}$ to the board $B_{2}$ within the game sequence that is a negative example for the task of winning. Then the expected outcome of $B_{1}$ is won and the expected outcome of $B_{2}$ is strictly lower with respect to the order $\succ$. Then, following Lemma \[lemma\] the outcome of the game is strictly lower than won, which leads to contradiction with the outcome observed.
\[theorem2\] We additionally assume an accurate strategy $S_{W}$ for winning has been learned by the learner $P_{1}$. If the outcome of the game is drawn and if the execution of $S_{W}$ from B fails, then any move played by $P_{1}$ or $P_{2}$ is a positive example for the task of drawing.
The initial position does not have an expected outcome of won for $P_{2}$ otherwise the outcome would be won for $P_{2}$ since it plays optimally. The initial position is not an expected outcome of win for $P_{1}$ by assumption. Therefore, the expected outcome of B is drawn. It follows from Lemma \[lemma\] that every position reached during the game has an expected outcome of drawn and that every move of both players is a positive example for the task of drawing.
Theorems \[theorem1\] and \[theorem2\] demonstrate that for an outcome of win for $P_{1}$ or drawn and because the opponent plays optimally, the expected outcome is necessarily maintained as won or drawn respectively. This cannot be further generalised to $P_{2}$’s moves: an outcome of won for $P_{2}$ might be the consequence of a mistake of $P_{1}$ who does not play optimally.
One should also highlight the fact that Theorems \[theorem1\] and \[theorem2\] do not provide any negative examples for *win/2* or *draw/2*, as these theorems do not help to evaluate moves for which the expected outcome decreases. Practically, the learning system considered learns from positive examples only.
Game evaluation
---------------
Given Theorems \[theorem1\] and \[theorem2\], the opponent chosen is an optimal player following the minimax algorithm. Both for Noughts-and-Crosses and Hexapawn, and more generally for most fair two player games, the opponent can always ensure a draw from the initial board, which leaves no opportunities for the learner to win. To ensure possibilities of winning, we start the game from a board randomly sampled from the set of one move ahead accessible boards; this set provides different expected outcomes for the games considered. Then, the actual outcome relies on both the initial board and the sequence of moves performed. We define the minimax regret as follows:
The *minimax regret* of a game is the difference between the minimax expected outcome of the initial board and the actual outcome of the game.
Practically, the minimax expected outcome of a board can be evaluated from a minimax database computed beforehand. Definition 3.4 provides an absolute measure to evaluate the performances of a learning algorithm as it does not rely on the choice of initial board. Thereafter, we evaluate the cumulative minimax regret to compare different learning systems.
Meta-Interpretive Learning (MIL)
--------------------------------
The system *MIGO* introduced in this work is a MIL system. MIL is a form of ILP [@[MIL]; @[MIL2]]. The learner is given a set of examples $E$ and background knowledge $B$ composed of a set of Prolog definitions $B_{p}$ and metarules $M$ such that $B = B_{p} \cup M$. The aim is to generate a hypothesis $H$ such that $B,H \models E$. The proof is based upon an adapted Prolog meta-interpreter. It first attempts to prove the examples considered deductively. Failing this, it unifies the head of a metarule with the goal, and saves the resulting meta-substitution. The body and then the other examples are similarly proved. The meta-substitutions recovered for each successful proofs are saved and can be used in further proofs by substituting them into their corresponding metarules. Key features of MIL are that it supports predicate invention, the learning of recursive programs and Abstraction [@[Abstraction]]. In the following, we use the MIL system *Metagol* [@metagol].
MIGO algorithm
--------------
We present within this section details of the MIGO algorithm.
#### Learning from positive examples
Theorems \[theorem1\] and \[theorem2\] provide a way of assigning positive labels to moves. Therefore, the learning is based upon positive examples only. This is possible because of Metagol’s strong language bias and ability to generalise from a few examples only. However, one pitfall is the risk of over-generalisation due to the absence of negative examples.
#### Dependent Learning
For successive values of $k$ a series of inter-related definitions are learned for predicates $\mbox{win}\_k(A,B)$ and $\mbox{draw}\_k(A,B)$. These predicates define maintenance of minimax win and draw in $k$-ply when moving from position $A$ to $B$. The learning algorithm is presented as Algorithm \[alg:learningprotocol\], each action ’learn’ represents a call to Metagol. This approach is related to Dependent Learning [@[Onsshot]]. The idea is to first learn low-level predicates. They are derived from single examples with limited complexity. The definitions are added into the background knowledge such that they can be used in further definitions. The process iterates until no further predicates can be learned.
**Input**: Positive examples for win$\_$k and draw$\_$k\
**Output**: Strategy for win$\_$k and draw$\_$k
one shot learn a rule and add it to the BK Learn win$\_$k/2 and add it to the BK one shot learn a rule and add it to the BK Learn draw$\_$k/2 and add it to the BK
#### Mixed Learning and Separated Learning
Theorem \[theorem2\] assigns positive labels to *draw/2* examples assuming a winning strategy $S_{W}$ has already been learned. In practice, we distinguish two variants of *MIGO*:
1. Separated Learning: *win/2* and *draw/2* are learned in two stages. *Win/2* is first learned. When a strategy for *win/2* is stable for a given number of iterations the learner starts learning *draw/2*.
2. Mixed Learning: *win/2* and *draw/2* are learned simultaneously. Examples of *draw/2* are first evaluated with the current strategy for *win/2*. If this latter is updated, examples of *draw/2* are re-tested against the new version of *win/2*.
Implementation
==============
Representation
--------------
A board $B$ is encoded as a 9-vector of marks from the set $\{O,X,Empty\}$. States $s(B,M)$ are atoms that represent the current board $B$ and the active player $M$.
Primitives and Metarules
------------------------
The language belongs to the language class $H_{2}^{2}$, which is the subset of Datalog logic programs with predicates of arity at most 2 and at most 2 literals in the body of each clause. Learned programs are formed of dyadic predicates, representing actions, and monadic predicates, representing fluents. The background knowledge contains a general move generator *move/2*, which is an action that modifies a state $s(B,M)$ by executing a move on board $B$ and updating the active player $M$. *Move/2* only holds for valid moves; in other words, the learner already knows the rules of the game. The background knowledge also contains two fluents: a won classifier *won/1* and a drawn classifier *drawn/1*. They hold when a board is respectively won or drawn.
We consider the metarules *postcond* and *negation* described in Table \[tab:metarules\]. The metarule *negation* expresses the logical negation for primitive predicates, and is implemented as negation as failure. This form of Negation does not introduce invented predicates in Metagol.
[width=0.35]{}
Name Metarule
------------ ------------------------------------------
*postcond* $P(A,B) \leftarrow Q(A,B),R(B).$
*negation* $P(A,B) \leftarrow Q(A,B), not(R(B,C)).$
: Metarules considered: the letters P,Q and R denote existentially quantified higher order variables. The letters A, B and C denote universally quantified first-order variables.
\[tab:metarules\]
Execution of the strategy
-------------------------
For each rule learned for *win$\_$i* and *draw$\_$i* a clause of the form below is added to the background knowledge:
win(A,B) :- win_i(A,B).
draw(A,B) :- draw_i(A,B).
When executing a strategy described with a hypothesis $H$, the move performed is the first one consistent with $H$. Practically, it first attempts to prove *win$\_{i}$/2* for increasing values for $i$. Failing that, it attempts to prove *draw$\_{i}$/2* for increasing values for $i$. If these proofs fail, a move is selected at random among the possible moves.\
The opponent plays a deterministic minimax strategy that yields the best outcome in the minimum number of moves.
Learning a strategy
-------------------
At the end of a game, the outcome is observed and the sequence of visited boards is divided into moves. The depth of each board is measured as the number of full moves until the end of the game in the observed sequence. Moves are added to the set of positive examples for win$\_k$/2 or draw$\_k$/2 if they satisfy Theorems \[theorem1\] or \[theorem2\]. Strategies are relearned from scratch after each game using the MIGO algorithm presented above. One additional constraint is added such that *draw/2* cannot be learned before *win/2* since this would cause the learner to always draw and never win.
Experiments
===========
Experimental Hypothesis
-----------------------
This section describes experiments which evaluate the performance of *MIGO* for the task of learning optimal two player game strategies[^2]. We use the games of Noughts-and-Crosses and a variant of the game of Hexapawn [@Hexapawn]. *MIGO* is compared against the reinforcement learning systems MENACE / HER, Q-learning and Deep Q-learning. Accordingly, we investigate the following null hypotheses: ***Null Hypothesis 1:*** **MIGO* cannot converge faster than MENACE / HER, Q-learning and Deep Q-learning for learning optimal two-player game strategies.* We additionally test the ability of *MIGO* to transfer learned strategies to more complex games, and thus verify the following null hypothesis: ***Null Hypothesis 2:*** **MIGO* cannot transfer the knowledge learned during a previous task to a more complex game.*
Convergence
-----------
### Materials and Methods
[width=0.4]{}
OX Hexapawn$_{3}$ Hexapawn$_{4}$
------------------------- --------------- ---------------- ----------------
MIGO mixed learning 1.5$.10^{-1}$ 3.0$.10^{-3}$ 3.9
MIGO separated learning 8.9$.10^{-2}$ 2.8$.10^{-3}$ 3.8
MENACE / HER 1.5$.10^{-3}$ 2.7$.10^{-4}$ /
Q-Learning 2.3$.10^{-1}$ 1.9 $.10^{-3}$ 2.7 $.10^{-1}$
Deep Q-Learning 2.4$.10^{-1}$ 1.7$.10^{-2}$ 2.1 $.10^{-1}$
: Average CPU time (seconds) of one iteration[]{data-label="fig:time"}
#### Common
We provide *MIGO*, Menace / HER and Q-learning with the same set of initial boards randomly sampled from the set of one-full-move-ahead positions - positions that result from one move of each player. The systems studied play games starting from these initial boards, and they face the same deterministic minimax player. Therefore, the only variable in the experiments is the learning system. It is assumed the learner always starts the game. The performance is evaluated in terms of cumulative minimax regret.\
We follow an implementation of Tabular Q-learning available from [@Qlearningcode] and used the parameter values which were provided for the Q-learning algorithm: the exploration rate is set to 0; the initial q-values are $1$; the discount factor is $\gamma = 0.9$ and the learning rate $\alpha = 0.3$.\
Similarly, we follow an implementation of Deep Q-learning available from [@DeepQlearningcode]. The provided parameters were used: the discount factor is set to $0.8$; the regularization strength to $0.01$; the target network update rate to $0.01$; the initial and final exploration rate are $0.6$ and $0.1$ respectively.\
The results presented here have been averaged oven 40 runs for Hexapawn$_{3}$ and 20 for Noughts and Crosses. Average running times are presented in Figure \[fig:time\].
#### Noughts-and-Crosses
The set of initial boards comprises 12 boards taking into account rotations and symmetries of the board. Among them 7 are expected win, and 5 are expected draw. Therefore the worst case regret of a random player is 1.58. The counter for starting learning *draw/2* is set to 10.
#### Hexapawn
Hexapawn’s initial board is represented in Figure \[fig:hexapawn\]. The goal of each player is to advance one of their pawns to the opposite end. Pawns can move one square forward if the next square is empty or capture another pawn one square diagonally ahead of it [@Hexapawn]. Rules have been modified: the game is said to be drawn when the current player has no legal move.Thereafter, we refer to Hexapawn$_{3}$ and Hexapawn$_{4}$ for the game of Hexapawn in dimensions 3 by 3 and 4 by 4 respectively. The set of initial boards comprises 5 boards taking into account the vertical symmetry. Among them, 3 are expected draw and 2 are expected win. Therefore the average worst case regret is 1.4. As the dimensions are smaller for Hexapawn$_{3}$ than for Noughts and Crosses, the counter for starting learning *draw/2* is set to 5.
### Results
\
Results are presented in Figure \[fig:samplecomp\] and show that *MIGO* converges faster than MENACE / HER, Q-learning and Deep Q-learning for both games, refuting null hypothesis 1. As the maximum depth is larger for Noughts-and-Crosses than for Hexapawn$_{3}$, all systems require more iterations to converge. Deep Q-learning performs worst for Hexapawn$_{3}$ as the parameters selected are the ones tuned for Noughts and Crosses and might not be adapted. For both games, mixed learning has lower cumulative regret than separated learning, because mixed learning does not waste any examples of *draw/2* from the initial period in which *win/2* is being learned and it does not stop learning *win/2* after the initial period.
Rules learned by *MIGO* are presented in Figure \[fig:rules\]. *MIGO* converges toward this full set of rules when playing Noughts-and-Crosses. Because the maximum depth of Hexapawn$_{3}$ is 2, *MIGO* learns up to the double line when playing Hexapawn$_{3}$. If unfolding, the first rule can be translated into English as: *State A is won at depth 1 if there exists a move from A to B such that B is won.* Similarly, winning at depth 2 can be described with the following statement: *State A is won at depth 2 if there exists a move of the current player from A to B such that B is not immediately won for the opponent and such that the opponent cannot make a move from B to C to prevent the current player from immediately winning.* This statement is similar to the one presented in section 1. Finally, winning at depth 3 can be explained as: *State A is won at depth 3 for the current player if there exists a move from A to B such that B is not won for the opponent in 1 or 2 moves and such that the opponent cannot make a move from B to C to prevent the current player from winning in 1 or 2 moves.* None of the other systems studied can provide similar explanation about the moves chosen. Rules are built on top on each other, the calling diagram in Figure \[diagram\] represents the dependencies between each learned predicates.
[width=0.45]{}
Depth Rule
------- --------------------------------------------------------
1 `win_1(A,B):-win_1_1_1(A,B),won(B).`
`win_1_1_1(A,B):-move(A,B),won(B).`
`draw_1(A,B):-draw_1_1_3(A,B),not(win_1(B,C)).`
`draw_1_1_3(A,B):-move(A,B),not(win_1(B,C)).`
2 `win_2(A,B):-win_2_1_1(A,B),not(win_2_1_1(B,C)).`
`win_2_1_1(A,B):-move(A,B),not(win_1(B,C)).`
`draw_2(A,B):-draw_2_1_1(A,B),not(win_1(B,C)).`
`draw_2_1_1(A,B):-draw_1(A,B),not(win_1(B,C)).`
3 `win_3(A,B):-win_3_1_1(A,B),not(win_3_1_1(B,C)).`
`win_3_1_1(A,B):-win_2_1_1(A,B),not(win_2(B,C)).`
`draw_3(A,B):-draw_3_1_10(A,B),not(draw_1_1_12(B,C))`.
`draw_3_1_10(A,B):-draw_2(A,B),not(draw_1_1_12(B,C))`.
4 `draw_4(A,B):-draw_4_1_2(A,B),not(draw_1_1_12(B,C))`.
`draw_4_1_2(A,B):-draw_3(A,B),not(draw_1_1_12(B,C))`.
: Example of rules learned for Noughts-and-Crosses (all) and Hexapawn$_{3}$ (above the double line)
\[fig:rules\]
\[diagram\]
### Discussion
MENACE, HER and Q-learning encode the knowledge into the parameters (number of beads or Q-values). The states and their parameters are unique for each board. This results in a weaker generalisation ability: knowledge cannot be transferred from one state to another. Deep Q-learning can provide some generalisation ability; however, it is only visible after a large number of iterations. Conversely, *MIGO* generalises the boards characteristics and each rule learned describes a set of states, which considerably reduces the number of parameters to learn and therefore the number of examples required.
The reinforcement learning systems tested have an implicit representation of the problem. For instance, no geometrical concepts have been encoded. Conversely, *MIGO* benefits from a background knowledge which describes the notion of winning, and from which it can extract a notion of alignment. This allows a degree of explanation.
The running time increases rapidly with the state dimensions for *MIGO*. This reflects the increasing execution time of the learned strategy which is not efficient since a deep evaluation requires extensive evaluation to decide whether a move leads to a win.
MENACE / HER are specifically tailored for theses games. Conversely, Q-learning and Deep Q-learning are a general approaches that can tackle a wide range of tasks, providing that parameters are tuned. *MIGO* benefits from underlying assumptions which reduce its range of applications. However, the primitives are abstract enough to allow playing a wide range of games and support transferring knowledge from one game to another as we will demonstrate in the next section.
Transferability
---------------
\
#### Materials and Methods
Strategies are first learned for Hexapawn$_{3}$ and Noughts-and-Crosses respectively. Strategies are learned with mixed learning and for 100 iterations for Hexapawn$_{3}$ and 200 iterations for Noughts and Crosses. The resulting learned program is transferred to the next learning task, which is learning a strategy for Hexapawn$_{4}$. Results have been averaged over 20 runs.
#### Results
The results presented in Figure \[fig:transfer\] show that transferring the knowledge learned in a previous task help to converge faster, thus refuting null hypothesis 2. Since the learner benefits from an initial knowledge, it is substantially improved compared to an initial random player.
Conclusion and Future Work
==========================
This article introduces a novel logical system named *MIGO* for learning two-player-game strategies. It is based upon the MIL framework. This system distinguishes itself from classical reinforcement learning by the way that it addresses the Credit Assignment Problem. Our experiments have demonstrated that *MIGO* achieves lower Cumulative Minimax Regret compared to Deep and classical Q-Learning. Moreover, we have demonstrated that strategies learned with *MIGO* are transferable to more complex games. Strategies have also been shown to be relatively easy to comprehend.
#### Future Work
One limitation of the system presented is the risk of over-generalisation, observable in the strategy learned. We will further extend the implementation to include a more thorough context for learning from positive examples such as the one presented in [@[Positive]].
The running time suggests that the execution time of the learned strategies increases with the dimensions of the states, which limits scalability. We will further extend *MIGO* to optimise the execution time for hypothesised programs. Selection of hypotheses could be performed following the idea described in [@[Metaopt]].
Another limitation to scalability is the restriction imposed by the initial assumptions. The current version of *MIGO* requires an optimal opponent, which is intractable in large dimensions. We will further extend this system by relaxing Theorems \[theorem1\] and \[theorem2\] and weakening the optimal opponent assumption. A solution could be to learn from self-play.
Because *MIGO* benefits from a strong declarative bias, the sample complexity is much improved compared to other approaches. However, most of the examples are wasted as no labels could be attributed. We plan to evaluate whether Active Learning could further help to reduce the sample complexity. The learner could choose an initial board to start the game, the choice being based upon an information gain criterion.
Although learned strategies provide a certain form of explanation, we will further study how comprehensible learned strategies are. We will evaluate whether MIGO can fullfill Michie’s Machine Learning Ultra Strong criterion, which requires the learner to be able to teach the learned hypothesis to a human [@UltraStrongML].
Despite these limitations, we believe the novel system introduced in this work opens exciting new avenues for machine learning game strategies.
[^1]: From the children’s game-playing phrase [*My go!*]{} and the literal translation into English of the French word *Ordinateur* which means computer.
[^2]: Code for these experiments available at\
<https://github.com/migo19/migo.git>
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We consider the operation of contraction of unitary irreducible representations of the de Sitter group $ SO(4,1) $. It is shown that a direct sum of unitary irreducible representations of the Poincaré group with different signs of the rest mass is obtained as a result of contraction. The results obtained are used to interpret the phenomena of “dark matter” and “dark energy” in terms of the elementary quantum systems of de Sitter’s world.
Keywords: de Sitter world, $SO(4,1)$ group, Wigner-Inönü limit, “dark matter”, “dark energy”, unitary irreducible representations, contraction, elementary systems. AMS Subject Classification: 20-20C, 83-83C, 83-83F, 81-81G
author:
- 'B.A. RAJABOV$^1$'
title: 'The contraction of the representations of the group $SO(4,1)$ and cosmological interpretation'
---
[^1]
Introduction
============
In this article, we will consider the application of the operation of contraction for representations of the group of movements of the de Sitter world. The operation of contraction was introduced in the paper of Wigner-Inönü for the Lie algebra of the relativistic Poincaré group, and the speed of light was used as the contraction parameter, [@Wigner-Inonu-1953]. As a result, a transition to the non-relativistic Galilean group was obtained. Subsequently, this operation was generalized to all Lie algebras, [@Barut-Raczka].
The contraction of the representations of the de Sitter group $SO(4,1)$ from the points of Lie algebra was considered in [@Strom1961]-[@Strom1965]. In the article [@Mic-Nied], the contraction operation is extended to representations of the continuous basic series of groups $ SO(p,1) $, and two types of contraction are considered: to representations of the non-homogeneous Lorentz group $ ISO(3,1) $ and to the representations of the group of Euclidean motions $M(p)$. Drechsler considered contraction in a fiber bundle with Cartan connection, in particular, with the de Sitter group bundle, [@Drechsler].
In [@Rajab], explicit expressions for the matrix elements of non-degenerate irreducible representations of the group $ M(4) $ are found using the operation of contraction of the matrix elements of unitary irreducible representations of the continuous fundamental series of $ SO(4,1) $ group.
The monograph [@Gromov] is devoted to multidimensional generalizations of contraction for classical and quantum groups.
The papers [@Nachtmann1967]-[@Borner-Durr] consider the quantum field theory in the de Sitter world. The problems of dark matter and dark energy are reviewed [@Ryabov]-[@DarkMatter-3].
The present paper is devoted to the contraction of unitary irreducible representations of the de Sitter group $SO(4,1)$. The scalar radius of curvature of space-time is chosen as the contraction parameter. Using the of Wigner-Inönü limit, a direct sum of unitary irreducible representations of the Poincaré group with different energy signs is obtained. Results obtained are used to interpret “dark matter” and “dark energy” using the elementary systems of de Sitter’s world.
The main results of this work were reported in the 3rd international Scientific Conference “Modern Problems of Astrophysics-III”, September 25-27, 2017, Georgia.
Background ... Spin is a purely quantum phenomenon
==================================================
The Stern-Gerlach experiment proved the existence of an intrinsic orbital angular momentum and the magnetic moment of the electron, which assume discrete values (1922), [@Stern].
The works of Goudsmit and Uhlenbeck showed that an attempt to explain the spin in the representations of classical physics leads to insuperable difficulties (appear speeds greater than the speed of light, 1925), [@Goudsmit]-[@Thomas].
V. Pauli proposed a non-relativistic Schrödinger equation for an electron with allowance for the spin variable (Pauli equation for a 2-component electron, 1927), [@Schiff].
P. Dirac developed the relativistic theory of the electron and introduced 4-component wave functions (Dirac equation, 1928), [@Bjorken].
The main conclusion was that the electron spin is a purely quantum phenomenon. The peak of these theories was the connection of spin with statistics and the prediction of the positron.
*Since then, in all reviews, monographs, and courses, this story is given to explain the quantum nature of spin...*
Wigner’s theory of elementary systems
=====================================
A real understanding of the nature of the spin appeared after the classical work of E. Wigner on the representations of the inhomogeneous Lorentz group (that is, the Poincaré group, 1937), [@Wigner1937]. He established that projective representation of this group acts in the Hilbert space of states. Wigner introduced the concept of an elementary system and showed that such systems correspond to irreducible representations, that is, the space of these representations does not have invariant subspaces.
Most importantly. Wigner proved that the rest mass and the spin of an elementary particle (!) are:
- invariants that uniquely characterize irreducible representations of the Poincaré group;
- consequences of the space-time symmetries and are not consequences of the equations of dynamics.[^2]
Invariants of the inhomogeneous Poincaré group (Casimir operators) are constructed using translation generators $ P_{\mu} $ and 4-dimensional rotations $ M_{\mu\nu} $ as follows: $$m^{2}=P^{\mu}P_{\mu};\qquad w^{2}=w^{\mu}w_{\mu}=m^{2}s(s+1);\qquad w_{\varrho}=\dfrac{1}{2}\epsilon _{\lambda\mu\nu\varrho}P^{\lambda}M^{\mu\nu},$$ where
$ m $ – mass, $ s $ – spin of the particle, and $ w_{\varrho} $ is the Pauli-Lubansky-Bargmann vector.
***Important!*** In the case $ m^{2}\geq 0 $, there is a third invariant - the sign of energy: $$\varepsilon = \dfrac{P_{0}}{|P_{0}|}=\pm 1$$
*So, spin is a purely quantum phenomenon, but it is associated with groups of space-time movements ...*[^3]
Einstein’s equations and de Sitter’s solutions
==============================================
According to Einstein’s general relativity, the metric properties of space-time are determined by the distribution and motion of matter, [@Einstein]: $$R_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu}R+\varLambda g_{\mu\nu}=-\dfrac{8\pi G}{c^{4}}T_{\mu\nu};\quad R=R_\mu^\mu$$ where
$ R_{\mu\nu} $ – Ricci tensor, $ g_{\mu\nu} $ – metric tensor, $ \varLambda $ – cosmological constant, $ G $ – the gravitational constant of Newton, $ T_{\mu\nu} $ – energy-momentum tensor and $ \mu,\nu=0,1,2,3. $.
This equation can be rewritten in an equivalent form: $$R_{\mu\nu}=\varLambda g_{\mu\nu}-\dfrac{8\pi G}{c^{4}}(T_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu}T);\quad T= T_\mu^\mu$$ It is clear from (2) that the $ \varLambda $-term, even in the absence of matter ($ T_{\mu\nu}=0 $), changes the space-time geometry and $ g_{kl}\Lambda $ is the energy-momentum tensor of the vacuum.[^4]
In the vacuum, the Einstein equations take the form: $$R_{\mu\nu} = \varLambda g_{\mu\nu}.$$ For $ \varLambda = 0 $, the solution of (3) is the Minkowski manifold with a group of Poincaré motions.
The histories of the cosmological constant are reviewed in [@Straumann]-[@One; @Hundred].
In the general case, the solutions of the Einstein (1)-(2) equations do not have a group of motions. But in 1917 Willem de Sitter found two solutions of (3) for $ \Lambda\neq 0 $ with different global groups of movements, [@Weinberg]–[@Tolman]: $$ds^{2}=\dfrac{dr^{2}}{1-r^{2}/R^{2}}+r^{2}(d\vartheta^{2}+\sin^{2}\vartheta d\varphi^{2})-\left({1-\dfrac{{r^{2}}}{{R^{2}}}}\right)c^{2}dt^{2},\qquad if \ \Lambda > 0;$$ $$ds^{2}=\dfrac{dr^{2}}{1+r^{2}/R^{2}}+r^{2}(d\vartheta^{2}+\sin^{2}\vartheta d\varphi^{2})-\left({1+\frac{r^{2}}{R^{2}}}\right)c^{2}dt^{2},\qquad if \ \Lambda < 0.$$ Here the radius of curvature $ R $ and the cosmological constant $ \Lambda $ are related by the following formula:[^5] $$\Lambda = \pm\dfrac{3}{R^{2}}$$
Using stereographic projections: $$\begin{aligned}
\xi_{1}=r\cos \varphi;\quad \xi_{2}=r\sin\vartheta\cos\varphi;\quad \xi_{3}=r\sin\vartheta\sin\varphi;\\\xi_{4}=R\sqrt{1-\dfrac{r^{2}}{R^{2}}}\cosh\left(\dfrac{ct}{R}\right);\quad \xi_{0}=R\sqrt{1-\dfrac{r^{2}}{R^{2}}}\sinh\left(\dfrac{ct}{R}\right)\end{aligned}$$ and $$\begin{aligned}
\eta_{1}=r\cos \varphi;\quad \eta_{2}=r\sin\vartheta\cos\varphi;\quad \eta_{3}=r\sin\vartheta\sin\varphi;\\\eta_{4}=R\sqrt{1+\dfrac{r^{2}}{R^{2}}}\cosh\left(\dfrac{ct}{R}\right);\quad \eta_{5}=R\sqrt{1+\dfrac{r^{2}}{R^{2}}}\sinh\left(\dfrac{ct}{R}\right)\end{aligned}$$ de Sitter solutions can be isometrically embedded as sub-manifolds in 5-dimensional pseudo-Euclidean spaces: $$\xi_{0}^{2}-\xi_{1}^{2}-\xi_{2}^{2}-\xi_{3}^{2}-\xi_{4}^{2}=-R^{2}\label{eq:07}$$ and $$\eta_{1}^{2}+\eta_{2}^{2}+\eta_{3}^{2}-\eta_{4}^{2}-\eta_{5}^{2}=-R^{2},\label{eq:8}$$ respectively.
These spaces have global symmetry groups $ SO(4,1) $ and $ SO(3,2) $ that leave the metrics (6)-(7) invariant. Groups $ SO(4,1) $ and $ SO(3,2) $ are called de Sitter groups. Spaces (4),(6) and (5),(7) are called de Sitter worlds of the 1st and 2nd kind or according to the modern terminology, de Sitter worlds $dS$ and anti–de Sitter $AdS.$
We restrict ourselves to the de Sitter world (5),(7) and the $ SO(4,1) $ group. The case of the anti-de Sitter world will be considered in a separate paper, because of the difficulties in interpreting the space-time coordinates.
The commutation relations for the generators of the group $ SO(4,1) $ have the form: $$\begin{aligned}
\left[ M_{\mu\nu},M_{\varrho\sigma} \right] =i\left( g_{\mu\varrho}M_{\nu\sigma}-g_{\mu\sigma}M_{\nu\varrho}-g_{\nu\varrho}M_{\mu\sigma}+g_{\nu\sigma}M_{\mu\varrho} \right);\\
\left[ M_{\mu\nu},P_{\varrho}\right] = i\left( g_{\mu\varrho}P_{\nu}-g_{\nu\varrho}P_{\mu}\right);\qquad \quad \left[ P_{\mu},P_{\nu}\right] =\dfrac{i}{R^{2}}M_{\mu\nu}; \end{aligned}$$ where $ P_\mu=(1/R)M_{4\mu} $.
The Casimir operators of the Lie algebra of the group $ SO(4,1) $ have the following form, [@Gursey]: $$C_{1}=-\dfrac{1}{2R^{2}}M_{ab}M^{ab}=-P_{\lambda}P^{\lambda}-\dfrac{1}{2R^{2}}M_{\mu\nu}M^{\mu\nu}=M^{2},\label{eq:9}$$ and $$C_{2}=-W_{a}W^{a},\quad W_{a}=\dfrac{1}{8R}\varepsilon_{abcde}M^{bc}M^{de}.\label{eq:10}$$
Here, the lifting and lowering of the indexes are carried out using the 5-dimensional metric tensor .
To consider the result contraction $ R\rightarrow\infty $, it is convenient to represent the operator $ C_{2} $ in the following form: $$C_{2}=-V_{\lambda}V^{\lambda}-\dfrac{1}{R^{2}}W_{4}^{2},$$ where $$\begin{aligned}
W_{\lambda}&=&-\dfrac{1}{2}\varepsilon_{\lambda\varrho\mu\nu 4}P^{\varrho}M^{\mu\nu},\\
W_{5}&=&\dfrac{1}{8}\varepsilon_{\lambda\mu\nu\varrho}M^{\lambda\mu}M^{\nu\varrho}.
\end{aligned}$$
From the last expressions, it is clear that when $ R\rightarrow \infty $ the Lie algebra of the group $ SO(4,1) $ becomes over to the Lie algebra of the Poincaré group, and the Casimir operators become: $$\begin{aligned}
C_{1}&\rightarrow&-P_{\lambda}P^{\lambda}= m^{2},\\
C_{2}&\rightarrow&-V_{\lambda}V^{\lambda}=m^{2}s(s+1).
\end{aligned}$$ where $ s,m $ – spin and rest mass, respectively.
From the limiting transition of the Casimir operators $ C_{1},C_{2} $ it follows that unlike the Minkowski world in the de Sitter world, elementary systems are identified not by mass and spin, but by some functions of spin and mass.
***Important!*** It is obvious that any function of invariants is invariant. Therefore, as invariants characterizing unitary irreducible representations of the group $ SO(4,1) $, any pair of functionally independent invariants can be chosen.
In particular, non-degenerate representations of the group $ SO(4,1) $ can be realized in the space of $ (2s+1) $-component vector-functions and the degree of homogeneity of $ \sigma $ on the upper field of the cone, [@Rajab; @Rajab-2; @Rajab-3]. Then the parameters $ s $ and $ \sigma $ will play the role of invariants characterizing the irreducible representations of the group $SO(4,1)$.
In addition, as will be shown in the next section, after the operation of contraction, the parameter $ s $ becomes into spin and the $ \sigma $ to the function of mass $ m $.
The contraction of the representations of the group $ SO(4,1) $
===============================================================
Representations of the group $ SO(4,1) $ will be constructed in the space of measurable vector-valued functions $\Phi:SO(4,1)\longrightarrow C^{2s+1}$ that satisfies the following conditions: $$\Phi_{\lambda}(gp)=\sum_{\lambda'=-s}^{s}\Delta_{\lambda\lambda'}^{(\sigma,s)}(p^{-1})\Phi_{\lambda'}(g),\qquad Re\,\sigma=-3/2;\label{eq:185}$$ $$\sum_{\lambda=-s}^{s}\int\left|\Phi_{\lambda}(p)\right|^{2}\varrho(g)dg<\infty.$$ Here, $ \rho (\cdot) $ is a continuous non-negative function on the group $ SO (4,1) $ such that $$\int\varrho(gp)d_{l}(p)=1\label{eq:186}$$ and supp$\,\rho$ has a compact intersection with each class of the contiguity $ gP $. Such a function exists for any locally compact group [@Kirillov]. We introduce in this space a scalar product: $$\left(\Phi^{(1)},\Phi^{(2)}\right)=\sum_{\lambda=-s}^{s}\overline{\Phi_{\lambda}^{(1)}(g)}\Phi_{\lambda}^{(2)}(g)\varrho(g)dg.\label{eq:187}$$
The Hilbert space obtained in this way will be denoted by $ \mathfrak{L}^2\left( SO(4,1); \Delta ^{(\sigma,S)}\right) $ and consider the representation of the group $ SO (4,1) $ acting in this space along formula: $$T^{(\sigma,s)}(g)\Phi_{\lambda}(g_{1})=\Phi_{\lambda}(g^{-1}g_{1}).\label{eq:188}$$
This is the induced representation in the Mackey sense of the group $ SO (4,1) $, [@Mackey].
We now construct another realization of the space used for studying problems of contraction of representations. For this, we consider on the cone: $$\left[ k,k\right] = k_{0}^{2}-k_{1}^{2}+k_{2}^{2}+ k_{3}^{2}+k_{4}^{2}=0,\quad k_{0}>0,$$ the new coordinate system: $$\begin{aligned}
k & = & \left|k_{4}\right|(\zeta,\varepsilon),\nonumber \\
\zeta_{0} & = & \cosh\beta,\nonumber \\
\zeta_{1} & = & \sinh\beta\sin\vartheta\sin\varphi,\label{eq:191}\\
\zeta_{2} & = & \sinh\beta\sin\vartheta\cos\varphi,\nonumber \\
\zeta_{3} & = & \cos\vartheta\sinh\beta;\nonumber
\end{aligned}$$ $$\left|k_{4}\right|\neq0,\quad\varepsilon=\pm1,\quad0\leq\beta<\infty,\quad0\leq\vartheta\leq\pi,\quad0\leq\varphi<2\pi.$$ $$\zeta_{0}^{2}-\zeta_{1}^{2}-\zeta_{2}^{2}-\zeta_{3}^{2}=1.\label{eq:192}$$
The set of points of the upper half of the cone that is not covered by this coordinate system forms a manifold of lower dimension. Since the upper half of the cone is a transitive surface, one can obtain parametrization of the elements of the group starting from this coordinate system.
First, we fix one of the points of the cone, namely, the point $ \mathring{k} $: $$\left|k_{4}\right|=1,\quad\varepsilon=1,\quad\beta=\vartheta=\varphi=0.$$ Next we note that the stationary group $\mathfrak{W} $ of the point $ \mathring{k} $ is isomorphic to the group motions $M(3)$ of 3-dimensional Euclidean space $ E_ {3} $. Each element $w\in \mathfrak{W}$ can be uniquely represented as: $$w=R(\rho)B(\vec{z}),\label{eq:16}$$ where $ \vec {z} $ is a 3-dimensional vector, and $ \rho $ is an orthogonal matrix $ 3 \times 3 $. The matrices $ R (\rho) $ and $ B (z) $ have the following form: $$R(\rho)=\begin{pmatrix}1 & \vec{0\,}^{\intercal} & 0\\
\vec{0} & \rho & \vec{0}\\
0 & \vec{0\,}^{\intercal} & 1
\end{pmatrix},\quad R(\rho)\in SO(3)\subset SO(4,1),\label{eq:17}$$ $$B(\vec{z})=\begin{pmatrix}1+z^{2}/2 & \vec{z\,}^{\intercal} & -z^{2}/2\\
\vec{z} & I_{3} & -\vec{z}\\
z^{2}/2 & \vec{z\,}^{\intercal} & 1-z^{2}/2
\end{pmatrix},\quad z^{2}=\vec{z\,}^{2}\label{eq:18}$$ The matrices $ B (\vec {z}) $ form an abelian subgroup that is normal division subgroup of the stationary subgroup $ \mathfrak{W}. $
The transitivity property allows an almost element $ g \in SO(4,1) $ to represent an expansion: $$g=h_{\varepsilon}(k)w,\quad\varepsilon=\pm 1,\quad w \in \mathfrak{W},\label{eq:195}$$ where $ h_{\varepsilon}(k) $ is the so-called Wigner operator (“boost”) having the property: $$k=h_{\varepsilon}(k)\mathring{k}.$$ The Wigner operator is defined by the following formulas: $$h_{1}(k)=g_{12}(\varphi)g_{23}(\vartheta)g_{03}(\beta)g_{04}(\tau),\qquad\tau=\ln\left|k_{4}\right|,\label{eq:193}$$ for the subset $ \varepsilon = 1 $, and $$h_{-1}(k)=g_{12}(\varphi)g_{23}(\vartheta)g_{03}(\beta)\eta g_{04}(\tau),\qquad\tau=\ln\left|k_{4}\right|,\label{eq:194}$$ for the subset $ \varepsilon = -1 $.
Here $ \eta $ is the diagonal $ 5\times 5 $ matrix: $ \eta =
\mathrm{diag}(1,-1,-1,-1,-1) $.
We now note that the action of $ g \in SO (4,1) $ on a cone induces nonlinear transformation of this element on the sections $\left | k_ {4} \right | = 1 $, which are the uppers of a two-sheeted hyperboloid.
It follows from - that the stationary subgroup of this variety is the minimal parabolic subgroup $ \mathsf {P} $, and for almost all $ g \in SO(4,1) $ the expansion is valid: $$g=h_{\varepsilon}(\zeta)p,\label{eq:196}$$ where $$h_{\varepsilon}(\zeta)=h_{\varepsilon}(k)|_{\tau=0},\quad\varepsilon=\pm1.$$
Let $ \mathfrak{H}_ {i}, \: i = 1,2 $ be the Hilbert spaces of vector functions on the hyperboloid with the scalar product: $$(F^{(1)},F^{(2)})=\int\overline{F^{(1)}(\zeta)}F^{(2)}(\zeta)d\mu(\zeta),\label{eq:197}$$ where $$\begin{aligned}
d\mu & = & \frac{\left(d\vec{\zeta}\right)}{2\sqrt{1+\zeta^{2}}},\label{eq:198}\\
\left(d\vec{\zeta}\right) & = & d\zeta_{1}d\zeta_{2}d\zeta_{2},\nonumber
\end{aligned}$$ is the usual Lorentz-invariant measure on the field of a two-sheeted hyperboloid.
We define an isometric mapping: $$\mathfrak{L}^{2}\left(SO(4,1);\Delta^{(\sigma,s)}\right)\longrightarrow\mathfrak{H}_{1}\oplus\mathfrak{H}_{2}$$ in the following way: $$F_{\lambda}(\zeta;\varepsilon)=\Phi_{\lambda}(h_{\varepsilon}(\zeta)).\label{eq:199}$$ The inverse mapping is found with the help of and : $$\Phi_{\lambda}(g)=\sum_{\lambda'=-s}^{s}\Delta_{\lambda\lambda'}^{(\sigma,s)}(p^{-1})F_{\lambda'}(\zeta;\varepsilon).\label{eq:200}$$
These mapping properties are easily verified. It follows from , that under the mapping - the representation of the group $ SO(4,1) $ in the space $ \mathfrak{L}^{2} \left (SO(4,1); \Delta^{(\sigma, s)} \right) $ becomes a unitary representation of this group in $ \mathfrak{H}_{1} \oplus \mathfrak{H}_{2} $: $$T^{(\sigma,s)}(g)F_{\lambda}(\zeta;\varepsilon)=\sum_{\lambda'=-s}^{s}\Delta_{\lambda\lambda'}^{(\sigma,s)}(p_{g}^{-1})F_{\lambda'}(\zeta_{g};\varepsilon_{g}),\label{eq:201}$$ where $ \zeta_{g}, \varepsilon_{g}, p_{g} $ - are determined by the equation: $$g^{-1}h_{\varepsilon}(\zeta)=h_{\varepsilon_{g}}(\zeta_{g})p_{g}.\label{eq:202}$$ Thus, we have found the realization of the $ UIR $’s of the continuous main series of the group $ SO(4,1) $ in the space $ \mathfrak{H}_{1} \oplus \mathfrak{H}_{2} $.
Consider now the Wigner-Inonu limit at which the $ UIR $’s of the group $ SO(4,1) $ goes into the $ UIR $’s of the group $ ISO(3,1) $. For this, it is necessary to know the limit of the parameters of the group $ SO(4,1) $, under which the de Sitter group passes to the Poincaré group $ ISO(3,1) $.
Since the $ SO(4,1) $ is a group of motions leaving an invariant quadratic form It is clear that to find the required limit it is sufficient to consider only transformations in the planes $ (\xi_{0}, \xi_{j}), \; j = 1,2,3,4 $. For example, consider the transformation: $$\xi_{0}^{'}=\xi_{0}\cosh\alpha+\xi_{4}\sinh\alpha,\qquad \xi_{4}^{'}=\xi_{0}\sinh\alpha+\xi_{4}\cosh\alpha.$$ Since the de Sitter space when $ R \rightarrow \infty $ transforms to Minkowski space, we redefine parameter $ \alpha $, putting $ \alpha = a_ {0} /R $. Since $$\xi_{4}=\sqrt{\xi_{0}^{2}-\xi_{1}^{2}-\xi_{2}^{2}-\xi_{3}^{2}+R^{2}},\label{eq:3119}$$ we obtain: $$\begin{aligned}
\xi_{0}' & = & \xi_{0}\cosh\left(\frac{a_{0}}{R}\right)+\sqrt{\xi_{0}^{2}-\xi_{1}^{2}-\xi_{2}^{2}-\xi_{3}^{2}+R^{2}}\sinh\left(\frac{a_{0}}{R}\right)\xrightarrow{R\rightarrow\infty}\xi_{0}+a_{0},\\
\xi_{4}' & \rightarrow & \xi_{4}.
\end{aligned}$$ Thus, we see that the transformations $ g_{04} \left (\alpha \right), \, g_{i4} \left (a_{i} \right), \, i = 1,2,3 $ in the planes $ \left (\xi_{n}, \, \xi_{4} \right), \, n = 0,1,2,3 $ go to the translation group if we assume: $$\alpha=a_{0}/R,\;\varphi_{i}=a_{i}/R,\;i=1,2,3\qquad\text{if}\quad R\rightarrow\infty.$$
Thus we have the statement: the transformations of the subgroup $ SO(3,1) \subset SO(4,1) $, leaving the vertex of the one-sheeted hyperboloid $ [k, k] = -R^{2} $ (i.e. the de Sitter world) transform into homogeneous Lorentz transformations of the Minkowski world, and the rotation $ g_ {n4} (\alpha_ {n}) $ on the planes $ (k_{4}, k_{n}), \; n = 0,1,2,3 $ translate into translations along the corresponding axes of a Cartesian coordinate system on the quantities: $ a_{n} = \alpha_{n} R $.
The result of contraction of the representation - of the group $SO(4,1)$ is formulated as a theorem:
The result of the contraction of the UIR’s, $ T ^ {(\sigma, s)} (g), \: g \in SO (4,1), \sigma = -3/2 + imR, $ for $ R \rightarrow \infty $ is the direct sum of UIR, $ U ^ {(m, s; \varepsilon)} (g), \: g \in ISO (3,1) $, with mass $ m $, spin $ s $ and differing in energy sign $ \varepsilon $.
The proof of the theorem is given in the full version of the article, [@Caucasus-3].
This conclusion is the same as the result of [@Strom1965], obtained using infinitesimal transformations.
Conclusion
==========
So let’s sum up ...
In the flat world of Minkowski, Wigner’s elementary systems are determined by the rest mass, the spin and the sign of their energy can be identified with elementary particles. Considerations of the stability of physical systems force us to limit the spectrum of energy from below and to exclude negative energies.
In the de Sitter world, elementary Wigner systems are identified by spin and by a parameter, which is the flat limit of a function of spin and mass, with different energy signs. But unlike the Minkowski world, we can not exclude negative energies from consideration.
That is, elementary systems on a cosmological scale can be in states with positive and negative energies. Elementary systems in a state with positive energy behave like a gravitating mass, and in a negative energy state as an anti-gravity mass.
Thus, we arrive at the following conclusions:
1. Mysterious “dark matter” and “dark energy” consist of such elementary systems.
2. “Dark matter” and “dark energy” are the first manifestations of quantum properties on the scale of the universe. Until now, quantum phenomena have been encountered in the micro-world, and also as macroscopic quantum effects in the theory of condensed matter.
3. “Dark matter” and “dark energy” are carriers of information about the first moments of the universe after the Big Bang.
The last conclusion follows from the fact that according to the standard cosmological model, the de Sitter world is a necessary phase of the evolution of the universe in the first instants ($ 10^{-34}-10^{-32} $ seconds) after the Big Bang.
Of course, our universe is not de Sitter’s world, although according to some data it is developing in the direction of this model. Considering the given phenomena of dark matter and dark energy in the general case is a difficult task because today there is no quantum theory of gravity. The solution of this problem in general for the gravitational field requires not only new physical concepts but also new mathematics.
Appendix: Parametrization of elements of the group $ SO(4,1) $ {#appendix-parametrization-of-elements-of-the-group-so41 .unnumbered}
==============================================================
The group $ SO (4,1) $ is a connected component of the unit group of motions 5-dimensional pseudo-Euclidean space that leaves invariant quadratic form, [@Gursey]: $$\left[k,k\right]=k_{0}^{2}-k_{1}^{2}-k_{2}^{2}-k_{3}^{2}-k_{4}^{2}.$$ It is a 10-parametric, as well as the Poincaré group, which describes symmetry of Minkowski space.
Elements of $ g \in SO (4,1) $ are represented by $ 5 \times5 $ by matrices that linear transformation in the space of points $ k = \left (k_ {0}, k_ {1}, k_ {2}, k_ {3}, k_ {4} \right) $. It follows from the definition that they satisfy the following relations: $$g^{\intercal}\eta g=\eta,\quad\det g=1,\quad g_{00}\geqslant1,\label{eq:6}$$ where the symbol $ \intercal $ denotes transposition and $ \eta $ is the diagonal $ 5\times 5 $ matrix: $$\eta = \mathrm{diag}(1,-1,-1,-1,-1).\label{eq:7}$$
The matrix elements will be indexed: $$\begin{aligned}
a,b,c & = & 0,1,2,3,4;\\
i,j,k & = & 0,1,2,3;\\
\alpha,\beta,\gamma & = & 1,2,3.
\end{aligned}$$ In the latter case, vector notation will also be used: $$\vec{a}=\left\{ a^{\alpha}:\:\alpha=1,2,3\right\} .$$
Elements of the Lie algebra $ SO (4,1) $ of the de Sitter group $ \Gamma_ {ab} $ are five-row matrices with elements of the form: $$\left(\Gamma_{ab}\right)_{d}^{c}=\delta_{\:a}^{c}\eta_{bd}-\delta_{\:b}^{c}\eta_{ad}$$ and satisfy the commutation relations: $$\left[\Gamma_{ab},\Gamma_{cd}\right]=\eta_{ac}\Gamma_{bd}-\eta_{bc}\Gamma_{ad}-\eta_{ad}\Gamma_{bc}+\eta_{bd}\Gamma_{ac}$$ Here $ \delta _ {\: a} ^ {b} $ is the Kronecker symbol: $$\delta _ {\: a} ^ {b}=\begin{cases}1, \quad \text{if}\quad a=b;\\
0,\quad \text{if} \quad a\neq b.
\end{cases}$$
Since the transformations $ g \in SO (4,1) $ preserve the form $ \left [k, k \right] = k_ {0} ^ {2} -k_ {1} ^ {2} -k_ {2} ^ { 2} -k_ {3}^{2} -k_ {4} ^ {3} $, then the surfaces are: $$k_{0}^{2}-k_{1}^{2}-k_{2}^{2}-k_{3}^{3}-k_{4}^{2}=const$$ transform into themselves under transformations from the de Sitter group.
There are three types of such surfaces, namely the upper (or lower) floor of the two-sheeted hyperboloid $ \left [k, k \right] = c> 0 $, the one-sheeted hyperboloid $ \left [k, k \right] = c <0 $, and finally the upper (or lower) floor of the cone $ \left [k, k \right] = 0 $ without a vertex, since the point $ k_ {0} = k_ {1} = k_ {2} = k_ {3} = k_ {4} = 0 $ itself forms a homogeneous space.
The action of the group on each of these surfaces is transitive. As is known, transitivity surfaces are characterized by their stationary subgroup, i.e. closed subgroup leaving the selected point fixed. Used on the different coordinate systems given on these surfaces, one can obtain different parameterizations of the elements of the group.
We will consider a spherical coordinate system on a cone $[k,k]=0,\;k_{0}>0:$ $$k=\omega(1,\upsilon),$$ where $ \upsilon $ is the point of a 3-dimensional unit sphere: $$\upsilon^{2}=\upsilon_{1}^{2}+\upsilon_{2}^{2}+\upsilon_{3}^{2}+\upsilon_{4}^{2}=1,\label{eq:14}$$ $$\upsilon =(\sin\chi\sin\vartheta\sin\varphi, \sin\chi\sin\vartheta\cos\varphi, \sin\chi\cos\vartheta, \cos\chi),$$ where $0<\omega<\infty,\quad0\leq\chi\leq\pi,\quad0\leq\vartheta\leq\pi,\quad0\leq\varphi<2\pi.$
Coordinates of the selected point $\overset{0}{k}=(1,0,0,0,1):\qquad\omega=1,\;\chi=\vartheta=\varphi=0.$
Wigner’s operator: $$h(k)=g_{12}(\varphi)g_{23}(\vartheta)g_{34}(\chi)g_{04}(\beta),\quad\beta=\ln\omega\label{eq:15}$$
The stationary group $ W $ of the point $ \overset {\circ} {k} $ is isomorphic to the group motions of $ M (3) $, 3-dimensional Euclidean space $ E_ {3} $. Each element $w \in \mathfrak{W}$ can be uniquely represented as: $$w=R(\rho)B(\vec{z}\,),\label{eq:11}$$ where $ \vec {z} $ is a 3-dimensional vector, and $ \rho $ is an orthogonal matrix $ 3 \times3: $ $$\rho^{\intercal}\rho=I.$$ The matrices $ R(\rho) $ and $ B(\vec{z}\,) $ have the following forms: $$R(\rho)=\begin{pmatrix}1 & \vec{0\,}^{\intercal} & 0\\
\vec{0} & \rho & \vec{0}\\
0 & \vec{0\,}^{\intercal} & 1
\end{pmatrix},\quad R(\rho)\in SO(3)\subset SO(4,1),\label{eq:12}$$ $$B(\vec{z})=\begin{pmatrix}1+z^{2}/2 & \vec{z\,}^{\intercal} & -z^{2}/2\\
\vec{z} & I_{3} & -\vec{z}\\
z^{2}/2 & \vec{z}\,^{\intercal} & 1-z^{2}/2
\end{pmatrix},\quad z^{2}=\vec{z\,}^{2}.\label{eq:13}$$ The matrices $ B (\vec {z}\,) $ form an abelian subgroup that is the normal subgroup of the stationary subgroup $ \mathfrak{W}. $
For $ g \in SO (4,1) $ we have: $$g=h(k)w.\label{eq:19}$$
Hence, using the formulas -, we obtain the Iwasawa expansion for the elements of the group $ SO(4,1): $ $$g=ug_{04}(\beta)B(\vec{z}\,),\quad u\in SO(4).\label{eq:20}$$
In doing so, we used the commutativity of the transformations $ g_ {04} (\beta) $ and $ R (\rho) $, and the fact that the group is locally $ SO (4) \simeq SO (3) \otimes SO(3) $.
Here it is necessary to make two comments:
1. The action of the group $ SO(4,1) $ on the cone induces a nonlinear mapping section of the cone $ \omega = 1 $, i.e. 3-dimensional unit sphere to itself. It is easy to verify that this mapping is a homeomorphism. Thus, the 3-sphere is a homogeneous space, where the group $ SO (4,1) $ acts nonlinearly. For this space Wigner operator $ h (\upsilon) $ can be defined as follows: $$h(\upsilon)=h(k)|_{\omega=1}.\label{eq:21}$$
2. The mappings and are Borel. Moreover, they are differentiable almost everywhere.
On the cone $ [k, k] = 0, \; k_ {0}> 0 $ we can introduce one more coordinate system: $$\begin{aligned}
k_{0} & = & \tau\frac{1+a^{2}}{2},\nonumber \\
\vec{k} & = & \tau\vec{a},\label{eq:22}\\
k_{4} & = & \tau\frac{1-a^{2}}{2},\nonumber
\end{aligned}$$ where $a=\vec{a\,}^{2},\quad\tau>0.$
Since $ k_ {0} + k_ {4}> 0 $, this coordinate system does not cover the whole cone, namely, outside the coordinate system there remains the intersection of the cone by a hyperplane $ k_ {0} + k_ {4} = 0 $, which forms a set of smaller dimension. It is obvious that the corresponding parametrization of the group will also not be global. The set of points $ k_ {0} + k_ {4}> 0 $ together with the map draws a map on the cone, which we denote by through $ \mathbb {C} _ {1}. $
The coordinates of the fixed point $ \overset {\circ} {k} = (1/2, \vec {0}, 1/2) $: $ \tau = 1, \; \vec {a} = \vec {0}. $
Wigner operator: $$h(k)=A(\vec{a})D(\tau),\label{eq:23}$$ where $$D(\tau)=g_{04}(\alpha)=\begin{pmatrix}\cosh\alpha & \vec{0\,}^{\intercal} & \sinh\alpha\\
\vec{0} & I_{3} & \vec{0}\\
\sinh\alpha & \vec{0\,}^{\intercal} & \cosh\alpha
\end{pmatrix},\qquad\tau=e^{\alpha}\label{eq:24}$$ $$A(\vec{a})=\begin{pmatrix}1+a^{2}/2 & \vec{a\,}^{\intercal} & a^{2}/2\\
\vec{a} & I_{3} & \vec{a}\\
-a^{2}/2 & -\vec{a\,}^{\intercal} & 1-a^{2}/2
\end{pmatrix}\label{eq:25}$$
The matrices $ A (\vec {a}\,) $, as well as the matrices $ B (\vec {z}\,) $ from form a 3-parameter abelian subgroup. Since, stationary subgroup is the group $\mathfrak{W}$ described in the previous subsection from and we get the expansion, which is valid *almost for all* $ g \in SO(4,1): $ $$g=A(\vec{a}\,)D(\tau)R(r)B(\vec{b}\,)\label{eq:26}$$
In order to obtain a parametrization for any element of the group, it is necessary to have an atlas on the cone without vertex.
The next map of $ \mathbb {C} _ {2} $, consisting of the region $k_ {0} -k_ {4}> 0$ and mapping: $$\begin{aligned}
k_{0} & = & \lambda\frac{1+x^{2}}{2},\label{eq:27}\\
\vec{k} & = & \lambda\vec{x},\nonumber \\
k_{4} & = & -\lambda\frac{1-x^{2}}{2};\nonumber \\
x^{2} & = & \vec{x\,}^{2},\quad\lambda>0.\nonumber
\end{aligned}$$ complements the map $ \mathbb {C} _ {1} $ to the atlas.
Now we see that outside the coordinate system $\mathbb {C} {} _ {2}$ remains the intersection of the cone with the hyperplane $k_{0}-k_{4} = 0$ , and that the maps $\mathbb {C} {}_{1}$ and $\mathbb {C} {} _ {2}$ make an atlas on the cone. The transition functions from one card to another are as follows: $$\begin{aligned}
\tau & = & \lambda x^{2};\qquad\:\lambda=\tau a^{2},\label{eq:28}\\
\vec{a} & = & \vec{x}/x^{2};\qquad\vec{x}=\vec{a}/a^{2}.\nonumber
\end{aligned}$$
The formulas show that we have established a smooth structure on the cone.
Now we need to construct the Wigner operator corresponding to the map . From - it follows that the matrix $ \eta \in SO(4,1) $, so you can enter the following boost: $$h_{2}(\vec{k\,})=B(\vec{x\,})\eta D(\lambda).\label{eq:29}$$
Thus, we obtain the following parametrization of $SO(4.1)$: $$g=B(\vec{x\,})\eta D(\lambda)R(r)B(\vec{y\,}).\label{eq:30}$$
It follows from the corresponding statements for the cone that the parameterizations and cover the entire group space. We also get that the expansion is valid for all $ g \in SO(4,1) $, except for the elements $ \left\{g \in SO(4,1): \: \vec {x} = 0 \right\} $, and is valid for all $ g \in SO (4,1) $, except for elements $ \left\{g \in SO(4,1): \: \vec {a} = 0 \right\} $.
Here it is necessary to make a few remarks:
1. The elements $ \vec {a} = 0 $, i.e. elements of the form $$p=D(\tau)R(r)B(\vec{b})\label{eq:31}$$ constitute a subgroup, namely, a minimal parabolic subgroup $ \mathsf{P} $. This subgroup plays an important role in the construction of induced representations, [@Mensky].
Accordingly, the expansions and can now be rewritten as: $$g=A(\vec{a})p_{1},\qquad\qquad p_{1}=D(\tau)R(r)B(\vec{b});\label{eq:33}$$ $$g=B(\vec{x})\eta p_{2};\qquad\qquad p_{2}=D(\lambda)R(r)B(\vec{y}).\label{eq:34}$$
2. Below are the formulas for the transition between these parameterizations: $$\begin{aligned}
p_{1} & = & D(x^{2})R\left(\pi;\overset{\circ}{\vec{x}}\right)B\left(-\frac{\vec{x}}{x^{2}}\right)p_{2};\qquad\qquad\vec{a}=\vec{x}/x^{2},\nonumber \\
p_{2} & = & D(a^{2})R\left(\pi;\overset{\circ}{\vec{a}}\right)B\left(\frac{\vec{a}}{a^{2}}\right)p_{1};\qquad\qquad\;\vec{x}=\vec{a}/a^{2}.\label{eq:35}
\end{aligned}$$
Here, we denote by $ R \left (\pi; \cdot \right) $ the rotation by an angle $ \pi $ around the vectors $ \overset {\circ} {\vec {x}} = \vec {x} / x $ and $ \overset {\circ} {\vec {a}} = \vec {a} / a $, respectively.
It is easy to see that $ \mathsf{P} $ is a stationary subgroup the north pole of the 3-sphere. Therefore: $$S^{3}\simeq SO(4,1)\diagup\mathsf{P}.$$
3. Maps $ \mathbb {C}_{i}, \: i = 1,2 $ induce on a submanifold $\omega = 1$ cone, i.e. on the 3-dimensional unit sphere, the local maps $ \mathbb{V} _ {i}, \, i = 1,2 $, respectively. The map $ \mathbb{V}_{1} $ (respectively, $ \mathbb{V}_{2} $) consists of points of a sphere with a punctured southern (respectively, northern) pole and coordinate systems: $$\upsilon_{\alpha}=\frac{2a_{\alpha}}{1+a^{2}},\qquad\qquad\upsilon_{4}=\frac{1-a^{2}}{1+a^{2}};\label{eq:36}$$ respectively, $$\upsilon_{\alpha}=\frac{2x_{\alpha}}{1+x^{2}},\qquad\qquad\upsilon_{4}=-\frac{1-x^{2}}{1+x^{2}}.\label{eq:37}$$
As can be seen, these coordinates coincide with the stereographic projection 3-dimensional sphere from its poles to $ E_ {3}. $
4. The Wigner operators $ h_ {i} (\upsilon), \: i = 1,2 $ corresponding to the maps $ \mathbb {V} _ {i} $, will be defined as follows: $$h_{1}(\upsilon)=A(\vec{a}),\qquad\qquad\qquad h_{2}(\upsilon)=B(\vec{x})\eta.\label{eq:38}$$
5. The mappings , and are infinitely differentiable.
In conclusion this section, we give some information about invariant measures on the group $ SO (4,1) $ and its minimally parabolic subgroup. The de Sitter group, like any connected semisimple group, unimodular, i.e. on it there exists a two-sided invariant measure Haar $ dg $. Below are the expressions for $ dg $ in different parameterizations (the numbers in parentheses on the left indicate the numbers of the corresponding parameterizations): $$\begin{aligned}
\eqref{eq:20}: \ \quad\qquad\qquad\qquad e^{3\beta}dud\beta(d\vec{z\,}),\nonumber\\
\eqref{eq:33}: \qquad\qquad\qquad\tau^{2}d\tau(d\vec{a})dr(d\vec{b\,}),\label{eq:39}\\
\eqref{eq:34}: \qquad\qquad\qquad\lambda^{2}d\lambda(d\vec{x})dr(d\vec{y\,}),\nonumber
\end{aligned}$$
where
$\qquad du $ is an invariant normalized measure on $ SO (4), $
$\qquad dr $ is an invariant normed measure on $ SO (3); $
$\qquad (d \vec {a\,}), \:( d \vec {b\,}), \:( d \vec {x\,}), \:( d \vec {y\,}) $ - volume elements in 3-dimensional Euclidean space.
The subgroup $ \mathsf {P} $, in contrast to the group $ SO(4,1) $, is not unimodular. Below are the expressions for the right-invariant measures $ d_ {r}(p) $, the left-invariant measure $ d_ {l}(p) $, and also the module $ \delta (p) $: $$\begin{aligned}
\delta(p) & = & \tau^{3},\nonumber \\
d_{l}(p) & = & \frac{1}{\tau}d\tau dr\left(d\vec{b}\right),\label{eq:40}\\
d_{r}(p) & = & \tau^{2}d\tau dr\left(d\vec{b}\right)=\delta(p)d_{l}(p).\nonumber
\end{aligned}$$
Finally, from the comparison with we get: $$\begin{aligned}
dg & = & dh_{i}d_{r}(p),\qquad i=1,2;\nonumber \\
dh_{1} & = & \left(d\vec{a}\right),\label{eq:42}\\
dh_{2} & = & \left(d\vec{x}\right).\nonumber
\end{aligned}$$
[99]{} Wigner-Inönü, Proc. Nat. Acad. Sci.(USA), **39**, 510, 1953; **40**, 119 (1954). A.O. Barut, R. Raczka, (1977), Theory of Group Representations and Applications, Warsaw. Ström S., Arkiv für Fysik, **40**, 1, 1961. Ström S., Arkiv für Fysik, **30**, 5, 1965. Michelsson J., Niederle J., Commun.Math.Phys., **27**, 3, 1972. Drechsler W. Group contraction in a fiber bundle with Cartan connection - J.Math.Phys., 1977, v.18, No.7, p.1358-1366. B.A. Rajabov, Trans. Nat. Acad. Sci., Azerbaijan, series of phys.-math. sciences, No.**6**, (1983), pp.58-63; (in Russian). N.A. Gromov, Contractions of the classical and quantum groups, Fizmatlit, Moscow, 2012 (in Russian) Nachtmann G., Commun.Math.Phys., **6**, 1, 1967. Nachtmann G., Zeit. für Physik, **208**, 113-115, 1968. Börner O., Dürr V., Nuovo Cimento, **64A**, 669, 1969. V.A. Ryabov, V.A. Tsarev, A.M. Tskhovrebov, The search for dark matter particles, Uspekhi Fizicheskikh Nauk 178 (11) 1129 – 1163 (2008) (in Russian) Martín López-Corredoira, Tests and problems of the standard model in Cosmology, arXiv:1701.08720v1 \[astro-ph.CO\], 2017, p.64 The XENON1T Dark Matter Experiment, arXiv:1708.0751v1 \[astro-ph.IM\], 2017, P.22 Dark Energy Survey Year 1 Results, arXiv: 1708.01530v1 \[astro-ph.CO\], 2017, p.31 New Ideas in Dark Matter 2017, arXiv: 1707.0459v1 \[hep-oh\], 2017, p.113 Gerlach W., Stern O. Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld. Zeitschrift für Physik, 9:349–352 (1922). G. E. Uhlenbeck, S. Goudsmit. Spinning Electrons and the Structure of Spectra, Nature,–1926, v.117, p.264–265. Thomas L. H. The motion of the spinning electron, Nature.–1926, v.117, p.514. Schiff, L.I.(1968), Quantum Mechanics, 3rd edition, McGraw-Hill, New York. J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics, Mc Graw – Hill Book Company, 1964, 312 p. E. Wigner, On unitary representations of the inhomogeneous Lorentz group, Nuclear Physics B(Proc.Suppl.), ****6, (1989), 9-64; Ann. Math., **40**, 149 (1939) V. Bargmann, E.P. Wigner, Group theoretical discussion of relativistic wave equations, Proc.Nat.Acad.Sci.(USA), **34**(5), (1948), 211-223. A. Einstein, Collected works, vol. 1-2, Nauka, Moscow, 1965-1966 (in Russian) N. Straumann, The history of the cosmological constant problem, arXiv:gr-qc/0208027v1, (2002), p.12 C. O’Raifeartaigh, M. O’Keeffe, W. Nahm and S. Mitton, One Hundred Years of the Cosmological Constant: from “Superfluous Stunt” to Dark Energy, arXiv:1711.06890 \[physics.hist-ph\], (2017), p.62 S. Weinberg, Gravitation and Cosmology, John Wiley and Sons, Inc., New York–London–Sydney–Toronto, 1972 Møller C., The theory of relativity, 2nd ed., Clarendon Press, Oxford, 1972 R.C. Tolman, Relativity, Thermodynamics and Cosmology, Clarendon Press, Oxford, 1969 F.Gürsey, Introduction to Group Theory, in “Relativity, Groups and Topology”, eds. C. DeWitt, B. DeWitt, New York-London, 1964. Rajabov B.A. Non-degenerate representations of the group de Sitter $SO(4,1)$ and the contraction of their matrix elements - Preprint **1**, Institute of Physics of Azerbaijan National Academy of Sciences, Baku, (1979), p.20, (in Russian) Rajabov B.A. Wigner coefficients of non-degenerate representations of the group de Sitter - Preprint **5**, Institute of Physics of Azerbaijan National Academy of Sciences, Baku, (1979), p.9, (in Russian) Kirillov A.A. Elements of the theory of representations, 2nd ed., Nauka, Moskow, 1978, p.344, (in Russian) Mackey G.W., On Induced Representations of Groups, American Journal of Mathematics, Vol. 73, No. 3 (Jul., 1951), pp. 576-592 M.B. Mensky, The method of induced representations: space-time and particle’s concept, Nauka, Moscow, 1976, p.288, (in Russian) B.A. Rajabov, Astronomy&Astrophysics (Caucasus), International Scientific Journal, **3**, (2018), Samtskhe-Javakheti State University Press, Georgia, (in the publication).
[^1]: $^1$ N.Tusi Shamakhi Astrophysics Observatory, National Academy of Sciences of Azerbaijan;\
e-mail: balaali.rajabov@mail.ru.
[^2]: Thus, the justification of the spin by the Pauli and Dirac equations is of historical significance. Rephrasing Oscar Wilde, we can say that if Wigner’s work had appeared earlier, the history of quantum electrodynamics would have evolved differently.
[^3]: In 1947, Wigner and Bargmann, on the basis of the theory of representations of the Poincaré group, classified the relativistic equations for spin particles, [@Bargmann]. That is, first the spin, and then the equations ...
[^4]: The values of $ G $ and $ \varLambda $ are constantly refined with the accumulation of observations:
$ G=6,67545\frac{sm^{3}}{g \cdot sec^{2}} $,(2013);$ \Lambda \sim 10^{-53}m^{-2},\quad (1998) $.
[^5]: It is easy to see that for $ \Lambda\rightarrow 0 $, i.e. $ R\rightarrow\infty $ both solutions (4)-(5) are transferred to the flat world of Minkowski.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Results from the HERMES experiment are presented on single-spin asymmetries in semi-inclusive hadron production from longitudinally polarized targets. The data are compared with a number of theoretical calculations which relate the azimuthal dependence of the asymmetries to the transversity structure function $h_1(x)$.'
address: 'University of Illinois, 1110 W Green St., Urbana, IL, USA 61801-3080'
author:
- 'N.C.R. Makins, for the HERMES Collaboration'
title: 'Transverse Spin: HERMES Results and Future Plans'
---
TRANSVERSITY AND FRIENDS
========================
It has come to light that a complete description of the quark distributions in the proton at leading twist requires not only the structure functions $f_1(x)$ and $g_1(x)$, but also the transversity distribution $h_1(x)$. This new structure function represents the degree to which the quarks are polarized along the proton’s spin direction when the proton is polarized transversely to the virtual photon. The unique properties of $h_1(x)$ were beautifully presented by R. Jaffe at this conference. The majority of this report is devoted to recent data sensitive to $h_1(x)$.
However, one may place transversity in a larger context. In 1996, Mulders and Tangerman performed a complete tree-level analysis of the semi-inclusive deep-inelastic scattering (SIDIS) cross-section, taking into consideration all measurable spin-degrees of freedom in the initial and final state [@Bible]. Their work identified a series of 8 structure functions of the proton at leading twist, along with an analogous set of 8 fragmentation functions. These functions are illustrated graphically in Fig. \[fig:stoplights\]a. In each picture, the virtual photon probe is assumed to be incident from the left, the large and small circles represent hadrons and quarks respectively, and the arrows indicate their spin directions. The notation for the fragmentation functions is obtained by replacing the letters $f$, $g$, and $h$ with $D$, $G$, and $H$ respectively.
Each of these functions describes qualitatively different information about hadronic structure and formation. The functions $f_1(x)$, $g_1(x)$, and $h_1(x)$ are the only ones which survive on integration over transverse momentum $k_T$. (The same is true of the analogous fragmentation functions.) The other functions are all implicitly dependent on intrinsic quark transverse motion, which is necessarily related to the unknown orbital angular momentum of quarks in the nucleon. New data are providing first glimpses of several functions in this table, and much more can be expected in the near future.
![ (a) Summary of the classification scheme of [@Bible] for leading-twist distribution functions. (b) Diagram of SIDIS kinematics.[]{data-label="fig:stoplights"}](mulders.green-alldist-mod.bw.eps){width="7cm"}
The Mulders decomposition of the cross-section reveals that experiments may access these new structure and fragmentation functions by measuring *azimuthal moments* in spin-dependent SIDIS. Fig. \[fig:stoplights\](b) illustrates the HERMES definition of $\phi$, the azimuthal angle of the measured final-state hadron, around the virtual photon direction, and relative to the lepton scattering plane. Specifically one must perform an azimuthal decomposition of various cross-section terms $d\sigma_\mathrm{ABC}$, where A, B, and C represent the spin-polarization of the lepton beam, proton target, and final-state hadron respectively. These subscripts take the values L, T, and U, denoting longitudinal, transverse, and no polarization. The spin of the final-state hadron can usually be accessed only by $\Lambda$-production measurements: the weak decay $\Lambda \rightarrow p {\ensuremath{\pi^-} }$ allows the hyperon’s spin to be determined from the angular distribution of its decay products.
THE HERMES EXPERIMENT
=====================
The HERMES experiment has been taking data at the HERA accelerator in Hamburg, Germany since 1995. HERMES scatters longitudinally polarized electron and positron beams of 27.6 GeV from polarized gas targets internal to the beam pipe. Pure atomic H, D, and $^3$He have been used (as well as a variety of unpolarized nuclear targets). Featuring polarized beams and targets, and an open-geometry spectrometer with good particle identification (PID), HERMES is well suited to a study of the spin-dependent azimuthal moments of the SIDIS cross-section. The PID capabilities of the experiment were significantly enhanced in 1998 when the threshold Čerenkov detector (used to identify pions above a momentum of 4 GeV) was upgraded to a Ring Imaging system (RICH). This new detector provides full separation between charged pions, kaons, and protons over essentially the entire momentum range of the experiment.
In September 2000 HERMES completed its first phase of data taking, using longitudinally-polarized targets. The 1998-2000 period was particularly successful, yielding a very large data set ($>$ 8 million DIS events) from polarized deuterium. HERMES is now entering its second running phase which will continue until 2006. A cornerstone of this period will be measurements from transversely-polarized targets, with the specific goal of exploring the transversity structure function. Prospects for Run 2 were described in the talk of K. Rith at this conference.
HERMES MEASUREMENTS OF [$A_\mathrm{UL}(\phi)$ ]{}
=================================================
HERMES has measured the azimuthal distribution of charged [@HERMEScharged] and neutral [@HERMESneutral] pions in the scattering of “unpolarized” positrons from a longitudinally-polarized hydrogen target. (As the HERA beam is always polarized, an “unpolarized” sample was achieved by the helicity-balancing of data collected with opposite beam polarization.) The measured quantity is the following single-spin asymmetry (SSA): $${\ensuremath{A_\mathrm{UL}(\phi)} }= \frac{1}{P_T}
\frac{N^+(\phi) - N^-(\phi)}{N^+(\phi) + N^-(\phi)}.$$ Here $P_T$ is the target polarization and $\phi$ is the azimuthal angle of the pion described above. $N^+$ and $N^-$ represent the pion yields in each of the two target spin orientations, with the superscript indicating the helicity of the target in the reaction’s center-of-mass frame (*i.e.* $N^+$ is collected when the target spin is antiparallel to the lepton beam momentum in the lab frame). This choice is motivated by the familiar definition of the double-spin asymmetry $A_\mathrm{LL}$. It is also important to note that the HERMES definition of $\phi$ is *not* the same as that of the “Collins angle” $\phi_C$ which appears in a number of publications.
The measured asymmetries show a significant $\sin\phi$ moment in the case of [$\pi^+$ ]{}and [$\pi^0$ ]{}production, while no $\phi$-dependence is seen in [$\pi^-$ ]{}production. The $\sin 2\phi$ moments of the asymmetries were also analyzed and found to be consistent with zero in all cases. The mean $Q^2$ of these measurements is around 1.6 GeV$^2$.
At “zeroth-order” in complexity, the asymmetry moment [$A_\mathrm{UL}^{\sin\phi}$ ]{}(also termed analyzing power) is related to the product of transversity $h_1(x)$ and the Collins fragmentation function $H_1^\perp(z)$. This latter function provides a “polarimeter” for initial-state quark polarization: it correlates the transverse spin of the struck quark with the angular distribution of hadrons in the jet it generates. Fig. \[fig:azimkinH\] shows the dependence of [$A_\mathrm{UL}^{\sin\phi}$ ]{}on the Bjorken scaling variable $x$, the energy fraction $z \equiv E_h/\nu$ of the pion, and its transverse momentum $p_T$ relative to the virtual photon direction. The dependences match the qualitative predictions of Collins [@Collins] that the effect should peak in the valence region $x \simeq 0.3$, rise with $p_T$ up to a maximum at some hadronic scale $0.3\,\textrm{--}\,0.9$ GeV, and be larger for [$\pi^+$ ]{}than [$\pi^-$ ]{}production.
![Kinematic dependences of [$A_\mathrm{UL}^{\sin\phi}$ ]{}measured from a hydrogen target [@HERMEScharged; @HERMESneutral].[]{data-label="fig:azimkinH"}](delia.May01.AULsinphi-allpi-alldep.eps){width="85.00000%"}
The reason for this last expectation can be shown by a simple calculation. It is reasonable to guess that the transverse quark polarization ($\delta q \equiv h_1^q$) is similar to the longitudinal quark polarization ($\Delta q \equiv g_1^q$) in the sense $\delta d / \delta u \approx \Delta d / \Delta u \approx -1/2$. One may also estimate that the favoured and disfavoured Collins fragmentation functions have a similar ratio as in the unpolarized case ($r \equiv D_d^{{\ensuremath{\pi^+} }} / D_u^{{\ensuremath{\pi^+} }} \approx (1-z)/(1+z) \approx 1/3$). One then arrives at these simple estimates: $$A_p^{\pi^+} \approx \frac{4 \,\delta u + r \,\delta d}{4\,u + r\,d}
\approx \frac{\delta u}{u},
\hspace*{1.5em}
A_p^{\pi^0} \approx \frac{4 \,\delta u + \,\delta d}{4\,u + d}
\lesssim \frac{\delta u}{u},
\hspace*{1.5em}
A_p^{\pi^-} \approx \frac{4\,r \,\delta u + \,\delta d}{4\,r\,u + d}
\approx 0
\label{equ:envelopeH}$$ The expectation of similar [$\pi^+$ ]{}and [$\pi^0$ ]{}asymmetries is also supported by the data.
HERMES has recently completed an analysis of [$A_\mathrm{UL}$ ]{}from the deuterium-target data collected in the years 1998 to 2000. In Fig. \[fig:azimkinD\] preliminary results are presented for the analyzing power [$A_\mathrm{UL}^{\sin\phi}$ ]{}for charged pion production. A simple calculation of the type given in Eq. \[equ:envelopeH\] leads to the expectation that [$A_\mathrm{UL}^{\sin\phi}$ ]{}from deuterium should be roughly the same for $\pi^+$, $\pi^-$, and $\pi^0$ production, and around half as large as for [$\pi^+$ ]{}production from hydrogen. These qualitative expectations are indeed borne out by the data. Spin-azimuthal asymmetries for charged kaon production have also been measured for the first time, as shown in the right-hand panel of Fig. \[fig:azimkinD\].
![HERMES preliminary results on the analyzing power [$A_\mathrm{UL}^{\sin\phi}$ ]{}for charged pion and kaon production from a deuterium target.[]{data-label="fig:azimkinD"}](delia.Feb02.AsinphiUL-Dtopi-x.eps "fig:"){width="31.00000%"} ![HERMES preliminary results on the analyzing power [$A_\mathrm{UL}^{\sin\phi}$ ]{}for charged pion and kaon production from a deuterium target.[]{data-label="fig:azimkinD"}](delia.Feb02.AsinphiUL-Dtopi-pt.eps "fig:"){width="31.00000%"} ![HERMES preliminary results on the analyzing power [$A_\mathrm{UL}^{\sin\phi}$ ]{}for charged pion and kaon production from a deuterium target.[]{data-label="fig:azimkinD"}](schill.Feb02.AsinphiUL-DtoK+-x.eps "fig:"){width="31.00000%"}
Modelling [$A_\mathrm{UL}(\phi)$ ]{}
------------------------------------
For a more sophisticated interpretation of the data, one must consider in detail which terms in the SIDIS cross-section of ref. [@Bible] contribute to the [$A_\mathrm{UL}(\phi)$ ]{}asymmetry. It is essential to realize that “longitudinal” and “transverse” target polarization have different meanings in experimental and theoretical contexts. In experimental papers, a longitudinal target is polarized along the *lepton beam* direction, but in theoretical contexts the relevant axis is the *virtual photon* direction. To distinguish the two, we use the notation $A_\mathrm{ABC}^w$ for moments $w$ of asymmetries in the experimental convention, and the notation $\langle w \rangle_\mathrm{ABC}$ of ref. [@Bible] to denote moments of the cross-section in the theoretical convention. The HERMES measurement of [$A_\mathrm{UL}^{\sin\phi}$ ]{}can thus be expressed as follows: $${\ensuremath{A_\mathrm{UL}^{\sin\phi}} }= \frac{
S_L\,\langle \sin\phi \rangle_\mathrm{UL} +
S_T\,\langle \sin\phi \rangle_\mathrm{UT}}
{S\,\langle 1 \rangle_\mathrm{UU}}.$$ Here $S_L$ and $S_T$ represent the longitudinal and transverse projections of the target polarization $S$ onto the virtual photon direction. At HERMES $S_L/S \approx 1$, and $S_T/S \approx 1/Q$.
The full theoretical decomposition of [$A_\mathrm{UL}^{\sin\phi}$ ]{}and [$A_\mathrm{UL}^{\sin 2\phi}$ ]{}involves a number of functions at both leading and higher twist: $$\langle 1 \rangle_\mathrm{UU} \sim f_1(x)\,D_1(z),
\hspace*{1cm}
\langle \sin\phi \rangle_\mathrm{UT} \sim
h_1(x)\,H_1^{\perp(1)}(z),
\hspace*{1cm}
\langle \sin 2\phi \rangle_\mathrm{UL} \sim
h_{1L}^{\perp(1)}(x)\,H_1^{\perp(1)}(z),$$ $$\langle \sin\phi \rangle_\mathrm{UL} \sim \frac{1}{Q} [
h_{1L}^{\perp(1)}(x)\,H_1^{\perp(1)}(z) \ \oplus\
\tilde{h_L}(x)\,H_1^{\perp(1)}(z) \ \oplus\
h_{1L}^{\perp(1)}(x)\,\tilde{H} ]
\label{eq:breakdown}$$ All kinematic prefactors have been suppressed in these expressions for brevity. The superscript $(1)$ appearing on several of the functions denotes a $k_T^2$-weighted integral over transverse momentum.
The $\langle \sin\phi \rangle_\mathrm{UT}$ moment is directly proportional to the product of transversity and the Collins function. However the present HERMES measurement is most directly related to $\langle \sin\phi \rangle_\mathrm{UL}$, which is much more complex: it is sub-leading in $Q$, and contains the interaction-dependent twist-3 functions $\tilde{h}_L$ and $\tilde{H}$. In addition the as-yet-unknown leading-twist distribution function $h_{1L}^\perp(x)$ makes an appearance (see Fig. \[fig:stoplights\]).
It is necessary to make a few assumptions in order to proceed with modelling of the asymmetry. The unknown functions $\tilde{h}_L$ and $h_{1L}^\perp$ are not unrelated; from Lorentz and rotational invariance one may derive the relation $$h_{1L}^{\perp(1)}(x) = x^2 \int_x^1 \frac{dy}{y^2} \left[
-h_1(y) + \tilde{h}_L(y) + \frac{m_q}{M_p}\frac{g_1(y)}{y} \right] .
\label{eq:Lorentz}$$ Two approximations are commonly made in the literature. The first is the “reduced twist-3 approximation”, which assumes that all twist-3 terms that are interaction-dependent or suppressed by the quark-mass $m_q$ are zero. The second choice is to set the distribution function $h_{1L}^\perp$ to zero. This last option is motivated by the HERMES measurement of ${\ensuremath{A_\mathrm{UL}^{\sin 2\phi}} }\approx 0$, as [$A_\mathrm{UL}^{\sin 2\phi}$ ]{}is proportional to $h_{1L}^\perp$ (Eq. \[eq:breakdown\]). With one of the functions $h_{1L}^\perp$ or $\tilde{h}_L$ set to zero, the other can be calculated using Eq. \[eq:Lorentz\] and a model for transversity.
The Collins fragmentation function is also unknown. Its kinematic shape is typically taken from the heuristic parametrization of ref. [@Collins], $$\frac{H_1^\perp(z,k_T)}{D_1(z)} = \eta \frac{M_C M_h |k_T|}{M_C^2 + k_T^2},
\label{equ:collinsparam}$$ with some normalization constant $\eta$ and hadronic scale $M_C \approx 2\,m_\pi\,\textrm{---}\,M_p$. These parameters may be constrained using hadron-production data from DELPHI: a quark and antiquark produced from $Z^0$ decay carry small but highly-correlated transverse polarizations. A pioneering analysis of these data [@DELPHI] has yielded the estimate ${\ensuremath{|H_1^\perp / D_1|} }= 6.3 \pm 1.7$ %. (Very recently, the value has been amended to $12.5 \pm 1.4$ % [@EfremovErratum].) Unfortunately this result is an average value integrated over a rather poorly defined interval in $z$. Nevertheless, it provides an important indication of the size of the Collins function, and one which is independent of uncertainties in the distribution function sector. As presented by A. Ogawa at this workshop, high statistics data from the BELLE experiment will soon be analyzed in a similar fashion to yield much more precise results on $H_1^\perp$.
Comparison with Model Calculations
----------------------------------
A variety of theoretical calculations have been performed to address the HERMES measurements of $A_\mathrm{UL}(\phi)$.
The work of ref. [@Ogan00] attempts to explain the data with a range of simple ansätze. In Fig. \[fig:ansatze\](a) and (b), the solid curves correspond to the assumption that $h_{1L}^\perp = 0$, while the dashed curves correspond to the “reduced twist-3” approximation $\tilde{h}_L = 0$. In each case, two guesses are made for the magnitude of the transversity distribution: $h_1 = g_1$, and saturation of the Soffer bound $h_1 = (f_1 + g_1)/2$. The calculations show that the majority of the asymmetry (around 75%) originates from the subleading term $\langle \sin\phi \rangle_\mathrm{UL}$ associated with the longitudinal component of the target polarization, with only a quarter coming from the leading twist term $\langle \sin\phi \rangle_\mathrm{UT}$. This fact is reflected in the curves: [$A_\mathrm{UL}^{\sin\phi}$ ]{}is more sensitive to the ansatz made for the higher-twist functions (solid vs dashed curves) than to the magnitude of $h_1$ itself. Also, the data appear to indicate that the interaction-dependent twist-3 function $\tilde{h}_L$ cannot be ignored, while the unknown twist-2 function $h_{1L}^\perp$ is likely of small magnitude.
![Calculations from ref. [@Ogan00] of (a) [$A_\mathrm{UL}^{\sin\phi}$ ]{}and (b) [$A_\mathrm{UL}^{\sin 2\phi}$ ]{}for [$\pi^+$ ]{}production from a hydrogen target. Panel (c) shows the [$\chi$QSM]{} calculation from ref. [@EfremovErratum] of [$A_\mathrm{UL}^{\sin\phi}$ ]{}for [$\pi^0$ ]{}production; the L and T curves show the contributions from the longitudinal and transverse components of the target polarization.[]{data-label="fig:ansatze"}](azimscan.AULsinphi-pi+-x.eps "fig:"){height="\xxht"} ![Calculations from ref. [@Ogan00] of (a) [$A_\mathrm{UL}^{\sin\phi}$ ]{}and (b) [$A_\mathrm{UL}^{\sin 2\phi}$ ]{}for [$\pi^+$ ]{}production from a hydrogen target. Panel (c) shows the [$\chi$QSM]{} calculation from ref. [@EfremovErratum] of [$A_\mathrm{UL}^{\sin\phi}$ ]{}for [$\pi^0$ ]{}production; the L and T curves show the contributions from the longitudinal and transverse components of the target polarization.[]{data-label="fig:ansatze"}](azimscan.AULsin2phi-pi+-x.eps "fig:"){height="\xxht"} ![Calculations from ref. [@Ogan00] of (a) [$A_\mathrm{UL}^{\sin\phi}$ ]{}and (b) [$A_\mathrm{UL}^{\sin 2\phi}$ ]{}for [$\pi^+$ ]{}production from a hydrogen target. Panel (c) shows the [$\chi$QSM]{} calculation from ref. [@EfremovErratum] of [$A_\mathrm{UL}^{\sin\phi}$ ]{}for [$\pi^0$ ]{}production; the L and T curves show the contributions from the longitudinal and transverse components of the target polarization.[]{data-label="fig:ansatze"}](xqsm-corr02.fig3c-AULsinphi-pi0-LT.eps "fig:"){height="\xxht"}
In ref. [@EfremovPion], $h_1(x)$ has been calculated directly in the Chiral-Quark Soliton Model ([$\chi$QSM]{}) and used to estimate the asymmetry moments [$A_\mathrm{UL}^{\sin\phi}$ ]{}and [$A_\mathrm{UL}^{\sin 2\phi}$ ]{}in the reduced twist-3 approximation $\tilde{h}_L = 0$. The calculations agree with the HERMES measurements for both charged and neutral pions, and also indicate that the longitudinal term $\langle \sin\phi \rangle_\mathrm{UL}$ makes the dominant contribution to the effect (see Fig. \[fig:ansatze\](c) for [$\pi^0$ ]{}production, taken from the corrected analysis of ref. [@EfremovErratum]). A second [$\chi$QSM]{} calculation can be found in ref. [@Wakamatsu01]. In this work, *all* distribution functions involved in the asymmetries were calculated: $h_1$, $\tilde{h}_L$, and $h_{1L}^\perp$. The higher-twist function $\tilde{h}_L$ is found to be of significant magnitude, especially in the region $x < 0.2$, while the unknown leading-twist distribution function $h_{1L}^\perp$ is predicted to be small but non-zero. The calculations for [$A_\mathrm{UL}^{\sin\phi}$ ]{}and [$A_\mathrm{UL}^{\sin 2\phi}$ ]{}are again in agreement with the HERMES data, within experimental accuracy.
Calculations in other models have been performed [@Ma02] and obtain reasonable agreement with the measurements. All calculations agree that [$A_\mathrm{UL}^{\sin\phi}$ ]{}is dominated by higher-twist effects, due to the longitudinal target polarization. For a better understanding of transversity, the next step is clear: high precision SIDIS data on a transversely-polarized target are required.
GLOBAL ANALYSIS OF SINGLE SPIN ASYMMETRY DATA
=============================================
Earlier data from the Fermilab E704 experiment, from DELPHI, and from SMC at CERN are also potentially related to transversity. It is worthwhile to consider whether a consistent picture has emerged from theoretical analyses of these data sets.
E704 and the Sivers Function
----------------------------
About 10 years ago, the Fermilab E704 experiment measured a large analyzing power $A_N$ in the inclusive production of pions from a transversely polarized proton beam of 200 GeV and an unpolarized target [@E704]. Unexpectedly, positive and neutral pions displayed a pronounced tendency to be produced to “beam-left” (when looking downstream with the beam polarization pointing upwards). Negative pions showed a similar analyzing power, but in the opposite “beam-right” direction.
The observables $A_N$ and [$A_\mathrm{UL}^{\sin\phi}$ ]{}are odd under the application of naive time reversal, and must arise from the non-perturbative part of the cross-section. In the factorization picture of ref. [@Bible], either a T-odd fragmentation function or a T-odd distribution function must play a role. In the first case, the E704 analyzing power is sensitive to the product of transversity $h_1$ and the T-odd Collins fragmentation function $H_1^\perp$. The second possibility involves the unknown T-odd distribution function $f_{1T}^\perp(x,k_T)$ first postulated by Sivers [@Sivers], together with the familiar unpolarized fragmentation function $D_1$. Fits performed in the Collins-only [@dualfits-Collins] and Sivers-only [@dualfits-Sivers] scenarios demonstrate that the E704 data may be described with equal success by either picture.
Considerable discussion occurred at this workshop concerning the nature of the Sivers function $f_{1T}^\perp$. Given the T-even nature of the strong and electromagnetic interactions, any T-odd function must involve an interference of amplitudes [@Todd]. The most obvious way to generate such an interference is via inital- or final-state interactions. Both are possible in a hadronic-beam experiment such as E704. In deep-inelastic scattering with lepton beams, initial state interactions (and so the Sivers mechanism) should be greatly suppressed. It may thus be said that the HERMES measurement of [$A_\mathrm{UL}$ ]{}provides the first conclusive evidence that the Collins function and transversity are both non-zero and of significant magnitude.
However, other suggestions exist for the origin of $f_{1T}^\perp$, including contributions from gluonic poles [@QiuSterman] and spin-isospin interactions [@Drago]. Further experimental and theoretical work is needed to determine the magnitude of this function, whether it plays any role in lepton DIS, and whether it exists at all at leading twist. As with all other distribution functions that vanish on integration over transverse momentum, the Sivers function must be related at some level to parton orbital motion. A tantalizing physical interpretation of this function in the context of the E704 asymmetry may be found in ref. [@ChouYang].
Turning briefly to the fragmentation aspect of these measurements, the existence of the Collins function is deeply interesting in its own right. As it must arise from some interference mechanism, it teaches us that the fragmentation process possesses a large degree of *phase coherence*. Older data on transverse hyperon polarization from high-energy unpolarized experiments have already provided evidence that this is the case [@Hyperon]. They are sensitive to the “polarizing” fragmentation function $D_{1T}^\perp(z,k_T)$ which is also T-odd. It is rather surprising that such interference effects persist at high energies, given the large number of amplitudes that must be involved in inclusive or semi-inclusive hadron production.
Global Analysis of Single-Spin Asymmetries
------------------------------------------
A tentative, qualitative concensus has begun to emerge among the various analyses of single-spin asymmetry data at high energies. The available data from HERMES, E704, and SMC may all be described by the transversity distribution calculated in the [$\chi$QSM]{}, a Sivers function of zero, and a Collins function of approximate magnitude ${\ensuremath{|H_1^\perp / D_1|} }\approx$ 7% [@EfremovPion; @dualfits-Collins]. This apparent concensus is qualitative at best: the uncertainties on the present experimental data and the rather poorly defined $z$-ranges over which the data are integrated certainly leave room for the existence of the Sivers function, for example. Also, a pair of sign errors was recently uncovered in the analyses of the HERMES and DELPHI data [@EfremovErratum]. The updated estimates for the Collins function are larger than before: ${\ensuremath{|H_1^\perp / D_1|} }= 12.5 \pm 1.4$ % from the DELPHI data and ${\ensuremath{|H_1^\perp / D_1|} }= 13.8 \pm 2.8$ % from the HERMES data. It is presently unclear to what degree such sign errors affect other theoretical analyses of transversity-related measurements.
FUTURE MEASUREMENTS OF TRANSVERSITY
===================================
It is clear from Eq. \[eq:breakdown\] that the most direct access to transversity in SIDIS lies in measurements with transversely polarized targets. A precise measurement of $A_\mathrm{UT}(\phi)$, sensitive at leading twist to the product of $h_1(x)$ and $H_1^\perp(z)$, will form a cornerstone of HERMES Run 2. Besides $A_\mathrm{UT}(\phi)$, HERMES has considered other observables that may be sensitive to $h_1$. A promising candidate is the correlation between the transverse polarization of the target and of final-state $\Lambda$ baryons, described by the spin-transfer fragmentation function $H_1(z)$. If this function is of significant size, $\Lambda$ polarization can serve as a “polarimeter” for the quark spin distribution in the target. However results from HERMES indicate that the *longitudinal* spin-transfer from struck quark to $\Lambda$ ($G_1(z)$) is small in DIS at intermediate energies [@HERMESDLL]. This likely indicates that the transverse spin transfer will also be small and that $\Lambda$ polarization will not serve as a useful tactic for future transversity measurements.
It is important to note that the next round of single-spin asymmetry measurements *will* be able to distinguish between the Collins and Sivers mechanisms. The SIDIS cross-section term $d\sigma_\mathrm{UT}$ has two contributions according to ref. [@Bible]: a $\sin(\phi_h^l + \phi_S^l)$ moment proportional to $h_1 \, H_1^\perp$, and a $\sin(\phi_h^l - \phi_S^l)$ moment proportional to $f_{1T}^\perp \, D_1$. The symbols $\phi_S^l$ and $\phi_h^l$ indicate the angle of the target spin and final-state hadron momentum respectively, with respect to the lepton scattering plane. The angular dependence of the Collins term may be understood by realizing that the spin directions of the struck quark ($\phi_q^l$) and final-state quark ($\phi_{q'}^l$) are related by $\phi_q^l = \pi - \phi_{q'}^l$ [@Collins]: when the virtual photon is absorbed by the struck quark, the component of the quark’s spin parallel to the lepton scattering plane is flipped, while the perpendicular component is left unaffected. The Collins fragmentation function, which correlates hadron direction with the spin of the primary quark, produces a $\sin(\phi_h^l - \phi_{q'}^l)$ behavior. Transversity relates $\phi_q^l$ to $\phi_S^l$, giving a net $\sin(\phi_h^l + \phi_S^l)$ dependence in the cross section. The Sivers function, on the other hand, correlates quark transverse (orbital) motion with target spin. This motion is transferred directly to the final-state hadrons by $D_1$, giving a net $\sin(\phi_h^l - \phi_S^l)$ behavior.
Current HERMES data cannot distinguish between the two: $\phi_S^l$ is always zero or $\pi$ for a target polarized longitudinally in the laboratory frame. Similarly, the E704 data could not distinguish between the moments as their experimental apparatus did not permit a measurement of the jet axis around which the pions were produced. But forthcoming HERMES data with a transversely-polarized target will permit the independent variation of the target spin angle $\phi_S^l$ and the hadron angle $\phi_h^l$.
In conclusion, data collected in recent years have provided a first glimpse of the transversity distribution and the Collins fragmentation function. The data are still of modest precision, but they “make sense”: the measurements can be described by a variety of theoretical models and resonable assumptions. This baseline understanding has provided clear guidance on which measurements to perform next. The upcoming round of results from HERMES, COMPASS, RHIC-spin, and BELLE will provide much more precise information on transversity and on other new structure and fragmentation functions.
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} |
---
abstract: 'Patterns often appear in a variety of large, real-world networks, and interesting physical phenomena are often explained by network topology as in the case of the bow-tie structure of the World Wide Web, or the small world phenomenon in social networks. The discovery and modelling of such regular patterns has a wide application from disease propagation to financial markets. In this work we describe a newly discovered regularly occurring striation pattern found in the PageRank ordering of adjacency matrices that encode real-world networks. We demonstrate that these striations are the result of well-known graph generation processes resulting in regularities that are manifest in the typical neighborhood distribution. The spectral view explored in this paper encodes a tremendous amount about the explicit and implicit topology of a given network, so we also discuss the interesting network properties, outliers and anomalies that a viewer can determine from a brief look at the re-ordered matrix.'
author:
- Corey Pennycuff
- Tim Weninger
title: 'Striations in PageRank-Ordered Matrices'
---
Introduction
============
Patterns and regularities found in nature frequently inform scientific theories with numerous applications. Regularities within networks give way to the analysis of a rich set of physical phenomenon including viral propagation patterns, social behavior, gene and protein interaction, transportation flow, and so on. These networks are typically understood by computing certain high-level characteristics like the degree and eigenvalue distribution. Deeper analysis of some networks may reveal the presence of community patterns or regularities in the average path length, [*i.e.*]{}, small world networks. In these and other cases it is the presence or absence of well-defined patterns that contribute to broadly applicable theories about the nature of the phenomenon present in the network [@Ugander2013]. In many cases, the presence of regular patterns makes the discovery of suspicious or anomalous actors easier leading to improvements in anomaly detection like spam filtering, intrusion and fraud detection, health monitoring, and ecosystem disturbances [@Chakrabarti2004b].
In this paper we describe the appearance of a newly discovered striation pattern that is present in the PageRank re-ordering of the adjacency matrices of real world networks.
Networks can be easily represented as an *adjacency matrix* $A$, where a cell $A_{ij}$ is set to 1 if nodes $i$ and $j$ are connected by an unweighted edge, or some weight $w$ in the weighted case, and 0 otherwise. If the graph is undirected then the matrix will be symmetric. The number of entries in a row in the adjacency matrix determines the [*outdegree*]{} of the corresponding node, and the number of entries in the column in the adjacency matrix determines the [*indegree*]{} of the corresponding node. In an undirected graph their is no concept of in- versus out-degree, therefore only the total [*degree*]{} of a node is relevant in a symmetric adjacency matrix. Edges define the topology of the network, therefore a visual rendering of the adjacency matrix often leads to meaningful analysis of the network topology [@bertin1973semiologie; @kang2011spectral; @mueller2007interpreting]. The network topology also informs a wide variety of graph processes that can be used to help rank and order the vertices in the network. One such graph process is random walk, which is a stationary process on undirected graphs, but may not be stationary on directed graphs because of the possible presence of nodes with 0 outdegree, [*i.e.*]{}, dangling nodes. In order to make a random walk stationary on a directed graph the random walker must have an opportunity to leave from a dangling end [@Perra2008].
The PageRank algorithm provides an elegant solution to this problem. In addition to the diffusion process modelled by randomly walking from node to node, PageRank adds a teleportation probability that stochastically jumps the walker to a random node. This function can be described as:
$$\label{eq:pr}
p(i) = \frac{1-d}{n} + d \sum_{j\in A_i}{\frac{p(j)}{k_{\textrm{out}(j)} }},$$
where $p(i)$ is the PageRank value of node $i$, $n$ is the number of nodes in the graph, $k_{\textrm{out}(j)}$ is the outdegree of node $j$, and the summation runs over all of the nodes with edges pointing towards $i$. The damping factor $d$ is the teleportation probability that weights the probability of walking an outgoing edge from $i$ or jumping to some other node. Teleportation effectively solves the dangling node problem by moving the walker to a new location when stuck at a dangling node [@page1999pagerank].
![Adjacency matrices of University of Oregon Internet routing network. Adjacency matrices are ordered by various importance measures: (A) Random, (B) Degree, (C) Eigenvector, (D) PageRank. In (B) the red cells that comprise the outer envelope of the shape denote nodes of degree=1[]{data-label="fig:ordering"}](./fig1.pdf){width="\textwidth"}
PageRank is related to the principal eigenvector of the adjacency matrix. The eigenvector similarly ranks the importance of a node based on the importance of its incoming neighbors:
$$\label{eq:ev}
\lambda x_i = \sum_{j\in A_i}{x_j} = \sum_{j}{A_{ij} x_j} = ({\textbf{A}^T\textbf{x}})_i,$$
meaning that $x_i$ is the $i^{\textrm{th}}$ component of the eigenvector of the transpose of the adjacency matrix with eigenvalue $\lambda$.
Solving Eq. \[eq:pr\] is said to be equivalent to solving Eq. \[eq:ev\] on a transition matrix $M$ defined as:
$$\label{eq:gm}
M_{ij} = \frac{1-d}{n} + d \frac{A_{ji}}{k_{\textrm{out}(j)}},$$
where $M$ is oftentimes called the [*Google Matrix*]{}. Thus, the PageRank score $p(i)$ from Eq. \[eq:pr\] is often said to be the principal eigenvector of $M$. We typically compute the eigenvector using power iteration, in which the Google matrix $M$ is repeatedly multiplied by some vector ($p(i)$ in the case of Eq. \[eq:pr\] or $x_i$ in the case of Eq. \[eq:ev\]) until convergence [@Langville2005].
Plotting and comparing graphs via PageRank or other importance measures, [*e.g.*]{}, degree, closeness, betweenness, has been a foundational representation of networks. Figure \[fig:ordering\] shows the adjacency matrix of Internet routing paths [@Leskovec2005] ordered randomly (A), by node-degree (B), by the principal eigenvector score (C) and by PageRank (D) score for each node.
Random ordering illustrates very little in terms of structural regularity, although the reader may be able to glean the general connectivity ([*i.e.*]{}, sparseness/denseness) from the proportion of whitespace. The degree-ordered matrix means that the node with the highest degree is listed at the top, and the edges that connect to other nodes of high degree are denoted with edges appearing towards the left-hand side of the matrix; ties occur frequently in degree-orderings and are broken arbitrarily. From this matrix we can begin to infer that the general degree distribution of the network.
Previously, Chakrabarti et al. discussed the “water droplet” pattern formed from adjacency matrices ordered by the principal eigenvector [@Chakrabarti2004b; @Chakrabarti2007] (illustrated in Fig. \[fig:ordering\]C). They initially found that the eigenvector-ordered adjacency matrix, called the A-plot, “has a clean and smooth oval-shaped boundary” [@Chakrabarti2004b]. They also found that the boundary corresponds to the 1-degree nodes in the graph; The shape and cleanness of this boundary is explained as follows: if $I_i$ denotes the network value of node $i$ and node $i$ connects only to node $j$, then the properties of spectral decomposition imply
$$\label{eq:wd}
I_i = 1/\lambda_1 \times I_j,$$
where $\lambda_1$ is the largest eigenvalue of the adjacency matrix of the graph. Therefore the boundary of edges in the A-plot can be calculated from the first eigenvalue and eigenvector [@Zhan2003]. Eq. \[eq:wd\] also shows that there cannot be edges at all outside the boundary (See appendix C of [@Zhan2003] for proof).
Although not illustrated in Fig. \[fig:ordering\]C, there exists a clean and smooth internal boundary that corresponds to 2-degree nodes. Specifically, if node $i$ is connected to two nodes of similar importance values $I_j$, then their edges will be plotted in a fashion similar to the 1-degree nodes:
$$\label{eq:wd2}
I_i = 2/\lambda_1 \times I_j,$$
In this same fashion there exist a boundary of 3-degree nodes, etc., but these boundaries are sparse because it becomes vanishingly difficult to find a 3, 4-degree nodes connected to 3 or 4 nodes, respectively, of similar eigenvector centrality.
Adjacency matrix orderings have been used previously to describe the topology of a network. Perhaps the most widely known matrix ordering technique is encompassed by Blockmodelling [@doreian2005generalized] in which a network’s adjacency matrix is organized such that edges are grouped into a set of ‘blocks’ that represent coherent clusters, typically situated along the diagonal, from which further downstream tasks and decisions can be improved. Aside from blockmodelling, there are several ways to determine an ordering of a matrix’s rows and columns in order to gather some insight into the network structure; a more thorough review of the related literature can be found in Section 2.
In this paper, we describe newly discovered [*Striation*]{} patterns that arise from PageRank orderings of the adjacency matrix of unweighted and undirected real world networks. Figure \[fig:ordering\]D illustrates one example of these striation patterns on the same real-world Internet routing network that is shown in Figs. \[fig:ordering\]A–\[fig:ordering\]C, the only difference being the ordering of the matrix. Throughout this work we use a wide variety of synthetic and real-world networks to describe the physical process underlying the striking visual patterns that are observed. Ultimately, we find that the shape of the patterns tell an interesting story about the hidden structures (both complex and predictable) that are present in everyday networks.
Related Work {#supp:related}
============
The origin of PageRank was rooted in the intent to rank web pages based on their link topology [@page1999pagerank]. Although there are alternative link topology based algorithms such as HITS [@kleinberg1999authoritative] and the SALSA [@lempel2000stochastic] (which combines PageRank and HITS), PageRank enjoys a brand recognition due to its early integration into and association with the Google search engine [@franceschet2011pagerank]. Many studies have examined the type and quality of output of the PageRank algorithm. Page [*et al.*]{} advised that $\alpha = .85$ based on empirical evidence [@page1999pagerank], and Bechetti and Castillo further showed that PageRank does not fit a power-law distribution for extreme $\alpha$ values [@Becchetti2006]. Pandurangan [*et al.*]{}showed that the PageRank values of the web follow a power law [@pandurangan2002using], and Volkovich [*et al.*]{} showed the correlation between various parameters of the network (in-degree, out-degree, and percentage of dangling nodes) and the overall log-log shape of the PageRank plot [@volkovich2007determining].
The adjacency matrix is a common tool for visualizing graph structure, and has been shown to perform well in providing an intuitive understanding of dense or large graphs, except in the task of path finding [@ghoniem2005readability]. The drawback of matrices, however, is that nodes can appear in any arbitrary order. Structures can often be revealed by re-ordering the nodes of the matrix to reveal clusters of related nodes. A study by Mueller [*et al.*]{} provided a comparison of different methods of ordering the matrices, using Random, BFS, DFS, King’s algorithm, Reverse Cuthill McKee, Degree, Spectral, Separator tree partitioning algorithm, and Sloan ordering [@mueller2007comparison]. They evaluated the different orderings in terms of their consistency and ability to reveal structure on graphs generated by different algorithms. They did not, however, investigate PageRank or any of the other ranking-models discussed in the present work.
The idea of finding order in visual representations of matrices, especially adjacency matrices of graphs, has a long history. McCormick [*et al.*]{} introduced the Bond Energy Algorithm [@mccormick1972problem], which provided a method of reordering the columns and rows of matrices so that larger values were grouped together. This method used a nearest neighbor heuristic to overcome the $N!$ possible permutations, and is among the earliest algorithms that may be used as a graph clustering algorithm.
One of the simplest graph-centric ordering algorithms is degree ordering, in which the *degree*, [*i.e.*]{}, the number of edges connected to a node, is used as a sorting metric. This is also the most naïve, in that there are usually many ties, in which case the ordering is undefined.
DFS and BFS are both standard and foundational graph theory algorithms. They both return orderings dependent on the starting node, as well as the arbitrary tie-breaking when choosing an edge to explore or in adding neighboring nodes. Reverse Cuthill McKee (RCM) [@george1971computer; @george1981computer] and King’s algorithm [@king1970automatic] provide a variation of BFS in which a priority queue chooses which node to visit next. The RCM priority queue is based on the node’s degree, while the King’s algorithm priority queue orders the nodes based on how many of the nodes connected to it by its outgoing edges have already been visited. As in the other methods, ties may occur within the priority queue, and choice of starting node affects the final order.
Sloan’s algorithm [@sloan1986algorithm] orders the nodes by defining a start and end node, then prioritizes the remaining nodes by their degree and distance from an end node. Nodes farthest from the end node and with a low degree have highest priority; as the highest priority nodes are processed, the priority of that node’s neighbors is increased, and the cycle continues until all nodes have been processed. Sloan’s algorithm is also dependent on the chosen start and end nodes, and like other variants does not have a global order.
The ordering of the Separator tree is based on an algorithm given by Blandford [*et al.*]{} [@blandford2003compact]. It removes vertices (*separators*) from the graph in order to partition the structure into subgraphs of a fixed size. Separator trees were a result of trying to reduce the size of storage required for large graphs by isolating and thereby removing the need to store empty parts of the adjacency matrix.
Mueller [*et al.*]{}, identified common shapes that emerge within ordered adjacency matrices and give explanations of the underlying structure which produces that shape [@mueller2007interpreting]. They identify diagonal lines, wedges, and blocks. Diagonal lines do not indicate connected components when the diagonal is only one cell wide, but wider lines identify chains of similarly connected components. Wedges and blocks indicate fully-connected cliques. They also identified three common *footprints*, or visual features, that were observed within the various ordered matrices, including *envelopes*, *horizons*, and *galaxies*. *Envelopes* are a border-type visual pattern that makes the adjacency matrix resemble a leaf; the edge, or envelope, gives the appearance of a solid border around the leaf shape. A *horizon* is an area along the diagonal of the adjacency matrix that is more dense than the surrounding area. A dense *horizon* may indicate chains of interconnected nodes. A *galaxy* is a sparse distribution of edges within a graph with no discernible structure. It is commonly observed in a randomly or poorly ordered adjacency matrix. A *galaxy* does not imply that there is no structure, but rather that the current ordering does not reveal any structure. The primary thrust of their work was to show that the idea of adjacency matrices could be extended to Visual Similarity Matrices, with which one could quickly observe the similarity in structure of various graphs. Mueller did not, however, identify the striation patterns of PageRank orderings demonstrated in the present work.
The traditional adjacency matrix is not the only method available to tease patterns from graph topology. Prakash [*et al.*]{} [@prakash2010eigenspokes] found a pattern called EigenSpokes, which is observed by creating a scatter plot of the singular vectors (an *EE-plot*) of the nodes against each other upon the presence of an edge. This plotting method often forms lines that align along specific vectors (*spokes*), which is then used in community detection. Kang [*et al.*]{} [@kang2011spectral] demonstrated that nodes which have high scores on the EE-plot will often form near-cliques or bipartite cores.
![PageRank ordered adjacency matrices of real world networks: (A) Dolphins, (B) Karate, (C) Football, (D) Les Miserables, (E) Power Grid, (F) ArXiv Relativity and Cosmology Collaborations, (G) CAIDA, (H) Notre Dame Web Site, (I) DBLP Collaborations, (J) Gowalla Social Network, (K) Brightkite Social Network, (L) Amazon Co-Purchases. Each dot/cell represents and edge connecting two nodes. Cell-color denotes the sum of connected nodes’ PageRank score.[]{data-label="fig:realworld"}](./fig2.pdf){width="\textwidth"}
The present work is the first to explore the PageRank-ordering of adjacency matrices of large networks. The next section presents the findings and explains the striation patterns as an inherent result of two fundamental properties of real-world networks.
Results
=======
We begin by illustrating the PageRank orderings of 12 real-world networks of various sizes and types. Figure \[fig:realworld\] shows that the striation patterns tend to originate from the high-PageRank vertices located in the top-right of each matrix downwards toward the lower ranked vertices at the bottom and towards the right. Whitespace in the adjacency matrix, denoting the absence of edges, is clearly correlated with lower-ranked vertices – an unsurprising finding because vertices with high PageRank also tend to have a high degree.
The adjacency matrices illustrated in Fig. \[fig:realworld\] are listed from smallest to largest, but we notice other differences in the size and shape of the various matrix patterns. For example, the football (\[fig:realworld\]C) and Amazon (\[fig:realworld\]L) networks have relatively few broad striations that are maintained throughout the entire matrix. On the other hand, the CAIDA (\[fig:realworld\]G), Power Grid (\[fig:realworld\]E) and, to a lesser extend, the Notre Dame Web Site (\[fig:realworld\]H), the Arxiv collaboration network (\[fig:realworld\]F) and the Les Miserables co-scene network (\[fig:realworld\]D) have thin striations that do not emanate throughout the entire matrix; these same matrices are also noticeably sparser in their low-ranking region than the former networks.
![PageRank ordered adjacency matrices of the undirected ClueWeb B data set if 50 million nodes. Each dot/cell represents and edge connecting two nodes. []{data-label="fig:500K"}](./fig3.pdf){width=".99\textwidth"}
Our largest network, the Clueweb crawl of English Web pages in 2009, contains full information of about 50 million nodes. Due to the tremendous size of the resulting adjacency matrix, we are limited in our ability to render the entire matrix. We were, however, able to partially render 1%-sized subsets of the matrix and stitch the 10,000 sub-matrices together to resolve the full adjacency matrix. It is difficult to view certain detail in the overall matrix, so Fig. \[fig:500K\] also shows the first 10% sub-matrix (left) in addition to the full matrix (right); we find the same striation pattern here as we see in the other adjacency matrices.
Throughout the remainder of this section we explore these and other differences using synthetic network simulations and point to known qualities of the real-world graphs in order to infer the cause of the various striation types.
Generating Scale-Free Striations
--------------------------------
![Scale Free network generation by preferential attachment $\gamma=3$. Top row illustrates the growth model where and the initial adjacency matrix of the right-most network containing 7 vertices. Each dot/cell represents and edge connecting two nodes. Cell-color denotes the sum of connected nodes’ PageRank score. The bottom row shows the continued growth of the network via preferential attachment 25, 50, 100, 200, 400 and 800 nodes. This generative model clearly grows networks with a certain striation pattern. Although preferential attachment is a stochastic model, repeated runs generate very similar PageRank-ordered adjacency matrices.[]{data-label="fig:growing_ba"}](./fig4.pdf){width="\textwidth"}
To investigate this phenomenon further we grow undirected graphs using preferential attachment [@Liben2007] as illustrated in Fig. \[fig:growing\_ba\] with $\gamma=3$. Each graph iteration is ordered and colored by their PageRank scores. The top row of this figure shows how the preferential attachment process grows the graph from steps 4 through 7 corresponding to graphs with sizes of $n=4$ through $n=7$. The PageRank rankings for the nodes in each graph demonstrate how symmetry arises from the graph generation model. The explicit PageRank ordered matrix in the bottom row of Fig. \[fig:growing\_ba\] shows the early stages of what grow to be striation patterns. Further growth is illustrated in the six graph snapshots of size $n=25$, $50$, $100$, $200$, $400$, and $800$ respectively. We also generated networks up to $n=$1,000,000 and found that the striation patterns remain consistent within a given $\gamma$ parameter.
![Degree (A,D), eigenvector centrality (B,E) and PageRank (C,F) values by edge attachments. Top row plots value association; bottom row plots value-to-order association. $Y$-axis indicate values, normalized between 0 and 1, and are consistent for each row. All figures plot exact same network generated by preferential attachment ($n=1000$, $\gamma=3$).[]{data-label="fig:value_ordering"}](./fig5.pdf){width="\textwidth"}
We are further interested in the explicit causes of the striation patterns that are apparently caused by regularities emanating from the generative process. As demonstrated in Figure \[fig:value\_ordering\]C and in related literature, we frequently find that PageRank [@Becchetti2006] and eigenvector centrality [@Chakrabarti2004] of individual nodes follow a power law distribution. These findings reflect the power laws found in degree distributions of real world graphs [@Barabasi1999] due to the inherent relationship between PageRank, eigenvector centrality and degree.
This investigation led us to plot the PageRank-values and orderings of the edges in the network. Figure \[fig:value\_ordering\] plots the degree (\[fig:value\_ordering\]A and \[fig:value\_ordering\]D), eigenvector centrality (\[fig:value\_ordering\]B and \[fig:value\_ordering\]E) and PageRank values (\[fig:value\_ordering\]C and \[fig:value\_ordering\]F) of edges in the top row, and the value-to-ordering association in the bottom row. The top row illustrates how the various importance values of the source nodes are associated with the values of the nodes it connects to. Figure \[fig:value\_ordering\]A indicates that nodes of high degree (normalized from 0 to 1) are attached to other nodes of various degree; the same can be said for the other value-to-value plots in Fig. \[fig:value\_ordering\]B and \[fig:value\_ordering\]C. The top row equates to a two-dimensional plot expressing two facets of graphs generated by preferential attachment:
1. PageRank scores follow a power law distribution [@Becchetti2006], and
2. PageRank values of a typical node’s neighborhood also follow a power law distribution.
These two facets result in the striation patterns demonstrated above. To understand why, consider the bottom row of Fig. \[fig:value\_ordering\] which plots the source nodes by their ranked-order rather than their values, essentially stretching the plots in the top row across the x-axis. Stretching the value-to-value plots on the top row into value-to-ranking plots on the bottom row show a half-way-there illustration of the striations; if we were to stretch the y-axis in the same way as the x-axis the resultant plots would be the order-by-order plots that show the striations in Fig. \[fig:ordering\]B–\[fig:ordering\]D according to their respective orderings.
Recall that the top row of Fig. \[fig:value\_ordering\] indicates the values of each node and its neighbor. The matrix is symmetric because the graph is undirected, therefore each source-to-target attachment will have a symmetric attachment in the opposite direction. The degree, eigenvector and PageRank values are spaced as expected due to the well known power law distribution of these three importance measures.
The major difference between Figs. \[fig:value\_ordering\]E and \[fig:value\_ordering\]F indicate two interesting dynamics in the attachment properties of eigenvector values and PageRank values. First, high values of eigenvector centrality tend to exclusively attach to nodes of a high value (therefore a high ordering), while the values of PageRank attachments are not distributed as cleanly.
We will investigate the neighborhood distributions more explicitly in Section \[sec:nodenbr\], but first we investigate the Watts-Strogatz process [@Watts1998], [*i.e.*]{}, the small world network generator, to demonstrate how differences in network generation result in different striation patterns. Different settings may help explain the underlying generation process of the real world networks.
Generating Small World Striations.
----------------------------------
![Small World network generation by Watts-Strogatz process ($n=1000$, $k=12$, and $p=0.20$). Top row illustrates the growth model and the initial PageRank ordered adjacency matrix of the right-most network containing 10 vertices; color represents relative PageRank score. The bottom row shows the continued growth of the network via the Watts-Strogatz process at 25, 50, 100, 200, 400 and 800 nodes. This generative model clearly grows networks with a certain striation pattern. Although Watts-Strogatz is a stochastic model, repeated runs generate very similar PageRank-ordered adjacency matrices.[]{data-label="fig:growing_ws"}](./fig6.pdf){width="\textwidth"}
Aside from preferential attachment, the small world networks of Watts and Strogatz [@Watts1998] show distinctly different striation patterns caused by their growth model. The Watts-Strogatz process initially creates a ring where each node is attached to $k$ neighbors symmetrically. From this ring an edge $e=[u,v]$ has a probability to reattach to some random node $x$, basically reassigning the end point of some node to $e=[u,x]$. Two iterations of this reassignment process is captured at the top of Fig. \[fig:growing\_ws\]. This process creates networks with a Poisson degree distribution. The PageRank ordered matrix of the top-center network is illustrated at the top-right and shows that the structural regularity of the small world network create the beginnings of a striation pattern. The example on the top row is purposefully drawn symmetrically, but the remaining six graph snapshots of sizes $n=25$, $50$, $100$, $200$, $400$, and $800$ (with an initial ring of 2 neighbors each) have a random rewiring probability of 20% and can rewrite a given edge to any other node. The larger graphs are rather unrealistic because they are created with an initial neighborhood ring of $k=2$, whereas the small world model typically calls for a neighborhood ring of size $k\gg\ln(n)$. More realistic graphs and real world graphs are investigated below.
During ordering, PageRank ties are broken arbitrarily, but we find that there are rarely ties in the resulting values, therefore the patterns are not a result of tiebreaking. Instead, we posit that the striation patterns are the result of the growth mechanisms on undirected graphs due to the structural regularity that these synthetic growth mechanisms produce.
![Degree (A,D), eigenvector centrality (B,E) and PageRank (C,F) values by edge attachments. Top row plots value association; bottom row plots value-to-order association. Y-axis indicate values, normalized between 0 and 1, and are consistent for each row. All figures plot exact same network generated by the Watts-Strogatz process ($n=1000$, $k=12$, and $p=0.20$). []{data-label="fig:sm_value_ordering"}](./fig7.pdf){width="\textwidth"}
Figure \[fig:sm\_value\_ordering\] shows the same progression of figures for the Watts-Strogatz process (which results in Poisson distributions for degree, eigenvector and PageRank values) as Fig. \[fig:value\_ordering\] does for the preferential attachment process (which results in power law distributions for degree, eigenvector and PageRank values). The top row shows value-by-value plots for degree, eigenvector and PageRank values. The bottom row shows corresponding value-by-rank plots where the x-axis is stretched by the ordering. Just as before, if we were to stretch the y-axis of the bottom row, then the plots from Fig. \[fig:growing\_ws\] would appear.
The degree plots in Fig. \[fig:sm\_value\_ordering\]A and \[fig:sm\_value\_ordering\]D show the Poisson distribution in discrete steps; this represents the distribution of whole-number degrees normalized between 0 and 1. Therefore each point in Fig. \[fig:sm\_value\_ordering\]A represents several overlapping edges of the same value (with a higher density towards the center), which are fanned out horizontally in \[fig:sm\_value\_ordering\]D. The eigenvector plots in Figs. \[fig:sm\_value\_ordering\]B and \[fig:sm\_value\_ordering\]E also show a Poisson distribution, but with with far fewer ties.
The PageRank results in Figs. \[fig:sm\_value\_ordering\]C and \[fig:sm\_value\_ordering\]F are particularly interesting. Figure \[fig:sm\_value\_ordering\]C shows that the PageRank distribution is a Poisson. However, we also find that the distribution is not continuous, [*i.e.*]{}, there are, in this case, 12 groupings of PageRank-densities across both axes. When stretched by ordering in Fig. \[fig:sm\_value\_ordering\]F we can see that the density of edge-points moves from the top-left to the bottom-right in a slightly diagonal fashion within each of the $12\times 12=144$ mini-sectors. When ordered along the y-axis, these top-left to bottom-right patterns become the striations present in Fig. \[fig:growing\_ws\] where the density in the value plots are translated into area in the ordered plots.
By varying the small world generation parameters we can see a how the striation patterns change. Changes to $k$ while keeping $p$ invariant generally affects the number of clusters. The number of clusters generally grows with $k$, but is not necessarily equal.
![Various Plots of Watts Strogatz Graphs. (A) Adjacency matrix of 1000-node small world graph generated by Watts-Strogatz process. Matrices are ordered by various importance measures. (B) Value-by-value and Order-by-Order PageRank adjacency matrices generated by 1000 node Watts-Strogatz model as p=0.20 and k varies. (C) Value-by-value and Order-by-Order PageRank adjacency matrices generated by 1000 node Watts-Strogatz model as $p$ varies and $k=12$[]{data-label="fig:sm_ordering"}](./fig8.pdf){width="\textwidth"}
Variations on the Watts-Strogatz Process
----------------------------------------
As with preferential attachment, the Watts-Strogatz process also produces a structural regularity within generated graphs. Figure \[fig:sm\_ordering\]A shows the result of ordering a small world graph with $n=1000$, $k=12$, and $p=0.20$. The closeness and betweenness plots do not exhibit any particular pattern, but rather a simple value gradient. The degree plot in Fig. \[fig:sm\_ordering\]A draws clear lines representing tie-breaking; and the PageRank ordered matrix shown in Fig. \[fig:sm\_ordering\]A shows a striation pattern similar to the striations in the preferential attachment graphs, where, instead of a power law distribution, these gaps clearly resemble the Poisson.
By varying the small world generation parameters we can see a how the striation patterns change. Changes to $k$ while keeping $p$ invariant generally affected the number of clusters. Although, $k=12$ typically resulted in 12 clusters this is likely a coincidence; the number of clusters generally grows with $k$, but is not necessarily equal. Figure \[fig:sm\_ordering\]B shows the plots as $k$ varies from $4$ to $8$ to $20$. In these plots, and in other experiments (not shown) the number of clusters tends to track with $k$, but not exactly. However, the size of the center clusters shrinks consistently as $k$ grows. As $k$ approaches $n$, [*e.g.*]{}, $k=n/10$, $k=n/5$, clusters are no longer evident because the graph becomes near completely connected.
Figure \[fig:sm\_ordering\]C shows the adjacency matrix plots as $k=12$ and the rewiring probability $p$ varies. The number of clusters increases and the size of each cluster decreases as the rewiring probability increases.
It is difficult to show all potential pairs as $p$ and $k$ vary together. In general, we find that a low $k$ value results in few, large clusters regardless of $p$. Similarly we find that, in general, a low $p$ value also results in few, large clusters.
Node Neighborhoods {#sec:nodenbr}
------------------
Here we discuss why these striation-patterns appear in the PageRank ordered matrices. It has been previously observed that PageRank distributions of Web graphs tend to follow a power law distribution [@pandurangan2002using]. We have also observed that, in directed graphs, the in-degree of a node and the its PageRank are highly correlated, meaning that the in-degree distribution of a graph also follows a power law distribution with the same exponent $\alpha$. Generally speaking, this means that the probability that the PageRank and/or in-degree of a node takes a value $x$ is approximately proportional to $x^{-\alpha}$. Aside from Web data, undirected uses of PageRank abound in related work [@abbassi2007recommender; @andersen2006local; @Perra2008], where the (undirected) degree and PageRank distributions are correlated in a variety of data sets, not just Web data. For example, if an undirected graph has a power law degree distribution, as in the case of Web data and other scale free networks, then the PageRank is very likely to also have a power law distribution; likewise, a graph with a Poisson degree distribution, as found in Small World graphs [@Barrat2000], is very likely to have a Poisson PageRank distribution.
![PageRank distributions of local neighborhoods in (A) a random network generated by preferential attachment ($n=1000$, $\gamma=3$), and (B) a random network generated by the Watts-Strogatz process ($n=1000$, $k=14$, $p=.20$). Color represents degree of the node, where each node occupies a point on the $y$-axis ordered by PageRank.[]{data-label="fig:synth_nbr"}](./fig9.pdf){width="\textwidth"}
With degree and PageRank distributions in mind, we ask: what does the degree and PageRank distribution of a single node’s neighborhood look like?
Previous work has found that the neighborhoods of individual nodes share similar characteristics as the overall graph. Work in spam detection, for example, finds that if the PageRank distribution of some Web page does not match the overall graph’s PageRank distribution, then that Web page is an anomaly and therefore likely to be spam. Furthermore, strong findings on self-similarity and assortativity further indicate that the PageRank in a neighborhood should have the same statistical properties as in the overall graph [@barabasi2000scale; @Newman2002]. The similarity between global PageRank and neighborhood PageRank distributions is the key to the presence of striation patterns.
Figure \[fig:synth\_nbr\] shows the PageRank distribution of each nodes’s local neighborhood on the x-axis. Note that the x-axis in the preferential attachment network (\[fig:synth\_nbr\]A) uses a log-scale, which, combined with the edge densities, indicates a power-law distribution in most node-neighborhoods. Similarly, the x-axis in the Watts-Strogatz network (\[fig:synth\_nbr\]B) uses a linear-scale, which, combined with the edge densities, indicates a Poisson distribution present in most node-neighborhoods [@Barrat2000].
Each row in Fig. \[fig:synth\_nbr\] shows a nodes’ neighborhood PageRank distribution. For example, each source node ([*i.e.*]{}, each row in the matrix) has $k$ neighbors drawn from a power law (Fig. \[fig:synth\_nbr\]A) or Poisson distribution (Fig. \[fig:synth\_nbr\]B), and these neighbors have PageRank values drawn from a the same distribution.
![PageRank ordered adjacency matrices of real world networks (A) Dolphins, (B) Karate, (C) Football, (D) Les Miserables, (E) Power Grid, (F) ArXiv Relativity and Cosmology Collaborations, (G) CAIDA, (H) Notre Dame Web Site, (I) DBLP Collaborations, (J) Gowalla Social Network, (K) Brightkite Social Network, (L) Amazon Co-Purchases. X-axis in log-scale. Cell color represents degree of the node (dark-red indicate degree $\ge$ 10), where each node occupies a point on the y-axis ordered by PageRank.[]{data-label="fig:realworld_nbr"}](./fig10.pdf){width="\textwidth"}
Taken together, we find that striation patterns tend to emerge from similarities in the PageRank-distribution of vertices and the PageRank-distribution of the typical node’s neighborhood. Preferential attachment creates networks whose global PageRank-distributions and typical node-neighborhood PageRank distributions follow a power law. Likewise, the Watts-Strogatz process creates networks with nodes and node-neighborhoods whose PageRank distributions follow Poisson distributions.
To reiterate the key point, striations appear in the PageRank-ordered adjacency matrix when the typical neighborhood’s PageRank distribution matches the global PageRank distribution.
The striation patterns are a direct consequence of the orderings on these self-similar distributions. Due to the connectivity mechanics of PageRank (and eigenvector centrality) low-valued nodes are connected to by other low valued nodes. Within the generative process the second lowest-valued node will be be connected to a neighborhood with a slightly higher overall PageRank than the lowest node, and so on. The gradual increase in PageRank on the x-axis, combined with similar PageRank distributions across the y-axis create the striations apparent in these plots.
![PageRank orderings of directed networks (A) C. Elegans, (B) Political Blogs, (C) directed preferential attachment network, (D) directed Watts Strogatz network, do not exhibit striation patterns.[]{data-label="fig:directed"}](./fig11){width="\textwidth"}
The neighborhoods of the real world networks originally illustrated in Fig. \[fig:realworld\] are shown in Fig. \[fig:realworld\_nbr\]. Here we find that the neighborhood distributions are not as clearly defined as in the synthetic networks of Fig. \[fig:synth\_nbr\], nevertheless, certain properties of the synthetic networks can be applied to understand the properties of the real-world networks. For example, the Football network (\[fig:realworld\_nbr\]C, on a log-scaled x-axis) appears to have neighborhoods with a Poisson PageRank distribution resembling the Watts-Strogatz networks from Fig. \[fig:synth\_nbr\]B. The CAIDA Internet routing network (\[fig:realworld\_nbr\]G) has a power law neighborhood distribution very similar to the preferential attachment network from Fig. \[fig:synth\_nbr\]A. Other neighborhood plots show similarities with one, both or neither of the synthetic generative processes indicating avenues for further research.
Directed Networks
-----------------
Here we look at some instances where striations do not appear. The previous graphs have all been undirected, as evident by their matrix-symmetry. Directed graphs do not exhibit the same striation patterns found in the undirected graphs shown above.
![PageRank orderings of undirected networks generated at random to conform to the scale free degree and Poisson distributions of preferential attachment (A) and Watts Strogatz (B) graphs respectively (at top). Cell-color denotes the sum of connected nodes’ PageRank score, and each dot/cell represents and edge connecting two nodes. Corresponding neighborhood plots (C and D) are illustrated below; where color corresponds to x-axis node degree.[]{data-label="fig:degseq"}](./fig12){width="\textwidth"}
Figure \[fig:directed\] illustrates PageRank ordered matrices from two real-world networks (A and B) and two (C and D) generated networks. The preferential attachment graph in Fig. \[fig:directed\]C was created with $n=1000$ and $\gamma=3$. The Watts-Strogatz (WS) graph in Fig. \[fig:directed\]D was created as an undirected graph with $n=1000$, $k=12$ and $\beta=.20$. After creation the graphs were converted into a directed graph using a fair coin flip to decide the directionality of each edge. Erdos-Reyni random graphs and many others, not shown, similarly do not exhibit striations in both the undirected and directed cases.
During ordering, PageRank ties are broken arbitrarily, but we find that there are rarely ties in the resulting values. Therefore the patterns are not a result of tiebreaking.
Clearly, the striation patterns are the result of the processes resulting in topology regularities that result in PageRank distributions unique in the undirected case only. We are unsure why the striation patterns do not appear in directed networks. It is important to remember that the damping factor in the PageRank algorithm was introduced to deal with the problem of sink-nodes, wherein a non-jumping random walker would get “stuck” in a sink node. Undirected networks, on the other hand, do not have sink-nodes because the random walker can exit any node by the link from which it arrived. Future work might explore this topic further by slowly introducing backward-edges to a directed graph to see if and when striation patterns emerge.
Random Networks
---------------
To explore the causality of the growth mechanism we fed a random graph generator a power-law degree sequence and a Poisson degree sequence representing the results of the preferential attachment and Watts-Strogatz mechanisms respectively. Rather than using these well defined graph processes, we can also generate random graphs where edges are randomly connected with the constraint that the final degree distribution matches some required distribution. Figure \[fig:degseq\] shows the PageRank ordered matrix of graphs randomly generated to conform to power law and Poisson degree sequences. We find that striation patterns are largely absent from the randomly generated graphs corresponding to a much smoother, [*i.e.*]{}, more random distribution of edges in the neighborhood plot.
![ PageRank ordered adjacency matrices of Kronecker approximations of real world graphs (A) ArXiv Relativity and Cosmology collaborations, (B) DBLP Collaborations, (C) Notre Dame Web Site, (D) CAIDA.[]{data-label="fig:kronapprox"}](./fig13.pdf){width="\textwidth"}
Kronecker Approximations
------------------------
Kronecker graphs are a new approach to modeling real-world networks as $n\times n$ adjacency matrices, where $n$ is relatively small, [*i.e.*]{}, typically between 2 and 4. The Kronecker graph model is based on a recursive construction that uses the Kronecker product $\otimes$ to multiply the initial $n\times n$ adjacency matrix by itself recursively [@Weichsel1962]. Leskovec [*et al.*]{}found efficient ways to learn the weights of the initiator matrix in order to recursively generate full-sized graphs that match some real world matrix. For example, the Notre Dame Web site matrix from Fig. \[fig:kronapprox\]C is generated by recursively taking the Kronecker product of the initiator matrix $[[0.999,0.414],[0.453,0.229]]$ 12 times resulting in an adjacency matrix with $2^{12}=4096$ rows and columns. The values of the initiator matrix are estimated by fitting an initiator matrix to the real-world network [@Leskovec2010].
Methods
=======
Network Vertices Edges Figure Source
--------------------------- ------------- ------------- --------- --------------------------
Dolphins 62 159 Fig. 2A [@Lusseau2003]$^\dag$
Karate 34 78 Fig.2B [@Zachary1977]$^\dag$
Football 115 613 Fig. 2C [@Girvan2002]$^\dag$
Les Miserables 77 254 Fig. 2D [@Knuth1993]$^\dag$
Power Grid 4,941 6,594 Fig. 2E [@Watts1998]$^\dag$
ArXiv Rel. & Cos. Collab. 5,242 14,496 Fig. 2F [@Leskovec2007]$^\ddag$
CAIDA 6,474 13,233 Fig. 2G [@Leskovec2005]$^\ddag$
Notre Dame Web Site 25,729 1,497,134 Fig. 2H [@Albert1999]$^\ddag$
DBLP Collaborations 317,080 1,049,866 Fig. 2I [@Yang2015]$^\ddag$
Gowalla Social Network 196,591 950,327 Fig. 2J [@Cho2011]$^\ddag$
Brightkite Social Network 58,228 214,078 Fig. 2K [@Cho2011]$^\ddag$
Amazon Co-Purchases 262,111 1,234,877 Fig. 2L [@Leskovec2007b]$^\ddag$
Clueweb2009 B 428,136,614 454,075,638 Fig. 3 $\ast$
: Sources of real networks from Fig. 2 and 3. Networks denoted with $^\dag$ were downloaded from Mark Newman’s network data collection <http://www-personal.umich.edu/~mejn/netdata/>. Networks denoted with $^\ddag$ were downloaded from the SNAP data collection <http://snap.stanford.edu/data/>. $\ast$ The ClueWeb B graph is available from <http://lemurproject.org/clueweb09/>.[]{data-label="tab:data"}
Table \[tab:data\] describes the various data sets used in Fig. \[fig:realworld\], Fig. \[fig:realworld\_nbr\], and throughout this paper. Directed or multigraph networks were converted to undirected, simple networks if needed. All edge weights, timestamps and other metadata (if any) were ignored. Networks sizes ranged from 34 vertices and 78 edges in the Karate network to 325,729 vertices and 1,497,134 edges in the Notre Dame network. Synthetic networks were created using the NetworkX toolkit [@Hagberg2008]. PageRank, eigenvector centrality and other measures were calculated using NetworkX and confirmed with matlab, numpy and scipy implementations.
Discussion
==========
We have shown the presence of certain striation patterns found in PageRank ordered adjacency matrices of synthetic and real-world networks. These striations arise in networks that are associated with the well known PageRank distributions combined with similar, regular neighborhood distributions. For example, if a network with a power law PageRank distribution also has vertices with neighborhoods that typically exhibit a power law PageRank distribution, then the resulting PageRank-ordered matrix will exhibit striations.
The total processing time is bounded by the time it takes to compute PageRank, sort the PageRank result and plot the edges in the graph. To compute the PageRank we use the power iteration method, which has a polynomial time complexity in the size of the matrix. In all cases we limited the number of power iterations to at most 50, however many datasets reach convergence before 50 iterations. Except in the ClueWeb dataset, the largest graphs’ PageRank scores could be computed, sorted and plotted in less than 5 minutes on a modern laptop. Once loaded into memory, the 50 million node ClueWeb subset took about an two hours for the submatrices to be computed and plotted and stitched together on a compute machine with 1TB of RAM.
The resulting plots provide a deep look into the topology of a given network, and the observed patterns may be helpful to researchers and practitioners when analysing large and complex networks, especially in the search for anomalous edges, vertices or whole networks. It is our intent to raise more questions that answer, and we solicit the communities help to further understand these patterns. We are working to find a closed form solution describing the striation patterns akin to Eq. \[eq:wd\] and Eq. \[eq:wd2\], although the results do not suggest that a strong boundary exists in PageRank-ordered matrices like those present in eigenvector centrality-ordered matrices.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Maintaining a consistent personality in conversations is quite natural for human beings, but is still a non-trivial task for machines. The persona-based dialogue generation task is thus introduced to tackle the personality-inconsistent problem by incorporating explicit persona text into dialogue generation models. Despite the success of existing persona-based models on generating human-like responses, their one-stage decoding framework can hardly avoid the generation of inconsistent persona words. In this work, we introduce a three-stage framework that employs a generate-delete-rewrite mechanism to delete inconsistent words from a generated response prototype and further rewrite it to a personality-consistent one. We carry out evaluations by both human and automatic metrics. Experiments on the Persona-Chat dataset show that our approach achieves good performance.'
author:
- |
Haoyu Song$^{1}$[^1], Yan Wang$^2$, Wei-Nan Zhang$^{1}$, Xiaojiang Liu$^2$, Ting Liu$^1$\
$^1$Research Center for Social Computing and Information Retrieval\
Harbin Institute of Technology, Heilongjiang, China\
$^2$ Tencent AI Lab, Shenzhen, China\
`{hysong,wnzhang,tliu}@ir.hit.edu.cn`\
`{brandenwang,kieranliu}@tencent.com`\
bibliography:
- 'acl2020.bib'
title: 'Generate, Delete and Rewrite: A Three-Stage Framework for Improving Persona Consistency of Dialogue Generation'
---
Introduction
============
In an open-domain conversation scenario, two speakers conduct open-ended chit-chat from the initial greetings and usually come to focus on their characteristics, such as hobbies, pets, and occupations, etc., in the course of the conversation. For humans, they can easily carry out conversations according to their personalities [@song-percvae], but fulfilling this task is still a challenge for recent neural dialogue models [@WelleckDNLI].
![ A common problem for persona-based dialogue models is that they can hardly avoid the generation of inconsistent persona words. Although the model generates a response which looks good, it is an inconsistent one. With further rewriting, the model can focus more on improving persona consistency. []{data-label="fig:1"}](./figures/figure1){width=".97\columnwidth"}
One main issue is that these models are typically trained over millions of dialogues from different speakers, and the neural dialogue models have a propensity to mimic the response with the maximum likelihood in the training corpus [@li-etal-2016-persona], which results in the frequent inconsistency in responses [@zhang-2018-persona]. Another issue is the user-sparsity problem [@qian2017assigning] in conventional dialogue corpora [@serban2015survey]. Some users have very few dialogue data, which makes it difficult for neural models to learn meaningful user representations [@li-etal-2016-persona].
To alleviate the above issues, @zhang-2018-persona introduced the Persona-Chat dataset to build more consistent dialogue models. Different from conventional dialogue corpora, this dataset endows dialogue models with predefined personas, which is in the form of textually described profile (as shown in the first line of Figure \[fig:1\]). The persona-based dialogue models also adopt an encoder-decoder architecture and are enhanced with persona encoding components, such as memory network [@memorynetworks] and latent variable [@kingma2013vae]. These models turn out to produce more consistent responses than the persona-free ones [@zhang-2018-persona; @song-percvae].
Despite the successful application of the encoder-decoder framework in persona-based dialogue models, one concern is that they lack extra attention to the key persona information. The model will learn to minimize the overall loss of every decoded word, but this may lead to the neglect of the key personas: change of one persona-related word may not significantly affect the overall loss, but could turn a good response into a totally inconsistent one. As shown in Stage 1 of Figure \[fig:1\], only one improper word “Colorado” leads the response to be inconsistent.
A desirable solution should be able to capture personas and automatically learn to avoid and refine inconsistent words before the response. In this paper, we present a Generate-Delete-Rewrite framework, GDR, to mitigate the generation of inconsistent personas. We design three stages specifically for the goal of generating persona consistent dialogues: The first [*Generate*]{} stage adopts a transformer-based generator to produce a persona-based response prototype. The second [*Delete*]{} stage employs a consistency matching model to identify inconsistencies and delete (by masking) the inconsistent words from the prototype. Finally, in the [*Rewrite*]{} stage, a rewriter polishes the masked prototype to be more persona consistent. To examine the effectiveness of our GDR model, we carried out experiments on the public available Persona-Chat dataset [@zhang-2018-persona].
We summarize the main contributions as follows:
- A three-stage end-to-end generative framework, GDR, was proposed for the generation of persona consistent dialogues.
- A matching model was integrated into the generation framework to detect and delete inconsistent words in response prototype.
- Experimental results show the proposed approach outperforms competitive baselines on both human and automatic metrics.
Related Work
============
End-to-end dialogue generation approaches are a class of models for building open-domain dialogue systems, which have seen growing interests in recent years [@vinyals2015neural; @shang2015neural; @serban2016building; @li_deeprl; @zhao2017dialoguecvae; @li2017adversarial]. These dialogue models adopted recurrent units in a sequence to sequence ([*seq2seq*]{}) fashion [@sutskever2014sequence]. Since the [*transformer*]{} has been shown to be on par with or superior to the recurrent units [@vaswani2017attention], some dialogue models began to take advantage of this architecture for better dialogue modeling [@dinan2018wizard; @su-rewriter].
Besides the advancements in dialogue models, the emergence of new dialogue corpus has also contributed to the research field. @zhang-2018-persona introduced the Persona-Chat dataset, with explicit persona texts to each dialogue. Based on seq2seq model and memory network, they further proposed a model named [*Generative Profile Memory Network*]{} for this dataset. Following this line, @yavuz2019deepcopy designed the [*DeepCopy*]{} model, which leverages copy mechanism to incorporate persona texts. @song-percvae integrated persona texts into the [*Per-CVAE*]{} model for generating diverse responses. However, the persona-based models still face the inconsistency issue [@WelleckDNLI]. To model the persona consistency, @WelleckDNLI annotated the Persona-Chat dataset and introduced the Dialogue Natural Language Inference (DNLI) dataset. This dataset converts the detection of dialogue consistency into a natural language inference task [@bowman2015large].
Personalized dialogue generation is an active research field [@li-etal-2016-persona; @qian2017assigning; @zhang-2018-persona; @zheng2019personalized; @zheng2019pre; @zhang2019neural]. In parallel with this work, @song2019generating leveraged adversarial training to enhance the quality of personalized responses. @liu-etal-2020-personachat incorporated mutual persona perception to build a more explainable [@liu-etal-2019-towards-explainable] dialogue agent. Other relevant work lies in the area of multi-stage dialogue models [@lei2020estimation]. Some retrieval-guided dialogue models [@weston2018retrieve; @wu2019response; @cai2019skeleton; @cai2019retrieval; @su2020prototype] also adopted a multi-stage framework, but the difference from our work is obvious: we generate the prototype rather than retrieve one. A high-quality retrieved response is not always available, especially under the persona-based setting.
Model
=====
Overview
--------
In this work, we consider learning a generative dialogue model to ground the response with explicit persona. We focus on the persona consistency of single-turn responses, and we leave the modeling of multi-turn persona consistency as future work.
{width="0.985\linewidth"}
Formally, we use uppercase letters to represent sentences and lowercase letters to represent words. Let $Q=q_1,q_2,...,q_n$ denotes the input query with $n$ words, and let $P=\{P^{(1)},P^{(2)},...,P^{(k)}\}$ be the $k$ different persona texts, where $P^{(i)}=p^{(i)}_1,p^{(i)}_2,...,p^{(i)}_{m_i}$ is the $i$-th persona text with ${m_i}$ words. Our goal is to learn a dialogue model $\mathbb M$ to generate a response $\hat Y=y_1,y_2,...,y_k$, which is consistent with the persona, based on both query $Q$ and persona $P$. In abbreviation, $\hat Y={\mathbb M}(Q,P)$.
More concretely, as shown in Figure \[fig:overall\_model\], the proposed model $\mathbb M$ consists of three parts:
1\) Prototype generator [$\text G$]{}. This component takes persona texts and query as input and generates a response prototype for further editing. It adopts an encoder-decoder architecture [@sutskever2014sequence], with the transformer [@vaswani2017attention] applied in both the encoder and the decoder.
2\) Consistency matching model [$\text D$]{}. This model is designed to detect and delete those words in the prototype that could lead to inconsistency. We train this model in a natural language inference fashion on the DNLI [@WelleckDNLI] dataset.
3\) Masked prototype rewriter [$\text R$]{}. The rewriter learns to rewrite the response prototype to a more consistent one. It is also a transformer decoder, which adopts a similar architecture as the decoder of [$\text G$]{}. The difference lies in that it takes the masked prototype, rather than the query, as input.
Generate: Prototype Generator {#sec:GTG}
-----------------------------
We apply the encoder-decoder structure to build our prototype generator $\text G$. For the encoder, we use the self-attentive encoder in the transformer. For the decoder, built upon the transformer decoder, we propose a tuple-interaction mechanism to model the relations among persona, query, and response.
### Self-Attentive Encoder {#sec:SAE .unnumbered}
As the persona $P$ is composed of several sentences, we unfold all words in $P$ into a sequence $p^{(1)}_1,p^{(1)}_2,...,p^{(i)}_{m_j},...,p^{(k)}_{m_k}$.
Then we use the self-attentive encoder [@vaswani2017attention] to compute the representations of the persona texts and query separately. The multi-head attention is defined as $\text{MultiHead}(Q,K,V)$, where $Q$,$K$,$V$ are query, key, and value, respectively. The encoder is composed of a stack of $N_G$ identical layers. Take the first stack encoding of $P$ for example: $$\begin{gathered}
\label{formula:4}
\text{V}^{(1)}_p = \text{MultiHead}(\textbf{I}(P),\textbf{I}(P),\textbf{I}(P)),\\
\text{O}^{(1)}_p = \text{FFN}(\text{V}^{(1)}_p),\\
\text{FFN}(x) = max(0, xW_1+b_1)W_2+b_2,\end{gathered}$$ where $\text{V}^{(1)}$ is the first layer result of the multi-head self-attention and $\text {I}(\cdot)$ is the embedding function of the input. The input embedding for word $w_i$ is the sum of its word embedding and position embedding. $\text{O}^{(1)}$ denotes the output of the first layer feed-forward network. For other layers: $$\begin{gathered}
\label{formula:7}
\text{V}^{(n)}_p = \text{MultiHead}(\text{O}^{(n-1)}_p),\text{O}^{(n-1)}_p),\text{O}^{(n-1)}_p),\\
\text{O}^{(n)}_p = \text{FFN}(\text{V}^{(n)}_p),\end{gathered}$$ where $\text{n}=$2,...,$N_G$. We applied layer normalization to each sublayer by $\text{LayerNorm}(x+\text{Sublayer}(x))$. $Q$ is encoded in the same way. After $N_G$ identical layers, we can get the final representations $\text{O}^{(N_G)}_p$ and $\text{O}^{(N_G)}_q$, where $\text{O}^{(N_G)}_p$ and $\text{O}^{(N_G)}_q$ are the encoded persona and encoded query, respectively.
### Tuple-Interaction Decoder {#sec:TLAD .unnumbered}
In the decoding phase, there are three types of information, persona $P$, query $Q$, and response $Y$, which make up a tuple ($P$,$Q$,$Y$). Accordingly, three inter-sentence relations need to be considered: (1) The alignment between $Q$ and $Y$ is beneficial to yield better results [@bahdanau2014attention]. (2) As the persona is composed of several sentences and describes different aspects, we need to find the most relevant persona information according to the relations between P and Y. (3) We also want to know whether the query needs to be answered with the given persona. Thus we should take the relations between $P$ and $Q$ into account.
Considering the above factors, we design a two-layer tuple-interaction mechanism in the decoder, as shown in the first part of Figure \[fig:overall\_model\]. There are three attentions in two layers: query attention ($\text {Q-Attn}$) and persona attention ($\text {P-Attn}$) in the first layer, and persona-query attention ($\text {PQ-Attn}$) in the second layer. $N_G$ such identical layers compose of the decoder. For the first layer: $$\begin{gathered}
\label{formula:9}
\text{V}^{(1)}_{y} = \text{MultiHead}(\textbf{I}(Y),\textbf{I}(Y),\textbf{I}(Y)),\\
\text{E}^{(1)} = \text{MultiHead}(\text{V}^{(1)}_y,\text{O}^{(N_G)}_p,\text{O}^{(N_G)}_p),\\
\text{F}^{(1)} = \text{MultiHead}(\text{V}^{(1)}_y,\text{O}^{(N_G)}_q,\text{O}^{(N_G)}_q),\\
\text{T}^{(1)} = \text{MultiHead}(\text{E}^{(1)},\text{F}^{(1)},\text{F}^{(1)}),\\
\text{O}^{(1)}_{dec} = \text{FNN}(\text{mean}(\text{E}^{(1)},\text{F}^{(1)},\text{T}^{(1)})),\end{gathered}$$ where $\text{E}^{(1)}$ and $\text{F}^{(1)}$ are the results of the first layer $\text{P-Attn}$ and $\text{Q-Attn}$. $\text{T}^{(1)}$ is the result of the first layer $\text{PQ-Attn}$. $\text{O}^{(1)}_{dec}$ denotes the first layer output. Note that the $Y$ here is masked to ensure depending only on the known words [@vaswani2017attention]. Repeatedly, for other layers: $$\begin{gathered}
\label{formula:14}
\text{V}^{(n)}_{y} = \text{MultiHead}(\text{O}^{(n-1)}_{dec}),\text{O}^{(n-1)}_{dec}),\text{O}^{(n-1)}_{dec}),\\
\text{O}^{(n)}_{dec} = \text{FNN}(\text{mean}(\text{E}^{(n)},\text{F}^{(n)},\text{T}^{(n)})),\end{gathered}$$ where $\text{n}=$2,...,$N_G$. After $N_G$ layers, the decoder output $\text{O}^{(N_G)}_{dec}$ is projected from hidden size to vocabulary size, then followed up by a $\text{softmax}$ function to get the words’ probabilities: $$\begin{gathered}
\label{formula:16}
{\text{Prob}}^{(1)}= \text{SoftMax}(\text{O}^{(N_G)}_{dec}W_3+b_3),\end{gathered}$$ where $W_3$ is a $\text{hidden size} \times \text{vocabulary size}$ weight matrix and $b_3$ is the bias term with $\text{vocabulary size}$ dimension. And $\text{Prob}^{(1)}$ denotes the output distribution of the first stage. Now we can get the response prototype $\hat{Y}^{(1)}$ from the $\text{Prob}^{(1)}$.
![ The architecture of our consistency matching model. “$\cdot$” and “$-$” denote element-wise product and difference. The dotted line shows inference process, including consistency matching and word deleting. []{data-label="fig:matching_model"}](./figures/figure3){width=".96\columnwidth"}
Delete: Consistency Matching Model
----------------------------------
The goal of the consistency matching model $\text{D}$ is to reveal word-level consistency between the persona texts and the response prototype, thus the inappropriate words can be deleted from the prototype.
This model is trained to estimate the sentence-level entailment category [@bowman2015large] of a response for the given persona texts, which includes [*entailment*]{}, [*neutral*]{} and [*contradiction*]{}. The key is that if the category is not [*entailment*]{}, we can delete the most contributing words by replacing them with a special mask token, thus giving the model a chance to rephrase. The attention weights can measure each word’s contribution.
The architecture of our consistency matching model is shown in Figure \[fig:matching\_model\]. From bottom to top are the self-attentive encoding layer, cross attention layer, and consistency matching layer.
As described in section \[sec:SAE\], the self-attentive encoder ($\text{SAE}(\cdot)$) performs self-attention over the input to get sentence representations. Because the task of consistency matching is quite different from dialogue generation, we did not share the encoders between the generator $\text G$ and matching model $D$: $$\begin{gathered}
\label{formula:17}
\bar{\text{A}}= \text{SAE}_D(P),\\
\bar{\text{B}}= \text{SAE}_D(\hat{Y}^{(1)}),\end{gathered}$$ where $\bar{\text{A}}$ is a $\text{hidden size} \times \text{n}$ matrix. $\bar{\text{A}}=[\bar a_1,\bar a_2,...,\bar a_n]$ and $\bar{\text{B}}=[\bar b_1,\bar b_2,...,\bar b_m]$. The $n$ and $m$ are the number of words in persona $P$ and response prototype $\hat{Y}^{(1)}$. Here we applied average pooling stragety [@liu2016learning; @chen2017enhanced] to get the summary representations: $$\begin{gathered}
\label{formula:19}
\bar{\text{a}}_{0}= \sum_{i=1}^{n}\frac{\bar{a}_i}{n},\end{gathered}$$ and we can get the response attention weights and attentive response representations by: $$\begin{gathered}
\label{formula:21}
{\text{W}_b}= \bar{\text{a}}_{0}^\top \bar{\text{B}},\\
\widetilde{\text{B}}= {\text{W}_b}\bar{\text{B}}^\top,\end{gathered}$$ where ${\text{W}_b}$ is attention weights and $\widetilde{\text{B}}$ is response representations. Similarly, we can get ${\text{W}_a}$ and $\widetilde{\text{A}}$.
Once $\widetilde{\text{A}}$ and $\widetilde{\text{B}}$ are generated, three matching methods [@chen2017enhanced] are applied to extract relations: concatenation, element-wise product, element-wise difference. The results of these matching methods are concatenated to feed into a multi-layer perceptron, which has three layers and tanh activation in between. The output is followed up by a SoftMax function to produce probabilities.
In the inference process, as shown in Figure \[fig:matching\_model\], the response attention weights $\text{W}_b$ is leveraged to illustrate the inconsistent words, which will be deleted[^2]. In practice, we use a simple heuristic rule for deleting words: as long as the category is not $entailment$, we will delete 10% of the words (at least one word)[^3], with the highest attention weight, in the prototype $\hat{Y}^{(1)}$. In this way, we get the masked prototype $\hat{Y}^{(2)}$.
Rewrite: Masked Prototype Rewriter
----------------------------------
The rewriter $\text{R}$ takes the masked prototype and persona texts as input and outputs the final response.
$\text{R}$ is also a transformer decoder, which is similar to the decoder of ${\text G}$ in section \[sec:TLAD\], but with a minor difference: the masked prototype is close to the target response, thus the direct attention between the prototype and target response is needless. The architecture of ${\text R}$ can be seen in the third part of Figure \[fig:overall\_model\], which can be formalized as: $$\begin{gathered}
\label{formula:22}
\text{O}^{(N_G)}_{mp} = \text{SAE}_G(\hat{Y}^{(2)}),\\
\text{V}^{(n)} = \text{MultiHead}(\text{O}^{(n-1)}_{rw}),\text{O}^{(n-1)}_{rw}),\text{O}^{(n-1)}_{rw}),\\
\text{S}^{(n)} = \text{MultiHead}(\text{V}^{(n)},\text{O}^{(N_G)}_p,\text{O}^{(N_G)}_p),\\
\text{K}^{(n)} = \text{MultiHead}(\text{S}^{(n)},\text{O}^{(N_G)}_{mp},\text{O}^{(N_G)}_{mp}),\\
\text{O}^{(n)}_{rw} = \text{FNN}(\text{mean}(\text{S}^{(n)},\text{K}^{(n)})),\end{gathered}$$ where $\text{O}^{(N_G)}_{mp}$ is the encoded masked prototype and $\text{SAE}_G$ is the self-attentive encoder of $\text G$. $\text{O}^{(N_G)}_p$ is the encoded persona. After $N_R$ identical layers, the same generation process as in $\text G$ is applied to the $\text{O}^{(N_R)}_{rw}$, and we can get the final response $\hat{Y}^{(3)}$.
Training
--------
The consistency matching model $\text D$ is trained separately from the prototype generator $\text G$ and rewriter $\text R$. As forementioned, the matching model $\text D$ is trained in a natural language inference fashion on the DNLI dataset [@WelleckDNLI], which has been well defined by the previous studies [@bowman2015large; @chen2017enhanced; @gong2018DIIN]. We minimize the CrossEntropy loss between the outputs of $\text D$ and the ground truth labels.
The $\text G$ and $\text R$ share the same training targets. We trained them by the standard maximum likelihood estimate. Notice that there are two different deleting strategies in training: (1) In the warm-up phase, because the $\text G$ can hardly generate high-quality prototypes at this period, we randomly delete each word, with a 10% probability, from the prototype. (2) After that, the trained consistency matching model $\text D$ is leveraged to delete words.
[**Data**]{} [**Train**]{} [**Valid**]{} [**Test**]{}
--------------- --------------- --------------- --------------
Persona Texts 74,522 5,843 4,483
Q-R Pairs 121,880 9,558 7,801
: Some statistics of Persona-Chat dataset. Valid denotes Validate and Q-R denotes Query-Response.[]{data-label="tab:dataset_personachat"}
[**Label**]{} [**Train**]{} [**Valid**]{} [**Test**]{}
--------------- --------------- --------------- --------------
Entailment 100,000 5,500 5,400
Neutral 100,000 5,500 5,400
Contradiction 110,110 5,500 5,700
: Key statistics of DNLI dataset.[]{data-label="tab:dataset_dnli"}
Experiments
===========
Datasets
--------
We carried out the persona-based dialogue generation experiments on the public available Persona-Chat dataset [@zhang-2018-persona]. Furthermore, we trained the consistency matching model on the recently released Dialogue Natural Language Inference (DNLI) dataset [@WelleckDNLI].
We show the statistics of the Persona-Chat dataset in Table \[tab:dataset\_personachat\]. The DNLI dataset [@WelleckDNLI] is an enhancement to the Persona-Chat. It is composed of [*persona-utterance*]{} pairs from the Persona-Chat, and these pairs are further labeled as [*entailment*]{}, [*neutral*]{}, and [*contradiction*]{}. Some statistics of this dataset are given in Table \[tab:dataset\_dnli\].
Compared Models
---------------
To the best of our knowledge, this is an early work in modeling explicit persona consistency. To show the effectiveness of our models, we mainly compare it with the persona-based dialogue models:
- [[**S2SA**]{}]{} S2SA is an RNN-based attentive seq2seq model [@bahdanau2014attention]. It only takes the query as input.
- [[**Per-S2SA**]{}]{} This is a seq2seq model that prepends all persona texts to the query as input [@zhang-2018-persona].
- [[**GPMN**]{}]{} Generative Profile Memory Network is an RNN-based model that encodes persona texts as individual memory representations in a memory network [@zhang-2018-persona].
- [[**DeepCopy**]{}]{} An RNN-based hierarchical pointer network, which leverages copy mechanism to integrate persona [@yavuz2019deepcopy].
- [[**Per-CVAE**]{}]{} This is a memory augmented CVAE model to exploit persona texts for diverse response generation [@song-percvae].
- [[**Transformer**]{}]{} Different from the RNN-based models, transformer is a self-attention based sequence transduction model [@vaswani2017attention]. The persona texts are concatenated to the query to serve as its input.
Experimental Settings
---------------------
For all the RNN-based baseline models, they are implemented by two-layer LSTM networks with a hidden size 512. For the Transformer, the hidden size is also set to 512, and the layers of both encoder and decoder are 3. The number of heads in multi-head attention is 8, and the inner-layer size of the feedforward network is 2048. The word embeddings are randomly initialized, and the embedding dimension of all models is set to 512.
Our model applies the same parameter settings as the transformer. The number of layers $\text N_G=\text N_D=\text N_R=3$. G and R share the word embeddings, but the matching model D uses independent embeddings. We use token-level batching with a size 4096. Adam is used for optimization, and the warm-up steps are set to 10,000. We implemented the model in [*OpenNMT-py*]{} [@klein-etal-2017-opennmt].
Evaluation Metrics
------------------
In the evaluation, there are two essential factors to consider: [**persona consistency**]{} and [**response quality**]{}. We apply both human evaluations and automatic metrics on these two aspects to compare different models.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[**Model**]{} [**Const.**]{} [**Fluc.**]{} [**Relv.**]{} [**Info.**]{} [**PPL**]{} [**Dist-1.**]{} [**Dist-2.**]{} [**$\text{Ent}_\text{diin}$**]{} [**$\text{Ent}_\text{bert}$**]{}
---------------- ---------------- --------------- ---------------- ---------------- ------------------------------------------------------------------- ------------------------ ----------------- ---------------------------------- ----------------------------------
S2SA 15.9% 3.17 2.84 2.63 34.8 1.92 4.86 9.80% 1.83%
GPMN 34.8% 3.78 3.57 3.76$^\dagger$ 34.1 1.89 7.53 14.5% 7.36%
Per-S2S 35.3% 3.43 3.22 3.32 36.1 2.01 7.31 13.5% 6.15%
DeepCopy 36.0% 3.26 3.08 2.87 41.2 2.35 8.93 16.7% 8.81%
Transformer 38.8% 3.46 3.65$^\dagger$ 3.54 27.9 3.12 15.8 14.2% 9.52%
Per-CVAE 42.7% 3.53 2.97 3.66 -$^*$ [**3.83**]{}$^\dagger$ 20.9 17.2% 7.36%
[GDR (ours)]{} [**49.2%**]{} [**3.86**]{} [**3.68**]{} [**3.77**]{} **[16.7]{} & 3.66 & [**22.7**]{} & [**21.5%**]{} & [**13.0%**]{}\
**
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
#### Human Evaluation
We recruit five professional annotators from a third-party company. These annotators have high-level language skills but know nothing about the models. We sampled 200 [*persona-query-response*]{} tuples per model for evaluation. Duplicated queries (such as greetings which appear more than once) will not be sampled twice.
First, we evaluate the persona consistency of a response. The annotators are asked to decide whether the response is consistent with the given persona. 0 indicates irrelevant or contradictory and 1 indicates [**consistent**]{} (Const.).
Second, we evaluate the quality of a response on three conventional criteria: [**fluency**]{} (Fluc.), [**relevance**]{} (Relv.), and [**informativeness**]{} (Info.). Each aspect is rated on five-scale, where 1, 3, and 5 indicate unacceptable, moderate, and excellent performance, respectively. 2 and 4 are used for unsure.
#### Automatic Metrics
@dziri2019evaluating has shown that natural language inference based [*entailment*]{} ratio can be used as an indicator of dialogue consistency. Here we trained two well-performed NLI models, DIIN [@gong2018DIIN] and BERT [@devlin2019bert], to automatically predict the category of [*persona-response*]{} pairs, and we calculated the ratio of [*entailment*]{} as an additional reference to the persona consistency. In our experiments, DIIN and BERT achieved 88.78% and 89.19% accuracy on the DNLI test set, respectively, compared with previous best results 88.20%. The trained models are then leveraged for calculating [*entailment*]{} ratios. Two model-based [*entailment*]{} ratios are abbreviated as [**$\text{Ent}_\text{diin}$**]{} and [**$\text{Ent}_\text{bert}$**]{}.
For dialogue quality, we follow @zhang-2018-persona to use [**perplexity**]{} (PPL) to measure the fluency of responses. Lower perplexity means better fluency. Besides, we also use [**Dist-1**]{} / [**Dist-2**]{} [@li2016diversity] to examine the model’s ability to generate diverse responses, which is the ratio of distinct uni-grams / bi-grams.
[[**GDR**]{} vs]{} [**Win(%)**]{} [**Tie(%)**]{} [**Lose(%)**]{}
-------------------- ---------------- ---------------- -----------------
S2SA 48.0 38.2 13.8
Per-CVAE 46.1 29.8 24.1
DeepCopy 43.8 35.5 20.7
Per-S2S 41.3 36.1 22.6
GPMN 35.0 31.0 34.0
Transformer 34.7 32.1 33.2
: GDR response quality gains over other baseline methods on a pairwise human judgment.[]{data-label="tab:pairwise"}
Main Results
------------
We report the main evaluation results in Table \[tab:main\]. Compared with baseline methods, our GDR model obtains the highest consistency score of 49.2% in human evaluation, which is significantly better than other methods. The target responses in the sampled data are also annotated, and 65.4% of them expressed persona information. Moreover, the two model-based entailment ratios, which are calculated on the whole test set, also prove the effectiveness of our GDR model. Although the two NLI models differ in results, our GDR model ranks first under the evaluation of both DIIN and BERT.
For dialogue quality, our proposed model has a remarkably lower perplexity of 16.7 than all other baseline methods. An analysis can be seen in Section \[sec:analysis\]. Besides, our distinct-2 metric is even significantly better than the Per-CVAE model, which is designed to generate diverse responses.
Additionally, we carried out pairwise response comparison to see the dialogue quality gains. We report the results in Table \[tab:pairwise\]. While the GDR model significantly improves persona consistency, it can still generate high-quality responses like the transformer and GPMN.
More Analysis {#sec:analysis}
-------------
As the proposed model achieves better performance than baseline methods, we turn to ablation tests to further quantify the contributions made by different components. We ablated our model through several different approaches:
- [[**GR**]{}]{} It removes the matching model D, i.e., generates a prototype and rewrites it directly.
- [[**GRdR**]{}]{} This approach replaces the matching model D with 10% random deleting (Rd), thus to see if the masked prototype, extracted by our matching model D, is beneficial.
- [[**G**]{}]{} Our model’s generator, without further consistency matching and rewriting.
- [[**T**]{}]{} It is a transformer generator but removes the tuple-interaction in section \[sec:TLAD\] and directly concatenates persona texts to the query. This model is equivalent to the vanilla transformer.
[**Model**]{} [**Const.**]{} [**Fluc.**]{} [**Relv.**]{} [**Info.**]{} [**PPL**]{}
--------------- ---------------- --------------- ----------------- --------------- -------------
GDR 49.2% 3.86 3.68 3.77 16.7
GR 42.4% 3.72 3.40 3.66 18.0
GRdR 40.0% 3.60 3.29 3.56 20.6
G 40.1% 3.69 3.35 3.55 26.3
T 38.8% 3.46 3.65$^\ddagger$ 3.54 27.9
: Results of the ablation study. GDR is significantly better than the ablated approaches, with an only exception marked by $\ddagger$.[]{data-label="tab:ablation_human"}
[[**GDR**]{} vs]{} [**Win(%)**]{} [**Tie(%)**]{} [**Lose(%)**]{}
-------------------- ---------------- ---------------- -----------------
GRdR 41.7 39.5 18.8
GR 39.9 40.9 19.2
G 38.1 35.8 26.1
: Pairwise human judgment on response quality.[]{data-label="tab:pairwise_ablated"}
We report the results in Table \[tab:ablation\_human\]. First, we look into which components contribute to the consistency. As seen, from T, G, GR to GDR, every step has an observable improvement in [*Const.*]{}, indicating the effectiveness of our model’s design. Both the tuple-interaction in G and the rewriting process in R contribute to the improvements of persona consistency. The GRdR approach, with nothing different from GDR but a random deleting strategy, serves as a foil to our GDR model, which indicates a well-learned consistency matching model is of great benefit to our three-stage generation framework to generate persona consistent dialogues.
Second, we investigated the improvement of our perplexity. As we can see, the one-stage transformer approaches G and T have a perplexity higher than 26. In contrast, after we add the rewriter R, the perplexity of all approaches has a significant decline, no matter whether there is a matching model D. Lower perplexity means lower cross-entropy, which indicates the responses from the models have more ground truth words. To some extent, perplexity verifies the human evaluation results of the consistency. One reason for this improvement could be that the rewriter works like a denoising autoencoder [@vincent2008denoising], and it can focus more on the reconstruction of the missing information of sequence itself, rather than learning to map a sequence to an entirely different one.
We observed that the relevance scores of GR, GRdR, and G are a little inferior to the T. Even the GDR model is not significantly better than T on the relevance score. One plausible explanation is that all these models are specially designed for integrating persona information, although they obtain much better consistency score, it may come at the cost of relevance score.
Moreover, we compared the GDR’s response quality with three ablated models and reported it in Table \[tab:pairwise\_ablated\]. As we can see, the deleting and rewriting, which are designed for improving consistency, also have a positive effect on the dialogue quality.
----------------- ----------------------------------------------------
[**Persona**]{} [i. My mother is a dentist]{}
[ii. I’m currently looking for a job]{}
[**Query**]{} [**I want to become a physical therapist.**]{}
[Gen&Del]{}
[Rewrite]{} [I want to be .]{}
[w/o Delete]{} [I want to become a nurse like you.]{}
Transfmr I hear you. I am a nurse and a mom.
Per-S2S What kind of business?
GPMN I am a professional painter, what do you do?
DeepCopy What do you do for a living?
Per-CVAE I guess I would buy you one
[**Persona**]{} [i. Gymnastics is my favorite sport]{}
[ii. I’m a gold medalist olympian]{}
[**Query**]{} [**I am a mini van driver and I drive around.**]{}
[Gen&Del]{}
[Rewrite]{} [That’s cool. I am a .]{}
[w/o Delete]{} [That’s awesome. I am going to ski.]{}
Transfmr Oh. I am still in high school .
Per-S2S I drive a big van, but around the house.
GPMN I drive a FIAT. I love it.
DeepCopy What do you do for a living?
Per-CVAE That is why you like dogs? Tell me.
[**Persona**]{} [i. I like to grill outdoors]{}
[ii. I have been retired for years]{}
[**Query**]{} [**Old enough to work in construction. you?**]{}
[Gen&Del]{}
[Rewrite]{} [I’m . I spend most of my time .]{}
[w/o Delete]{} [I’m retired. I spend most of my time outdoors.]{}
Transfmr I am a boy and work in the construction. you?
Per-S2S No, I am a student.
GPMN I am a teacher, I love to be a teacher.
DeepCopy I work in the construction industry.
Per-CVAE I am a retired officer I love my bike ride.
----------------- ----------------------------------------------------
: Example responses from different models, with a visualization of the consistency matching weights. Strikethrough words are the masked words in Delete stage. The [*w/o Delete*]{} is the ablated model GR in section \[sec:analysis\], and [*Transfmr*]{} is short for Transformer. []{data-label="tab:cases"}
At last, we presented some generated examples in Table \[tab:cases\], together with the visualization of attention weights from match module D. In the first case, although the generated prototype is [*neutral*]{} regarding the persona, the word “nurse” is still masked according to our strategy. And after the rewriting stage, the final response expresses persona. In the second case, the prototype is potentially contradictory to the persona, and the keyword is successfully deleted by the matching model D. In the third case, the prototype is consistent with the persona, and no word is deleted. As a result, the final output response is the same as the output of no deletion model GR. In these cases, both consistency and quality are improved after the final rewriting.
Conclusion and Future Work
==========================
In this paper, we presented a three-stage framework, Generate-Delete-Rewrite, for persona consistent dialogue generation. Our method adopts transformer architecture and integrates a matching model to delete the inconsistent words. Experiments are carried out on public-available datasets. Both human evaluations and automatic metrics show that our method achieves remarkably good performance. In the future, we plan to extend our approach to improve the consistency of multi-turn dialogues.
Acknowledgments {#acknowledgments .unnumbered}
===============
This paper is supported by the National Natural Science Foundation of China under Grant No.61772153 and No.61936010. Besides, we want to acknowledge the Heilongjiang Province Art Planning Project 2019C027 and the Heilongjiang Province Social Science Research Project 18TQB100. We also would like to thank all the anonymous reviewers for their helpful comments and suggestions.
[^1]: This work was done when Haoyu Song was an intern at Tencent AI Lab.
[^2]: In this paper, “delete” a word means replacing this word with a special mask token.
[^3]: In our experiments, we found that deleting more words made it difficult for rewriter R to learn.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
A cut on the maximum lifetime in a lifetime fit does not only reduce the number of events, but it also, in some circumstances dramatically, decreases the statistical significance of each event. The upper impact parameter cut in the hadronic B trigger at CDF [@CDF] [@SVT] [@XFT], which is due to technical limitations, has the same effect. In this note we describe and quantify the consequences of such a cut on lifetime measurements. We find that even moderate upper lifetime cuts, leaving event numbers nearly unchanged, can dramatically increase the statistical uncertainty of the fit result.\
Keywords: [*lifetime fit; lifetime cuts; impact parameter cuts; lifetime bias; hadronic B trigger; statistical power per event; CDF; B Physics*]{}\
PACs: [ 21.10.Tg; 02.50.-r; 14.40.Nd]{}
author:
- |
[Jonas Rademacker]{}\
*University of Bristol*
title: '[Reduction of the Statistical Power Per Event Due to Upper Lifetime Cuts in Lifetime Measurements ]{}'
---
[99]{} D. Acosta [*et al.*]{} \[CDF Collaboration\], Phys. Rev. D [**71**]{} (2005) 032001 \[arXiv:hep-ex/0412071\]. B. Ashmanskas [*et al.*]{} \[CDF-II Collaboration\], Nucl. Instrum. Meth. A [**518**]{} (2004) 532 \[arXiv:physics/0306169\]. E. J. Thomson [*et al.*]{}, IEEE Trans. Nucl. Sci. [**49**]{} (2002) 1063.
| {
"pile_set_name": "ArXiv"
} |
---
title: 'Disconnected contributions to D-meson semi-leptonic decay form factors'
---
Introduction
============
Semileptonic decays of $D$-mesons contain rich physics. Lattice calculations of the form factors for these decays are important for the search for hints of new physics through the determination of CKM matrix elements. These form factors have been well-studied on the lattice. Previously, some of us tested a stochastic method to measure 3-point functions needed to calculate the semileptonic decay form factor [@Evans:2010tg]. The advantage of stochastic methods is that we have access to a greater range of momenta at fixed cost. This enables us to extract the form factor more reliably from results for the three point functions at different momentum transfers.
In particular, the $D_s$ meson is interesting for flavor physics. Its major semi-leptonic decay is to $\eta$ and $\eta'$, which has a contribution from a disconnected loop diagram (Fig. \[fig:diagrams\]). The loop runs over three light flavors so the effect is enhanced by a factor three, and thus may be large. The purpose of this work is to test the feasibility of measuring the disconnected diagram, and to quantify its contribution to the form factor.
We extract the scalar form factor $f_0$ from the relation [@Na:2009au]: $$f_0(q^2)
=\frac{m_c - m_l}{m_{D_s}^2 - m_\eta^2}\langle \eta |S| D_s \rangle,
\label{eq:f0}$$ where $S= \bar{l}c $ is a scalar current made from charm and light quarks. $m_i$ are the masses of the quarks and mesons. The matrix element can be extracted from the following ratio of 3-point over 2-point functions: $$\langle \eta({{\Vec{k}}},t_i) |S({{\Vec{q}}},t)| D_s({{\Vec{p}}},t_f) \rangle
= Z_\eta Z_{D_s} \frac{C_3(t_f-t_i,t-t_i; {{\Vec{p}}},{{\Vec{q}}})}
{ C_2^{\eta}(t-t_i;{{\Vec{k}}}) C_2^{D_s}(t_f-t;{{\Vec{p}}})}
= Z_\eta Z_{D_s} R({{\Vec{k}}},{{\Vec{q}}},{{\Vec{p}}},t-t_i,t_f-t_i),
\label{eq:matrixelement}$$ and similarly for $\eta'$. For large $t_f-t_i$ and $t-t_i$ this ratio should approach a constant. $Z_{\eta}$ and $Z_{D_s}$ are the overlap factors between the meson state and the interpolating operator, which can be extracted from the two point functions $C_2^{\eta}(t-t_i;{{\Vec{k}}})$ and $C_2^{D_s}(t_f-t;{{\Vec{p}}})$, respectively. The two point functions for $\eta$ and $\eta'$ also have a disconnected part, however, at $m_{\rm PS} \simeq 445\ {\rm MeV}$ we expect its contribution to the mass to be small and we neglect it in this first exploratory study.
We use QCDSF $24^3 \times 48$ $n_f=2+1$ configurations [@Bietenholz:2011qq]. So far we only use the ${\rm SU}(3)$ symmetric set ($\kappa_l=\kappa_s=0.1209$) with lattice spacing $a\simeq0.08 {\ \rm fm}$. This was generated using the tree-level Symanzik-improved gluonic action and non-perturbatively improved Wilson fermions with stout links in the derivative terms (SLiNC action). We use the same relativistic quark action for the (quenched) charm quark with $\kappa_{\rm charm}=0.11$. Note that since we use the flavor ${\rm SU}(3)$ symmetric configurations, the disconnected contributions in the $D_s \to \eta$ 3-point function cancel, when we identify $\eta=\eta_8$. The Chroma software package [@Edwards:2004sx] is used for some of the analysis.
![Connected (left) and disconnected (right) diagrams which contribute to $C_3(t_f,t;{{\Vec{p}}},{{\Vec{q}}})$. We use a stochastic method to estimate the all-to-all propagators, denoted by blue lines. []{data-label="fig:diagrams"}](diagram.eps "fig:"){width="0.3\linewidth"} ![Connected (left) and disconnected (right) diagrams which contribute to $C_3(t_f,t;{{\Vec{p}}},{{\Vec{q}}})$. We use a stochastic method to estimate the all-to-all propagators, denoted by blue lines. []{data-label="fig:diagrams"}](diagram-disconnected.eps "fig:"){width="0.3\linewidth"}\
Noise Reduction techniques
==========================
In order to calculate the disconnected loop, all-to-all propagators are required. These are estimated using stochastic methods, which involve performing $N$ inversions of the light quark Dirac operator for each configuration; $N$ should be large enough to give sufficiently small stochastic errors relative to the gauge noise. For some quantities the stochastic noise dominates the overall uncertainty and it is important to use efficient noise reduction techniques.
We measure the disconnected “loop” $$\begin{aligned}
C_1(t;{{\Vec{p}}})
&=\sum_{{{\Vec{x}}},{{\Vec{x}}}',{{\Vec{x}}}''}e^{i{{\Vec{p}}}\cdot{{\Vec{x}}}}
{\mathop{\rm tr}\nolimits}\left[\gamma_5 \phi({{\Vec{x}}},{{\Vec{x}}}') { M^{-1}({{\Vec{x}}}',t;{{\Vec{x}}}'',t)}
{\phi({{\Vec{x}}}'',{{\Vec{x}}})} \right],\end{aligned}$$ where $M$ is the Dirac operator for a light quark and $\phi$ is a smearing function. The stochastic estimation of the all-to-all propagator $M^{-1}({{\Vec{x}}}',t;{{\Vec{x}}}'',t)$ involves the following approximation: $$M^{-1}
=\frac{1}{N}\sum_{i=1}^N |s_i\rangle\langle \eta_i|
+\mathcal{O}\left(\frac{1}{\sqrt{N}} \right),$$ where $|\eta_i\rangle$ is a random noise vector and $|s_i\rangle = M^{-1} |\eta_i\rangle$. We use $\frac{1}{\sqrt{2}}( \mathbb{Z}_2 + i \mathbb{Z}_2$) complex random numbers for the noise vector. For each $i$ we need to smear both $|\eta_i\rangle$ and $|s_i\rangle$ ($|\eta_i\rangle$ must be smeared *after* solving for $|s_i\rangle$) so we need $2N$ applications of the smearing operator. This significantly increases the computer time needed to calculate the disconnected loop. Time dilution (partitioning) [@Bernardson:1993yg] is implemented: the noise vector is only non-zero on one or two time slices.
We test the following three noise reduction techniques.
Spin dilution/partitioning
: This uses projected noise vectors on a single spinor component and sums over the projections afterwards [@Bernardson:1993yg]: $$\frac{1}{N}\sum_{a=1}^4\sum_{i=1}^N
|s_i^{(a)}\rangle \langle \eta_i^{(a)}|,$$ where $|\eta_i^{(a)}\rangle = P^{(a)}|\eta_i\rangle$ is the projected noise vector. It requires $4N$ inversions but for some quantities the stochastic error is reduced by a factor greater than $2$. In addition, we can reduce the cost of smearing because the spin projection $P^{(a)}$ commutes with the smearing of our choice. A naive scaling gives $8N$ smearing operations, but we only need $5N$ applications: $4N$ for $|s_i^{(a)}\rangle$ and $N$ for $|\eta_i\rangle$.
Hopping Parameter Acceleration (HPA)
: [@Thron:1997iy] This is based on the following identity $$(\kappa D)^n M^{-1}
= M^{-1} -\kappa D - (\kappa D)^2 - \cdots - (\kappa D)^{n-1},
\label{eq:hpa}$$ where $\kappa D$ is the hopping part of the Dirac operator. Note that the derivative operator satisfies ${\mathop{\rm tr}\nolimits}[ \gamma_5 \kappa D] = 0$ due to the spinor structure so that this term only contributes to the noise. This means that $(\kappa D)^2 M^{-1}$ represents an improved estimate of $M^{-1}$ (we call it $n=2$ HPA). As long as the smearing is diagonal in spinor space, this is also true for the smeared all-to-all propagator.
Truncated Solver Method (TSM)
: For some quantities the ultra violet modes dominate. In these cases, using a small number of CG iterations in the solver for the solution vector $|s_i\rangle$ provides a good approximation, for example, to the disconnected loop [@Collins:2007mh; @Bali:2009hu]. To arrive at an unbiased estimate, a correction term needs to be added to the truncated part: $$M^{-1} =
\frac{1}{N_1}\sum_{i=1}^{N_1}
|s_{\rm trunc,}{}_i \rangle\langle \eta_i|
+ \frac{1}{N_2} \sum_{j=N_1+1}^{N_1+N_2}
|s_{\rm bias,}{}_j \rangle
\langle\eta_j|.
\label{eq:tsm}$$ The first term uses the truncated solution $|s_{\rm trunc,}{}_i \rangle$, which is cheap to calculate and typically causes the main part of the stochastic error. The second term contains $|s_{\rm bias,}{}_j \rangle
= |s_{\rm conv,}{}_j \rangle -|s_{\rm trunc,}{}_j \rangle$, where $|s_{\rm conv,}{}_j \rangle$ is a converged solution. $|s_{\rm conv,}{}_j \rangle$ is expensive, and only accounts for a small part of the stochastic error if $|s_{\rm bias,}{}_j\rangle $ does not contribute significantly to the observable. Therefore, by tuning parameters — $n$: number of CG-iterations for the truncated part, $N_1$: number of stochastic noises for the truncated part, $N_2$: number of stochastic noises for the bias part — we can reduce the total calculation cost. We use a CG solver for the truncated solutions and a BiCGstab solver for the converged solutions.
Comparisons
===========
We investigate the noise reduction techniques using one configuration. We use Wuppertal smearing [@Gusken:1989ad] for the quarks, with parameters which are tuned to minimize the contributions from the excited states to the effective mass.
In Figs. \[fig:err-p000\] and \[fig:err-p100\] we plot the stochastic errors for various combinations of the noise reduction techniques. In each case, the computational cost is fixed. The horizontal axes correspond to $n$, the number of iterations of the solver in the TSM. The data at $n=-100$ indicate the results without the TSM. In particular, the red plus symbols (“+”) show the results without any noise reduction techniques. For a fixed $n$, we have optimized $N_1$ and $N_2$ to give the smallest stochastic error under the cost condition $$N_1 ( n \tau_{\rm CG} + \tau_{\rm smear})
+ N_2( n \tau_{\rm CG} + n_{\rm conv} \tau_{\rm BiCGstab} + \tau_{\rm smear})
= \text{constant},$$ assuming the square of error, $\sigma_{\rm stoch.}^2$, to scale according to $$\sigma^2_{\rm stoch.} = \frac{f_1}{N_1} + \frac{f_2}{N_2},$$ where $f_1$ and $f_2$ are the variances of the first and second terms in eq. (\[eq:tsm\]), respectively. $n_{\rm conv}$ is the number of iterations needed to obtain the converged solution. $\tau_{\rm CG},\
\tau_{\rm BiCGStab}$ and $\tau_{\rm smear}$ represent the computer time needed for 1 CG iteration, 1 BiCGstab iteration, and smearing, respectively. The optimal ratios of $N_1/N_2$ are around $1$ ($10$), with (without) smearing.
Although small differences between the results are not significant due to the uncertainty on the stochastic errors, in all cases spin dilution together with HPA (purple squares), gives the minimum error when combined with TSM. Therefore we use this combination in the following analysis.
The gain factor, $$g=
\frac{\sigma^2(\text{without noise reduction})}
{ \sigma^2(\text{with noise reduction})},$$ strongly depends on the smearing. Without smearing (left panels), we obtain maximum gain factors of 16 – 25, which translates into a reduction of the computational cost of the same magnitude. With smearing, it is only about a factor 2. This is because the contribution to the error from the bias part (i.e., $f_2$) is larger than or of the same magnitude as the truncated part ($f_1$).
![Estimated stochastic errors at fixed cost for ${{\Vec{p}}}=(0,0,0)$. The horizontal axes are $n$ for the TSM. Data at $n=-100$ are without TSM. Left panel: without smearing. Right panel: with smearing.[]{data-label="fig:err-p000"}](opterr_g15p000smear0.eps "fig:"){width="0.45\linewidth"} ![Estimated stochastic errors at fixed cost for ${{\Vec{p}}}=(0,0,0)$. The horizontal axes are $n$ for the TSM. Data at $n=-100$ are without TSM. Left panel: without smearing. Right panel: with smearing.[]{data-label="fig:err-p000"}](opterr_g15p000smear130.eps "fig:"){width="0.45\linewidth"}
![The same as Fig. \[fig:err-p000\] but for ${{\Vec{p}}}=(1,0,0)$.[]{data-label="fig:err-p100"}](opterr_g15p100smear0.eps "fig:"){width="0.45\linewidth"} ![The same as Fig. \[fig:err-p000\] but for ${{\Vec{p}}}=(1,0,0)$.[]{data-label="fig:err-p100"}](opterr_g15p100smear130.eps "fig:"){width="0.45\linewidth"}
Results
=======
Having optimized the noise reduction, we can now measure the disconnected contribution to the form factor. For the TSM, we truncate after $n=20$ CG iterations and the numbers of noise vectors are $N_1=10$ and $N_2=20$. A total of $939$ configurations were used in the analysis.
Following our previous study [@Evans:2010tg], we use stochastic techniques for the connected contribution as well. The noise vectors are placed at the sink of the $D_s$ meson (denoted by a red circle in Fig. \[fig:diagrams\]). For each configuration, $24\times 4$ spin diluted noise vectors were computed for the charm quark. In terms of momenta, $57$ different combinations of ${{\Vec{p}}}$ for the $D_s$ meson were calculated. Note that a similar calculation with the sequential method would require $57\times 12$ inversions.
In order to extract the matrix elements in eq. (\[eq:matrixelement\]), we fixed the time separation between the $\eta$ source and the $D_s$ sink separately for the connected ($t_f=24$, $t_i=0$) and the disconnected ($t_f=24$, $t_i=16$) matrix elements. We combine the two contributions afterwards. For the connected part, taking the maximum separation $t_f-t_i=T/2=24$ enables us to average over the forward and backward propagations. For the disconnected part, in order to average the forward and backward propagations, the noise vector has a non-zero value at two time-slices separated by $16$ time-slices ($t_f \pm 8$). The usage of different $t_f-t_i$ for the connected and disconnected 3-point functions is allowed because we have assumed $m_\eta=m_{\eta'}$ (remember that $m_u=m_d=m_s$).
Fig. \[fig:plateau\] shows the ratio of the correlation functions, which corresponds to ${f_0(q^2)}/{Z_\eta Z_{D_s}}$. The disconnected part is multiplied by $3$ because of the $3$ light flavors. The errors for the disconnected contribution are small enough to obtain signals, significantly different from zero.
In Fig. \[fig:formfactor\] we show the form factors for the octet ($\eta_8$) and singlet ($\eta_1$) $\eta$s: $$\begin{aligned}
|\eta_8 \rangle
&=\frac{1}{\sqrt{6}}
(|\bar{u}u\rangle + |\bar{d}d\rangle - 2|\bar{s}s\rangle)
&&\text{connected only,} \\
|\eta_1 \rangle
&=\frac{1}{\sqrt{3}}
(|\bar{u}u\rangle + |\bar{d}d\rangle + |\bar{s}s\rangle)
&& \text{connected${}-3\times\;$disconnected.}\end{aligned}$$ Preliminary fits to $f_0(q^2)$ of the form $f_0(q^2)=\frac{f_0(0)}{1-b q^2}$ give $f_0(0) = 0.75(3)$ and $f_0(0) = 0.52(5)$, for $D_s \to \eta_8$ and $D_s \to \eta_1$, respectively. Also included in Fig. \[fig:formfactor\] is a value from light cone QCD sum rules for the decay into $\eta$ [@Azizi:2010zj], $f_0(0)=0.45(14)$. Due to ${\rm SU}(3)$ flavor symmetry the $D_s\to \eta_8$ form factor also represents the form factor of $D\to l\nu \pi$ and $D\to l\nu K$. Note that $f_0(0)$ for $\eta_1$ is smaller than that for $\eta_8$. This is consistent with the form factors for $B\to \eta,\eta'$ [@Ball:2007hb], which is the heavy quark limit, can be compared to our calculation.
![Ratios of 3-point over 2-point functions, $R$ in eq. \[eq:matrixelement\], for connected and disconnected parts. ${{\Vec{k}}}={{\Vec{q}}}=(0,0,0)$ for the left panel and $-{{\Vec{k}}}={{\Vec{q}}}=(1,0,0)$ for the right panel.[]{data-label="fig:plateau"}](conn-disconn_p000q000k000.eps "fig:"){width="0.48\linewidth"} ![Ratios of 3-point over 2-point functions, $R$ in eq. \[eq:matrixelement\], for connected and disconnected parts. ${{\Vec{k}}}={{\Vec{q}}}=(0,0,0)$ for the left panel and $-{{\Vec{k}}}={{\Vec{q}}}=(1,0,0)$ for the right panel.[]{data-label="fig:plateau"}](conn-disconn_p000q100k-100.eps "fig:"){width="0.48\linewidth"}
Conclusions
===========
We tested three methods (and their combinations) of noise reduction techniques for measuring the disconnected contributions to the $D_s$ meson semi-leptonic decay form factor. The combination of spin dilution, hopping parameter acceleration and truncated solver method was found to give the biggest gain in computer time. These noise reduction techniques allowed us to measure non-zero contributions to the form factor, on ${\rm SU}(3)$ flavor symmetric QCDSF $n_f=2+1$ configurations. Further studies with non-${\rm SU}(3)$ symmetric $n_f=2+1$ configurations are planned.
This work was supported by the EU ITN STRONGnet (grant number 238353) and the DFG SFB/Transregio 55. SC acknowledges support from the Claussen-Simon-Foundation (Stifterverband für die Deutsche Wissenschaft). JZ is supported by the Australian Research Council under grant FT100100005. The calculations were performed on the Athene HPC cluster at the University of Regensburg.
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| {
"pile_set_name": "ArXiv"
} |
LPENSL - 2014\
IMB - 2014
[Open spin chains with generic integrable boundaries:\
Baxter equation and Bethe ansatz completeness from SOV]{}
[**N. Kitanine**, **J.-M. Maillet**, **G. Niccoli** ]{}
**Abstract**
Introduction
============
The functional characterization of the complete transfer matrix spectrum associated to the most general spin-1/2 representations of the 6-vertex reflection algebra on general inhomogeneous chains is a longstanding open problem. It has attracted much attention in the framework of quantum integrability producing so far only partial results. The interest in the solution of this problem is at least twofold. On the one hand, the quantum integrable system associated to the limit of the homogeneous chain, i.e. the open spin-1/2 XXZ quantum chain with arbitrary boundary magnetic fields, is an interesting physical quantum model. It appears, in particular, in the context of out-of-equilibrium physics ranging from the relaxation behavior of some classical stochastic processes, as the asymmetric simple exclusion processes [@EssD05; @EssD06], to the transport properties of the quantum spin systems [@SirPA09; @Pro11]. Their solution can lead to non-perturbative physical results and a complete and manageable functional characterization of their spectrum represents the first fundamental steps in this direction. On the other hand, it is important to remark that the analysis of the spectral problem of these integrable quantum models turned out to be quite involved by standard Bethe ansatz [@Bet31; @FadST79] techniques. Therefore, these quantum models are natural laboratories where to define alternative non-perturbative approach to their exact solution. Indeed, the algebraic Bethe ansatz, introduced for open systems by Sklyanin [@Skl88] based on the Cherednik’s reflection equation [@Che84], in the case of open XXZ quantum spin chains can be applied directly only in the case of parallel z-oriented boundary magnetic fields. Under these special boundary conditions the spectrum is naturally described by a finite system of Bethe ansatz equations. Moreover the dynamics of such systems can be studied by exact computation of correlation functions [@KitKMNST07; @KitKMNST08], derived from a generalisation of the method introduced in [@KitMT99; @KitMT00; @MaiT00] for periodic spin chains.
Introducing a Baxter $T$-$Q$ equation, Nepomechie [@Nep02; @Nep04] first succeeded to describe the spectrum of the XXZ spin chain with non-diagonal boundary terms in the case of an anisotropy parameter associated to the roots of unity; furthermore, the result was obtained there only if the boundary terms satisfied a very particular constraint relating the magnetic fields on the two boundaries. This last constraint was also used in [@CaoLSW03] to introduce a generalized algebraic Bethe ansatz approach to this problem inspired by papers of Baxter [@Bax72; @Bax72a] and of Faddeev and Takhtadjan [@FadT79] on the XYZ spin chain. This method has led to the first construction of the eigenstates of the XXZ spin chain with non z-oriented boundary magnetic fields and this construction has been obtained for a general anisotropy parameter, i.e., not restricted to the roots of unity cases[^1]. In [@YanZ07] a different version of this technique based on the vertex-IRF transformation was proposed but in fact it required one additional constraint on the boundary parameters to work. It is worth mentioning that even if these constrained boundary conditions are satisfied and generalized Bethe ansatz method gives a possibility to go beyond the spectrum, as it was done for the diagonal boundary conditions, no representation for the scalar product of Bethe vectors[^2] and hence for the correlation functions were obtained.
This spectral problem in the most general setting has then been also addressed by other approaches. It is worth mentioning a new functional method leading to nested Bethe ansatz equations presented in [@Gal08] for the eigenvalue characterization and analogous to those previously introduced in [@MurN05] by a generalized $T$-$Q$ formalism. The eigenstate construction has been considered in these general settings in [BasK05a,Bas06]{} by developing the so-called $q$-Onsager algebra formalism. In this last case the characterization of the spectrum is given by classifying the roots of some characteristic polynomials. More recently, in [@CaoYSW13-4] an ansatz $T$-$Q$ functional equations for the spin chains with non-diagonal boundaries has been proposed[^3].
It is extremely important to remark that in general all methods based on Bethe ansatz (or generalized Bethe ansatz) are lacking proofs of the completeness of the spectrum and in most cases the only evidences of completeness are based on numerical checks for short length chains. This is the case for the XXZ chain with non-diagonal boundary matrices with the boundary constraint for which the completeness of the spectrum description by the associated system of Bethe ansatz equations has been studied numerically [@Nep-R-2003; @Nep-R-2003add]. In the case of the XXZ chain with completely general non-diagonal boundary matrices some numerical analysis is also presented in [@CaoYSW13-4]. Further numerical analysis have been developed in a much simpler case of the isotropic XXX spin chain where the most general boundary conditions can be always reduced by using the $SU(2)$ symmetry to one diagonal and one non-diagonal boundary matrices. For the XXX chains the ansatz introduced in [@CaoYSW13-2] was also applied and the completeness of the Bethe ansatz spectrum was checked numerically [@CaoJYW2013]. It is also important to mention a simplified ansatz proposed by Nepomechie based on a standard second order difference functional $T$-$Q$ equation with an additional inhomogeneous term. The completeness of the Bethe ansatz spectrum has been verified numerically for small XXX chains in [@Nep-2013] while in [@Nep-W-2013] the problem of the description of some thermodynamical properties has been addressed.
These interesting developments attracted our attention in connection to the quantum separation of variables (SOV) method pioneered by Sklyanin [@Skl85; @Skl92]. The first analysis of the spin chain in the classical limit from this point of view was performed in [@Skl89a; @Skl89b]. This alternative approach allows to obtain (mainly by construction) the complete set of eigenvalues and eigenvectors of quantum integrable systems. In particular, it was recently developed [NicT10,N10-1,N10-2,GroMN12,GroN12,Nic12b,Nic13a,N13-1,Nic13b,Nic13c,Fald-KN13,FaldN13,GroMN13]{} for a large variety of quantum models not solvable by algebraic Bethe ansatz. Moreover it has been shown first in [@GroMN12] that once the SOV spectrum characterization is achieved manageable and rather universal determinant formulae can be derived for matrix elements of local operators between transfer matrix eigenstates. In particular, this SOV method was first developed in [@Nic12b] for the spin-1/2 representations of the 6-vertex reflection algebra with quite general non-diagonal boundaries and then generalized to the most general boundaries in [@Fald-KN13]. There, it gives the complete spectrum (eigenvalues and eigenstates) and already allows to compute matrix elements of some local operators within this most general boundary framework. However, it is important to remark that this SOV characterization of the spectrum is somehow unusual in comparison to more traditional characterizations like those obtained from Bethe ansatz techniques. More precisely, the spectrum is described not in terms of the set of solutions to a standard system of Bethe ansatz equations but is given in terms of sets of solutions to a characteristic system of $\mathsf{N}$ quadratic equations in $\mathsf{N}$ unknowns, $\mathsf{N}$ being the number of sites of the chain. While the clear advantage of this SOV characterization is that it permits to characterize completely the spectrum without introducing any ansatz one has to stress that the classification of the sets of solutions of the SOV system of quadratic equations represents a new problem in quantum integrability which requires a deeper and systematic analysis.
The aim of the present article is to show that the SOV analysis of the transfer matrix spectrum associated to the most general spin-1/2 representations of the 6-vertex reflection algebra on general inhomogeneous chains is strictly equivalent to a system of *generalized* Bethe ansatz equations. This ensures that this system of Bethe equations characterizes automatically the entire spectrum of the transfer matrix. More in detail, we prove that the SOV characterization is equivalent to a second order finite difference functional equation of Baxter type: $$\tau (\lambda )Q(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}
(-\lambda )Q(\lambda +\eta )+F(\lambda ),$$ which contains an inhomogeneous term $F(\lambda )$ independent on the $\tau $ and $Q$-functions and entirely fixed by the boundary parameters. It vanishes only for some special but yet interesting non-diagonal boundary conditions (corresponding to the boundary constraints mentioned above). One central requirement in our construction of this functional characterization is the polynomial character of the $Q$-function. Indeed, it is this requirement that allows then to show that a finite system of equations of generalized Bethe ansatz type can be used to describe the complete transfer matrix spectrum. Note that similar results on the reformulation of the SOV spectrum characterization in terms of functional $T$-$Q$ equations with $Q$-function solutions in a well defined model dependent set of polynomials were previously derived [@N10-1; @N10-2; @GroN12] for the cases of transfer matrices associated to cyclic representations of the Yang-Baxter algebra. The analysis presented here is also interesting as it introduces the main tools to generalize this type of reformulation to other classes of integrable quantum models. The article is organized as follows. In Section 2 we set the main notations and we recall the main results of previous papers on SOV necessary for our purposes. Section 3 contains the main results of the paper with the reformulation of the SOV characterization of the transfer matrix spectrum in terms of the inhomogeneous Baxter functional equation and the associated finite system of generalized Bethe ansatz equations. In Section 4 we define the boundary conditions for which the inhomogeneity in the Baxter equation identically vanishes, in this way deriving the completeness of standard Bethe ansatz equations. There, we moreover derive the SOV spectrum functional reformulation for the remaining boundary conditions compatibles with homogeneous Baxter equations. Section 5 contains the description of a set of discrete transformations which leave unchanged the SOV characterization of the spectrum in this way proving the isospectrality of the transformed transfer matrices. These symmetries are used to find equivalent functional equation characterizations of the spectrum which allow to generalize the results described in Section 3 and 4. In Section 6 we present the SOV characterization of the spectrum for the rational 6-vertex representation of the reflection algebra and the reformulation of the spectrum by inhomogeneous Baxter equation. Finally, in Section 7, we present a comparison with the known numerical results in the literature for both the XXZ and XXX chains; the evidenced compatibility suggests that even in the homogenous chains our spectrum description is still complete.
Separation of variable for spin-1/2 representations of the reflection algebra
=============================================================================
Spin-1/2 representations of the reflection algebra and open XXZ quantum chain
-----------------------------------------------------------------------------
The representation theory of the reflection algebra can be studied in terms of the solutions $\mathcal{U}(\lambda )$ (monodromy matrices) of the following reflection equation:$$R_{12}(\lambda -\mu )\,\mathcal{U}_{1}(\lambda )\,R_{21}(\lambda +\mu -\eta
)\,\mathcal{U}_{2}(\mu )=\mathcal{U}_{2}(\mu )\,R_{12}(\lambda +\mu -\eta )\,\mathcal{U}_{1}(\lambda )\,R_{21}(\lambda -\mu ). \label{bYB}$$Here we consider the reflection equation associated to the 6-vertex trigonometric $R$ matrix $$R_{12}(\lambda )=\left(
\begin{array}{cccc}
\sinh (\lambda +\eta ) & 0 & 0 & 0 \\
0 & \sinh \lambda & \sinh \eta & 0 \\
0 & \sinh \eta & \sinh \lambda & 0 \\
0 & 0 & 0 & \sinh (\lambda +\eta )\end{array}\right) \in \text{End}(\mathcal{H}_{1}\otimes \mathcal{H}_{2}),$$where $\mathcal{H}_{a}\simeq \mathbb{C}^{2}$ is a 2-dimensional linear space. The 6-vertex trigonometric $R$-matrix is a solution of the Yang-Baxter equation:$$R_{12}(\lambda -\mu )R_{13}(\lambda )R_{23}(\mu )=R_{23}(\mu )R_{13}(\lambda
)R_{12}(\lambda -\mu ).$$The most general scalar solution ($2\times 2$ matrix) of the reflection equation reads$$K(\lambda ;\zeta ,\kappa ,\tau )=\frac{1}{\sinh \zeta }\left(
\begin{array}{cc}
\sinh (\lambda -\eta /2+\zeta ) & \kappa e^{\tau }\sinh (2\lambda -\eta ) \\
\kappa e^{-\tau }\sinh (2\lambda -\eta ) & \sinh (\zeta -\lambda +\eta /2)\end{array}\right) \in \text{End}(\mathcal{H}_{0}\simeq \mathbb{C}^{2}), \label{ADMFKK}$$where $\zeta ,$ $\kappa $ and $\tau $ are arbitrary complex parameters. Using it and following [@Skl88] we can construct two classes of solutions to the reflection equation (\[bYB\]) in the 2$^{\mathsf{N}}$-dimensional representation space:$$\mathcal{H}=\otimes _{n=1}^{\mathsf{N}}\mathcal{H}_{n}.$$Indeed, starting from$$K_{-}(\lambda )=K(\lambda ;\zeta _{-},\kappa _{-},\tau _{-}),\text{ \ \ \ \ }K_{+}(\lambda )=K(\lambda +\eta ;\zeta _{+},\kappa _{+},\tau _{+}),$$where $\zeta _{\pm },\kappa _{\pm },\tau _{\pm }$ are the boundary parameters, the following boundary monodromy matrices can be introduced $$\begin{aligned}
\mathcal{U}_{-}(\lambda ) &=&M_{0}(\lambda )K_{-}(\lambda )\widehat{M}_{0}(\lambda )=\left(
\begin{array}{cc}
\mathcal{A}_{-}(\lambda ) & \mathcal{B}_{-}(\lambda ) \\
\mathcal{C}_{-}(\lambda ) & \mathcal{D}_{-}(\lambda )\end{array}\right) \in \text{End}(\mathcal{H}_{0}\otimes \mathcal{H}), \\
\mathcal{U}_{+}^{t_{0}}(\lambda ) &=&M_{0}^{t_{0}}(\lambda
)K_{+}^{t_{0}}(\lambda )\widehat{M}_{0}^{t_{0}}(\lambda )=\left(
\begin{array}{cc}
\mathcal{A}_{+}(\lambda ) & \mathcal{C}_{+}(\lambda ) \\
\mathcal{B}_{+}(\lambda ) & \mathcal{D}_{+}(\lambda )\end{array}\right) \in \text{End}(\mathcal{H}_{0}\otimes \mathcal{H}).\end{aligned}$$ These matrices $\mathcal{U}_{-}(\lambda )$ and $\mathcal{V}_{+}(\lambda )=\mathcal{U}_{+}^{t_{0}}(-\lambda )$ define two classes of solutions of the reflection equation (\[bYB\]). Here, we have used the notations:$$M_{0}(\lambda )=R_{0\mathsf{N}}(\lambda -\xi _{\mathsf{N}}-\eta /2)\dots
R_{01}(\lambda -\xi _{1}-\eta /2)=\left(
\begin{array}{cc}
A(\lambda ) & B(\lambda ) \\
C(\lambda ) & D(\lambda )\end{array}\right) \label{T}$$and $$\widehat{M}(\lambda )=(-1)^{\mathsf{N}}\,\sigma _{0}^{y}\,M^{t_{0}}(-\lambda
)\,\sigma _{0}^{y}, \label{Mhat}$$where $M_{0}(\lambda )\in $ End$(\mathcal{H}_{0}\otimes \mathcal{H})$ is the bulk inhomogeneous monodromy matrix (the $\xi _{j}$ are the arbitrary inhomogeneity parameters) satisfing the Yang-Baxter relation:$$R_{12}(\lambda -\mu )M_{1}(\lambda )M_{2}(\mu )=M_{2}(\mu )M_{1}(\lambda
)R_{12}(\lambda -\mu ). \label{YB}$$The main interest of these boundary monodromy matrices is the property shown by Sklyanin [@Skl88] that the following family of transfer matrices:$$\mathcal{T}(\lambda )=\text{tr}_{0}\{K_{+}(\lambda )\,M(\lambda
)\,K_{-}(\lambda )\widehat{M}(\lambda )\}=\text{tr}_{0}\{K_{+}(\lambda )\mathcal{U}_{-}(\lambda )\}=\text{tr}_{0}\{K_{-}(\lambda )\mathcal{U}_{+}(\lambda
)\}\in \text{\thinspace End}(\mathcal{H}), \label{transfer}$$defines a one parameter family of commuting operators in End$(\mathcal{H})$. The Hamiltonian of the open XXZ quantum spin 1/2 chain with the most general integrable boundary terms can be obtained in the homogeneous limit ($\xi _{m}=0$ for $m=1,\ldots ,\mathsf{N}$) from the following derivative of the transfer matrix (\[transfer\]):$$H=\frac{2(\sinh \eta )^{1-2\mathsf{N}}}{\text{tr}\{K_{+}(\eta /2)\}\,\text{tr}\{K_{-}(\eta /2)\}}\frac{d}{d\lambda }\mathcal{T}(\lambda )_{\,\vrule height13ptdepth1pt\>{\lambda =\eta /2}\!}+\text{constant,} \label{Ht}$$and its explicit form reads: $$\begin{aligned}
H& =\sum_{i=1}^{\mathsf{N}-1}(\sigma _{i}^{x}\sigma _{i+1}^{x}+\sigma
_{i}^{y}\sigma _{i+1}^{y}+\cosh \eta \sigma _{i}^{z}\sigma _{i+1}^{z})
\notag \\
& +\frac{\sinh \eta }{\sinh \zeta _{-}}\left[ \sigma _{1}^{z}\cosh \zeta
_{-}+2\kappa _{-}(\sigma _{1}^{x}\cosh \tau _{-}+i\sigma _{1}^{y}\sinh \tau
_{-})\right] \notag \\
& +\frac{\sinh \eta }{\sinh \zeta _{+}}[(\sigma _{\mathsf{N}}^{z}\cosh \zeta
_{+}+2\kappa _{+}(\sigma _{\mathsf{N}}^{x}\cosh \tau _{+}+i\sigma _{\mathsf{N}}^{y}\sinh \tau _{+}). \label{H-XXZ-Non-D}\end{aligned}$$Here $\sigma _{i}^{a}$ are local spin $1/2$ operators (Pauli matrices), $\Delta =\cosh \eta $ is the anisotropy parameter and the six complex boundary parameters $\zeta _{\pm }$, $\kappa _{\pm }$ and $\tau _{\pm }$ define the most general integrable magnetic interactions at the boundaries.
Some relevant properties
------------------------
The following quadratic linear combination of the generators $\mathcal{A}_{-}(\lambda ),$ $\mathcal{B}_{-}(\lambda ),$ $\mathcal{C}_{-}(\lambda )$ and $\mathcal{D}_{-}(\lambda )$ of the reflection algebra: $$\begin{aligned}
\frac{\mathrm{det}_{q}\,\mathcal{U}_{-}(\lambda )}{\sinh (2\lambda -2\eta )}& =\mathcal{A}_{-}(\epsilon \lambda +\eta /2)\mathcal{A}_{-}(\eta /2-\epsilon
\lambda )+\mathcal{B}_{-}(\epsilon \lambda +\eta /2)\mathcal{C}_{-}(\eta
/2-\epsilon \lambda ) \label{q-detU_1} \\
& =\mathcal{D}_{-}(\epsilon \lambda +\eta /2)\mathcal{D}_{-}(\eta
/2-\epsilon \lambda )+\mathcal{C}_{-}(\epsilon \lambda +\eta /2)\mathcal{B}_{-}(\eta /2-\epsilon \lambda ), \label{q-detU_2}\end{aligned}$$where $\epsilon =\pm 1$, is the *quantum determinant* . It was shown by Sklyanin that it is a central element of the reflection algebra$$\lbrack \mathrm{det}_{q}\,\mathcal{U}_{-}(\lambda ),\mathcal{U}_{-}(\mu )]=0.$$The quantum determinant plays a fundamental role in the characterization of the transfer matrix spectrum and it admits the following explicit expressions:$$\begin{aligned}
\mathrm{det}_{q}\,\mathcal{U}_{-}(\lambda ) &=&\mathrm{det}_{q}K_{-}(\lambda )\mathrm{det}_{q}M_{0}(\lambda )\mathrm{det}_{q}M_{0}(-\lambda )
\label{q-detU_-exp} \\
&=&\sinh (2\lambda -2\eta )\mathsf{A}_{-}(\lambda +\eta /2)\mathsf{A}_{-}(-\lambda +\eta /2),\end{aligned}$$where: $$\mathrm{det}_{q}M(\lambda )=a(\lambda +\eta /2)d(\lambda -\eta /2),
\label{bulk-q-det}$$is the bulk quantum determinant and$$\mathrm{det}_{q}K_{\pm }(\lambda )=\mp \sinh (2\lambda \pm 2\eta )g_{\pm
}(\lambda +\eta /2)g_{\pm }(-\lambda +\eta /2).$$Here, we used the following notations:$$\mathsf{A}_{-}(\lambda )=g_{-}(\lambda )a(\lambda )d(-\lambda ),\text{ \ }d(\lambda )=a(\lambda -\eta ),\text{ \ \ }a(\lambda )=\prod_{n=1}^{\mathsf{N}}\sinh (\lambda -\xi _{n}+\eta /2), \label{eigenA}$$and$$g_{\pm }(\lambda )=\frac{\sinh (\lambda +\alpha _{\pm }-\eta /2)\cosh
(\lambda \mp \beta _{\pm }-\eta /2)}{\sinh \alpha _{\pm }\cosh \beta _{\pm }}, \label{g_PM}$$where $\alpha _{\pm }$ and $\beta _{\pm }$ are defined in terms of the boundary parameters by:$$\sinh \alpha _{\pm }\cosh \beta _{\pm }=\frac{\sinh \zeta _{\pm }}{2\kappa
_{\pm }},\text{ \ \ \ \ \ }\cosh \alpha _{\pm }\sinh \beta _{\pm }=\frac{\cosh \zeta _{\pm }}{2\kappa _{\pm }}. \label{alfa-beta}$$
\[normality\]The transfer matrix $\mathcal{T}(\lambda )$ is an even function of the spectral parameter $\lambda $:$$\mathcal{T}(-\lambda )=\mathcal{T}(\lambda ), \label{even-transfer}$$and it is central for the following special values of the spectral parameter: $$\begin{aligned}
\lim_{\lambda \rightarrow \pm \infty }e^{\mp 2\lambda (\mathsf{N}+2)}\mathcal{T}(\lambda ) &=&2^{-(2\mathsf{N}+1)}\frac{\kappa _{+}\kappa
_{-}\cosh (\tau _{+}-\tau _{-})}{\sinh \zeta _{+}\sinh \zeta _{-}},
\label{Central-asymp} \\
\mathcal{T}(\pm \eta /2) &=&(-1)^{\mathsf{N}}2\cosh \eta \mathrm{det}_{q}M(0),
\label{Central-1} \\
\mathcal{T}(\pm (\eta /2-i\pi /2)) &=&-2\cosh \eta \coth \zeta _{-}\coth
\zeta _{+}\mathrm{det}_{q}M(i\pi /2). \label{Central-2}\end{aligned}$$Moreover, the monodromy matrix $\mathcal{U}_{\pm }(\lambda )$ satisfy the following transformation properties under Hermitian conjugation:
- Under the condition $\eta \in i\mathbb{R}$ (massless regime), it holds: $$\mathcal{U}_{\pm }(\lambda )^{\dagger }=\left[ \mathcal{U}_{\pm }(-\lambda
^{\ast })\right] ^{t_{0}}, \label{ml-Hermitian_U}$$ for $\{i\tau _{\pm },i\kappa _{\pm },i\zeta _{\pm },\xi
_{1},...,\xi _{\mathsf{N}}\}\in \mathbb{R}^{\mathsf{N}+3}.$
- Under the condition $\eta \in \mathbb{R}$ (massive regime), it holds: $$\mathcal{U}_{\pm }(\lambda )^{\dagger }=\left[ \mathcal{U}_{\pm }(\lambda
^{\ast })\right] ^{t_{0}}, \label{m-Hermitian_U}$$for $\{\tau _{\pm },\kappa _{\pm },\zeta _{\pm },i\xi
_{1},...,i\xi _{\mathsf{N}}\}\in \mathbb{R}^{\mathsf{N}+3}.$
So under the same conditions on the parameters of the representation it holds: $$\mathcal{T}(\lambda )^{\dagger }=\mathcal{T}(\lambda ^{\ast }),
\label{I-Hermitian_T}$$i.e. $\mathcal{T}(\lambda )$ defines a one-parameter family of normal operators which are self-adjoint both for $\lambda $ real and purely imaginary.
SOV representations for $\mathcal{T}(\protect\lambda )$-spectral problem
------------------------------------------------------------------------
Let us recall here the characterization obtained in [@Nic12b; @Fald-KN13] by SOV method of the spectrum of the transfer matrix $\mathcal{T}(\lambda )$. First we introduce the following notations:$$X_{k,m}^{(i,r)}(\tau _{\pm },\alpha _{\pm },\beta _{\pm })\equiv \left(
-1\right) ^{i}\left( 1-r\right) \eta +\tau _{-}-\tau_{+}+(-1)^{k}(\alpha
_{-}+\beta _{-})-(-1)^{m}(\alpha _{+}-\beta _{+})+i\pi (k+m),
\label{SOV-cond-}$$and by using these linear combinations of the boundary parameters we introduce the set $N_{SOV}\subset\mathbb{C}^6$ of boundary parameters for which the separation of variables cannot be applied directly. More precisely $$(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha_{-},\beta _{-})\in N_{SOV},$$ if $\exists (k,h,m,n)\in \left\{ 0,1\right\} $ such that $$X_{k,m}^{(0,\mathsf{N})}(\tau _{\pm },\alpha _{\pm },\beta _{\pm })=0 \quad\text{and}\quad
X_{h,n}^{(1,\mathsf{N})}(\tau _{\pm },\alpha _{\pm },\beta _{\pm })=0.$$All the results in the following will be obtained for the generic values of the boundary parameters, not belonging to this set. The SOV method applicability can be further extended applying the discrete symmetries discussed in the Section \[sect-descretesym\].
Following [@Fald-KN13] we define the functions:$$\begin{aligned}
g_{a}(\lambda ) &=&\frac{\cosh ^{2}2\lambda -\cosh ^{2}\eta }{\cosh
^{2}2\zeta _{a}^{(0)}-\cosh ^{2}\eta }\,\prod_{\substack{ b=1 \\ b\neq a}}^{\mathsf{N}}\frac{\cosh 2\lambda -\cosh 2\zeta _{b}^{(0)}}{\cosh 2\zeta
_{a}^{(0)}-\cosh 2\zeta _{b}^{(0)}}\quad \text{ \ for }a\in \{1,...,\mathsf{N}\}, \\
\mathbf{A}(\lambda ) &=&(-1)^{\mathsf{N}}\frac{\sinh (2\lambda +\eta )}{\sinh 2\lambda }g_{+}(\lambda )g_{-}(\lambda )a(\lambda )d(-\lambda ),\end{aligned}$$and$$\begin{aligned}
f(\lambda )=& \frac{\cosh 2\lambda +\cosh \eta }{2\cosh \eta }\prod_{b=1}^{\mathsf{N}}\frac{\cosh 2\lambda -\cosh 2\zeta _{b}^{(0)}}{\cosh \eta -\cosh
2\zeta _{b}^{(0)}}\mathbf{A}(\eta /2) \notag \\
& -(-1)^{\mathsf{N}}\frac{\cosh 2\lambda -\cosh \eta }{2\cosh \eta }\prod_{b=1}^{\mathsf{N}}\frac{\cosh 2\lambda -\cosh 2\zeta _{b}^{(0)}}{\cosh
\eta +\cosh 2\zeta _{b}^{(0)}}\mathbf{A}(\eta /2+i\pi /2) \notag \\
& +2^{(1-\mathsf{N})}\frac{\kappa _{+}\kappa _{-}\cosh (\tau _{+}-\tau _{-})}{\sinh \zeta _{+}\sinh \zeta _{-}}(\cosh ^{2}2\lambda -\cosh ^{2}\eta
)\prod_{b=1}^{\mathsf{N}}(\cosh 2\lambda -\cosh 2\zeta _{b}^{(0)}),
\label{f-function}\end{aligned}$$where$$\zeta _{n}^{(h_{n})}=\xi _{n}+(h_{n}-\frac{1}{2})\eta \quad \forall n\in
\{1,...,\mathsf{N}\},\text{ }h_{n}\in \{0,1\}\text{.}$$We can now recall the main result on the characterization of the set $\Sigma
_{\mathcal{T}}$ formed by all the eigenvalue functions of the transfer matrix $\mathcal{T}(\lambda )$.
\[C:T-eigenstates-\]Let $(\tau _{+},\alpha _{+},\beta _{+},\tau
_{-},\alpha _{-},\beta _{-})\in \mathbb{C}^{6}\backslash N_{SOV}$ and let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ be generic:$$\xi _{a}\neq \pm\xi _{b}+r\eta\text{\ \ mod\,}2\pi \text{ \ }\forall a\neq b\in \{1,...,\mathsf{N}\}\,\,\text{and\thinspace \thinspace }r\in \{-1,0,1\}, \label{xi-conditions}$$then $\mathcal{T}(\lambda )$ has simple spectrum and the set of its eigenvalues $\Sigma _{\mathcal{T}}$ is characterized by:$$\Sigma _{\mathcal{T}}=\left\{ \tau (\lambda ):\tau (\lambda )=f(\lambda
)+\sum_{a=1}^{\mathsf{N}}g_{a}(\lambda )x_{a},\text{ \ \ }\forall
\{x_{1},...,x_{\mathsf{N}}\}\in \Sigma _{T}\right\} ,
\label{Interpolation-Form-T}$$where $\Sigma _{T}$ is the set of solutions to the following inhomogeneous system of $\mathsf{N}$ quadratic equations:$$x_{n}\sum_{a=1}^{\mathsf{N}}g_{a}(\zeta _{n}^{(1)})x_{a}+x_{n}f(\zeta
_{n}^{(1)})=q_{n},\text{ \ \ \ }q_{n}=\frac{\mathrm{det}_{q}K_{+}(\xi _{n})\mathrm{det}_{q}\,\mathcal{U}_{-}(\xi _{n})}{\sinh (\eta +2\xi _{n})\sinh (\eta
-2\xi _{n})},\text{ \ \ }\forall n\in \{1,...,\mathsf{N}\},
\label{Quadratic System}$$in $\mathsf{N}$ unknowns $\{x_{1},...,x_{\mathsf{N}}\}$.
Inhomogeneous Baxter equation
==============================
Here we show that the SOV characterization of the spectrum admits an equivalent formulation in terms of a second order functional difference equation of Baxter type:$$\tau (\lambda )Q(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )+F(\lambda ), \label{Inhom-Baxter-Eq}$$which contains a non-zero inhomogeneous term $F(\lambda )$ non-zero for generic integrable boundary conditions and the $Q$-functions are [*trigonometric polynomials*]{}. In this paper we will call $f(\lambda)$ a trigonometric polynomial of degree $\mathsf{M}$ if $e^{\mathsf{M}\lambda}\,f(\lambda)$ is a polynomial of $e^{2\lambda}$ of degree $\mathsf{M}$. Most trigonometric polynomials we will consider in the following sections will be even functions of $\lambda$ and will satisfy an additional condition $f({\lambda}+i\pi)=f({\lambda})$. It is easy to see in this situation that such functions can be written as polynomials of $\cosh 2\lambda$.
Main functions in the functional equation
-----------------------------------------
Let $Q(\lambda )$ be an even trigonometric polynomial of degree $2\mathsf{N}$. It can be written in the following form:$$\begin{aligned}
Q(\lambda )& =\sum_{a=1}^{\mathsf{N}}\prod_{\substack{ b=1 \\ b\neq a}}^{\mathsf{N}}\frac{\cosh 2\lambda -\cosh 2\zeta _{b}^{(0)}}{\cosh 2\zeta
_{a}^{(0)}-\cosh 2\zeta _{b}^{(0)}}Q(\zeta _{a}^{(0)})+2^{\mathsf{N}}\prod_{a=1}^{\mathsf{N}}\left( \cosh 2\lambda -\cosh 2\zeta
_{a}^{(0)}\right) \label{Q-form1} \\
& =2^{\mathsf{N}}\prod_{a=1}^{\mathsf{N}}\left( \cosh 2\lambda -\cosh
2\lambda _{a}\right) , \label{Q-form2}\end{aligned}$$where from now on the $Q(\zeta _{a}^{(0)})$ are arbitrary complex numbers or similarly the $\lambda _{a}$ are arbitrary complex numbers. Then, introducing the function:$$Z_{Q}(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda
)Q(\lambda +\eta )$$we can prove the following Lemma
Let $Q(\lambda )$ be any function of the form $\left( \ref{Q-form2}\right) $ then the associated function $Z_{Q}(\lambda )$ is an even trigonometric polynomial of degree $4\mathsf{N}+4$ of the following form:$$Z_{Q}(\lambda )=\sum_{a=0}^{2(\mathsf{N}+1)}z_{a}\cosh ^{a}2\lambda ,\text{
with }z_{2(\mathsf{N}+1)}=\frac{2\kappa _{+}\kappa _{-}\cosh (\alpha
_{+}+\alpha _{-}-\beta _{+}+\beta _{-}-(\mathsf{N}+1)\eta )}{\sinh \zeta
_{+}\sinh \zeta _{-}}.$$
The fact that the function $Z_{Q}(\lambda )$ is even in $\lambda $ is a trivial consequence of the fact that $Q(\lambda )$ is even; in fact, it holds:$$\begin{aligned}
Z_{Q}(-\lambda ) &=&\mathbf{A}(-\lambda )Q(-\lambda -\eta )+\mathbf{A}(\lambda )Q(-\lambda +\eta ) \notag \\
&=&\mathbf{A}(-\lambda )Q(\lambda +\eta )+\mathbf{A}(\lambda )Q(\lambda
-\eta )=Z_{Q}(\lambda ).\end{aligned}$$The fact that $Z_{Q}(\lambda )$ is indeed a trigonometric polynomial follows from its definition once we observe that $\lambda=0$ is not a singular point and the following identity holds:$$\lim_{\lambda \rightarrow 0}Z_{Q}(\lambda )=2g_{+}(0)g_{-}(0)a(0)a(-\eta
)Q(0)\cosh \eta .$$Now the functional form of $Z_{Q}(\lambda )$ is a consequence of the following identities:$$Z_{Q}(\lambda +i\pi )=Z_{Q}(\lambda ),\text{ \ }\lim_{\lambda \rightarrow
\pm \infty }\frac{Z_{Q}(\lambda )}{e^{\pm 4(\mathsf{N}+1)\lambda }}=\frac{\kappa _{+}\kappa _{-}\cosh (\alpha _{+}+\alpha _{-}-\beta _{+}+\beta _{-}-(\mathsf{N}+1)\eta )}{2^{(2\mathsf{N}+1)}\sinh \zeta _{+}\sinh \zeta _{-}},$$where the second identity follows from:$$\begin{aligned}
\lim_{\lambda \rightarrow \pm \infty }e^{\mp (2\mathsf{N}+4)\lambda }\mathbf{A}(\lambda )& =2^{-2(\mathsf{N}+1)}\frac{\kappa _{+}\kappa _{-}\exp \pm
(\alpha _{+}+\alpha _{-}-\beta _{+}+\beta _{-}+(\mathsf{N}-1)\eta )}{\sinh
\zeta _{+}\sinh \zeta _{-}}, \\
\lim_{\lambda \rightarrow \pm \infty }e^{\mp 2\mathsf{N}\lambda }Q(\lambda
)& =1.\end{aligned}$$
On the need of an inhomogeneous term in the functional equation
---------------------------------------------------------------
Here, we would like to point out that it is simple to define the boundary conditions for which one can prove that the homogeneous version of the Baxter equation $\left( \ref{Inhom-Baxter-Eq}\right) $ does not admit trigonometric polynomial solutions for $\tau (\lambda )\in \Sigma _{\mathcal{T}}$.
\[impossible\_hom\] Assume that the boundary parameters satisfy the following conditions:$$\kappa _{+}\neq 0,\kappa _{-}\neq 0,\text{ \ }Y^{(i,r)}(\tau _{\pm },\alpha
_{\pm },\beta _{\pm })\neq 0\text{ \ }\forall i\in \left\{ 0,1\right\} ,r\in
\mathbb{Z} \label{Inhomogeneous-boundary conditions}$$where we have defined:$$Y^{(i,r)}(\tau _{\pm },\alpha _{\pm },\beta _{\pm })\equiv \tau _{-}-\tau
_{+}+\left( -1\right) ^{i}\left[ \left( \mathsf{N}-1-r\right) \eta +(\alpha
_{-}+\alpha _{+}+\beta _{-}-\beta _{+})\right] ,$$then for any $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ the homogeneous Baxter equation:$$\tau (\lambda )Q(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta ),$$does not admit any (non identically zero) $Q(\lambda )$ of Laurent polynomial form in $e^{\lambda }$.
If we consider the following function:$$Q(\lambda )=\sum_{a=-s}^{r}y_{a}e^{a\lambda },\text{ \ with }r,s\in \mathbb{N}$$we can clearly always chose the coefficients $y_{a}$ such that the r.h.s. of the homogeneous Baxter equation has no poles as required. However, it is enough to consider now the asymptotics:$$\begin{aligned}
\lim_{\lambda \rightarrow +\infty }\frac{\left[ \mathbf{A}(\lambda
)Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )\right] }{e^{(2\mathsf{N}+4+r)\lambda }}& =\frac{y_{r}\kappa _{+}\kappa _{-}\cosh (\alpha
_{+}+\alpha _{-}-\beta _{+}+\beta _{-}+(\mathsf{N}-1-r)\eta )}{2^{2(\mathsf{N}+1)}\sinh \zeta _{+}\sinh \zeta _{-}} \\
\lim_{\lambda \rightarrow +\infty }e^{-(2\mathsf{N}+4+r)\lambda }\tau
(\lambda )Q(\lambda )& =\frac{y_{r}\kappa _{+}\kappa _{-}\cosh (\tau
_{+}-\tau _{-})}{2^{2(\mathsf{N}+1)}\sinh \zeta _{+}\sinh \zeta _{-}}\end{aligned}$$and use the conditions $\left( \ref{Inhomogeneous-boundary conditions}\right) $ to observe that for any $r\in \mathbb{Z}$ the asymptotic of the homogeneous Baxter equation cannot be satisfied which implies the validity of the lemma.
SOV spectrum in terms of the inhomogeneous Baxter equation
----------------------------------------------------------
We introduce now the following function of the boundary parameters:$$F_{0}=\frac{2\kappa _{+}\kappa _{-}\left( \cosh (\tau _{+}-\tau _{-})-\cosh
(\alpha _{+}+\alpha _{-}-\beta _{+}+\beta _{-}-(\mathsf{N}+1)\eta )\right) }{\sinh \zeta _{+}\sinh \zeta _{-}},$$and then the function:$$\begin{aligned}
F(\lambda ) &=&2^{\mathsf{N}}\,F_{0}\,(\cosh ^{2}2\lambda -\cosh ^{2}\eta
)a(\lambda )a(-\lambda )d(-\lambda )d(\lambda ) \\
&=&F_{0}\, (\cosh ^{2}2\lambda -\cosh ^{2}\eta )\prod_{b=1}^{\mathsf{N}}\prod_{i=0}^{1}(\cosh 2\lambda -\cosh 2\zeta _{b}^{(i)}).\end{aligned}$$We introduce also the set of functions $\Sigma _{\mathcal{Q}}$ such that $Q(\lambda)\in\Sigma _{\mathcal{Q}}$ if it has a form $\left( \ref{Q-form2}\right) $ and $$\tau(\lambda)=\frac{Z_{Q}(\lambda )+F(\lambda )}{Q(\lambda )}$$ is a trigonometric polynomial. We are now ready to prove the main theorem of this article:
\[T-eigenvalue-F-eq\]Let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ be generic [(\[xi-conditions\])]{} and let the boundary parameters $(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta
_{-})\in \mathbb{C}^{6}\backslash N_{SOV}$ satisfy the following conditions:$$\kappa _{+}\neq 0,\kappa _{-}\neq 0,\text{ \ }Y^{(i,2r)}(\tau _{\pm },\alpha
_{\pm },\beta _{\pm })\neq 0\text{ \ }\forall i\in \left\{ 0,1\right\} ,r\in
\left\{ 0,...,\mathsf{N}-1\right\} , \label{Inhom-cond-BaxEq}$$then $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ if and only if $\exists
!Q(\lambda )\in \Sigma _{\mathcal{Q}}$ such that $$\tau (\lambda
)Q(\lambda )=Z_{Q}(\lambda )+F(\lambda ).$$
First we prove that if $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ then there is a trigonometric polynomial $Q(\lambda )\in \Sigma _{\mathcal{Q}}$ satisfying the inhomogeneous functional Baxter equation: $$\tau (\lambda )Q(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )+F(\lambda ).$$To prove it we will show that there is the unique set of values $Q(\zeta _{b}^{(0)})$ such that $Q(\lambda)$ of the form (\[Q-form1\]) satisfies this equation. It is straightforward to verify that if $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ and $Q(\lambda )$ has the form $\left( \ref{Q-form2}\right) $ then the left and right hand sides of the above equation are both even trigonometric polynomials of $\lambda
$ and both can be written (using the asymptotic behavior) in the form:$$\frac{2\kappa _{+}\kappa _{-}\cosh (\tau _{+}-\tau _{-})\prod_{b=1}^{2\mathsf{N}+2}(\cosh 2\lambda -\cosh 2y_{b}^{\left( lhs/rhs\right) })}{\sinh
\zeta _{+}\sinh \zeta _{-}}.$$Then to prove that we can introduce a $Q(\lambda )$ of the form $\left( \ref{Q-form2}\right) $ which satisfies the inhomogeneous Baxter equation $\left( \ref{Inhom-Baxter-Eq}\right) $ with $\tau (\lambda )\in \Sigma _{\mathcal{T}} $, we have only to prove that $\left( \ref{Inhom-Baxter-Eq}\right) $ is satisfied in $4\mathsf{N}+4$ different values of $\lambda $. As the [*r.h.s*]{} and [*l.h.s*]{} of $\left( \ref{Inhom-Baxter-Eq}\right) $ are even functions we need to check this identity only for $2N+2$ non-zero points $\mu_j$ such that $\mu_j\neq \pm \mu_k$. It is a simple exercise verify that the equation $\left( \ref{Inhom-Baxter-Eq}\right) $ is satisfied automatically for any $Q(\lambda )$ of the form $\left( \ref{Q-form2}\right) $ in the following two points, $ \eta /2$ and $ \eta
/2+i\pi /2$:$$\tau (\eta /2)Q(\eta /2)=\mathbf{A}(\eta /2)Q(\eta /2-\eta )=\mathbf{A}(\eta /2)Q(\eta /2), \label{System-A}$$and:$$\tau (\eta /2+i\pi /2)Q(\eta /2+i\pi /2)=\mathbf{A}(\eta /2+i\pi /2)Q(i\pi /2-\eta /2) \\
=\mathbf{A}(\eta /2+i\pi /2)Q(\eta /2+i\pi /2) . \label{System-B}$$Indeed, these equations reduce to:$$\tau (\eta /2)=\mathbf{A}(\eta /2),\text{ \ \ \ }\tau (\eta /2+i\pi /2)=\mathbf{A}(\eta /2+i\pi /2)$$and so they are satisfied by definition for any $\tau (\lambda )\in \Sigma _{\mathcal{T}}$. Then we check the explicit form of the equation $\left( \ref{Inhom-Baxter-Eq}\right) $ in the $2\mathsf{N}$ points $ \zeta _{b}^{(0)}$ and $\zeta _{b}^{(1)}$:$$\tau (\zeta _{b}^{(0)})Q(\zeta _{b}^{(0)})=\mathbf{A}(-\zeta _{b}^{(0)})Q(\zeta _{b}^{(0)}+\eta )=\mathbf{A}(-\zeta
_{b}^{(0)})Q(\zeta _{b}^{(1)}),$$and:$$\tau (\zeta _{b}^{(1)})Q(\zeta _{b}^{(1)})=\mathbf{A}(\zeta _{b}^{(1)})Q(\zeta _{b}^{(1)}-\eta )=\mathbf{A}(\zeta
_{b}^{(1)})Q(\zeta _{b}^{(0)}).$$They are equivalent to the following system of equations:$$\begin{aligned}
\frac{\mathbf{A}(\zeta _{b}^{(1)})}{\tau (\zeta _{b}^{(1)})}& =\frac{\tau
(\zeta _{b}^{(0)})}{\mathbf{A}(-\zeta _{b}^{(0)})}\text{ \ \ \ \ \ }\forall
b\in \{1,...,\mathsf{N}\} \label{System1} \\
\frac{Q(\zeta _{b}^{(0)})\tau (\zeta _{b}^{(0)})}{\mathbf{A}(-\zeta
_{b}^{(0)})}& =\sum_{a=1}^{\mathsf{N}}\prod_{\substack{ c=1 \\ c\neq a}}^{\mathsf{N}}\frac{\cosh 2\zeta _{b}^{(1)}-\cosh 2\zeta _{c}^{(0)}}{\cosh
2\zeta _{a}^{(0)}-\cosh 2\zeta _{c}^{(0)}}Q(\zeta _{a}^{(0)})+2^{\mathsf{N}}\prod_{a=1}^{\mathsf{N}}\left( \cosh 2\zeta _{b}^{(1)}-\cosh 2\zeta
_{a}^{(0)}\right) \label{System2}\end{aligned}$$Now using the following quantum determinant identity $$\frac{\det_{q}K_{+}(\lambda-\eta/2)\det_{q}
\mathcal{U}_{-}(\lambda -\eta /2)}{\sinh (2\lambda +\eta )\sinh (2\lambda -\eta )}=\mathbf{A}(\lambda )\mathbf{A}(-\lambda +\eta ).\label{Tot-q-det-tt}$$ it is easy to see that the system of equations $\left( \ref{System1}\right) $ is certainly satisfied as $\tau (\lambda )\in \Sigma _{\mathcal{T}}$, once we recall the SOV characterization (\[Interpolation-Form-T\]) of $\Sigma _{\mathcal{T}}$. Indeed there is a set $\{x_1,\dots,x_n\}$ satisfying the equations (\[Quadratic System\]) and $\tau(\zeta _{b}^{(0)})=x_b$.
So we are left with $\left( \ref{System2}\right) $ a linear system of $\mathsf{N}$ inhomogeneous equations with $\mathsf{N}$ unknowns $Q(\zeta _{a}^{(0)})$. Here, we prove that the matrix of this linear system$$c_{a b}\equiv \prod_{\substack{ c=1 \\ c\neq a}}^{\mathsf{N}}\frac{\cosh
2\zeta _{b}^{(1)}-\cosh 2\zeta _{c}^{(0)}}{\cosh 2\zeta _{a}^{(0)}-\cosh
2\zeta _{c}^{(0)}}-\delta _{a b}\frac{\tau (\zeta _{b}^{(0)})}{\mathbf{A}(-\zeta _{b}^{(0)})}\text{ \ \ \ \ \ }\forall a,b\in \{1,...,\mathsf{N}\}$$ has nonzero determinant for the given $\tau (\lambda
)\in \Sigma _{\mathcal{T}}$. Indeed, let us suppose that for some $\tau (\lambda )\in \Sigma _{\mathcal{T}}$:$$\mathrm{det}_{\mathsf{N}}\left[ c_{a b}\right] =0. \label{det-coeff}$$Then there is at least one nontrivial solution $\{Q(\zeta
_{1}^{(0)}),...,Q(\zeta _{\mathsf{N}}^{(0)})\}\neq \{0,...,0\}$ to the homogeneous system of equations:$$\frac{Q(\zeta _{b}^{(0)})\tau (\zeta _{b}^{(0)})}{\mathbf{A}(-\zeta
_{b}^{(0)})}=\sum_{a=1}^{\mathsf{N}}\prod_{\substack{ c=1 \\ c\neq a}}^{\mathsf{N}}\frac{\cosh 2\zeta _{b}^{(1)}-\cosh 2\zeta _{c}^{(0)}}{\cosh
2\zeta _{a}^{(0)}-\cosh 2\zeta _{c}^{(0)}}Q(\zeta _{a}^{(0)})
\label{System2-homo}$$and hence we can define:$$Q_{\mathsf{M}}(\lambda )=\sum_{a=1}^{\mathsf{N}}\prod_{\substack{ b=1 \\ b\neq a}}^{\mathsf{N}}\frac{\cosh 2\lambda -\cosh 2\zeta _{b}^{(0)}}{\cosh
2\zeta _{a}^{(0)}-\cosh 2\zeta _{b}^{(0)}}Q(\zeta _{a}^{(0)})=\lambda _{\mathsf{M}+1}^{(\mathsf{M})}\prod_{b=1}^{\mathsf{M}}\left( \cosh 2\lambda
-\cosh 2\lambda _{b}^{(\mathsf{M})}\right) .$$It is an even trigonometric polynomial of degree $2\mathsf{M}$ such that $0\leq
\mathsf{M}\leq \mathsf{N}-1$ fixed by the solution $\{Q(\zeta
_{1}^{(0)}),...,Q(\zeta _{\mathsf{N}}^{(0)})\}$. Now using the $Q_{\mathsf{M}}(\lambda )$ and $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ we can define two functions:$$W_{1}(\lambda )=Q_{\mathsf{M}}(\lambda )\tau (\lambda )\text{ \, and \, }W_{2}(\lambda )=\mathbf{A}(\lambda )Q_{\mathsf{M}}(\lambda -\eta )+\mathbf{A}(-\lambda )Q_{\mathsf{M}}(\lambda +\eta )$$which are both even trigonometric polynomials of degree $2\mathsf{M}+2\mathsf{N}+4$. Then it is straightforward to observe that the systems of equations $\left( \ref{System1}\right) $ and $\left( \ref{System2-homo}\right) $ plus the conditions $\left( \ref{System-A}\right) $ and $\left( \ref{System-B}\right)
$, which are also satisfied with the function $Q_{\mathsf{M}}(\lambda )$, imply that $W_{1}(\lambda )$ and $W_{2}(\lambda )$ coincide in $4\mathsf{N}+4 $ different values of $\lambda $ ($\pm \eta /2$, $\pm (\eta /2+i\pi /2)$, $\pm \zeta _{b}^{(0)}$ and $\pm \zeta _{b}^{(1)}$). It means that $W_{1}(\lambda )\equiv W_{2}(\lambda )$, as these are two polynomials of maximal degree $4\mathsf{N}+2$. So, we have shown that from the assumption $\exists \tau (\lambda )\in \Sigma _{\mathcal{T}}$ such that $\left( \ref{det-coeff}\right) $ holds it follows that $\tau (\lambda )$ and $Q_{\mathsf{M}}(\lambda )$ have to satisfy the following homogeneous Baxter equations: $$\tau (\lambda )Q_{\mathsf{M}}(\lambda )=\mathbf{A}(\lambda )Q_{\mathsf{M}}(\lambda -\eta )+\mathbf{A}(-\lambda )Q_{\mathsf{M}}(\lambda +\eta ).
\label{Baxter-eq-homo}$$Now we can apply the Lemma \[impossible\_hom\] which implies that $Q_{\mathsf{M}}(\lambda )=0$ for any $\lambda$, which contradicts the hypothesis of the existence of a nontrivial solution to the homogeneous system [(\[System2-homo\])]{}. Hence, we have proven that $\mathrm{det}_{\mathsf{N}}\left[ c_{a b}\right] \neq 0.$ Therefore there is a unique solution $\{Q(\zeta
_{1}^{(0)}),...,Q(\zeta _{\mathsf{N}}^{(0)})\}$ of the inhomogeneous system $\left( \ref{System2}\right) $ which defines one and only one $Q(\lambda )$ of the form $\left( \ref{Q-form1}\right) $ satisfying the functional inhomogeneous Baxter’s equation $\left( \ref{Inhom-Baxter-Eq}\right) $.
We prove now that if $Q(\lambda )\in \Sigma _{\mathcal{Q}}$ then $\tau (\lambda )=\left( Z_{Q}(\lambda )+F(\lambda )\right) /Q(\lambda )\in
\Sigma _{\mathcal{T}}$. By definition of the functions $Z_{Q}(\lambda ),$ $F(\lambda )$ and $Q(\lambda )$ the function $\tau (\lambda )$ has the desired form:$$\tau (\lambda )=f(\lambda )+\sum_{a=1}^{\mathsf{N}}g_{a}(\lambda )\tau
(\zeta _{a}^{(0)}).$$To prove now that $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ we have to write the inhomogeneous Baxter equation $\left( \ref{Inhom-Baxter-Eq}\right) $ in the $2\mathsf{N}$ points $ \zeta _{b}^{(0)}$ and $ \zeta
_{b}^{(1)}$. Indeed, we have already proved that this reproduce the systems $\left( \ref{System1}\right) $ and $\left( \ref{System2}\right) $ and it is simple to observe that the system of equations $\left( \ref{System1}\right) $ just coincides with the inhomogeneous system of $\mathsf{N}$ quadratic equations:$$x_{n}\sum_{a=1}^{\mathsf{N}}g_{a}(\zeta _{n}^{(1)})x_{a}+x_{n}f(\zeta
_{n}^{(1)})=q_{n},\text{ \ \ \ }\forall n\in \{1,...,\mathsf{N}\},$$once we define $x_{a}=\tau (\zeta _{a}^{(0)})$ for any $a\in \{1,...,\mathsf{N}\}$ and we write $\tau (\zeta _{n}^{(1)})$ in terms of the $x_{a}$. Thus we show that $$\tau (\lambda )=\left( Z_{Q}(\lambda )+F(\lambda )\right)
/Q(\lambda )\in \Sigma _{\mathcal{T}},$$ completing the proof of the theorem.
Completeness of the Bethe ansatz equations
------------------------------------------
In the previous section we have shown that to solve the transfer matrix spectral problem associated to the most general representations of the trigonometric 6-vertex reflection algebra we have just to classify the set of functions $Q(\lambda )$ of the form $\left( \ref{Q-form2}\right) $ for which $\left( Z_{Q}(\lambda )+F(\lambda )\right) /Q(\lambda )$ is a trigonometric polynomial; i.e. the set of functions $\Sigma _{\mathcal{Q}}$ completely fixes the set $\Sigma _{\mathcal{T}}$. We can show now that the previous characterization of the transfer matrix spectrum allows to prove that $\Sigma _{InBAE}\subset \mathbb{C}^{\mathsf{N}}$ the set of all the solutions of inhomogeneous Bethe equations $$\{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in \Sigma _{InBAE}$$ if $$\mathbf{A}(\lambda _{a})Q_{\mathbf{\lambda }}(\lambda _{a}-\eta )+\mathbf{A}(-\lambda _{a})Q_{\mathbf{\lambda }}(\lambda
_{a}+\eta )=-F(\lambda _{a}),\text{ \ }\forall a\in \{1,...,\mathsf{N}\} , \label{I-BAE}$$ defines the complete set of transfer matrix eigenvalues. In particular, the following corollary follows:
\[Theo-InBAE\] Let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ be generic $\left( \ref{xi-conditions}\right) $ and let the boundary parameters $(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha
_{-},\beta _{-})\in \mathbb{C}^{6}\backslash N_{SOV}$ satisfy $\left( \ref{Inhom-cond-BaxEq}\right) $ then $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ if and only if $\exists
!\{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in \Sigma _{InBAE}$ such that:$$\tau (\lambda )=\frac{Z_{Q}(\lambda )+F(\lambda )}{Q(\lambda )}\text{ \ \
with \ \ }Q(\lambda )=2^{\mathsf{N}}\prod_{a=1}^{\mathsf{N}}\left( \cosh
2\lambda -\cosh 2\lambda _{a}\right) .$$Moreover, under the condition of normality defined in Proposition [normality]{}, the set $\Sigma _{InBAE}$ of all the solutions to the inhomogeneous system of Bethe equations $\left( \ref{I-BAE}\right) $ contains $2^{\mathsf{N}}$ elements.
Homogeneous Baxter equation
===========================
Boundary conditions annihilating the inhomogeneity of the Baxter equation
-------------------------------------------------------------------------
The description presented in the previous sections can be applied to completely general integrable boundary terms including as a particular case the boundary conditions for which the inhomogeneous term in the functional Baxter equation vanishes. As these are still quite general boundary conditions it is interesting to point out how the previous general results explicitly look like in these cases.
\[homogeneousBE\_N\] Let $(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta _{-})\in
\mathbb{C}^{6}\backslash N_{SOV}$ satisfying the condition:$$\kappa _{+}\neq 0,\kappa _{-}\neq 0,\text{ \ }\exists i\in \left\{
0,1\right\} \text{\ }:Y^{(i,2\mathsf{N})}(\tau _{\pm },\alpha _{\pm },\beta
_{\pm })=0 \label{ond-homo-boundary}$$and let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}
$ $^{\mathsf{N}}$ be generic [(\[xi-conditions\])]{}, then $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ if and only if $ \exists
!Q(\lambda )\in \Sigma _{\mathcal{Q}}$ such that $$\tau (\lambda
)Q(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda
)Q(\lambda +\eta ).$$Or equivalently, $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ if and only if $\exists !\{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in \Sigma _{BAE}$ such that:$$\tau (\lambda )=\frac{\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )}{Q(\lambda )}\text{ \ \ with \ \ }Q(\lambda
)=2^{\mathsf{N}}\prod_{a=1}^{\mathsf{N}}\left( \cosh 2\lambda -\cosh
2\lambda _{a}\right) .$$where:$$\Sigma _{BAE}=\left\{ \{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in \mathbb{C}^{\mathsf{N}}:\mathbf{A}(\lambda _{a})Q_{\mathbf{\lambda }}(\lambda
_{a}-\eta )+\mathbf{A}(-\lambda _{a})Q_{\mathbf{\lambda }}(\lambda _{a}+\eta
)=0,\text{ \ }\forall a\in \{1,...,\mathsf{N}\}\right\} . \label{BAE}$$Moreover, under the condition of normality defined in Proposition [normality]{}, the set $\Sigma _{BAE}$ of the solutions to the homogeneous system of Bethe ansatz type equations $\left( \ref{BAE}\right) $ contains $2^{\mathsf{N}}$ elements.
This theorem is just a rewriting of the results presented in the Theorem \[T-eigenvalue-F-eq\] and Corollary \[Theo-InBAE\] for the case of vanishing inhomogeneous term. Indeed if the conditions $\left( \ref{BAE}\right) $ are satisfied then automatically the conditions of the main theorem $\left( \ref{Inhom-cond-BaxEq}\right) $ are satisfied too that implies that the map from the $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ to the $\{\lambda
_{1},...,\lambda _{\mathsf{N}}\}\in \Sigma _{BAE}$ is indeed an isomorphism.
More general boundary conditions compatibles with homogeneous Baxter equations
------------------------------------------------------------------------------
We address here the problem of describing the boundary conditions:$$\kappa _{+}\neq 0,\kappa _{-}\neq 0,\text{ \ }\exists i\in \left\{
0,1\right\} ,\mathsf{M}\in \left\{ 0,...,\mathsf{N}-1\right\} :Y^{(i,2\mathsf{M})}(\tau _{\pm },\alpha _{\pm },\beta _{\pm })=0,
\label{Cond-homo-M}$$for which the conditions $\left( \ref{Inhom-cond-BaxEq}\right) $ are not satisfied and then the Theorem \[T-eigenvalue-F-eq\] cannot be directly applied. In these $2\mathsf{N}$ hyperplanes in the space of the boundary parameters we have just to modify this theorem to take into account that the Baxter equation associated to the choice of coefficient $\mathbf{A}(\lambda
) $ is indeed compatible with the homogeneous Baxter equation for a special choice of the polynomial $Q(\lambda )$. First we define the following functions$$Q_{\mathsf{M}}(\lambda )=2^{\mathsf{M}}\prod_{b=1}^{\mathsf{M}}\left( \cosh
2\lambda -\cosh 2\lambda _{b}^{(\mathsf{M})}\right) . \label{Q-form-M}$$We introduce also the set of polynomials $\Sigma _{\mathcal{Q}}^{\mathsf{M}}$ such that $Q_{\mathsf{M}}(\lambda )\in \Sigma _{\mathcal{Q}}^{\mathsf{M}}$ if $Q_{\mathsf{M}}(\lambda )$ has a form $\left( \ref{Q-form-M}\right)$ and $$\tau (\lambda )=\frac{\mathbf{A}(\lambda )Q_{\mathsf{M}}(\lambda -\eta )+\mathbf{A}(-\lambda )Q_{\mathsf{M}}(\lambda +\eta )}{Q_{\mathsf{M}}(\lambda )}$$is a trigonometric polynomial. Then we can define the corresponding set $\Sigma _{\mathcal{T}}^{\mathsf{M}}$$$\Sigma _{\mathcal{T}}^{\mathsf{M}}=\left\{ \tau (\lambda ):\tau (\lambda
)\equiv \frac{\mathbf{A}(\lambda )Q_{\mathsf{M}}(\lambda -\eta )+\mathbf{A}(-\lambda )Q_{\mathsf{M}}(\lambda +\eta )}{Q_{\mathsf{M}}(\lambda )}\text{
\, if }Q_{\mathsf{M}}(\lambda )\in \Sigma _{\mathcal{Q}}^{\mathsf{M}}\right\}
.$$It is simple to prove the validity of the following:
\[mixed-condition\] Let the boundary conditions $\left( \ref{Cond-homo-M}\right) $ be satisfied, then $\Sigma _{\mathcal{T}}^{\mathsf{M}}\subset\Sigma _{\mathcal{T}}$ and moreover for any $\tau (\lambda )\in \Sigma _{\mathcal{T}}^{\mathsf{M}}$ there exists one and only one $Q_{\mathsf{M}}(\lambda )\in \Sigma _{\mathcal{Q}}^{\mathsf{M}}$ such that:$$\label{homogen-Bax-eq-M}
\tau (\lambda )Q_{\mathsf{M}}(\lambda )=\mathbf{A}(\lambda )Q_{\mathsf{M}}(\lambda -\eta )+\mathbf{A}(-\lambda )Q_{\mathsf{M}}(\lambda +\eta ),$$and for any $\tau (\lambda )\in \Sigma _{\mathcal{T}}\backslash \Sigma _{\mathcal{T}}^{\mathsf{M}}$ there exists one and only one $Q(\lambda )\in
\Sigma _{\mathcal{Q}}$ such that:$$\label{inhomogen-Bax-eq-M}
\tau (\lambda )Q(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )+F(\lambda ).$$
The proof follows the one given for the main Theorem \[T-eigenvalue-F-eq\] we have just to observe that thanks to the boundary conditions $\left( \ref{Cond-homo-M}\right) $ the set $\Sigma _{\mathcal{T}}^{\mathsf{M}}$ is formed by transfer matrix eigenvalues as the Baxter equation implies that for any $\tau (\lambda )\in \Sigma _{\mathcal{T}}^{\mathsf{M}}$ the systems of equations $\left( \ref{System-A}\right) ,$ $\left( \ref{System-B}\right) $ and $\left( \ref{System1}\right) $ are satisfied and moreover that the asymptotics of the $\tau (\lambda )\in \Sigma _{\mathcal{T}}^{\mathsf{M}}$ is exactly that of the transfer matrix eigenvalues.
Finally, it is interesting to remark that under the boundary conditions $\left( \ref{Inhom-cond-BaxEq}\right) $ the complete characterization of the spectrum of the transfer matrix is given in terms of the even polynomials $Q(\lambda )$ all of fixed degree $2\mathsf{N}$ and form $\left( \ref{Q-form2}\right) $ which are solutions of the inhomogeneous/homogeneous Baxter equation. However, in the cases when the boundary parameters satisfy the constraints $\left( \ref{Cond-homo-M}\right) $ for a given $\mathsf{M}\in \left\{ 0,...,\mathsf{N}-1\right\} $ a part of the transfer matrix spectrum can be defined by polynomials of smaller degree; i.e. the $Q_{\mathsf{M}}(\lambda
)\in \Sigma _{\mathcal{Q}}^{\mathsf{M}}$ for the fixed $\mathsf{M}\in
\left\{ 0,...,\mathsf{N}-1\right\} $.
Discrete symmetries and equivalent Baxter equations {#sect-descretesym}
===================================================
It is important to point out that we have some large amount of freedom in the choice of the functional reformulation of the SOV characterization of the transfer matrix spectrum. We have reduced it looking for trigonometric polynomial solutions $Q(\lambda )$ of the second order difference equations with coefficients $\mathbf{A}(\lambda )$ which are rational trigonometric functions. It makes the finite difference terms $\mathbf{A}(\lambda
)Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )$ in the functional equation a trigonometric polynomial. Indeed, this assumption reduces the possibility to use the following gauge transformations of the coefficients allowed instead by the SOV characterization:$$\mathbf{A}_{\alpha }(\lambda )=\alpha (\lambda )\mathbf{A}(\lambda ),\text{
\ }\mathbf{D}_{\alpha }(\lambda )=\frac{\mathbf{A}(-\lambda )}{\alpha
(\lambda +\eta )}.$$In the following we discuss simple transformations that do not modify the functional form of the coefficients allowing equivalent reformulations of the SOV spectrum by Baxter equations.
Discrete symmetries of the transfer matrix spectrum
---------------------------------------------------
It is not difficult to see that the spectrum (eigenvalues) of the transfer matrix presents the following invariance:
\[Lem-invariance\]We denote explicitly the dependence from the boundary parameters in the set of boundary parameters $\Sigma _{\mathcal{T}}^{(\tau _{+},\alpha
_{+},\beta _{+},\tau _{-},\alpha _{-},\beta _{-})}$ of the eigenvalue functions of the transfer matrix $\mathcal{T}(\lambda )$, then this set is invariant under the following $Z_{2}^{\otimes 3}$ transformations of the boundary parameters:$$\begin{aligned}
&\Sigma _{\mathcal{T}}^{(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha
_{-},\beta _{-})}\equiv \Sigma _{\mathcal{T}}^{(\epsilon _{\tau }\tau
_{+},\epsilon _{\alpha }\alpha _{+},\epsilon _{\beta }\beta _{+},\epsilon
_{\tau }\tau _{-},\epsilon _{\alpha }\alpha _{-},\epsilon _{\beta }\beta
_{-})}\ \\ &
\forall (\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta
})\in \{-1,1\}\times \{-1,1\}\times \{-1,1\}.\nonumber\end{aligned}$$
To prove this statement it is enough to look at the SOV characterization which defines completely the transfer matrix spectrum, i.e. the set $\Sigma _{\mathcal{T}}$, and to prove that it is invariant under the above considered $Z_{2}^{\otimes 3}$ transformations of the boundary parameters. We have first to remark that the central values $\left( \ref{Central-asymp}\right) $-$\left( \ref{Central-2}\right) $ of the transfer matrix $\mathcal{T}(\lambda )$ are invariant under these discrete transformations and then the function $f(\lambda )$, defined in $\left( \ref{f-function}\right) $, is invariant too and the same is true for the form $\left( \ref{Interpolation-Form-T}\right) $ of the interpolation polynomial describing the elements of $\Sigma _{\mathcal{T}}$. Then the invariance of the SOV characterization $\left( \ref{Quadratic System}\right) $ follows from the invariance of the quantum determinant$$\begin{aligned}
\mathrm{det}_{q}K_{+}(\lambda )\mathrm{det}_{q}\,\mathcal{U}_{-}(\lambda )
&=&\sinh (2\eta -2\lambda )\sinh (2\lambda +2\eta )g_{+}(\lambda +\eta
/2)g_{+}(-\lambda +\eta /2)g_{-}(\lambda +\eta /2) \notag \\
&&\times g_{-}(-\lambda +\eta /2)a(\lambda +\eta /2)d(\lambda -\eta
/2)a(-\lambda +\eta /2)d(-\lambda -\eta /2)\end{aligned}$$ under these discrete transformations.
It is important to underline that the above $Z_{2}^{\otimes 3}$ transformations of the boundary parameters do indeed change the transfer matrix $\mathcal{T}(\lambda )$ and the Hamiltonian and so this invariance is equivalent to the statement that these different transfer matrices are all isospectral. In particular, it is simple to find the similarity matrices implementing the following $Z_{2}$ transformations of the boundary parameters:$$\begin{aligned}
\mathcal{T}(\lambda |-\tau _{+},-\zeta _{+},\kappa _{+},-\tau _{-},-\zeta
_{-},\kappa _{-}) &=&\Gamma _{y}\mathcal{T}(\lambda |\tau _{+},\zeta
_{+},\kappa _{+},\tau _{-},\zeta _{-},\kappa _{-})\Gamma _{y},\text{ \ \ \ }\Gamma _{y}\equiv \otimes _{n=1}^{\mathsf{N}}\sigma _{n}^{y}, \\
\mathcal{T}(\lambda |\tau _{+},\zeta _{+},-\kappa _{+},\tau _{-},\zeta
_{-},-\kappa _{-}) &=&\Gamma _{z}\mathcal{T}(\lambda |\tau _{+},\zeta
_{+},\kappa _{+},\tau _{-},\zeta _{-},\kappa _{-})\Gamma _{z},\text{ \ \ \ }\Gamma _{z}\equiv \otimes _{n=1}^{\mathsf{N}}\sigma _{n}^{z}.\end{aligned}$$
Equivalent Baxter equations and the SOV spectrum
------------------------------------------------
The invariance of the spectrum $\Sigma _{\mathcal{T}}$ under these $Z_{2}^{\otimes 3}$ transformations of the boundary parameters can be used to define equivalent Baxter equation reformulation of $\Sigma _{\mathcal{T}}$. More precisely, let us introduce the following functions $\mathbf{A}_{(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta
})}(\lambda )$ and $F_{(\epsilon _{\tau },\epsilon _{\alpha },\epsilon
_{\beta })}(\lambda )$ obtained respectively by implementing the $Z_{2}^{\otimes 3}$ transformations:$$(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta
_{-})\rightarrow (\epsilon _{\tau }\tau _{+},\epsilon _{\alpha }\alpha
_{+},\epsilon _{\beta }\beta _{+},\epsilon _{\tau }\tau _{-},\epsilon
_{\alpha }\alpha _{-},\epsilon _{\beta }\beta _{-}),$$then the following characterizations hold for any fixed $(\epsilon _{\tau
},\epsilon _{\alpha },\epsilon _{\beta })\in \{-1,1\}\times \{-1,1\}\times
\{-1,1\}$:
\[T-eigenvalue-F-eq-gen\]Let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ be generic [(\[xi-conditions\])]{} and let the boundary parameters $(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta
_{-})\in \mathbb{C}^{6}\backslash N_{SOV}$ satisfy the following conditions:$$\kappa _{+}\neq 0,\kappa _{-}\neq 0,\text{ \ }Y^{(i,2r)}(\epsilon _{\tau
}\tau _{\pm },\epsilon _{\alpha }\alpha _{\pm },\epsilon _{\beta }\beta
_{\pm })\neq 0\text{ \ }\forall i\in \left\{ 0,1\right\} ,r\in \left\{ 0,...,\mathsf{N}-1\right\} , \label{Inhom-cond-BaxEq-gen}$$then $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ if and only if $\exists
!Q(\lambda )\in \Sigma _{\mathcal{Q}}$ such that $$\tau (\lambda
)Q(\lambda )=Z_{Q,(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta
})}(\lambda )+F_{(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta
})}(\lambda ),$$where:$$Z_{Q,(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta })}(\lambda )=\mathbf{A}_{(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta
})}(\lambda )(\lambda )Q(\lambda -\eta )+\mathbf{A}_{(\epsilon _{\tau
},\epsilon _{\alpha },\epsilon _{\beta })}(-\lambda )Q(\lambda +\eta ).$$
The proof follows step by step the one given for the main Theorem \[T-eigenvalue-F-eq\].
General validity of the inhomogeneous Baxter equations
------------------------------------------------------
The previous reformulations of the spectrum in terms of different inhomogeneous Baxter equations and the observation that the conditions under which the Theorem does not apply are related to the choice of the $(\epsilon
_{\tau },\epsilon _{\alpha },\epsilon _{\beta })\in \{-1,1\}\times
\{-1,1\}\times \{-1,1\}$ allow us to prove that unless the boundary parameters are lying on a finite lattice of step $\eta $ we can always use an inhomogeneous Baxter equations to completely characterize the spectrum of the transfer matrix. More precisely, let us introduce the following hyperplanes in the space of the boundary parameters:$$M\equiv \left\{
\begin{array}{l}
(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta _{-})\in
\mathbb{C}^{6}:\exists (r_{+,+},r_{-,+},r_{-,-})\in \{0,...,\mathsf{N}-1\}
\\
\text{ such that: \ \ \ }\left\{
\begin{array}{l}
r_{+,+}+r_{-,-}-r_{-,+}\in \{0,...,\mathsf{N}-1\} \\
\alpha _{+}+\alpha _{-}=(r_{-,+}-r_{+,+})\eta \\
\beta _{-}-\beta _{+}=(r_{-,-}-r_{-,+})\eta \\
\tau _{-}-\tau _{+}=(\mathsf{N}-1+r_{-,-}-3r_{+,+})\eta\end{array}\right.\end{array}\right\} \label{Def-M}$$then the following theorem holds:
Let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ satisfy the conditions [(\[xi-conditions\])]{} and let $(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta
_{-})\in \mathbb{C}^{6}\backslash \left( M\cup N_{SOV}\right) $ then we can always find a $(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta })\in
\{-1,1\}\times \{-1,1\}\times \{-1,1\}$ such that $\tau (\lambda )\in\Sigma _{\mathcal{T}}$ if and only if $\exists
!Q(\lambda )\in \Sigma _{\mathcal{Q}}$ such that $$\tau (\lambda
)Q(\lambda )=Z_{Q,(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta
})}(\lambda )+F_{(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta
})}(\lambda ).$$
The Theorem \[T-eigenvalue-F-eq-gen\] does not apply if $\exists i\in
\left\{ 0,1\right\} $ and $\exists r\in \left\{ 0,...,\mathsf{N}-1\right\} $ such that the following system of conditions on the boundary parameters are satisfied:$$Y^{(i,2r)}(\epsilon _{\tau }\tau _{\pm },\epsilon _{\alpha }\alpha _{\pm
},\epsilon _{\beta }\beta _{\pm })=0\text{ \ }\forall (\epsilon _{\tau
},\epsilon _{\alpha },\epsilon _{\beta })\in \{-1,1\}^{\otimes 3}
\label{Cond-general-homo}$$ then by simple computations it is possible to observe that the set $M$ defined in $\left( \ref{Def-M}\right) $ indeed coincides with the following set:$$\left\{ (\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta
_{-})\in \mathbb{C}^{6}:\text{ }\exists i\in \left\{ 0,1\right\} ,r\in
\left\{ 0,...,\mathsf{N}-1\right\} \text{ such that }\left( \ref{Cond-general-homo}\right) \text{ is satisfied}\right\} ,$$from which the theorem clearly follows.
Homogeneous Baxter equation
---------------------------
The discrete symmetries of the transfer matrix allow also to define the general conditions on the boundary parameters for which the spectrum can be characterized by a homogeneous Baxter equation. In particular the following corollary holds:
Let $(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta _{-})\in
\mathbb{C}^{6}\backslash N_{SOV}$ satisfy the condition:$$\begin{aligned}
&\kappa _{+}\neq 0,\kappa _{-}\neq 0,\nonumber\\
&\exists i\in \left\{
0,1\right\} ,\text{\ }\exists (\epsilon _{\tau },\epsilon _{\alpha
},\epsilon _{\beta })\in \{-1,1\}\times \{-1,1\}\times \{-1,1\}:Y^{(i,2\mathsf{N})}(\epsilon _{\tau }\tau _{\pm },,\epsilon _{\alpha }\alpha _{\pm
},\epsilon _{\beta }\beta _{\pm })=0\end{aligned}$$and let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}
$ $^{\mathsf{N}}$ be generic [(\[xi-conditions\])]{}, then $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ if and only if $\exists
!Q(\lambda )\in \Sigma _{\mathcal{Q}}$ such that $$\tau (\lambda
)Q(\lambda )=\mathbf{A}_{(\epsilon _{\tau },\epsilon _{\alpha },\epsilon
_{\beta })}(\lambda )(\lambda )Q(\lambda -\eta )+\mathbf{A}_{(\epsilon
_{\tau },\epsilon _{\alpha },\epsilon _{\beta })}(-\lambda )Q(\lambda +\eta
).$$Or equivalently we can define the set of all the solutions of the Bethe equations $$\Sigma _{BAE}=\left\{ \{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in \mathbb{C}^{\mathsf{N}}:\mathbf{A}(\lambda _{a})Q_{\mathbf{\lambda }}(\lambda
_{a}-\eta )+\mathbf{A}(-\lambda _{a})Q_{\mathbf{\lambda }}(\lambda _{a}+\eta
)=0,\text{ \ }\forall a\in \{1,...,\mathsf{N}\}\right\} .$$Then $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ if and only if $\exists !\{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in \Sigma _{BAE}$ such that:$$\tau (\lambda )=\frac{\mathbf{A}_{(\epsilon _{\tau },\epsilon _{\alpha
},\epsilon _{\beta })}(\lambda )Q(\lambda -\eta )+\mathbf{A}_{(\epsilon
_{\tau },\epsilon _{\alpha },\epsilon _{\beta })}(-\lambda )Q(\lambda +\eta )}{Q(\lambda )},$$with $$Q(\lambda )=2^{\mathsf{N}}\prod_{a=1}^{\mathsf{N}}\left( \cosh 2\lambda -\cosh 2\lambda _{a}\right).$$ Moreover, under the condition of normality defined in Proposition [normality]{}, the set $\Sigma _{BAE}$ of the solutions to the homogeneous system of Bethe ansatz type equations $\left( \ref{BAE}\right) $ contains $2^{\mathsf{N}}$ elements.
XXX chain by SOV and Baxter equation
====================================
The construction of the SOV characterization can be naturally applied in the case of the rational 6-vertex $R$-matrix, which in the homogeneous limit reproduces the XXX open quantum spin-1/2 chain with general integrable boundary conditions[^4]. Let us define:$$R_{12}(\lambda )=\left(
\begin{array}{cccc}
\lambda +\eta & 0 & 0 & 0 \\
0 & \lambda & \eta & 0 \\
0 & \eta & \lambda & 0 \\
0 & 0 & 0 & \lambda +\eta\end{array}\right) \in \text{End}(\mathcal{H}_{1}\otimes \mathcal{H}_{2}).$$Due to the $SU(2)$ invariance of the bulk monodromy matrix the boundary matrices defining the most general integrable boundary conditions can be always recasted in the following form:$$K_{-}(\lambda ;p)=\left(
\begin{array}{cc}
\lambda -\eta /2+p & 0 \\
0 & p-\lambda +\eta /2\end{array}\right) ,\text{ \ \ \ }K_{+}(\lambda ;q,\xi )=\left(
\begin{array}{cc}
\lambda +\eta /2+q & \xi (\lambda +\eta /2) \\
\xi (\lambda +\eta /2) & q-(\lambda +\eta /2)\end{array}\right) ,$$leaving only three arbitrary complex parameters here denoted with $\xi ,$ $p$ and $q$. Then the one parameter family of commuting transfer matrices:$$\mathcal{T}(\lambda )=\text{tr}_{0}\{K_{+}(\lambda )\,M(\lambda
)\,K_{-}(\lambda )\hat{M}(\lambda )\}\in \text{\thinspace End}(\mathcal{H}),$$in the homogeneous limit leads to the following Hamiltonian:$$H=\sum_{n=1}^{\mathsf{N}}\left( \sigma _{n}^{x}\sigma _{n+1}^{x}+\sigma
_{n}^{y}\sigma _{n+1}^{y}+\sigma _{n}^{z}\sigma _{n+1}^{z}\right) +\frac{\sigma _{\mathsf{N}}^{z}}{p}+\frac{\sigma _{1}^{z}+\xi \sigma _{1}^{x}}{q}.$$It is simple to show that the following identities hold:$$\mathrm{det}_{q}K_{+}(\lambda )\mathrm{det}_{q}\,\mathcal{U}_{-}(\lambda
)=4(\lambda ^{2}-\eta ^{2})(\lambda ^{2}-p^{2})((1+\xi ^{2})\lambda
^{2}-q^{2})\prod_{b=1}^{\mathsf{N}}(\lambda ^{2}-(\xi _{n}+\eta
)^{2})(\lambda ^{2}-(\xi _{n}-\eta )^{2}).$$We define:$$\mathbf{A}(\lambda )=(-1)^{\mathsf{N}}\frac{2\lambda +\eta }{2\lambda }(\lambda -\eta /2+p)(\sqrt{(1+\xi ^{2})}(\lambda -\eta /2)+q)\prod_{b=1}^{\mathsf{N}}(\lambda -\zeta _{b}^{(0)})(\lambda +\zeta _{b}^{(1)}),$$then it is easy to derive the following quantum determinant identity:$$\frac{\mathrm{det}_{q}K_{+}(\lambda )\mathrm{det}_{q}\,\mathcal{U}_{-}(\lambda )}{(4\lambda ^{2}-\eta ^{2})}=\mathbf{A}(\lambda +\eta /2)\mathbf{A}(-\lambda
+\eta /2).$$ From the form of the boundary matrices it is clear that for the rational 6-vertex case one can directly derive the SOV representations using the method developed in [@Nic12b] without any need to introduce Baxter’s gauge transformations. Some results in this case also appeared in [@FraSW08; @FraGSW11] based on a functional version of the separation of variables of Sklyanin, a method which allows to define the eigenvalues and wave-functions but which does not allow to construct in the original Hilbert space of the quantum chain the transfer matrix eigenstates.
The separation of variable description in this rational 6-vertex case reads:
\[C:T-eigenstates- copy(1)\]Let the inhomogeneities $\{\xi _{1},...,\xi
_{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ be generic: $$\xi _{a}\neq \pm\xi _{b}+r\eta \text{ \ }\forall a\neq b\in \{1,...,\mathsf{N}\}\,\,\text{and\thinspace \thinspace }r\in \{-1,0,1\}, \label{xi-conditions-xxx}$$ then $\mathcal{T}(\lambda )$ has simple spectrum and $\Sigma _{\mathcal{T}}$ is characterized by:$$\Sigma _{\mathcal{T}}=\left\{ \tau (\lambda ):\tau (\lambda )=f(\lambda
)+\sum_{a=1}^{\mathsf{N}}g_{a}(\lambda )x_{a},\text{ \ \ }\forall
\{x_{1},...,x_{\mathsf{N}}\}\in \Sigma _{T}\right\} ,$$where:$$g_{a}(\lambda )=\frac{4\lambda ^{2}-\eta ^{2}}{4{\zeta
_{a}^{(0)}} ^{2}-\eta ^{2}}\,\prod_{\substack{ b=1 \\ b\neq a}}^{\mathsf{N}}\frac{\lambda ^{2}-{ \zeta _{b}^{(0)}} ^{2}}{{
\zeta _{a}^{(0)}} ^{2}-{ \zeta _{b}^{(0)}} ^{2}}\quad \text{
\ for }a\in \{1,...,\mathsf{N}\},$$and$$f(\lambda )=\prod_{b=1}^{\mathsf{N}}\frac{\lambda ^{2}-{ \zeta
_{b}^{(0)}}^{2}}{{ \zeta _{a}^{(0)}} ^{2}-{\zeta
_{b}^{(0)}} ^{2}}\mathbf{A}(\eta /2)+2\left( 4\lambda ^{2}-\eta
^{2}\right) \,\prod_{b=1}^{\mathsf{N}}\lambda ^{2}-{ \zeta
_{b}^{(0)}} ^{2},$$$\Sigma _{T}$ is the set of solutions to the following inhomogeneous system of $\mathsf{N}$ quadratic equations:$$x_{n}\sum_{a=1}^{\mathsf{N}}g_{a}(\zeta _{n}^{(1)})x_{a}+x_{n}f(\zeta
_{n}^{(1)})=q_{n},\text{ \ \ \ }q_{n}=\frac{\mathrm{det}_{q}K_{+}(\xi _{n})\mathrm{det}_{q}\,\mathcal{U}_{-}(\xi _{n})}{\eta -4\xi _{n}^{2}},\text{ \ \ }\forall n\in \{1,...,\mathsf{N}\},$$in $\mathsf{N}$ unknowns $\{x_{1},...,x_{\mathsf{N}}\}$.
We are now ready to present the following equivalent characterization of the transfer matrix spectrum:
\[T-eigenvalue-F-eq copy(1)\]Let the inhomogeneities $\{\xi _{1},...,\xi
_{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ be generic [(\[xi-conditions-xxx\])]{}, then for $\xi \neq 0$ the set of transfer matrix eigenvalue functions $\Sigma _{\mathcal{T}}$ is characterized by:$$\tau (\lambda )\in \Sigma _{\mathcal{T}}\text{ \ if and only if }\exists
!Q(\lambda )=\prod_{b=1}^{\mathsf{N}}\left( \lambda ^{2}-\lambda
_{b}^{2}\right) \text{ such that }\tau (\lambda )Q(\lambda )=Z_{Q}(\lambda
)+F(\lambda ),$$with$$F(\lambda )=2(1-\sqrt{(1+\xi ^{2})})\left( 4\lambda ^{2}-\eta ^{2}\right)
\,\prod_{b=1}^{\mathsf{N}}\prod_{i=0}^{1}\left( \lambda ^{2}- {�\zeta
_{b}^{(i)}} ^{2}\right) .$$
The proof presented in Theorem \[T-eigenvalue-F-eq\] applies with small modifications also to present rational case.
The previous characterization of the transfer matrix spectrum allows to prove that the set $\Sigma _{InBAE}\subset\mathbb{C}^\mathsf{N}$ of all the solutions of the Bethe equations$$\{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in\Sigma _{InBAE}$$ if $$\mathbf{A}(\lambda _{a})Q_{\mathbf{\lambda }}(\lambda _{a}-\eta )+\mathbf{A}(-\lambda _{a})Q_{\mathbf{\lambda }}(\lambda
_{a}+\eta )=-F(\lambda _{a}),\text{ \ }\forall a\in \{1,...,\mathsf{N}\} , \label{I-BAE-XXX}$$ define the complete set of transfer matrix eigenvalues. In particular, the following corollary can be proved:
\[Theo-InBAE-XXX\]Let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ satisfy the following conditions [(\[xi-conditions\])]{}, then $\mathcal{T}(\lambda )$ has simple spectrum and for $\xi \neq 0$ then $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ if and only if $\exists
!\{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in \Sigma _{InBAE}$ such that:$$\tau (\lambda )=\frac{Z_{Q}(\lambda )+F(\lambda )}{Q(\lambda )}\text{ \ \
with \ \ }Q(\lambda )=\prod_{b=1}^{\mathsf{N}}\left( \lambda ^{2}-\lambda
_{b}^{2}\right) .$$
Homogeneous chains and existing numerical analysis
==================================================
It is important to stress that the spectrum construction together with the corresponding statements of completeness presented in this paper strictly work for the most general spin 1/2 representations of the 6-vertex reflection algebra only for generic inhomogeneous chains. However, it is worth mentioning that the transfer matrix as well as the coefficients and the inhomogeneous term in our functional equation characterization of the SOV spectrum are analytic functions of the inhomogeneities $\{\xi _{j}\}$ so we can take without any problem the homogeneous limit ($\xi _{a}\rightarrow
0$ $\forall a\in \{1,...,\mathsf{N}\}$) in the functional equations. The main problem to be addressed then is the completeness of the description by this functional equations. Some first understanding of this central question can be derived looking at the numerical analysis [@Nep-R-2003; @Nep-R-2003add] of the completeness of Bethe Ansatz equations when the boundary constraints are satisfied and for the open XXX chain with general boundary terms [@Nep-2013].
Comparison with numerical results for the XXZ chain
---------------------------------------------------
The numerical checks of the completeness of Bethe Ansatz equations for the open XXZ quantum spin 1/2 chains were first done in [@Nep-R-2003] for the chains with non-diagonal boundaries satisfying boundary constraints: $$\kappa _{+}\neq 0,\kappa _{-}\neq 0,\text{ \ }\exists i\in \left\{
0,1\right\} ,\mathsf{M}\in \mathbb{N}\text{\ }:Y^{(i,2\mathsf{M})}(\tau _{\pm },\alpha _{\pm
},\beta _{\pm })=0.$$Indeed, under these conditions some generalizations of algebraic Bethe Ansatz can be used and so the corresponding Bethe equations can be defined.
In particular, the Nepomechie-Ravanini’s numerical results reported in [Nep-R-2003,Nep-R-2003add]{} suggest that the Bethe ansatz equations $\left( \ref{BAE}\right) $ in the homogeneous limit for the roots of the $Q$ function:$$Q(\lambda )=2^{\mathsf{M}}\prod_{a=1}^{\mathsf{M}}\left( \cosh 2\lambda -\cosh 2\lambda
_{a}\right) ,$$with the degree $\mathsf{M}$ obtained from the boundary constraint
- for $\mathsf{M}=\mathsf{N}$ they define the complete transfer matrix spectrum.
- for $\mathsf{M}<\mathsf{N}$ the complete spectrum of the transfer matrix contains two parts described by different Baxter equations. The first one has trigonometric polynomial solutions of degree $2\mathsf{M}$ the second one has a trigonometric polynomial solutions of degree $2\mathsf{N}-2-2\mathsf{M}$.
- for $\mathsf{M}>\mathsf{N}$ the complete spectrum of the transfer matrix spectrum plus $\tau (\lambda )$ functions which do not belong to the spectrum of the transfer matrix.
These results seem to be compatible with our characterization for the inhomogeneous chains. Indeed, the case $\mathsf{M}=\mathsf{N}$ coincides with the case in which our Baxter functional equation becomes homogeneous. Theorem \[homogeneousBE\_N\] states that in this case for generic inhomogeneities the Bethe ansatz is complete so we can expect (from the numerical analysis) that completeness will survive in the homogeneous limit. In the case $\mathsf{M}<$, our description of the spectrum by Lemma [mixed-condition]{} separates the spectrum in two parts. A first part of the spectrum is described by trigonometric polynomial solutions of degree $2\mathsf{M}$ to the homogeneous Baxter equation [(\[homogen-Bax-eq-M\])]{} and a second part is instead described by trigonometric polynomial solutions of degree 2 of the inhomogeneous Baxter equation [(\[inhomogen-Bax-eq-M\])]{}. However, by implementing the following discrete symmetry transformations $\alpha _{\pm
}\rightarrow -\alpha _{\pm }$, $\beta _{\pm }\rightarrow -\beta _{\pm }$, $\tau _{\pm }\rightarrow -\tau _{\pm }$ and applying the same Lemma [mixed-condition]{} w.r.t. the Baxter equations with coefficients $\mathbf{A}_{(-,-,-)}(\lambda )$ we get an equivalent description of the spectrum separated in two parts. One part of the spectrum is described in terms of the solutions of the transformed homogeneous Baxter equation which should be trigonometric polynomials of degree $2\mathsf{M}^{\prime }$, with $\mathsf{M}^{\prime }=\mathsf{N}-1-\mathsf{M}$ and the second part by the inhomogeneous Baxter equation. The comparison with the numerical results then suggests that, at least in the limit of homogeneous chains, the part of the spectrum generated by the trigonometric polynomial solutions of degree 2 of the inhomogeneous Baxter equation [(\[inhomogen-Bax-eq-M\])]{} coincides with the part generated by the trigonometric polynomial solutions of degree $2\mathsf{M}^{\prime }$ of the transformed homogeneous Baxter equation.
Finally, in the case $\mathsf{M}>\mathsf{N}$ we have a complete characterization of the spectrum given by an inhomogeneous Baxter functional equation however nothing prevent to consider solutions to the homogeneous Baxter equation once we take the appropriate $Q$-function with $\mathsf{M}>\mathsf{N}$ Bethe roots. The numerical results however seem to suggest that considering the homogeneous Baxter equations is not the proper thing to do in the homogeneous limit.
The previous analysis seems to support the idea that in the limit of homogeneous chain our complete characterization still describe the complete spectrum of the homogeneous transfer matrix.
Comparison with numerical results for the XXX chain
---------------------------------------------------
In the case of the open spin 1/2 XXX chain an ansatz based on two $Q$-functions and an inhomogeneous Baxter functional equation has been first introduced in [@CaoYSW13-2], the completeness of the spectrum obtained by that ansatz has been later verified numerically for small chains [@CaoJYW2013]. Using these results Nepomechie has introduced a simpler ansatz and developed some further numerical analysis in [@Nep-2013] confirming once again that the ansatz defines the complete spectrum for small chains. Here, we would like to point out that our complete description of the transfer matrix spectrum in terms of a inhomogeneous Baxter functional equation obtained for the inhomogeneous chains has the following well defined homogeneous limit: $$\tau (\lambda )Q(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )+F(\lambda )$$where:$$\begin{aligned}
F(\lambda ) &=&8(1-\sqrt{(1+\xi ^{2})})\left( \lambda ^{2}-\left( \eta
/2\right) ^{2}\right) ^{2\mathsf{N}+1}, \\
\mathbf{A}(\lambda ) &=&(-1)^{\mathsf{N}}\frac{2\lambda +\eta }{2\lambda }\left(\vphantom{\sqrt{(1+\xi ^{2})}}\lambda -\eta /2+p\right)\left(\sqrt{(1+\xi ^{2})}(\lambda -\eta /2)+q\right)\left(\lambda ^{2}-\left( \eta /2\right) ^{2}\right)^{\mathsf{N}}.\end{aligned}$$Taking into account the shift in our definition of the monodromy matrix which insures that the transfer matrix is an even function of the spectral parameter, the limit of our inhomogeneous Baxter functional equation coincides with the ansatz proposed by Nepomechie in [@Nep-2013]. Then the numerical evidences of completeness derived by Nepomechie in [@Nep-2013] suggest that the exact and complete characterization that we get for the inhomogeneous chain is still valid and complete in the homogeneous limit.
Conclusion and outlook {#conclusion-and-outlook .unnumbered}
======================
In this paper we have shown that the transfer matrix spectrum associated to the most general spin-1/2 representations of the 6-vertex reflection algebras (rational and trigonometric), on general inhomogeneous chains is completely characterized in terms of a second order difference functional equations of Baxter $T$-$Q$ type with an inhomogeneous term depending only on the inhomogeneities of the chain and the boundary parameters. This functional $T$-$Q$ equation has been shown to be equivalent to the SOV complete characterization of the spectrum when the $Q$-functions belong to a well defined set of polynomials. The polynomial character of the $Q$-function is a central feature of our characterization which allows to introduce an equivalent finite system of generalized Bethe ansatz equations. Moreover, we have explicitly proven that our functional characterization holds for all the values of the boundary parameters for which SOV works, clearly identifying the only 3-dimensional hyperplanes in the 6-dimensional space of the boundary parameters where our description cannot be applied. We have also clearly identified the 5-dimensional hyperplanes in the space of the boundary parameters where the spectrum (or a part of the spectrum) can be characterized in terms of a homogeneous $T$-$Q$ equation and the polynomial character of the $Q$-functions is then equivalent to a standard system of Bethe equations. Completeness of this description is a built in feature due to the equivalence to the SOV characterization. The equivalence between our functional $T$-$Q$ equation and the SOV characterization holds for generic values of the $\xi _{a}$ in the $\mathsf{N}$-dimensional space of the inhomogeneity parameters however there exist hyperplanes for which the conditions [(\[xi-conditions\])]{} are not satisfied and so a direct application of the SOV approach is not possible (at least for the separate variables described in [@Fald-KN13]) and the limit of homogeneous chains ($\xi _{a}\rightarrow 0$ $\forall a\in \{1,...,\mathsf{N}\}$) clearly belong to these hyperplanes. From the analyticity of the transfer matrix eigenvalues, of the coefficients of the functional $T$-$Q$ equation and of the inhomogeneous term in it w.r.t. the inhomogeneity parameters it is possible to argue that these functional equations still describes transfer matrix eigenvalues on the hyperplanes where SOV method cannot be applied and, in particular, in the homogeneous limit. However, in all these cases the statements about the simplicity of the transfer matrix spectrum and the completeness of the description by our functional $T$-$Q$ equation are not anymore granted and they require independent proofs. These fundamental issues will be addressed in a future publication. Here we want just to recall that the comparison with the few existing numerical results on the subject seems to suggests that the statement of completeness should be satisfied even in the homogeneous limit of special interest as it allows to reproduce the spectrum of the Hamiltonian of the spin-1/2 open XXZ quantum chains under the most general integrable boundary conditions.
Finally, it is important to note that the form of the Baxter functional equation for the most general spin-1/2 representations of the 6-vertex reflection algebras and in particular the necessity of an inhomogeneous term are mainly imposed by the requirement that the set of solutions is restricted to polynomials. Then the problem to get homogeneous Baxter equations relaxing this last requirement remains an interesting open problem.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank E. Sklyanin and V. Terras for discussions. J.M.M. and G. N. are supported by CNRS. N.K and J.M.M. are supported by ANR grant “DIADEMS”. N. K. would like to thank LPTHE, University Paris VI and Laboratoire de Physique, ENS-Lyon for hospitality.
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[^1]: Different methods leading to Bethe ansatz equations have been also proposed under the same boundary conditions by using the framework of the Temperley-Lieb algebra in [@deGP04; @NicRd05] and by making a combined use of coordinate Bethe ansatz and matrix ansatz in [@CraRS10; @CraR12].
[^2]: Some partial results in this direction were achieved in [@FilK11] but only in the special case of double boundary constrains introduced in [YanZ07]{}.
[^3]: See also the papers [@CaoYSW13-1; @CaoYSW13-2; @CaoYSW13-3] for the application of the same method to different models.
[^4]: Here we use notations similar to those introduced in the papers [@CaoJYW2013] and [@Nep-2013] where some inhomogeneous Baxter equation ansatzs appear with the aim to make simpler for the reader a comparison when the limit of homogeneous chain is implemented.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
Ling Bao$^{a}$[^1] $\,$, Elli Pomoni$^{b}$[^2] $\,$, Masato Taki$^{c}$[^3] $\,$ and Futoshi Yagi$^{d}$[^4]\
\
*$^a$ Fundamental Physics, Chalmers University of Technology, 41296 Göteborg, Sweden\
*$^b$ Institut für Mathematik und Institut für Physik, Humboldt-Universität zu Berlin\
Johann von Neumann-Haus, Rudower Chaussee 25, 12489 Berlin, Germany\
*$^c$ Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwake-Cho, Sakyo-Ku, Kyoto, Japan\
*$^d$ International School of Advanced Studies (SISSA) via Bonomea 265, 34136 Trieste, Italy and INFN, Sezione di Trieste****
title: 'M5-branes, toric diagrams and gauge theory duality'
---
Introduction
============
$\mathcal N=2$ gauge theories have been of great interest in the past twenty-five years. While $\mathcal N=4$ SYM has trivial non-perturbative physics the more realistic $\mathcal N=1$ gauge theories are yet to be solved. $\mathcal N=2$ gauge theories exhibit many interesting phenomena, such as confinement and monopole condensation. Moreover, their topological sector gives access to their non-perturbative regime.
Seiberg and Witten derived the Wilsonian low energy effective action of the $\mathcal N=2$ $SU(2)$ gauge theory by encoding the problem into a two-dimensional (2D) holomorphic curve [@Seiberg:1994rs]. Their work was soon after generalized to other gauge groups and matter contents [@Argyres:1994xh; @Klemm:1994qs; @Seiberg:1994aj; @Argyres:1995wt]. Although for the paradigmatic $SU(2)$ case the Seiberg-Witten (SW) curve was derived from first principles [@Seiberg:1994rs], its construction becomes difficult for generic quiver gauge theories. Therefore, other methods have been employed, e.g., integrability [@Donagi:1995cf], geometric engineering [@Katz:1996fh; @Katz:1997eq] and the type IIA/M-theory brane constructions [@Witten:1997sc; @Kol:1997fv; @Brandhuber:1997ua]. The SW curve was initially introduced as an auxiliary space [@Seiberg:1994rs], however, it was later understood that it is part of the M-theory target space [@Witten:1997sc]. Using string theory, $\mathcal{N}=2$ gauge theories can be realized as world volume theories on D4-branes, which are suspended between NS5-branes. Uplifting this brane setup to M-theory, all the branes can be seen as one single M5-brane with a non-trivial topology. The geometry of this M5-brane is encoded in the SW curve. Therefore, the SW curve can also be derived by studying the minimal surface of the M5-brane [@Witten:1997sc].
An alternative way to derive the Seiberg-Witten results was discovered by Nekrasov [@Nekrasov:2002qd]. He succeeded in finding the instanton partition functions of the $\mathcal N=2$ gauge theories by introducing a special deformation called the $\Omega$ background. The deformed theory should in fact be interpreted as a five-dimensional (5D) $\mathcal N=1$ gauge theory defined on the space $\mathcal{M}_4 \times S^1$. This class of 5D gauge theories was first studied by Seiberg [@Seiberg:1996bd] and their relation to the four-dimensional (4D) $\mathcal N=2$ gauge theories on $\mathcal{M}_4$ was explored in [@Nekrasov:1996cz]. Further, it was found that the 5D $\mathcal N=1$ gauge theories can be realized using D5- and NS5-branes [@Aharony:1997ju; @Aharony:1997bh]. This D5/NS5 brane construction is T-dual to the D4/NS5 system discussed above [@Witten:1997sc] as well as the original D3/NS5 Hanany-Witten set-up [@Hanany:1996ie]. The 5D extension of the SW curve has been studied in [@Kol:1997fv; @Brandhuber:1997ua]. The curve was obtained by compactifying one of the directions along which the NS5-branes extend in the D4/NS5 setup. After T-duality along the compactified direction, D4-branes turn into D5-branes, whose world volume theory is a 5D $\mathcal{N}=1$ gauge theory.
An intriguing relation between the gauge theory partition function and topological string theory was conjectured by Nekrasov [@Nekrasov:2002qd]. String theory compactified on Calabi-Yau threefold (CY$_3$) yields $\mathcal{N}=2$ gauge theory on the 4D transverse space. The partition function of this gauge theory is equivalent to the field theory limit of the topological string partition function. This relation has been tested and verified by several authors [@Iqbal:2003ix; @Iqbal:2003zz; @Eguchi:2003sj]. The topological string theory computation leads to a special case of $\Omega$ deformed gauge theories. The generic $\Omega$ deformation of gauge theories is obtained by considering an extension called refined topological string partition function [@Iqbal:2007ii; @Awata:2008ed; @Taki:2007dh]. Topological strings without field theory limit gives the generating function of the BPS states coming from M2-branes wrapped on two-cycles inside CY$_3$. This means that the topological string theory describes the holomorphic sector of M-theory on CY$_3$. The topological string partition function is then equivalent to the Nekrasov partition function for 5D gauge theory via M-theory lift of the geometric engineering.
In the present article, we consider the 5D $\mathcal{N}=1$ $SU(N)^{M-1}$ liner quivers depicted in Figure \[quiver\]. Their type IIA string theory description involves $N$ D4-branes and $M$ NS5-branes. In this set-up, the NS5-branes wrap a coordinate circle $S^1$. From the M-theory point of view, there is, in addition, an M-theory circle around which the M5-branes that lead to D4-branes wrap. We have thus two compact circles, whose roles can be exchanged. In other words, all we have is an M5-brane with non-trivial topology, which yields the SW curve of either $SU(N)^{M-1}$ theory or $SU(M)^{N-1}$ theory. In this sense, $SU(N)^{M-1}$ gauge theory is dual to $SU(M)^{N-1}$ gauge theory. Although the conceptual understanding of this duality has been discussed previously [@Katz:1997eq; @Aharony:1997bh][^5], the explicit duality map was not known.
We take a first step toward understanding this duality in detail. The strategy we adopt is to compare the low energy effective theories of 5D $SU(N)^{M-1}$ and $SU(M)^{N-1}$ gauge theories (Figure \[quiver\]). This is achieved by independently using both the Seiberg-Witten formalism and Nekrasov’s partition function. We derive the map between the ultraviolet (UV) parameters of the two gauge theories, through which they are dual to each other.
![The circle $SU(N)_{(i)}$ corresponds to the $i$-th gauge group, the segments between two circles are bifundamental hypermultiplets, and the flavor symmetries are illustrated by the two blue boxes $SU(N)_{(0)}$ and $SU(N)_{(M)}$ at the ends of the quiver.[]{data-label="quiver"}](LinQuiver.eps){width="150mm"}
The Seiberg-Witten curves are obtained by minimizing the worldvolume of an M5-brane with nontrivial geometry. Nekrasov partition functions are computed using topological string theory. Both in the M-theory and the topological string theory descriptions the duality is geometrically realized simply as a rotation of the M5-brane and toric diagram respectively. We would also like to mention that there is another duality for 4D $\mathcal{N}=1$ gauge theories that is based on performing (non-trivial in this case) operations on the toric diagrams. The $\mathcal{N}=1$ toric duality is studied in [@Feng:2000mi].
This article is organized as follows. In Section \[sec:Review\], we review well known tools and notions that will be used for our study of the duality. In particular, we will describe the Seiberg-Witten framework adopted for the 5D gauge theories, as well as the derivation of Nekrasov’s partition function using topological string theory. In Section \[sec:MtheoryDeriv\], we compute the duality map for the gauge theory parameters based on analysis of the SW curve. The same map will then be re-derived independently in Section \[sec:TopStringDeriv\], where we calculate Nekrasov’s partition function via the topological string partition function for toric CY$_3$. Starting from the toric diagram, one notices that the duality is manifest. The consequences of the duality for 2D CFTs through AGTW are discussed in Section \[sec:GaugeToCFT\] together with the simplest extension to the generic $\Omega$ background. Section \[sec:Discussion\] is devoted to discussions of our results and possible future applications.
Background material {#sec:Review}
===================
Seiberg-Witten formalism {#sec:SWReview}
------------------------
We begin by summarizing the Seiberg-Witten solution for 4D $\mathcal{N}=2$ gauge theories. Nice reviews of this topic can be found in [@Bilal:1995hc; @Lerche:1996xu; @Klemm:1997gg; @Peskin:1997qi]. The complete description of the low energy effective theory (up to two derivatives and quartic fermion terms) is encoded in the prepotential $\mathcal{F}(a)$ according to $$S_{eff} = \int d^4x d^4 \theta \mathcal{F} (a) + \int d^4x d^4 \bar{\theta} \bar{\mathcal{F}} (\bar{a}) \, .$$ The prepotential is a holomorphic function of the vacuum expectation values (${a}_i$) of the scalar fields in the $\mathcal{N} = 2$ vector multiplet. The holomorphic gauge couplings are obtained as $$\tau_{ij} =\frac{ \partial^2\mathcal{F}(a)}{\partial {a}_i \partial {a}_j} \, ,$$ while the expectation values of the scalar fields in the dual (magnetic) theory are given by $${a_D}^i = \frac{\partial \mathcal{F}(a)}{\partial {a}_i} \, .
{\label{eqn:PrepotentialDeriv}}$$ The electromagnetic duality acts on the Coulomb moduli as the modular transformation $$\left( \begin{array}{c}
{a_D}^i \\
{a}_i \\
\end{array}
\right) \rightarrow
\left( \begin{array}{cc}
a & b \\
c & d \\
\end{array}
\right)
\left( \begin{array}{c}
{a_D}^i \\
{a}_i \\
\end{array}
\right)
\quad \mbox{with} \quad \left( \begin{array}{cc}
a & b \\
c & d \\
\end{array}
\right) \in SL(2, \mathbb{Z}) \, .$$ The prepotential is determined using an auxiliary curve called the SW curve $$F_{4D}(t,v) = 0$$ together with a meromorphic differential $\lambda_{SW}$. The derivatives of the meromorphic one-form with respect to the moduli of the SW curve[^6] are the holomorphic differentials of the curve. The Coulomb moduli are then computed according to $$a_i = \oint_{A_i} \lambda_{SW} \quad \mbox{and} \quad {a_D}^i = \oint_{B^i} \lambda_{SW} \, ,
\label{a_aD}$$ where $A_i$ and $B_i$ are the basic cycles of the algebraic curve with intersection number $A_i \cdot B^j = \delta_i^j$. The prepotential itself can be found by integrating [(\[eqn:PrepotentialDeriv\])]{}. Moreover, contour integrals of the meromorphic differential $\lambda_{SW}$ around its poles give linear combinations of the bare quark masses ($m_i$).
The SW curve and one-form can also be derived from M-theory [@Witten:1997sc]. To do this we consider the brane setup in Table \[config\], where $N$ D4-branes are suspended between $M$ NS5-branes. We introduce also $2N$ flavor branes attached to the two outermost NS5-branes and extended to infinity. The theory described by this setup is 4D $\mathcal{N}=2$ $SU(N)^{M-1}$ gauge theory, which is asymptotically conformal. The rotation of $x^{4}$ and $x^{5}$ coordinates corresponds to $U(1)_R$ symmetry, while rotation of $x^{7}$, $x^{8}$, and $x^{9}$ corresponds to $SU(2)_R$ symmetry.
$x^0$ $x^1$ $x^2$ $x^3$ $x^4$ $x^5$ $x^6$ $x^7$ $x^8$ $x^9$ ($x^{10}$)
---------------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------------
$M$ NS5-branes $-$ $-$ $-$ $-$ $-$ $-$ . . . . .
$N$ D4-branes $-$ $-$ $-$ $-$ . . $-$ . . . $-$
: Brane configuration in type IIA string theory[]{data-label="config"}
Table \[config\] is a classical configuration from the gauge theory point of view. Taking the tension of the branes into account, the configuration has to be modified to include the quantum effects. Uplifting to M-theory and minimizing the world volume of the corresponding M5-brane under fixed boundary condition yields the SW curve. This curve describes a 2D subsurface inside the space spanned by the coordinates $\{x^4, x^5, x^6, x^{10}\}$, where $x^{10}$ is the direction of the M-theory circle.
To obtain 5D $\mathcal{N}=1$ gauge theory we compactify the $x^{5}$ coordinate. After T-duality along $x_5$, the system becomes an D5/NS5 brane system in type IIB string theory with a 5D $\mathcal N=1$ gauge theory living on the D5-branes (Table \[configIIB\]). This is the 5D $\mathcal N=1$ gauge theory for which we are constructing the SW curve. The spacetime of this gauge theory is $\mathcal{M}_4 \times S^1$ with the circumference of the IIB circle being $$\beta =\frac{ 2 \pi \alpha'}{R_5} = \frac{ 2 \pi \ell^3_{p}}{R_5 R_{10}} \, ,$$ where $\alpha' = \ell_{s}^2 = \frac{\ell_{\text{p}}^3}{R_{10}}$. Going back to the type IIA description, we define the complex coordinates $v$ and $s$ according to $$v \equiv x^4 + i x^5 \quad \text{and} \quad s \equiv x^6 + i x^{10} \, .$$ Due to the periodic nature of $x^5$ and $x^{10}$ it is natural to introduce another pair of complex coordinates $$w \equiv e^{-\frac{v}{R_5}} \quad \text{and} \quad t \equiv e^{-\frac{s}{R_{10}}} \, .
\label{def_tw}$$ The radius of the $x^{5}$ circle is denoted as $R_5$ and that of the M-theory circle as $R_{10}$.
$x^0$ $x^1$ $x^2$ $x^3$ $x^4$ $x^5$ $x^6$ $x^7$ $x^8$ $x^9$ ($x^{10}$)
---------------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------------
$M$ NS5-branes $-$ $-$ $-$ $-$ $-$ $-$ . . . . .
$N$ D5-branes $-$ $-$ $-$ $-$ . $-$ $-$ . . . $-$
: Brane configuration in type IIB string theory[]{data-label="configIIB"}
The SW curve of the 5D $SU(N)^{M-1}$ theory is now written as a polynomial of degree $N$ in $w$ and degree $M$ in $t$ as $$\begin{aligned}
F (t,w) \equiv \sum_{i=0}^N \sum_{j=0}^M C_{p,q} w^p t^q \, .
\label{SW_curve}\end{aligned}$$ The periodic boundary condition along the $x^5$ coordinate makes the curve invariant under a shift of the positions of the color branes ($a'$) and flavor branes ($m'$) by $2 \pi R_5$. Therefore, the coefficients of the curve $C_{p,q}$ depend only on the gauge coupling $q$ and $$\label{tildema}
\begin{split}
\tilde{m} &\equiv e^{-m' / R_5} = e^{- \beta m} \, , \\
\tilde{a} &\equiv e^{-a' / R_5} = e^{- \beta a} \, ,
\end{split}$$ in which periodicity is manifest. Note that quantities that have dimension of mass are related to the ones with dimension of length (primed) as $$a = \frac{a'}{2\pi \ell^2_s} \quad \text{and} \quad
m = \frac{m'}{2\pi \ell^2_s} \, .$$ The coefficients $C_{p,q}$ will be determined explicitly in Section \[sec:MtheoryDeriv\].
The M-theory derivation of the SW one-form can be found in [@Fayyazuddin:1997by; @Henningson:1997hy; @Mikhailov:1997jv]. We summarize it for pure $SU(2)$ theory here. The extension to generic quiver theories is straightforward. The idea is to relate two different expressions of the masses of BPS states. On one hand, the mass of a BPS particle is given by $$\begin{aligned}
{m_{\text{BPS}}}^2 = |n_e a + n_m a_D|^2 \, ,
\label{BPS}\end{aligned}$$ where $n_e$ and $n_m$ are the electric and magnetic charges of the BPS state respectively. This formula can be rewritten using the SW one-form as $$\begin{aligned}
{m_{\text{BPS}}}^2 = \left| \int_{n_e A + n_m B} \lambda_{\rm SW} \right| ^2 \, .
\label{mass1}\end{aligned}$$ On the other hand, a BPS state is interpreted as an open M2-brane attached to an M5-brane whose volume is minimized. The boundary of a such minimal M2-brane with charge $(n_e,n_m)$ is the cycle $n_e A + n_m B$. Finally, the mass of this BPS state is calculated using the volume-form of the M2-brane $$\begin{aligned}
\omega = ds \wedge dv = d \left[ \log t \, (d \log w) \right]\end{aligned}$$ and reads $$\begin{aligned}
{m_{\text{BPS}}}^2 = \left| \frac{1}{(2 \pi)^2 \ell_p{}^3} \int_{M_2} \omega \right| ^2 \, ,
\label{mass2}\end{aligned}$$ where $1/ (2 \pi)^2 \ell_p{}^3 $ is the tension of the M2-brane. Comparing (\[mass1\]) with (\[mass2\]), we find that the SW one-form takes the form $$\begin{aligned}
\lambda_{\rm SW} = - \frac{i}{(2 \pi)^2 \ell_p{}^3} \log t \, (d \log w) \, .
\label{SW_1-form}\end{aligned}$$
Partition function and topological vertex
-----------------------------------------
The microscopic way to obtain the prepotential is via Nekrasov’s partition function [@Nekrasov:2002qd; @Nekrasov:2003rj] $$\label{ZF}
Z(a;\epsilon_1 ,\epsilon_2) = e^{ \frac{\mathcal{F} (a)}{\epsilon_1 \epsilon_2} + \cdots} \, ,$$ which contains the full low energy effective description of $\mathcal{N} = 2$ gauge theories in a deformed background. More details can be found in [@Nekrasov:2005wg; @Bruzzo:2002xf; @Marino:2004cn; @Tachikawa:2004; @Shadchin:2005mx]. The starting point of Nekrasov’s derivation is 5D $\mathcal{N} = 1$ gauge theory on $\mathcal{M}_4 \times S^1$. This theory depends on two deformation parameters $(\epsilon_1, \epsilon_2)$ and the circumference of the circle $\beta$. Taking the limit $\beta \rightarrow 0$ leads to the so called $\Omega$ deformed 4D $\mathcal{N}=2$ gauge theory. The deformation parameterized by $\epsilon_1$ and $\epsilon_2$ breaks the $SO(4)$ Lorentz symmetry down to $SO(2){\times}SO(2)$. In this way the path integral is localized to one point on $\mathcal{M}_4$ and the computation of the partition function is simplified to supersymmetric quantum mechanics along $S^1$.
Nekrasov’s partition function $Z(a,m,q;\epsilon_1,\epsilon_2)$ of 4D $\mathcal{N}=2$ gauge theory is a function of the set of moduli $a$ parameterizing the Coulomb branch, the masses $m$ of all the flavor and bifundamental fields, the coupling constants $q=e^{2\pi i \tau}$ and the two parameters $\epsilon_1$ and $\epsilon_2$. It can be factorized as $$\label{Partition}
Z = Z_{\rm pert}\,Z_{\rm inst} \, ,$$ where $Z_{\rm pert}$ is the perturbative part containing tree-level and one-loop contributions, while $Z_{\rm inst}$ is the contribution from the instantons. The instanton part can be expanded with respect to the instanton number $k$ $$\label{Instantons}
Z_{\rm inst} = \sum_{k} q^{k} Z_{k} \, .$$
As discussed previously, one way to realize 4D $\mathcal{N}=2$ gauge theories is the Hanany-Witten setup in Table \[config\]. Another way is to consider CY$_3$ compactification of type IIA string theory. These two different points of view are connected by a series of duality transformations [@Karch:1998yv]. Starting from the Hanany-Witten setup, the transformations consist of a T-duality along the $x^6$ coordinate, followed by an S-duality involving $x^6$ and $x^{10}$ and lastly another T-duality along the new $x^6$ coordinate. The resulting theory is IIA string theory on non-compact CY$_3$ without any branes. The gauge symmetry of the 4D theory is geometrically realized by the vanishing cycles inside CY$_3$. A special class of CY$_3$ which yields $\mathcal{N}=2$ gauge theories is the toric type [@Aganagic:2003db]. Its generic configuration is a fibration of special Lagrangian $T^2 \times \mathbb{R}$ over the base $\mathbb{R}^3$. For $SU(N)$ gauge symmetry it is further required that the CY$_3$ manifold is a non-trivial fibration of $A_{N-1}$ singularity over the space $\mathbb{P}^1$ [@Iqbal:2003zz].
Already in [@Nekrasov:2002qd] Nekrasov suggested that the partition function of $\mathcal{N}=2$ gauge theories is the field theory limit of the topological string partition function on toric CY$_3$. For toric CY$_3$ the topological string partition function can be computed graphically using the so called toric diagram[^7], which characterizes the toric Calabi-Yau manifold. The toric diagram consists of a collection of trivalent vertices, which are joined together by oriented straight lines.
Writing down the topological string partition function is simple using the topological vertex formalism. The procedure is very similar to computing Feynman diagrams for usual field theory, where the internal momentum integrals are replaced by sums over the Young diagrams $R$ $$\int d p \quad \longrightarrow \quad \sum_{R} \nonumber \, .$$ Schematically, it takes the form $$\mathcal{Z} = \sum_{R} \, (\text{three-vertices}) \times (\text{oriented lines}) \, .
{\label{eqn:PartitionFuncSum}}$$ The topological three-vertex describes the open string amplitude on a local $\mathbb{C}^3$ coordinate patch. In the case $\hbar = \epsilon_1 = - \epsilon_2$, the contribution from the topological vertex is given by [@Aganagic:2003db] $$C_{R_1 R_2 R_3}(\mathfrak{q}) = \mathfrak{q}^{\frac{\kappa_{R_3}}{2}}S_{R_2}(\mathfrak{q}^\rho) \sum_\eta S_{R_1/\eta}(\mathfrak{q}^{R_2^T+\rho})\, S_{R_3^T/\eta}(\mathfrak{q}^{R_2+\rho}) \, ,
\label{def_topv}$$ where $S_{\alpha}$ and $S_{\alpha / \eta}$ are the Schur and skew-Schur functions, respectively. We also introduce the symbol $\mathfrak{q}=e^{ - \beta \hbar}$ for the exponentiated $\Omega$ background. The three free indices represent the three straight lines going out from the vertex. Each line is labeled by an infinite set of all possible Young tableaux associated with the group $U(\infty)$. In the Feynman diagram analogy, the vertex
 (17,21)[$\longrightarrow$]{}  (13,21)[$= \quad C_{R_1 R_2 R_3}$]{} (-19,17)[$R_1$]{} (-50,32)[$R_3$]{} (-50,13)[$R_2$]{}
is replaced by the topological vertex (\[def\_topv\]),\
while for the propagator $$G(p)=\frac{1}{p^2} \quad \longrightarrow \quad (-Q)^{|R|} (-1)^m \mathfrak{q}^{-\frac{m}{2}\kappa_{R}} \, , \nonumber$$ where $Q=e^{-t}$ is the exponentiated Kähler moduli (size) of the two-cycle represented by the segment. The framing factor ($(-1)^m \mathfrak{q}^{-\frac{m}{2}\kappa_{R}}$) of the “propagator” contains the second Casimir $\kappa_R$ of the representation $R$, which is defined as $\kappa_R = \sum_j R_j (R_j-2j-1)=-\kappa_{R^T}$ where $R_j$ is the number of boxes in the $j$-th line of the tableu and $R^T$ is the transposed Young tableu. The integer $(-m-1)$ is the self-intersection number of the two-cycle and is illustrated in Figure \[mfigure\] together with two examples.
![In the definition of the framing factor we have $m=\det \left(\vec{v}_{\text{in}} \cdot \vec{v}_{\text{out}}\right)$. We graphically clarify its definition and give two examples.[]{data-label="mfigure"}](framing_factor.eps "fig:") ![In the definition of the framing factor we have $m=\det \left(\vec{v}_{\text{in}} \cdot \vec{v}_{\text{out}}\right)$. We graphically clarify its definition and give two examples.[]{data-label="mfigure"}](framing_factor_uss.eps "fig:") ![In the definition of the framing factor we have $m=\det \left(\vec{v}_{\text{in}} \cdot \vec{v}_{\text{out}}\right)$. We graphically clarify its definition and give two examples.[]{data-label="mfigure"}](framing_factor_paral.eps "fig:") (-336,8)[$ \vec{v}_{\text{in}}$]{} (-258,32)[$ \vec{v}_{\text{out}}$]{} (-296,21)[$R$]{} (-296,10)[$m$]{} (-180,8)[$m=1$]{} (-48,10)[$m=0$]{}
The closed string amplitude on the full CY$_3$ is obtained by gluing together the local $\mathbb{C}^3$ patches as in [(\[eqn:PartitionFuncSum\])]{}. The sum in [(\[eqn:PartitionFuncSum\])]{} is taken over all the Young tableaux sets attached to the internal lines of the toric diagram. After carrying out the summation explicitly, it is straightforward to compare with the gauge theory partition function given by Nekrasov. The topological vertex formalism gives thus an alternative derivation for Nekrasov’s partition function based on the geometric shape of the corresponding toric diagram.
When $\epsilon_1 \neq - \epsilon_2$, the topological vertex function above should be replaced by the *refined* topological vertex function [@Iqbal:2007ii; @Awata:2008ed] $$\begin{aligned}
\begin{split}
C_{R_1 R_2 R_3} (\mathfrak{t},\mathfrak{q}) = & {\left( {\frac{\mathfrak{q}}{\mathfrak{t}}} \right)}^{\frac{{\left\| R_1 \right\|^2 + \left\| R_2 \right\|^2 }}{2}} \mathfrak{t}^{\frac{{{\kappa}_{R_1}}}{2}}
P_{{R_2}^T } (\mathfrak{t}^{ - \rho } ;\mathfrak{q},\mathfrak{t}) \, \times \\
& \sum_\eta {{\left( {\frac{\mathfrak{q}}{\mathfrak{t}}} \right)}^{\frac{{\left| \eta \right| + \left| R_3 \right| - \left| R_1 \right|}}{2}}
S_{R_1 /\eta } (\mathfrak{t}^{ - {R_2}^T } \mathfrak{q}^{ - \rho } )}
S_{{R_3}^T /\eta } (\mathfrak{t}^{ - \rho } \mathfrak{q}^{ - R_2 } ) \, ,
\end{split}\end{aligned}$$ where $\mathfrak{q}=e^{-\beta \epsilon_1},\, \mathfrak{t}=e^{\beta \epsilon_2}$, and $P_{R} (\mathop \mathfrak{t}\nolimits^{ - \rho } ;\mathfrak{q},\mathfrak{t})$ is the principal specialization of the Macdonald function $$\begin{aligned}
P_{R^T } (\mathfrak{t}^{ - \rho } ;\mathfrak{q},\mathfrak{t})=\mathfrak{t}^{\frac{1}{2}||R||^2} \prod_{(i,j)\in R}(1-\mathfrak{t}^{{R_j}^T-i+1} \mathfrak{q}^{R_i-j})^{-1} \, .\end{aligned}$$ The refined topological vertex function is a generalization which reduces to the ordinary vertex function when choosing $\epsilon_1 = - \epsilon_2$. It has slightly different properties compared to the ordinary topological vertex. E.g., instead of being entirely cyclic symmetric, one of its legs indicates a preferred direction. Slicing invariance is a conjecture claiming that the full partition function should be invariant under a change of the choice of the preferred direction.
Introducing the duality {#subsec:review_duality}
-----------------------
The first hint toward a duality between the 5D gauge theories with gauge groups $SU(N)^{M-1}$ and $SU(M)^{N-1}$ is given by counting the physical parameters of these two theories. Indeed, we find that the number of parameters matches exactly. For this counting we can ignore the infinite tower of Kaluza-Klein modes and count only the zero modes[^8]. The zero modes coincide with the parameters of the corresponding 4D gauge theories on $\mathcal{M}_4$, we will therefore use 4D terminology in the rest of this section.
The infrared (IR) physics of $SU(N)^{M-1}$ and $SU(M)^{N-1}$ gauge theories at generic points on the Coulomb branch are both described by the $U(1)^{(N-1)(M-1)}$ theory. They are thus described by $(N-1)(M-1)$ IR effective coupling constants $$\tau_{IR}^i = \tau_{IR}^i\left(\tau_{UV} , m_{\text{f}} , m_{\text{bif}} , a \right) \, ,$$ which depend holomorphically on the gauge theory parameters. $\tau_{UV}$ are the UV coupling constants, $m_{\text{f}}$ are the mass parameters of the flavor hypermultiplets, $m_{\text{bif}}$ are the mass parameters of the bifundamental hypermultiplets and $a$ are the Coulomb moduli parameters. The counting of the parameters for asymptotically superconformal $SU(N)^{M-1}$ and $SU(M)^{N-1}$ gauge theories is summarized in Table \[tab:CountParam\]. Summing all the parameters shows that there are in total $[(N+1)(M+1)-3]$ parameters in both theories, allowing the possibility to derive a map between them.
$SU(N)^{M-1}$ $SU(M)^{N-1}$
------------------ ---------------- ----------------
$\tau_{UV}$ $M-1$ $N-1$
$m_{\text{f}}$ $2N$ $2M$
$m_{\text{bif}}$ $M-2$ $N-2$
$a$ $(N-1)(M-1)$ $(M-1)(N-1)$
Total $(N+1)(M+1)-3$ $(M+1)(N+1)-3$
: Counting of the gauge theory parameters[]{data-label="tab:CountParam"}
One of approaches we use is to match the coefficients of the SW curves and the SW one-form of the two dual theories. Before attempting that, we first count the degrees of freedom that are encoded in the SW curve. The SW curve of the 5D $SU(N)^{M-1}$ gauge theory is a polynomial of degree $M$ in the variable $t$ and $N$ in the variable $w$. We have therefore $[(M+1)(N+1)-1]$ complex coefficients, where one has been subtracted to allow an overall coefficient. Moreover, there is the freedom to set the origins of the coordinates $s$ and $v$. Removing two more coefficients we find $[(M+1)(N+1)-3]$ degrees of freedom in total. Thus, the number of coefficients in the SW curve is always the same as the number of physical parameters.
If we exchange the role of the variables $t$ and $w$, the SW curve (\[SW\_curve\]) of the original $SU(N)^{M-1}$ theory can be read as the SW curve of the dual $SU(M)^{N-1}$ theory. The coefficients $C_{p,q}$ in the original curve get reinterpreted as the coefficients in the curve of the dual theory $(C_{q,p})_d$. In addition, the SW one-form (\[SW\_1-form\]) also remains the same up to a minus sign (\[equalSWoneform\]). Using (\[a\_aD\]) the IR effective coupling constant is given by $$\tau_{IR}
= \frac{ \frac{\partial a_D}{\partial u} }{ \frac{\partial a}{\partial u} }
= \frac{ \int_B \omega }{ \int_A \omega } \, ,$$ where $\omega$ is the holomorphic differential. Since the holomorphic differential does not distinguish[^9] the cycle $A$ (or $B$) of the original theory from $A_d$ (or $B_d$) of the dual theory, we get that the dual IR effective coupling constant is equal to the original one. Therefore, once the relation between the gauge theory parameters and the coefficients $C_{p,q}$ in the SW curve is established, it is straightforward to find the duality map. The map is obtained by equating the coefficients $C_{p,q}$, written in terms of the gauge theory parameters of the original $SU(N)^{M-1}$ theory, with the coefficients $(C_{q,p})_d$, written in terms of the parameters of the dual $SU(M)^{N-1}$ theory.
The interpretation of this duality in the context of the brane setup in IIA/M and IIB theories [@Aharony:1997bh] is the following. Consider M-theory compactified on $T^2$. The cycles of the torus correspond to the two phases of the variables $t$ and $w$. Exchanging $t$ and $w$ is the holomorphic extension of a particular $SL(2,\mathbb{Z})$ modular transformation on the compactification torus, where the M-theory circle is exchanged with the $x^5$ circle. This modular transformation is equivalent to S-duality in IIB theory compactified on $S^1$ via T-duality along the $x^5$ circle. The modular transformation in the IIA theory limit exchanges D4-branes with NS5-branes compactified along the $x^5$ circle, while S-duality in IIB theory exchanges D5-branes with NS5-branes. Rigorously, the 90 degree rotation of the brane configuration ($w\rightarrow t$ and $t\rightarrow w^{-1}$) corresponds to this S-duality, but in the main body of this paper we study the $t\leftrightarrow w$ reflection that it is technically simpler. The 90 degree rotation case is presented in detail in the appendix \[app:90Rotation\]. Note that this S-duality is different from the one which appears as the electric-magnetic duality in the 4D Seiberg-Witten theory. As we will see in the following sections, it acts on the gauge theory parameters in a totally different manner. The difference of these two types of S-dualities is due to the difference of the brane setup. It is known that the Montonen-Olive duality, which is the extension of the electric-magnetic duality for 4D $\mathcal{N}=4$ theory, is obtained by compactifying the M5-branes on a torus [@Vafa:1997mh; @Tachikawa:2011ch]. In the brane setup of Table \[config\] the $x^6$ direction has to be compactified instead of $x^5$ (as in our case).
The duality described here was originally found in the context of geometric engineering [@Katz:1997eq]. On the IIB string theory side, the SW curve is embedded in the CY manifold and the duality can be seen in a similar way as the M-theory analysis above. In the mirror IIA theory, on the other hand, the duality is most clear from the toric diagram. Indeed, the toric diagram for the $SU(N)^{M-1}$ theory is exactly the same as the one for the $SU(M)^{N-1}$ theory, up to a simple reflection or 90 degree rotation. This duality is therefore manifest at the level of the topological string partition function. Depending on which sums are carried out explicitly in [(\[eqn:PartitionFuncSum\])]{}, we obtain Nekrasov’s partition function of $SU(N)^{M-1}$ theory or $SU(M)^{N-1}$ theory. The topological vertex formalism provides the extension of the duality for the non-zero self-dual $\Omega$ background.
M-theory derivation {#sec:MtheoryDeriv}
===================
In this section we present the first derivation of the duality map using the Seiberg-Witten formalism reviewed in Section \[sec:SWReview\]. Another, independent derivation based on the topological vertex formalism is given in \[sec:TopStringDeriv\]. The map between the gauge theory parameters of the 5D $\mathcal{N}=1$ $SU(N)^{M-1}$ and $SU(M)^{N-1}$ liner quiver gauge theories compactified on $S^1$ is obtained by comparing their Seiberg-Witten curves. The SW curves are derived using the M-theory approach [@Witten:1997sc]. We, firstly, warm up with the self-dual case of $SU(2)$ gauge theory with four flavors and then turn to the generic duality between $SU(N)^{M-1}$ and $SU(M)^{N-1}$. The special case ($M=2$) between $SU(N)$ and $SU(2)^{N-1}$ is given at the end of this section.
$SU(2)$ gauge theory with four flavors {#subsec:Msu2}
--------------------------------------
We begin by deriving the SW curve for the simplest case; the compactified 5D $SU(2)$ gauge theory with four flavors.[^10] The brane setup is described in Table \[config\] together with Figure \[su2branesetup\] and includes $M=2$ NS5-branes with $N=2$ D4-branes suspended between them.
(1,1)(0,0) (0.5,0)[(0,1)[1]{}]{} (0.44,1.15)[NS5$_1$]{} (0.9,0)[(0,1)[1]{}]{} (0.81,1.15)[NS5$_{2}$]{}
(0.1,0.2)[(1,0)[0.4]{}]{} (0.27,0.23)[$m'_{1}$]{} (0.1,0.7)[(1,0)[0.4]{}]{} (0.27,0.73)[$m'_{2}$]{}
(-0.2,0.18)[D4$_1$]{} (-0.2,0.68)[D4$_2$]{}
(0.5,0.25)[(1,0)[0.4]{}]{} (0.67,0.28)[$a'{}_{1}$]{} (0.5,0.75)[(1,0)[0.4]{}]{} (0.67,0.78)[$a'{}_{2}$]{}
(0.9,0.15)[(1,0)[0.4]{}]{} (1.07,0.18)[$m'_{3}$]{} (0.9,0.8)[(1,0)[0.4]{}]{} (1.07,0.83)[$m'_{4}$]{}
The asymptotic behavior of the NS5-branes is determined by the holomorphic extension of the equations of motion $\nabla^2 s=0$, which minimizes its worldvolume. If the $x^5$ direction is not compactified the asymptotic behavior of an NS5-brane at large $|v|$ is given by $$\begin{aligned}
\frac{s}{R_{10}} = \sum_{i=1}^2 \log (v-a'_i)
- \sum_{i=1}^{2} \log (v-b'_i)
+ {\rm const} \, ,\end{aligned}$$ where $a'_i$ and $b'_i$ are the classical positions on the $v$-plane of the D4-branes attached to the NS5-brane from the left and the right respectively. Compactifying the $x^5$ direction is equivalent to periodically attaching D4-branes on a non-compact $x^5$ coordinate. Firstly, we concentrate on the first NS5-brane. The two flavor D4-branes that are attached to it at $v=m'_1$ and $v=m'_2$ can be reinterpreted as infinitely many D4-branes attached at $v = m'_1 + 2 \pi i R_5 n$ and $v=m'_2 + 2 \pi i R_5 n$, with $n=\cdots, -1,0,1,\cdots$. Similarly, two color D4-branes are attached at $v = a'_1 + 2 \pi i R_5 n$ and $v=a'_2 + 2 \pi i R_5 n$ from the other side. The asymptotic behavior of the first NS5-brane is, therefore, given by $$\begin{split}
\frac{s_{(1)}}{R_{10}}
=& \sum_{n=-\infty}^{\infty} \left( \log (v - m'_1 - 2 \pi i R_5 n)
+ \log(v - m'_2 - 2 \pi i R_5 n) \right) - \\
& \sum_{n=-\infty}^{\infty} \left( \log (v - a'_1 - 2 \pi i R_5 n)
+ \log(v - a'_2 - 2 \pi i R_5 n) \right)
+ {\rm const} \, .
\end{split}
\nonumber$$ Using the definitions of the periodic coordinates (\[def\_tw\]) and gauge theory parameters (\[tildema\]) we can write the position of the first NS5-brane as $$\begin{aligned}
t_{(1)} = & \, C \,
\frac{\sinh\left( \frac{v-a'_1}{2R_5} \right)\sinh\left( \frac{v-a'_2}{2R_5} \right)}
{\sinh\left( \frac{v-m'_1}{2R_5} \right)\sinh\left( \frac{v-m'_2}{2R_5} \right)} \quad \longrightarrow \quad C
\left\{
\begin{array}{l}
\sqrt{
\frac{ \tilde{m}_1 \, \tilde{m}_2
}{
\tilde{a}_1\, \tilde{a}_2
}
}
\quad (w \to \infty) \\
\sqrt{ \frac{
\tilde{a}_1 \, \tilde{a}_2
}
{
\tilde{m}_1 \, \tilde{m}_2
}
}
\quad (w \to 0)
\end{array}
\right. \, ,
\label{asym_t1}\end{aligned}$$ where the expressions after the arrow are the asymptotic behaviors in the $w \to \infty$ and $w \to 0$ regions. Similarly, for the second NS5-brane we have $$\begin{aligned}
t_{(2)}
= & C' \frac{\sinh\left( \frac{v-m'_3}{2R_5} \right)\sinh\left( \frac{v-m'_4}{2R_5} \right)}
{\sinh\left( \frac{v-a'_1}{2R_5} \right)\sinh\left( \frac{v-a'_2}{2R_5} \right)} \quad \longrightarrow \quad C' \left\{
\begin{array}{l}
\sqrt{ \frac{
\tilde{a}_1 \, \tilde{a}_2
}
{
\tilde{m}_3 \, \tilde{m}_4
} }
\quad (w \to \infty) \\
\sqrt{ \frac{
\tilde{m}_3 \, \tilde{m}_4
}{
\tilde{a}_1 \, \tilde{a}_2
}
}
\quad (w \to 0)
\end{array}
\right. \, .\end{aligned}$$ Following [@Witten:1997sc], the distance between the two NS5-branes should give the 4D bare coupling constant $q \equiv \exp \left( 2 \pi i \tau_{\rm bare} \right)$ in the limit $R_5 \to \infty$. However, since we are studying the compactified 5D case, $$\begin{aligned}
\frac{t_{(2)}}{t_{(1)}}
= & \exp \left( \frac{s_{(1)}-s_{(2)}}{R_{10}} \right)
= \frac{C'}{C}
\frac{\prod_{\mathfrak{i}=1}^4 \sinh\left( \frac{v-m'_\mathfrak{i}}{2R_5} \right)}{\sinh^2 \left( \frac{v-a'_1}{2R_5} \right) \sinh^2 \left( \frac{v-a'_2}{2R_5} \right)} \,
\label{1lp}\end{aligned}$$ and the asymptotic distance between the NS5-branes at $w \to 0$ is different from the distance at $w \to \infty$ by a factor[^11] $\prod_\mathfrak{i} \tilde{m}_\mathfrak{i} \prod_i \tilde{a}_i^{-2}$. Thus, relating the constants $C$ and $C'$ to the 4D gauge theory parameters is subtle. In the rest of this section, we assume that these constants do not depend on the radius $R_5$ and that $$\begin{aligned}
\frac{C'}{C} = q
\label{CCq}\end{aligned}$$ is an exact relation for arbitrary $R_5$. This assumption indicates that the bare coupling constant is identified as the average of the two asymptotic distances, which is one of the most natural possibilities. Indeed, as discussed in section \[subsec:topsu2\], this identification is justified by comparing the topological string partition function with Nekrasov partition function.
We continue by writting the 5D SW curve as a polynomial of degree two in $w$ $$\begin{aligned}
q_1(t) w^2 + q_2(t) w + q_3(t) =0 \, .
\label{WPolynomial}\end{aligned}$$ In the $w \to \infty$ region, having two NS5-branes at $t=C \left(\frac{ \tilde{m}_1 \, \tilde{m}_2 }{ \tilde{a}_1\, \tilde{a}_2 }\right)^{\frac{1}{2}}$ and $t=Cq \left( \frac{ \tilde{a}_1 \, \tilde{a}_2 }{ \tilde{m}_3 \, \tilde{m}_4 } \right)^{\frac{1}{2}}$ leads to $$\begin{aligned}
q_1(t) = \left(t-C \left(\frac{ \tilde{m}_1 \, \tilde{m}_2 }{ \tilde{a}_1\, \tilde{a}_2 }\right)^{\frac{1}{2}}
\right)
\left(
t-Cq
\left( \frac{ \tilde{a}_1 \, \tilde{a}_2 }{ \tilde{m}_3 \, \tilde{m}_4 } \right)^{\frac{1}{2}}
\right) \, .\end{aligned}$$ Similarly, in the $w \to 0$ region, we have the two NS5-branes at $t=C \left(\frac{ \tilde{a}_1\, \tilde{a}_2 }{ \tilde{m}_1 \, \tilde{m}_2 }\right)^{\frac{1}{2}}$ and $t=Cq \left( \frac{ \tilde{m}_3 \, \tilde{m}_4 }{ \tilde{a}_1 \, \tilde{a}_2 } \right)^{\frac{1}{2}}$, so we obtain $$\begin{aligned}
q_3(t) = d' \left(t-C \left(\frac{ \tilde{a}_1\, \tilde{a}_2 }{ \tilde{m}_1 \, \tilde{m}_2 }\right)^{\frac{1}{2}}
\right)
\left(
t-Cq
\left( \frac{ \tilde{m}_3 \, \tilde{m}_4 }{ \tilde{a}_1 \, \tilde{a}_2 } \right)^{\frac{1}{2}}
\right) \, ,\end{aligned}$$ where $d'$ is a temporarily undetermined constant. If we, now, write the 5D SW curve as a polynomial of degree two in $t$ and consider the asymptotic behavior of the flavor D4-branes we can determine some more coefficients. In the $t \to \infty$ ($s \to -\infty$) region there are two flavor D4-branes at $w=\tilde{m}_1$ and $w=\tilde{m}_2$ and in the $t \to 0$ ($s \to \infty$) region two flavor D4-branes at $w=\tilde{m}_3$ and $w=\tilde{m}_4$. These boundary conditions constrain the SW curve to be of the form $$\begin{aligned}
(w-\tilde{m}_1)(w-\tilde{m}_2) t^2 + P_2(w) t + d (w-\tilde{m}_3)(w-\tilde{m}_4) = 0 \, ,
\label{TPolynomial}\end{aligned}$$ where $d$ is another undetermined constant that we will now fix. The two forms (\[WPolynomial\]) and (\[TPolynomial\]) of the SW curve are simultaneously satisfied if we write $$\begin{split}
& (w-\tilde{m}_1)(w-\tilde{m}_2) t^2 \\
& - \left( \left[
C \left(\frac{ \tilde{m}_1 \, \tilde{m}_2 }{ \tilde{a}_1\, \tilde{a}_2 }\right)^{\frac{1}{2}}
+ Cq
\left( \frac{ \tilde{a}_1 \, \tilde{a}_2 }{ \tilde{m}_3 \, \tilde{m}_4 } \right)^{\frac{1}{2}}
\right]w^2
- b \, w
\right. \\
& \quad \left.
+ \, \tilde{m}_1 \, \tilde{m}_2 \left[
C \left(\frac{ \tilde{a}_1\, \tilde{a}_2 }{ \tilde{m}_1 \, \tilde{m}_2 }\right)^{\frac{1}{2}}
+ Cq
\left( \frac{ \tilde{m}_3 \, \tilde{m}_4 }{ \tilde{a}_1 \, \tilde{a}_2 } \right)^{\frac{1}{2}}
\right]
\right) t \\
& + C^2 q \left( \frac{ \tilde{m}_1 \, \tilde{m}_2 }{ \tilde{m}_3 \, \tilde{m}_4 } \right)^{\frac{1}{2}} (w-\tilde{m}_3)(w-\tilde{m}_4) \, =0 \, ,
\label{curveC}
\end{split}$$ or equivalently $$\begin{split}
&\left(t-C \left(\frac{ \tilde{m}_1 \, \tilde{m}_2 }{ \tilde{a}_1\, \tilde{a}_2 }\right)^{\frac{1}{2}}
\right)
\left(
t-Cq
\left( \frac{ \tilde{a}_1 \, \tilde{a}_2 }{ \tilde{m}_3 \, \tilde{m}_4 } \right)^{\frac{1}{2}}
\right) w^2 \\
& + \left( - \left( \tilde{m}_1 + \tilde{m}_2 \right) t^2
+ b \, t
- \, C^2q \left( \frac{ \tilde{m}_1 \, \tilde{m}_2 }{ \tilde{m}_3 \, \tilde{m}_4 } \right)^{\frac{1}{2}}
\left( \tilde{m}_3 + \tilde{m}_4 \right)
\right) w \\
& + \,
\tilde{m}_1 \, \tilde{m}_2
\left(t-C \left(\frac{ \tilde{a}_1\, \tilde{a}_2 }{ \tilde{m}_1 \, \tilde{m}_2 }\right)^{\frac{1}{2}}
\right)
\left(
t-Cq
\left( \frac{ \tilde{m}_3 \, \tilde{m}_4 }{ \tilde{a}_1 \, \tilde{a}_2 } \right)^{\frac{1}{2}}
\right)
\, =0 \, .
\label{curveCD}
\end{split}$$ We have, thus, determined all the coefficients in the curve except for $b$, which is related to the Coulomb moduli parameter.
A comment on the weak coupling limit ($q \equiv C'/C \ll 1$) of the obtained curve is in order. In this limit, the curve (\[curveC\]) reduces to $$\begin{split}
& (w-\tilde{m}_1)(w-\tilde{m}_2) t^2
- \left(
C \left(\frac{ \tilde{m}_1 \, \tilde{m}_2 }{ \tilde{a}_1\, \tilde{a}_2 }\right)^{\frac{1}{2}} w^2
- b \, w
+ \,
C \left( \tilde{m}_1 \, \tilde{m}_2 \tilde{a}_1\, \tilde{a}_2 \right)^{\frac{1}{2}}
\right) t \\
& + C^2 q \left( \frac{ \tilde{m}_1 \, \tilde{m}_2 }{ \tilde{m}_3 \, \tilde{m}_4 } \right)^{\frac{1}{2}} (w-\tilde{m}_3)(w-\tilde{m}_4) = 0 \, .
\label{reduced_C}
\end{split}$$ If we choose $C=1$ and assume that $b=\tilde{a}_1+\tilde{a}_2$ with $\tilde{a}_1 = \tilde{a}_2^{-1}$ the curve (\[reduced\_C\]) coincides with the one previously given in [@Nekrasov:1996cz; @Brandhuber:1997cc; @Eguchi:2000fv]. However, we want to emphasize that this expression is valid only under the weak coupling approximation. Moreover, we want to briefly comment on the 4D limit ($R_5 \to \infty$) of our 5D curve (\[curveC\]). Details are provided in Appendix \[app:4DLimit\]. In the 4D limit the curve (\[curveC\]) reduces to the one obtained in [@Eguchi:2009gf]. This is an additional check of our result.
We are now ready to derive the duality map that corresponds to the exchange of the coordinates $$\begin{aligned}
t_d = w \, , \qquad w_d = t \, ,\end{aligned}$$ where $d$ stands for dual. Without any loss of generality we pick $|\tilde{m}_1| \geq |\tilde{m}_2|$, $|\tilde{m}_3| \geq |\tilde{m}_4|$, $|(\tilde{m}_1)_d| \geq |(\tilde{m}_2)_d|$ and $|(\tilde{m}_3)_d| \geq |(\tilde{m}_4)_d|$. Then, simply by comparing the two expressions (\[curveC\]) and (\[curveCD\]) of the SW curve we obtain the duality transformation $$\begin{split}
(\tilde{m}_1)_d = C \left(\frac{ \tilde{m}_1 \, \tilde{m}_2 }{ \tilde{a}_1 \, \tilde{a}_2 }\right)^{\frac{1}{2}} \, ,&
\qquad
(\tilde{m}_2)_d = \, C q
\left( \frac{ \tilde{a}_1 \, \tilde{a}_2 }{ \tilde{m}_3 \, \tilde{m}_4 } \right)^{\frac{1}{2}}
\, ,
\\
(\tilde{m}_3)_d = C \left(\frac{ \tilde{a}_1 \, \tilde{a}_2 }{ \tilde{m}_1 \, \tilde{m}_2 }\right)^{\frac{1}{2}} \, ,&
\qquad
(\tilde{m}_4)_d = \, C q
\left( \frac{ \tilde{m}_3 \, \tilde{m}_4 }{ \tilde{a}_1 \, \tilde{a}_2 } \right)^{\frac{1}{2}} \, ,
\\
b_d = b \, ,& \qquad
q_d = \left( \frac{\tilde{m}_2\tilde{m}_4}{\tilde{m}_1 \tilde{m}_3} \right)^{\frac{1}{2}} \, ,
\\
C_d = \tilde{m}_1^{\frac{1}{2}} \tilde{m}_3^{\frac{1}{2}}
\, ,&
\qquad
(\tilde{a}_1)_d{} (\tilde{a}_2)_d{} = \, C^2 q
\left( \frac{\tilde{m}_2\tilde{m}_3}{\tilde{m}_1 \tilde{m}_4} \right)^{\frac{1}{2}}
\, .
\label{Map_SU(2)_C}
\end{split}$$
So far we have not specified the $v$; a natural choice is to set it at the center of mass of the two D4-branes, where $$\begin{aligned}
a_1 = - a_2 \quad \Rightarrow \quad \tilde{a}_1 = \tilde{a}_2^{-1} \, .
\label{U(1)}\end{aligned}$$ Similarly, we pick the $s=v_d$ so that $$\begin{aligned}
(\tilde{a}_1)_d = (\tilde{a}_2)_d^{-1}
\label{U(1)D}\end{aligned}$$ is realized. This condition is satisfied when the constant $C$ is $$\begin{aligned}
C = \left( \frac{\tilde{m}_1 \tilde{m}_4}{\tilde{m}_2\tilde{m}_3} \right)^{\frac{1}{4}}
q^{-\frac{1}{2}}
\, = \, (\tilde{m}_1)_d^{\frac{1}{2}} (\tilde{m}_3)_d^{\frac{1}{2}} \, .\end{aligned}$$
![In this figure the configuration of the M5-brane that leads to 5D $SU(2)$ four flavor is depicted.[]{data-label="web"}](cyclesSU2.eps "fig:"){width="6cm"} (-210,-20)[(0,1)[190]{}]{} (-220,170)[(0,-1)[190]{}]{} (-210,-20)[(1,0)[240]{}]{} (30,-30)[(-1,0)[240]{}]{} (50,-25)[$s$]{} (50,-35)[$t$]{} (-223,180)[$w$]{} (-213,180)[$v$]{} (-200,50)[$w=\tilde{m}_1$]{} (-200,120)[$w=\tilde{m}_2$]{} (-5,50)[$w=\tilde{m}_3$]{} (-5,120)[$w=\tilde{m}_4$]{} (-153,170)[$t=\sqrt[4]{ \frac{ \tilde{m}_4}{ \tilde{m}_1 \tilde{m}_2^3 \tilde{m}_3 q^{2}} } $]{} (-73,170)[$t=\sqrt[4]{ \frac{\tilde{m}_1 \tilde{m}_3 \tilde{m}_4^3 q^{2}}{ \tilde{m}_2 } } $]{} (-150,-5)[$t=\sqrt[4]{ \frac{\tilde{m}_1^3 \tilde{m}_2 \tilde{m}_4}{ \tilde{m}_3 \, q^{2}} } $]{} (-73,-5)[$t=\sqrt[4]{ \frac{\tilde{m}_1 \, q^{2}}{ \tilde{m}_2 \tilde{m}_3^3 \tilde{m}_4} } $]{} (-90,30)[$A_1$]{} (-90,100)[$A_2$]{} (-150,30)[$M_1$]{} (-150,100)[$M_2$]{} (-25,30)[$M_3$]{} (-25,100)[$M_4$]{} (-32,145)[$T_4$]{} (-102,145)[$T_3$]{} (-32,20)[$T_2$]{} (-102,20)[$T_1$]{} (-37,80)[$\left(A_2\right)_d$]{} (-107,80)[$\left(A_1\right)_d$]{}
The transformation rule $b_d=b$ contains the duality transformation of the Coulomb moduli implicitly. We could in principle calculate $b$ explicitly in terms of the Coulomb moduli, as in [@Eguchi:2009gf]. However, for our purpose it is enough to consider the contour integrals of the SW one-form around the $A$ and the $A_d$ cycles. We depict all the cycles on the M5-brane in Figure \[web\]. For the four junctions in Figure \[web\] we find the following topological relations $$\begin{split}
A_1 - T_1 - M_1 + \left(A_1\right)_d = 0 \, , \qquad
& - \left(A_1\right)_d - M_2 + T_3 + A_2 = 0 \, , \\
-A_2 + T_4 + M_4 - \left(A_2\right)_d = 0 \, , \qquad
& \left(A_2\right)_d+ M_3 - T_2 - A_1 = 0
\end{split}
\label{cycle_cons}$$ among the cycles. In our conventions the integrals are positive when we go around $w = \infty$ and $t=\infty$ clock-wise. The relations (\[cycle\_cons\]) among the cycles can also be read as relations among cycle integrals by replacing $A \to \int_{A} \lambda_{SW}$. Using the expression of the SW one-form given in (\[SW\_1-form\]) $$\begin{aligned}
\lambda_{SW}
= \frac{i}{(2\pi)^2 \ell_p{}^3} v ds
= \frac{i R_5 R_{10}}{(2\pi)^2 \ell_p{}^3} \log w \frac{dt}{t} \, ,\end{aligned}$$ where the factor $1/(2\pi)^2 l_s{}^3$ is the tension of the M2-brane, we can calculate the cycle integrals. The integral around $M_1$ is obtained by considering the limit $t \to \infty$ and regarding the coordinate $w$ as a multivalued function of $t$. The curve around this region is approximately given by $$(w(t) - \tilde{m}_1)(w(t) - \tilde{m}_2) t^2 \approx 0
\qquad \Rightarrow \qquad
w(t) = \left\{
\begin{array}{l}
\tilde{m}_1 + {\cal O}(t^{-1}) \\
\tilde{m}_2 + {\cal O}(t^{-1})
\end{array}
\right. \, .$$ The first branch contributes to the integral around the cycle $M_1$. The contour integral can be carried out as $$\oint_{M_1} \log w(t) \frac{dt}{t}
= - \oint_{t=\infty} \left( \log \tilde{m}_1 + {\cal O}(t^{-1}) \right) \frac{dt}{t} = 2 \pi i \log \tilde{m}_1 \, .$$ Similarly, we obtain the rest of the integrals around the cycles $M_\mathfrak{i}$ and $T_\mathfrak{i}$ $$\begin{split}
\oint_{M_\mathfrak{i}} \lambda_{SW} &= - \frac{R_5 R_{10}}{2 \pi \ell_p{}^3} \log\tilde{m_\mathfrak{i}} = m_\mathfrak{i} \, ,
\\
\oint_{T_\mathfrak{i}} \lambda_{SW} &= \frac{R_5 R_{10}}{2 \pi \ell_p{}^3} \log (\tilde{m_\mathfrak{i}})_d = \left( m_\mathfrak{i} \right)_d \, ,
\end{split}
\label{cint}$$ where $\mathfrak{i}=1,\dots,4$ is the flavor index. What is more, the contour integrals around the cycles $A_i$ and $\left(A_i\right)_d$, by definition, give the Coulomb moduli: $$\begin{split}
\oint_{A_1} \lambda_{SW}
&= - \oint_{A_2} \lambda_{SW}
= - \frac{R_5 R_{10}}{2 \pi \ell_p{}^3} \log \tilde{a} = a \, ,
\\
\oint_{A'_1} \lambda_{SW}
&= - \oint_{A'_2} \lambda_{SW}
= \frac{R_5 R_{10}}{2 \pi \ell_p{}^3} \log \tilde{a}_d = - a_d \, ,
\end{split}
\label{aint}$$ where $i=1,2$ is the color index. The first equality in each line is ensured by (\[cycle\_cons\]). The sign of $a_d$ in the second line is inverted because $$\begin{aligned}
\label{equalSWoneform}
(\lambda_{SW})_d
= \frac{i (R_5)_d (R_{10})_d}{(2\pi)^2 \ell_p{}^3} \log (w)_d \frac{d(t)_d}{(t)_d}
= \frac{i R_{10} R_5}{(2\pi)^2 l_p{}^3} \log t \frac{dw}{w}
= - \lambda_{SW} \, .\end{aligned}$$ The four conditions in (\[cycle\_cons\]) consistently lead to the duality relation $$\begin{aligned}
\tilde{a}_d = \left( \frac{\tilde{m}_2 \tilde{m}_4}
{\tilde{m}_1\tilde{m}_3} \right)^{\frac{1}{4}} q^{-\frac{1}{2}} \tilde{a} \, .
\label{dual_a}\end{aligned}$$ The positions $a_1$ and $a_2$ of the color D4-branes were originally defined “classically” in the D4/NS5 brane setup and are not necessarily [equal]{} to $a$ defined by the cycle integral (\[aint\]). However, the first equality of each line in (\[aint\]) indicates that the classical conditions (\[U(1)\]) and (\[U(1)D\]) are satisfied even when we include the quantum effects.
Summarizing, the duality map for the self-dual $SU(2)$ case is $$\begin{split}
\label{map_reflection_su2}
(\tilde{m}_1)_d
= \tilde{m}_1^{\frac{3}{4}} \tilde{m}_2^{\frac{1}{4}}
\tilde{m}_3^{-\frac{1}{4}} \tilde{m}_4^{\frac{1}{4}} q^{-\frac{1}{2}} \, ,&
\qquad
(\tilde{m}_2)_d
= \tilde{m}_1^{\frac{1}{4}} \tilde{m}_2^{-\frac{1}{4}}
\tilde{m}_3^{-\frac{3}{4}} \tilde{m}_4^{-\frac{1}{4}} q^{\frac{1}{2}} \, ,
\\
(\tilde{m}_3)_d
= \tilde{m}_1^{-\frac{1}{4}} \tilde{m}_2^{-\frac{3}{4}}
\tilde{m}_3^{-\frac{1}{4}} \tilde{m}_4^{\frac{1}{4}} q^{-\frac{1}{2}} \, ,&
\qquad
(\tilde{m}_4)_d
= \tilde{m}_1^{\frac{1}{4}} \tilde{m}_2^{-\frac{1}{4}}
\tilde{m}_3^{\frac{1}{4}} \tilde{m}_4^{\frac{3}{4}} q^{\frac{1}{2}} \, ,
\\
q_d = \left( \frac{\tilde{m}_2\tilde{m}_4}{\tilde{m}_1 \tilde{m}_3} \right)^{\frac{1}{2}} \, ,&
\qquad
\tilde{a}_d = \left( \frac{\tilde{m}_2 \tilde{m}_4}{\tilde{m}_1\tilde{m}_3} \right)^{\frac{1}{4}} q^{-\frac{1}{2}} \tilde{a} \,
\end{split}$$ and can, alternatively, be reorganized as $$\begin{split}
(\tilde{m}_1)_d (\tilde{m}_2)_d (\tilde{m}_3)_d (\tilde{m}_4)_d
= \frac{\tilde{m}_1 \tilde{m}_4}{\tilde{m}_2 \tilde{m}_3} \, ,&
\qquad
\frac{(\tilde{m}_1)_d (\tilde{m}_4)_d}{(\tilde{m}_2)_d (\tilde{m}_3)_d}
= \tilde{m}_1 \tilde{m}_2 \tilde{m}_3 \tilde{m}_4 \, ,
\\
\frac{(\tilde{m}_2)_d (\tilde{m}_4)_d}{(\tilde{m}_1)_d (\tilde{m}_3)_d}
= q^2 \, ,&
\qquad
{q_d}^2
= \frac{\tilde{m}_2 \tilde{m}_4}{\tilde{m}_1 \tilde{m}_3} \, ,
\\
\frac{(\tilde{m}_1)_d (\tilde{m}_2)_d}{(\tilde{m}_3)_d (\tilde{m}_4)_d}
= \frac{\tilde{m}_1 \tilde{m}_2}{\tilde{m}_3 \tilde{m}_4} \, ,&
\qquad
q_d^{-\frac{1}{2}} \tilde{a}_d = q^{-\frac{1}{2}} \tilde{a} \, .
\end{split}
\label{Map_5DC}$$
$SU(N)^{M-1} \leftrightarrow SU(M)^{N-1}$ duality
-------------------------------------------------
The extension of the analysis in the previous subsection to the generic linear quiver gauge theory is straightforward. The asymptotics of the NS5-branes and the D4-branes constrain the form of the SW curve of $SU(N)^{M-1}$ gauge theory to $$\begin{aligned}
\prod^{N}_{\alpha=1} (w-\tilde{m}_\alpha) t^M + \cdots + d \prod^{2N}_{\alpha=N+1} (w-\tilde{m}_\alpha)
=0
\label{NMw}\end{aligned}$$ and $$\begin{aligned}
\prod^{M}_{i=1}\left[t-C_{(i)}\left(
\frac{ \prod_{\alpha=1}^N \tilde{a}_\alpha^{(i-1)} }{ \prod_{\beta=1}^N \tilde{a}^{(i)}_\beta}
\right)^{1/2}
\right] w^N + \cdots
+ d' \prod^{M}_{i=1}\left[ t-C_{(i)}\left(
\frac{ \prod_{\alpha=1}^N \tilde{a}^{(i)}_\alpha}{ \prod_{\beta=1}^N \tilde{a}_\beta^{(i-1)} }
\right)^{1/2}
\right]
=0 \, ,
\label{NMt}\end{aligned}$$ where we have defined $$\begin{aligned}
\label{mass-a}
\tilde{a}_{\alpha}^{(0)} \equiv \tilde{m}_\alpha
\quad \text{and} \quad
\tilde{a}_{\alpha}^{(M)} \equiv \tilde{m}_{N+\alpha} \, .\end{aligned}$$ The index $\alpha=1,\dots,N$ is used to count colors inside each single $SU(N)$ factor, whereas the index $i=1,\dots,M$ counts hypermultiplets along the quiver gauge group. This notation is further clarified in Figure \[branesetup\].
(1,1)(0,0) (-0.3,0)[(0,1)[1]{}]{} (-0.36,1.15)[NS5$_1$]{} (0.1,0)[(0,1)[1]{}]{} (0.06,1.15)[NS5$_2$]{} (0.5,0)[(0,1)[1]{}]{} (0.44,1.15)[NS5$_i$]{} (0.9,0)[(0,1)[1]{}]{} (0.81,1.15)[NS5$_{i+1}$]{} (1.3,0)[(0,1)[1]{}]{} (1.19,1.15)[NS5$_{M-1}$]{} (1.7,0)[(0,1)[1]{}]{} (1.63,1.15)[NS5$_M$]{}
(-0.7,0.1)[(1,0)[0.4]{}]{} (-0.65,0.13)[$m'_{1} = a'{}_{1}^{(0)}$]{} (-0.7,0.25)[(1,0)[0.4]{}]{} (-0.65,0.28)[$m'_{2} = a'{}_{2}^{(0)}$]{} (-0.5,0.45) (-0.5,0.55) (-0.5,0.65) (-0.7,0.75)[(1,0)[0.4]{}]{} (-0.78,0.78)[$m'_{N-1} = a'{}_{N-1}^{(0)}$]{} (-0.7,0.9)[(1,0)[0.4]{}]{} (-0.66,0.93)[$m'_{N} = a'{}_{N}^{(0)}$]{}
(-1.1,0.08)[D4$_1$]{} (-1.1,0.23)[D4$_2$]{} (-1.1,0.73)[D4$_{N-1}$]{} (-1.1,0.88)[D4$_{N}$]{}
(-0.3,0.15)[(1,0)[0.4]{}]{} (-0.17,0.18)[$a'{}_{1}^{(1)}$]{} (-0.3,0.3)[(1,0)[0.4]{}]{} (-0.17,0.33)[$a'{}_{2}^{(1)}$]{} (-0.1,0.47) (-0.1,0.53) (-0.1,0.59) (-0.3,0.7)[(1,0)[0.4]{}]{} (-0.19,0.73)[$a'{}_{N-1}^{(1)}$]{} (-0.3,0.85)[(1,0)[0.4]{}]{} (-0.17,0.88)[$a'{}_{N}^{(1)}$]{}
(0.22,0.5) (0.3,0.5) (0.38,0.5)
(0.5,0.1)[(1,0)[0.4]{}]{} (0.63,0.13)[$a'{}_{1}^{(i)}$]{} (0.7,0.27) (0.7,0.34) (0.7,0.41) (0.5,0.5)[(1,0)[0.4]{}]{} (0.63,0.53)[$a'{}_{\alpha}^{(i)}$]{} (0.7,0.67) (0.7,0.74) (0.7,0.81) (0.5,0.9)[(1,0)[0.4]{}]{} (0.63,0.93)[$a'{}_{N}^{(i)}$]{}
(1.02,0.5) (1.1,0.5) (1.18,0.5)
(1.3,0.1)[(1,0)[0.4]{}]{} (1.39,0.13)[$a'{}_{1}^{(M-1)}$]{} (1.3,0.3)[(1,0)[0.4]{}]{} (1.39,0.33)[$a'{}_{2}^{(M-1)}$]{} (1.5,0.48) (1.5,0.54) (1.5,0.6) (1.3,0.7)[(1,0)[0.4]{}]{} (1.39,0.73)[$a'{}_{N-1}^{(M-1)}$]{} (1.3,0.9)[(1,0)[0.4]{}]{} (1.39,0.93)[$a'{}_{N}^{(M-1)}$]{}
(1.7,0.05)[(1,0)[0.4]{}]{} (1.73,0.08)[$m'_{N+1} = a'{}_{1}^{(M)}$]{} (1.7,0.2)[(1,0)[0.4]{}]{} (1.73,0.23)[$m'_{N+2} = a'{}_{2}^{(M)}$]{} (1.9,0.41) (1.9,0.51) (1.9,0.61) (1.7,0.75)[(1,0)[0.4]{}]{} (1.73,0.78)[$m'_{2N-1} = a'{}_{N-1}^{(M)}$]{} (1.7,0.87)[(1,0)[0.4]{}]{} (1.73,0.9)[$m'_{2N} = a'{}_{N}^{(M)}$]{}
Similarly, the curve for $SU(M)^{N-1}$ can be written in two forms: $$\begin{aligned}
\prod^{M}_{i=1} (w-\tilde{m}_i) t^N + \cdots
+ D \prod^{2M}_{i=M+1} (w-\tilde{m}_i)
= 0
\label{MNw}\end{aligned}$$ and $$\begin{aligned}
\prod^{N}_{\alpha=1}\left[t - C_{(\alpha)}\left(
\frac{ \prod_{i=1}^M \tilde{a}_i^{(\alpha-1)} }{ \prod_{j=1}^M \tilde{a}^{(\alpha)}_j}
\right)^{1/2}
\right] w^M + \cdots
+ D' \prod^{N}_{\alpha=1}\left[t-C_{(\alpha)}\left(
\frac{ \prod_{i=1}^M \tilde{a}^{(\alpha)}_i}{ \prod_{j=1}^M \tilde{a}_j^{(\alpha-1)} }
\right)^{1/2}
\right] = 0 \, ,
\label{MNt}\end{aligned}$$ where, now, the index $i=1,\dots,M$ is used to count colors inside a single $SU(M)$ factor and the index $\alpha=1,\dots,N$ counts hypermultiplets along the product gauge group. Also, we define $$\begin{aligned}
\tilde{a}_{i}^{(0)} \equiv \tilde{m}_i
\quad \text{and} \quad
\tilde{a}_{i}^{(N)} \equiv \tilde{m}_{M+i} \, .\end{aligned}$$ As in the previous subsection, we now have to express the constants $C_{(i)}$ of the $SU(N)^{M-1}$ SW curve in terms of the gauge coupling constants $q^{(i)}$. The educated assumption $$\begin{aligned}
q^{(i)} = \frac{C_{(i+1)}}{C_{(i)}}
\quad \Rightarrow \quad
C_{(i)}= C \prod_{k=1}^{i-1} q^{(k)} \, ,
\label{CK}\end{aligned}$$ turns out to be the correct one, where $C \equiv C_{(1)}$ is some common constant that corresponds to the ambiguity of the rescaling of the coordinate $t$. The same relation (\[CK\]) holds for the constants $C_{(\alpha)}$ of the $SU(M)^{N-1}$ SW curve in terms of the gauge coupling constants $q^{(\alpha)}$. We are now ready to derive the duality map for the exchange $w \leftrightarrow t$. By comparing the SW curves (\[NMw\]) and (\[MNt\]) we obtain[^12] $$\begin{aligned}
\tilde{m}_\alpha
= \left( C_{(\alpha)}\left(
\frac{ \prod_{i=1}^M \tilde{a}_i^{(\alpha-1)} }{ \prod_{j=1}^M \tilde{a}^{(\alpha)}_j}
\right)^{1/2} \right)_d \, ,
\quad
\tilde{m}_{N+\alpha}
= \left( C_{(\alpha)}\left(
\frac{ \prod_{i=1}^M \tilde{a}^{(\alpha)}_i}{ \prod_{j=1}^M \tilde{a}_j^{(\alpha-1)} }
\right)^{1/2} \right)_d \, .
\label{comp1}\end{aligned}$$ Furthermore, by comparing (\[NMt\]) with (\[MNw\]), we obtain $$\begin{aligned}
C_{(i)}\left(
\frac{ \prod_{\alpha=1}^N \tilde{a}_\alpha^{(i-1)} }{ \prod_{\beta=1}^N\tilde{a}^{(i)}_\beta}
\right)^{1/2}
= ( \tilde{m}_i )_d \, ,
\quad
C_{(i)}\left(
\frac{ \prod_{\alpha=1}^N \tilde{a}^{(i)}_\alpha}{ \prod_{\beta=1}^N \tilde{a}_\beta^{(i-1)} }
\right)^{1/2}
= ( \tilde{m}_{M+i} )_d \, .
\label{comp2}\end{aligned}$$
{width="5cm"} (-87,-5)[$ \left( A^{(\alpha-1)}_i \right)_d$]{} (-11,70)[$A^{(i)}_\alpha$]{} (-165,70)[ $A^{(i-1)}_\alpha$]{} (-82,135)[$\left( A^{(\alpha)}_i \right)_d$]{} (-200,-20)[(0,1)[160]{}]{} (-200,-20)[(1,0)[240]{}]{} (50,-25)[$s$]{} (-215,140)[$v$]{}
Again as in the previous subsection we have to impose the “conservation” of the cycle integrals, $$\begin{aligned}
\left( A^{(\alpha)}_i \right)_d
- A^{(i)}_\alpha
- \left( A^{(\alpha-1)}_i \right)_d
+ A^{(i-1)}_\alpha = 0 \, ,
\label{cycle_cons_gen}\end{aligned}$$ which leads to the map $$\begin{aligned}
\label{aa_aa}
\frac{ (\tilde{a}^{(\alpha)}_i )_d}{(a^{(\alpha-1)}_i )_d}
= \frac{\tilde{a}^{(i)}_\alpha}{\tilde{a}^{(i-1)}_\alpha} \, .\end{aligned}$$ Combining the equations above we get $$\begin{split}
\left( \tilde{a}^{(\alpha)}_{i} \right)_d
&= C \left(
\frac{\prod_{\gamma=1}^\alpha \tilde{a}^{(i)}_\gamma \prod_{\delta=\alpha+1}^N \tilde{a}^{(i-1)}_\delta}
{ \prod_{\gamma=1}^\alpha \tilde{a}_\gamma^{(i-1)} \prod_{\delta=\alpha+1}^N \tilde{a}_\delta^{(i)} }\right)^{1/2} \prod_{k=1}^{i-1} q^{(k)} \, , \\
\left( q^{(\alpha)} \right)_d
&= \left( \frac{\tilde{m}_{\alpha+1} \tilde{m}_{N+\alpha+1}}{\tilde{m}_{\alpha} \tilde{m}_{N+\alpha}} \right)^{1/2} \, .
\end{split}
\label{gen_map}$$ The duality map as derived above still includes one unknown coefficient $C$ that reflects the freedom to rescale the coordinate $t$. Moreover, the Coulomb moduli parameters are defined up to the choice of the $v$. A natural way to fix both is to impose $$\begin{aligned}
\prod_{\alpha=1}^N \tilde{a}^{(0)}_\alpha = 1
\quad \text{and} \quad
\prod_{i=1}^M (\tilde{a}^{(0)}_i)_d = 1 \, .
\label{prod_a}\end{aligned}$$ The latter condition determines the constant $C$ to be $$\begin{aligned}
C =
{\prod_{\alpha=1}^N \left( \tilde{a}^{(M)}_{\alpha} \right)^{\frac{1}{2M}}}
\prod_{i=1}^{M-1} \left( q^{(i)} \right)^{- \frac{M-i}{M}} \, .
\label{Def_C}\end{aligned}$$ At this point we have to stress that the constant $C$ depends on the choice of the origin (\[prod\_a\]) for the coordinates $v$ and $s$. Naively, one may think that the duality map depends on this choice. However, in terms of the [*physical*]{} gauge theory parameters the map is independent of this choice. A detailed description of the physical gauge theory parameters is given in Appendix \[phys-par\].
In terms of the physical gauge theory parameters the duality map is $$\begin{aligned}
& \left( \hat{a}_{i}^{(\alpha)} \right)_d
= \left( \tilde{m}_{\text{bif}}^{(i-1,i)} \right)^{\alpha-\frac{N}{2}}
\prod_{\gamma=1}^\alpha
\left(
\frac{\hat{a}_{\gamma}^{(i)} }{\hat{a}_{\gamma}^{(i-1)} } \right)
\left(
\frac{\hat{a}_\gamma^{(0)}}{\hat{a}_\gamma^{(M)}}
\right) ^{\frac{1}{M}}
\prod_{k=1}^{M}
\left( \tilde{m}_{\text{bif}}^{(k-1,k)} \right)^{\frac{N-2\alpha}{2M}}
\prod_{\ell=1}^{i-1} \left( q^{(\ell)} \right)^{\frac{i}{M}}
\prod_{\ell=i}^{M-1} \left( q^{(\ell)} \right) ^{- \frac{M-\ell}{M}} \, , \nonumber
\\
&\left( \tilde{m}^{(\alpha-1,\alpha)}_{\text{bif}} \right)_d
= \left(
\frac{\hat{a}_\alpha^{(M)}}{\hat{a}_\alpha^{(0)}}
\prod_{k=1}^{M} \tilde{m}_{\text{bif}}^{(k-1,k)}
\right)^{\frac{1}{M}} \, ,
\\
& \left( q^{(\alpha)} \right)_d
= \left( \frac{\hat{a}^{(0)}_{\alpha+1} \hat{a}^{(M)}_{\alpha+1}}
{\hat{a}^{(0)}_{\alpha} \hat{a}^{(M)}_{\alpha}} \right)^{1/2} \, . \nonumber
$$
### $SU(M)$ $\leftrightarrow$ $SU(2)^{M-1}$ case {#sum-leftrightarrow-su2m-1-case .unnumbered}
Before ending this section we wish to consider the special case with $N=2$. This duality between $SU(M)$ and $SU(2)^{M-1}$ gauge theories is of particular interest due to its implications in 2D CFTs. Through the AGTW conjecture this duality relates four-point correlation functions of $q$-deformed $W_M$ Toda theories to $(M+2)$-correlation functions of $q$-deformed Liouville theories. This topic will be discussed in Section \[sec:GaugeToCFT\]. For now we just give the map $$\begin{split}
&(\tilde{m}_{1}^{\text{f}})_d
= \left( \frac{(\tilde{m}^{\text{f}}_1)^{1+M} \tilde{m}^{\text{f}}_3}
{(\tilde{m}^{\text{f}}_2)^{1-M} \tilde{m}^{\text{f}}_4} \right) ^{\frac{1}{2M}}
\prod_{k=1}^{M-1} \left( q^{(k)} \right)^{\frac{M-k}{M}} \, ,
\\
&(\tilde{m}_{i}^{\text{f}})_d
= \left( \frac{\tilde{m}^{\text{f}}_1 \tilde{m}^{\text{f}}_3}{\tilde{m}^{\text{f}}_2 \tilde{m}^{\text{f}}_4} \right) ^{\frac{1}{2M}}
\tilde{m}_{ \text{bif} }^{ (i-1,i)}
\prod_{k=1}^{i-1} \left( q^{(k)} \right)^{-\frac{k}{M}}
\prod_{k=i}^{M-1} \left( q^{(k)} \right)^{\frac{M-k}{M}} \, ,
\\
&(\tilde{m}_{M}^{\text{f}})_d
= \left( \frac{\tilde{m}^{\text{f}}_1 (\tilde{m}^{\text{f}}_3)^{1+M}}
{\tilde{m}^{\text{f}}_2 (\tilde{m}^{\text{f}}_4)^{1-M} } \right) ^{\frac{1}{2M}}
\prod_{k=1}^{M-1} \left( q^{(k)} \right)^{-\frac{k}{M}} \, ,
\\
&(\tilde{m}_{M+1}^{\text{f}})_d
= \left( \frac{(\tilde{m}_2^{\text{f}})^{1+M} \tilde{m}_4^{\text{f}}}
{(\tilde{m}_1^{\text{f}})^{1-M} \tilde{m}_3^{\text{f}}}
\right) ^{\frac{1}{2M}}
\prod_{k=1}^{M-1} \left( q^{(k)} \right)^{-\frac{M-k}{M}} \, ,
\\
&(\tilde{m}_{M+i}^{\text{f}})_d
= \left( \frac{\tilde{m}_2^{\text{f}} \tilde{m}_4^{\text{f}}}
{\tilde{m}_1^{\text{f}} \tilde{m}_3^{\text{f}}}
\right) ^{\frac{1}{2M}}
\tilde{m}_{ \text{bif} }^{ (i-1,i)}
\prod_{k=1}^{i-1} \left( q^{(k)} \right)^{\frac{k}{M}}
\prod_{k=i}^{M-1} \left( q^{(k)} \right)^{-\frac{M-k}{M}} \, ,
\\
&(\tilde{m}_{2M}^{\text{f}})_d
= \left( \frac{\tilde{m}_2^{\text{f}} (\tilde{m}_4^{\text{f}})^{1+M}}
{\tilde{m}_1^{\text{f}} (\tilde{m}_3^{\text{f}})^{1-M}}
\right) ^{\frac{1}{2M}}
\prod_{k=1}^{M-1} \left( q^{(k)} \right)^{\frac{k}{M}} \, ,
\end{split}$$ and $$\begin{split}
&\left( \tilde{a}_{1}^{\text{f}} \right)_d
= \tilde{a}^{(1)}_{\text{f}}
\left(
\frac{(\tilde{m}_2^{\text{f}})^{1-M} \tilde{m}_4^{\text{f}}}
{(\tilde{m}_1^{\text{f}})^{1-M} \tilde{m}_3^{\text{f}}}
\right) ^{\frac{1}{2M}}
\prod_{i=1}^{M-1} \left( q^{(i)} \right)^{-\frac{M-k}{M}} \, ,
\\
&\left( \tilde{a}_{i}^{\text{f}} \right)_d
= \frac{\tilde{a}^{(i)}_{\text{f}} }{\tilde{a}^{(i-1)}_{\text{f}} }
\left( \frac{\tilde{m}_2^{\text{f}} \tilde{m}_4^{\text{f}}}
{\tilde{m}_1^{\text{f}} \tilde{m}_3^{\text{f}}}
\right) ^{\frac{1}{2M}}
\prod_{k=1}^{i-1} \left( q^{(k)} \right)^{\frac{k}{M}}
\prod_{k=i}^{M-1} \left( q^{(k)} \right)^{-\frac{M-k}{M}} \, ,
\\
&\left( \tilde{a}_{M}^{\text{f}} \right)_d
= \frac{1}{\tilde{a}^{(M-1)}_{\text{f}} }
\left( \frac{\tilde{m}_2^{\text{f}} \tilde{m}_4^{\text{f}}}
{\tilde{m}_1^{\text{f}} \tilde{m}_3^{\text{f}}}
\right) ^{\frac{1}{2M}}
\prod_{i=1}^{M-1} \left( q^{(i)} \right)^{\frac{k}{M}} \, ,
\\
&q_d
= \left( \frac{ \tilde{m}_{1}^{\text{f}} \tilde{m}_{4}^{\text{f}}}
{\tilde{m}_{2}^{\text{f}} \tilde{m}_{3}^{\text{f}}}
\right)^{1/2} \, .
\end{split}$$ It is interesting to note that the mass parameters and the gauge coupling constant of the dual $SU(M)$ theory are completely independent of the Coulomb moduli parameters of the original $SU(2)^{M-1}$ theory.
Topological string derivation {#sec:TopStringDeriv}
=============================
In the previous section we presented a derivation of the duality map using the Seiberg-Witten formalism. Here, we will present an independent derivation (or check) using Nekrasov’s partition function. We compute Nekrasov’s partition functions for 5D $\mathcal{N}=1$ $SU(N)^{M-1}$ and $SU(M)^{N-1}$ linear quivers and show that they are equal when we relate their gauge theory parameters with the duality map (\[gen\_map\]).
The computation of Nekrasov’s partition functions is performed using topological string theory. As we reviewed in Section \[sec:Review\], topological string theory offers an alternative derivation of the gauge theory partition functions and most importantly provides [*a rewriting of the partition function*]{} in a form in which the duality is manifest. It is unlikely that gauge theory reasoning alone would lead to this rewriting. However, from the string theory point of view it is natural. Due to the fact that the partition function is read off from a toric diagram, symmetries that arise from the CY geometry (and are obscured otherwise) are manifest in this formalism.
In the previous section we used the type IIA D4/NS5 brane setup to realize the linear quiver gauge theories. As we discussed in Section \[sec:Review\], the D4/NS5 brane configuration is dual to type IIA string theory compactified on CY$_3$. We are interested in the special class of Calabi-Yau manifolds that satisfy the toric condition and lead to $SU(N)$ gauge theory. Theses CY$_3$ are completely specified by their toric diagrams. In the case of linear quivers the toric diagram is essentially same as the brane diagram.
![The D4/NS5 system is T-dual to IIB $(p,q)$ 5-brane system. The M/IIB duality relates it to M-theory on the corresponding toric CY.[]{data-label="BraneToToric"}](BraneToToric.eps){width="130mm"}
Following [@Aharony:1997bh], the D4/NS5 brane setup in IIA theory is T-dual to IIB $(p,q)$-brane web system (D5/NS5). When uplifting this system to M-theory via M/IIB duality, we obtain M-theory on $M^4\times \textrm{CY}_3\times S^1$ where CY is a toric three-fold whose $(p,q)$-cycle shrinks. In this way the D4/NS5 system is equivalent to M-theory on toric CY, or IIA on CY which is the usual geometric engineering setup. This connection is illustrated in Figure \[BraneToToric\].
Given the toric diagram we can use the topological vertex formalism to calculate Nekrasov’s partition function of 4D $\mathcal{N}=2$ gauge theories. We should stress again that in this paper we study the Nekrasov partition function for the 5D uplift of the 4D gauge theory. The 5D Nekrasov partition function is precisely equal to the topological string partition function[^13]; of course after the appropriate identification of the gauge theory parameters with the string theory parameters.
Writing down the topological string partition function is simple using the topological vertex formalism. The procedure was reviewed in Section \[sec:Review\]. What is quite tedious is to bring the topological string partition function in the form given by Nekrasov. For that we have to perform the sums. Such calculations have previously been done by [@Aganagic:2002qg; @Iqbal:2003zz; @Eguchi:2003sj; @Iqbal:2004ne; @Taki:2007dh]. They involve summations over Young diagrams. The summand contains Schur and skew-Schur functions. The calculation is quite technical so we hide most of the details in Appendix \[app:NekrasovTop\]. We first warm up with the $SU(2)$ case and then present the general $\mathcal{N}=2$ $SU(N)^{M-1}$ linear quiver in its full glory. Once we bring the topological string partition function in the form that was given by Nekrasov, we obtain the identification between the gauge theory and the string theory parameters. Using this identification we finally write down the duality map that is identical to the one found in Section \[sec:MtheoryDeriv\].
$SU(2)$ gauge theory with four flavors {#subsec:topsu2}
--------------------------------------
In this subsection we compute the topological string partition function for $SU(2)$ SQCD with four flavors using the topological vertex formalism. The toric diagram from which we read off the partition functions is depicted in Figure \[fig:4flavSQCD\]. Due to the symmetry of the diagram, it is convenient to first consider the “half-geometry” of the corresponding toric CY shown in Figure \[fig:bifund\].
![The sub-diagram that engineers the bifundamental hypermultiplet of $SU(2)$ quiver gauge theories, where $R_i,Y_i,\mu_i$ denote Young diagrams. The parameters $Q_1,Q_2,Q_3$ are associated with the line labeled by the Young diagram $\mu_1, \mu_2, \mu_3$, respectively.[]{data-label="fig:bifund"}](BifundGeom2v2.eps){width="40mm"}
This sub-diagram is dual to two horizontal D4-branes crossing a vertical NS5-brane. The vertical sequence of closed loops describes a combination of the two-cycles in CY$_3$ which give a vector multiplet and two fundamental hypermultiplets. As we will see, the Kähler parameters of the three two-cycles inside the CY geometry correspond to the Coulomb moduli of the $SU(2)$ gauge group and two of the hypermultiplet masses. After gluing the two “half-geometries” according to Figure \[fig:4flavSQCD\] we obtain a toric CY$_3$ with six two-cycles, which correspond to the Coulomb moduli parameter $a$, the four flavor masses $m_{\mathfrak{i}}$ ($\mathfrak{i}=1,\dots,4$) and the gauge coupling constant $q$.
First, we will focus on the computation of the contribution from this “half-geometry” to the topological partition function. The Young diagram $R_i$ is kept to be arbitrary for as long as possible so that this computation can be used also for more generic cases like $SU(2)^{M-1}$ gauge theory. For the $SU(2)$ gauge theory with four flavors we then set $R_1=R_2=\emptyset$ in order to get the partition function.
Using the topological vertex formalism we read off the following sub-amplitude for the local geometry depicted in Figure \[fig:bifund\] $$\begin{split}
L^{\,R_1\,Y_1}_{\,R_2\,Y_2}
(Q_{1},\,Q_{2},\,Q_{3})
&\equiv
\sum_{\mu_1,\mu_2,\mu_3}
(-1)^{|\mu_2|}\mathfrak{q}^{\frac{1}{2}\kappa_{\mu_2} }\,
\prod_{i=1}^3(-Q_i)^{|\mu_i|}
\\
&\rule{0pt}{3ex}
\qquad \times
C_{\emptyset R_1 \mu_1}(\mathfrak{q})\,
C_{\mu_2 Y_1^T \mu_1^T}(\mathfrak{q})\,
C_{\mu_3 Y_2^T\mu_2^T}(\mathfrak{q})\,
C_{\mu_3^T R_2 \emptyset}(\mathfrak{q})
\\
&\rule{0pt}{4ex}=
\,S_{R_1}(\mathfrak{q}^\rho)\,S_{R_2}(\mathfrak{q}^\rho)\,S_{Y_1^T}(\mathfrak{q}^\rho)\,S_{Y_2^T}(\mathfrak{q}^\rho)\\
&\rule{0pt}{4ex}
\qquad \times
\sum_{\mu_1,\mu_2,\mu_3}
\sum_{\zeta,\eta}
S_{\mu_1^T}(-Q_1\mathfrak{q}^{R_1+\rho})\,
S_{\mu_1/\zeta}(\mathfrak{q}^{Y_1^T+\rho})\,
S_{\mu_2/\zeta}(\mathfrak{q}^{Y_1+\rho})\\
&\qquad
\times
S_{\mu_2/\eta}(Q_2\mathfrak{q}^{Y_2^T+\rho})\,
S_{\mu_3/\eta}(Q_2^{-1}\mathfrak{q}^{Y_2+\rho})\,
S_{\mu_3^T}(-Q_2Q_3\mathfrak{q}^{R_2^T+\rho})\, .
\end{split}$$ The second line of the equation is obtained by inserting the definition of the vertex function (\[def\_topv\]). In order to get a closed form of the topological string amplitude we have to perform the summation explicitly. For that we employ the Cauchy formulas $$\begin{split}
\sum_\eta
S_{\eta/R_1}(x)
S_{\eta/R_2}(y)
&=
\prod_{i,j}
(1-x_iy_j)^{-1}
\sum_\eta
S_{R_1/\eta}(y)
S_{R_2/\eta}(x) \, , \\
\sum_\eta
S_{\eta^T/R_1}(x)
S_{\eta/R_2}(y)
&=
\prod_{i,j}
(1+x_iy_j)
\sum_\eta
S_{R_1^T/\eta}(y)
S_{R_2^T/\eta^T}(x) \, .
\end{split}$$ Notice that $S_{R/\emptyset}=S_{R}$ and $S_{\emptyset/R}=\delta_{R,\emptyset}$. By using these formulas repeatedly, we obtain the following closed form of the amplitude $$\begin{split}
L^{\,R_1\,Y_1}_{\,R_2\,Y_2}
(Q_{1},\,Q_{2},\,Q_{3})
&=
\,S_{R_1}(\mathfrak{q}^\rho)\,S_{R_2}(\mathfrak{q}^\rho)\,S_{Y_1^T}(\mathfrak{q}^\rho)\,S_{Y_2^T}(\mathfrak{q}^\rho)
\\
&\rule{0pt}{4ex}\times
\frac{
\left[R_1, Y_1^T \right]_{Q_1}
\left[Y_2, R_2^T \right]_{Q_3}
\left[R_1, Y_2^T \right]_{Q_1Q_2}
\left[Y_1, R_2^T, \right]_{Q_2Q_3}
}
{
\left[ Y_1, Y_2^T \right]_{Q_2}
\left[ R_1, R_2^T \right]_{Q_1Q_2Q_3}
} \, ,
\end{split}$$ where $[*,*]_Q$ is defined as $$\begin{aligned}
\left[Y_1, Y_2 \right]_{Q}
\equiv
\prod_{i,j=1}^\infty
(1-Q\mathfrak{q}^{Y_{1i}+Y_{2j}-i-j+1})
=\left[Y_2, Y_1 \right]_{Q} \, .\end{aligned}$$ The instanton contribution of the gauge theory partition function is given by the normalized amplitude $$\begin{split}
&\tilde{L}^{\,R_1\,Y_1}_{\,R_2\,Y_2}
\equiv
\frac{L^{\,R_1\,Y_1}_{\,R_2\,Y_2}}{L^{\,\emptyset\,\emptyset}_{\,\emptyset\,\emptyset}}\\
&=\rule{0pt}{4ex}
2^{|R_1|+|R_2|+|Y_1|+|Y_2|}
\left(\sqrt{\frac{Q_1}{Q_3}}\right)^{|R_1|-|R_2|}
\left(\sqrt{{Q_1}{Q_3}}\right)^{|Y_1|+|Y_2|}
\mathfrak{q}^{-\frac{1}{4}(\kappa_{R_1}-\kappa_{R_2}-\kappa_{Y_1}+\kappa_{Y_2})}
\\
&\times\rule{0pt}{4ex}
\label{bifundamp}
S_{R_1}(\mathfrak{q}^\rho)\,S_{R_2}(\mathfrak{q}^\rho)\,S_{Y_1^T}(\mathfrak{q}^\rho)\,S_{Y_2^T}(\mathfrak{q}^\rho)
\frac{
P^{-1}_{Y_1R_1}(Q_1)
P^{-1}_{Y_2R_1}(Q_1Q_2)
P^{-1}_{R_2Y_1}(Q_2Q_3)
P^{-1}_{R_2Y_2}(Q_3)
}
{P^{-1}_{R_2R_1}(Q_1Q_2Q_3)P^{-1}_{Y_2Y_1}(Q_2)} \, ,
\end{split}$$ where we define the function $P$ as follows [@Konishi:2003qq]: $$\begin{split}
&\frac{1
}{P_{Y_1Y_2}(\mathfrak{q},Q)}
\equiv
\prod_{(i,j)\in Y_1}
\sinh \frac{\beta}{2}
\left(
a+\hbar(Y_{1\,i}+Y^T_{2\,j}-i-j+1)
\right)
\\
&\rule{0pt}{5ex}
\quad\qquad\qquad\quad
\times
\prod_{(i,j)\in Y_2}
\sinh \frac{\beta}{2}
\left(a+\hbar
(-Y^T_{1\,j}-Y_{2\,i}+i+j-1)\right)
\end{split}$$ for $\mathfrak{q}=e^{-\beta \hbar}$ and $Q=e^{-\beta a}$. To get this expression we have used the formulas (\[NPrelation\]) and (\[bracket\_N\]).
![The toric diagram that gives $SU(2)$ SQCD with four fundamental hypermultiplets. Since the eight external lines are semi-infinite half-lines, we assign the empty Young diagram $\emptyset$ to them.[]{data-label="fig:4flavSQCD"}](4fSQCDv2.eps){width="70mm"}
With this sub-amplitude at hand we move on to the computation of the full partition function of $SU(2)$ SQCD with four flavors. The associated toric diagram is depicted in Figure \[fig:4flavSQCD\]. The partition function for this toric diagram is obtained by gluing two sub-diagrams $\tilde{L}$ according to $$\begin{aligned}
Z_{\,\textrm{inst}}=
\sum_{Y_1,Y_2}
Q_B^{|Y_1|+|Y_2|}\,
\mathfrak{q}^{\frac{\kappa_{Y_1}}{2}-\frac{\kappa_{Y_2}}{2}}\,
\tilde{L}^{\,\emptyset\,Y_1}_{\,\emptyset\,Y_2}\,
(Q_{m1},Q_{F},Q_{m2})\,
\tilde{L}^{\,\emptyset\,Y_2^T}_{\,\emptyset\,Y_1^T}\,
(Q_{m4},Q_{F},Q_{m3}) \, .\end{aligned}$$ This expression is written in terms of the string theory parameters used in geometric engineering. By comparing it with the Nekrasov partition function in [@Nekrasov:2002qd] we obtain the identifications $$\begin{split}
&Q_{m1}=e^{-\beta(m_1^{\text{f}} - a)}
=\frac{\tilde{m}_1}{\tilde{a}},\quad
Q_{m2}=e^{-\beta(-m_2^{\text{f}}-a)}
= \frac{1}{\tilde{m}_2\tilde{a}}
,\quad\cr
&Q_{m3}=e^{-\beta(m_3^{\text{f}}-a)}
=\frac{\tilde{m}_3}{\tilde{a}}
,\quad
Q_{m4}=e^{-\beta(-m_4^{\text{f}}-a)}
= \frac{1}{\tilde{m}_4 \tilde{a}}, \quad
Q_{F}=e^{-2a\beta}
=\tilde{a}^{2} \, ,
\label{Q_Def}
\end{split}$$ where the second equality is written in the M-theoretical parametrization from Section \[sec:MtheoryDeriv\].
In particular, the “numerator contribution" of the left sub-diagram $
\tilde{L}^{\,\emptyset\,Y_1}_{\,\emptyset\,Y_2}\,
(Q_{m1},Q_{F},Q_{m2})
$ takes the form $$\begin{split}
&P^{-1}_{Y_1\emptyset}(Q_{m1})
P^{-1}_{Y_2\emptyset}(Q_{m1}Q_F)
P^{-1}_{\emptyset Y_1}(Q_FQ_{m2})
P^{-1}_{\emptyset Y_2}(Q_{m2})
\cr
&\rule{0pt}{4ex} \qquad\qquad\qquad
= (-1)^{|Y_1|+|Y_2|}
\prod_{f=1,2}
Z_{\,\textrm{fund}}(\,a,\vec{Y},m_f,\hbar;\beta\,) \, ,
\end{split}$$ where we have used (\[Pinverse\]) together with Nekrasov’s expresion (\[Nek5Dfund\]). This is precisely the contribution from the two fundamental hypermultiplets with masses $m_1$ and $m_2$. The sub-diagram $\tilde{L}^{Y_2^T\emptyset}_{Y_1^T\emptyset}$ gives the contribution of the two fundamental hypermultiplets with masses $m_3$ and $m_4$ in a similar fashion. Moreover, the remaining part has the interpretation of contribution from the vector multiplet $$\begin{aligned}
\frac{S_{Y_1}(\mathfrak{q}^\rho)\,S_{Y_2}(\mathfrak{q}^\rho)}{P^{-1}_{Y_1 Y_2}(Q_F)}
\frac{S_{Y_2^T}(\mathfrak{q}^\rho)\,S_{Y_1^T}(\mathfrak{q}^\rho)}{P^{-1}_{Y_2^T Y_1^T}(Q_F)}
= (-4)^{-|Y_1|-|Y_2|} Z_{\textrm{vector}} (\,a,\vec{Y},\hbar;\beta\,)\end{aligned}$$ where we have used (\[specializedSchur\]), (\[Ptranspose\]) and (\[Nek5Dvect2\]). The details of the computation can be found in Appendix \[app:NekrasovTop\]. We have thus exactly reproduced the Nekrasov partition function [@Nekrasov:2002qd], where the instanton factor is given by $$\begin{aligned}
q
=Q_B\sqrt{Q_{m1}Q_{m2}Q_{m3}Q_{m4}}
=Q_B\, \tilde{a}^{-2} \sqrt{\frac{\tilde{m}_1\tilde{m}_3}{\tilde{m}_2\tilde{m}_4}} \, .
\label{q_Def}\end{aligned}$$ It is remarkable that the parametrization (\[q\_Def\]) does not depend on the $\Omega$ background parameter $\mathfrak{q}=e^{ - \beta \hbar}$.
We can interpret the identification of the parameters (\[Q\_Def\]) and (\[q\_Def\]) in the context of the brane setup. Taking into account that $\tilde{a}_{\alpha}$ and $\tilde{m}_\mathfrak{i}$, correspond to the positions of the color branes and the flavor branes respectively, the ratio of them corresponds to the distance between the corresponding branes as in (\[Q\_Def\]). In the similar way, by rewriting (\[q\_Def\]) as $$q = \sqrt{(Q_{m1} Q_B Q_{m3}) \times (Q_{m2} Q_B Q_{m4})}$$ we see that $q$ can be interpreted as the average distance between the two NS5-branes in the $v \rightarrow \pm \infty$ asymptotic regions, as discussed in Section \[sec:MtheoryDeriv\]. This observation justifies the identification (\[CCq\]).
### Reflection symmetry {#reflection-symmetry .unnumbered}
The topological string partition function $Z=\left(L_{\,\emptyset\emptyset}^{\,\emptyset\emptyset}\right)^2 Z_{\textrm{inst}}$ (without normalization) has the same symmetries as the toric diagram it is based on. The normalization factor $\left(L_{\,\emptyset\emptyset}^{\,\emptyset\emptyset}\right)^2$ gives the perturbative contribution of the Nekrasov partition function, while $Z$ is equivalent to the full Nekrasov partition function. Therefore, a graphical symmetry of the toric diagram is also a symmetry of the full quantum gauge theory, including perturbative and instanton corrections.
Typical examples are the reflection symmetries of Figure \[fig:4flavSQCD\]. The partition function is invariant under reflection along the diagonal axis when it is performed together with the transformation $$\begin{aligned}
Q_{m2}\leftrightarrow Q_{m3}\, , \quad
Q_{B}\leftrightarrow Q_{F} \, .\end{aligned}$$ This reflection symmetry implies the following duality relations $$\begin{split}
(Q_{m1})_d=Q_{m1}\, ,\quad
(Q_{m2})_d=Q_{m3}\, ,&\quad
(Q_{m3})_d=Q_{m2}\, ,\quad
(Q_{m4})_d=Q_{m4}\, ,\cr
(Q_{B})_d=Q_{F}\, ,&\quad
(Q_{F})_d=Q_{B}\, .
\end{split}$$ In the M-theory language, it is an invariance of the Nekrasov partition function under the transformation $$\begin{split}
\frac{(\tilde{m}_1)_d}{(\tilde{a})_d}=
\frac{\tilde{m}_1}{\tilde{a}} \, , \quad
\frac{1}{(\tilde{m}_2)_d (\tilde{a})_d}
=\frac{\tilde{m}_3}{\tilde{a}} \, , \quad
\rule{0pt}{4ex}
& \frac{(\tilde{m}_3)_d}{(\tilde{a})_d}
= \frac{1}{\tilde{m}_2\tilde{a}} \, , \quad
\frac{1}{(\tilde{m}_4)_d (\tilde{a})_d} = \frac{1}{\tilde{m}_4 \tilde{a}} \, ,
\cr
\rule{0pt}{4ex}
q_d\,{(\tilde{a})_d^{2}}\sqrt{\frac{(\tilde{m}_2)_d(\tilde{m}_4)_d}{(\tilde{m}_1)_d(\tilde{m}_3)_d}}
=\tilde{a}^{2} \, , \quad
\rule{0pt}{4ex}
& (\tilde{a})_d^{2}=q\,{\tilde{a}^{2}}\sqrt{\frac{\tilde{m}_2\tilde{m}_4}{\tilde{m}_1\tilde{m}_3}} \, .
\label{top_su2_map1}
\end{split}$$ This is the self-duality of the holomorphic sector of the 5D gauge theory in the Coulomb branch.
Note that if we combine this duality map with a known symmetry of the Nekrasov partition function, we obtain another expression for this self-duality. In particular, we can combine with a simultaneous change of the sign of the Coulomb moduli and the masses discussed at the end of Appendix \[app:NekrasovTop\], which correspond to $\tilde{m}_{\mathfrak{i}} \to \tilde{m}_{\mathfrak{i}}^{-1}$, $\tilde{a} \to \tilde{a}^{-1}$. Acting this symmetry transformation on both the original and the dual theory in (\[top\_su2\_map1\]) we obtain $$\begin{split}
\frac{(\tilde{a})_d}{(\tilde{m}_1)_d}=
\frac{\tilde{a}}{\tilde{m}_1} \, , \quad
(\tilde{m}_2)_d (\tilde{a})_d
=\frac{\tilde{a}}{\tilde{m}_3} \, ,\quad
\rule{0pt}{4ex}
& \frac{(\tilde{a})_d}{(\tilde{m}_3)_d}
=\tilde{m}_2\tilde{a} \, , \quad
(\tilde{m}_4)_d (\tilde{a})_d = \tilde{m}_4 \tilde{a} \, ,
\cr
\rule{0pt}{4ex}
q_d\,{(\tilde{a})_d^{-2}}\sqrt{\frac{(\tilde{m}_1)_d(\tilde{m}_3)_d}{(\tilde{m}_2)_d(\tilde{m}_4)_d}}
=\tilde{a}^{-2} \, , \quad
\rule{0pt}{4ex}
& (\tilde{a})_d^{-2}=q\,{\tilde{a}^{-2}}\sqrt{\frac{\tilde{m}_1\tilde{m}_3}{\tilde{m}_2\tilde{m}_4}} \, .
\label{top_su2_map2}
\end{split}$$ It is now straightforward to see that (\[top\_su2\_map2\]) is equivalent to the duality map (\[Map\_5DC\]) which was derived using the M5-brane construction in the previous section.[^14] The point here is that the self-dual $\Omega$-background deformation $\hbar$ maintains this duality, since we have shown that not only the Seiberg-Witten solution but also the Nekrasov partition function is invariant under the duality transformation. This result is due to the nontrivial fact that the duality map does not depend on the $\Omega$-background parameter $\hbar$. In the following we will see that this duality for high rank gauge theories also satisfy this non-trivial property.
$SU(N)^{M-1}\leftrightarrow SU(M)^{N-1}$ duality
------------------------------------------------
![The sub-diagram of the toric diagram for $SU(N)$ quiver gauge theories. The parameters $Q_{m\alpha}$ and $Q_{F\alpha}$ are associated with the lines labeled by the Young diagrams $\mu_{\alpha}$ and $\nu_{\alpha}$, respectively.[]{data-label="fig:quiv2"}](QuiverSub2.eps){width="110mm"}
We will now generalize to $SU(N)$ quiver gauge theories. As in the previous subsection, we divide the toric diagram into sub-diagrams along its symmetry lines. The sub-diagram of the generic ladder geometry we choose to compute is shown in Figure \[fig:quiv2\]. Using the topological vertex formalism, the contribution coming from this sub-diagram is $$\begin{split}
&H_{\,Y_1Y_2\cdots Y_N}^{\,R_1R_2\cdots R_N}\,(\,\mathfrak{q},Q_{m1},\cdots,Q_{mN},Q_{F1},\cdots,Q_{FN})
\\
&=\sum_{\mu_{1,\cdots ,N},\nu_{1,\cdots ,N}}
\prod_{\alpha=1}^N\,
(-Q_{m\alpha})^{|\mu_\alpha|}\,
\prod_{\alpha=1}^{N-1}\,
(-Q_{F\alpha})^{|\nu_\alpha|}\,
\\
&\times
C_{ \emptyset R_1\mu_1^T}\,
C_{ \nu_1^TY_1^T \mu_1 }\,
C_{ \nu_1 R_2 \mu_2^T}\,
C_{ \nu_2^T Y_2^T \mu_2}\,
C_{ \nu_2 R_3\mu_3^T}\,
C_{ \nu_3^TY_3^T \mu_3}\,
\cdots
C_{ \nu_{N-1}R_N \mu_N^T}\,
C_{ \emptyset Y_N^T \mu_N} \, .
\end{split}$$ By substituting the definition of the topological vertex, we obtain the following expression $$\begin{split}
&H_{\,Y_1Y_2\cdots Y_N}^{\,R_1R_2\cdots R_N}
\\
&=\rule{0pt}{4ex}
\prod_{\alpha=1}^N\,
S_{R_\alpha}(\mathfrak{q}^{\rho})\,
S_{Y_\alpha^T}(\mathfrak{q}^{\rho})\,
\sum_{\mu,\nu,\eta,\zeta}
\prod_{\alpha=1}^N\,
(-Q_{m\alpha})^{|\mu_\alpha|}\,
(-Q_{F\alpha})^{|\nu_\alpha|}
\\
&\quad\times
\prod_{\alpha=1}^N\,
S_{ \nu_{\alpha-1}/\zeta_{\alpha-1}}(\mathfrak{q}^{R_\alpha^T+\rho})\,
S_{ \mu_\alpha/\zeta_{\alpha-1}}(\mathfrak{q}^{R_\alpha+\rho})\,
S_{ \mu_\alpha^T/\eta_\alpha}(\mathfrak{q}^{Y_\alpha^T+\rho})\,
S_{ \nu_\alpha^T/\eta_\alpha}(\mathfrak{q}^{Y_\alpha+\rho}) \, .
\end{split}$$ Note that the lines on the left and right edges are associated with a singlet or empty $\nu_0=\nu_N=\emptyset$ tableu. We can take the summation since all the $\kappa$-factors from the framing factors are canceled out in the partition function. This type of subdiagram is called “the vertex on a strip geometry" and is studied extensively in [@Iqbal:2004ne]. By using the formula (B.1) from [@Taki:2007dh] we can compute it explicitly: $$\begin{split}
H_{\,Y_1Y_2\cdots Y_N}^{\,R_1R_2\cdots R_N}
&=\frac{\prod_{\alpha=1}^N\,
S_{R_\alpha}(\mathfrak{q}^{\rho})\,
S_{Y_\alpha^T}(\mathfrak{q}^{\rho})}
{\prod_{1\leq \alpha<\beta\leq N}\left[R_\alpha, R_\beta^T\right]_{Q_{\alpha\beta}}
\left[Y_\alpha,Y_\beta^T \right]_{Q_{m\alpha}^{-1}Q_{\alpha\beta}Q_{m\beta}}
}
\\
&\qquad\times\rule{0pt}{4ex}
\prod_{1\leq \alpha<\beta\leq N}
\left[Y_\alpha,R_\beta^T \right]_{Q_{m\alpha}^{-1}Q_{\alpha\beta}}
\prod_{1\leq \alpha\leq\beta\leq N}
\left[R_\alpha, Y_\beta^T\right]_{Q_{\alpha\beta}Q_{m\beta}} \, ,
\end{split}$$ where $Q_{\alpha\beta} = \prod_{a=\alpha}^{\beta-1}Q_{m\,a}Q_{F\,a}$. Normalizing this sub-diagram by dividing with $H_{\emptyset\cdots}^{\emptyset\cdots}$ we obtain $$\begin{split}
\tilde{H}_{\,Y_1Y_2\cdots Y_N}^{\,R_1R_2\cdots R_N}
&=\frac{\prod_{\alpha=1}^N\,
S_{R_\alpha}(\mathfrak{q}^{\rho})\,
S_{Y_\alpha^T}(\mathfrak{q}^{\rho})}
{\prod_{1\leq \alpha<\beta\leq N}
N_{R_\beta R_\alpha}(Q_{\alpha\beta})\,
N_{Y_\beta Y_\alpha}(Q_{m\alpha}^{-1}Q_{\alpha\beta}Q_{m\beta})
}
\\
&\qquad\times\rule{0pt}{4ex}
\prod_{1\leq \alpha<\beta\leq N}
N_{R_\beta Y_\alpha}(Q_{m\alpha}^{-1}Q_{\alpha\beta})
\prod_{1\leq \alpha\leq\beta\leq N}
N_{Y_\beta R_\alpha}(Q_{\alpha\beta}Q_{m\beta}) \, ,
\end{split}
\label{Htilde}$$ where $$\begin{aligned}
N_{Y_1Y_2}(\mathfrak{q},Q)
\equiv \frac{\left[Y_1^T, Y_2 \right]_{Q}}
{\left[\emptyset, \emptyset \right]_{Q}}
=N_{Y_2^TY_1^T}(\mathfrak{q},Q) \, .\end{aligned}$$
The generic $SU(N)^{M-1}$ linear quiver theories with fundamental and bifundamental hypermultiplets are engineered using CY$_3$ with linear toric diagrams that are obtained by gluing local structures depicted in Figure \[fig:quiv2\]. The partition function for $SU(N)^{M-1}$ quivers is read off from Figure \[fig:large\] and is written in terms of the local structure of the geometry that is illustrated in Figure \[fig:genQuiver\],
![The local structure of the toric diagrams for the $SU(N)$ linear quivers.[]{data-label="fig:genQuiver"}](genericQuiverVer3.eps){width="100mm"}
$$\label{genQuivPartFuncmain}
\begin{split}
Z_{\,\textrm{inst}} &=
\sum
\cdots
\sum_{Y_1^{(i)},\cdots,Y_N^{(i)}}
\cdots
\prod_{\alpha}(-Q_{B\alpha}^{(i)})^{|Y_\alpha|}
\cr
&\quad \cdots \tilde{H}_{\,Y_1^{(i+1)} Y_2^{(i+1)} \cdots Y_N^{(i+1)}}^{\,Y_1^{(i)} Y_2^{(i)} \cdots Y_N^{(i)}}\,
(\,Q^{(i)}_{m1}, \cdots,Q^{(i)}_{mN},Q^{(i)}_{F1},\cdots,Q^{(i)}_{FN})
\cr
&\quad \times
\tilde{H}^{\,Y_1^{(i-1)} Y_2^{(i-1)} \cdots Y_N^{(i-1)}}_{\,Y_1^{(i)} Y_2^{(i)} \cdots Y_N^{(i)}}\,
(\,Q^{(i-1)}_{m1},\cdots,Q^{(i-1)}_{mN},Q^{(i-1)}_{F1},\cdots,Q^{(i-1)}_{FN})
\cdots.
\end{split}$$
This expression is written in terms of the string theory parameters. In order to make contact with Nekrasov’s partition function we introduce the following identification for the Kähler parameters $$\begin{aligned}
Q^{(i)}_{\alpha\beta}=e^{-\beta(a^{(i)}_\alpha-a^{(i)}_\beta)}
= \frac{\tilde{a}_{\alpha}^{(i)}}{\tilde{a}_{\beta}^{(i)}} \, , \qquad
Q^{(i)}_{m\alpha}=e^{-\beta(a^{(i)}_\alpha-{a}^{(i+1)}_\alpha-m^{(i,i+1)})}
= \frac{\tilde{a}^{(i)}_{\alpha}}{\tilde{a}^{(i+1)}_{\alpha}} \, ,
\label{Qab_Qm}\end{aligned}$$ where $\tilde{a}_{\alpha}^{(i)}$ are the M-theory parameters from Section \[sec:MtheoryDeriv\]. Here, we have defined $$\begin{aligned}
Q_{\alpha \, \alpha+1}^{(i)} \equiv Q_{m \, \alpha}^{(i)} Q_{F\alpha}^{(i)}
\label{QF_Qab}\end{aligned}$$ (see Figure \[fig:genQuiver\]), which leads to the identification $$\begin{aligned}
Q_{F\alpha}^{(i)}
{= \frac{\tilde{a}_{\alpha}^{(i+1)}}{\tilde{a}_{\alpha+1}^{(i)}}}
= \exp \left[ - \beta (a_{\alpha}^{(i+1)}-a_{\alpha+1}^{(i)} + m^{(i,i+1)}) \right] \, .
\label{QF_param}\end{aligned}$$ Note that the parameters above satisfy the following relations $$\begin{split}
Q^{(i)}_{\alpha\beta}
&= Q^{(i-1)}_{\alpha\beta} \frac{Q^{(i-1)}_{m\beta}}{Q^{(i-1)}_{m\alpha}} \, ,
\\
Q_{F\alpha}^{(i)}
&= Q_{F \alpha}^{(i-1)} \frac{Q_{m \, \alpha+1}^{(i-1)}}{Q_{m\alpha}^{(i)}} \, .
\label{rec_QF}
\end{split}$$
When comparing with the expression (\[GluingOfQuiver\]) of the Nekrasov partition function [@Nekrasov:2002qd; @Fucito:2004gi], we can show that the topological string partition function (\[genQuivPartFuncmain\]) is almost the same as the partition function for the quiver gauge theory.
![The quiver diagram for the $SU(N)$ quiver gauge theory associated with Figure \[fig:genQuiver\]. []{data-label="fig:genQuiverDiag"}](genericQuiverDiag.eps){width="120mm"}
The remaining problem is to find the identification between the base Kähler parameters $Q_{B}^{(i)}$ and the gauge coupling constants $q^{(i)}$. For the purpose, let us compute the corresponding part of the partition function (\[genQuivPartFuncmain\]). The Kähler parameters $Q_{B\alpha}$ of the two-cycles $B+m_\alpha F+n_\alpha F'$ are given by $$\begin{aligned}
Q_{B1}^{(i)}=Q_B^{(i)}
\qquad \text{and} \qquad
Q_{B\alpha}^{(i)}
= Q^{(i)}_{B\,\alpha-1} \frac{Q^{(i)}_{m \, \alpha-1}}{ Q^{(i-1)}_{m\alpha}} \, .
\label{rec_QB}\end{aligned}$$ The part of the partition function (\[genQuivPartFuncmain\]) that contains these parameters is $$\begin{split}
\prod_{\alpha}(-Q_{B\alpha}^{(i)})^{|Y_\alpha^{(i)}|}
&=(-Q_B^{(i)})^{\sum{|Y_\alpha^{(i)}|}}
\frac
{
\prod_{1\leq\alpha<\beta\leq N}
(Q^{(i)}_{m\alpha})^{|Y_\beta^{(i)}|}
}
{
\prod_{2\leq\alpha\leq\beta\leq N}
(Q^{(i-1)}_{m\alpha})^{|Y_\beta^{(i)}|}
}\\
&=(-Q_B ^{(i)}Q^{(i-1)}_{m1})^{\sum{|Y_\alpha^{(i)}|}}
\frac
{
\prod_{1\leq\alpha<\beta\leq N}
(Q^{(i)}_{m\alpha})^{|Y_\beta^{(i)}|}
}
{
\prod_{1\leq\alpha\leq\beta\leq N}
(Q^{(i-1)}_{m\alpha})^{|Y_\beta^{(i)}|}
}
\, .
\label{QB_q}
\end{split}$$ By comparing (\[QB\_q\]) with (\[GaugeCouplingKahlers\]), we find the following relation between the gauge coupling constants and the Kähler parameters of the base $\mathbb{P}^1$ $$\begin{split}
Q_B^{(i)}
= &
q^{(i)}
\frac{1}{Q^{(i-1)}_{m1}}
\prod_{\alpha=1}^N
\sqrt{\frac{Q^{(i-1)}_{m\alpha}}
{Q^{(i)}_{m\alpha}}}
=
q^{(i)}
\frac{\tilde{a}_1^{(i)}}{\tilde{a}_1^{(i-1)}}
\prod_{\alpha=1}^N
\frac{\sqrt{\tilde{a}_{\alpha}^{(i-1)} \tilde{a}_{\alpha}^{(i+1)}}}{\tilde{a}_{\alpha}^{(i)}}
\cr
= &
q^{(i)}
\exp \left[
- \beta
\left(
a_1^{(i)} - a_1^{(i-1)} - \frac{N-2}{2} m^{(i-1,i)} + \frac{N}{2} m^{(i,i+1)}
\right)
\right] \, .
\end{split}
\label{QB_param}$$ Inserting (\[Qab\_Qm\]), (\[QF\_param\]) and (\[QB\_param\]) into the topological partition function (\[genQuivPartFuncmain\]) gives precisely the Nekrasov partition function for the quiver theory in Figure \[fig:genQuiverDiag\].
From the relations (\[QF\_Qab\]), (\[rec\_QF\]), and (\[rec\_QB\]) we see that all the $Q_{\alpha \beta}^{(i)}$, $Q_{F\alpha}^{(i)}$ for $1 \le i \le M-1$ and $Q_{B\alpha}^{(i)}$ for $2 \le \alpha \le N$ are not independent. The toric diagram in Figure \[fig:genQuiverDiag\] shows that the remaining parameters $Q_{m\alpha}^{(i)}$, $Q_{B}^{(i)}$ and $Q_{F \alpha}^{(0)}$ are independent; this can also be deduced from the relations (\[Qab\_Qm\]), (\[QF\_param\]) and (\[QB\_q\]). Moreover, the number of parameters add up to $(M+1)(N+1)-3$, which is the same as the number of parameters of the $SU(N)^{M-1}$ gauge theory as discussed in Section \[subsec:review\_duality\]. Therefore, they are one-to-one correspondent with the gauge theory parameters.
![The toric diagram for the linear $SU(N)$ quiver gauge theory. $Q_{B}^{(i)}$ is related to the gauge coupling constant $q^{(i)}$ of the $i$-th gauge group $SU(N)_{(i)}$. The Coulomb moduli of the $i$-th gauge group are given by $Q_{\alpha\beta}^{(i)}$. Since $SU(N)_{(0)}$ and $SU(N)_{(M)}$ are in fact not gauge groups but global, flavor symmetries, the Kähler parameters $Q_{\alpha\beta}^{(0)}$ and $Q_{\alpha\beta}^{(M)}$ on the edges encode the masses of the (anti-) fundamental hypermultiplets living on the endpoints of the corresponding quiver diagram.[]{data-label="fig:large"}](largefigure.eps){width="140mm"}
The duality map of the reflection transformation is given by $$\begin{split}
(Q_{mi}^{(\alpha-1)})_d = Q_{m \alpha}^{(i-1)} \, , & \quad
(Q_{Fi}^{(\alpha-1)})_d = Q_{B \alpha}^{(i)} \, , \cr
\qquad(Q_{Bi}^{(\alpha)})_d = Q_{F \alpha}^{(i-1)} \, , & \quad
(Q_{\alpha \, \alpha+1}^{(i)})_d = Q_{m \, i+1}^{(\alpha-1)} Q_{B \, i+1}^{(\alpha)} \, .
\label{top_gen_map}
\end{split}$$ Again, by taking into account (\[QF\_Qab\]), (\[rec\_QF\]), and (\[rec\_QB\]), we see that in (\[top\_gen\_map\]) the second map (in the first line) for $2 \le \alpha \le N$, the third (in the second line) for $2 \le i \le M$ and the fourth can be derived from the remaining maps are redundant. Therefore, the independent ones are the first map, the second with $\alpha=1$ and the third with $i=1$.
Finally, we show that the duality map obtained here is equivalent to the one we found using the M5-brane analysis. To do so, it is enough to show that the independent duality relations in (\[top\_gen\_map\]) can also be derived from the relations (\[gen\_map\]). Just like the $SU(2)$ case, we combine this duality map with a [simultaneous]{} transformation $\tilde{a}_{\alpha}^{(i)} \to \tilde{a}_{\alpha}^{(i)} {}^{-1}$ and $(\tilde{a}_{\alpha}^{(i)})_d \to (\tilde{a}_{\alpha}^{(i)})_d^{-1}$, which is a symmetry of the Nekrasov partition function. Then, the first map, the second map (in the first line) for $\alpha=1$, and the third map (in the second line) for $i=1$ in (\[top\_gen\_map\]) respectively become $$\begin{split}
\left( \frac{\tilde{a}^{(\alpha-1)}_{i}}{\tilde{a}^{(\alpha)}_{i}} \right)_d
&= \frac{\tilde{a}^{(i-1)}_{\alpha}}{\tilde{a}^{(i)}_{\alpha}} \, ,
\\
\left( \frac{\tilde{a}^{(1)}_{i}}{\tilde{a}^{(0)}_{i+1}} \right)_d
&= q^{(i)}
\frac{\tilde{a}_1^{(i-1)}}{\tilde{a}_1^{(i)}}
\prod_{\alpha=1}^N
\frac{\sqrt{\tilde{a}_{\alpha}^{(i-1)} \tilde{a}_{\alpha}^{(i+1)}}}{\tilde{a}_{\alpha}^{(i)}} \, ,
\\
\frac{\tilde{a}^{(1)}_{i}}{\tilde{a}^{(0)}_{i+1}}
&= \left(
q^{(i)}
\frac{\tilde{a}_1^{(i-1)}}{\tilde{a}_1^{(i)}}
\prod_{\alpha=1}^N
\frac{\sqrt{\tilde{a}_{\alpha}^{(i-1)} \tilde{a}_{\alpha}^{(i+1)}}}{\tilde{a}_{\alpha}^{(i)}}
\right)_d
\label{top_final_map}
\end{split}$$ after inserting (\[Qab\_Qm\]), (\[QF\_param\]) and (\[QB\_param\]). The first line in (\[top\_final\_map\]) is precisely the relation (\[aa\_aa\]), from which the duality map (\[gen\_map\]) is derived. The second line can be derived from (\[gen\_map\]), while the third line in (\[top\_final\_map\]) is the same as the second line in (\[top\_final\_map\]) with the parameters of the original theory exchanged with the ones of the dual theory. Since the role of the original and the dual theory can be exchanged the third line of (\[top\_final\_map\]) is also correct. We have thus shown that the duality map obtained from the topological string analysis is identical to the one obtained from the M-theory analysis.
From 5D $\mathcal{N}=1$ gauge theory to 2D CFT {#sec:GaugeToCFT}
==============================================
In this section we discuss the implications of the 5D $SU(N)^{M-1} \leftrightarrow SU(M)^{N-1}$ duality in 2D CFTs and we propose that the DOZZ three-point function of $q$-deformed Toda theory is obtained from the topological string partition function of $U(1)$ linear quivers. We rewrite the $U(1)$ gauge theory partition function into the DOZZ three-point function of $q$-deformed Liouville theory that is given in [@Kozcaz:2010af]. What is more, we extend it to $q$-deformed Toda theory and then conjecture that $q$-deformed Heisenberg free CFT on a multi-punctured sphere is dual to $q$-deformed Toda CFT on a three-punctured sphere. We begin with a short review of the AGTW duality [@Alday:2009aq; @Wyllard:2009hg] between 4D $\mathcal{N}=2$ $SU(N)$ conformal gauge quivers and 2D $A_{N-1}$ conformal Toda field theories and then turn to its 5D generalization between $\mathcal{N}=1$ gauge theories and $q$-deformed Virasoro and $W_N$ algebra [@Awata:2010yy]. The 5D gauge theory duality studied in this article then implies relations between correlation functions (conformal blocks) of the $q$-deformed Virasoro algebra and those of the $q$-deformed $W_N$ algebra. Ultimately, the 4D version of this duality should lead to relations between Liouville and Toda theories.
In [@Gaiotto:2009we] Gaiotto was able to obtain a large class of $\mathcal{N}=2$ superconformal field theories in four dimensions by compactifying (a twisted version of) the six-dimentional $(2,0)$ SCFT on a Riemann surface with genus $g$ and $n$ punctures. The parameter space of the exactly marginal gauge couplings of the 4D gauge theory is identified with the complex structure moduli space $\mathcal{C}_{g,n} /\Gamma_{g,n}$ of the Riemann surface. The discrete group $\Gamma_{g,n}$ is the generalized S-duality transformations of the 4D theory.
Soon after, Alday, Gaiotto and Tachikawa conjectured [@Alday:2009aq] that the instanton partition function of a $\mathcal{N}=2$ $SU(2)$ quiver gauge theory in $\Omega$ background is equal to the conformal block of the conformal Liouville theory on a certain Riemann surface $\mathcal{C}_{g,n}$. This Riemann surface can be found in a systematic way from the quiver diagram of the 4D gauge theory[^15]. The two theories are equal under the following identificaton between their parameters $$\epsilon_1 = b \, , \qquad \epsilon_2 = \frac{1}{b} \, ,$$ with the central charge of the Virasoro algebra being $c=1+6\left(b+\frac{1}{b}\right)^2$. The coupling constants $q$ are identified with the cross-ratios $z$, the hypermultiplet masses $m$ (both flavor and bifundamental) correspond to the external momenta in the Liouville theory and the Coulomb moduli $a$ correspond to the internal momenta in the conformal block. Both external and internal momenta are denoted by $\alpha$ here. The AGTW conjecture has been proved for a special case in [@Fateev:2009aw; @Hadasz:2010xp; @Mironov:2009qn; @Mironov:2010pi], and attempts for proof in more generic settings have been made by using a new basis of the Verma module [@Alba:2010qc; @Fateev:2011hq; @Belavin:2011js].
The one-loop contribution in the partition function precisely reproduces the product of the so called DOZZ three-point function of the Liouville theory [@Dorn:1994xn; @Zamolodchikov:1995aa; @Teschner:2001rv; @Nakayama:2004vk] $$\begin{split}
\label{dozz}
&C_{\,{}_{\textrm{DOZZ}}}(\alpha_1,\alpha_2,\alpha_3)=
\left[
\pi\, \mu\, \gamma\,\left(b^2\right)\, b^{2-2b^2}
\right]^{\frac{b+1/b-\sum_{i}\alpha_i}{b}}
\\
&\quad \quad \rule{0pt}{5ex}
\times\frac{\Upsilon_0\,\Upsilon_b (2\alpha_1)\Upsilon_b (2\alpha_2)\Upsilon_b (2\alpha_3)}
{\Upsilon_b (\alpha_1+\alpha_2+\alpha_3-b-1/b)
\Upsilon_b (\alpha_1+\alpha_2-\alpha_3)
\Upsilon_b (\alpha_2+\alpha_3-\alpha_1)
\Upsilon_b (\alpha_3+\alpha_1-\alpha_2)} \,,
\end{split}$$ where the special function $\Upsilon_b (x)$ is defined by $$\begin{aligned}
&\Upsilon_b (x)=\frac{1}{\Gamma_b(x)\Gamma_b(\epsilon_1+\epsilon_2-x)},
\\
&\Gamma_b(x)
=
\exp \frac{d}{ds}
\frac{1}{\Gamma(s)}
\int_0^\infty
\frac{dt}{t}\,
\frac{t^se^{-tx}}{(1-x^{-\epsilon_1t})(1-e^{-\epsilon_2t})}
\Big|_{s=0}
\propto
\sum_{m,n=0}^\infty
(x+\epsilon_1m+\epsilon_2n)^{-1}.\end{aligned}$$ Finally, the partition function of the 4D SCFT on $S^4$, $$\int d a \, a^2 \, |\,Z_{\,\textrm{Nek}}(a)\,|^2$$ with $Z_{\textrm{Nek}} =Z_{\textrm{tree}} Z_{\textrm{1-loop}}Z_{\textrm{inst}}$ being the full partition function, is equal to the correlation function of primary fields $V_\alpha =e^{2 \alpha \phi}$ in the Liouville theory with conformal dimension $\Delta =\alpha\left(b+\frac{1}{b} - \alpha \right)$. Take the $SU(2)$ gauge theory with four flavors as an example, this theory corresponds to the Liouville CFT on the Riemann sphere with four punctures $\mathcal{C}_{0,4}$. Quantitatively, the AGTW conjecture states $$\int d \mu(\alpha)\,C_{\,\textrm{DOZZ}}C_{\,\textrm{DOZZ}} \, | \,q^{\Delta-\Delta_1-\Delta_2} \mathcal{B}_{0,4}(\alpha)\,|^2
\propto
\int d a \, a^2 \, |\,Z_{\,\textrm{Nek}}(a)\,|^2 \, ,$$ where the two DOZZ factors come from the decomposition of the four punctured sphere into two pants. The conformal block $\mathcal{B}$ is then equal to the instanton part of the Nekrasov partition function, and the “square root” of the DOZZ part gives the perturbative correction of the partition function.
The 5D extension of the conjecture suggests that the instanton part of the 5D Nekrasov partition function is equal to the conformal block of a $q$-deformed CFT. Schematically this conjectured duality is the following equality $$\mathcal{B}^{\,q-\textrm{Liouville}}(\alpha)=
Z^{\,\textrm{5D}}_{\,\textrm{Nek}}(a) \, .$$ In [@Awata:2010yy] the authors studied the case of $SU(2)$ pure SYM, which is the simplest setup of the AGT duality, and they found that the partition function coincides with the irregular conformal block of the $q$-deformed Virasoro algebra. Although the 5D extension of the instanton counting is established, the theoretical framework of $q$-deformed CFT’s is not well developed. Therefore, we cannot establish the duality for the full sector yet. The $q$-deformation of conformal field theory should first be developed to reveal the scope of the AGTW duality. However, by assuming the 5D AGTW conjecture, we will now illustrate how the gauge theory duality studied in Section \[sec:MtheoryDeriv\] and \[sec:TopStringDeriv\] can be used to make predictions about $q$-deformed CFT’s. Although we have mostly reviewed the $SU(2)$ quiver case, the ideas can be generalized to $SU(N)$ quivers.
![The $SU(3) \leftrightarrow SU(2)\times SU(2)$ duality implies that the four-point $W_{3}$ Toda correlator on a sphere (left) should be equal to the five-point Liouville correlator on a sphere (right). The black points denote $U(1)$ punctures and the encircled ones $SU(3)$ punctures in the $W_{3}$ Toda theory respectively, whereas the grey points correspond to $SU(2)$ punctures in the Liouville theory.[]{data-label="SU(3)example"}](Todacorrelator.eps "fig:") ![The $SU(3) \leftrightarrow SU(2)\times SU(2)$ duality implies that the four-point $W_{3}$ Toda correlator on a sphere (left) should be equal to the five-point Liouville correlator on a sphere (right). The black points denote $U(1)$ punctures and the encircled ones $SU(3)$ punctures in the $W_{3}$ Toda theory respectively, whereas the grey points correspond to $SU(2)$ punctures in the Liouville theory.[]{data-label="SU(3)example"}](liouvilecorrelator.eps "fig:")
5D quiver $U(1)$ gauge theories and $q$-deformation of DOZZ
-----------------------------------------------------------
In this subsection we give an example involving $U(1)$ gauge theory, whose instanton partition function is given by $$Z^{\,\textrm{5D inst}}_{\,U(1)}=
\sum_{Y}\,
{q}^{|Y|}
\frac{\prod_{(i,j)\in Y}\sinh \frac{\beta}{2}(m_1+\hbar(i-j))\sinh \frac{\beta}{2}(-m_2+\hbar(i-j))}
{\prod_{(i,j)\in Y}\sinh \frac{\beta}{2}(\hbar(Y_i+Y^T_j-i-j+1))\sinh \frac{\beta}{2}(-\hbar(Y_i+Y^T_j-i-j+1))}
\, ,$$ with one fundamental and one anti-fundamental hypermultiplet. Moreover, we introduce the perturbative part of the partition function: $$Z^{\,\textrm{5D pert}}_{\,U(1)}={[\emptyset,\emptyset]_{e^{-\beta m_1}}[\emptyset,\emptyset]_{e^{-\beta m_2}}} \, ,$$ where the bracket is defined in (\[bracket\]). The full Nekrasov partition function is the product of the two: $Z^{\,\textrm{5D}}_{\,U(1)}=Z^{\,\textrm{5D pert}}_{\,U(1)}Z^{\,\textrm{5D inst}}_{\,U(1)}$. By using the techniques from the topological vertex formalism, we can perform the summation inside the full partition function to obtain $$\label{eq:U1PartFunc}
Z^{\,\textrm{5D}}_{\,U(1)}=
\frac{[\emptyset,\emptyset]_{Q_1}[\emptyset,\emptyset]_{Q_F}[\emptyset,\emptyset]_{Q_2}[\emptyset,\emptyset]_{Q_1Q_FQ_2}}
{[\emptyset,\emptyset]_{Q_1{Q_F}}[\emptyset,\emptyset]_{{Q_F}Q_2}} \, .$$ The right hand side of this equation has appeared already in [@Iqbal:2004ne; @Kozcaz:2010af]. The parameters are defined as $$Q_i=e^{-\beta m_i}\,\,(i=1,2)
\quad \text{and} \quad
-Q_F{\sqrt{Q_1Q_2}}=q \, .$$
What is interesting here is that the expression (\[eq:U1PartFunc\]) corresponds to the $q$-deformed DOZZ function [@Kozcaz:2010af] $$\mid[\emptyset,\emptyset]_{Q_1}[\emptyset,\emptyset]_{Q_F}[\emptyset,\emptyset]_{Q_2}[\emptyset,\emptyset]_{Q_1Q_FQ_2}\mid^2
\propto
C_{\,\textrm{DOZZ}}^{\,\mathfrak{q}} \, ,$$ where the $q$-deformation parameter is $\mathfrak{q}=e^{-\beta \hbar}$ and the identification of parameters takes the form $$Q_1=e^{-\beta(-\alpha_1-\alpha_2+\alpha_3)} \, , \quad
Q_F=e^{-\beta(-\alpha_1+\alpha_2-\alpha_3)} \, , \quad
Q_2=e^{-\beta(\alpha_1-\alpha_2-\alpha_3)} \, .$$ By using the rotational duality described in Appendix \[app:90Rotation\], the $q$-deformed DOZZ function is expected to be given by the following replacement of the $\Upsilon$-function in the definition (\[dozz\]): $$\begin{aligned}
\Upsilon_b (x)=\frac{1}{\Gamma_b(x)\Gamma_b(\epsilon-x)}
\quad \longrightarrow \quad
\Upsilon_b^{\,\mathfrak{q}} (x)
=\frac{1}{\Gamma_b^{\,\mathfrak{q}}(x)\Gamma_b^{\,\mathfrak{q}}(\epsilon-x)} \, ,\end{aligned}$$ where $\Gamma_b^{\,\mathfrak{q}}(x)\propto \prod_{i,j} \left( \sinh \frac{\beta}{2}(x+\epsilon_1 i+\epsilon_2j) \right)^{-1}$. The idea is illustrated using the toric diagrams in Figure \[fig:abelquiv\], where the $U(1)^{N-1}$ quiver gauge theory is on the left[^16]. The rotated diagram on the right depicts the so-called 4D Gaiotto theory for the sphere with two full punctures and one simple puncture.
![The toric diagram for $U(1)^2$ linear quiver (left) and the free hypermultiplets (right).[]{data-label="fig:abelquiv"}](AbelianQuiver.eps){width="120mm"}
The AGT dual of this $U(1)$ gauge theory partition function is the DOZZ three-point function of the rank-$N$ Toda field theory. In response to Gaiotto’s construction of the 4D gauge theory, the DOZZ function is the three-point function for two full primary fields and one semi-degenerate field. Above we studied $U(1)$ gauge theory with two flavors, which is dual to Liouville theory on the sphere with 3 punctures. Since we consider the 5D uplift of the gauge theory, 2D CFT is replaced by the $q$-analogue of it. It is straightforward to extend this argument to generic $\Omega$ background, in which case the 2D CFT with generic central charge appears.
Using the idea above we can conjecture the $q$-analogue of the Toda DOZZ function. We consider the $U(1)^{N-1}$ linear quiver gauge theory. The toric diagram for this theory is shown on the left in Figure \[fig:abelquiv\]. With the formalism of the *refined* topological vertex, we can compute the closed form of the full Nekrasov partition function $$\label{5Dabelianquiver}
Z^{\,\textrm{5D}}_{\,U(1)^{N-1}}=
\prod_{i,j=1}^\infty
\frac{\prod_{1\leq a\leq b\leq N}\left(1-Q_{ab}Q_{m\,b}\mathfrak{t}^{i-\frac{1}{2}}\mathfrak{q}^{j-\frac{1}{2}} \right)
\prod_{1\leq a\le b\leq N}\left(1-Q_{m\,a}^{-1}Q_{ab}\mathfrak{t}^{i-\frac{1}{2}}\mathfrak{q}^{j-\frac{1}{2}} \right)
}
{
\prod_{1\leq a\le b\leq N}\left(1-Q_{ab}\mathfrak{t}^{i-\frac{3}{2}}\mathfrak{q}^{j+\frac{1}{2}} \right)
\left(1-Q_{m\,a}^{-1}Q_{ab}Q_{ab}\mathfrak{t}^{i+\frac{1}{2}}\mathfrak{q}^{j-\frac{3}{2}} \right)
} \, ,$$ where $Q_{ab}=\prod_{i=a}^{b-1}Q_{m\,i}{q}^{(i)}$. In order to relate this expression to the combinatorial form of the instanton part of the partition function, we have to assume the slicing invariance of the refined topological vertex (see [@Iqbal:2008ra] for details). Our claim is that the square of (\[5Dabelianquiver\]) gives the major portion of the DOZZ three-point function of the “$q$-deformed $sl(N)$ Toda field theory” on sphere with two full primary fields and one semi-degenerate field. This result would be a powerful guide to formulate a yet-unknown $q$-deformation of the Toda field theory.
We can also recast our proposal as a duality between the $(M+2)$-point function of $W_{N}$-algebra and the $(N+2)$-point function of $W_{M}$-algebra. See Figure \[SU(3)example\] for an example. The $q$-deformed conformal blocks for the Heisenberg algebra are defined in the form of the Dotsenko-Fateev integral representation [@Mironov:2011dk; @Awata:2010yy], and we can see that these conformal blocks give the 5D Nekrasov partition functions for $U(1)$ quiver gauge theories[^17]. The simplest situation we have studied in this subsection is thus the equivalence between the $(N+2)$-point function of “$W_1$” (Heisenberg) algebra and the three-point function of $W_{N}$ algebra. This conjecture for $W_{N}$ algebras is the direct consequence of combining the duality from Section \[sec:TopStringDeriv\] and the AGTW conjecture. It gives a CFT analogue of this duality, which can be valuable in the studies of 2D CFT.
Summary and discussion {#sec:Discussion}
======================
In this paper, we have studied the duality between two 5D ${\cal N}=1$ linear quiver gauge theories compactified on $S^1$ with gauge groups $SU(N)^{M-1}$ and $SU(M)^{N-1}$ respectively. We have found the explicit map between the gauge theory parameters of these two theories, under which they describe the same low energy effective theory on the Coulomb branch. We have derived the map both by considering the M5-brane configuration and by calculating the topological string partition function. There are several interesting extensions and applications of this duality.
The implications of this duality in 2D CFT through the 5D extension of the AGTW conjecture have been discussed above. We conjuctured the three-point function of $q$-deformed Toda theory from the topological string partition function of the U(1) linear quiver. Moreover, the duality between $(M+2)$-point function of $q$-deformed $W_{N}$-algebra and $(N+2)$-point function of $q$-deformed $W_{M}$-algebra is proposed. An interesting future direction is to study in detail the duality we have proposed here between Liouville and Toda correlation functions.
Although it is natural and interesting to consider the 4D limit of this duality, it seems to be subtle. In an upcoming paper [@index] we follow a simple path to the 4D version of this duality, where the 4D superconformal index [@Kinney:2005ej; @Romelsberger:2005eg] is used to study the duality between the 4D conformal $\mathcal{N}=2$ $SU(N)^{M-1}$ and $SU(M)^{N-1}$ line quivers. The superconformal index counts the multiplets that obey shortening conditions, up to equivalence relations that set to zero all the short multiplets that can recombine into long multiplets. Basically, it knows the complete list of protected operators in a superconformal theory. Together with one-loop computations, the analysis of the chiral ring and representation theory arguments it was used to study the spectrum of $\mathcal{N}=2$ superconformal QCD at large $N$ in [@Gadde:2009dj; @Gadde:2010zi]. What is more, there is a relation between the 4D superconformal index and topological quantum field theories in 2D [@Gadde:2009kb; @Gadde:2010te; @Gadde:2011ik], which provides a simpler version of the AGTW relation between 4D partion functions and 2D CFT correlators. The index is the partition function on $S^3 \times S^1$ [@Festuccia:2011ws], it is coupling-independent and easier to calculate than Pestun’s partition function on $S^4$. It is related to a 2D TQFT correlation function [@Gadde:2009kb] as opposed to the full-fledged CFT correlation function that is required in AGTW. The superconformal index has been successfully used to test ${\mathcal N}=1$ Seiberg duality [@Romelsberger:2005eg; @Romelsberger:2007ec] and ${\mathcal N}=1$ toric duality [@Gadde:2010en] (as well as AdS/CFT [@Kinney:2005ej]).
Low energy physics of supersymmetric gauge theories can also be captured by matrix models. Different types of matrix models have been studied in this context. First, the (“old”) Dijkgraaf-Vafa matrix model [@Dijkgraaf:2002fc; @Dijkgraaf:2002vw; @Dijkgraaf:2002dh] gives the low energy effective superpotential of 4D ${\cal N}=1$ gauge theory that is obtained by deforming ${\cal N}=2$ with the addition of superpotential terms of polynomial type. The action of this matrix model is given by its tree-level superpotential. Another matrix model was later proposed by the same authors in [@Dijkgraaf:2009pc]. The “new” Dijkgraaf-Vafa matrix model gives Nekrasov’s partition function of 4D ${\cal N}=2$ gauge theory, and though the AGTW conjecture, the conformal block of the Liouville/Toda CFT [@Itoyama:2009sc; @Awata:2010yy; @Eguchi:2009gf; @Mironov:2011dk]. Since the prepotential of the ${\cal N}=2$ gauge theory can be reproduced from the low energy effective superpotential [@Cachazo:2002pr], these two matrix models should be closely related even if they are computing different quantities. Indeed, both of them are introduced in the context of topological string theory in such a way that the spectral curves of these matrix models reproduce the Seiberg-Witten curve.
However, at first sight they look quite different in the following way. On the one hand, in the “old” Dijkgraaf-Vafa matrix model the matrix corresponds to the zero-modes of the adjoint scalar fields. Therefore, $SU(N)$ theory is studied using a single matrix while $SU(2)^{M-1}$ a quiver matrix (multi-matrix) model. On the other hand, in the “new” Dijkgraaf-Vafa matrix model $SU(2)^{M-1}$ theory corresponds to the single matrix model with multi-Penner type action, while $SU(N)$ theory corresponds to the quiver matrix model with $N-1$ adjoint matrices [@Schiappa:2009cc]. As was already pointed out in [@Dijkgraaf:2009pc], the role of the base and the fiber of the Calabi-Yau geometry is inverted in the second matrix model compared to the first one. Since the structure of the base and the fiber are related to the numbers $N$ and $M$ of the $SU(N)^{M-1}$ gauge theory, it implies that these matrix models are related by the duality studied in this paper. We expect that it will play an important role to understand the relation between these matrix models.
Several other kinds of extensions of the duality we study here are also possible. In this article we focus on the duality between theories which are 5D uplifts of 4D superconformal field theories. It should be possible to extend the duality to the theories which are uplifts of asymptotically free theories. In such cases it is known that we can introduce the Chern-Simons term [@Seiberg:1996bd] in the action. The configuration of the M5-brane curve depends on the Chern-Simons level [@Brandhuber:1997ua] and thus the duality will also act on it. Considering such an effect would be interesting.
The extension to the elliptic quiver gauge theories, including $\mathcal{N}=2^*$ theory, is another future direction. Such quiver gauge theories are obtained by further compactifying the $x^6$ direction in addition to the $x^5$ direction in Table \[config\]. Following [@Vafa:1997mh; @Tachikawa:2011ch], the S-duality corresponding to the electric-magnetic duality appears by compactifying the $x^6$ direction in the special case where no NS5-branes are placed. The duality studied in this article can be also interpreted as S-duality, but it acts on the gauge theories in a totally different manner than the conventional electric-magnetic duality. The elliptic quiver gauge theories will offer an interesting playground to understand these two different types of S-dualities in a unified manner.
In the present article, we have studied the duality in the self-dual $\Omega$ background. The extension to the generic $\Omega$ background would be an important direction related to the existence of a preferred direction in the refined topological vertex. The conjectured slicing invariance would then be crucial for extending our result to the refined case. The duality maps we have derived are maintained even after switching on the self-dual $\Omega$ background. However, it is non-trivial whether the generic $\Omega$ background modifies the maps.
Considering this duality for generic $\Omega$ background in the context of the integrable system would be also interesting, where the “quantum Seiberg-Witten curve" appears as the Hamiltonian of the Schrödinger equation. If we manage to find the explicit expression of the 5D Hamiltonian [@Aganagic:2011mi], then it would be straightforward to obtain the duality map by using the same method we employed in this paper. The Nekrasov-Schatashivilli [@Nekrasov:2009ui; @Nekrasov:2009rc; @Nekrasov:2009zz] limit is especially interesting because the time-dependent terms[^18] in the Schrödinger equation are expected to vanish there. We then get a simple eigenvalue problem as an alternative way to solve quantum gauge theory.
Acknowledgements {#acknowledgements .unnumbered}
================
It is a pleasure to thank Giulio Bonelli, Nadav Drukker, Tohru Eguchi, Amihay Hanany, Kazunobu Maruyoshi, Sara Pasquetti, Filippo Passerini, Leonardo Rastelli, Kazuhiro Sakai, Yuji Tachikawa, Alessandro Tanzini, Niclas Wyllard and Konstantinos Zoubos for useful discussions and correspondence. L.B., E.P. and F.Y. would like to thank IHES for providing a stimulating atmosphere during the course of this work. E.P. wishes to thank IHP for its warm hospitality as this work was in progress. The work of E.P. is supported by the Humboldt Foundation. The research of M.T. is supported in part by JSPS Grant-in-Aid for Creative Scientific Research No. 19GS0219. F.Y. is partially supported by the INFN project TV12.
Physical parameters {#phys-par}
===================
The duality map (\[gen\_map\]), as discussed in Section \[sec:MtheoryDeriv\], seems to depend on the choice of coordinates when written in terms of the [*position parameters*]{}[^19]. However, when rewritten in terms of the [*physical parameters*]{} it is manifestly independent of the choice of coordinates. In this appendix we define the [physical gauge theory parameters]{} in terms of the position parameters. We also introduce the “traceless” and “trace” parameters that are more natural in the context of the AGTW conjecture, where these two kinds of mass parameters correspond to the $SU(N)$ and the $U(1)$ punctures respectively.
The [*position parameters*]{} $\tilde{a}^{(i)}_\alpha$, introduced in (\[tildema\]) and depicted in Figure \[branesetup\], denote the positions of the D4-branes in the $w$ coordinate. On the other hand, the [*physical parameters*]{} are defined as distances between the endpoints of open strings and relative distances do not depend on the choice of coordinates. As we will see, the definitions of the physical masses and the physical Coulomb moduli are such that it is difficult to define them in a unified way as we do for the position parameters (\[mass-a\]) or the “traceless” and “trace” parameters (\[trace\_and\_traceless\]).
First, we turn to the [*physical flavor mass*]{} that correspond to the distance (along $v$) between a flavor D4-brane position and the center of mass position of the adjacent color branes. D4-branes attached from the right to an NS5-brane correspond to fundamental masses, whereas D4-branes attached from the left to an NS5-brane correspond to anti-fundamental masses. The first $N$ flavor masses ($m_1,\cdots,m_N$) on the left of the quiver are, thus, anti-fundamental under the first gauge group. Moreover, they are fundamental under the “$0$-th gauge group”, which is in fact a global symmetry. According to these conventions, we define the [*anti-fundamental flavor mass*]{} $$\begin{aligned}
m_{\alpha}^{\text{af}} = \frac{1}{N}\sum_{\beta=1}^N a^{(1)}_{\beta} - m_{\alpha} \, ,\end{aligned}$$ where $m_{\alpha}$ is the position of the semi-infinite flavor D4-brane on the left of the quiver, see Figure \[fig:PhyMass\].
The anti-fundamental flavor mass can then be exponentiated, following (\[tildema\]), as $$\begin{aligned}
\tilde{m}_{\alpha}^{\text{af}}
= \left( \tilde{a}^{(0)}_{\alpha} \right)^{-1}
\prod_{\beta=1}^N \left( \tilde{a}^{(1)}_{\beta} \right)
^{\frac{1}{N}} \, .\end{aligned}$$ We can, moreover, think of the flavor masses on the left of the quiver as fundamental masses if we define $$\begin{aligned}
\tilde{m}_{\alpha}^{\text{f}}
= \tilde{a}^{(0)}_{\alpha} \prod_{\beta=1}^N \left( \tilde{a}^{(1)}_{\beta} \right)
^{-\frac{1}{N}}
= \frac{1}{\tilde{m}_{\alpha}^{\text{af}} } \, .\end{aligned}$$ In this paper we use this convention when there is only one gauge group factor. However, we find the anti-fundamental definition more natural for a generic quiver. In addition, the last $N$ masses ($m_{N+1}, \cdots m_{2N}$) on the right of the quiver are fundamental under the $(M-1)$-th gauge group and anti-fundamental under the “$M$-th gauge group”, with the latter being a global symmetry. Following the “right minus left” convention, we have $$\begin{aligned}
m_{N+\alpha}^{\text{f}} = m_{N+\alpha} - \frac{1}{N}\sum_{\beta=1}^N a^{(M-1)}_{\beta} \, ,\end{aligned}$$ which becomes $$\begin{aligned}
\tilde{m}_{N+\alpha}^{\text{f}}
= \tilde{a}^{(M)}_{\alpha} \prod_{\beta=1}^N \left( \tilde{a}^{(M-1)}_{\beta} \right)^{-\frac{1}{N}}\end{aligned}$$ after exponentiation (\[tildema\]).
Next, we turn to the definition of the [*physical Coulomb moduli parameter*]{}. This should be thought of as the distance between a color D4-brane position and the center of mass position of the color branes within a single gauge group factor, see Figure \[fig:CoulombModuli\].
(1,1)(0,0) (0.5,0)[(0,1)[1]{}]{} (0.45,-0.1)[NS5]{} (0.9,0)[(0,1)[1]{}]{} (0.85,-0.1)[NS5]{} (0.5,0.15)[(1,0)[0.4]{}]{} (0.95,0.15)[$a^{(i)}_{1}$]{} (0.5,0.3)[(1,0)[0.4]{}]{} (0.95,0.3)[$a^{(i)}_{2}$]{} (0.5,0.65)[(1,0)[0.4]{}]{} (0.95,0.65)[$a^{(i)}_{\alpha}$]{} (0.5,0.85)[(1,0)[0.4]{}]{} (0.95,0.85)[$a^{(i)}_{N}$]{} (0.5,0.5)(0.04,0)[10]{}[(1,0)[0.025]{}]{} (0.95,0.5)[$a^{(i)}_{\textrm{cm}}$]{} (0.7,0.40) (0.7,0.575) (0.7,0.75)
In other words, this is the [*“traceless part”*]{} of the Coulomb moduli $$\begin{aligned}
{a}^{(i)}_\alpha - \frac{1}{N}\sum_{\beta=1}^N {a}^{(i)}_\beta \, ,\end{aligned}$$ which in terms of the 5D parameters (\[mass-a\]) is defined as $$\begin{aligned}
\hat{a}^{(i)}_{\alpha}
= \frac{\tilde{a}_{\alpha}^{(i)}}{\prod_{\beta=1}^N \left( \tilde{a}_{\beta}^{(i)} \right)^{\frac{1}{N}}} \, .\end{aligned}$$
One last definition is in order, that of the [*bi-fundamental masses*]{} ${m}^{(i-1,i)}_{\text{bif}}$. Recall that a conformal quiver gauge theory has not only the overall $U(1)$ factored out[^20], but also all the relative $U(1)$s so that each factor in the quiver is $SU(N)$ and not $U(N)$. However, we want to study not just the conformal quiver, but the more general asymptotically conformal quiver with non-zero bi-fundamental masses. The bi-fundamental masses ${m}^{(i-1,i)}$ (for $2 \le i \le M-1$) are related to the relative $U(1)$s between the $(i)$-th and $(i-1)$-th $SU(N)$ gauge factors. They are equal to the distance (along $v$) between the center of mass positions of the color D4-branes that correspond to these two adjacent gauge group factors. As above, we use the notation that the bi-fundamental fields are fundamental under the gauge group on the left ($(i-1)$ -th gauge group) and anti-fundamental under the gauge group on the right ($i$-th gauge group) $$\label{biff-def}
{m}^{(i-1,i)}_{\text{bif}} = \frac{1}{N}\sum_\alpha a^{(i)}_\alpha - \frac{1}{N}\sum_\alpha a^{(i-1)}_\alpha \, .$$ This definition (\[biff-def\]) can be extended to include $1 \le i \le M$ as long as we keep in mind that ${m}_{\text{bif}}^{(0,1)}$ and ${m}_{\text{bif}}^{(M-1,M)}$ are not bifundamental masses, although the subscript suggests otherwise. In terms of the 5D parameters the definition takes the form $$\begin{aligned}
\tilde{m}_{\text{bif}}^{(i-1,i)}
= \frac{\prod_{\beta=1}^N ( \tilde{a}^{(i)}_{\beta} )^{\frac{1}{N}}}
{\prod_{\alpha=1}^N ( \tilde{a}^{(i-1)}_{\alpha} )^{\frac{1}{N}}} \, .\end{aligned}$$
In the context of the AGTW conjecture mass parameters that correspond to the $U(1)$ and the $SU(N)$ punctures are introduced. To make contact with Gaiotto’s quiver diagrams we define $$\label{trace_and_traceless}
\begin{split}
\hat{a}^{(0)}_{\alpha}
= \frac{\prod_{\beta=1}^N (\tilde{m}_{\beta}^{\text{af}})^{\frac{1}{N}} }
{\tilde{m}_{\alpha}^{\text{af}} } \, ,&
\qquad
\tilde{m}_{\text{bif}}^{(0,1)}
= \prod_{\alpha=1}^N \left(
\tilde{m}_{\alpha}^{\text{af}}
\right)^{\frac{1}{N}} \, ,
\\
\hat{a}^{(M)}_{\alpha}
= \frac{\tilde{m}_{\alpha}^{\text{f}} }
{\prod_{\beta=1}^N (\tilde{m}_{\beta}^{\text{f}})^{\frac{1}{N}} } \, ,&
\qquad
\tilde{m}_{\text{bif}}^{(M-1,M)}
= \prod_{\alpha=1}^N \left(
\tilde{m}_{\alpha}^{\text{f}}
\right)^{\frac{1}{N}} \, ,
\end{split}$$ where $\hat{a}^{(0)}_{\alpha}$ and $\hat{a}^{(M)}_{\alpha}$ are the [*traceless parts*]{} of the flavor masses while $\tilde{m}_{\text{bif}}^{(0,1)}$ and $\tilde{m}_{\text{bif}}^{(M-1,M)}$ the [*trace parts*]{}. In the AGTW language, the traceless masses $\hat{a}^{(0)}_{\alpha}$ and $\hat{a}^{(M)}_{\alpha}$ correspond to $SU(N)$ punctures, whereas the trace part $\tilde{m}_{\text{bif}}^{(0,1)}$ and $\tilde{m}_{\text{bif}}^{(M-1,M)}$ together with all the bifindamental masses $\tilde{m}_{\text{bif}}^{(i-1,i)}$ ($2 \le i \le M-1$) correspond to $U(1)$ punctures.
4D limit of the SW curve for $SU(2)$ gauge theory with four flavors {#app:4DLimit}
===================================================================
We consider the 4D limit of the Seiberg-Witten curve (\[curveC\]) of the $\mathcal{N}=2$ $SU(2)$ gauge theory with four flavors. First, by multiplying $\tilde{m}_1^{-1/2} \tilde{m_2}^{-1/2} w^{-1}$ to the curve and imposing $a_1=-a_2=a$ as in (\[U(1)\]), it can be rewritten as $$\begin{split}
0 \,\, = \,\,
&4 \sinh \left( \frac{\beta}{2} (v-m_1) \right) \sinh \left( \frac{\beta}{2} (v-m_2) \right)
t^2
\\
& + \left(
- 2C \cosh \left( \beta v \right)
- 2Cq \cosh \left( \frac{\beta}{2}
\left( 2v - \sum_{i=1}^4 m_i \right) \right)
+ \frac{b}{\tilde{m}_1^{\frac{1}{2}} \tilde{m}_2^{\frac{1}{2}}}
\right) t
\\
& + 4 C^2 q \sinh \left( \frac{\beta}{2} (v-m_3) \right)
\sinh \left( \frac{\beta}{2} (v-m_4) \right) \, .
\label{failedC}
\end{split}$$ Further, by expanding the coefficients in powers of the circumference of the 5D $\beta$ $$\begin{split}
C &= C_{(0)} + C_{(1)} \beta + C_{(2)} \beta^2 + \cdots \, ,
\\
\frac{b}{\tilde{m}_1^{\frac{1}{2}}\tilde{m}_2^{\frac{1}{2}}}
&= b_{(0)} + b_{(1)} \beta + b_{(2)} \beta^2 + \cdots \, ,
\end{split}$$ the leading and the next-to-leading order of (\[failedC\]) lead to the relations $$\begin{split}
- 2 C_{(0)} (1+q) + b_{(0)} &= 0 \, , \\
- 2 C_{(1)} (1+q) + b_{(1)} &= 0 \, ,
\end{split}$$ respectively. In other words, the expansion coefficients are related to each other. The next-to-next-to-leading order gives the following non-trivial result: $$\begin{split}
0
= &(v-m_1)(v-m_2) t^2
\\
& + \left(
- C_{(0)} (1+q) v^2
+ C_{(0)}q \sum_{i=1}^4 m_i v
- 2 C_{(2)} (1+q)
- \frac{C_{(0)} q}{4} \left( \sum_{i=1}^{4} m_i \right)^2
+ b_{(2)}
\right) t
\\
& + C_{(0)}{}^2 q (v-m_3)(v-m_4) \, .
\end{split}$$ By defining the parameter $U$ as $$\begin{aligned}
C_{(0)}U \equiv - 2 C_{(2)} (1+q)
- \frac{C_{(0)}q}{4} \left( \sum_{i=1}^{4} m_i \right)^2
+ b_{(2)}\end{aligned}$$ and rescaling the coordinate $t$ as $t \to C_{(0)} t$, we obtain $$\begin{aligned}
0
= (v-m_1)(v-m_2) t^2
+ \left( - (1+q) v^2 + q \sum_{i=1}^{4} m_i \, v + U \right) t
+ q (v- m_3)(v- m_4) \, ,\end{aligned}$$ which is precisely the SW curve for the 4D superconformal $SU(2)$ gauge theory found in [@Eguchi:2009gf]. In conclusion, we find that the 5D SW curve (\[curveC\]) correctly reproduces the known 4D SW curve in the 4D limit $\beta \to 0$.
Details of Nekrasov partition function and topological strings {#app:NekrasovTop}
==============================================================
In this appendix we provide the details that are needed for the computation of the Nekrasov partition function using the topological vertex formalism. In particular, we give the Nekrasov partition function in a way that is convenient for the comparison with the topological string partition function in Section \[sec:TopStringDeriv\].
Young diagrams and combinatorial relations
------------------------------------------
Before writing down the Nekrasov partition function itself, we will provide some useful formulas. We start by proving a few combinatorial relations for Young diagrams. Let $Y$ be a Young diagram, which can be viewed as a decreasing sequence of non-negative integers $Y_1\geq Y_2\geq \cdots Y_{d(Y)}>Y_{d(Y)+1}=Y_{d(Y)+2}=\cdots=0$. Taking the summation over the boxes $(i,j)\in Y$ we find $$\label{eq:SumYoungDiagram}
\begin{split}
\sum_{(i,j)\in Y}i
&=
\sum_{j=1}^{Y_1}
\sum_{i=1}^{Y^T_j}i
=
\frac{\| Y^T\|^2}{2}
+
\frac{| Y|}{2} \, ,
\\
\sum_{(i,j)\in Y}j
&=
\sum_{i=1}^{Y^T_1}
\sum_{j=1}^{Y_i}j
=
\frac{\| Y\|^2}{2}
+
\frac{| Y|}{2} \, ,
\\
\sum_{(i,j)\in Y}Y_i
&=
\| Y\|^2 \, ,
\end{split}$$ where $Y^T$ is the transpose of the diagram $Y$, $|Y| \equiv \sum_{i=1}^{Y^T_1} Y_i = \sum_{i=1}^{Y_1} Y^T_i$ is the total number of boxes in the diagram, and $\| Y\|^2 \equiv \sum_{i=1}^{Y^T_1} (Y_i)^2$. By combining the formulas in (\[eq:SumYoungDiagram\]) we obtain $$\begin{aligned}
\sum_{(i,j)\in Y}\left(
Y_i-i-j+1
\right)
=\frac{\kappa_{Y}}{2} \, ,\end{aligned}$$ where the second Casimir is defined as $\kappa_Y\equiv \|Y\|^2-\|Y^T\|^2$.
Next, we introduce the 5D “Nekrasov factor" $$\begin{aligned}
N_{Y_1Y_2}(\mathfrak{q},Q)
\equiv
\prod_{(i,j)\in Y_1}
\left(1-Q
\mathfrak{q}^{Y_{1\,i}+Y^T_{2\,j}-i-j+1}
\right)
\prod_{(i,j)\in Y_2}
\left(1-Q
\mathfrak{q}^{-Y^T_{1\,j}-Y_{2\,i}+i+j-1}
\right) \, .
\label{def_Nfac}\end{aligned}$$ Moreover, following [@Konishi:2003qq] we also define a function $P$ as $$\begin{split}
&\frac{1
}{P_{Y_1Y_2}(\mathfrak{q},Q)}
\equiv
\prod_{(i,j)\in Y_1}
\sinh \frac{\beta}{2}
\left(
a+\hbar(Y_{1\,i}+Y^T_{2\,j}-i-j+1)
\right)
\\
&\rule{0pt}{5ex}
\quad\qquad\qquad\quad
\times
\prod_{(i,j)\in Y_2}
\sinh \frac{\beta}{2}
\left(a+\hbar
(-Y^T_{1\,j}-Y_{2\,i}+i+j-1)\right)
\end{split}$$ with $\mathfrak{q}=e^{-\beta \hbar}$ and $Q=e^{-\beta a}$. After using the identity $1 - e^{x} = 2 e^{x/2} \sinh (x/2)$ we find an important relation between these two ubiquitous factors: $$\begin{aligned}
N_{Y_1Y_2}(\mathfrak{q},Q)
=\frac{(2\,Q^{\frac{1}{2}})^{|Y_1|+|Y_2|}\,\mathfrak{q}^{\frac{\kappa_{Y_1}}{4}-\frac{\kappa_{Y_2}}{4}}
}{P_{Y_1\,Y_2}(Q)} \, .
\label{NPrelation}\end{aligned}$$ In order to obtain (\[NPrelation\]) we have used the identity[^21] $$\sum_{(i,j)\in Y_2}Y^T_{1\,j}
=
\sum_{(i,j)\in Y_1}Y^T_{2\,j} \, .$$ The combinatorial properties of these functions are essential when we rewrite the topological partition function to the form of Nekrasov’s partition function. A basic formula we use frequently is $$\begin{aligned}
P_{R_1R_2}(\mathfrak{q},Q)
=(-1)^{|R_1|+|R_2|}P_{R_2R_1}(\mathfrak{q},Q^{-1}) \, ,
\label{Pinverse}\end{aligned}$$ which follows immediately from the definition of $P$. Moreover, the following infinite product expressions of the Nekrasov factor $N_{Y_1Y_2}$ $$\label{InfinFin}
\prod_{i,j=1}^\infty
\frac{1-Q
\mathfrak{q}^{-Y_{1\,i}-Y^T_{2\,j}+i+j-1}}
{1-Q
\mathfrak{q}^{i+j-1}}
=\prod_{i,j=1}^\infty
\frac{1-Q
\mathfrak{q}^{Y^T_{1\,i}+Y_{2\,j}-i-j+1}}
{1-Q
\mathfrak{q}^{-i-j+1}}
=
N_{Y_1Y_2}(\mathfrak{q},Q) \,$$ taken from (3.9) and (3.10) in [@Taki:2007dh], will also be useful. By defining a bracket $[*,*]_Q$ $$\begin{aligned}
\label{bracket}
\left[Y_1, Y_2 \right]_{Q}
\equiv
\prod_{i,j=1}^\infty
(1-Q\mathfrak{q}^{Y_{1i}+Y_{2j}-i-j+1})
=\left[Y_2, Y_1 \right]_{Q} \, ,\end{aligned}$$ we can recast the relation (\[InfinFin\]) into a simple form $$\begin{aligned}
\label{bracket_N}
\frac{
\left[Y_1^T, Y_2 \right]_{Q}}
{\left[\emptyset, \emptyset \right]_{Q}}
=N_{Y_1Y_2}(\mathfrak{q},Q)
=N_{Y_2^TY_1^T}(\mathfrak{q},Q) \, .\end{aligned}$$ This formula now implies $$\begin{aligned}
P_{Y_1 Y_2}(\mathfrak{q},Q)=P_{Y_2^T Y_1^T}(\mathfrak{q},Q) \, .
\label{Ptranspose}\end{aligned}$$
The Schur functions play a special role in our derivation, since they describe the “topological vertex decomposition" of Nekrasov’s partition function. The specialized Schur function $S_{R}(\mathfrak{q}^\rho)$ takes the form $$\begin{aligned}
S_{R}(\mathfrak{q}^{\rho})
=(-1)^{|R|}S_{R^T}(\mathfrak{q}^{-\rho})
=\mathfrak{q}^{-n(R)-\frac{|R|}{2}}
\prod_{(i,j)\in R}
(1-\mathfrak{q}^{-R_i-R^T_j+i+j-1})^{-1} \, ,\end{aligned}$$ where $n(R)=\sum_{R}(i-1)$ satisfies $n(R^T)-n(R)=\kappa_R/2$. Taking into account the definition (\[def\_Nfac\]) and the relation (\[NPrelation\]), we obtain the following relation $$\begin{split}
S_{R}(\mathfrak{q}^{\rho})
S_{R^T}(\mathfrak{q}^{\rho})
=
(-1)^{|R|}
N_{RR}^{-1}(\mathfrak{q},1)
=
(-4)^{-|R|}
P_{RR}(\mathfrak{q},1) \, .
\label{specializedSchur}
\end{split}$$
Nekrasov partition function
---------------------------
The Nekrasov partition function for the linear quiver gauge theories that we have investigated in this article is given by [@Nekrasov:2002qd; @Fucito:2004gi] $$\begin{split}
Z
= &
\sum_{\vec{Y}^{(1)}} \cdots \sum_{\vec{Y}^{(M-1)}}\,
\left( q^{(1)} \right)^{|\vec{Y}^{(1)}|} \cdots
\left( q^{(M-1)} \right)^{|\vec{Y}^{(M-1)}|}
\cr
& \times
\prod_{i=1}^{M-1} Z_{\,\textrm{vect}}(\vec{a}^{(i)},\vec{Y}^{(i)}, \hbar; \beta)\,
\prod_{i=1}^{M-2} Z_{\,\textrm{bifund}}
(\vec{a}^{(i)}, \vec{Y}^{(i)}, \vec{a}^{(i+1)}, \vec{Y}^{(i+1)}, m_{\text{bif}}^{(i,i+1)}, \hbar; \beta)\,
\cr
& \times
\prod_{\gamma=1}^N Z_{\,\textrm{antifund}} (\vec{a}^{(1)} ,\vec{Y}^{(1)}, m^{\text{af}}_\gamma, \hbar; \beta)
\prod_{\delta=1}^N Z_{\,\textrm{fund}} (\vec{a}^{(M-1)} ,\vec{Y}^{(M-1)}, m^{\text{f}}_{N+\delta}, \hbar; \beta) \, ,
\label{Nek_formula}
\end{split}$$ where $$\vec{a}^{(i)} = (a^{(i)}_1, \cdots, a^{(i)}_N)
\qquad \text{and} \qquad
\vec{Y}^{(i)} = (Y^{(i)}_1, \cdots, Y^{(i)}_N)$$ denote the Coulomb moduli parameters and the Young diagrams for the corresponding gauge group factors, respectively. The Young diagrams describe the fixed points of the localization computation. The explicit forms and the basic properties of the factors $Z_{\textrm{vect}}$, $Z_{\textrm{bifund}}$, $Z_{\textrm{fund}}$ and $Z_{\textrm{antifund}}$ will now be described separately.
### Vector multiplet contribution {#vector-multiplet-contribution .unnumbered}
The contribution from a vector multiplet is the following product of $\sinh$ functions $$\begin{split}
Z_{\,\textrm{vect}}(\vec{a},\vec{Y},\hbar ; \beta)
&=
\prod_{\alpha,\beta=1}^N
\prod_{(i,j)\in Y_\alpha}
\sinh^{-1} \frac{\beta}{2}
\left(
a_\alpha-a_\beta
-\hbar(
Y_{\alpha i}+Y^T_{\beta j}-i-j+1
)
\right) \\
&\qquad\qquad\qquad
\times
\prod_{(i,j)\in Y_\beta}
\sinh^{-1} \frac{\beta}{2}
\left(
a_\alpha-a_\beta
+\hbar(
Y_{\beta i}+Y^T_{\alpha j}-i-j+1
)
\right) \\
&
=\prod_{\alpha,\beta=1}^N
P_{Y_\beta Y_\alpha}(\mathfrak{q},Q_{\alpha\beta}) \, ,
\label{Nek5Dvect}
\end{split}$$ where the argument $Q_{\alpha\beta}$ is defined as $$\begin{aligned}
Q_{\alpha\beta}=e^{-\beta(a_\alpha-a_\beta)} \, .\end{aligned}$$ By separating the products in (\[Nek5Dvect\]) into three parts ($\alpha=\beta$, $\alpha < \beta$, $\beta < \alpha$) and applying (\[Pinverse\]), we obtain $$\begin{aligned}
Z_{\,\textrm{vect}}(\vec{a},\vec{Y},\hbar ; \beta)
=
(-1)^{(N-1)\sum_\alpha |Y_\alpha|}
\prod_{\alpha=1}^N\,
P_{Y_\alpha Y_\alpha}(\mathfrak{q},1)\
\prod_{1\leq\alpha<\beta\leq N}
P_{Y_\beta Y_\alpha}(\mathfrak{q},Q_{\alpha\beta})
P_{Y_\beta Y_\alpha}(\mathfrak{q},Q_{\alpha\beta}) \, .
\label{Nek5Dvect2}\end{aligned}$$ Further, when combining this result with (\[NPrelation\]) and (\[specializedSchur\]) it follows that $$\begin{aligned}
Z_{\,\textrm{vect}}(\vec{a},\vec{Y},\hbar ; \beta)
&= C( Q_{\alpha\beta}, \vec{Y} )\,
\prod_{\alpha=1}^N\,
S_{Y_\alpha}(\mathfrak{q}^\rho)\,
S_{Y_\alpha^T}(\mathfrak{q}^\rho)
\prod_{1\leq\alpha<\beta\leq N}
\left( N_{Y_\beta Y_\alpha}(\mathfrak{q},Q_{\alpha\beta})\right)^{-2} \, ,\end{aligned}$$ where the coefficient $C$ is defined as $$\begin{aligned}
C ( Q_{\alpha\beta}, \vec{Y} ) =
(-4)^{N\sum_\alpha |Y_\alpha|}
\prod_{1\leq\alpha<\beta\leq N}
(Q_{\alpha\beta})^{|Y_\alpha|+|Y_\beta|}
\mathfrak{q}^{-\frac{\kappa_{Y_\alpha}}{2}+\frac{\kappa_{Y_\beta}}{2}} \, .
\label{defC}\end{aligned}$$
### Hypermultiplet contribution {#hypermultiplet-contribution .unnumbered}
The hypermultiplets appearing in this article transform as fundamental and bifundamental representations. We start with studying the bifundamental one. Their contribution to the 5D Nekrasov partition function is $$\begin{split}
Z_{\,\textrm{bifund}}(\vec{a},\vec{R},\vec{\tilde{a}},\vec{Y},m,\hbar ; \beta)
=&
\prod_{\alpha,\beta=1}^N
\prod_{(i,j)\in R_\alpha}
\sinh \frac{\beta}{2}
\left(
a_\alpha-\tilde{a}_\beta-m
-\hbar(
R_{\alpha i}+Y^T_{\beta j}-i-j+1
)
\right)\\
&\qquad
\times
\prod_{(i,j)\in Y_\beta}
\sinh\frac{\beta}{2}
\left(
a_\alpha-\tilde{a}_\beta-m
+\hbar (
Y_{\beta i}+R^T_{\alpha j}-i-j+1
)
\right)\\
=&\prod_{\alpha,\beta=1}^N
P^{-1}_{Y_\beta R_\alpha}(\mathfrak{q},e^{-\beta (a_\alpha-\tilde{a}_\beta-m )}) \, .
\label{Nek5Dbifund}
\end{split}$$ We introduce the variable $Q_{m\alpha}$ for later convenience $$\begin{aligned}
Q_{m\alpha}=e^{-\beta(a_\alpha-\tilde{a}_\alpha-m)} \, .\end{aligned}$$ The arguments of $P^{-1}$ in (\[Nek5Dbifund\]) can then be written as $$\begin{aligned}
e^{-\beta(a_\alpha-\tilde{a}_\beta-m)}
={Q}_{\alpha\beta}Q_{m\beta} \qquad \text{and} \qquad
e^{-\beta(-a_\beta+\tilde{a}_\alpha+m)}=
Q_{m\alpha}^{-1}{Q}_{\alpha\beta} \, .\end{aligned}$$ In terms of these new variables (\[Nek5Dbifund\]) reads $$\begin{aligned}
Z_{\,\textrm{bifund}} & (\vec{a},\vec{R},\vec{\tilde{a}},\vec{Y},m,\hbar ; \beta)
\cr
= &
\prod_{1\leq\alpha\leq\beta\leq N}
P^{-1}_{Y_\beta R_\alpha}(\mathfrak{q},{Q}_{\alpha\beta}Q_{m\beta})
\prod_{1\leq\alpha<\beta\leq N}
(-1)^{|R_\beta|+|Y_\alpha|}\,
P^{-1}_{R_\beta Y_\alpha }(\mathfrak{q},Q_{m\alpha}^{-1}{Q}_{\alpha\beta}) \, ,\end{aligned}$$ where we have separated the product into $\alpha \le \beta$ and $\beta < \alpha$ and applied (\[Pinverse\]). After also applying (\[NPrelation\]) we obtain $$\begin{aligned}
Z_{\,\textrm{bifund}} & (\vec{a},\vec{R},\vec{\tilde{a}},\vec{Y},m,\hbar ; \beta)
\cr
& = D\,
\prod_{1\leq\alpha\leq\beta\leq N}
N_{Y_\beta R_\alpha}(\mathfrak{q},{Q}_{\alpha\beta}Q_{m\beta})
\prod_{1\leq\alpha<\beta\leq N}
N_{R_\beta Y_\alpha }(\mathfrak{q},Q_{m\alpha}^{-1}{Q}_{\alpha\beta}) \, ,\end{aligned}$$ where $D$ is defined as $$D =
\prod_{1\leq\alpha\leq\beta\leq N}
\left(-2\sqrt{Q_{\alpha\beta}Q_{m\beta}}\right)^{-|R_\alpha|-|Y_\beta|}
\mathfrak{q}^{-\frac{\kappa_{Y_\beta}}{4}+\frac{\kappa_{R_\alpha}}{4}}
\prod_{1\leq\alpha<\beta\leq N}
\left(2\sqrt{Q_{m\alpha}^{-1}{Q}_{\alpha\beta}}\right)^{-|R_\beta|-|Y_\alpha|}
\mathfrak{q}^{-\frac{\kappa_{R_\beta}}{4}+\frac{\kappa_{Y_\alpha}}{4}} \, .$$ By decomposing this expression into the $\vec{R}$ dependent and the $\vec{Y}$ dependent part, and separating out the $\alpha=\beta$ term in the first products, we find $$D = D_{L}(Q_{\alpha \beta}, Q_{m\alpha}, \vec{R})\,D_{R}(Q_{\alpha \beta}, Q_{m\alpha}, \vec{Y})$$ with $$\label{defDLR}
\begin{split}
&D_{L}(Q_{\alpha \beta}, Q_{m\alpha}, \vec{R})=
\prod_{\alpha=1}^N
\frac{
(-)^{(N-\alpha+1)|R_\alpha|}
\mathfrak{q}^{\frac{1}{4} \kappa_{R_\alpha}}}
{2^{N |R_\alpha|} \left( Q_{m\alpha} \right)^{\frac{|R_\alpha|}{2}}}
\prod_{1\leq\alpha<\beta\leq N}
(Q_{\alpha\beta}^{-\frac{1}{2}})^{|R_\alpha|+|R_\beta|}
\frac
{\left(Q_{m\alpha}\right)^{\frac{|R_\beta|}{2}}}
{\left(Q_{m\beta}\right)^{\frac{|R_\alpha|}{2}}}
\mathfrak{q}^{
\frac{\kappa_{R_\alpha}}{4}-\frac{\kappa_{R_\beta}}{4}} \, ,
\\
&D_{R}(Q_{\alpha \beta}, Q_{m\alpha}, \vec{Y})=
\prod_{\alpha=1}^N
\frac{
(-)^{\alpha|Y_\alpha|}
\mathfrak{q}^{ - \frac{1}{4} \kappa_{Y_\alpha}}}
{ 2^{N |Y_\alpha|}\, \left( Q_{m\alpha} \right)^{\frac{|Y_\alpha|}{2}} }
\prod_{1\leq\alpha<\beta\leq N}
(Q_{\alpha\beta}^{-\frac{1}{2}})^{|Y_\alpha|+|Y_\beta|}
\frac{\left(Q_{m\alpha}\right)^{\frac{|Y_\alpha|}{2}}}
{\left(Q_{m\beta}\right)^{\frac{|Y_\beta|}{2}} }
\mathfrak{q}^{\frac{\kappa_{Y_\alpha}}{4}-\frac{\kappa_{Y_\beta}}{4}} \, .
\end{split}$$
The contribution from a fundamental hypermultiplet $$\begin{split}
Z_{\,\textrm{fund}}(\vec{a},\vec{Y}, m, \hbar ; \beta)
&= \prod_{\alpha=1}^N \prod_{(i,j) \in Y_{\alpha}}
\sinh \frac{\beta}{2} (a_{\alpha} - m + \hbar (i-j ) )
\cr
&= \prod_{\alpha=1}^N P^{-1}_{Y_{\alpha \emptyset}} \left( \mathfrak{q}, e^{-\beta(a_{\alpha}-m)} \right)
\label{Nek5Dfund}
\end{split}$$ is related to that from an anti-fundamental hypermultiplet as $$\begin{aligned}
Z_{\textrm{antifund}}(\vec{a},\vec{R},m)=Z_{\textrm{fund}}(\vec{a},\vec{R},-m) \, .
\label{Nek5Dantifund}\end{aligned}$$ Moreover, the product of contributions from all the $N$ fundamental hypermultiplets obeys $$\begin{aligned}
\prod_{\alpha=1}^N
Z_{\,\textrm{fund}}(\vec{a},\vec{R},{m}_\alpha,\hbar ; \beta)
=
Z_{\,\textrm{bifund}}(\vec{a},\vec{R},\vec{m}-m\vec{1},\vec{\emptyset},m,\hbar ; \beta) \, ,
\label{fund_bif}\end{aligned}$$ where $m \equiv \frac{1}{N}\sum_\alpha m_\alpha$. A similar relation exists for the anti-fundamental contribution: $$\begin{aligned}
\prod_{\alpha=1}^N
Z_{\,\textrm{antifund}}(\vec{a},\vec{R},{m}_\alpha,\hbar ; \beta)
=
(-1)^{N\sum |R_\alpha|}
Z_{\,\textrm{bifund}}(-\vec{m}+m\vec{1},\vec{\emptyset},\vec{a},\vec{R},-m,\hbar ; \beta) \, .
\label{antifund_bif}\end{aligned}$$
### Gluing the multiplets {#gluing-the-multiplets .unnumbered}
We will now go back to the full Nekrasov partition function in (\[Nek\_formula\]) and rewrite it into a form which is convenient for the comparison with the topological string partition function. To do this, we focus on the $i$-th vector multiplet corresponding to $\vec{Y}^{(i)}$ together with the bifundamental hypermultiplets which are charged under the associated vector field. The local structure of the Nekrasov partition function for $\vec{Y}^{(i)}$ $$\begin{split}
\cdots
\sum_{\vec{Y}^{(i)}}
& \left( q^{(i)} \right) ^{\sum_{\alpha} |Y^{(i)}_\alpha|}\,
Z_{\,\textrm{bifund}}(\vec{a}^{(i-1)},\vec{Y}^{(i-1)},\vec{a}^{(i)}, \vec{Y}^{(i)},{m}_{\text{bif}}^{(i-1,i)},\hbar ; \beta)\,
\cr
& \times Z_{\,\textrm{vect}}(\vec{a}^{(i)},\vec{Y}^{(i)},\hbar ; \beta)\,
Z_{\,\textrm{bifund}}(\vec{a}^{(i)},\vec{Y}^{(i)},\vec{a}^{(i+1)}, \vec{Y}^{(i+1)}, {m}_{\text{bif}}^{(i,i+1)}, \hbar ; \beta)
\cdots
\end{split}$$ dictates the contribution from these multiplets. By employing the relations (\[fund\_bif\]) and (\[antifund\_bif\]), this expression becomes valid also for $i=1$ and $i=M-1$. After collecting the factors which depend on $\vec{Y}^{(i)}$ we find the following contribution in Nekrasov’s formula $$\begin{split}
&
D_{R}({Q}_{\alpha\beta}^{(i-1)}, {Q}_{m\alpha}^{(i-1)}, \vec{Y}^{(i)})\,
C(Q_{\alpha\beta}^{(i)}, \vec{Y}^{(i)})\,
D_{L}(Q_{\alpha\beta}^{(i)}, {Q}_{m\alpha}^{(i)}, \vec{Y}^{(i)})\,
q^{\sum |Y^{(i)}_\alpha|}
\cr
&\times
\prod_{\alpha=1}^N\,
S_{Y^{(i)}_\alpha}(\mathfrak{q}^\rho)\,
S_{Y^{(i) \, T}_\alpha}(\mathfrak{q}^\rho)
\prod_{1\leq\alpha<\beta\leq N}
\left( N_{Y^{(i)}_\beta, Y^{(i)}_\alpha} (\mathfrak{q},Q_{\alpha\beta}^{(i)}) \right)^{-2}
\cr
& \times
\prod_{1\leq\alpha\leq\beta\leq N}
N_{Y^{(i)}_\beta Y^{(i-1)}_\alpha}
\left( \mathfrak{q},{Q}_{\alpha\beta}^{(i-1)} {Q}_{m\beta}^{(i-1)} \right)
\prod_{1\leq\alpha<\beta\leq N}
N_{Y_\beta^{(i-1)} Y^{(i)}_\alpha }
\left( \mathfrak{q}, ({Q}_{m\alpha}^{(i-1)})^{-1} {Q}_{\alpha\beta}^{(i-1)} \right)
\cr
& \times
\prod_{1\leq\alpha\leq\beta\leq N}
N_{Y^{(i+1)}_\beta Y^{(i)}_\alpha}
\left( \mathfrak{q},{Q}_{\alpha\beta}^{(i)} Q_{m\beta}^{(i)} \right)
\prod_{1\leq\alpha<\beta\leq N}
N_{Y^{(i)}_\beta Y^{(i+1)}_\alpha }
\left( \mathfrak{q},(Q_{m\alpha}^{(i)}) ^{-1}{Q}_{\alpha\beta}^{(i)} \right)\, .
\label{GluingOfQuiver}
\end{split}$$ The instanton factor $q^{(i)}$ will absorb most of the coefficients $C$, $D_L$ and $D_R$ in (\[GluingOfQuiver\]) $$\begin{split}
&D_{R}({Q}^{(i-1)}_{\alpha\beta}, {Q}^{(i-1)}_{m\alpha}, \vec{Y}^{(i)})\,
C(Q^{(i)}_{\alpha\beta}, \vec{Y}^{(i)})\,
D_{L}(Q^{(i)}_{\alpha\beta}, {Q}^{(i)}_{m\alpha}, \vec{Y}^{(i)})\,
\left( q^{(i)} \right) ^{\sum |\vec{Y}^{(i)}_\alpha|}
\\
&=
\prod_\alpha
\left(\frac{-q^{(i)}}{\sqrt{Q^{(i)}_{m\alpha}Q^{(i-1)}_{m\alpha}}}\right)^{|Y^{(i)}_\alpha|}
\prod_{1\leq\alpha<\beta\leq N}
\sqrt{\frac{(Q^{(i)}_{m\alpha})^{|Y^{(i)}_\beta|}}{(Q^{(i)}_{m\beta})^{|Y^{(i)}_\alpha|}}}
\sqrt{
\frac
{
(Q^{(i-1)}_{m\beta})^{|Y^{(i)}_\alpha|}
}
{
(Q^{(i-1)}_{m\alpha})^{|Y^{(i)}_\beta|}
}
}\\
&=
\prod_\alpha
\label{GaugeCouplingKahlers}
\left(
-q^{(i)} \prod_\beta
\sqrt{\frac
{
Q^{(i-1)}_{m\beta}
}
{
Q^{(i)}_{m\beta}
}}
\right)^{|Y^{(i)}_\alpha|}
\frac
{
\prod_{1\leq\alpha<\beta\leq N}
(Q^{(i)}_{m\alpha})^{|Y^{(i)}_\beta|}
}
{
\prod_{1\leq\alpha\leq\beta\leq N}
(Q^{(i-1)}_{m\alpha})^{|Y^{(i)}_\beta|}
} \, ,
\end{split}$$ where in the first equality we have used the definitions (\[defC\]) and (\[defDLR\]) together with the relation (\[rec\_QF\]). This result plays a key role in identifying the relation between the gauge coupling $q$ and the Kähler parameter $Q_B$ of the base $\mathbb{P}^1$ of the corresponding CY$_3$. The last line of (\[GaugeCouplingKahlers\]) is thus related to the product $\prod (-Q_{B \alpha}^{(i)})^{|Y^{(i)}_\alpha|}$.
Comparison with the topological string partition function
---------------------------------------------------------
In the topological vertex computation in Section \[sec:TopStringDeriv\], first we calculated a sub-diagram corresponding to a decomposed toric diagram. To get the full string partition function, we had to glue together the sub-diagrams. That procedure is similar to the gluing construction of the Nekrasov partition functions in the previous subsection. Here, we demonstrate the gluing construction for the $SU(N)^{M-1}$ linear quiver theory. Since the gluing of the toric sub-diagrams is done by taking the summation over the Young diagrams, the full partition function corresponding to Figure \[fig:large\] is given by $$\begin{split}
Z_{\,\textrm{inst}}&=
\label{genQuivPartFunc}
\sum\cdots
\sum_{Y^{(i)}_1,\cdots,Y^{(i)}_N}
\prod_{\alpha}(-Q^{(i)}_{B\alpha})^{|Y^{(i)}_\alpha|}\\
& \times \tilde{H}_{\,Y^{(i+1)}_1Y^{(i+1)}_2\cdots Y^{(i+1)}_N}^{\,Y^{(i)}_1Y^{(i)}_2\cdots Y^{(i)}_N}\,(\,Q^{(i)}_{m1},\cdots,Q^{(i)}_{mN},Q^{(i)}_{F1},\cdots,Q^{(i)}_{FN})\\
& \times \tilde{H}^{\,Y^{(i-1)}_1Y^{(i-1)}_2\cdots Y^{(i-1)}_N}_{\,Y^{(i)}_1Y^{(i)}_2\cdots Y^{(i)}_N}\,(\,Q^{(i-1)}_{m1},
\cdots,Q^{(i-1)}_{mN},Q^{(i-1)}_{F1},\cdots,Q^{(i-1)}_{FN})
\cdots.
\end{split}$$ Using the explicit expression for the sub-diagram $\tilde{H}$ in (\[Htilde\]), which resembles the Nekrasov factor very much, we can rewrite (\[genQuivPartFunc\]) as the Nekrasov partition function for a linear quiver gauge theory.
We will again focus on the contribution from the $i$-th gauge group in the full partition function (\[genQuivPartFunc\]). In the topological vertex computation, the $i$-th color D4-branes are assigned with the Young diagrams $Y^{(i)}$ and correspond to the chopped lines of the web-diagram in Figure \[fig:genQuiver\]. The contribution from the $i$-th gauge group depends therefore on $Y^{(i)}$. By collecting such factors, we obtain $$\label{eq:GluingOfVertex}
\begin{split}
&\prod_{\alpha}(-Q^{(i)}_{B\alpha})^{|Y^{(i)}_\alpha|}\\
&\times\frac{\prod_{\alpha=1}^N\,
S_{Y^{(i)}_\alpha}(\mathfrak{q}^{\rho})\,
S_{{Y^{(i)\,T}_\alpha}}(\mathfrak{q}^{\rho})}
{\prod_{1\leq \alpha<\beta\leq N}
N_{Y^{(i)}_\beta Y^{(i)}_\alpha}\left(Q_{\alpha\beta}^{(i)}\right)\,
N_{Y^{(i)}_\beta Y^{(i)}_\alpha}\left((Q_{m\alpha}^{(i-1)})^{-1}Q_{\alpha\beta}^{(i-1)}Q_{m\beta}^{(i-1)}\right)
}\\
&\times
\rule{0pt}{4ex}
\prod_{1\leq \alpha<\beta\leq N}
N_{Y^{(i)}_\beta Y^{(i+1)}_\alpha}\left((Q^{(i)}_{m\alpha})^{-1}Q^{(i)}_{\alpha\beta}\right)
N_{Y^{(i-1)}_\beta Y^{(i)}_\alpha}\left((Q^{(i-1)}_{m\alpha})^{-1}Q^{(i-1)}_{\alpha\beta}\right)
\\
&\times
\rule{0pt}{4ex}
\prod_{1\leq \alpha\leq\beta\leq N}
N_{Y^{(i+1)}_\beta Y^{(i)}_\alpha}\left(Q^{(i)}_{\alpha\beta}Q^{(i)}_{m\beta}\right)
N_{Y^{(i)}_\beta Y^{(i-1)}_\alpha}\left(Q^{(i-1)}_{\alpha\beta}Q^{(i-1)}_{m\beta}\right) \, .
\end{split}$$ From the web-diagram in Figure \[fig:genQuiver\] we see that the arguments of the Nekrasov factors in denominator of the second line satisfy $$\begin{aligned}
Q_{\alpha\beta}^{(i)}=(Q_{m\alpha}^{(i-1)})^{-1} Q_{\alpha\beta}^{(i-1)}Q_{m\beta}^{(i-1)} \, .\end{aligned}$$ After inserting the known factor $\prod_{\alpha}(Q^{(i)}_{B\alpha})^{|Y^{(i)}_\alpha|}$ from (\[QB\_q\]) together with the gauge theory parametrization (\[Qab\_Qm\]) of the string parameters, we are now ready to compare the topological string partition function (\[eq:GluingOfVertex\]) with the Nekrasov partition function (\[GluingOfQuiver\]). By inspection these two expressions are identical.
For the full Nekrasov partition function, the above argument is applied successively for each index $i$. We find that $Z_{\,\textrm{inst}}$ exactly matches the instanton partition function for the linear quiver gauge theory, provided that $Q^{(i)}_B$ and ${q}^{(i)}$ are related as $$\begin{aligned}
Q_B^{(i)}
=
{q}^{(i)}
\frac{1}{Q^{(i-1)}_{m1}}
\prod_{\alpha=1}^N
\sqrt{\frac{Q^{(i-1)}_{m\alpha}}
{Q^{(i)}_{m\alpha}}} \, .\end{aligned}$$ This is the relation between the Kähler parameters of the base $\mathbb{P}^1$ and the gauge couplings. Together with (\[Qab\_Qm\]), they provide the complete identification rules between the gauge theory parameters and the Kähler parameters of CY$_3$. Using this identification, the topological vertex computation gives precisely the Nekrasov partition function, which proves the geometric engineering for the quiver gauge theories.
Symmetry of the Nekrasov partition function
-------------------------------------------
We end this section by commenting on a specific symmetry of the Nekrasov partition function. By using the identities of $P_{Y_1Y_2}(Q)$ in (\[Pinverse\]) and (\[Ptranspose\]), the contributions from the vector multiplet (\[Nek5Dvect\]) and the hypermultiplets (\[Nek5Dbifund\]) (\[Nek5Dfund\]) (\[Nek5Dantifund\]) respectively satisfy the following properties $$\begin{split}
&Z_{\,\textrm{vect}}(\vec{a},\vec{Y},\hbar ; \beta)
=Z_{\,\textrm{vect}}(-\vec{a},\vec{Y^T},\hbar ; \beta) \, ,
\\
&Z_{\,\textrm{bifund}}(\vec{a},\vec{R},\vec{\tilde{a}},\vec{Y},m,\hbar ; \beta)
=(-1)^{N\sum_{\alpha}|R_\alpha|+|Y_\alpha|}
Z_{\,\textrm{bifund}}(-\vec{a},\vec{R^T},-\vec{\tilde{a}},\vec{Y^T},-m,\hbar ; \beta) \, ,
\\
&Z_{\,\textrm{fund}}(\vec{a},\vec{R},{m},\hbar ; \beta)
= (-1)^{N\sum_{\alpha}|R_\alpha|} Z_{\,\textrm{fund}}(-\vec{a},\vec{R}^T,-{m},\hbar ; \beta) \, ,
\\
&Z_{\,\textrm{antifund}}(\vec{a},\vec{R},{m},\hbar ; \beta)
= (-1)^{N\sum_{\alpha}|R_\alpha|} Z_{\,\textrm{antifund}}(-\vec{a},\vec{R}^T,-{m},\hbar ; \beta) \, .
\end{split}$$ All the signs will cancel out if we substitute them into the Nekrasov partition function. When we sum over all the Young diagrams to obtain the expression (\[Nek\_formula\]), we find a symmetry with the signs of all the Coulomb moduli and the masses being inverted: $$Z(\vec{a}, \vec{m}, q, \hbar, \beta) = Z(-\vec{a}, -\vec{m}, q, \hbar, \beta) \, .$$
Rotation of 90 degrees {#app:90Rotation}
======================
In Section \[subsec:Msu2\] and \[subsec:topsu2\] we analyze the symmetry coming from the reflection in a diagonal axis of the toric diagram for $SU(2)$ SQCD. The invariance of the topological string partition function reproduces the duality transformation found in the M-theory setup. However, both the M5-brane configuration and the toric web-diagram are also invariant under the rotation by 90 degrees. Rigorously, it is this 90 degree rotation which corresponds to part of the $SL(2,Z)$ S-duality transformations, and not the reflection as mentioned in Section \[subsec:review\_duality\]. We will now give the duality map of the 90 degree rotation.
In the M-theory setup, this rotation leads to the coordinate transformation $$\begin{aligned}
w_d = t, \qquad t_d^{-1} = w
\label{coord_rot}\end{aligned}$$ of the SW curve. Contrary to the reflection transformation in Section \[subsec:Msu2\], the dual of the SW one-form is given by $$(\lambda_{SW})_d = \lambda_{SW} \, .$$ However, due to the convention of the direction of the cycle, we have $(A_1)_d = - A'_1$. The second relation in (\[aint\]) and thus also (\[dual\_a\]) are the same in this case. Taking these into account, we obtain the following duality map for the rotation $$\begin{split}
(\tilde{m}_1)_d
= \tilde{m}_1^{-\frac{1}{4}} \tilde{m}_2^{-\frac{3}{4}}
\tilde{m}_3^{-\frac{1}{4}} \tilde{m}_4^{\frac{1}{4}} q^{-\frac{1}{2}} \, ,&
\qquad
(\tilde{m}_2)_d
= \tilde{m}_1^{\frac{1}{4}} \tilde{m}_2^{-\frac{1}{4}}
\tilde{m}_3^{\frac{1}{4}} \tilde{m}_4^{\frac{3}{4}} q^{\frac{1}{2}} \, ,
\cr
(\tilde{m}_3)_d
= \tilde{m}_1^{\frac{3}{4}} \tilde{m}_2^{\frac{1}{4}}
\tilde{m}_3^{-\frac{1}{4}} \tilde{m}_4^{\frac{1}{4}} q^{-\frac{1}{2}} \, ,&
\qquad
(\tilde{m}_4)_d
= \tilde{m}_1^{\frac{1}{4}} \tilde{m}_2^{-\frac{1}{4}}
\tilde{m}_3^{-\frac{3}{4}} \tilde{m}_4^{-\frac{1}{4}} q^{\frac{1}{2}} \, ,
\cr
\tilde{a}_d = \left( \frac{\tilde{m}_2 \tilde{m}_4}{\tilde{m}_1\tilde{m}_3} \right)^{\frac{1}{4}} q^{-\frac{1}{2}} \tilde{a} \, ,&
\qquad
q_d = \left( \frac{\tilde{m}_2\tilde{m}_4}{\tilde{m}_1 \tilde{m}_3} \right)^{\frac{1}{2}} \, .
\label{map_rotation}
\end{split}$$
On the other hand, in the topological string setup we the full partition function to be invariant under the transformation $$\begin{aligned}
(Q_{m1})_d= Q_{m2} \, ,\quad
(Q_{m2})_d =Q_{m4} \, ,&\quad
(Q_{m4})_d =Q_{m3} \, ,\quad
(Q_{m3})_d =Q_{m1} \, ,\quad
\cr
(Q_{B})_d = Q_F \, ,&\quad
(Q_{F})_d = Q_B \, ,\end{aligned}$$ where all the parameters are given by Figure \[fig:4flavSQCD\]. Moreover, the relation between the Kähler parameters and the gauge theory parameters are found in (\[Q\_Def\]) and (\[q\_Def\]). After combining with the symmetry transformation $\tilde{m}_{i}\to\tilde{m}^{-1}_{i}$, $(\tilde{m}_{i})_d\to(\tilde{m}_{i})_d^{-1}$, $\tilde{a}\to\tilde{a}^{-1}$ and $(\tilde{a})_d\to(\tilde{a})_d^{-1}$ of the Nekrasov partition function, we confirm that it exactly reproduces the duality map (\[map\_rotation\]). We have thus proved that the $\Omega$-background does not break this duality either.
The difference between the reflection and the rotation symmetry can be understood as a simple parity transformation, which corresponds to the coordinate transformation $$\begin{aligned}
w_d = w^{-1} \, , \qquad t_d = t \, .
\label{w_inv}\end{aligned}$$ The corresponding duality map of the gauge theory parameters is $$\begin{split}
(\tilde{m}_1)_d = \tilde{m}_2^{-1} \, , \qquad
(\tilde{m}_2)_d = \tilde{m}_1^{-1} \, ,& \qquad
(\tilde{m}_3)_d = \tilde{m}_4^{-1} \, , \qquad
(\tilde{m}_4)_d = \tilde{m}_3^{-1} \, ,
\cr
\tilde{a}_d = \tilde{a}^{-1} \, ,& \qquad
q_d = q \, .
\label{map_parity}
\end{split}$$ It is straightforward to show that the duality map (\[map\_reflection\_su2\]) for the reflection can be obtained by sequentially acting with the transformations (\[map\_parity\]) and (\[map\_rotation\]).
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[^1]: Email: ling.bao@chalmers.se
[^2]: Email: pomoni@mathematik.hu-berlin.de
[^3]: Email: taki@yukawa.kyoto-u.ac.jp
[^4]: Email: fyagi@sissa.it
[^5]: For related work see also [@Muneyuki:2011qu].
[^6]: The moduli of the SW curve $u$ for the SU(2) example is the gauge invariant Coulomb moduli $u=\langle \text{tr}\phi^2 \rangle + \dots$.
[^7]: In this article we use the word toric diagram for the dual graph of the toric data. This is also called web-diagram [@Aharony:1997bh].
[^8]: To include all the Kaluza-Klein modes only the circumference of the circle $\beta$ is added as an extra parameter. However, the circumference $\beta$ always appears multiplied with the gauge parameters whose mass dimension is $1$, so it does not actually introduce any new degree of freedom.
[^9]: This is true because $\omega$ has no poles as opposed to the meromorphic $\lambda_{SW}$ that does have poles.
[^10]: An alternative derivation of the SW curve is given in [@Minahan:1997ch] using a different point of view that exploits the enhancement of the global symmetry to $E_5 = SO(10)$ [@Seiberg:1996bd].
[^11]: Note that the index $i=1,2$ counts the color, while the index $\mathfrak{i}=1,\dots,4$ counts the flavor.
[^12]: We use the notation $(A B)_d\equiv A_d B_d$.
[^13]: To be precise, the obtained topological string partition function is the Nekrasov partition function for the $U(N)$ gauge theory whose Coulomb moduli parameters are constrained as $a_1=-a_2=a$. According to [@Alday:2009aq], this constrained partition function is still not precisely $SU(N)$. The difference is the overall factor which in [@Alday:2009aq] is called the U(1) factor and is independent of the Coulomb moduli. This $U(1)$ factor does not affect the low-energy effective coupling constants which we studied in the previous section.
[^14]: It is possible to obtain (\[top\_su2\_map2\]) directly if we define the toric diagram in Figure \[fig:4flavSQCD\] with all the arrows reversed. In that case, the parametrization of the geometric engineering parameters in (\[Q\_Def\]) gets inverted. In this article, we use the standard parametrization from [@Iqbal:2004ne].
[^15]: The quiver diagram drawn à la Gaiotto [@Gaiotto:2009we] looks identical to the diagram associated with the conformal block.
[^16]: See [@Tai:2010ps] for work related to this idea.
[^17]: See [@Losev:2003py; @Marshakov:2009gs; @Alba:2009ya] for the 4D version of the AGTW “Heisenberg/$U(1)$” duality, which implies the equality between the free conformal block for the Heisenberg algebra and the Nekrasov partition function for $U(1)$ quiver gauge theory. This is a simplified toy model for the original AGT duality of Virasoro/$SU(2)$.
[^18]: The time coordinates are interpreted as gauge couplings.
[^19]: In this appendix we set $\ell_{s}=1$.
[^20]: The overall $U(1)$ corresponds to the center of mass in the $v$ coordinate when all the flavor masses are identically equal to zero.
[^21]: See p.12 of [@Taki:2007dh].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We consider graphene on monolayer WSe$_2$ and the spin-orbit coupling induced by the transition-metal dichalcogenide substrate for application to spin-active devices. We study quantum dots and graphene quantum rings as tunable spin filters and inverters. We use an atomistic tight-binding model as well as the Dirac equation to determine stationary states confined in quantum dots and rings. Next we solve the spin-transport problem for dots and rings connected to nanoribbon leads. The systems connected to zigzag nanoribbons at low magnetic fields act as spin filters and provide strongly spin polarized current.'
author:
- 'A. Mreńca-Kolasińska'
- 'B. Rzeszotarski'
- 'B. Szafran'
bibliography:
- 'RingTMDC.bib'
title: ' Spin-active devices based on graphene / WSe$_2$ heterostructure '
---
=4
Introduction
============
Graphene [@Neto] has inspired many ideas for applications in electronics and valleytronics and was hoped to be useful in spintronics [@ERashba2009; @Bercioux2010; @Tse2011; @Inglot2015]. However, the spin-orbit coupling (SOC) in pristine graphene turns out to be too weak [@Gmitra2009] for spintronic applications. SOC in graphene can be enhanced by doping [@Rocha2010; @Sheng2010] or adsorption [@Leenaerts2009; @Balakrishnan2013; @Gmitra2013; @Gargiulo2014; @Hong2011; @Hong2012; @Santos2014; @Irmer2015; @Avsar2015] of light-element atoms or deposition of heavy ones [@Weeks2011; @Brey2015]. However, the adatoms and dopants introduce disorder, enhance the scattering, and limit the carrier mobility. To circumvent this problem, SOC can be proximity-induced in graphene combined with two-dimensional transition-metal dichalcogenides (TMDCs) as calculated theoretically [@Kaloni2014; @Gmitra2015] and shown experimentally [@Avsar2014; @Wang2015; @Wang2016; @Yang2016; @Volkl2017; @Yang2017; @Ghiasi2017; @Benitez2017; @Zihlmann2018; @Wakamura2018]. Upon contact the Dirac point of graphene falls within the energy gap of TMDCs, which preserves the linear band structure of graphene at the Fermi level [@Kaloni2014; @Gmitra2015]. Moreover, in graphene coupled to TMDCs, the electronic bands acquire a spin texture [@Cummings2017] which can lead to modification of the electron spin, and as a consequence, to spin operations.
Previous proposals for spin-active graphene devices, in particular for spin filters, relied on the ferromagnetic substrates [@Varykhalov2008; @Munarriz2012; @JunFeng2012; @Rybkina2013] or doping with nonmagnetic atoms [@Rocha2010; @Sheng2010] like boron, nitride, oxygen, and fluorine. However, the proximity of the metal is problematic for the construction of electronic devices exploiting the graphene transport properties. In this paper we consider the graphene/WSe$_2$ heterostructure with internal magnetic field due to the spin-orbit coupling instead of the exchange field, and the substrate has a nonmetallic nature. We consider using the graphene/WSe$_2$ heterostructure to produce spin-active elements. We propose devices built of quantum rings with attached leads that can be used as a spin filter or spin inverter. The spin-orbit (SO) interaction for a graphene/TMDC structure implies a perpendicular component of the effective magnetic field of opposite orientation for both valleys [@Cummings2017]. The perpendicular component introduces spin splitting of energy bands provided that the valley degeneracy is lifted and intervalley scattering is weak. The valley degeneracy is lifted by external magnetic field in closed systems, including quantum rings, dots, and antidots [@Thomsen2017; @Grujic2011; @Recher2007]. The lifted valley degeneracy with the valley-dependent spin-orbit field leads to the polarization of the spin states. Moreover, the in-plane component of the SO effective magnetic field is considered for the spin inversion. In this work we study a quantum dot and quantum ring as spin-active elements for the spin currents fed by graphene nanoribbons. We calculate the spectrum of an isolated system and the transport properties of an open system. For the transport calculations we consider a ring with semi-infinite leads attached (Fig. \[systemWSe\]).
Theory
======
Dirac equation
--------------
We focus on the electronic properties near the Dirac point. The low-energy Hamiltonian for graphene on TMDCs is [@Gmitra2016] $$\begin{aligned}
\begin{aligned}
H =& H_{orb}+H_\Delta+ H_{SO} ,\\
H_{SO}=&H_{I}+H_{R}+H_{PIA} ,
\label{eq:hamLow}
\end{aligned}\end{aligned}$$ with $$\begin{aligned}
H_{orb} =& \hbar v_F ( \kappa \sigma_x k_x + \sigma_y k_y ), \\
H_\Delta =& \Delta \sigma_z,
\label{eq:hamLow2}
\end{aligned}$$ and the SOC terms $$\begin{aligned}
H_{I} =& \frac{1}{2} \left[ \lambda_I^{A}(\sigma_z+\sigma_0) + \lambda_I^{B}(\sigma_z-\sigma_0) \right]\kappa s_z, \\
H_{R} =& \lambda_R ( \kappa\sigma_x s_y-\sigma_y s_x ), \\
H_{PIA}=& \frac{a}{2} \left[ \lambda_{PIA}^A (\sigma_z+\sigma_0) \right.+\\
& +\left. \lambda_{PIA}^B(\sigma_z-\sigma_0) \right] ( k_x s_y - k_y s_x ),
\end{aligned}
\label{eq:hamLow3}$$ where $\kappa=1(-1)$ for the $K$ ($K'$) valley, $\sigma_i$ are the sublattice Pauli matrices, $s_i$ are the spin Pauli matrices, $k_i$ are the components of the wave vector with respect to the $K$ or $K'$ valley, $v_F$ is the Fermi velocity, and $a=0.246$ nm is the graphene lattice constant. The $H_{orb}$ term describes freestanding graphene, and $H_\Delta$ is the staggered potential induced by the TMDC substrate giving rise to an energy gap. $H_I$ describes the intrinsic SOC, $H_R$ is the Rashba SOC, and $H_{PIA}$ is the pseudoinversion asymmetry [@Gmitra2016].
As shown by Cummings [*et al.*]{} [@Cummings2017], graphene coupled to a TMDC acquires a spin texture. The effective SO field $\hbar\boldsymbol{\omega}$ can be described by the Hamiltonian written in the basis of the eigenstates of $H_{orb}$, $$\begin{aligned}
H_{SO} &= \frac{1}{2} \hbar \boldsymbol{\omega} \mathbf{\hat s},\\
\hbar \omega_x &= -2(ak \Delta_{PIA} \pm \lambda_R) \sin(\theta), \\
\hbar \omega_y &= 2(ak \Delta_{PIA} \pm \lambda_R) \cos(\theta), \\
\hbar \omega_z &= 2 \kappa \lambda_{VZ},
\end{aligned}
\label{eq:SOfield}$$ where $\theta$ is the direction of $k$ relative to $k_x$, $\boldsymbol{\omega}$ is the spin precession frequency, $\Delta_{PIA} = \tfrac{1}{2}( \lambda_{PIA}^{A}-\lambda_{PIA}^{B} )$, $\lambda_{VZ} = \tfrac{1}{2}( \lambda_I^{A}-\lambda_I^{B} )$ parametrizes the valley Zeeman SOC, and the $+(-)$ sign corresponds to the conductance (valence) band. The $\Delta_{PIA}$ and $\lambda_R$ terms contribute to the effective field in the plane of graphene and are both perpendicular to the $k$ vector, while the $\lambda_{VZ}$ term gives an out-of-plane component, which is opposite for the $K$ and $K'$ valleys.
Confined states
---------------
For the discussion of the low-energy spectrum of graphene systems we solve the Dirac equation only with the dominant (spin diagonal) terms. In particular, we focus on the intrinsic SOC, which is responsible for the out-of-plane effective field that leads to the spin polarization in the $z$ direction \[see Eq. (\[eq:hamLow3\])\]: $$\begin{aligned}
\nonumber &H' = H_{orb}+H_\Delta +H_{I} .
\label{eq:hamLowKM}
\end{aligned}$$ The $2\times 2$ Hamiltonian for the valley with the index $\kappa$ takes the form $$\begin{aligned}
H_\kappa = \left( \begin{array}{c c}
\lambda_I^A \kappa s_z +\Delta & \hbar v_F( \kappa k_x -i k_y ) \\ % \otimes \mathbb{I}
\hbar v_F( \kappa k_x +i k_y ) & -\lambda_I^B \kappa s_z -\Delta \\
\end{array}\right),
% \left( \begin{array}{c}
% \psi_1(\mathbf{r}) \\
% \psi_2(\mathbf{r})
% \end{array}\right)
%= \varepsilon
% \left( \begin{array}{c}
% \psi_1(\mathbf{r}) \\
% \psi_2(\mathbf{r})
%\end{array}\right),
\label{eq:h2x2}
\end{aligned}$$ where $\mathbf{k}=-i\mathbf{\nabla} + \tfrac{e}{\hbar} \mathbf{A}$, with $\mathbf{A}$ being the vector potential. We use $\mathbf{A}=(-\tfrac{By}{2},\tfrac{Bx}{2},0)$. This Hamiltonian acts on a two-component wave function , where $\psi_1(\mathbf{r})$ and $\psi_2(\mathbf{r})$ describe the A and B sublattices, respectively. For a circularly symmetric system, these are also eigenstates of the total angular momentum operator $J_z = L_z +\hbar \sigma_z/2$; thus one can write the wave function as $$\begin{aligned}
\psi(r,\phi) = e^{im\phi}
\left( \begin{array}{c}
\chi_1(r) \\
e^{i\kappa\phi} \chi_2(r)
\end{array}\right),
\label{eq:wf}
\end{aligned}$$ where $m=0,\pm 1,\pm 2,...$ is the total angular momentum quantum number and $\chi_1$ and $\chi_2$ are radial wave functions corresponding to sublattices A and B, respectively. These satisfy the eigenequation
$$\begin{aligned}
E \chi_1 =& ( \lambda_I^A \kappa \hat s_z + \Delta ) \chi_1
-i \hbar v_F \left(\kappa \partial_r + \frac{(m+\kappa)}{\rho}+ \frac{Be}{2\hbar} r \right)\chi_2, \\
E \chi_2 =& -i\hbar v_F \left(\kappa\partial_r - \frac{ m}{r}-\frac{Be}{2\hbar} r \right)\chi_1%(r)
-( \lambda_I^B \kappa \hat s_z + \Delta ) \chi_2 , %(r)
\label{eq:eqDirak}
\end{aligned}$$
which we solve with the finite-difference method.
![The schematic of the considered system: quantum ring etched out of graphene on WSe$_2$. Two leads are attached to the ring, where electrons enter through the left lead and exit through the right one. []{data-label="systemWSe"}](quarin-tmdc.png){width="\columnwidth"}
Tight-binding approximation
---------------------------
For the atomistic modeling, we use the tight-binding Hamiltonian [@Gmitra2016]: $$\begin{aligned}
\nonumber & H= \sum\limits_{\langle i,j\rangle,s } t c_{i s}^\dagger c_{j s}+\sum\limits_{i,s} \Delta \xi_{c_i} c_{i s}^\dagger c_{is} \\
&+ \frac{2i}{3} \sum\limits_{\langle i,j\rangle } \sum\limits_{s,s'} c_{i s}^\dagger c_{j s'} \left[\lambda_R ( \mathbf{\hat s} \times\mathbf{d}_{ij} )_z \right]_{s,s'} \\
\nonumber &+ \frac{i}{3} \sum\limits_{\langle\langle i,j\rangle\rangle } \sum\limits_{s,s'} c_{is}^\dagger c_{j s'} \left[ \frac{\lambda_I^{c_i}}{\sqrt{3}} \nu_{ij} \hat s_z
+ 2\lambda_{PIA}^{c_i} ( \mathbf{\hat s} \times\mathbf{D}_{ij} )_z \right]_{s,s'},
\label{eq:dh}\end{aligned}$$ where $c_{i s}^\dagger = (a_i^\dagger,b_i^{\dagger} )$ and $c_{is} = (a_i,b_i )$ are the creation and annihilation operators for an electron with spin $s$ in sublattice-A or -B site $i$. The summation $\langle i,j\rangle$ runs over the first-nearest neighbors, and $\langle\langle i,j\rangle\rangle$ runs over the second-nearest neighbors. In the first sum $t$ is the first-nearest-neighbor hopping parameter, and the second sum describes the staggered on-site potential with effective energy difference $\Delta$ with $\xi_{a_i}=1$ ($\xi_{b_i}=-1$) on the A (B) sublattice. The following terms describe the spin-orbit coupling: the Rashba SOC parametrized by $\lambda_R$, the intrinsic SOC term with the lattice-resolved parameter $\lambda_I^{c_i}=\lambda_I^{A(B)}$, and the pseudospin-inversion asymmetry (PIA) with $\lambda_{PIA}^{c_i}=\lambda_{PIA}^{A(B)}$ for $c_i$ in sublattice A(B). $\mathbf{d}_{ij}$ are the unit vectors from site $j$ to $i$ for the nearest neighbors, and $\mathbf{D}_{ij}$ are those for the next-nearest neighbors. $\mathbf{\hat s}$ is the vector of Pauli matrices acting on the spin state, and $\nu_{ij}=+ 1 (-1)$ for the clockwise (anticlockwise) path between sites $j$ and $i$.
We use the tight-binding parametrization of Ref. for graphene coupled to WSe$_2$. We introduce the magnetic field by the Peierls phase: a general hopping parameter described by $H=\sum_{i,j,s,s' } h_{i s j s'} c_{is}^\dagger c_{j s'} $ is modified by $h_{i s js'}\rightarrow h_{is js'}e^{\phi_{ij}}$, where .
### Transport calculation
![The schematic of the quantum ring with two leads attached with a (a) a zigzag edge and (b) an armchair edge. The zigzag leads are oriented along the $x$ axis, while the armchair leads are oriented along the $y$ axis. The ring inner (outer) radius is $R_1$ ($R_2$). []{data-label="modelTransp"}](RingLeadsZig.pdf "fig:"){width="0.55\columnwidth"} ![The schematic of the quantum ring with two leads attached with a (a) a zigzag edge and (b) an armchair edge. The zigzag leads are oriented along the $x$ axis, while the armchair leads are oriented along the $y$ axis. The ring inner (outer) radius is $R_1$ ($R_2$). []{data-label="modelTransp"}](RingLeadsArm.pdf "fig:"){width="0.42\columnwidth"}
We perform the transport calculations in the tight-binding formalism. The considered system is shown in Fig. \[modelTransp\]. It consists of a quantum ring of inner radius $R_1=7.3$ nm and outer radius $R_2=25$ nm, centered at $(x_0,y_0)=(61.5,25)$ nm. The quantum ring has two leads attached for the incoming and outgoing electrons. The leads are in the form of narrow ribbons with a zigzag or armchair edge with a width of 17.7 nm, which corresponds to 84 (71) atoms across the zigzag (armchair) ribbon. The edge of the armchair ribbon induces strong intervalley scattering which is absent for the zigzag ribbon. We maintain the same quantum ring orientation but attach the leads at different angles. The zigzag leads are oriented along the $x$ axis, while the armchair leads are oriented along the $y$ axis (see Fig. \[modelTransp\]). We use the gauge appropriate for each terminal using the approach from Ref. . We take $\mathbf{A_0}=(By,0,0)$ and apply the transformation $\mathbf{A}=\mathbf{A_0}+\mathbf{\nabla}[f(x,y)m(x,y)]$, with $f(x,y)=-xyB$, and $m(x,y)$ being a smooth steplike function that is 0 for -5 nm$<y<$55 nm and 1 elsewhere.
For the evaluation of the transmission probability, we use the wave-function-matching (WFM) technique [@Kolacha]. The spin direction of the $m$th mode with wave function $\psi^m$ is determined by the quantum expectation values of the Pauli matrices $\langle \mathbf{\hat s} \rangle = \langle \psi^m | \mathbf{\hat s} | \psi^m \rangle$. We label the positive (negative) spin $\langle \mathbf{\hat s} \rangle$ by $s= \uparrow$ ($\downarrow$). The transmission probability from the input lead to mode $m$ with spin direction $s$ in the output lead is $$\begin{aligned}
T^m_s = \sum_{ n,s' } |t^{mn}_{s,s'}|^2,
\label{eq:transprob}
\end{aligned}$$ where $t^{mn}_{s,s'}$ is the probability amplitude for the transmission from mode $n$ with spin $s'$ in the input lead to mode $m$ with spin direction $s$ in the output lead. The summed transmission to spin $s$ is $$T_s = \sum_{ m } T^m_s.
\label{eq:transprob_spin}$$ We evaluate the summed conductance as $G={G_0}\sum_{s} T_s$, with $G_0={e^2}/{h}$. We consider the spin-conserving $G_{ss}$ and spin-flipping $G_{ss'}$ components, given respectively by $G_{ss} = G_0 \sum_{ n,m } |t^{mn}_{s,s}|^2$ and $G_{ss'} = G_0\sum_{ n,m } |t^{mn}_{s,s'}|^2$. For discussion of the spin filtering we use the spin polarization defined as $$P = \frac{ G_{\uparrow\uparrow}+G_{\downarrow\uparrow} - (G_{\downarrow \downarrow}+G_{\uparrow\downarrow}) }{G}.
\label{eq:polarization}$$ For the system filtering out the spin-up (spin-down) electrons this gives $P=-1$ ($P=1$). The transport properties below are discussed within the energy range in which only subbands with opposite spin appear at the Fermi level. The orientation of the spin depends on the type of the ribbon feeding the current to the lead and on the external magnetic field. The $\uparrow$ and $\downarrow$ in formula (\[eq:polarization\]) and below in the discussion stand for the orthogonal spin eigenstates which depend on the case.
Results
=======
![The schematic of the (a) dot and (b) ring induced by a staggered potential in the hexagonal graphene flake with side length $d$. In the shaded area the staggered potential is zero, and outside of it the on-site potential is given by Eq. (\[eq:deltaDot\]) for the dot and (\[eq:deltaRing\]) for the ring. The dot radius is $R_2$, and the ring inner (outer) radius is $R_1$ ($R_2$). []{data-label="schemeIMBC"}](IMBC.pdf){width="0.95\columnwidth"}
Effective mass-induced closed quantum ring
------------------------------------------
{width="46.00000%"} {width="46.00000%"} {width="46.00000%"} {width="46.00000%"}
As a proof of concept for the spin filtering by quantum dots and rings in external magnetic field, we focus on the low-energy properties of graphene systems with proximity-induced spin-orbit coupling. For this purpose, we calculate the spectrum of Hamiltonian (\[eq:hamLowKM\]) in the continuum approximation for a quantum dot and ring defined by infinite-mass boundary conditions [@Berry53]. The considered dot has a radius of $R_2=25$ nm, and the ring has an outer radius $R_2=25$ nm and inner radius $R_1=7.3$ nm.
For comparison, we calculate the tight-binding spectrum of an analogous system presented schematically in Fig. \[schemeIMBC\], where the dot or ring is defined by a staggered potential $\delta( r )$ that introduces a mass term: $$H= H_{orb}+H_\Delta+H_I
+ \sum\limits_{i,s} \delta( r) \xi_{c_i} c_{is}^\dagger c_{is}.
\label{eq:dhKM}$$ For the confined states we consider only the intrinsic SOC, and the radial potential $\delta( r )$ for the dot is $$\begin{aligned}
\delta(r) = \left\{ \begin{array}{c c}
V_g, & r>R_2,\\
0, & r<R_2,
\end{array}\right.
\label{eq:deltaDot}
\end{aligned}$$ and for the ring it is $$\begin{aligned}
\delta(r) = \left\{ \begin{array}{c c}
V_g, & r>R_2,\\
0, & R_1<r<R_2,\\
V_g, & r<R_1,
\end{array}\right.
\label{eq:deltaRing}
\end{aligned}$$ with $V_g=5 $ eV. The staggered-potential-defined system, dot or ring is defined in a hexagonal graphene flake with a side length $d=$30.5 nm (see Fig. \[schemeIMBC\]). The results for the continuum and tight-binding approximations are presented in Fig. \[eigenDiracTB\]. In Figs. \[eigenDiracTB\](a) and \[eigenDiracTB\](c) for pristine graphene without SOC in the continuum approximation, one can determine the valley to which the levels belong. At $B=0$ the energy levels are valley degenerate \[see orange (blue) lines for $K$ $(K')$ valley in Figs. \[eigenDiracTB\](a) and \[eigenDiracTB\](c)\] and in finite magnetic field the levels split for different valleys. The energy levels obtained in the continuum approximation and tight-binding approach agree, especially for a low energy range and high magnetic field.
In Figs. \[eigenDiracTB\](b) and \[eigenDiracTB\](d) the spectrum for systems with intrinsic SOC is shown. The levels are no longer spin degenerate in both the continuum and tight-binding approximations. For levels of the $K'$ valley spin-down states are lower in energy than spin-up states due to the out-of-plane valley-Zeeman SOC field. For the $K$ valley this field has the opposite sign; thus the spin-up states have lower energy. The energy splitting of the levels due to intrinsic SOC for WSe$_2$ is $E_{VZ} = 2\hbar \omega_{z} = 2.38$ meV.
In the systems confined by the position-dependent mass the intervalley scattering is absent, and the valley degeneracy is lifted by a finite magnetic field. The effective SO magnetic field $\hbar \omega_z$ is activated in Eq. (\[eq:SOfield\]), and due to the $\kappa$ term, it is opposite for both valleys. However, in systems that contain sections of armchair edges, mixing of both valley states occurs, which leads to a reduction of the effective magnetic field and the resulting spin splitting (see the next section).
{width="48.50000%"} {width="50.00000%"}
{width="52.50000%"} {width="46.00000%"}
Closed quantum ring with etched edges
--------------------------------------
We now turn our attention to etched systems. In order to reduce the numerical cost of the tight-binding calculations, we focus on quantum rings, which have a smaller number of atomic sites than the quantum dots. However, quantum rings and dots will have similar spin-splitting properties, the dots having the advantage of a smaller disorder caused by etching.
Here the simple continuum model with finite-mass confinement does not apply since the edge contains short zigzag and armchair sections. Due to the zigzag edge, states close to zero energy can occur, in comparison with the results with infinite-mass boundary conditions, which generate no zero-energy states.
The ring has inner radius $R_1=7.3$ nm and outer radius $R_2=25$ nm. In Fig. \[eigenring\] we present the spectra of the ring as a function of magnetic flux through one carbon hexagon $\phi/ \phi_0$ (with $\phi_0=h/e$ and $\phi=3\sqrt{3}a_{CC}^2 B$, with $a_{CC}=0.142$ nm) in suspended graphene \[Fig. \[eigenring\](a)\] and graphene deposited on WSe$_2$ \[Fig. \[eigenring\](b)\]. Figure \[eigenringLow\] shows the low-energy zoom of the spectra.
The spectrum is different from the one obtained with the finite-mass confinement. We note that in contrast to the finite-mass quantum dots and rings \[Figs. \[eigenDiracTB\] (a) and \[eigenDiracTB\](c)\], for low magnetic field the spectrum of the etched system contains energy levels close to zero (Fig. \[eigenring\]). Such states were shown in Ref. to be localized in the zigzag segments of the quantum dot edges. The staggered potential produces armchair and zigzag boundaries, but they do not act as a physical edge and do not support the zero-energy levels. Moreover, it was shown in Ref. that the spectrum in the etched systems strongly depends on the details of the edge structure, whereas in the finite-mass-induced systems it is less sensitive to the imperfections of the circular shape.
For high energies the spectra in the ring of suspended graphene and graphene on WSe$_2$ are very close to each other. The spectrum has two series of periodic levels, with energies decreasing (growing) with magnetic field, which correspond to the states associated with a clockwise (anticlockwise) current in the ring and are localized near the outer (inner) edge of the ring [@Bahamon2009; @Poniedzialek] and having angular momentum parallel (antiparallel) to the external magnetic field [@Poniedzialek; @MrencaLorentz]. In both series of states the spin is parallel to the $z$ direction.The width of those states is of the order of 10 nm at $\phi=0.0008 \phi_0$ and decreases with growing magnetic field.
For low energy (below approximately 0.01 eV) the spectra start to differ. Various effects can be seen that result from different SOC terms: $\lambda_{I}^{A,B}$ causes the spin polarization out of plane, and splits the levels of opposite spin in energy. On the other hand, due to the terms dependent on $\lambda_R$ and $\lambda_{PIA}^{A,B}$ an in-plane spin component arises. This is especially pronounced at low energy. Figure \[eigenringLow\] shows the low-energy zoom of the spectrum in Fig. \[eigenring\]. The eigenstates have spin predominantly in the $z$ direction \[Fig. \[eigenringLow\] (a)\], with the exception of the states with energy weakly dependent on the magnetic field close to the Dirac point. In the former case the spin is in the $x-y$ plane \[Fig. \[eigenringLow\](b)\], which suggests that those states are governed by the Rashba-like SOC terms. These states are mostly localized on the segments of the ring that contain the Klein edge [@KleinEdge; @He].
The states that carry clockwise or anticlockwise current, also for low energy, have spin almost perfectly polarized in the $z$ direction. The opposite spin levels are split in energy due to the $\lambda_{VZ}$ term in Eq. (\[eq:SOfield\]). The splitting of the order of 1.14 meV can be seen in Fig. \[eigenringLow\], as highlighted by the black arrow, lower than the maximum of $2.38$ meV because the ring contains short armchair segments which lead to intervalley scattering and, as a consequence, to partial cancellation of the energy splittings of the two valleys.
Magnetotransport of the quantum ring
--------------------------------------
### Zeeman splitting neglected
In this section we present the results of the transport in a quantum ring the same size as in the previous section with the leads attached. The spin transport depends on the properties of the ring and the leads. From formula (\[eq:SOfield\]) it is evident that the $z$ component of the SOC field is opposite for the two valleys. It affects the spin direction depending on the edge type of the graphene system. An armchair edge leads to the intervalley scattering, whereas for a zigzag edge the valleys are well defined [@pcc; @Wurm2012]. Figures \[dispRibbon\] and \[dispRibbonArm\] show band structures of zigzag and armchair nanoribbons of graphene on WSe$_2$, respectively. For the zigzag nanoribbon (Fig. \[dispRibbon\]) in the dispersion relation the $K$ and $K'$ valleys are around $k_x=\pm2\pi/3a$. The bands have spin aligned almost perfectly in the $z$ direction \[Figs. \[dispRibbon\](a) and \[dispRibbon\](b)\], with the spin direction in the lowest subband being opposite for the two valleys \[Fig. \[dispRibbon\](c)\]. This is still the case in finite external magnetic field \[Fig. \[dispRibbon\](d)\].
The armchair edge, on the other hand, mixes valleys, and in zero external field the contributions of the spin field for the $K$ and $K'$ valleys cancel out. Thus the spin is polarized in the nanoribbon plane (Fig. \[dispRibbonArm\]), perpendicular to the direction of motion. In Figs. \[dispRibbonArm\](a) and \[dispRibbonArm\](b) the bands are clearly polarized in the $x$ direction (for the nanoribbon aligned along the $y$ axis). Only in finite external magnetic field does the spin get tilted out of plane \[Figs. \[dispRibbonArm\](c) and \[dispRibbonArm\](d)\].
Figure \[transpBoth\] shows the summed conductance \[Figs. \[transpBoth\](a) and \[transpBoth\](b)\] and the spin-flipping conductance \[Figs. \[transpBoth\](c) and \[transpBoth\](d)\] as a function of magnetic field and Fermi energy for the system with zigzag and armchair leads. The spin inversion is highest close to the Dirac point and corresponds to the ring eigenstates that have spin aligned in the graphene plane \[see Fig. \[eigenringLow\](b)\]. This is most pronounced in the system with zigzag leads in which the transport gap is smaller and the transport is mediated by the lowest-lying states with spin almost entirely in the plane \[see Fig. \[transpRibbonZoom\](a) for the low-energy zoom\]. The incoming states have out-of-plane spin and can flip via those states with the in-plane spin. The spin inversion probability for higher energy is generally low \[see Fig. \[transpBoth\](c) and \[transpBoth\](d)\]. The exception is the narrow resonances with energy increasing with magnetic field, indicated by black arrows in Figs. \[transpBoth\](c) and \[transpBoth\](d). These resonances correspond to the quantum ring states with current circulating around the inner edge of the ring [@MrencaLorentz]. Such states have a long lifetime, and the electrons remain a long time in the system, taking many turns around the ring [@MrencaLorentz]. The SO in-plane effective magnetic field for the long-living resonances eventually leads to the spin flips of the Fermi level electron. On the other hand, the resonances that circulate close to the outer edge of the ring have a short lifetime because the magnetic field steers the current out of the ring to the right lead. The cumulated phase is not large enough for the spin flip to occur.
![The band structure of the zigzag nanoribbon with WSe$_2$ for (a)-(c) $\phi=0$ and (d) $\phi=0.0005\phi_0$. The color scale shows (a) the mean spin $y$ or (b)-(d) $z$ component. []{data-label="dispRibbon"}](dysp0.pdf "fig:"){width="0.47\columnwidth"} ![The band structure of the zigzag nanoribbon with WSe$_2$ for (a)-(c) $\phi=0$ and (d) $\phi=0.0005\phi_0$. The color scale shows (a) the mean spin $y$ or (b)-(d) $z$ component. []{data-label="dispRibbon"}](dysp0z.pdf "fig:"){width="0.4\columnwidth"} ![The band structure of the zigzag nanoribbon with WSe$_2$ for (a)-(c) $\phi=0$ and (d) $\phi=0.0005\phi_0$. The color scale shows (a) the mean spin $y$ or (b)-(d) $z$ component. []{data-label="dispRibbon"}](dysp0zzoom.pdf "fig:"){width="0.47\columnwidth"} ![The band structure of the zigzag nanoribbon with WSe$_2$ for (a)-(c) $\phi=0$ and (d) $\phi=0.0005\phi_0$. The color scale shows (a) the mean spin $y$ or (b)-(d) $z$ component. []{data-label="dispRibbon"}](dysp40zzoom.pdf "fig:"){width="0.4\columnwidth"}
![The band structure of the armchair nanoribbon with WSe$_2$ for (a) and (b) $\phi=0$ and (c) and (d) $\phi=0.0005 \phi_0$. []{data-label="dispRibbonArm"}](dysp0_arm.pdf "fig:"){width="0.5\columnwidth"} ![The band structure of the armchair nanoribbon with WSe$_2$ for (a) and (b) $\phi=0$ and (c) and (d) $\phi=0.0005 \phi_0$. []{data-label="dispRibbonArm"}](dysp0zoom_arm.pdf "fig:"){width="0.48\columnwidth"} ![The band structure of the armchair nanoribbon with WSe$_2$ for (a) and (b) $\phi=0$ and (c) and (d) $\phi=0.0005 \phi_0$. []{data-label="dispRibbonArm"}](dysp40zoom_arm.pdf "fig:"){width="0.5\columnwidth"} ![The band structure of the armchair nanoribbon with WSe$_2$ for (a) and (b) $\phi=0$ and (c) and (d) $\phi=0.0005 \phi_0$. []{data-label="dispRibbonArm"}](dysp40zzoom_arm.pdf "fig:"){width="0.48\columnwidth"}
In addition to the spin flip, we expect that the system can be used for spin filtering. In Fig. \[transpRibbonZoom\](c) the low-energy zoom of the spin polarization in a quantum ring with zigzag leads is shown. At energy below approximately $1.7$meV the modes carrying spin-up states are almost entirely blocked. This is the result of the lack, in this low energy range, of spin-up states supporting clockwise or anticlockwise current. The system exhibits a range of magnetic field in which the current is spin polarized. The energy window of $P\approx -1$ is indicated by the black arrow in Fig. \[transpRibbonZoom\](c).
On the other hand, for armchair leads the filtering is more challenging because the armchair lead acquires an energy gap that at high magnetic field saturates at about 3 meV \[see Figs. \[transpRibbonZoom\](b) and \[transpRibbonZoom\](d)\], already above the onset of the ring spin-up levels that carry current around the ring. In Fig. \[transpRibbonZoom\](d) the low-energy zoom of the spin polarization in a quantum ring with armchair leads is shown. In high magnetic field, the spin subband splitting is only about 0.7 meV. The spin polarization is shown with only a narrow range of $P\approx -1$. The system with armchair leads would require a more accurate tuning of the parameters for the spin filtering.
In Figs. \[transpRibbonZoom\](a) and \[transpRibbonZoom\](b) the zoomed spin-flip probability is shown. The transition probability between modes with opposite spin directions is highest close to the narrow resonant states, especially for the zigzag leads \[Fig. \[transpRibbonZoom\](a)\]; however, obtaining such a high inversion probability would require fine tuning of the back-gate potential and magnetic field. On the other hand, the energy range where the spin filtering occurs is much broader. Therefore we conclude that the quantum rings are more suitable for spin filtering than for spin inversion.
![The band structure of zigzag nanoribbon with WSe$_2$ for $\phi=0.00025 \phi_0$. The color scale shows the mean (a) spin $y$ component and (b) spin $z$ component. []{data-label="zigZeemRel"}](dysp20.pdf){width="0.49\columnwidth"}
### Transport with the Zeeman effect
So far we have considered the transport without Zeeman splitting in order to understand the pure effect of SOC. For completeness, we study the influence of the Zeeman effect on the transport properties of the considered systems. For magnetic field used in experiment, in the range of a few teslas, the Zeeman splitting is of the order $2\times \tfrac{1}{2}g\mu_B B=0.06-0.6$ meV for $B=1-10$ T, with $\mu_B$ being the Bohr magneton, and $g=2$. This is small compared to the maximum SOC-induced splitting of around $2.4$ meV for graphene contacted with WSe$_2$, but both effects add up, and the splitting can reach, for example, 3 meV for $B=10$ T. The Zeeman splitting influences the spin inversion characteristics, as we present below. Figure \[transpBothZeem\] shows the summed conductance and the spin-flipping components of conductance as a function of the Fermi energy of the incident electrons and the external magnetic field. In the system with zigzag leads a vertical strip of higher spin flip emerges \[Fig. \[transpBothZeem\](c)\]. This is when the Zeeman energy coincides with the opposite-spin subbands and the subbands have spin in the plane (see Fig. \[zigZeemRel\]) around the value $\phi= 0.00025 \phi_0$. Second, the spin flip in the clockwise-current resonances disappear because the Zeeman splitting separates the resonant states for both spins. Without the overlap of these states no spin transfer can occur. Along the ring an effective SO field occurs, shown schematically in the inset of Fig. \[transpBoth\](a). In high magnetic field the spin of the incoming electrons from both types of leads is oriented more in the $z$ direction, close to the precession axis \[see the inset of Figs. \[transpBothZeem\](a) and \[transpBothZeem\](b)\]; therefore no spin-flip occurs. For zigzag ribbons \[Figs. \[transpBothZeem\](a) and \[transpBothZeem\](c)\] the effective magnetic field due to the SOC superposes with the Zeeman effect. For the armchair ribbons the incident spins at low magnetic field are deflected to the in-plane orientation by the Rashba and PIA SOC \[see the inset of Fig. \[transpBothZeem\](b)\], and the variation of the spin within the ring appears via precession in the effective intrinsic SO magnetic field oriented in the $z$ direction, which is missing in the leads due to the intervalley scattering. At higher magnetic field the spin-flipping transport disappears when the Zeeman interaction dominates over the effective SO interaction.
Summary and Conclusions
=======================
We considered the application of a graphene-TMDC heterostructure for building spin-active elements. We studied the properties of quantum rings produced by graphene in contact with WSe$_2$. The induced valley Zeeman SO coupling leads to energy splitting of the ring levels of opposite spin. In magnetic field the system has spin-filtering properties when tuned to the Fermi energy between the split levels. For this purpose the zigzag leads are especially promising because the zigzag edge does not mix the $K$ and $K'$ valleys. Quantum rings can also be used as a spin-inverting element for building a spin transistor; however, for the complete spin inversion high precision of the electron energy or external magnetic field would be required.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the National Science Centre (NCN) according to decision DEC-2015/17/B/ST3/01161 and by AGH UST budget with the subsidy of the Ministry of Science and Higher Education, Poland with Grant No. 15.11.220.718/6 for young researchers and Statutory Task No. 11.11.220.01/2. The calculations were performed on PL-Grid Infrastructure.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The geometric formulation of quantum mechanics is a very interesting field of research which has many applications in the emerging field of quantum computation and quantum information, such as schemes for optimal quantum computers. In this work we discuss a geometric formulation of mixed quantum states represented by density operators. Our formulation is based on principal fiber bundles and purifications of quantum states. In our construction, the Riemannian metric and symplectic form on the total space are induced from the real and imaginary parts of the Hilbert-Schmidt Hermitian inner product, and we define a mechanical connection in terms of a locked inertia tensor and moment map. We also discuss some applications of our geometric framework.'
address: 'Department of Physics, Stockholm University, SE-106 91 Stockholm, Sweden'
author:
- Ole Andersson and Hoshang Heydari
title: Geometry of quantum evolution for mixed quantum states
---
Introduction
============
Ever since the advent of general relativity, scientists have been looking for geometrical principles underlying physical laws. Nowadays it is well known that geometry affects the physics on all length scales, and physical theory building consists to a large extent of geometrical considerations. This paper concerns geometric quantum mechanics, a branch of quantum physics that has received much attention lately – largely due to the crucial role geometry plays in quantum information and quantum computing. Here we equip the phase spaces for unitarily evolving finite level quantum systems with natural Riemannian and symplectic structures, and establish remarkable but fundamental relations between these and quantum theory. Important previous works on the geometrical formulation of quantum mechanics that should be mentioned in this context are the book [@Bengtsson_etal2008] by Bengtsson and Życzkowski and the papers [@Grabowski_etal2005] by J. Grabowski, M. Kús and G. Marmo and [@Ashtekar_etal1998] by Ashtekar and Schilling.
A quantum system prepared in a pure state is usually modeled on a projective Hilbert space, and if the system is closed its state will evolve unitarily in this space. The state of an experimentally prepared quantum system generally exhibits classical uncertainty, and is most appropriately described as a probabilistic mixture of pure states. It is common to represent mixed states by density operators, and many metrics on spaces of density operators have been developed to capture various physical, mathematical, or information theoretical aspects of quantum mechanics [@Bengtsson_etal2008; @Nielsen_etal2010].
In this paper we discuss a geometrical framework for general quantum states represented by density operators on finite dimensional quantum systems. We show that our geometrical framework enable us to establish very important relations between abstract geometrical structures and general quantum systems which gives breakthrough insight in our understanding of the foundations of quantum mechanics and quantum information with many applications. In section \[GfW\] we give an introduction to our geometric framework for mixed quantum states, and in section \[applications\] we discuss some recent applications of the framework such as operational geometric phases [@GP], a geometric uncertainty relation [@GUR], and a dynamic distance measure [@DD]. This paper is based on [@GP; @GUR; @DD; @ED; @GS].
Geometry of orbits of isospectral density operators {#GfW}
===================================================
In this paper we will only be interested in finite dimensional quantum systems that evolve unitarily. They will be modeled on a Hilbert space $\mathcal{H}$ of unspecified dimension $n$, and their states will be represented by density operators. Recall that a density operator is a Hermitian, nonnegative operator with unit trace. We write $\mathcal{D}(\mathcal{H})$ for the space of density operators on $\mathcal{H}$.
Riemannian structure on orbits of density operators
---------------------------------------------------
A density operator whose evolution is governed by a von Neumann equation remains in a single orbit of the left conjugation action of the unitary group of $\mathcal{H}$ on $\mathcal{D}(\mathcal{H})$. The orbits of this action are in one-to-one correspondence with the possible spectra for density operators on $\mathcal{H}$, where by the *spectrum* of a density operator of rank $k$ we mean the decreasing sequence $$\sigma=(p_1,p_2,\dots,p_k)
\label{spectrum}$$ of its, not necessarily distinct, *positive* eigenvalues. Throughout this paper we fix $\sigma$, and write $\mathcal{D}(\sigma)$ for the corresponding orbit.
To furnish $\mathcal{D}(\sigma)$ with a geometry let $\mathcal{L}(\mathbf{C}^k,\mathcal{H})$ be the space of linear maps from $\mathbf{C}^k$ to $\mathcal{H}$, $P(\sigma)$ be the diagonal $k\times k$ matrix that has $\sigma$ as its diagonal, set $$\mathcal{S}(\sigma)=\{\Psi\in\mathcal{L}(\mathbf{C}^k,\mathcal{H}):\Psi^\dagger \Psi=P(\sigma)\},$$ and define $$\pi:\mathcal{S}(\sigma)\to\mathcal{D}(\sigma),\quad \Psi\mapsto\Psi\Psi^\dagger.
\label{bundle}$$ Then $\pi$ is a principal fiber bundle with right acting gauge group $$\mathcal{U}(\sigma)
=\{U\in\mathcal{U}(k):UP(\sigma)=P(\sigma)U\},$$ whose Lie algebra is $$\mathbf{u}(\sigma)
=\{\xi\in\mathbf{u}(k):\xi P(\sigma)=P(\sigma)\xi\}.$$ We equip $\mathcal{L}(\mathbf{C}^k,\mathcal{H})$ with the Hilbert-Schmidt Hermitian product, and ${\mathcal{S}}(\sigma)$ with the Riemannian metric $G$ and the symplectic form $\Omega$ given by $2\hbar$ times the real and imaginary parts, respectively, of this product: $$G(X,Y)=\hbar\mathrm{Tr}(X^\dagger Y+Y^\dagger X), \qquad
\Omega(X,Y)=-i\hbar\mathrm{Tr}(X^\dagger Y-Y^\dagger X). $$ We also equip $\mathcal{D}(\sigma)$ with the unique metric $g$ that makes $\pi$ a Riemannian submersion.
Mechanical connection
---------------------
The *vertical* and *horizontal bundles* over $\mathcal{S}(\sigma)$ are the subbundles $\mathrm{V}\mathcal{S}(\sigma)=\mathrm{Ker} \pi_*$ and $\mathrm{H}\mathcal{S}(\sigma)=\mathrm{V}\mathcal{S}(\sigma)^\bot$ of the tangent bundle of $\mathcal{S}(\sigma)$. Here $\pi_*$ is the differential of $\pi$ and $^\bot$ denotes orthogonal complement with respect to $G$. Vectors in $\mathrm{V}\mathcal{S}(\sigma)$ and $\mathrm{H}\mathcal{S}(\sigma)$ are called vertical and horizontal, respectively, and a curve in $\mathcal{S}(\sigma)$ is called horizontal if its velocity vectors are horizontal. Recall that for every curve $\rho$ in $\mathcal{D}(\sigma)$ and every $\Psi_0$ in the fiber over $\rho(0)$ there is a unique horizontal lift of $\rho$ to $\mathcal{S}(\sigma)$ that extends from $\Psi_0$. This lift and $\rho$ have the same lengths because $\pi$ is a Riemannian submersion.
The infinitesimal generators of the gauge group action yield canonical isomorphisms between $\mathbf{u}(\sigma)$ and the fibers in $\mathrm{V}\mathcal{S}(\sigma)$: $$\label{eq:inf gen}
\mathbf{u}(\sigma)\ni\xi\mapsto \Psi\xi\in\mathrm{V}_\Psi\mathcal{S}(\sigma).$$ Furthermore, $\mathrm{H}\mathcal{S}(\sigma)$ is the kernel bundle of the gauge invariant *mechanical connection form* $\mathcal{A}_{\Psi}=\mathcal{I}_{\Psi}^{-1}J_{\Psi}$, where $\mathcal{I}_{\Psi}:\mathbf{u}(\sigma)\to \mathbf{u}(\sigma)^*$ and $J_{\Psi}:\mathrm{T}_{\Psi}{\mathcal{S}(\sigma)}\to \mathbf{u}(\sigma)^*$ are the *moment of inertia* and *moment map*, respectively, $$\mathcal{I}_{\Psi}\xi\cdot \eta=G(\Psi\xi,\Psi\eta),\qquad
J_{\Psi}(X)\cdot\xi=G(X,\Psi\xi).$$ The moment of inertia is of *constant bi-invariant type* since it is an adjoint-invariant form on $\mathbf{u}(\sigma)$ which is independent of $\Psi$ in $\mathcal{S}(\sigma)$. To be exact, $$\label{eq1}
\mathcal{I}_{\Psi}\xi\cdot \eta=\frac{1}{2}\mathrm{Tr}\left(\left(\xi^\dagger \eta+\eta^\dagger \xi\right)P(\sigma)\right).$$ Using equation (\[eq1\]) we can derive an explicit formula for the connection form. Indeed, if $m_1, m_2, \dots , m_l$ are the multiplicities of the different eigenvalues in $\sigma$, with $m_1$ being the multiplicity of the greatest eigenvalue, $m_2$ the multiplicity of the second greatest eigenvalue, etc., and if for $j=1,2,\dots,l$, $$E_j=\mathrm{diag}(0_{m_1},\dots,0_{m_{j-1}},1_{m_j},0_{m_{j+1}},\dots,0_{m_l}),$$ then $$\begin{aligned}
\mathcal{I}_\Psi(\sum_jE_j\Psi^\dagger XE_jP(\sigma)^{-1})\cdot\xi
&=&
\frac{1}{2}\mathrm{Tr}(\sum_jE_jX^\dagger\Psi E_j\xi-\xi E_j\Psi^\dagger XE_j)\\\nonumber&=&
\frac{1}{2}\mathrm{Tr}(X^\dagger\Psi\xi-\xi\Psi^\dagger X)\\\nonumber&=&
J_\Psi(X)\cdot\xi\end{aligned}$$ for every $X$ in $\mathrm{T}_\Psi\mathcal{S}(\sigma)$ and every $\xi$ in $\mathbf{u}(\sigma)$. Thus, $$\label{eq:explicit}
\mathcal{A}_\Psi(X)=\sum_jE_j\Psi^\dagger XE_jP(\sigma)^{-1}.$$ Observe that the orthogonal projection of $\mathrm{T}_\Psi\mathcal{S}(\sigma)$ onto $\mathrm{V}_\Psi\mathcal{S}(\sigma)$ is given by the connection form followed by the infinitesimal generator given by equation (\[eq:inf gen\]). Therefore, the *vertical* and *horizontal projections* of $X$ in $\mathrm{T}_\Psi\mathcal{S}(\sigma)$ are $X^\bot=\Psi\mathcal{A}_\Psi(X)$ and $X^{||}=X-\Psi\mathcal{A}_\Psi(X)$, respectively.
Applications of geometric framework for mixed quantum states {#applications}
============================================================
We have introduced a geometrical framework for mixed quantum states represented by density operators which has so far resulted in an operational geometric phase and higher order geometric phases, a geometric uncertainty relation, a dynamic distance measure, and an energy estimate and a classification of optimal Hamiltonians for mixed quantum states. In this section we will briefly discuss these applications of our framework.
Operational geometric phases for non-degenerate and degenerate mixed quantum states
-----------------------------------------------------------------------------------
Geometric phases are very important tools both in classical and quantum physics. Uhlmann [@Uhlmann_1976; @Uhlmann1986; @Uhlmann1989; @Uhlmann1991] was among the first to develop a theory for geometric phase for parallel transported mixed states. The theory is based on the concept of purification. Another approach to geometric phase for parallel transported non-degenerate mixed states, based on quantum interferometry, was proposed by Sjöqvist *et al.* [@Sjoqvist_etal2000]. This phase has been verified in several experiments, and according to Slater [@Slater2002] it generally yields different outcomes than that of Uhlmann.
Recently, we have introduced an operational geometric phase [@GP] for mixed quantum states based on spectral weighted traces of holonomies, and we have shown that it generalizes the interferometric definition of Sjöqvist *et al.* The operational geometric phase is a direct application of the Riemannian structure of our geometric framework. We also introduce higher order geometric phases for mixed quantum states. Our operational geometric phase applies to general unitary evolutions of nondegenerate and also degenerate mixed states. The operational geometric phase is defined by $$\gamma_{g}(\rho)=
\arg \mathrm{Tr}(\Psi^{\dagger}_{0}\Pi[\rho]\Psi_{0})\nonumber=
\arg \mathrm{Tr}(\Psi^{\dagger}_{\|}(0)\Psi_{||}(\tau)),$$ where $\Pi[\rho]\Psi_{0}=\Psi_{\|}(\tau)$ and $\Psi_{\|}$ is the horizontal lift of $\rho$ extending from $\Psi_{0}$: $$\label{horiz}
\Psi_{||}(t)=\Psi(t)\exp_{+}\left(-\int_{0}^{t}\mathcal{A}_\Psi(\dot \Psi)\, dt\right).$$ Here $\exp_{+}$ is the positive time-ordered exponential. For more details about the construction of the operational geometric phase and higher order geometric phases we refer the reader to our recent paper [@GP].
Dynamic distance measure
------------------------
Distance measures are very important tools in quantum information processing. Recently we have proposed a new distance measure for mixed quantum states that we call the dynamic distance measure. The dynamic distance measure is defined in terms of a measurable quantity, which make it very suitable for applications.
Let $\rho_0$ and $\rho_1$ be isospectral density operators and consider a von Neumann equation $$\label{von}
i\dot\rho=[H,\rho],\qquad \rho(t_0)=\rho_0,\quad\rho(t_1)=\rho_1.$$ We define $$\mathcal{D}(H,\rho_0,\rho_1)=\int_{t_0}^{t_1}
\sqrt{\mathrm{Tr}(H^2\rho)-\mathrm{Tr}(H\rho)^2}\mathrm{dt}$$ provided $H$ is such that a solution curve $\rho$ to (\[von\]) exists, and we define the dynamic distance between $\rho_0$ and $\rho_1$ by $$\label{distance}
\mathrm{Dist}(\rho_0,\rho_1)=\inf_{H}\mathcal{D}(H,\rho_0,\rho_1),$$ where the infimum is taken over all Hamiltonian $H$ for which the boundary value problem (\[von\]) is solvable. In [@DD] we have shown that (\[distance\]) is a proper distance measure between isospectral density operators. In fact, it is the distance function associated with the metric $g$. We have also compared our dynamic distance measure with the well-known Bures distance [@Bures1969; @Uhlmann1992; @Dittmann1993; @Dittmann1999], and it turns out that the dynamic measure is bounded from below, but is in general not equal to, the Bures distance. The reason is that Uhlmann’s definition of parallel transport is different from ours.
Conclusion
==========
Recently, we have used the framework presented in section \[GfW\] to derive a geometric uncertainty relation for observables acting on mixed quantum states. For pure states the uncertainty relation reduces to the geometric interpretation of the Robertson-Schrödinger uncertainty relation by Ashtekar and Schilling [@Ashtekar_etal1998]. But in general the two relations are not equivalent. This is due to the multiple dimensions of the gauge group for general mixed states. More information about our result, especially a comparison with the Robertson-Schrödinger uncertainty relation, can be found in [@GUR].
We have introduced a geometrical framework for general quantum states represented by density operators on finite dimensional quantum systems in mixed states that evolve unitarily. Our geometrical framework enable us to establish relation between geometrical structures and general quantum systems. This correspondence between geometry and quantum physics gives new insight in our understanding of the foundations of quantum mechanics and quantum information with many applications. We have shown that our geometric framework has already resulted in a new operational geometric phase and higher order geometric phases and a new dynamic distance measure. There are other applications of our framework that worth mentioning such as quantum speed limit and optimal quantum control for mixed quantum states. Unfortunately, there is no space left here to discuss these issues. Interested reader may see our paper [@ED] for further information and a detailed discussion of the subject. We believe our geometric framework could results in many other interesting applications in the field of quantum dynamics, quantum information, quantum computations, quantum control, and quantum optics.
References {#references .unnumbered}
==========
[10]{}
I. Bengtsson and K. Życzkowski. . Cambridge University Press, 2008.
Janusz Grabowski, Marek KuÅ, and Giuseppe Marmo. Geometry of quantum systems: density states and entanglement. , 38(47):10217, 2005.
A. Ashtekar and T. A. Schilling. Geometrical formulation of quantum mechanics. In Alex Harvey, editor, [*On Einstein’s Path*]{}, pages 23–65. Springer-Verlag, 1998.
M. A. Nielsen and I. L. Chuang. . Cambridge University Press, Cambridge, 2010.
O Andersson and H Heydari. Operational geometric phase for mixed quantum states. , 15(5):053006, 2013.
O. Andersson and H. Heydari. Geometric uncertainty relation for mixed quantum states. , 2013.
Ole Andersson and Hoshang Heydari. Dynamic distance measure on spaces of isospectral mixed quantum states. , 15(9):3688–3697, 2013.
O. Andersson and H. Heydari. Geometry of quantum dynamics and a time-energy uncertainty relation for mixed states. , 2013.
O Andersson and H Heydari. Geometrical structures of quantum phase space of mixed quantum states. , 2013.
A. Uhlmann. The transition probability in the state space of a \*-algebra. , 9(2):273 – 279, 1976.
A. Uhlmann. Parallel transport and “quantum holonomy” along density operator. , 74(2):229–240, 1986.
A. Uhlmann. On berry phases along mixtures of states. , 501(1):63–69, 1989.
A. Uhlmann. A gauge field governing parallel transport along mixed states. , 21(3):229–236, 1991.
E. Sjöqvist, A. K. Pati, A. Ekert, J. S. Anandan, M. Ericsson, D. K. L. Oi, and V. Vedral. Geometric phases for mixed states in interferometry. , 85:2845–2849, 2000.
P. B. Slater. Mixed state holonomies. , 60:123–133, 2002.
D. Bures. An extension of kakutani’s theorem on infinite product measures to the tensor product of semifinite w\*-algebras. , 135:199–212, 1969.
A. Uhlmann. The metric of [B]{}ures and the geometric phase. In [*Quantum groups and related topics ([W]{}rocław, 1991)*]{}, volume 13 of [*Math. Phys. Stud.*]{}, pages 267–274. Kluwer Acad. Publ., Dordrecht, 1992.
J. Dittmann. On the riemannian geometry of finite dimensional mixed states. , 3:73–87, 1993.
J. Dittmann. Explicit formulae for the bures metric. , 32(14):2663–2670, 1999.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We obtain the global well-posedness to the 3D incompressible magnetohydrodynamics (MHD) equations in Besov space with negative index of regularity. Particularly, we can get the global solutions for a new class of large initial data. As a byproduct, this result improves the corresponding result in [@HHW]. In addition, we also get the global result for this system in $\mathcal{\chi}^{-1}({\mathbb{R}}^3)$ originally developed in [@LL]. More precisely, we only assume that the norm of initial data is exactly smaller than the sum of viscosity and diffusivity parameters.'
address: '$^1$ School of Mathematics Sciences, Zhejiang University, Hanzhou 310027, China'
author:
- 'Renhui Wan$^{1}$'
title: 'Global well-posedness to the 3D incompressible MHD equations with a new class of large initial data'
---
.2in
.2in
Introduction {#s1}
============
We are concerned with the 3D incompressible MHD equations: $$\label{1.1}
\left\{
\begin{aligned}
& \partial_t u + u\cdot\nabla u -\mu_1 \Delta u +\nabla p = B\cdot\nabla B, \\
& \partial_t B + u\cdot\nabla B-B\cdot\nabla u -\mu_2 \Delta B=0, \\
& {\rm div} u={\rm div}B=0,\\
& u(0,x) =u_0(x),\ B(0,x)=B_0(x),
\end{aligned}
\right.$$ here $(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^3$, $u,p,B$ stand for velocity vector, scalar pressure and magnetic vector, respectively, $\mu_1$ and $\mu_2$ are nonnegative viscosity and diffusivity parameters, respectively.
.1in For $\mu_1>0$ and $\mu_2>0$, the local well-posedness and global existence with small data for (\[1.1\]) were obtained by Duvaut and Lions [@DL] in $d$ dimensional Sobolev space $H^s({\mathbb{R}}^d)$, $s\ge d$. Then Sermange and Termam [@ST] studied the regularity of weak solutions $(u,B)\in L^\infty(0,T;H^1({\mathbb{R}}^3))$. And some regularity criteria were established in [@Wu1; @Wu2; @Wu3]. For $\mu_1>0$ and $\mu_2=0$ (so-called non-resistive MHD equations), by the new Kato-Ponce commutator estimate, $$\|\Lambda^s(u\cdot\nabla B)-u\cdot\nabla \Lambda^s B\|_{L^2({\mathbb{R}}^d)}\le C\|\nabla u\|_{H^s({\mathbb{R}}^d)}\|B\|_{H^s({\mathbb{R}}^d)},\ s>\frac{d}{2},\ d=2,3,$$ Fefferman et al. [@Fefferman] proved the low regularity local well-posedness of strong solutions, which was extended to general inhomogeneous Besov space with initial data $(u_0,B_0)\in B_{2,1}^{\frac{d}{2}-1}({\mathbb{R}}^d)\times B_{2,1}^\frac{d}{2}({\mathbb{R}}^d)$ in the recent works [@Chemin; @1] and [@Wan]. Furthermore, for the non-resistive version with smooth initial data near some nontrivial steady state, we refer [@PZhnag1; @LT; @ZZhang; @PZhang2] for the related works.
.1in Due to a new observation that the velocity field plays a more important role than magnetic field. The new regularity criteria only involving the velocity were proved, see [@CMZ; @HX; @YZ1; @YZ2] and references therein.
.1in One can easily get a new formation of (\[1.1\]) by the following: $$W^+:=u+B,\ W^-:=u-B,\ \nu_{+}=\frac{\mu_1+\mu_2}{2},\ \nu_{-}=\frac{\mu_1-\mu_2}{2}$$ with initial data $W^{\pm}_0(x):=u_0(x)\pm B_0(x),$ that is, $$\label{1.12}
\left\{
\begin{aligned}
& \partial_t W^+ + W^-\cdot\nabla W^+ -\nu_{+}\Delta W^+ +\nabla p = \nu_{-}\Delta W^-, \\
& \partial_t W^- + W^+\cdot\nabla W^- -\nu_{+}\Delta W^- +\nabla p = \nu_{-}\Delta W^+, \\
& {\rm div} W^+={\rm div}W^-=0,\\
& W^+(0,x) =W^+_0(x),\ W^-(0,x)=W^-_0(x).
\end{aligned}
\right.$$
Very recently, He et al. [@HHW] obtained the global well-posedness for (\[1.12\]) with initial data $(u_0,B_0)$ satisfying:\
($i$) $\nu_-=0$ and $$\nu_+^{-3}\|W_0^-\|_{L^3}^3\exp\{C\nu_+^{-3}\|W_0^+\|_{L^3}^3\}<\epsilon_0$$ or $$\nu_+^{-3}\|W_0^+\|_{L^3}^3\exp\{C\nu_+^{-3}\|W_0^-\|_{L^3}^3\}<\epsilon_0;$$ ($ii$) $\nu_-\neq0$ and $$\label{1.121}
\left(\nu_+^{-2}\|W^-_0\|_{\dot{H}^\frac{1}{2}}^2+\frac{\nu_-^2}{\nu_+^2}(\nu_+^{-2}\|W_0^+\|_{\dot{H}^\frac{1}{2}}^2+\frac{\nu_-^2}{\nu_+^2})\right)
\exp\left\{C\nu_+^{-4}(\|W_0^+\|_{\dot{H}^\frac{1}{2}}^4+\nu_-^4)\right\}<\epsilon_0$$ or $$\left(\nu_+^{-2}\|W^+_0\|_{\dot{H}^\frac{1}{2}}^2+\frac{\nu_-^2}{\nu_+^2}(\nu_+^{-2}\|W_0^-\|_{\dot{H}^\frac{1}{2}}^2+\frac{\nu_-^2}{\nu_+^2})\right)
\exp\left\{C\nu_+^{-4}(\|W_0^-\|_{\dot{H}^\frac{1}{2}}^4+\nu_-^4)\right\}<\epsilon_0.$$ Here $\epsilon_0$ ia a sufficiently small positive constant. .1in
In this paper, we will prove the global well-posedness of (\[1.1\]) $(\mu_1>0,\mu_2>0)$ in generalized space, $\dot{B}_{p,r}^{\frac{3}{p}-1}({\mathbb{R}}^3)$, by make full use of the harmonic analysis tools. The details can be given as follows:
\[t1\] Consider (\[1.1\]) with initial data $(u_0,B_0)\in \dot{B}_{p,r}^{\frac{3}{p}-1}({\mathbb{R}}^3),$ $(p,r)\in (1,\infty)\times [1,\infty),$ satisfying ${\rm div}u_0={\rm div}B_0=0$. There exists a constant $C$ and a small constant $\eta>0$ such that if $$\label{1.14}
\left(\|W^-_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\frac{\nu_-}{\nu_+}(\|W_0^+\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\nu_-)\right)
\exp\left\{C\nu_+^{-\frac{2}{1-\epsilon}}(\nu_-+\|W_0^+\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}})^\frac{2} {1-\epsilon} \right\}<\eta \nu_+$$ or $$\label{1.15}
\left(\|W^+_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\frac{\nu_-}{\nu_+}(\|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\nu_-)\right)
\exp\left\{C\nu_+^{-\frac{2}{1-\epsilon}}(\nu_-+\|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}})^\frac{2}{1-\epsilon} \right\}<\eta\nu_+,$$ where $(\epsilon,r)$ satisfies
$$\label{1.16}
\left\{
\begin{aligned}
0\le \epsilon<1&, \ {\rm if }\ r=1; \\
0< \epsilon<1&, \ {\rm if }\ 1<r\le 2; \\
1-\frac{2}{r}\le \epsilon<1&,\ {\rm if }\ 2<r<\infty.
\end{aligned}
\right.$$
Then (\[1.1\]) admits a unique global solution $(u,B)$ satisfying $$(u,B)\in \tilde{C}([0,\infty); \dot{B}_{p,r}^{\frac{3}{p}-1}({\mathbb{R}}^3))\cap \tilde{L}^1([0,\infty); \dot{B}_{p,r}^{\frac{3}{p}+1}({\mathbb{R}}^3)).$$
.1in If $\nu_-=0$, i.e., $\mu_1=\mu_2=\nu_+$, we have a corollary immediately.
\[c1\] Consider (\[1.1\]) with initial data $(u_0,B_0)\in \dot{B}_{p,r}^{\frac{3}{p}-1}({\mathbb{R}}^3),$ $(p,r)\in (1,\infty)\times [1,\infty),$ satisfying ${\rm div}u_0={\rm div}B_0=0$. There exists a constant $C$ and a small constant $\eta>0$ such that if $$\label{1.141}
\|W^-_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}
\exp\left\{C\nu_+^{-\frac{2}{1-\epsilon}}\|W_0^+\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}^\frac{2}{1-\epsilon} \right\}<\eta \nu_+$$ or $$\label{1.151}
\|W^-_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}
\exp\left\{C\nu_+^{-\frac{2}{1-\epsilon}}\|W_0^+\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}^\frac{2}{1-\epsilon} \right\}<\eta \nu_+,$$ where $(\epsilon,r)$ satisfies (\[1.16\]). Then (\[1.1\]) admits a unique global solution $(u,B)$ satisfying $$(u,B)\in \tilde{C}([0,\infty); \dot{B}_{p,r}^{\frac{3}{p}-1}({\mathbb{R}}^3))\cap \tilde{L}^1([0,\infty); \dot{B}_{p,r}^{\frac{3}{p}+1}({\mathbb{R}}^3)).$$
\[r1\] ($i$) We will construct the global solution with a new class of large initial data. More precisely, assume that $\phi$ satisfies the condition in Proposition \[l2\], let $$u_0=(\partial_2\phi,-\partial_1\phi,0),\ \ B_0=2\sin^2\frac{x_3}{2\epsilon}(\partial_2\phi,-\partial_1\phi,0),$$ then ${\rm div}u_0={\rm div}B_0=0$ and $\|(u_0,B_0)\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\le \mathfrak{M}$ $(p>3)$, which is independent of $\epsilon$. Moreover, thanks to Proposition \[l2\], there exists a positive constant $C_1$ and $C_2$, $$\|u_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\ge C_1,\ \ \|B_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\ge \frac{C_1}{2},$$ $$\|u_0-B_0\|_{\dot{B_{p,r}^{\frac{3}{p}-1}}}\le C_2\epsilon^{1-\frac{3}{p}},$$ which ensures the conditions (\[1.14\])($\nu_+\gg \nu_-$) and (\[1.141\]) hold. Additionally, the assumption $\nu_+\gg\nu_-$ is reasonable in astrophysical magnetic phenomena, see Remark 2.3 in [@HHW]. Combining with the above explanations, this class of large data can lead the global well-posedness to (\[1.1\]).\
($ii$) One can easily check that condition (\[1.14\]) is equal to (\[1.121\]) when $p=r=2$ and choosing $\epsilon=\frac{1}{2}$. By Bernstein’s inequality, we have the following embedding relationship: $$\dot{H}^\frac{1}{2}\hookrightarrow \dot{B}_{p,r}^{\frac{3}{p}-1},\ \ p> 2, r\ge2.$$ So our result improves the corresponding work under (\[1.121\]) in [@HHW]. By the same way, similar improvements can also be obtained under (\[1.15\]) , (\[1.141\]) and (\[1.151\]).
We shall point out that the above result can not be extended to $p=\infty.$ As a matter of fact, by these works [@ill2] and [@ill1] concerning the well-known Navier-Stokes equations, (\[1.1\]) may ill-posedness in this endpoint Besov space. .1in Next, we consider the space $\chi^{-1}({\mathbb{R}}^3)$, which is smaller than $\dot{B}_{\infty,r}^{-1}$ due to Proposition \[l33\]. It was originally developed in [@LL] and applied to get the global well-posedness for the Navier-Stokes equations under $$\|u_0\|_{\chi^{-1}}<\mu.$$ For MHD equations (\[1.1\]), similar result holds under $$\label{1.2}
\|u_0\|_{\chi^{-1}}+\|B_0\|_{\chi^{-1}}<\min\{\mu,\eta\},$$ see [@KWang] for details. .1in We have some new result in $\chi^{-1}({\mathbb{R}}^3).$
\[t2\] Consider (\[1.1\]) with initial data $(u_0,B_0)\in \chi^{-1}({\mathbb{R}}^3)$ satisfying ${\rm div}u_0={\rm div}B_0=0$. There exists a constant $C$ such that if $$\label{1.21}
\left(\|W_0^-\|_{\chi^{-1}}+\frac{C\nu_-}{\nu_+}(\nu_-+\|W_0^+\|_{\chi^{-1}})\right)\exp\left\{\frac{C}{\nu_+^2}(\nu_-+\|W_0^+\|_{\chi^{-1}})^2\right\}<2\nu_+$$ or $$\label{1.22}
\left(\|W_0^+\|_{\chi^{-1}}+\frac{C\nu_-}{\nu_+}(\nu_-+\|W_0^-\|_{\chi^{-1}})\right)\exp\left\{\frac{C}{\nu_+^2}(\nu_-+\|W_0^-\|_{\chi^{-1}})^2\right\} <2\nu_+.$$ Then (\[1.1\]) admits a unique global solution $(u,B)$ satisfying $$(u,B)\in C([0,\infty); \chi^{-1}({\mathbb{R}}^3))\cap L^1([0,\infty);\chi^1({\mathbb{R}}^3)).$$
Similarly, we also have a corollary immediately when $\nu_-=0$.
\[c2\] Consider (\[1.1\]) with initial data $(u_0,B_0)\in \chi^{-1}({\mathbb{R}}^3)$ satisfying ${\rm div}u_0={\rm div}B_0=0$. There exists a constant $C$ such that if $$\|W_0^-\|_{\chi^{-1}}\exp\left\{\frac{C}{\nu_+^2}\|W_0^+\|_{\chi^{-1}}^2\right\} <2\nu_+$$ or $$\|W_0^+\|_{\chi^{-1}}\exp\left\{\frac{C}{\nu_+^2}\|W_0^-\|_{\chi^{-1}}^2\right\}<2\nu_+.$$ Then (\[1.1\]) admits a unique global solution $(u,B)$ satisfying $$(u,B)\in C([0,\infty); \chi^{-1}({\mathbb{R}}^3))\cap L^1([0,\infty);\chi^1({\mathbb{R}}^3)).$$
\[r2\] The authors in [@LL] proved the global well-posedness for Navier-Stokes equations by using $$\|u\cdot\nabla u\|_{\chi^{-1}}\le \|u\|_{\chi^{-1}}\|u\|_{\chi^1},$$ while we shall use the new estimate below in our proof, i.e., $$\|u\cdot\nabla v\|_{\chi^{-1}}\le \|u\|_{\chi^0}\|v\|_{\chi^0}.$$
\[r3\] Due to the symmetric structure of (\[1.12\]), we only give the proof of Theorem \[t1\] and Theorem \[t2\] under (\[1.14\]) and (\[1.21\]), respectively.
The present paper is structured as follows:\
In section \[s2\], we provide some definitions of spaces, establish several lemmas. The third section proves Theorem \[t1\], while the last section gives the proof of Theorem \[t2\].
.1in Let us complete this section by describing the notations we shall use in this paper.\
[**Notations**]{} The uniform constant $C$ is different on different lines. We also use $L^p$, $\dot{B}_{p,r}^s$ and $\chi^s$ to stand for $L^p(\mathbb{R}^d)$, $\dot{B}_{p,r}^s(\mathbb{R}^d)$ and $\chi^s({\mathbb{R}}^d)$ in somewhere, respectively. We use $A:=B$ to stands for $A$ is defined by $B$, and ${\bf 1}$ is the characteristic function.
.4in
Preliminaries {#s2}
==============
In this section, we give some necessary definitions, propositions and lemmas. .1in The Fourier transform is given by $$\widehat{f}(\xi)=\int_{{\mathbb{R}}^d}e^{-ix\cdot\xi}f(x)dx.$$
Let $\mathfrak{B}=\{\xi\in\mathbb{R}^d,\ |\xi|\le\frac{4}{3}\}$ and $\mathfrak{C}=\{\xi\in\mathbb{R}^d,\ \frac{3}{4}\le|\xi|\le\frac{8}{3}\}$. Choose two nonnegative smooth radial function $\chi,\ \varphi$ supported, respectively, in $\mathfrak{B}$ and $\mathfrak{C}$ such that $$\sum_{j\in\mathbb{Z}}\varphi(2^{-j}\xi)=1,\ \ \xi\in\mathbb{R}^d\setminus\{0\}.$$ We denote $\varphi_{j}=\varphi(2^{-j}\xi),$ $h=\mathfrak{F}^{-1}\varphi$ and $\tilde{h}=\mathfrak{F}^{-1}\chi,$ where $\mathfrak{F}^{-1}$ stands for the inverse Fourier transform. Then the dyadic blocks $\Delta_{j}$ and $S_{j}$ can be defined as follows $$\Delta_{j}f=\varphi(2^{-j}D)f=2^{jd}\int_{\mathbb{R}^d}h(2^jy)f(x-y)dy,\ \ S_{j}f=\sum_{k\le j-1}\Delta_{k}f$$
Formally, $\Delta_{j}$ is a frequency projection to annulus $\{\xi:\ C_{1}2^j\le|\xi|\le C_{2}2^j\}$, and $S_{j}$ is a frequency projection to the ball $\{\xi:\ |\xi|\le C2^j\}$. One easily verifies that with our choice of $\varphi$ $$\Delta_{j}\Delta_{k}f=0\ {\rm if} \ |j-k|\ge2\ \ {\rm and}\ \ \Delta_{j}(S_{k-1}f\Delta_{k}f)=0\ {\rm if}\ |j-k|\ge5.$$ Let us recall the definition of the Besov space.
\[HB\] Let $s\in \mathbb{R}$, $(p,q)\in[1,\infty]^2,$ the homogeneous Besov space $\dot{B}_{p,q}^s({\mathbb{R}}^d)$ is defined by $$\dot{B}_{p,q}^{s}({\mathbb{R}}^d)=\{f\in \mathfrak{S}'({\mathbb{R}}^d);\ \|f\|_{\dot{B}_{p,q}^{s}({\mathbb{R}}^d)}<\infty\},$$ where $$\|f\|_{\dot{B}_{p,q}^s({\mathbb{R}}^d)}=\left\{\begin{aligned}
&\displaystyle (\sum_{j\in \mathbb{Z}}2^{sqj}\|\Delta_{j}f\|_{L^p({\mathbb{R}}^d)}^{q})^\frac{1}{q},\ \ \ \ {\rm for} \ \ 1\le q<\infty,\\
&\displaystyle \sup_{j\in\mathbb{Z}}2^{sj}\|\Delta_{j}f\|_{L^p({\mathbb{R}}^d)},\ \ \ \ \ \ \ \ {\rm for}\ \ q=\infty,\\
\end{aligned}
\right.$$ and $\mathfrak{S}'({\mathbb{R}}^d)$ denotes the dual space of $\mathfrak{S}({\mathbb{R}}^d)=\{f\in\mathcal{S}(\mathbb{R}^d);\ \partial^{\alpha}\hat{f}(0)=0;\ \forall\ \alpha\in \ \mathbb{N}^d $ [multi-index]{}} and can be identified by the quotient space of $\mathcal{S'}/\mathcal{P}$ with the polynomials space $\mathcal{P}$.
The norm of the space $\tilde{L}^{r_1}_t(\dot{B}_{p,r}^s)$ and $\tilde{L}^{r_1}_{t,\omega}(\dot{B}_{p,r}^s)$ is defined by $$\|f\|_{\tilde{L}^{r_1}_t(\dot{B}_{p,r}^s)}:=\|2^{js}\|\Delta_j f\|_{L^{r_1}_tL^p}\|_{l^r({\mathbb{Z}})}$$ and $$\|f\|_{\tilde{L}^{r_1}_{t,\omega}(\dot{B}_{p,r}^s)}:=\|2^{js}\left(\int_0^t \omega(\tau)^{r_1}\|\Delta_j f(\tau)\|_{L^p}^{r_1} d\tau\right)^\frac{1}{r_1}\|_{l^r({\mathbb{Z}})}.$$ $f\in \tilde{C}(0,t;\dot{B}_{p,r}^s)$ means $f\in \tilde{L}^{\infty}_t(\dot{B}_{p,r}^s)$ and $\|f(t)\|_{\dot{B}_{p,r}^s}$ is continuous in time.
.1in The following proposition provide Bernstein type inequalities.
Let $1\le p\le q\le \infty$. Then for any $\beta,\gamma\in (\mathbb{N}\cup \{0\})^3$, there exists a constant $C$ independent of $f,j$ such that
1. If $f$ satisfies $$\mbox{supp}\, \widehat{f} \subset \{\xi\in \mathbb{R}^d: \,\, |\xi|
\le \mathcal{K} 2^j \},$$ then $$\|\partial^\gamma f\|_{L^q(\mathbb{R}^d)} \le C 2^{j|\gamma| +
j d(\frac{1}{p}-\frac{1}{q})} \|f\|_{L^p(\mathbb{R}^d)}.$$
2. If $f$ satisfies $$\label{spp}
\mbox{supp}\, \widehat{f} \subset \{\xi\in \mathbb{R}^d: \,\, \mathcal{K}_12^j
\le |\xi| \le \mathcal{K}_2 2^j \}$$ then $$\|f\|_{L^p(\mathbb{R}^d)} \le C2^{-j|\gamma|}\sup_{|\beta|=|\gamma|} \|\partial^\beta f\|_{L^p(\mathbb{R}^d)}.$$
For more details about Besov space such as some useful embedding relations, see [@BCD; @Grafakos; @Stein].
\[l1\][@Danchin] Let $1<p<\infty,$ $supp \widehat{u}\subset C(0,R_{1},R_{2})$ (with $0<R_{1}<R_{2}$). There exists a constant $c$ depending on $\frac{R_{2}}{R_{1}}$ and such that $$\label{Ber}
c\frac{R_{1}^2}{p^2}\int_{\mathbb{R}^3}|u|^p dx\le -\frac{1}{p-1}\int_{\mathbb{R}^3} \Delta u |u|^{p-2}u dx.$$
\[l2\] Let $\phi\in \mathcal{S}({\mathbb{R}}^3)$, whose Fourier transform supported in annulus contained in ${\mathbb{R}}^3\setminus\{0\}$, and $p>3$. If $u_0=(\partial_2 \phi, -\partial_1\phi,0)$ and $B_0=2\sin^2\frac{x_3}{2\epsilon}(\partial_2 \phi, -\partial_1\phi,0),$ then there exists a constant $C_1,C_2>0$ such that $$\|u_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\ge C_1,\ \ \|B_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\ge \frac{C_1}{2}$$ and $$\|u_0-B_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\le C_2\epsilon^{1-\frac{3}{p}},$$ here $\epsilon$ is sufficiently small.
The last estimate can be obtained by following the proof of Lemma 3.1 in [@Chemin; @2]. So we suffice to show both $\|u_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}$ and $\|B_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}$ has positive lower bound. With this $\phi$, there exists a finite $j_0\in{\mathbb{Z}}$, such that $\Delta_{j_0}\partial_2\phi\neq0$, which implies $$\|\Delta_{j_0} \partial_2\phi\|_{L^\infty}\ge \epsilon_0$$ for some positive constant $\epsilon_0$. Thanks to this, by Bernstein’s inequality, we have $$\|u_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\ge \|u_0\|_{\dot{B}_{\infty,\infty}^{-1}}\ge 2^{-j_0}\|\Delta_{j_0}\partial_2\phi\|_{L^\infty}\ge 2^{-j_0}\epsilon_0,$$ and by triangle inequality $$\|B_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\ge \|u_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}-\|u_0-B_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\ge
2^{-j_0}\epsilon_0-C_2\epsilon^{1-\frac{3}{p}}\ge 2^{-j_0-1}\epsilon_0$$ due to the sufficient small $\epsilon$. Choosing $C_1=2^{-j_{0}}\epsilon_0$ yields the desired result.
For some convenience, we provide the following definition of $\chi^{s}({\mathbb{R}}^d)$, $$\|f\|_{\chi^s}:=\int_{{\mathbb{R}}^d} |\xi|^s |\hat{f}(\xi)| d\xi,$$ and we refer [@LL] for some details.
\[l33\] Let $f\in \chi^{-1}$, then we have $$\|f\|_{\dot{B}_{\infty,r}^{-1}}\le \|f\|_{\dot{B}_{\infty,1}^{-1}}\le \|f\|_{\mathbb{{B}}_{1,1}^{-1}}\thickapprox \|f\|_{\chi^{-1}},$$ where $$\|f\|_{\mathbb{B}_{1,1}^{-1}}:=\sum_{j\in{\mathbb{Z}}}2^{-j}\|\widehat{\Delta_j f}\|_{L^1}.$$
The first inequality is obvious, while the second inequality can be proved by using $\|f\|_{L^\infty}\le \|\hat{f}\|_{L^1}.$ Now, we prove $\|f\|_{\mathbb{{B}}_{1,1}^{-1}}\thickapprox \|f\|_{\chi^{-1}}$. By the definition of $\Delta_j$, and using Monotone Convergence Theorem, $$\begin{aligned}
\|f\|_{\mathbb{{B}}_{1,1}^{-1}}=&\sum_{j\in {\mathbb{Z}}}2^{-j}\| \varphi(2^{-j}\xi)\hat{f}(\xi)\|_{L^1}\\
\thickapprox&
\sum_{j\in {\mathbb{Z}}}\| |\xi|^{-1}\varphi(2^{-j}\xi)\hat{f}(\xi)\|_{L^1}\\
=&\||\xi|^{-1}|\hat{f}(\xi)|\|_{L^1}\\
=&\|f\|_{\chi^{-1}},
\end{aligned}$$ where we have used $\sum_{j\in Z}\varphi(2^{-j}\xi)=1$ and $\varphi\ge 0$.
\[l3\] ($i$) Let $(p,r)\in [1,\infty)\times[1,\infty],$ ${\rm div}u=0$, then $$\label{SKP1}
\|u\cdot\nabla v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}\le C\left(\|u\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}\|v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}+
\|v\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}\|u\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}\right);$$ ($ii$) Let $(p,r)\in [1,\infty)\times[1,\infty],$ ${\rm div}u=0$, then $$\label{SKP2}
\|u\cdot\nabla v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}\le
C\|v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}^\frac{1+\epsilon}{2}\|v\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}^\frac{1-\epsilon}{2},$$ where $0<\epsilon<1$ and $f=\|u\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-1}}^\frac{2}{1-\epsilon}.$ In particular, (\[SKP2\]) also holds when $(\epsilon,r)=(0,1)$ and $f=\|u\|_{\dot{B}_{p,1}^\frac{3}{p}}^2.$
For the proof, we shall use homogeneous Bony’s decomposition: $$uv=T_uv+T_vu+R(u,v),$$ where $$T_uv=\sum_{j\in \mathbb{Z}}S_{j-1}u\Delta_{j}v,\ \ T_vu=\sum_{j\in \mathbb{Z}}\Delta_{j}uS_{j-1}v,\ \
R(u,v)=\sum_{j\in\mathbb{Z}}\Delta_{j}u\tilde{\Delta}_{j}v,$$ here $\tilde{\Delta}_{j}=\Delta_{j-1}+\Delta_{j}+\Delta_{j+1}.$
The estimate of (\[SKP1\]) can be established by using $$\|u\cdot\nabla v\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\le C\{\|u\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\|v\|_{\dot{B}_{p,r}^{\frac{3}{p}+1}}+
\|v\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}\|u\|_{\dot{B}_{p,r}^{\frac{3}{p}+1}}\},$$ whose proof is standard. Thus the goal is the estimate of (\[SKP2\]). By homogeneous Bony’s decomposition, $$\label{2.4}
\begin{aligned}
\|u\cdot\nabla v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}\le& \|T_{u_i}\partial_i v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}
+\|T_{\partial_i v}u_i\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}+\|R(u,\nabla v)\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}\\
:=&I_1+I_2+I_3.
\end{aligned}$$ Let $\theta=\frac{1-\epsilon}{2}$, $0<\epsilon<1$. For $I_1$, using Hölder’s inequality and Bernstein’s inequality, $$\begin{aligned}
I_1\le& \left\|2^{j(\frac{3}{p}-1)}\sum_{|k-j|\le 4}\|\Delta_j(S_{k-1}u\cdot\nabla \Delta_k v)\|_{L^1_tL^p}\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j(\frac{3}{p}-1)}\|S_{j-1}u\cdot\nabla \Delta_j v\|_{L^1_tL^p}\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j(\frac{3}{p}-1)}\int_0^t \|S_{j-1}u\|_{L^\infty}\|\nabla \Delta_j v\|_{L^p}d\tau\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j(\frac{3}{p}+\epsilon)}\int_0^t \|u\|_{\dot{B}_{\infty,\infty}^{-\epsilon}}\| \Delta_j v\|_{L^p}d\tau\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j(\frac{3}{p}+\epsilon)}\int_0^t \|u\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-\epsilon}}\| \Delta_j v\|_{L^p}d\tau\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j(\frac{3}{p}+1)(1-\theta)}\|\Delta_j v\|_{L^1_tL^p}^{1-\theta}
(\int_0^t 2^{j(\frac{3}{p}-1)} \|u\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-\epsilon}}^\frac{1}{\theta}\| \Delta_j v\|_{L^p}d\tau)^\theta\right\|_{l^r({\mathbb{Z}})}\\
\le& C\|v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}^{1-\theta}\|v\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}^\theta
= C\|v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}^\frac{1+\epsilon}{2}\|v\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}^\frac{1-\epsilon}{2},
\end{aligned}$$ here $f=\|u\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-1}}^\frac{2}{1-\epsilon}$ and we have used $$\left\|2^{js}\|S_j u\|_{L^p}\right\|_{l^r({\mathbb{Z}})}\approx \|u\|_{\dot{B}_{p,r}^{s}},\ \forall\ s<0.$$ Similarly, for $I_2$, by Hölder’s inequality and Bernstein’s inequality, $$\begin{aligned}
I_2\le& \left\|2^{j(\frac{3}{p}-1)}\sum_{|k-j|\le 4}\|\Delta_j(\Delta_{k}u\cdot\nabla S_{k-1} v)\|_{L^1_tL^p}\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j(\frac{3}{p}-1)}\|\Delta_{j}u\cdot\nabla S_{j-1} v\|_{L^1_tL^p}\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j(\frac{3}{p}-1)}\int_0^t \|\Delta_{j}u\|_{L^p}\|\nabla S_{j-1} v\|_{L^\infty} d\tau\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j(\epsilon-1)}\int_0^t \|u\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-\epsilon}}\sum_{j'\le j-2}2^{j'(\frac{3}{p}+1)}\| \Delta_{j'} v\|_{L^p} d\tau\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|\sum_{j'\le j-2}2^{(j-j')(\epsilon-1)}\int_0^t \|u\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-\epsilon}}2^{j'(\frac{3}{p}+\epsilon)}\| \Delta_{j'} v\|_{L^p} d\tau\right\|_{l^r({\mathbb{Z}})}\\
\le& C \left\|2^{j(\frac{3}{p}+\epsilon)}\int_0^t \|u\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-\epsilon}}\| \Delta_{j} v\|_{L^p} d\tau\right\|_{l^r({\mathbb{Z}})}\\
\end{aligned}$$ where we have used Young’s inequality for series for the last inequality, i.e., $$\left\|\sum_{j'\le j-2}2^{(j-j')(\epsilon-1)}c_{j'}\right\|_{l^r({\mathbb{Z}})}\le C\|2^{j(\epsilon-1)}{\bf 1}_{j\ge 2}\|_{l^1({\mathbb{Z}})}\|c_j\|_{l^r({\mathbb{Z}})}\le C\|c_j\|_{l^r({\mathbb{Z}})}.$$ Following the same argument as $I_1$, one gets $$I_2\le C\|v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}^\frac{1+\epsilon}{2}\|v\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}^\frac{1-\epsilon}{2}.$$ Finally, we bound $I_3$. By Bernstein’s inequality, Young’s inequality for series and Hölder’s inequality, we have $$\begin{aligned}
I_3\le& \left\|2^{j(\frac{3}{p}-1)}\sum_{k\ge j-3}\|\Delta_j(\Delta_{k}u\cdot\nabla \tilde{\Delta}_k v)\|_{L^1_tL^p}\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j\frac{3}{p}}\sum_{k\ge j-3}\|\Delta_j(\Delta_{k}u\otimes \tilde{\Delta}_k v)\|_{L^1_tL^p}\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|\sum_{k\ge j-3}2^{(j-k)\frac{3}{p}}2^{k\frac{3}{p}}\|\Delta_j(\Delta_{k}u\otimes \tilde{\Delta}_k v)\|_{L^1_tL^p}\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{k\frac{3}{p}}\int_0^t \|\Delta_k u\|_{L^p}\|\tilde{\Delta}_k v\|_{L^\infty} d\tau \right\|_{l^r({\mathbb{Z}})}\ (p<\infty)\\
\le& C\left\|2^{k(\frac{3}{p}+\epsilon)}\int_0^t \| u\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-\epsilon}}\|\tilde{\Delta}_k v\|_{L^p} d\tau \right\|_{l^r({\mathbb{Z}})},
\end{aligned}$$ and using the same way as the estimate of $I_1$ derives $$I_3\le C\|v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}^\frac{1+\epsilon}{2}\|v\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}^\frac{1-\epsilon}{2}.$$ Plugging the above estimates into (\[2.4\]) leads the desired result (\[SKP2\]).\
In addition, if $r=1$, the estimate of $I_1$ can be replaced as follows: $$\begin{aligned}
I_1\le& C\left\|2^{j(\frac{3}{p}-1)}\int_0^t \|S_{j-1}u\|_{L^\infty}\|\nabla \Delta_j v\|_{L^p}d\tau\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j\frac{3}{p}}\int_0^t \|u\|_{\dot{B}_{p,1}^\frac{3}{p}}\|\Delta_j v\|_{L^p}d\tau\right\|_{l^r({\mathbb{Z}})}\\
\le& C\left\|2^{j(\frac{3}{p}+1)\frac{1}{2}}\|\Delta_j v\|_{L^1_tL^p}^\frac{1}{2}(2^{j(\frac{3}{p}-1)}\int_0^t \|u\|_{\dot{B}_{p,1}^\frac{3}{p}}^2\|\Delta_j v\|_{L^p}d\tau)^\frac{1}{2}\right\|_{l^r}\\
\le& C\|v\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}^\frac{1}{2}\|v\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}^\frac{1}{2},
\end{aligned}$$ here $f=\|v\|_{\dot{B}_{p,1}^\frac{3}{p}}^2$. At the same time, one can get the new estimates of $I_2$ and $I_3$ with the similar procedure. Thus we complete the proof of this lemma.
.3in
Proof of Theorem \[t1\] {#s3}
=======================
As the Remark \[r3\], it suffices to prove the Theorem \[t1\] under (\[1.14\]). One can get the local existence and uniqueness for (\[1.1\]) by using the standard argument on the Navier-Stokes equations, namely, there exists a $T^\star>0$, such that $$(u,B)\in \tilde{C}([0,T^\star); \dot{B}_{p,r}^{\frac{3}{p}-1})\cap \tilde{L}^1([0,T^\star); \dot{B}_{p,r}^{\frac{3}{p}+1}).$$ Since the equivalence between (\[1.1\]) and (\[1.12\]), we will consider (\[1.12\]) and suffice to prove $T^\star=\infty$.\
Now, we begin the proof. Let us consider $0<\epsilon<1$ and $r\le \frac{2}{1-\epsilon}$, containing all cases in (\[1.16\]) except $(\epsilon,r)=(0,1).$ Define $$\label{3.1}
\bar{T}:=\sup\left\{t\in (0,T^\star):\ \|W^-\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}+\nu_+\|W^-\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}\le \epsilon_o \nu_+ \right\},$$ where $\epsilon_0$ is small positive constant and will be determined later on.\
[**Step 1. The estimate of $W^+$.**]{} Consider the first equation in (\[1.12\]), using (\[Ber\]), we get $$\frac{d}{dt}\|\Delta_j W^+\|_{L^p}+c\nu_+ 2^{2j}\|\Delta_j W^+\|_{L^p}\le C\|\Delta_j (W^-\cdot\nabla W^+)\|_{L^p}+C\nu_- 2^{2j}\|\Delta_j W^-\|_{L^p},$$ which yields by a standard procedure $$\begin{aligned}
\|W^+\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}&+c\nu_+\|W^+\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}\\
\le& 2\|W^+_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+C\|W^-\cdot\nabla W^+\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}
+C\nu_-\|W^-\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}.
\end{aligned}$$ By (\[SKP1\]) and (\[3.1\]), we have for all $t\in (0,\bar{T}],$ $$\begin{aligned}
\|W^+\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}+&c\nu_+\|W^+\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}
\le 2\|W^+_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+C\nu_-\|W^-\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}\\
&+C(\|W^-\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}
\|W^+\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}+\|W^+\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}
\|W^-\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})})\\
\le& 2\|W^+_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+C\epsilon_0 ( \nu_+\|W^+\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}+
\|W^+\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})})+C\epsilon_0\nu_-,\\
\end{aligned}$$ with the selection of $\epsilon_0<\min\{\frac{c}{2C},\frac{1}{2C}\}$ leads $$\label{3.2}
\|W^+\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}+c\nu_+\|W^+\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}
\le 4\|W^+_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+2c\nu_-.$$
[**Step 2. The estimate of $W^-$.**]{} Denote $$f(t):=\|W^+(t)\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-\epsilon}}^\frac{2}{1-\epsilon},\ W_\lambda^\pm:=W^\pm\exp\{-\lambda\int_0^t f(\tau) d\tau\},
\ p_\lambda:=p\exp\{-\lambda\int_0^t f(\tau) d\tau\},$$ where $\lambda$ is large enough constant and will be determined later on. So we can rewrite the second equation in (\[1.12\]) as $$\partial_t W^-_\lambda+\lambda f(t)W^-_\lambda+W^+\cdot\nabla W^-_\lambda+\nabla p_\lambda-\nu_+\Delta W^-_\lambda=\nu_-\Delta W^+_\lambda.$$ By a similar procedure, we have $$\begin{aligned}
\|\Delta_j W^-_\lambda\|_{L^\infty_t L^p}+&\lambda \int_0^t f(\tau)\|\Delta_j W^-_\lambda\|_{L^p} d\tau+c\nu_+2^{2j}\|\Delta_j W^-_\lambda\|_{L^1_tL^p}\\
&\le \|\Delta_j W_0^-\|_{L^p}+C\|\Delta_j(W^+\cdot\nabla W^-_\lambda)\|_{L^1_tL^p}+C\nu_-2^{2j}\|\Delta_j W^+_\lambda\|_{L^1_tL^p}.
\end{aligned}$$ Then we obtain $$\begin{aligned}
\|W^-_\lambda\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}+&c\nu_+\|W^-_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}
+\lambda \|W^-_\lambda\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}\\
&\le \|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+C\nu_-\|W^+_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}
+C\|W^+\cdot\nabla W^-_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}.
\end{aligned}$$ Thanks to (\[SKP2\]), and by Young’s inequality, we obtain $$\label{3.3}
\begin{aligned}
&\ \ \ \|W^-_\lambda\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}+c\nu_+\|W^-_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}
+\lambda \|W^-_\lambda\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}\\
\le& \|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+C\nu_-\|W^+_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}
+C\|W^-_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}^\frac{1+\epsilon}{2}
\|W^-_\lambda\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}^\frac{1-\epsilon}{2}\\
\le& \|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+C\nu_-\|W^+_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}
+\frac{c\nu_+}{8}\|W^-_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}\\
&+C\nu_+^{-\frac{1+\epsilon}{1-\epsilon}}\|W^-_\lambda\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}.
\end{aligned}$$ Choosing $\lambda>2C\nu_+^{-\frac{1+\epsilon}{1-\epsilon}},$ absorbing the third and fourth term on the right hand side of last inequality by the left hand side in (\[3.3\]) follows $$\begin{aligned}
\|W^-_\lambda\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}&+\frac{7c}{8}\nu_+\|W^-_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}
+C\nu_+^{-\frac{1+\epsilon}{1-\epsilon}}\|W^-_\lambda\|_{\tilde{L}^1_{t,f}(\dot{B}_{p,r}^{\frac{3}{p}-1})}\\
\le& \|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+C\nu_-\|W^+_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}.
\end{aligned}$$ Obviously, using (\[3.2\]), we have $$\begin{aligned}
\|W^-_\lambda\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}+c\nu_+\|W^-_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}
\le&2\|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+C\nu_-\|W^+_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}\\
\le& C\left( \|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\frac{\nu_-}{\nu_+}(\|W_0^+\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\nu_-)\right).
\end{aligned}$$ This yields, after using (\[3.2\]) again, for all $t\in(0,\bar{T}),$ $$\begin{aligned}
&\ \ \ \|W^-_\lambda\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}+c\nu_+\|W^-_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}\\
\le& C\left( \|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\frac{\nu_-}{\nu_+}(\|W_0^+\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\nu_-)\right)
\exp\left\{C\nu_+^{-\frac{1+\epsilon}{1-\epsilon}}\int_0^t \|W^+(\tau)\|_{\dot{B}_{p,\infty}^{\frac{3}{p}-\epsilon}}^\frac{2}{1-\epsilon}d\tau
\right\}\\
\le& C\left( \|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\frac{\nu_-}{\nu_+}(\|W_0^+\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\nu_-)\right)
\exp\left\{C\nu_+^{-\frac{1+\epsilon}{1-\epsilon}}\|W^+\|_{\tilde{L}^\frac{2}{1-\epsilon}_t(\dot{B}_{p,r}^{\frac{3}{p}-\epsilon})}^\frac{2}{1-\epsilon}\right\}\\
\le& C\left( \|W_0^-\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\frac{\nu_-}{\nu_+}(\|W_0^+\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}}+\nu_-)\right)
\exp\left\{C\nu_+^{-\frac{2}{1-\epsilon}}(\nu_-+\|W^+_0\|_{\dot{B}_{p,r}^{\frac{3}{p}-1}})^\frac{2}{1-\epsilon}\right\},
\end{aligned}$$ which implies that if we take $\eta$ small enough in (\[1.14\]), there holds for all $t\le \bar{T}$, $$\|W^-_\lambda\|_{\tilde{L}^\infty_t(\dot{B}_{p,r}^{\frac{3}{p}-1})}+\nu_+\|W^-_\lambda\|_{\tilde{L}^1_t(\dot{B}_{p,r}^{\frac{3}{p}+1})}\le C\eta \nu_+<\frac{\epsilon_0}{2}\nu_+.$$ Then by a standard continuous method, we get $\bar{T}=T^\star=\infty.$ .1in The remainder is $r=1, \epsilon=0$, by a similar arguments, using (\[SKP2\]) for this case and let $(\epsilon,r)=(0,1)$, $f=\|W^+\|_{\dot{B}_{p,1}^\frac{3}{p}}^2$ in (\[3.3\]), the desired result can be otained. Hence, we complete the proof of Theorem \[t1\].
.3in
Proof of Theorem \[t2\] {#s4}
=======================
One can easily get the local well-posedness of (\[1.1\]), that is, there exists a $T^\star>0$ such that $$(u,B)\in C([0,T^\star); \chi^{-1}({\mathbb{R}}^3))\cap L^1([0,T^\star); \chi^1({\mathbb{R}}^3)).$$ So we suffices to show $T^\star=\infty$. .1in Now, we begin the proof. (\[1.21\]) is indeed equal to $$\label{4.1}
\left(\|W_0^-\|_{\chi^{-1}}+\frac{C\nu_-}{\nu_+}(\nu_-+\|W_0^+\|_{\chi^{-1}})\right)\exp\left\{\frac{C}{\nu_+^2}(\nu_-+\|W_0^+\|_{\chi^{-1}})^2\right\}\le(2-\epsilon_0)\nu_+$$ for some $\epsilon_0>0.$ And next we suffices to prove the desired result under (\[4.1\]). Let $C_1,C_2\in (0,2)$ satisfying $2(2-\epsilon_0)^2<C_2(2-C_1)^2$ and $$a=C_2 \nu_+,\ \ a_1=C_1 \nu_+,\ b\in (\frac{2(2-\epsilon_0)}{2-C_1}\nu_+,\sqrt{2C_2}\nu_+).$$ Then consider the first equation in (\[1.12\]), by the procedure as [@LL], and using interpolation inequality, we have $$\begin{aligned}
\frac{d}{dt}\|W^+\|_{\chi^{-1}}+&\nu_+\|W^+\|_{\chi^1}\le \|W^+\cdot\nabla W^-\|_{\chi^{-1}}+\nu_-\|W^-\|_{\chi^1}\\
\le& \|W^+\|_{\chi^0}\|W^-\|_{\chi^0}+\nu_-\|W^-\|_{\chi^1}\\
\le& \|W^-\|_{\chi^0}^2\|W^+\|_{\chi^{-1}}^\frac{1}{2}\|W^+\|_{\chi^1}^\frac{1}{2}+\nu_-\|W^-\|_{\chi^1}\\
\le & \frac{1}{2a}\|W^-\|_{\chi^0}^2\|W^+\|_{\chi^{-1}}+\frac{a}{2}\|W^+\|_{\chi^1}+\nu_- \|W^-\|_{\chi^1},
\end{aligned}$$ which derives by integrating in time, $$\label{4.12}
\begin{aligned}
\|W^+\|_{L^\infty_t(\chi^{-1})}+&(\nu_+-\frac{a}{2})\|W^+\|_{L^1_t(\chi^1)}\\
\le& \frac{1}{2a}\|W^-\|_{L^2_t(\chi^0)}^2\|W^+\|_{L^\infty_t(\chi^{-1})}
+\nu_-\|W^-\|_{L^1_t(\chi^1)}+\|W_0^+\|_{\chi^{-1}}.
\end{aligned}$$ Define $$\label{4.2}
\bar{T}:=\sup\left\{t\in (0,T^\star):\ \|W^-\|_{L^\infty_t(\chi^{-1})}+\nu_+\|W^-\|_{L^1_t(\chi^1)}\le b \right\}.$$ Then we will prove $T^\star=\bar{T}=\infty$ under (\[4.1\]). Using (\[4.2\]), combining with (\[4.12\]), we have $$\label{4.3}
(1-\frac{b^2}{2a\nu_+})\|W^+\|_{L^\infty_t(\chi^{-1})}+(\nu_+-\frac{a}{2})\|W^+\|_{L^1_t(\chi^1)}\le \|W_0^+\|_{\chi^{-1}}+\frac{b\nu_-}{\nu_+}.$$ Following the similar way as (\[4.12\]), one gets $$\frac{d}{dt}\|W^-\|_{\chi^{-1}}+\nu_+\|W^-\|_{\chi^1}
\le \frac{1}{2a_1}\|W^+\|_{\chi^0}^2\|W^-\|_{\chi^{-1}}+\frac{a_1}{2}\|W^-\|_{\chi^1}+\nu_- \|W^+\|_{\chi^1}$$ and thanks to (\[4.3\]), $$\begin{aligned}
\|W^-(t)\|_{\chi^{-1}}+&(\nu_+-\frac{a_1}{2})\|W^-\|_{L^1_t(\chi^1)}\\
\le& \frac{1}{2a_1}\int_0^t \|W^+\|_{\chi^0}^2\|W^-\|_{\chi^{-1}} d\tau
+\frac{\nu_-}{\nu_+-\frac{a}{2}}(\frac{b\nu_-}{\nu_+}+\|W_0^+\|_{\chi^{-1}})+\|W^{-}_0\|_{\chi^{-1}},
\end{aligned}$$ with the application of Gronwall’s lemma, by interpolation’s inequality and (\[4.3\]) leads $$\begin{aligned}
&\ \ \ \ \ \ \|W^-\|_{L^\infty_t(\chi^{-1})}+(\nu_+-\frac{a_1}{2})\|W^-\|_{L^1_t(\chi^1)}\\
\le& (\|W_0^-\|_{\chi^{-1}}+\frac{\nu_-}{\nu_+-\frac{a}{2}}(\frac{b\nu_-}{\nu_+}+\|W_0^+\|_{\chi^{-1}})\exp\left\{\frac{1}{2a_1}\int_0^t \|W^+\|_{\chi^0}^2 d\tau \right\}\\
\le& (\|W_0^-\|_{\chi^{-1}}+\frac{\nu_-}{\nu_+-\frac{a}{2}}(\frac{b\nu_-}{\nu_+}+\|W_0^+\|_{\chi^{-1}})\exp\left\{\frac{1}{2a_1}\|W^+\|_{L^\infty_t(\chi^{-1})}
\|W^+\|_{L^1_t(\chi^1)}\right\}\\
\le& (\|W_0^-\|_{\chi^{-1}}+\frac{\nu_-}{\nu_+-\frac{a}{2}}(\frac{b\nu_-}{\nu_+}+\|W_0^+\|_{\chi^{-1}})\exp\left\{\frac{2a\nu_+}{a_1(2a\nu_+-b^2)(2\nu_+-a_1)}
(\frac{b\nu_-}{\nu_+}+\|W^+_0\|_{\chi^{-1}})^2
\right\}.
\end{aligned}$$ which indicates that there exists constant $C$ such that $$\begin{aligned}
\|W^-\|_{L^\infty_t(\chi^{-1})}+&(1-\frac{C_1}{2})\nu_+\|W^-\|_{L^1_t(\chi^1)}\\
\le& \left(\|W_0^-\|_{\chi^{-1}}+\frac{C\nu_-}{\nu_+}(\nu_-+\|W_0^+\|_{\chi^{-1}})\right)\exp\left\{\frac{C}{\nu_+^2}(\nu_-+\|W_0^+\|_{\chi^{-1}})^2\right\}\\
\le& (2-\epsilon_0)\nu_+.
\end{aligned}$$ This implies that $$\|W^-\|_{L^\infty_t(\chi^{-1})}+\nu_+\|W^-\|_{L^1_t(\chi^1)}<\frac{2(2-\epsilon_0)}{2-C_1}<b.$$ Therefore, by standard continuous method, we get $T^\star=\bar{T}=\infty.$ This concludes the proof of Theorem \[t2\].
.4in
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Since the onset of the ‘space revolution’ of high-precision high-cadence photometry, asteroseismology has been demonstrated as a powerful tool for informing Galactic archaeology investigations. The launch of the NASA TESS mission has enabled seismic-based inferences to go full sky – providing a clear advantage for large ensemble studies of the different Milky Way components. Here we demonstrate its potential for investigating the Galaxy by carrying out the first asteroseismic ensemble study of red giant stars observed by TESS. We use a sample of 25 stars for which we measure their global asteroseimic observables and estimate their fundamental stellar properties, such as radius, mass, and age. Significant improvements are seen in the uncertainties of our estimates when combining seismic observables from TESS with astrometric measurements from the Gaia mission compared to when the seismology and astrometry are applied separately. Specifically, when combined we show that stellar radii can be determined to a precision of a few percent, masses to 5-10% and ages to the 20% level. This is comparable to the precision typically obtained using end-of-mission [*Kepler*]{} data.'
author:
- Víctor Silva Aguirre
- Dennis Stello
- Amalie Stokholm
- 'Jakob R. Mosumgaard'
- 'Warrick H. Ball'
- Sarbani Basu
- Diego Bossini
- Lisa Bugnet
- Derek Buzasi
- 'Tiago L. Campante'
- Lindsey Carboneau
- 'William J. Chaplin'
- Enrico Corsaro
- 'Guy R. Davies'
- Yvonne Elsworth
- 'Rafael A. García'
- Patrick Gaulme
- 'Oliver J. Hall'
- Rasmus Handberg
- Marc Hon
- Thomas Kallinger
- Liu Kang
- 'Mikkel N. Lund'
- Savita Mathur
- Alexey Mints
- Benoit Mosser
- Zeynep Çelik Orhan
- 'Thaíse S. Rodrigues'
- Mathieu Vrard
- 'Mutlu Y[i]{}ld[i]{}z'
- 'Joel C. Zinn'
- Sibel Örtel
- 'Paul G. Beck'
- 'Keaton J. Bell'
- Zhao Guo
- Chen Jiang
- 'James S. Kuszlewicz'
- 'Charles A. Kuehn'
- Tanda Li
- 'Mia S. Lundkvist'
- Marc Pinsonneault
- Jamie Tayar
- 'Margarida S. Cunha'
- Saskia Hekker
- Daniel Huber
- Andrea Miglio
- 'Mario J. P. F. G. Monteiro'
- Ditte Slumstrup
- 'Mark L. Winther'
- George Angelou
- Othman Benomar
- Attila Bódi
- 'Bruno L. De Moura'
- Sébastien Deheuvels
- Aliz Derekas
- Maria Pia Di Mauro
- 'Marc-Antoine Dupret'
- Antonio Jiménez
- Yveline Lebreton
- Jaymie Matthews
- Nicolas Nardetto
- 'Jose D. do Nascimento, Jr.'
- Filipe Pereira
- 'Luisa F. Rodríguez Díaz'
- 'Aldo M. Serenelli'
- Emanuele Spitoni
- Edita Stonkutė
- Juan Carlos Suárez
- Robert Szabó
- Vincent Van Eylen
- Rita Ventura
- Kuldeep Verma
- Achim Weiss
- Tao Wu
- Thomas Barclay
- 'Jørgen Christensen-Dalsgaard'
- 'Jon M. Jenkins'
- Hans Kjeldsen
- 'George R. Ricker'
- Sara Seager
- Roland Vanderspek
bibliography:
- 'TESSwg7.bib'
title: 'Detection and characterisation of oscillating red giants: first results from the TESS satellite'
---
Introduction {#sec:intro}
============
Asteroseismology of red giant stars has been one of the major successes of the CoRoT and [*Kepler*]{} missions. The unambiguous detection of non-radial oscillations has fundamentally widened our understanding of the inner workings of red giants, including the conditions in their core [e.g., @2011Natur.471..608B]. The observed frequency spectra have allowed the determination of the physical properties of thousands of red giants to an unprecedented level of precision [e.g., @2013MNRAS.429..423M], paving the way for the emergence of asteroseismology as a powerful tool for Milky Way studies and Galactic archaeology [e.g., @Miglio09; @2016MNRAS.455..987C; @Anders17; @2018MNRAS.475.5487S; @2019arXiv190412444S]. The [*Transiting Exoplanet Survey Satellite*]{} [TESS, @2015JATIS...1a4003R] is on the path of continuing this legacy with its all-sky survey that is expected to increase the number of detected oscillating red giants by an order of magnitude compared to the tens of thousands reported by its predecessors CoRoT and [*Kepler*]{}.
In the nominal TESS mission, the ecliptic northern and southern hemispheres are each observed during thirteen 27-day-long sectors, and most (92%) of the surveyed sky will be monitored for just 1-2 sectors. Except for the 20,000 targets pre-selected in each sector for 2-min cadence observations, all stars are observed as part of the full frame images obtained in 30-min cadence, similar to the long cadence sampling of the [*Kepler*]{} satellite. The length of the observations sets the lower limit on the oscillation frequencies one can resolve, and the sampling sets the upper frequency limit. We know from previous [*Kepler*]{} observations that one month of 30-min cadence data should be well suited to detect oscillations in the low red-giant branch and sufficient to measure the global oscillation properties characterising the frequency spectrum, in particular, its frequency of maximum power, [$\nu_{\mathrm{max}}$]{}, and the frequency separation between overtone modes, [$\Delta\nu$]{} [@Bedding:2010ki]. These in turn can be used in combination with complementary data such as the effective temperature, [$T_{\mathrm{eff}}$]{}, the relative iron abundance, \[Fe/H\], and parallax, to obtain precise stellar properties (including ages) when applying asteroseismic-based grid modelling approaches [see e.g., @2017MNRAS.467.1433R; @2018ApJS..239...32P].
Due to the large sky coverage, approximately 97% of asteroseismic detections in red giants from the TESS nominal mission data are expected to come from stars observed for only one or two sectors[^1]. Here we set out to explore the capability of TESS to detect the oscillations in giants ranging from the base of the red giant branch to the red clump, determine their stellar properties, and use that to assess the prospects for Galactic archaeology studies using one to two sectors of TESS data.
Target selection {#sec:targets}
================
![‘Asteroseismic HR diagram’ showing (predicted) [$\nu_{\mathrm{max}}$]{} instead of luminosity. Red dots show the selected targets inside the black selection box. For reference, the Sun is shown as well as all [*Hipparcos*]{} stars brighter than 6th magnitude (grey dots). Solar metalicity MESA tracks from @Stello:2013jz are shown to guide the eye with masses in solar units indicated (pre- and post- helium core-ignition phases are shown separately).\[Fig:hrd\]](hrd.eps){width="\linewidth"}
Our goal is to have a representative sample of giants including the types of stars in which we can expect to detect oscillations from one sector 30-min cadence TESS data. We selected red-giant candidates observed during sectors 1 and/or 2 that were deemed viable for asteroseismic detections according to their predicted properties based on the [*Hipparcos*]{} catalogue [@VanLeeuwen:2007dc]. We first estimated the stellar [$T_{\mathrm{eff}}$]{} and luminosity using $B-V$ color, $V$-band, and [*Hipparcos*]{} parallax, and the color-temperature and bolometric correction relations of @1996ApJ...469..355F. We then obtained a prediction of [$\nu_{\mathrm{max}}$]{} ($\propto$ [$T_{\mathrm{eff}}$]{}$^{3.5} M/L$; solar scaled, e.g. @2018ApJS..236...42Y) for each star assuming a mass of 1.2 [M$_\odot$]{}, which is representative of a typical red giant as observed by [*Kepler*]{} [and unlikely to be more than a factor of two from the true value of each star, e.g. @2018ApJS..236...42Y]. We note that one of our targets (TIC 129649472) is a known exoplanet host star recently analysed by @2019arXiv190905961C.
To ensure that the selected targets were amenable to asteroseismic detection from one sector of 30-min cadence data, we required that they would have an expected [$\nu_{\mathrm{max}}$]{} in the range 30-220[$\mu$Hz]{} and [$T_{\mathrm{eff}}$]{} in the typical range of red giants of 4500-5200$\,$K. In addition, we applied a narrower [$T_{\mathrm{eff}}$]{} range of 4500-4700$\,$K for the stars with [$\nu_{\mathrm{max}}$]{} between 30[$\mu$Hz]{} and 70[$\mu$Hz]{}, to avoid having red clump stars dominating our sample. The resulting sample of stars span evolutionary phases from the base of the red giant branch to the red giant branch bump, as well as some clump stars.
From this sample, we selected the 25 brightest targets for light curve extraction and asteroseismic analysis. The faintest stars in our sample turned out to be $\sim$6-7th magnitude in $V$ band (see Table \[tab:props\]). Under the assumption that the photometric performance of TESS is similar to [*Kepler’s*]{}, apart from its smaller aperture, this magnitude limit is equivalent to 11-12th magnitude for [*Kepler*]{}. Because single-quarter observations from [*Kepler’s*]{} second life, K2, showed no oscillation detection bias for red giants brighter than around 12th magnitude [@2017ApJ...835...83S] we would expect to detect oscillations in all 25 giants with TESS.
Figure \[Fig:hrd\] illustrates the location of the selected stars in the HR-diagram and the applied selection criteria. We confirmed that the stars were in sectors 1-2 using the Web TESS Viewing tool (WTV)[^2].
Data processing and asteroseismic analysis {#sec:data_analysis}
==========================================
![Power spectra sample of our targets representative of the [$\nu_{\mathrm{max}}$]{} range that they cover from around the red clump (top) to the low luminosity red giant branch (bottom). [*Left*]{}: Spectra shown in log-log space (smoothed in red) showing the location of the oscillation power excess, [$\nu_{\mathrm{max}}$]{}, indicated by red arrows on top of a frequency-dependent granulation background and flat white noise component. [*Right*]{}: Close-up of spectra showing locations of the roughly equally-spaced radial modes (using red equally-spaced vertical lines to guide the eye) and their average separation, [$\Delta\nu$]{} (red horizontal arrows). In the three bottom panels multiple dipole ($l=1$) mixed modes are resolved in between consecutive radial modes as indicated by the black brackets. \[Fig:PS\]](powerspec.eps){width="\linewidth"}
The stars selected were included in an early release of processed data from the TASOC pipeline[^3]. The calibrated full frame images were produced by the TESS Science Processing Operations Center (SPOC) at NASA Ames Research Center [@jenkinsSPOC2016], and processed by combining the methodology from the K2P2 pipeline [@2015ApJ...806...30L] for extracting the flux from target pixel data with the KASOC filter for systematics correction [@2014MNRAS.445.2698H]. The resulting TASOC light curves were high-pass filtered using a filter width of 4 days, corresponding to a cut-off frequency of approximately 3[$\mu$Hz]{}, and $4\sigma$ outliers were removed. Finally, we used linear interpolation to fill gaps that lasted up to three consecutive cadences and derived the Fourier transforms (power frequency spectra) of each light curve.
The light curves for the seven stars observed in both sectors were merged. To follow the approach anticipated for the millions of light curves from the TESS full frame images in the future, we first applied the neural network-based detection algorithm by @Hon:2018jr resulting in detection of oscillations in the power spectra of all stars except one. The non-detection (TIC 204314449) is listed as an A2 dwarf and a ’Visual Double’ in the University of Michigan Catalogue of two-dimensional spectral types for the HD stars [@1994ASPC...60..285H], and hence possibly too hot to show solar-like oscillations, or potentially contaminated. For the current test case, the number of stars was small enough that we visually checked the results, which confirmed all detections and the non-detection. The power spectra of a representative sample of the stars are shown in Figure \[Fig:PS\] showing clear oscillation excess power and the frequency pattern required to measure both [$\nu_{\mathrm{max}}$]{} and [$\Delta\nu$]{}.
The neural network also supplies a rough estimate for [$\nu_{\mathrm{max}}$]{}, which we provided as a prior to 13 independent groups analysing the power spectra to extract high-precision values of both [$\nu_{\mathrm{max}}$]{}, [$\Delta\nu$]{}, and their respective uncertainties using their preferred method. These methods have been thoroughly tested and described in the literature .
From the 13 independent determinations of the global asteroseismic parameters we adopted as central reference value for [$\Delta\nu$]{} and [$\nu_{\mathrm{max}}$]{} the results from the pipeline by , as this method was on average closest to the ensemble mean after applying a 2-$\sigma$ outlier rejection. Uncertainties in the global asteroseismic parameters obtained by the selected pipeline are at the 1.9% and 2.4% level for [$\Delta\nu$]{} and [$\nu_{\mathrm{max}}$]{}, respectively. These uncertainties are of comparable magnitude to those obtained from a single campaign with the K2 mission [see appendix in @2017ApJ...835...83S] and about twice as large as those extracted from 50 days of [*Kepler*]{} observations [see Figs. 3 and 4 in @Hekker:2012ic]. We report the central values and statistical uncertainties in [$\Delta\nu$]{} and [$\nu_{\mathrm{max}}$]{} from the selected pipeline for all targets in Table \[tab:props\].
For each star, we take into account the scatter across the different methods by adding in quadrature the standard deviation among the central values retained after the 2-$\sigma$ outlier rejection procedure to the formal uncertainty reported by the selected reference method. This consolidation process yields median uncertainties of 3.9% in [$\Delta\nu$]{} and 2.6% in [$\nu_{\mathrm{max}}$]{}, where the individual contribution arising from this systematic component to the total uncertainty is listed in Table \[tab:props\]. We note that we could decrease the level of uncertainties resulting from our ‘blind’ statistical consolidation approach by for example checking the [$\Delta\nu$]{} and [$\nu_{\mathrm{max}}$]{} results against the power spectra and/or échelle diagrams [see Fig. 5 in @Stello:2011hu]. However, we want to draw a realistic picture of the uncertainties one can expect when dealing with large ensembles of stars (as expected from TESS) where detailed ’boutique’ analysis/checking on a star-by-star basis is not practically feasible. Hence, our quoted uncertainties are conservative, but representative for analysis of TESS red giants where several pipelines are involved.
Derived stellar properties {#sec:stel_prop}
==========================
We have determined stellar properties for a subsample of 17 stars that had spectroscopic measurements of effective temperature and chemical composition available in the literature. Since one of our goals is to follow the same analysis procedure expected for large ensembles of stars, we assumed fixed uncertainties in [$T_{\mathrm{eff}}$]{}and \[Fe/H\] of 80 K and 0.08 dex, which are at the level of those provided by current large-scale spectroscopic surveys. To extract the physical properties of our sample, the atmospheric information was complemented with the asteroseismic scaling relations: $$\label{eqn:sca_dnu}
\left(\frac{\Delta\nu}{\Delta\nu_{\sun}}\right)^{2} \simeq
\frac{\overline{\rho}}{\overline{\rho}_{\sun}}$$ $$\label{eqn:sca_num}
\left(\frac{\nu_{\mathrm{max}}}{\nu_{\mathrm{max},\sun}}\right) \simeq
\frac{M}{M_{\sun}}
\left(\frac{R}{R_{\sun}}\right)^{-2}
\left(\frac{T_{\mathrm{eff}}}{T_{\mathrm{eff},\sun}}\right)^{1/2} \,,$$ where we adopted $\Delta\nu_{\sun}=135.5$ ($\mu$Hz) and $\nu_{\mathrm{max},\sun}=3140$ ($\mu$Hz) as obtained by our reference pipeline from the analysis of solar data.
Seven teams independently applied grid-based modelling pipelines based on stellar evolution models or isochrones to determine the main physical properties of the targets . When matching the models to the atmospheric properties and the global asteroseismic parameters [$\Delta\nu$]{} and [$\nu_{\mathrm{max}}$]{} the pipelines yielded median uncertainties of $\sim$6% in radius, $\sim$14% in mass, and $\sim$50% in age. These statistical uncertainties are of the same magnitude to those obtained with the K2 mission [@2019arXiv190412444S], as expected from the similar resulting errors in the global seismic parameters described in Section \[sec:data\_analysis\], and about a factor of two larger than what can be achieved with the full duration of the [*Kepler*]{} observations [@2018ApJS..239...32P].
In addition to the asteroseismic information, five of the pipelines can include parallaxes from Gaia DR2 [@GaiaCollaboration:2018dt] coupled with Tycho-2 observed $V$-magnitudes in their fitting algorithm to further constrain the stellar properties. As a consequence of having the additional constraint on stellar radius from the astrometry, the resulting uncertainties decrease to a level of $\sim$3% in radius, $\sim$6% in mass, and $\sim$20% in age. This level of precision resembles that obtained with the use of the full length of asteroseismic observations from the nominal [*Kepler*]{} mission, and emphasizes the potential of TESS for Galactic studies using red giants given its larger sky coverage, simple and reproducible selection function, and one order of magnitude higher expected yield of asteroseismic detections than any other previous mission.
![Comparison of stellar radii obtained with [BASTA]{} when fitting different combinations of input parameters: Gaia DR2 parallax and $V$-band magnitude ($\varpi$), global asteroseismic parameters ([$\Delta\nu$]{},[$\nu_{\mathrm{max}}$]{}), and all combined ($\varpi$,[$\Delta\nu$]{},[$\nu_{\mathrm{max}}$]{}). Effective temperature and composition are also fitted in all cases. See text for details.\[Fig:BASTA\_props\]](Radius_comp_sigma.pdf){width="\linewidth"}
To illustrate the differences in the obtained stellar properties arising from the selection of fitted observables, Fig. \[Fig:BASTA\_props\] shows the stellar radius obtained with one of the pipelines [[BASTA]{}, @2015MNRAS.452.2127S] when fitting different combinations of input parameters. The figure uses as the reference value the case when, in addition to the atmospheric properties, only the Gaia DR2 parallax and observed $V$-band magnitude are included in the fit. For the majority of the targets the results are consistent across the three sets within their formal statistical uncertainties. A summary of the measured and derived stellar properties for our targets can be found in Table \[tab:props\], where we have listed the central values and statistical uncertainties obtained with the [BASTA]{} pipeline, and determined the systematic contribution as the standard deviation across the results reported by all pipelines.
Two targets (TIC 141280255 and TIC 149347992) present a larger disagreement between the radii obtained with parallax and the seismic set ([$\Delta\nu$]{}, [$\nu_{\mathrm{max}}$]{}). We investigated if these discrepancies were due to the quality of the astrometric data by computing the re-normalised unit weight error (RUWE[^4]) for our sample of stars. In the case of TIC 141280255 we obtained a RUWE=1.98, which is above the value recommended by the Gaia team as a criterion for a good astrometric solution (RUWE$\leq1.4$). Therefore, we adopt for this star the stellar properties obtained from fitting the asteroseismic input only ([$\Delta\nu$]{}, [$\nu_{\mathrm{max}}$]{}).
In the case of TIC 149347992 the discrepancy is the result of predicted evolutionary phases: while the parallax-only solution suggests that the star in the clump phase, the asteroseismic fit favours a star in the red-giant branch. The combined fit therefore presents a bimodal distribution that encompasses these two families of solutions. A similar situation occurs in the fit of TIC 175375523, which shows agreement in the radius determined from different sets of input but has a fractional age uncertainty above unity when only ([$\Delta\nu$]{}, [$\nu_{\mathrm{max}}$]{}) are included in the fit. Its resulting age distribution is bimodal in this set as both red-giant branch and clump models can reproduce the observations, but the inclusion of parallax information favours the red giant branch solution and accounts for the $\sim17$% statistical uncertainty reported in Table \[tab:props\]. The availability of evolutionary classifications from deep neural networks trained on short [*Kepler*]{} data [@Hon:2018jr] would further decrease the obtained uncertainties by clearly disentangling these two scenarios.
![Distribution of fractional age uncertainties for our sample of stars determined by the [BASTA]{} pipeline fitting different combinations of available observables. The points indicate the individual values used to construct the Gaussian kernel density estimation. For better visualization we have excluded TIC 175375523 from the figure. See text for details.\[Fig:BASTA\_unc\]](Age_unc_BASTA.pdf){width="\linewidth"}
In Fig. \[Fig:BASTA\_unc\] we plot the distribution of fractional age uncertainties obtained with [BASTA]{} for the three considered cases of input, showing the clear improvement in precision when asteroseismic information and parallax are simultaneously included in the fit. For visualization purposes we have excluded the target TIC 175375523 from the figure. Our stellar ages at the 20% level are significantly more precise than what is obtained by data-driven and neural-network methods trained using asteroseismic ages from [*Kepler*]{} [above the 30% level, see e.g., @2019MNRAS.489..176M]. As a final remark, we note that asteroseismically derived properties of red giants are accurate to at least a similar level than our statistical uncertainties [below $\sim5$% and $\sim10$% for radii and masses, respectively. See discussion in e.g., @2018ApJS..239...32P and references therein]. We have made emphasis on our achieved precision instead of accuracy as our results could still be affected by a systematic component arising from uncertainties in evolutionary calculations, although recent investigations quantifying these effects at solar metallicity suggest that they are smaller than our statistical uncertainties [@2019arXiv191204909S].
Conclusions {#sec:conc}
===========
We presented the first ensemble analysis of red giants stars observed with the TESS mission. We selected a sample of 25 stars where we expected to detect oscillations based on their magnitude and parallax value, and analysed the extracted light curves in search for asteroseismic signatures in the power spectra. Our main findings can be summarized as follows:
- We detected oscillations in all the stars (except one that was likely incorrectly listed as a red giant). Despite the modest number of stars in our sample, our detection yield supports that the TESS photometric performance is similar to that of [*Kepler*]{} and K2 except shifted by about 5 magnitudes towards brighter stars due to its smaller aperture.
- Individual pipelines retrieve the global asteroseismic parameters with uncertainties at the $\sim$2% level in [$\Delta\nu$]{} and $\sim$2.5% in [$\nu_{\mathrm{max}}$]{}, which respectively increase to $\sim$4% and $\sim$3.5% when we take into account the scatter across results. We consider these uncertainties to be representative for the forthcoming ensemble analysis of TESS targets observed in 1-2 sectors, as individual validation of the results will not be feasible due to the large number of targets observed.
- Grid-based modelling techniques applying asteroseismic scaling relations were used to retrieve stellar properties for the 17 targets with spectroscopic information. Radii, masses, and ages were obtained with uncertainties at the 6%, 14%, and 50% level, and decrease to 3%, 6%, and 20% when parallax information from Gaia DR2 is included.
The expected number of red giants with detected oscillations by TESS ($\sim$500,000[^5]) greatly surpasses the final yield of [*Kepler*]{} ($\sim$20,000). In this respect, the combination of TESS observations, Gaia astrometry, and large scale spectroscopic surveys holds a great potential for studies of Galactic structure where precise stellar properties (particularly ages) are of key importance. We note that the recently approved extended TESS mission will change the 30-min cadence to 10 minutes, making it possible to detect oscillations of stars of smaller radii using the full frame images. This will enable more rigorous investigations of the asteroseismic mass scale for giants when anchored to empirical mass determinations (e.g., from eclipsing binaries) of turn-off and subgiant stars. This paper includes data collected by the TESS mission, which are publicly available from the Mikulski Archive for Space Telescopes (MAST). Funding for the TESS mission is provided by NASA’s Science Mission directorate. Funding for the TESS Asteroseismic Science Operations Centre is provided by the Danish National Research Foundation (Grant agreement no.: DNRF106), ESA PRODEX (PEA 4000119301) and Stellar Astrophysics Centre (SAC) at Aarhus University. VSA acknowledges support from the Independent Research Fund Denmark (Research grant 7027-00096B). DB is supported in the form of work contract FCT/MCTES through national funds and by FEDER through COMPETE2020 in connection to these grants: UID/FIS/04434/2019; PTDC/FIS-AST/30389/2017 & POCI-01-0145-FEDER-030389. LB, RAG and BM acknowledge the support from the CNES/PLATO grant. DB acknowledges NASA grant NNX16AB76G. TLC acknowledges support from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 792848 (PULSATION). This work was supported by FCT/MCTES through national funds (UID/FIS/04434/2019). EC is funded by the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 664931. RH and MNL acknowledge the support of the ESA PRODEX programme. T.S.R acknowledges financial support from Premiale 2015 MITiC (PI B. Garilli). KJB is supported by the National Science Foundation under Award AST-1903828. MSL is supported by the Carlsberg Foundation (Grant agreement no.: CF17-0760). MC is funded by FCT//MCTES through national funds and by FEDER through COMPETE2020 through these grants: UID/FIS/04434/2019, PTDC/FIS-AST/30389/2017 & POCI-01-0145-FEDER-030389, CEECIND/02619/2017. The research leading to the presented results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no 338251 (StellarAges). AM acknowledges support from the European Research Council Consolidator Grant funding scheme (project ASTEROCHRONOMETRY, G.A. n. 772293, http://www. asterochronometry.eu). AMS is partially supported by MINECO grant ESP2017-82674-R. JCS acknowledges funding support from Spanish public funds for research under projects ESP2017-87676-2-2, and from project RYC-2012-09913 under the ’Ramón y Cajal’ program of the Spanish Ministry of Science and Education. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products.
[lllllllBlllll]{} 13097379&114842&$59.10 \pm1.50\pm1.01$&$6.02 \pm0.03\pm0.24$&$6.646\pm0.010$&$4634\pm80 $&$0.04 \pm0.08$&13097379&$8.49 \pm0.28\pm0.17$&$1.22\pm0.08\pm0.06$&$6.10 \pm1.06\pm0.97$&\
38574220&19805&$29.40 \pm0.90\pm0.72$&$4.06 \pm0.20\pm0.26$&$5.577\pm0.009$&–&–&38574220&–&–&–&–\
38828538&21253&$189.90\pm1.60\pm0.42$&$14.90\pm0.10\pm0.13$&$5.896\pm0.009$&$4828\pm80 $&$0.11 \pm0.08$&38828538&$4.66 \pm0.10\pm0.06$&$1.21\pm0.05\pm0.03$&$6.20 \pm0.50\pm1.02$&\
39082723&4293&$49.30 \pm2.10\pm1.99$&$5.20 \pm0.10\pm0.06$&$5.574\pm0.009$&$4706\pm80 $&$-0.05\pm0.08$&39082723&$9.30 \pm0.27\pm0.17$&$1.19\pm0.09\pm0.07$&$5.90 \pm1.20\pm1.37$&\
47424090&112612&$28.30 \pm1.80\pm1.90$&$3.40 \pm0.10\pm0.29$&$6.930\pm0.010$&–&–&47424090&–&–&–&–\
70797228&655&$31.80 \pm1.50\pm0.75$&$4.37 \pm0.20\pm0.37$&$5.787\pm0.009$&$4750\pm80 $&$0.12 \pm0.08$&70797228&$11.27\pm0.61\pm0.47$&$1.19\pm0.13\pm0.11$&$6.80 \pm2.20\pm2.20$&\
77116701&103071&$48.30 \pm7.60\pm29.85$&$5.64 \pm0.20\pm3.37$&$8.568\pm0.018$&–&–&77116701&–&–&–&–\
111750740&113148&$142.60\pm2.70\pm1.11$&$11.80\pm0.10\pm0.23$&$5.658\pm0.009$&$4688\pm80 $&$0.16 \pm0.08$&111750740&$5.11 \pm0.16\pm0.09$&$1.06\pm0.07\pm0.05$&$10.80\pm1.78\pm1.71$&\
115011683&103836&$58.80 \pm1.20\pm0.97$&$6.10 \pm0.10\pm0.01$&$6.057\pm0.010$&$4590\pm80 $&$-0.13\pm0.08$&115011683&$7.92 \pm0.21\pm0.22$&$1.04\pm0.07\pm0.07$&$9.90 \pm1.63\pm2.01$&\
129649472&105854&$31.80 \pm1.20\pm1.39$&$4.20 \pm0.20\pm0.11$&$5.755\pm0.009$&$4748\pm80 $&$0.28 \pm0.08$&129649472&$10.85\pm0.60\pm0.24$&$1.13\pm0.12\pm0.06$&$8.50 \pm2.76\pm1.87$&\
139756492&106566&$27.60 \pm0.90\pm0.35$&$4.16 \pm0.20\pm0.65$&$6.819\pm0.010$&–&–&139756492&–&–&–&–\
141280255&25918&$150.40\pm1.00\pm0.57$&$12.52\pm0.02\pm0.10$&$5.307\pm0.009$&$4630\pm80 $&$0.33 \pm0.08$&141280255&$4.98 \pm0.12\pm0.05$&$1.07\pm0.06\pm0.03$&$11.70\pm2.62\pm2.39$&\
144335025&117075&$68.50 \pm1.60\pm0.64$&$7.35 \pm0.20\pm0.39$&$6.194\pm0.010$&–&–&144335025&–&–&–&–\
149347992&26190&$165.80\pm4.10\pm16.12$&$11.10\pm0.40\pm0.60$&$6.405\pm0.010$&$5132\pm80 $&$-0.17\pm0.08$&149347992&$7.20 \pm0.38\pm0.20$&$2.17\pm0.22\pm0.06$&$0.80 \pm0.30\pm0.10$&\
155940286&1766&$73.20 \pm1.30\pm0.32$&$7.40 \pm0.02\pm0.11$&$6.810\pm0.010$&$4630\pm80 $&$0.03 \pm0.08$&155940286&$6.95 \pm0.18\pm0.14$&$1.01\pm0.06\pm0.04$&$12.00\pm1.78\pm36.07$&\
175375523&114775&$60.00 \pm1.10\pm0.30$&$5.80 \pm0.10\pm0.14$&$5.899\pm0.009$&$4660\pm80 $&$0.26 \pm0.08$&175375523&$9.00 \pm0.30\pm0.58$&$1.38\pm0.09\pm0.21$&$4.50 \pm0.76\pm1.35$&\
183537408&117659&$57.90 \pm1.10\pm0.80$&$6.20 \pm0.20\pm0.42$&$6.781\pm0.010$&–&–&183537408&–&–&–&–\
204313960&113801&$106.00\pm3.30\pm1.47$&$9.40 \pm0.50\pm0.43$&$6.083\pm0.010$&$4897\pm80 $&$-0.20\pm0.08$&204313960&$6.50 \pm0.18\pm0.13$&$1.32\pm0.07\pm0.05$&$3.80 \pm0.57\pm0.57$&\
220517490&12871&$117.30\pm1.20\pm0.60$&$10.87\pm0.02\pm0.15$&$5.846\pm0.009$&$4961\pm80 $&$-0.26\pm0.08$&220517490&$5.61 \pm0.11\pm0.10$&$1.10\pm0.04\pm0.04$&$6.10 \pm0.50\pm1.18$&\
237914586&17440&$47.00 \pm1.40\pm1.33$&$5.74 \pm0.20\pm0.08$&$3.959\pm0.009$&–&–&237914586&–&–&–&–\
270245797&109584&$72.20 \pm1.70\pm0.64$&$6.60 \pm0.10\pm0.27$&$6.239\pm0.009$&$4824\pm80 $&$-0.10\pm0.08$&270245797&$8.73 \pm0.23\pm0.39$&$1.60\pm0.09\pm0.13$&$2.10 \pm0.22\pm0.64$&\
281597433&2789&$73.30 \pm1.00\pm1.37$&$7.20 \pm0.03\pm0.19$&$6.163\pm0.010$&$4700\pm80 $&$-0.41\pm0.08$&281597433&$6.77 \pm0.15\pm0.24$&$0.95\pm0.05\pm0.08$&$11.00\pm1.35\pm2.73$&\
439399563&343&$44.30 \pm1.40\pm0.66$&$4.54 \pm0.06\pm0.11$&$5.892\pm0.009$&$4778\pm80 $&$0.11 \pm0.08$&439399563&$10.69\pm0.34\pm0.40$&$1.44\pm0.11\pm0.14$&$3.50 \pm0.71\pm0.78$&\
441387330&102014&$46.60 \pm0.80\pm0.86$&$5.25 \pm0.10\pm0.25$&$5.592\pm0.009$&$4710\pm80 $&$-0.02\pm0.08$&441387330&$10.18\pm0.34\pm0.68$&$1.40\pm0.09\pm0.20$&$3.40 \pm0.57\pm1.19$&\
[^1]: Based on a preliminary simulation of the full TESS sky (TESS GI Proposal No G011188).
[^2]: <https://heasarc.gsfc.nasa.gov/cgi-bin/tess/webtess/wtv.py>
[^3]: T’DA Data Release Notes - Data Release 3 for TESS Sectors 1$+$2 (<https://doi.org/10.5281/zenodo.2510028>)
[^4]: see Gaia technical note GAIA-C3-TN-LU-LL-124-01 (<https://www.cosmos.esa.int/web/gaia/dr2-known-issues>)
[^5]: Based on a preliminary simulation of the full TESS sky (TESS GI Proposal No G011188).
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"pile_set_name": "ArXiv"
} |
---
abstract: 'We discuss non-perturbative aspects of string effective field theories with $N=1$ supersymmetry in four dimensions. By the use of a scalar potential, which is on-shell invariant under the supersymmetric duality of the dilaton, we study gaugino condensation in $(2,2)$ symmetric $Z_{N}$ orbifold compactifications. The duality under consideration relates a two-form antisymmetric tensor to a pseudoscalar. We show, that our approach is independent of the superfield-representation of the dilaton and preserves the $U(1)_{PQ}$ Peccei-Quinn symmetry exactly.'
---
15.5cm -4.5mm -10mm 0em
HUB–EP–95/19\
hep-th/9510022\
**On-Mass-Shell Gaugino Condensation\
in $Z_{N}$ Orbifold Compactifications**
Ingo Gaida$^{\hbox{\footnotesize{1,2}}}$ and Dieter Lüst$^{\hbox{\footnotesize{1}}}$,\
[*Institut für Physik, Humboldt–Universität,\
Invalidenstrasse 110, D–10115 Berlin, Germany*]{}
We discuss effective quantum field theories (EQFT’s) of strings with local N=1 supersymmetry in four dimensions. These theories are effective in the sense, that they are low-energy limits of a given higher dimensional string theory after dimensional reduction and integrating out all heavy modes. We restrict ourselves to the case of EQFT’s, which are only of second order in derivatives in the bosonic fields. In these EQFT’s the tree level gauge coupling constant is dynamical and can be expressed by the vacuum expectation value of the dilaton superfield. The dilaton superfield can be represented by the more familiar chiral superfield $S$ or the linear superfield $L$: $g_{tree}^{2} = 2 \ {\langle S+ \bar S \rangle}^{-1} = {\langle L \rangle}$. Throughout this paper $S+ \bar S$ will be denoted as the chiral representation of the dilaton (S-representation) and $L$ as the linear representation (L-representation).
It has been shown that these two superfield representations of the dilaton are connected via a supersymmetric legendre transformation called [*supersymmetric duality*]{} \[\[linear\], \[Grimm1\], \[Chern\_Simons\], \[Cardoso\], \[derendinger\], \[derque\]\]. This duality transformation destroys the holomorphic structure of the tree level gauge coupling constant. We will focus on the supersymmetric duality of the dilaton as an on-shell duality. In the simplest form it relates the two-form antisymmetric tensor $b_{mn}(x)$ to a pseudoscalar, the so called axion $a(x)$, via the following algebraic equation:
$$\begin{aligned}
\label{comp_duality_trafo}
\partial_{m} \ a(x) &=& - \ {\varepsilon}_{mnpq} \ \partial^{n} \ b^{pq}(x)\end{aligned}$$
The two-form antisymmetric tensor is a physical degree of freedom and plays an important role in the four dimensional formulation of the Green-Schwarz anomaly cancellation mechanism \[\[Cardoso\],\[derendinger\]\].
Since one integrates out only the massive states to go to the effective theory, the universal degrees of freedom, namely the graviton, the antisymmetric tensor and the dilaton, appear in the low energy supergravity action of four-dimensional, $N=1$ supersymmetric heterotic strings. Due to phenomenological reasons $N=1$ supersymmetry must be broken and gaugino condensation provides a promising mechanism for spontaneous supersymmetry breaking \[\[gaugino\]\]. At the level of an string effective supergravity action this has been studied extensively in the $S$-representation of global and local supersymmetry \[\[filq\],\[fmtv\],\[dual\]\].
Because the $L$-representation contains a two-form antisymmetric tensor this formulation seems to be the more natural EQFT of strings. It has been shown, that this EQFT most directly reproduces results calculated in the underlying string theory \[\[mayr\], \[antoniadis\_1\]\]. Thus, it is an important task to formulate supersymmetry breaking by gaugino condensation in the linear representation of the dilaton. This issue was recently discussed by us \[\[gaida\]\], for some other interesting considerations see also \[\[derque\], \[binetruy\_gaillard\]\].
In this context we have to face a puzzle: Gaugino condensation is known to be a non-perturbative effect producing an effective scalar potential $V \sim e^{-1/g^{2}}$. This can be achieved in the $S$-representation at tree level by a non-perturbative superpotential ${\omega}_{np} \sim e^{-S}$, because $S$ is a chiral superfield. This is - first of all - impossibel in the $L$-representation, because the linear multiplet is not chiral and can therefore not enter the superpotential. On the other hand the duality (\[comp\_duality\_trafo\]) is independent of the dilaton at component level and contains derivatives. How can then the $e^{-1/g_{tree}^{2}}$ dependence of the effective scalar potential be spoilt by the duality transformation?
We will show in the following that the formation of gaugino condensates can be consistently formulated in both representations of the dilaton. However, our approach only works on-shell. That is to say, the off-shell structure of the discussion given here is still an open problem in many aspects. One important result of our approach is, that the well-known $U(1)_{PQ}$ Peccei-Quinn symmetry is exactly preserved:
$$\begin{aligned}
\label{pq_symmetry_1}
b_{mn}(x) \ &\rightarrow& \ b_{mn}(x)
\ + \ \partial_{m} \ b_{n} (x)
\ - \ \partial_{n} \ b_{m} (x)
\\
\label{pq_symmetry_2}
a(x) \ &\rightarrow& \ a(x) \ + \ \Theta \hspace{2cm} , \ \ \Theta \in {\bf R}
\\
\label{pq_symmetry_3}
b_{mn}(x) \ &\rightarrow& \ b_{mn}(x)
\ + \ c_{mn} \hspace{1,2cm} , \ \ c_{mn} \in {\bf R}\end{aligned}$$
Note that (\[pq\_symmetry\_1\]) is known to be the gauge symmetry of a two-form antisymmetric tensor, whereas (\[pq\_symmetry\_2\]) and (\[pq\_symmetry\_3\]) are global shift symmetries of the axion and the antisymmetric tensor respectively. The on-shell duality transformation (\[comp\_duality\_trafo\]) is invariant under the $U(1)_{PQ}$ symmetry.
The paper is organized as follows: After a short introduction of various $N=1$ off-shell multiplets we discuss the supersymmetric duality of the dilaton. These results are already well-known. Then we derive a duality-invariant scalar potential and study gaugino condensation in $(2,2)$ symmetric $Z_{N}$ orbifold compactifications. We present a duality-invariant discussion of gaugino condensation for several gauge groups[^1].
A general supersymmetric multiplet $Z$ has the following structure at component level: $Z \sim (\mbox{Bosons} \ | \ \mbox{Fermions} \ || \ \mbox{Auxiliary-Fields} )$. One of the basic objects in any superspace formulation \[\[Grimm1\],\[wess\_and\_bagger\],\[Grimm\_2\]\] are chiral $4_{B} + 4_{F}$ multiplets $ {\Sigma}\sim ( A \ | \ {\chi}_{{\alpha}} \ || \ F ) $, because their lowest components are scalar fields parametrizing a Kähler manifold \[\[zumino\]\]. These chiral superfields obey the constraint $\bar{{\cal D}}^{\dot{\alpha}} \ {\Sigma}= 0$ and are defined at component level as
$$\begin{aligned}
{\Sigma}_{|} &=& A(x)
\hspace{1cm}
{{\cal D}}_{{\alpha}} {\Sigma}_{|} = \sqrt{2} \ {\chi}_{{\alpha}}(x)
\hspace{1cm}
{{\cal D}}^{2} {\Sigma}_{|} = - 4 \ F(x).\end{aligned}$$
We will denote in the following all possible chiral superfields by ${\Sigma}$ except the dilaton in the chiral representation. The $4_{B} + 4_{F} $ Yang Mills multiplet $W_{{\alpha}}^{ \ (r)} \sim ( a_{m}^{ \ (r)} \ | \ \lambda_{{\alpha}}^{ \ (r)} \ || \
D^{(r)} )$ with the index $r$ belonging to the internal gauge group $G_{(r)}$ is defined as
$$\begin{aligned}
W_{{\alpha}|}^{ \ (r)} &=& -i \ \lambda_{{\alpha}}^{ \ (r)}(x)
\hspace{1cm}
{{\cal D}}_{{\beta}} W_{{\alpha}|}^{ \ (r)} = - {\varepsilon}_{{\beta}{\alpha}} D^{(r)}(x)
-i \ ({\sigma}^{mn}{\varepsilon})_{{\beta}{\alpha}} \ f_{mn}^{ \ \ (r)}(x).\end{aligned}$$
The Yang-Mills prepotential $V$ satisfies $W_{{\alpha}} = - \frac{1}{4} \ (\bar{{\cal D}}^{2} - 8 R) \
e^{-2V} \ {{\cal D}}_{{\alpha}} \ e^{2V} $. By constructing the chiral density ${{\cal E}}= e + i e {\theta}{\sigma}^{a} \bar\psi_{a} - e {\theta}^{2} (\bar M +
\bar\psi_{a}\bar{\sigma}^{ab}\bar\psi_{b})$ one finds the $12_{B} + 12_{F} $ minimal multiplet for the supergravity sector \[\[minimal\_multiplet\]\], which we denote by the supercurvature $R \sim ( e_{m}^{ \ a} \ | \ \psi_{m}^{ \ {\alpha}} \ || M, \ b_{a} )$, namely the graviton, the gravitino and two auxiliary fields \[\[wess\_and\_bagger\]\]. The reducibel $8_{B} + 8_{F} $ linear multiplet
$$\begin{aligned}
L &\sim& ( C, b_{mn}, a_{m}^{ \ (r)} \ | \
\varphi_{{\alpha}}, \lambda_{{\alpha}}^{ \ (r)} \ || \
D^{(r)} )\end{aligned}$$
is the difference of the $4_{B} + 4_{F} $ Chern-Simons superfield $ {\Omega}\sim ( a_{m}^{ \ (r)} \ | \ \lambda_{{\alpha}}^{ \ (r)} \ || \ D^{(r)} ) $ and the $4_{B} + 4_{F} $ real linear multiplet $ l \sim ( C, b_{mn} \ | \ \varphi_{{\alpha}} \ || \ -)$. It satisfies the following two constraints:
$$\begin{aligned}
\label{linear_multiplet_constraint}
(\bar{{\cal D}}^{2} - 8 R) \ L = -2 \ k_{(r)} \ \ W^{{\alpha}(r)} W_{{\alpha}(r)}
\hspace{2cm}
L = l - \ k_{(r)} \ {\Omega}^{(r)} = L^{+}\end{aligned}$$
The parameter $k_{(r)}$ denotes the normalization of the gauge group generators $tr \ T_{(r)} T_{(s)} = k_{(r)} \ \delta_{(r)(s)}$ and is in the context of string theory the level of the Kac-Moody current algebra of $G_{(r)}$. The linear mutliplet $L$ is the $N=1$ limit of the $N=2$ vector-tensor multiplet \[\[vector\_tensor\]\]. Both multiplets have the same field content, but the $N=2$ vector-tensor multiplet is irreducibel because of the extended supersymmetry. The reducibel parts of the linear superfield, namely the real linear superfield $l$ and the Chern-Simons superfield ${\Omega}$, satisfy $( \bar{{\cal D}}^{2} - 8 R ) \ l = 0$ and $(\bar{{\cal D}}^{2} - 8 R) \ {\Omega}= 2 \ \mbox{tr} \ W^{{\alpha}} W_{{\alpha}}$ respectively. This can be used to write a local F-density into a local D-density up to total derivatives: Consider the following Yang Mills action with an arbitrary chiral function $F({\Sigma})$ $$\begin{aligned}
\label{test_action}
\int d^{2} {\theta}\ F({\Sigma}) \ W^{{\alpha}} W_{{\alpha}}
+ \mbox{total derivatives} + h.c.
= -2 \int d^{4} {\theta}\left \{
F({\Sigma}) + \bar F( \bar{\Sigma})
\right \} {\Omega}\end{aligned}$$ The RHS of (\[test\_action\]) is [*exactly*]{} invariant under the following shift $ F({\Sigma}) \rightarrow F({\Sigma}) \ + \ i \ \Theta $. On the LHS of (\[test\_action\]) this holds only if one takes the boundary terms into account. The LHS of (\[test\_action\]) [*without*]{} boundary terms contains at component level the CP-odd term $f \tilde f$ coupled to the axion. In perturbation theory these boundary terms can be ignored, but non-perturbatively this is not obvious. This has to be taken into account in the discussion of gaugino condensation. Furthermore we want to mention, that the real Yang-Mills Chern-Simons superfield ${\Omega}$ and the chiral Yang-Mills superfield $ W^{{\alpha}} $ have the same field content. That is why the superfield representation of the component fields $( a_{m}^{ \ (r)} \ | \ \lambda_{{\alpha}}^{ \ (r)} \ || \ D^{(r)} ) $ is not unique.
The linear multiplet $L$ contains a real scalar $C$, which is called dilaton in this framework, its supersymmetric partner, the dilatino $\varphi_{{\alpha}}$, a two-form antisymmetric tensor $b_{mn}$ and the Yang-Mills Chern-Simons three-form ${\omega}_{3Ynml} = - \mbox{tr} (a_{[l}\partial_{m}a_{n]}
- \frac{2i}{3} \ a_{[l} a_{m} a_{n]} )$ :
$$\begin{aligned}
\mbox{ln} L_{|} &=& C(x)
\nonumber\\
\nonumber\\
{{\cal D}}_{{\alpha}} \ \mbox{ln} L_{|} &=& \varphi_{{\alpha}}(x)
\nonumber\\
\nonumber\\
{\left[ {{\cal D}}_{{\alpha}},\bar{{\cal D}}_{\dot{\alpha}} \right]} L_{|}
&=& - \frac{4}{3} e^{C} b_{{\alpha}\dot{\alpha}}
+ 4 \ k_{(r)} \ \lambda^{(r)}_{{\alpha}} \bar\lambda_{\dot{\alpha}(r)}
+ {\sigma}_{k {\alpha}\dot{\alpha}}
\left \{
{\varepsilon}^{klmn} (\partial_{n} b_{ml}
- \frac{1}{3} \ k_{(r)} \ {\omega}_{3Ynml}^{(r)}
\right .
\nonumber\\ & &
\left .
+ i \ e^{C} \psi_{n} {\sigma}_{m} \bar\psi_{l})
+2i \ e^{C} ( \psi_{m} {\sigma}^{mk} \varphi -
\bar\psi_{m} \bar{\sigma}^{mk} \bar\varphi)
\right \} (x)\end{aligned}$$
Note that $L$ is invariant under the $U(1)_{PQ}$ symmetry. The duality transformed linear multiplet will be denoted as $S_{R} = S + \bar S$, where $S$ is a chiral multiplet. We define
$$\begin{aligned}
S_{|} &=& (e^{-C} \ + \ i \ a )(x)
\hspace{1cm}
{{\cal D}}_{{\alpha}} S_{|} = \sqrt{2} \ \rho_{{\alpha}}(x)
\hspace{1cm}
{{\cal D}}^{2} S_{|} = - 4 \ f(x)\end{aligned}$$
In general the gauge group G has a product structure $G = \prod_{(r)} G_{(r)}$. Nevertheless for $G_{(r)}$ the gauge coupling is defined at tree level as $g_{(r)}^{-2} = k_{(r)} \ {\langle e^{-C} \rangle}$. So these physical parameters of the different gauge groups $G_{(r)}$ are related to each other at tree level \[\[ginsparg\],\[GHMR\]\]. The combination $S + \bar S$ is invariant under the $U(1)_{PQ}$ symmetry because it depends on the axion only due to derivative terms. On shell the supersymmetric duality of the dilaton transforms the $2_{B} + 2_{F}$ real linear multiplet $ l \sim ( C, b_{mn} \ | \ \varphi_{{\alpha}} \ || \ - ) $ to the $2_{B} + 2_{F}$ multiplet $ S_{R} \sim ( C, \partial_{m} a \ | \ \rho_{{\alpha}} \ || \ - )$ without any difficulties. However, the off-shell structure of this duality relating two inequivalent off-shell theories to each other is still an open problem in many aspects, although the duality transformation can be performed off-shell.
All the multiplets we have introduced so far, except the linear multiplet, are irreducible. In the context of string compactifications it is interesting to build reducibel multiplets out of them and study their relationship to EQFT’s with extended supersymmetry. We have already mentioned, that the linear multiplet is associated to the $N=2$ vector-tensor multiplet. Furthermore the minimal multiplet and the real linear multiplet form the $16_{B} + 16_{F}$ multiplet $ (R + l) \sim ( e_{m}^{ \ a}, C, b_{mn} \ | \
\psi_{m}^{ \ {\alpha}}, \varphi_{{\alpha}} \ || \
M, \ b_{a} ) $, which was shown to be the $N = 1$ limit of a $N = 4$ EQFT \[\[nicolai\_1\]\]. In the end the minimal multiplet and the linear multiplet can combine to the $20_{B} + 20_{F}$ multiplet
$$\begin{aligned}
\label{enlarged_multiplet}
(R + L) &\sim& ( e_{m}^{ \ a}, a_{m}^{ \ (r)}, C, b_{mn} \ | \
\psi_{m}^{ \ {\alpha}}, \lambda_{{\alpha}}^{ \ (r)}, \varphi_{{\alpha}} \ || \
M, \ b_{a}, D^{(r)} ).\end{aligned}$$
We will couple the irreducible parts of this $20_{B} + 20_{F}$ multiplet to matter via the Kähler potential $ K({\Sigma}, \bar{\Sigma})$. The Kähler potential is as usual a real function depending on the supersymmetric matter multiplets ${\Sigma}$ of the underlying EQFT.
In the $U_{K}(1)$-superspace formulation of $N = 1$ supergravity \[\[Grimm1\],\[Grimm\_2\]\] the action contains three parts
$$\begin{aligned}
\label{U1_lagrangian_general}
{{\cal L}}&=& {{\cal L}}_{matter} +
{{\cal L}}_{pot} +
{{\cal L}}_{YM}\end{aligned}$$
with the three basic functions, namely the Kähler potential $K$, the superpotential ${\omega}$ and the gauge kinetic function $f_{(r)(s)}$:
$$\begin{aligned}
\label{U1_lagrangian}
{{\cal L}}_{matter}&=& m_{i} \int \ E[K_{i}]
\nonumber\\
{{\cal L}}_{pot} &=&\frac{1}{2} \int \frac{E}{R} \ e^{K/2} \ {\omega}({\Sigma})
+ h.c.
\nonumber\\
{{\cal L}}_{YM} &=& \frac{1}{2} \int \frac{E}{R}
\ W^{(r)} \ f_{(r)(s)} \ W^{(s)}
+ h.c.\end{aligned}$$
The parameter $m_{i}$ depends on the representation of the dilaton: It is useful in the following to introduce the parameter $n = 1/4$. Then $m_{i}$ is given as $ m_{linear} = 4n - 3 $ and $m_{chiral} = - 3 $.
Moreover we will use the following variations in $U_{K}(1)$-superspace for a general superfield $Z$ :
$$\begin{aligned}
\delta_{U} Z &=& -\frac{{{\cal W}}_{K}(Z)}{2m} \ \frac{\partial K}{\partial U} \
\delta U \ Z,\end{aligned}$$
The Kähler weights are given as ${{\cal W}}_{K}(E,L,l,\Omega,S,{\Sigma}, Y^{3}) = (-2,2,2,2,0,0,2)$, whereas the superfield $Y^{3} = W^{{\alpha}} W_{{\alpha}}$ plays an important role in the context of gaugino condensation, because its lowest component is given by gaugino bilinears. In our superspace formulation the following identity holds: $ \delta_{Y^{3}} \left ( Y^{3} \ e^{-K/2} \ \right ) = \delta Y^{3} e^{-K/2}$. And by the use of (\[linear\_multiplet\_constraint\]) we find
$$\begin{aligned}
\label{key_variation}
\delta_{Y^{3}} \int d^{4}\theta \ E[K] \ f({\Omega})
= -\frac{1}{2} \int d^{2}\theta \ E[K] \ \delta Y^{3}
\left \{
\frac{\partial f({\Omega})}{ \partial {\Omega}}
- \frac{{{\cal W}}_{K}(E)}{2m} \frac{\partial K}{ \partial {\Omega}} f({\Omega})
\right \}.\end{aligned}$$
It turns out, that this is a very helpful identity in the discussion of gaugino condensation.
Before we will derive a general, duality-invariant scalar potential, we want to have a first look at the duality (\[comp\_duality\_trafo\]) for an ordinary bosonic QFT in the presence of Chern-Simons forms: We start with an unconstrained lagrangian
$$\begin{aligned}
{{\cal L}}_{u}&=& \frac{1}{2} \ H^{m} H_{m}
\ + \
\left (
H^{m} \ - \ k_{(r)} {\Omega}^{(r) m}
\right ) \partial_{m} a\end{aligned}$$
with $ {\Omega}^{(r) m} = \frac{1}{3} {\varepsilon}^{mnpq} {\omega}_{3Ynpq}$. Variation with respect to $a(x)$ yields $ \partial_{m} ( H^{m} - k_{(r)} {\Omega}^{(r) m} ) = 0$ with the general solution $H^{m} = {\varepsilon}^{mnpq} \partial_{n} b_{pq} + k_{(r)} {\Omega}^{(r) m}$. This leads to the following action:
$$\begin{aligned}
\label{a_action_1}
{{\cal L}}( b_{pq}, a ) &=& \frac{1}{2} \ H^{m} H_{m}
+ \partial_{m}
\left (
a \ {\varepsilon}^{mnpq} \ \partial_{n} \ b_{pq}
\right )\end{aligned}$$
Note that (\[a\_action\_1\]) is an action of the antisymmetric tensor only, if one omits the boundary term. Furthermore ${{\cal L}}(b_{pq})$ is invariant under the $U(1)_{PQ}$ symmetry. Variation of ${{\cal L}}_{u}$ with respect to $H_{m}(x)$ yields $ H_{m} = - \partial_{m} a $. This leads to a lagrangian, which contains a pseudoscalar instead of the antisymmetric tensor:
$$\begin{aligned}
{{\cal L}}( \partial_{m} a ) &=&
- \frac{1}{2} \ \partial_{m} a \ \partial^{m} a
- \ a \ k_{(r)} \ f_{mn}^{(r)} \ \tilde f^{mn}_{(r)}
- \ \partial_{m} \left ( a \ k_{(r)} {\Omega}^{(r) m} \right )\end{aligned}$$
Here we used the usual definition $ \tilde f_{mn}^{ \ \ (r)} = \frac{1}{2} {\varepsilon}_{mnpq} f^{pq (r)} $ and the well-known identity $ \partial_{m} {\Omega}^{(r) m} = - f_{mn}^{(r)} \tilde f^{mn}_{(r)} $. Again the resulting lagrangian is invariant under the $U(1)_{PQ}$ symmetry. If we consider a constrained lagrangian ${{\cal L}}_{c}$, which is ${{\cal L}}_{u}$ satisfying the on-shell constraint: $ H^{m} = {\varepsilon}^{mnpq} \ \partial_{n} \ b_{pq}
\ + \ k_{(r)} {\Omega}^{(r) m}
= - \ \partial^{m} a $, then we can relate the two models to each other directly - one containing an antisymmetric tensor and the other a pseudoscalar.
So far we have only summarized well-known results that are useful in the following. We want to discuss the duality now in the framework of $N=1$ superspace: The tree-level Kähler potential under discussion in the linear representation of the dilaton will be $\tilde K = 4n \ \mbox{ln} (L/2) + K({\Sigma}, \bar {\Sigma})$. It already includes the tree level gauge kinetic function, but therefore it is not a well-defined Kähler potential in the sense, that it does not fulfill the Kähler condition \[\[gaida\]\]. Furthermore at one loop the Wilsonian gauge coupling function is determined by a holomorphic function of the chiral fields ${\Sigma}$: $f_{(r)(s)}^{[1]} = \delta_{(r)(s)} \ f^{[1]}({\Sigma}) $. We assume here, that in perturbation theory the Wilsonian gauge coupling function is beyond tree level independent of the dilaton \[\[DKL\]\]. The lagrangian considered here is evaluated at component level in \[\[Grimm1\]\]. The part of the lagrangian containing the auxiliary fields reads
$$\begin{aligned}
{{\cal L}}_{aux}/e = \frac{4n-3}{9} M \bar M
+ G_{i \bar j} F^{i} {\bar F}^{\bar j}
+ e^{{\Gamma}/2} \left (
\frac{4n-3}{3}(M+\bar M)
+ F^{i} G_{i}
+ \bar F^{\bar j} \bar G_{\bar j}
\right ),\end{aligned}$$
where G is the well-defined dual G-function: $ G = K({\Sigma}, \bar {\Sigma}) + \mbox{ln} |{\omega}|^{2}$. So the scalar potential from this part after elimination of the auxiliary fields is
$$\begin{aligned}
\label{linear_potential_1}
V_{1} &=& e^{{\Gamma}} \ (G_{i} \ G^{i \bar j} G_{\bar j} + 4n - 3 )\end{aligned}$$
The second part of the potential is the sum of all monomials coupling the dilaton $C$ only to tr $\lambda^2$:
$$\begin{aligned}
V_{2} &=&
(n \ \mbox{tr}\lambda^2 - e^{{\Gamma}/2} {\Gamma}_{L} )
\ {\Gamma}^{LL} \
(n \ \mbox{tr}\bar \lambda^2 - e^{{\Gamma}/2} {\Gamma}_{L} )
- 4n \ e^{{\Gamma}}\end{aligned}$$
Derivatives with respect to the linear multiplet are defined as $\partial_{L} = \frac{\partial}{\partial(2/L)}$. The whole potential can be rewritten as
$$\begin{aligned}
\label{linear_potential_1}
V &=& \left(
n \ \delta_{iL} \ tr \ \lambda^{{\alpha}} \lambda_{{\alpha}}
- e^{{\Gamma}/2} {\Gamma}_{i}
\right)
\ {\Gamma}^{i \bar j}
\left( n \ \delta_{\bar j L} \
tr \ \bar\lambda_{\dot{\alpha}} \bar\lambda^{\dot{\alpha}}
- e^{{\Gamma}/2} {\Gamma}_{\bar j}
\right)
- 3 e^{{\Gamma}} .\end{aligned}$$
The indices $i,\bar j$ include derivatives with respect to the linear muliplet now. And $\delta_{iL}$ is the Kronecker-delta. The result reduces to the known potential in the chiral limit $n=0$.
Performing now the duality transformation our lagrangian changes to an unconstrained lagrangian ${{\cal L}}_{u}$ by changing $l$ to $U$, where $U$ is unconstrained, and by adding a lagrange multiplier ${{\cal L}}_{lm}$. The lagrange multiplier contains the unconstrained field $U$ and the $S+\bar S$ multiplet.
$$\begin{aligned}
{{\cal L}}_{u} &=&
m \ \int \ E \
\left (
1 - \frac{2n}{m} \ U \ (S+\bar S)
\right )
+ {{\cal L}}_{pot}\end{aligned}$$
Variation with respect to $S+\bar S$ yields the old theory in the linear representation of the dilaton. Variation with respect to $U$ yields the chiral representation of the dilaton. Note that the variations depend on the representation of the dilaton. In the $U_{K}(1)$-superspace the torsion constraints and consequently the solution of the Bianchi-identities depend on the representation of the dilaton: In the notation of \[\[Grimm\_2\]\] the superdeterminant of the vielbein $E$, for instance, gets rescaled as $ E^{\prime} = E \ (X \bar X)^{2}$ with $X = \bar X = e^{K/4m}$. The rescalings are chosen in such a way, that the whole theory is automatically Einstein normalized. [^2] Thus, we find with $\tilde K(U) = 4n \ \mbox{ln} (U - k_{(r)} \Omega^{(r)} )$ the [*duality relation*]{} $ S + \bar S = 2/L$. Inserting this relation one ends up with the action in the chiral representation of the dilaton:
$$\begin{aligned}
\label{dual_action}
{{\cal L}}&=&
- 3 \ \int E[K] \left (
1 + \frac{2n}{3} k_{(r)} \Omega^{(r)} \ ( S + \bar S)
\right )
\nonumber\\ & &
+ \left\{
\frac{1}{2} \int \frac{E}{R} \ e^{K/2} \ {\omega}({\Sigma})
+ \frac{1}{2} \int \frac{E}{R} \ W^{(r)} \
f_{(r)(s)}^{ \ \ [1]} \ W^{(s)} \ + h.c.
\right\}\end{aligned}$$
This action is manifestly invariant under the $U(1)_{PQ}$ symmetry of the dilaton in the chiral representation. It depends only on $\partial_{m} a (x)$, because the theory in the $L$-representation only knows about the field strength of the antisymmetric tensor. One open problem is encoded in the fact, that off-shell we have in the $S$-representation the corresponding auxiliary field $f$, which is absent in the linear representation. But since we are discussing the supersymmetric duality only on-shell, we have to eliminate the auxiliary fields via their equations of motion. It is quite interesting, that the remaining scalar potential is the same than the one in the $L$-picture as we will show now: The Kähler potential and the gauge coupling function are given in the $S$-representation by $K = -4n \ \mbox{ln} (S + \bar S) + K({\Sigma}, \bar {\Sigma})$ and $f_{(r)(s)} = n \ \delta_{(r)(s)} \ \left( f^{[0]} + f^{[1]} \right)$ respectively. The tree-level gauge coupling function $ f^{[0]} = (S + \bar S) \ k_{(r)} = 2 \ k_{(r)}/L$ is part of a D-density. This point of view was already very successful in the discussion of non-holomorphic field-dependent contributions to the gauge coupling function \[\[derendinger\]\]. After performing the usual Kähler transformation to go to the G-function the auxiliary part of the lagrangian is given by
$$\begin{aligned}
{{\cal L}}_{aux}/e &=& - \frac{1}{3} M \bar M
+ G_{i \bar j} F^{i} {\bar F}^{\bar j}
- n \ F^{i} \ \delta_{iS} \ tr \ \lambda^{{\alpha}} \lambda_{{\alpha}}
- n \ \bar F^{\bar j} \ \delta_{\bar j \bar S}
\ tr \ \bar\lambda_{\dot{\alpha}} \bar\lambda^{\dot{\alpha}}
\nonumber\\ & &
+ e^{G/2}
\left (
F^{i} G_{i}
+ \bar F^{\bar j} \bar G_{\bar j}
- M - \bar M
\right ),\end{aligned}$$
with $ F^{i} \ \delta_{iS} = f $. Eliminating the auxiliary fields via their equation of motion leads to the scalar potential:
$$\begin{aligned}
\label{chiral_potential_1}
V &=& \left(
n \ \delta_{iS} \ tr \ \lambda^{{\alpha}} \lambda_{{\alpha}}
- e^{G/2} G_{i}
\right)
\ G^{i \bar j}
\left( n \ \delta_{\bar j \bar S} \
tr \ \bar\lambda_{\dot{\alpha}} \bar\lambda^{\dot{\alpha}}
- e^{G/2} G_{\bar j}
\right)
- 3 e^{G}\end{aligned}$$
This potential includes the usual scalar potential of ordinary matter fields of \[\[sugra\_1\]\] and is precisely (\[linear\_potential\_1\]) - the scalar potential derived in the linear representation of the dilaton.
Now we can use the duality-invariant scalar potential to study gaugino condensation in the two superfield-representations of the dilaton. We will restrict ourselves first of all to the case of one gauge group and take $k_{(r)} = 1$ for simplicity. We start in the linear representation of the dilaton: The superpotential consists of two parts: The first part is the so called quantum part and has its origin in chiral and conformal anomalies \[\[taylor1\]\]. The second part represents the one-loop threshold corrections to the gauge coupling function. Note that the superpotential is explicitly dilaton-free, because it is defined to be a chiral function. However it depends implicitly on the dilaton through the Kähler potential, which can enter the superpotential[^3]. The two parts combine to the following effective superpotential
$$\begin{aligned}
\label{effective_superpotential}
{\omega}({\Sigma}) &=& \frac{1}{{\beta}} \ Y^{3} e^{- \tilde K/2} \
\mbox{ln}
\left \{
c^{6} \ e^{ {\beta}n f^{[1]} } \ Y^{3} \ e^{- \tilde K/2}
\right \}\end{aligned}$$
with $ {\beta}= -24 \pi^{2}/b \ n $, where $b$ denotes the $N=1$ ${\beta}$-function coefficient. By the use of (\[key\_variation\]) we find the equations of motion for $Y^{3}$. Because the action contains the Yang-Mills Chern-Simons superfield ${\Omega}$ and its chiral projection $Y^{3}$ at the same time, the equation of motion splits into a non-holomorphic and a holomorphic part. This property can be used to introduce non-holomorphic terms into the superpotential. After an appropriate Kähler transformation with $\tilde K \rightarrow \Gamma = \tilde K + \mbox{ln} |{\omega}|^{2}$ we have
$$\begin{aligned}
\label{gaugino_constraint_01}
\lambda^{{\alpha}}\lambda_{{\alpha}} &=& e^{\Gamma/2} \ \frac{1}{1/ {\beta}+ n f^{[0]} }.\end{aligned}$$
So we find immediately, that in the weak coupling limit $f^{[0]} \rightarrow \infty$ the vacuum expectation value of the gaugino bilinears vanishes and consequently gaugino condensation does not take place. The scalar potential is given by (\[linear\_potential\_1\]) and by the use of the equations of motion for the gaugino bilinears we can integrate them out. At this point we want to show, that the results given here are invariant under the duality transformation: Performing the duality transformation the action is given by (\[dual\_action\]). Again the tree-level gauge coupling function is part of a D-density. Analogous to the linear representation of the dilaton the effective superpotential is given by (\[effective\_superpotential\]). Using (\[key\_variation\]) we find the equation of motion for $Y^{3}$ in the $S$-representation. This yields again (\[gaugino\_constraint\_01\]) with $G$ instead of $\Gamma$, of course. Again it is now straightforward to integrate out the gaugino bilinears using (\[chiral\_potential\_1\]). In the end the procedure of integrating out the gaugino bilinears via their equations of motion is not affected by the duality transformation.
As a concrete example we turn now to the discussion of orbifold models describing the compactification of the heterotic string from ten dimensions down to four \[\[orbifold\_ref\]\]. We will focus on (2,2) symmetric $Z_{N}$ orbifold compactifications without Wilson lines. In these models occurs the generic gauge group $E_{8} \ \bigotimes \ E_{6} \ \bigotimes \ H$ with $H = \{SU(3), SU(2) \times U(1), U(1)^{2} \}$. Orbifold compactifications posses various continuous parameters, called moduli, corresponding to marginal deformations of the underlying conformal field theory. These moduli enter the EQFT and they take their values in a manifold $\cal M$ called moduli space. For models with $N = 1$ space-time supersymmetry $\cal M$ is, locally, a Kählerian manifold. Thus, the corresponding Kähler potential describes the coupling of the moduli to $N = 1$ supergravity in the EQFT under consideration. The EQFT must respect target space modular symmetries (for a review see \[\[gpr\]\]) induced by the target space duality group. For orbifold compactifications the target space duality group is often given by the modular group $PSL(2,{\bf Z})$, acting on one chiral field $T$ as
$$\begin{aligned}
\label{modular_transformation_of_moduli}
T^{\prime} &=& \frac{a \ T - i \ b}{ i \ c \ T + d } \hspace{1cm}
ad - bc = 1 \hspace{1cm} a,b,c,d \in {\bf Z}\end{aligned}$$
where $T$ corresponds to an internal, overall modulus: $T=R^2+iB$. For simplicity we will discuss (2,2) symmetric $Z_{N}$ orbifolds without (1,2) moduli. At the massless level of these orbifolds one finds (1,1) moduli and matter fields. These fields can be in general (un-) twisted and (un-) charged. In the following the three diagonal elements of the untwisted uncharged (1,1) moduli are denoted by $T^{A}$, whereas all other charged (uncharged) fields are given by $Q_{ch}^{I} $ ($Q_{uch}^{I}$). We will only be interested in the lowest components of the superfields in order to study the vacuum structure of our theory. Since in the Wess-Zumino gauge we have $ V_{| WZ} = 0 $, it is not necessary for us to investigate the role of the charged and uncharged matter separately. So we define $Q^{I} = \{ Q_{ch}^{I}, Q_{uch}^{I} \}$ and use this definition in an obvious way.
Now we must specify the Kähler potential $\tilde K$ and the effective superpotential ${\omega}$. The Kähler potential $\tilde K$ can be generically expanded in powers of $Q^{I}$:
$$\begin{aligned}
\label{orbifold_kaehlerpotential}
\tilde K &=& \mbox{ln}(L/2) \ + \ \hat K(T) \ + \ Z_{I \bar J} (T , \bar T)
\ \bar Q^{\bar J} \ e^{2V} \ Q^{I}
\ + \ {\cal O}
\left(
( \bar Q Q)^{2}
\right )\end{aligned}$$
We choose the matter metric in the following diagonal form
$$\begin{aligned}
\label{orb_kaehler_2}
Z_{I \bar J} (T, \bar T) = \delta_{I \bar J} \ Z^{I} (T, \bar T)
\hspace{2cm}
Z_{I} (T, \bar T) = \prod_{A} \ ( T^{A} + \bar T^{A} )^{-q_{I}^{ \ A}},\end{aligned}$$
whereas the metric for the moduli is given by $\hat K_{i \bar j}$ with $\hat K = - \sum_{A=1}^{3} \ \mbox{ln}(T^{A}+\bar T^{A} )$. We will use the usual notation $K^{i \bar j} = K_{i \bar j}^{-1}$ also for the matter metric. The parameters $q_{I}^{ \ A}$ are the so called modular weights \[\[mod\_weights\]\]. The superpotential has the following structure
$$\begin{aligned}
\label{orbifold_superpotential}
{\omega}({\Sigma}) &=& P(Q,T) \ + \ \sum_{(r),A} {\omega}_{(r)}^{ \ A} ({\Sigma}).\end{aligned}$$
More precisely we have superpotentials of the following form in mind
$$\begin{aligned}
\label{orbifold_superpotential_2}
P(Q,T) &=& \frac{1}{3} \ Y_{IJK}(T) \ Q^{I} \ Q^{J} \ Q^{K}
\\
\label{orbifold_superpotential_3}
{\omega}_{(r)}^{ \ A} ({\Sigma}) &=& \frac{1}{ {\beta}_{(r)}^{A} }
\ Y^{3} e^{-\tilde K/2} \
\mbox{ln}
\left \{
c_{(r)}^{ \ A 6} \ Y^{3} \
e^{-\tilde K/2} \ e^{n {\beta}_{(r)}^{A} f^{[1] A}_{(r)}}
\right \},\end{aligned}$$
with the definition ${\beta}_{(r)}^{A} = - 24 \pi^{2} / b_{(r)}^{A} n $. The canonical dimensions of the fields are $dim (T^{A}, l, {\Omega}, Y, Q^{I}) = (0, 0, 2, 1, 1)$ and the transformation properties under the target space duality group (\[modular\_transformation\_of\_moduli\]) with $F_{A} = \mbox{ln} (icT_{A} + d)$ read
$$\begin{aligned}
\label{orbifold_kaehlerpotential_2}
T^{A} + \bar T^{\bar A}
&\rightarrow&
(T^{A} + \bar T^{\bar A}) \ e^{- (F_{A} + \bar F_{A})}
\hspace{1cm}
\mbox{(no sum over $A$)}
\nonumber\\
\nonumber\\
Z_{I \bar J} (T, \bar T)
&\rightarrow&
Z_{I \bar J} (T, \bar T) \ e^{q_{I}^{ \ A}(F_{A} + \bar F_{A})}
\hspace{1cm}
\mbox{(sum over $A$)}
\nonumber\\
\nonumber\\
Q_{I}
&\rightarrow&
Q_{I} \ e^{- q_{I}^{ \ A} F_{A} }
\hspace{2,8cm}
\mbox{(sum over $A$)}\end{aligned}$$
Therefore the target space duality transformations act just as Kähler transformations.
The potential is given in general by (\[linear\_potential\_1\]). Note that this is a ‘closed’ formula without specifying the superpotential:
$$\begin{aligned}
\label{z_n_potential_2}
G_{i} \ G^{i \bar j} G_{\bar j} &=& \frac{1}{t} \
\left \{
\frac{T_{R}^{ \ A 2}}{t} \ | t \ {\partial_{T^{A}}} \mbox{ln} \ {\omega}\ - \ \frac{1}{T_{R}^{ \ A}} |^{2}
\ - \ \frac{1}{t} \ |q_{I}^{ \ A 2} \ Q^{2} |^{2}
\right .
\nonumber\\ & &
+ \ Z^{I \bar J } \
\left (
q_{I}^{ \ A 2} \ Q^{2} \ | {\partial_{Q^{I}}} \mbox{ln} \ {\omega}|_{I \bar J}^{2}
\ + \ t \ | Z_{I \bar K} \bar Q^{\bar K} \
+ \ {\partial_{Q^{I}}} \ \mbox{ln} \ {\omega}|_{I \bar J}^{2}
\right .
\nonumber\\ & &
- \ | q_{I}^{ \ A} Z_{I \bar K} \bar Q^{\bar K}
\ + \ {\partial_{Q^{I}}} \ \mbox{ln} \ {\omega}|_{I \bar J}^{2}
\nonumber\\ & &
\left .
\left .
+ \ | q_{I}^{ \ A} Z_{I \bar K} \bar Q^{\bar K} \
T_{R}^{ \ A} {\partial_{T^{A}}} \mbox{ln} \ {\omega}\ + \ {\partial_{Q^{I}}} \mbox{ln} \ {\omega}|_{I \bar J}^{2}
\right )
\right \}\end{aligned}$$
All indices are contracted and the following definitions have been used: $ Q^{2} = Z_{I \bar J} (T , \bar T) \bar Q^{\bar J} Q^{I}$ and $ t = 1 - q_{I}^{ \ A 2} \ Q^{2} $. The equation of motion for the gaugino bilinears yields now
$$\begin{aligned}
\label{gaugino_constraint_orbifold_1}
tr \ \lambda^{{\alpha}}\lambda_{{\alpha}} &=&
\frac{1}{3} \ e^{\Gamma/2} \ tr \ \sum_{A=1}^{3} \
\left (
\frac{1}{ {\beta}_{(r)}^{A} } \ + \ n \ f_{(r)}^{[0]A}
\right )^{-1}\end{aligned}$$
with $f_{(r)}^{[0]A} = f_{(r)}^{[0]} /3 = 2k_{(r)}/3L = k_{(r)} S_{R} /3$. In the weak coupling limit gaugino condensation still disappears. Using the equation of motion for the gaugino bilinears leads to non-holomorphic contributions to the superpotential:
$$\begin{aligned}
\label{truncated_superpotential_orbifold}
{\omega}^{A}_{(r)} &=& - e^{-1} \ c^{A \ -6}_{(r)} \
e^{- n {\beta}^{A}_{(r)} ( f_{(r)}^{[0]A} + f_{(r)}^{[1]A} ) }
\left (
\frac{1}{ {\beta}_{(r)}^{A} } \ + \ n \ f_{(r)}^{[0]A}
\right )\end{aligned}$$
It is important to stress, that $f_{(r)}^{[0]A}$ can be expressed by the use of the equations of motion for $Y^{3}$ in a pure holomorphic way. This property is directly related to the fact, that we have to deal with the non-holomorphic Chern-Simons superfield ${\Omega}$ and its holomorphic projection $Y^{3}$ at the same time. As a consequence the remaining scalar potential has the desired non-perturbative structure $ V \sim e^{-(S + \bar S)} = e^{-2/L} \sim e^{-1/g^{2}_{tree}} $, if the theory is asymptotical free (${\beta}_{(r)}^{A} > 0$). Our approach produces this functional dependence on the dilaton as an overall factor in (\[truncated\_superpotential\_orbifold\]). So the $U(1)_{PQ}$ symmetry is still unbroken. The matter fields we have introduced are quantum fields with vanishing vacuum expectation value. Because we are interested in the vacuum structure of our theory, we take the limit $Q^{I} \rightarrow 0$.
Up to now it was not necessary to specify the one-loop contribution to the gauge-coupling function. Following \[\[DKL\]\] we have
$$\begin{aligned}
\label{one-loop_gauge_function}
f^{[1]A}_{(r)}({\Sigma}) &=& - \ \frac{b^{A}_{ \ (r)}}{ 8 \pi^{2}} \
\mbox{ln} \ \eta^{2}(T^{A}) ,\end{aligned}$$
where $ \eta(T^{A})$ is the well-known Dedekind function and reflects the one-loop threshold contributions of momentum and winding states of the underlying string theory. For simplicity we have assumed, that the coefficient which appears in the threshold correction is the $N=1$ ${\beta}$-function coefficient. This is the case for a hidden gauge group of pure Yang-Mills theory like $E_{8}$ in the absence of Green-Schwarz terms. The inclusion of a Green-Schwarz term was discussed in the last reference of \[\[dual\]\].
Using (\[gaugino\_constraint\_orbifold\_1\]) the scalar potential is given as
$$\begin{aligned}
\label{full_linear_potential_0}
V(C,T^{A}) =
\frac{ e^{C} }{2} \prod_{A=1}^{3} (T_{R}^{ \ A})^{-1}
| {{\omega}}|^{2}
\left \{
\sum_{A=1}^{3} T^{ \ A 2}_{R} \
|\frac{3{\hat G}_{2}(T^{A})}{2 \pi} + \frac{2}{ T_{R}^{ \ A}}|^{2}
+ k(C) - 3
\right \}\end{aligned}$$
with the Eisenstein function $\hat{G}_{2}(T) = G_{2}(T) \ - \ 2\pi/T_{R}$ and
$$\begin{aligned}
k(C) &=& | \ tr \ \sum_{A=1}^{3}
\frac{ {\beta}_{(r)}^{A} }{ 6e^{C} \ + \ {\beta}_{(r)}^{A} \ k_{(r)}}
+ 1 \ |^{2}.
\nonumber\\\end{aligned}$$
We have calculated the effective scalar potential for factorizable Kählerian moduli spaces ${\cal M}$ of the form ${\cal M} = {\cal M}^{(1,1)}_{dilaton} \ \bigotimes \ {\cal M}^{\prime}$ with ${\cal M}^{(1,1)} = SU(1,1)/U(1)$. It is well-known that supersymmetry can be broken via effective scalar potentials of the form (\[full\_linear\_potential\_0\]) with (\[one-loop\_gauge\_function\]) \[\[filq\]\]. This property does not depend on the representation of the dilaton \[\[gaida\]\].
To conclude, we have shown in this paper, that there exists a consistent on-mass-shell formulation of gaugino condensation in local $N=1$ string effective field theories in four dimensions.
We studied the one-loop anomalous contribution to the Wilsonian gauge coupling, discussed by Dixon, Louis and Kaplunovsky \[\[DKL\]\], and the anomalous contribution to the effective action of Taylor, Veneziano and Yankielowicz \[\[taylor1\]\]. Both can combine to an effective superpotential. Using this effective superpotential we studied supersymmetry breaking via gaugino condensation in (2,2) symmetric $Z_{N}$ orbifolds. In our approach we have integrated out the gaugino bilinears via their equations of motion. By the use of the Yang-Mills Chern-Simons superfield and its chiral projection the equations of motion split into a holomorphic and a non-holomorphic part. This result, which clearly only holds on-shell, leads to the non-perturbative structure of the effective scalar potential with the typical $e^{-1/g^{2}_{tree}}$ behaviour. Moreover we have shown that our approach is independent of the superfield-representation of the dilaton and preserves the $U(1)_{PQ}$ symmetry.
Finally, we want to mention that the duality invariant off-shell formulation of gaugino condensation is still an open problem in many aspects.
[**Acknowledgement:**]{} We would like to thank G. Lopes-Cardoso and C. Preitschopf for the many conversations and J.P. Derendinger for helpful discussions.
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[^1]: We use the convention $\kappa^{2} = 8 \pi / M_{pl}^{2} = 1$. Moreover we use the usual superspace notations $\int \equiv \int d^{4}\theta$ and $X_{|} \equiv X_{| \underline\theta = 0}$ with $\int d^{4}\theta = - \frac{1}{4} \int d^{2}\theta (\bar {{\cal D}}^{2} - 8R)$. We will refer to $ \int d^{4}\theta$ as a D-density and to $ \int d^{2}\theta$ as a F-density.
[^2]: In Lorentz-superspace the Weyl-rescaling of the graviton depends on the representation of the dilaton \[\[gaida\]\].
[^3]: This is also the case in the superconformal approach, where the compensators are functions of the Kähler potential.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Systemic risk is concerned with the instability of a financial system whose members are interdependent in the sense that the failure of a few institutions may trigger a chain of defaults throughout the system. Recently, several systemic risk measures are proposed in the literature that are used to determine capital requirements for the members subject to joint risk considerations. We address the problem of computing systemic risk measures for systems with sophisticated clearing mechanisms. In particular, we consider the Eisenberg-Noe network model and the Rogers-Veraart network model, where the former one is extended to the case where operating cash flows in the system are unrestricted in sign. We propose novel mixed-integer linear programming problems that can be used to compute clearing vectors for these models. Due to the binary variables in these problems, the corresponding (set-valued) systemic risk measures fail to have convex values in general. We associate nonconvex vector optimization problems to these systemic risk measures and solve them by a recent nonconvex variant of Benson’s algorithm which requires solving two types of scalar optimization problems. We provide a detailed analysis of the theoretical features of these problems for the extended Eisenberg-Noe and Rogers-Veraart models. We test the proposed formulations on computational examples and perform sensitivity analyses with respect to some model-specific and structural parameters.'
author:
- 'Çağın Ararat[^1]'
- 'Nurtai Meimanjanov[^2]'
bibliography:
- 'bibliography.bib'
date: 'March 19, 2019'
nocite: '[@*]'
title: 'Computation of systemic risk measures: a mixed-integer linear programming approach'
---
**Keywords and phrases:** systemic risk measure, aggregation function, set-valued risk measure, systemic risk, capital requirement, Eisenberg-Noe model, Rogers-Veraart model, Benson’s algorithm, nonconvex vector optimization.\
**Mathematics Subject Classification (2010):** 26E25, 90C11, 90C29, 91B30.
Introduction
============
Financial contagion is usually associated with a chain of failures in a financial system triggered by external correlated shocks as well as direct or indirect interdependencies among the members of the system leading to, from an economic point of view, undesirable consequences such as financial crisis, necessity for bailout loans, economic regression, rise in national debt and so on. A good example is a bank run, when a large number of holders withdraw their money from a bank due to panic or decrease in confidence in the bank, causing insolvency of the bank. In turn, the bank may call its claims from the other banks, decreasing confidence in them and causing new bank runs. Being unable to meet their liabilities, some of the banks may become bankrupt and, thus, aggravate the contagion even further. Unlike the usual notion of risk, when it is associated with a single entity, systemic risk is related to the strength of an entire financial system against financial contagions.
In this paper, we consider financial systems in which members have direct links to each other through contractual liabilities. When the members realize their operating cash flows, the actual interbank payments are determined through a clearing procedure. As an example of such systems, [@eisenberg-noe] models a financial system as a static directed network of banks where interbank liabilities are attached to the arcs. Assuming a positive operating cash flow for each bank, the paper develops two approaches to calculate a clearing vector, that is, a vector of payments to meet interbank liabilities. The first is a simple algorithm, called the *fictitious default algorithm*, which gradually calculates a clearing vector by finitely many updates. The second is a laconic mathematical programming problem with linear constraints determined by the liabilities, the operating cash flows, and an arbitrary strictly increasing objective function. In particular, one can choose a linear objective function so that a clearing vector is calculated as an optimal solution of a *linear programming problem*.
The former algorithmic approach is preferred by most of the scholars that work in network models of systemic risk. [@suzuki] introduces a similar approach to evaluate clearing vectors as in [@eisenberg-noe]. In addition, [@suzuki] considers cross-holdings of stock among members of a financial system. [@cifuentes] investigates systemic risk in terms of liquidity of institutions in a financial system and considers unsteadiness of asset prices as well. Unlike earlier works, the network model in this paper differentiates between liquid and illiquid assets. [@elsinger] extends the work in [@eisenberg-noe] by introducing a cross-holdings structure similar to the one in [@suzuki]. Additionally, [@elsinger] relaxes the positivity assumption on operating cash flows of the members in a system and studies the model by imposing some seniority assumptions. [@rogers-veraart] introduces default costs to the model in [@eisenberg-noe]. In addition, one of the main focuses in [@rogers-veraart] is devoted to the investigation of the necessity of bailing out procedures for the defaulting institutions. It is shown that under strictly positive default costs, it might be beneficial for some of the solvent institutions to take over insolvent institutions. [@weber-weske] integrates many of the factors that contribute to systemic risk into one network model. These factors include cross-holdings introduced in [@suzuki] and [@elsinger], file sales investigated in [@cifuentes], and bankruptcy costs viewed in [@elsinger] and [@rogers-veraart]. [@weber-weske] takes the model in [@eisenberg-noe] as a base and introduces all the above factors simultaneously, making it more realistic and compex at the same time. For a detailed review of network models of systemic risk, the reader is referred to the survey [@kabanov], which focuses on the existence and uniqueness of clearing vectors in the models mentioned above as well as their calculations by certain variations of the fictitious default algorithm in [@eisenberg-noe]. However, none of the above works builds on the second mathematical programming approach of [@eisenberg-noe].
On the other hand, the operating cash flows of the members of a network are typically subject to uncertainty due to correlated risk factors. Hence, these cash flows can be modeled as one possible realization of a random vector with possibly correlated components. Then, the resulting clearing vector is a deterministic function of the operating cash flow random vector, where the deterministic function is defined through the underlying clearing mechanism. Based on the random clearing vector, one can define various *systemic risk measures* to calculate the necessary capital allocations for the members of the network in order to control some (nonlinear) averages over different scenarios. This is the main focus of a recent stream of research started with [@chen]. Using the clearing mechanism, one defines a random aggregate quantity associated to the clearing vector, such as the total debt paid in the system or the total equity made by all members as a result of clearing. This aggregate quantity can be seen as a deterministic and scalar function, called the *aggregation function*, of the operating cash flow vector. In [@chen], a systemic risk measure is defined as a scalar functional of the operating cash flow vector that measures the risk of the random aggregate quantity through a convex risk measure [@stoch.finance Chapter 4] such as negative expected value, average value-at-risk or entropic risk measure.
The value of the systemic risk measure in [@chen] can be seen as the total capital requirement for the system to keep the risk of the aggregate quantity at an acceptable level. However, since the total capital is used only after the shock is aggregated, the allocation of this total back into the members of the system remains as a question to be addressed by an additional procedure. To that end, set-valued and scalar systemic risk measures that are considered “sensitive" to capital levels are proposed in [@feinstein] and [@biagini], respectively. These systemic risk measures look for deterministic capital allocation vectors that are directly used to augment the random operating cash flow vector. Hence, the new augmented cash flow vector is aggregated and the risk of the resulting random aggreagate quantity is controled by a convex risk measure as in [@chen]. In particular, the value of the set-valued systemic risk measure in [@feinstein] is the set of all “feasible" capital allocation vectors, which addresses the measurement and allocation of systemic risk as a joint problem.
The sensitive systemic risk measures studied in [@feinstein] and [@biagini] have convenient theoretical properties when the underlying aggregation function is simple enough. In [@ararat], assuming a monotone and concave aggregation function, it has been shown that the set-valued sensitive systemic risk measure is a convex set-valued risk measure in the sense of @hamel-heyde-rudloff and dual representations are obtained in terms of the conjugate function of the aggregation function. In particular, the aggregation function for the Eiseberg-Noe model, assuming positive operating cash flows as in the original formulation in [@eisenberg-noe], is monotone and concave, and an explicit dual representation is obtained for the corresponding systemic risk measure of this model.
In this paper, we are concerned with the *computation* of a sensitive systemic risk measure discussed above. We relate the value of this systemic risk measure to a vector (multiobjective) optimization problem whose “efficient frontier" corresponds to the boundary of the systemic risk measure. The vector optimization problem has a risk constraint written in terms of the aggregation function. The main challenge in solving this problem is that the aggregation function needs to be evaluated for every scenario of the underlying probability space as well as for every choice of the capital allocation vector, which is the decision variable of the optimization problem. For the standard Eisenberg-Noe model, thanks to the linear programming characterization of the clearing vectors, one can formulate the aggregation function in terms of a linear programming problem parametrized by the scenario and the capital allocation vector. Hence, the ultimate vector optimization problem can be seen as a nested optimization problem.
We focus particularly on models beyond the standard Eisenberg-Noe framework with positive operating cash flows. In particular, we consider an extension of the Eisenberg-Noe model by relaxing the positivity assumption as well as the Rogers-Veraart model with default costs. It turns out that both models have a common type of singularity that can be formulated in terms of *binary variables*, a novel feature studied in this paper. One of our main contributions is to develop mixed-integer linear programming problems that calculate clearing vectors in these models. We fix the objective functions of these optimization problems in such a way that the optimal values give the total debts paid at clearing in the corresponding models. Hence, we calculate the aggregation functions as the optimal values of these optimization problems.
The existence of binary variables in optimization problems results in *lack of concavity* for the corresponding aggregation functions. Consequently, the sensitive systemic risk measures for the two models do not possess the nice theoretical features such as convexity and dual representations studied in the earlier papers on systemic risk measures. Indeed, we even have that the values of these systemic risk measures fail to be convex sets, in general. Therefore, one of our fundamental observations is that *binary variables* and the accompanying *lack of concavity/convexity* show up naturally at the cost of using more sophisticated aggregation mechanisms beyond the standard Eisenberg-Noe framework.
Going back to the computations of systemic risk measures, the associated vector optimization problems are consequently nonconvex, in general. We use the Benson-type algorithm for such problems developed recently in [@non-conv.benson]. The algorithm (as well as its original version in [@benson]) has two “blackbox" subroutines for the following two scalar optimization problems: weighted-sum scalarization problem and the problem of calculating the minimum step-length to hit the efficient frontier from an outside point. It should be noted that the algorithm in [@non-conv.benson] assumes that these scalar problems are solvable by some unspecified methods and the convergence of the algorithm is guaranteed based on this assumption. In our context, we formulate these problems as mixed-integer linear programming problems for a generic aggregation function that is formulated in terms of a mixed-integer linear programming problem. We address further questions regarding the the finiteness of the optimal values and existence of feasible/optimal solutions separately for each of the extended Eisenberg-Noe model and the Rogers-Veraart model.
We perform a detailed computational study for both models as well as sensitivity analyses with respect to some model parameters such as the default cost parameters in the Rogers-Veraart model, the threshold level used in the risk constraint, and also some parameters determining the interconnectedness of the network.
The rest of this paper is organized as follows. We study the Eisenberg-Noe and Rogers-Veraart network models in detail together with the mathematical programming characterizations of clearing vectors in Section \[systemic\_risk\_models\]. In Section \[systemic\_risk\_measures\], we study the sensitive systemic risk measures and their associated nonconvex vector optimization problems. The proofs of some results in Sections \[systemic\_risk\_models\], \[systemic\_risk\_measures\] are deferred to Appendices \[appendixA\], \[appendixB\], respectively. We present the computational results in Section \[computational\_results\].
Network models of systemic risk {#systemic_risk_models}
===============================
In this section, after reviewing the original Eisenberg-Noe network model in Section \[original\_model\], we propose a seniority-based extension of this model by allowing signed operating cash flows in Section \[signed\_model\] and provide a novel mixed-integer linear programming (MILP) formulations of clearing vectors in Theorem \[EN\_theorem\]. Then, in Section \[rv\_model\], we consider the Rogers-Veraart network model and provide a novel MILP formulation of clearing vectors in Theorem \[RV\_theorem\].
Let us introduce the related notation. Let $n\in{\mathbb{N}}={\ensuremath{ \left\{ 1,2,\ldots \right\} }}$. Given $a, b\in{\mathbb{R}}$, we write $a\wedge b = \min{\ensuremath{ \left\{ a,b \right\} }}$, $a\vee b = \max{\ensuremath{ \left\{ a,b \right\} }}$, $a^+ = 0\vee a$, and $a^- = 0 \vee {\ensuremath{\left( -a \right)}}$. Similarly, given $\bm{a}={{(a_1,\ldots,a_n)}^\mathsf{T}}, \bm{b}={{(b_1,\ldots,b_n)}^\mathsf{T}}\in{\mathbb{R}}^n$, we write $$\bm{a}\wedge\bm{b} = {{(a_1\wedge b_1, \ldots, a_n\wedge b_n)}^\mathsf{T}}, \quad \bm{a}\vee\bm{b} = {{(a_1\vee b_1, \ldots, a_n\vee b_n)}^\mathsf{T}}$$ as well as $\bm{a}^+ = {\mathbf{0}}\vee \bm{a}$, and $\bm{a}^- =0\vee (-\bm{a})$, where ${\mathbf{0}}= {{(0,\ldots,0)}^\mathsf{T}}\in{\mathbb{R}}^n$. We sometimes use ${\bm{\mathbbm{1}}}= {{{\ensuremath{\left( 1,\ldots,1 \right)}}}^\mathsf{T}}\in{\mathbb{R}}^n$ as well. The vector $\bm{a}\odot\bm{b} = {{(a_1 b_1,\ldots,a_n b_n)}^\mathsf{T}}$ denotes the Hadamard product of $\bm{a},\bm{b}$. We write $\bm{a}\le\bm{b}$ if and only if $a_i\le b_i$ for each $i\in{\ensuremath{ \left\{ 1,\ldots,n \right\} }}$. In this case, we also define the rectangle $[\bm{a},\bm{b}] = [a_1,b_1]\times\ldots\times[a_n,b_n]\subseteq {\mathbb{R}}^n$. Using $\leq$ on ${\mathbb{R}}^n$, we define ${\mathbb{R}}^n_+={\ensuremath{ \left\{ \bm{x}\in{\mathbb{R}}^n\mid \bm{0}\leq \bm{x} \right\} }}$, whose elements are said to be *positive*. Finally, $${\ensuremath{ \left\Vert \bm{a} \right\Vert }}_\infty = \underset{i\in{\ensuremath{ \left\{ 1,\ldots,n \right\} }}}{\max}{\ensuremath{ \left| a_i \right| }}$$ is the $\ell^\infty$-norm of $\bm{a}$.
Eisenberg-Noe network model {#original_model}
---------------------------
In this section, the original Eisenberg-Noe network model in [@eisenberg-noe] and its corresponding aggregation function are provided for completeness.
\[ENsystem\] A quadruple ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}} \right)}}}$ is called an Eisenberg-Noe network if ${\mathcal{N}}= {\ensuremath{ \left\{ 1,\ldots,n \right\} }}$ for some $n\in{\mathbb{N}}$, $\bm{\pi} = {\ensuremath{\left( \pi_{ij} \right)}}_{i,j\in{\mathcal{N}}}\in{\mathbb{R}}^{n\times n}_+$ is a stochastic matrix with $\pi_{ii} = 0$ and $\sum_{j=1}^{n}\pi_{ji} < n$ for each $i\in{\mathcal{N}}$, $\bm{{\bar{p}}}={{{\ensuremath{\left( \bar{p}_1,\ldots,\bar{p}_n \right)}}}^\mathsf{T}}\in{\mathbb{R}}^n_{++}$, and $\bm{x}={{{\ensuremath{\left( x_1,\ldots,x_n \right)}}}^\mathsf{T}}\in{\mathbb{R}}^n_+$.
In Definition \[ENsystem\], ${\mathcal{N}}$ is the index set of nodes in a network that represents a financial system of $n$ institutions. For every $i\in{\mathcal{N}}$, $\bar{p}_i > 0$ denotes the total amount of liabilities of node $i$. We call $\bm{{\bar{p}}}$ the *total obligation vector*.
For every $i,j\in{\mathcal{N}}$ such that $i\neq j$, $\pi_{ij}>0$ denotes the fraction of the total liability of node $i$ owed to node $j$. We call $\bm{\pi}$ the *relative liabilities matrix*. For every $i\in{\mathcal{N}}$, the assumption $\pi_{ii} = 0$ means that node $i$ cannot have liabilities to itself. By $\sum_{j=1}^{n}\pi_{ji} < n$ for every $i\in{\mathcal{N}}$, we assume that no node owns all the claims in the network. Note that, given $\bm{{\bar{p}}}$ and $\bm{\pi}$, for every $i,j\in{\mathcal{N}}$, the nominal liability $l_{ij}$ of node $i$ to node $j$ can be calculated as $l_{ij} = \pi_{ij}\bar{p}_i$.
For each $i\in{\mathcal{N}}$, $x_i\ge0$ denotes the operating cash flow of node $i$. We call $\bm{x}$ the *operating cash flow vector*.
Let ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}} \right)}}}$ be an Eisenberg-Noe network. For each $i\in{\mathcal{N}}$, let $p_i\ge0$ be the sum of all payments made by node $i$ to the other nodes in the network. Then, $\bm{p}={{{\ensuremath{\left( p_1,\ldots,p_n \right)}}}^\mathsf{T}}\in{\mathbb{R}}^n_+$ is called a *payment vector*.
\[clearing\_vector\_defn\] A vector $\bm{p}\in{\ensuremath{ \left[ {\mathbf{0}},\bm{{\bar{p}}} \right] }}$ is called a *clearing vector* for ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}} \right)}}}$ if it satisfies the following properties:
- *Limited liability:* for each $i \in {\mathcal{N}}, p_i \le \sum_{j=1}^n\pi_{ji}p_j + x_i$, which implies that node $i$ cannot pay more than it has.
- *Absolute priority:* for each $i \in {\mathcal{N}}$, either $p_i = \bar{p}_i$ or $p_i = \sum_{j=1}^n\pi_{ji}p_j + x_i$, which implies that node $i$ either meets its obligations in full or else it defaults by paying as much as it has.
Let $\Phi^{\text{EN}_+}:{\ensuremath{ \left[ {\mathbf{0}},\bm{{\bar{p}}} \right] }}\to{\ensuremath{ \left[ {\mathbf{0}},\bm{{\bar{p}}} \right] }}$ be defined by $$\label{fixed_point_original}
\Phi^{\text{EN}_+}{\ensuremath{\left( \bm{p} \right)}} \coloneqq {\ensuremath{\left( {{\bm{\pi}}^\mathsf{T}}\bm{p} + \bm{x} \right)}}\wedge\bm{{\bar{p}}}.$$ It is shown in [@eisenberg-noe] that a clearing vector $\bm{p}$ for ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}} \right)}}}$ is a fixed point of $\Phi^{\text{EN}_+}$, that is, $\Phi^{\text{EN}_+}(\bm{p})=\bm{p}$.
Next, we recall the programming characterization of clearing vectors shown in [@eisenberg-noe], which is the basis of our generalizations to follow. We say that a function $f:{\mathbb{R}}^n\to{\mathbb{R}}$ is strictly increasing if $\bm{a}\le\bm{b}$ and $\bm{a}\neq\bm{b}$ imply $f{\ensuremath{\left( \bm{a} \right)}}<f{\ensuremath{\left( \bm{b} \right)}}$ for every $\bm{a},\bm{b}\in{\mathbb{R}}^n$.
\[eisenberg\_noe\_Lemma\] [@eisenberg-noe Lemma 4] Let $f: {\mathbb{R}}^n\to{\mathbb{R}}$ be a strictly increasing function. Consider the following optimization problem with linear constraints: $$\begin{aligned}
\label{EN_LP}
\begin{split}
\max \quad & f{\ensuremath{\left( \bm{p} \right)}} \\
\text{s.t.} \quad & \bm{p} \le {{\bm{\pi}}^\mathsf{T}}\bm{p} + \bm{x}, \\
& \bm{p} \in {\ensuremath{ \left[ {\mathbf{0}}, \bm{{\bar{p}}} \right] }}.
\end{split}
\end{aligned}$$ If $\bm{p}\in{\mathbb{R}}^n_+$ is an optimal solution to this optimization problem, then it is a clearing vector for ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}} \right)}}}$.
Each member in a network has its impact on economy. As in [@chen], [@biagini], [@feinstein], [@ararat], we use aggregation functions to summarize these individual effects and provide a total impact of the network on economy. They play a significant role in evaluating systemic risks and in the computation of systemic risk measures. The aggregation function $\Lambda: {\mathbb{R}}^n\to{\mathbb{R}}$ for the Eisenberg-Noe network ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}} \right)}}}$ is defined as $$\Lambda{\ensuremath{\left( \bm{x} \right)}} \coloneqq \sup{\ensuremath{ \left\{ f{\ensuremath{\left( \bm{p} \right)}} \mid \bm{p} \le {{\bm{\pi}}^\mathsf{T}}\bm{p} + \bm{x},\ \bm{p} \in {\ensuremath{ \left[ {\mathbf{0}}, \bm{{\bar{p}}} \right] }} \right\} }},$$ where $f: {\mathbb{R}}^n\to{\mathbb{R}}$ is a strictly increasing function, namely, $\Lambda(\bm{x})$ is the optimal value of the problem in .
Signed Eisenberg-Noe network model {#signed_model}
----------------------------------
In the original Eisenberg-Noe network model, it is assumed that the operating cash flow vector is positive. In reality, however, it may happen that an institution has liabilities to external entities not modeled as part of the network resulting in a negative *operating cash flow* or a positive *operating cost*.
\[signedENsystem\] A quadruple ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}} \right)}}}$ is called a signed Eisenberg-Noe network if ${\mathcal{N}}$, $\bm{\pi}$ and $\bm{{\bar{p}}}$ are as in Definition \[ENsystem\], and $\bm{x}={{{\ensuremath{\left( x_1,\ldots,x_n \right)}}}^\mathsf{T}}\in{\mathbb{R}}^n$.
Note that Definition \[signedENsystem\] removes the positivity assumption on the operating cash flow vector $\bm{x}$. Our aim is to provide a new definition of clearing vector by extending Definition \[clearing\_vector\_defn\] with an additional seniority assumption for negative operating cash flows. Based on this definition, we prove a fixed-point and a mathematical programming characterization of clearing vectors. Finally, we introduce an associated aggregation function through a MILP problem.
Let ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}} \right)}}}$ be a signed Eisenberg-Noe network. We assume that the nodes that have obligations outside the network, that is, the nodes with negative operating cash flows have to meet these obligations first, and if they do not default in this “first round," then they should meet their obligations to the other nodes inside the network. At this “second round", as in the original Eisenberg-Noe network model, they either meet their obligations to the other nodes in full or pay as much as they have at hand and default. This motivates the following definition.
\[clearing\_vector\_defn\_modified\] A vector $\bm{p}\in{\ensuremath{ \left[ {\mathbf{0}},\bm{{\bar{p}}} \right] }}$ is called a *clearing vector* for ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}} \right)}}}$ if it satisfies the following properties:
- *Immediate default:* for each $i\in{\mathcal{N}}$, if $\sum_{j=1}^n\pi_{ji}p_j + x_i \le 0$, then $p_i = 0$.
- *Limited liability:* for each $i \in {\mathcal{N}}$, if $\sum_{j=1}^n\pi_{ji}p_j + x_i > 0$, then $p_i \le \sum_{j=1}^n\pi_{ji}p_j + x_i$, which implies that if node $i$ has a strictly positive operating cash flow, then it cannot pay more than it has.
- *Absolute priority:* for each $i \in {\mathcal{N}}$, if $\sum_{j=1}^n\pi_{ji}p_j + x_i > 0$, then either $p_i = \bar{p}_i$ or $p_i = \sum_{j=1}^n\pi_{ji}p_j + x_i$, which implies that if node $i$ has a strictly positive operating cash flow, then it either meets its obligations in full or else it defaults by paying as much as it has.
Let $\Phi^\text{EN}: {\ensuremath{ \left[ {\mathbf{0}},\bm{{\bar{p}}} \right] }}\to{\ensuremath{ \left[ {\mathbf{0}},\bm{{\bar{p}}} \right] }}$ be defined by $$\label{fixed_point_EN_modified}
\Phi^\text{EN}{\ensuremath{\left( \bm{p} \right)}} \coloneqq {\ensuremath{\left( \bm{{\bar{p}}}\wedge{\ensuremath{\left( {{\bm{\pi}}^\mathsf{T}}\bm{p} + \bm{x} \right)}} \right)}}^+,$$ or more explicitly, for each $i\in{\mathcal{N}}$, $$\begin{aligned}
\label{fixed_point_EN_modified_explicit}
\Phi^\text{EN}_i{\ensuremath{\left( \bm{p} \right)}} = \begin{cases}
0 &\quad\text{if }\sum_{j=1}^n\pi_{ji}p_j + x_i \le 0, \\
\sum_{j=1}^n\pi_{ji}p_j + x_i &\quad\text{if }0 < \sum_{j=1}^n\pi_{ji}p_j + x_i \le \bar{p}_i,\\
\bar{p}_i &\quad\text{if }\sum_{j=1}^n\pi_{ji}p_j + x_i > \bar{p}_i. \\
\end{cases}
\end{aligned}$$ Observe that, if $\bm{x}\in{\mathbb{R}}^n_+$, then $\Phi^\text{EN}$ coincides with the function $\Phi^{\text{EN}_+}$ in defined for the original Eisenberg-Noe network model.
We establish the fixed point characterization of clearing vectors next.
\[clearing\_vector\_fixed\_point\] A vector $\bm{p}\in{\ensuremath{ \left[ {\mathbf{0}},\bm{{\bar{p}}} \right] }}$ is a clearing vector for ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}} \right)}}}$ if and only if it is a fixed point of $\Phi^\text{EN}$.
To prove the “only if" part, let $\bm{p} = {{{\ensuremath{\left( p_1,\ldots,p_n \right)}}}^\mathsf{T}}\in{\ensuremath{ \left[ {\mathbf{0}},\bm{{\bar{p}}} \right] }}$ be a clearing vector. To show that $\bm{p}$ is a fixed point of $\Phi^\text{EN}$, let $i\in{\mathcal{N}}$.
If $\sum_{j=1}^n\pi_{ji}p_j + x_i \le 0$, then $p_i = 0$, by *immediate default*, and $\Phi^\text{EN}_i{\ensuremath{\left( \bm{p} \right)}} = 0$, by . Hence, $\Phi^\text{EN}_i{\ensuremath{\left( \bm{p} \right)}} = p_i$.
If $\sum_{j=1}^n\pi_{ji}p_j + x_i > 0$, then, by *absolute priority*, either $p_i = \bar{p}_i$ or $p_i = \sum_{j=1}^n\pi_{ji}p_j + x_i$. If $p_i = \bar{p}_i$, then, by *limited liability*, $\bar{p}_i \le \sum_{j=1}^n\pi_{ji}p_j + x_i$ and, thus, by , $\Phi^\text{EN}_i{\ensuremath{\left( \bm{p} \right)}} = \bar{p}_i$. Hence, $\Phi^\text{EN}_i{\ensuremath{\left( \bm{p} \right)}} = p_i$. On the other hand, if $p_i = \sum_{j=1}^n\pi_{ji}p_j + x_i < \bar{p}_i$, then, by , $\Phi^\text{EN}_i{\ensuremath{\left( \bm{p} \right)}} = \sum_{j=1}^n\pi_{ji}p_j + x_i$. Hence, again $\Phi^\text{EN}_i{\ensuremath{\left( \bm{p} \right)}} = p_i$. Thus, $\bm{p}$ is a fixed point of $\Phi^\text{EN}$.
To prove the “if" part, let $\bm{p} = {{{\ensuremath{\left( p_1,\ldots,p_n \right)}}}^\mathsf{T}}$ be a fixed point of $\Phi^\text{EN}$. In other words, for every $i\in{\mathcal{N}}$, $\Phi^\text{EN}_i{\ensuremath{\left( \bm{p} \right)}} = p_i$. To show that $\bm{p}$ is a clearing vector, let $i\in{\mathcal{N}}$.
If $\sum_{j=1}^n\pi_{ji}p_j + x_i \le 0$, then $\Phi^\text{EN}_i{\ensuremath{\left( \bm{p} \right)}} = p_i = 0$, by . Hence, *immediate default* holds.
If $\sum_{j=1}^n\pi_{ji}p_j + x_i > 0$, then $\Phi^\text{EN}_i{\ensuremath{\left( \bm{p} \right)}} = p_i \le \sum_{j=1}^n\pi_{ji}p_j + x_i$, by . Hence, *limited liability* holds.
Now assume $\sum_{j=1}^n\pi_{ji}p_j + x_i > 0$. If $\sum_{j=1}^n\pi_{ji}p_j + x_i \le \bar{p}_i$, then $\Phi^\text{EN}{\ensuremath{\left( \bm{p} \right)}} = p_i = \sum_{j=1}^n\pi_{ji}p_j + x_i$. If $\sum_{j=1}^n\pi_{ji}p_j + x_i > \bar{p}_i$, then $\Phi^\text{EN}{\ensuremath{\left( \bm{p} \right)}} = p_i = \bar{p}_i$, by . Hence, *absolute priority* holds as well. Hence, $\bm{p}$ is a clearing vector.
The next theorem is the main result of Section \[signed\_model\]. It extends Proposition \[eisenberg\_noe\_Lemma\] for the signed Eisenberg-Noe network model by showing that a clearing vector can be calculated as an optimal solution of a certain MILP. Hence, relaxing the positivity assumption on the operating cash flow vector is at the cost of using binary variables in the mathematical programming characterization of clearing vectors, hence, adding a discrete feature to the originally continuous optimization problem.
\[EN\_theorem\] Let $\Lambda^\text{EN}: {\mathbb{R}}^n\to{\mathbb{R}}$ be a MILP aggregation function defined by $$\label{aggregation_EN_signed}
\begin{split}
\Lambda^\text{EN}{\ensuremath{\left( \bm{y} \right)}} \coloneqq \sup \Big\{ f{\ensuremath{\left( \bm{p} \right)}} \mid \ & \bm{p} \le {\ensuremath{ \left[ {{\bm{\pi}}^\mathsf{T}}\bm{p} + \bm{y} + M{\ensuremath{\left( {\bm{\mathbbm{1}}}- \bm{s} \right)}} \right] }} \wedge {\ensuremath{\left( \bm{{\bar{p}}}\odot\bm{s} \right)}}, \\
& {{\bm{\pi}}^\mathsf{T}}\bm{p} + \bm{y}\le M\bm{s}, \bm{p} \in {\ensuremath{ \left[ {\mathbf{0}}, \bm{{\bar{p}}} \right] }}, \bm{s} \in {\ensuremath{ \left\{ 0,1 \right\} }}^n \Big\},
\end{split}$$ where $f: {\mathbb{R}}^n\to{\mathbb{R}}$ is a strictly increasing linear function and $M = n{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{y} \right\Vert }_\infty}$. If ${\ensuremath{\left( \bm{p}, \bm{s} \right)}}$ is an optimal solution to MILP for $\Lambda^\text{EN}{\ensuremath{\left( \bm{x} \right)}}$, then $\bm{p}$ is a clearing vector for ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}} \right)}}}$.
\[noconcave1\] $\Lambda^\text{EN}$ fails to be concave in general.
Observe that $\Lambda^\text{EN}{\ensuremath{\left( \bm{x} \right)}}$ can be written more explicitly as $$\begin{aligned}
\label{EN_MILP_explicit}
\text{maximize} \quad & f{\ensuremath{\left( \bm{p} \right)}} && \\
\text{subject to} \quad & p_i \le \sum_{j=1}^{n}\pi_{ji}p_j + x_i + M{\ensuremath{\left( 1-s_i \right)}}, &&i\in{\mathcal{N}}, \label{EN_MILP_constraint_1} \\
& p_i \le \bar{p}_is_i,&&i\in{\mathcal{N}}, \label{EN_MILP_constraint_2}\\
& \sum_{j=1}^{n}\pi_{ji}p_j + x_i \le Ms_i,&&i\in{\mathcal{N}}, \label{EN_MILP_constraint_3}\\
& 0 \le p_i \le \bar{p}_i,&&i\in{\mathcal{N}}, \label{EN_MILP_constraint_4} \\
& s_i \in {\ensuremath{ \left\{ 0,1 \right\} }},&&i\in{\mathcal{N}}. \label{EN_MILP_constraint_5}\end{aligned}$$
Let $\bm{u} = {{{\ensuremath{\left( u_1,\ldots,u_n \right)}}}^\mathsf{T}}\in{\ensuremath{ \left\{ 0,1 \right\} }}^n$ be a binary vector, where $u_i = 0$ if $x_i<0$, and $u_i = 1$ if $x_i\ge0$, for each $i\in{\mathcal{N}}$. Then ${\ensuremath{\left( \bm{p},\bm{s} \right)}} = {\ensuremath{\left( {\mathbf{0}},\bm{u} \right)}}\in{\mathbb{R}}^n\times{\mathbb{Z}}^n$ is a feasible solution to the MILP in . Moreover, since $f$ is a bounded function on the rectangle ${\ensuremath{ \left[ {\mathbf{0}},\bar{\bm{p}} \right] }}\subseteq{\mathbb{R}}^n_+$, by @meyer [Theorem 2.1], the MILP has an optimal solution. Observe that, by Theorem \[EN\_theorem\], the existence of an optimal solution to the MILP in proves the existence of a clearing vector for the network ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}} \right)}}}$.
In Theorem \[EN\_theorem\], $M = n{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{x} \right\Vert }_\infty}$ is taken to ensure the feasibility of the constraint . In other words, it is enough to choose $M$ such that $\sum_{j=1}^{n}\pi_{ji}p_j + x_i \le M$, for each $i\in{\mathcal{N}}$ and for every $\bm{p}\in{\ensuremath{ \left[ {\mathbf{0}},\bm{{\bar{p}}} \right] }}$. Furthermore, for each $i\in{\mathcal{N}}$ and for every $\bm{p}\in{\ensuremath{ \left[ {\mathbf{0}},\bm{{\bar{p}}} \right] }}$, since $\sum_{j=1}^{n}\pi_{ji} < n$, it holds $\sum_{j=1}^{n}\pi_{ji}p_j < n{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}$. Hence, $\sum_{j=1}^{n}\pi_{ji}p_j + x_i \le n{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{x} \right\Vert }_\infty}= M$.
The linearity of $f$ is not a necessary condition for Theorem \[EN\_theorem\] to hold.
The proof of Theorem \[EN\_theorem\] is based on the following lemma.
\[EN\_lemma\_1\] Let ${\ensuremath{\left( \bm{p}, \bm{s} \right)}}$ be an optimal solution to the MILP for $\Lambda^\text{EN}{\ensuremath{\left( \bm{x} \right)}}$. Let $i\in{\mathcal{N}}$ such that $0 < \sum_{j=1}^n\pi_{ji}p_j + x_i$. Then, $p_i = \min{\ensuremath{ \left\{ \sum_{j=1}^n\pi_{ji}p_j + x_i, \bar{p}_i \right\} }}$.
The proof of Lemma \[EN\_lemma\_1\] can be found in Appendix \[lemma\_proof\_EN\].
Let ${\ensuremath{\left( \bm{p}, \bm{s} \right)}}$ be an optimal solution to the MILP for $\Lambda^\text{EN}{\ensuremath{\left( \bm{x} \right)}}$. To prove that $\bm{p}$ is a clearing vector, by Proposition \[clearing\_vector\_fixed\_point\], we equivalently show that $\Phi^\text{EN}{\ensuremath{\left( \bm{p} \right)}} = \bm{p}$. Let $i\in{\mathcal{N}}$. Recalling , we consider three cases: $${\ensuremath{\left( 1 \right)}} \sum_{j=1}^n\pi_{ji}p_j + x_i \le 0,
\quad {\ensuremath{\left( 2 \right)}} 0 < \sum_{j=1}^n\pi_{ji}p_j + x_i \le \bar{p}_i,
\quad {\ensuremath{\left( 3 \right)}} \sum_{j=1}^n\pi_{ji}p_j + x_i > \bar{p}_i.$$
1. Assume that $\sum_{j=1}^n\pi_{ji}p_j + x_i \le 0$. Then, by , $\Phi^\text{EN}_i{\ensuremath{\left( \bm{p} \right)}} = 0$. By the arguments from the proof of the Lemma \[EN\_lemma\_1\] for this case, $p_i = 0$. Hence, $p_i = 0 = \Phi^\text{EN}_i{\ensuremath{\left( \bm{p} \right)}}$.
2. Assume that $0 < \sum_{j=1}^n\pi_{ji}p_j + x_i \le \bar{p}_i$. Then, by , $\Phi^\text{EN}_i{\ensuremath{\left( \bm{p} \right)}} = \sum_{j=1}^n\pi_{ji}p_j + x_i$. Since $0 < \sum_{j=1}^n\pi_{ji}p_j + x_i$, by Lemma \[EN\_lemma\_1\], $$p_i =\min{\ensuremath{ \left\{ \sum_{j=1}^{n}\pi_{ji}p_j + x_i, \bar{p}_i \right\} }} =\sum_{j=1}^{n}\pi_{ji}p_j + x_i.$$ Hence, $p_i = \sum_{j=1}^n\pi_{ji}p_j + x_i = \Phi^\text{EN}_i{\ensuremath{\left( \bm{p} \right)}}$.
3. Assume $\sum_{j=1}^n\pi_{ji}p_j + x_i > \bar{p}_i$. Then, by , $\Phi^\text{EN}_i{\ensuremath{\left( \bm{p} \right)}} = \bar{p}_i$. Since $\sum_{j=1}^n\pi_{ji}p_j + x_i > \bar{p}_i > 0$, again by Lemma \[EN\_lemma\_1\], $$p_i =\min{\ensuremath{ \left\{ \sum_{j=1}^{n}\pi_{ji}p_j + x_i, \bar{p}_i \right\} }} =\bar{p}_i.$$ Hence, $p_i = \bar{p}_i = \Phi^\text{EN}_i{\ensuremath{\left( \bm{p} \right)}}$.
Therefore, $\bm{p}$ is a clearing vector for ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}} \right)}}}$.
Instead of the seniority-based approach developed above, a naive approach would be to introduce an additional node and consider negative operational cash flows of the nodes as liabilities to this additional node, which itself has neither obligations nor an operating cash flow, as suggested in [@eisenberg-noe]. This approach is valid for the fictitious default algorithm described in [@eisenberg-noe] and this way a clearing vector for the original network can be found. However, the modified network lacks a solid interpretation in terms of the original network since the relative liabilities matrix of the new network depends on the operational cash flow vector. Hence, we do not follow this route here.
Rogers-Veraart network model {#rv_model}
----------------------------
In [@rogers-veraart], the original Eisenberg-Noe network model is extended by including default costs. It is assumed that a defaulting node is not able to use all of its liquid assets to meet its obligations. Unlike the Eisenberg-Noe model, the possibility of a mathematical programming formulation for clearing vectors seems to be an open problem for the Rogers-Veraart. We fill up this gap by proposing a MILP whose optimal solution includes a clearing vector for the Rogers-Veraart network model. Based on this characterization, we define an aggregation function and provide its relationship to the network model. Finally, inspired by Definition \[clearing\_vector\_defn\], we propose a weaker definition of a clearing vector for the Rogers-Veraart network model.
\[RVsystem\] A sextuple ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}}, \alpha, \beta \right)}}}$ is called a Rogers-Veraart network if ${\mathcal{N}}= {\ensuremath{ \left\{ 1,\ldots,n \right\} }}$ for some $n\in{\mathbb{N}}$, $\bm{\pi} = {\ensuremath{\left( \pi_{ij} \right)}}_{i,j\in{\mathcal{N}}}\in{\mathbb{R}}^{n\times n}_+$ is a stochastic matrix with $\pi_{ii} = 0$ and $\sum_{j=1}^{n}\pi_{ji} < n$ for each $i\in{\mathcal{N}}$, $\bm{{\bar{p}}}={{{\ensuremath{\left( \bar{p}_1,\ldots,\bar{p}_n \right)}}}^\mathsf{T}}\in{\mathbb{R}}^n_{++}$, $\bm{x}={{{\ensuremath{\left( x_1,\ldots,x_n \right)}}}^\mathsf{T}}\in{\mathbb{R}}^n_+$ and $\alpha,\beta\in\left.{\left(0,1\right.}\right]$.
As in Definition \[ENsystem\], ${\mathcal{N}}$ is the set of nodes in a network with $n$ institutions, $\bm{{\bar{p}}}$ is the *total obligation vector*, $\bm{\pi}$ is the *matrix of relative liabilities* and $\bm{x}$ is the *operating cash flow vector*. It is assumed that a defaulting node may not be able to use all of its liquid assets to meet its obligations. For this purpose, we use $\alpha$ as the fraction of the operating cash flow and $\beta$ as the fraction of the cash inflow from other nodes that can be used by a defaulting node to meet its obligations.
Let ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}}, \alpha, \beta \right)}}}$ be a Rogers-Veraart network. For each $i\in{\mathcal{N}}$, let $p_i\ge0$ be the sum of all payments made by node $i$ to the other nodes in the network. Then, $\bm{p} = {{{\ensuremath{\left( p_1, \ldots, p_n \right)}}}^\mathsf{T}}\in{\mathbb{R}}^n_+$ is called a *payment vector*.
Motivated by Definition \[clearing\_vector\_defn\] of a clearing vector for an Eisenberg-Noe network, we suggest the following similar definition of a clearing vector for the Rogers-Veraart network ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}}, \alpha, \beta \right)}}}$.
\[clearing\_vector\_defn\_RV\] A vector $\bm{p}\in{\ensuremath{ \left[ {\mathbf{0}},\bm{{\bar{p}}} \right] }}$ is called a *clearing vector* for ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}}, \alpha, \beta \right)}}}$ if it satisfies the following properties:
- *Limited liability:* for each $i \in {\mathcal{N}}, p_i \le x_i + \sum_{j=1}^n\pi_{ji}p_j$, which implies that node $i$ cannot pay more than it has.
- *Absolute priority:* for each $i \in {\mathcal{N}}$, either $p_i = \bar{p}_i$ or $p_i = \alpha x_i + \beta\sum_{j=1}^n\pi_{ji}p_j$, which implies that node $i$ either has to meet its obligations in full or else it defaults by paying as much as it can.
Let $\Phi^{\text{RV}_+}:{\ensuremath{ \left[ {\mathbf{0}},\bm{{\bar{p}}} \right] }}\to{\ensuremath{ \left[ {\mathbf{0}},\bm{{\bar{p}}} \right] }}$ be defined by $$\label{fixed_point_RV_explicit}
\Phi^{\text{RV}_+}_i{\ensuremath{\left( \bm{p} \right)}} \coloneqq
\begin{cases}
\bar{p}_i & \quad\text{if }\bar{p}_i\le x_i+\sum_{j=1}^{n}\pi_{ji}p_j, \\
\alpha x_i+\beta\sum_{j=1}^{n}\pi_{ji}p_j & \quad\text{if }\bar{p}_i> x_i+\sum_{j=1}^{n}\pi_{ji}p_j,
\end{cases}$$ for each $i\in{\mathcal{N}}$.
Observe that, if $\alpha=1$ and $\beta=1$, then the function $\Phi^{\text{RV}_+}$ becomes the usual $\Phi^{\text{EN}_+}$ in from the original Eisenberg-Noe network model.
\[clearing\_vector\_fixed\_point\_RV\] A fixed point $\bm{p}\in{\ensuremath{ \left[ {\mathbf{0}},\bm{{\bar{p}}} \right] }}$ of $\Phi^{\text{RV}_+}$ is a clearing vector for ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}}, \alpha, \beta \right)}}}$.
Let $\bm{p} = {{{\ensuremath{\left( p_1,\ldots,p_n \right)}}}^\mathsf{T}}$ be a fixed point of $\Phi^{\text{RV}_+}$. To show that $\bm{p}$ is a clearing vector for ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}}, \alpha, \beta \right)}}}$, let $i\in{\mathcal{N}}$.
If $\bar{p}_i \le x_i + \sum_{j=1}^n\pi_{ji}p_j$, then $\Phi^{\text{RV}_+}_i{\ensuremath{\left( \bm{p} \right)}} = \bar{p}_i = p_i \le x_i + \sum_{j=1}^n\pi_{ji}p_j$, and if $\bar{p}_i > x_i + \sum_{j=1}^n\pi_{ji}p_j$, then $\Phi^{\text{RV}_+}_i{\ensuremath{\left( \bm{p} \right)}} = \alpha x_i + \beta\sum_{j=1}^n\pi_{ji}p_j = p_i \le x_i + \sum_{j=1}^n\pi_{ji}p_j$, by the definition of $\Phi^{\text{RV}_+}$ in and since $\bm{p}$ is a fixed point of $\Phi^{\text{RV}_+}$. Hence, both *limited liability* and *absolute priority* in Definition \[clearing\_vector\_defn\_RV\] hold. Hence, $\bm{p}$ is a clearing vector for ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}}, \alpha, \beta \right)}}}$.
The converse of Proposition \[clearing\_vector\_fixed\_point\_RV\] fails to hold in general. Here is a counterexample. Consider a Rogers-Veraart network ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}}, \alpha, \beta \right)}}}$ and payment vector $\bm{p}$, where ${\mathcal{N}}= {\ensuremath{ \left\{ 1,2 \right\} }}$, $$\bm{\pi} = \begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}, \quad \bm{{\bar{p}}} = \begin{bmatrix} 20 \\ 25 \end{bmatrix},\quad \bm{x} = \begin{bmatrix} 10 \\ 10 \end{bmatrix}, \quad \bm{p} = \begin{bmatrix} 20 \\ 15 \end{bmatrix},$$ $\alpha = 0.5$, $\beta = 0.5$. According to Definition \[clearing\_vector\_defn\_RV\], $\bm{p}$ is a clearing vector for ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}}, \alpha, \beta \right)}}}$ since it satisfies *absolute priority* and *limited liability*. However, by , $\Phi^{\text{RV}_+}_2{\ensuremath{\left( \bm{p} \right)}} = 25 > p_2 = 15$. Hence, $\bm{p}$ is not a fixed point of $\Phi^{\text{RV}_+}$.
The next theorem is the main result of Section \[rv\_model\]. In the spirit of Theorem \[EN\_theorem\] for a signed Eisenberg-Noe network, it provides a MILP characterization of clearing vectors for the Rogers-Veraart nework ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}}, \alpha, \beta \right)}}}$.
\[RV\_theorem\] Let $\Lambda^{\text{RV}_+}: {\mathbb{R}}^n\to{\mathbb{R}}$ be a MILP aggregation function defined by $$\label{aggregation_RV}
\Lambda^{\text{RV}_+}{\ensuremath{\left( \bm{y} \right)}} \coloneqq
\begin{cases}
\!\begin{aligned}
\sup \Big\{f{\ensuremath{\left( \bm{p} \right)}} \mid \ & \bm{p} \le \alpha\bm{y} + \beta{{\bm{\pi}}^\mathsf{T}}\bm{p} + \bm{{\bar{p}}}\odot\bm{s}, \\
& \bm{{\bar{p}}}\odot\bm{s} \le \bm{y} + {{\bm{\pi}}^\mathsf{T}}\bm{p}, \bm{p} \in {\ensuremath{ \left[ {\mathbf{0}}, \bm{{\bar{p}}} \right] }}, \bm{s} \in {\ensuremath{ \left\{ 0,1 \right\} }}^n\Big\},
\end{aligned} &\text{if }\bm{y}\in{\mathbb{R}}^n_+, \\
-\infty, &\text{if }\bm{y}\notin{\mathbb{R}}^n_+,
\end{cases}$$ where $f: {\mathbb{R}}^n\to{\mathbb{R}}$ is a strictly increasing linear function. If ${\ensuremath{\left( \bm{p}, \bm{s} \right)}}$ is an optimal solution to the MILP for $\Lambda^{\text{RV}_+}{\ensuremath{\left( \bm{x} \right)}}$, then $\bm{p}$ is a clearing vector for ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}}, \alpha, \beta \right)}}}$.
\[noconcave2\] $\Lambda^{\text{RV}_+}$ fails to be concave in general.
Let us comment on the MILP problems in Theorem \[EN\_theorem\] and Theorem \[RV\_theorem\]. While both problems have a discrete feature through the binary variables, the natures of this feature are quite different from each other. In Theorem \[EN\_theorem\], the binary variables serve for quantifying the switch from the “first round" to the “second round" in the definition of $\Phi^{\text{EN}}$, which is described above Definition \[clearing\_vector\_defn\_modified\]. In this case, in addition to the binary variables, one also uses a large constant $M$ in the problem formulation. On the other hand, binary variables are used in Theorem \[RV\_theorem\] to model the *discontinuity* in $\Phi^{\text{RV}_+}$ which occurs when $\alpha<1$ or $\beta<1$. In this case, a formulation without using a large constant $M$ is possible.
Note that $\Lambda^{\text{RV}_+}{\ensuremath{\left( \bm{x} \right)}}$ can be written more explicitly as $$\begin{aligned}
\label{RV_MILP_explicit}
\text{maximize} \quad & f{\ensuremath{\left( \bm{p} \right)}} && \\
\text{subject to} \quad & p_i \le \alpha x_i + \beta\sum_{j=1}^{n}\pi_{ji}p_j + \bar{p}_is_i, &&i\in{\mathcal{N}}, \label{RV_MILP_constraint_1} \\
& \bar{p}_is_i \le x_i + \sum_{j=1}^{n}\pi_{ji}p_j,&&i\in{\mathcal{N}}, \label{RV_MILP_constraint_2}\\
& 0 \le p_i \le \bar{p}_i,&& i\in{\mathcal{N}},\label{RV_MILP_constraint_3}\\
& s_i \in {\ensuremath{ \left\{ 0,1 \right\} }}&&i\in{\mathcal{N}}. \label{RV_MILP_constraint_4}\end{aligned}$$
It is easy to check that ${\ensuremath{\left( \bm{p},\bm{s} \right)}} = {\ensuremath{\left( {\mathbf{0}},{\mathbf{0}}\right)}}\in{\mathbb{R}}^n\times{\mathbb{Z}}^n$ is a feasible solution to the MILP in . Moreover, since $f$ is a bounded function on the interval ${\ensuremath{ \left[ {\mathbf{0}},\bar{\bm{p}} \right] }}\subseteq{\mathbb{R}}^n_+$, by @meyer [Theorem 2.1], the MILP in has an optimal solution. Observe that, by Theorem \[RV\_theorem\], the existence of an optimal solution to the MILP in proves the existence of a clearing vector for ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{x}}, \alpha, \beta \right)}}}$. Hence, Theorem \[RV\_theorem\] provides an alternative argument for the proof of @rogers-veraart [Theorem 3.1] on the existence of a clearing vector.
The proof of Theorem \[RV\_theorem\] relies on the following three lemmata.
\[RV\_lemma\_1\] Let ${\ensuremath{\left( \bm{p}, \bm{s} \right)}}$ be an optimal solution to the MILP for $\Lambda^{\text{RV}_+}{\ensuremath{\left( \bm{x} \right)}}$. Let $i\in{\mathcal{N}}$ such that $$\alpha x_i+ \beta\sum_{j=1}^{n}\pi_{ji}p_j < \bar{p}_i \le x_i+\sum_{j=1}^{n}\pi_{ji}p_j.$$ Then, $s_i = 1$.
\[RV\_lemma\_2\] Let ${\ensuremath{\left( \bm{p}, \bm{s} \right)}}$ be an optimal solution to the MILP for $\Lambda^{\text{RV}_+}{\ensuremath{\left( \bm{x} \right)}}$. Let $i\in{\mathcal{N}}$ with $\bar{p}_i \le x_i+ \sum_{j=1}^{n}\pi_{ji}p_j$. Then, $p_i = \bar{p}_i$.
\[RV\_lemma\_3\] Let ${\ensuremath{\left( \bm{p}, \bm{s} \right)}}$ be an optimal solution to the MILP for $\Lambda^{\text{RV}_+}{\ensuremath{\left( \bm{x} \right)}}$. Let $i\in{\mathcal{N}}$ with $\bar{p}_i > x_i+\sum_{j=1}^{n}\pi_{ji}p_j$. Then, $p_i = \alpha x_i+\beta \sum_{j=1}^{n}\pi_{ji}p_j$.
The proofs of Lemmata \[RV\_lemma\_1\], \[RV\_lemma\_2\], \[RV\_lemma\_3\] can be found in Appendices \[lemma\_proof\_RV1\], \[lemma\_proof\_RV2\], \[lemma\_proof\_RV3\], respectively.
Let ${\ensuremath{\left( \bm{p}, \bm{s} \right)}}$ be an optimal solution to the MILP for $\Lambda^{\text{RV}_+}{\ensuremath{\left( \bm{x} \right)}}$. To prove that $\bm{p}$ is a clearing vector, thanks to Proposition \[clearing\_vector\_fixed\_point\_RV\], it suffices to show $\Phi^{\text{RV}_+}{\ensuremath{\left( \bm{p} \right)}} = \bm{p}$.
Let us fix $i\in{\mathcal{N}}$. Recalling , we consider two cases: $${\ensuremath{\left( 1 \right)}} \bar{p}_i \le x_i+\sum_{j=1}^{n}\pi_{ji}p_j, \quad {\ensuremath{\left( 2 \right)}} \bar{p}_i > x_i+\sum_{j=1}^{n}\pi_{ji}p_j.$$
1. Assume that $\bar{p}_i \le x_i+\sum_{j=1}^{n}\pi_{ji}p_j$. Then, by , $\Phi^{\text{RV}_+}_i{\ensuremath{\left( \bm{p} \right)}} = \bar{p}_i$. By Lemma \[RV\_lemma\_2\], $p_i = \bar{p}_i$. Hence, $p_i = \bar{p}_i = \Phi^{\text{RV}_+}_i{\ensuremath{\left( \bm{p} \right)}}$.
2. Assume that $\bar{p}_i > x_i+\sum_{j=1}^{n}\pi_{ji}p_j$. Then, by Definition , $\Phi^{\text{RV}_+}_i{\ensuremath{\left( \bm{p} \right)}} = \alpha x_i + \beta\sum_{j=1}^{n}\pi_{ji}p_j$. By Lemma \[RV\_lemma\_3\], $p_i = \alpha x_i + \beta\sum_{j=1}^{n}\pi_{ji}p_j$. Hence, $p_i = \alpha x_i + \beta\sum_{j=1}^{n}\pi_{ji}p_j = \Phi^{\text{RV}_+}_i{\ensuremath{\left( \bm{p} \right)}}$.
Therefore, $\bm{p}$ is a clearing vector.
The MILP aggregation functions $\Lambda^\text{EN}$ and $\Lambda^{\text{RV}_+}$ developed in this section are used in Section \[systemic\_risk\_measures\] to define and calculate systemic risk measures.
Optimization problems for systemic risk measures {#systemic_risk_measures}
================================================
In this section, we consider the computation of the *(sensitive) systemic risk measures* studied in [@feinstein], [@biagini], [@ararat], which are set-valued functionals of a random operating cash flow vector and defined in terms of the aggregation function of the underlying network model. While the aforementioned articles focus mainly on the case where the aggregation function is concave which results in the convex-valuedness of the corresponding systemic risk measure, the aggregation functions we use are not concave and the corresponding systemic risk measures fail to have convex values, in general.
We are mainly interested in the systemic risk measures for the signed Eisenberg-Noe and Rogers-Veraart network models. We follow a unifying approach by using a general aggregation function defined in terms of a mixed-integer optimization problem. To be able to approximate the nonconvex values of the corresponding systemic risk measure, we associate a (generally nonconvex) vector optimization problem to it. We solve this problem by the recent Benson-type algorithm in [@non-conv.benson] (Section \[algorithm\]). For this purpose, we study two families of (scalar) optimization problems: the weighted-sum scalarization problem (Section \[secP1\]) and the problem of calculating the minimum step-length to enter a set with a fixed direction (Section \[secP2\]). We prove that both problems in both models can be formulated as MILP problems. We also prove some results related to the feasibility and boundedness of these MILP problems.
Without specifying a particular network model, we consider a financial network with $n\in{\mathbb{N}}$ institutions. As in Section \[systemic\_risk\_models\], we write ${\mathcal{N}}={\ensuremath{ \left\{ 1,\ldots,n \right\} }}$. Similarly, let ${\mathcal{K}}={\ensuremath{ \left\{ 1,\ldots,K \right\} }}$ for some $K\in{\mathbb{N}}$. We consider a finite probability space $(\Omega,{\mathcal{F}},\mathbb{P})$, where $\Omega = {\ensuremath{ \left\{ \omega^1,\ldots,\omega^K \right\} }}$, ${\mathcal{F}}$ is the power set of $\Omega$, and ${\mathbb{P}}$ is a probability measure determined by the elementary probabilities $q^k \coloneqq {\mathbb{P}}{\ensuremath{ \left\{ \omega^k \right\} }}>0$, $k\in{\mathcal{K}}$. We denote by $L{\ensuremath{\left( {\mathbb{R}}^n \right)}}$ the linear space of all random vectors $\bm{X}\colon\Omega\to{\mathbb{R}}^n$. For every $\bm{X}\in L{\ensuremath{\left( {\mathbb{R}}^n \right)}}$, let $${\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}\coloneqq \underset{i\in{\mathcal{N}},~k\in{\mathcal{K}}}{\max}{\ensuremath{ \left| X_i(\omega^k) \right| }}.$$
We use the notion of grouping, also discussed in [@feinstein], to keep the dimension of the systemic risk measure at a reasonable level for computational purposes. This notion allows one to categorize the members of the network into groups and assign the same capital level for all the members of a group. To that end, let $G\ge1$ be an integer denoting the number of groups and ${\mathcal{G}}={\ensuremath{ \left\{ 1,\ldots,G \right\} }}$ the set of groups in the network. For the computations in Section \[computational\_results\], we will use $G=2$ or $G=3$ groups. Let ${\ensuremath{\left( {\mathcal{N}}_\ell \right)}}_{\ell\in{\mathcal{G}}}$ be a partition of ${\mathcal{N}}$, where ${\mathcal{N}}_\ell$ denotes the set of all institutions that belong to group $\ell\in{\mathcal{G}}$. For each $\ell\in{\mathcal{G}}$, let $n_\ell\coloneqq|{\mathcal{N}}_\ell|$ and $B_\ell\in{\mathbb{R}}^{G\times n_\ell}$ the matrix having 1’s in the $\ell$^th^ row and 0’s elsewhere: $$B_\ell \coloneqq
\left[
\begin{array}{ccc}
0 & \ldots & 0 \\
\vdots & \ddots & \vdots \\
1 & \ldots & 1 \\
\vdots & \ddots & \vdots \\
0 & \ldots & 0 \\
\end{array}
\right].$$ Let $B\in{\mathbb{R}}^{G\times n}$ be the *grouping matrix* defined by $$\label{grouping_matrix_B}
B \coloneqq
\left[
\begin{array}{cccc}
B_1 & B_2 & \ldots & B_G \\
\end{array}
\right].$$
We consider the *(grouped) sensitive systemic risk measure* ${R^{\text{OPT}}}\colon L({\mathbb{R}}^n)\to 2^{{\mathbb{R}}^n}$ defined by $$\label{sensitive_set-valued}
{R^{\text{OPT}}}{\ensuremath{\left( \bm{X} \right)}}\coloneqq{\ensuremath{ \left\{ \bm{z}\in{\mathbb{R}}^n\mid \Lambda^\text{OPT}(\bm{X}+{{B}^\mathsf{T}}\bm{z})\in{\mathcal{A}}\right\} }},$$ where $\Lambda^\text{OPT}\colon {\mathbb{R}}^n\to{\mathbb{R}}\cup{\ensuremath{ \left\{ -\infty \right\} }}$ is an aggregation function and ${\mathcal{A}}\subseteq L({\mathbb{R}}^n)$ is an *acceptance set*, that is, the set of all random aggregate outputs that are at an acceptable level of risk. We assume that $\Lambda^\text{OPT}$ is a general optimization aggregation function of the form $$\label{aggregation_OPT}
\Lambda^\text{OPT}{\ensuremath{\left( \bm{x} \right)}} \coloneqq \sup{\ensuremath{ \left\{ f{\ensuremath{\left( \bm{p} \right)}}\mid {\ensuremath{\left( \bm{p}, \bm{s} \right)}}\in{\mathcal{Y}}{\ensuremath{\left( \bm{x} \right)}}, \bm{p}\in{\mathbb{R}}^{n}, \bm{s}\in{\mathbb{Z}}^{n} \right\} }},$$ where $f:{\mathbb{R}}^{n}\to{\mathbb{R}}$ is a strictly increasing and continuous function, and ${\mathcal{Y}}:{\mathbb{R}}^n\to 2^{{\mathbb{R}}^{n}\times{\mathbb{Z}}^{n}}$ is a set-valued constraint function such that ${\mathcal{Y}}{\ensuremath{\left( \bm{x} \right)}}$ is either the empty set or a nonempty compact set for every $\bm{x}\in{\mathbb{R}}^n$. In particular, this general structure covers the aggregation functions $\Lambda^\text{OPT} = \Lambda^\text{EN}$ and $\Lambda^\text{OPT} = \Lambda^{\text{RV}_+}$ defined in Section \[systemic\_risk\_models\]. On the other hand, we assume that ${\mathcal{A}}$ is a halfspace-type acceptance set defined by $$\label{acceptance_set}
{\mathcal{A}}= {\ensuremath{ \left\{ Y \in L{\ensuremath{\left( {\mathbb{R}}^n \right)}} \mid {\mathbb{E}}{\ensuremath{ \left[ Y \right] }}\ge\gamma \right\} }},$$ where $\gamma\in{\mathbb{R}}$ is some suitable threshold. Hence, the corresponding systemic risk measure ${R^{\text{OPT}}}$ becomes $$\label{senSystemicRiskMeasureOPT}
{R^{\text{OPT}}}{\ensuremath{\left( \bm{X} \right)}} = {\ensuremath{ \left\{ \bm{z}\in{\mathbb{R}}^G\mid{\mathbb{E}}{\ensuremath{ \left[ \Lambda^\text{OPT}(\bm{X}+{{B}^\mathsf{T}}\bm{z}) \right] }}\ge\gamma \right\} }}.$$
We write ${R^{\text{OPT}}}={R^\text{EN}}$ when $\Lambda^{\text{OPT}}=\Lambda^\text{EN}$ and ${R^{\text{OPT}}}={R^{\text{RV}_+}}$ when $\Lambda^{\text{OPT}} = \Lambda^{\text{RV}_+}$, and refer to them as the Eisenberg-Noe and Rogers-Veraart systemic risk measures, respectively.
For ${R^{\text{RV}_+}}(\bm{X})$, the definition of $\Lambda^{\text{RV}_+}$ in implies the *implicit* condition $\bm{X} + {{B}^\mathsf{T}}\bm{z} \ge 0$.
Next, we introduce a vector optimization problem associated to each value of ${R^{\text{OPT}}}$. Let us fix $\bm{X}\in L({\mathbb{R}}^n)$ and consider the vector optimization problem $$\begin{aligned}
\label{vector_optimization_OPT}
\begin{split}
\text{minimize} &\quad \bm{z}\in{\mathbb{R}}^G \text{ with respect to } \le \\
\text{subject to}&\quad {\mathbb{E}}{\ensuremath{ \left[ \Lambda^\text{OPT}(\bm{X}+{{B}^\mathsf{T}}\bm{z}) \right] }}\ge\gamma,
\end{split}\end{aligned}$$ where $\le$ denotes the usual componentwise ordering in ${\mathbb{R}}^n$. Note that ${R^{\text{OPT}}}{\ensuremath{\left( \bm{X} \right)}}$ coincides with the so-called *upper image* of this vector optimization problem in the sense that $${R^{\text{OPT}}}{\ensuremath{\left( \bm{X} \right)}} = {\ensuremath{ \left\{ \bm{z}+{\mathbb{R}}^G_+ \mid {\mathbb{E}}{\ensuremath{ \left[ \Lambda^\text{OPT}(\bm{X}+{{B}^\mathsf{T}}\bm{z}) \right] }}\ge\gamma \right\} }}.$$ In general, due to the lack of concavity for $\Lambda^{\text{OPT}}$, the set ${R^{\text{OPT}}}{\ensuremath{\left( \bm{X} \right)}}$ may fail to be convex (see Remarks \[noconcave1\],\[noconcave2\]). While the majority of the Benson-type approximation algorithms in the literature (@benson, @hamel-lohne-rudloff, @lohne-rudloff-ulus) work for linear/convex vector optimization problems and are based on creating supporting halfspaces for the upper image, we use the more recent Benson-type algorithm proposed in @non-conv.benson, which works for nonconvex upper images and is based on creating sets of the form “point plus cone." The algorithm relies on the assumption that the associated weighted-sum scalarization and minimum step-length problems can be solved to optimality. In the next two subsections, we propose methods to solve these problems in our case by exploiting the structure of the optimization aggregation function $\Lambda^\text{OPT}$.
Weighted-sum scalarizations of systemic risk measures {#secP1}
-----------------------------------------------------
For each $\bm{w}\in{\mathbb{R}}^G_+\backslash{\ensuremath{ \left\{ {\mathbf{0}}\right\} }}$, we consider the weighted-sum scalarization problem $$\label{P1_OPT}
{\mathcal{P}}_1{\ensuremath{\left( \bm{w} \right)}} = \inf_{z\in{R^{\text{OPT}}}(\bm{X})}{{\bm{w}}^\mathsf{T}}\bm{z}=\underset{\bm{z}\in{\mathbb{R}}^G}{\inf} {\ensuremath{ \left\{ {{\bm{w}}^\mathsf{T}}\bm{z} \mid {\mathbb{E}}{\ensuremath{ \left[ \Lambda^\text{OPT}(\bm{X}+{{B}^\mathsf{T}}\bm{z}) \right] }}\ge\gamma \right\} }}.$$
The following theorem provides an alternative formulation for ${\mathcal{P}}_1{\ensuremath{\left( \bm{w} \right)}}$.
\[theorem\_P1\_OPT\] Let $\bm{w}\in{\mathbb{R}}^G_+\backslash{\ensuremath{ \left\{ {\mathbf{0}}\right\} }}$. Consider the problem in and let $$\label{z_OPT}
\begin{split}
{\mathcal{Z}}_1{\ensuremath{\left( \bm{w} \right)}} \coloneqq \underset{\bm{z}\in{\mathbb{R}}^G}{\inf} \Bigg\{{{\bm{w}}^\mathsf{T}}\bm{z} \mid\ & \sum_{k\in{\mathcal{K}}}q^k f(\bm{p}^k) \ge \gamma, \\
&(\bm{p}^k, \bm{s}^k)\in{\mathcal{Y}}(\bm{X}(\omega^k)+{{B}^\mathsf{T}}\bm{z}), \bm{p}^k\in{\mathbb{R}}^{n}, \bm{s}^k\in{\mathbb{Z}}^{n}\ \forall k\in{\mathcal{K}}\Bigg\}.
\end{split}$$ Then, ${\mathcal{P}}_1{\ensuremath{\left( \bm{w} \right)}} = {\mathcal{Z}}_1{\ensuremath{\left( \bm{w} \right)}}$. In particular, if one of the problems in and has a finite optimal value, then so does the other one and the optimal values coincide.
Let $(\bm{z},(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ be a feasible solution for the problem in . Then, for each $k\in{\mathcal{K}}$, $(\bm{p}^k, \bm{s}^k)$ is a feasible solution to $\Lambda^\text{OPT}(\bm{X}(\omega^k)+{{B}^\mathsf{T}}\bm{z})$ in because the optimization problem in includes the constraints of . Hence, for every $k\in{\mathcal{K}}$, $$\Lambda^\text{OPT}(\bm{X}(\omega^k)+{{B}^\mathsf{T}}\bm{z})\ge f(\bm{p}^k),$$ which implies $${\mathbb{E}}{\ensuremath{ \left[ \Lambda^\text{OPT}(\bm{X}+{{B}^\mathsf{T}}\bm{z}) \right] }} \ge \sum_{k=1}^{K}q^k f(\bm{p}^k) \ge \gamma,$$ where the second inequality holds by feasibility of $(\bm{z},(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$. Hence, $\bm{z}$ is a feasible solution for the problem in . So ${\mathcal{P}}_1{\ensuremath{\left( \bm{w} \right)}} \le {\mathcal{Z}}_1{\ensuremath{\left( \bm{w} \right)}}$.
Conversely, let ${\accentset{\bullet}{\bm{z}}}$ be a feasible solution for the problem in . For each $k\in{\mathcal{K}}$, there exists an optimal solution $({\accentset{\bullet}{\bm{p}}}^k, {\accentset{\bullet}{\bm{s}}}^k)$ to the problem for $\Lambda^\text{OPT}(\bm{X}(\omega^k)+{{B}^\mathsf{T}}{\accentset{\bullet}{\bm{z}}})$. Then, $$\sum_{k=1}^{K}q^k f({\accentset{\bullet}{\bm{p}}}^k) = {\mathbb{E}}{\ensuremath{ \left[ \Lambda^\text{OPT}(\bm{X}+{{B}^\mathsf{T}}{\accentset{\bullet}{\bm{z}}}) \right] }}\ge \gamma,$$ by the definition of ${\mathcal{P}}_1{\ensuremath{\left( \bm{w} \right)}}$. Hence, $({\accentset{\bullet}{\bm{z}}},({\accentset{\bullet}{\bm{p}}}^k, {\accentset{\bullet}{\bm{s}}}^k)_{k\in{\mathcal{K}}})$ is a feasible solution for the problem in . So ${\mathcal{P}}_1{\ensuremath{\left( \bm{w} \right)}} \ge {\mathcal{Z}}_1{\ensuremath{\left( \bm{w} \right)}}$.
Let $\ell\in{\mathcal{G}}$ and $\bm{e}^\ell$ the corresponding standard unit vector in ${\mathbb{R}}^G$. Observe that the weighted-sum scalarization problem $${\mathcal{P}}_1(\bm{e}^\ell) = \underset{\bm{z}\in{\mathbb{R}}^G}{\inf} {\ensuremath{ \left\{ z_\ell \mid {\mathbb{E}}{\ensuremath{ \left[ \Lambda^\text{OPT}(\bm{X}+{{B}^\mathsf{T}}\bm{z}) \right] }}\ge\gamma \right\} }}$$ is a single-objective optimization problem of the vector optimization problem in . By Theorem \[theorem\_P1\_OPT\], ${\mathcal{P}}_1{\ensuremath{\left( \bm{e}^\ell \right)}} = {\mathcal{Z}}_1{\ensuremath{\left( \bm{e}^\ell \right)}}$.
\[ideal\] Let $\bm{z}^\text{ideal}\in{\mathbb{R}}^G$ be the ideal point of the vector optimization problem in in the sense that the entries of $\bm{z}^\text{ideal}$ minimize each of the objective functions of the vector optimization problem. In other words, one can define $$\bm{z}^\text{ideal} \coloneqq {{{\ensuremath{\left( {\mathcal{P}}_1{\ensuremath{\left( \bm{e}^1 \right)}},\ldots,{\mathcal{P}}_1{\ensuremath{\left( \bm{e}^G \right)}} \right)}}}^\mathsf{T}}\in{\mathbb{R}}^G$$ assuming that ${\mathcal{P}}_1{\ensuremath{\left( \bm{e}^\ell \right)}}$ is finite for each $\ell\in{\mathcal{G}}$. Theorem \[theorem\_P1\_OPT\] allows one to solve $G$ optimization problems with compact feasible sets, namely, the problems ${\ensuremath{\left( {\mathcal{Z}}_1{\ensuremath{\left( \bm{e}^\ell \right)}} \right)}}_{\ell\in{\mathcal{G}}}$, to obtain the ideal point of the vector optimization problem in .
In the following two subsections, we apply Theorem \[theorem\_P1\_OPT\] to the special cases $\Lambda^\text{OPT} = \Lambda^\text{EN}$ and $\Lambda^\text{OPT} = \Lambda^{\text{RV}_+}$, respectively. For this purpose, we fix the function $f:{\mathbb{R}}^n\to{\mathbb{R}}$ in the objective functions of the MILP aggregation functions as $$f(\bm{p}) \coloneqq {{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{p}.$$ It is clear that $f$ is a strictly increasing continuous linear function bounded on the interval ${\ensuremath{ \left[ {\mathbf{0}}, \bm{{\bar{p}}} \right] }}\subseteq{\mathbb{R}}^n$. Moreover, since the vector optimization algorithm in Section \[algorithm\] requires solving weighted-sum scalarization problems only for the standard unit vectors, we state our results for such direction vectors.
### Weighted-sum scalarizations of Eisenberg-Noe systemic risk measure
Let ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{X}} \right)}}}$ be a signed Eisenberg-Noe network.
\[corollary\_P1\_EN\] Let $\ell\in{\mathcal{G}}$. Consider the single-objective optimization problem $$\label{P1_EN}
{\mathcal{P}}_1^\text{EN}(\bm{e}^\ell) \coloneqq \underset{\bm{z}\in{\mathbb{R}}^G}{\inf} {\ensuremath{ \left\{ z_\ell \mid {\mathbb{E}}{\ensuremath{ \left[ \Lambda^\text{EN}(\bm{X}+{{B}^\mathsf{T}}\bm{z}) \right] }}\ge\gamma \right\} }},$$ and let $$\label{P1_MILP_EN}
\begin{split}
{\mathcal{Z}}^\text{EN}_1(\bm{e}^\ell) \coloneqq \underset{\bm{z}\in{\mathbb{R}}^G}{\inf} \Bigg\{ z_\ell \mid\ & \sum_{k\in{\mathcal{K}}}q^k{{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{p}^k \ge \gamma, \\
& \bm{p}^k \le {\ensuremath{\left( {{\bm{\Pi}}^\mathsf{T}}\bm{p}^k + (\bm{X}(\omega^k) + {{B}^\mathsf{T}}\bm{z}) + M({\bm{\mathbbm{1}}}- \bm{s}^k) \right)}} \wedge (\bm{{\bar{p}}}\odot\bm{s}^k), \\
& {{\bm{\Pi}}^\mathsf{T}}\bm{p}^k + (\bm{X}(\omega^k) + {{B}^\mathsf{T}}\bm{z})\le M\bm{s}^k, \\
& \bm{p}^k \in {\ensuremath{ \left[ {\mathbf{0}}, \bm{{\bar{p}}} \right] }}, \bm{s}^k \in {\ensuremath{ \left\{ 0,1 \right\} }}^n\ \forall k\in{\mathcal{K}}\Bigg\},
\end{split}$$ where $M = {2{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{\left( n+1 \right)}}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}$. Then, ${\mathcal{P}}_1^\text{EN}(\bm{e}^\ell) = {\mathcal{Z}}^\text{EN}_1(\bm{e}^\ell)$. In particular, if one of the problems in and has a finite optimal value, then so does the other one and the optimal values coincide.
Let ${\mathcal{Y}}_\text{EN}: {\mathbb{R}}^n\to 2^{{\mathbb{R}}^n\times{\mathbb{Z}}^n}$ be a set-valued function defined by $$\label{YYY_EN}
\begin{split}
{\mathcal{Y}}_\text{EN}{\ensuremath{\left( \bm{x} \right)}} \coloneqq \Bigg\{ (\bm{p}, \bm{s}) \in{\mathbb{R}}^n\times{\mathbb{Z}}^n \mid\ & \bm{p} \le {\ensuremath{ \left[ {{\bm{\Pi}}^\mathsf{T}}\bm{p} + \bm{x} + M{\ensuremath{\left( {\bm{\mathbbm{1}}}- \bm{s} \right)}} \right] }} \wedge {\ensuremath{\left( \bm{{\bar{p}}}\odot\bm{s} \right)}}, \\
& {{\bm{\Pi}}^\mathsf{T}}\bm{p} + \bm{x} \le M\bm{s},\ \bm{p} \in {\ensuremath{ \left[ {\mathbf{0}}, \bm{{\bar{p}}} \right] }},\ \bm{s} \in {\ensuremath{ \left\{ 0,1 \right\} }}^n
\Bigg\}.
\end{split}$$ Then, applying Theorem \[theorem\_P1\_OPT\] with ${\mathcal{Y}}= {\mathcal{Y}}_\text{EN}$ gives ${\mathcal{P}}_1^\text{EN}(\bm{e}^\ell) = {\mathcal{Z}}^\text{EN}_1(\bm{e}^\ell)$.
The next three propositions present some boundedness and feasibility results for the MILP problem of computing ${\mathcal{Z}}^\text{EN}_1(\bm{e}^\ell)$, $\ell\in{\mathcal{G}}$, in .
\[P1\_EN\_upperbound\] Let $\ell\in{\mathcal{G}}$. If the problem in has an optimal solution, then $${\mathcal{P}}_1^\text{EN}(\bm{e}^\ell)={\mathcal{Z}}^\text{EN}_1(\bm{e}^\ell)\leq {\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}.$$
\[P1\_EN\_boundedness\] Let $\ell\in{\mathcal{G}}$. If the problem in has a feasible solution, then it has a finite optimal value, that is, ${\mathcal{Z}}^\text{EN}_1{\ensuremath{\left( \bm{e}^\ell \right)}}\in{\mathbb{R}}$.
\[P1\_EN\_feasibility\] Let $\ell\in{\mathcal{G}}$. The problem in has a feasible solution if and only if $\gamma \le {{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}}$.
The proofs of Propositions \[P1\_EN\_upperbound\], \[P1\_EN\_boundedness\], \[P1\_EN\_feasibility\] can be found in Appendices \[P1\_EN\_upperbound\_proof\], \[P1\_EN\_boundedness\_proof\], \[P1\_EN\_feasibility\_proof\], respectively.
Let $\ell\in{\mathcal{G}}$. Suppose that there exists an optimal solution $(\bm{z},(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ of the MILP problem in . By the structure of the matrix $B$, for each $i\in{\mathcal{N}}$, it holds ${\ensuremath{\left( {{B}^\mathsf{T}}\bm{z} \right)}}_i = z_t$ for some $t\in{\mathcal{G}}$. Hence, by Proposition \[P1\_EN\_upperbound\], $({{B}^\mathsf{T}}\bm{z})_i \le {\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}$ holds for each $i\in{\mathcal{N}}$. In addition, for every $i\in{\mathcal{N}}$, $k\in{\mathcal{K}}$, and $\bm{p}^k\in{\ensuremath{ \left[ {\mathbf{0}},\bm{{\bar{p}}} \right] }}$, it holds $\sum_{j=1}^{n}\pi_{ji}p_j^k < n{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}$ and $X_i{\ensuremath{\left( \omega^k \right)}} \le {\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}$. Hence, the choice of $M = {2{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{\left( n+1 \right)}}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}$ in Corollary \[corollary\_P1\_EN\] is justified, since, to ensure the feasibility of the third constraint in , it is enough to choose $M$ such that $$\sum_{j=1}^{n}\pi_{ji}p_j^k + (X_i(\omega^k) + ({{B}^\mathsf{T}}\bm{z})_i)\le M$$ for every $i\in{\mathcal{N}}$, $k\in{\mathcal{K}}$ and $\bm{p}^k\in{\ensuremath{ \left[ {\mathbf{0}},\bm{{\bar{p}}} \right] }}$.
### Weighted-sum scalarizations of Rogers-Veraart systemic risk measure
Let ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{X}}, \alpha, \beta \right)}}}$ be a Rogers-Veraart network.
\[corollary\_P1\_RV\] Let $\ell\in{\mathcal{G}}$. Consider the single-objective optimization problem $$\label{P1_RV}
{\mathcal{P}}_1^{\text{RV}_+}(\bm{e}^\ell) \coloneqq \underset{\bm{z}\in{\mathbb{R}}^G}{\inf} {\ensuremath{ \left\{ z_\ell \mid {\mathbb{E}}{\ensuremath{ \left[ \Lambda^{\text{RV}_+}(\bm{X}+{{B}^\mathsf{T}}\bm{z}) \right] }}\ge\gamma \right\} }},$$ and let $$\label{P1_MILP_RV}
\begin{split}
{\mathcal{Z}}_1^{\text{RV}_+}(\bm{e}^\ell) \coloneqq \underset{\bm{z}\in{\mathbb{R}}^G}{\inf} \Bigg\{ z_\ell \mid\ & \sum_{k\in{\mathcal{K}}}q^k{{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{p}^k \ge \gamma, \\
& \bm{p}^k \le \alpha{\ensuremath{\left( \bm{X}(\omega^k)+{{B}^\mathsf{T}}\bm{z} \right)}} + \beta{{\bm{\Pi}}^\mathsf{T}}\bm{p}^k + \bm{{\bar{p}}}\odot\bm{s}^k, \\
& \bm{{\bar{p}}}\odot\bm{s}^k \le {\ensuremath{\left( \bm{X}(\omega^k)+{{B}^\mathsf{T}}\bm{z} \right)}} + {{\bm{\Pi}}^\mathsf{T}}\bm{p}^k, \\
& \bm{X}(\omega^k)+{{B}^\mathsf{T}}\bm{z} \ge 0, \ \bm{p}^k \in {\ensuremath{ \left[ {\mathbf{0}}, \bm{{\bar{p}}} \right] }}, \bm{s}^k \in {\ensuremath{ \left\{ 0,1 \right\} }}^n\ \forall k\in{\mathcal{K}}\Bigg\}.
\end{split}$$ Then, ${\mathcal{P}}_1^{\text{RV}_+}{\ensuremath{\left( \bm{e}^\ell \right)}} = {\mathcal{Z}}_1^{\text{RV}_+}{\ensuremath{\left( \bm{e}^\ell \right)}}$. In particular, if one of the problems in and has a finite optimal value, then so does the other one and the optimal values coincide.
Let ${\mathcal{Y}}_{\text{RV}_+}\colon {\mathbb{R}}^n\to 2^{{\mathbb{R}}^n\times{\mathbb{Z}}^n}$ be a set-valued function defined by $$\label{YYY_RV}
\begin{split}
{\mathcal{Y}}_{\text{RV}_+}{\ensuremath{\left( \bm{x} \right)}} \coloneqq \Bigg\{ (\bm{p}, \bm{s})\in{\mathbb{R}}^n\times{\mathbb{Z}}^n \mid\ & \bm{p} \le \alpha\bm{x} + \beta{{\bm{\Pi}}^\mathsf{T}}\bm{p} + \bm{{\bar{p}}}\odot\bm{s}, \\
& \bm{{\bar{p}}}\odot\bm{s} \le \bm{x} + {{\bm{\Pi}}^\mathsf{T}}\bm{p}, \bm{p} \in {\ensuremath{ \left[ {\mathbf{0}}, \bm{{\bar{p}}} \right] }}, \bm{s} \in {\ensuremath{ \left\{ 0,1 \right\} }}^n \Bigg\}.
\end{split}$$ for each $\bm{x}\in{\mathbb{R}}^n_+$ and ${\mathcal{Y}}_{\text{RV}_+}(\bm{x})=\emptyset$ for each $\bm{x}\in {\mathbb{R}}^n\setminus {\mathbb{R}}^n_+$. Then, applying Theorem \[theorem\_P1\_OPT\] with ${\mathcal{Y}}= {\mathcal{Y}}_{\text{RV}_+}$ gives ${\mathcal{P}}_1^{\text{RV}_+}{\ensuremath{\left( \bm{e}^\ell \right)}} = {\mathcal{Z}}_1^{\text{RV}_+}{\ensuremath{\left( \bm{e}^\ell \right)}}$.
The next three propositions present some boundedness and feasibility results for the MILP problem of computing ${\mathcal{Z}}^{\text{RV}_+}_1(\bm{e}^\ell)$, $\ell\in{\mathcal{G}}$, in .
\[P1\_RV\_upperbound\] Let $\ell\in{\mathcal{G}}$. If the problem in has an optimal solution, then $${\mathcal{P}}_1^{\text{RV}_+}(\bm{e}^\ell)={\mathcal{Z}}_1^{\text{RV}_+}(\bm{e}^\ell)\leq {{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ \frac{1}{\alpha}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}.$$
\[P1\_RV\_boundedness\] Let $\ell\in{\mathcal{G}}$. If the problem in has a feasible solution, then it has a finite optimal value, that is, ${\mathcal{Z}}_1^{\text{RV}_+}{\ensuremath{\left( \bm{e}^\ell \right)}}\in{\mathbb{R}}$.
\[P1\_RV\_feasibility\] Let $\ell\in{\mathcal{G}}$. The problem in has a feasible solution if and only if $\gamma \le {{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}}$.
The proofs of Propositions \[P1\_RV\_upperbound\], \[P1\_RV\_boundedness\] and \[P1\_RV\_feasibility\] can be found in Appendices \[P1\_RV\_upperbound\_proof\], \[P1\_RV\_boundedness\_proof\] and \[P1\_RV\_feasibility\_proof\], respectively.
Minimum step-length function {#secP2}
----------------------------
Weighted-sum scalarizations are used to calculate supporting halfspaces for the value of a systemic risk measure and they can be sufficient to characterize the entire risk set when the set is convex. In our nonconvex case, we make use of additional scalarizations that are used to calculate the minimum step-lengths to the enter the risk set from possibly outside points. Such scalarizations are well-known in vector optimization; see @p2_reference_1, @p2_reference_2, for instance.
For each $\bm{v}\in{\mathbb{R}}^G$, we consider $$\begin{aligned}
\label{P2_OPT}
{\mathcal{P}}_2{\ensuremath{\left( \bm{v} \right)}} &\coloneqq\inf{\ensuremath{ \left\{ \mu\in{\mathbb{R}}\mid {{B}^\mathsf{T}}\bm{v}+\mu{\bm{\mathbbm{1}}}\in {R^{\text{OPT}}}{\ensuremath{\left( \bm{X} \right)}} \right\} }}\notag\\
&= \inf {\ensuremath{ \left\{ \mu\in{\mathbb{R}}\mid{\mathbb{E}}{\ensuremath{ \left[ \Lambda^\text{OPT}(\bm{X}+{{B}^\mathsf{T}}\bm{v}+\mu{\bm{\mathbbm{1}}}) \right] }}\ge\gamma \right\} }},\end{aligned}$$ which can be interpreted as the minimum step-length in the direction ${\bm{\mathbbm{1}}}$ from the point $\bm{v}$ to the boundary of the set ${R^{\text{OPT}}}{\ensuremath{\left( \bm{X} \right)}}$.
The following theorem provides an alternative formulation for ${\mathcal{P}}_2{\ensuremath{\left( \bm{v} \right)}}$.
\[theorem\_P2\_OPT\] Let $\bm{v}\in{\mathbb{R}}^n$. Consider the problem in and let $$\label{P2_OPT_lin}
\begin{split}
{\mathcal{Z}}_2{\ensuremath{\left( \bm{v} \right)}} \coloneqq \inf\Bigg\{ \mu \in{\mathbb{R}}\mid\ & \sum_{k\in{\mathcal{K}}}q^k f(\bm{p}^k) \ge \gamma,\\
& (\bm{p}^k, \bm{s}^k)\in{\mathcal{Y}}(\bm{X}(\omega^k)+{{B}^\mathsf{T}}\bm{v}+\mu{\bm{\mathbbm{1}}}), \
\bm{p}^k\in{\mathbb{R}}^n, \bm{s}^k\in{\mathbb{Z}}^n\ \forall k\in{\mathcal{K}}\Bigg\}.
\end{split}$$ Then, ${\mathcal{P}}_2{\ensuremath{\left( \bm{v} \right)}} = {\mathcal{Z}}_2{\ensuremath{\left( \bm{v} \right)}}$. In particular, if one of the problems in and has a finite optimal value, then so does the other one and the optimal values coincide.
Let $(\mu,(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ be a feasible solution of the problem in . For each $k\in{\mathcal{K}}$, $(\bm{p}^k, \bm{s}^k)$ is a feasible solution to $\Lambda^\text{OPT}(\bm{X}{\ensuremath{\left( \omega^k \right)}}+{{B}^\mathsf{T}}\bm{v}+\mu{\bm{\mathbbm{1}}})$ in because the problem in includes the constraints of . Hence, for each $k\in{\mathcal{K}}$, $$\Lambda^\text{OPT}(\bm{X}(\omega^k)+{{B}^\mathsf{T}}\bm{v}+\mu{\bm{\mathbbm{1}}}) \ge f(\bm{p}^k),$$ which implies $${\mathbb{E}}{\ensuremath{ \left[ \Lambda^\text{OPT}(\bm{X}+{{B}^\mathsf{T}}\bm{v}+\mu{\bm{\mathbbm{1}}}) \right] }} \ge \sum_{k=1}^{K}q^kf(\bm{p}^k) \ge \gamma,$$ where the second inequality holds by feasibility of $(\mu,(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$. Then, $\mu$ is a feasible solution for the problem in . Hence, ${\mathcal{P}}_2{\ensuremath{\left( \bm{v} \right)}} \leq {\mathcal{Z}}_2{\ensuremath{\left( \bm{v} \right)}}$.
Conversely, let ${\accentset{\bullet}{\mu}}\in{\mathbb{R}}$ be a feasible solution for the problem in . Then, for each $k\in{\mathcal{K}}$, $\Lambda^\text{OPT}(\bm{X}(\omega^k)+{{B}^\mathsf{T}}\bm{v}+{\accentset{\bullet}{\mu}}{\bm{\mathbbm{1}}})\in{\mathbb{R}}$ and, by the compactness of ${\mathcal{Y}}(\bm{X}(\omega^k)+{{B}^\mathsf{T}}\bm{v}+{\accentset{\bullet}{\mu}}{\bm{\mathbbm{1}}})$, there exists an optimal solution $({\accentset{\bullet}{\bm{p}}}^k, {\accentset{\bullet}{\bm{s}}}^k)$ for the problem $\Lambda^\text{OPT}(\bm{X}(\omega^k)+{{B}^\mathsf{T}}\bm{v}+{\accentset{\bullet}{\mu}}{\bm{\mathbbm{1}}})$ in . Then, $$\sum_{k=1}^{K}q^k f({\accentset{\bullet}{\bm{p}}}^k) = {\mathbb{E}}{\ensuremath{ \left[ \Lambda^\text{OPT}(\bm{X}+{{B}^\mathsf{T}}\bm{v}+{\accentset{\bullet}{\mu}}{\bm{\mathbbm{1}}}) \right] }}\ge \gamma$$ by the definition of ${\mathcal{P}}_2{\ensuremath{\left( \bm{v} \right)}}$. Hence, $({\accentset{\bullet}{\mu}},({\accentset{\bullet}{\bm{p}}}^k, {\accentset{\bullet}{\bm{s}}}^k)_{k\in{\mathcal{K}}})$ is a feasible solution for the problem in . Hence, ${\mathcal{P}}_2{\ensuremath{\left( \bm{v} \right)}} \geq {\mathcal{Z}}_2{\ensuremath{\left( \bm{v} \right)}}$.
The following two sections apply Theorem \[theorem\_P2\_OPT\] to the special cases $\Lambda^\text{OPT} = \Lambda^\text{EN}$ and $\Lambda^\text{OPT} = \Lambda^{\text{RV}_+}$, respectively.
### Minimum step-length function for Eisenberg-Noe systemic risk measure
Let ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{X}} \right)}}}$ be an Eisenberg-Noe network.
\[corollary\_P2\_EN\] Let $\bm{v}\in{\mathbb{R}}^G$. Consider the problem $$\label{P2_EN}
{\mathcal{P}}_2^\text{EN}{\ensuremath{\left( \bm{v} \right)}} \coloneqq \inf {\ensuremath{ \left\{ \mu\in{\mathbb{R}}\mid{\mathbb{E}}{\ensuremath{ \left[ \Lambda^\text{EN}(\bm{X}+{{B}^\mathsf{T}}\bm{v}+\mu{\bm{\mathbbm{1}}}) \right] }}\ge\gamma \right\} }},$$ and let $$\label{P2_MILP_EN}
\begin{split}
{\mathcal{Z}}_2^\text{EN}{\ensuremath{\left( \bm{v} \right)}} \coloneqq \inf\Bigg\{ \mu\in{\mathbb{R}}&\mid \sum_{k\in{\mathcal{K}}}q^k{{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{p}^k \ge \gamma, \\
&\ \ \bm{p}^k \le {\ensuremath{\left( {{\bm{\Pi}}^\mathsf{T}}\bm{p}^k + (\bm{X}(\omega^k) + {{B}^\mathsf{T}}\bm{v}+\mu{\bm{\mathbbm{1}}}) + M({\bm{\mathbbm{1}}}- \bm{s}^k) \right)}} \wedge (\bm{{\bar{p}}}\odot\bm{s}^k), \\
&\ \ {{\bm{\Pi}}^\mathsf{T}}\bm{p}^k + (\bm{X}(\omega^k) + {{B}^\mathsf{T}}\bm{v}+\mu{\bm{\mathbbm{1}}})\le M\bm{s}^k, \\
&\ \ \bm{p}^k \in {\ensuremath{ \left[ {\mathbf{0}}, \bm{{\bar{p}}} \right] }}, \bm{s}^k \in {\ensuremath{ \left\{ 0,1 \right\} }}^n\ \forall k\in{\mathcal{K}}\Bigg\},
\end{split}$$ where $M={2{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+2{\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}+{\ensuremath{\left( n+1 \right)}}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}$. Then, ${\mathcal{P}}_2^\text{EN}{\ensuremath{\left( \bm{v} \right)}} = {\mathcal{Z}}_2^\text{EN}{\ensuremath{\left( \bm{v} \right)}}$. In particular, if one of the problems in and has a finite optimal value, then so does the other one and the optimal values coincide.
Let ${\mathcal{Y}}= {\mathcal{Y}}_\text{EN}$ as in the proof of Corollary \[corollary\_P1\_EN\]. Then, applying Theorem \[theorem\_P2\_OPT\] gives ${\mathcal{P}}_2^\text{EN}{\ensuremath{\left( \bm{v} \right)}} = {\mathcal{Z}}_2^\text{EN}{\ensuremath{\left( \bm{v} \right)}}$.
The next three propositions present some boundedness and feasibility results for the MILP problem in .
\[P2\_EN\_upperbound\] Let $\bm{v}\in{\mathbb{R}}^G$. If the problem in has an optimal solution, then $${\mathcal{P}}_2^\text{EN}{\ensuremath{\left( \bm{v} \right)}} = {\mathcal{Z}}_2^\text{EN}{\ensuremath{\left( \bm{v} \right)}}\leq {{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}.$$
\[P2\_EN\_boundedness\] Let $\bm{v}\in{\mathbb{R}}^G$. If the problem in has a feasible solution, then it has a finite optimal value, that is, ${\mathcal{Z}}_2^\text{EN}{\ensuremath{\left( \bm{v} \right)}}\in{\mathbb{R}}$.
\[P2\_EN\_feasibility\] Let $\bm{v}\in{\mathbb{R}}^G$. The problem in has a feasible solution if and only if $\gamma \le {{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}}$.
The proofs of Propositions \[P2\_EN\_upperbound\], \[P2\_EN\_boundedness\], \[P2\_EN\_feasibility\] are presented in Appendices \[P2\_EN\_upperbound\_proof\], \[P2\_EN\_boundedness\_proof\], \[P2\_EN\_feasibility\_proof\], respectively.
Let $\bm{v}\in{\mathbb{R}}^G$ and $(\mu,(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ an optimal solution of the MILP problem in . By Proposition , $\mu \le {{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}$. By the structure of the matrix $B$, for each $i\in{\mathcal{N}}$, it holds $({{B}^\mathsf{T}}\bm{v})_i = \bm{v}_t$ for some $t\in{\mathcal{G}}$. Hence, for every $\bm{v}\in{\mathbb{R}}^G$, $({{B}^\mathsf{T}}\bm{v})_i \le {\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}$. In addition, for every $i\in{\mathcal{N}}$, $k\in{\mathcal{K}}$, and $\bm{p}^k\in{\ensuremath{ \left[ {\mathbf{0}}, \bm{{\bar{p}}} \right] }}$, it holds $\sum_{j=1}^{n}\pi_{ji}p_j^k < n{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}$ and $X_i(\omega^k) \le {\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}$. Hence, the choice of $M$ as $M= {2{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+2{\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}+{\ensuremath{\left( n+1 \right)}}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}$ in Corollary \[corollary\_P2\_EN\] is justified, since, to ensure the feasibility of the third constraint in , it is enough to choose $M$ such that $$\sum_{j=1}^{n}\pi_{ji}p_j^k + {\ensuremath{\left( X_i(\omega^k) + ({{B}^\mathsf{T}}\bm{v})_i+\mu \right)}}\le M$$ for every $i\in{\mathcal{N}}$, $k\in{\mathcal{K}}$, $\bm{v}\in{\mathbb{R}}^G$ and $\bm{p}^k\in{\ensuremath{ \left[ {\mathbf{0}},\bm{{\bar{p}}} \right] }}$.
Proposition \[P1\_EN\_boundedness\] shows that if the MILP problem ${\mathcal{Z}}^\text{EN}_1{\ensuremath{\left( \bm{e}^\ell \right)}}$ in is feasible for every $\ell\in{\mathcal{G}}$, then the ideal point $z^\text{ideal}\in{\mathbb{R}}^n$ exists for the vector optimization problem in with $\Lambda^\text{OPT} = \Lambda^\text{EN}$. Proposition \[P1\_RV\_boundedness\] provides the same result for the vector optimization problem in with $\Lambda^\text{OPT} = \Lambda^{\text{RV}_+}$. In addition, the results of Propositions \[P1\_EN\_upperbound\], \[P1\_EN\_boundedness\], \[P2\_EN\_upperbound\] and \[P2\_EN\_boundedness\] allow one to choose the exact value for the upper bound $M$ in the corresponding MILP problems instead of assuming some heuristic values.
### Minimum step-length function for Rogers-Veraart systemic risk measure
Let ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{X}}, \alpha, \beta \right)}}}$ be a Rogers-Veraart network.
\[corollary\_P2\_RV\] Let $\bm{v}\in{\mathbb{R}}^G$. Consider the problem $$\label{P2_RV}
{\mathcal{P}}_2^{\text{RV}_+}{\ensuremath{\left( \bm{v} \right)}} \coloneqq \inf {\ensuremath{ \left\{ \mu\in{\mathbb{R}}\mid {\mathbb{E}}{\ensuremath{ \left[ \Lambda^{\text{RV}_+}(\bm{X}+{{B}^\mathsf{T}}\bm{v}+\mu{\bm{\mathbbm{1}}}) \right] }}\ge\gamma \right\} }},$$ and let $$\label{P2_MILP_RV}
\begin{split}
{\mathcal{Z}}_2^{\text{RV}_+}(\bm{v}) \coloneqq \inf\Bigg\{ \mu\in{\mathbb{R}}\mid \ & \sum_{k\in{\mathcal{K}}}q^k{{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{p}^k \ge \gamma, \\
& \bm{p}^k \le \alpha{\ensuremath{\left( \bm{X}(\omega^k) + {{B}^\mathsf{T}}\bm{v}+\mu{\bm{\mathbbm{1}}}\right)}} + \beta{{\bm{\Pi}}^\mathsf{T}}\bm{p}^k + \bm{{\bar{p}}}\odot\bm{s}^k, \\
& \bm{{\bar{p}}}\odot\bm{s}^k \le (\bm{X}(\omega^k) + {{B}^\mathsf{T}}\bm{v}+\mu{\bm{\mathbbm{1}}})+ {{\bm{\Pi}}^\mathsf{T}}\bm{p}^k, \\
& \bm{X}(\omega^k) + {{B}^\mathsf{T}}\bm{v}+\mu{\bm{\mathbbm{1}}}\ge0, \\
& \bm{p}^k \in {\ensuremath{ \left[ {\mathbf{0}}, \bm{{\bar{p}}} \right] }}, \bm{s}^k \in {\ensuremath{ \left\{ 0,1 \right\} }}^n\ \forall k\in{\mathcal{K}}\Bigg\}.
\end{split}$$ Then, ${\mathcal{P}}_2^{\text{RV}_+}(\bm{v}) = {\mathcal{Z}}_2^{\text{RV}_+}(\bm{v})$. In particular, if one of the problems in and has a finite optimal value, then so does the other one and the optimal values coincide.
Let ${\mathcal{Y}}= {\mathcal{Y}}_{\text{RV}_+}$ as in the proof of Corollary \[corollary\_P1\_RV\]. By Theorem \[theorem\_P2\_OPT\], the result follows.
The next three propositions present some boundedness and feasibility results for the problem in .
\[P2\_RV\_upperbound\] Let $\bm{v}\in{\mathbb{R}}^G$. If the problem in has an optimal solution, then $${\mathcal{P}}_2^{\text{RV}_+}(\bm{v}) = {\mathcal{Z}}_2^{\text{RV}_+}(\bm{v})\leq {{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+{\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}+ \frac{1}{\alpha}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}.$$
\[P2\_RV\_boundedness\] Let $\bm{v}\in{\mathbb{R}}^G$. If the problem in has a feasible solution, then it has a finite optimal value, that is ${\mathcal{Z}}_2^{\text{RV}_+}(\bm{v})\in{\mathbb{R}}$.
\[P2\_RV\_feasibility\] Let $\bm{v}\in{\mathbb{R}}^G$. The problem in has a feasible solution if and only if $\gamma \le {{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}}$.
The proofs of Propositions \[P2\_RV\_upperbound\], \[P2\_RV\_boundedness\], \[P2\_RV\_feasibility\] can be found in Appendices \[P2\_RV\_upperbound\_proof\], \[P2\_RV\_boundedness\_proof\], \[P2\_RV\_feasibility\_proof\], respectively
\[remark\_gamma\_threshold\] For $\ell\in{\mathcal{G}}$ and $\bm{v}\in{\mathbb{R}}^G$, the threshold $\gamma$ appearing in $R^\text{EN}, R^{\text{RV}_+}$ can be taken as some percentage of ${{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}}$, the sum of the debts of all nodes in the network. Then this threshold ensures that the expected total amount of payments exceeds this fraction of the total debt in the system. Indeed, Corollaries \[P1\_EN\_feasibility\], \[P1\_RV\_feasibility\], \[P2\_EN\_feasibility\], \[P2\_RV\_feasibility\] show that the MILP problems for calculating ${\mathcal{Z}}_1^\text{EN}{\ensuremath{\left( \bm{e}^\ell \right)}}$, ${\mathcal{Z}}_1^{\text{RV}_+}{\ensuremath{\left( \bm{e}^\ell \right)}}$, ${\mathcal{Z}}_2^\text{EN}{\ensuremath{\left( \bm{v} \right)}}$, ${\mathcal{Z}}_2^{\text{RV}_+}{\ensuremath{\left( \bm{v} \right)}}$ are feasible if and only if $\gamma\le{{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}}$. Hence, this choice of $\gamma$ is justified.
The nonconvex Benson-type algorithm {#algorithm}
-----------------------------------
In this section, we present an algorithm that approximates the Eisenberg-Noe and Rogers-Veraart systemic risk measures. The risk measures are approximated with respect to a user-defined approximation error $\epsilon>0$ and an upper bound point $\bm{z}^\text{UB}\in{\mathbb{R}}^G$. The algorithm is based on the Benson-type algorithm for nonconvex multi-objective programming problems described in @non-conv.benson. The following definitions are borrowed from [@non-conv.benson].
Let ${\mathcal{L}}\subseteq{\mathbb{R}}^G$. A point $\bm{v}\in {\mathcal{L}}$ is called a *vertex* of ${\mathcal{L}}$ if there exists a neighborhood $N$ of $\bm{v}$ for which $\bm{v}$ cannot be expressed as a strict convex combination of two distinct points in ${\mathcal{L}}\cap N$. The set of all vertices of ${\mathcal{L}}$ is denoted by $\text{vert}{\mathcal{L}}$. The notation $\text{int}{\mathcal{L}}$ denotes the interior of ${\mathcal{L}}$. Given a point $\bm{z}\in{\mathbb{R}}^G$ and ${\mathcal{L}}\subseteq{\mathbb{R}}^G$, we define ${\mathcal{L}}|_{\bm{z}} \coloneqq {\ensuremath{ \left\{ \bm{v}\in {\mathcal{L}}\mid \bm{v}\leq\bm{z} \right\} }}$.
Let $R,{\mathcal{L}},{\mathcal{U}}\subseteq{\mathbb{R}}^G$, $\bm{z}\in{\mathbb{R}}^G$ and $\epsilon>0$ be given. The set ${\mathcal{L}}$ is called an *outer approximation* for $R$ with respect to $\epsilon$ and $\bm{z}$, if $R \subseteq {\mathcal{L}}$ and ${\mathcal{L}}|_{\bm{z}} \subseteq R + B{\ensuremath{\left( {\mathbf{0}},\epsilon \right)}}$, where $B{\ensuremath{\left( {\mathbf{0}},\epsilon \right)}}$ is the closed ball in ${\mathbb{R}}^G$ centered at ${\mathbf{0}}$ with radius $\epsilon$. The set ${\mathcal{U}}$ is called an *inner approximation* for $R$ with respect to $\epsilon$ and $\bm{z}$ if $R$ is an outer approximation for ${\mathcal{U}}$ with respect to $\epsilon$ and $\bm{z}$.
The algorithm that calculates inner and outer approximations of a systemic risk measure works as follows. It is provided in detail only for the Eisenberg-Noe systemic risk measures, since it works similarly for the Rogers-Veraart systemic risk measures. Let ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{X}} \right)}}}$ be a signed Eisenberg-Noe network. Let $G$ be the number of groups in the network and ${\mathcal{G}}={\ensuremath{ \left\{ 1,\ldots,G \right\} }}$. Consider the corresponding Eisenberg-Noe systemic risk measure ${R^\text{EN}}{\ensuremath{\left( \bm{X} \right)}}$. Let $\bm{z}^\text{ideal}\in{\mathbb{R}}^G$ be the ideal point of the vector optimization problem in with $\Lambda^\text{OPT} = \Lambda^\text{EN}$, see Remark \[ideal\] for its definition. One can calculate $\bm{z}^\text{ideal} = {{{\ensuremath{\left( {\mathcal{Z}}_1^\text{EN}{\ensuremath{\left( \bm{e}^1 \right)}},\ldots,{\mathcal{Z}}_1^\text{EN}{\ensuremath{\left( \bm{e}^G \right)}} \right)}}}^\mathsf{T}}$ by Corollary \[corollary\_P1\_EN\]. In addition, for $\bm{v}\in{\mathbb{R}}^G$, the minimum step-length ${\mathcal{P}^\text{EN}_2}{\ensuremath{\left( \bm{v} \right)}}$ can be obtained by solving the MILP problem ${\mathcal{Z}}_2^\text{EN}{\ensuremath{\left( \bm{v} \right)}}$ in , by Corollary \[corollary\_P2\_EN\].
The algorithm starts with the initial inner approximation ${\mathcal{U}}^0 \coloneqq \bm{z}^\text{UB} + {\mathbb{R}}^G_+$ and the initial outer approximation ${\mathcal{L}}^0 \coloneqq \bm{z}^\text{ideal} + {\mathbb{R}}^G_+$, which satisfy ${\mathcal{U}}^0 \subseteq {R^\text{EN}}{\ensuremath{\left( \bm{X} \right)}} \subseteq {\mathcal{L}}^0$. Let $\varepsilon = \epsilon{\bm{\mathbbm{1}}}$ and initially set $t\leftarrow0$. At the $t^\text{th}$ iteration, for a vertex $\bm{v}^t \in {{\ensuremath{\left( \text{vert}{\mathcal{L}}^{t}|_{\bm{z}^\text{UB}} \right)}}}$ such that $\bm{v}^t + \varepsilon \notin \text{int }{\mathcal{U}}^t$, the algorithm solves ${\mathcal{Z}}_2^\text{EN}{\ensuremath{\left( \bm{v}^t \right)}}$ to obtain the minimum step-length $\mu^t$ from the point $\bm{v}^t$ to the boundary of ${R^\text{EN}}{\ensuremath{\left( \bm{X} \right)}}$ in the direction ${\bm{\mathbbm{1}}}\in{\mathbb{R}}^G$. In other words, $\bm{y}^t = \bm{v}^t + \mu^t{\bm{\mathbbm{1}}}$ is a boundary point of the set ${R^\text{EN}}{\ensuremath{\left( \bm{X} \right)}}$. Then the algorithm excludes the cone $\bm{y}^t - {\mathbb{R}}^G_+$ from ${\mathcal{L}}^t$ to obtain ${\mathcal{L}}^{t+1}$ by ${\mathcal{L}}^{t+1}\coloneqq{\mathcal{L}}^t\backslash{\ensuremath{\left( \bm{y}^t - {\mathbb{R}}^G_+ \right)}}$, and adds the cone $\bm{y}^t + {\mathbb{R}}^G_+$ to ${\mathcal{U}}^t$ to obtain ${\mathcal{U}}^{t+1}$ as follows: ${\mathcal{U}}^{t+1}\coloneqq{\mathcal{U}}^t\cup{\ensuremath{\left( \bm{y}^t + {\mathbb{R}}^G_+ \right)}}$. Therefore, at each step of the algorithm, we have ${\mathcal{U}}^t \subseteq {\mathcal{U}}^{t+1} \subseteq {R^\text{EN}}{\ensuremath{\left( \bm{X} \right)}} \subseteq {\mathcal{L}}^{t+1} \subseteq {\mathcal{L}}^t$. At the end of the $t^\text{th}$ iteration, ${\text{vert}{\mathcal{L}}^{t+1}}$ is computed. The computation of ${\text{vert}{\mathcal{L}}^{t+1}}$ is described in detail in @gourion-luc. The above process repeats for $t\leftarrow t+1$. The algorithm stops at $T^\text{th}$ iteration, when ${{\ensuremath{\left( \text{vert}{\mathcal{L}}^{T}|_{\bm{z}^\text{UB}} \right)}}} + \varepsilon \subseteq \text{int }{\mathcal{U}}^T$. The sets ${\mathcal{U}}^T$ and ${\mathcal{L}}^T$ are the inner and outer approximations for ${R^\text{EN}}(\bm{X})$ with respect to $\epsilon>0$ and $\bm{z}^\text{UB}\in{\mathbb{R}}^G$. Note that $\bm{z}^\text{UB}$ has to be chosen such that $\bm{z}^\text{UB}\in{R^\text{EN}}{\ensuremath{\left( \bm{X} \right)}}$ to get nonempty approximations. The pseudocode of the algorithm for the Eisenberg-Noe systemic risk measures is provided in Algorithm \[alg1\].
Let $\bm{z}^\text{UB}\in{R^\text{EN}}{\ensuremath{\left( \bm{X} \right)}}$, ${\mathcal{L}}^0 = \bm{z}^\text{ideal} + {\mathbb{R}}^G_+$, ${\mathcal{U}}^0 = \bm{z}^\text{UB} + {\mathbb{R}}^G_+$ and $\epsilon>0$.
\[i1\]
Put $\varepsilon = \epsilon{\bm{\mathbbm{1}}}$ and set $t \leftarrow 0$, $S\leftarrow\emptyset$.
\[i3\]
If ${{\ensuremath{\left( \text{vert}{\mathcal{L}}^{t}|_{\bm{z}^\text{UB}} \right)}}} \subseteq S$, then set $T=t$ and stop. Otherwise, choose $\bm{v}^t \in {{\ensuremath{\left( \text{vert}{\mathcal{L}}^{t}|_{\bm{z}^\text{UB}} \right)}}\backslash S} $.
\[k1\]
If $\bm{v}^t + \varepsilon \in \text{int }{\mathcal{U}}^t$, then set $S\leftarrow S\cup{\ensuremath{ \left\{ \bm{v}^t \right\} }}$ and go to \[k1\].
\[k2\]
Suppose that $\mu^t = {\mathcal{P}^\text{EN}_2}{\ensuremath{\left( \bm{v}^t \right)}}$. Define $\bm{y}^t = \bm{v}^t + \mu^t{\bm{\mathbbm{1}}}$.
\[k3\]
Define ${\mathcal{L}}^{t+1} \coloneqq {\mathcal{L}}^t \backslash {\ensuremath{\left( \bm{y}^t - {\mathbb{R}}^G_+ \right)}}$ and ${\mathcal{U}}^{t+1} \coloneqq {\mathcal{U}}^t \cup {\ensuremath{\left( \bm{y}^t+{\mathbb{R}}^G_+ \right)}}$.
\[k4\]
Determine ${\text{vert}{\mathcal{L}}^{t+1}}$ and set $t \leftarrow t + 1$. Go to \[k1\].
\[k5\]
${\mathcal{L}}^T$ is an outer approximation and ${\mathcal{U}}^T$ is an inner approximation for ${R^\text{EN}}{\ensuremath{\left( \bm{X} \right)}}$.
\[r1\]
Computational results and analysis {#computational_results}
==================================
In this section, we present some computational results to illustrate the approximation of the Eisenberg-Noe and Rogers-Veraart systemic risk measures by the Benson-type algorithm described in Section \[algorithm\]. We implement the algorithm on Java Photon (Release 4.8.0) calling Gurobi Interactive Shell (Version 7.5.2) and run it on an Intel(R) Core(TM) i7-4790 processor with 3.60 GHz and 4 GB RAM. We approximate the Eisenberg-Noe and Rogers-Veraart systemic risk measures within a two-group framework and perform a detailed sensitivity analysis. In the last part, we present several computational results for three-group networks.
Recall that $n$ is the number of institutions in a financial system, refered to as banks here, $n_\ell$ is the number of nodes in a group $\ell\in{\mathcal{G}}$, $K$ is the number of scenarios, $\epsilon$ is a user-defined approximation error and $\bm{z}^\text{UB}$ is a user-defined upper-bound vector that limits the approximated region of a systemic risk measure. Throughout the computation of systemic risk measures, except for the Rogers-Veraart case in a three-group framework (Section \[RV3\]), $\bm{z}^\text{UB}$ is taken as $$\bm{z}^\text{UB} = \bm{{z}}^\text{ideal} + 2{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty},$$ where $\bm{{z}}^\text{ideal}$ is the ideal point of the corresponding systemic risk measure (Remark \[ideal\]) for the case $\gamma = {{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}}$, that is, when it is required that the expected total value of payments is at least as much as the total amount of liabilities in the network.
For convenience, let us write $\gamma = \gamma^\text{p}{\ensuremath{\left( {{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}} \right)}}$, where $\gamma^\text{p}\in{\ensuremath{ \left[ 0,1 \right] }}$.
Data generation
---------------
We consider a network with $n$ banks forming $G=2$ or $G=3$ groups. Recall that ${\mathcal{G}}= {\ensuremath{ \left\{ 1,\ldots,G \right\} }}$, ${\mathcal{N}}= \bigcup_{\ell\in{\mathcal{G}}}{{\mathcal{N}}_\ell} = {\ensuremath{ \left\{ 1,\ldots,n \right\} }}$, and $n_\ell = |{\mathcal{N}}_\ell|$. When $G=2$, the groups $\ell = 1$ and $\ell = 2$ correspond to big and small banks, respectively. When $G=3$, the groups $\ell = 1$, $\ell = 2$ and $\ell = 3$ correspond to big, medium and small banks, respectively.
In order to construct a signed Eisenberg-Noe network ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{X}} \right)}}}$ and a Rogers-Veraart network ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{X}}, \alpha, \beta \right)}}}$, the corresponding interbank liabilities matrix $\bm{l}\coloneqq{\ensuremath{\left( l_{ij} \right)}}_{i,j\in{\mathcal{N}}}\in{\mathbb{R}}^{n\times n}_+$ and the random operating cash flow vector $\bm{X}$ are generated in the following fashion. For $\bm{l}$, we use an Erd[ö]{}s-R[é]{}nyi random graph model [@ersod-renyi], [@gilbert]. First, we fix a *connectivity probabilities matrix* $\bm{q}^\text{con}\coloneqq(q^\text{con}_{{\ell,\hat{\ell}}})_{{\ell,\hat{\ell}}\in{\mathcal{G}}}\in{\mathbb{R}}^{G\times G}$ and an *intergroup liabilities matrix* $\bm{l}^\text{gr}\coloneqq(l^\text{gr}_{{\ell,\hat{\ell}}})_{{\ell,\hat{\ell}}\in{\mathcal{G}}}\in{\mathbb{R}}^{G\times G}$. For any two banks $i,j\in{\mathcal{N}}$ with $i\in{\mathcal{N}}_\ell$, $j\in{\mathcal{N}}_{\hat{\ell}}$ and $\ell,\hat{\ell}\in{\mathcal{G}}$, $q^\text{con}_{{\ell,\hat{\ell}}}$ is interpreted as a probability that bank $i$ owes $l^\text{gr}_{{\ell,\hat{\ell}}}$ amount to bank $j$. Then, the liability $l_{ij}$ is generated by the Bernoulli trial $$l_{ij} = \begin{cases}
l^\text{gr}_{{\ell,\hat{\ell}}}, &\quad\text{if } U_{ij} < q^\text{con}_{{\ell,\hat{\ell}}}, \\
0, &\quad\text{otherwise},
\end{cases}$$ where $U_{ij}$ is the realization of a continuous random variable with a standard uniform distribution on a separate probability space. Then, the relative liabilities matrix $\bm{\pi}$ and the total obligation vector $\bm{{\bar{p}}}$ are calculated accordingly.
Recall that the operating cash flow vector $\bm{X} = {\ensuremath{\left( X_1,\ldots, X_n \right)}}\in L{\ensuremath{\left( {\mathbb{R}}^n \right)}}$ is a multivariate random vector and $\Omega$ is a finite set of $K$ scenarios. It is assumed that all scenarios are equally likely to happen, the operating cash flows have a common standard deviation $\sigma$, and there is a common correlation $\varrho$ between any two operating cash flows. Then, each entry $X_i$, $i\in{\mathcal{N}}$, is generated as a random sample of size $K$ as described below.
For the Eisenberg-Noe network, the mean values of operating cash flows in each group, $\bm{\nu} \coloneqq {\ensuremath{\left( \nu_\ell \right)}}_{\ell\in{\mathcal{G}}}$, are fixed and the random vector $\bm{X}$ is generated from $K$ instances of a Gaussian random vector. For the Rogers-Veraart network, first, shape parameters $\bm{\kappa} \coloneqq {\ensuremath{\left( \kappa_\ell \right)}}_{\ell\in{\mathcal{G}}}$ and scale parameters $\bm{\theta} \coloneqq {\ensuremath{\left( \theta_\ell \right)}}_{\ell\in{\mathcal{G}}}$ are fixed in accordance with the choices of $\sigma,\varrho$ and then, $\bm{X}$ is generated from $K$ instances of a random vector whose cumulative distribution function is stated in terms of a Gaussian copula with gamma marginal distributions with the chosen parameters. In particular, $\nu_\ell = \kappa_\ell \theta_\ell$ and $\sigma = \sqrt{\kappa_\ell} \theta_\ell$ for each $\ell\in{\mathcal{G}}$.
A two-group signed Eisenberg-Noe network with $50$ nodes {#two-group_n=50}
--------------------------------------------------------
We consider a two-group Eisenberg-Noe network with $n = 50$ banks that consists of $n_1 = 15$ big banks, $n_2 = 35$ small banks. We take $K = 100$, $\sigma = 100$, $\varrho = 0.05$, $$\bm{q}^\text{con} =
\begin{bmatrix}
0.9 & 0.3 \\
0.7 & 0.5
\end{bmatrix},\quad
\bm{l}^\text{gr} =
\begin{bmatrix}
10 & 5 \\
8 & 5
\end{bmatrix},\quad
\bm{\nu} =
\begin{bmatrix}
-50 & -100
\end{bmatrix}.$$ In the corresponding Eisenberg-Noe systemic risk measure, we take $\gamma^p = 0.7$.
[||m[0.5cm]{}|m[1.2cm]{}|m[1.2cm]{}|m[1.3cm]{}|m[2cm]{}|m[2.4cm]{}|m[2.4cm]{}||]{} $\epsilon$ & Inner approx. vertices & Outer approx. vertices & ${\mathcal{P}}_2$ problems & Avg. time per ${\mathcal{P}}_2$ prob. (seconds) & Total algorithm time (seconds) & Total algorithm time (hours)\
20 & 18 & 19 & 18 & 663.546 & 11944 & 3.318\
10 & 35 & 36 & 35 & 541.419 & 18950 & 5.264\
5 & 73 & 74 & 73 & 512.998 & 37449 & 10.403\
1 & 394 & 395 & 394 & 492.597 & 194083 & 53.912\
The Benson-type algorithm is run with four different approximation errors $\epsilon$ to demonstrate different inner approximation levels. Table \[table\_general\_2D\] presents the computational performance of the algorithm for $\epsilon\in {\ensuremath{ \left\{ 1, 5, 10, 20 \right\} }}$. Figure \[figure\_general\_2D\_inner\_zoom\] consists of the zoomed inner approximations.
[0.4]{} ![Zoomed inner approximations of the Eisenberg-Noe systemic risk measure for $\epsilon\in {\ensuremath{ \left\{ 1, 5, 10, 20 \right\} }}$.[]{data-label="figure_general_2D_inner_zoom"}](inner_epsilon_20_zoom.eps "fig:"){width="\textwidth"}
[0.4]{} ![Zoomed inner approximations of the Eisenberg-Noe systemic risk measure for $\epsilon\in {\ensuremath{ \left\{ 1, 5, 10, 20 \right\} }}$.[]{data-label="figure_general_2D_inner_zoom"}](inner_epsilon_10_zoom_nolegend.eps "fig:"){width="\textwidth"}
[0.4]{} ![Zoomed inner approximations of the Eisenberg-Noe systemic risk measure for $\epsilon\in {\ensuremath{ \left\{ 1, 5, 10, 20 \right\} }}$.[]{data-label="figure_general_2D_inner_zoom"}](inner_epsilon_5_zoom_nolegend.eps "fig:"){width="\textwidth"}
[0.4]{} ![Zoomed inner approximations of the Eisenberg-Noe systemic risk measure for $\epsilon\in {\ensuremath{ \left\{ 1, 5, 10, 20 \right\} }}$.[]{data-label="figure_general_2D_inner_zoom"}](inner_epsilon_1_zoom_nolegend.eps "fig:"){width="\textwidth"}
One can easily observe from Figure \[figure\_general\_2D\_inner\_zoom\] that as $\epsilon$ decreases the algorithm gives a more precise inner approximation of the systemic risk measure. In addition, as the number of ${\mathcal{P}}_2$ problems increases, the average computation time per ${\mathcal{P}}_2$ problem decreases. This may be attributed to the warm start feature of the Gurobi solver. When a sequence of mixed-integer programming problems are solved, the solver constructs an initial solution out of the previously obtained optimal solution. This feature is explained in detail in @gurobi [Chapter 10.2, pp. 594-595].
In the rest of this section, we perform sensitivity analyses on this network with respect to the connectivity probabilities between big and small banks and on the number of scenarios.
### Connectivity probabilities
Connectivity probabilities play a major role in determining the topology of the network because they define the existence of liabilities between the banks. We would like to identify the sensitivity of the systemic risk measure with respect to the changes in the connectivity probability $q^\text{con}_{1,2}$ corresponding to the liabilities of big banks to small banks, and the probability $q^\text{con}_{2,1}$ corresponding to the liabilities of small banks to big banks.
For the sensitivity analysis with respect to $q^\text{con}_{1,2}$, originally taken as $q^\text{con}_{1,2} = 0.3$, we present in Table \[table\_pbigsmall\_2D\] the computational performance of the algorithm for $q^\text{con}_{1,2} \in {\ensuremath{ \left\{ 0.1, 0.3, 0.5, 0.7, 0.9 \right\} }}$. Figure \[figure\_pbigsmall\_2D\_inner\] consists of the corresponding inner approximations.
[||m[0.5cm]{}|m[1.2cm]{}|m[1.2cm]{}|m[1.3cm]{}|m[2cm]{}|m[2.4cm]{}|m[2.4cm]{}||]{} $q^\text{con}_{1,2}$ & Inner approx. vertices & Outer approx. vertices & ${\mathcal{P}}_2$ problems & Avg. time per ${\mathcal{P}}_2$ prob. (seconds) & Total algorithm time (seconds) & Total algorithm time (hours)\
0.1 & 279 & 280 & 358 & 294.07 & 105 277 & 29.244\
0.3 & 394 & 395 & 394 & 492.597 & 194 083 & 53.912\
0.5 & 360 & 361 & 360 & 556.795 & 200 447 & 55.680\
0.7 & 364 & 365 & 364 & 633.644 & 230 647 & 64.069\
0.9 & 377 & 378 & 377 & 772.76 & 291 331 & 80.925\
Observe from Table \[table\_pbigsmall\_2D\] that the average time per ${\mathcal{P}}_2$ problem increases with $q^\text{con}_{1,2}$. This is the case because as $q^\text{con}_{1,2}$ increases, big and small banks in the network become more connected in terms of liabilities. Hence, the corresponding MILP formulations of ${\mathcal{P}}_2$ problems need more time to be solved. This seems to be the only factor behind the increase because most of the algorithm runtime is devoted to solving ${\mathcal{P}}_2$ problems and the number of ${\mathcal{P}}_2$ problems in each case does not change much.
![Inner approximations of the Eisenberg-Noe systemic risk measure for $q^\text{con}_{1,2} \in {\ensuremath{ \left\{ 0.1, 0.3, 0.5, 0.7, 0.9 \right\} }}$.[]{data-label="figure_pbigsmall_2D_inner"}](pbigsmall_all_zoom.eps){width="70.00000%"}
It can be observed that, as $q^\text{con}_{1,2}$ increases, the corresponding inner approximations of systemic risk measures in Figure \[figure\_pbigsmall\_2D\_inner\] shift from the top left corner towards the bottom right corner. It can be interpreted as follows: as $q^\text{con}_{1,2}$ increases, the first group, the group of big banks, loses capital allocation options, while the second group, the group of small banks, gains a wider range of capital allocation options. It can also be observed from Figure \[figure\_pbigsmall\_2D\_inner\] that generating a network with $q^\text{con}_{1,2} = 0.1$ results in a nonconvex Eisenberg-Noe systemic risk measure. However, for the values $q^\text{con}_{1,2}\in{\ensuremath{ \left\{ 0.3, 0.5, 0.7, 0.9 \right\} }}$, the corresponding Eisenberg-Noe systemic risk measures seem to be convex sets. For these cases, there might be some breakpoint between $0.1$ and $0.3$ that switches these Eisenberg-Noe systemic risk measures from a nonconvex shape to a convex one, meaning that, whenever the probability $q^\text{con}_{1,2}$ is less than this breakpoint, big banks are less likely to be liable to small banks and have even more capital allocation options than they have in the other cases.
Next, for the sensitivity analysis with respect to $q^\text{con}_{2,1}$, we present in Table \[table\_psmallbig\_2D\] the computational performance of the algorithm for $q^\text{con}_{2,1} \in {\ensuremath{ \left\{ 0.1, 0.3, 0.5, 0.7, 0.9 \right\} }}$. Figure \[figure\_psmallbig\_2D\_inner\] consists of the corresponding inner approximations.
[||m[0.5cm]{}|m[1.2cm]{}|m[1.2cm]{}|m[1.3cm]{}|m[2cm]{}|m[2.4cm]{}|m[2.4cm]{}||]{} $q^\text{con}_{2,1}$ & Inner approx. vertices & Outer approx. vertices & ${\mathcal{P}}_2$ problems & Avg. time per ${\mathcal{P}}_2$ prob. (seconds) & Total algorithm time (seconds) & Total algorithm time (hours)\
0.1 & 257 & 258 & 257 & 233.243 & 59943 & 16.651\
0.3 & 294 & 295 & 294 & 319.511 & 93936 & 26.093\
0.5 & 328 & 329 & 328 & 377.398 & 123787 & 34.385\
0.7 & 394 & 395 & 394 & 492.597 & 194083 & 53.912\
0.9 & 435 & 436 & 512 & 487.547 & 249624 & 69.340\
As in the previous sensitivity analysis, observe from Table \[table\_psmallbig\_2D\] that the average time per ${\mathcal{P}}_2$ problem increases with $q^\text{con}_{2,1}$. Hence, it is another justification of the presumption that this happens because with higher connectivity probabilities the network becomes more connected in terms of liabilities and the corresponding MILP formulations of ${\mathcal{P}}_2$ problems need more time to be solved.
![Inner approximations of the Eisenberg-Noe systemic risk measure for $q^\text{con}_{2,1} \in {\ensuremath{ \left\{ 0.1, 0.3, 0.5, 0.7, 0.9 \right\} }}$.[]{data-label="figure_psmallbig_2D_inner"}](psmallbig_all_zoom.eps){width="70.00000%"}
Note that as $q^\text{con}_{2,1}$ increases, the inner approximations of the corresponding Eisenberg-Noe systemic risk measures in Figure \[figure\_psmallbig\_2D\_inner\] shift from the bottom right corner towards the top left corner. Conversely to the previous sensitivity analysis, it can be interpreted as follows: as $q^\text{con}_{2,1}$ increases, the first group gains a wider range of capital allocation options, while the second group loses capital allocation options. It can also be observed from Figure \[figure\_psmallbig\_2D\_inner\] that generating a network with $q^\text{con}_{2,1} = 0.9$ results in a nonconvex Eisenberg-Noe systemic risk measure. However, for the values $q^\text{con}_{2,1}\in{\ensuremath{ \left\{ 0.1, 0.3, 0.5, 0.7 \right\} }}$, the corresponding Eisenberg-Noe systemic risk measures seem to be convex sets. As in the previous sensitivity analysis, it can be presumed that for these cases there is some breakpoint between $0.7$ and $0.9$ that switches these Eisenberg-Noe systemic risk measures from a convex shape to a nonconvex one, meaning that, whenever the probability $q^\text{con}_{2,1}$ is higher than this breakpoint, small banks are more likely to be liable to big banks and the latter have even more capital allocation options than they have in the other cases.
### Number of scenarios {#scsen1}
[||m[0.5cm]{}|m[1.2cm]{}|m[1.2cm]{}|m[1.3cm]{}|m[2cm]{}|m[2.4cm]{}|m[2.4cm]{}||]{} $K$ & Inner approx. vertices & Outer approx. vertices & ${\mathcal{P}}_2$ problems & Avg. time per ${\mathcal{P}}_2$ prob. (seconds) & Total algorithm time (seconds) & Total algorithm time (hours)\
10 & 376 & 377 & 376 & 3.088 & 1 161 & 0.323\
20 & 380 & 381 & 380 & 11.977 & 4 551 & 1.264\
30 & 389 & 390 & 389 & 28.134 & 10 944 & 3.040\
40 & 381 & 382 & 381 & 56.685 & 21 597 & 5.999\
50 & 373 & 374 & 373 & 96.488 & 35 990 & 9.997\
60 & 381 & 382 & 381 & 151.635 & 57 773 & 16.048\
70 & 385 & 386 & 385 & 206.924 & 79 666 & 22.129\
80 & 390 & 391 & 390 & 293.155 & 114 330 & 31.758\
90 & 381 & 382 & 381 & 378.346 & 144 150 & 40.042\
100 & 394 & 395 & 394 & 492.597 & 194 083 & 53.912\
![Inner approximations of the Eisenberg-Noe systemic risk measure for $K \in {\ensuremath{ \left\{ 10, 20, \ldots, 100 \right\} }}$.[]{data-label="figure_scenarios_2D"}](scenarios_zoom.eps){width="60.00000%"}
Next, we analyze how computation times and the corresponding systemic risk measures change with the number $K$ of scenarios. Since the network structure remains the same all the time, it is expected that there will be no major changes in Eisenberg-Noe systemic risk measures. However, since each scenario adds $n$ continuous and $n$ binary variables to the corresponding ${\mathcal{P}}_2$ problem and its MILP formulation ${\mathcal{Z}}_2^\text{EN}$, given in , one would expect major changes in computation times.
Table \[table\_scenarios\_2D\] shows the computational performance of the algorithm for $K\in{\ensuremath{ \left\{ 10, 20, \ldots, 100 \right\} }}$ and Figure \[figure\_scenarios\_2D\] provides the inner approximations of the corresponding Eisenberg-Noe systemic risk measures. Finally, the plots in Figure \[figure\_scenarios\_KvsAvg\_KvsTotal\] suggest that the average time per ${\mathcal{P}}_2$ problem and the total algorithm time increase faster than linearly with $K$. At the same time, it can be observed from Figure \[figure\_scenarios\_2D\] that the corresponding inner approximations of the Eisenberg-Noe systemic risk measures do not change much. Hence, the results obtained justify the expectations.
[0.49]{} ![Scenarios-average time per ${\mathcal{P}}_2$ problem and scenarios-total algorithm time plots for the signed Eisenberg-Noe network of 50 banks.[]{data-label="figure_scenarios_KvsAvg_KvsTotal"}](scenarios_vs_averP2time.eps "fig:"){width="80.00000%"}
[0.49]{} ![Scenarios-average time per ${\mathcal{P}}_2$ problem and scenarios-total algorithm time plots for the signed Eisenberg-Noe network of 50 banks.[]{data-label="figure_scenarios_KvsAvg_KvsTotal"}](scenarios_vs_totalAlgTime.eps "fig:"){width="80.00000%"}
A two-group signed Eisenberg-Noe network with $70$ nodes
--------------------------------------------------------
In this section, we consider an Eisenberg-Noe network ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{X}} \right)}}}$ with $n = 70$, $n_1 = 10$, $n_2 = 60$, $K = 50$, $\sigma = 100$, $\varrho = 0.05$ and $$\bm{q}^\text{con} =
\begin{bmatrix}
0.7 & 0.1 \\
0.5 & 0.5
\end{bmatrix},\quad
\bm{l}^\text{gr} =
\begin{bmatrix}
10 & 5 \\
8 & 5
\end{bmatrix},\quad
\bm{\nu} =
\begin{bmatrix}
-50 & -100
\end{bmatrix}.$$ In the corresponding Eisenberg-Noe systemic risk measure, we take $\gamma^\text{p} = 0.9$. The approximation error in the algorithm is taken as $\epsilon = 1$.
On this network, we perform sensitivity analyses with respect to the threshold $\gamma^p$, the distribution of nodes among groups, and the number of scenarios.
[||m[0.6cm]{}|m[1.2cm]{}|m[1.2cm]{}|m[1.3cm]{}|m[2cm]{}|m[2.4cm]{}|m[2.4cm]{}||]{} $\gamma^\text{p}$ & Inner approx. vertices & Outer approx. vertices & ${\mathcal{P}}_2$ problems & Avg. time per ${\mathcal{P}}_2$ prob. (seconds) & Total algorithm time (seconds) & Total algorithm time (hours)\
0.01 & 376 & 377 & 376 & 3.088 & 1 161 & 0.323\
0.1 & 210 & 210 & 437 & 305.389 & 133 455 & 37.071\
0.2 & 145 & 146 & 727 & 492.418 & 357 988 & 99.441\
0.3 & 90 & 91 & 893 & 560.268 & 500 320 & 138.978\
0.4 & 87 & 88 & 1037 & 494.65 & 512 952 & 142.487\
0.5 & 91 & 95 & 1099 & 448.063 & 492 421 & 136.784\
0.6 & 94 & 95 & 1065 & 240.982 & 256 646 & 71.291\
0.7 & 96 & 97 & 927 & 97.501 & 90 383 & 25.106\
0.8 & 141 & 142 & 719 & 45.546 & 32 748 & 9.097\
0.9 & 234 & 235 & 461 & 15.285 & 7 047 & 1.957\
0.95 & 217 & 218 & 217 & 11.622 & 2 522 & 0.701\
0.99 & 136 & 137 & 136 & 2.504 & 341 & 0.095\
1.00 & 1 & 1 & 1 & 0.203 & 0.204 & 0\
### Threshold level
We investigate how the Eisenberg-Noe systemic risk measures and their computation times change when the requirement that some fraction of the total amount of liabilities in the network should be met on average gets more strict. Table \[table\_gamma\_2D\_EN\] illustrates the computational performance of the algorithm for $\gamma^\text{p}\in \{0.01, 0.1, 0.2, \ldots, 0.9, 0.95, 0.99, 1\}$ and Figure \[figure\_gamma\_2D\_EN\_inner\] represents the corresponding inner approximations of the Eisenberg-Noe systemic risk measures.
It can be noted from Table \[table\_gamma\_2D\_EN\] that the average times per ${\mathcal{P}}_2$ problem are high for the values of $\gamma^\text{p}$ around $0.3$, and the number of ${\mathcal{P}}_2$ problems are high for the values of $\gamma^\text{p}$ around $0.5$. These two factors result in high total algorithm times for the values of $\gamma^\text{p}$ around $0.4$. In addition, it can be observed that the difference between the number of inner and outer approximation vertices and the number of ${\mathcal{P}}_2$ problems increase drastically for the values of $\gamma^\text{p}$ around $0.5$. This happens because the boundaries of the corresponding Eisenberg-Noe systemic risk measures in Figure \[figure\_gamma\_2D\_EN\_inner\] contain “flat" regions, which makes the algorithm solve more ${\mathcal{P}}_2$ problems without actually improving the approximation. Observe from Figure \[figure\_gamma\_2D\_EN\_inner\] that as $\gamma^\text{p}$ increases, each subsequent Eisenberg-Noe systemic risk measure is contained in the previous one. This result is fully consistent with the corresponding Eisenberg-Noe systemic risk measure since capital allocations that are feasible at a particular $\gamma^\text{p}$ level are also feasible for any level lower than $\gamma^\text{p}$.
![Inner approximations of the Eisenberg-Noe systemic risk measure for $\gamma^\text{p}\in{\ensuremath{ \left\{ 0.01, 0.1, \ldots, 0.9, 0.95, 0.99, 1 \right\} }}$.[]{data-label="figure_gamma_2D_EN_inner"}](gamma_all_zoom2.eps){width="70.00000%"}
### Distribution of nodes among groups
In this part, we perform a sensitivity analysis with respect to the distribution of nodes among the groups for a fixed total number of nodes $n=70$. We take the number of big banks $n_1$ in the set ${\ensuremath{ \left\{ 5, 10, 20, \ldots, 60, 65 \right\} }}$. Then, the number of small banks is $n_2 = n - n_1$. The generated random operating cash flows remain the same all the time, while the network structure changes at each run. Hence, the corresponding Eisenberg-Noe systemic risk measures are expected to vary significantly.
[||m[0.5cm]{}|m[1.2cm]{}|m[1.2cm]{}|m[1.3cm]{}|m[2cm]{}|m[2.4cm]{}|m[2.4cm]{}||]{} $n_1$ & Inner approx. vertices & Outer approx. vertices & ${\mathcal{P}}_2$ problems & Avg. time per ${\mathcal{P}}_2$ prob. (seconds) & Total algorithm time (seconds) & Total algorithm time (hours)\
5 & 93 & 94 & 1096& 16.88 & 18 501 & 5.139\
10 & 234 & 235 & 461 & 15.285 & 7 047 & 1.957\
20 & 209 & 210 & 209 & 38.512 & 8 049 & 2.236\
30 & 201 & 202 & 201 & 45.225 & 9 090 & 2.525\
40 & 213 & 214 & 213 & 55.444 & 11 809 & 3.280\
50 & 250 & 251 & 250 & 61.329 & 15 332 & 4.259\
60 & 403 & 404 & 639 & 79.577 & 50 850 & 14.125\
65 & 205 & 206 & 1092& 131.431 & 143 523 & 39.867\
Table \[table\_nodes\_2D\] shows the computational performance of the algorithm for $n_1\in{\ensuremath{ \left\{ 5, 10, 20, \ldots, 60, 65 \right\} }}$ and Figure \[figure\_nodes\_2D\_inner\] represents the corresponding inner approximations of the Eisenberg-Noe systemic risk measures.
![Inner approximations of the Eisenberg-Noe systemic risk measure for $n_1\in{\ensuremath{ \left\{ 5, 10, 20, \ldots, 60, 65 \right\} }}$.[]{data-label="figure_nodes_2D_inner"}](nodes_all_zoom2.eps){width="70.00000%"}
Note that the average time per ${\mathcal{P}}_2$ problem in Table \[table\_nodes\_2D\] tends to increase as the number of big banks increases. This happens because the highest connectivity probability, $q^\text{con}_{1,1} = 0.7$, is the probability that one big bank is liable to another big bank. Hence, as the number of big banks increases, the nodes in the network become more connected with liabilities and it takes more time to solve a ${\mathcal{P}}_2$ problem because the MILP formulations of ${\mathcal{P}}_2$ problems get more complex in terms of constraints. In addition, it can be observed that the difference between the numbers of inner and outer approximation vertices and the number of ${\mathcal{P}}_2$ problems increases as the distribution of nodes changes toward the two extreme cases: $5$ big banks and $65$ big banks. As in the previous sensitivity analysis, this happens because the boundaries of the Eisenberg-Noe systemic risk measures around these extreme cases in Figure \[figure\_nodes\_2D\_inner\] contain “flat" regions, which makes the algorithm solve more ${\mathcal{P}}_2$ problems without actually improving the approximation.
We observe from Figure \[figure\_nodes\_2D\_inner\] that as the number of big banks increases and the number of small banks decreases, the small banks get a wider range of capital allocation options, as opposed to the big banks. This happens because the total number of banks is fixed and the group with less number of banks has a wider range of capital allocation options since it has more claims to the other group’s banks. When the number of banks in each group is evenly distributed, the group of big banks has a wider range of capital allocation options. The reason lies behind connectivity probabilities. Recall that for this set-up it is assumed that the connectivity probability from big banks to small banks is $q^\text{con}_{12} = 0.1$, while the connectivity probability from small banks to big banks is $q^\text{con}_{21} = 0.5$. It means that small banks are more likely to be liable to big banks and, since big banks have more claims compared to small banks, they have a wider range of capital allocation options.
A two-group Rogers-Veraart networks with $45$ nodes
---------------------------------------------------
In this section, we consider a Rogers-Veraart network ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{X}}, \alpha, \beta \right)}}}$ generated with the following parameters: $n = 45$, $n_1 = 15$, $n_2 = 30$, $K = 50$, $\varrho = 0.05$ and $$\bm{q}^\text{con} =
\begin{bmatrix}
0.5 & 0.1 \\
0.3 & 0.5
\end{bmatrix},\quad
\bm{l}^\text{gr} =
\begin{bmatrix}
200 & 100 \\
50 & 50
\end{bmatrix}.$$ In addition, the liquid fraction of the random operating cash flows available to a defaulting node is fixed as $\alpha = 0.7$, and the liquid fraction of the realized claims available to a defaulting node is fixed as $\beta = 0.9$. The shape and scale parameters of gamma distributions of the random operating cash flows $X_i$, $i\in{\mathcal{N}}_\ell$, $\ell\in{\mathcal{G}}$, are chosen as $$\kappa = \begin{bmatrix} 100 & 64 \end{bmatrix}, \quad \theta = \begin{bmatrix} 1 & 1.25 \end{bmatrix}.$$ Then the mean values of the random operating cash flows in the corresponding groups are $$\bm{\nu} = \begin{bmatrix} 100 & 80 \end{bmatrix}$$ and the common standard deviation is $\sigma = 10$. In the corresponding Rogers-Veraart systemic risk measure, we take $\gamma^p = 0.9$. The approximation error in the algorithm is taken as $\epsilon = 1$.
### Rogers-Veraart $\alpha$ parameter
[||m[0.5cm]{}|m[1.2cm]{}|m[1.2cm]{}|m[1.3cm]{}|m[2cm]{}|m[2.4cm]{}|m[2.4cm]{}||]{} $\alpha$ & Inner approx. vertices & Outer approx. vertices & ${\mathcal{P}}_2$ problems & Avg. time per ${\mathcal{P}}_2$ prob. (seconds) & Total algorithm time (seconds) & Total algorithm time (hours)\
0.1 & 273 & 274 & 333 & 12.165 & 4 051 & 1.125\
0.3 & 461 & 462 & 484 & 10.572 & 5 117 & 1.421\
0.5 & 592 & 593 & 602 & 5.231 & 3 149 & 0.875\
0.7 & 583 & 584 & 584 & 3.876 & 2 264 & 0.629\
0.9 & 589 & 590 & 589 & 3.395 & 2 000 & 0.555\
In this part, we perform a sensitivity analysis with respect to $\alpha$, the liquid fraction of the operating cash flow that can be used by a defaulting node to meet its obligations. The generated network ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{X}}, \alpha, \beta \right)}}}$ remains the same in all cases. Table \[table\_alpha\_2D\] illustrates the computational performance of the algorithm for $\alpha\in{\ensuremath{ \left\{ 0.1, 0.3, 0.5, 0.7, 0.9 \right\} }}$ and Figure \[figure\_alpha\_2D\_inner\] consists of the inner approximations of the corresponding Rogers-Veraart systemic risk measures.
![Inner approximations of the Rogers-Veraart systemic risk measures for $\alpha\in{\ensuremath{ \left\{ 0.1, 0.3, 0.5, 0.7, 0.9 \right\} }}$.[]{data-label="figure_alpha_2D_inner"}](alpha_all_zoom.eps){width="70.00000%"}
Note from Table \[table\_alpha\_2D\] that the average time per ${\mathcal{P}}_2$ problem decreases with $\alpha$. It can be presumed that this happens because of the following observation: as $\alpha$ parameter increases, the discontinuity in the fixed-point characterization of clearing vectors in the Rogers-Veraart model in decreases and it gets easier to solve the corresponding MILP formulation of a ${\mathcal{P}}_2$ problem because it contains the constraints of , the MILP characterization of clearing vectors in the Rogers-Veraart model.
Observe from Figure \[figure\_alpha\_2D\_inner\] that the Rogers-Veraart systemic risk measures expand significantly as $\alpha$ increases. It means that both big and small banks get less strict capital requirements as default costs decrease. One can also observe that in each case allocating zero capital requirement to the groups is not an available option. In addition, in each case big banks can be allocated a negative amount of capital requirement given that the capital requirements for small banks are high enough. On the other hand, small banks do not have this privilege.
[||m[0.5cm]{}|m[1.2cm]{}|m[1.2cm]{}|m[1.3cm]{}|m[2cm]{}|m[2.4cm]{}|m[2.4cm]{}||]{} $\beta$ & Inner approx. vertices & Outer approx. vertices & ${\mathcal{P}}_2$ problems & Avg. time per ${\mathcal{P}}_2$ prob. (seconds) & Total algorithm time (seconds) & Total algorithm time (hours)\
0.1 & 187 & 189 & 214 & 5.014 & 1 073 & 0.298\
0.3 & 223 & 225 & 270 & 5.561 & 1 502 & 0.417\
0.5 & 323 & 324 & 350 & 3.733 & 1 307 & 0.363\
0.7 & 394 & 395 & 401 & 3.710 & 1 488 & 0.413\
0.9 & 583 & 584 & 584 & 3.876 & 2 264 & 0.629\
### Rogers-Veraart $\beta$ parameter
In this part, we perform a sensitivity analysis with respect to $\beta$, the liquid fraction of the realized claims from the other nodes that can be used by a defaulting node to meet its obligations. The generated network ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{X}}, \alpha, \beta \right)}}}$ remains the same in all cases. Table \[table\_beta\_2D\] shows the computational performance of the algorithm for $\beta\in{\ensuremath{ \left\{ 0.1, 0.3, 0.5, 0.7, 0.9 \right\} }}$ and Figure \[figure\_beta\_2D\_inner\] provides the inner approximations of the corresponding Rogers-Veraart systemic risk measures.
Note from Table \[table\_beta\_2D\] that the total number of ${\mathcal{P}}_2$ problems increases with $\beta$. We can observe smaller average times per ${\mathcal{P}}_2$ problem for higher values of $\beta$. As in the case of the $\alpha$ parameter, it can be presumed that this happens because of the following observation: as $\beta$ parameter increases, the discontinuity in the fixed-point characterization of clearing vectors in the Rogers-Veraart model in decreases, which makes it easier to solve the MILP formulation of a ${\mathcal{P}}_2$ problem.
![Inner approximations of the Rogers-Veraart systemic risk measures for $\beta\in{\ensuremath{ \left\{ 0.1, 0.3, 0.5, 0.7, 0.9 \right\} }}$.[]{data-label="figure_beta_2D_inner"}](beta_all_zoom.eps){width="70.00000%"}
Observe from Figure \[figure\_beta\_2D\_inner\] that the Rogers-Veraart systemic risk measures expand significantly as $\beta$ increases. It means that both big and small banks get less strict capital requirements if defaulting banks are able to use larger fractions of realized claims. It can also be observed that in each case allocating zero capital requirement to the groups is not an available option. In addition, if $\beta = 0.9$ then big banks can be allocated a negative amount of capital requirement given that the capital requirements for small banks are high enough. On the other hand, small banks do not have this privilege.
[||m[0.5cm]{}|m[1.2cm]{}|m[1.2cm]{}|m[1.3cm]{}|m[2cm]{}|m[2.4cm]{}|m[2.4cm]{}||]{} $\gamma^\text{p}$ & Inner approx. vertices & Outer approx. vertices & ${\mathcal{P}}_2$ problems & Avg. time per ${\mathcal{P}}_2$ prob. (seconds) & Total algorithm time (seconds) & Total algorithm time (hours)\
0.1 & 1 & 1 & 1 & 0.384 & 0.384 & 0\
0.2 & 13 & 14 & 13 & 13.809 & 180 & 0.050\
0.3 & 51 & 52 & 51 & 30.273 & 1 544 & 0.429\
0.4 & 94 & 95 & 94 & 36.645 & 3 445 & 0.957\
0.5 & 165 & 166 & 165 & 98.625 & 16 273 & 4.520\
0.6 & 223 & 224 & 223 & 138.532 & 30 893 & 8.581\
0.7 & 389 & 390 & 389 & 204.288 & 79 468 & 22.075\
0.8 & 395 & 396 & 395 & 91.600 & 36 182 & 10.051\
0.9 & 583 & 584 & 584 & 3.876 & 2 264 & 0.629\
0.95 & 418 & 419 & 431 & 2.946 & 1 270 & 0.353\
0.99 & 66 & 67 & 74 & 1.639 & 121 & 0.034\
1.00 & 1 & 1 & 1 & 0.132 & 0.132 & 0\
### Threshold level
In this part, different $\gamma^\text{p}$ levels are compared. Table \[table\_gamma\_2D\_RV\] shows the computational performance of the algorithm for $\gamma^\text{p}\in{\ensuremath{ \left\{ 0.1, 0.2, \ldots, 0.9, 0.95, 0.99, 1 \right\} }}$ and Figure \[figure\_gamma\_2D\_RV\_inner\] consists of the inner approximations of the corresponding Rogers-Veraart systemic risk measures.
![Inner approximations of the Rogers-Veraart systemic risk measure for $\gamma^\text{p}\in{\ensuremath{ \left\{ 0.1, 0.2, \ldots, 0.9, 0.95, 0.99, 1 \right\} }}$.[]{data-label="figure_gamma_2D_RV_inner"}](gamma_RV_all_zoom1.eps){width="70.00000%"}
It can be noted from Table \[table\_gamma\_2D\_RV\] that the average time per ${\mathcal{P}}_2$ problem and the total algorithm time are high for $\gamma^\text{p}$ values around $0.7$. In addition, the number of ${\mathcal{P}}_2$ problems increases up to $\gamma^\text{p} = 0.9$ and then decreases. Similar to the structure in Figure \[figure\_gamma\_2D\_EN\_inner\], we observe in Figure \[figure\_gamma\_2D\_RV\_inner\] that the Rogers-Veraart systemic risk measures with smaller $\gamma^\text{p}$ values contain the ones that have higher $\gamma^\text{p}$ values, which is consistent with the definition of these risk measures.
### Distribution of nodes among groups
[||m[0.5cm]{}|m[1.2cm]{}|m[1.2cm]{}|m[1.3cm]{}|m[2cm]{}|m[2.4cm]{}|m[2.4cm]{}||]{} $n_1$ & Inner approx. vertices & Outer approx. vertices & ${\mathcal{P}}_2$ problems & Avg. time per ${\mathcal{P}}_2$ prob. (seconds) & Total algorithm time (seconds) & Total algorithm time (hours)\
5 & 6 & 7 & 6 & 1.006 & 6 & 0.002\
10 & 436 & 437 & 436 & 3.994 & 1 742 & 0.484\
15 & 583 & 584 & 584 & 3.876 & 2 264 & 0.629\
20 & 516 & 517 & 517 & 7.887 & 4 078 & 1.133\
25 & 557 & 558 & 557 & 6.118 & 3 408 & 0.947\
30 & 371 & 372 & 371 & 5.786 & 2 147 & 0.596\
35 & 187 & 188 & 187 & 6.100 & 1 141 & 0.317\
40 & 106 & 107 & 108 & 5.196 & 561 & 0.156\
In this part, we perform a sensitivity analysis by changing the distribution of nodes among the groups for a fixed total number of nodes $n=45$ where the number of big banks $n_1$ takes values in ${\ensuremath{ \left\{ 5, 10, 15, 20, 25, 30, 35, 40 \right\} }}$. Then, the number of small banks is $n_2 = n - n_1$. Table \[table\_nodesRV\_2D\] shows the computational performance of the algorithm and Figure \[figure\_nodesRV\_2D\_inner\] provides the inner approximations of the corresponding Rogers-Veraart systemic risk measures.
Note that the average time per ${\mathcal{P}}_2$ problem in Table \[table\_nodesRV\_2D\] is relatively high for the values $n_1\in{\ensuremath{ \left\{ 20, 25, 30, 35, 40 \right\} }}$. In addition, the number of ${\mathcal{P}}_2$ problems is greater for the values around $n_1 = 20$. Observe from Figure \[figure\_nodesRV\_2D\_inner\] that as the number of big banks increases and the number of small banks decreases, the small banks get a wider range of capital allocation options, as opposed to the big banks. This happens because the total number of banks is fixed and the group with less number of banks has a wider range of capital allocation options since it has more claims to the other group’s banks in the scope of this set-up.
![Inner approximations of the Rogers-Veraart systemic risk measure for $n_1\in{\ensuremath{ \left\{ 5, 10, 15, 25, 30, 35, 40 \right\} }}$.[]{data-label="figure_nodesRV_2D_inner"}](nodes_RV_all_zoom1.eps){width="70.00000%"}
[||m[1.2cm]{}|m[1.2cm]{}|m[1.3cm]{}|m[2cm]{}|m[2.4cm]{}|m[2.4cm]{}||]{} Inner approx. vertices & Outer approx. vertices & ${\mathcal{P}}_2$ problems & Avg. time per ${\mathcal{P}}_2$ prob. (seconds) & Total algorithm time (seconds) & Total algorithm time (hours)\
413 & 516 & 1250 & 2.904 & 3631 & 1.009\
A three-group signed Eisenberg-Noe network with 60 nodes
--------------------------------------------------------
In this section, we consider a three-group signed Eisenberg-Noe network ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{X}} \right)}}}$ generated with $n = 60$, $n_1 = 10$, $n_2 = 20$, $n_3 = 30$, $K = 50$, $\sigma = 100$, $\varrho = 0.05$ and $$\bm{q}^\text{con} =
\begin{bmatrix}
0.4 & 0.2 & 0.1 \\
0.3 & 0.4 & 0.1 \\
0.2 & 0.3 & 0.4
\end{bmatrix},\quad
\bm{l}^\text{gr} =
\begin{bmatrix}
20 & 15 & 8 \\
15 & 10 & 6 \\
8 & 6 & 5
\end{bmatrix},\quad
\bm{\nu} =
\begin{bmatrix}
-50 & -100 & -150
\end{bmatrix}.$$ In the corresponding Eisenberg-Noe systemic risk measure, we take $\gamma^p = 0.95$.
Table \[table\_3D\_EN\] shows the computational performance of the algorithm for $\epsilon = 20$. Figure \[figure\_3D\_EN\] represents the inner approximation of the corresponding three-group Eisenberg-Noe systemic risk measure. It can be presumed that the value of this Eisenberg-Noe systemic risk measure is convex.
![Inner approximation of the three-group Eisenberg-Noe systemic risk measure with 60 nodes, 50 scenarios and approximation error $\epsilon = 20$.[]{data-label="figure_3D_EN"}](EN_10_20_30.eps){width="60.00000%"}
A three-group Rogers-Veraart network with 60 nodes {#RV3}
--------------------------------------------------
In this section, we consider a Rogers-Veraart network ${{\ensuremath{\left( {\mathcal{N}}, \bm{{\pi}}, \bm{{\bar{p}}}, \bm{{X}}, \alpha, \beta \right)}}}$ generated with $n = 60$, $n_1 = 10$, $n_2 = 20$, $n_3 = 30$, $K = 50$, $\varrho = 0.05$, and $$\bm{q}^\text{con} =
\begin{bmatrix}
0.4 & 0.2 & 0.1 \\
0.2 & 0.3 & 0.2 \\
0.1 & 0.2 & 0.2
\end{bmatrix},\quad
\bm{l}^\text{gr} =
\begin{bmatrix}
200 & 190 & 180 \\
190 & 190 & 180 \\
180 & 180 & 170
\end{bmatrix}.$$
[||m[1.2cm]{}|m[1.2cm]{}|m[1.3cm]{}|m[2cm]{}|m[2.4cm]{}|m[2.4cm]{}||]{} Inner approx. vertices & Outer approx. vertices & ${\mathcal{P}}_2$ problems & Avg. time per ${\mathcal{P}}_2$ prob. (seconds) & Total algorithm time (seconds) & Total algorithm time (hours)\
975 & 1323 & 19382 & 0.427 & 8284 & 2.301\
In addition, the liquid fraction of the random operating cash flows and the liquid fraction of the realized claims available to defaulting banks are fixed as $\alpha = \beta = 0.9$. The shape and scale parameters of gamma distributions of $X_i$, $i\in{\mathcal{N}}_\ell$, $\ell\in{\mathcal{G}}$, are chosen as $$\kappa = \begin{bmatrix} 100 & 81 & 64 \end{bmatrix}, \quad
\theta = \begin{bmatrix} 1 & \frac{10}{9} & 1.25 \end{bmatrix}.$$ In the corresponding Rogers-Veraart systemic risk measure, we take $\gamma^p = 0.99$. The upper bound point in the approximation is taken as $\bm{z}^\text{UB} = \bm{{\hat{z}}}^\text{ideal} + \frac{1}{5}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}$.
Table \[table\_3D\_RV\] shows the computational performance of the algorithm for $\epsilon = 40$. Figure \[figure\_3D\_RV\] provides the inner approximation of the corresponding three-group Rogers-Veraart systemic risk measure. It can be observed that the value of this Rogers-Veraart systemic risk measure is not convex.
![Inner approximation of the three-group Rogers-Veraart systemic risk measure with 60 nodes, 50 scenarios and approximation error $\epsilon = 40$.[]{data-label="figure_3D_RV"}](RV_10_20_30.eps){width="60.00000%"}
Proofs of some results in Section \[systemic\_risk\_models\] {#appendixA}
============================================================
Proof of Lemma \[EN\_lemma\_1\] {#lemma_proof_EN}
-------------------------------
If $s_i = 0$, then constraint is infeasible by assumption. Hence, $s_i = 1$, and this yields $p_i \le \sum_{j=1}^n\pi_{ji}p_j + x_i$ and $p_i \le \bar{p}_i$, by constraints and , respectively. Hence, $$p_i \le \min{\ensuremath{ \left\{ \sum_{j=1}^n\pi_{ji}p_j + x_i, \bar{p}_i \right\} }}.$$ To get a contradiction to the claim of the lemma, suppose that $p_i < \min{\ensuremath{ \left\{ \sum_{j=1}^n\pi_{ji}p_j + x_i, \bar{p}_i \right\} }}$. Now let $\bm{{p^{\epsilon}}}\in{\mathbb{R}}^n_+$ be equal to $\bm{p}$ in all components except the $i\textsuperscript{th}$ one, and let ${p^{\epsilon}}_i = p_i + \epsilon$, where $$\epsilon \coloneqq \min{\ensuremath{ \left\{ \min{\ensuremath{ \left\{ \sum_{j=1}^n\pi_{ji}p_j + x_i, \bar{p}_i \right\} }} - p_i, M -\underset{l\in{\mathcal{N}}}{\max} {\ensuremath{\left( \sum_{j=1}^{n}\pi_{jl}p_j + x_l \right)}}, \epsilon' \right\} }}>0,$$ and $$\epsilon' \coloneqq {\min} {\ensuremath{ \left\{ {\ensuremath{ \left| \sum_{j=1}^{n}\pi_{jl}p_j+x_l \right| }}\mid \sum_{j=1}^{n}\pi_{jl}p_j + x_l < 0,\ l\in{\mathcal{N}}\right\} }}.$$ (Here, we assume that $\epsilon' = +\infty$ if there is no qualifying $l\in{\mathcal{N}}$ in the above definition.) This choice of $\epsilon$ ensures $${p^{\epsilon}}_i \le \bar{p}_i \quad\text{and}\quad {p^{\epsilon}}_i \le \sum_{j=1}^{n}\pi_{ji}{p^{\epsilon}}_j + x_i,$$ and will also be justified by other technical details later in this proof.
Let $\bm{{s^{\epsilon}}}\in{\ensuremath{ \left\{ 0,1 \right\} }}^n$ be a vector of binaries, where ${s^{\epsilon}}_l = 0$ if $\sum_{j=1}^{n}\pi_{jl}{p^{\epsilon}}_j + x_l < 0$ and ${s^{\epsilon}}_l = 1$ if $\sum_{j=1}^{n}\pi_{jl}{p^{\epsilon}}_j + x_l \ge 0$, for each $l\in{\mathcal{N}}$. We show that ${\ensuremath{\left( \bm{{p^{\epsilon}}}, \bm{{s^{\epsilon}}} \right)}}$ is a feasible solution to $\Lambda^\text{EN}{\ensuremath{\left( \bm{x} \right)}}$ by showing that all constraints in are satisfied. First, for fixed $k\in{\mathcal{N}}\backslash{\ensuremath{ \left\{ i \right\} }}$, we verify the $k\textsuperscript{th}$ constraints in for ${\ensuremath{\left( \bm{{p^{\epsilon}}}, \bm{{s^{\epsilon}}} \right)}}$. We consider three cases: $${\ensuremath{\left( 1 \right)}}~ \sum_{j=1}^n\pi_{jk}p_j + x_k < 0,
\quad {\ensuremath{\left( 2 \right)}}~ \sum_{j=1}^n\pi_{jk}p_j + x_k = 0,
\quad {\ensuremath{\left( 3 \right)}}~ 0 < \sum_{j=1}^n\pi_{jk}p_j + x_k.$$
1. Assume that $\sum_{j=1}^n\pi_{jk}p_j + x_k < 0$. If $s_k = 1$, then, by constraint , $$p_k \le \sum_{j=1}^{n}\pi_{jk}p_j + x_k + M{\ensuremath{\left( 1-1 \right)}} = \sum_{j=1}^{n}\pi_{jk}p_j + x_k < 0,$$ which is a contradiction to the feasibility of ${\ensuremath{\left( \bm{p}, \bm{s} \right)}}$ in constraint . Hence, $s_k = 0$, which in its turn implies $p_k = 0$ by and .
By the definitions of $\bm{{p^{\epsilon}}}$ and $\bm{{s^{\epsilon}}}$, it holds that ${p^{\epsilon}}_k = p_k = 0$ since $k\neq i$, and ${s^{\epsilon}}_k = 0$. Constraint holds as $${p^{\epsilon}}_k = p_k = 0 \le \sum_{j=1}^{n}\pi_{jk}{p^{\epsilon}}_j + x_k + M{\ensuremath{\left( 1-{s^{\epsilon}}_k \right)}} = \sum_{j=1}^{n}\pi_{jk}p_j + x_k + M + \epsilon\pi_{ik}$$ by the feasibility of $p_k =0$ and $s_k=0$, and since $\epsilon>0$ and $\pi_{ik}\ge0$. Constraint holds as $$\sum_{j=1}^{n}\pi_{jk}{p^{\epsilon}}_j + x_k = \sum_{j=1}^{n}\pi_{jk}p_j + x_k + \epsilon\pi_{ik} \le \sum_{j=1}^{n}\pi_{jk}p_j + x_k + \epsilon \le 0 = M{s^{\epsilon}}_k$$ since $\sum_{j=1}^n\pi_{jk}p_j + x_k < 0$, $\pi_{ik}\le 1$ and since a small enough $\epsilon>0$ is taken to ensure $\sum_{j=1}^n\pi_{jk}p_j + x_k + \epsilon \le 0$. Constraints , , and for node $k$ hold trivially by the feasibility of $p_k =0$ and $s_k=0$. Hence, ${p^{\epsilon}}_k=0$ and ${s^{\epsilon}}_k=0$ satisfy the corresponding constraints in .
2. Assume that $\sum_{j=1}^n\pi_{jk}p_j + x_k = 0$. Now, either $s_k = 0$ or $s_k = 1$ holds. If $s_k = 0$, then $p_k = 0$ by constraints and . If $s_k = 1$, then, by the assumption of this case and , $p_k \le \sum_{j=1}^{n}\pi_{jk}p_j + x_k + M{\ensuremath{\left( 1-1 \right)}} = 0$, which, together with , implies $p_k = 0$.
Also, ${p^{\epsilon}}_k = p_k = 0$ and ${s^{\epsilon}}_k = 1$, by the definitions of $\bm{{p^{\epsilon}}}$ and $\bm{{s^{\epsilon}}}$. Constraint holds as $$\begin{aligned}
{p^{\epsilon}}_k = p_k = 0 &\le \sum_{j=1}^{n}\pi_{jk}{p^{\epsilon}}_j + x_k + M{\ensuremath{\left( 1-{s^{\epsilon}}_k \right)}} \\
&= \sum_{j=1}^{n}\pi_{jk}p_j + x_k + M{\ensuremath{\left( 1-1 \right)}} + \epsilon\pi_{ik} = \epsilon\pi_{ik},
\end{aligned}$$ since $\sum_{j=1}^n\pi_{jk}p_j + x_k = 0$, $\epsilon>0$ and $\pi_{ik}\ge0$. Constraint holds as $$\sum_{j=1}^{n}\pi_{jk}{p^{\epsilon}}_j + x_k = \sum_{j=1}^{n}\pi_{jk}p_j + x_k + \epsilon\pi_{ik} = \epsilon\pi_{ik} \le M{s^{\epsilon}}_k = M$$ since $\sum_{j=1}^n\pi_{jk}p_j + x_k = 0$, $\epsilon\le\underset{l\in{\mathcal{N}}}{\min}{\ensuremath{ \left\{ M - {\ensuremath{\left( \sum_{j=1}^{n}\pi_{jl}p_j + x_l \right)}} \right\} }}\le M$ by the definition of $\epsilon$, and $0\le\pi_{ik}\le 1$. It is easy to observe that all other constraints in for node $k$ are satisfied trivially by ${p^{\epsilon}}_k=0$ and ${s^{\epsilon}}_k=1$.
3. Assume that $0 < \sum_{j=1}^n\pi_{jk}p_j + x_k$. If $s_k = 0$, then, by constraint , $$\sum_{j=1}^{n}\pi_{jk}p_j + x_k \le Ms_k = 0,$$ which is a contradiction to the assumption. Hence, $s_k = 1$. Also, ${s^{\epsilon}}_k = 1$, by the definition of $\bm{{s^{\epsilon}}}$.
Since $s_k = 1$, and hold by the feasibility of $p_k$ since ${p^{\epsilon}}_k = p_k$ for $k\neq i$. Also, holds since $\epsilon>0$ is taken small enough to ensure $$\label{A1_1}
\sum_{j=1}^{n}\pi_{jk}{p^{\epsilon}}_j + x_k = \sum_{j=1}^{n}\pi_{jk}p_j + x_k + \epsilon\pi_{ik} \le M.$$ Indeed, recall the assumption $\sum_{j=1}^{n}\pi_{jl} < n$, for each $l\in{\mathcal{N}}$. Hence, for each $l\in{\mathcal{N}}$ and for every $\bm{p}\in{\ensuremath{ \left[ {\mathbf{0}}, \bar{{\bm{p}}} \right] }}$, $\sum_{j=1}^{n}\pi_{jl}p_j + x_l < M$, where $M=n{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{x} \right\Vert }_\infty}$. So, is guaranteed by the choice of $\epsilon$. (This is the reason behind including the term $M-\underset{l\in{\mathcal{N}}}{\max} {\ensuremath{\left( \sum_{j=1}^{n}\pi_{jl}p_j + x_l \right)}}$ in the definition of $\epsilon$.)
Note that, since $s_k = 1$, $p_k \le \sum_{j=1}^{n}\pi_{jk}p_j + x_k$ holds. Then constraint is satisfied since $$\begin{aligned}
{p^{\epsilon}}_k = p_k &\le \sum_{j=1}^{n}\pi_{jk}p_j + x_k \le \sum_{j=1}^{n}\pi_{jk}p_j + x_k + \epsilon\pi_{ik} \\
&= \sum_{\substack{j\in{\mathcal{N}}\\j\neq i}}\pi_{jk}p_j + \pi_{ik}{\ensuremath{\left( p_i + \epsilon \right)}} + x_k = \sum_{j=1}^{n}\pi_{jk}{p^{\epsilon}}_j + x_k.
\end{aligned}$$ Constraint is satisfied trivially. Hence, ${p^{\epsilon}}_k$ and ${s^{\epsilon}}_k$ satisfy the corresponding constraints in .
Next, we show that ${p^{\epsilon}}_i$ and ${s^{\epsilon}}_i$ satisfy the constraints in for $i$. It holds ${s^{\epsilon}}_i = 1$, since $\sum_{j=1}^n\pi_{ji}p_j + x_i > 0$ by the assumption of Lemma \[EN\_lemma\_1\]. Then, constraints and hold since ${p^{\epsilon}}_i = p_i + \epsilon > 0$ and ${p^{\epsilon}}_i = p_i + \epsilon \le p_i + \bar{p}_i - p_i\le \bar{p}_i$, where $\epsilon \le \bar{p}_i - p_i$ holds since $\epsilon \le \min{\ensuremath{ \left\{ \sum_{j=1}^n\pi_{ji}p_j + x_i, \bar{p}_i \right\} }} - \bar{p}_i$. Constraint holds as $$\begin{aligned}
{p^{\epsilon}}_i &= p_i + \epsilon \le p_i + \sum_{j=1}^{n}\pi_{ji}p_j + x_i - p_i = \sum_{j=1}^{n}\pi_{ji}p_j + x_i \\
&\le\sum_{j=1}^{n}\pi_{jk}p_j + x_k + \epsilon\pi_{ik} = \sum_{\substack{j\in{\mathcal{N}}\\j\neq i}}\pi_{jk}p_j + \pi_{ik}{\ensuremath{\left( p_i + \epsilon \right)}} + x_k = \sum_{j=1}^{n}\pi_{jk}{p^{\epsilon}}_j + x_k,
\end{aligned}$$ where $\epsilon \le \sum_{j=1}^{n}\pi_{ji}p_j + x_i - p_i$ holds since $\epsilon \le \min{\ensuremath{ \left\{ \sum_{j=1}^n\pi_{ji}p_j + x_i, \bar{p}_i \right\} }} - \bar{p}_i$. Constraint holds as $$\sum_{j=1}^{n}\pi_{ji}{p^{\epsilon}}_j + x_i = \sum_{j=1}^{n}\pi_{ji}p_j + x_i + \epsilon\pi_{ii} = \sum_{j=1}^{n}\pi_{ji}p_j + x_i\le M$$ by the feasibility of $\bm{p}$ and since $\pi_{ll} = 0$, for each $l\in{\mathcal{N}}$. Constraint is satisfied trivially. Hence, ${p^{\epsilon}}_i$ and ${s^{\epsilon}}_i$ satisfy the corresponding constraints in .
Hence, ${\ensuremath{\left( \bm{{p^{\epsilon}}}, \bm{{s^{\epsilon}}} \right)}}$ is a feasible solution to $\Lambda^\text{EN}{\ensuremath{\left( \bm{x} \right)}}$. However, since $\bm{{p^{\epsilon}}}\ge\bm{p}$ with $\bm{{p^{\epsilon}}}\neq\bm{p}$ and $f$ is a strictly increasing function, it holds that $f{\ensuremath{\left( \bm{{p^{\epsilon}}} \right)}} > f{\ensuremath{\left( \bm{p} \right)}}$, which is a contradiction to the optimality of $\bm{p}$. Hence, $p_i = \min{\ensuremath{ \left\{ \sum_{j=1}^{n}\pi_{ji}p_j + x_i, \bar{p}_i \right\} }}$.
Proof of Lemma \[RV\_lemma\_1\] {#lemma_proof_RV1}
-------------------------------
To get a contradiction, suppose that $s_i = 0$. Then $p_i \le \alpha x_i+ \beta\sum_{j=1}^{n}\pi_{ji}p_j < \bar{p}_i$ by constraint and the assumption. Let $\bm{{p'}}\in{\mathbb{R}}^n_+$ be equal to $\bm{p}$ in all components except the $i\textsuperscript{th}$ one, and let ${p'}_i = \bar{p}_i$. Also, let $\bm{{s'}}\in{\mathbb{R}}^n_+$ be equal to $\bm{s}$ in all components except the $i\textsuperscript{th}$ one, and let ${s'}_i = 1$.
We show that ${\ensuremath{\left( \bm{{p'}}, \bm{{s'}} \right)}}$ is a feasible solution to $\Lambda^{\text{RV}_+}{\ensuremath{\left( \bm{x} \right)}}$ by checking that all constraints in are satisfied. First, for fixed $k\in{\mathcal{N}}\backslash{\ensuremath{ \left\{ i \right\} }}$, we verify the $k\textsuperscript{th}$ constraints in for ${\ensuremath{\left( \bm{{p'}}, \bm{{s'}} \right)}}$. Constraints and hold as $$\begin{aligned}
{p'}_k &= p_k \le \alpha x_k + \beta \sum_{j=1}^{n}\pi_{jk}p_j + \bar{p}_k s_k \\
& \le \alpha x_k + \beta \sum_{j=1}^{n}\pi_{jk}p_j + \bar{p}_k s_k + \pi_{ik}{\ensuremath{\left( \bar{p}_i - p_i \right)}} = \alpha x_k + \beta \sum_{j=1}^{n}\pi_{jk}{p'}_j + \bar{p}_k {s'}_k,
\end{aligned}$$ and $$\begin{aligned}
\bar{p}_k {s'}_k = \bar{p}_k s_k \le x_k + \sum_{j=1}^{n}\pi_{jk}p_j \le x_k + \sum_{j=1}^{n}\pi_{jk}p_j + \pi_{ik}{\ensuremath{\left( \bar{p}_i - p_i \right)}} = x_k + \sum_{j=1}^{n}\pi_{jk}{p'}_j,
\end{aligned}$$ since ${p'}_k = p_k$, ${s'}_k = s_k$ for every $k\in{\mathcal{K}}$ such that $k\neq i$, $\bar{p}_i - p_i > 0$, $\pi_{ik}\ge0$, and by the feasibility of ${\ensuremath{\left( \bm{p}, \bm{s} \right)}}$. Constraints , hold trivially by the feasibility of ${\ensuremath{\left( \bm{p}, \bm{s} \right)}}$.
Next, we verify the $i\textsuperscript{th}$ constraints in for ${p'}_i = \bar{p}_i$, ${s'}_i = 1$. Constraints and hold as $$\begin{aligned}
&{p'}_i = \bar{p}_i \le \alpha x_i + \beta \sum_{j=1}^{n}\pi_{ji}p_j + \bar{p}_i {s'}_i \\
&\qquad = \alpha x_i + \beta \sum_{j=1}^{n}\pi_{ji}p_j + \bar{p}_i + \pi_{ii}{\ensuremath{\left( \bar{p}_i - p_i \right)}} = \alpha x_i + \beta \sum_{j=1}^{n}\pi_{ji}{p'}_j + \bar{p}_i,
\end{aligned}$$ and $$\begin{aligned}
\bar{p}_i {s'}_i = \bar{p}_i \le x_i + \sum_{j=1}^{n}\pi_{ji}p_j = x_i + \sum_{j=1}^{n}\pi_{ji}p_j + \pi_{ii}{\ensuremath{\left( \bar{p}_i - p_i \right)}} = x_i + \sum_{j=1}^{n}\pi_{ji}{p'}_j,
\end{aligned}$$ since $\alpha x_i + \beta \sum_{j=1}^{n}\pi_{ji}{p'}_j \ge 0$, $\pi_{ii}=0$ and by the assumption of Lemma \[RV\_lemma\_1\]. Constraints , are satisfied trivially.
Hence, ${\ensuremath{\left( \bm{{p'}}, \bm{{s'}} \right)}}$ is a feasible solution to $\Lambda^{\text{RV}_+}{\ensuremath{\left( \bm{x} \right)}}$. However, since $\bm{{p'}}\ge\bm{p}$ with $\bm{{p'}}\neq\bm{p}$ and $f$ is a strictly increasing function, it holds that $f{\ensuremath{\left( \bm{{p'}} \right)}} > f{\ensuremath{\left( \bm{p} \right)}}$, which is a contradiction to the optimality of $\bm{p}$. Hence, $s_i = 1$.
Proof of Lemma \[RV\_lemma\_2\] {#lemma_proof_RV2}
-------------------------------
To get a contradiction, suppose that $p_i < \bar{p}_i$. Let $\bm{{p'}}\in{\mathbb{R}}^n_+$ be equal to $\bm{p}$ in all components except the $i\textsuperscript{th}$ one, and let ${p'}_i = \bar{p}_i$
We show that ${\ensuremath{\left( \bm{{p'}}, \bm{s} \right)}}$ is a feasible solution to $\Lambda^{\text{RV}_+}{\ensuremath{\left( \bm{x} \right)}}$ by showing that all constraints in are satisfied. First, for fixed $k\in{\mathcal{N}}\backslash{\ensuremath{ \left\{ i \right\} }}$, we verify the $k\textsuperscript{th}$ constraints in for ${\ensuremath{\left( \bm{{p'}}, \bm{s} \right)}}$. Constraints and hold as $$\begin{aligned}
{p'}_k &= p_k \le \alpha x_k + \beta \sum_{j=1}^{n}\pi_{jk}p_j + \bar{p}_k s_k \\
& \le \alpha x_k + \beta \sum_{j=1}^{n}\pi_{jk}p_j + \bar{p}_k s_k + \pi_{ik}{\ensuremath{\left( \bar{p}_i - p_i \right)}} = \alpha x_k + \beta \sum_{j=1}^{n}\pi_{jk}{p'}_j + \bar{p}_k s_k,
\end{aligned}$$ and $$\begin{aligned}
\bar{p}_k s_k \le x_k + \sum_{j=1}^{n}\pi_{jk}p_j \le x_k + \sum_{j=1}^{n}\pi_{jk}p_j + \pi_{ik}{\ensuremath{\left( \bar{p}_i - p_i \right)}} = x_k + \sum_{j=1}^{n}\pi_{jk}{p'}_j,
\end{aligned}$$ since ${p'}_k = p_k$ for every $k\in{\mathcal{K}}$ such that $k\neq i$, $\bar{p}_i - p_i > 0$, $\pi_{ik}\ge0$ and by the feasibility of ${\ensuremath{\left( \bm{p}, \bm{s} \right)}}$. Constraints , hold trivially by the feasibility of ${\ensuremath{\left( \bm{p}, \bm{s} \right)}}$.
Next, we verify the $i\textsuperscript{th}$ constraints in for ${p'}_i=\bar{p}_i $, $s_i$. We consider two cases: $${\ensuremath{\left( 1 \right)}}~ \bar{p}_i \le \alpha x_i+ \beta\sum_{j=1}^{n}\pi_{ji}p_j, \quad {\ensuremath{\left( 2 \right)}}~ \alpha x_i+ \beta\sum_{j=1}^{n}\pi_{ji}p_j < \bar{p}_i.$$
- If the first case holds, then constraints and hold for both $s_i=0$ and $s_i=1$ as $$\begin{aligned}
{p'}_i &= \bar{p}_i \le \alpha x_i + \beta \sum_{j=1}^{n}\pi_{ji}p_j + \bar{p}_i s_i \\
&= \alpha x_i + \beta \sum_{j=1}^{n}\pi_{ji}p_j + \bar{p}_i s_i + \pi_{ii}{\ensuremath{\left( \bar{p}_i - p_i \right)}} = \alpha x_i + \beta \sum_{j=1}^{n}\pi_{ji}{p'}_j + \bar{p}_i s_i,
\end{aligned}$$ and $$\begin{aligned}
\bar{p}_i s_i \le x_i + \sum_{j=1}^{n}\pi_{ji}p_j = x_i + \sum_{j=1}^{n}\pi_{ji}p_j + \pi_{ii}{\ensuremath{\left( \bar{p}_i - p_i \right)}} = x_i + \sum_{j=1}^{n}\pi_{ji}{p'}_j,
\end{aligned}$$ since $\pi_{ii}=0$ and by the assumption of Lemma \[RV\_lemma\_2\]. Constraints , hold trivially.
- If the second case holds, then, by Lemma \[RV\_lemma\_1\], $s_i=1$. Then constraints and hold as $$\begin{aligned}
{p'}_i& = \bar{p}_i \le \alpha x_i + \beta \sum_{j=1}^{n}\pi_{ji}p_j + \bar{p}_i s_i \\
& = \alpha x_i + \beta \sum_{j=1}^{n}\pi_{ji}p_j + \bar{p}_i + \pi_{ii}{\ensuremath{\left( \bar{p}_i - p_i \right)}} = \alpha x_i + \beta \sum_{j=1}^{n}\pi_{ji}{p'}_j + \bar{p}_i,
\end{aligned}$$ and $$\begin{aligned}
\bar{p}_i s_i = \bar{p}_i \le x_i + \sum_{j=1}^{n}\pi_{ji}p_j = x_i + \sum_{j=1}^{n}\pi_{ji}p_j + \pi_{ii}{\ensuremath{\left( \bar{p}_i - p_i \right)}} = x_i + \sum_{j=1}^{n}\pi_{ji}{p'}_j,
\end{aligned}$$ since $\pi_{ii}=0$ and by the assumption of Lemma \[RV\_lemma\_2\]. Constraints , are satisfied trivially.
Hence, ${\ensuremath{\left( \bm{{p'}}, \bm{s} \right)}}$ is a feasible solution to $\Lambda^{\text{RV}_+}{\ensuremath{\left( \bm{x} \right)}}$. However, since $\bm{{p'}}\ge\bm{p}$ with $\bm{{p'}}\neq\bm{p}$ and $f$ is a strictly increasing function, it holds that $f{\ensuremath{\left( \bm{{p'}} \right)}} > f{\ensuremath{\left( \bm{p} \right)}}$, which is a contradiction to the optimality of $\bm{p}$. Hence, $p_i = \bar{p}_i$.
Proof of Lemma \[RV\_lemma\_3\] {#lemma_proof_RV3}
-------------------------------
To get a contradiction, suppose that $p_i \neq \alpha x_i+\beta \sum_{j=1}^{n}\pi_{ji}p_j$. If $s_i = 1$, then constraint is not satisfied by assumption. Hence, $s_i = 0$ and $p_i < \alpha x_i + \beta \sum_{j=1}^n\pi_{ji}p_j$ by constraint . Let $\bm{{p'}}\in{\mathbb{R}}^n_+$ be equal to $\bm{p}$ in all components except the $i\textsuperscript{th}$ one, and let ${p'}_i = \alpha x_i+\beta \sum_{j=1}^{n}\pi_{ji}p_j$.
We show that ${\ensuremath{\left( \bm{{p'}}, \bm{s} \right)}}$ is a feasible solution to $\Lambda^{\text{RV}_+}{\ensuremath{\left( \bm{x} \right)}}$ by checking that all constraints in are satisfied. First, for fixed $k\in{\mathcal{N}}\backslash{\ensuremath{ \left\{ i \right\} }}$, we verify the $k\textsuperscript{th}$ constraints in for ${\ensuremath{\left( \bm{{p'}}, \bm{s} \right)}}$. Constraints and hold as $$\begin{aligned}
{p'}_k &= p_k \le \alpha x_k + \beta \sum_{j=1}^{n}\pi_{jk}p_j + \bar{p}_k s_k \\
&\le \alpha x_k + \beta \sum_{j=1}^{n}\pi_{jk}p_j + \bar{p}_k s_k + \pi_{ik}{\ensuremath{\left( \bar{p}_i - p_i \right)}} = \alpha x_k + \beta \sum_{j=1}^{n}\pi_{jk}{p'}_j + \bar{p}_k s_k,
\end{aligned}$$ and $$\begin{aligned}
\bar{p}_k s_k \le x_k + \sum_{j=1}^{n}\pi_{jk}p_j \le x_k + \sum_{j=1}^{n}\pi_{jk}p_j + \pi_{ik}{\ensuremath{\left( \bar{p}_i - p_i \right)}} = x_k + \sum_{j=1}^{n}\pi_{jk}{p'}_j,
\end{aligned}$$ since ${p'}_k = p_k$ for every $k\in{\mathcal{K}}$ such that $k\neq i$, $\bar{p}_i - p_i > 0$, $\pi_{ik}\ge0$ and by the feasibility of ${\ensuremath{\left( \bm{p}, \bm{s} \right)}}$. Constraints , hold trivially by the feasibility of ${\ensuremath{\left( \bm{p}, \bm{s} \right)}}$.
Next, we verify the $i\textsuperscript{th}$ constraints in for ${p'}_i=\alpha x_i+\beta \sum_{j=1}^{n}\pi_{ji}p_j$, $s_i=0$. Constraints and hold as $$\begin{aligned}
{p'}_i &= \alpha x_i+\beta \sum_{j=1}^{n}\pi_{ji}p_j \le \alpha x_i + \beta \sum_{j=1}^{n}\pi_{ji}p_j + \bar{p}_i s_i \\
& = \alpha x_i + \beta \sum_{j=1}^{n}\pi_{ji}p_j + \pi_{ii}{\ensuremath{\left( \bar{p}_i - p_i \right)}} = \alpha x_i + \beta \sum_{j=1}^{n}\pi_{ji}{p'}_j ,
\end{aligned}$$ and $$\begin{aligned}
\bar{p}_i s_i = 0 \le x_i + \sum_{j=1}^{n}\pi_{ji}p_j = x_i + \sum_{j=1}^{n}\pi_{ji}p_j + \pi_{ii}{\ensuremath{\left( \bar{p}_i - p_i \right)}} = x_i + \sum_{j=1}^{n}\pi_{ji}{p'}_j,
\end{aligned}$$ since $\pi_{ii}=0$ and $x_i + \sum_{j=1}^{n}\pi_{ji}p_j\ge0$. Constraints , are satisfied trivially.
Hence, ${\ensuremath{\left( \bm{{p'}}, \bm{s} \right)}}$ is a feasible solution to $\Lambda^{\text{RV}_+}{\ensuremath{\left( \bm{x} \right)}}$. However, since $\bm{{p'}}\ge\bm{p}$ with $\bm{{p'}}\neq\bm{p}$ and $f$ is a strictly increasing function, it holds that $f{\ensuremath{\left( \bm{{p'}} \right)}} > f{\ensuremath{\left( \bm{p} \right)}}$, which is a contradiction to the optimality of $\bm{p}$. Hence, $p_i = \bar{p}_i$.
Proofs of some results in Section \[systemic\_risk\_measures\] {#appendixB}
==============================================================
For convenience, let us rewrite the mixed-integer linear programming problem in more explicitly as $$\begin{aligned}
\text{minimize}\quad &z_\ell \label{P1_MILP_EN_explicit}\\
\text{subject to}\quad& \sum_{k\in{\mathcal{K}}}q^k{{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{p}^k \ge \gamma, \label{P1_MILP_EN_constraint_1}\\
& p_i^k \le \sum_{j=1}^{n}\pi_{ji}p_j^k + (X_i(\omega^k) + ({{B}^\mathsf{T}}\bm{z})_i) + M(1-s_i^k), &&\forall i\in{\mathcal{N}}, k\in{\mathcal{K}}, \label{P1_MILP_EN_constraint_2}\\
& p_i^k \le \bar{p}_is_i^k, &&\forall i\in{\mathcal{N}}, k\in{\mathcal{K}}, \label{P1_MILP_EN_constraint_3}\\
& \sum_{j=1}^{n}\pi_{ji}p_j^k + (X_i(\omega^k) + ({{B}^\mathsf{T}}\bm{z})_i)\le Ms_i^k, &&\forall i\in{\mathcal{N}}, k\in{\mathcal{K}}, \label{P1_MILP_EN_constraint_4}\\
&0 \le p_i^k \le \bar{p}_i, &&\forall i\in{\mathcal{N}}, k\in{\mathcal{K}}, \label{P1_MILP_EN_constraint_5}\\
&s_i^k\in{\ensuremath{ \left\{ 0,1 \right\} }}, && \forall i\in{\mathcal{N}}, k\in{\mathcal{K}}, \label{P1_MILP_EN_constraint_6} \\
& \bm{z}\in{\mathbb{R}}^G.\end{aligned}$$
Proof of Proposition \[P1\_EN\_upperbound\] {#P1_EN_upperbound_proof}
-------------------------------------------
Let $(\bm{z},(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ be an optimal solution of the problem in . To get a contradiction, suppose that $z_\ell > {\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}$. Let $\bm{z'}\in{\mathbb{R}}^G$ be the vector such that $z'_\ell = {\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}$ and $z'_{\hat{\ell}} = z_{\hat{\ell}}$ for each $\hat{\ell}\in{\mathcal{G}}\setminus{\ensuremath{ \left\{ \ell \right\} }}$. We claim that $(\bm{z'},(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ is a feasible solution of the problem in . Indeed, for each $i\in{\mathcal{N}}$, $k\in{\mathcal{K}}$ such that ${\ensuremath{\left( {{B}^\mathsf{T}}\bm{z'} \right)}}_i = {\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}$, constraint holds as $$\begin{aligned}
p_i^k &\le \sum_{j=1}^{n}\pi_{ji}p_j^k + {\ensuremath{\left( X_i(\omega^k)+({{B}^\mathsf{T}}\bm{z'})_i \right)}}+M(1-s_i^k)\\
& = \sum_{j=1}^{n}\pi_{ji}p_j^k + X_i(\omega^k)+{{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}+ M(1-s_i^k)
\end{aligned}$$ since $$\sum_{j=1}^{n}\pi_{ji}p_j^k \ge 0, \quad X_i(\omega^k) + {\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}\ge 0, \quad p_i^k \le \bar{p}_i \le {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}, \quad M(1-s_i^k)\ge 0.$$ Also, for each $i\in{\mathcal{N}}$, $k\in{\mathcal{K}}$ such that $({{B}^\mathsf{T}}\bm{z'})_i = {\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}$, constraint holds as $$\begin{aligned}
\sum_{j=1}^{n}\pi_{ji}p_j^k + {\ensuremath{\left( X_i(\omega^k)+({{B}^\mathsf{T}}\bm{z'})_i \right)}} &= \sum_{j=1}^{n}\pi_{ji}p_j^k + X_i(\omega^k)+{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}\\
&< \sum_{j=1}^{n}\pi_{ji}p_j^k + X_i(\omega^k)+z_{\ell} \le Ms_i^k,
\end{aligned}$$ which holds by the supposition ${\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}< z_\ell$ and the feasibility of $(\bm{z},(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$. All the other constraints in hold by the feasibility of $(\bm{z},(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$, since they are free of ${\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}$. Hence, the claim follows, which yields $z_\ell ={\mathcal{Z}}_1^\text{EN}{\ensuremath{\left( \bm{e}^\ell \right)}}\leq z'_\ell ={{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}$. As this is a contradiction, the result follows.
Proof of Proposition \[P1\_EN\_boundedness\] {#P1_EN_boundedness_proof}
--------------------------------------------
To get a contradiction, suppose that the problem in has a feasible solution but ${\mathcal{Z}}_1^\text{EN}(\bm{e}^\ell)=-\infty$. Since ${\mathcal{Z}}_1^\text{EN}(\bm{e}^\ell)=-\infty$, there exist $\epsilon>0$ and $(\bm{z},(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$, where $\bm{z}\in{\mathbb{R}}^G$ and $(\bm{p}^k, \bm{s}^k)\in{\mathbb{R}}^n\times{\mathbb{Z}}^n$ for each $k\in{\mathcal{K}}$, such that ${{\bm{e}^\ell}^\mathsf{T}}\bm{z} = z_\ell = -2M$ and $(\bm{z}-\epsilon\bm{e}^\ell,(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ is a feasible solution for the problem in . Fix $i\in{\mathcal{N}}$, $k\in{\mathcal{K}}$ such that $({{B}^\mathsf{T}}\bm{z})_i = z_\ell = -2M$. Then, constraint contradicts constraint as $$\begin{aligned}
p_i^k &\le \sum_{j=1}^{n}\pi_{ji}p_j^k + (X_i(\omega^k)+({{B}^\mathsf{T}}(\bm{z}-\epsilon\bm{e}^\ell))_i) + M(1-s^k_i) \\
&\le\sum_{j=1}^{n}\pi_{ji}p_j^k + X_i(\omega^k)- 2M - \epsilon+M \\
&=\sum_{j=1}^{n}\pi_{ji}p_j^k + X_i(\omega^k)-\epsilon - 2{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}-{\ensuremath{\left( n+1 \right)}}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}\\
&={\ensuremath{\left( \sum_{j=1}^{n}\pi_{ji}p_j^k-n{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}\right)}} + {\ensuremath{\left( X_i(\omega^k)-2{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}\right)}}- {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}-\epsilon < 0
\end{aligned}$$ since $$\sum_{j=1}^{n}\pi_{ji}p_j^k < n{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}, \quad X_i(\omega^k) \le 2{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}, \quad -{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}< 0, \quad -\epsilon < 0.$$ Hence, $(\bm{z}-\epsilon\bm{e}_\ell,(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ is infeasible, which is a contradiction to the assumption. Hence, ${\mathcal{Z}}_1^\text{EN}(\bm{e}_\ell)>-\infty$. In addition, the existence of a feasible solution implies that ${\mathcal{Z}}_1^\text{EN}(\bm{e}_\ell)<+\infty$. Hence, ${\mathcal{Z}}_1^\text{EN}(\bm{e}_\ell)\in{\mathbb{R}}$.
Proof of Proposition \[P1\_EN\_feasibility\] {#P1_EN_feasibility_proof}
--------------------------------------------
Assume that $\gamma \le {{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}}$. Let $\bm{z} = ({{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}){\bm{\mathbbm{1}}}$, $\bm{p}^k = \bm{{\bar{p}}}$, $\bm{s}^k = {\bm{\mathbbm{1}}}$ for each $k\in{\mathcal{K}}$. We show that $(\bm{z},(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ is a feasible solution for the problem in . Since $\bm{p}^k = \bm{{\bar{p}}}$ for each $k\in{\mathcal{K}}$, it is clear that $\sum_{k\in{\mathcal{K}}}q^k{{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{p}^k = {{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}} \ge \gamma$. Hence, constraint holds. Let $i\in{\mathcal{N}}$, $k\in{\mathcal{K}}$. Constraint holds as $$\begin{aligned}
& \sum_{j=1}^{n}\pi_{ji}p_j^k + {\ensuremath{\left( X_i(\omega^k) + ({{B}^\mathsf{T}}\bm{z})_i \right)}} + M(1-s_i^k) \\
&= \sum_{j=1}^{n}\pi_{ji}p_j^k + X_i(\omega^k) + ({{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}})({{B}^\mathsf{T}}{\bm{\mathbbm{1}}})_i + M(1-1)\\
& = \sum_{j=1}^{n}\pi_{ji}p_j^k + X_i(\omega^k) + {{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}\ge \bar{p}_i = p_i^k
\end{aligned}$$ since $$\begin{aligned}
&\sum_{j=1}^{n}\pi_{ji}p_j^k \ge 0, \quad X_i(\omega^k) + {\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}\ge 0,\quad ({{B}^\mathsf{T}}{\bm{\mathbbm{1}}})_i = 1,\quad s_i^k = 1.
\end{aligned}$$ Constraint holds as $$\begin{aligned}
\sum_{j=1}^{n}\pi_{ji}p_j^k + {\ensuremath{\left( X_i(\omega^k) + ({{B}^\mathsf{T}}\bm{z})_i \right)}} &= \sum_{j=1}^{n}\pi_{ji}p_j^k + X_i(\omega^k) + {{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}\\
& \le {2{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{\left( n+1 \right)}}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}= M = Ms_i^k,
\end{aligned}$$ since $\sum_{j=1}^{n}\pi_{ji}p_j^k \le n{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}$. All the other constraints in hold trivially by the choice of $\bm{z}$, $\bm{p}^k$ and $\bm{s}^k$, for each $k\in{\mathcal{K}}$. Hence, $(\bm{z},(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ is a feasible solution of the problem in .
Conversely, if $\gamma>{{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}}$, then constraint is infeasible, since $\sum_{k\in{\mathcal{K}}}q^k{{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{p}^k \le {{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}} < \gamma$ by constraint . Hence, the problem in is infeasible, which concludes the proof.
The mixed-integer linear programming problem of calculating ${\mathcal{Z}}_1^{\text{RV}_+}{\ensuremath{\left( \bm{e}^\ell \right)}}$ in can be written more explicitly as $$\begin{aligned}
\text{minimize}\quad &z_\ell \label{P1_MILP_RV_explicit}\\
\text{subject to}\quad& \sum_{k\in{\mathcal{K}}}q^k{{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{p}^k \ge \gamma, \label{P1_MILP_RV_constraint_1}\\
& p_i^k \le \alpha {\ensuremath{\left( X_i(\omega^k) + ({{B}^\mathsf{T}}\bm{z})_i \right)}} + \beta\sum_{j=1}^{n}\pi_{ji}p_j^k + \bar{p}_is_i^k, && \forall i\in{\mathcal{N}}, k\in{\mathcal{K}}, \label{P1_MILP_RV_constraint_2}\\
& \bar{p}_is_i^k \le {\ensuremath{\left( X_i(\omega^k) + ({{B}^\mathsf{T}}\bm{z})_i \right)}} + \sum_{j=1}^{n}\pi_{ji}p_j^k, && \forall i\in{\mathcal{N}}, k\in{\mathcal{K}}, \label{P1_MILP_RV_constraint_3}\\
& X_i(\omega^k) + ({{B}^\mathsf{T}}\bm{z})_i \ge 0, && \forall i\in{\mathcal{N}}, k\in{\mathcal{K}}, \label{P1_MILP_RV_constraint_4}\\
&0 \le p_i^k \le \bar{p}_i, &&\forall i\in{\mathcal{N}}, k\in{\mathcal{K}}, \label{P1_MILP_RV_constraint_5}\\
&s_i^k\in{\ensuremath{ \left\{ 0,1 \right\} }}, && \forall i\in{\mathcal{N}}, k\in{\mathcal{K}}, \label{P1_MILP_RV_constraint_6} \\
& \bm{z}\in{\mathbb{R}}^G.\end{aligned}$$ Here, constraint ensures $\bm{X}+{{B}^\mathsf{T}}\bm{z}\ge0$ so that $\Lambda^{\text{RV}_+}{\ensuremath{\left( \bm{X}{\ensuremath{\left( \omega^k \right)}}+{{B}^\mathsf{T}}\bm{z} \right)}}\neq-\infty$ for every $k\in{\mathcal{K}}$.
Proof of Proposition \[P1\_RV\_upperbound\] {#P1_RV_upperbound_proof}
-------------------------------------------
Let $(\bm{z},(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ be an optimal solution of the problem in . To get a contradiction, suppose that $z_\ell > {{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ \frac{1}{\alpha}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}$. Let $\bm{z'}\in{\mathbb{R}}^n$ be the vector such that $z'_\ell = {{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ \frac{1}{\alpha}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}$ and $z'_{\hat{\ell}} = z_{\hat{\ell}}$ for each $\hat{\ell}\in{\mathcal{G}}\setminus{\ensuremath{ \left\{ \ell \right\} }}$. Similar to the argument in the proof of Proposition \[P1\_EN\_upperbound\], it can be checked that $(\bm{z'},(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ is a feasible solution of the problem in . Hence, $z_\ell = {\mathcal{Z}}_1^{\text{RV}_+}{\ensuremath{\left( \bm{e}^\ell \right)}}\leq z'_\ell={{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ \frac{1}{\alpha}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}$. As this is a contradiction, the result follows.
Proof of Proposition \[P1\_RV\_boundedness\] {#P1_RV_boundedness_proof}
--------------------------------------------
To get a contradiction, suppose that the problem in has a feasible solution but ${\mathcal{Z}}_1^{\text{RV}_+}(\bm{e}^\ell)=-\infty$. Let $M={{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty} + \frac{1}{\alpha}{\ensuremath{\left( n+1 \right)}}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}$. Since ${\mathcal{Z}}_1^{\text{RV}_+}(\bm{e}^\ell)=-\infty$, there exist $\epsilon>0$ and $(\bm{z},(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$, where $\bm{z}\in{\mathbb{R}}^n$ and $(\bm{p}^k, \bm{s}^k)\in{\mathbb{R}}^n\times{\mathbb{Z}}^n$ for each $k\in{\mathcal{K}}$, such that ${{\bm{e}^\ell}^\mathsf{T}}\bm{z} = z_\ell = -M$ and $(\bm{z}-\epsilon\bm{e}^\ell,(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ is a feasible solution for the problem in . Fix $i\in{\mathcal{N}}$, $k\in{\mathcal{K}}$ such that ${\ensuremath{\left( {{B}^\mathsf{T}}\bm{z} \right)}}_i = z_\ell = -M$. Similar to the argument in the proof of Proposition \[P1\_EN\_boundedness\], it can be checked that constraint contradicts constraint . Hence, $(\bm{z}-\epsilon\bm{e}_\ell,(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ is infeasible, which is a contradiction to the assumption. Hence, ${\mathcal{Z}}_1^{\text{RV}_+}{\ensuremath{\left( \bm{e}_\ell \right)}}>-\infty$. In addition, the existence of a feasible solution implies that ${\mathcal{Z}}_1^{\text{RV}_+}{\ensuremath{\left( \bm{e}_\ell \right)}}<+\infty$. Hence, ${\mathcal{Z}}_1^{\text{RV}_+}{\ensuremath{\left( \bm{e}_\ell \right)}}\in{\mathbb{R}}$.
Proof of Proposition \[P1\_RV\_feasibility\] {#P1_RV_feasibility_proof}
--------------------------------------------
Assume that $\gamma \le {{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}}$. Let $\bm{z} = {\ensuremath{\left( {\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ \frac{1}{\alpha}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}\right)}}{\bm{\mathbbm{1}}}$, $\bm{p}^k = \bm{{\bar{p}}}$, $\bm{s}^k = {\bm{\mathbbm{1}}}$ for each $k\in{\mathcal{K}}$. As in the proof of Proposition \[P1\_EN\_feasibility\], it can be checked that $(\bm{z},(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ is a feasible solution for the problem in . Conversely, if $\gamma>{{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}}$, then constraint is infeasible, since $\sum_{k\in{\mathcal{K}}}q^k{{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{p}^k \le {{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}} < \gamma$ by constraint . Hence, the problem in is infeasible, which concludes the proof.
The mixed-integer linear programming problem of computing ${\mathcal{Z}}_2^\text{EN}{\ensuremath{\left( \bm{v} \right)}}$ in can be written more explicitly as $$\begin{aligned}
\text{minimize}\quad &\mu \label{P2_MILP_EN_explicit}\\
\text{subject to}\quad& \sum_{k\in{\mathcal{K}}}q^k{{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{p}^k \ge \gamma, \label{P2_MILP_EN_constraint_1}\\
& p_i^k \le \sum_{j=1}^{n}\pi_{ji}p_j^k + (X_i(\omega^k) + ({{B}^\mathsf{T}}\bm{v})_i+\mu) + M(1-s_i^k), &&\forall i\in{\mathcal{N}}, k\in{\mathcal{K}}, \label{P2_MILP_EN_constraint_2}\\
& p_i^k \le \bar{p}_is_i^k, &&\forall i\in{\mathcal{N}}, k\in{\mathcal{K}}, \label{P2_MILP_EN_constraint_3}\\
& \sum_{j=1}^{n}\pi_{ji}p_j^k + (X_i(\omega^k) + ({{B}^\mathsf{T}}\bm{v})_i+\mu)\le Ms_i^k, &&\forall i\in{\mathcal{N}}, k\in{\mathcal{K}}, \label{P2_MILP_EN_constraint_4}\\
&0 \le p_i^k \le \bar{p}_i, &&\forall i\in{\mathcal{N}}, k\in{\mathcal{K}}, \label{P2_MILP_EN_constraint_5}\\
&s_i^k\in{\ensuremath{ \left\{ 0,1 \right\} }}, && \forall i\in{\mathcal{N}}, k\in{\mathcal{K}}. \label{P2_MILP_EN_constraint_6}\end{aligned}$$
Proof of Proposition \[P2\_EN\_upperbound\] {#P2_EN_upperbound_proof}
-------------------------------------------
Let $(\mu,(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ be an optimal solution of the problem in . To get a contradiction, suppose that $\mu>{{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}$. We claim that $(\mu^{\max},(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ is a feasible solution of the problem in . Let $i\in{\mathcal{N}}$, $k\in{\mathcal{K}}$. Note that constraint holds as $$\begin{aligned}
p_i^k &\le \sum_{j=1}^{n}\pi_{ji}p_j^k + (X_i(\omega^k)+({{B}^\mathsf{T}}\bm{v})_i+\mu^{\max})+M(1-s_i^k)\\
& = \sum_{j=1}^{n}\pi_{ji}p_j^k + X_i(\omega^k)+({{B}^\mathsf{T}}\bm{v})_i+{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+{\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}+{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}+ M(1-s_i^k) \\
& = \sum_{j=1}^{n}\pi_{ji}p_j^k + (X_i(\omega^k)+{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}) + (({{B}^\mathsf{T}}\bm{v})_i +{\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}) +{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}+ M(1-s_i^k),
\end{aligned}$$ since $$\begin{aligned}
&\sum_{j=1}^{n}\pi_{ji}p_j^k \ge 0, \quad X_i(\omega^k) + {\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}\ge 0, \quad
({{B}^\mathsf{T}}\bm{v})_i + {\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}\ge 0,\\
&p_i^k \le {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}, \quad M(1-s_i^k)\ge0.
\end{aligned}$$ Constraint holds as $$\begin{aligned}
&\sum_{j=1}^{n}\pi_{ji}p_j^k + {\ensuremath{\left( X_i(\omega^k)+({{B}^\mathsf{T}}\bm{v})_i+{{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}\right)}}\\
& < \sum_{j=1}^{n}\pi_{ji}p_j^k + {\ensuremath{\left( X_i(\omega^k)+({{B}^\mathsf{T}}\bm{v})_i+\mu \right)}} \le Ms_i^k = M
\end{aligned}$$ by the assumption ${{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}< \mu$ and the feasibility of $(\mu,(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$. All the other constraints in hold by the feasibility of $(\mu,(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$, since they are free of ${{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}$. Hence, the claim follows, which yields $\mu={\mathcal{Z}}_2^\text{EN}{\ensuremath{\left( \bm{v} \right)}}\leq {{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}$. As this is a contradiction, we obtain the desired result.
Proof of Proposition \[P2\_EN\_boundedness\] {#P2_EN_boundedness_proof}
--------------------------------------------
To get a contradiction, suppose that the problem in has a feasible solution but ${\mathcal{Z}}_2^\text{EN}{\ensuremath{\left( \bm{v} \right)}}=-\infty$. Then, there exist $\epsilon>0$ and $(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}}$, where $(\bm{p}^k, \bm{s}^k)\in{\mathbb{R}}^n\times{\mathbb{Z}}^n$ for each $k\in{\mathcal{K}}$, such that $(-2M-\epsilon,(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ is a feasible solution of the problem in . Fix $i\in{\mathcal{N}}$, $k\in{\mathcal{K}}$. Then constraint violates constraint as $$\begin{aligned}
p_i^k &\le \sum_{j=1}^{n}\pi_{ji}p_j^k + {\ensuremath{\left( X_i(\omega^k)+({{B}^\mathsf{T}}\bm{v})_i-2M-\epsilon \right)}} + M(1-s_i^k) \\
&\le\sum_{j=1}^{n}\pi_{ji}p_j^k + X_i(\omega^k)+({{B}^\mathsf{T}}\bm{v})_i-\epsilon-M \\
&= \sum_{j=1}^{n}\pi_{ji}p_j^k + X_i(\omega^k)+({{B}^\mathsf{T}}\bm{v})_i - \epsilon - 2{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}-2{\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}-{\ensuremath{\left( n+1 \right)}}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}\\
&= {\ensuremath{\left( \sum_{j=1}^{n}\pi_{ji}p_j^k -{\ensuremath{\left( n+1 \right)}}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}\right)}}+{\ensuremath{\left( X_i(\omega^k)-2{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}\right)}} + {\ensuremath{\left( ({{B}^\mathsf{T}}\bm{v})_i-2{\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}\right)}} - \epsilon < 0,
\end{aligned}$$ since $$\sum_{j=1}^{n}\pi_{ji}p_j^k < {\ensuremath{\left( n+1 \right)}}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}, \quad X_i(\omega^k) \le 2{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}, \quad({{B}^\mathsf{T}}\bm{v})_i \le 2{\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}, \quad -\epsilon < 0.$$ Hence, $(-2M-\epsilon,(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ is infeasible, which is a contradiction to the assumption. Hence, ${\mathcal{Z}}_2^\text{EN}(\bm{v})>-\infty$. On the other hand, ${\mathcal{Z}}_2^\text{EN}(\bm{v})<+\infty$ by the existence of a feasible solution. So ${\mathcal{Z}}_2^\text{EN}(\bm{v})\in{\mathbb{R}}$.
Proof of Proposition \[P2\_EN\_feasibility\] {#P2_EN_feasibility_proof}
--------------------------------------------
Assume that $\gamma \le {{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}}$. Let $\mu = {{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}$, $\bm{p}^k = \bm{{\bar{p}}}$, $\bm{s}^k = {\bm{\mathbbm{1}}}$ for each $k\in{\mathcal{K}}$. We show that $(\mu,(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ is a feasible solution for the problem in . Since $\bm{p}^k = \bm{{\bar{p}}}$ for each $k\in{\mathcal{K}}$, it holds that $\sum_{k\in{\mathcal{K}}}q^k{{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{p}^k = {{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}} \ge \gamma$. Hence, constraint holds. Now fix $i\in{\mathcal{N}}$, $k\in{\mathcal{K}}$. Constraint holds as $$\begin{aligned}
& \sum_{j=1}^{n}\pi_{ji}p_j^k + {\ensuremath{\left( X_i(\omega^k) + ({{B}^\mathsf{T}}\bm{v})_i+\mu \right)}} + M(1-s_i^k) \\
& = \sum_{j=1}^{n}\pi_{ji}p_j^k + X_i(\omega^k) + ({{B}^\mathsf{T}}\bm{v})_i + \mu + M(1-1)\\
& = \sum_{j=1}^{n}\pi_{ji}p_j^k + X_i(\omega^k) + ({{B}^\mathsf{T}}\bm{v})_i + {\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+{\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}+{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}\ge \bar{p}_i = p_i^k,
\end{aligned}$$ since $$\sum_{j=1}^{n}\pi_{ji}p_j^k \ge 0, \quad X_i(\omega^k) + {\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}\ge 0, \quad ({{B}^\mathsf{T}}\bm{v})_i + {\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}\ge 0,$$ and $s_i^k = 1$, by the choice of $\bm{s}^k$. Constraint holds as $$\begin{aligned}
& \sum_{j=1}^{n}\pi_{ji}p_j^k + {\ensuremath{\left( X_i(\omega^k) + ({{B}^\mathsf{T}}\bm{v})_i+\mu \right)}} \\
& = \sum_{j=1}^{n}\pi_{ji}p_j^k + X_i(\omega^k) + ({{B}^\mathsf{T}}\bm{v})_i + {\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+{\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}+{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}\\
& \le {2{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+2{\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}+{\ensuremath{\left( n+1 \right)}}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}= M = Ms_i^k,
\end{aligned}$$ since $\sum_{j=1}^{n}\pi_{ji}p_j^k \le n{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}$. All the other constraints hold trivially by the choice of $\mu$, $\bm{p}^k$ and $\bm{s}^k$, for each $k\in{\mathcal{K}}$. Hence, $(\mu,(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ is a feasible solution for the problem in .
Conversely, if $\gamma>{{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}}$, then constraint is infeasible, since $\sum_{k\in{\mathcal{K}}}q^k{{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{p}^k \le {{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}} < \gamma$, by constraint . Hence, the problem in is infeasible, which finishes the proof.
The mixed-integer linear programming problem for calculating ${\mathcal{Z}}_2^{\text{RV}_+}{\ensuremath{\left( \bm{v} \right)}}$ in can be written more explicitly as $$\begin{aligned}
\text{minimize}\quad &\mu \label{P2_MILP_RV_explicit}\\
\text{subject to}\quad& \sum_{k\in{\mathcal{K}}}q^k{{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{p}^k \ge \gamma, \label{P2_MILP_RV_constraint_1}\\
& p_i^k \le \alpha {\ensuremath{\left( X_i(\omega^k) + ({{B}^\mathsf{T}}\bm{v})_i+\mu \right)}} + \beta\sum_{j=1}^{n}\pi_{ji}p_j^k + \bar{p}_is_i^k, && \forall i\in{\mathcal{N}}, k\in{\mathcal{K}}, \label{P2_MILP_RV_constraint_2}\\
& \bar{p}_is_i^k \le {\ensuremath{\left( X_i(\omega^k) + ({{B}^\mathsf{T}}\bm{v})_i+\mu \right)}} + \sum_{j=1}^{n}\pi_{ji}p_j^k, && \forall i\in{\mathcal{N}}, k\in{\mathcal{K}}, \label{P2_MILP_RV_constraint_3}\\
& X_i(\omega^k) + ({{B}^\mathsf{T}}\bm{v})_i +\mu \ge 0, && \forall i\in{\mathcal{N}}, k\in{\mathcal{K}}, \label{P2_MILP_RV_constraint_4} \\
&0 \le p_i^k \le \bar{p}_i, &&\forall i\in{\mathcal{N}}, k\in{\mathcal{K}}, \label{P2_MILP_RV_constraint_5}\\
&s_i^k\in{\ensuremath{ \left\{ 0,1 \right\} }}, && \forall i\in{\mathcal{N}}, k\in{\mathcal{K}}. \label{P2_MILP_RV_constraint_6}\end{aligned}$$ Here, constraint ensures $\bm{X}+{{B}^\mathsf{T}}\bm{v}+\mu{\bm{\mathbbm{1}}}\ge0$ so that $\Lambda^{\text{RV}_+}(\bm{X}(\omega^k)+{{B}^\mathsf{T}}\bm{v}+\mu{\bm{\mathbbm{1}}})\neq+\infty$ for every $k\in{\mathcal{K}}$.
Proof of Proposition \[P2\_RV\_upperbound\] {#P2_RV_upperbound_proof}
-------------------------------------------
Let $(\mu, (\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ be an optimal solution for the problem in . To get a contradiction, suppose that $\mu>{{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+{\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}+ \frac{1}{\alpha}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}$. Following similar arguments as in the proof of Proposition \[P2\_EN\_upperbound\], it can be shown that $({{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+{\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}+ \frac{1}{\alpha}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}},(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ is a feasible solution for the problem in . Hence, $\mu={\mathcal{Z}}_2^{\text{RV}_+}(\bm{v})\leq {{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+{\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}+ \frac{1}{\alpha}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}$, which is a contradiction. Hence, the result follows.
Proof of Proposition \[P2\_RV\_boundedness\] {#P2_RV_boundedness_proof}
--------------------------------------------
To get a contradiction, suppose that the problem in has a feasible solution but ${\mathcal{Z}}_2^{\text{RV}_+}(\bm{v})=-\infty$. Let $M = {\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+ {\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}+\frac{1}{\alpha}{\ensuremath{\left( n+1 \right)}}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}$. Then, there exist $\epsilon>0$ and $(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}}$, where ${\ensuremath{\left( \bm{p}^k, \bm{s}^k \right)}}\in{\mathbb{R}}^n\times{\mathbb{Z}}^n$ for each $k\in{\mathcal{K}}$, such that $(-M-\epsilon,(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ is a feasible solution for the problem in . Fix $i\in{\mathcal{N}}$, $k\in{\mathcal{K}}$. As in the proof of Corollary \[P2\_EN\_boundedness\], it can be checked that constraint violates constraint . Hence, $(-M-\epsilon,(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ is infeasible, which is a contradiction to the assumption. Hence, ${\mathcal{Z}}_2^{\text{RV}_+}(\bm{v})>-\infty$. Together with the feasibility of the problem, it follows that ${\mathcal{Z}}_2^{\text{RV}_+}(\bm{v})\in{\mathbb{R}}$.
Proof of Proposition \[P2\_RV\_feasibility\] {#P2_RV_feasibility_proof}
--------------------------------------------
Assume that $\gamma \le {{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}}$. Let $\mu = {{\ensuremath{ \left\Vert \bm{X} \right\Vert }_\infty}+{\ensuremath{ \left\Vert \bm{v} \right\Vert }_\infty}+ \frac{1}{\alpha}{\ensuremath{ \left\Vert \bm{{\bar{p}}} \right\Vert }_\infty}}$, $\bm{p}^k = \bm{{\bar{p}}}$, $\bm{s}^k = {\bm{\mathbbm{1}}}$ for each $k\in{\mathcal{K}}$. As in the proof of Corollary \[P2\_RV\_feasibility\], it can be shown that $(\mu,(\bm{p}^k, \bm{s}^k)_{k\in{\mathcal{K}}})$ is a feasible solution for the problem in . Conversely, if $\gamma>{{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}}$, then constraint is infeasible, since $\sum_{k\in{\mathcal{K}}}q^k{{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{p}^k \le {{{\bm{\mathbbm{1}}}}^\mathsf{T}}\bm{{\bar{p}}} < \gamma$, by constraint . Hence, the problem in is infeasible, which concludes the proof.
[^1]: Bilkent University, Department of Industrial Engineering, Ankara, Turkey, cararat@bilkent.edu.tr.
[^2]: Bilkent University, Department of Industrial Engineering, Ankara, Turkey, nurtai@bilkent.edu.tr.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The question raised by \[Bastin and Martin 2003 J. Phys. B: At. Mol. Opt. Phys. [**36**]{}, 4201\] is examined and used to explain in more detail a key point of our calculations. They have sought to rebut criticisms raised by us of certain techniques used in the calculation of the off-resonance case. It is also explained why this result is not a problem for the off-resonance case, but, on the contrary, opens the door to a general situation. Their comment is based on a blatant misunderstanding of our proposal an d as such is simply wrong.'
address: |
$^1$Mathematics Department, Faculty of Science, South Valley University, 82524 Sohag, Egypt.\
$^2$Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City, Cairo, Egypt.
author:
- 'Mahmoud Abdel-Aty$^{1,}$, A.-S. F. Obada$^2$'
title: '**Reply to Comment on “Quantum inversion of cold atoms in a microcavity: spatial dependence”** '
---
[**Submitted to:**]{}
In our paper \[2\] we have used the Hamiltonian (in the mesa mode case)
$$\hat{H}=\frac{p_{z}^{2}}{2M}+\frac{\Delta }{2}\sigma_{z}+\omega (a^{\dagger
}a+\frac{1}{2}\sigma_{z})+\lambda f(z)\{\sigma ^{-}a^{\dagger }+a\sigma
^{+}\}. \label{1}$$
Let us write equation (1) in the following form (in the mesa mode case $%
f(z)=1)$ $$\begin{aligned}
\hat{H} &=&\frac{p_{z}^{2}}{2M}+\hat{V}
\nonumber
\\
\hat{V} &=&\frac{\Delta }{2}\sigma_{z}+\omega (a^{\dagger }a+\frac{1}{2}%
\sigma_{z})+\lambda \{\sigma ^{-}a^{\dagger }+a\sigma ^{+}\}.\end{aligned}$$ It is easy to show that in the $2\times 2$ atomic-photon space the eigenvalues and eigenfunction of the interaction Hamiltonian $\hat{V}$ ($%
\hat{V}|\Phi_{n}^{\pm }\rangle =E_{n}^{\pm }|\Phi_{n}^{\pm }\rangle )$ are given by \[3\], $$E_{n}^{\pm }=\omega (n+\frac{1}{2})\pm \sqrt{\frac{\Delta ^{2}}{4}+\lambda
^{2}(n+1)},$$ $$\begin{aligned}
|\Phi_{n}^{+}\rangle &=&\cos \theta_{n}|n+1,g\rangle +\sin \theta
_{n}|n,e\rangle,
\nonumber
\\
|\Phi_{n}^{-}\rangle &=&-\sin \theta_{n}|n+1,g\rangle +\cos \theta
_{n}|n,e\rangle,\end{aligned}$$ where $$\theta_{n}=\tan ^{-1}\left( \frac{\lambda \sqrt{n+1}}{\sqrt{\frac{\Delta
^{2}}{4}+\lambda ^{2}(n+1)}-\frac{\Delta }{2}}\right),\qquad$$ We write the wave function $|\Psi (z,t)\rangle =\sum\limits_{n}C_{n}^{\pm
}(z,t)|\Phi_{n}^{\pm }\rangle .$ Then using the total Hamiltonian (1) we have $$\begin{aligned}
\hat{H}|\Psi (z,t)\rangle &=&\sum\limits_{n}\left( \frac{p^{2}}{2M}%
+V\right) C_{n}^{\pm }(z,t)|\Phi_{n}^{\pm }\rangle
\nonumber
\\
&=&\sum\limits_{n}\frac{p^{2}}{2M}C_{n}^{\pm }(z,t)|\Phi_{n}^{\pm }\rangle
+\sum\limits_{n}VC_{n}^{\pm }(z,t)|\Phi_{n}^{\pm }\rangle
\nonumber
\\
&=&\sum\limits_{n}\frac{p^{2}}{2M}C_{n}^{\pm }(z,t)|\Phi_{n}^{\pm }\rangle
+\sum\limits_{n}C_{n}^{\pm }(z,t)E_{n}^{\pm }|\Phi_{n}^{\pm }\rangle
\nonumber
\\
&=&\sum\limits_{n}\left( \frac{p^{2}}{2M}C_{n}^{\pm }(z,t)+E_{n}^{\pm
}C_{n}^{\pm }(z,t)\right) |\Phi_{n}^{\pm }\rangle ,\end{aligned}$$ because of the orthonormality of the wavefunctions $|\Phi_{n}^{\pm }\rangle $ then $$i\frac{\partial }{\partial t}C_{n}^{\pm }=\left( -\frac{1}{2M}\frac{\partial
^{2}}{\partial z^{2}}+E_{n}^{\pm }\right) C_{n}^{\pm },$$ with no coupling even in the presence of the detuning (equation (13) in BM comment \[1\]).
It may be worthwhile for the authors to consult some papers that have been published previously (see for example Refs. \[4-6\]) where the detuning has been considered and similar results have been obtained. To be more precisely:
- We have used the interaction picture, so that the term $(n+\frac{1%
}{2})\omega $ does not appear, it can be used as a phase only. Bearing in mind the case of mesa mode is being treated in our paper i.e. $f(z)=1$.
- The most serious point is that Bastin and Martin have overlooked the formulae for $$\cos 2\theta_{n}\qquad and\qquad \sin 2\theta_{n}.$$ From equation (10) raised in Bastin and Martin comment, it is easy to write $$\tan \theta_{n}=\frac{\lambda \sqrt{n+1}}{\sqrt{\frac{\Delta ^{2}}{4}%
+\lambda ^{2}(n+1)}-\frac{\Delta }{2}}=\frac{\sqrt{\frac{\Delta ^{2}}{4}%
+\lambda ^{2}(n+1)}+\frac{\Delta }{2}}{\lambda \sqrt{n+1}}$$ then $$\tan 2\theta_{n}=\frac{\lambda \sqrt{n+1}}{-\frac{\Delta }{2}}$$ Also, it is easy to prove that $$\cos 2\theta_{n}=\frac{-\Delta /2}{\sqrt{\frac{\Delta ^{2}}{4}+\lambda
^{2}(n+1)}}\qquad and\qquad \sin 2\theta_{n}=\frac{\lambda \sqrt{n+1}}{%
\sqrt{\frac{\Delta ^{2}}{4}+\lambda ^{2}(n+1)}}.$$ Once these formulae inserted in equations (18) and (19) of Bastin and Martin comment, we find that, the second terms vanish identically.
Now let us look more carefully at the general case when we take f(z) no longer a constant, i.e. we go beyond the mesa mode case. In this case the orthonormal functions $|\Phi_{n}^{\pm }\rangle $ in the $2\times 2$ system diagonalize the Hamiltonian $V$ and its elements are diagonal in this set of functions with $$\begin{aligned}
V_{n}^{\pm } &=&(n+\frac{1}{2})\omega \pm \sqrt{\frac{\Delta ^{2}}{4}%
+\lambda ^{2}f^{2}(z)(n+1)} \nonumber \\
\tan 2\theta_{n} &=&\frac{\lambda f(z)\sqrt{n+1}}{-\frac{\Delta }{2}}.\end{aligned}$$ The states $|\Phi_{n}^{\pm }\rangle $ are z-dependent through the trigonometric functions, they satisfy $$\begin{aligned}
\frac{\partial }{\partial z}|\Phi_{n}^{\pm }\rangle &=&\pm |\Phi_{n}^{\pm
}\rangle \frac{d\theta }{dz}, \nonumber \\
\frac{\partial ^{2}}{\partial z^{2}}|\Phi_{n}^{\pm }\rangle &=&\pm |\Phi
_{n}^{\pm }\rangle \frac{d^{2}\theta }{dz^{2}}-|\Phi_{n}^{\pm }\rangle
\left( \frac{d\theta }{dz}\right) ^{2}.\end{aligned}$$ Then $|\Psi (z,t)\rangle $ can be expanded in the form $|\Psi (z,t)\rangle
=\sum\limits_{n}C_{n}^{\pm }(z,t)|\Phi_{n}^{\pm }\rangle $ and it satisfies the Schrodinger equation $$i\frac{\partial }{\partial z}|\Psi (z,t)\rangle =H|\Psi (z,t)\rangle .$$
Hence the coefficients $C_{n}^{\pm }(z,t)$ satisfy the coupled equations $$\begin{aligned}
i\frac{\partial C_{n}^{+}}{\partial z} &=&\left( -\frac{1}{2M}\frac{\partial
^{2}}{\partial z^{2}}+V_{n}^{+}-\left( \frac{d\theta }{dz}\right)
^{2}\right) C_{n}^{+}-\left( 2\frac{\partial C_{n}^{-}}{\partial z}\left(
\frac{d\theta }{dz}\right) +C_{n}^{-}\left( \frac{d\theta }{dz}\right)
^{2}\right) ,
\nonumber
\\
i\frac{\partial C_{n}^{-}}{\partial z} &=&-\left( -\frac{1}{2M}\frac{%
\partial ^{2}}{\partial z^{2}}+V_{n}^{-}-\left( \frac{d\theta }{dz}\right)
^{2}\right) C_{n}^{-}+\left( 2\frac{\partial C_{n}^{+}}{\partial z}\left(
\frac{d\theta }{dz}\right) +C_{n}^{+}\left( \frac{d\theta }{dz}\right)
^{2}\right) ,\end{aligned}$$ These equations should replace equations (18) and (19) of the comment of \[1\]. But once $f(z)$ is taken to be constant, then $\frac{d\theta }{dz}$ will vanish and we get equations (7) and the results of \[2,3\].
[9]{} Bastin T and Martin J 2003 J. Phys. B: At. Mol. Opt. Phys. [**36**]{}, 4201.
Abdel-Aty M and Obada A-S F 2002 J. Phys. B: At. Mol. Opt. Phys. **35** 807.
Abdel-Aty M and Obada A-S F 2002 Modern Physics Letters B **16** 117.
Battocletti M and Englert B-G, 1994 J. Phys. II France **4** 1939.
Zhang, Z-M, et al 2000 J.Phys.B: At. Mol. Opt. Phys. **33** 2125.
Zhang, Z-M, He L-S 1998 Opt. Commun. **157** 77.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The orbital boundary value problem, also known as Lambert Problem, is revisited. Building upon Lancaster and Blanchard approach, new relations are revealed and a new variable representing all problem classes, under L-similarity, is used to express the time of flight equation. In the new variable, the time of flight curves have two oblique asymptotes and they mostly appear to be conveniently approximated by piecewise continuous lines. We use and invert such a simple approximation to provide an efficient initial guess to an Householder iterative method that is then able to converge, for the single revolution case, in only two iterations. The resulting algorithm is compared, for single and multiple revolutions, to Gooding’s procedure revealing to be numerically as accurate, while having a significantly smaller computational complexity.'
author:
- Dario Izzo
bibliography:
- 'main.bib'
date: 'Received: date / Accepted: date'
title: 'Revisiting Lambert’s Problem'
---
Introduction {#intro}
============
Lambert’s problem, sometimes referred to as orbital boundary value problem, is a fascinating problem in astrodynamics that intrigued, over the years, most famous mathematicians. Just like Kepler’s equation, its solution is at the very heart of fundamental astrodynamical and space engineering questions [@celmech1; @celmech2; @izzolambert]. Following the fundamental work laid down, among others, by Euler, Lambert, Lagrange and Gauss, the need of having one robust algoritmic procedure able to function for a wide set of conditions led to revisit the Lambert’s problem during the space era. Among the many contributions made during that period, the work of Lancaster and Blanchard [@lancaster] is to be highlighted as it reduced the solution to Lambert’s problem to performing iterations each one requiring the computation of one only inverse trigonometric or hyperbolic function. Later, Gooding [@gooding] built upon these results and published a procedure achieving high precision in only three iterations for all geometries. Gooding’s algorithm makes use of Halley’s iterations sided to well designed heuristics to set the initial guess of the iterated variable. His methodology to reconstruct the terminal velocity vectors is also remarkable as it is purely algebraic. The resulting procedure is extremely efficient having low computational cost and high accuracy. A number of studies [@peterson], [@klumpp] and [@parrish] have tested Gooding approach extensively, suggesting its superiority with respect to other Lambert solvers. His procedure is most accurate and considered as the fastest existing approach to solve Lambert’s problem [@arora]. Aside from Gooding’s algorithm, many other proposal have been put forward to design Lambert solvers, they all differ in the details of at least one of three fundamental ingredients: a) the iteration variable (directly connected to the time of flight equation), b) the iteration algorithm c) the initial guess and d) the reconstruction of the terminal velocity vectors. More recently iimprovements on the original Gooding algorithm were also claimed [@arora] making use of the universal variable formulation [@bate] and an original cosine transformation. At the same time, a number of works recently addressed the possibility of deploying a large number of Lambert’s algorithms on modern GPU architectures [@parrish], [@arora2] and [@wie]. Interestingly, in the first of these works, a comparison is also made between Gooding procedure, a universal variable Lambert’s solver and an early (slow) version of the algorithm here described (unpublished at that time) showing already its promising nature.
In this paper, we build upon Lancaster and Blanchard work, first deriving some new results, and then proposing and testing a new algorithm. The new algorithm a) iterates on the Lancaster-Blanchard variable $x$ using b) a Householder iteration scheme c) feeded by a simple initial guess found exploiting new analytical results found. The resulting procedure is simple to implement, does not make use of heuristics for the initial guess generation and is able to converge, on average, in only 2 iterations for the single revolution case and 3 in the multirevolution case, introducing a significant reduction in the overall solver complexity.
Background {#sec:1}
==========
From Lambert to Gauss {#sec:2}
---------------------
Lambert’s theorem states that the time of flight $t$ to travel along a keplerian orbit from $\mathbf r_1$ to $\mathbf r_2$ is a function of the orbit semi-major axis $a$, of the sum $r_1+r_2$ and of the chord $c$ of the triangle having $\mathbf r_1$ and $\mathbf r_2$ as sides. The complete formal proof was first delivered by Lagrange and is here sketched briefly in the form reported by Battin [@battin] as some of the quantities and equations involved will prove to be useful in our later developments. We start introducing the eccentric anomaly $E$ and the hyperbolic anomaly $H$ via the corresponding Sundmann transformations $rdE = ndt$ $rdH = Ndt$. The mean motion $n = \sqrt{\mu / a^3}$ and its hyperbolic equivalent $N = \sqrt{-\mu / a^3}$ are also introduced. As we do not make use of universal variables we will be forced to give all our arguments twice: one for the elliptic case $a>0$ and one for the hyperbolic case $a<0$. To this purpose some of the equations will be split in two lines, in which case the line above holds for the elliptic case and the line below holds in the hyperbolic case. We also make use of the reduced eccentric anomaly $E_r \in [0,2\pi]$ so that when $\tilde M$ full revolutions are made $E = E_r + 2\tilde M\pi$. To ease the notation, in the following, we will drop the subscript $r$ so that $E$ will be the reduced eccentric anomaly. The following relations are then valid for an elliptic orbit ($a > 0$): $$\label{eq:ellipse}
\begin{array}{c}
r = a(1-e \cos E) \\
nt = E - e \sin E +2\tilde M\pi\\
r \cos f = a (\cos E - e) \\
r \sin f = a \sqrt{(1-e^2)} \sin E
\end{array}$$ The first one relates the orbital radius $r$ to the eccentric anomaly $E$, the second one is the famous Kepler’s equation relating the eccentric anomaly to the time of flight and the following two relation define the relations between true anomaly $f$ and eccentric anomaly $E$. Similar equations hold in the case of hyperbolic motion: $$\label{eq:hyperbola}
\begin{array}{c}
r = a(1-e \cosh H) \\
Nt = e \sinh H - H\\
r \cos f = a (\cosh H - e) \\
r \sin f = -a \sqrt{(e^2-1)} \sinh H
\end{array}$$ The above equations are valid along a Keplerian orbit, including $\mathbf r_1$ and $\mathbf r_2$. The time of flight can thus be written as: $$\begin{array}{c}
\sqrt{\mu}(t_2-t_1)= \left\{ \begin{array}{l} a^{3/2} \left(E_2-E_1 + e\cos E_1 - e\cos E_2+2M\pi\right) \\ -a^{3/2} \left(e\cosh H_2 - e\cosh H_1 - (H_2 - H_1)\right) \end{array} \right.
\end{array}$$ where $M = \tilde M_2 - \tilde M_1$ is the number of complete revolutions made during the transfer from $r_1$ to $r_2$. We may then define two new quantities such that: $$\begin{array}{c}
\psi = \left\{ \begin{array}{l} \frac{E_2-E_1}2 \\ \frac{H_2-H_1}2 \end{array} \right.,
\cos\varphi = \left\{ \begin{array}{l} e\cos\frac{E_2+E_1}2 \\ e\cosh\frac{H_2+H_1}2 \end{array} \right.
\end{array}$$ so that, by construction, in both the elliptic and hyperbolic motion case $\psi\in[0,\pi]$. We also restrict $\varphi\in[0,\pi]$ (elliptc case) and $\varphi\ge0$ (hyperbolic case) as to avoid ambiguity in the definition of the new angle. The time of flight equation is then written as: $$\label{eq:tof_psi}
\sqrt{\mu}(t_2-t_1)= \left\{ \begin{array}{l} 2 a^{3/2} \left(\psi - \cos\varphi\sin\psi + M\pi)\right. \\ -2a^{3/2} \left(\cosh\varphi\sinh\psi - \psi\right) \end{array} \right.$$ The two new quantities introduced, $\varphi$ and $\psi$ only depend on the problem geometry and the sami-major axis $a$ as can be easily found by computing $c^2 = (r_2\cos f_2 - r_1\cos f_1)^2 - (r_2\sin f_2 - r_1\sin f_1)^2$ and $r_1 + r_2$ from Eq.(\[eq:ellipse\]) and Eq.(\[eq:hyperbola\]), holding: $$\label{eq:g1}
r_1+r_2 = \left\{ \begin{array}{l} 2a(1-\cos\psi\cos\varphi) \\ 2a(1-\cosh\psi\cosh\varphi) \end{array} \right.$$ $$\label{eq:g2}
c = \left\{ \begin{array}{l} 2a\sin\psi\sin\varphi \\ -2a\sinh\psi\sinh\varphi \end{array} \right.$$ Thus, one can conclude that the time of flight, given in Eq.(\[eq:tof\_psi\]), is a function of $a$, $c$ and $r_1+r_2$. To further investigate the functional relation of the time of flight to these quantities it is convenient to introduce two new angles: $$\alpha = \varphi+\psi, \beta = \varphi-\psi$$ Clearly, in the elliptic case $\alpha\in[0.2\pi]$ and $\beta\in[-\pi,\pi]$ while for the hyperbolic case $\alpha\ge0$ and $\beta\ge-\pi$. These bounds are very important, as we shall see, in solving a quadrant ambiguity of the newly defined quantities. The time of flight equation now takes the elegant form: $$\label{eq:tof_alpha}
\sqrt{\mu}(t_2-t_1)= \left\{ \begin{array}{l} a^{3/2} \left(\left(\alpha-\sin\alpha)-(\beta-\sin\beta\right) + 2M\pi\right) \\ -2a^{3/2} \left(\left(\sinh\alpha-\alpha\right)-\left(\sinh\beta-\beta\right)\right) \end{array} \right.$$ and computing $r_1+r_2 \pm c$ from Eq.(\[eq:g1\]) and Eq.(\[eq:g2\]) one easily finds: $$\label{eq:g5}
\frac s{2a} = \left\{ \begin{array}{l} \sin^2\frac\alpha 2 \\ -\sinh^2\frac\alpha 2 \end{array} \right.$$ $$\label{eq:g6}
\frac {s-c}{2a} = \left\{ \begin{array}{l} \sin^2\frac\beta 2 \\ -\sinh^2\frac\beta 2 \end{array} \right.$$ These last three equations were first derived by Lagrange and used in his proof of the Lambert’s theorem. The angles $\alpha$ and $\beta$ cannot be determined univoquely from the equations above as their quadrant is not defined. We thus appear to have two possible solutions for $\alpha$ and $\beta$. The quadrant of $\beta$ can actually be resolved by expanding $\cos \theta /2 = \cos (f_2-f_1)/2$ using trigonometric identities and eventaully showing that the following holds: $$\label{eq:cost2}
\sqrt{r_1r_2}\cos\frac\theta 2 =
\left\{
\begin{array}{l}
2a\sin\frac\alpha 2\sin \frac\beta 2 \\
-2a\sinh\frac\alpha 2\sinh \frac\beta 2
\end{array}
\right.$$ since $\sin\frac\alpha 2,\sinh\frac\alpha 2 \ge 0$ the above equations dictate that $\sin\frac \beta 2, \sinh\frac \beta 2$ have the same sign as $\cos\frac\theta 2$, thus $\beta \in [-\pi,0]$ when $\theta \ge \pi$ and $\beta > 0$ when $\theta \in[0,\pi]$. The ambiguity on the $\alpha$ angle, instead, cannot be resolved as it derives from the fact that exactly two different ellipses, having the same semi-major axis $a$, link $\mathbf r_1$ and $\mathbf r_2$ and thus two different time of flights exist that satisfy Eq.(\[eq:tof\_alpha\]). From Eq.(\[eq:g5\]) and Eq.(\[eq:g6\]) one can also derive the useful relation: $$\label{eq:g7}
\sin\frac\alpha 2 = \lambda\sin\frac\beta 2$$
The Lambert’s problem revival
-----------------------------
During the 18th-19th century, the work on the orbital boundary value problem culminated with Gauss masterpiece Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium [@gauss] where the prince of mathematicians conceives was is probably the first procedure able to accurately solve the Lambert’s problem (see Battin [@battin] for an excellent account of Gauss method). In the following years science drifted slowly away this topic, only to revisit it in the second half of the 20th century when the orbital boundary value problem received more attention in the context of Moon exploration. Hence, the work of Lancaster, Blanchard, Battin, Bate and many others introduced several advances on the topic. We here follow the approach from Lancaster and Blanchard that inspred most of our developments and we will thus rederive some of their relations which are needed to explain our new ideas. Consider the parameter $\lambda$ defined as: $$s \lambda = \sqrt{r_1r_2}\cos\frac\theta 2$$ using Eq.(\[eq:cost2\]) and substituting the expressions in Eq.(\[eq:g5\]) and Eq.(\[eq:g6\]) it is simple to show that: $$\lambda^2 = \frac{s-c} s$$ The parameter $\lambda \in [-1,1]$ is positive when $\theta \in[0,\pi]$ and negative when $\theta\in[\pi,2\pi]$. Values of $\lambda^2$ close to unity indicate a chord of zero length, a case which is indeed extremely interesting in interplanetary trajectory design as it is linked to the design of resonant transfers. We also introduce a non dimensional time-of-flight defined as: $$T = \frac 12 \sqrt{\frac \mu{a_m^3}} (t_2-t_1) = \sqrt{2\frac \mu{s^3}} (t_2-t_1)$$ where $a_m = s/2$ is the minimum energy ellipse semi-major axis [@battin]. The advantage of using $\lambda$ and $T$ derives from the fact that $T$ is a function of $a / a_m$ and $\lambda$ alone, which allows to greatly simplify the taxonomy of possible Lambert’s problems. In Gooding’s words [@gooding], all the triangles having equal $c / s$ ratio form a large equivalence class and can be described as L-similar. For them, all Lambert solutions are the same in terms of $a / a_m$ and $T$.
![Non-dimensional time of flight curve for $\lambda=-0.9$ parametrized using $a/a_m$ . \[fig:tof\_a\]](tof_a.png){width="75.00000%"}
If we now plot the time of flight given by Eq.(\[eq:tof\_alpha\]) as a function of the ratio between the semi-major axis and the minimum energy ellipse semi-major axis, for a particular value of $\lambda$ and for single and multiple revolution cases, we get Figure \[fig:tof\_a\]. It is evident how, in order to invert the time of flight relation iterating over $a / a_m$, while possible, is not a good choice. To avoid these problems we follow Lancaster and Blanchard in some further derivations introducing the new quantities: $$\label{eq:def_x}
\begin{array}{cc}
x = \left\{ \begin{array}{l} \cos\frac\alpha 2 \\ \cosh\frac\alpha 2 \end{array}\right., &
y = \left\{ \begin{array}{l} \cos\frac\beta 2 \\ \cosh\frac\beta 2 \end{array}\right.
\end{array}$$ which imply: $$\label{eq:def_x}
\begin{array}{cc}
\left\{ \begin{array}{l} \sqrt{1-x^2} = \sin\frac\alpha 2 \\ \sqrt{x^2-1} = \sinh\frac\alpha 2 \end{array}\right., &
\left\{ \begin{array}{l} \lambda \sqrt{1-x^2} = \sin\frac\beta 2 \\ \lambda \sqrt{x^2-1} = \sinh\frac\beta 2 \end{array}\right.
\end{array}$$ and $y = \sqrt{1-\lambda^2(1-x^2)}$. Using these relations it is possible to relate the auxiliary angles $\varphi$ and $\psi$ directly to $x$: $$\label{eq:g8}
\begin{array}{l}
\cos\varphi = xy - \lambda(1-x^2) \\
\cosh\varphi = xy + \lambda(x^2-1)
\end{array}
, \hskip1cm
\begin{array}{l}
\sin\varphi = (y+x\lambda)\sqrt{1-x^2} \\
\sinh\varphi = (y+x\lambda)\sqrt{x^2-1}
\end{array}$$ and, $$\label{eq:g9}
\begin{array}{l}
\cos\psi = xy + \lambda(1-x^2) \\
\cosh\psi = xy - \lambda(x^2-1)
\end{array}
, \hskip1cm
\begin{array}{l}
\sin\psi = (y - x\lambda)\sqrt{1-x^2} \\
\sinh\psi = (y - x\lambda)\sqrt{x^2-1}
\end{array}$$ which allows to derive the relations $\cos\varphi\sin\psi = (x-\lambda y)\sqrt{1-x^2}$, $\cosh\varphi\sinh\psi = (x-\lambda y)\sqrt{x^2-1}$ and thus have the following time of flight equation valid in all cases: $$\label{eq:tof_x}
T = \frac{1}{1-x^2}\left( \frac{\psi+M\pi}{\sqrt{|1-x^2|}} - x + \lambda y\right)$$ where we must set $M=0$ in the case of hyperbolic motion where unbounded motion prevents complete revolutions to happen. The auxiliary angle $\psi$ is computed using Eq.(\[eq:g9\]) by the appropriate inverse function and thus, the time of flight evaluation is reduced to one only inverse function computation. Given the bounds on $\alpha$, from the definition of $x$, we can see how $x \in [-1,\infty]$. Also, $x>1$ implies hyperbolic motion, while $x<1$ elliptic motion. Since $1-x^2 = \sin^2\frac\alpha 2 = \frac s{2a} = \frac{a_m}a$, we see how $x=0$ corresponds to the minimum energy ellipse. Note that different Lambert’s problems having identical $\lambda$ values (i.e. same $c / s$), result in the same $x$, we then say that $x$ is a Lambert invariant parameter.
Computing Eq.(\[eq:tof\_x\]) in $x=0$ we get: $$\label{eq:t0}
T(x=0)=T_{0M} = \arccos\lambda + \lambda\sqrt{1-\lambda^2} + M\pi = T_{00} + M\pi$$ where we have introduced $T_0$ as the value of $T$ in $x=0$ and $T_{00}$ as the value in the single revolution case $M=0$.
When computing Eq.(\[eq:tof\_x\]) in the single revolution case, a loss of precision is encountered due to numerical cancellation for values of $x\approx 1$ where both $1-x^2$ and $\psi$ tend to zero. In these cases we compute the time of flight equation by series expansion using the elegant result from Battin [@battin] setting: $$\label{eq:tof_x_series}
\begin{array}{l}
\eta = y - \lambda x \\
S_1 = \frac 12 (1-\lambda - x\eta) \\
Q = \frac 43 {}_1F_2(3,1,\frac 52,S_1) \\
2T = \eta^3Q + 4\lambda\eta
\end{array}$$ where ${}_1F_2(a,b,c,d)$ is the Gaussian or ordinary hypergeometric function. This can be evaluated by direct computation of the associated hypergeometric series. Noting that $S_1\to 0$ when $x\to 1$ the number of terms to retain in the series is small whenever the series is used in the neighbourhood of $x = 1$. Departing from Battin, we study the parabolic case substituting $x=1, y=1$ into Eq.(\[eq:tof\_x\_series\]) and thus obtaining the following remarkable expression: $$\label{eq:t1}
T(x=1) = T_1 = \frac{2}{3}(1-\lambda^3)$$ relating the geometry of the triangle created by two different observations of an object on a parabolic keplerian orbit to the non-dimensional time elapsed between them. It is also possible to derive the following formulas for the time of flight derivatives: $$\label{eq:derivatives}
\begin{array}{l}
(1-x^2)\frac {dT}{dx} = 3Tx-2+2\lambda^3\frac xy\\
(1-x^2)\frac {d^2T}{dx^2} = 3T + 5x\frac{dT}{dx}+2(1-\lambda^2)\frac{\lambda^3}{y^3}\\
(1-x^2)\frac{d^3T}{dx^3} = 7x\frac{d^2T}{dx^2} + 8 \frac{dT}{dx} - 6(1-\lambda^2)\lambda^5\frac x{y^5}
\end{array}$$ which are valid in all cases (single and multiple revolutions, elliptic and hyperbolic) except in $\lambda^2=1, x=0$ and $x=1,\forall \lambda$. We then apply de l’Hôpital rule to the first of the above equations, and using the expression derived for $T_1$ we are also able to find the value of the derivative of the time of flight curves in the case of a parabola: $$\label{eq:dt1dx}
\left. \frac{dT}{dx}\right|_{x=1} = \frac 25 (\lambda^5-1)$$ which is valid for $M=0$. By direct substitution, one can also easily show: $$\left. \frac{dT}{dx}\right|_{x=0} = -2$$ A great advantage of the time of flight equation in the form of Eq.(\[eq:tof\_x\]) as derived by Lancaster and Blanchard [@lancaster] is in the low computational cost of computing $T$ and its derivatives, up to the third order. Only one trigonometric (or hyperbolic) function inversion, two square roots and a few multiplications, divisions and sums are indeed necessary to compute these numerical values. Other approaches based on geomerical considerations or on a universal variables formulation are, at best, only able to match such a simple representation. We now summarize all the information relative to all possible Lambert problems in one single graph as done in Figure \[fig:tof\_x\].
![Time of flight curves parametrized using $x$ for different $\lambda$ and $M$ values. \[fig:tof\_x\]](tof_x.png){width="75.00000%"}
A new Lambert invariant variable
================================
Let us consider the following new variables: $$\label{eq:new_variables}
\xi = \left\{
\begin{array}{ll}
\log(1+x), &M=0 \\
\log(\frac{1+x}{1-x}), &M>0
\end{array} \right., \hskip1cm
\tau = \log(T)$$ The domain of the time-of flight curve is now extended to $[-\infty,\infty]$. In the case of $M=0$ the co-domain is also extended similarly. Let us study the resulting time of flight equation $\tau(\xi,\lambda,M)$. In Figure \[fig:tof\_xi\] we plot $\tau$ against $\xi$ for $M=0$ and $M=1$ and thirty equally spaced values of $\lambda\in[-0.9,0.9]$. In the case $M=0$ the curves appear to have two asymptotes having negative inclination coefficient. For the multiple revolution case ($M>0$) the curves have two symmetric asymptotes. The new introduced parameter $\xi$ is Lambert invariant according to Gooding’s definition [@gooding] as it essentially is a transformation of the Lambert invariant variable $x$. We study the differential properties of the new curves, we have:
$$\label{eq:new_variables_differentials}
d\xi = \left\{
\begin{array}{ll}
\frac{1}{1+x}dx, &M=0 \\
\frac{2}{1-x^2}dx, &M>0
\end{array} \right., \hskip1cm
d\tau = \frac 1T dT$$
Substituting these relations into Eq.(\[eq:derivatives\]), after some manipulations we may derive the following hybrid expressions for the derivatives in the case $M=0$: $$\label{eq:derivatives_tau}
\begin{array}{l}
\frac {d\tau}{d\xi} = \frac{1+x}T\frac{dT}{dx}\\
\frac {d^2\tau}{d\xi^2} = \frac{(x+1)^2}{T}\frac{d^2T}{dx^2}+\frac{d\tau}{d\xi}-\left(\frac{d\tau}{d\xi}\right)^2\\
\frac{d^3\tau}{d\xi^3} = \frac{(1+x)^3}{T}\frac{d^3T}{dx^3} + \left(\frac{d^2\tau}{d\xi^2}-\frac{d\tau}{d\xi}+\left(\frac{d\tau}{d\xi}\right)^2\right)\left(2-\frac{d\tau}{d\xi}\right)+\frac{d^2\tau}{d\xi^2} - 2\frac{d\tau}{d\xi}\frac{d^2\tau}{d\xi^2}
\end{array}$$
![Time of flight curves ($\tau$) parametrized using $\xi$ for 30 $\lambda$ values equally spaced in $[-0.9,0.9]$. \[fig:tof\_xi\]](tof_xi_M0.png "fig:"){width="49.00000%"} ![Time of flight curves ($\tau$) parametrized using $\xi$ for 30 $\lambda$ values equally spaced in $[-0.9,0.9]$. \[fig:tof\_xi\]](tof_xi_M1.png "fig:"){width="49.00000%"}
Note that these expressions can be computed, sequentially, after Eq.(\[eq:derivatives\]). The following holds for the $M=0$ case: $$\label{eq:asympt1}
\begin{array}{l}
\lim_{\xi\to\infty} \tau = -\xi+\log(1-\lambda|\lambda|)\\
\lim_{\xi\to-\infty} \tau = -\frac 32 \xi + \log\left(\frac{\pi}{4}\sqrt{2}\right)
\end{array}$$ which describes the asymptotic behaviour of the time of flight as visualized in Figure \[fig:tof\_xi\]. The two asymptotes are thus revealed to have negative coefficients $-1$ and $-3 / 2$. For the multirevolution cases the derivatives are found to be: $$\label{eq:derivatives_tau_mr}
\begin{array}{l}
\frac {d\tau}{d\xi} = \frac{1-x^2}{2T}\frac{dT}{dx}\\
\frac{d^2\tau}{d\xi^2}=\frac{(1-x^2)^2}{4T}\frac{d^2T}{dx^2} - x\frac{d\tau}{d\xi} - \left(\frac{d\tau}{d\xi}\right)^2\\
\begin{split}
\frac{d^3\tau}{d\xi^3}=\frac{(1-x^2)^3}{8T}\frac{d^3T}{dx^3}-\left(\frac{d^2\tau}{d\xi^2}+x\frac{d\tau}{d\xi}+\left(\frac{d\tau}{d\xi}\right)^2\right)\left(2x+\frac{d\tau}{d\xi}\right) - \\
-\frac{1-x^2}{2}\frac{d\tau}{d\xi}-x\frac{d^2\tau}{\xi^2}-2\frac{d\tau}{d\xi}\frac{d^2\tau}{d\xi^2}
\end{split}
\end{array}$$ again computable in cascade and the following asymptotic behavior can be derived: $$\label{eq:asympt2}
\begin{array}{l}
\lim_{\xi\to-\infty} \tau = \log\left(\frac{\pi+M\pi}8\right) - \frac 32 \xi\\
\lim_{\xi\to\infty} \tau = \log\left(\frac{M\pi}8\right) + \frac 32 \xi
\end{array}$$ revealing two symmetric asymptotes having inclination $\pm 3 / 2$.
![Time of flight curves parametrized using $\xi$ for different $\lambda$ and $M$ values. \[fig:tof\_xi\_tot\]](tof_xi.png){width="95.00000%"}
In Figure \[fig:tof\_xi\_tot\] we report the time of flight curves in the new variables for different values of $\lambda$ and $M$. The reader can then compare this $\xi$-$\tau$ plane to the $x$-$T$ plane visualised in Figure \[fig:tof\_x\].
Lambert Solver
==============
A Lambert solver can be defined as a procedure that returns, for a gravitational field of strength $\mu$, all the possible velocity vectors $\mathbf v_1$ and $\mathbf v_2$ along keplerian orbits linking $\mathbf r_1$, $\mathbf r_2$ in a transfer time $T^*$. The ingredient of such an algorithm are, essentially a) the choice of a variable to iterate upon and thus invert the time of flight curve, b) the iteration method, c) the starting guess to use with the iteration method and d) the reconstruction methodology to compute $\mathbf v_1$ and $\mathbf v_2$ from the value returned by the iterations. As we will detail, our Lambert solver a) iterates on the Lancaster-Blanchard variable $x$ using b) a Householder iteration scheme c) feeded by initial guesses found exploiting the curve shape in the $\tau$-$\xi$ plane and the new analytical results found above. Eventually the velocity vectors are reconstructed following, again, the methodology proposed by Gooding [@gooding]. The final pseudo-code of the proposed Lambert solver is reported in Algorithm \[alg:main\], \[alg:findxy\]. Note how we detect the maximum number of revolutions $M_{max}$ at the beginning by computing $T_{min}$ in one case. All other cases (i.e. $M<=M_{max}$) will not require the evaluation of a $T_{min}$ via an iterative procedure. By doing so we do not bound the roots (short and long period) and thus risk cases where the solution jumps between the long and short period branches. While this did not happen in our tests of the new routine, it is a possibility we are not safeguarding against.
The code, written in C++ and exposed to python, is made available as part of the open source project PyKEP from the European Space Agency github repository <https://github.com/esa/pykep/>. The final algorithm is the final result of many different trials to exploit the newly found results detailed above, and in particular the $T_0,T_1$ expressions (and their derivatives) and the $\xi$-$\tau$ plane. It is worth reporting how one very robust set-up, not selected as our final proposed algorithm, was that of iterating directly with a simple derivative free method (regula-falsi) on the $\xi$-$\tau$ plane using constant initial guesses (i.e $x_l = -0.7, x_r=0.7$). We ended up choosing a different set-up (Algorithm \[alg:main\], \[alg:findxy\]) which turned up to be faster in our computational tests.
The Householder iterations
--------------------------
One of the main differences of the proposed Lamber solver with respect to previous work is the use of the Householder iterative scheme as a root finder for the time-of-flight curves $T(x)-T^* = 0$. Householder iterations are not used widely as the added computational effort of computing higher order derivatives is not worth the gain whenever these request further function evaluations. In our case, as the derivatives computation is done using equations \[eq:derivatives\], Householder iterations are able to provide a significant benefit. We report the exact form used to implement the iterations as it is known how different numerical form can produce different behaviours. After experimenting with different implementations the following was used:
$$x_{n+1} = x_n - f(x_n) \frac{f'^2(x_n) - f(x_n)f''(x_n) / 2}{f'(x_n) (f'^2(x_n) - f(x_n)f''(x_n)) + f'''(x_n)f^2(x_n) / 6 }$$ where $f$ is, in our case, $T(x)-T^*$ and the derivatives are indicated as $f',f'',f'''$.
Initial Guess
-------------
To generate an initial guess for $x$ we use the newly introduced $\xi$-$\tau$ variables and the values $T_0$ and $T_1$ as computed from Eq.(\[eq:t0\]) and Eq,(\[eq:t1\]). Our initial guess is obtained inverting the following linear approxmation to the time of flight curves: $$\tau = c\xi + d$$ where we vary the $c$ and $d$ values according to the value of $\tau$ and $M$.
### Single revolution
Let us start form the single revolution case. Clearly, for high values of $\tau$, we must set $c = -1.0$, while for low values $c=-3/2$ so that the asymptotic behavior derived in Eq.(\[eq:asympt1\]) is reproduced. We then consider the following piece-wise linear approximation: $$\begin{array}{ll}
\tau = -\frac 32 \xi + \tau_0, & T \ge T_0 \\
\tau = - \xi + \log 2 +\tau_1, & T < T_1 \\
\tau = \tau_0 + \frac{\xi}{\log 2}(\tau_1 - \tau_0) & T_1 < T < T_0
\end{array}$$ where $\tau_1=\log T_1$ and $\tau_0$ = $\log T_0$. We have basically enforced the lines to pass through the points $(x_0,T_0)$ and $(x_1,T_1)$ and have the desired asymptotic behaviour.
The above relation is easily inverted and thus the following simple starter $\xi_0$ is derived: $$\begin{array}{ll}
\xi_0 = \frac 23(\tau_0-\tau), & T \ge T_0 \\
\xi_0 = \log 2+\tau_1-\tau, & T < T_1 \\
\xi_0 = \frac{\tau-\tau_0}{\tau_1-\tau_0} \log2 & T_1 < T < T_0
\end{array}$$ Transforming these relations back to the $x$-$T$ plane we find the following expression for the starter $x_0$: $$\label{eq:x0}
\begin{array}{ll}
x_0 = \left(\frac{T_0}{T}\right)^{\frac 23} -1, & T \ge T_0 \\ \\
x_0 = 2\frac{T_1}{T} -1, & T < T_1 \\
x_0 = \left(\frac{T_0}{T}\right)^{\log_2\left(\frac {T_1}{T_0}\right)} -1& T_1 < T < T_0
\end{array}$$ having an extremely low computational cost also in view of the fact $\log_2$ admits efficient implementaions. We can improve the expression for the $T<T_1$ case making use of the newly found result expressed in Eq.(\[eq:dt1dx\]). We thus set $$\begin{array}{ll}
x_0 = \left(\frac{T_0}{T}\right)^{\frac 23} -1, & T \ge T_0 \\ \\
x_0 = \frac 52 \frac{T_1(T_1-T)}{T (1-\lambda^5) } + 1, & T < T_1 \\
x_0 = \left(\frac{T_0}{T}\right)^{\log_2\left(\frac {T_1}{T_0}\right)} -1& T_1 < T < T_0
\end{array}$$ where we enforced the derivatives and values at $x=1$ and $x=\infty$. We report the error introduced by using these expressions in Figure \[fig:errors\] where we also show, for comparison, the same plot relative to the Gooding initial guess. The error is defined by the difference between the initial guess computed for a given $T^*$ and the actual value of $x$ resulting in a time of flight $T^*$.
![Absolute errors introduced by the Gooding initial guess (left) and the proposed initial guess (right) for the single revolution $M=0$ case. Each line correspond to a different $\lambda$ value ranging from -0.99 to 0.99 \[fig:errors\]](ic_err_gooding.png "fig:"){width="45.00000%"} ![Absolute errors introduced by the Gooding initial guess (left) and the proposed initial guess (right) for the single revolution $M=0$ case. Each line correspond to a different $\lambda$ value ranging from -0.99 to 0.99 \[fig:errors\]](ic_err_mine.png "fig:"){width="45.00000%"}
### Multiple revolutions
For the multiple revolution case, assuming a solution exists, there are two possible values of $x$ and thus we will need two starters. We obtain these by direct inversion of Eq.(\[eq:asympt2\]) that define the two asymptotes. $$\begin{array}{l}
\xi_{0l} = \frac 23(\log{\frac{\pi+M\pi}8}-\tau) \\
\xi_{0r} = \frac 23(\tau-\log{\frac{M\pi}8})
\end{array}$$ The above equations may then be transformed back to the $x$-$T$ plane so that the following simple starters are derived: $$\label{eq:x0lr}
\begin{array}{l}
x_{0l} = \frac{\left(\frac{M\pi + \pi}{8T}\right)^{\frac 23}-1}{\left(\frac{M\pi + \pi}{8T}\right)^{\frac 23}+1} \\
x_{0r} = \frac{\left(\frac{8T}{M\pi}\right)^{\frac 23}-1}{\left(\frac{8T}{M\pi}\right)^{\frac 23}+1}
\end{array}$$ The above expressions approximate well the time of flight curves as $|x| \to 1$. A great advantage of these expressions is that they do not make use of $T_{min}$, $x_{min}$ (i.e. the minimum of the time of flight curve and its extremal value) which would require a distinct set of iterations to be found. We then avoid to pass to the root solver solution bounds at each $M$, at the risk of allowing, during the iterations, switches between short and long period solutions, a theoretical occurence, though, that was never encountered in our experimental tests.
$t > 0$, $\mu > 0$ $\mathbf c = \mathbf r_2 - \mathbf r_1$ $c=|\mathbf c|$, $r_1=|\mathbf r_1|$, $r_2=|\mathbf r_2|$ $s=\frac 12 (r_1+r_2+c)$ $\hat {\mathbf i}_{r,1} = \mathbf r_1 / r_1$, $\hat {\mathbf i}_{r,2} = \mathbf r_2/ r_2$ $\hat {\mathbf i}_{h} = \hat {\mathbf i}_{r,1} \times \hat {\mathbf i}_{r,2}$ $\lambda^2 = 1-c/s$, $\lambda = \sqrt{\lambda^2}$ $\lambda = -\lambda$ $\hat {\mathbf i}_{t,1} = \hat {\mathbf i}_{r,1} \times \hat{\mathbf i}_h $, $\hat {\mathbf i}_{t,2} = \hat {\mathbf i}_{r,2} \times \hat
{\mathbf i}_{r,2}$ $\hat {\mathbf i}_{t,1} = \hat{\mathbf i}_h \times \hat {\mathbf i}_{r,1}$, $\hat {\mathbf i}_{t,2} = \hat{\mathbf i}_h \times \hat
{\mathbf i}_h$ $T = \sqrt{\frac{2\mu}{s^3}} t$ $x_{list},y_{list} = $findxy($\lambda$, $T$) $\gamma = \sqrt{\frac{\mu s}{2}}$, $\rho = \frac{r_1-r_2}{c}$, $\sigma = \sqrt{(1-\rho^2)}$ $V_{r,1} = \gamma [(\lambda y - x) - \rho(\lambda y + x)] / r_1$ $V_{r,2} = -\gamma [(\lambda y - x) + \rho(\lambda y + x)] / r_2$ $V_{t,1} = \gamma\sigma(y+\lambda x) / r_1$ $V_{t,2} = \gamma\sigma(y+\lambda x) / r_2$ $\mathbf v_1 = V_{r,1} \hat {\mathbf i}_{r,1} + V_{t,1} \hat {\mathbf i}_{t,1}$ $\mathbf v_2 = V_{r,2} \hat {\mathbf i}_{r,2} + V_{t,2} \hat {\mathbf i}_{t,2}$
$|\lambda| < 1$, $T < 0$ $M_{max} = \mbox{floor} (T / \pi)$ $T_{00} = \arccos \lambda + \lambda\sqrt{1-\lambda^2}$ start Halley iterations from $x=0,T=T_0$ and find $T_{min}(M_{max})$ $M_{max} = M_{max} - 1$ $T_1 = \frac 23 (1-\lambda^3)$ compute $x_0$ from Eq.(\[eq:x0\]) start Householder iterations from $x_0$ and find $x,y$ compute $x_{0l}$ and $x_{0r}$ from Eq.(\[eq:x0lr\]) with $M=M_{max}$ start Householder iterations from $x_{0l}$ and find $x_r,y_r$ start Householder iterations from $x_{0r}$ and find $x_l,y_l$ $M_{max}=M_{max}-1$
New solver performances
=======================
To test the performances of our new algorithm we start by assessing its accuracy. We consider, for an assigned $M$, a random $\lambda \in [-0.999,0.999]$ and a random $x_{true} \in [-0.99,3]$ (or $x_{true} \in [-0.999,0.999]$ whenever $M>0$) and we compute the resulting time of flight $T$. We then use Housholder iterations starting from the appropriate initial guess to find back the $x$ value. We find that stopping the iterations whenever the difference between the $x$ value computed at two successive iterations is less than $10^{-5}$ ($10^{-8}$ whenever $M>0$) is a good setting. We record, for each of such trials, the number of iterations made by the Householder method $it$, and the error defined as $\epsilon = |x_{true}-x|$. This is repeated 1,000,000 times for $M=0$ and then 100,000 times for each $M=1,2,....,50$. The result is shown in Figure \[fig:accuracy\]. In the vast majority of cases we obtain an error $<10^{-13}$. Few cases have a slightly larger error (up to $<10^{-11}$) and these are mainly corresponding to multirevolution cases where $T\approx T_{min}$. We also note, on the proximity of $\lambda=1$ values, a distinct rise in the absolute error. This is due to the $M=0$ case and the loss of precision in the computation of $y$ from $\lambda$, a problem that can be avoided computing $y$ directly from the problem geometry, but it is here deemed as not necessary. Looking then at the number of iterations, we compute the mean over all instances having the same $M$ value. We obtain, for the single revolution case, an average of 2.1 iterations while, in the multiple revolution case, we get an average of 3.3 iterations to convergence. Note how in these tests we do not find a case where a switch occurs between the short period and long period solution during the root solving procedure. Such a switch would infact immediately appear in Figure \[fig:accuracy\] as a point with a large absolute error $\epsilon$.
![Absolute error $\epsilon = |x-x_{true}|$ of our Lambert solver as a function of $\lambda$. This is achieved, on average over the $M=0$ cases, in 2 iterations \[fig:accuracy\]](accuracy.png){width="95.00000%"}
We then turn to the anaysis of our algorithm complexity with respect to the known Gooding algorithm, considered by many as the most accurate and efficient Lambert solver up to date. First we note that in terms of accuracy, Gooding algorithm is comparable to ours. We then run a speed test. For the purose of this test we reimplement both our and Gooding algorithm in pure Python language (i.e. no C++ bindings) and we record the execution time to solve the same 100,000 randomly generated problems (using the same bounds as above). In the case $M=0$, the proposed algorithm resulted to be faster by a factor $1.25$, while in the multi revolution cases by a factor $1.5$. This type of test is very sensitive to implementation details and to the underlying computing architecture and even if we did our best to pay as much attention to them in both cases, we support our result with more general considerations. The main difference between our algorithm and Gooding’s is in the initial guess generation and in the iteration method. Gooding algorithm employs Halley iterations, while we make use of Householder iterative scheme. While Halley’s method has a slighlty lower complexity and does not need to compute also the third derivative from Eq.(\[eq:derivatives\]), our iterative scheme reaches, in the case $M=0$, a comparable accuracy in only 2 iterations on average compared to the 3 iterations needed for the Gooding case. For $M>0$ the number of required iterations is comparable in both cases but the initial Guess used in Gooding algorithm has, in general, a higher complexity as it makes use of a higher number of square roots and exponentiations. In the $M>0$ case Gooding initial guess also requires the determination of $x_{min},T_{min}$ via a further Halley iterative scheme, while the initial guess we use does not make use of any particular value, while still allowing the Householder method to converge within a few iterations and in all cases. We must, though, note once more that as we do not compute $x_{min},T_{min}$ we also cannot bound the solution during the root solver iterations and thus allow for the theoretical possibilty of a switch between short and long period solutions. Such a rare event never appeared in our extensive testing of the new routine
Finally, we measure the error also in terms of the computed terminal velocities by comparing all $\mathbf v_2$, returned by our Lambert solver, to the same values as computed via numerical propagation (using Lagrange coefficients) from $\mathbf r_1,\mathbf v_1$. We do this by instantiating at random $\mathbf r_1, \mathbf r_2$ with each component in the range $[-4,4]$ and $t\in[0.1,100]$. For the purpose of this test we consider $\mu=1$ and we measure the norm of the resulting vector of the velocity difference. Repeating this experiment for a total cumulative 10,000,000 Lambert’s Problems, an average error of $10^{-13}$ is obtained, with a maximum error measured to be $10^{-8}$.
Conclusion
==========
We revisit Lambert’s problem building upon the results of Lancaster and Blanchard and finding some new properties of the time of flight curves. We propose a new transformation of such curves able to further simplify the problem suggesting efficient approximations to the final solution. Using our results to design a new procedure to solve the Lambert problem we are able to build a low complexity algorithm that we find able to provide accurate solutions in a shorter time when compared to the state of the art Gooding’s algorithm.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this work we carry out an analysis of the observed times of primary and secondary eclipses of the post-common envelope binary NSVS14256825. Recently, [@Almeida2013] proposed that two circumbinary companions orbit this short-period eclipsing binary, in order to explain observed variations in the timing of mutual eclipses between the two binary components. Using a standard weighted least-squares minimisation technique, we have extensively explored the topology of $\chi^2$ parameter space of a single planet model. We find the data set to be insufficient to reliably constrain a one-companion model. Various models, each with similar statistical significance, result in substantially different orbital architectures for the additional companion. No evidence is seen for a second companion of planetary nature. We suspect insufficient coverage (baseline) of timing data causing the best-fit parameters to be unconstrained.'
author:
- 'Tobias Cornelius Hinse & Jae Woo Lee'
- 'Krzysztof Go[ź]{}dziewski'
- 'Jonathan Horner & Robert A. Wittenmyer'
date: 'MNRAS, Accepted 2013 November 7. Received 2013 November 5; in original form 2013 October 11'
title: 'Revisiting the proposed circumbinary multi-planet system NSVS14256825'
---
Introduction
============
The discovery of planets within binary star systems has recently sparked an increased interest in their formation, occurrence frequency and dynamical evolution [@Zwart2013]. Several techniques exist to detect additional bodies accompanying binary stars. In addition to the traditional radial velocity technique, [@Han2008] outlines the possibility to infer such planets from microlensing observations. Recently, transiting circumbinary planets have been detected using the Kepler space telescope [@Doyle2011; @Welsh2012; @Orosz2012a; @Orosz2012b; @Schwamb2013; @Kostov2013]. Furthermore, companions can be detected from pulsar timing measurements [@WolszczanFrail1992]. The formation and dynamical evolution of planets around binary star systems have been the subject of recent theoretical studies [@QuintanaLissauer2006; @HaghighipourRaymond2007; @Marzari2009; @Shi2012].
Utilising ground-based observations, a number of multi-planet systems around short-period eclipsing binary stars have been proposed in recent years [@Lee2009; @Beuermann2010; @Potter2011; @Qian2011]. From measuring the times at minimum light (either primary and/or secondary eclipse) one can use the light-travel time (LTT) effect to detect additional companions by measuring periodic changes in the binary period [@Irwin1952; @Hinse2012a; @Horner2012a]. In contrast to other detection methods (radial velocity, microlensing and transit) the LTT technique is sensitive to massive companions on a long-period orbit: the semi-amplitude $K$ of the LTT signal scales with the companions mass and period as $K \sim M_3$ and $K \sim P_{3}^{2/3}$, respectively. In addition, low-mass binary components will favour the detection of low-mass companions on short-period orbits [@Pribulla2012].
From ground-based photometric observations, the first two-planet circumbinary system (HW Virginis, a.k.a HW Vir) was proposed by [@Lee2009]. Additional multi-body systems of planetary nature were subsequently proposed by [@Beuermann2010; @Marsh2013] (NN Serpentis, a.k.a NN Ser), [@Potter2011] (UZ Fornacis, a.k.a. UZ For) and [@Qian2011] (HU Aquarii, a.k.a. HU Aqr). Recently, [@Lee2012] proposed a quadruple system with two circumbinary sub-stellar companions orbiting the Algol-type binary SZ Hercules (a.k.a. SZ Her).
For a secure detection of a multi-planet circumbinary system, at least two criteria need to be satisfied. First, any period variation, due to additional companions, must be recurring and periodic in time. The data should extend over at least two complete cycles of the longest period. Second, the proposed system should be dynamically stable on time scales comparable to the age of the binary components. [@Horner2011] first studied the dynamical stability of the two planets in HU Aqr. Their study allowed them to conclude that the system is highly unstable with disruption times of a few hundred years. Subsequent studies of the same system were carried out by [@Hinse2012a], [@Wittenmyer2012] and [@Gozdziewski2012]. The overall conclusion of these studies is that the planets, as proposed in the discovery work, are simply not feasible. More observational data is necessary before any further constraints can be imposed on the orbital parameters of any companions in the HU Aqr system.
The proposed planets orbiting the close binary system HW Vir [@Lee2009] is another case where the proposed planets do not stand up to dynamical scrutiny [@Horner2012b]. In that case, the dynamical character of the HW Vir system was studied, and the planets proposed were found to follow highly unstable orbits most likely due to their crossing orbit architecture and relatively high masses. However, [@Beuermann2012b] presented new timing measurements of HW Vir allowing them to conclude stable orbits under the assumption of fixing some of the orbital elements in their least-square analysis.
The NN Ser system was also recently studied by [@Horner2012a]. These authors found stable orbits for the [@Beuermann2010] solutions, if the planets are locked in a mean-motion resonant (MMR) configuration. However, an in-depth remodeling of the timing data renders the system unstable when all parameters are allowed to vary freely. Very recently, [@Beuermann2013] published additional timing data of NN Ser. Their re-analysis allowed them to conclude the existence of two companions orbiting the binary pair involved in a 2:1 MMR.
Unstable orbits in proposed multi-body circumbinary systems have not only been found among companions of planetary nature. The SZ Her system with two sub-stellar mass companions was recently investigated within a dynamical analysis [@Hinse2012b]. Here, the authors also found that the proposed companions followed highly unstable orbits.
In a recent work, [@Almeida2013] interpreted observed eclipse timing variations of the post-common envelope binary NSVS14256825 as being the result of a pair of light-travel time effect introduced by two unseen circumbinary companions. The proposed companions are of planetary nature, with orbital periods $\simeq 3.5$ and $\simeq 6.7$ years, and masses of $3~M_{jup}$ and $8~M_{jup}$, respectively. Once again, however, a recent study [@Wittenmyer2013] reveals that the proposed planetary system would be dynamically unstable on very short timescales - with most plausible orbital architectures being unstable on timescales of just a few hundred years, and only a small fraction of systems surviving on timescales of $10^5$ years.
The aim of this paper is as follows. In section 2 we present the available timing data of NSVS14256825, which forms the basis of our analysis. In particular we augment the timing measurements presented in [@Almeida2013] with three additional data points presented in [@Beuermann2012a]. We also introduce the light-travel time model using Jacobian coordinates and outline the derivation of the minimum mass and projected semi-major axis for a single circumbinary companion along with a short description of our least-squares minimisation methodology. In section 3, we carry out a data analysis and perform a period analysis based on Fourier techniques and present our results describing the main properties of our best-fit solutions. In particular, we present results from finding a best-fit linear, best-fit quadratic and best-fit one-companion model. Finally, we summarise our results and discuss our conclusion in section 4.
Data acquisition and Jacobian light-travel time model {#dataacquisition}
=====================================================
As the basis of this work we consider the same timing data set as published in [@Almeida2013]. However, we noticed that three timing measurements published in [@Beuermann2012a] were not included in [@Almeida2013]. We have therefore carried out two independent analysis based on the following data sets. *Dataset I*: Data as presented in Table 3 in [@Almeida2013]. This data set spans the period from June 22, 2007 to August 13, 2012, corresponding to an observing baseline of around 5 years. *Dataset II*: Data as presented in Table 3 in [@Almeida2013] *plus three data points* (primary eclipse) from [@Beuermann2012a]. The additional points are as follows. BJD $2,451,339.803273 \pm 0.000429$ days, BJD $2,452,906.673899 \pm 0.000541$ days and BJD $2,453,619.579776 \pm 0.000537$ days. The second data set spans the period from June 10, 1999 to August 13, 2012, corresponding to an observing baseline of around 13 years (i.e., doubling the time window). The aim of considering the second data set (Dataset II) is to investigate the effect of the additional timing data on the overall best-fit solution and compare the results obtained from considering the first data set (Dataset I) since it covers a longer observing baseline.
The time stamps in [@Beuermann2012a] are stated using the terrestrial time (TT) standard while the times in [@Almeida2013] states timing measurements in the barycentric dynamical time (TDB) standard. However, the difference between these time standards (TT vs TDB) introduces timing differences on a milli-second (approx. 0.002s) level due to relativistic effects [@Eastman2010]. In light of the quoted measurement uncertainties (from the literature) of the eclipse timings in the two data sets, the two time stamps (TT and TDB) can be combined and no further transformation of one time standard to the other is necessary.
Considering the binary as an isolated two-body system and in the absence of mechanisms that cause period variations, the linear ephemeris of future (or past) eclipses $T_{ecl}$ is given by [@Hilditch2001]
$$T_{ecl}(E) = T_{0} + E \times P_{0},
\label{linearephemerisequation}$$
where $E$ denotes the cycle number, $T_0$ is the reference epoch and $P_0$ is the nominal binary period. Additional effects that cause variations of the binary period would be observed as a systematic residual about this best-fit line.
We use the formulation of the light-travel time effect based on Jacobian coordinates [@Gozdziewski2012]. In the general case a circumbinary $N$-body system is a hierarchical system and employing Jacobian elements therefore seems natural. This is particularly true for the case of a single companion (the first object in a hierarchical multi-body ensemble), where the Jacobian coordinate is equivalent to astrocentric coordinates and readily returns the geometric osculating orbital elements of the companion relative to the binary. Here we assume the binary to be a single massive object with mass equivalent to the sum of the two component masses. For a single circumbinary companion the LTT signal can be expressed as [@Gozdziewski2012]
$$\tau(t) = -\frac{\zeta_1}{c},
\label{equation2}$$
where $c$ is the speed of light and $\zeta_{1}$ is given as $$\zeta_{1}(t) = K_{1}\Bigg[ \sin\omega_{1} (\cos E_{1}(t) - e_{1}) + \cos\omega_{1} \sqrt{1-e_{1}^2} \sin E_{1}(t)\Bigg],$$ Here $e_{1}$ denotes the orbital eccentricity and $\omega_1$ measures the argument of pericentre of the companion relative to the combined binary representing the dynamical centre. The eccentric anomaly is given as $E_{1}$. Following [@Gozdziewski2012] the semi-amplitude of the LTT signal is given as $$K_{1} = \Bigg( \frac{1}{c}\Bigg)~\frac{m_1}{m_{*} + m_1}~a_{1}\sin I_{1},$$ with $c$ measuring the speed of light, $a_{1}$ the semi-major axis, $I_{1}$ the inclination of the orbit relative to the skyplane. The quantities $m_{*}$ and $m_{1}$ denote the masses of the combined binary and companion, respectively.
In summary, the set $(K_{1},P_{1},e_{1},\omega_{1}, T_{1})$ represent the five free osculating orbital parameters for the companion with $P_{1}$ and $T_{1}$ denoting the orbital period and time of pericentre passage, respectively. These latter two quantities are introduced implicitly via Kepler’s equation and the eccentric anomaly [@Gozdziewski2012; @Hinse2012a]
Deriving minimum mass and projected semi-major axis
---------------------------------------------------
Once a weighted least-squares best-fit model has been found the minimum mass of the companion is obtained from solving the following transcendental function
$$f(m_{1}) = \gamma_{1}(m_{1}+m_{*})^2 - m_{1}^3 = 0,$$
where $$\gamma_{1} = \Bigg( \frac{c^3}{k^2}\Bigg) \Bigg( \frac{4\pi^2}{P_1^2}\Bigg)K_{1}^3.$$ The projected minimum (with $\sin I_{1} = 1$) semi-major axis ($a_1$) is then found from Kepler’s third law $$\frac{P_{1}^2}{a_{1}^3} = \frac{4\pi^2}{\mu_1},$$ where the gravitational parameter is given by $\mu_1 = k^2 (m_{*}+m_{1})$ with $k$ denoting Gauss’ gravitational constant. The combined mass of the two binary components is assumed to be $m_{*} = 0.528 M_{\odot}$ [@Almeida2013].
Considering only the case of a single circumbinary companion, the timings of minimum light for primary eclipses is given as $$T_{ecl}(E) = T_{0} + E \times P_{0} + \tau(K_{1}, P_{1}, e_{1}, \omega_{1}, T_{1}).$$ We therefore have a total of seven model parameters describing the light-travel time effect caused by a single circumbinary companion. For a description of two companions we refer to [@Gozdziewski2012]. The LTT signal is a one-dimensional problem similar to the radial velocity technique. We therefore only derive the minimum mass and minimum (projected) semi-major axis of the companion. For simplicity, we henceforth write $m_{1}$ and $a_{1}$ for the minimum masses and minimum semi-major axis of the companion[^1].
It is worth pointing out that no gravitational interactions have been taken into account in the above formulation of the LTT signal. Only Keplerian motion is considered. It is possible to include additional effects (such as mutual gravitational interactions) that can cause period variations and we refer to [@Gozdziewski2012] for more details.
Finally, we stress that the case of a [*single*]{} companion the Jacobian-based description of the one-companion LTT effect is equivalent to the formulation given in [@Irwin1952; @Irwin1959]. Hence, the $P_{1}, e_{1}, \omega_{1}, T_{1}$ should be identical to those parameters obtained using the [@Irwin1952] LTT model along with the derived minimum mass. The only parameter which is different is the semi-major axis of the binary due to the different reference systems used and we refer the reader to [@Gozdziewski2012] for details. For consistency, we tested our results for the presently (Jacobian) derived LTT formulation using the procedure detailed in [@Irwin1952], and obtained identical results. However, one complication could arise in the argument of pericentre which can differ depending on the defined direction of the line-of-sight axis. Either this axis can point towards or away from the observer. The difference will affect the argument of pericentre and can be rectified using the relation $\omega_1 = \omega^{'} + \pi$, where $\omega^{'}$ is the argument of pericentre defined in a reference system with opposite line-of-sight direction compared to the formulation outlined in [@Gozdziewski2012]. Hence the difference is only a matter of convention and does not affect the quantitative results obtained from the two formulations.
Weighted least-squares fitting
------------------------------
We have implemented the Jacobian-based Kepler-kinematic LTT model in IDL[^2]. The Levenberg-Marquardt (LM) least-square minimisation algorithm was used to find a best-fit model and is available via the `MPFIT` routine [@Markwardt2009]. We quantify the goodness of fit statistic as
$$\chi^2 = \sum_{i=1}^{N}\Bigg( \frac{O_{i}-C_{i}}{\sigma_{i}}\Bigg)^2,
\label{chisquare}$$
where $N$ is the number of data points, $O_{i}-C_{i}$ measures the vertical difference between the observed data and the computed model at the $i$th cycle, and $\sigma_{i}$ measures the 1-sigma timing uncertainty (usually obtained formally). However, in this work we will quote the *reduced* $\chi^2$ defined as $\chi^2_{\nu} = \chi^2/\nu$ with $\nu = N-n$ denoting the degree of freedom. The `MPFIT` routine attempts to minimise $\chi^2$ iteratively using $n$ free parameters.
In the search for a global minimum $\chi^2_{\nu,0}$ of the underlying $\chi^2$ space we utilise a Monte Carlo approach by generating a large number $(5 \times 10^5)$ of random initial guesses. Two approaches can be used to explore the $\chi^2$ space for a global minimum. The first involves generating random initial guesses in a relatively narrow region of a given parameter and may be applied when information about the periodicity and amplitude of the LTT signal is inferred from other means (e.g. Fourier analysis). For example, if a Fourier analysis reveals a given frequency within the data one can then generate random initial guesses from a normal distribution centred at that period with some (more or less narrow) standard deviation for the variance. In the second approach, random initial guesses are generated from a uniform distribution defined over a broad interval for a given parameter. However, in both approaches we randomly choose the eccentricity from a uniform distribution within $e_1 \in [0,0.99]$ with the argument of pericentre chosen from $\omega_1 \in [-\pi,\pi]$. In all our searches we recorded the initial guess and final parameters along with the goodness-of-fit value, the corresponding root-mean-square (RMS) statistic and formal 1-sigma uncertainties. A single LM iteration sequence is terminated following default values of accuracy parameters within `MPFIT` or after a maximum of 3000 iterations (rarely encountered with the average number of iterations required being just 11).
Data analysis & results
=======================
Period analysis and linear ephemeris
------------------------------------
As a starting point for our analysis, we first determined the parameters of the linear ephemeris ($T_{0}, P_{0}$) by calculating a linear least-squares regression line to the same data (Dataset I) as considered by [@Almeida2013]. A best fit line resulted in a $\chi^2_{65} \simeq 13$ with $n=2$ free parameters and $N=67$ data points. The corresponding $\chi^2$ value was found to be 853 and the (rounded) linear ephemeris was determined to be
$$\begin{aligned}
T_{ecl}^{I}&=& T_{0} + E \times P_{0} \\
&=& \textnormal{BJD}~2455408.744502 \pm 3 \times 10^{-6} + E \times 0.1103741881 \pm 8 \times 10^{-10}~\textnormal{days} \\
&=& \textnormal{BJD}~2455408.744502^{505}_{499} + E \times 0.1103741881^{89}_{73}~\textnormal{days}\end{aligned}$$
For Dataset II we obtained the slightly different ephemeris, with little improvement in the precision of the binary period $$\begin{aligned}
T_{ecl}^{II}&=& T_{0} + E \times P_{0} \\
&=& \textnormal{BJD}~2455408.744504 \pm 3 \times 10^{-6} + E \times 0.1103741759 \pm 8 \times 10^{-10}~\textnormal{days} \\
&=& \textnormal{BJD}~2455408.744504^{507}_{501} + E \times 0.1103741759^{67}_{51}~\textnormal{days}\end{aligned}$$ We applied the `PERIOD04`[^3] [@LenzBreger2005] Lomb-Scargle algorithm on the residual data (Fig. \[periodstudy\]) obtained from subtracting the best-fit line, and compared two fits to the residual data. The first had a single Fourier component, whilst the second had two Fourier components as shown in Fig. \[periodstudy\]. The two-component fit was found to provide a better description of the data. We show the corresponding power spectra in Fig. \[powerspec\], and find the 6.9 year period to be in agreement with the period found by [@Almeida2013]. However, the algorithm was unable to detect the 3.5 year cycle (inner proposed planet) as determined in [@Almeida2013]. Instead, we found a 20.6 year cycle with a detection six times above the noise level.
Quadratic ephemeris model - Dataset I {#quadraticephemerissection}
-------------------------------------
In some cases a change of the binary period can be caused by non-gravitational interaction between the two components of a short-period eclipsing binary. Often the period change is described by a quadratic ephemeris (linear plus secular) with the times of primary eclipses given by [@Hilditch2001] $$T_{ecl}(E) = T_{0} + P_{0} \times E + \beta \times E^{2},$$ where $\beta$ is a period damping factor [@Gozdziewski2012] which can account for mass-transfer, magnetic braking, gravitational radiation and/or the influence of a distant companion on a long-period orbit. Following [@Brinkworth2006] the rate of period change, in the case of mass-transfer, is then given by $$\dot{P} = \frac{2\beta}{P},$$ with $P$ denoting the currently measured binary period. In Fig. \[SecularBestFitModel\] we show the best-fit quadratic ephemeris given as $$\begin{aligned}
T_{ecl}(E) = (\textnormal{BJD}~2,455,408.744485 \pm 3.4 \times 10^{-6}) &+& (0.1103741772 \pm 8.9 \times 10^{-10}) \times E \\ &+& (3.1 \times 10^{-12} \pm 1.4 \times 10^{-13}) \times E^{2}\end{aligned}$$ with unreduced $\chi^2 = 360$ for (67-3) degrees of freedom. In Fig. \[2DMapSecularBestFitModel\] we show the location of the best-fit surrounded by the $1\sigma$ (68.3%, $\Delta\chi^2 = 2.3$), $2\sigma$ (95.4%, $\Delta\chi^2 = 6.2$) and $3\sigma$ (99.7%, $\Delta\chi^2 = 18.4$) joint-confidence contours [@Press2002; @BevingtonRobinson2003; @HughesHase2010] for the $(P_0, \beta)$ parameter space. Similar results were obtained for the remaining two parameter combinations. Considering Dataset I we found the average period change to be $\dot{P} = 5.6 \times 10^{-11}~\textnormal{s}~\textnormal{s}^{-1}$. This value is about one order of magnitude smaller than the period decrease reported in [@Almeida2013].
Single companion model - Dataset I
----------------------------------
To reliably assess the validity of a two-companion model we first considered a one-companion model. Our period analysis yielded a shortest period of around $P_1 \simeq 7$ years (2557 days) with a semi-amplitude of $K_{1} \simeq 0.000231$ days (20 seconds). We therefore searched for a best-fit solution in a narrow interval around these values by seeding 523,110 initial guesses. The best-fit solution with $\chi^2_{60,0} = 1.98$ is shown in Fig. \[1bodyLTTJacBestFit\]. In Table \[bestfitparam\] we show the corresponding best-fit parameters and derived quantities for the companion along with their formal (derived from the covariance matrix) $1\sigma$ uncertainties as obtained from `MPFIT`. Formal errors in the derived quantities were obtained from numerical error propagation, as described in [@BevingtonRobinson2003]. The residual plot in Fig. \[1bodyLTTJacBestFit\] (middle panel) shows no obvious trend above the 5 seconds level. The average timing uncertainty in the Almeida et al. (2013) data set is 5.5 seconds. An additional signal associated with a light-travel time effect should be detected on a $3\sigma$ level equivalent to a timing semi-amplitude of $\simeq 15$ seconds. Usually timing measurement are assumed to distribute normally around the expected model. We have therefore also plotted the normalised residuals $(O_{i}-C_{i})/\sigma_i$ [@HughesHase2010] as shown in the bottom panel of Fig. \[1bodyLTTJacBestFit\]. The corresponding histogram is shown in Fig. \[histogram\]. Whether the timing residuals follow a Gaussian distribution is unclear at the moment.
Again, we have explored the $\chi^2_{60}$ function in the vicinity of the best-fit parameters and determined two-dimensional joint-confidence intervals. We show all 21 two-parameter combinations in Fig. \[2DScansFig1\] and Fig. \[2DScansFig2\]. While the two considered parameters in a given panel were kept fixed, we allowed all the remaining parameters to re-optimise (with an initial guess given by the best-fit values listed in Table \[bestfitparam\]) during a LM iteration [@BevingtonRobinson2003].
We note that several of the parameters correlate with each other. This is especially true for the $(T_1,\omega_1)$ pair shown in Fig. \[2DScansFig2\]. Choosing our reference epoch $T_0$ to be close to the middle of the data set results in almost no correlation between $T_0$ and $P_0$ (see top left panel in Fig. \[2DScansFig1\]). In addition, we note that the $\chi^2$ topology around the best-fit parameters deviates from its expected parabolic form. This is most readily apparent in the $(\omega_1,e_1)$ panel in Fig. \[2DScansFig2\].
Finally, we note that the $3\sigma$ confidence level in the $(P_{1},K_{1})$ (bottom-right) panel of Fig. \[2DScansFig1\] appears open, and stretches toward longer periods $(P_1)$ and larger semi-amplitudes ($K_1$). With this in mind, we then recalculated the $\chi^2$ space of $(P_{1},K_{1})$ considering a larger interval in the two parameters. The result is shown in Fig. \[2DScansFig16Zoom\], demonstrating that the 3-sigma joint-confidence contour remains open for orbital periods larger than around 22 years. We therefore suspect that our best-fit model resides within a local minimum.
To test whether we are dealing with a local minimum we explore the $\chi^2$ parameter space on a wider search grid by following the approach as outlined previously. Surprisingly, we found a marginally improved solution with a smaller best-fit $\chi^2_{60,0*}$ value of 1.96, a reduction by $2\%$ compared to the first best-fit solution of 1.98. Computing the $\chi^2_{60}$ space around the new best-fit solution over a large interval in the parameters $K_1, e_1$ and $P_1$ resulted in Fig. \[2DScansThreeFigures\].
In each panel, our (new) improved best-fit solution is marked by a cross-hair. The corresponding model parameters are shown in Table \[bestfitparam\_extended\]. We have omitted quoting the formal uncertainties for reasons that will become apparent shortly. In Fig. \[2DScansThreeFigures\] we also show the 1-sigma (68.3%) joint-confidence contour of $\Delta\chi^2_{60} = 1.993$ (black line) encompassing our best-fit model. Our results suggest that a plethora of models, with $\chi^2_{60}$ within the 1-sigma confidence level, are equally capable of explaining the timing data. Statistically, within the 1-sigma uncertainty region, essentially no differences in the $\chi^2$ exist between the various solutions.
For this reason, the considered parameters (semi-amplitude, eccentricity and period) span a vast range, making it impossible to place firm confidence intervals on them. From Fig. \[2DScansThreeFigures\] possible periods span from $\simeq 2500$ days (6.8 years) to at least 80,000 days (219 years) chosen as our upper cut-off limit in the search procedure. We have tested this result by selecting three significantly different pairs of $(P_{1},e_{1})$ in Fig. \[2DScansThreeFigures\]a. We label them as follows: *Example 1*): $(P_{1},e_{1})=(3973~\textnormal{days},0.40)$. *Example 2*): $(P_{1},e_{1})=(15769~\textnormal{days},0.77)$. *Example 3*): $(P_{1},e_{1})=(75318~\textnormal{days},0.91)$. We re-calculated a best-fit model with the $(P_{1}, e_{1})$ parameters held fixed, and remaining parameters $(T_{0}, P_{0}, K_{1}, \omega_{1}, T_{1})$ allowed to vary freely (starting from the best-fit solution given by the cross-hair in Fig. \[2DScansThreeFigures\]a) to find new optimum values. We show the results of this experiment in Fig. \[BestFitSolutions4Figs\]. All models have $\chi^2_{60}$ within the 1-sigma confidence level (1.993) but differ significantly in their orbital periods, eccentricities and semi-amplitudes. Our best-fit model (cross-hair) is shown in Fig. \[BestFitSolutions4Figs\]d and Table \[bestfitparam\_extended\]. We calculated the following values for the companion’s minimum mass and semi-major axis for our three examples. *Example 1*): $m_1\sin I_1 = 7.6~M_{jup}$, $a_{1}\sin I_1 = 4.0$ au. *Example 2*): $m_1\sin I_1 = 8.5~M_{jup}$, $a_{1}\sin I_1 = 10.0$ au. *Example 3*): $m_1\sin I_1 = 9.7~M_{jup}$, $a_{1}\sin I_1 = 28.4$ au. In light of the large range of possible parameters we omit quoting parameter uncertainties. Minimum mass and semi-major axis for our improved best-fit solution (Fig. \[BestFitSolutions4Figs\]d) are given in Table \[bestfitparam\_extended\].
Up to this point our analysis allows us to conclude that the data is not spanning a sufficiently long observing baseline to firmly constrain the parameters of a single companion model. We stress that the model itself could still be valid. With the data currently at hand it is impossible to establish firm confidence intervals on the parameters. Our first solution (comparable with the solution presented in [@Almeida2013]) likely represents a local minimum in the $\chi^2_{\nu}$ parameter space, or appears to be a solution within the $1\sigma$ confidence interval characterised by a shallow topology of $\chi^2$ space. In such a case we cannot distinguish isolated models in the continuum of possible solutions. All three panels in Fig. \[2DScansThreeFigures\] indicate the existence of local minima with $\chi^2_{\nu}$ statistics close to our first best-fit solution with $\chi^2_{60,0} = 1.98$ (Table \[bestfitparam\]). In fact, Fig. \[2DScansThreeFigures\] suggests the existence of multiple local minima in the $\chi^2_{\nu}$ space. Since in Fig. \[2DScansThreeFigures\] we have not found the 1-sigma confidence level to render as a closed-loop contour line, we suspect that the data can be fit to an infinite number of models each having the *same* statistical significance, but exhibiting significant differences in their orbital architectures. In light of this result any efforts to search for a second companion in Dataset I seems unfruitful.
Single companion model - Dataset II
-----------------------------------
We have noted that three datapoints from [@Beuermann2012a] were not included in the analysis presented in [@Almeida2013]. Although they are accurate (placing them well on the linear ephemeris) their timing precision is lower. However, the large timing uncertainty for these points should not disqualify them from being included in the analysis. In principle, the precision of the eclipsing period $P_0$ should increase for a dataset of increased baseline, and could eventually help to constrain any long-period trend. We have repeated our search procedure as outlined previously to find a best-fit model based on dataset II. We show our best-fit solution in Fig. \[BestFitSolutionDataSet2\] and state the best-fit parameters within the figure area.
For dataset II, the main characteristics of the Keplerian orbit for the companion are similar to the parameters shown in Table \[bestfitparam\_extended\]. The period, minimum semi-major axis and eccentricity are comparable in both cases. We also explored the topology of $\chi^2$ space for a large region around the best-fit solution and found similar results as discussed previously by generating two-dimensional joint-confidence interval maps. The 1-sigma confidence contour around the best-fit solution extends over a large interval in the period, eccentricity and semi-amplitude.
From examining the residual plot in Fig. \[BestFitSolutionDataSet2\] we are not convienced about any additional light-travel time periodicity above the RMS level of about six seconds. A light-travel time signal with amplitude of around six seconds would require a dataset with RMS of about one second or less. Hence, from a qualitative judgment, the data in Dataset II does not currently support the inclusion of five additional parameters corresponding to a second companion. The results from examining Dataset II reinforces insufficient coverage of the orbit as presented in [@Almeida2013]. Because Dataset II covers two-times the best-fit period found for Dataset I, one would expect Dataset II to constrain the orbital period to a higher degree than for Dataset I. However, this is not the case for the present situation.
Summary and conclusions
=======================
In this work we have carried out a detailed data analysis of timing measurements of the short-period eclipsing binary NSVS14256825. In particular we have examined the one-companion model bearing in mind that additional valid companions should be readily visible in the resulting residuals. On the basis of Dataset I, we first carried out an initial local search for a weighted least-squares best-fit solution. A best-fit model (Table \[bestfitparam\]) with $\chi^{2}_{\nu} \simeq 1.98$ resulted in an inner circumbinary companion with orbital characteristics comparable to the short-period companion presented in [@Almeida2013]. Extending our search grid of $\chi^2$ parameter space resulted in a similar best-fit $\chi^2_{\nu}$ statistic with significantly different orbital characteristics (Table \[bestfitparam\_extended\]). We were able to show quantitatively that the present timing data does not allow us to firmly constrain a particular model with well-established parameter confidence limits. In light of this, quoting formal errors for the model parameters seems meaningless. We concluded that the best-fit solution found by [@Almeida2013] most likely represents a local minimum. We explain the lack of constraint in the parameters by the limited monitoring baseline over which timing data was acquired. Dataset I represented a baseline of about 5 years. If a periodicity is present, the principle of recurrance should apply, requiring two full orbital periods to be covered in order to establish firm evidence for the presence of a companion. This would correspond to a light-travel time period of at most 2.5 years for Dataset I and 6 years for Dataset II (spanning about 13 years). However, for Dataset I, the data did not allow models with periods shorter than $\simeq 1000$ days. Simultaneously Dataset II does not constrain the period any better than Dataset I.
Our analysis did not allow us to find convincing evidence of a second light-travel time signal. The RMS scatter of timing data around the best-fit model was found to be around 5 seconds. Signals with a semi-amplitude comparable with the measurement uncertainties seem unlikely to be supported by the present data. The claimed second companion in [@Almeida2013] has a semi-amplitude of $K_2 \simeq 4.9$ seconds. It is likely that noise was wrongly interpreted as a light-travel time signal. We recommend that a secure detection requires a signal semi-amplitude of at least three times above the noise level (i.e $K \simeq 3 \times \textnormal{RMS}$). Future timing data [@Pribulla2012; @ParkKMTnet2012] of this system will be important to help constraining the parameters significantly.
### Acknowledgements {#acknowledgements .unnumbered}
Research by T. C. H is carried out at the Korea Astronomy and Space Science Institute (KASI) under the 2012 KRCF (Korea Research Council for Science and Technology) Young Scientist Research Fellowship Program. K. G. is supported by Polish NSC, grant N/N203/402739. Numerical computations were partly carried out using the SFI/HEA Irish Centre for High-End Computing (ICHEC) and the PLUTO computing cluster at KASI. Astronomical research at Armagh Observatory is funded by the Department of Culture, Arts and Leisure (DCAL). T. C. H and J. W. L acknowledges support from KASI grant 2013-9-400-00. J. H. gratefully acknowledges financial support of the Australian government through ARC Grant DP0774000. R.A.W is supported by a UNSW Vice-Chancellor’s Fellowship.
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Dataset I
------------------- ------------------------------------------------------- ----------- -- --
$\chi^2_{60,0} = 1.98$, $N = 67$, $n = 7$, $\nu = 60$
RMS 5.4 seconds
$T_0$ 2,455,408.74450(36) BJD
$P_0$ 0.11037415(5) days
$K_{1}$ $0.00023 \pm 0.00005$ AU
$e_{1}$ $0.3 \pm 0.1$ -
$\omega_{1}$ $1.7 \pm 0.3$ radians
$T_{1}$ 2,455,197(67) BJD
$P_{1}$ $2921 \pm 258$ days
$m_{1}\sin I_{1}$ $6.7 \pm 0.9$ $M_{jup}$
$a_{1}\sin I_{1}$ $3.3 \pm 0.6$ AU
$e_{1}$ $0.3 \pm 0.1$ -
$\omega_{1}$ $(1.7 + \pi) \pm 0.3$ radians
$P_{1}$ $2921 \pm 258$ days
Dataset I
------------------- ----------------------------------------------------- ----------- -- --
$\chi_{60,0}^2 = 1.96$, $N = 67$, $n = 7$, $\nu=60$
RMS 5.3 seconds
$T_0$ 2,455,408.74455(41) BJD
$P_0$ 0.11037411(6) days
$K_{1}$ $0.00169$ AU
$e_{1}$ $0.85$ -
$\omega_{1}$ $2.33$ radians
$T_{1}$ 2,455,330 BJD
$P_{1}$ $34263$ days
$m_{1}\sin I_{1}$ $9.8$ $M_{jup}$
$a_{1}\sin I_{1}$ $16.8$ AU
$e_{1}$ $0.85$ -
$\omega_{1}$ $(2.33 + \pi)$ radians
$P_{1}$ $34263$ days
[^1]: Technically, the values obtained represent the minimum possible values of $m_{1}\sin I_{1}$ and $a_{1}\sin I_{1}$ - but in standard papers dealing with eclipse timing or radial velocity studies authors use the shortened versions, for brevity.
[^2]: http://www.exelisvis.com/ProductsServices/IDL.aspx
[^3]: http://www.univie.ac.at/tops/Period04/
| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
Thibault Damour\
\
[*Institut des Hautes Etudes Scientifiques*]{}\
[*35 route de Chartres, 91440 Bures-sur-Yvette, France*]{}
date:
title: '**General Relativity Today[^1] [^2]**'
---
**Abstract:** After recalling the conceptual foundations and the basic structure of general relativity, we review some of its main modern developments (apart from cosmology) : (i) the post-Newtonian limit and weak-field tests in the solar system, (ii) strong gravitational fields and black holes, (iii) strong-field and radiative tests in binary pulsar observations, (iv) gravitational waves, (v) general relativity and quantum theory.
Introduction {#sec1}
============
The [*theory of general relativity*]{} was developed by Einstein in work that extended from 1907 to 1915. The starting point for Einstein’s thinking was the composition of a review article in 1907 on what we today call the [*theory of special relativity*]{}. Recall that the latter theory sprang from a new kinematics governing length and time measurements that was proposed by Einstein in June of 1905 [@E05], [@oeuvres], following important pioneering work by Lorentz and Poincaré. The theory of special relativity essentially poses a new fundamental framework (in place of the one posed by Galileo, Descartes, and Newton) for the formulation of physical laws: this framework being the chrono-geometric space-time structure of Poincaré and Minkowski. After 1905, it therefore seemed a natural task to formulate, reformulate, or modify the then known physical laws so that they fit within the framework of special relativity. For Newton’s law of gravitation, this task was begun (before Einstein had even supplied his conceptual crystallization in 1905) by Lorentz (1900) and Poincaré (1905), and was pursued in the period from 1910 to 1915 by Max Abraham, Gunnar Nordström and Gustav Mie (with these latter researchers developing [*scalar*]{} relativistic theories of gravitation).
Meanwhile, in 1907, Einstein became aware that gravitational interactions possessed particular characteristics that suggested the necessity of [*generalizing*]{} the framework and structure of the 1905 theory of relativity. After many years of intense intellectual effort, Einstein succeeded in constructing a [*generalized theory of relativity*]{} (or [*general relativity*]{}) that proposed a profound modification of the chrono-geometric structure of the space-time of special relativity. In 1915, in place of a simple, neutral arena, given a priori, independently of all material content, space-time became a physical “field” (identified with the gravitational field). In other words, it was now a dynamical entity, both influencing and influenced by the distribution of mass-energy that it contains.
This radically new conception of the structure of space-time remained for a long while on the margins of the development of physics. Twentieth century physics discovered a great number of new physical laws and phenomena while working with the space-time of special relativity as its fundamental framework, as well as imposing the respect of its symmetries (namely the Lorentz-Poincaré group). On the other hand, the theory of general relativity seemed for a long time to be a theory that was both poorly confirmed by experiment and without connection to the extraordinary progress springing from application of quantum theory (along with special relativity) to high-energy physics. This marginalization of general relativity no longer obtains. Today, general relativity has become one of the essential players in cutting-edge science. Numerous high-precision experimental tests have confirmed, in detail, the pertinence of this theory. General relativity has become the favored tool for the description of the macroscopic universe, covering everything from the big bang to black holes, including the solar system, neutron stars, pulsars, and gravitational waves. Moreover, the search for a consistent description of fundamental physics in its entirety has led to the exploration of theories that unify, within a general quantum framework, the description of matter and all its interactions (including gravity). These theories, which are still under construction and are provisionally known as string theories, contain general relativity in a central way but suggest that the fundamental structure of space-time-matter is even richer than is suggested separately by quantum theory and general relativity.
Special Relativity {#sec2}
==================
We begin our exposition of the theory of general relativity by recalling the chrono-geometric structure of space-time in the theory of [*special*]{} relativity. The structure of Poincaré-Minkowski space-time is given by a generalization of the Euclidean geometric structure of ordinary space. The latter structure is summarized by the formula $L^2 = (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2$ (a consequence of the Pythagorean theorem), expressing the square of the distance $L$ between two points in space as a sum of the squares of the differences of the (orthonormal) coordinates $x,y,z$ that label the points. The symmetry group of Euclidean geometry is the group of coordinate transformations $(x,y,z) \to
(x',y',z')$ that leave the quadratic form $L^2 = (\Delta x)^2 +
(\Delta y)^2 + (\Delta z)^2$ invariant. (This group is generated by translations, rotations, and “reversals” such as the transformation given by reflection in a mirror, for example: $x' =
-x$, $y' = y$, $z'=z$.)
The Poincaré-Minkowski space-time is defined as the ensemble of [*events*]{} (idealizations of what happens at a particular point in space, at a particular moment in time), together with the notion of a [*(squared) interval*]{} $S^2$ defined between any two events. An event is fixed by four coordinates, $x,y,z$, and $t$, where $(x,y,z)$ are the spatial coordinates of the point in space where the event in question “occurs,” and where $t$ fixes the instant when this event “occurs.” Another event will be described (within the same reference frame) by four different coordinates, let us say $x + \Delta x$, $y + \Delta y$, $z +
\Delta z$, and $t + \Delta t$. The points in space where these two events occur are separated by a distance $L$ given by the formula above, $L^2 = (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2$. The moments in time when these two events occur are separated by a time interval $T$ given by $T = \Delta t$. The squared interval $S^2$ between these two events is given as a function of these quantities, by definition, through the following generalization of the Pythagorean theorem: $$\label{rg1} S^2 = L^2 - c^2 \, T^2 = (\Delta x)^2 + (\Delta y)^2 +
(\Delta z)^2 - c^2 (\Delta t)^2 \, ,$$ where $c$ denotes the speed of light (or, more precisely, the maximum speed of signal propagation).
Equation (\[rg1\]) defines the [*chrono-geometry*]{} of Poincaré-Minkowski space-time. The symmetry group of this chrono-geometry is the group of coordinate transformations $(x,y,z,t) \to (x',y',z',t')$ that leave the quadratic form (\[rg1\]) of the interval $S$ invariant. We will show that this group is made up of linear transformations and that it is generated by translations in space and time, spatial rotations, “boosts” (meaning special Lorentz transformations), and reversals of space and time.
It is useful to replace the time coordinate $t$ by the “light-time” $x^0 \equiv ct$, and to collectively denote the coordinates as $x^{\mu} \equiv (x^0 , x^i)$ where the Greek indices $\mu , \nu , \ldots = 0,1,2,3$, and the Roman indices $i,j,\ldots = 1,2,3$ (with $x^1 = x$, $x^2 = y$, and $x^3 = z$). Equation (\[rg1\]) is then written $$\label{rg2}
S^2 = \eta_{\mu\nu} \, \Delta x^{\mu} \, \Delta x^{\nu} \, ,$$ where we have used the Einstein summation convention[^3] and where $\eta_{\mu\nu}$ is a diagonal matrix whose only non-zero elements are $\eta_{00} = -1$ and $\eta_{11} = \eta_{22} = \eta_{33} = +1$. The symmetry group of Poincaré-Minkowski space-time is therefore the ensemble of Lorentz-Poincaré transformations, $$\label{rg3} x'^{\mu} = \Lambda_{\nu}^{\mu} \, x^{\nu} + a^{\mu} \,
,$$ where $\eta_{\alpha \beta} \, \Lambda_{\mu}^{\alpha} \,
\Lambda_{\nu}^{\beta} = \eta_{\mu\nu}$.
The chrono-geometry of Poincaré-Minkowski space-time can be visualized by representing, around each point $x$ in space-time, the locus of points that are separated from the point $x$ by a unit (squared) interval, in other words the ensemble of points $x'$ such that $S_{xx'}^2 = \eta_{\mu\nu} (x'^{\mu} - x^{\mu})
(x'^{\nu} - x^{\nu}) = + 1$. This locus is a one-sheeted (unit) hyperboloid.
If we were within an ordinary Euclidean space, the ensemble of points $x'$ would trace out a (unit) sphere centered on $x$, and the “field” of these spheres centered on each point $x$ would allow one to completely characterize the Euclidean geometry of the space. Similarly, in the case of Poincaré-Minkowski space-time, the “field” of unit hyperboloids centered on each point $x$ is a visual characterization of the geometry of this space-time. See Figure \[fig1\]. This figure gives an idea of the symmetry group of Poincaré-Minkowski space-time, and renders the rigid and homogeneous nature of its geometry particularly clear.
$$\includegraphics[width=50mm]{Fig1.eps}$$
The essential idea in Einstein’s article of June 1905 was to impose the group of transformations (\[rg3\]) as a symmetry group of the fundamental laws of physics (“the principle of relativity”). This point of view proved to be extraordinarily fruitful, since it led to the discovery of new laws and the prediction of new phenomena. Let us mention some of these for the record: the relativistic dynamics of classical particles, the dilation of lifetimes for relativistic particles, the relation $E
= mc^2$ between energy and inertial mass, Dirac’s relativistic theory of quantum ${\rm spin} \, \frac{1}{2}$ particles, the prediction of antimatter, the classification of particles by rest mass and spin, the relation between spin and statistics, and the CPT theorem.
After these recollections on special relativity, let us discuss the special feature of gravity which, in 1907, suggested to Einstein the need for a profound generalization of the chrono-geometric structure of space-time.
The Principle of Equivalence {#sec3}
============================
Einstein’s point of departure was a striking experimental fact: all bodies in an external gravitational field fall with the same acceleration. This fact was pointed out by Galileo in 1638. Through a remarkable combination of logical reasoning, thought experiments, and real experiments performed on inclined planes,[^4] Galileo was in fact the first to conceive of what we today call the “universality of free-fall” or the “weak principle of equivalence.” Let us cite the conclusion that Galileo drew from a hypothetical argument where he varied the ratio between the densities of the freely falling bodies under consideration and the resistance of the medium through which they fall: “Having observed this I came to the conclusion that in a medium totally devoid of resistance all bodies would fall with the same speed” [@G70]. This universality of free-fall was verified with more precision by Newton’s experiments with pendulums, and was incorporated by him into his theory of gravitation (1687) in the form of the identification of the inertial mass $m_i$ (appearing in the fundamental law of dynamics ${\mbox{\boldmath${F}$}}= m_i \, {\mbox{\boldmath${a}$}}$) with the gravitational mass $m_g$ (appearing in the gravitational force, $F_g = G \, m_g \, m'_g / r^2$): $$\label{rg4}
m_i = m_g \, .$$
At the end of the nineteenth century, Baron Roland von Eötvös verified the equivalence (\[rg4\]) between $m_i$ and $m_g$ with a precision on the order of $10^{-9}$, and Einstein was aware of this high-precision verification. (At present, the equivalence between $m_i$ and $m_g$ has been verified at the level of $10^{-12}$ [@tests].) The point that struck Einstein was that, given the precision with which $m_i = m_g$ was verified, and given the equivalence between inertial mass and energy discovered by Einstein in September of 1905 [@oeuvres] ($E = m_i \, c^2$), one must conclude that all of the various forms of energy that contribute to the inertial mass of a body (rest mass of the elementary constituents, various binding energies, internal kinetic energy, etc.) do contribute in a strictly identical way to the gravitational mass of this body, meaning both to its capacity for reacting to an external gravitational field and to its capacity to create a gravitational field.
In 1907, Einstein realized that the equivalence between $m_i$ and $m_g$ implicitly contained a deeper equivalence between inertia and gravitation that had important consequences for the notion of an inertial reference frame (which was a fundamental concept in the theory of special relativity). In an ingenious thought experiment, Einstein imagined the behavior of rigid bodies and reference clocks within a freely falling elevator. Because of the universality of free-fall, all of the objects in such a “freely falling local reference frame” would appear not to be accelerating with respect to it. Thus, with respect to such a reference frame, the exterior gravitational field is “erased” (or “effaced”). Einstein therefore postulated what he called the “principle of equivalence” between gravitation and inertia. This principle has two parts, that Einstein used in turns. The first part says that, for any external gravitational field whatsoever, it is possible to locally “erase” the gravitational field by using an appropriate freely falling local reference frame and that, because of this, the non-gravitational physical laws apply within this local reference frame just as they would in an inertial reference frame (free of gravity) in special relativity. The second part of Einstein’s equivalence principle says that, by starting from an inertial reference frame in special relativity (in the absence of any “true” gravitational field), one can create an apparent gravitational field in a local reference frame, if this reference frame is accelerated (be it in a straight line or through a rotation).
Gravitation and Space-Time Chrono-Geometry {#sec4}
==========================================
Einstein was able (through an extraordinary intellectual journey that lasted eight years) to construct a new theory of gravitation, based on a rich generalization of the 1905 theory of relativity, starting just from the equivalence principle described above. The first step in this journey consisted in understanding that the principle of equivalence would suggest a profound modification of the chrono-geometric structure of Poincaré-Minkowski space-time recalled in Equation (\[rg1\]) above.
To illustrate, let $X^{\alpha}$, $\alpha = 0,1,2,3$, be the space-time coordinates in a local, freely-falling reference frame (or [*locally inertial reference frame*]{}). In such a reference frame, the laws of special relativity apply. In particular, the infinitesimal space-time interval $ds^2 = dL^2 - c^2 \, dT^2$ between two neighboring events within such a reference frame $X^{\alpha}$, $X'^{\alpha} =
X^{\alpha} + dX^{\alpha}$ (close to the center of this reference frame) takes the form $$\label{rg5} ds^2 = dL^2 - c^2 \, dT^2 = \eta_{\alpha\beta} \,
dX^{\alpha} \, dX^{\beta} \, ,$$ where we recall that the repeated indices $\alpha$ and $\beta$ are summed over all of their values ($\alpha , \beta = 0,1,2,3$). We also know that in special relativity the local energy and momentum densities and fluxes are collected into the ten components of the [*energy-momentum tensor*]{} $T^{\alpha\beta}$. (For example, the energy density per unit volume is equal to $T^{00}$, in the reference frame described by coordinates $X^{\alpha} = (X^0 ,
X^i)$, $i=1,2,3$.) The conservation of energy and momentum translates into the equation $\partial_{\beta} \, T^{\alpha\beta}
= 0$, where $\partial_{\beta} = \partial / \partial \, X^{\beta}$.
The theory of special relativity tells us that we can change our locally inertial reference frame (while remaining in the neighborhood of a space-time point where one has “erased” gravity) through a Lorentz transformation, $X'^{\alpha} =
\Lambda_{\beta}^{\alpha} \, X^{\beta}$. Under such a transformation, the infinitesimal interval $ds^2$, Equation (\[rg5\]), remains invariant and the ten components of the (symmetric) tensor $T^{\alpha\beta}$ are transformed according to $T'^{\alpha\beta} = \Lambda^{\alpha}_{\gamma} \,
\Lambda^{\beta}_{\delta} \, T^{\gamma \delta}$. On the other hand, when we pass from a [*locally*]{} inertial reference frame (with coordinates $X^{\alpha}$) to an [*extended*]{} non-inertial reference frame (with coordinates $x^{\mu}$; $\mu = 0,1,2,3$), the transformation connecting the $X^{\alpha}$ to the $x^{\mu}$ is no longer a [*linear*]{} transformation (like the Lorentz transformation) but becomes a [*non-linear*]{} transformation $X^{\alpha} = X^{\alpha} (x^{\mu})$ that can take any form whatsoever. Because of this, the value of the infinitesimal interval $ds^2$, when expressed in a general, extended reference frame, will take a more complicated form than the very simple one given by Equation (\[rg5\]) that it had in a reference frame that was locally in free-fall. In fact, by differentiating the non-linear functions $X^{\alpha} = X^{\alpha} (x^{\mu})$ we obtain the relation $dX^{\alpha} = \partial X^{\alpha} / \partial x^{\mu}
\, dx^{\mu}$. By substituting this relation into (\[rg5\]) we then obtain $$\label{rg6}
ds^2 = g_{\mu\nu} (x^{\lambda}) \, dx^{\mu} \, dx^{\nu} \, ,$$ where the indices $\mu , \nu$ are summed over $0,1,2,3$ and where the ten functions $g_{\mu\nu} (x)$ (symmetric over the indices $\mu$ and $\nu$) of the four variables $x^{\lambda}$ are defined, point by point (meaning that for each point $x^{\lambda}$ we consider a reference frame that is locally freely falling at $x$, with local coordinates $X_x^{\alpha}$) by $g_{\mu\nu} (x) =
\eta_{\alpha\beta} \, \partial X_x^{\alpha} (x) / \partial x^{\mu}
\, \partial X_x^{\beta} (x) / \partial x^{\nu}$. Because of the nonlinearity of the functions $X^{\alpha} (x)$, the functions $g_{\mu\nu} (x)$ generally depend in a nontrivial way on the coordinates $x^{\lambda}$.
The local chrono-geometry of space-time thus appears to be given, not by the simple Minkowskian metric (\[rg2\]), with constant coefficients $\eta_{\mu\nu}$, but by a quadratic metric of a much more general type, Equation (\[rg6\]), with coefficients $g_{\mu\nu} (x)$ that vary from point to point. Such general metric spaces had been introduced and studied by Gauss and Riemann in the nineteenth century (in the case where the quadratic form (\[rg6\]) is positive definite). They carry the name [*Riemannian spaces*]{} or [*curved spaces*]{}. (In the case of interest for Einstein’s theory, where the quadratic form (\[rg6\]) is not positive definite, one speaks of a pseudo-Riemannian metric.)
We do not have the space here to explain in detail the various geometric structures in a Riemannian space that are derivable from the data of the infinitesimal interval (\[rg6\]). Let us note simply that given Equation (\[rg6\]), which gives the distance $ds$ between two infinitesimally separated points, we are able, through integration along a curve, to define the length of an arbitrary curve connecting two widely separated points $A$ and $B$: $L_{AB} = \int_A^B ds$. One can then define the “straightest possible line” between two given points $A$ and $B$ to be the shortest line, in other words the curve that minimizes (or, more generally, extremizes) the integrated distance $L_{AB}$. These straightest possible lines are called [*geodesic curves*]{}. To give a simple example, the geodesics of a spherical surface (like the surface of the Earth) are the great circles (with radius equal to the radius of the sphere). If one mathematically writes the condition for a curve, as given by its parametric representation $x^{\mu} = x^{\mu} (s)$, where $s$ is the length along the curve, to extremize the total length $L_{AB}$ one finds that $x^{\mu}
(s)$ must satisfy the following second-order differential equation: $$\label{rg7}
\frac{d^2 \, x^{\lambda}}{ds^2} + \Gamma_{\mu\nu}^{\lambda} (x) \,
\frac{dx^{\mu}}{ds} \, \frac{dx^{\nu}}{ds} = 0 \, ,$$ where the quantities $ \Gamma_{\mu\nu}^{\lambda}$, known as the [*Christoffel coefficients* ]{} or [*connection coefficients*]{}, are calculated, at each point $x$, from the [*metric components*]{} $g_{\mu\nu} (x)$ by the equation $$\label{rg8}
\Gamma_{\mu\nu}^{\lambda} \equiv \frac{1}{2} \, g^{\lambda\sigma}
(\partial_{\mu} \, g_{\nu\sigma} + \partial_{\nu} \, g_{\mu\sigma} -
\partial_{\sigma} \, g_{\mu\nu}) \, ,$$ where $g^{\mu\nu}$ denotes the matrix inverse to $g_{\mu\nu}$ ($g^{\mu\sigma} \, g_{\sigma\nu} = \delta_{\nu}^{\mu}$ where the Kronecker symbol $\delta_{\nu}^{\mu}$ is equal to $1$ when $\mu =
\nu$ and $0$ otherwise) and where $\partial_{\mu} \equiv \partial
/ \partial x^{\mu}$ denotes the partial derivative with respect to the coordinate $x^{\mu}$. To give a very simple example: in the Poincaré-Minkowski space-time the components of the metric are constant, $g_{\mu\nu} = \eta_{\mu\nu}$ (when we use an inertial reference frame). Because of this, the connection coefficients (\[rg8\]) vanish in an inertial reference frame, and the differential equation for geodesics reduces to $d^2 \, x^{\lambda}
/ ds^2 = 0$, whose solutions are ordinary straight lines: $x^{\lambda} (s) = a^{\lambda} \, s + b^{\lambda}$. On the other hand, in a general “curved” space-time (meaning one with components $g_{\mu\nu}$ that depend in an arbitrary way on the point $x$) the geodesics cannot be [*globally*]{} represented by straight lines. One can nevertheless show that it always remains possible, for any $g_{\mu\nu} (x)$ whatsoever, to change coordinates $x^{\mu} \to X^{\alpha} (x)$ in such a way that the connection coefficients $\Gamma_{\beta\gamma}^{\alpha}$, in the new system of coordinates $X^{\alpha}$, vanish [*locally*]{}, at a given point $X_0^{\alpha}$ (or even along an arbitrary curve). Such [*locally geodesic*]{} coordinate systems realize Einstein’s equivalence principle mathematically: up to terms of second order, the components $g_{\alpha\beta} (X)$ of a “curved” metric in locally geodesic coordinates $X^{\alpha}$ ($ds^2 = g_{\alpha\beta}
(X) \, dX^{\alpha} \, dX^{\beta}$) can be identified with the components of a “flat” Poincaré-Minkowski metric: $g_{\alpha\beta} (X) = \eta_{\alpha\beta} + {\mathcal O}
((X-X_0)^2)$, where $X_0$ is the point around which we expand.
Einstein’s Equations: Elastic Space-Time {#sec5}
========================================
Having postulated that a consistent relativistic theory of the gravitational field should include the consideration of a far-reaching generalization of the Poincaré-Minkowski space-time, Equation (\[rg6\]), Einstein concluded that the same ten functions ${\mbox{\boldmath${g}$}}_{\mu\nu} (x)$ should describe both the [**g**]{}eometry of space-time as well as [**g**]{}ravitation. He therefore got down to the task of finding which equations must be satisfied by the “geometric-gravitational field” $g_{\mu\nu} (x)$. He was guided in this search by three principles. The first was the [*principle of general relativity*]{}, which asserts that in the presence of a gravitational field one should be able to write the fundamental laws of physics (including those governing the gravitational field itself) in the same way in any coordinate system whatsoever. The second was that the “source” of the gravitational field should be the energy-momentum tensor $T^{\mu\nu}$. The third was a principle of [*correspondence*]{} with earlier physics: in the limit where one neglects gravitational effects, $g_{\mu\nu} (x) = \eta_{\mu\nu}$ should be a solution of the equations being sought, and there should also be a so-called [*Newtonian*]{} limit where the new theory reduces to Newton’s theory of gravity.
Note that the principle of general relativity (contrary to appearances and contrary to what Einstein believed for several years) has a different physical status than the principle of special relativity. The principle of special relativity was a symmetry principle for the structure of space-time that asserted that physics is [*the same*]{} in a particular class of reference frames, and therefore that certain “corresponding” phenomena occur in exactly the same way in different reference frames (“active” transformations). On the other hand, the principle of general relativity is a [*principle of indifference*]{}: the phenomena do not (in general) take place in the same way in different coordinate systems. However, none of these (extended) coordinate systems enjoys any privileged status with respect to the others.
The principle asserting that the energy-momentum tensor $T^{\mu\nu}$ should be the source of the gravitational field is founded on two ideas: the relations $E = m_i \, c^2$ and the weak principle of equivalence $m_i = m_g$ show that, in the Newtonian limit, the source of gravitation, the gravitational mass $m_g$, is equal to the total energy of the body considered, or in other words the integral over space of the energy density $T^{00}$, up to the factor $c^{-2}$. Therefore at least one of the components of the tensor $T^{\mu\nu}$ must play the role of source for the gravitational field. However, since the gravitational field is encoded, according to Einstein, by the ten components of the metric $g_{\mu\nu}$, it is natural to suppose that the source for $g_{\mu\nu}$ must also have ten components, which is precisely the case for the (symmetric) tensor $T^{\mu\nu}$.
In November of 1915, after many years of conceptually arduous work, Einstein wrote the final form of the theory of general relativity [@livres]. [*Einstein’s equations*]{} are non-linear, second-order partial differential equations for the geometric-gravitational field $g_{\mu\nu}$, containing the energy-momentum tensor $T_{\mu\nu}
\equiv g_{\mu\kappa} \, g_{\nu\lambda} \, T^{\kappa\lambda}$ on the right-hand side. They are written as follows: $$\label{rg9}
R_{\mu\nu} - \frac{1}{2} \, R \, g_{\mu\nu} = \frac{8\pi \, G}{c^4} \,
T_{\mu\nu}$$ where $G$ is the (Newtonian) gravitational constant, $c$ is the speed of light, and $R \equiv g^{\mu\nu} \, R_{\mu\nu}$ and the [*Ricci tensor*]{} $R_{\mu\nu}$ are calculated as a function of the connection coefficients $\Gamma_{\mu\nu}^{\lambda}$ (\[rg8\]) in the following way: $$\label{rg10}
R_{\mu\nu} \equiv \partial_{\alpha} \, \Gamma_{\mu\nu}^{\alpha} - \partial_{\nu}
\, \Gamma_{\mu\alpha}^{\alpha} + \Gamma_{\beta\alpha}^{\alpha} \,
\Gamma_{\mu\nu}^{\beta} - \Gamma_{\beta\nu}^{\alpha} \,
\Gamma_{\mu\alpha}^{\beta} \, .$$
One can show that, in a four-dimensional space-time, the three principles we have described previously uniquely determine Einstein’s equations (\[rg9\]). It is nevertheless remarkable that these equations may also be developed from points of view that are completely different from the one taken by Einstein. For example, in the 1960s various authors (in particular Feynman, Weinberg and Deser; see references in [@tests]) showed that Einstein’s equations could be obtained from a purely [*dynamical*]{} approach, founded on the consistency of interactions of a long-range spin $2$ field, without making any appeal, as Einstein had, to the [*geometric*]{} notions coming from mathematical work on Riemannian spaces. Let us also note that if we relax one of the principles described previously (as Einstein did in 1917) we can find a generalization of Equation (\[rg9\]) in which one adds the term $+ \, \Lambda \, g_{\mu\nu}$ to the left-hand side, where $\Lambda$ is the so-called [*cosmological constant*]{}. Such a modification was proposed by Einstein in 1917 in order to be able to write down a globally homogeneous and [*stationary*]{} cosmological solution. Einstein rejected this additional term after work by Friedmann (1922) showed the existence of [*expanding*]{} cosmological solutions of general relativity and after the observational discovery (by Hubble in 1929) of the expanding motion of galaxies within the universe. However, recent cosmological data have once again made this possibility fashionable, although in the fundamental physics of today one tends to believe that a term of the type $\Lambda \,
g_{\mu\nu}$ should be considered as a particular physical contribution to the right-hand side of Einstein’s equations (more precisely, as the stress-energy tensor of the [*vacuum*]{}, $T_{\mu\nu}^V = - \frac{c^4}{8\pi G} \, \Lambda \, g_{\mu\nu}$), rather than as a universal geometric modification of the left-hand side.
Let us now comment on the physical meaning of Einstein’s equations (\[rg9\]). The essential new idea is that the chrono-geometric structure of space-time, Equation (\[rg6\]), in other words the structure that underlies all of the measurements that one could locally make of duration, $dT$, and of distance, $dL$, (we recall that, locally, $ds^2 = dL^2 - c^2 \, dT^2$) is no longer a rigid structure that is given a priori, once and for all (as was the case for the structure of Poincaré-Minkowski space-time), but instead has become a [*field*]{}, a dynamical or [*elastic*]{} structure, which is created and/or deformed by the presence of an energy-momentum distribution. See Figure \[fig2\], which visualizes the “elastic” geometry of space-time in the theory of general relativity by representing, around each point $x$, the locus of points (assumed to be infinitesimally close to $x$) separated from $x$ by a constant (squared) interval: $ds^2 =
\varepsilon^2$. As in the case of Poincaré-Minkowski space-time (Figure \[fig1\]), one arrives at a “field” of hyperboloids. However, this field of hyperboloids no longer has a “rigid” and homogeneous structure.
$$\includegraphics[width=50mm]{Fig2.eps}$$
The [*space-time field*]{} $g_{\mu\nu} (x)$ describes the variation from point to point of the chrono-geometry as well as all gravitational effects. The simplest example of space-time chrono-geometric [*elasticity*]{} is the effect that the proximity of a mass has on the “local rate of flow for time.” In concrete terms, if you separate two twins at birth, with one staying on the surface of the Earth and the other going to live on the peak of a very tall mountain (in other words farther from the Earth’s center), and then reunite them after 100 years, the “highlander” will be older (will have lived longer) than the twin who stayed on the valley floor. Everything takes place as if time flows more slowly the closer one is to a given distribution of mass-energy. In mathematical terms this effect is due to the fact that the coefficient $g_{00} (x)$ of $(dx^0)^2$ in Equation (\[rg6\]) is deformed with respect to its value in special relativity, $g_{00}^{\rm Minkowski} = \eta_{00} = -1$, to become $g_{00}^{\rm
Einstein} (x) \simeq -1 + 2GM / c^2 r$, where $M$ is the Earth’s mass (in our example) and $r$ the distance to the center of the Earth. In the example considered above of terrestrial twins the effect is extremely small (a difference in the amount of time lived of about one second over 100 years), but the effect is real and has been verified many times using atomic clocks (see the references in [@tests]). Let us mention that today this “Einstein effect” has important practical repercussions, for example in aerial or maritime navigation, for the piloting of automobiles, or even farm machinery, etc. In fact, the GPS (Global Positioning System), which uses the data transmitted by a constellation of atomic clocks on board satellites, incorporates the Einsteinian deformation of space-time chrono-geometry into its software. The effect is only on the order of one part in a billion, but if it were not taken into account, it would introduce an unacceptably large error into the GPS, which would continually grow over time. Indeed, GPS performance relies on the high stability of the orbiting atomic clocks, a stability better than $10^{-13}$, or in other words 10,000 times greater than the apparent change in frequency($\sim 10^{-9}$) due to the Einsteinian deformation of the chrono-geometry.
The Weak-Field Limit and the Newtonian Limit {#sec6}
============================================
To understand the physical consequences of Einstein’s equations (\[rg9\]), it is useful to begin by considering the limiting case of [*weak*]{} geometric-gravitational fields, namely the case where $g_{\mu\nu} (x) = \eta_{\mu\nu} + h_{\mu\nu} (x)$, with perturbations $h_{\mu\nu} (x)$ that are very small with respect to unity: $\vert h_{\mu\nu} (x) \vert \ll 1$. In this case, a simple calculation (that we encourage the reader to perform) starting from Definitions (\[rg8\]) and (\[rg10\]) above, leads to the following explicit form of Einstein’s equations (where we ignore terms of order $h^2$ and $hT$): $$\label{rg11}
\Box \, h_{\mu\nu} - \partial_{\mu} \, \partial^{\alpha} \, h_{\alpha \nu} -
\partial_{\nu} \, \partial^{\alpha} \, h_{\alpha\mu} + \partial_{\mu\nu} \,
h_{\alpha}^{\alpha} = - \frac{16 \, \pi \, G}{c^4} \, \tilde T_{\mu\nu} \, ,$$ where $\Box = \eta^{\mu\nu} \, \partial_{\mu\nu} = \Delta -
\partial_0^2 =
\partial^2 / \partial x^2 + \partial^2 / \partial y^2 + \partial^2 / \partial
z^2 - c^{-2} \, \partial^2 / \partial t^2$ denotes the “flat” d’Alembertian (or wave operator; $x^{\mu} = (ct , x, y , z)$), and where indices in the upper position have been raised by the inverse $\eta^{\mu\nu}$ of the flat metric $\eta_{\mu\nu}$ (numerically $\eta^{\mu\nu} = \eta_{\mu\nu}$, meaning that $-\eta^{00} = \eta^{11} = \eta^{22} = \eta^{33} = +1$). For example $\partial^{\alpha} \, h_{\alpha\nu}$ denotes $\eta^{\alpha\beta} \, \partial_{\alpha} \, h_{\beta\nu}$ and $h_{\alpha}^{\alpha} \equiv \eta^{\alpha\beta} \, h_{\alpha\beta}
= -h_{00} + h_{11} + h_{22} + h_{33}$. The “source” $\tilde
T_{\mu\nu}$ appearing on the right-hand side of (\[rg11\]) denotes the combination $\tilde T_{\mu\nu} \equiv T_{\mu\nu} -
\frac{1}{2} \, T_{\alpha}^{\alpha} \, \eta_{\mu\nu}$ (when space-time is four-dimensional).
The “linearized” approximation (\[rg11\]) of Einstein’s equations is analogous to Maxwell’s equations $$\label{rg12}
\Box \, A_{\mu} - \partial_{\mu} \, \partial^{\alpha} \, A_{\alpha} = -4\pi \,
J_{\mu} \, ,$$ connecting the electromagnetic four-potential $A_{\mu} \equiv
\eta_{\mu\nu} \, A^{\nu}$ (where $A^0 = V$, $A^i = {\mbox{\boldmath${A}$}}$, $i =
1,2,3$) to the four-current density $J_{\mu} \equiv \eta_{\mu\nu}
\, J^{\nu}$ (where $J^0 = \rho$ is the charge density and $J^i =
{\mbox{\boldmath${J}$}}$ is the current density). Another analogy is that the structure of the left-hand side of Maxwell’s equations implies that the “source” $J_{\mu}$ appearing on the right-hand side must satisfy $\partial^{\mu} \, J_{\mu} = 0$ ($\partial^{\mu}
\equiv \eta^{\mu\nu} \,
\partial_{\nu}$), which expresses the conservation of electric charge. Likewise, the structure of the left-hand side of the linearized form of Einstein’s equations (\[rg11\]) implies that the “source” $T_{\mu\nu} = \tilde T_{\mu\nu} - \frac{1}{2} \, \tilde
T_{\alpha}^{\alpha} \, \eta_{\mu\nu}$ must satisfy $\partial^{\mu}
\, T_{\mu\nu} = 0$, which expresses the conservation of energy and momentum of matter. (The structure of the left-hand side of the exact form of Einstein’s equations (\[rg9\]) implies that the source $T_{\mu\nu}$ must satisfy the more complicated equation $\partial_{\mu} \, T^{\mu\nu} + \Gamma_{\sigma\mu}^{\mu} \,
T^{\sigma\nu} + \Gamma_{\sigma\mu}^{\nu} \, T^{\mu\sigma} = 0$, where the terms in $\Gamma T$ can be interpreted as describing an exchange of energy and momentum between matter and the gravitational field.) The major difference is that, in the case of electromagnetism, the field $A_{\mu}$ and its source $J_{\mu}$ have a single space-time index, while in the gravitational case the field $h_{\mu\nu}$ and its source $\tilde T_{\mu\nu}$ have two space-time indices. We shall return later to this analogy/difference between $A_{\mu}$ and $h_{\mu\nu}$ which suggests the existence of a certain relation between gravitation and electromagnetism.
We recover the Newtonian theory of gravitation as the limiting case of Einstein’s theory by assuming not only that the gravitational field is a weak deformation of the flat Minkowski space-time ($h_{\mu\nu} \ll 1$), but also that the field $h_{\mu\nu}$ is slowly varying ($\partial_0 \, h_{\mu\nu} \ll
\partial_i \, h_{\mu\nu}$) and that its source $T_{\mu\nu}$ is non-relativistic ($T_{ij} \ll T_{0i} \ll T_{00}$). Under these conditions Equation (\[rg11\]) leads to a Poisson-type equation for the purely temporal component, $h_{00}$, of the space-time field, $$\label{rg13}
\Delta \, h_{00} = - \frac{16 \, \pi \, G}{c^4} \, \tilde T_{00} = - \frac{8 \,
\pi \, G}{c^4} \, (T_{00} + T_{ii}) \simeq - \frac{8 \, \pi \, G}{c^4} \, T_{00}
\, ,$$ where $\Delta = \partial_x^2 + \partial_y^2 + \partial_z^2$ is the Laplacian. Recall that, according to Laplace and Poisson, Newton’s theory of gravity is summarized by saying that the gravitational field is described by a single potential $U(x)$, produced by the mass density $\rho (x)$ according to the Poisson equation $\Delta
U = - 4 \, \pi \, G \rho$, that determines the acceleration of a test particle placed in the exterior field $U(x)$ according to the equation $d^2 \, x^i / dt^2 =
\partial_i \, U(x) \equiv
\partial U / \partial x^i$. Because of the relation $m_i = m_g = E/c^2$ one can identify $\rho = T^{00} / c^2$. We therefore find that (\[rg13\]) reproduces the Poisson equation if $h_{00} = + \, 2
\, U / c^2$. It therefore remains to verify that Einstein’s theory indeed predicts that a non-relativistic test particle is accelerated by a space-time field according to $d^2 \, x^i / dt^2
\simeq \frac{1}{2} \, c^2 \, \partial_i \, h_{00}$. Einstein understood that this was a consequence of the equivalence principle. In fact, as we discussed in Section \[sec4\] above, the principle of equivalence states that the gravitational field is (locally) erased in a locally inertial reference frame $X^{\alpha}$ (such that $g_{\alpha\beta} (X) = \eta_{\alpha\beta}
+ {\mathcal O} ((X-X_0)^2)$). In such a reference frame, the laws of special relativity apply at the point $X_0$. In particular an isolated (and electrically neutral) body must satisfy a principle of inertia in this frame: its center of mass moves in a straight line at constant speed. In other words it satisfies the equation of motion $d^2 \, X^{\alpha} / ds^2 = 0$. By passing back to an arbitrary (extended) coordinate system $x^{\mu}$, one verifies that this equation for inertial motion transforms into the geodesic equation (\[rg7\]). Therefore (\[rg7\]) describes falling bodies, such as they are observed in arbitrary extended reference frames (for example a reference frame at rest with respect to the Earth or at rest with respect to the center of mass of the solar system). From this one concludes that the relativistic analog of the Newtonian field of gravitational acceleration, ${\mbox{\boldmath${g}$}}(x) = {\mbox{\boldmath${\nabla}$}}\, U(x)$, is $g^{\lambda} (x)
\equiv - c^2 \, \Gamma_{\mu\nu}^{\lambda} \, dx^{\mu} /ds \,
dx^{\nu} / ds$. By considering a particle whose motion is slow with respect to the speed of light ($dx^i / ds \ll dx^0 / ds
\simeq 1$) one can easily verify that $g^i (x) \simeq -c^2 \,
\Gamma_{00}^i$. Finally, by using the definition (\[rg8\]) of $\Gamma_{\mu\nu}^{\alpha}$, and the hypothesis of weak fields, one indeed verifies that $g^i (x) \simeq \frac{1}{2} \, c^2 \,
\partial_i \, h_{00}$, in perfect agreement with the identification $h_{00} = 2 \, U/c^2$ anticipated above. We encourage the reader to personally verify this result, which contains the very essence of Einstein’s theory: gravitational motion is no longer described as being due to a force, but is identified with motion that is “as inertial as possible” within a space-time whose chrono-geometry is deformed in the presence of a mass-energy distribution.
Finding the Newtonian theory as a limiting case of Einstein’s theory is obviously a necessity for seriously considering this new theory. But of course, from the very beginning Einstein explored the observational consequences of general relativity that go beyond the Newtonian description of gravitation. We have already mentioned one of these above: the fact that $g_{00} = \eta_{00} +
h_{00} \simeq -1 + 2 U (x) / c^2$ implies a distortion in the relative measurement of time in the neighborhood of massive bodies. In 1907 (as soon as he had developed the principle of equivalence, and long before he had obtained the field equations of general relativity) Einstein had predicted the existence of such a distortion for measurements of time and frequency in the presence of an external gravitational field. He realized that this should have observable consequences for the frequency, as observed on Earth, of the spectral rays emitted from the surface of the Sun. Specifically, a spectral ray of (proper local) frequency $\nu_0$ emitted from a point $x_0$ where the (stationary) gravitational potential takes the value $U ({\mbox{\boldmath${x}$}}_0)$ and observed (via electromagnetic signals) at a point $x$ where the potential is $U({\mbox{\boldmath${x}$}})$ should appear to have a frequency $\nu$ such that $$\label{rg14}
\frac{\nu}{\nu_0} = \sqrt{\frac{g_{00} (x_0)}{g_{00} (x)}} \simeq 1 +
\frac{1}{c^2} \, [U({\mbox{\boldmath${x}$}}) - U ({\mbox{\boldmath${x}$}}_0)] \, .$$ In the case where the point of emission $x_0$ is in a gravitational potential well deeper than the point of observation $x$ (meaning that $U ({\mbox{\boldmath${x}$}}_0)
> U({\mbox{\boldmath${x}$}})$) one has $\nu < \nu_0$, in other words a [*reddening*]{} effect on frequencies. This effect, which was predicted by Einstein in 1907, was unambiguously verified only in the 1960s, in experiments by Pound and collaborators over a height of about twenty meters. The most precise verification (at the level of $\sim 10^{-4}$) is due to Vessot and collaborators, who compared a hydrogen maser, launched aboard a rocket that reached about 10,000 km in altitude, to a clock of similar construction on the ground. Other experiments compared the times shown on clocks placed aboard airplanes to clocks remaining on the ground. (For references to these experiments see [@tests].) As we have already mentioned, the “Einstein effect” (\[rg14\]) must be incorporated in a crucial way into the software of satellite positioning systems such as the GPS.
In 1907, Einstein also pointed out that the equivalence principle would suggest that light rays should be deflected by a gravitational field. Indeed, a generalization of the reasoning given above for the motion of particles in an external gravitational field, based on the principle of equivalence, shows that light must itself follow a trajectory that is “as inertial as possible,” meaning a geodesic of the curved space-time. Light rays must therefore satisfy the geodesic equation (\[rg7\]). (The only difference from the geodesics followed by material particles is that the parameter $s$ in Equation (\[rg7\]) can no longer be taken equal to the “length” along the geodesic, since a “light” geodesic must also satisfy the constraint $g_{\mu\nu} (x) \, dx^{\mu} \,
dx^{\nu} = 0$, ensuring that its speed is equal to $c$, when it is measured in a locally inertial reference frame.) Starting from Equation (\[rg7\]) one can therefore calculate to what extent light is deflected when it passes through the neighborhood of a large mass (such as the Sun). One nevertheless soon realizes that in order to perform this calculation one must know more than the component $h_{00}$ of the gravitational field. The other components of $h_{\mu\nu}$, and in particular the spatial components $h_{ij}$, come into play in a crucial way in this calculation. This is why it was only in November of 1915, after having obtained the (essentially) final form of his theory, that Einstein could predict the total value of the deflection of light by the Sun. Starting from the linearized form of Einstein’s equations (\[rg11\]) and continuing by making the “non-relativistic” simplifications indicated above ($T_{ij} \ll
T_{0i} \ll T_{00}$, $\partial_0 \, h \ll
\partial_i \, h$) it is easy to see that the spatial component $h_{ij}$, like $h_{00}$, can be written (after a helpful choice of coordinates) in terms of the Newtonian potential $U$ as $h_{ij}
(x) \simeq + \, 2 \, U(x) \, \delta_{ij} / c^2$, where $\delta_{ij}$ takes the value $1$ if $i=j$ and $0$ otherwise ($i,j
= 1,2,3$). By inserting this result, as well as the preceding result $h_{00} = + \, 2 \, U / c^2$, into the geodesic equation (\[rg7\]) for the motion of light, one finds (as Einstein did in 1915) that general relativity predicts that the Sun should deflect a ray of light by an angle $\theta = 4GM / (c^2 b)$ where $b$ is the impact parameter of the ray (meaning its minimum distance from the Sun). As is well known, the confirmation of this effect in 1919 (with rather weak precision) made the theory of general relativity and its creator famous.
The Post-Newtonian Approximation and Experimental Confirmations in the Regime of Weak and Quasi-Stationary Gravitational Fields {#sec7}
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We have already pointed out some of the experimental confirmations of the theory of general relativity. At present, the extreme precision of certain measurements of time or frequency in the solar system necessitates a very careful account of the modifications brought by general relativity to the Newtonian description of space-time. As a consequence, general relativity is used in a great number of situations, from astronomical or geophysical research (such as very long range radio interferometry, radar tracking of the planets, and laser tracking of the Moon or artificial satellites) to metrological, geodesic or other applications (such as the definition of international atomic time, precision cartography, and the G.P.S.). To do this, the so-called [*post-Newtonian*]{} approximation has been developed. This method involves working in the Newtonian limit sketched above while keeping the terms of higher order in the small parameter $$\varepsilon \sim \frac{v^2}{c^2} \sim \vert h_{\mu\nu} \vert \sim \vert
\partial_0 \, h / \partial_i \, h \vert^2 \sim \vert T^{0i} / T^{00} \vert^2
\sim \vert T^{ij} / T^{00} \vert \, ,$$ where $v$ denotes a characteristic speed for the elements in the system considered.
For all present applications of general relativity to the solar system it suffices to include the [*first post-Newtonian approximation*]{}, in other words to keep the relative corrections of order $\varepsilon$ to the Newtonian predictions. Since the theory of general relativity was poorly verified for a long time, one found it useful (as in the pioneering work of A. Eddington, generalized in the 1960s by K. Nordtvedt and C.M. Will) to study not only the precise predictions of the equations (\[rg9\]) defining Einstein’s theory, but to also consider possible deviations from these predictions. These possible deviations were parameterized by means of several non-dimensional “post-Newtonian” parameters. Among these parameters, two play a key role: $\gamma$ and $\beta$. The parameter $\gamma$ describes a possible deviation from general relativity that comes into play starting at the linearized level, in other words one that modifies the linearized approximation given above. More precisely, it is defined by writing that the difference $h_{ij} \equiv g_{ij} -
\delta_{ij}$ between the spatial metric and the Euclidean metric can take the value $h_{ij} = 2 \gamma \, U \, \delta_{ij} / c^2$ (in a suitable coordinate system), rather than the value $h_{ij}^{\rm GR} = 2 \, U \, \delta_{ij} / c^2$ that it takes in general relativity, thus differing by a factor $\gamma$. Therefore, by definition $\gamma$ takes the value $1$ in general relativity, and $\gamma -1$ measures the possible deviation with respect to this theory. As for the parameter $\beta$ (or rather $\beta - 1$), it measures a possible deviation (with respect to general relativity) in the value of $h_{00} \equiv g_{00} -
\eta_{00}$. The value of $h_{00}$ in general relativity is $h_{00}^{\rm GR} = 2 \, U / c^2 - 2 \, U^2 / c^4$, where the first term (discussed above) reproduces the Newtonian approximation (and cannot therefore be modified, as the idea is to parameterize gravitational physics beyond Newtonian predictions) and where the second term is obtained by solving Einstein’s equations (\[rg9\]) at the second order of approximation. One then writes an $h_{00}$ of a more general parameterized type, $h_{00} = 2 \, U
/ c^2 - 2 \, \beta \, U^2 / c^4$, where, by definition, $\beta$ takes the value $1$ in general relativity. Let us finally point out that the parameters $\gamma - 1$ and $\beta - 1$ completely parameterize the post-Newtonian regime of the simplest theoretical alternatives to general relativity, namely the tensor-scalar theories of gravitation. In these theories, the gravitational interaction is carried by two fields at the same time: a massless tensor (spin 2) field coupled to $T^{\mu\nu}$, and a massless scalar (spin 0) field $\varphi$ coupled to the trace $T_{\alpha}^{\alpha}$. In this case the parameter $-(\gamma -1)$ plays the key role of measuring the ratio between the scalar coupling and the tensor coupling.
All of the experiments performed to date within the solar system are compatible with the predictions of general relativity. When they are interpreted in terms of the post-Newtonian (and “post-Einsteinian”) parameters $\gamma - 1$ and $\beta - 1$, they lead to strong constraints on possible deviations from Einstein’s theory. We make note of the following among tests performed in the solar system: the deflection of electromagnetic waves in the neighborhood of the Sun, the gravitational delay (‘Shapiro effect’) of radar signals bounced from the Viking lander on Mars, the global analysis of solar system dynamics (including the advance of planetary perihelia), the sub-centimeter measurement of the Earth-Moon distance obtained from laser signals bounced off of reflectors on the Moon’s surface, etc. At present (October of 2006) the most precise test (that has been published) of general relativity was obtained in 2003 by measuring the ratio $1+y \equiv
f/f_0$ between the frequency $f_0$ of radio waves sent from Earth to the Cassini space probe and the frequency $f$ of coherent radio waves sent back (with the same local frequency) from Cassini to Earth and compared (on Earth) to the emitted frequency $f_0$. The main contribution to the small quantity $y$ is an effect equal, in general relativity, to $y_{GR} = 8(GM/c^3 \, b) \, db/dt$ (where $b$ is, as before, the impact parameter) due to the propagation of radio waves in the geometry of a space-time deformed by the Sun: $ds^2 \simeq - (1-2 \, U / c^2) \, c^2 \, dt^2 + (1+2 \, U / c^2)
(dx^2 + dy^2 + dz^2)$, where $U = GM/r$. The maximum value of the frequency change predicted by general relativity was only $\vert
y_{\rm GR} \vert \lesssim 2 \times 10^{-10}$ for the best observations, but thanks to an excellent frequency stability $\sim
10^{-14}$ (after correction for the perturbations caused by the solar corona) and to a relatively large number of individual measurements spread over 18 days, this experiment was able to verify Einstein’s theory at the remarkable level of $\sim 10^{-5}$ [@Cassini]. More precisely, when this experiment is interpreted in terms of the post-Newtonian parameters $\gamma - 1$ and $\beta - 1$, it gives the following limit for the parameter $\gamma - 1$ [@Cassini] $$\label{rg15}
\gamma - 1 = (2.1 \pm 2.3) \times 10^{-5} \, .$$ As for the best present-day limit on the parameter $\beta-1$, it is smaller than $10^{-3}$ and comes from the non-observation, in the data from lasers bounced off of the Moon, of any eventual polarization of the Moon’s orbit in the direction of the Sun (‘Nordtvedt effect’; see [@tests] for references) $$\label{rg16}
4 (\beta - 1) - (\gamma - 1) = -0.0007 \pm 0.0010 \, .$$
Although the theory of general relativity is one of the best verified theories in physics, scientists continue to design and plan new or increasingly precise tests of the theory. This is the case in particular for the space mission Gravity Probe B (launched by NASA in April of 2004) whose principal aim is to directly observe a prediction of general relativity that states (intuitively speaking) that space is not only “elastic,” but also “fluid.” In the nineteenth century Foucalt invented both the gyroscope and his famous pendulum in order to render Newton’s absolute (and rigid) space directly observable. His experiments in fact showed that, for example, a gyroscope on the surface of the Earth continued, despite the Earth’s rotation, to align itself in a direction that is “fixed” with respect to the distant stars. However, in 1918, when Lense and Thirring analyzed some of the consequences of the (linearized) Einstein equations (\[rg11\]), they found that general relativity predicts, among other things, the following phenomenon: the rotation of the Earth (or any other ball of matter) creates a particular deformation of the chrono-geometry of space-time. This deformation is described by the “gravito-magnetic” components $h_{0i}$ of the metric, and induces an effect analogous to the “rotation drag” effect caused by a ball of matter turning in a fluid: the rotation of the Earth (minimally) drags all of the space around it, causing it to continually “turn,” as a fluid would.[^5] This “rotation of space” translates, in an observable way, into a violation of the effects predicted by Newton and confirmed by Foucault’s experiments: in particular, a gyroscope no longer aligns itself in a direction that is “fixed in absolute space,” rather its axis of rotation is “dragged” by the rotating motion of the local space where it is located. This effect is much too small to be visible in Foucalt’s experiments. Its observation by Gravity Probe B (see [@gpb] and the contribution of John Mester to this Poincaré seminar) is important for making Einstein’s revolutionary notion of a fluid space-time tangible to the general public.
Up till now we have only discussed the regime of weak and slowly varying gravitational fields. The theory of general relativity predicts the appearance of new phenomena when the gravitational field becomes strong and/or rapidly varying. (We shall not here discuss the cosmological aspects of relativistic gravitation.)
Strong Gravitational Fields and Black Holes {#sec8}
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The regime of strong gravitational fields is encountered in the physics of [*gravitationally condensed bodies*]{}. This term designates the final states of stellar evolution, and in particular neutron stars and black holes. Recall that most of the life of a star is spent slowly burning its nuclear fuel. This process causes the star to be structured as a series of layers of differentiated nuclear structure, surrounding a progressively denser core (an “onion-like” structure). When the initial mass of the star is sufficiently large, this process ends into a catastrophic phenomenon: the core, already much denser than ordinary matter, collapses in on itself under the influence of its gravitational self-attraction. (This implosion of the central part of the star is, in many cases, accompanied by an explosion of the outer layers of the star—a supernova.) Depending on the quantity of mass that collapses with the core of a star, this collapse can give rise to either a neutron star or a black hole.
A [*neutron star*]{} condenses a mass on the order of the mass of the Sun inside a radius on the order of 10 km. The density in the interior of a neutron star (named thus because neutrons dominate its nuclear composition) is more than 100 million tons per cubic centimeter ($10^{14}$ g/cm$^3$)! It is about the same as the density in the interior of atomic nuclei. What is important for our discussion is that the deformation away from the Minkowski metric in the immediate neighborhood of a neutron star, measured by $h_{00} \sim h_{ii} \sim 2GM/c^2 R$, where $R$ is the radius of the star, is no longer a small quantity, as it was in the solar system. In fact, while $h \sim 2GM/c^2 R$ is on the order of $10^{-9}$ for the Earth and $10^{-6}$ for the Sun, one finds that $h \sim 0.4$ for a typical neutron star ($M \simeq
1.4 \, M_{\odot}$, $R \sim 10$ km). One thus concludes that it is no longer possible, as was the case in the solar system, to study the structure and physics of neutron stars by using the post-Newtonian approximation outlined above. One must consider the exact form of Einstein’s equations (\[rg9\]), with all of their non-linear structure. Because of this, we expect that observations concerning neutron stars will allow us to confirm (or refute) the theory of general relativity in its strongly non-linear regime. We shall discuss such tests below in relation to observations of binary pulsars.
A [*black hole*]{} is the result of a [*continued*]{} collapse, meaning that it does not stop with the formation of an ultra-dense star (such as a neutron star). (The physical concept of a black hole was introduced by J.R. Oppenheimer and H. Snyder in 1939. The global geometric structure of black holes was not understood until some years later, thanks notably to the work of R. Penrose. For a historical review of the idea of black holes see [@israel].) It is a particular structure of curved space-time characterized by the existence of a boundary (called the “black hole surface” or “horizon”) between an exterior region, from which it is possible to emit signals to infinity, and an interior region (of space-time), within which any emitted signal remains trapped. See Figure \[fig3\].
$$\includegraphics[width=85mm]{Fig3.eps}$$
The cones shown in this figure are called “light cones.” They are defined as the locus of points (infinitesimally close to $x$) such that $ds^2 = 0$, with $dx^0 = c dt \geq 0$. Each represents the beginning of the space-time history of a flash of light emitted from a certain point in space-time. The cones whose vertices are located outside of the horizon (the shaded zone) will evolve by spreading out to infinity, thus representing the possibility for electromagnetic signals to reach infinity.
On the other hand, the cones whose vertices are located inside the horizon (the grey zone) will evolve without ever succeeding in escaping the grey zone. It is therefore impossible to emit an electromagnetic signal that reaches infinity from the grey zone. The horizon, namely the boundary between the shaded zone and the unshaded zone, is itself the history of a particular flash of light, emitted from the center of the star over the course of its collapse, such that it asymptotically stabilizes as a space-time cylinder. This space-time cylinder (the asymptotic horizon) therefore represents the space-time history of a bubble of light that, viewed locally, moves outward at the speed $c$, but which globally “runs in place.” This remarkable behavior is a striking illustration of the “fluid” character of space-time in Einstein’s theory. Indeed, one can compare the preceding situation with what may take place around the open drain of an emptying sink: a wave may move along the water, away from the hole, all the while running in place with respect to the sink because of the falling motion of the water in the direction of the drain.
Note that the temporal development of the interior region is limited, terminating in a [*singularity*]{} (the dark gray surface) where the curvature becomes infinite and where the classical description of space and time loses its meaning. This singularity is locally similar to the temporal inverse of a cosmological singularity of the big bang type. This is called a [*big crunch*]{}. It is a space-time frontier, beyond which space-time ceases to exist. The appearance of singularities associated with regions of strong gravitational fields is a generic phenomenon in general relativity, as shown by theorems of R. Penrose and S.W. Hawking.
Black holes have some remarkable properties. First, a [*uniqueness*]{} theorem (due to W. Israel, B. Carter, D.C. Robinson, G. Bunting, and P.O. Mazur) asserts that an isolated, stationary black hole (in Einstein-Maxwell theory) is completely described by three parameters: its mass $M$, its angular momentum $J$, and its electric charge $Q$. The exact solution (called the Kerr-Newman solution) of Einstein’s equations (\[rg11\]) describing a black hole with parameters $M,J,Q$ is explicitly known. We shall here content ourselves with writing the space-time geometry in the simplest case of a black hole: the one in which $J=Q=0$ and the black hole is described only by its mass (a solution discovered by K. Schwarzschild in January of 1916): $$\label{rg17}
ds^2 = - \left( 1-\frac{2GM}{c^2 r} \right) c^2 \, dt^2 + \frac{dr^2}{1 -
\frac{2GM}{c^2 r}} + r^2 (d\theta^2 + \sin^2 \theta \, d\varphi^2) \, .$$ We see that the purely temporal component of the metric, $g_{00} =
- (1 - 2GM / c^2 r)$, vanishes when the radial coordinate $r$ takes the value $r = r_H \equiv 2GM / c^2$. According to the earlier equation (\[rg14\]), it would therefore seem that light emitted from an arbitrary point on the sphere $r_0 = r_H$, when it is viewed by an observer located anywhere in the exterior (in $r
> r_H$), would experience an infinite reddening of its emission frequency ($
\nu / \nu_0 = 0$). In fact, the sphere $r_H = 2GM/c^2$ is the [*horizon*]{} of the Schwarzschild black hole, and no particle (that is capable of emitting light) can remain at rest when $r=r_H$ (nor, a fortiori, when $r < r_H$). To study what happens at the horizon ($r = r_H$) or in the interior ($r < r_H$) of a Schwarzschild black hole, one must use other space-time coordinates than the coordinates $(t,r,\theta , \varphi)$ used in Equation (\[rg17\]). The “big crunch” singularity in the interior of a Schwarzschild black hole, in the coordinates of (\[rg17\]), is located at $r=0$ (which does not describe, as one might believe, a point in space, but rather an instant in time).
The space-time metric of a black hole space-time, such as Equation (\[rg17\]) in the simple case $J=Q=0$, allows one to study the influence of a black hole on particles and fields in its neighborhood. One finds that a black hole is a gravitational potential well that is so deep that any particle or wave that penetrates the interior of the black hole (the region $r < r_H$) will never be able to come out again, and that the total energy of the particle or wave that “falls” into the black hole ends up augmenting the total mass-energy $M$ of the black hole. By studying such black hole “accretion” processes with falling particles (following R. Penrose), D. Christodoulou and R. Ruffini showed that a black hole is not only a potential well, but also a physical object possessing a significant [*free energy*]{} that it is possible, in principle, to extract. Such [*black hole energetics*]{} is encapsulated in the “mass formula” of Christodoulou and Ruffini (in units where $c=1$) $$\label{rg18}
M^2 = \left(M_{\rm irr} + \frac{Q^2}{4 \, GM_{\rm irr}} \right)^2 + \frac{J^2}{4
\, G^2M_{\rm irr}^2} \, ,$$ where $M_{\rm irr}$ denotes the [*irreducible mass*]{} of the black hole, a quantity [*that can only grow, irreversibly*]{}. One deduces from (\[rg18\]) that a rotating $(J \ne 0$) and/or charged ($Q \ne 0$) black hole possesses a free energy $M - M_{\rm
irr} > 0$ that can, in principle, be extracted through processes that reduce its angular momentum and/or its electric charge. Such black hole energy-extraction processes may lie at the origin of certain ultra-energetic astrophysical phenomena (such as quasars or gamma ray bursts). Let us note that, according to Equation (\[rg18\]), (rotating or charged) black holes are the largest reservoirs of free energy in the Universe: in fact, 29% of their mass energy can be stored in the form of rotational energy, and up to 50% can be stored in the form of electric energy. These percentages are much higher than the few percent of nuclear binding energy that is at the origin of all the light emitted by stars over their lifetimes. Even though there is not, at present, irrefutable proof of the existence of black holes in the universe, an entire range of very strong presumptive evidence lends credence to their existence. In particular, more than a dozen X-ray emitting binary systems in our galaxy are most likely made up of a black hole and an ordinary star. Moreover, the center of our galaxy seems to contain a very compact concentration of mass $\sim 3 \times 10^6 M_{\odot}$ that is probably a black hole. (For a review of the observational data leading to these conclusions see, for example, Section 7.6 of the recent book by N. Straumann [@livres].)
The fact that a quantity associated with a black hole, here the irreducible mass $M_{\rm irr}$, or, according to a more general result due to S.W. Hawking, the total area $A$ of the surface of a black hole ($A = 16 \, \pi \, G^2 M_{\rm irr}^2$), can evolve only by irreversibly growing is reminiscent of the second law of thermodynamics. This result led J.D. Bekenstein to interpret the horizon area, $A$, as being proportional to the [*entropy*]{} of the black hole. Such a thermodynamic interpretation is reinforced by the study of the growth of $A$ under the influence of external perturbations, a growth that one can in fact attribute to some local dissipative properties of the black hole surface, notably a surface viscosity and an electrical resistivity equal to 377 ohm (as shown in work by T. Damour and R.L. Znajek). These “thermodynamic” interpretations of black hole properties are based on simple analogies at the level of classical physics, but a remarkable result by Hawking showed that they have real content at the level of quantum physics. In 1974, Hawking discovered that the presence of a horizon in a black hole space-time affected the definition of a quantum particle, and caused a black hole to continuously emit a flux of particles having the characteristic spectrum (Planck spectrum) of thermal emission at the temperature $T = 4 \, \hbar \, G \, \partial M / \partial A$, where $\hbar$ is the reduced Planck constant. By using the general thermodynamic relation connecting the temperature to the energy $E=M$ and the entropy $S$, $T =
\partial M / \partial S$, we see from Hawking’s result (in conformity with Bekenstein’s ideas) that a black hole possesses an [*entropy*]{} $S$ equal (again with $c=1$) to $$\label{rg19}
S = \frac{1}{4} \, \frac{A}{\hbar \, G} \, .$$ The Bekenstein-Hawking formula (\[rg19\]) suggests an unexpected, and perhaps profound, connection between gravitation, thermodynamics, and quantum theory. See Section \[sec11\] below.
Binary Pulsars and Experimental Confirmations in the Regime of Strong and Radiating Gravitational Fields {#sec9}
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[*Binary pulsars*]{} are binary systems made up of a pulsar (a rapidly spinning neutron star) and a very dense companion star (either a neutron star or a white dwarf). The first system of this type (called PSR B1913$ + $16) was discovered by R.A. Hulse and J.H. Taylor in 1974 [@hulse]. Today, a dozen are known. Some of these (including the first-discovered PSR B1913$ + $16) have revealed themselves to be remarkable probes of relativistic gravitation and, in particular, of the regime of strong and/or radiating gravitational fields. The reason for which a binary pulsar allows for the probing of strong gravitational fields is that, as we have already indicated above, the deformation of the space-time geometry in the neighborhood of a neutron star is no longer a small quantity, as it is in the solar system. Rather, it is on the order of unity: $h_{\mu\nu} \equiv g_{\mu\nu} -
\eta_{\mu\nu} \sim 2GM / c^2 R \sim 0.4$. (We note that this value is only 2.5 times smaller than in the extreme case of a black hole, for which $2GM / c^2 R = 1$.) Moreover, the fact that the gravitational interaction propagates at the speed of light (as indicated by the presence of the wave operator, $\Box = \Delta -
c^{-2}
\partial^2 / \partial t^2$ in (\[rg11\])) between the pulsar and its companion is found to play an observationally significant role for certain binary pulsars.
Let us outline how the observational data from binary pulsars are used to probe the regime of strong ($h_{\mu\nu}$ on the order of unity) and/or radiative (effects propagating at the speed $c$) gravitational fields. (For more details on the observational data from binary pulsars and their use in probing relativistic gravitation, see Michael Kramer’s contribution to this Poincaré seminar.) Essentially, a pulsar plays the role of an extremely stable [*clock*]{}. Indeed, the “pulsar phenomenon” is due to the rotation of a bundle of electromagnetic waves, created in the neighborhood of the two magnetic poles of a strongly magnetized neutron star (with a magnetic field on the order of $10^{12}$ Gauss, $10^{12}$ times the size of the terrestrial magnetic field). Since the magnetic axis of a pulsar is not aligned with its axis of rotation, the rapid rotation of the pulsar causes the (inner) magnetosphere as a whole to rotate, and likewise the bundle of electromagnetic waves created near the magnetic poles. The pulsar is therefore analogous to a lighthouse that sweeps out space with two bundles (one per pole) of electromagnetic waves. Just as for a lighthouse, one does not see the pulsar from Earth except when the bundle sweeps the Earth, thus causing a flash of electromagnetic noise with each turn of the pulsar around itself (in some cases, one even sees a secondary flash, due to emission from the second pole, after each half-turn). One can then measure the time of arrival at Earth of (the center of) each flash of electromagnetic noise. The basic observational data of a pulsar are thus made up of a regular, discrete sequence of the [*arrival times*]{} at Earth of these flashes or “pulses.” This sequence is analogous to the signal from a clock: tick, tick, tick, $\ldots$. Observationally, one finds that some pulsars (and in particular those that belong to binary systems) thus define clocks of a stability comparable to the best atomic clocks [@taylor]. In the case of a solitary pulsar, the sequence of its arrival times is (in essence) a regular “arithmetic sequence,” $T_N = aN + b$, where $N$ is an integer labelling the pulse considered, and where $a$ is equal to the period of rotation of the pulsar around itself. In the case of a binary pulsar, the sequence of arrival times is a much richer signal, say $T_N = aN + b + \Delta_N$, where $\Delta_N$ measures the deviation with respect to a regular arithmetic sequence. This deviation (after the subtraction of effects not connected to the orbital period of the pulsar) is due to a whole ensemble of physical effects connected to the orbital motion of the pulsar around its companion or, more precisely, around the center of mass of the binary system. Some of these effects could be predicted by a purely [*Keplerian*]{} description of the motion of the pulsar in space, and are analogous to the “Rœmer effect” that allowed Rœmer to determine, for the first time, the speed of light from the arrival times at Earth of light signals coming from Jupiter’s satellites (the light signals coming from a body moving in orbit are “delayed” by the time taken by light to cross this orbit and arrive at Earth). Other effects can only be predicted and calculated by using a [*relativistic*]{} description, either of the orbital motion of the pulsar, or of the propagation of electromagnetic signals between the pulsar and Earth. For example, the following facts must be accounted for: (i) the “pulsar clock” moves at a large speed (on the order of 300 km/s $\sim
10^{-3} c$) and is embedded in the varying gravitational potential of the companion; (ii) the orbit of the pulsar is not a simple Keplerian ellipse, but (in general relativity) a more complicated orbit that traces out a “rosette” around the center of mass; (iii) the propagation of electromagnetic signals between the pulsar and Earth takes place in a space-time that is curved by both the pulsar and its companion, which leads to particular effects of relativistic delay; etc. Taking relativistic effects in the theoretical description of arrival times for signals emitted by binary pulsars into account thus leads one to write what is called a [*timing formula*]{}. This timing formula (due to T. Damour and N. Deruelle) in essence allows one to parameterize the sequence of arrival times, $T_N = aN + b +
\Delta_N$, in other words to parameterize $\Delta_N$, as a function of a set of “phenomenological parameters” that include not only the so-called “Keplerian” parameters (such as the orbital period $P$, the projection of the semi-major axis of the pulsar’s orbit along the line of sight $x_A = a_A \sin i$, and the eccentricity $e$), but also the [*post-Keplerian*]{} parameters associated with the relativistic effects mentioned above. For example, effect (i) discussed above is parameterized by a quantity denoted $\gamma_T$; effect (ii) by (among others) the quantities $\dot\omega$, $\dot P$; effect (iii) by the quantities $r,s$; etc.
The way in which observations of binary pulsars allow one to test relativistic theories of gravity is therefore the following. A (least-squares) fit between the observational timing data, $\Delta_N^{\rm obs}$, and the parameterized theoretical timing formula, $\Delta_N^{\rm th} (P , x_A , e ; \gamma_T ,
\dot\omega , \dot P , r , s)$, allows for the determination of the observational values of the Keplerian $(P^{\rm obs} , x_A^{\rm
obs} , e^{\rm obs})$ and post-Keplerian $(\gamma_T^{\rm obs} ,
\dot\omega^{\rm obs} , \dot P^{\rm obs} , r^{\rm obs} , s^{\rm
obs})$ parameters. The theory of general relativity predicts the value of each post-Keplerian parameter as a function of the Keplerian parameters and the two masses of the binary system (the mass $m_A$ of the pulsar and the mass $m_B$ of the companion). For example, the theoretical value predicted by general relativity for the parameter $\gamma_T$ is $\gamma_T^{\rm GR} (m_A , m_B) =
en^{-1} (GMn/c^3)^{2/3} \, m_B (m_A + 2 \, m_B)/ M^2$, where $e$ is the eccentricity, $n = 2\pi / P$ the orbital frequency, and $M
\equiv m_A + m_B$. We thus see that, if one assumes that general relativity is correct, the observational measurement of a post-Keplerian parameter, for example $\gamma_T^{\rm obs}$, determines a [*curve*]{} in the plane $(m_A , m_B)$ of the two masses: $\gamma_T^{\rm GR} (m_A , m_B) = \gamma_T^{\rm obs}$, in our example. The measurement of two post-Keplerian parameters thus gives two curves in the $(m_A , m_B)$ plane and generically allows one to determine the values of the two masses $m_A$ and $m_B$, by considering the intersection of the two curves. We obtain a test of general relativity as soon as one observationally measures three or more post-Keplerian parameters: if the three (or more) curves all intersect at one point in the plane of the two masses, the theory of general relativity is confirmed, but if this is not the case the theory is refuted. At present, four distinct binary pulsars have allowed one to test general relativity. These four “relativistic” binary pulsars are: the first binary pulsar PSR B1913$ + $16, the pulsar PSR B1534$ +$12 (discovered by A. Wolszczan in 1991), and two recently discovered pulsars: PSR J1141$ - $6545 (discovered in 1999 by V.M. Kaspi et al., whose first timing results are due to M. Bailes et al. in 2003), and PSR J0737$ - $3039 (discovered in 2003 by M. Burgay et al., whose first timing results are due to A.G. Lyne et al. and M. Kramer et al.). With the exception of PSR J1141$ - $6545, whose companion is a white dwarf, the companions of the pulsars are neutron stars. In the case of PSR J0737$ -
$3039 the companion turns out to also be a pulsar that is visible from Earth.
In the system PSR B1913$ + $16, [*three*]{} post-Keplerian parameters have been measured $(\dot\omega , \gamma_T , \dot P)$, which gives [*one*]{} test of the theory. In the system PSR J1141$
- $65, [*three*]{} post-Keplerian parameters have been measured $(\dot\omega , \gamma_T , \dot P)$, which gives [*one*]{} test of the theory. (The parameter $s$ is also measured through scintillation phenomena, but the use of this measurement for testing gravitation is more problematic.) In the system PSR B1534$
+ $12, [*five*]{} post-Keplerian parameters have been measured, which gives [*three*]{} tests of the theory. In the system PSR J0737$ - $3039,[*six*]{} post-Keplerian parameters,[^6] which gives [*four*]{} tests of the theory. It is remarkable that all of these tests have confirmed general relativity. See Figure \[fig4\] and, for references and details, [@tests; @taylor; @stairs; @gef], as well as the contribution by Michael Kramer.
$$\includegraphics[width=80mm]{Fig4.eps}$$
Note that, in Figure \[fig4\], some post-Keplerian parameters are measured with such great precision that they in fact define very thin curves in the $m_A , m_B$ plane. On the other hand, some of them are only measured with a rough fractional precision and thus define “thick curves,” or “strips” in the plane of the masses (see, for example, the strips associated with $\dot P$, $r$ and $s$ in the case of PSR B1534$ + $12). In any case, the theory is confirmed when all of the strips (thick or thin) have a non-empty common intersection. (One should also note that the strips represented in Figure \[fig4\] only use the “one sigma” error bars, in other words a 68% level of confidence. Therefore, the fact that the $\dot P$ strip for PSR B1534$ + $12 is a little bit disjoint from the intersection of the other strips is not significant: a “two sigma” figure would show excellent agreement between observation and general relativity.)
In view of the arguments presented above, all of the tests shown in Figure \[fig4\] confirm the validity of general relativity in the regime of strong gravitational fields ($h_{\mu\nu} \sim 1$). Moreover, the four tests that use measurements of the parameter $\dot P$ (in the four corresponding systems) are direct experimental confirmations of the fact that the gravitational interaction propagates at the speed $c$ between the companion and the pulsar. In fact, $\dot P$ denotes the long-term variation $\langle dP / dt \rangle$ of the orbital period. Detailed theoretical calculations of the motion of two gravitationally condensed objects in general relativity, that take into account the effects connected to the propagation of the gravitational interaction at finite speed[@motion], have shown that one of the observable effects of this propagation is a long-term decrease in the orbital period given by the formula $$\dot P^{\rm GR} (m_A , m_B) = - \frac{192 \, \pi}{5} \, \frac{1 + \frac{73}{24}
\, e^2 + \frac{37}{96} \, e^4}{(1-e^2)^{7/2}} \left( \frac{GM n}{c^3}
\right)^{5/3} \, \frac{m_A \, m_B}{M^2} \, .$$ The direct physical origin of this decrease in the orbital period lies in the modification, produced by general relativity, of the usual Newtonian law of gravitational attraction between two bodies, $F_{\rm Newton} = G \, m_A \, m_B / r_{AB}^2$. In place of such a simple law, general relativity predicts a more complicated force law that can be expanded in the symbolic form $$\label{rgn19}
F_{\rm Einstein} = \frac{G \, m_A \, m_B}{r_{AB}^2} \left( 1 + \frac{v^2}{c^2} +
\frac{v^4}{c^4} + \frac{v^5}{c^5} + \frac{v^6}{c^6} + \frac{v^7}{c^7} + \cdots
\right) \, ,$$ where, for example, “$v^2 / c^2$” represents a whole set of terms of order $v_A^2 / c^2$, $v_B^2 / c^2$, $v_A \, v_B / c^2$, or even $G \, m_A / c^2 \, r$ or $G \, m_B / c^2 \, r$. Here $v_A$ denotes the speed of body $A$, $v_B$ that of body $B$, and $r_{AB}$ the distance between the two bodies. The term of order $v^5 / c^5$ in Equation (\[rgn19\]) is particularly important. This term is a direct consequence of the finite-speed propagation of the gravitational interaction between $A$ and $B$, and its calculation shows that it contains a component that is opposed to the relative speed ${\mbox{\boldmath${v}$}}_A - {\mbox{\boldmath${v}$}}_B$ of the two bodies and that, consequently, slows down the orbital motion of each body, causing it to evolve towards an orbit that lies closer to its companion (and therefore has a shorter orbital period). This “braking” term (which is correlated with the emission of gravitational waves), $\delta F_{\rm Einstein} \sim v^5 / c^5 \, F_{\rm
Newton}$, leads to a long-term decrease in the orbital period $\dot P^{\rm GR} \sim - (v/c)^5 \sim -10^{-12}$ that is very small, but whose reality has been verified with a fractional precision of order $10^{-3}$ in PSR B1913$ + $16 and of order 20% in PSR B1534$ + $12 and PSR J1141$ - $6545 [@tests; @taylor; @gef].
To conclude this brief outline of the tests of relativistic gravitation by binary pulsars, let us note that there is an analog, for the regime of strong gravitational fields, of the formalism of parametrization for possible deviations from general relativity mentioned in Section \[sec6\] in the framework of weak gravitational fields (using the post-Newtonian parameters $\gamma - 1$ and $\beta - 1$). This analog is obtained by considering a two-parameter family of relativistic theories of gravitation, assuming that the gravitational interaction is propagated not only by a tensor field $g_{\mu\nu}$ but also by a scalar field $\varphi$. Such a class of tensor-scalar theories of gravitation allows for a description of possible deviations in both the solar system and in binary pulsars. It also allows one to explicitly demonstrate that binary pulsars indeed test the effects of strong fields that go beyond the tests of the weak fields of the solar system by exhibiting classes of theories that are compatible with all of the observations in the solar system but that are incompatible with the observations of binary pulsars, see [@tests; @gef].
Gravitational Waves: Propagation, Generation, and Detection {#sec10}
===========================================================
As soon as he had finished constructing the theory of general relativity, Einstein realized that it implied the existence of waves of geometric deformations of space-time, or “gravitational waves” [@E16; @oeuvres]. Mathematically, these waves are analogs (with the replacement $A_{\mu} \to h_{\mu\nu}$) of electromagnetic waves, but conceptually they signify something remarkable: they exemplify, in the purest possible way, the “elastic” nature of space-time in general relativity. Before Einstein space-time was a rigid structure, given a priori, which was not influenced by the material content of the Universe. After Einstein, a distribution of matter (or more generally of mass-energy) that changes over the course of time, let us say for concreteness a binary system of two neutron stars or two black holes, will not only deform the chrono-geometry of the space-time in its immediate neighborhood, but this deformation will propagate in every possible direction away from the system considered, and will travel out to infinity in the form of a wave whose oscillations will reflect the temporal variations of the matter distribution. We therefore see that the study of these gravitational waves poses three separate problems: that of generation, that of propagation, and, finally, that of detection of such gravitational radiation. These three problems are at present being actively studied, since it is hoped that we will soon detect gravitational waves, and thus will be able to obtain new information about the Universe [@thorne]. We shall here content ourselves with an elementary introduction to this field of research. For a more detailed introduction to the detection of gravitational waves see the contribution by Jean-Yves Vinet to this Poincaré seminar.
Let us first consider the simplest case of very weak gravitational waves, outside of their material sources. The geometry of such a space-time can be written, as in Section \[sec6\], as $g_{\mu\nu} (x) = \eta_{\mu\nu} + h_{\mu\nu} (x)$, where $h_{\mu\nu} \ll 1$. At first order in $h$, and outside of the source (namely in the domain where $T_{\mu\nu} (x) = 0$), the perturbation of the geometry, $h_{\mu\nu} (x)$, satisfies a homogeneous equation obtained by replacing the right-hand side of Equation (\[rg11\]) with zero. It can be shown that one can simplify this equation through a suitable choice of coordinate system. In a [*transverse traceless*]{} (TT) coordinate system the only non-zero components of a general gravitational wave are the spatial components $h_{ij}^{\rm TT}$, $i,j = 1,2,3$ (in other words $h_{00}^{\rm TT} = 0 = h_{0i}^{\rm TT}$), and these components satisfy $$\label{rg20}
\Box \, h_{ij}^{\rm TT} = 0 \, , \ \partial_j \, h_{ij}^{\rm TT} = 0 \, , \
h_{jj}^{\rm TT} = 0 \, .$$ The first equation in (\[rg20\]), where the wave operator $\Box
= \Delta - c^{-2} \, \partial_t^2$ appears, shows that gravitational waves (like electromagnetic waves) propagate at the speed $c$. If we consider for simplicity a monochromatic plane wave ($h_{ij}^{\rm TT} = \zeta_{ij} \, \exp (i \, {\mbox{\boldmath${k}$}}\cdot {\mbox{\boldmath${x}$}}-
i \, \omega \, t) \, +$ complex conjugate, with $\omega = c \,
\vert {\mbox{\boldmath${k}$}}\vert$), the second equation in (\[rg20\]) shows that the (complex) tensor $\zeta_{ij}$ measuring the polarization of a gravitational wave only has non-zero components in the plane orthogonal to the wave’s direction of propagation: $\zeta_{ij} \,
k^j = 0$. Finally, the third equation in (\[rg20\]) shows that the polarization tensor $\zeta_{ij}$ has vanishing trace: $\zeta_{jj} = 0$. More concretely, this means that if a gravitational wave propagates in the $z$-direction, its polarization is described by a $2 \times 2$ matrix, $\begin{pmatrix} \zeta_{xx} &\zeta_{xy}
\\ \zeta_{yx} &\zeta_{yy}
\end{pmatrix}$, which is symmetric and traceless. Such a polarization matrix therefore only contains two independent (complex) components: $\zeta_+ \equiv \zeta_{xx} = - \zeta_{yy}$, and $\zeta_{\times}
\equiv \zeta_{xy} = \zeta_{yx}$. This is the same number of independent (complex) components that an electromagnetic wave has. Indeed, in a transverse gauge, an electromagnetic wave only has spatial components $A_i^T$ that satisfy $$\label{rg21}
\Box \, A_i^T = 0 \, , \ \partial_j \, A_j^T = 0 \, .$$ As in the case above, the first equation (\[rg21\]) means that an electromagnetic wave propagates at the speed $c$, and the second equation shows that a monochromatic plane electromagnetic wave ($A_i^T = \zeta_i \, \exp (i \, {\mbox{\boldmath${k}$}}\cdot {\mbox{\boldmath${x}$}}- i \, \omega
\, t) +$ c.c., $\omega = c \, \vert {\mbox{\boldmath${k}$}}\vert$) is described by a (complex) polarization vector $\zeta_i$ that is orthogonal to the direction of propagation: $\zeta_j \, k^j = 0$. For a wave propagating in the $z$-direction such a vector only has two independent (complex) components, $\zeta_x$ and $\zeta_y$. It is indeed the same number of components that a gravitational wave has, but we see that the two quantities measuring the polarization of a gravitational wave, $\zeta_+ = \zeta_{xx} = - \zeta_{yy}$, $\zeta_{\times} =
\zeta_{xy} = \zeta_{yx}$ are mathematically quite different from the two quantities $\zeta_x , \zeta_y$ measuring the polarization of an electromagnetic wave. However, see Section \[sec11\] below.
We have here discussed the propagation of a gravitational wave in a background space-time described by the Minkowski metric $\eta_{\mu\nu}$. One can also consider the propagation of a wave in a curved background space-time, namely by studying solutions of Einstein’s equations (\[rg9\]) of the form $g_{\mu\nu} (x) =
g_{\mu\nu}^B (x) + h_{\mu\nu} (x)$ where $h_{\mu\nu}$ is not only small, but varies on temporal and spatial scales much shorter than those of the background metric $g_{\mu\nu}^B (x)$. Such a study is necessary, for example, for understanding the propagation of gravitational waves in the cosmological Universe.
The problem of [*generation*]{} consists in searching for the connection between the tensorial amplitude $h_{ij}^{\rm TT}$ of the gravitational radiation in the radiation zone and the motion and structure of the source. If one considers the simplest case of a source that is sufficiently diffuse that it only creates waves that are everywhere weak ($g_{\mu\nu} - \eta_{\mu\nu} = h_{\mu\nu}
\ll 1$), one can use the linearized approximation to Einstein’s equations (\[rg9\]), namely Equations (\[rg11\]). One can solve Equations (\[rg11\]) by the same technique that is used to solve Maxwell’s equations (\[rg12\]): one fixes the coordinate system by imposing $\partial^{\alpha} \, h_{\alpha\mu} -
\frac{1}{2} \,
\partial_{\mu} \, h_{\alpha}^{\alpha} = 0$ (analogous to the Lorentz gauge condition $\partial^{\alpha} \, A_{\alpha} = 0$), then one inverts the wave operator by using retarded potentials. Finally, one must study the asymptotic form, at infinity, of the emitted wave, and write it in the reduced form of a transverse and traceless amplitude $h_{ij}^{\rm TT}$ satisfying Equations (\[rg20\]) (analogous to a transverse electromagnetic wave $A_i^T$ satisfying (\[rg21\])). One then finds that, just as charge conservation implies that there is no monopole type electro-magnetic radiation, but only dipole or higher orders of polarity, the conservation of energy-momentum implies the absence of monopole [*and*]{} dipole gravitational radiation. For a slowly varying source $(v/c \ll 1$), the dominant gravitational radiation is of [*quadrupole*]{} type. It is given, in the radiation zone, by an expression of the form $$\label{rg22}
h_{ij}^{TT} (t,r,{\mbox{\boldmath${n}$}}) \simeq \frac{2G}{c^4 \, r} \, \frac{\partial^2}{\partial
t^2} \, [ I_{ij} (t-r/c)]^{\rm TT} \, .$$ Here $r$ denotes the distance to the center of mass of the source, $I_{ij} (t) \equiv \int d^3 x \, c^{-2}$ $T^{00} (t,{\mbox{\boldmath${x}$}})
\left(x^i x^j - \frac{1}{3} \, {\mbox{\boldmath${x}$}}^2 \delta^{ij} \right)$ is the quadrupole moment of the mass-energy distribution, and the upper index TT denotes an algebraic projection operation for the quadrupole tensor $I_{ij}$ (which is a $3 \times 3$ matrix) that only retains the part orthogonal to the local direction of wave propagation $n^i \equiv x^i / r$ with vanishing trace ($I_{ij}^{\rm TT}$ is therefore locally a (real) $2 \times 2$ symmetric, traceless matrix of the same type as $\zeta_{ij}$ above). Formula (\[rg22\]) (which was in essence obtained by Einstein in 1918 [@E16]) is only the first approximation to an expansion in powers of $v/c$, where $v$ designates an internal speed characteristic of the source. The prospect of soon being able to detect gravitational waves has motivated theorists to improve Formula (\[rg22\]): (i) by describing the terms of higher order in $v/c$, up to a very high order, and (ii) by using new approximation methods that allow one to treat sources containing regions of strong gravitational fields (such as, for example, a binary system of two black holes or two neutron stars). See below for the most recent results.
Finally, the problem of [*detection*]{}, of which the pioneer was Joseph Weber in the 1960s, is at present giving rise to very active experimental research. The principle behind any detector is that a gravitational wave of amplitude $h_{ij}^{\rm TT}$ induces a change in the distance $L$ between two bodies on the order of $\delta L \sim hL$ during its passage. One way of seeing this is to consider the action of a wave $h_{ij}^{\rm TT}$ on two free particles, at rest before the arrival of the wave at the positions $x_1^i$ and $x_2^i$ respectively. As we have seen, each particle, in the presence of the wave, will follow a geodesic motion in the geometry $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$ (with $h_{00} =
h_{0i} = 0$ and $h_{ij} = h_{ij}^{\rm TT}$). By writing out the geodesic equation, Equation (\[rg7\]), one finds that it simply reduces (at first order in $h$) to $d^2 x^i / ds^2 = 0$. Therefore, particles that are initially at rest ($x^i =$ const.) remain at rest in a transverse and traceless system of coordinates! This does not however mean that the gravitational wave has no observable effect. In fact, since the spatial geometry is perturbed by the passage of the wave, $g_{ij} (t,{\mbox{\boldmath${x}$}}) =
\delta_{ij} + h_{ij}^{\rm TT} (t,{\mbox{\boldmath${x}$}})$, one finds that the physical distance between the two particles $x_1^i$, $x_2^i$ (which is observable, for example, by measuring the time taken for light to make a round trip between the two particles) varies, during the passage of the wave, according to $L^2 = (\delta_{ij} +
h_{ij}^{\rm TT}) (x_2^i - x_1^i) (x_2^j - x_1^j)$.
The problem of detecting a gravitational wave thus leads to the problem of detecting a small relative displacement $\delta L / L
\sim h$. By using Formula (\[rg22\]), one finds that the order of magnitude of $h$, for known or hoped for astrophysical sources (for example,a very close system of two neutron stars or two black holes), situated at distances such that one may hope to see several events per year ($r \gtrsim 600$ million light-years), is in fact extremely small: $h \lesssim 10^{-22}$ for signals whose characteristic frequency is around 100 Hertz. Several types of detectors have been developed since the pioneering work of J. Weber [@thorne]. At present, the detectors that should succeed in the near future at detecting amplitudes $h \sim \delta
L / L \sim 10^{-22}$ are large interferometers, of the Michelson or Fabry-Pérot type, having arms that are many kilometers in length into which a very powerful monochromatic laser beam is injected. Such terrestrial interferometric detectors presently exist in the U.S.A. (the LIGO detectors [@ligo]), in Europe (the VIRGO [@virgo] and GEO 600 [@geo] detectors) and elsewhere (such as the TAMA detector in Japan). Moreover, the international space project LISA [@lisa], made up of an interferometer between satellites that are several million kilometers apart, should allow one to detect low frequency ($\sim$ one hundredth or one thousandth of a Hertz) gravitational waves in a dozen years or so. This collection of gravitational wave detectors promises to bring invaluable information for astronomy by opening a new “window” on the Universe that is much more transparent than the various electromagnetic (or neutrino) windows that have so greatly expanded our knowledge of the Universe in the twentieth century.
The extreme smallness of the expected gravitational signals has led a number of experimentalists to contribute, over many years, a wealth of ingenuity and know-how in order to develop technology that is sufficiently precise and trustworthy (see [@ligo; @virgo; @geo; @lisa]). To conclude, let us also mention how much concerted theoretical effort has been made, both in calculating the general relativistic predictions for gravitational waves emitted by certain sources, and in developing methods adapted to the extraction of the gravitational signal from the background noise in the detectors. For example, one of the most promising sources for terrestrial detectors is the wave train for gravitational waves emitted by a system of two black holes, and in particular the final (most intense) portion of this wave train, which is emitted during the last few orbits of the system and the final coalescence of the two black holes into a single, more massive black hole. We have seen above (see Section \[sec9\]) that the finite speed of propagation of the gravitational interaction between the two bodies of a binary system gives rise to a progressive acceleration of the orbital frequency, connected to the progressive approach of the two bodies towards each other. Here we are speaking of the final stages in such a process, where the two bodies are so close that they orbit around each other in a spiral pattern that accelerates until they attain (for the final “stable” orbits) speeds that become comparable to the speed of light, all the while remaining slightly slower. In order to be able to determine, with a precision that is acceptable for the needs of detection, the dynamics of such a binary black hole system in such a situation, as well as the gravitational amplitude $h_{ij}^{\rm TT}$that it emits, it was necessary to develop a whole ensemble of analytic techniques to a very high level of precision. For example, it was necessary to calculate the expansion (\[rgn19\]) of the force determining the motion of the two bodies to a very high order and also to calculate the amplitude $h_{ij}^{\rm TT}$ of the gravitational radiation emitted to infinity with a precision going well beyond the quadrupole approximation (\[rg22\]). These calculations are comparable in complexity to high-order calculations in quantum field theory. Some of the techniques developed for quantum field theory indeed proved to be extremely useful for these calculations in the (classical) theory of general relativity (such as certain resummation methods and the mathematical use of analytic continuation in the number of space-time dimensions). For an entryway into the literature of these modern analytic methods, see [@BDEI04], and for an early example of a result obtained by such methods of direct interest for the physics of detection see Figure \[fig5\] [@BD], which shows a component of the gravitational amplitude $h_{ij}^{\rm TT} (t)$ emitted during the final stages of evolution of a system of two black holes of equal mass. The first oscillations shown in Figure \[fig5\] are emitted during the last quasi-circular orbits (accelerated motion in a spiral of decreasing radius). The middle part of the signal corresponds to a phase where, having moved past the last stable orbit, the two black holes “fall” toward each other while spiraling rapidly. In fact, contrarily to Newton’s theory, which predicts that two condensed bodies would be able to orbit around each other with an orbit of arbitrarily small radius (basically up until the point that the two bodies touch), Einstein’s theory predicts a modified law for the force between the two bodies, Equation (\[rgn19\]), whose analysis shows that it is so attractive that it no longer allows for stable circular orbits when the distance between the two bodies becomes smaller than around $6 \, G (m_A + m_B) / c^2$. In the case of two black holes, this distance is sufficiently larger than the black hole “radii” ($2 \, G \, m_A /
c^2$ and $2 \, G \, m_B / c^2$) that one is still able to analytically treat the beginning of the “spiralling plunge” of the two black holes towards each other. The final oscillations in Figure \[fig5\] are emitted by the rotating (and initially highly deformed) black hole formed from the merger of the two initial, separate black holes.
$$\includegraphics[width=80mm]{Fig5.eps}$$
Up until quite recently the analytic predictions illustrated in Figure \[fig5\] concerning the gravitational signal $h(t)$ emitted by the spiralling plunge and merger of two black holes remained conjectural, since they could be compared to neither other theoretical predictions nor to observational data. Recently, worldwide efforts made over three decades to attack the problem of the coalescence of two black holes by [*numerically*]{} solving Einstein’s equations (\[rg9\]) have spectacularly begun to bear fruit. Several groups have been able to numerically calculate the signal $h(t)$ emitted during the final orbits and merger of two black holes [@NR]. In essence, there is good agreement between the analytical and numerical predictions. In order to be able to detect the gravitational waves emitted by the coalescence of two black holes, it will most likely be necessary to properly combine the information on the structure of the signal $h(t)$ obtained by the two types of methods, which are in fact complementary.
General Relativity and Quantum Theory: From Supergravity to String Theory {#sec11}
=========================================================================
Up until now, we have discussed the [*classical*]{} theory of general relativity, neglecting any quantum effects. What becomes of the theory in the quantum regime? This apparently innocent question in fact opens up vast new prospects that are still under construction. We will do nothing more here than to touch upon the subject, by pointing out to the reader some of the paths along which contemporary physics has been led by the challenge of unifying general relativity and quantum theory. For a more complete introduction to the various possibilities “beyond” general relativity suggested within the framework of [*string theory*]{} (which is still under construction) one should consult the contribution of Ignatios Antoniadis to this Poincaré Seminar.
Let us recall that, from the very beginning of the quasi-definitive formulation of quantum theory (1925–1930), the creators of quantum mechanics (Born, Heisenberg, Jordan; Dirac; Pauli; etc.) showed how to “quantize” not only systems with several particles (such as an atom), but also [*fields*]{}, continuous dynamical systems whose classical description implies a continuous distribution of energy and momentum in space. In particular, they showed how to [*quantize*]{} (or in other words how to formulate within a framework compatible with quantum theory) the electromagnetic field $A_{\mu}$, which, as we have recalled above, satisfies the Maxwell equations (\[rg12\]) at the classical level. They nevertheless ran into difficulty due to the following fact. In quantum theory, the physics of a system’s evolution is essentially contained in the [*transition amplitudes*]{} $A (f,i)$ between an initial state labelled by $i$ and a final state labelled by $f$. These amplitudes $A(f,i)$ are complex numbers. They satisfy a “transitivity” property of the type $$\label{rg23}
A(f,i) = \sum_n A(f,n) \, A(n,i) \, ,$$ which contains a sum over all possible intermediate states, labelled by $n$ (with this sum becoming an integral when there is a continuum of intermediate possible states). R. Feynman used Equation (\[rg23\]) as a point of departure for a new formulation of quantum theory, by interpreting it as an analog of [*Huygens’ Principle*]{}: if one thinks of $A(f,i)$ as the amplitude, “at the point $f$,” of a “wave” emitted “from the point $i$,” Equation (\[rg23\]) states that this amplitude can be calculated by considering the “wave” emitted from $i$ as passing through all possible intermediate “points” $n$ ($A(n,i)$), while reemitting “wavelets” starting from these intermediate points ($A(f,n)$), which then superpose to form the total wave arriving at the “final point $f$.”
Property (\[rg23\]) does not pose any problem in the quantum mechanics of discrete systems (particle systems). It simply shows that the amplitude $A(f,i)$ behaves like a wave, and therefore must satisfy a “wave equation” (which is indeed the case for the Schrödinger equation describing the dependence of $A(f,i)$ on the parameters determining the final configuration $f$). On the other hand, Property (\[rg23\]) poses formidable problems when one applies it to the quantization of continuous dynamical systems (fields). In fact, for such systems the “space” of intermediate possible states is infinitely larger than in the case of the mechanics of discrete systems. Roughly speaking, the intermediate possible states for a field can be described as containing $\ell
= 1,2,3,\ldots$ quantum excitations of the field, with each quantum excitation (or pair of “virtual particles”) being described essentially by a plane wave, $\zeta \exp (i \, k_{\mu}
\, x^{\mu})$, where $\zeta$ measures the polarization of these virtual particles and $k^{\mu} = \eta^{\mu\nu} \, k_{\nu}$, with $k^0 = \omega$ and $k^i = {\mbox{\boldmath${k}$}}$, their angular frequency and wave vector, or (using the Planck-Einstein-de Broglie relations $E =
\hbar \, \omega$, ${\mbox{\boldmath${p}$}}= \hbar \, {\mbox{\boldmath${k}$}}$) their energy-momentum $p^{\mu} = \hbar \, k^{\mu}$. The quantum theory shows (basically because of the uncertainty principle) that the four-frequencies (and four-momenta) $p^{\mu} = \hbar \, k^{\mu}$ of the intermediate states cannot be constrained to satisfy the classical equation $\eta_{\mu\nu} \, p^{\mu} \, p^{\nu} = -m^2$ (or in other words $E^2 = {\mbox{\boldmath${p}$}}^2 + m^2$ ; we use $c=1$ in this section). As a consequence, the sum over intermediate states for a quantum field theory has the following properties (among others): (i) when $\ell
= 1$ (an intermediate state containing only one pair of virtual particles, called a [*one-loop contribution*]{}), there is an integral over a four-momentum $p^{\mu}$, $\int d^4 p = \int d E
\int d {\mbox{\boldmath${p}$}}$; (ii) when $\ell = 2$ (two pairs of virtual particles; a [*two-loop contribution*]{}), there is an integral over two four-momenta $p_1^{\mu}$, $p_2^{\mu}$, $\int d^4 p_1 \, d^4 p_2$; etc. The delicate point comes from the fact that the energy-momentum of an intermediate state can take arbitrarily high values. This possibility is directly connected (through a Fourier transform) to the fact that a field possesses an infinite number of degrees of freedom, corresponding to configurations that vary over arbitrarily small time and length scales.
The problems posed by the necessity of integrating over the infinite domain of four-momenta of intermediate virtual particles (or in other words of accounting for the fact that field configurations can vary over arbitrarily small scales) appeared in the 1930s when the quantum theory of the electromagnetic field $A_{\mu}$ (called quantum electrodynamics, or QED) was studied in detail. These problems imposed themselves in the following form: when one calculates the transition amplitude for given initial and final states (for example the collision of two light quanta, with two photons entering and two photons leaving) by using (\[rg23\]), one finds a result given in the form of a [*divergent integral*]{}, because of the integral (in the one-loop approximation, $\ell = 1$) over the arbitrarily large energy-momentum describing virtual electron-positron pairs appearing as possible intermediate states. Little by little, theoretical physicists understood that the types of divergent integrals appearing in QED were relatively benign and, after the second world war, they developed a method ([*renormalization theory*]{}) that allowed one to unambiguously isolate the infinite part of these integrals, and to subtract them by expressing the amplitudes $A(f,i)$ solely as a function of observable quantities [@weinberg] (work by J. Schwinger, R. Feynman, F. Dyson etc.).
The preceding work led to the development of consistent quantum theories not only for the electromagnetic field $A_{\mu}$ (QED), but also for generalizations of electromagnetism (Yang-Mills theory or non-abelian gauge theory) that turned out to provide excellent descriptions of the new interactions between elementary particles discovered in the twentieth century (the electroweak theory, partially unifying electromagnetism and weak nuclear interactions, and quantum chromodynamics, describing the strong nuclear interactions). All of these theories give rise to only relatively benign divergences that can be “renormalized” and thus allowed one to compute amplitudes $A(f,i)$ corresponding to observable physical processes [@weinberg] (notably, work by G. ’t Hooft and M. Veltman).
What happens when we use (\[rg23\]) to construct a “perturbative” quantum theory of general relativity (namely one obtained by expanding in the number $\ell$ of virtual particle pairs appearing in the intermediate states)? The answer is that the integrals over the four-momenta of intermediate virtual particles are not at all of the benign type that allowed them to be renormalized in the simpler case of electromagnetism. The source of this difference is not accidental, but is rather connected with the basic physics of relativistic gravitation. Indeed, as we have mentioned, the virtual particles have arbitrarily large energies $E$. Because of the basic relations that led Einstein to develop general relativity, namely $E = m_i$ and $m_i = m_g$, one deduces that these virtual particles correspond to arbitrarily large gravitational masses $m_g$. They will therefore end up creating intense gravitational effects that become more and more intense as the number $\ell$ of virtual particle pairs grows. These gravitational interactions that grow without limit with energy and momentum correspond (by Fourier transform) to field configurations concentrated in arbitrarily small space and time scales. One way of seeing why the quantum gravitational field creates much more violent problems than the quantum electromagnetic field is, quite simply, to go back to dimensional analysis. Simple considerations in fact show that the relative (non-dimensional) one-loop amplitude $A_1$ must be proportional to the product $\hbar \, G$ and must contain an integral $\int d^4 k$. However, in 1900 Planck had noticed that (in units where $c=1$) the dimensions of $\hbar$ and $G$ were such that the product $\hbar \, G$ had the dimensions of length (or time) squared: $$\label{rg24}
\ell_P \equiv \sqrt{\frac{\hbar \, G}{c^3}} \simeq 1.6 \times 10^{-33} \, {\rm
cm} , \ t_P \equiv \sqrt{\frac{\hbar \, G}{c^5}} \simeq 5.4 \times 10^{-44} \,
{\rm s} \, .$$ One thus deduces that the integral $\int d^4 k \, f(k)$ must have the dimensions of a squared frequency, and therefore that $A_1$ must (when $k \to \infty$) be of the type, $A_1 \sim \hbar \, G
\int d^4 k / k^2$. Such an integral diverges quadratically with the upper limit $\Lambda$ of the integral (the cutoff frequency, such that $\vert k \vert \leq \Lambda$), so that $A_1
\sim \hbar \, G \, \Lambda^2 \sim t_P^2 \, \Lambda^2$. The extension of this dimensional analysis to the intermediate states with several loops ($\ell > 1$) causes even more severe polynomial divergences to appear, of a type such that the power of $\Lambda$ that appears grows without limit with $\ell$.
In summary, the essential physical characteristics of gravitation ($E = m_i = m_g$ and the dimension of Newton’s constant $G$) imply the impossibility of generalizing to the gravitational case the methods that allowed a satisfactory quantum treatment of the other interactions (electromagnetic, weak, and strong). Several paths have been explored to get out of this impasse. Some researchers tried to quantize general relativity non-perturbatively, without using an expansion in intermediate states (\[rg23\]) (work by A. Ashtekar, L. Smolin, and others). others have tried to generalize general relativity by adding a fermionic field to Einstein’s (bosonic) gravitational field $g_{\mu\nu} (x)$, the gravitino field $\psi_{\mu} (x)$. It is indeed remarkable that it is possible to define a theory, known as [*supergravity*]{}, that generalizes the geometric invariance of general relativity in a profound way. After the 1974 discovery (by J. Wess and B. Zumino) of a possible new global symmetry for interacting bosonic and fermionic fields, [*supersymmetry*]{} (which is a sort of global rotation transforming bosons to fermions and vice versa), D.Z. Freedman, P. van Nieuwenhuizen, and S. Ferrara; and S. Deser and B. Zumino; showed that one could generalize global supersymmetry to a [*local supersymmetry*]{}, meaning that it varies from point to point in space-time. Local supersymmetry is a sort of fermionic generalization (with anti-commuting parameters) of the geometric invariance at the base of general relativity (the invariance under any change in coordinates). The generalization of Einstein’s theory of gravitation that admits such a local supersymmetry is called [*supergravity theory*]{}. As we have mentioned, in four dimensions this theory contains, in addition to the (commuting) bosonic field $g_{\mu\nu} (x)$, an (anti-commuting) fermionic field $\psi_{\mu}
(x)$ that is both a space-time vector (with index $\mu$) and a spinor. (It is a massless field of spin $3/2$, intermediate between a massless spin $1$ field like $A_{\mu}$ and a massless spin $2$ field like $h_{\mu\nu} = g_{\mu\nu} - \eta_{\mu\nu}$.) Supergravity was extended to richer fermionic structures (with many gravitinos), and was formulated in space-times having more than four dimensions. It is nevertheless remarkable that there is a maximal dimension, equal to $D = 11$, admitting a theory of supergravity (the maximal supergravity constructed by E. Cremmer, B. Julia, and J. Scherk). The initial hope underlying the construction of these supergravity theories was that they would perhaps allow one to give meaning to the perturbative calculation (\[rg23\]) of quantum amplitudes. Indeed, one finds for example that at one loop, $\ell = 1$, the contributions coming from intermediate fermionic states have a sign opposite to the bosonic contributions and (because of the supersymmetry, bosons $\leftrightarrow$ fermions) exactly cancel them. Unfortunately, although such cancellations exist for the lowest orders of approximation, it appeared that this was probably not going to be the case at all orders[^7]. The fact that the gravitational interaction constant $G$ has “a bad dimension” remains true and creates non-renormalizable divergences starting at a certain number of loops $\ell$.
Meanwhile, a third way of defining a consistent quantum theory of gravity was developed, under the name of [*string theory*]{}. Initially formulated as models for the strong interactions (in particular by G. Veneziano, M. Virasoro, P. Ramond, A. Neveu, and J.H. Schwarz), the string theories were founded upon the quantization of the relativistic dynamics of an extended object of one spatial dimension: a “string.” This string could be closed in on itself, like a small rubber band (a closed string), or it could have two ends (an open string). Note that the point of departure of string theory only includes the Poincaré-Minkowski space-time, in other words the metric $\eta_{\mu\nu}$ of Equation (\[rg2\]), and quantum theory (with the constant $\hbar
= h/2\pi$). In particular, the only symmetry manifest in the classical dynamics of a string is the Poincaré group (\[rg3\]). It is, however, remarkable that (as shown by T. Yoneya, and J. Scherk and J.H. Schwarz, in 1974) one of the quantum excitations of a closed string reproduces, in a certain limit, all of the non-linear structure of general relativity (see below). Among the other remarkable properties of string theory [@strings], let us point out that it is the first physical theory to determine the space-time dimension $D$. In fact, this theory is only consistent if $D=10$, for the versions allowing fermionic excitations (the purely bosonic string theory selects $D=26$). The fact that $10 > 4$ does not mean that this theory has no relevance to the real world. Indeed, it has been known since the 1930s (from work of T. Kaluza and O. Klein) that a space-time of dimension $D > 4$ is compatible with experiment if the supplementary (spatial) dimensions close in on themselves (meaning they are [*compactified*]{}) on very small distance scales. The low-energy physics of such a theory seems to take place in a four-dimensional space-time, but it contains new (a priori massless) fields connected to the geometry of the additional compactified dimensions. Moreover, recent work (due in particular to I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. Dvali) has suggested the possibility that the additional dimensions are compactified on scales that are small with respect to everyday life, but very large with respect to the Planck length. This possibility opens up an entire phenomenological field dealing with the eventual observation of signals coming from string theory (see the contribution of I. Antoniadis to this Poincaré seminar).
However, string theory’s most remarkable property is that it seems to avoid, in a radical way, the problems of divergent (non-renormalizable) integrals that have weighed down every direct attempt at perturbatively quantizing gravity. In order to explain how string theory arrives at such a result, we must discuss some elements of its formalism.
Recall that the classical dynamics of any system is obtained by minimizing a functional of the time evolution of the system’s configuration, called the [*action*]{} (the principle of least action). For example, the action for a particle of mass $m$, moving in a Riemannian space-time (\[rg6\]), is proportional to the length of the line that it traces in space-time: $S = -m \int
ds$. This action is minimized when the particle follows a geodesic, in other words when its equation of motion is given by (\[rg7\]). According to Y. Nambu and T. Goto, the action for a string is $S = -T \iint dA$, where the parameter $T$ (analogous to $m$ for the particle) is called the string [*tension*]{}, and where $\iint dA$ is the area of the [*two-dimensional*]{} surface traced out by the evolution of the string in the ($D$-dimensional) space-time in which it lives. In quantum theory, the action functional serves (as shown by R. Feynman) to define the transition amplitude (\[rg23\]). Basically, when one considers two intermediate configurations $m$ and $n$ (in the sense of the right-hand side of (\[rg23\])) that are close to each other, the amplitude $A(n,m)$ is proportional to $\exp ( i \, S (n,m) /
\hbar)$, where $S(n,m)$ is the minimal classical action such that the system considered evolves from the configuration labelled by $n$ to that labelled by $m$. Generalizing the decomposition in (\[rg23\]) by introducing an infinite number of intermediate configurations that lie close to each other, one ends up (in a generalization of Huygens’ principle) expressing the amplitude $A(f,i)$ as a multiple sum over all of the “paths” (in the configuration space of the system studied) connecting the initial state $i$ to the final state $f$. Each path contributes a term $e^{i\phi}$ where the phase $\phi = S/\hbar$ is proportional to the action $S$ corresponding to this “path,” or in other words to this possible evolution of the system. In string theory, $\phi
= - (T/\hbar) \iint dA$. Since the phase is a non-dimensional quantity, and $\iint dA$ has the dimension of an area, we see that the quantum theory of strings brings in the quantity $\hbar / T$, having the dimensions of a length squared, at a fundamental level. More precisely, the fundamental length of string theory, $\ell_s$, is defined by $$\label{rg25}
\ell_s^2 \equiv \alpha' \equiv \frac{\hbar}{2 \, \pi \, T} \, .$$
This fundamental length plays a central role in string theory. Roughly speaking, it defines the characteristic “size” of the quantum states of a string. If $\ell_s$ is much smaller than the observational resolution with which one studies the string, the string will look like a point-like particle, and its interactions will be described by a quantum theory of relativistic particles, which is equivalent to a theory of relativistic fields. It is precisely in this sense that general relativity emerges as a limit of string theory. Since this is an important conceptual point for our story, let us give some details about the emergence of general relativity from string theory.
The action functional that is used in practice to quantize a string is not really $-T \iint dA$, but rather (as emphasized by A. Polyakov) $$\label{rg26}
\frac{S}{\hbar} = - \frac{1}{4 \, \pi \, \ell_s^2} \iint d^2 \sigma \,
\sqrt{-\gamma} \, \gamma^{ab} \, \partial_a \, X^{\mu} \, \partial_b \, X^{\nu}
\, \eta_{\mu\nu} + \cdots \, ,$$ where $\sigma^a$, $a=0,1$ are two coordinates that allow an event to be located on the space-time surface (or ‘world-sheet’) traced out by the string within the ambient space-time; $\gamma_{ab}$ is an auxiliary metric ($d \, \Sigma^2 = \gamma_{ab} (\sigma) \, d\sigma^a \,
d\sigma^b$) defined on this surface (with $\gamma^{ab}$ being its inverse, and $\gamma$ its determinant); and $X^{\mu} (\sigma^a)$ defines the embedding of the string in the ambient (flat) space-time. The dots indicate additional terms, and in particular terms of fermionic type that were introduced by P. Ramond, by A. Neveu and J.H. Schwarz, and by others. If one separates the two coordinates $\sigma^a = (\sigma^0 , \sigma^1)$ into a temporal coordinate, $\tau \equiv \sigma^0$, and a spatial coordinate, $\sigma \equiv \sigma^1$, the configuration “at time $\tau$” of the string is described by the functions $X^{\mu} (\tau ,
\sigma)$, where one can interpret $\sigma$ as a curvilinear abscissa describing the spatial extent of the string. If we consider a closed string, one that is topologically equivalent to a circle, the function $X^{\mu} (\tau , \sigma)$ must be periodic in $\sigma$. One can show that (modulo the imposition of certain constraints) one can choose the coordinates $\tau$ and $\sigma$ on the string such that $d \, \Sigma^2 = -d\tau^2 + d \sigma^2$. Then, the dynamical equations for the string (obtained by minimizing the action (\[rg26\])) reduce to the standard equation for waves on a string: $-\partial^2 X^{\mu} /
\partial \tau^2 +
\partial^2 X^{\mu} / \partial \sigma^2 = 0$. The general solution to this equation describes a superposition of waves travelling along the string in both possible directions: $X^{\mu} = X_L^{\mu}
(\tau + \sigma) + X_R^{\mu} (\tau - \sigma)$. If we consider a closed string (one that is topologically equivalent to a circle), these two types of wave are independent of each other. For an open string (with certain reflection conditions at the endpoints of the string) these two types of waves are connected to each other. Moreover, since the string has a finite length in both cases, one can decompose the left- or right-moving waves $X_L^{\mu} (\tau +
\sigma)$ or $X_R^{\mu} (\tau - \sigma)$ as a Fourier series. For example, for a closed string one may write $$\label{rg27}
X^{\mu} (\tau , \sigma) = X_0^{\mu} (\tau) + \frac{i}{\sqrt 2} \, \ell_s
\sum_{n=1}^{\infty} \left( \frac{a_n^{\mu}}{\sqrt n} \, e^{-2in(\tau - \sigma)}
+ \frac{\tilde a_n^{\mu}}{\sqrt n} \, e^{-2 in (\tau + \sigma)} \right) + {\rm
h.c.}$$ Here $X_0^{\mu} (\tau) = x^{\mu} + 2 \, \ell_s^2 \, p^{\mu} \tau$ describes the motion of the string’s center of mass, and the remainder describes the decomposition of the motion around the center of mass into a discrete set of oscillatory modes. Like any vibrating string, a relativistic string can vibrate in its fundamental mode ($n=1$) or in a “harmonic” of the fundamental mode (for an integer $n > 1$). In the classical case the complex coefficients $a_n^{\mu}$, $\tilde a_n^{\mu}$ represent the (complex) amplitudes of vibration for the modes of oscillation at frequency $n$ times the fundamental frequency. (with $a_n^{\mu}$ corresponding to a wave travelling to the right, while $\tilde
a_n^{\mu}$ corresponds to a wave travelling to the left.) When one quantizes the string dynamics the position of the string $X^{\mu}
(\tau , \sigma)$ becomes an operator (acting in the space of quantum states of the system), and because of this the quantities $x^{\mu} , p^{\mu} , a_n^{\mu}$ and $\tilde a_n^{\mu}$ in (\[rg27\]) become operators. The notation h.c. signifies that one must add the hermitian conjugates of the oscillation terms, which will contain the operators $(a_n^{\mu})^{\dagger}$ and $(\tilde a_n^{\mu})^{\dagger}$. (The notation $\dagger$ indicates hermitian conjugation, in other words the operator analog of complex conjugation.) One then finds that the operators $x^{\mu}$ and $p^{\mu}$ describing the motion of the center of mass satisfy the usual commutation relations of a relativistic particle, $[x^{\mu} , p^{\mu}] = i \,
\hbar \, \eta^{\mu\nu}$, and that the operators $a_n^{\mu}$ and $\tilde a_n^{\mu}$ become annihilation operators, like those that appear in the quantum theory of any vibrating system: $[a_n^{\mu}
, (a_m^{\nu})^{\dagger}] = \hbar \, \eta^{\mu\nu} \, \delta_{nm}$, $[\tilde a_n^{\mu} , (\tilde a_m^{\nu})^{\dagger}] = \hbar \,
\eta^{\mu\nu} \, \delta_{mn}$. In the case of an open string, one only has [*one*]{} set of oscillators, let us say $a_n^{\mu}$. The discussion up until now has neglected to mention that the oscillation amplitudes $a_n^{\mu} , \tilde a_n^{\mu}$ must satisfy an infinite number of constraints (connected with the equation obtained by minimizing (\[rg26\]) with respect to the auxiliary metric $\gamma_{ab}$). One can satisfy these by expressing [*two*]{} of the space-time components of the oscillators $a_n^{\mu} ,
\tilde a_n^{\mu}$ (for each $n$) as a function of the other. Because of this, the physical states of the string are described by oscillators $a_n^i , \tilde a_n^i$ where the index $i$ only takes $D-2$ values in a space-time of dimension $D$. Forgetting this subtlety for the moment (which is nevertheless crucial physically), let us conclude this discussion by summarizing the [*spectrum*]{} of a quantum string, or in other words the ensemble of quantum states of motion for a string.
For an open string, the ensemble of quantum states describes the states of motion (the momenta $p^{\mu}$) of an infinite collection of relativistic particles, having squared masses $M^2 = -
\eta_{\mu\nu} \, p^{\mu} \, p^{\nu}$ equal to $(N-1) \, \, m_s^2$, where $N$ is a non-negative integer and $m_s \equiv \hbar /
\ell_s$ is the fundamental mass of string theory associated to the fundamental length $\ell_s$. For a closed string, one finds another “infinite tower” of more and more massive particles, this time with $M^2 = 4 (N-1) \, m_s^2$. In both cases the integer $N$ is given, as a function of the string’s oscillation amplitudes (travelling to the right), by $$\label{rg28}
N = \sum_{n=1}^{\infty} n \, \eta_{\mu\nu} (a_n^{\mu})^{\dagger} \, a_n^{\nu} \, .$$ In the case of a closed string one must also satisfy the constraint $N = \tilde N$ where $\tilde N$ is the operator obtained by replacing $a_n^{\mu}$ by $\tilde a_n^{\mu}$ in (\[rg28\]).
The preceding result essentially states that the (quantized) internal energy of an oscillating string defines the squared mass of the associated particle. The presence of the additional term $-1$ in the formulae given above for $M^2$ means that the quantum state of minimum internal energy for a string, that is, the “vacuum” state $\vert 0 \rangle$ where all oscillators are in their ground state, $a_n^{\mu} \mid 0 \rangle = 0$, corresponds to a negative squared mass ($M^2 = -m_s^2$ for the open string and $M^2 = - 4 \, m_s^2$ for the closed string). This unusual quantum state (a [*tachyon*]{}) corresponds to an instability of the theory of bosonic strings. It is absent from the more sophisticated versions of string theory (“superstrings”) due to F. Gliozzi, J. Scherk, and D. Olive, to M. Green and J.H. Schwarz, and to D. Gross and collaborators. Let us concentrate on the other states (which are the only ones that have corresponding states in superstring theory). One then finds that the first possible physical quantum states (such that $N=1$) describe some massless particles. In relativistic quantum theory it is known that any particle is the quantized excitation of a corresponding field. Therefore the massless particles that appear in string theory must correspond to long-range fields. To know which fields appear in this way one must more closely examine which possible combinations of oscillator excitations $a_1^{\mu} , a_2^{\mu} , a_3^{\mu} ,
\ldots$, appearing in Formula (\[rg28\]), can lead to $N=1$. Because of the factor $n$ in (\[rg28\]) multiplying the harmonic contribution of order $n$ to the mass squared, only the oscillators of the fundamental mode $n=1$ can give $N=1$. One then deduces that the internal quantum states of massless particles appearing in the theory of [*open strings*]{} are described by a string oscillation state of the form $$\label{rg29}
\zeta_{\mu} (a_1^{\mu})^{\dagger} \mid 0 \rangle \, .$$ On the other hand, because of the constraint $N = \tilde N = 1$, the internal quantum states of the massless particles appearing in the theory of [*closed strings*]{} are described by a state of excitation containing both a left-moving oscillation and a right-moving oscillation: $$\label{rg30}
\zeta_{\mu\nu} (a_1^{\mu})^{\dagger} \, (\tilde a_1^{\nu})^{\dagger} \mid 0 \rangle \, .$$ In Equations (\[rg29\]) and (\[rg30\]) the state $\vert 0
\rangle$ denotes the ground state of all oscillators ($a_n^{\mu}
\mid 0 \rangle = \tilde a_n^{\mu} \mid 0 \rangle = 0$).
The state (\[rg29\]) therefore describes a massless particle (with momentum satisfying $\eta_{\mu\nu} \, p^{\mu} \, p^{\nu} =
0$), possessing an “internal structure” described by a vector polarization $\zeta_{\mu}$. Here we recognize exactly the definition of a photon, the quantum state associated with a wave $A_{\mu} (x) = \zeta_{\mu} \exp (i \, k_{\lambda} \,
x^{\lambda})$, where $p^{\mu} = \hbar \, k^{\mu}$. The theory of open strings therefore contains Maxwell’s theory. (One can also show that, because of the constraints briefly mentioned above, the polarization $\zeta_{\mu}$ must be transverse, $k^{\mu} \,
\zeta_{\mu} = 0$, and that it is only defined up to a gauge transformation: $\zeta'_{\mu} = \zeta_{\mu} + a \, k_{\mu}$.) As for the state (\[rg30\]), this describes a massless particle ($\eta_{\mu\nu} \, p^{\mu} \, p^{\nu} = 0$), possessing an “internal structure” described by a tensor polarization $\zeta_{\mu\nu}$. The plane wave associated with such a particle is therefore of the form $\bar h_{\mu\nu} (x) = \zeta_{\mu\nu}
\exp (i \, k_{\lambda} \, x^{\lambda})$, where $p^{\mu} = \hbar \,
k^{\mu}$. As in the case of the open string, one can show that $\zeta_{\mu\nu}$ must be transverse, $\zeta_{\mu\nu} \, k^{\nu} =
0$ and that it is only defined up to a gauge transformation, $\zeta'_{\mu\nu} = \zeta_{\mu\nu} + k_{\mu} \, a_{\nu} + k_{\nu}
\, b_{\mu}$. We here see the same type of structure appear that we had in general relativity for plane waves. However, here we have a structure that is richer than that of general relativity. Indeed, since the state (\[rg30\]) is obtained by combining two independent states of oscillation, $(a_1^{\mu})^{\dagger}$ and $(\tilde a_1^{\mu})^{\dagger}$, the polarization tensor $\zeta_{\mu\nu}$ is not constrained to be symmetric. Moreover it is not constrained to have vanishing trace. Therefore, if we decompose $\zeta_{\mu\nu}$ into its possible irreducible parts (a symmetric traceless part, a symmetric part with trace, and an antisymmetric part) we find that the field $\bar h_{\mu\nu} (x)$ associated with the massless states of a closed string decomposes into: (i) a field $h_{\mu\nu} (x)$ (the [*graviton*]{}) representing a weak gravitational wave in general relativity, (ii) a scalar field $\Phi (x)$ (called the [*dilaton*]{}), and (iii) an antisymmetric tensor field $B_{\mu\nu} (x) = - B_{\nu\mu} (x)$ subject to the gauge invariance $B'_{\mu\nu} (x) = B_{\mu\nu} (x)
+ \partial_{\mu} \, a_{\nu} (x) -
\partial_{\nu} \, a_{\mu} (x)$. Moreover, when one studies the non-linear interactions between these various fields, as described by the transition amplitudes $A(f,i)$ in string theory, one can show that the field $h_{\mu\nu} (x)$ truly represents a deformation of the flat geometry of the background space-time in which the theory was initially formulated. Let us emphasize this remarkable result. We started from a theory that studied the quantum dynamics of a string in a rigid background space-time. This theory predicts that [*certain quantum excitations of a string*]{} (that propagate at the speed of light) [*in fact represent waves of deformation of the space-time geometry*]{}. In intuitive terms, the “elasticity” of space-time postulated by the theory of general relativity appears here as being due to certain internal vibrations of an elastic object extended in one spatial dimension.
Another suggestive consequence of string theory is the link suggested by the comparison between (\[rg29\]) and (\[rg30\]). Roughly, Equation (\[rg30\]) states that the internal state of a closed string corresponding to a graviton is constructed by taking the (tensor) product of the states corresponding to photons in the theory of open strings. This unexpected link between Einstein’s gravitation $(g_{\mu\nu})$ and Maxwell’s theory $(A_{\mu})$ translates, when we look at interactions in string theory, into remarkable identities (due to H. Kawai, D.C. Lewellen, and S.-H.H. Tye) between the transition amplitudes of open strings and those of closed strings. This affinity between electromagnetism, or rather Yang-Mills theory, and gravitation has recently given rise to fascinating conjectures (due to A. Polyakov and J. Maldacena) connecting quantum Yang-Mills theory in flat space-time to quasi-classical limits of string theory and gravitation in curved space-time. Einstein would certainly have been interested to see how classical general relativity is used here to clarify the limit of a [*quantum*]{} Yang-Mills theory.
Having explained the starting point of string theory, we can outline the intuitive reason for which this theory avoids the problems with divergent integrals that appeared when one tried to directly quantize gravitation. We have seen that string theory contains an infinite tower of particles whose masses grow with the degree of excitation of the string’s internal oscillators. The gravitational field appears in the limit that one considers the low energy interactions ($E \ll m_s$) between the massless states of the theory. In this limit the graviton (meaning the particle associated with the gravitational field) is treated as a “point-like” particle. When we consider more complicated processes (at one loop, $\ell = 1$, see above), virtual elementary gravitons could appear with arbitrarily high energy. It is these virtual high-energy gravitons that are responsible for the divergences. However, in string theory, when we consider any intermediate process whatsoever where high energies appear, it must be remembered that this high intermediate energy can also be used to excite the internal state of the virtual gravitons, and thus reveal that they are “made” from an extended string. An analysis of this fact shows that string theory introduces an effective truncation of the type $E \lesssim m_s$ on the energies of exchanged virtual particles. In other words, the fact that there are no truly “point-like” particles in string theory, but only string excitations having a characteristic length $\sim
\ell_s$, eliminates the problem of infinities connected to arbitrarily small length and time scales. Because of this, in string theory one can calculate the transition amplitudes corresponding to a collision between two gravitons, and one finds that the result is given by a finite integral [@strings].
Up until now we have only considered the starting point of string theory. This is a complex theory that is still in a stage of rapid development. Let us briefly sketch some other aspects of this theory that are relevant for this exposé centered around relativistic gravitation. Let us first state that the more sophisticated versions of string theory ([*superstrings*]{}) require the inclusion of fermionic oscillators $b_n^{\mu}$, $\tilde b_n^{\mu}$, in addition to the bosonic oscillators $a_n^{\mu}$, $\tilde a_n^{\mu}$ introduced above. One then finds that there are no particles of negative mass-squared, and that the space-time dimension $D$ must be equal to 10. One also finds that the massless states contain more states than those indicated above. In fact, one finds that the fields corresponding to these states describe the various possible theories of supergravity in $D=10$. Recently (in work by J. Polchinski) it has also been understood that string theory contains not only the states of excitation of strings (in other words of objects extended in one spatial direction), but also the states of excitation of objects extended in $p$ spatial directions, where the integer $p$ can take other values than $1$. For example, $p=2$ corresponds to a [*membrane*]{}. It even seems (according to C. Hull and P. Townsend) that one should recognize that there is a sort of “democracy” between several different values for $p$. An object extended in $p$ spatial directions is called a [*$p$-brane*]{}. In general, the masses of the quantum states of these $p$-branes are very large, being parametrically higher than the characteristic mass $m_s$. However, one may also consider a limit where the mass of certain $p$-branes tends towards zero. In this limit, the fields associated with these $p$-branes become long-range fields. A surprising result (by E. Witten) is that, in this limit, the infinite tower of states of certain $p$-branes (in particular for $p=0$) corresponds exactly to the infinite tower of states that appear when one considers the maximal supergravity in $D=11$ dimensions, with the eleventh (spatial) dimension compactified on a circle (that is to say with a periodicity condition on $x^{11}$). In other words, in a certain limit, a theory of superstrings in $D=10$ transforms into a theory that lives in $D=11$ dimensions! Because of this, many experts in string theory believe that the true definition of string theory (which is still to be found) must start from a theory (to be defined) in 11 dimensions (known as “$M$-theory”).
We have seen in Section \[sec8\] that one point of contact between relativistic gravitation and quantum theory is the phenomenon of thermal emission from black holes discovered by S.W. Hawking. String theory has shed new light upon this phenomenon, as well as on the concept of black hole “entropy.” The essential question that the calculation of S.W. Hawking left in the shadows is: what is the physical meaning of the quantity $S$ defined by Equation (\[rg19\])? In the thermodynamic theory of ordinary bodies, the entropy of a system is interpreted, since Boltzmann’s work, as the (natural) logarithm of the number of microscopic states $N$ having the same macroscopic characteristics (energy, volume, etc.) as the state of the system under consideration: $S = \log N$. Bekenstein had attempted to estimate the number of microscopic internal states of a macroscopically defined black hole, and had argued for a result such that $\log N$ was on the order of magnitude of $A / \hbar \, G$, but his arguments remained indirect and did not allow a clear meaning to be attributed to this counting of microscopic states. Work by A. Sen and by A. Strominger and C. Vafa, as well as by C.G. Callan and J.M. Maldacena has, for the first time, given examples of black holes whose microscopic description in string theory is sufficiently precise to allow for the calculation (in certain limits) of the number of internal quantum states, $N$. It is therefore quite satisfying to find a final result for $N$ whose logarithm is [*precisely*]{} equal to the expression (\[rg19\]). However, there do remain dark areas in the understanding of the quantum structure of black holes. In particular, the string theory calculations allowing one to give a precise statistical meaning to the entropy (\[rg19\]) deal with very special black holes (known as [*extremal*]{} black holes, which have the maximal electric charge that a black hole with a regular horizon can support). These black holes have a Hawking temperature equal to zero, and therefore do not emit thermal radiation. They correspond to [*stable*]{} states in the quantum theory. One would nevertheless also like to understand the detailed internal quantum structure of [*unstable*]{} black holes, such as the Schwarzschild black hole (\[rg17\]), which has a non-zero temperature, and which therefore loses its mass little by little in the form of thermal radiation. What is the final state to which this gradual process of black hole “evaporation” leads? Is it the case that an initial pure quantum state radiates all of its initial mass to transform itself entirely into incoherent thermal radiation? Or does a Schwarzschild black hole transform itself, after having obtained a minimum size, into something else? The answers to these questions remain open to a large extent, although it has been argued that a Schwarzschild black hole transforms itself into a highly massive quantum string state when its radius becomes on the order of $\ell_s$ [@bh].
We have seen previously that string theory contains general relativity in a certain limit. At the same time, string theory is, strictly speaking, infinitely richer than Einstein’s gravitation, for the graviton is nothing more than a particular quantum excitation of a string, among an infinite number of others. What deviations from Einstein’s gravity are predicted by string theory? This question remains open today because of our lack of comprehension about the connection between string theory and the reality observed in our everyday environment (4-dimensional space-time; electromagnetic, weak, and strong interactions; the spectrum of observed particles; $\ldots$). We shall content ourselves here with outlining a few possibilities. (See the contribution by I. Antoniadis for a discussion of other possibilities.) First, let us state that if one considers collisions between gravitons with energy-momentum $k$ smaller than, but not negligible with respect to, the characteristic string mass $m_s$, the calculations of transition amplitudes in string theory show that the usual Einstein equations (in the absence of matter) $R_{\mu\nu} = 0$ must be modified, by including corrections of order $(k/m_s)^2$. One finds that these modified Einstein equations have the form (for bosonic string theory) $$\label{rg31}
R_{\mu\nu} + \frac{1}{4} \, \ell_s^2 \, R_{\mu\alpha\beta\gamma} \,
R_{\nu}^{\centerdot\alpha\beta\gamma} + \cdots = 0 \, ,$$ where $$\label{rg32}
R_{\centerdot\nu\alpha\beta}^{\mu} \equiv \partial_{\alpha} \,
\Gamma_{\nu\beta}^{\mu} + \Gamma_{\sigma\alpha}^{\mu} \,
\Gamma_{\nu\beta}^{\sigma} - \partial_{\beta} \, \Gamma_{\nu\alpha}^{\mu} -
\Gamma_{\sigma\beta}^{\mu} \, \Gamma_{\nu\alpha}^{\sigma} \, ,$$ denotes the “curvature tensor” of the metric $g_{\mu\nu}$. (the quantity $R_{\mu\nu}$ defined in Section \[sec5\] that appears in Einstein’s equations in an essential way is a “trace” of this tensor: $R_{\mu\nu} = R_{\centerdot\mu\sigma\nu}^{\sigma}$.) As indicated by the dots in (\[rg31\]), the terms written are no more than the two first terms of an infinite series in growing powers of $\ell_s^2 \equiv \alpha'$. Equation (\[rg31\]) shows how the fact that the string is not a point, but is rather extended over a characteristic length $\sim \ell_s$, modifies the Einsteinian description of gravity. The corrections to Einstein’s equation shown in (\[rg31\]) are nevertheless completely negligible in most applications of general relativity. In fact, it is expected that $\ell_s$ is on the order of the Planck scale $\ell_p$, Equation (\[rg24\]). More precisely, one expects that $\ell_s$ is on the order of magnitude of $10^{-32}$ cm. (Nevertheless, this question remains open, and it has been recently suggested that $\ell_s$ is much larger, and perhaps on the order of $10^{-17}$ cm.)
If one assumes that $\ell_s$ is on the order of magnitude of $10^{-32}$ cm (and that the extra dimensions are compactified on distances scales on the order of $\ell_s$), the only area of general relativistic applications where the modifications shown in (\[rg31\]) should play an important role is in primordial cosmology. Indeed, close to the initial singularity of the Big Bang (if it exists), the “curvature” $R_{\mu\nu\alpha\beta}$ becomes extremely large. When it reaches values comparable to $\ell_s^{-2}$ the infinite series of corrections in (\[rg31\]) begins to play a role comparable to the first term, discovered by Einstein. Such a situation is also found in the interior of a black hole, when one gets very close to the singularity (see Figure \[fig3\]). Unfortunately, in such situations, one must take the infinite series of terms in (\[rg31\]) into account, or in other words replace Einstein’s description of gravitation in terms of a [*field*]{} (which corresponds to a [*point-like*]{} (quantum) particle) by its exact stringy description. This is a difficult problem that no one really knows how to attack today.
However, a priori string theory predicts more drastic low energy ($k \ll m_s$) modifications to general relativity than the corrections shown in (\[rg31\]). In fact, we have seen in Equation (\[rg30\]) above that Einsteinian gravity does not appear alone in string theory. It is always necessarily accompanied by other long-range fields, in particular a scalar field $\Phi (x)$, the [*dilaton*]{}, and an antisymmetric tensor $B_{\mu\nu} (x)$. What role do these “partners” of the graviton play in observable reality? This question does not yet have a clear answer. Moreover, if one recalls that (super)string theory must live in a space-time of dimension $D=10$, and that it includes the $D=10$ (and eventually the $D=11$) theory of supergravity, there are many other supplementary fields that add themselves to the ten components of the usual metric tensor $g_{\mu\nu}$ (in $D=4$). It is conceivable that all of these supplementary fields (which are massless to first approximation in string theory) acquire masses in our local universe that are large enough that they no longer propagate observable effects over macroscopic scales. It remains possible, however, that one or several of these fields remain (essentially) massless, and therefore can propagate physical effects over distances that are large enough to be observable. It is therefore of interest to understand what physical effects are implied, for example, by the dilaton $\Phi (x)$ or by $B_{\mu\nu} (x)$. Concerning the latter, it is interesting to note that (as emphasized by A. Connes, M. Douglas, and A. Schwartz), in a certain limit, the presence of a background $B_{\mu\nu} (x)$ has the effect of deforming the space-time geometry in a “non-commutative” way. This means that, in a certain sense, the space-time coordinates $x^{\mu}$ cease to be simple real (commuting) numbers in order to become non-commuting quantities: $x^{\mu} x^{\nu} - x^{\nu} x^{\mu} =
\varepsilon^{\mu\nu}$ where $\varepsilon^{\mu\nu} = -
\varepsilon^{\nu\mu}$ is connected to a (uniform) background $B_{\mu\nu}$. To conclude, let us consider the other obligatory partner of the graviton $g_{\mu\nu} (x)$, the dilaton $\Phi (x)$. This field plays a central role in string theory. In fact, the average value of the dilaton (in the vacuum) determines the string theory coupling constant, $g_s = e^{\Phi}$. The value of $g_s$ in turn determines (along with other fields) the physical coupling constants. For example, the gravitational coupling constant is given by a formula of the type $\hbar \, G = \ell_s^2 (g_s^2 +
\cdots)$ where the dots denote correction terms (which can become quite important if $g_s$ is not very small). Similarly, the fine structure constant, $\alpha = e^2 / \hbar c \simeq 1/137$, which determines the intensity of electromagnetic interactions is a function of $g_s^2$. Because of these relations between the physical coupling constants and $g_s$ (and therefore the value of the dilaton; $g_s = e^{\Phi}$), we see that if the dilaton is massless (or in other words is long-range), its value $\Phi (x)$ at a space-time point $x$ will depend on the distribution of matter in the universe. For example, as is the case with the gravitational field (for example $g_{00} (x) \simeq -1 + 2 GM /
c^2 r$), we expect that the value of $\Phi (x)$ depends on the masses present around the point $x$, and should be different at the Earth’s surface than it is at a higher altitude. One may also expect that $\Phi (x)$ would be sensitive to the expansion of the universe and would vary over a time scale comparable to the age of the universe. However, if $\Phi (x)$ varies over space and/or time, one concludes from the relations shown above between $g_s =
e^{\Phi}$ and the physical coupling constants that the latter must also vary over space and/or time. Therefore, for example, the value, here and now, of the fine structure constant $\alpha$ could be slightly different from the value it had, long ago, in a very distant galaxy. Such effects are accessible to detailed astronomical observations and, in fact, some recent observations have suggested that the interaction constants were different in distant galaxies. However, other experimental data (such as the fossil nuclear reactor at Oklo and the isotopic composition of ancient terrestrial meteorites) put very severe limits on any variability of the coupling “constants.” Let us finally note that if the fine structure “constant” $\alpha$, as well as other coupling “constants,” varies with a massless field such as the dilaton $\Phi (x)$, then this implies a violation of the basic postulate of general relativity: the principle of equivalence. In particular, one can show that the universality of free fall is necessarily violated, meaning that bodies with different nuclear composition would fall with different accelerations in an external gravitational field. This gives an important motivation for testing the principle of equivalence with greater precision. For example, the MICROSCOPE space mission [@micro] (of the CNES) should soon test the universality of free fall to the level of $10^{-15}$, and the STEP space project (Satellite Test of the Equivalence Principle) [@step] could reach the level $10^{-18}$.
Another interesting phenomenological possibility is that the dilaton (and/or other scalar fields of the same type, called [*moduli*]{}) acquires a non-zero mass that is however very small with respect to the string mass scale $m_s$. One could then observe a modification of Newtonian gravitation over small distances (smaller than a tenth of a millimeter). For a discussion of this theoretical possibility and of its recent experimental tests see, respectively, the contributions by I. Antoniadis and J. Mester to this Poincaré seminar.
Conclusion {#sec12}
==========
For a long time general relativity was admired as a marvellous intellectual construction, but it only played a marginal role in physics. Typical of the appraisal of this theory is the comment by Max Born [@B56] made upon the fiftieth anniversary of the [*annus mirabilis*]{}: “The foundations of general relativity seemed to me then, and they still do today, to be the greatest feat of human thought concerning Nature, the most astounding association of philosophical penetration, physical intuition, and mathematical ability. However its connections to experiment were tenuous. It seduced me like a great work of art that should be appreciated and admired from a distance.”
Today, one century after the [*annus mirabilis*]{}, the situation is quite different. General relativity plays a central role in a large domain of physics, including everything from primordial cosmology and the physics of black holes to the observation of binary pulsars and the definition of international atomic time. It even has everyday practical applications, via the satellite positioning systems (such as the GPS and, soon, its European counterpart Galileo). Many ambitious (and costly) experimental projects aim to test it (G.P.B., MICROSCOPE, STEP, $\ldots$), or use it as a tool for deciphering the distant universe (LIGO/VIRGO/GEO, LISA, $\ldots$). The time is therefore long-gone that its connection with experiment was tenuous. Nevertheless, it is worth noting that the fascination with the structure and physical implications of the theory evoked by Born remains intact. One of the motivations for thinking that the theory of strings (and other extended objects) holds the key to the problem of the unification of physics is its deep affinity with general relativity. Indeed, while the attempts at “Grand Unification” made in the 1970s completely ignored the gravitational interaction, string theory necessarily leads to Einstein’s fundamental concept of a dynamical space-time. At any rate, it seems that one must more deeply understand the “generalized quantum geometry” created through the interaction of strings and $p$-branes in order to completely formulate this theory and to understand its hidden symmetries and physical implications. Einstein would no doubt appreciate seeing the key role played by symmetry principles and gravity within modern physics.
[99999]{} , [*Zur Elektrodynamik bewegter K[ö]{}rper*]{}, [*Annalen der Physik*]{} [**17**]{}, 891 (1905). See http://www.einstein.caltech.edu for an entry into the Einstein Collected Papers Project. The French reader will have access to Einstein’s main papers in [*Albert Einstein, Œuvres choisies*]{}, Paris, Le Seuil/CNRS, 1993, under the direction of F. Balibar. See in particular Volumes 2 (Relativités I) and 3 (Relativités II). One can also consult the 2005 Poincaré seminar dedicated to Einstein (http://www.lpthe.jussieu.fr/poincare): [*Einstein, 1905-2005, Poincaré Seminar 2005*]{}, edited by T. Damour, O. Darrigol, B. Duplantier and V. Rivasseau (Birkh" auser Verlag, Basel, Suisse, 2006). See also the excellent summary article by D. Giulini and N. Straumann, “Einstein’s impact on the physics of the twentieth century,” Studies in History and Philosophy of Modern Physics [**37**]{}, 115-173 (2006). For online access to many of Einstein’s original articles and to documents about him, see http://www.alberteinstein.info/. We also note that most of the work in progress on general relativity can be consulted on various archives at http://xxx.lanl.gov, in particular the archive gr-qc. Review articles on certain sub-fields of general relativity are accessible at http://relativity.livingreviews.org. Finally, see T. Damour, [*Once Upon Einstein*]{}, A K Peters Ltd, Wellesley, 2006, for a recent non-technical account of the formation of Einstein’s ideas. , [*Dialogues Concerning Two New Sciences*]{}, translated by Henry Crew and Alfonso di Salvio, Macmillan, New York, 1914. The reader interested in learning about recent experimental tests of gravitational theories may consult, on the internet, either the highly detailed review by C.M. Will in Living Reviews (http://relativity.livingreviews.org/Articles/lrr-2001-4) or the brief review by T. Damour in the Review of Particle Physics (http://pdg.lbl.gov/). See also John Mester’s contribution to this Poincaré seminar. , [*Die Feldgleichungen der Gravitation*]{}, [*Sitz. Preuss. Akad. Wiss.*]{}, 1915, p. 844. The reader wishing to study the formalism and applications of general relativity in detail can consult, for example, the following works: L. Landau and E. Lifshitz, [*The Classical Theory of Fields*]{}, Butterworth-Heinemann, 1995; S. Weinberg, [*Gravitation and Cosmology*]{}, Wiley, New York, 1972; H.C. Ohanian and R. Ruffini, [*Gravitation and Spacetime*]{}, Second Edition, Norton, New York, 1994; N. Straumann, [*General Relativity, With Applications to Astrophysics*]{}, Springer Verlag, 2004. Let us also mention detailed course notes on general relativity by S.M. Carroll, available on the internet: http://pancake.uchicago.edu/$\sim$carroll/notes/ ; as well as at gr-qc/9712019. Finally, let us mention the recent book (in French) on the history of the discovery and reception of general relativity: J. Eisenstaedt, [*Einstein et la relativité générale*]{}, CNRS, Paris, 2002. , [*A Test of General Relativity Using Radio Links with the Cassini Spacecraft*]{}, [*Nature*]{} [**425**]{}, 374 (2003). http://einstein.stanford.edu , [*Dark stars: the evolution of an idea*]{}, in [*300 Years of Gravitation*]{}, edited by S.W. Hawking and W. Israel, Cambridge University Press, Cambridge, 1987, Chapter 7, pp. 199-276. The discovery of binary pulsars is related in Hulse’s Nobel Lecture: R.A. Hulse, [*Reviews of Modern Physics*]{} [**66**]{}, 699 (1994). For an introduction to the observational characteristics of pulsars, and their use in testing relativistic gravitation, see Taylor’s Nobel Lecture: J.H. Taylor, [*Reviews of Modern Physics*]{} [**66**]{}, 711 (1994). See also Michael Kramer’s contribution to this Poincaré seminar. For an update on the observational characteristics of pulsars, and their use in testing general relativity, see the Living Review by I.H. Stairs, available at http://relativity.livingreviews.org/Articles/lrr-2003-5/ and the contribution by Michael Kramer to this Poincaré seminar. For a recent update on tests of relativistic gravitation (and of tensor-scalar theories) obtained through the chronometry of binary pulsars, see G. Esposito-Farèse, gr-qc/0402007 (available on the general relativity and quantum cosmology archive at the address http://xxx.lanl.gov), and T. Damour and G. Esposito-Farèse, in preparation. Figure \[fig4\] is adapted from these references. For a review of the problem of the motion of two gravitationally condensed bodies in general relativity, up to the level where the effects connected to the finite speed of propagation of the gravitational interaction appear, see T. Damour, [*The problem of motion in Newtonian and Einsteinian gravity*]{}, in [*300 Years of Gravitation*]{}, edited by S.W. Hawking and W. Israel, Cambridge University Press, Cambridge, 1987, Chapter 6, pp. 128-198. , [*N[ä]{}herungsweise Integration der Feldgleichungen der Gravitation*]{}, [*Sitz. Preuss. Akad. Wiss.*]{}, 1916, p. 688 ; [*ibidem*]{}, [*[Ü]{}ber Gravitationswellen*]{}, 1918, p. 154. For a highly detailed introduction to these three problems, see K.S. Thorne [*Gravitational radiation*]{}, in [*300 Years of Gravitation*]{}, edited by S.W. Hawking and W. Israel, Cambridge University Press, Cambridge, 1987, Chapter 9, pp. 330-458. http://www.ligo.caltech.edu/ http://www.virgo.infn.it/ http://www/geo600.uni-hanover.de/ http://lisa.jpl.nasa.gov/ , gr-qc/0406012 ; see also the Living Review by L. Blanchet, available at http://relativity.livingreviews.org/Articles. Figure \[fig5\] is adapted from work by A. Buonanno and T. Damour, gr-qc/0001013. , Phys. Rev. Lett. [**95**]{}, 121101 (2005), gr-qc/0507014; [M. Campanelli et al.]{}, Phys. Rev. Lett. [**96**]{}, 111101 (2006), gr-qc/0511048 [J. Baker et al.]{}, Phys. Rev. D [**73**]{}, 104002 (2006), gr-qc/0602026. For a particularly clear exposé of the development of the quantum theory of fields, see, for example, the first chapter of S. Weinberg, [*The Quantum Theory of Fields*]{}, volume 1, Foundations, Cambridge University Press, Cambridge, 1995. For an introduction to the theory of (super)strings see http://superstringtheory.com/. For a detailed (and technical) introduction to the theory see the books: K. Becker, M. Becker, and J.H. Schwarz, [*String Theory and M-theory: An Introduction*]{}, Cambridge University Press, Cambridge, 2006; B. Zwiebach, [*A First Course in String Theory*]{}, Cambridge University Press, Cambridge, 2004; M.B. Green, J.H. Schwarz et E. Witten, [*Superstring theory*]{}, 2 volumes, Cambridge University Press, Cambridge, 1987 ; and J. Polchinski, [*String Theory*]{}, 2 volumes, Cambridge University Press, Cambridge, 1998. To read review articles or to research this theory as it develops see the hep-th archive at http://xxx.lanl.gov. To search for information on the string theory literature (and more generally that of high-energy physics) see also the site http://www.slac.stanford.edu/spires/find/hep. For a detailed introduction to black hole physics see P.K. Townsend, gr-qc/9707012; for an entry into the vast literature on black hole entropy, see, for example, T. Damour, hep-th/0401160 in [*Poincaré Seminar 2003*]{}, edited by Jean Dalibard, Bertrand Duplantier, and Vincent Rivasseau (Birkh" auser Verlag, Basel, 2004), pp. 227-264. http://www.onera.fr/microscope/ http://www.sstd.rl.ac.uk/fundphys/step/. , [*Physics and Relativity*]{}, in [*Fünfzig Jahre Relativitätstheorie, Bern, 11-16 Juli 1955, Verhandlungen*]{}, edited by A. Mercier and M. Kervaire, Helvetica Physica Acta, Supplement [**4**]{}, 244-260 (1956).
[^1]: Talk given at the Poincaré Seminar “Gravitation et Expérience” (28 October 2006, Paris); to appear in the proceedings to be published by Birkhäuser.
[^2]: Translated from the French by Eric Novak.
[^3]: Every repeated index is supposed to be summed over all of its possible values.
[^4]: The experiment with falling bodies said to be performed from atop the Leaning Tower of Pisa is a myth, although it aptly summarizes the essence of Galilean innovation.
[^5]: Recent historical work (by Herbert Pfister) has in fact shown that this effect had already been derived by Einstein within the framework of the provisory relativistic theory of gravity that he started to develop in 1912 in collaboration with Marcel Grossmann.
[^6]: In the case of PSR J0737$ - $3039, one of the six measured parameters is the ratio $x_A / x_B$ between a Keplerian parameter of the pulsar and its analog for the companion, which turns out to also be a pulsar.
[^7]: Recent work by Z. Bern et al. and M. Green et al., has, however, suggested that such cancellations take place at all orders for the case of maximal supergravity, dimensionally reduced to $D=4$ dimensions.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Tidal disruption events (TDEs) of stars by single or binary supermassive black holes (SMBHs) brighten galactic nuclei and reveal a population of otherwise dormant black holes. Adopting event rates from the literature, we aim to establish general trends in the redshift evolution of the TDE number counts and their observable signals. We pay particular attention to (i) jetted TDEs whose luminosity is boosted by relativistic beaming, and (ii) TDEs around binary black holes. We show that the brightest (jetted) TDEs are expected to be produced by massive black hole binaries if the occupancy of intermediate mass black holes (IMBHs) in low mass galaxies is high and if the TDE luminosity is proportional to the black hole mass. The same binary population will also provide gravitational wave sources for eLISA. In addition, we find that the shape of the X-ray luminosity function of TDEs strongly depends on the occupancy of IMBHs and could be used to constrain scenarios of SMBH formation. Finally, we make predictions for the expected number of TDEs observed by future X-ray telescopes finding that a 50 times more sensitive instrument than the Burst Alert Telescope (BAT) on board the [*Swift*]{} satellite is expected to trigger $\sim 10$ times more events than BAT, while 6-20 TDEs are expected in each deep field observed by a telescope 50 times more sensitive than the [*Chandra X-ray Observatory*]{} if the occupation fraction of IMBHs is high. Because of their long decay times, high-redshift TDEs can be mistaken for fixed point sources in deep field surveys and targeted observations of the same deep field with year-long intervals could reveal TDEs.'
author:
- |
Anastasia Fialkov$^{1}$[^1], Abraham Loeb$^{1}$\
$^{1}$ Institute for Theory and Computation, Harvard University, 60 Garden Street, Cambridge, MA, 02138, USA\
title: Jetted Tidal Disruptions of Stars as a Flag of Intermediate Mass Black Holes at High Redshifts
---
\[firstpage\]
cosmology: theory–galaxies: jets–X-rays: bursts
Introduction {#Sec:Intro}
============
Supermassive black holes (SMBHs) with masses between $10^6$ and $10^{10}~M_\odot$ [@Ghisellini:2010; @Thomas:2016] are observed to reside at the centers of dark matter halos with masses $\gtrsim 10^{12}~M_\odot$. Smaller dark matter halos, such as hosts of present day dwarf galaxies or galaxies at high redshifts, are expected to be populated with intermediate mass black holes (IMBHs) with masses in the range $\sim 10^2- 10^6~M_\odot$ [@Greene:2012]. The origin of SMBHs and IMBHs is still not well understood. In hierarchical structure formation these black holes are expected to grow from initial seeds as a result of galaxy mergers in which black holes coalesce [see @Graham:2015 for review]. Gas-rich mergers fuel AGN emitting energy in the optical, ultraviolet and X-ray bands. Roughly 10% of low-redshift AGN ($z\lesssim 5$) are radio-loud [@Jiang:2007], producing a pair of collimated relativistic jets which could be observed to greater distances because of the relativistic beaming effect. The fraction of radio-loud quasars at higher redshifts ($z\sim6$) was shown to be $8^{+5.0}_{-3.2}$% [@Banados:2015], suggesting no evolution of the radio-loud fraction with $z$. On the other hand, if the merger is dry and the merging galaxies do not have enough gas to feed the black hole, a dormant massive black hole (MBH) results without observable electromagnetic signature.
Even though the samples of both SMBHs and IMBHs build up, the percentage of galaxies hosting central black holes (the so-called occupation fraction) is still unclear, especially in low mass galaxies [@Greene:2007], and until recently IMBHs were considered hypothetical. However, recent observations have shown that some of dwarf galaxies in the local Universe could indeed be populated by IMBHs [@Farrell:2009; @Reines:2013; @Moran:2014; @BALDASSARE:2015; @BALDASSARE:2016]. The growing observational evidence includes 151 dwarf galaxies with candidate black holes in the mass range $10^5-10^6~M_\odot$ as identified from optical emission line ratios and/or broad H$\alpha$ emission [@Reines:2013]; 28 active galactic nuclei (AGN) with black hole masses in $10^3- 10^4~M_\odot$ range in nearby low-mass, low-luminosity dwarf galaxies found with the Sloan Digital Sky Survey [@Moran:2014]. In addition, @Lemons:2015 showed that large fraction of hard X-ray sources in dwarf galaxies are ultra-luminous, suggesting that they actually are IMBHs; @Yuan:2014 describe four dwarf Seyferts with masses $< 10^6~M_\odot$; @BALDASSARE:2015 reported observations of a $5\times 10^4~M_\odot$ black hole in the dwarf galaxy RGG 118; while @BALDASSARE:2016 list eleven additional black holes with masses between $7\times 10^4 - 1\times 10^6~M_\odot$. Finally, it was shown that more than $20$% of early-type galaxies with a stellar mass less that $ 10^{10}~M_\odot$ are expected to have massive black holes in their cores [@Miller:2015]. All this observational evidence suggests that IMBHs in dwarf galaxies are not as exotic as previously thought. However, dynamical mass measurements suggest a decline in the occupation fraction of black holes in host galaxies with velocity dispersion below 40 km s$^{-1}$ @Stone:2016KO.
One way to improve our constraints on the black hole occupation fraction is by probing the population of quiescent black holes when they are temporary illuminated by a tidal disruption event (TDE) in which a star passing too close to the black hole is shredded by a gravitational tide which exceeds the self-gravity of the star. Theoretical work on TDEs spans several decades, including works by @Hill:1975 [@Frank:1976; @Lacy:1982; @Carter:1983; @Rees:1988; @Evans:1989; @Phinney:1989; @Magorrian:1999; @Wang:2004; @Perets:2006; @Guillochon:2013; @Stone:2013; @Stone:2016; @Roth:2016] and others. When a TDE occurs, part of the stellar mass is ejected away, forming an elongated stream and heating the ambient medium [@Guillochon:2016], while the bound debris are accreted by the black hole emitting bright observable flare at a wide range of wavelengths from radio to $\gamma$-rays [@Rees:1990]. However, for very massive black holes [$\sim 1\times 10^8 M_\odot$ for a solar mass star @Kesden:2012] the tidal disruption distance is smaller than the Schwarzschild radius and stars are swallowed whole without exhibiting TDE flares. The emission peaks in the UV or soft X-rays with typical peak luminosity in the soft X-rays band being $L_{X}\sim 10^{42}-10^{44}$ erg s$^{-1}$. The flare decays on the timescale of months or years as a power law with a typical index $-5/3$, which is often considered to be the telltale signature of a tidal disruption of a star by a massive black hole.
Following the first detection by ROSAT [@Bade:1996; @Komossa:1999], about 50 TDEs have been observed [@Komossa:2015] in hard X-ray [@Bloom:2011; @Burrows:2011; @Cenko:2012; @Pasham:2015], soft X-ray [@Bade:1996; @Komossa:1999; @Donley:2002; @Esquej:2008; @Maksym:2010; @Saxton:2012; @Saxton:2016], UV [@Stern:2004; @Gezari:2006; @Gezari:2008; @Gezari:2009] and optical [@vanVelzen:2011; @Gezari:2012; @Arcavi:2014; @Chornock:2014; @Holoien:2014; @Vinko:2015] wavelengths. Some of the observed TDEs exhibit unusual properties. In particular, one of the detected TDEs shows an excess of variability in its light curve [@Saxton:2012] which can be explained if the black hole is actually a binary with a mass of $10^6~M_\odot$, mass ratio of 0.1 and semimajor axis of 0.6 milliparsecs [@Liu:2014]. This candidate appears to have one of the most compact orbits among the known SMBH binaries and has overcome the “final parsec problem” [@Colpi:2014]. Upon coalescence, it will be a strong source of gravitational wave emission in the sensitivity range of eLISA. Three other TDEs appear to be very bright in X-rays with peak soft X-ray isotropic luminosity being highly super-Eddington [@Burrows:2011; @Cenko:2012; @Brown:2015], while followup observations showed that these events were also associated with bright, compact, variable radio synchrotron emission [@Zauderer:2011; @Cenko:2012]. The observed high X-ray luminosity can be explained if the tidal disruption of stars in these cases powered a highly beamed relativistic jet pointed at the observer [@Tchekhovskoy:2014]. Based on these three observations, @Kawamuro:2016 concluded that $0.0007\% - 34\%$ of all TDE source relativistic jets, while @Bower:2013 and @vanVelzen:2013 estimate that $\lesssim 10\%$ of TDEs produce jetted emission at the observed level. Formation of jets in TDE is a topic of active research, e.g., works by @Metzger:2011 [@Mimica:2015; @Generozov:2016].
Based on the observations the TDE rate was derived to be $10^{-4}-10^{-5}$ per year per galaxy [@Donley:2002; @Khabibullin:2014; @Esquej:2008; @Luo:2008; @Maksym:2010; @Gezari:2008; @Wang:2012; @vanVelzen:2014] and is consistent with order-of magnitude theoretical predictions when IMBHs are ignored. The TDE rates were shown to be sensitive to the density profile and relaxation processes taking place in galactic nuclei [@Magorrian:1999; @Wang:2004; @Stone:2016]. The simplest and most commonly used estimate of the TDE rates is based on the steady-state solution of the Fokker-Planck equation describing the diffusion of stars in angular momentum and energy space driven by two-body relaxation. This process re-populates stellar orbits along which stars are disrupted by MBH, the so-called “loss cone”. With the two-body relaxation being the main process to refill the loss cone, other processes that may contribute to the stellar budget were also discussed in literature [@Rauch:1996; @Hopman:2006; @Merritt:2015; @Bar-Or:2016; @Perets:2006; @Magorrian:1999; @Merritt:2004; @Vasiliev:2013; @Vasiliev:2014; @Chen:2009; @Li:2015; @Wegg:2011; @Liu:2013; @Ivanov:2005; @Chen:2011; @Merritt:2005; @Lezhnin:2015; @Lezhnin:2016].
It is still unclear why IMBHs do not contribute to the observed TDEs, and most of the related theoretical studies show that black holes with masses smaller than $10^6~M_\odot$ can disrupt stars at rates higher than those of higher masses [@Wang:2004; @Stone:2016]. Therefore, if small halos are occupied by IMBHs, most disruptions are expected to occur in these systems making TDEs particularly good probes of the poorly-understood, low-mass end of MBH mass function [@Stone:2016]. Moreover, once detected in large quantities, TDEs will offer insight into physics of quiescent black holes, probe extreme accretion physics near the last stable orbit, provide the means to measure the spin of black holes and probe general relativity in the strong-field limit [@Hayasaki:2016; @Guillochon:2015c]. In addition, jetted TDEs will allow us to explore processes through which relativistic jets are born.
In this paper we extrapolate the population of TDEs to high redshifts (out to $z=20$), and predict their detectability with the next-generation X-ray telescopes. We propose a new way to test the occupation fraction of IMBHs through their unique contribution to the X-ray luminosity function. The paper is organized as follows: We summarize our approach in Section \[Sec:Methods\], deriving TDE rates as a function and outlining the TDE luminosity prescriptons. We showing the intrinsic X-ray luminosity function in Section \[Sec:LF\]. Next, we make predictions for realistic next generation X-ray surveys in Section \[Sec:Obs\] focusing on upgrades of [*Swift*]{} and [*Chandra*]{}. We summarize our conclusions in Section \[Sec:sum\].
Model Components {#Sec:Methods}
================
Even though TDEs have been extensively studied, the predicted rates are not in good agreement with observations. Therefore, we adopt simple assumptions for the event rates from literature with the aim to establish general trends in the redshift evolution of the TDE number counts and their observable signals. After defining the population of galaxies and black holes in Section \[Sec:BH\], we start by considering rates in a given galaxy of halo mass $M_h$ (Section \[Sec:TDE\]) and generalize for a cosmological population of galaxies in Section \[Sec:Cosm\]. We discuss the luminosity of TDE flares in Section \[Sec:Lum\].
Galaxies and Black Holes {#Sec:BH}
------------------------
One of the key model ingredients that determines the TDE rates is the distribution of stars in galactic nuclei [@Magorrian:1999; @Wang:2004]. Depending on the merger history of the galaxy and the efficiency of feedback on star formation, the stellar density profile can develop either a core or a cusp. For simplicity we adopt a singular isothermal sphere (SIS) density profile $\rho(r) = \sigma^2/2\pi G R^2$ with $\sigma$ being the constant velocity dispersion and $R$ the halo virial radius. For a galaxy of halo mass $M_h$, the relation between the halo mass and the velocity dispersion is simply $M_h =2\sigma^2R /G$; while the velocity dispersion can be directly related to the black hole mass using the $M_{BH}-\sigma$ relation [@Kormendy:2013; @McConnell:2013; @Saglia:2016; @BALDASSARE:2015; @Thomas:2016] $$M_{BH} = 0.309\times 10^9\times \left(\sigma /200~\textrm{km s$^{-1}$}\right)^{4.38}~M_\odot,
\label{Eq:Msigma}$$ which holds for a wide range of black hole masses from $5\times 10^4$ M$_\odot$ [@BALDASSARE:2015] to $1.7\times 10^{10}$ M$_\odot$ [@Thomas:2016] in galaxies with a bulge [@Guillochon:2015b]. Assuming the isothermal stellar distribution, @Wang:2004 derived TDE rates for galaxies with a single central black hole, while @Chen:2009 report the rates in a case of a black hole binary. As we discuss in Section \[Sec:TDE\], for MBH with masses in the range $M_{BH}\sim 10^5-10^{8}~M_{\odot}$ the TDE rates per halo computed using the isothermal stellar distribution are similar (within tens of percent) to more realistic estimates based on a large galaxy sample [@Stone:2016], which justifies our assumption. The error in the rate estimation due to the idealized stellar density profile is small compared to other uncertainties, e.g., introduced by the poorly constrained occupation fraction of IMBHs in low mass galaxies, which amounts to one-two orders of magnitude uncertainty in the derived volumetric TDE rates.
In order to address the uncertainty in the occupation fraction, $f_{occ}$, of MBH we consider two extreme cases: (i) complete black hole occupation of all halos that form stars, and (ii) assume that there are no black holes with masses below $10^6~M_{\odot}$, which is equivalent to the vanishing occupation fraction in halos below $10^{10}-10^{11}~M_\odot$ (depending on redshift). We refer to the former case as MBHs (or $f_{occ}=1$) and latter case as SMBHs (or $f_{occ}=0$). The two cases can be considered as an upper (former case) and lower (latter case) limits for the occupation fraction yielding, respectively, upper and lower limits for the expected TDE rates.
Even though black hole seeds could exist in ever smaller halos in the hierarchical picture of structure formation, one also needs stars to fuel a TDE flare. The lowest mass of a halo in which stars can form at high redshifts before the end of hydrogen reionization at $z\sim 6$ is determined by the cooling condition, which involves either molecular or atomic hydrogen [@Tegmark:1997; @LoebFurlanetto:2013; @Barkana:2016]. The lowest temperature coolant, molecular hydrogen, forms stars in dark matter halos as tiny as $\sim 10^5 ~M_\odot$. However, hydrogen molecules are easily destroyed by radiative feedback [@Machacek:2001] in which case star formation proceeds via atomic cooling in halos of $10^7-10^8~M_{\odot}$. Here we neglect the molecular cooling channel and assume that before reionization galaxies can form in halos down to a velocity dispersion of $\sim 12$ km s$^{-1}$, which host black holes of mass $10^{3.1}~M_{\odot}$. After reionization is complete, the smallest star forming halos are sterilized by photoheating feedback which evaporates gas out of all halos with velocity dispersion less than $\sim 25$ km s$^{-1}$. As a result, small galaxies stop producing many stars and the loss cone of stars around the central black hole is most likely not refilled efficiently enough to support the equilibrium TDE rates. Therefore, we assume in the post-reionization era that all black holes below $\sim 10^{4.5}~M_{\odot}$ remain without fuel and do not source TDEs. In a realistic reionization scenario, the minimal halo mass which efficiently forms stars would gradually rise with redshift [@Bacchic:2013; @Cohen:2016]. However, because the reionization history is poorly constrained at present, we adopt the simplest scenario of instantaneous reionization at $z_{re}=8$, consistent with latest constraints by the [*Planck*]{} satellite [@Planck:2016]. The minimal black hole mass in our MBHs scenario is thus $$M_{BH,min} = \left\{
\begin{array}{lr}
10^{3.1}~M_{\odot}, & z \ge 8\\
10^{4.5}~M_{\odot}, & z < 8
\end{array}
\right.$$
In our second, conservative, scenario we assume that $M_{BH, min} =10^6~M_\odot$. Several effects can contribute to low TDE rates from IMBHs, justifying our SMBHs scenario: black holes could be kicked out of halos as a result of merger; radiation from AGN could have negative feedback on star formation (AGN feedback), in this case the loss cone would not be replenished. Another possible feedback mechanism is the stellar feedback from supernova explosions which can expel gas from the halo making star formation less efficient [@Wyithe:2013].
An additional model ingredient that determines the TDE rate is the merger history of a halo which we incorporate in Section \[Sec:Cosm\]. As discussed in Section \[Sec:TDE\], the TDE rate in a recently merged galaxy is boosted by several orders of magnitude for $\sim 10^5$ years compared to a galaxy with a quiet merger history, e.g., works by @Ivanov:2005 [@Chen:2009; @Chen:2011]. The enhanced TDE rates are explained by the fact that the dynamics of the system are changed by the presence of a black hole binary produced as a result of a merger.
TDE rates per halo {#Sec:TDE}
------------------
In the case of a single black hole, the most secure way to feed stars into the loss cone around the black hole is via the standard two-body relaxation, which sets a lower limit on the TDE rates between $10^{-4}$ and $10^{-6}$ yr$^{-1}$ [@Frank:1976; @Lightman:1977; @Cohn:1978; @Magorrian:1999; @Wang:2004; @Stone:2016]. Other processes that may contribute to the stellar budget include resonant relaxation [@Rauch:1996; @Hopman:2006; @Merritt:2015; @Bar-Or:2016], presence of massive perturbers such as stellar clusters or gas clouds [@Perets:2006], nonspherical geometry [@Magorrian:1999; @Merritt:2004; @Vasiliev:2013; @Vasiliev:2014], black hole binaries [@Chen:2009; @Li:2015; @Wegg:2011; @Liu:2013; @Ivanov:2005; @Chen:2011], and anisotropy in the initial conditions [@Merritt:2005; @Lezhnin:2015; @Lezhnin:2016].
Assuming that the loss cone is refilled via two-body relaxation, the rate of tidal disruptions per halo per year for an isothermal stellar distribution @Wang:2004 reads $$\dot N^{1h}_{TDE} \sim 2.47\times 10^{-4} \left(\frac{\sigma}{100~\textrm{km s$^{-1}$}}\right)^{7/2}\left(\frac{M_{BH}}{10^7~M_{\odot}}\right)^{-1}~\textrm{yr$^{-1}$}
\label{Eq:TDE1}$$ where we only considered the disruption of solar mass stars[^2]. Despite the fact that the stellar density used to derive Eq. (\[Eq:TDE1\]) is idealized, the rates are similar to those derived by @Stone:2016 for stellar profiles from a real galaxy sample. @Stone:2016 estimated TDE rates due to two-body relaxation from $\sim 200$ galaxies with $M_{BH}\sim 10^5-10^{10}~M_{\odot}$ at $z\sim 0$ and got $\dot N_{TDE}\sim 1.2\times 10^{-5}\left(M_{BH}/10^8 M_\odot \right)^{-0.247}$ for galaxies with a core and $\dot N_{TDE}\sim 6.5 \times 10^{-5}\left(M_{BH}/10^8 M_{\odot}\right)^{-0.223}$ for galaxies with a cusp. We checked that for our choice of $M_{BH}-\sigma$ relation Eq. (\[Eq:TDE1\]) gives similar rates in normalization (up to several tens of percents) and comparable slope of $\sim -0.2$ when compared to the cusp fit of @Stone:2016.
Modeling TDE rates in merging system is more challenging, and rates are less understood than disruptions by a single MBH. Even though all theoretical studies point in the direction of TDE rates boosted for $\sim 10^5$ years by $\sim 2$ orders of magnitude compared to the disruption by single MBH, there is no consensus on details and it is unclear at present what is the leading process that replenishes the loss cone. When two galaxies merge, the black holes in the galactic cores first inspiral toward each other due to the dynamical friction. Next, when the mass in gas and stars enclosed within the black hole orbit is smaller than the total mass of the two black holes, the black holes become gravitationally bound and evolve as a binary. For black hole masses of $\sim 10^6~M_{\odot}$ this occurs when the typical separation between the two black holes is $\sim$parsec. Gradually, the binary hardens. If the binary reaches separations $\lesssim 0.001$ pc, gravitational waves are emitted as the two MBHs coalesce. Each one of the stages in the evolution of the binary has its own rate of TDEs. Using N-body simulations to model dry major mergers, @Li:2015 conclude that in the first stage, the tidal disruption rate for two well separated MBHs in merging system has similar levels to the sum of the rates of two individual MBHs in two isolated galaxy. In their fiducial model @Li:2015 find that after two MBHs get close enough to form a bound binary, the disruption rate is enhanced by a factor of 80 within a short time lasting for 13 Myr. This boosted disruption stage finishes after the SMBH binary evolves to a compact binary system, corresponding to a drop back of the disruption rate to a level few times higher than for an individual MBH. Other studies also point in the direction of enhanced rates from binaries. In particular, @Ivanov:2005 considered secular evolution of stellar orbits in the gravitational potential of an unequal mass binary and derived rates of $10^{-2}-1$ TDEs per year per galaxy for a $10^6-10^7$ M$_\odot$ primary black hole and a binary mass ratio $q> 0.01$. The duration of this boosted disruption stage was determined by the dynamical friction time scale $$T_{dyn}\sim \frac{2\times 10^2(1+q)}{q} \left(\frac{10^7~ \textrm{M}_\odot}{\textrm{M}_{\textrm{BH}}}\right)^{1/2} ~\textrm{yr}.$$ @Chen:2009 used scattering experiments to show that gravitational slingshot interactions between hardened binaries and a bound spherical isothermal stellar cusp will be accompanied by a burst of TDEs. It appears that a significant fraction of stars initially bound to the primary black hole will be scattered into the loss cone by resonant interaction with the secondary black hole. @Chen:2009 provide a fitting formula for the maximal TDE rates per halo with a binary MBH system $$\dot N^{2h}_{TDE} \sim (1+q)^{1/2} \left(\frac{\sigma}{100~\textrm{km s$^{-1}$}}\right)^4\left(\frac{M_{BH}}{10^7 ~M_\odot}\right)^{-1/3} ~\textrm{yr}^{-1}.
\label{Eq:TDE2}$$ and show that the enhancement lasts for $\sim 10^4$ years. @Chen:2011 included the Kozai-Lidov effect, chaotic three-body orbits, the evolution of the binary and the non-Keplerian stellar potentials and found that for masses of $ 10^7$ M$_\odot$ and $10^5$ M$_\odot$, TDE rates 0.2 events per year last for $\sim 3\times 10^5$ years which is three orders of magnitude larger than for a single black hole and broadly agrees with the conclusions of @Chen:2009. @Wegg:2011 included the same processes as @Chen:2011 and arrived at similar rates. They found that the majority of TDEs are due to chaotic orbits in agreement with @Chen:2009, showing that the Kozai-Lidov effect plays a secondary role. Their rates are somewhat smaller than in @Chen:2009 largely because the authors consider less cuspy stellar profiles. @LiNaoz:2015 considered the evolution of stellar disruption around a binary with MBHs masses of 10$^7$ M$_\odot$ and 10$^8$ M$_\odot$ due to the eccentric Kozai-Lidov mechanism yielding rate of 10$^{-2}$ TDE per year for $5\times 10^5$ years. Finally, @Liu:2013 concluded that the TDE rates of stars by SMBHs in the early phase of galaxy merger when galactic dynamical friction dominates could also be enhanced by several orders of magnitude (up to $10^{-2}$ events per year per galaxy) as a result of the perturbation by companion galactic core and the triaxial gravitational potential of the galactic nucleus.
To accommodate tidal disruptions induced by binary MBHs we adopt the fitting function given by Eq. (\[Eq:TDE2\]) assuming that this enhanced rate last for a dynamical time $T_{dyn}$, while for the rest of the time the TDE rate is simply $\dot N^{1h}_{TDE}$. As we can derive from Eqs. (\[Eq:TDE1\]) and (\[Eq:TDE2\]) the scaling of TDE rates with $M_{BH}$ is different for single and binary MBHs. Applying the $M_{BH}-\sigma$ relation to Eqs. (\[Eq:TDE1\]) and (\[Eq:TDE2\]) we find that the TDE rate scales as $\dot N^{2h}_{TDE}\propto M_{BH}^{0.6}$ for binaries and $\dot N^{1h}_{TDE} \propto M_{BH}^{-0.2}$ for single black holes. This property has immediate implications to the total observable TDE rates that will be discussed in Section \[Sec:Obs\].
Number of TDEs across cosmic time {#Sec:Cosm}
---------------------------------
The observed TDE number counts per unit time that originate from redshift $z$ depends on several factors with the dominant factor being the amount of halos of each mass and their merger history. To determine the halo abundance we make use of the Sheth-Tormen mass function [@Sheth:1999] in calculating the comoving number density of halos in each mass bin $dN_h/dM_h$ in units of $M_{\odot}^{-1}$ Mpc$^{-3}$. Next, adopting the merger rates of @Fakhouri:2010 we calculate the dimensionless average merger rate $dN_m/d\zeta /dz$ (in units of mergers per halo per unit redshift per unit halo mass ratio, $\zeta$), given by a fitting formula $$\frac{dN_m}{d\zeta dz}(\textrm{M},\zeta,z) =A\left(\frac{\textrm{M}}{10^{12}\textrm{M}_{\odot}}\right)^{\alpha} \zeta^{\beta}\exp\left[\left(\frac{\zeta}{\bar \zeta}\right)^{\gamma}\right](1+z)^{\eta},$$ where $(\alpha,\beta,\gamma,\eta) = (0.133,-1.995,0.263,0.0993$ and $(A,\bar \zeta)=(0.0104,9.72\times10^{-3})$.
For each halo we assign a TDE rate of $\dot N^{2h}_{TDE}$ according to the probability, $P_{2}$, of it to encounter a recent merger, and $\dot N^{1h}_{TDE}$ with a probability $P_1 = 1-P_{2}$. The probability, $P_2$, is determined using the following criterion: if the time between mergers is larger than the dynamical time, the TDE rates are those of single MBH, while if the time between mergers is smaller than $T_{dyn}$, there is an enhancement due to binaries. Given the merger rates, we estimate the probability of a halo of mass $M_h$ at redshift $z$ to be a result of a recent merger as follows $$P_{2}(M_h,z) = 1-\exp\left[- T_{dyn}\lambda\right],$$ where $$\lambda = \int d \zeta \frac{dN_m}{dz d\zeta}\frac{dz}{dt}.
\label{Eq:lambda}$$ For very light halos, mergers are frequent and halos typically undergo several mergers within $T_{dyn}$, in which case the probability for a merger is near unity. We assume that probability for merger with black hole mass ratio $q$ is flat for $q = 10^{-3}-10^{-1}$, and $q$ is related to the halo mass ratio, $\zeta$, through Eq. (\[Eq:Msigma\]). We ignore mergers with $0.1<q<1$ as they are expected to be rare. The top panel of Figure \[Fig:1\] shows the halo mass weighted number density of mergers $\int dM_h P_{2} dN_h/dM_h$ that yield enhanced TDE rates per unit volume in cases of $f_{occ} = 0$ (SMBHs only) and $f_{occ} = 1$ (all MBHs). In each case, the integral is over halos whose progenitors have both large enough black holes and gas to form stars. Such mergers are rare when the minimal mass is high (SMBH case), especially at high redshifts. In the case of $f_{occ}=1$ we can clearly see the turn on of photoheating feedback at $z=8.8$ which shuts down star formation in galaxies below $M_h\sim 10^9~M_{\odot}$ at lower redshifts leading to suppressed TDE rate.
![Top: Halo mass averaged number density of systems with mergers that occurred at $t<T_{dyn}$. We show the case of SMBHs only ($f_{occ} = 0$, black) and all MBHs ($f_{occ} = 1$, red). Bottom: Intrinsic TDE rates in the observer’s frame per comoving volume as a function of redshift are shown for SMBHs only (black) and all MBHs (red). In each case we show the contribution of single black holes with rates from @Wang:2004 (solid, $\dot N^{1h}_{TDE}$) and contribution of binaries (dashed, $\dot N^{2h}_{TDE}$) assuming solar mass stars. In all cases with mergers we assume that the enhancement due to binaries lasts for $T_{dyn}$.[]{data-label="Fig:1"}](Nm_feed "fig:"){width="3.4in"} ![Top: Halo mass averaged number density of systems with mergers that occurred at $t<T_{dyn}$. We show the case of SMBHs only ($f_{occ} = 0$, black) and all MBHs ($f_{occ} = 1$, red). Bottom: Intrinsic TDE rates in the observer’s frame per comoving volume as a function of redshift are shown for SMBHs only (black) and all MBHs (red). In each case we show the contribution of single black holes with rates from @Wang:2004 (solid, $\dot N^{1h}_{TDE}$) and contribution of binaries (dashed, $\dot N^{2h}_{TDE}$) assuming solar mass stars. In all cases with mergers we assume that the enhancement due to binaries lasts for $T_{dyn}$.[]{data-label="Fig:1"}](Rates0b_feed "fig:"){width="3.4in"}
Having the proper probabilities we can now calculate the expected TDE rates for an observer as a function of redshift. At every given redshift $P_{1}$ halos in each mass bin host a single MBH yielding TDE rate $$\dot N^{1}_{TDE} = \int dM_h \frac{dN_h}{dM_h}P_{1}\frac{ \dot N^{1h}_{TDE}}{(1+z)}.$$ The factor $(1+z)$ compensates for the time dilation in the apparent rate. The contribution from binaries is given by $$\dot N^{2}_{TDE} = \int dM_h \frac{dN_h}{dM} \int d q \frac{dP_{2}}{dq} \frac{\dot N^{2h}_{TDE}}{(1+z)}.$$ The total number density of TDE per year per unit comoving volume, $\dot N_{TDE} = \dot N^{1}_{TDE}+\dot N^{2}_{TDE}$, is shown on the bottom panel of Figure \[Fig:1\].
As was pointed out above, TDE rates induced by binary black holes are higher in the high black hole mass end, while the single black hole systems are more efficient in the low black hole mass end. To demonstrate this feature, we show on Figure \[Fig:2\] the fraction of intrinsic events (with no (1+z) factor) sourced by single black holes out of total number of TDEs at $z=0$ and $z=5$ as a function of the black hole mass (total mass in the case of binaries) in solar mass units for $f_{occ} =0$ and 1. As expected from the TDE scaling with the black hole mass ($\propto M_{BH}^{0.6}$ for binaries and $\propto M_{BH}^{-0.2}$ for single black holes), binary systems dominate at large black hole masses and at high redshifts (because of the increased merger rates). The mass dependence determines contribution of each component to the overall luminosity function of the TDEs which we consider in the next section.
![Fraction of intrinsic disruption events sourced by single black holes versus black hole mass for SMBHs only ($f_{occ} = 0$, black curves) and all MBHs ($f_{occ} = 1$, red curves). We show the fraction at $z = 0$ (solid) and $z=5$ (dashed). The dark grey region marks the occupation of SMBH, whereas the pale grey refers to the occupation of IMBHs.[]{data-label="Fig:2"}](Fr0_feed){width="3.4in"}
Finally, in our cosmological model we assume that 10% of all TDE source jets with the Lorenz factor of order $\Gamma = 10$ based on X-ray observations of jetted TDE [@Burrows:2011]. In the spirit of our simple approach we ignore parameters, such as stellar magnetic fields [@Guillochon:2016], which could affect fraction of jetted TDEs and assume a constant jet fraction over the halo mass and redshift range. Because the luminosity is channeled into a collimated beam of an opening angle $\theta \sim 1/\Gamma$, only a small fraction of the jetted TDEs will be actually observable. For an observer, the fraction of sky covered by the jets pointing toward the observer is, thus, $ f_{jet} = 10\%\times 2\times \pi \theta^2/4\pi =5\times 10^{-2}$% where we accounted for two jets emitted by every system. Overall, the observed TDE rates will include 90% of non-jetted TDE and $ f_{jet}$ of TDE with jets where we account only for the events that point toward the observer.
Luminosity of TDE flares {#Sec:Lum}
------------------------
The TDE rates shown in Figure \[Fig:1\] are not the ones we would actually observe. Observable rates depend on the luminosity (observed flux) of each event as well as on the sensitivity of a telescope (discussed in the next Section). In this section we outline our assumptions for the TDE luminosity and use them in Section \[Sec:Obs\] to calculate the observable signals. Because we are mainly interested in the high-redshift events which could be observed in X-rays when relativistic jets are produced, we will focus on the TDE signature in X-rays ignoring their UV and optical counterparts. We are also largely ignoring radio signals because it is unclear whether or not the radio emission from the TDE sourced jets arises from the same regions as their X-ray emission. The three X-ray observations of TDEs with jets include: (1) SwiftJ164449.3+573451, hereafter Sw1644+57 [@Bloom:2011; @Burrows:2011; @Levan:2011], of peak X-ray isotropic luminosity $L_X\sim 4\times 10^{48}$ erg s$^{-1}$ which originated from a galaxy at $z=0.353$ and was discovered in March 2011 by the [*Swift*]{} Burst Alert Telescope [BAT, 15-150 keV, @Barthelmy:2005]; (2) SwiftJ2058.4+0516, hereafter SwJ2058+05 [@Cenko:2012], of peak X-ray isotropic luminosity equivalent to $L_X\sim 4\times 10^{48}$ erg s$^{-1}$, which was discovered at $z=1.185$ in May 2011 by the BAT as part of the hard X-ray transient search; and (3) SwiftJ1112.2-8238, Sw1112-82 hereafter [@Brown:2015], which was detected by BAT in June 2011 as an unknown, long-lived gamma-ray transient source in a host identified at $z=0.89$ and with $L_X\sim 6\times 10^{48}$ erg s$^{-1}$. Estimates of the SMBH mass in each one of these events yield $M_{BH}\sim 10^6- 10^7~M_\odot$. Because the Eddington luminosity for a black hole of mass $M_{BH}$ is only $L_{Edd} = 1.3\times 10^{38} \left( M_{BH}/M_{\odot}\right)$ erg s$^{-1}$, these events either are intrinsically highly super-Eddington or the emitted energy is channeled in tightly collimated jets and the luminosity is boosted by a factor of $\sim 10^3-10^4$.
Theory predicts that a flare is produced when debris return to the vicinity of a black hole $t_{fall} \approx 0.1 \left(M_{BH}/10^6~M_\odot\right)^{1/2}$ years after the disruption and forms an accretion disc. If a solar mass star is completely disrupted, its debris fallback rate is [@Rees:1988; @Phinney:1989; @Stone:2013] $$\dot M_{fall}\sim \frac{1}{3t_{fall}}\left(\frac{t}{t_{fall}}\right)^{-5/3}~M_\odot~\textrm{yr$^{-1}$}
\label{Eq:Mfall}$$ with the peak mass accretion rate value of $$\frac{\dot M_{peak}}{\dot M_{Edd}} = 133 \left(\frac{M}{10^6~M_\odot}\right)^{-3/2},
\label{Eq:Mac}$$ where the Eddington accretion rate is $\dot M_{Edd} = L_{Edd}/\eta c^2$ and $\eta \sim 0.1$ is a typical radiative efficiency. It is likely that the mass fallback rate can be directly related to the observed X-ray luminosity of the source and, thus, can be used to determine the total emitted energy. In particular, if the accretion rate is fully translated to the bolometric luminosity, the peak luminosity is $L_{peak} = \eta c^2 \dot M_{peak}$. However, it is still not clear what is the efficiency of this process especially for intermediate black holes with mass less than 50 million solar masses for which $L_{peak}$ is highly super-Eddington for efficient circularization of the debris [@Guillochon:2015; @Dai:2015; @Shiokawa:2015].
Super-Eddington accretion fueled by a tidal disruption of a star was both detected in Nature [@Kara:2016] and studied in numerical simulations [@McKinney:2014; @Sadowski:2015a; @Sadowski:2015b; @Jiang:2014; @Inayoshi:2016; @Sakurai:2016]. Based on observations of reverberation in the redshifted iron K$\alpha$ line [@Kara:2016], super-Eddington accretion was detected in one of the jetted TDE events, Sw1644+57. From the reverberation timescale, the authors estimate the mass of the black hole to be a few million solar masses, suggesting an accretion rate of at least 100 times the Eddington limit. Simulations suggest that relativistic beaming can explain observed super-Eddington luminosities indicating that, once the accretion is super-Eddington, relativistic jets can be produced [@McKinney:2014]. Although the overall radiative efficiency and luminosity are still debated, in simulations a strong outflow is generated and radiation can leak through a narrow funnel along the polar direction. Close to the black hole, a jet carves out the inner accretion flow, exposing the X-ray emitting region of the disk. @Sadowski:2015a found that if a source with moderate accretion rate is observed down the funnel, the apparent luminosity of such a source will be orders of magnitude higher than the non-jetted luminosity. @Sadowski:2015b show that for an observer viewing down the axis, the isotropic equivalent luminosity is as high as $10^{48}$ erg s$^{-1}$ for a $10^7~M_\odot$ black hole accreting at $10^3$ the Eddington rate, which agrees with the observations of jetted TDEs. Independent of the accretion rate in simulations, super-Eddington disks around black holes exhibit a surprisingly large efficiency of $\eta \sim 4\%$ for non-rotating black holes; while spinning black holes yield the maximal efficiency of jets of 130% [@Piran:2015].
In other simulated systems such as stellar black holes, observed super-Eddington luminosities are also inferred: @Jiang:2014 studied super-Eddington accretion flows onto black holes using a global three dimensional radiation magneto-hydrodynamical simulation and found mass accretion rate of $\dot M \sim 220 L_{Edd}/c^2$ with outflows along the rotation axis, and radiative luminosity of $10 L_{Edd}$; $\dot M \sim 400 L_{Edd}/c^2$ was measured for a $10~M_\odot$ black hole with peak luminosity of $50 L_{Edd}$ [@McKinney:2015]; @Inayoshi:2016 argued that $\dot M \sim 5000 L_{Edd}/c^2$ is limited to the Eddington luminosity in a metal poor environment, but @Sakurai:2016 find $1 < L/L_{Edd} < 100$.
Because the TDE flares in jetted events are not fully understood [although see @Crumley:2016], we use two simple approaches to derive first the bolometric, and then the X-ray, luminosity for each event. Our first approach (Model A) is to simply assign the Eddington luminosity to each event according to the black hole mass, $L_{TDE}^A = L_{Edd}$. The second approach (Model B) assumes that the TDE bolometric luminosity is proportional to the mass accretion rate. However, for IMBHs the peak accretion rate exceeds both the observed rates and the simulated ones by few orders of magnitude. As studies have shown, TDE luminosity is not likely to exceed few hundreds $L_{Edd}$ [@McKinney:2014; @Sadowski:2015a; @Sadowski:2015b; @Jiang:2014; @Inayoshi:2016; @Sakurai:2016]. Therefore, we adopt an upper limit of $300 L_{Edd}$ and the luminosity of each event reads $$L_{TDE}^B = \textrm{min}\left[L_{peak},300 L_{Edd}\right].
\label{Eq:Lacc}$$ The major distinction between Models A and B is that in Model A the brightest events are produced by the biggest black holes which also are the rarest ones, especially at high redshifts; while when the luminosity scales as the accretion rate with a ceiling (Model B), the most luminous events are produced by black holes of mass $M_{BH}\sim 2.5\times 10^{6} ~M_{\odot}$ which are more common.
Observations show that the X-ray luminosity of the three jetted TDEs has a spectral energy distribution (SED) well fitted by a power law $S_\nu \propto \nu^{-\alpha}$ with a spectral index $\alpha$ in the range of $0.3-1$ with $\alpha = 0.33$ for Sw1112-82, $\alpha \sim 0.8$ for Sw1644+57 and $\alpha \sim 0.6$ for SwJ2058+05. Therefore, in our modeling we adopt power-law SED with a unique spectral index of $\alpha = 0.5$ to describe all the jetted events. The SED of a non-jetted TDE is expected to be a combination of a power-law and a black body, where the latter is negligible at high enough energies [$\sim 1$ keV and above, @Kawamuro:2016]. We follow @Kawamuro:2016 assuming that the spectral index, $\alpha$, is the same for non-jetted events as for the jetted ones ($\alpha =0.5$). The intrinsic spectral luminosity of an event is thus $L_\nu = L_0\nu^{-\alpha}$ where $L_0$ is the normalization constant. Assuming that the SED of these objects over a wide wavelength range is similar to that of AGN, we can calculate the X-ray luminosity of each event based on its bolometric luminosity. For the soft X-ray band ($2-10$ keV) we adopt a bolometric correction factor of $k_{2-10}\sim 50$ for the Eddington and $k_{2-10} = 70$ for the super-Eddington accretion rates [@Kawamuro:2016; @Vasudevan:2007]. Given these numbers we normalize our power law spectra in the soft X-ray band so that $L_0 = k_{2-10}(1-\alpha)L_{TDE}\left[10^{1-\alpha}-2^{1-\alpha}\right]^{-1}$. Using this prescription we can calculate the observed spectral flux for non-jetted TDEs at redshift $z $ $$S_\nu = \frac{L_0 \nu^{-\alpha} (1+z)^{1-\alpha}}{4\pi D_L^2},$$ where $D_L$ is the luminosity distance to the source. In a jet, the observed flux at an observed frequency $\nu$ is boosted by the factor of $\mathcal{D }^{3+\alpha}$ where $\mathcal{D }= \left[\Gamma (1-B\mu_{obs})\right]^{-1}$ is the Doppler factor and $\mu_{obs} = \cos\theta$ is the angle of the jet with respect to the observer [@Burrows:2011].
Simulations show that TDEs occurring around MBH binaries have similar peak luminosity in X-rays as TDEs sourced by single black holes; however, the light curve has stronger variability in time due to the perturbations introduced by the secondary black hole [@Liu:2009; @Liu:2014; @Coughlin:2016; @Ricarte:2016]. Therefore, we adopt similar prescription as described above to asigned X-ray luminosity to TDEs sourced by binaries.
Intrinsic Luminosity Function {#Sec:LF}
=============================
We can now make predictions for the intrinsic X-ray luminosity function of TDEs in the two cases of SMBHs and MBHs. It appears that, because TDE rates from single and binary black holes scale differently with $M_{BH}$, binaries dominate TDE production in the most massive halos and, as a result, contribute the brightest TDE flares in Model A. However, this contribution is significant only when small dark matter halos are occupied by IMBHs providing enough progenitors to form SMBH binaries. In the case of Model B, the most luminous events happen in systems with $M_{BH}\sim 2.5 \times 10^6~M_{\odot}$ which are dominated by single black holes. In case only SMBHs populate halos, TDEs from binaries occur only in systems with both black holes of $M_{BH}>10^6~M_\odot$ which are extremely rare and contribute at most few percent of the brightest TDE flares.
The fraction of events brighter than $ 10^{45}$ erg s$^{-1}$ (which is close to the Eddington luminosity of a $M_{BH}=10^7~M_\odot$ black hole) is shown on Figure \[Fig:3\]. To reinforce this point, we list in Table \[Table:1\] the percentage of events brighter than $10^{45}$ erg s$^{-1}$ produced by binary black holes at redshifts $z=$0, 0.5, 1, 2, 5, 7, 10, and 15. The importance of binaries grows toward higher redshifts where mergers are more frequent. In the case of fully occupied halos, binaries start dominating the bright events at $z=1$ in Model A and their contribution increases with redshift; while in Model B the maximal fraction of bright TDEs sourced by binaries is only $\sim 11\%$ in the post-reionization era ($z\lesssim 8$). In both Models A & B with $f_{occ}=1$ we find a sudden increase in the binary contribution at $z>8$ (pre-reionization era) when the photoheating feedback is not active. In the case of $f_{occ} =0$, as expected, the fraction of binaries is at most few percents and varies smoothly with redshift as this population is not affected by the photoheating feedback. With next generation X-ray telescopes which could statistically analyze high redshift TDEs, the change in TDE number counts with redshift could be a smoking gun of feedback processes or a marker of the black hole occupation fraction.
![Fraction of TDEs brighter than an intrinsic luminosity $L> 10^{45}$ erg s$^{-1}$ (close to the Eddington luminosity of $M_{BH}=10^7~M_\odot$) that are sourced by single MBHs, $N^{1}_{TDE}/(N^{1}_{TDE}+N^{2}_{TDE})$, as a function of $M_{BH}$ at $z=0$ (top) add $z=5$ (bottom). We show the case of Model A (solid) and B (dashed) for SMBH (black) and MBHs (red). The dark grey region marks the occupation of SMBH, with pale grey referring to IMBHs.[]{data-label="Fig:3"}](F0_feed "fig:"){width="3.4in"} ![Fraction of TDEs brighter than an intrinsic luminosity $L> 10^{45}$ erg s$^{-1}$ (close to the Eddington luminosity of $M_{BH}=10^7~M_\odot$) that are sourced by single MBHs, $N^{1}_{TDE}/(N^{1}_{TDE}+N^{2}_{TDE})$, as a function of $M_{BH}$ at $z=0$ (top) add $z=5$ (bottom). We show the case of Model A (solid) and B (dashed) for SMBH (black) and MBHs (red). The dark grey region marks the occupation of SMBH, with pale grey referring to IMBHs.[]{data-label="Fig:3"}](F5_feed "fig:"){width="3.4in"}
Model Redshift All Bright (A) Bright (B)
------------- ---------- --------- ------------ ------------ -- -- -- -- --
$f_{occ}=1$ $z=0$ 0.4% 25.2% 2.1%
$z=0.5$ 0.7% 38.3% 3.6%
$z=1$ 1.0% 50.8% 5.1%
$z=2$ 1.6% 66.7% 8.7%
$z=5$ 1.5% 83.8% 11.1%
$z=7$ 0.9% 85.7% 8.3%
$z=8$ 0.9% 98.2% 68.6%
$z=10$ 0.5% 98.4% 69.8%
$z=15$ 0.1% 98.8% 74.4%
$f_{occ}=0$ $z=0$ $<0.1$% 0.2% $<0.1$%
$z=0.5$ $<0.1$% 0.3% $<0.1$%
$z=1$ $<0.1$% 0.5% $<0.1$%
$z=2$ 0.2% 0.9% 0.2%
$z=5$ 0.2% 2.2% 0.2%
$z=7$ 0.1% 2.2% 0.1%
$z=8$ 0.1% 2.2% 0.1%
$z=10$ $<0.1$% 1.8% $<0.1$%
$z=15$ $<0.1$% 0.5% $<0.1$%
: Percentage of TDE produced by binary black holes: all events (3rd column), only bright events with intrinsic luminosity $L> 10^{45}$ erg s$^{-1}$ for Model A (4th column) and Model B (5th column). []{data-label="Table:1"}
We conclude this section by showing the expected cumulative X-ray luminosity function for Model A (top panel of Figure \[Fig:4\]) and Model B (bottom panel of Figure \[Fig:4\]) versus X-ray luminosity in the observed 1-150 keV band. Our results are presented for both SMBHs and MBHs. In the case of MBHs there are four distinct terms (shown separately on the figure) that affect the luminosity function and contributes a “knee”, i.e., the contributions from single and binary black holes of jetted and non-jetted events. When the TDE luminosity scales with the halo mass (Model A) each on eof these terms dominates specific luminosity regime, while the impact of binaries is less evident in the case of Model B. On the other hand, our SMBHs model has only two distinct features because in this case mergers have a negligible contribution and the knees in the luminosity function result from the jetted and non-jetted population of TDEs produced by single black holes. The shape of the luminosity function alone could be used to place limits on the occupancy of the IMBHs once a complete compilation of TDEs is available.
![Number of events per year detected by an ideal instrument of a field of view of 1 deg$^2$ versus X-ray luminosity in the observed 1-150 keV band. We show the cumulative luminosity functions assuming sources with Eddington luminosity, i.e., Model A, (top) and luminosity that scales as the mass accretion rate, i.e., Model B, (bottom), with $f_{occ}=1$ (red, thick solid) and $f_{occ}=0$ (black, thick solid). For the $f_{occ}=1$ case, we also show contributions due to various components: non-jetted events sourced by single black holes (thick dashed red) and binary black holes (thin dashed red), jetted events sourced by single black holes (thick dotted red) and binary black holes (thin dotted red). The brightest events are dominated by jetted TDE sourced by binary black holes, but these are very rare. []{data-label="Fig:4"}](LFint_feed "fig:"){width="3.4in"} ![Number of events per year detected by an ideal instrument of a field of view of 1 deg$^2$ versus X-ray luminosity in the observed 1-150 keV band. We show the cumulative luminosity functions assuming sources with Eddington luminosity, i.e., Model A, (top) and luminosity that scales as the mass accretion rate, i.e., Model B, (bottom), with $f_{occ}=1$ (red, thick solid) and $f_{occ}=0$ (black, thick solid). For the $f_{occ}=1$ case, we also show contributions due to various components: non-jetted events sourced by single black holes (thick dashed red) and binary black holes (thin dashed red), jetted events sourced by single black holes (thick dotted red) and binary black holes (thin dotted red). The brightest events are dominated by jetted TDE sourced by binary black holes, but these are very rare. []{data-label="Fig:4"}](LFint_feed2 "fig:"){width="3.4in"}
Observational signature {#Sec:Obs}
=======================
Number of disruption events that are actually detected by a telescope depend on its flux limit and field of view. Here we will focus on telescopes such as [*Swift*]{} and [*Chandra*]{} and explore signals which next-generation X-ray missions could probe. Bright X-ray transients such as GRBs or jetted TDEs are detected when they first trigger the BAT on [*Swift*]{}. The trigger occurs if the signal’s flux rises above $28.8\times 10^{-11}$ erg cm$^{-2} s^{-1}$ in the hard X-ray band (15-150 keV), i.e., reaches the 6-$\sigma$ statistical significance of BAT [@Barthelmy:2005]. Interestingly, all three jetted TDEs were detected by [*Swift*]{} over a period of three consecutive months, which suggests the possibility that further examples may be uncovered by detailed searches of the BAT archives. The X-ray Telescope (XRT) is another instrument on board [*Swift*]{} observing in the soft X-ray band (0.2-10 keV) and reaching $2\times 10^{-14}$ erg cm$^{-2}$ s$^{-1}$ sensitivity in $10^4$ seconds with a $23.6\times 23.6$ arcmin$^2$ field of view. (Because soft X-ray photons below $\sim 1$ keV can be absorbed by dust, we will quote numbers in the observed $2-10$ keV band when referring to soft X-rays.) As we show below, a telescope with such field of view and sensitivity as XRT is good for follow up observations of TDEs; while either a larger field of view or sensitivity are required to detect TDEs in large quantities. In fact, a telescope such as [*Chandra*]{} with its high point source sensitivity of $\sim 4\times 10^{-15}$ erg cm$^{-2}$ s$^{-1}$ in $10^4$ s (or $\sim 4\times 10^{-17}$ erg cm$^{-2}$ s$^{-1}$ in $10^6$ s) over 0.4-6 keV band and field of view of $\sim 15\times 15$ arcmin$^2$, could have many TDEs per frame, as we argue below.
Following @Woods:1998, the observed number of new events per year seen by BAT in the 15-150 keV band with peak flux larger than the flux limit $S_{lim}$ is given by $$\dot N_{TDE}^{S>S_{\lim}} = \int_0^{z_{max}} \int_{S_{15-150}>S_{lim}} \frac{ \dot N_{TDE}}{(1+z)}\frac{dV}{dz}dz dS,
\label{Eq:Surv}$$ where $S_{15-150}$ is the observed peak flux produced by each event. This equation is appropriate for threshold experiments, such as BAT, observing a population of transient sources that are standard candles in a peak flux. Figure \[Fig:5\] shows the total rates of events with observed flux greater than $S_{lim}$ produced at all redshfits including jetted and non-jetted TDEs produced by both single and binary black holes. The black coordinate system in the Figure shows the total number of hard X-ray events observed per year over the entire sky (field of view of $4\pi$) and for a 100% duty cycle as a function of the telescope flux limit $S_{lim}$, in the cases of our Model A and B and for $f_{occ} =0$ and 1.
{width="3.4in"}{width="3.4in"}
As seen from Figure \[Fig:5\] where the contribution of non-jetted TDEs is labeled by a dotted line for each scenario, the expected number counts are dominated by jetted TDE at the BAT sensitivity limit since the non-jetted contribution is negligible. To compare our predictions to BAT observations we need to re-scale the rates correcting for the limited field of view and duty cycle of the telescope. First, assuming that three jetted TDEs were detected by BAT in 9 years of [*Swift*]{} lifetime with the duty cycle of 75% over $4\pi /7$ of the sky we get $\dot N_{TDE} = 3$ yr$^{-1}$, while making use of the fact that the events were detected in three consecutive months (i.e., BAT sees 1 TDE per month) we get 112 TDEs per year. The latter number can be interpreted as a reasonable lower limit on the occurrence rate of jets and is just a factor of $\sim 2$ lower than our predictions for $f_{occ}=0$ (Model A) and a factor of $\sim 8$ for $f_{occ}=1$. The discrepancy could be explained by both observational limitations and modeling uncertainties, e.g., the assumed jetted fraction of 10% might be overestimated. For a next-generation survey with 50 times better sensitivity than BAT, i.e. going from the BAT configuration to “$50\times$BAT”, our model predicts 11 times more sources for $f_{occ}=1$ and 3-9 more sources for $f_{occ}=0$ (see Table \[Table:2\] for details).
------------- --------------- --------------------------------------- --------------------------------------- --------------------------------------- --------------------------------------- -- -- -- --
Model Flux Limit $\frac{\dot N_{TDE}^{4\pi, A}}{10^3}$ $\frac{\dot N_{TDE}^{4\pi, A}}{10^3}$ $\frac{\dot N_{TDE}^{4\pi, B}}{10^3}$ $\frac{\dot N_{TDE}^{4\pi, B}}{10^3}$
All $z<3$ All $z<3$
$f_{occ}=1$ BAT $0.93 $ $0.62$ $4.3$ $3.9$
$50\times$BAT $11$ $9$ $47$ $24$
$f_{occ}=0$ BAT $0.27 $ $0.25 $ $1.3 $ $0.99 $
$50\times$BAT $2.4$ $1.5 $ $4.6 $ $3.6 $
------------- --------------- --------------------------------------- --------------------------------------- --------------------------------------- --------------------------------------- -- -- -- --
: For each model we show the statistics of the observed events depending on the telescope sensitivity. TDE rates per sky per year (and divided by a factor of $10^3$, $\dot N_{TDE}^{4\pi, A}/10^3$) are shown for Model A for sources at all redshifts (3rd column) and at $0<z<3$ (4rd column); for Model B ($\dot N_{TDE}^{4\pi, B}/10^3$) for all source redshifts (5rd column) and $0<z<3$ (6rd column).[]{data-label="Table:2"}
In Table \[Table:3\] we list the fraction of observable TDEs produced by binaries (in percents) for each model and telescope sensitivity limit. At BAT sensitivity limit and in the case of high occupancy of IMBHs considerable fraction of observable TDEs are sourced by binary systems ($\sim 60\%$ for Model A and $\sim 5\%$ for Model B). As expected, because most of the faint systems are contributed by single MBHs, the fraction decreases as the sensitivity of the telescope improves. However, the decrease is non-monotonic as a function of $S_{lim}$ (as evident from the right panel of Figure \[Fig:5\]), because of the contributions from different components (with/without jets, single and binary MBHs). In the case of a low occupancy, the contribution of binaries to the TDE sample is always below $2\%$. The contribution of binaries, and thus the occupation fraction of IMBHs, could be verified observationally by analyzing the variability of each event in and comparing to the models available in literature [@Liu:2009; @Liu:2014; @Coughlin:2016; @Ricarte:2016].
Model Flux Limit $F_{Bin, A}^{BAT}$ $F_{Bin, B}^{BAT}$
------------- ----------------------- -------------------- -------------------- -- -- -- -- -- --
$f_{occ}=1$ [*Swift*]{} 64% 4.6%
$50\times$[*Swift*]{} 22% 3.6%
Ideal 0.72% 0.72%
$f_{occ}=0$ [*Swift*]{} 1.3% 0.05%
$50\times$[*Swift*]{} 0.5% 0.09%
Ideal 0.17% 0.17%
: For each model we show the fraction of TDEs sourced by binaries at each flux limit for Model A ($F_{Bin, A}^{BAT}$, column 3) and Model B ($F_{Bin, B}^{BAT}$, column 4). []{data-label="Table:3"}
Another observational mode is when the telescope takes a snapshot of the same part of the sky with a long exposure (integration time). The snapshot mode allows to probe a smaller portion of the sky with greater sensitivity than is done in the survey mode. A telescope with integration time $t_{int}$ will measure the following number of new events per frame [@Woods:1998] $$N_{TDE}^{S>S_{\lim}} = f_{sky}\int_0^{z_{max}}\int_{S_{2-10}>S_{lim}} \frac{\dot N_{TDE}}{(1+z)}\frac{dV}{dz}dz dS
\label{Eq:Snap}$$ $$\times\min\left[t_{int}, t_{dur}(1+z)\right],$$ where $t_{dur}$ is the time during which the event is above the sensitivity limit of the telescope, and $f_{sky}$ is the sky fraction covered by the telescope. The typical integration time of a telescope such as XRT, $\sim 10^4$ seconds, is much shorter than the typical duration of a TDE event ($\sim 10^6$ seconds, fall-back time), and thus, $t_{dur}$ can be ignored compared to $t_{int}$. To complete our discussion of [*Swift*]{} capabilities in detecting TDEs we show in the blue coordinate system of Figure \[Fig:5\] the expected number of events per typical exposure time of $10^4$ seconds in a field of view of 1 deg$^2$ assuming that all events shine at their peak luminosity. The number counts expected for the XRT sensitivity are much smaller than unity (see Figure \[Fig:5\] for details) meaning that XRT is good for follow-up missions but not to detect new TDEs, unless larger integration times are chosen.
The snapshot regime also applies to telescopes such as [*Chandra*]{} which observe one patch of the sky ($15\times15$ deg$^2$ in the case of [*Chandra*]{}) for a long time (more than $10^6$ seconds). The next generation upgrade of [*Chandra*]{}, called the [*X-ray Surveyor*]{}, is proposed to have $\sim 30$ times bigger collecting area than [*Chandra*]{} and, therefore, better sensitivity [@Weisskopf:2015]. A small fraction of point sources in each snapshot taken by [*Chandra*]{} (or the future [*X-ray Surveyor*]{}) could be TDEs and mistakenly identified as steady sources because of their long decay times. To single them out, a succession of snapshots of the same field should be taken within a time interval longer than a year. Because of the very long integration time of deep field survey, the TDE flux would decline below the flux limit in the course of the observation. To estimate the expected number of hidden TDEs in a [*Chandra*]{} deep field, we use Eq. (\[Eq:Snap\]) and assume that the light curve of each source fades according to Eq. (\[Eq:Mfall\]). Figure \[Fig:6\] shows the increment in the observed number counts per one snapshot as a telescope sensitivity limit improves. The number counts of SMBHs saturate at sensitivities $S_{lim} \sim 10^{-17}-10^{-16}$ erg s$^{-1}$, while the number counts of MBHs keep rising with decreasing $S_{lim}$. Comparing observed TDE luminosity function to the 6-$\sigma$ detection level ($\sim 2.4\times 10^{-16}$ erg cm$^{-2}$ s$^{-1}$ in $10^6$ s), we estimate number of TDEs in one [*Chandra*]{} deep field of $15\times 15$ arcmin$^2$ to be $0.3-0.7$ for SMBHs and 0.7-4 for MBHs. For a future mission such as the [*X-ray Surveyor*]{} TDE number counts remain $\sim 0.7$ for SMBHs, while they evolve to $\sim 6-20$ for MBHs. Therefore, non-detection of TDEs in a deep field would be a strong evidence for either a low occupation fraction of IMBHs, e.g., if they are kicked out of their parent halos as a result of mergers [@OLeary:2012] or a direct collapse scenario, e.g., works by @Bromm:2003 [@Taeho:2016; @Latif:2016; @Chon:2016].
![Number counts per one deep exposure with a telescope such as the [*Chandra X-ray Observatory*]{}. The grey vertical lines show $6-\sigma$ sensitivity of $2.4\times 10^{-16}$ erg s$^{-1}$ and a 50 times better sensitivity, for an integration time of 4Ms.[]{data-label="Fig:6"}](Chandra1){width="3.4in"}
Interestingly, current surveys with [*Chandra*]{} and [*XMM-Newton*]{} find an exponential decline in the space density of luminous AGNs at $z > 3$ [@Brandt:2016] suggesting that MBHs might not exist at higher redshifts. As Table \[Table:2\] shows, in our Models A & B for current BAT sensitivity, 67% & 92% of all observable TDEs are expected originate at $z<3$ for MBHs (and 94% & 77% for SMBHs); while for a 50 times more sensitive telescope, the corresponding numbers for MBHs are 83% & 52% (66% & 78% for SMBHs). For [*Chandra*]{} the corresponding numbers are 68% & 70% ($f_{occ}=1$) and 83% & 50% ($f_{occ}=0$), while for its successor 52% & 35% for MBHs and 49% for SMBHs.
The maximal redshift out to which TDEs can be detected depends on the telescope sensitive. Figure \[Fig:7\] shows the TDE rates in the survey mode (BAT, left) and number of TDEs observed per snapshot ([*Chandra*]{}, right) for several choices of $S_{lim}$ including present day instrument, a $50\times$ more sensitive telescope and an ideal detector which identifies both jetted and non-jetted TDE accounting only for redshifts $z>z_{min}$. In other words, for each telescope sensitivity, we only account for the events which originate at $z_{min}$ or above. As we probe higher redshifts, the expected number of sources drops because there are no sufficiently massive halos to source sufficiently bright flares. The left panel of Figure \[Fig:6\] focuses on a BAT-like survey which is primarily sensitive to bright (mainly jetted) events. It is evident that if TDEs do not source jets, prospects for observations with BAT would not be as bright, and TDEs would be observable only out to $z \sim 0.1$ with BAT and out to $z = 0.4$ with a 50 times more sensitive telescope than BAT. The redshifts at which number of observed TDE per year drops by a factor of 2 ($z_{50\%}^{BAT}$) and 10 ($z_{10\%}^{BAT}$) are listed in Table \[Table:4\] for all models under consideration. Note that in some cases, $z_{50\%}^{BAT}$ does not change monotonically as a function of the sensitivity. This is because the non-jetted events become unobservable despite being more numerous while few jetted events are seen out to greater distances. The right panel of Figure \[Fig:7\] focuses on a [*Chandra*]{}-like instrument which has more sensitivity and integration time but a smaller field of view. If the IMBHs occupancy is high, 50% of events observed by [*Chandra*]{} would originate from $z\lesssim 2$ (and $z\lesssim 2-4$ if observed by [*Chandra*]{} successors with the uncertainty arising from our luminosity modeling). For completeness we also consider XRT. As expected, XRT is mainly sensitive to non-jetted TDEs and $50$% of TDEs that can be observed by XRT are predicted to originate from $z\lesssim 0.4-1$. This is broadly consistent with the current sample [@Komossa:2015] in which all non-jetted TDEs are identified to be at $z\sim 0.405$ while the rare jetted events are observed at $z=0.353$, 0.89 and 1.186.
{width="3.4in"}{width="3.4in"}
Model Flux Limit $z_{50\%, A}^{BAT}$ $z_{10\%, A}^{BAT}$ $z_{50\%, B}^{BAT}$ $z_{10\%, B}^{BAT}$
------------- ----------------------- --------------------- --------------------- --------------------- --------------------- -- -- -- --
$f_{occ}=1$ [*Swift*]{} 2.2 4.4 1.2 2.6
$50\times$[*Swift*]{} 1.3 3.8 2.8 5.3
Ideal 9.0 12.6 9.0 12.6
$f_{occ}=0$ [*Swift*]{} 1.1 2.5 2.0 3.2
$50\times$[*Swift*]{} 2.2 4.6 0.2 4.8
Ideal 3.0 5.9 3.0 5.9
: The redshift $z_{min}$ so that 50% (columns 3 and 5 for Model A and B respectively), and 10% (columns 4 and 6) of observed BAT TDEs arrive from $z>z_{min}$. []{data-label="Table:4"}
The signature of reionization, due to the evolving $M_{BH, min}$ in star-forming halos, is evident in MBH cases and manifests itself as a cusp around $z_{re}=8.8$. Because we assume instantaneous reionization, the feature is sharp. In a more realistic case of gradual reionization, the signature is expected to show as a mild enhancement of the TDE rates at $z\gtrsim z_{re}$.
Conclusions {#Sec:sum}
===========
Current observations pose only poor constraints on massive black hole growth at high redshifts as well as on the occupation fraction of IMBHs. Flares from TDEs could reveal the population of otherwise dormant black holes allowing us to constrain the contribution of IMBHs. In this paper we have considered evolution of the observable TDE number counts with X-ray telescopes including predictions for future missions. Our discussion of the black hole mass distribution included a model with $f_{occ}=1$ (all star forming halos are occupied by black holes) and $f_{occ}=0$ (only heavy halos host black holes of $M_{BH}>10^6 ~M_{\odot}$). These two scenarios provide an upper and lower limits for the expected number counts respectively. In addition, we considered two different prescriptions for the TDE luminosity: (i) Eddington luminosity, and (ii) luminosity proportional to the accretion rate. Even though current TDE observations suggest that the occupation fraction of IMBHs is very low with the majority of TDEs being produced by black holes of masses $\sim 10^6-10^8~M_\odot$, the results are far from being conclusive. Our study offers new ways to constrain the occupation fraction of IMBHs at different cosmological redshifts, and our main conclusions are as follows
1. We show that jetted TDEs can be observed out to high redshifts and offer a unique probe of the occupancy of IMBHs. Earlier works have demonstrated that TDE rates in merging systems are enhanced due to gravitational interactions of stars with binary black holes. Using this result we find that the higher is the occupation fraction of IMBHs the stronger is the impact of binaries on the total observed TDE rates. This is because with high IMBHs occupation there are enough progenitors to form binary systems.
2. We show that TDEs sourced by binary black holes dominate the bright end of the X-ray luminosity function if the occupation fraction of IMBHs is high and if the TDE luminosity scales as Eddington. The shape of the TDE X-ray luminosity function is expected to show a unique signature of IMBHs in the form of two additional “knees”, compared to the case with low IMBHs occupation. These features arise from the jetted and non-jetted contribution of black hole binaries and are independent of our luminosity prescription (although the features are more evident when the TDE luminosity scales as the Eddington luminosity). Therefore, for a complete TDE sample, the shape of the luminosity function could be used to set an upper limit on the occupation fraction of IMBHs. Our results imply that, if $f_{occ} =1$ and the TDE luminosity scales as Eddington, the brightest events detected by BAT could be associated with massive binary black holes. In this case the X-ray luminosity of TDE flares is expected to have excess of variability due to the binary interaction in addition to the typical power-law decay.
3. The fraction of observable TDE that are generated by binaries depends on the luminosity prescription as well as on the sensitivity of the telescope. With current X-ray telescopes, we expect to see $>2\%$ and up to 64% of TDEs produced by binary black holes if the occupation fraction of IMBHs is high; while the fraction is at most 1.3% if the occupation fraction of IMBHs is low. Since dimmer events are mainly contributed by single black holes, the binary fraction drops with the telescope sensitivity.
4. Detection of TDEs in deep field observations by [*Chandra*]{} and future missions would provide a smoking gun signature of IMBHs. We find that in the case when only SMBHs contribute, TDEs are not expected in [*Chandra*]{} deep fields; while if the IMBHs occupation fraction is high, some point sources in the archival data of X-ray deep field surveys may be TDEs. To identify such events one should compare two images of the same deep field separated by an interval of at least a year. Non-detection of TDEs from high redshifts can set upper limits on the occupation fraction of IMBHs and constrain direct collapse scenarios of SMBH formation, e.g., works by @Bromm:2003 [@Taeho:2016; @Latif:2016; @Chon:2016].
5. Increasing sensitivity of X-ray telescopes by a factor of 50 comparing to current instruments will increase the expected number counts by a factor of $4- 10$ for a BAT-like mission and a factor of $20-40$ for an XRT-like mission with 1ks itegration time. For a deep field survey the improvement strongly depends on the occupation fraction of IMBHs. Current sensitivity is enough to resolve most TDEs if $f_{occ}=0$, and, therefore, improvement in sensitivity would not yield new events in this case. However, if $f_{occ}=1$, improving the sensitivity by a factor of 50 would increase the number of TDEs per snapshot by a factor of 5-10.
Comparing our model to existing observations, a low occupation fraction is suggested (see also Stone & Metzger 2016). However, observations are far from being conclusive and it is still unclear why TDEs sourced by IMBHs with masses below $10^6~M_\odot$ are not observed. Several possible explanations can follow: IMBHs are kicked out of their dark matter halos as a result of mergers [@OLeary:2012], SMBHs are formed from massive seeds in massive halos, observational selection effects exclude TDEs around IMBHs, the assumption of an isothermal density distribution is less suitable for smaller galaxies, or low mass systems are more sensitive to AGN feedback which expels gas from halo and limits star formation leading to inefficient replenishing of the loss cone.
If $f_{occ}$ is high at low black hole masses and IMBH binaries indeed play a role in sourcing TDEs, these binaries would also produce gravitational waves on their approach to coalescence. eLISA should be sensitive to MBH binaries over a wide range of total masses and mass ratios, e.g., systems with total mass $ \gtrsim 10^5$ M$_\odot$ and mass ratios of $\gtrsim 0.1$ will yield signal to noise ratio of $> 20$ out to $z = 4$ [@Amaro:2012]. Therefore, in the future one could use a cross-correlation between the stochastic gravitational wave background and the spatial distribution of brightest TDEs to constrain the role of IMBHs in TDE production. If the two quantities correlate, IMBHs must make a significant contribution to TDEs.
In addition to the X-ray observations discussed here, jetted TDEs may also be bright in the radio band [@Zauderer:2011; @Levan:2011]. Snapshot rates of jetted TDEs in radio band have been computed by @vanVelzen:2011, and prospects for the detection of jetted TDEs by the Square Killometer Array out to $z\sim 2$ as well as the synergy between radio and X-ray observations was discussed [@Donnarumma:2015b; @Donnarumma:2015; @Rossi:2015]. At present there is no consensus as to whether X-ray emission by jetted AGN should correlate with their radio emission. In particular, the peak emissivity of Sw1644+57 appeared 100 days after the BAT trigger and the Lorentz factor of radio jet was found to be $\Gamma \sim 2$, much lower than what was observed in X-rays right after detection ($\Gamma \sim 10$). The radio data require a different energy injection mechanism such as an outflow with a distribution of Lorentz factors [@Berger:2012; @Generozov:2016]. Therefore, we did not use our simplex model to predict radio emission from TDEs.
Tidal disruptions occurring at high redshifts can reveal the seeds of quasars. In this paper we have assumed that the high-redshift population resembles that of today; however, its properties might evolve with redshift. In particular, simulations show that first stars were much more massive (up to $10^3~M_\odot$) than present-day stars and could serve as an additional population of seeds. Star formation in high-redshift, low-mass halos strongly depends on feedback processes such as photoheating feedback, as well as AGN and supernovae feedback. The evolution of the TDE number counts with redshift could serve as a smoking gun for these processes.
Acknowledgments
===============
We thank J. Guillochon and N. Stone for helpful comments on the manuscript. We also thank A. Sadowski, Y.-F. Jiang, and A. Siemiginowska for useful discussions. This work was supported in part by Harvard’s Black Hole Initiative, which is supported by a grant from the John Templeton Foundation.
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\[lastpage\]
[^1]: E-mail: anastasia.fialkov@cfa.harvard.edu
[^2]: The dependence of Eq. (\[Eq:TDE1\]) on stellar mass can be re-introduced by adding a factor $\left(m_*/M_{\odot}\right)^{-1/3}\left(r_*/R_{\odot}\right)^{1/4}$ with $r_* = R_{\odot}(m_*/M_{\odot})^{0.8}$ for stars along the lower main sequence.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'J. Kube[^1]'
- 'B. T. Gänsicke'
- 'F. Euchner'
- 'B. Hoffmann'
date: 'Received 29 November 2002 / Accepted 12 March 2003'
title: 'CVcat: an interactive database on cataclysmic variables'
---
Introduction
============
CVcat is an interactive database or “online catalogue” that offers a number of features so far unknown to scientific catalogues. It was developed as a tool for the research community working on cataclysmic variables (CVs), a class of close interacting binaries, and as a case study for some of the concepts to be used in the development of a general catalogue software, AstroCat. CVcat can be accessed online at `http://www.cvcat.org`. It was first presented to the public in August 2001 during a CV conference held in Göttingen [@KGH]. Since then, the number of users of CVcat has increased to more than one hundred, the daily average of requests is around fifty (total number of delivered pages).
The concept
===========
CVcat has been developed in order to overcome major conceptual shortcomings of existing CV catalogues: @Ritter include only systems with known orbital period, which limits their catalogue to $\approx 1/3$ of all CVs and related objects. @Downes do list all known CVs, but their catalogue provides only very limited information on each individual system, i.e., the only binary parameter included is the orbital period. Our aim was to develop an online data base that combines the information of the existing catalogues *and* allows the users to actively contribute to the content of the data base, implementing a first version of an “open catalogue”.
CVcat differs from other CV catalogues and other astronomical databases in the concept of the data input. So far, the majority of astronomical catalogues have been compiled by *relatively small editorial teams* consisting of scientists knowledgeable in the fields covered by the catalogues [e.g. @Downes; @Ritter; @mccook+sion99-1; @liuetal01-1]. These catalogues typically contain more or less detailed information on a specific class of astronomical objects. Updates are published, if at all, only on a very irregular basis. The catalogues contain just *one value* for each listed property (e.g. distance, orbital period) of a given object. While this is helpful for non-specialist users to obtain a quick overview of the properties of an individual object, or of the statistical properties of a given group of objects, the more expert user will certainly benefit if different and possibly competing values for a given parameter are referenced in such catalogues. This is particularly useful if the information has been obtained by different methods.
Some of the aforementioned catalogues moved from “classical” printed publication to online web-based publication, which allows shorter update cycles \[e.g. the “living edition” of the @Downes CV catalogue, @LivingDownes\], however, the overall concepts remained unchanged. In addition to these specialized catalogues for a specific object class there exists huge data bases like SIMBAD [@SIMBAD], which provide very basic properties for an enormously large number of objects. However, due to the very global coverage of astronomical objects, data contained in SIMBAD are prone to be incomplete and/or inaccurate.
The concept of an “open catalogue” implemented in CVcat permits *every registered user* to add data to the catalogue, which is *instantly* visible to all other users. The quality control is performed by an editorial team (Sect.2.1), which may alter or remove erroneous data. For every property of an object, an *arbitrary number of values* can be stored (e.g. several published values for the distance). CVcat returns one of these values as the “best available” value, selected as such by the editors. However, as such a selection process often involves some subtle subjective view of the editor, the more expert user may decide to inspect the original sources for the competing values, and, thereafter, decide based on his/her experience which value is best suited for a given purpose.
A sketch of the different concepts of data input and validation is given in Fig. \[f:concept\] (a) for classical catalogues and (b) for the CVcat concept. (c) introduces the concept that will be implemented in the next release of CVcat, which is adressed in Sect. \[s:prospects\]. In the future version, newly-added data will be instantly visible as it is now \[Fig. \[f:concept\], (b)\], but will be tagged as “unapproved” until an editor cross-checks the data. In the current implementation (b), the user cannot see if a database entry has been approved by an editor.
Distributed editorial team
--------------------------
The editorial team of CVcat is recruited from the CV research community, and consists (ideally) of one expert per CV subclass. Since these editors are typically familiar with the publications on their “favourite” objects anyway, the amount of work to cross-check newly entered data is lowered. We estimate that for a typical subclass of CVs, say, polars, the time to be spent on editorial duties in CVcat is of the order of two hours per week or less. Each editor has the privilege to remove or change erroneous data for that person’s object class only. For data on all other object classes, the editor is a non-privileged user who may add new data and browse the catalogue. Note that every user may add data, in contrast to, e.g., SIMBAD, where only the editorial team can directly modify the catalogue content.
Database content
----------------
In contrast to the existing CV catalogues, CVcat is designed to contain a great variety of information for each object. Currently, the following object properties can be included, with the number in brackets being the number of entries on September 17, 2002 (note that because CVcat allows multiple entries for a given property of an object these numbers do not represent the number of unique objects for which a particular property is known):
High state magnitude (922), low state magnitude (794), optical spectrum exists (665), orbital period (509), distance (221), general magnitude (209), primary mass (156), inclination (153), secondary mass (142), superhump period (128), secondary spectral type (121), mass ratio (93), Doppler tomogram exists (80), primary temperature (76), optical light curve exists (61), hydrogen column density (58), orbital ephemeris (55), eclipsing (48), spin period (36), primary radius (33), secondary radius (29), uv data exists (22), general magnetic field strength of primary star (18), field strength of primary magnetic pole (17), x-ray data exists (14), eclipse map exists (11), secondary temperature (8), colatitude of primary magnetic pole (5), field strength of secondary magnetic pole (3), spin ephemeris (3), azimuth of primary magnetic pole (2), magnitude in eclipse (2), azimuth of secondary magnetic pole (1).
Every value stored in CVcat is linked to its original publication, either in the NASA ADS using the ADS bibcode of the paper [@ADS], to the astro-ph/arXiv e-print archive, or to the VSNET messages [@VSNET]. Besides references to the publications from which the data entries contained in CVcat are taken, a list of articles with general information on a given object can be stored in CVcat.
We allow inclusion of data from astro-ph, which is not necessarily identical to the data published in the final refereed version (or which may in some cases never make it through the refereeing stage). Most of the astro-ph data, however, is promoted to refereed information at some point. It is a typical task of the editors to track such updates and to conduct the appropriate changes to the database, i.e. changing the source from an astro-ph to an ADS bibcode.
Searching the database
----------------------
Data retrieval from CVcat works in two ways: (i) the user obtains all available data for a specific object, which can be found using its object type, its coordinates, or one of its names, (ii) the user creates a table containing selected properties for a list of objects. The latter method allows the user to create data tables suitable for easy graphics generation as well as ready-to-publish LaTeX-tables.
Searching in CVcat is organized as a two-step process. In the first step, the user can enter the search pattern, which can be the name or a substring of the name, a set of object classes, and the coordinate range of the object (Fig. \[f:search\]). The object class is selected using a grid of logical operators (“and”, “or”, and “not”). It is possible, e.g., to look for dwarf novae, which have also been observed as novae: choose “and DN” and “and Nova”. Another example is to look for nova-likes which do not show the SWSex phenomenon: the corresponding selection would be “or NL” and “not SW”.
After submitting this search request a list with all objects matching the search criteria is returned (not shown). In this list, all object names are hyperlinks to the page showing all results for the specific object (Fig. \[f:result\]). Alternatively, a set of objects can be chosen from this list to generate a user-configurable list with certain properties of the objects. This list may be adjusted in a way that objects without a published value for a certain property are not included. By iterative calls of the list generator it is possible to distill a table containing e.g. all objects with known masses and periods.
An example of the results of such a list creation process is given in the following section.
Example: Orbital periods and donor masses
-----------------------------------------
Using the data contained in the CVcat database, we have plotted the secondary star masses as a function of the orbital period, Fig. \[f:introcv.m2overp\], for all CVs. A linear trend with $$\frac{M_2}{M_{\sun}}\approx 0.11 P/{\rm h}-0.06
\label{e:introcv.m2overp}$$ is clearly visible. This is predicted by theoretical considerations, where the reasoning is roughly this [@Frank92]: Using an approximation for the Roche geometry, valid for $1.3\la q\la 10$, and Kepler’s law, one finds that the mean density of a Roche lobe filling star is a function of only the orbital period. With the knowledge of the lower main-sequence $M/R$ relation [@Kippenhahn90; @BCA98], one then finds $$\frac{M_2}{M_{\sun}}\approx 0.11 P/{\rm h}
\label{e:introcv.m2overptheory}$$
A more detailed analysis of secondary star masses leads to slightly different period-mass-relations [@1998MNRAS.301..767S]: $$\begin{aligned}
\frac{M_2}{M_{\sun}}&=&(0.038\pm0.003)(P/{\rm h})^{(1.58\pm0.09)}\quad\mbox{or}
\label{e:introcv.m2overp_a}\\
\frac{M_2}{M_{\sun}}&=&(0.126\pm0.011)P/{\rm h}-(0.11\pm0.04)
\label{e:introcv.m2overp_b}\end{aligned}$$ The data currently available in CVcat are consistent with the results from Smith & Dhillon (Fig. \[f:introcv.m2overp\]). Note that some of the published secondary star masses may be derived from the orbital period using some theoretical models, hence artificially stabilizing the fit close to the theoretical predictions.
Usage statistics
================
The log file analysis of CVcat usage over a one year period shows that CVcat has a stable user community which is still slowly increasing. The typical usage of CVcat is to request all available data on a specific object, which is normally queried by its name. List generation of objects selected by their class ranks second in the usage of the database. The current growth rate of the catalogue is around 50 entries per month, mostly from the CVcat editors.
The data flow into CVcat originating from outside the CVcat core team is not yet satisfying. It is unclear why most users refrain from adding data from their own publications. Users should bear in mind that the probability of having their papers cited increases if the information from their publications can be found in the database.
Prospects {#s:prospects}
=========
The concept of CVcat has demonstrated the benefits of a public scientific catalogue with a globally-shared expertise of its users and editors. The general structure of CVcat is also applicable to other fields of astronomy. Hence, a more general software based on the experiences with CVcat which will allow the implemenation of catalogues for arbitrary object classes, called “AstroCat”, is currently being developed. This software will also include additional features to improve the knowledge management of astronomical results and data:
- Besides the “single number data” stored so far, the infrastructure for storing more complex data products, e.g., light curves, spectra, and finding charts will be added. This feature should help to overcome a major shortcoming in the present method of scientific publishing: most authors publish their reduced data only in the form of plots, making follow-up work on these results rather unattractive. While these data *may* be obtained directly from the authors, experience shows that in many cases the data have been lost due to faulty hardware or storage media, with the probability of loss increasing dramatically with time since the publication of the original work. In addition, the authors may have left astronomy, preventing access to their data. If the data is available in CVcat, the re-examination of observations would be much easier and independent of contact with the original author and the corresponding data archive. The storage of reduced and published data in a usable form at e.g. CDS, Strasbourg, is already promoted by journals like A&A.
- Information agents will allow users to be informed automatically if new data is entered into the database for their objects of interest. This is a service that keeps the user informed about new publications on a given object or class of objects without having to log in to CVcat frequently.
- An elaborated validation system will allow the users to add their comments to published data. This unique feature will add personal communication aspects to the database. Public discussion on details of the published data may arise from this.
- To allow exact quoting of a specific state of the ever-changing catalogue, a “versioning´´ method using a global identification number (GID) will be used: the GID is incremented with each change of the content of AstroCat (e.g. adding new data, marking data as correct, marking data as the best available etc.). Every output resulting from the use of AstroCat will include the current GID number. An older state of the database can be exactly reproduced by issuing a given GID (lower than the current one), and using AstroCat with such a specific GID will return always *precisely* the same results, independent of the actual state of the data base. Any statistical analysis based on data from CVcat should therefore include the GID of the CVcat state at the time of analysis, permitting a quantitative comparison with future studies. It is not necessary to rely on a fixed or frozen state so far only available in printed catalogues.
- Data that have not yet been cross-checked by the editorial team will be tagged as such. This method combines both the unique speed of the CVcat concept and the expertise of the editorial team. As long as a new datum is visible but marked as unapproved, the user of the database can use the new entry under his own responsibility, while an approved value is authoritative.
- Hyperlinks to a large number of available web resources will be included for every object. This includes, e.g., links to other CV cataloges like SIMBAD, the ADS, arXiv.org, the different variable star observers archives, and many more.
The development of the new CVcat and the AstroCat software is done in close cooperation with the users of the current implementation. In summer 2003, the data from the current CVcat will be transferred to the new database. By the end of 2003, the AstroCat/CVcat framework will be completed. Ideas from the editors and users of other interactive catalogues, e.g. the high-$z$ database [@HighZ], are highly appreciated. The AstroCat project is hosted at `http://astrocat.uni-goettingen.de`.
Technical realization
=====================
CVcat is implemented as a Perl script which runs on a Linux PC. This Perl script processes the user requests (HTTP GET/POST requests) and creates HTML pages that are delivered via the Apache web server. Hence, CVcat presents itself as an interactive web page.
The data is stored in a MySQL data base to which the Perl script communicates using the standard Perl DBI/DBD interface.
For the implementation of AstroCat, an XML layer will be included between the data base and the HTML layer. This XML layer can be used to automatically include larger data sets into the database. Another possible application will be to use the AstroCat framework as an archive for reduced data from robotic telescopes that use RTML [@RTML] for their observation requests, accomplished observations, and some of the metadata. A technical advantage of the usage of XML is the easy availability of many very good tools specialized in the processing of XML data and the transformation of XML documents into HTML pages. This will improve the quality and the speed of the implementation. PHP might be used in addition to Perl as the scripting language for the AstroCat programs.
Summary
=======
With CVcat we have implemented an online catalogue for cataclysmic variable stars, which – for the first time – allows its users to add data instantly visible to all other users. The quality control is realized by a team of experts who share the responsibility for the different object classes. From the experiences of one year of public usage of the catalogue we have compiled a number of new concepts which will be implemented in the next generation of CVcat in the context of a more general framework for astronomical databases, AstroCat.
We would like to thank all the colleagues who volunteered to take the responsibility for one of the object classes included in CVcat: Domitilla de Martino, Don Hoard, Steve Howell, Tom Marsh, Klaus Reinsch, Matthias Schreiber, and John Thorstensen. The AstroCat/CVcat project is funded by the *Deutsche Forschungsgemeinschaft (DFG)* project number LIS 4 – 554 95 (1) Göttingen. The development of the feasibility study has partially been supported by the DLR under grant 50OR99036. BTG also acknowledges support from a PPARC Advanced Fellowship.
We appreciate the helpful comments from the referee, Dr. Ochsenbein, to both this paper and the CVcat database itself.
[15]{} natexlab\#1[\#1]{}
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, F. V. 2001, Astronomische Nachrichten, 322, 303
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, J., [G[" a]{}nsicke]{}, B. T., & [Hoffmann]{}, B. 2002, in ASP Conf. Ser. 261: The Physics of Cataclysmic Variables and Related Objects, 678
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[^1]: *Present affiliation: Alfred-Wegener-Institute for Polar- and Marine Research, Telegrafenberg A43, D-14473 Potsdam, Germany; Koldewey-Station, N-9173 Ny-Ålesund, Norway*
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We use the action-angle variables to describe the geodesic motions in the $5$-dimensional Sasaki-Einstein spaces $Y^{p,q}$. This formulation allows us to study thoroughly the complete integrability of the system. We find that the Hamiltonian involves a reduced number of action variables. Therefore one of the fundamental frequency is zero indicating a chaotic behavior when the system is perturbed.
[*Keywords:*]{} Sasaki-Einstein spaces, complete integrability, action-angle variables.
[*PACS Nos:*]{} 11.30-j; 11.30.Ly; 02.40.Tt
author:
- 'Mihai Visinescu[^1]'
title: 'Action-angle variables for geodesic motions in Sasaki-Einstein spaces $Y^{p,q}$'
---
Introduction
============
There has been considerable interest recently in Sasaki-Einstein (SE) geometry [@JS]. In dimension five, an infinite family of explicit SE metrics $Y^{p,q}$ on $S^2 \times S^3$ has been constructed, where $p$ and $q$ are two coprime positive integers, with $q < p$ [@GMSW2].
A $(2n-1)$-dimensional manifold $M$ is a *contact manifold* if there exists a $1$-form $\eta$ (called a contact $1$-form) on $M$ such that $$\eta \wedge (d \eta)^{n-1} \neq 0\,.$$ The *Reeb vector field* $\xi$ dual to $\eta$ satisfies: $$\eta (\xi) = 1 \quad \text{and} \quad \xi {\raisebox{-0.35ex}{\makebox[0.6em][r]
{\scriptsize $-$}}\hspace{-0.15em}\raisebox{0.25ex}
{\makebox[0.4em][l]{\tiny $|$}}}d\eta = 0\,,$$ where ${\raisebox{-0.35ex}{\makebox[0.6em][r]
{\scriptsize $-$}}\hspace{-0.15em}\raisebox{0.25ex}
{\makebox[0.4em][l]{\tiny $|$}}}$ is the operator dual to the wedge product.
A contact Riemannian manifold $(Y_{2n-1}, g_{Y_{2n-1}})$ is Sasakian if its metric cone $C(Y_{2n-1}) = Y_{2n-1} \times \mathbb{R}_+$ with the metric $$ds^2( C(Y_{2n-1})) = dr^2 + r^2 ds^2 (Y_{2n-1})\,,$$ is Kähler [@BG]. Here $r\in (0,\infty)$ may be considered as a coordinate on the positive real line $\mathbb{R}_+$. If the Sasakian manifold is Einstein, the metric cone is Ricci-flat and Kähler, i.e. Calabi-Yau.
The orbits of the Reeb vector field $\xi$ may or may not close. If the orbits of the Reeb vector field $\xi$ are all closed, then $\xi$ integrates to an isometric $U(1)$ action on $(Y_{2n-1}, g_{Y_{2n-1}})$. Since $\xi$ is nowhere zero this action is locally free. If the $U(1)$ action is in fact free, the Sasakian structure is said to be *regular*. Otherwise it is said to be *quasi-regular*. If the orbits of $\xi$ are not all closed, the Sasakian structure is said to be *irregular* and the closure of the $1$-parameter subgroup of the isometry group of $(Y_{2n-1}, g_{Y_{2n-1}})$ is isomorphic to a torus $\mathbb{T}^n$ [@JS].
The homogeneous SE metric on $S^2 \times S^3$, known as $T^{1,1}$, represents an example of regular Sasakian strucure with $SU(2) \times
SU(2) \times U(1)$ isometry. The $Y^{p,q}$ spaces have isometry $SU(2) \times
U(1) \times U(1)$ and for $4p^2 - 3 q^2$ a square they are examples of quasi-regular SE manifolds. The geometries $Y^{p,q}$ with $4p^2 - 3 q^2$ not a square are irregular SE spaces.
In a recent paper [@BV] the constants of motion for geodesic motions in the five-dimensional spaces $Y^{p,q}$ have been explicitly constructed. This task was achieved using the complete set of Killing vectors and Killing-Yano tensors of these toric SE spaces. A multitude of constants of motion have been generated, but only five of them are functionally independent implying the complete integrability of geodesic flow on $Y^{p,q}$ spaces.
The complete integrability of geodesics permits us to construct explicitly the action-angle variables. The formulation of an integrable system in these variables represents a useful tool for developing perturbation theory. The action-angle variables define an $n$-dimensional surface which is a topological torus (Kolmogorov-Arnold-Moser (KAM) tori) [@VIA].
Our motivation for studying the action-angle parametrization of the phase space for geodesic motions in SE spaces comes from recent studies of non-integrability and chaotic behavior of some classical configuration of strings in the context of AdS/CFT correspondence. It was shown that certain classical string configurations in $AdS_5 \times T^{1,1}$ [@BZ1] or $AdS_5 \times Y^{p,q}$ [@BZ2] are chaotic. There were used numerical simulations or an analytic approach through the Kovacic’s algorithm [@JJK]
The purpose of this paper is to describe the geodesic motions in the SE spaces $Y^{p,q}$ in the action-angle formulation. We find that the Hamiltonian (energy) involves only four action variables which have the corresponding frequencies different of zero. One of the fundamental frequency is zero foreshadowing a chaotic behavior when the system is perturbed.
The paper is organized as follows. In the next Section we give the necessary preliminaries regarding the metric and the constants of motion for geodesics on $Y^{p,q}$ spaces. In Sec. 3 we perform the separation of variables and give an action-angle parametrization of the phase space. The paper ends with conclusions in Sec. 4.
$Y^{p,q}$ spaces
================
The AdS/CFT correspondence represents an important advancement in string theory. A large class of examples consists of type $IIB$ string theory on the background $AdS_5 \times Y_5$ with $Y_5$ a $5$-dimensional SE space. In the frame of AdS/CFT correspondence $Y^{p,q}$ spaces have played a central role as they provide an infinite class of dualities.
We write the metric of the 5-dimensional $Y^{p,q}$ spaces [@GMSW2; @GMSW1; @BK] as $$\label{Ypq}
\begin{split}
ds^2_{Y^{p,q}} & = \frac{1-c\, y}{6}( d \theta^2 + \sin^2 \theta\, d \phi^2)
+ \frac{1}{w(y)q(y)} dy^2
+ \frac{q(y)}{9} ( d\psi - \cos \theta \, d \phi)^2 \\
& \quad +
w(y)\left[ d\alpha + \frac{ac -2y+ c\, y^2}{6(a-y^2)}
(d\psi - \cos\theta \, d\phi)\right]^2\,,
\end{split}$$ where $$w(y) = \frac{2(a-y^2)}{1-cy} \,, \quad
q(y) = \frac{a-3y^2 + 2c y^3}{a-y^2}\,.$$ This metric is Einstein with $\Ric g_{Y^{p,q}} = 4 g_{Y^{p,q}}$ for all values of the constants $a,c$. For $c=0$ the metric takes the local form of the standard homogeneous metric on $T^{1,1}$ [@MS]. Otherwise the constant $c$ can be rescaled by a diffeomorphism and in what follows we assume $c=1$.
A detailed analysis of the SE metric $Y^{p,q}$ [@GMSW2] showed that for $0 \leq \theta \leq \pi$ and $0 \leq \phi \leq 2\pi$ the first two terms of give the metric on a round two-sphere. The two-dimensional $(y, \psi)$-space defined by fixing $\theta$ and $\phi$ is fibred over this two-sphere. The range of $y$ is fixed so that $1-y >0\,,\, a-y^2 >0$ which implies $w(y) > 0$. Also it is demanded that $q(y) \geq 0$ and that $y$ lies between two zeros of $q(y)$, i.e. $y_1 \leq y \leq y_2$ with $q(y_i)=0$. To be more specific, the roots $y_i$ of the cubic equation $$a-3y^2 + 2 y^3 = 0 \,,$$ are real, one negative $(y_1)$ and two positive, the smallest being $y_2$. All of these conditions are satisfied if the range of $a$ is $$0 < a < 1 \,.$$ Taking $\psi$ to be periodic with period $2 \pi$, the $(y,\psi)$-fibre at fixed $\theta$ and $\phi$ is topologically a two-sphere. Finally, the period of $\alpha$ is chosen so as to describe a principal $S^1$ bundle over $B_4 = S^2 \times S^2$. For any $p$ and $q$ coprime, the space $Y^{p,q}$ is topologically $S^2 \times S^3$ and one may take [@MS; @GMSW2] $$0 \leq \alpha \leq 2 \pi \ell\,,$$ where $$\ell = \frac{q}{ 3q^2 - 2 p^2 + p(4 p^2 - 3 q^2 )^{1/2}}\,.$$
To put the formulas in a simpler forms, in that follows we introduce also $$f(y)= \frac{a-2y +y^2}{6(a-y^2)}\,,$$ $$p(y)= \frac{w(y) q(y)}{6} =\frac{a-3y^2 + 2y^3}{3(1-y)}\,.$$
The conjugate momenta to the coordinates $(\theta,\phi, y, \alpha, \psi)$ are: $$\label{momenta}
\begin{split}
&P_{\theta} =
\frac{1-y}{6} \dot{\theta}\,,\\
&P_y = \frac{1}{6 p(y)} \dot{y}\,,\\
&P_{\alpha}=w(y) \left(\dot{\alpha} + f(y) \left(\dot{\psi} - \cos\theta
\dot{\phi}\right)\right) \,,\\
&P_{\psi} = w(y) f(y) \dot{\alpha} +
\left[ \frac{q(y)}{9} + w(y) f^2(y)\right]\left(\dot{\psi} - \cos\theta
\dot{\phi}\right)\,,\\
&P_{\phi} = \frac{1-y}{6} \sin^2\theta \dot{\phi}
- \cos\theta P_{\psi}\\
&~~~~= \frac{1-y}{6} \sin^2\theta \dot{\phi} -
\cos\theta w(y) f(y) \dot{\alpha}
- \cos\theta\left[\frac{q(y)}{9} +w(y)f^2(y) \right]\dot{\psi}\\
&~~~~~~~+\cos^2\theta\left[ \frac{q(y)}{9} + w(y) f^2(y) \right] \dot{\phi}\,,
\end{split}$$ with overdot denoting proper time derivative.
The Hamiltonian describing the motion of a free particle is $$\label{freeHam}
H = \frac12 g^{\mu\nu} P_\mu P_\nu \,,$$ which for the $Y^{p,q}$ metric and using the momenta has the form: $$\label{HYpq}
\begin{split}
H=&\frac12 \Biggl\{ 6 p(y) P_y^2 + \frac{6}{1-y}\biggl(P_\theta^2 +
\frac{1}{\sin^2 \theta}(P_\phi + \cos\theta P_\psi)^2\biggr) +
\frac{1-y}{2(a-y^2)}P^2_\alpha\Biggr.\\
& \Biggl.+ \frac{9(a-y^2)}{a-3y^2 +2 y^3}\biggl(P_\psi -
\frac{a -2y +y^2}{6(a-y^2)} P_\alpha \biggr)^2\Biggr\}\,.
\end{split}$$
Starting with the complete set of Killing vectors and Killing-Yano tensors of the SE spaces $Y^{p,q}$ it is possible to find quite a lot of integrals of motions [@BV; @SVV1; @SVV2]. However the number of functionally independent constants of motion is only five implying the complete integrability of geodesic flow on $Y^{p,q}$ spaces. For example we can choose as independent conserved quantities the energy $$\label{E}
E=H\,,$$ the momenta corresponding to the cyclic coordinates $(\phi\,,\, \psi\,,\,\alpha)$ $$\label{Pcyc}
\begin{split}
&P_{\phi} = c_{\phi}\,,\\
&P_{\psi} = c_{\psi}\,,\\
&P_{\alpha} = c_{\alpha}\,,
\end{split}$$ where $(c_{\phi}\,,\,c_{\psi}\,,c_{\alpha})$ are some constants, and the total $SU(2)$ angular momentum $$\label{J2}
\vec{J}^{~2} =P_{\theta}^2 + \frac{1}{\sin^2\theta} \left(P_{\phi}+
\cos\theta P_{\psi}\right)^2 + P_{\psi}^2 \,.$$
Action-angle variables
======================
The connection between completely integrable systems and toric geometry in the symplectic setting is described by the classical Liouville-Arnold theorem [@VIA; @GPS]. A dynamical system defined by a given Hamiltonian $H$ on a $2n$-dimensional symplectic manifold $(M^{2n},\omega)$ is called Liouville integrable if it admits $n$ functionally independent first integrals in involution. In other words, there are $n$ functions $\mathbf{F} = (f_1 =H, f_2,\dots,f_n)$ such that $df_1\wedge \dots \wedge f_n \neq 0$ almost everywhere and $$\{f_i,f_j\} = 0 \quad, \quad \forall i,j \,.$$
Let $\mathbf{F}_\mathbf{c} = (f_1 =E, f_2= c_2,\dots,f_n= c_n)$ by a common invariant level set. If $\mathbf{F}_\mathbf{c}$ is regular, compact and connected, then it is diffeomorphic to the $n$-dimensional Lagrangian torus. For $n$ degrees of freedom the motion is confined to an $n$-torus $$\label{it}
\mathbb{T}^n = \underbrace{S^1 \times S^1 \times \cdots \times S^1}_{\text{n~times}} \,.$$ These are called *invariant tori* and never intersects taking into account the uniqueness of the solution to the dynamical system, expressed as a set of coupled ordinary differential equations.
In a neighborhood of $\mathbf{F}_\mathbf{c}$ there are action-angle variables $\mathbf{J},\mathbf{w}$ mod $2\pi$, such that the symplectic form becomes $$\omega = \sum_{i=1}^{n} d J_i \wedge d w_i\,,$$ and the Hamiltonian $H$ depends only on actions $J_1,\dots,J_n$. An action variable $J_i$ specifies a particular $n$-torus $\mathbb{T}^n$ and is constant since the tori are invariant. The location on the torus is specified by $n$ angle variables $w_i$. Even the system is integrable, the dynamics on the singular set (where the differentials of the integrals $f_1,\dots,f_n$ are dependent) can be quite complicated [@JJD].
In the case of the geodesic motions on $Y^{p,q}$, for the beginning, we fix a level surface $\mathbf{F}=(H, P_\phi,P_\psi,P_\alpha,\vec{J}^{~2})=\mathbf{c}$ of the mutually commuting constants of motion –. The differentials of the chosen first integrals are real analytic [@BV]. Then it suffices to require their functional independence at least at one point [@BJ] to apply the Liouville-Arnold theorem. Further we introduce the generating function for the canonical transformation from the coordinates $(\mathbf{p},\mathbf{q})$, where $\mathbf{p}$ are the conjugate momenta to the coordinates $\mathbf{q}=(\theta,\phi,y,\alpha,\psi)$, to the action-angle variables $(\mathbf{J},\mathbf{w})$ as the indefinite integral $$S(\mathbf{q},\mathbf{c}) = \int_{\mathbf{F}=\mathbf{c}} \mathbf{p}\cdot d\mathbf{q}\,.$$
Since the Hamiltonian has no explicit time dependence, we can write $$S(\mathbf{q},\mathbf{c}) = W(\mathbf{q},\mathbf{c}) - Et\,,$$ with the Hamilton’s characteristic function $$\label{Hcf}
W =\sum_i \int p_i d q_i\,.$$
In the case of geodesic motions in SE spaces $Y^{p,q}$ the variables in the Hamilton-Jacobi equation are separable and consequently we seek a solution of the Hamilton’s characteristic function of the form $$\label{W}
W(y,\theta,\phi, \psi, \alpha)= W_{y}(y) + W_{\theta}(\theta) + W_{\phi}(\phi)
+ W_{\psi}(\psi) + W_{\alpha}(\alpha)\,.$$
The *action variables* $\mathbf{J}$ are defined as integrals over complete period of the orbit in the $(p_i,q_i)$ plane $$J_i = \oint p_i d q_i = \oint \frac{\partial W_i(q_i;c)}{\partial q_i}
dq_i \qquad \mbox{(no summation)}\,.$$ $J_i$’s form $n$ independent functions of $c_i$’s and can be taken as a set of new constant momenta.
Conjugate *angle variables* $w_i$ are defined by the equations: $$\label{av}
w_i = \frac{\partial W}{\partial J_i} =
\sum_{j=1}^n \frac{\partial W_j(q_j;J_1,\cdots,J_n)}{\partial J_i}$$ having a linear evolution in time $$\label{ff}
w_i = \omega_i t + \beta_i$$ with $\beta_i$ other constants of integration and $\omega_i$ are frequencies associated with the periodic motion of $q_i$.
Hamilton characteristic functions associated with cyclic variables are $$\begin{split}
&W_{\phi} = P_{\phi} \phi = c_{\phi} \phi \,,\\
&W_{\psi} = P_{\psi} \psi = c_{\psi} \psi \,,\\
&W_{\alpha} = P_{\alpha} \alpha = c_{\alpha} \alpha \,,
\end{split}$$ where $c_\phi,c_\psi,c_\alpha$ are the constants introduced in .
The corresponding action variables are $$\begin{split}
&J_{\phi} = 2 \pi c_{\phi} \,,\\
&J_{\psi} = 2 \pi c_{\psi} \,,\\
&J_{\alpha} = 2 \pi \ell c_{\alpha} \,.
\end{split}$$
Taking into account and , the Hamilton-Jacobi equation becomes $$\label{EJYpq}
\begin{split}
E=&\frac12 \Biggl\{ 6 p(y) \left(\frac{\partial W_y}{\partial y}\right)^2 +
\frac{6}{1-y}\biggl[\left(\frac{\partial W_\theta}{\partial \theta}\right)^2 +
\frac{1}{\sin^2 \theta}(c_\phi + \cos\theta c_\psi)^2\biggr]
\Biggr.\\
& \Biggl.\frac{1-y}{2(a-y^2)}c^2_\alpha +
\frac{9(a-y^2)}{a-3y^2 +2 y^3}\biggl[c_\psi -
\frac{a -2y +y^2}{6(a-y^2)} c_\alpha \biggr]^2\Biggr\}\,.
\end{split}$$
This equation can be written as follows $$\label{sepJYpq}
\begin{split}
&\left(\frac{\partial W_\theta}{\partial \theta}\right)^2 +
\frac{1}{\sin^2 \theta}(c_\phi + \cos\theta _\psi)^2\\
&~~=\frac{1-y}{3}E - p(y)(1-y) \left(\frac{\partial W_y}{\partial y}\right)^2
- \frac{(1-y)^2}{12(a-y^2)}c^2_\alpha\\
&~~~~-\frac{3(a-y^2)(1-y)}{2(a-3y^2 +2 y^3)}\biggl[c_\psi -
\frac{a -2y +y^2}{6(a-y^2)} c_\alpha \biggr]^2\,.
\end{split}$$
We observe that the LHS of this equation depends only $\theta$ and independent of $y$. Therefore we may set $$\left(\frac{\partial W_\theta}{\partial \theta}\right)^2 +
\frac{1}{\sin^2 \theta}(c_\phi + \cos\theta c_\psi)^2 =
c_{\theta}^2\,,$$ with $c_{\theta}$ another constant. From the last equation we can evaluate the action variable $$\label{J_}
J_{\theta} = \oint d\theta \sqrt{c^2_{\theta} -
\frac{(c_{\phi} + c_\psi \cos\theta )^2}
{\sin^2\theta}}\,.$$
The limits of integrations are defined by the roots $\theta_{-}$ and $\theta_{+}$ of the expressions in the square root sign and a complete cycle of $\theta$ involves going from $\theta_{-}$ to $\theta_{+}$ and back to $\theta_{-}$.
This integral can be evaluated by elementary means or using the complex integration method of residues which turns out to be more efficient [@GPS; @MV2016]. For the evaluation of the integral we put $\cos \theta = t$, extend $t$ to a complex variable $z$ and interpret the integral as a closed contour integral in the complex $z$-plane. Consider the integrand in $$\label{integr}
\frac{\sqrt{-(c^2_\theta + c^2_\psi)z^2 - 2c_\phi c_\psi z + c^2_\theta-
c^2_\phi}}{z^2-1} = \frac{\sqrt{-(c^2_\theta + c^2_\psi)}}{z^2-1}
\sqrt{(z-t_+) (z-t_-)}\,,$$ where the roots $$t_{\pm} = \frac{-c_{\phi}c_\psi \pm
c_{\theta}\sqrt{c^2_{\theta} +
c^2_\psi -c^2_{\phi}}}{c^2_{\theta} + c^2_\psi}\,,$$ are the turning points of the $t$-motion. They are real for $$\label{constr}
c^2_{\theta} + c^2_\psi
-c^2_{\phi}\geq 0 \,,$$ and situated in the interval $(-1,+1)$.
For $z>t_+$ we specify the right side of the square root from as positive. We cut the complex $z$-plane from $t_{-}$ to $t_{+}$ and the closed contour integral of the integrand is a loop enclosing the cut in a clockwise sense. The contour can be deformed to a large circular contour plus two contour integrals about the poles at $z= \pm 1$. After simple evaluation of the residues and the contribution of the large contour integral we finally get: $$J_{\theta} = 2\pi\Biggl[\sqrt{c^2_{\theta} + c^2_\psi} -
c_{\phi} \Biggr] \,.$$
For the action variable corresponding to $y$ coordinate we have from $$\label{JyYpq}
\begin{split}
\frac{\partial W_y}{\partial y} = & \Biggl\{\frac{1-y}{a - 3y^2 +2 y^3} E -
\frac{3}{a-3y^2 +2y^3} c_{\theta}^2 \Biggr.\\
& \Biggl. ~-
\frac{9(a-y^2)(1-y)}{2(a-3y^2+2 y^3)^2} c_{\psi}^2
+ \frac{3(a-2y +y^2)(1-y)}{2 (a-3y^2 +2y^3)^2} c_{\psi}c_{\alpha}
\Biggr.\\
&\Biggl.
~- \frac{(1-y)(2a+a^2 -6ay -2y^2 +2ay^2 +6y^3 -3y^4)}{8(a-3y^2 +2y^3)^2(a-y^2)}
c_{\alpha}^2
\Biggr\}^{\frac12}\,.
\end{split}$$
It is harder to evaluate the action variable $J_y$ in a closed analytic form taking into account the complicated expression . In fact the closed-form of $J_y$ is not at all illuminating. More important is the fact that $J_y$ depends only of four constants of motion: $E, J_\theta, J_\alpha, J_\psi$. In consequence the energy depends only on four action variables $J_y, J_\theta, J_\alpha, J_\psi$ representing a reduction of the number of action variables entering the expression of the energy of the system.
For the angular variable $w_{\phi}$ we have $$w_{\phi} = \frac{1}{2\pi} J_{\phi} + \frac{\partial W_\theta}{\partial
J_\phi}\,.$$
Putting $\cos\theta = t$ the second term is $$\label{wphi}
\begin{split}
\frac{\partial W_\theta}{\partial J_\phi}
&=- \frac{1}{2\pi} \int dt\frac{(J_\phi + J_\theta) t^2 + J_\psi t}
{(1-t^2)\sqrt{-(J_\phi + J_\theta)^2 t^2- 2 J_\phi J_\psi t + (J^2_\theta +
2 J_\theta J_\phi - J^2_\psi)}}\\
&= \frac{1}{2\pi}\int \frac{dt}{1- t^2} \frac{\mathfrak{d} t^2 + \mathfrak{e} t}
{\sqrt{\mathfrak{a} + \mathfrak{b}t + \mathfrak{c}t^2}}\,,
\end{split}$$ where $$\begin{split}
\mathfrak{a}= & J^2_\theta + 2 J_\theta J_\phi - J^2_\psi\,,\\
\mathfrak{b}= & - 2 J_\theta J_\psi \,,\\
\mathfrak{c}= & - (J_\theta + J_\phi)^2\,,\\
\mathfrak{d}= & J_\theta + J_\phi \,,\\
\mathfrak{e}= & J_\psi\,.
\end{split}$$
We necessitate the following integrals [@GR]: $$\begin{split}
I_1(\mathfrak{a},\mathfrak{b},\mathfrak{c};t) =
&\int \frac{dt}{\sqrt{\mathfrak{a} + \mathfrak{b}t + \mathfrak{c}t^2}}\\
=&\frac{-1}{\sqrt{-\mathfrak{c}}}\arcsin
\Biggl(\frac{2 \mathfrak{c} t + \mathfrak{b}}{\sqrt{-\Delta}}\Biggr)
\end{split}$$ evaluated for $\mathfrak{c} <0\,,\,\Delta = 4\mathfrak{a}\mathfrak{c} - \mathfrak{b}^2 <0$, and $$\begin{split}
I_2(\mathfrak{a},\mathfrak{b},\mathfrak{c};t)
+&\int \frac{dt}{t\sqrt{\mathfrak{a} + \mathfrak{b}t + \mathfrak{c}t^2}}\\
=&\frac{1}{\sqrt{-\mathfrak{a}}}\arctan \Biggl(\frac{2\mathfrak{a}+
\mathfrak{b}t}{2\sqrt{-\mathfrak{a}}
\sqrt{\mathfrak{a} +\mathfrak{b}t +\mathfrak{c}t^2}}\Biggr)
\end{split}$$ evaluated for $\mathfrak{a}<0$. That is the case of the constants $\mathfrak{a},\mathfrak{b},\mathfrak{c}$ taking into account the constraint .
Using these integrals we get for the angular variable $w_\phi$ $$\begin{split}
w_{\phi} =& \frac{1}{2\pi} J_{\phi} -\frac{\mathfrak{d}}{2\pi}
I_1 (\mathfrak{a},\mathfrak{b},\mathfrak{c};\cos \theta)\\
&- \frac{\mathfrak{d}+\mathfrak{e}}{4 \pi}
I_2(\mathfrak{a}+\mathfrak{b}+\mathfrak{c}, \mathfrak{b} +
2\mathfrak{c},\mathfrak{c};\cos\theta -1)\\
&- \frac{\mathfrak{e}-\mathfrak{d}}{4 \pi}I_2(\mathfrak{a}-\mathfrak{b}+\mathfrak{c},
\mathfrak{b}-2\mathfrak{c},\mathfrak{c};\cos\theta +1)\,.
\end{split}$$
The explicit evaluation of the angular variables $w_\theta, w_\psi, w_\alpha, w_y$ is again intricate due to the absence of a simple closed-form for the action variable $J_y$. However, it is remarkable the fact that one of the fundamental frequencies $$\omega_i = \frac{\partial H}{\partial J_i}\,,$$ is zero, namely $$\label{fzero}
\omega_\phi = \frac{\partial H}{\partial J_\phi} =0\,,$$ since the action $J_\phi$ does not enter the expression of the energy.
The topological nature of the flow of each invariant torus depends on the properties of the frequencies $\omega_i$ . There are essentially two cases [@JP]:
1. The frequencies $\omega_i$ are nonresonant $$k_i \omega_i \neq 0 \quad \text{for all} \quad 0\neq k_i \in \mathbb{Z}^n \,.$$ Then, on this torus each orbit is dense and the flow is ergodic.
2. The frequencies $\omega_i$ are resonant or rational dependent $$k_i \omega_i = 0 \quad \text{for some} \quad 0\neq k_i \in \mathbb{Z}^n \,.$$ The prototype is $\mathbf{\omega} = (\omega_1, \cdots , \omega_{n-m}, 0,\cdots,0)$ with $1\leq m\leq n-1$ zero frequencies and $(\omega_1, \cdots , \omega_{n-m})$ nonresonant frequencies.
The KAM theorem [@VIA] describes how an integrable system reacts to small non-integrable deformations. The KAM theorem states that for nearly integrable systems, i.e. integrable systems plus sufficiently small conservative Hamiltonian perturbations, most tori survive, but suffer a small deformation. However the resonant tori which have rational ratios of frequencies get destroyed and motion on them becomes chaotic.
In the case of geodesics on $Y^{p,q}$ space, the frequencies are resonant giving way to chaotic behavior when the system is perturbed. The analysis performed in [@BZ2] confirms the present results produced in the action-angle approach.
Conclusions
===========
The action-angle formulation for $Y^{p,q}$ spaces gives us a better understanding of the dynamics of the geodesic motions in these spaces. In spite of the complexity of the evaluation of some variables, we are able to prove that the energy of the system depends on a reduced number of action variables signaling a degeneracy of the system.
This fact corroborates a similar result obtained in the case of geodesic motions in the homogeneous SE space $T^{1,1}$ [@MV2016]. The metric on $T^{1,1}$ may be written by utilizing the fact that it is a $U(1)$ bundle over $S^2 \times S^2$. The evaluations of all action and angle variables was completely done putting them in closed analytic forms. In the case of the space $T^{1,1}$ the isometry is $SU(2)^2 \times U(1)$ and there are two pairs of fundamental frequencies which are resonant. The degeneracy of these two pairs of frequencies may be removed by a canonical transformation to new action-angle variables. Finally the Hamiltonian governing the motions on $T^{1,1}$ can be written in terms of only *three* action variables for which the corresponding frequencies are different from zero.
In conclusion, the action-angle approach offers a strong support for the assertion that certain classical string configurations in $AdS_5 \times Y_5$ with $Y_5$ in a large class of Einstein spaces is non-integrable [@BDG; @ZE]. It would be interesting to extend the action-angle formulations to other five-dimensional SE spaces as well as to their higher dimensional generalizations relevant for the predictions of the AdS/CFT correspondence.
Acknowledgements {#acknowledgements .unnumbered}
================
The author would like to thank the referee for valuable comments and suggestions which helped to improve the manuscript. This work has been partly supported by the project [*CNCS-UEFISCDI PN–II–ID–PCE–2011–3–0137*]{} and partly by the project [*NUCLEU 16 42 01 01/2016*]{}.
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[^1]: mvisin@theory.nipne.ro
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We use global three dimensional radiation magneto-hydrodynamic simulations to study the properties of inner regions of accretion disks around a $5\times 10^8\mbh$ black hole with mass accretion rates reaching $7\%$ and $20\%$ of the Eddington value. This region of the disk is supported by magnetic pressure with surface density significantly smaller than the values predicted by the standard thin disk model but with a much larger disk scale height. The disks do not show any sign of thermal instability over many thermal time scales. More than half of the accretion is driven by radiation viscosity in the optically thin corona region for the lower accretion rate case, while accretion in the optically thick part of the disk is driven by the Maxwell and Reynolds stresses from MRI turbulence. Coronae with gas temperatures $\gtrsim 10^8{\rm K}$ are generated only in the inner $\approx 10$ gravitational radii in both simulations, being more compact in the higher accretion rate case. In contrast to the thin disk model, surface density increases with increasing mass accretion rate, which causes less dissipation in the optically thin region and a relatively weaker corona. The simulation results may explain the formation of X-ray coronae in Active Galactic Nuclei (AGNs), the compact size of such coronae, and the observed trend of optical to X-ray luminosity with Eddington ratio for many AGNs.'
author:
- 'Yan-Fei Jiang(姜燕飞), Omer Blaes, James M. Stone & Shane W. Davis'
bibliography:
- 'SubEddAGN.bib'
title: 'Global Radiation Magneto-hydrodynamic Simulations of sub-Eddington Accretion Disks around Supermassive black Holes'
---
[UTF8]{}[gbsn]{}
Introduction
============
Active Galactic Nuclei (AGNs) are believed to be powered by accretion onto the central black holes. Luminosity of most AGNs is observed to be smaller than the Eddington limit as defined by the electron scattering opacity $\Ledd=1.5\times 10^{46}\mbh/\left(10^8\msun\right)$ erg s$^{-1}$. The standard thin disk model (@ShakuraSunyaev1973) is usually adopted to describe the accretion disks in this regime when the AGN luminosity is larger than $\approx 0.01\Ledd$. The key assumptions in this model are that the disk is optically thick and supported by thermal pressure (gas or radiation), and all the dissipation is radiated away locally. The detailed structures of the disk are then determined by the famous $\alpha$ assumption, which assumes that the stress responsible for the angular momentum transport in the disk is proportional to the total pressure with a constant value.
Comparing the predictions from the standard thin model with the observed properties of AGNs has raised many questions on the assumptions in this model (@KoratkarBlaes1999). The standard thin disk model subjects to the well known thermal and inflow instabilities (@ShakuraSunyaev1976 [@lightmanEardley1974; @Piran1978]), which should cause the disk to evolve away from the equilibrium state in a few thermal time scales (@Jiangetal2013c [@Fragileetal2018]) and probably show some kind of limit-cycle behavior (@Honmaetal1991 [@Janiuketal2002]). However, this is not observed for most AGNs. The predicted spectrum from this model is also not consistent with the observed energy distribution (@Zhengetal1997 [@Davisetal2007; @LaorDavis2014]). Particularly, the predicted edge feature in the spectrum is also not observed (@SincellKrolik1997 [@Shulletal2012; @Tiltonetal2016]).
The standard thin disk model assumes that thermal pressure (gas or radiation) is dominant in the disk. Alternative possibility that the vertical component of gravity in the accretion disk may be primarily balanced by magnetic pressure has been proposed (@Shibataetal1990 [@Parievetal2003]). [@BegelmanPringle2007] studied the structures and stability of magnetic pressure dominated disks by assuming the toroidal magnetic fields will saturate to a level so that the associated Alfvén velocity reaches $\sqrt{c_sV_k}$ (@PessahPsaltis2005 [@Dasetal2018]), where $c_s$ is the gas sound speed and $V_k$ is the disk rotation speed. Amplification of magnetic fields near the disk midplane is thought to be balanced by the escape of magnetic fields away from the midplane due to buoyancy. This magnetic elevated model is found to have a larger pressure scale height and does not subject to the thermal and viscous instabilities (@Sadowski2016), which have interesting implications for both X-ray binaries and AGNs (@Begelmanetal2015 [@BegelmanSilk2017; @DexterBegelman2019]). However, assumptions in the magnetic elevated disk model have not been checked numerically. It is also unclear how the strong radiation pressure in AGN accretion disks will modify the structures, which is typically neglected in these models. It is known that in order to reach the magnetic pressure dominated state, large scale poloidal or radial magnetic fields are required (@Salvesenetal2016 [@Salvesenetal2016b; @FragileSadowski2017]). The advantage of our simulations is that gas, radiation and magnetic pressure are all included self-consistently. Our goal is to study the structures of the disk that will be formed with these magnetic field configurations for realistic parameters of AGNs. Our simulations form the a disk in a region that is initially vacuum by accreting gas from larger radius. Although the resulting accretion disk depends on our initial condition for the magnetic fields, the disk forms self-consistently. This distinguishes it from previous work that initialized the simulation with a standard thin disk model and study its long term evolution (@Fragileetal2018).
Formation of coronae in AGNs, as well as the dependence of corona properties on the accretion flow, remain a puzzle. The effective temperature of the accretion disks in AGNs (typically $\sim 10^4-10^5$ K) is too low to produce the widely observed X-rays in AGNs via thermal emission. It is proposed that high temperature corona, as inspired by the solar corona, is formed on top of the accretion disk (@BisnovatyiBlinnikov1976 [@HaardtMaraschi1991; @HaardtMaraschi1993; @SvenssonZdziarski1994; @Zdziarskietal1999]). X-rays are produced via Compton scattering of the seed photons emitted by the disk with the hot electrons in the corona, which is infereed to exist in a compact region near the black hole ($\sim 10$ gravitational radii, @ReisMiller2013 [@Uttleyetal2014]). The amount of X-rays that can be produced via this mechanism compared with the thermal emission from the disk is determined by the fraction of accretion power that is dissipated in the corona region. This is basically a free parameter in these models since the mechanism to produce the corona in AGNs is unknown. Since the early isothermal simulations of vertically stratified accretion disk with turbulence generated by the Magneto-rotational instability (MRI) (@MillerStone2000), it is commonly observed that magnetic fields amplified by MRI near the midplane of the disk can buoyantly rise to the surface and create a magnetic pressure dominated low density region, which is suspected to be the corona. However, most local simulations that determine the thermal properties of the disks based on radiative cooling (using both the flux-limited diffusion and VET method) typically find that the gas in the magnetic pressure dominated region is not heated to a very high temperature due to insignificant dissipation (@Kroliketal2007 [@Blaesetal2007; @Blaesetal2011; @Hiroseetal2009; @Jiangetal2013c; @Jiangetal2016a]). An exception is [@Jiangetal2014], which shows that the amount of dissipation in the magnetic pressure dominated region can be increased by reducing the surface density of the disk, which can increase the gas temperature in this region. Surface density is a free parameter in these local shearing box simulations and the value required to produce the high temperature corona is smaller than what the thin disk model will predict for the same accretion rate. It is therefore necessary to use global simulations to determine the surface density of the disk as well as the properties of coronae self-consistently for a given mass accretion rate, which is one goal of the paper.
[@Jiangetal2016a] shows that for typical density and temperature expected for AGN accretion disks based on the standard thin disk model, the Rosseland mean opacity should be larger than the electron scattering value due to irons. The density and temperature dependences of the iron opacity peak can modify the thermal stability and structures of AGN accretion disks significantly. However, the local shearing box simulations done by [@Jiangetal2016a] adopt the surface density as given by the standard thin disk model. Whether this opacity peak will show up or not depends on the actual disk structure we will get. We will include the full opacity table in the simulations to capture its potential importance.
The remainder of this paper is organized as follows. In Section \[sec:setup\], we describe the simulation setup. Detailed structures of the disk are described in Section \[sec:result\]. We discuss the implications of our simulations in Section \[sec:discussion\].
Simulation Setup {#sec:setup}
================
We solve the same set of ideal MHD equations coupled with the time dependent radiative transfer equation for specific intensities as in [@Jiangetal2018] using the code [Athena++]{} (Stone et al, in preparation). We carry out two simulations, [AGN0.2]{} and [AGN0.07]{}, for a $\mbh=5\times 10^8\msun$ black hole with accretion rates smaller than the Eddington value $\Medd\equiv 10\Ledd/c^2=8.22\times 10^{26}\ {\rm g/s}$. The simulations are performed with the pseudo-Newtonian potential [@PaczynskiWiita1980] $\phi=-G\mbh/(r-2r_g)$ to mimic the general relativity effects around a Schwarzschild black hole, where $G$ is the gravitational constant while gravitational radius $r_g\equiv G\mbh/c^2=7.42\times 10^{13}\ {\rm cm}$. We initialize a torus centered at $80r_g$ with the maximum density $\rho_0=10^{-8}\ {\rm g}\ {\rm cm}^{-3}$ and temperature $3.18T_0$, where $T_0=2\times 10^5 {\rm K}$. The shape of the torus is the same as the ones used in [@Jiangetal2018]. The inner edge of the torus is at $40r_g$ and the region inside that radius is filled with density floor $10^{-9}\rho_0$ initially. The initial properties of the torus, including the ratios between the averaged radiation pressure $P_r$, gas pressure $P_g$ and magnetic pressure $P_B$, are summarized in Table \[Table:parameters\]. The main difference between the two simulations is the initial magnetic field in the torus. The run [AGN0.2]{} uses a single loop of poloidal magnetic field while the run [AGN0.07]{} adopts multiple magnetic field loops as described in [@Jiangetal2018]. The different setups lead to different mass accretion rates in the disks that are formed near the black hole.
We use four levels of static mesh refinement to cover the whole simulation domain $\left(r,\theta,\phi\right)\in\left(4r_g, 1600r_g\right)\times \left(0,\pi\right)\times \left(0,2\pi\right)$. The level with the highest resolution reaches $\Delta r/r=\Delta \theta=\Delta\phi=6.1\times 10^{-3}$ for the region $\left(6r_g,200r_g\right)\times \left(1.48,1.66\right)\times\left(0,2\pi\right)$, which covers most of the mass near the disk midplane. The equivalent resolution is $1024\times 512\times 1024$, which is necessary to resolve these sub-Eddington accretion disks.
We use $80$ discrete angles in each cell to resolve the angular distribution of the radiation field, which is in thermal equilibrium initially in the torus. We calculate the Rosseland mean opacity in each cell by using local density and temperature based on the OPAL opacity table with solar metallicity (@Paxtonetal2013 [@Jiangetal2015]). Planck mean free-free absorption opacity is also included as in [@Jiangetal2018]. Each simulation takes $\approx 20-30$ millions CPU time in the ALCF machine [Mira]{}.
Variables/Units [AGN0.2]{} [AGN0.07]{}
-------------------------------------- ---------------------- ---------------------- -- --
$r_i/r_g$ 80 80
$\rho_i/\rho_0$ 1 1
$T_i/T_0$ 3.18 3.18
$ \langle P_{r}/ P_{g}\rangle$ $4.47\times 10^5$ $4.61\times 10^5$
$ \langle P_{r}/P_{g}\rangle_{\rho}$ $4.40\times 10^2$ $4.39\times 10^2$
$ \langle P_{B}/P_{g}\rangle$ $1.27\times 10^{-2}$ $2.32\times 10^{-4}$
$ \langle P_{B}/P_{g}\rangle_{\rho}$ $7.70\times 10^{-3}$ $7.32\times 10^{-5}$
$\Delta r/r$ $6.1\times 10^{-3}$ $6.1\times 10^{-3}$
$\Delta \theta$ $6.1\times 10^{-3}$ $6.1\times 10^{-3}$
$\Delta \phi$ $6.1\times 10^{-3}$ $6.1\times 10^{-3}$
$N_n$ 80 80
\[Table:parameters\]
Note: The center of the initial torus is located at $r_i$ with density and temperature to be $\rho_i$ and $T_i$. The fiducial density and temperature are $\rho_0=10^{-8}\ {\rm g}\ {\rm cm}^{-3}$ and $T_0=2\times 10^5 {\rm K}$. For any quantity $a$, $\langle a \rangle$ is the volume averaged value over the torus while $\langle a \rangle_{\rho}$ is the averaged value weighted by the mass in each cell. The grid sizes $\Delta r, \Delta \theta, \Delta\phi$ are for the finest level at the center of the torus. The number of angles for the radiation grid is $N_n$ in each cell.
Results {#sec:result}
=======
Resolution for MRI
------------------
To quantify how well the MRI is resolved in our simulations, we calculate the quality factors $Q_{\theta}$ and $Q_{\phi}$, which are the ratios between the wavelength of the fastest growing MRI mode $\lambda=2\pi\sqrt{16/15}|v_{A}|/\Omega$ and cell sizes $r\Delta \theta, r\sin\theta\Delta \phi$ along $\theta$ and $\phi$ directions, where the Alfvén velocity $v_{A}$ is calculated for $B_{\theta}$ and $B_{\phi}$ respectively. Resolution studies for non-radiative ideal MHD simulations [@Hawleyetal2011; @Sorathiaetal2012] find that when $Q_{\phi} \gtrapprox 25, Q_{\theta}\gtrapprox 6$ or both $Q_{\phi}$ and $Q_{\theta}$ are larger than $10$, properties of MRI turbulence is converged with respect to resolution. Although there is no general criterion on the convergence of radiation MHD simulations of accretion disks, we use it as a way to compare our resolutions with these calculations. We calculate the azimuthally averaged quality factors during the final turbulent states for both of the calculations. For the simulation [AGN0.07]{} at the disk midplane, $Q_{\phi}=100$ at $r=5r_g$ and decreases to 80 at $r=30r_g$, while $Q_{\theta}$ varies from 10 to 8 in the same radial range. For a fixed radius, the combined effects of increasing Alfvén velocity and reduced resolution with height from the disk midplane cause the quality factors to decrease by a factor of $\approx 2$ at the photosphere. For the simulation [AGN0.2]{}, the quality factors are larger as the disk is thicker. The midplane $Q_{\phi}$ changes from 200 at $5r_g$ to 40 at $40r_g$ while $Q_{\theta}$ changes from $\approx 20$ to $\approx 10$ between $5$ and $40r_g$ at the disk midplane. For this run, the quality factors increase by a factor of $\approx 2$ from the midplane to the photosphere. Therefore, the resolutions are sufficient to resolve the MRI turbulence according to the criterion found by non-radiative MHD simulations. This is possible for these sub-Eddington accretion disks because magnetic pressure is the dominant pressure, which makes the wavelength of the fastest growing MRI mode be comparable to the disk scale height and easier to resolve.
{width="0.49\hsize"} {width="0.49\hsize"}
{width="0.49\hsize"} {width="0.49\hsize"}
![Time and azimuthally averaged spatial structures for the inner $40r_g$ of the disks from the two simulations [AGN0.07]{} (top panels) and [AGN0.2]{} (bottom panels). The left column is for density (the color) and flow velocity (the streamlines) while the right column is for radiation energy density (the color) and magnetic field lines (the streamlines). The dashed black line in the left column indicates the location where optical depth to the rotation axis for Rosseland mean opacity is $1$.[]{data-label="ave_disk_structure"}](AGN3B.pdf "fig:"){width="1.0\hsize"} ![Time and azimuthally averaged spatial structures for the inner $40r_g$ of the disks from the two simulations [AGN0.07]{} (top panels) and [AGN0.2]{} (bottom panels). The left column is for density (the color) and flow velocity (the streamlines) while the right column is for radiation energy density (the color) and magnetic field lines (the streamlines). The dashed black line in the left column indicates the location where optical depth to the rotation axis for Rosseland mean opacity is $1$.[]{data-label="ave_disk_structure"}](AGNGlobal3.pdf "fig:"){width="1.0\hsize"}
Simulation Histories
--------------------
Gas in the initial torus flows towards the black hole and the accretion disk is slowly built up. Histories of the mass accretion rate $\dot{M}\equiv \int_0^{2\pi}\int_0^{\pi}\rho v_r r^2\sin\theta d\theta d\phi$ at $10r_g$ for the two runs are shown in the left panel of Figure \[mdot\_hist\]. After a period of initial transient, the simulation [AGN0.07]{} reaches an accretion rate $7\%\Medd$ when averaged after time $t=2.4\times 10^4r_g/c$, which roughly corresponds to $13$ thermal time scales at $10r_g$. For the run [AGN0.2]{}, the averaged accretion rate is $22\%\Medd$ after $t=3\times 10^4r_g/c$, which also corresponds to more than $\approx 6$ thermal time scales at $10r_g$. All the analysis for the time averaged properties of the two simulations are also performed for the same time intervals. The mass accretion rates show significant fluctuations due to turbulence. The standard deviation of $\dot{M}$ during the steady state for the run [AGN0.07]{} is only $1.2\%$ of the averaged value while the corresponding value is $12.3\%$ for the run [AGN0.2]{}. The main difference of the two simulations is the magnetic field topology as described in Section \[sec:setup\]. The run [AGN0.2]{} adopts a single loop of magnetic field in the initial torus and there are net poloidal magnetic fields $\overline{B}_{\theta}$ through the inner region of the disk. The ratio between radiation pressure and $\overline{B}_{\theta}^2/2$ near the disk midplane is $\approx 10^3$. For the run [AGN0.07]{}, quadrupole magnetic fields are used in the initial condition, which result in net radial magnetic fields $\overline{B}_r$ near the disk midplane, although the shell averaged $B_r$ across the whole disk is still zero. The net $\overline{B}_r$ near the disk midplane is sheared into toroidal magnetic field within an orbital time scale and quickly builds up strong magnetic pressure, which then escapes from the midplane due to buoyancy. MRI turbulence is still developed in this case (@PessahPsaltis2005 [@Dasetal2018]), but it shows less variability compared with the other run. Detailed investigations of magnetic pressure dominated disks resulting from this magnetic field configuration are described in Section \[sec:ave\_structure\].
In most MHD simulations of accretion disks where radiative transfer is not calculated self-consistently, density is usually used as the proxy to understand the observed properties of these systems. Since our simulations calculate the photons emitted by the disk directly, we can calculate the frequency-integrated lightcurves for the time scale that our simulations can cover. In order to avoid contamination from photons generated by the torus at large radii where the disk has not reached steady state, we convert the radiation flux ${{{\mbox{\boldmath $F$}}}_r}$ to the radial $F_R$ and vertical components $F_z$ in cylindrical coordinates and then integrate the total radiative luminosity leaving the cylindrical surface at $R=10r_g$ and height $z=100r_g$, which is well beyond the photosphere. The resulting lightcurves are shown in the right column of Figure \[mdot\_hist\], which have much weaker short time scale variabilities compared with what the variability in $\dot{M}$ might suggest. This is likely due to the scattering of photons through the optically thick disks. In particular, the power spectrum of the lightcurve from the lower accretion rate run ([AGN0.07]{}) has more power at high frequencies compared with the lightcurve power spectrum from the run with higher accretion rate ([AGN0.2]{}). This is because the midplane optical depth increases with increasing mass accretion rate (see Figure \[surface\_density\]). The changes of variability amplitude with luminosity are also different in the two lightcurves, revealing different dynamo actions in the disk, which will be discussed in Section \[sec:discussion\].
Histories of poloidal profiles of azimuthally averaged density $\rho$, gas temperature $T_g$ and toroidal magnetic field $B_{\phi}$ at $10r_g$ for the two runs are shown in Figure \[STplot\]. The disk photosphere only extends to $\approx 6^{\circ}$ away from the disk midplane for the run [AGN0.07]{} while in the simulation [AGN0.2]{}, the disk photosphere covers more than $14^{\circ}$. Once above the photosphere as indicated by the blue lines in the top panels, gas temperature increases rapidly to $\approx 10^9$ K, which is the corona region. More detailed properties of the corona are discussed in Section \[sec:corona\]. In the optically thick part of the disk, gas and radiation are in thermal equilibrium with a temperature around $10^5$ K. The grid scale variation of gas temperature is unrealistic because gas pressure is smaller than $0.1\%$ of radiation and magnetic pressure and it suffers from numerical noise. The toroidal magnetic field switches signs near the disk midplane after $\approx 10^4r_g/c$ for the run [AGN0.2]{}, which is the well known butterfly diagram caused by the MRI dynamo [@Stoneetal1996; @MillerStone2000; @Davisetal2010; @ONeilletal2011; @Simonetal2012; @Jiangetal2013c; @Jiangetal2014c]. This does not happen for the run [AGN0.07]{}, although MRI turbulence has also developed there. This is because shearing of the net radial magnetic field near the disk midplane for the run [AGN0.07]{} always forms $B_{\phi}$ with the same sign in addition to the $B_{\phi}$ generated by MRI (see Section \[sec:vertical\]).
![Radial profiles of the time and shell averaged gas pressure ($P_g$, dashed red lines), radiation pressure ($P_r$, solid red lines), total magnetic pressure ($P_B$, solid black lines) and magnetic pressure due to the azimuthally averaged magnetic field ($\overline{B}^2/2$, dashed black lines) in the two simulations [AGN0.07]{} (top panel) and [AGN0.2]{} (bottom panel). []{data-label="pressure_profile"}](pressure.pdf){width="1.0\hsize"}
![Radial profiles of the time averaged net mass accretion rate $\dot{M}$ for the two runs [AGN0.07]{} (top panel) and [AGN0.2]{} (bottom panel). The solid black lines are $\dot{M}$ integrated over all polar angles, while the solid red and dotted black lines are $\dot{M}$ in the optically thin and thick regions respectively.[]{data-label="Mdot_profile"}](Mdot_profile.pdf){width="1.0\hsize"}
![Radial profiles of the total Rosseland mean ($\kappa_R\Sigma/2$) and effective absorption ($\sqrt{\kappa_s\kappa_a}\Sigma/2$) optical depth from the disk midplane to the rotation axis. Here $\kappa_a$ and $\kappa_s$ are the absorption and scattering opacities while $\kappa_R\equiv \kappa_a+\kappa_s$ is the Rosseland mean opacity. The top and bottom panels are for the two runs [AGN0.07]{} and [AGN0.2]{} respectively.[]{data-label="surface_density"}](surface_density.pdf){width="1.0\hsize"}
Disk Structure {#sec:ave_structure}
--------------
The time and azimuthally averaged spatial structure inside $30r_g$ for the density, flow velocity, radiation energy density and magnetic field lines of the two simulations are shown in Figure \[ave\_disk\_structure\]. Radial profiles of different shell averaged pressure components, the mass accretion rates as well as total optical depth are shown in Figures \[pressure\_profile\], \[Mdot\_profile\] and \[surface\_density\]. The disk is thinner in the run [AGN0.07]{} compared with [AGN0.2]{} due to lower accretion rate and the thickness of the disk is different from standard accretion disk model predictions. In a radiation pressure and electron scattering dominated $\alpha$ disk model in a spherically symmetric gravitational potential, vertical hydrostatic equilibrium implies that the thickness of the disk $H=[\kappa_s\dot{M}/(4\pi c)](d\ln\Omega/d\ln r)$ away from the inner boundary. Near the inner boundary $r_{\rm in}$, it is smaller by $1-\left(r_{\rm in}/r\right)^{1/2}$. This is nearly independent of radius (and of $\alpha$), and implies $H/r_g\approx 1.1$ [@Franketal2002] for $\dot{M}=7\%\Medd$ and $H/r_g\approx 3.0$ for $\dot{M}=20\%\Medd$. It will be even smaller if we take into account the boundary effect. As shown in Figure \[ave\_disk\_structure\], the heights of the photosphere clearly increase rapidly with radius in the two runs, and become larger than the values in the standard $\alpha$ disk model beyond $\approx 10r_g$. The contrast of radiation energy density inside and above the photosphere is much smaller in [AGN0.07]{} compared with the run [AGN0.2]{} because of the reduced total optical depth as well as different spatial distributions of dissipation. Although the shell averaged radiation pressure is still larger or comparable to the magnetic pressure (Figure \[pressure\_profile\]), the vertical gradient of radiation pressure is actually smaller, particularly in the run [AGN0.07]{}. The disk is actually supported by magnetic pressure gradient in this region (section \[sec:vertical\]). There are two magnetic field loops above and below the disk midplane in the run [AGN0.07]{} and they are configured in such a way to have a net radial magnetic field near the midplane. In the run [AGN0.2]{}, there are net poloidal magnetic fields threaded through the disk by design. With both magnetic field configurations, we have confirmed that the strong magnetic pressure is dominated by the turbulent component, since the magnetic pressure due to the azimuthally averaged mean magnetic field $\overline{B}^2/2$ is smaller than the total magnetic pressure by a factor of $\approx 10$ (Figure \[pressure\_profile\]).
Radial profiles of mass accretion rate averaged over the whole spherical shell reach constant values inside $\approx 15r_g$ for [AGN0.07]{} and $\approx 13r_g$ for [AGN0.2]{} (Figure \[Mdot\_profile\]). We decompose the net mass accretion rate into two parts based on the location of the photosphere shown in Figure \[ave\_disk\_structure\]. At each radius, we integrate $\rho v_r r^2$ for $\theta$ angles either above or below the photosphere and radial profiles of the two components are shown as red and dashed black lines in Figure \[Mdot\_profile\]. For the run [AGN0.2]{}, the majority of accretion happens in the optically thick part of the disk until $r<7r_g$. While for the run [AGN0.07]{}, more than half of the mass is accreted in the optically thin surface of the disk over the entire radial range where the disk has reached steady state. The spatial distribution of mass inflow is consistent with the location of stress for angular momentum transport (section \[sec:stress\]). Significant surface accretion has also been found in non-radiative ideal MHD simulations of accretion disks with a net poloidal magnetic field [@ZhuStone2018]. In fact, the resulting magnetic field configuration from the run [AGN0.2]{} shown in Figure \[ave\_disk\_structure\] is very similar to what [@ZhuStone2018] found (for example, Figure 25 of that paper). The poloidal magnetic fields are advected inwards above the disk midplane. Since we can determine the thermal properties of the disk self-consistently, for the accretion rate of $26\%\Medd$, accretion still happens in the optically thick part of the disk with little accretion in the corona region. When the accretion rate drops to $7\%$ with a reduced optical depth, mass is primarily accreted in the optically thin region. However, for these AGN disks, the mechanism for angular momentum transport responsible for the corona accretion is due to radiation instead of magnetic field as found in non-radiative ideal MHD simulations (section \[sec:stress\]).
Radial profiles of time averaged Rosseland mean optical depth $\tau_R=\kappa_R\Sigma/2$ for the two runs are shown in Figure \[surface\_density\]. Here the Rosseland mean opacity $\kappa_R\equiv \kappa_a+\kappa_s$ is the sum of absorption $\kappa_a$ and scattering $\kappa_s$ opacity. Optical depths for effective absorption $\tau_{\rm eff}=\sqrt{\kappa_s\kappa_a}\Sigma/2$ are also shown in the same Figure. The Rosseland mean optical depth is only 2 at $6r_g$ for [AGN0.07]{} and it increases to $100$ at $14r_g$, while the effective absorption optical depth drops below $1$ near the ISCO. When the accretion rate is increased to $26\%\Medd$, the optical depth is increased by a factor of $2-3$ and the entire disk is optically thick even for effective absorption. The optical depth in the disk is smaller than what the standard $\alpha$ disk model predicts by a factor of $\approx 10$ for the same mass accretion rate due to increased disk scale height and inflow velocity because of strong magnetic pressure support (Section \[sec:vertical\]).
![Radial profiles of the time and shell averaged ratios between Reynolds ($\alpha_h$, dashed red lines), Maxwell ($\alpha_m$, solid black lines), radiation ($\alpha_{\rm rad}$, solid red lines) stresses and total pressure for the run [AGN0.07]{} (top) and [AGN0.2]{} (bottom). The dotted black lines ($\alpha_{\overline{m}}$) are for stresses due to the azimuthally averaged mean $\langle B_r\rangle$ and mean $\langle B_{\phi}\rangle$.[]{data-label="compare_alpha"}](Compare_alpha.pdf){width="1.0\hsize"}
{width="1.0\hsize"}
Stresses for Angular Momentum Transport {#sec:stress}
---------------------------------------
In accretion disks, transport of angular momentum can be provided by the sum of Maxwell, Reynolds and radiation (the off-diagonal components of the radiation pressure) stresses. Since radiation pressure is so significant in AGN accretion disks, radiation stress can potentially play an important role as an effective viscosity (@LoebLaor1992). In the optically thick regime, radiation viscosity is proportional to $E_r/(c\rho\kappa_R)$ multiplied by the shear rate of the velocity field (@Weinberg1971 [@MihalasMihalas1984; @KaufmanBlaes2016]). If we only consider the shear rate for a Keplerian disk near the disk midplane, the ratio between the radiation stress and radiation pressure at radius $r$ is $\sim \mathcal{O}\left(\sqrt{r_g/r}/(r\rho\kappa_R)\right)$. For a standard thin accretion disk model where $r\rho\kappa_R \gtrapprox 100$, the radiation stress will be much smaller than the typical Maxwell and Reynolds stresses produced by MRI turbulence. In other words, radiation viscosity is small because the mean free path of the photons is too short. This is also true for super-Eddington accretion disks around AGNs due to large optical depth [@Jiangetal2018], even though radiation pressure is significantly larger than the gas and magnetic pressure there. However, this is not the case for these sub-Eddington simulations.
![Comparison between the vertical profiles of the azimuthally averaged radiation stress $P_r^{r\phi}$ (solid lines) calculated by the simulation [AGN0.07]{} with the analytical formula for radiation viscosity $P_{r,\rm vis}^{r\phi}$ (dashed lines), based on equations (\[eq:vis1\]) and (\[eq:vis2\]). The top and bottom panels are for radii $10r_g$ and $13r_g$ respectively. This comparison is done for the snapshot at time $4.5\times 10^4r_g/c$.[]{data-label="compare_vis_stress"}](compare_rad_stress_10rg_13rg.pdf){width="1.0\hsize"}
![Time and azimuthally averaged spatial distributions of gas temperature $T_g$ for the two runs [AGN0.07]{} (left panel) and [AGN0.2]{} (right panel). The gas temperature is $\approx 10^5-2\times 10^5$ K in the optically thick part of the disk but increases to $10^8-10^9$ K rapidly in the optically thin corona regions. []{data-label="Corona"}](AGN3B_Tgas.pdf "fig:"){width="0.49\hsize"} ![Time and azimuthally averaged spatial distributions of gas temperature $T_g$ for the two runs [AGN0.07]{} (left panel) and [AGN0.2]{} (right panel). The gas temperature is $\approx 10^5-2\times 10^5$ K in the optically thick part of the disk but increases to $10^8-10^9$ K rapidly in the optically thin corona regions. []{data-label="Corona"}](AGNGlobal3_Tgas.pdf "fig:"){width="0.49\hsize"}
{width="0.49\hsize"} {width="0.49\hsize"}
![Time and azimuthally averaged vertical profiles of the poloidal components of Poynting flux (black line) and radiation flux (red line) at $10r_g$ for the run [AGN0.07]{}. The energy fluxes are scaled with the critical value $c G\mbh|\cos(\theta)|/\left(\kappa_{\rm es}(r-2r_g)^2\right)$.[]{data-label="vertical_energy"}](theta_EBflux.pdf){width="1.0\hsize"}
Radial profiles of the shell averaged Maxwell ($\alpha_m$), Reynolds ($\alpha_h$) and radiation stresses (only the $r-\phi$ component remains, $\alpha_{\rm rad}$) scaled with the shell averaged total pressure for the two runs are shown in Figure \[compare\_alpha\]. The stresses are calculated in the same way as in [@Jiangetal2018]. For both runs, we have also calculated the Maxwell stress due to the azimuthally averaged mean $\langle B_r \rangle$ and $\langle B_{\phi} \rangle$, which are shown as the dashed red lines in Figure \[compare\_alpha\]. It is smaller than the total Maxwell stress by a factor of 6 in [AGN0.2]{} and a factor of 2 in [AGN0.07]{}, despite the fact that the latter has shearing net radial magnetic field in the midplane. The total stress in the run [AGN0.2]{} is dominated by the Maxwell stress with the Reynolds and radiation stresses smaller by a factor of $\approx 5$, which is consistent with the results reported by previous both local and global simulations of MRI turbulence with net vertical magnetic fields [@BaiStone2013; @ZhuStone2018]. However, for the run [AGN0.07]{}, the radiation stress is larger than the others by more than a factor of $2$ for all the radial range $r>5r_g$, while the Maxwell and Reynolds stresses vary from $2\%$ to $10\%$ of the total pressure, which are typical values found by MRI turbulence without net vertical magnetic fields [@HGB1995; @MillerStone2000; @Hiroseetal2006; @Davisetal2010; @Jiangetal2013c; @Jiangetal2014].
Radiation stress and Maxwell stress dominate different locations of the disk in the run [AGN0.07]{} as they have different spatial distributions. The time and azimuthally averaged spatial distributions of the $r-\phi$ and $\theta-\phi$ components of radiation stress $P_{r}^{r\phi}$ and $P_{r}^{\theta\phi}$ for the run [AGN0.07]{} are shown in the first two panels of Figure \[AGN3Bstress\]. For comparison, the Maxwell stress $S_m$ and Reynolds stress $S_h$ are shown in the third and fourth panels. Inside the photosphere of the disk, the radiation stress is actually much smaller than the Maxwell and Reynolds stresses. This is consistent with our estimate that radiation stress is proportional to $1/(r\rho\kappa_R)$, while the Maxwell and Reynolds stresses roughly follow the distribution of mass. However, once near the photosphere, because of the increased photon mean free path as well the large velocity gradient, significant radiation stress shows up while Maxwell and Reynolds stresses drop significantly. This suggests that in the run [AGN0.07]{}, angular momentum is transported by the Maxwell and Reynolds stresses in the optically thick part of the disk. But once near the photosphere, angular momentum is transported by the radiation stress. Since total radiation stress dominates, more accretion happens in the optically thin region, which is consistent with the mass accretion rates shown in Figure \[Mdot\_profile\]. This is not the case for the run [AGN0.2]{}, where most accretion still happens in the optically thick region until $r \lessapprox 8r_g$. This also causes a stronger corona component at large radii for the run [AGN0.07]{} (Section \[sec:corona\]), since dissipation associated with the accretion in the optically thin region can heat up the gas easily without cooling efficiently.
The amount of radiation stress we observe in the simulations can also be compared with analytic expectations for radiation viscosity. When the optical depth per cell is large, the $r-\phi$ component of the co-moving frame radiation viscosity can be calculated analytically as [@Masaki1971; @KaufmanBlaes2016] $$\begin{aligned}
P_{r,{\rm vis}}^{r\phi}=-\frac{8}{27}\frac{E_r}{\rho\kappa_R c}D_{r\phi},
\label{eq:vis1}\end{aligned}$$ where the $r-\phi$ shear rate in spherical polar coordinates is $D_{r\phi}=r\partial \left(v_{\phi}/r\right)/\partial r$+$1/\left(r\sin\theta\right)\partial v_r/\partial \phi$. However, when cells become optically thin, this formula needs to be modified to account for the nonzero mean free path of photons. Under optically thin conditions, the viscous stress tensor associated with a shear velocity difference across some given length scale should be proportional to the optical depth across that scale times the velocity difference [@Socratesetal2004; @KaufmanBlaes2016]. In order to ensure continuous behavior between the optically thin and thick regimes, we therefore write the stress tensor as $$\begin{aligned}
P_{r,{\rm vis}}^{r\phi}=-\frac{8E_r}{27c}k(\tau)D^{\prime}_{r\phi},
\label{eq:vis2}\end{aligned}$$ Here $D^{\prime}_{r\phi}\equiv r\Delta (v_{\phi}/r)+\Delta v_r$ is just the difference of shear rate across distances with optical depth $\tau$ along radial (for $v_{\phi}$) and azimuthal (for $v_r$) directions. and $k(\tau)=\tau^{-1}$ for $\tau>1$ and $\tau$ for $\tau<1$. In practice, we first determine the optical depth per cell at each location. If that value is larger than one, we calculate the difference of shear rate between neighboring cells. If that value is smaller than 1, we extend the distance until $\tau=1$ is reached and calculate the difference of shear rate there. The radiation stress tensor calculated based on the above formula is transformed back to the lab frame and then averaged azimuthally to compare with the $r-\phi$ radiation stress returned by the simulations directly. Comparisons for two different radii are shown in Figure \[compare\_vis\_stress\]. The numerically calculated radiation stress nicely follow the prediction for radiation viscosity in both optically thick and optically thin regimes.
Vertical Structure of the Disk {#sec:vertical}
------------------------------
Time and azimuthally averaged poloidal profiles of various quantities at $10r_g$ for the two runs are shown in Figure \[vertical\]. Density drops faster with height in the run [AGN0.07]{} due to a smaller pressure scale height. Shapes of density profiles are also more centrally peaked compared with gas or radiation pressure dominated disks found by both local shearing box and global simulations [@Turner2004; @Hiroseetal2006; @Hiroseetal2009; @Jiangetal2016a; @Jiangetal2018]. Although the shell averaged radiation pressure is comparable to the shell averaged magnetic pressure in the run [AGN0.07]{} (Figure \[pressure\_profile\]), the radiation pressure is relatively flat with height and the whole disk is supported by the magnetic pressure gradient. In fact there is a small enhancement of $P_r$ near the photosphere (see also Figure \[ave\_disk\_structure\]) due to the significant dissipation there caused by radiation viscosity. In the run [AGN0.2]{}, radiation pressure partially supports the disk near the midplane and the disk becomes completely magnetic pressure supported around $10^{\circ}$ away from the midplane. This also causes the density to drop more slowly with height around the same location as shown in the right panel of Figure \[vertical\]. Gas pressure is always smaller than the other pressure components by more than a factor of 1000.
For the two magnetic field configurations used in the simulations, besides the Maxwell stress from the turbulence, there are always significant azimuthally averaged mean $\langle B_r\rangle$ and $\langle B_{\phi}\rangle$, although the product of these components never becomes the dominant stress (Figure \[compare\_alpha\]). This is different from the magnetic pressure supported disk as found by [@Gaburovetal2012], where the Maxwell stress is primarily due to $-\langle B_r\rangle \langle B_{\phi}\rangle$. They also have different vertical distributions compared with the turbulent stress as shown in the third panels of Figure \[vertical\]. The Maxwell stress generated by the turbulence (the dashed red lines) shows double peaks away from the midplane, which is consistent with previous simulations [@Blaesetal2011; @Jiangetal2016a]. The stress due to the azimuthally averaged mean magnetic field (the difference between the solid black lines and the dashed red lines) is peaked at the midplane and drops quickly with height.
Inside the photosphere near the midplane, the rotation speed of the disk is pretty close to the Keplerian value with negligible radial inflow speed (the fourth and bottom panels of Figure \[vertical\]). Inside the corona region, the rotation speed drops to only $40\%$ of the Keplerian value and significant radial velocity is present.
Despite the fact that the disk is supported by magnetic pressure, the energy dissipated in the disk is transported away vertically by both the radiation flux and Poynting flux near the midplane. Vertical profiles of the poloidal components of the energy fluxes at $10r_g$ for the run [AGN0.07]{} are shown in Figure \[vertical\_energy\]. They are scaled with the critical energy flux $F_c\equiv cG\mbh|\cos(\theta)|/\left(\kappa_{\rm es}(r-2r_g)^2\right)$, which is the radiation flux if the vertical component of gravity is completely balanced by the radiation force with the electron scattering opacity $\kappa_{\rm es}$. The radiation flux is less than $\approx 5\%$ of the critical value inside the disk, which confirms that radiation pressure plays a negligible role to support the disk. But the radiation flux is comparable to the Poynting flux within $\approx 3^{\circ}$ away from the midplane and it completely dominates the energy transport beyond that. The sign of Poynting flux also suggests that magnetic energy amplified in the disk is transported away from the midplane likely due to magnetic buoyancy [@Blaesetal2011; @Begelmanetal2015], although this simulation does not show any butterfly diagram (Figure \[STplot\]). The existence of non-zero azimuthally averaged radial magnetic field $\langle B_r\rangle$ in localized region of the disk will also keep generating $B_{\phi}$ due to the differential rotation of the disk, which is balanced by the buoyant escape of the field. Since the shell averaged $B_r$ is zero, $B_{\phi}$ generated at different locations in the disk will reconnect and cause dissipation.
Properties of the Coronae {#sec:corona}
-------------------------
Local shearing box radiation MHD simulations [@Jiangetal2014] have found that coronae with high gas temperature are only formed above the disk when the surface density is low enough so that a significant fraction of the total dissipation happens in optically thin regions. For the surface density at $30r_g$ adopted by [@Jiangetal2014] based on the standard thin disk model, only $3.4\%$ of the dissipation happens in the corona region. This fraction is increased to more than $50\%$ inside $15r_g$ for the run [AGN0.07]{} with significantly reduced surface density in the magnetic pressure supported disk. This results in a high gas temperature of $\approx 10^8$ K covering all the radial range $\lesssim 15r_g$ with height $20r_g$ away from the disk midplane as shown in the left panel of Figure \[Corona\]. When the accretion rate is increased to $26\%\Medd$ in the run [AGN0.2]{}, the disk becomes thicker and the fraction of accretion as well as the associated dissipation in the optically thin region are also reduced. Coronae with high gas temperature ($\gtrapprox 10^8$ K) only show up inside $\lesssim 10r_g$ when the surface density drops to a value such that $\kappa_R\Sigma/2\lesssim 100$. Inside this region, the corona is also more vertically extended covering more than $40r_g$ away from the disk midplane.
Coronae produced by these simulations are pretty consistent with many observational inferred properties of AGN corona. The simulations only produce high temperature corona (gas temperature $>10^8$ K) inside $\approx 10r_g$ and the temperature decreases with increasing distance from the black hole. This is because surface density of the disk increases with radius and the fraction of dissipation in the optically thin region drops, which is necessary to maintain the high temperature corona. This agrees with the constrains from both reflection and microlensing modeling of AGN coronae [@Chartasetal2009; @Daietal2010; @ReisMiller2013; @JimenezVicenteetal2015], which suggest that they are compact with roughly the same size as we find.
Surface density increases for the same region of the disk when the accretion rate increases from [AGN0.07]{} to [AGN0.2]{}. This is in contrast to the standard thin disk model, for which the surface density decreases linearly with increasing mass accretion rate in the radiation pressure dominated regime. When the surface density increases, the fraction of dissipation in the optically thin regime is reduced and the radial regime where the corona is formed is also smaller. This will result in a smaller ratio between X-ray luminosity from the corona and luminosity of the thermal emission from the disk, which is in excellent agreement with the observational fact that the spectral index of AGNs gets harder as luminosity decreases [@Steffenetal2006; @Justetal2007; @Ruanetal2019].
Discussion {#sec:discussion}
==========
It is interesting to compare the structures of the disk produced by the simulations with proposed models of magnetic pressure dominated disks in the literature. [@Parievetal2003] modified the standard $\alpha$ disk model by including a magnetic pressure component in the disk, which was assumed to be larger than the thermal pressure at all radii. Magnetic pressure is assumed to be a constant at each radius and decreases with increasing radius following a power law, the slope of which is a free parameter. The strength of the magnetic field is also a free parameter in this model. The disk structures in this model are very similar to the original $\alpha$ disk model, which predicts that the surface density is proportional to $1/\dot{M}$. This is clearly in contrast to what we find from the simulations. [@BegelmanPringle2007] proposed that the toroidal magnetic field should saturate to the level so that the associated Alfvén velocity is the geometric mean of gas isothermal sound speed $c_s$ and Keplerian speed $V_k$ . Under this assumption, the disk surface density is found to be proportional to $ \dot{M}^{7/9}$, which has the same trend as what we find, although their dependence on $\dot{M}$ is weaker. The Alfvén velocity in our simulations is found to be larger than $\sqrt{c_sV_k}$ by a factor of $2-3$ in the midplane and much smaller than that near and above the photosphere. Our simulation results can be used to constrain the assumptions and improve these analytical models.
The relatively uniform spatial distribution of $E_r$ in the run [AGN0.07]{} is caused by the significant dissipation near the photosphere. In fact, $E_r$ near the photosphere is slightly larger than $E_r$ at the midplane as shown in the left panel of Figure \[vertical\]. This is also consistent with the rapid increase of radiation flux at these locations as shown in Figure \[vertical\_energy\]. For comparison, when more dissipation happens in the optically thick part of the disk as in the run [AGN0.2]{}, radiation energy density decreases with increasing height monotonically. This is also the reason why radiation viscosity can be significant in the run [AGN0.07]{}. Since radiation viscosity is proportional to $E_r/\tau$ in the optically thick regime, if $E_r$ at the photosphere is smaller than $E_r$ in the midplane by a factor of $1/\tau$ as in the standard thin disk model, radiation viscosity will not be significant near the photosphere even mean free path is larger there.
These simulations adopt the gray approximation for radiation transport and Compton scattering is included based on the effective radiation temperature to roughly estimate the energy exchange between photons and gas. Therefore, we cannot produce the expected spectra from these simulations directly. This approximation will also cause underestimate of gas temperature in the corona region. Monte Carlo technique has been developed to post process the simulation data and generate spectra from the disk. Preliminary results show that the coronae in these simulations are indeed able to produce significant X-ray emission with X-ray luminosity consistent with the total dissipation we have in these region. Detailed spectrum properties will be presented in a separate publication. Since corona only exists inside $\approx 10r_g$, for which general relativity may play an important role, simulations with fully self-consistent radiation transport in general relativity will be carried out in the near future.
The total luminosity from the inner $10r_g$ of the disk is only $1\%\Ledd$ with $\Medd=7\%\Medd$ for the run [AGN0.07]{}, which means the radiative efficiency is only $1.4\%$. The radiative efficiency is slightly increased to $\approx 3.5\%$ for the run [AGN0.07]{}. The efficiency from these disks is significantly smaller than the prediction of the standard thin disks for the same radial range. One reason for the lower efficiency is the significant dissipation in the optically thin region, which cannot convert to thermal radiation efficiently. Another reason for the lower radiative efficiency in the run [AGN0.07]{} is the stress $-\< B_r\>\<B_{\phi}\>$ near the disk midplane (Figure \[vertical\]). This component of stress can cause mass accretion without dissipation.
Both of the simulations in this paper established long-lived thermal equilibria between turbulent dissipation of gravitational binding energy in the flow and radiative cooling for more than $10$ thermal time scales. This is despite the fact that radiation dominates the thermal pressure, a situation which generally leads to thermal instability in $\alpha$-disk models [@ShakuraSunyaev1976], at least when electron scattering dominates the Rosseland mean opacity. Iron opacity can be significant in disks around supermassive black holes, and has been shown to stabilize the thermal instability in shearing box simulations of MRI turbulence [@Jiangetal2016a]. However, while we have included this source of opacity in the simulations here, it actually does not play a significant role in the inner regions near the black hole. Instead, it is likely that the fact that these simulations are supported by magnetic, not thermal pressure, leads to their stability [@BegelmanPringle2007]. Indeed, @Sadowski2016 found that global simulations of geometrically thin disks are stabilized if the magnetic pressure exceeds half the total pressure, a criterion which we more than satisfy in our simulations here.
Light curves of a variety of accretion-powered sources in astrophysics, including active galactic nuclei, show a linear relationship between rms variability amplitude and flux (see e.g. @Uttleyetal2005 and references therein). Thereis a hint of this in the [AGN0.2]{} light curve shown in Figure \[mdot\_hist\], where the variability amplitude is clearly larger at higher luminosities than at lower luminosities. Unfortunately, the simulation was not run long enough and the light curve is too short to produce a clean rms-flux relation. On the other hand, there is no indication of any relationship between variability amplitude and flux in [AGN0.07]{}. @HoggReynolds2016 have suggested that the rms-flux relation is related to the presence of the characteristic butterfly diagram of quasi-periodic azimuthal field reversals. It is therefore noteworthy that these are present only in [AGN0.2]{}, and not in [AGN0.07]{} because of its much stonger magnetic pressure support. The presence of an rms-flux relation may be a way of observationally ruling out magnetic fields which are so strong that the butterfly dynamo is suppressed, but longer simulations will need to be run to confirm this.
Note that the shape and magnetic field in the initial tori we use for the two simulations [AGN0.07]{} and [AGN0.2]{} are very similar to the ones adopted by [@Jiangetal2017b]. We only change the density of the torus to achieve different mass accretion rates, while the ratio between the initial magnetic pressure and radiation pressure is kept fixed. However, the accretion disks with super-Eddington accretion rates shown in [@Jiangetal2017b] do not evolve to a magnetic pressure dominated regime. When the surface density as well as the optical depth are increased, radiation pressure becomes more important in supporting the disk. Since the amount of magnetic field that in the disk is determined by a balance between buoyant escape and amplification by the MRI, this suggests that stronger radiation support in the disk may either increase the magnetic buoyancy or reduce the saturation amplitude of MRI [@Jiangetal2013b]. We intend to investigate this further in future work.
Summary
=======
We have successfully simulated two accretion disks around a $5\times 10^8\mbh$ black hole with mass accretion rates reaching $0.07\Medd$ and $0.2\Medd$ up to $\sim 15r_g$. The disks do not show any sign of thermal instability over many thermal timescales. The structure of the disk differs markedly from the standard thin disk model [@ShakuraSunyaev1973] for the same mass accretion rate in the following ways.
- The disk is supported vertically by magnetic pressure rather than thermal pressure for these accretion rates.
- The surface density and total optical depth are reduced by a factor of $\approx 10$. The disk scale height increases with radius significantly instead of being constant.
- A significant fraction of dissipation as well as the associated mass accretion (more than $50\%$ for $\dot{M}=0.07\Medd$) happen away from the disk midplane in the optically thin region, resulting in a high temperature corona inside $10r_g$. The fraction of dissipation in the corona region decreases with increasing mass accretion rate.
- The transport of angular momentum is due to Maxwell and Reynolds stress inside the photosphere. But near the photosphere, radiation viscosity is the dominant mechanism for angular momentum transport, and this can dominate the vertically averaged stress.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Julian Krolik for helpful comments on an earlier draft. This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1125915, 17-48958 and AST-1715277. S.W.D. is supported by a Sloan Foundation Research Fellowship. An award of computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program. This research used resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357. Resources supporting this work were also provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center.
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