RR-field tadpole cancellation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
  {#Idea}

In the presence of [[D-branes]], plain [[type II string theory]] in fact has a [[quantum anomaly]] reflected on the [[worldsheet]] by [[tadpole]] [[Feynman diagrams]] in the [[string perturbation series]] for [[RR-fields]]

\begin{center}
\begin{imagefromfile}
  "file_name": "TadpoleCancellationFactorization.jpg",
  "width": 600
\end{imagefromfile}
\end{center}

> graphics grabbed from [Blumenhagen-Lüst-Theisen 13](#BlumenhagenLustTheisen13)

and reflected in [[target spacetime]] by non-trivial total [[RR-field]] [[flux]] on [[compact topological space|compact spaces]]

\begin{center}
\begin{imagefromfile}
  "file_name": "RRTadpoleCancellation.jpg",
  "width": 600
\end{imagefromfile}
\end{center}

> graphics grabbed from [Ibanez-Uranga 12](#IbanezUranga12)

This anomaly cancels if the [[D-branes]] are accompanied by a suitable collection of [[O-planes]], hence if one considers [[orientifold]] backgrounds ([Sagnotti 88, pp. 5](#Sagnotti88), [Gimon-Polchinski 96, section 3](#GimonPolchinski96)). (For space-filling [[O-planes]] this means to consider [[type I string theory]] instead.)

Accordingly, tadpole cancellation via [[orientifolds|orientifolding]] is a key consistency condition in the construction of [[intersecting D-brane models]] for [[string phenomenology]].

Traditionally RR-tadpole cancellation is discussed in [[ordinary cohomology]], the common arguments notwithstanding that [[D-brane charge]] should be in [[K-theory]].

Discussion of tadpole cancellation with [[D-brane charge]] regarded in [[K-theory]] was initated in [Uranga 00, Section 5](#Uranga00), see also [Garcia-Uranga 05](#GarciaUranga05), [Marchesano 03, Section 4](#Marchesano03), [Marchesano-Shiu 04](#MarchesanoShiu04), [CKMNW 05, Section 2.2](#CKMNW05), [Maiden-Shiu-Stefanski 06, Section 5](#MaidenShiuStefanski06), [DFM 09, p. 6-7](#DFM09).

But the situation remains somewhat inconclusive (see also [Moore 14, p. 21-22](#Moore14)).

\linebreak

## In plane and toroidal orientifolds
 {#ForFractionalDBranes}

More details are understood in the special case of plane orbifolds $V^{cpt} \!\sslash\! G$ and [[toroidal orbifold|toroidal orientifolds]] $\mathbb{T}^V \!\sslash\! G$ where [[fractional D-branes]] may be stuck at [[orbifold]]/[[orientifold]] singularities, whose [[D-brane charge]] is supposed to be in the [[equivariant K-theory]] of the point, hence the [[representation ring]] of the given [[isotropy group]].

### In terms of equivariant K-theory / the representation ring
 {#InTermsOfEquivariantKTheory}

In this case tadpole cancellation conditions are given by [[representation theory|representation theoretic]] [[equations]], constraining the [[character of a linear representation|characters]] of the [[linear representations]] corresponding to the [[fractional D-branes]].

Let $G$ be a [[finite group]]. Let

$$
  [1]
    \subset
  [H_1]
    \subset
  [H_2]
    \subset
    \cdots
    \subset
  [G]
$$

be a [[linear extension of a partial order|linear extension]] of its [[partially ordered set|partially ordered]] [[lattice of subgroups|lattice of]] [[conjugacy classes]] of [[subgroups]], with [[subset|sub-]] [[linear order]] of [[cyclic group|cyclic]] [[subgroups]]

$$
  [1]
    \subset
  \left[
   \left\langle
     g_1
   \right\rangle
  \right]
    \subset
  \left[
   \left\langle
     g_2
   \right\rangle
  \right]
    \subset
    \cdots
    \subset
  \left[
   \left\langle
     g_{\vert ConjCl(G)\vert}
   \right\rangle
  \right]
  \,.
$$

This way every [[virtual representation]] $[V] \in RU(G) = KU_G(\ast)$ (the [[D-brane charge]] of a [[bound state]] of [[fractional D-branes]]/[[anti-branes]]) has a [[character of a linear representation|character]] which is a list of [[complex numbers]] of the form

| $[H] = $ | $\left[\langle e\rangle\right]$ | $\left[\langle g_1\rangle\right]$ |  $\left[\langle g_2\rangle\right]$ | $\cdots$ |  $\left[\langle g_{\vert ConjCl(G)\vert}\rangle\right]$ |
|--|--|--|--|--|--|
| $\chi_V =$ | $dim(V)$ |  $tr_V\left( g_1\right)$ |  $tr_V\left(g_2\right)$ | $\cdots$ |  $tr_V\left(g_{\vert ConjCl(G)\vert}\right)$ |
| ${{\text{fractional} \atop \text{D-brane/anti-brane}} \atop \text{bound state}}$ |  ${ {\text{mass} =} \atop {{\text{net number} \atop \text{of branes}}}}$ | ${\text{RR-charge in} \atop {g_1\text{-twisted sector}}}$  | ${\text{RR-charge in} \atop {g_2\text{-twisted sector}}}$  | $\cdots$  | $\cdots$  |

Here $dim(V) \in \mathbb{Z}$ is the [[mass]], hence the net number of [[fractional D-branes]]/[[anti-branes]] in the [[bound state]], while $tr_V\left(g_k\right)$ is (up to a global [[rational number]]-factor $1/{\vert G \vert}$) supposed to be its  [[charge]] as seen by the [[RR-fields]] in the $g_k$-[[twisted sector]].

In fact, since we are dealing with fractional D-branes, both the charge and mass in the above table are in factional units $1/{\vert G\vert}$ of the [[order of a group|order]] of the [[isotropy group]] $G$ (by [this formula](fractional+D-brane#eq:RRChargeOfFractionalDBraneInGTwistedSector)), so that normalized [[mass]] and [[charge]] is

\[
  \label{ChargeMass}
  M
  \;=\;
  \tfrac{1}{{\vert G\vert}}
  dim(V)
  \,,
  \phantom{AAA}
  Q_V(g)
  \;=\;
  \tfrac{1}{\vert G\vert}
  \chi_V(g)
  \coloneqq
    \tfrac{1}{\vert G\vert}
   tr_V\left( g\right)
  \,.
\]

#### Local/twisted tadpole cancellation
 {#TwistedTadpoleCancellation}

The **twisted (local) tadpole cancellation condition** for [[fractional D-branes]] at orbifold singularities is that the RR-charges in all non-trivially twisted sectors vanish:

\[
  \label{VanishingOfCharacterValuesOnNontrivialSubgroups}
  Q_V(g)
  =
  0
  \phantom{AA}\text{hence equivalently} \phantom{AA}
  \chi_{V}\left(g\right) \;=\; 0
  \,,
  \phantom{AAA}
  g \neq e
\]

+-- {: .num_example #UnitRegularRepresentation}
###### Example
**([[regular representation]] solves tadpole cancellation for [[fractional D-branes]])**

For every [[finite group]] $G$, the homogeous tadpole cancellation  condition (eq:VanishingOfCharacterValuesOnNontrivialSubgroups) is satisfied by all multiples $n \cdot k[G/1]$ of the [[regular representation]] $k[G/1]$ (since no non-trivial element $g \in G$ has [[fixed points]] when acting on $G$, and using [this Prop.](table+of+marks#MarkHomomorphismIsCharactersOfPermutationRepresentation)).  Hence the [[mass]] and [[charge]] (eq:ChargeMass) of the [[fractional D-brane]] corresponding to the [[regular representation]] is

$$
  M_{{}_{k[G/1]}}
  \;=\;
  1
  \,,
  \phantom{AA}
  Q_{{}_{k[G/1]}}(g)
  \;=\;
  0
  \,.
$$

These multiples of the [[regular representation]] are regarded as trivial solutions to (eq:VanishingOfCharacterValuesOnNontrivialSubgroups).

=--

+-- {: .num_prop #RegularRepSpansSolutionToHomogeneousTadpoleCancellation}
###### Proposition

In fact, the multiples of the [[regular representation]] (Example \ref{UnitRegularRepresentation}) are the _only_ solutions to the local/twisted tadpole cancellation condition (eq:VanishingOfCharacterValuesOnNontrivialSubgroups) for [[fractional D-branes]].

=--

+-- {: .proof}
###### Proof

Consider the truncated [[character morphism]]

$$
  Q \cdot {\vert G \vert}
  \;\colon\;
  Rep_k(G)
  \overset{\chi}{\longrightarrow}
  k^{\left\vert ConjCl(G)  \right\vert}
  \overset{ \text{forget dimension/mass}  }{\longrightarrow}
  k^{\left\vert ConjCl(G)\right\vert -1 }
  \,.
$$

We have to show that the [[kernel]] of this map is the [[free abelian group]] generated by the [[regular representation]]:

$$
  ker\big(
    Q \cdot {\vert G \vert}
  \big)
  \;\simeq\;
  \mathbb{Z} \cdot k[G/1]
  \,.
$$

Now over a [[ground field]] $k$ of [[characteristic zero]] (such as the [[real numbers]] or [[complex numbers]], in the case at hand) we have (from [this Example](character+of+a+linear+representation#NormalizedSumOfCharacters)) that

1. for $\rho \neq \mathbf{1}$ a non-[[trivial representation|trivial]] [[irreducible representation]] we have

   $$
     \underset{g \in G \setminus \{e\}}{\sum} Q_{\rho}(g) \cdot {\vert G \vert}
     \;\coloneqq\;
     \underset{g \in G \setminus \{e\}}{\sum} \chi_\rho(g)
     \;=\;
     - dim(\rho)
   $$

1. for $\rho = \mathbf{1}$ the [[trivial representation|trivial]] [[irreducible representation]] we have

   $$
     \underset{g \in G \setminus \{e\}}{\sum} Q_{\rho}(g) \cdot {\vert G \vert}
     \;\coloneqq\;
     \underset{g \in G \setminus \{e\}}{\sum} \chi_\rho(g)
     \;=\;
     {\left\vert G\right \vert}  - 1
     \;=\;
     - dim(\mathbf{1}) \;mod\; {\vert G\vert}
   $$

Since every $V \in R_{k}(G)$ is a $\mathbb{Z}$-[[linear combination]] of these [[irreps]], it follows generally that the _fractional part_ of the [[mass]] of a [[fractional D-brane]] is recovered from its [[charges]]:

$$
   dim(V)
   \;mod\;
   {\vert G \vert}
   \;=\;
   -
   \underset{g \in G \setminus \{e\}}{\sum} Q_{V}(g) \cdot {\vert G \vert}
   \;\coloneqq\;
   -
   \underset{g \in G \setminus \{e\}}{\sum} \chi_V(g)
   \,.
$$

But this means that all $V$ in the [[kernel]] of $Q \cdot {\vert G \vert}$ must have

$$
  dim(V) \;=\; 0 \;mod\; {\vert G \vert}
  \,.
$$

This is indeed the case for the multiples $V = n\cdot k[G/1]$ of the [[regular representation]] (Example \ref{UnitRegularRepresentation}). Conversely, the injectivity of the full [[character morphism]] $\chi$ ([this Prop.](character+of+a+linear+representation#InCharZeroCharacterMorphismIsInjective)) says that every $V$ with $dim(V) = n \cdot {\vert G\vert }$ and $Q_V(g) = 0$ must be the $n$th multiple of the regular representation.

=--

#### Global/untwisted tadpole cancellation
 {#UntwistedTadpoleCancellation}

On the other hand, at an [[orientifold]] singularity, the [[O-plane]] itself carries such charge -- [[O-plane charge]] (see [there](orientifold+plane#OPlaneCharge)):

| $[H] = $ | $\left[\langle e\rangle\right]$ | $\left[\langle g_1\rangle\right]$ |  $\left[\langle g_2\rangle\right]$ | $\cdots$ |  $\left[\langle g_{\vert ConjCl(G)\vert}\rangle\right]$ |
|--|--|--|--|--|--|
| $\chi_O =$ | $dim(O)$ |  $tr_O\left( g_1\right)$ |  $tr_O\left(g_2\right)$ | $\cdots$ |  $tr_O\left(g_{\vert ConjCl(G)\vert}\right)$ |
| $\text{O-plane}$ |   | ${\text{O-plane charge in} \atop {g_1\text{-twisted sector}}}$  | ${\text{O-plane-charge in} \atop {g_2\text{-twisted sector}}}$  | $\cdots$  | $\cdots$  |

(These are $O^-$-plane charges. There may also be $O^+$-plane charges. Alternatively, these are $O^-$-branes with a fractional D-brane stuck on them.)

Now the **untwisted (global) tadpole cancellation condition** is that (all representations are real and) this [[O-plane charge]] is cancelled against the [[D-brane charge]]:

\[
  \label{InhomogeneousTadpoleCancellation}
  \chi_{V}\left(g_{k \geq 1}\right)  \;=\; 0
  \phantom{AA}
  \text{and}
  \phantom{AA}
  dim(V) = dim(O)
  \,.
\]

By Prop. \ref{RegularRepSpansSolutionToHomogeneousTadpoleCancellation} the only possible solution of this is the $n$th multiple of the [[regular representation]], if $dim(O)$ is $n$ times the dimension of the [[regular representation]]:

\[
  \label{VIsMultipleOfRegularRepresentation}
  V = N \cdot k[G/1]
  \,.
\]

In basic examples the [[O-plane]]-charge

$$
  O = 2^{p-4} n \cdot  \mathbf{1}
$$

is for $n_O$ coincident [[O-planes]] is the corresponding multiple by the [[O-plane charge]] $\mu_{Op} = -2^{8-4}$ ([here](orientifold+plane#eq:OpPlaneCharge)) of the [[trivial representation|trivial]] [[irrep]], whence a solution to the tadpole cancellation exists if $ \frac{2^{p-4}}{\vert G\vert } \in \mathbb{N} \subset \mathbb{Q}$ and is then given by

$$
  V \;=\; \frac{2^{p-4}}{\vert G\vert } \cdot k[G/1]
  \,.
$$

Sometimes the condition (eq:VIsMultipleOfRegularRepresentation) is found with an offset by a trivial representation

\[
  \label{VIsMultipleOfRegularRepresentation}
  V = N \cdot k[G/1] + \mathbf{p}_{triv}
  \,.
\]

This corresponds to single fractional D-branes sitting on top of the O-planes, turning $O^-$-planes into $O^+$-planes.

### Examples for toroidal orientifolds
 {#ExamplesForToroidalOrientifolds}

[[!include RR-field tadpole cancellation on toroidal orientifolds -- table]]

\linebreak

> graphics grabbed from [Sati-Schreiber 19](#SatiSchreiber19)

See also at _[[equivariant Hopf degree theorem]]_.

> graphics grabbed from [Sati-Schreiber 19](#SatiSchreiber19)

### Examples of non-compact singularities
 {#ExamplesForFractionalDBranes}

We discuss more explicitly the solutions to the local/twisted tadpole cancellation condition (eq:VanishingOfCharacterValuesOnNontrivialSubgroups) for [[fractional D-branes]] at [[orbifold]] [[singularities]] for [[isotropy group]] one of  the [[nonabelian group|non-abelian]] [[finite subgroups of SU(2)]],

$$
  G_{DE}
  \;\subset\;
  SU(2)
$$

hence those in the [[ADE-classification|D- and E-series]], hence the [[binary dihedral groups]] $2 D_{2n}$ and the three exceptional cases: [[2T]], [[2O]] and [[2I]].

For these groups, by [BSS 18, Theorem 4.1](#BurtonSatiSchreiber18) the [[virtual representation|virtual]] [[permutation representations]] span precisely the sub charge lattice of integral (non-irrational) characters/RR-charges in the [[orientifold]] charge lattice of the corresponding [[ADE-singularity]], namely of the [[equivariant KO-theory]]=[[representation ring|real representation ring]]

$$
  KO^0_{G_{DE}}(\ast)
  \;=\;
  RO\left( G_{DE} \right)
  \,.
$$

Since the tadpole cancellation condition (eq:VanishingOfCharacterValuesOnNontrivialSubgroups) in particular requires the characters/charges to be integral (specifically: zero) the general solution to the tadpole cancellation condition is indeed in this sub-lattice, and so that is where we may and do solve it, below.

In accord with the general Prop. \ref{RegularRepSpansSolutionToHomogeneousTadpoleCancellation}
we find that in each case there is precisely a 1-dimensional (i.e. $\simeq \mathbb{Z}$) sublattice of the charge lattice (the [[representation ring]]) which solves the twisted tadpole cancellation condition (eq:VanishingOfCharacterValuesOnNontrivialSubgroups), hence a sublattice given by the [[integer]]-multiples $N \cdot V_0$ of one single [[fractional D-brane]] [[bound state]] $V_0 \in KO^0_G(\ast)$. There are then necessarily two of these generators $\pm V_0$. We check below that in all cases the normalized [[mass]] of these is $\pm$ unity, as it must be for the [[regular representation]], by Prop. \ref{RegularRepSpansSolutionToHomogeneousTadpoleCancellation}.

#### At a $\mathbb{Z}_2$-orientifold singularity

For $G = \mathbb{Z}_2$ the [[cyclic group]] of [[order of a group|order]] ${\vert \mathbb{Z}_2\vert} = 2$, the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are (e.g. [here](https://people.maths.bris.ac.uk/~matyd/GroupNames/1/C2.html))

| $[H] = $ | $\left[\langle e\rangle\right]$ | $\left[\langle g_1\rangle\right]$ |
|--|--|--|
| $\chi_{V_1} =$ | $\phantom{-}1$ | $\phantom{-}1$ |
| $\chi_{V_2} =$ | $\phantom{-}1$ | $-1$ |

One sees immediately that the general solution to the local/twisted tadpole cancellation condition (eq:VanishingOfCharacterValuesOnNontrivialSubgroups) for $G = \mathbb{Z}_2$ is

$$
  V
  \;=\;
    n \cdot
  \Big(
    1 \cdot V_1 + 1 \cdot V_2
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
  \,.
$$

whose minimal [[positive number|positive]]  [[mass]] (net brane number) is

$$
  \begin{aligned}
    M_{\mathbb{Z}_2}
    & =
    dim(V)  / {\vert \mathbb{Z}_2 \vert }
    \\
      & =
    \chi_V([\langle e\rangle]) / {\vert \mathbb{Z}_2 \vert}
    \\
      & =
      \big(
      1 \cdot 1
        +
      1 \cdot 1
     \big) / {2}
      &
      \\
      &
      =
      2 / 2
      \\
      & = 1
  \end{aligned}
$$

#### At a $\mathbb{Z}_4$-orientifold singularity

For $G = \mathbb{Z}_4$ the [[cyclic group of order 4]], the [[character of a linear representation|characters]]/[[D-brane charges]] of the complex [[irreducible representations]]/[[fractional D-branes]] are (e.g. [here](https://people.maths.bris.ac.uk/~matyd/GroupNames/1/C4.html))

| $[H] = $ | $\left[\langle e\rangle\right]$ | $\left[\langle g_1\rangle\right]$ | $\left[\langle g_2\rangle\right]$ | $\left[\langle g_3\rangle\right]$ |
|--|--|--|--|--|
| $\chi_{V_1} =$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ |
| $\chi_{V_2} =$ | $\phantom{-}1$ | $\phantom{-}1$ | $-1$ | $-1$ |
| $\chi_{V_3} =$ | $\phantom{-}1$ | $-1$ | $-i$ | $\phantom{-}i$ |
| $\chi_{V_4} =$ | $\phantom{-}1$ | $-1$ | $\phantom{-}i$ | $-i$ |

One sees immediately that the general solution to the local/twisted tadpole cancellation condition (eq:VanishingOfCharacterValuesOnNontrivialSubgroups) for $G = \mathbb{Z}_3$ is

$$
  V
  \;=\;
    n \cdot
  \Big(
    1 \cdot V_1
    +
    1 \cdot V_2
    +
    1 \cdot V_3
    +
    1 \cdot V_4
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
  \,.
$$

whose minimal [[positive number|positive]]  [[mass]] (net brane number) is

$$
  \begin{aligned}
    M_{\mathbb{Z}_4}
    & =
    dim(V)  / {\vert \mathbb{Z}_4 \vert }
    \\
      & =
    \chi_V([\langle e\rangle]) / {\vert \mathbb{Z}_4 \vert}
    \\
      & =
      \big(
      1 \cdot 1
        +
      1 \cdot 1
        +
      1 \cdot 1
        +
      1 \cdot 1
     \big) / {4}
      &
      \\
      &
      =
      4 / 4
      \\
      & = 1
  \end{aligned}
$$

#### At a $2 D_4$-orientifold singularity
 {#At2d4Singularity}

For $G = 2 D_4 = Q_8$ the [[binary dihedral group]] of [[order of a group|order]] ${\vert 2 D_4\vert}$ (equivalently: the [[quaternion group]]), the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are ([BSS 18, 4.1](#BurtonSatiSchreiber18)):

| $[H] = $ | $\left[\langle e\rangle\right]$ | $\left[\langle g_1\rangle\right]$ |  $\left[\langle g_2\rangle\right]$ | $\left[\langle g_3\rangle\right]$ | $\left[\langle g_4\rangle\right]$ |
|--|--|--|--|--|--|
| $\chi_{V_1} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $\phantom{-}1$ | $\phantom{-}1$ |  $\phantom{-}1$ |
| $\chi_{V_2} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $-1$ | $\phantom{-}1$ |  $-1$ |
| $\chi_{V_3} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $-1$ | $-1$ |  $\phantom{-}1$ |
| $\chi_{V_4} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $\phantom{-}1$ | $-1$ |  $-1$ |
| $\chi_{V_5} =$ | $\phantom{-}4$ |  $-4$ |  $\phantom{-}0$ | $\phantom{-}0$ |  $\phantom{-}0$ |

One sees ([here](https://www.wolframalpha.com/input/?i=Nullspace+Transpose%5B%7B%7B1,1,1,1%7D,%7B1,%E2%88%921,1,-1%7D,%7B1,-1,-1,1%7D,%7B1,1,-1,-1%7D,%7B-4,0,0,0%7D%7D%5D)) that the general solution to the local/twisted tadpole cancellation condition (eq:VanishingOfCharacterValuesOnNontrivialSubgroups) for $G =2 D_4$ is

$$
  V
  \;=\;
  n
  \Big(
    1 \cdot V_1
      +
    1 \cdot V_2
      +
    1 \cdot V_3
      +
    1 \cdot V_4
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
$$

whose minimal [[positive number|positive]] [[mass]] (net brane number) is

$$
  \begin{aligned}
    M_{2 D_4}
    & =
    dim(V)/ {\vert 2 D_4\vert}
    \\
    & =
    \chi_V\left( [\langle e\rangle]\right) / {\vert 2 D_4\vert}
    & =
    \big( 1 + 1 + 1 + 1 + 4 \big) / 8
    \\
    & = 8 / 8
    \\
    & = 1
  \end{aligned}
$$

#### At a $2 D_6$-orientifold singularity
  {#At2D6Singularity}

For $G = 2 D_6$ the [[binary dihedral group]] of [[order of a group|order]] ${\vert 2 D_6\vert} = 12$, the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are ([BSS 18, 4.2](#BurtonSatiSchreiber18)):

| $[H] = $ | $\left[\langle e\rangle\right]$ | $\left[\langle g_1\rangle\right]$ |  $\left[\langle g_2\rangle\right]$ | $\left[\langle g_3\rangle\right]$ | $\left[\langle g_4\rangle\right]$ | $\left[\langle g_5\rangle\right]$ |
|--|--|--|--|--|--|--|
| $\chi_{V_1} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $\phantom{-}1$ | $\phantom{-}1$ |  $\phantom{-}1$ | $\phantom{-}1$ |
| $\chi_{V_2} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $\phantom{-}1$ | $-1$ |  $-1$ | $\phantom{-}1$ |
| $\chi_{V_3} =$ | $\phantom{-}2$ |  $\phantom{-}2$ |  $-1$ | $\phantom{-}0$ |  $\phantom{-}0$ | $-1$ |
| $\chi_{V_4} =$ | $\phantom{-}2$ |  $-2$ |  $\phantom{-}2$ | $\phantom{-}0$ |  $\phantom{-}0$ | $-2$ |
| $\chi_{V_5} =$ | $\phantom{-}4$ |  $-4$ |  $-2$ | $\phantom{-}0$ |  $\phantom{-}0$ | $\phantom{-}2$ |

One finds ([here](https://www.wolframalpha.com/input/?i=Nullspace+Transpose%5B%7B%7B1,1,1,1,1%7D,%7B1,1,%E2%88%921,%E2%88%921,1%7D,%7B2,%E2%88%921,0,0,%E2%88%921%7D,%7B%E2%88%922,2,0,0,%E2%88%922%7D,%7B%E2%88%924,%E2%88%922,0,0,2%7D%7D%5D)) that the general solution to the local/twisted tadpole cancellation condition (eq:VanishingOfCharacterValuesOnNontrivialSubgroups) for $G =2 D_6$ is

$$
  V
  \;=\;
  n
  \Big(
    1 \cdot V_1
      +
    1 \cdot V_2
      +
    2 \cdot V_3
      +
    1 \cdot V_4
      +
    1 \cdot V_5
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
$$

whose minimal [[positive number|positive]]  [[mass]] (net brane number) is

$$
  \begin{aligned}
    M_{2 D_6}
    & =
    dim(V)  / {\vert 2 D_6 \vert }
    \\
      & =
    \chi_V([\langle e\rangle]) / {\vert 2 D_6\vert}
    \\
      & =
      \big(
      1 \cdot 1
        +
      1 \cdot 1
        +
      2 \cdot 2
        +
      1 \cdot 2
        +
      1 \cdot 4
     \big) / {12}
      &
      \\
      &
      =
      12 / 12
      \\
      & = 1
  \end{aligned}
$$

#### At a $2 D_8$-orientifold singularity
  {#At2D8Singularity}

For $G = 2 D_8$ the [[binary dihedral group]] of [[order of a group|order]] ${\vert 2 D_8\vert} = 16$, the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are ([BSS 18, 4.3](#BurtonSatiSchreiber18)):

| $[H] = $ | $\left[\langle e\rangle\right]$ | $\left[\langle g_1\rangle\right]$ |  $\left[\langle g_2\rangle\right]$ | $\left[\langle g_3\rangle\right]$ | $\left[\langle g_4\rangle\right]$ | $\left[\langle g_5\rangle\right]$ |  $\left[\langle g_6\rangle\right]$ |
|--|--|--|--|--|--|--|--|
| $\chi_{V_1} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $\phantom{-}1$ | $\phantom{-}1$ |  $\phantom{-}1$ | $\phantom{-}1$ |$\phantom{-}1$ |
| $\chi_{V_2} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $\phantom{-}1$ | $-1$ |  $\phantom{-}1$ | $-1$ | $-1$ |
| $\chi_{V_3} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $\phantom{-}1$ | $-1$ |  $-1$ | $\phantom{-}1$ | $\phantom{-}1$ |
| $\chi_{V_4} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $\phantom{-}1$ | $\phantom{-}1$ |  $-1$ | $-1$ | $-1$ |
| $\chi_{V_5} =$ | $\phantom{-}2$ |  $\phantom{-}2$ |  $-2$ | $\phantom{-}0$ |  $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}0$ |
| $\chi_{V_6} =$ | $\phantom{-}8$ |  $-8$ |  $\phantom{-}0$ | $\phantom{-}0$ |  $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}0$ |

One finds ([here](https://www.wolframalpha.com/input/?i=Nullspace+Transpose%5B%7B%7B1,1,1,1,1,1%7D,%7B1,1,-1,1,-1,-1%7D,%7B1,1,-1,-1,1,1%7D,%7B1,1,1,-1,-1,-1%7D,%7B2,-2,0,0,0,0%7D,%7B-8,0,0,0,0,0%7D%7D%5D)), that the general solution to the local/twisted tadpole cancellation condition (eq:VanishingOfCharacterValuesOnNontrivialSubgroups) for $G =2 D_8$ is

$$
  V
  \;=\;
  n
  \Big(
    1 \cdot V_1
      +
    1 \cdot V_2
      +
    1 \cdot V_3
      +
    1 \cdot V_4
      +
    2 \cdot V_5
      +
    1 \cdot V_6
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
$$

whose minimal [[positive number|positive]]  [[mass]] (net brane number) is

$$
  \begin{aligned}
    M_{2 D_8}
     & =
    dim(V)  / {\vert 2 D_8\vert}
    \\
      & =
    \chi_V([\langle e\rangle]) / { \vert 2 D_8\vert }
    \\
      & =
      \big(
      1 \cdot 1
        +
      1 \cdot 1
        +
      1 \cdot 1
        +
      1 \cdot 1
        +
      2 \cdot 2
        +
      1 \cdot 8
      \big) / 16
      &
      \\
      &
      =
      16 / 16
      \\
      & =
      1
  \end{aligned}
$$

#### At a $2 D_{10}$-orientifold singularity
  {#At2D10Singularity}

For $G = 2 D_{10}$ the [[binary dihedral group]] of [[order of a group|order]] ${\vert 2 D_{10}\vert} = 20$, the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are ([BSS 18, 4.4](#BurtonSatiSchreiber18)):

| $[H] = $ | $\left[\langle e\rangle\right]$ | $\left[\langle g_1\rangle\right]$ |  $\left[\langle g_2\rangle\right]$ | $\left[\langle g_3\rangle\right]$ | $\left[\langle g_4\rangle\right]$ | $\left[\langle g_5\rangle\right]$ |  $\left[\langle g_6\rangle\right]$ |  $\left[\langle g_7\rangle\right]$ |
|--|--|--|--|--|--|--|--|--|
| $\chi_{V_1} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ |
| $\chi_{V_2} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $-1$ | $-1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ |
| $\chi_{V_3} =$ | $\phantom{-}2$ |  $-2$ |  $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}2$ | $\phantom{-}2$ | $-2$ | $-2$ |
| $\chi_{V_4} =$ | $\phantom{-}4$ |  $\phantom{-}4$ |  $\phantom{-}0$ | $\phantom{-}0$ | $-1$ | $-1$ | $-1$ | $-1$ |
| $\chi_{V_5} =$ | $\phantom{-}8$ |  $-8$ |  $\phantom{-}0$ | $\phantom{-}0$ | $-2$ | $-2$ | $\phantom{-}2$ | $\phantom{-}2$ |

One finds ([here](https://www.wolframalpha.com/input/?i=Nullspace+Transpose%5B%7B%7B1,1,1,1,1,1,1%7D,%7B1,-1,-1,1,1,1,1%7D,%7B-2,0,0,2,2,-2,-2%7D,%7B4,0,0,-1,-1,-1,-1%7D,%7B-8,0,0,-2,-2,2,2%7D%7D%5D)) that the general solution to the local/twisted tadpole cancellation condition (eq:VanishingOfCharacterValuesOnNontrivialSubgroups) for $G =2 D_{10}$ is

$$
  V
  \;=\;
  n
  \Big(
    1 \cdot V_1
      +
    1 \cdot V_2
      +
    1 \cdot V_3
      +
    2 \cdot V_4
      +
    1 \cdot V_5
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
$$

whose minimal [[positive number|positive]]  [[mass]] (net brane number) is

$$
  \begin{aligned}
    M_{2 D_{10}}
      & =
    dim(V) / {\vert 2 D_{10}\vert}
    \\
      & =
    \chi_V([\langle e\rangle]) /  {\vert 2 D_{10}\vert}
    \\
      & =
      \big(
      1 \cdot 1
        +
      1 \cdot 1
        +
      1 \cdot 2
        +
      2 \cdot 4
        +
      1 \cdot 8
      \big) / 20
      &
      \\
      &
      =
      20 / 20
      \\
      & = 1
  \end{aligned}
$$

#### At a $2 D_{12}$-orientifold singularity
  {#At2D12Singularity}

For $G = 2 D_{12}$ the [[binary dihedral group]] of [[order of a group|order]] ${\vert 2 D_{12}\vert} = 24$, the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are ([BSS 18, 4.5](#BurtonSatiSchreiber18)):

| $[H] = $ | $\left[\langle e\rangle\right]$ | $\left[\langle g_1\rangle\right]$ |  $\left[\langle g_2\rangle\right]$ | $\left[\langle g_3\rangle\right]$ | $\left[\langle g_4\rangle\right]$ | $\left[\langle g_5\rangle\right]$ |  $\left[\langle g_6\rangle\right]$ |  $\left[\langle g_7\rangle\right]$ | $\left[\langle g_8\rangle\right]$ |
|--|--|--|--|--|--|--|--|--|--|
| $\chi_{V_1} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ |
| $\chi_{V_2} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $\phantom{-}1$ | $-1$ | $-1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ |
| $\chi_{V_3} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $\phantom{-}1$ | $-1$ | $\phantom{-}1$ | $-1$ | $\phantom{-}1$ | $-1$ | $-1$ |
| $\chi_{V_4} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $\phantom{-}1$ | $\phantom{-}1$ | $-1$ | $-1$ | $\phantom{-}1$ | $-1$ | $-1$ |
| $\chi_{V_5} =$ | $\phantom{-}2$ |  $\phantom{-}2$ |  $-1$ | $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}2$ | $-1$ | $-1$ | $-1$ |
| $\chi_{V_6} =$ | $\phantom{-}2$ |  $\phantom{-}2$ |  $-1$ | $\phantom{-}0$ | $\phantom{-}0$ | $-2$ | $-1$ | $\phantom{-}1$ | $\phantom{-}1$ |
| $\chi_{V_7} =$ | $\phantom{-}4$ |  $-4$ |  $\phantom{-}4$ | $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}0$ | $-4$ | $\phantom{-}0$ | $\phantom{-}0$ |
| $\chi_{V_8} =$ | $\phantom{-}8$ |  $-8$ |  $-4$ | $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}4$ | $\phantom{-}0$ | $\phantom{-}0$ |

One sees ([here](https://www.wolframalpha.com/input/?i=Nullspace+Transpose%5B%7B%7B1,1,1,1,1,1,1,1%7D,%7B1,1,-1,-1,1,1,1,1%7D,%7B1,1,-1,1,-1,1,-1,-1%7D,%7B1,1,1,-1,-1,1,-1,-1%7D,%7B2,-1,0,0,2,-1,-1,-1%7D,%7B2,-1,0,0,-2,-1,1,1%7D,%7B-4,4,0,0,0,-4,0,0%7D,%7B-8,-4,0,0,0,4,0,0%7D%7D%5D)) that the general solution to the local/twisted tadpole cancellation condition (eq:VanishingOfCharacterValuesOnNontrivialSubgroups) for $G =2 D_{12}$ is

$$
  V
  \;=\;
  n
  \Big(
    1 \cdot V_1
      +
    1 \cdot V_2
      +
    1 \cdot V_3
      +
    1 \cdot V_4
      +
    2 \cdot V_5
      +
    2 \cdot V_6
      +
    1 \cdot V_7
      +
    1 \cdot V_8
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
$$

whose minimal [[positive number|positive]]  [[mass]] (net brane number) is

$$
  \begin{aligned}
    M_{2 D_{12}}
      & =
    dim(V)  / {\vert 2 D_{12}\vert }
    \\
      & =
    \chi_V([\langle e\rangle]) / {\vert 2 D_{12}\vert }
    \\
      & =
      \big(
      1 \cdot 1
        +
      1 \cdot 1
        +
      1 \cdot 1
        +
      1 \cdot 1
        +
      2 \cdot 2
        +
      2 \cdot 2
        +
      1 \cdot 4
        +
      1 \cdot 8
      \big) / 24
      &
      \\
      &
      =
      24 / 24
      \\
      & = 1
  \end{aligned}
$$

#### At a $2 D_{14}$-orientifold singularity
  {#At2D14Singularity}

For $G = 2 D_{14}$ the [[binary dihedral group]] of [[order of a group|order]] ${\vert 2 D_{14}\vert} = 28$, the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are ([BSS 18, 4.6](#BurtonSatiSchreiber18)):

| $[H] = $ | $\left[\langle e\rangle\right]$ | $\left[\langle g_1\rangle\right]$ |  $\left[\langle g_2\rangle\right]$ | $\left[\langle g_3\rangle\right]$ | $\left[\langle g_4\rangle\right]$ | $\left[\langle g_5\rangle\right]$ |  $\left[\langle g_6\rangle\right]$ |  $\left[\langle g_7\rangle\right]$ | $\left[\langle g_8\rangle\right]$ | $\left[\langle g_9\rangle\right]$ |
|--|--|--|--|--|--|--|--|--|--|--|
| $\chi_{V_1} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ |
| $\chi_{V_2} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $-1$ | $-1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ |
| $\chi_{V_3} =$ | $\phantom{-}2$ |  $-2$ |  $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}2$ | $\phantom{-}2$ | $\phantom{-}2$ | $-2$ | $-2$ | $-2$ |
| $\chi_{V_4} =$ | $\phantom{-}6$ |  $\phantom{-}6$ |  $\phantom{-}0$ | $\phantom{-}0$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ | $-1$ |
| $\chi_{V_5} =$ | $\phantom{-1}\mathllap{12}$ |  $\phantom{-1}\mathllap{-12}$ |  $\phantom{-}0$ | $\phantom{-}0$ | $-2$ | $-2$ | $-2$ | $\phantom{-}2$ | $\phantom{-}2$ | $\phantom{-}2$ |

One sees by immediate inspection, that the general solution to the local/twisted tadpole cancellation condition (eq:VanishingOfCharacterValuesOnNontrivialSubgroups) for $G =2 D_{14}$ is

$$
  V
  \;=\;
  n
  \Big(
    1 \cdot V_1
      +
    1 \cdot V_2
      +
    1 \cdot V_3
      +
    2 \cdot V_4
      +
    1 \cdot V_5
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
$$

whose minimal [[positive number|positive]]  [[mass]] (net brane number) is

$$
  \begin{aligned}
    M_{2 D_{14}}
      & =
    dim(V)  / {\vert 2 D_{14}\vert }
    \\
      & =
    \chi_V([\langle e\rangle]) / {\vert 2 D_{14}\vert }
    \\
      & =
      \big(
      1 \cdot 1
        +
      1 \cdot 1
        +
      1 \cdot 2
        +
      2 \cdot 6
        +
      1 \cdot 12
      \big) / 28
      \\
      &
      =
      28 /28
      \\
      & = 1
  \end{aligned}
$$

#### At a $2 D_{16}$-orientifold singularity
  {#At2D16Singularity}

For $G = 2 D_{16}$ the [[binary dihedral group]] of [[order of a group|order]] ${\vert 2 D_{16}\vert} = 32$, the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are ([BSS 18, 4.7](#BurtonSatiSchreiber18)):

| $[H] = $ | $\left[\langle e\rangle\right]$ | $\left[\langle g_1\rangle\right]$ |  $\left[\langle g_2\rangle\right]$ | $\left[\langle g_3\rangle\right]$ | $\left[\langle g_4\rangle\right]$ | $\left[\langle g_5\rangle\right]$ |  $\left[\langle g_6\rangle\right]$ |  $\left[\langle g_7\rangle\right]$ | $\left[\langle g_8\rangle\right]$ | $\left[\langle g_9\rangle\right]$ |$\left[\langle g_9\rangle\right]$ |
|--|--|--|--|--|--|--|--|--|--|--|--|
| $\chi_{V_1} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ |
| $\chi_{V_2} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $\phantom{-}1$ | $-1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $-1$ | $-1$ | $-1$ | $-1$ |
| $\chi_{V_3} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $-1$ | $-1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ |
| $\chi_{V_4} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $-1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $-1$ | $-1$ | $-1$ | $-1$ |
| $\chi_{V_5} =$ | $\phantom{-}2$ |  $\phantom{-}2$ |  $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}2$ | $-2$ | $-2$ | $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}0$ |
| $\chi_{V_6} =$ | $\phantom{-}4$ |  $\phantom{-}4$ |  $\phantom{-}0$ | $\phantom{-}0$ | $-4$ | $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}0$ |
| $\chi_{V_7} =$ | $\phantom{-1}\mathllap{16}$ |  $\phantom{-1}\mathllap{-16}$ |  $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}0$ | $\phantom{-}0$ |

One sees by immediate inspection, that the general solution to the local/twisted tadpole cancellation condition (eq:VanishingOfCharacterValuesOnNontrivialSubgroups) for $G =2 D_{16}$ is

$$
  V
  \;=\;
  n
  \Big(
    1 \cdot V_1
      +
    1 \cdot V_2
      +
    1 \cdot V_3
      +
    1 \cdot V_4
      +
    2 \cdot V_5
      +
    2 \cdot V_6
      +
    1 \cdot V_7
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
$$

whose minimal [[positive number|positive]]  [[mass]] (net brane number) is

$$
  \begin{aligned}
    M_{2 D_{16}}
      & =
    dim(V) / {\vert 2 D_{16}\vert}
    \\
      & =
    \chi_V([\langle e\rangle]) / {\vert 2 D_{16}\vert}
    \\
      & =
      \big(
      1 \cdot 1
        +
      1 \cdot 1
        +
      1 \cdot 1
        +
      1 \cdot 1
        +
      2 \cdot 2
        +
      2 \cdot 4
        +
      1 \cdot 16
      \big) /32
      \\
      &
      =
      32 / 32
      \\
      & = 1
  \end{aligned}
$$

#### At a $2 T$-orientifold singularity
  {#At2TSingularity}

For $G = 2 T$ the [[binary tetrahedral group]] (whose [[order of a group|order]] is ${\vert 2T \vert} =24$), the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are ([BSS 18, 4.8](#BurtonSatiSchreiber18)):

| $[H] = $ | $\left[\langle e\rangle\right]$ | $\left[\langle g_1\rangle\right]$ |  $\left[\langle g_2\rangle\right]$ | $\left[\langle g_3\rangle\right]$ | $\left[\langle g_4\rangle\right]$ | $\left[\langle g_5\rangle\right]$ |  $\left[\langle g_6\rangle\right]$ |
|--|--|--|--|--|--|--|--|
| $\chi_{V_1} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $\phantom{-}1$ | $\phantom{-}1$ |  $\phantom{-}1$ | $\phantom{-}1$ |$\phantom{-}1$ |
| $\chi_{V_2} =$ | $\phantom{-}2$ |  $\phantom{-}2$ |  $-1$ | $-1$ |  $\phantom{-}2$ | $-1$ | $-1$ |
| $\chi_{V_3} =$ | $\phantom{-}3$ |  $\phantom{-}3$ |  $\phantom{-}0$ | $\phantom{-}0$ |  $-1$ | $\phantom{-}0$ | $\phantom{-}0$ |
| $\chi_{V_4} =$ | $\phantom{-}4$ |  $-4$ |  $\phantom{-}1$ | $\phantom{-}1$ |  $\phantom{-}0$ | $-1$ | $-1$ |
| $\chi_{V_5} =$ | $\phantom{-}4$ |  $-4$ |  $-2$ | $-2$ |  $\phantom{-}0$ | $\phantom{-}2$ | $\phantom{-}2$ |

One finds ([here](https://www.wolframalpha.com/input/?i=Nullspace+Transpose%5B%7B%7B1,1,1,1,1,1%7D,%7B2,-1,-1,2,-1,-1%7D,%7B3,0,0,-1,0,0%7D,%7B-4,1,1,0,-1,-1%7D,%7B-4,-2,-2,0,2,2%7D%7D%5D)) that the general solution to the local/twisted tadpole cancellation condition (eq:VanishingOfCharacterValuesOnNontrivialSubgroups) for $G = 2T$ is

$$
  V
  \;=\;
  n
  \Big(
    1 \cdot V_1
    +
    1 \cdot V_2
    +
    3 \cdot V_3
    +
    2 \cdot V_4
    +
    1 \cdot V_5
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
$$

whose minimal [[positive number|positive]] [[mass]] (net brane number) is

$$
  \begin{aligned}
    M_{2I}
      & =
    dim(V) / {\vert 2 I \vert }
    \\
      & =
    \chi_V([\langle e\rangle]) / {\vert 2 I \vert }
    \\
      & =
      \big(
        1 \cdot 1
        +
        1 \cdot 2
        +
        3 \cdot 3
        +
        2 \cdot 4
        +
        1 \cdot 4
      \big) / 24
      &
      \\
      &
      =
      24 / 24
      \\
      & = 1
  \end{aligned}
$$

#### At a $2 O$-orientifold singularity
  {#At2OSingularity}

For $G = 2 O$ the [[binary octahedral group]] (whose [[order of a group|order]] is ${\vert 2O \vert} = 48$), the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are ([BSS 18, 4.9](#BurtonSatiSchreiber18)):

| $[H] = $ | $\left[\langle e\rangle\right]$ | $\left[\langle g_1\rangle\right]$ |  $\left[\langle g_2\rangle\right]$ | $\left[\langle g_3\rangle\right]$ | $\left[\langle g_4\rangle\right]$ | $\left[\langle g_5\rangle\right]$ |  $\left[\langle g_6\rangle\right]$ | $\left[\langle g_7\rangle\right]$ |
|--|--|--|--|--|--|--|--|--|
| $\chi_{V_1} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $\phantom{-}1$ | $\phantom{-}1$ |  $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ |
| $\chi_{V_2} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $\phantom{-}1$ | $\phantom{-}1$ |  $-1$ | $\phantom{-}1$ | $-1$ | $-1$ |
| $\chi_{V_3} =$ | $\phantom{-}2$ |  $\phantom{-}2$ |  $-1$ | $\phantom{-}2$ |  $\phantom{-}0$ | $-1$ | $\phantom{-}0$ | $\phantom{-}0$ |
| $\chi_{V_4} =$ | $\phantom{-}3$ |  $\phantom{-}3$ |  $\phantom{-}0$ | $-1$ |  $\phantom{-}1$ | $\phantom{-}0$ | $-1$ | $-1$ |
| $\chi_{V_5} =$ | $\phantom{-}3$ |  $\phantom{-}3$ |  $\phantom{-}0$ | $-1$ |  $-1$ | $\phantom{-}0$ | $\phantom{-}1$ | $\phantom{-}1$ |
| $\chi_{V_6} =$ | $\phantom{-}8$ |  $-8$ |  $\phantom{-}2$ | $\phantom{-}0$ |  $\phantom{-}0$ | $-2$ | $\phantom{-}0$ | $\phantom{-}0$ |
| $\chi_{V_7} =$ | $\phantom{-}8$ |  $-8$ |  $-4$ | $\phantom{-}0$ |  $\phantom{-}0$ | $\phantom{-}4$ | $\phantom{-}0$ | $\phantom{-}0$ |

One finds ([here](https://www.wolframalpha.com/input/?i=Nullspace+Transpose%5B%7B%7B1,1,1,1,1,1,1%7D,%7B1,1,1,-1,1,-1,-1%7D,%7B2,-1,2,0,-1,0,0%7D,%7B3,0,-1,1,0,-1,-1%7D,%7B3,0,-1,-1,0,1,1%7D,%7B-8,2,0,0,-2,0,0%7D,%7B-8,-4,0,0,4,0,0%7D%7D%5D)) that the general solution to the local/twisted tadpole cancellation condition (eq:VanishingOfCharacterValuesOnNontrivialSubgroups) for $G = 2O$ is

$$
  V
  \;=\;
  n
  \Big(
    1 \cdot V_1
    +
    1 \cdot V_2
    +
    2 \cdot V_3
    +
    3 \cdot V_4
    +
    3 \cdot V_5
    +
    2 \cdot V_6
    +
    1 \cdot V_7
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
$$

whose minimal [[positive number|positive]] [[mass]] (net brane number) is

$$
  \begin{aligned}
    M_{2O}
      & =
    dim(V) / {\vert 2 O \vert }
    \\
      & =
    \chi_V([\langle e\rangle]) / {\vert 2 O \vert }
    \\
      & =
      \big(
      1 \cdot 1
      +
      1 \cdot 1
      +
      2 \cdot 2
      +
      3 \cdot 3
      +
      3 \cdot 3
      +
      2 \cdot 8
      +
      1 \cdot 8
      \big) / 48
      &
      \\
      &
      =
      48 / 48
      \\
      & = 1
  \end{aligned}
$$

#### At a $2 I$-orientifold singularity
  {#At2ISingularity}

For $G = 2 I$ the [[binary icosahedral group]] (whose [[order of a group|order]] is ${\vert 2I \vert} = 120$), the [[character of a linear representation|characters]]/[[D-brane charges]] of the elementary [[virtual representation|virtual]] [[permutation representations]]/[[fractional D-branes]] are ([BSS 18, 4.10](#BurtonSatiSchreiber18)):

| $[H] = $ | $\left[\langle e\rangle\right]$ | $\left[\langle g_1\rangle\right]$ |  $\left[\langle g_2\rangle\right]$ | $\left[\langle g_3\rangle\right]$ | $\left[\langle g_4\rangle\right]$ | $\left[\langle g_5\rangle\right]$ |  $\left[\langle g_6\rangle\right]$ | $\left[\langle g_7\rangle\right]$ | $\left[\langle g_8\rangle\right]$ |
|--|--|--|--|--|--|--|--|--|--|
| $\chi_{V_1} =$ | $\phantom{-}1$ |  $\phantom{-}1$ |  $\phantom{-}1$ | $\phantom{-}1$ |  $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}1$ |
| $\chi_{V_2} =$ | $\phantom{-}4$ |  $\phantom{-}4$ |  $\phantom{-}1$ | $\phantom{-}0$ |  $-1$ | $-1$ | $\phantom{-}1$ | $-1$ | $-1$ |
| $\chi_{V_3} =$ | $\phantom{-}5$ |  $\phantom{-}5$ |  $-1$ | $\phantom{-}1$ |  $\phantom{-}0$ | $\phantom{-}0$ | $-1$ | $\phantom{-}0$ | $\phantom{-}0$ |
| $\chi_{V_4} =$ | $\phantom{-}6$ |  $\phantom{-}6$ |  $\phantom{-}0$ | $-2$ |  $\phantom{-}1$ | $\phantom{-}1$ | $\phantom{-}0$ | $\phantom{-}1$ | $\phantom{-}1$ |
| $\chi_{V_5} =$ | $\phantom{-1}\mathllap{12}$ |  $\phantom{-1}\mathllap{-12}$ |  $\phantom{-}0$ | $\phantom{-}0$ |  $\phantom{-}2$ | $\phantom{-}2$ | $\phantom{-}0$ | $-2$ | $-2$ |
| $\chi_{V_6} =$ | $\phantom{-}8$ |  $-8$ |  $\phantom{-}2$ | $\phantom{-}0$ |  $-2$ | $-2$ | $-2$ | $\phantom{-}2$ | $\phantom{-}2$ |
| $\chi_{V_7} =$ | $\phantom{-}8$ |  $-8$ |  $-4$ | $\phantom{-}0$ |  $-2$ | $-2$ | $\phantom{-}4$ | $\phantom{-}2$ | $\phantom{-}2$ |

One finds ([here](https://www.wolframalpha.com/input/?i=Nullspace+Transpose%5B%7B%7B1,1,1,1,1,1,1,1%7D,%7B4,1,0,-1,-1,1,-1,-1%7D,%7B5,-1,1,0,0,-1,0,0%7D,%7B6,0,-2,1,1,0,1,1%7D,%7B-12,0,0,2,2,0,-2,-2%7D,%7B-8,2,0,-2,-2,-2,2,2%7D,%7B-8,-4,0,-2,-2,4,2,2%7D%7D%5D)) that the general solution to the local/twisted tadpole cancellation condition (eq:VanishingOfCharacterValuesOnNontrivialSubgroups) for $G = 2I$ is

$$
  V
  \;=\;
  n
  \Big(
    1 \cdot V_1
    +
    4 \cdot V_2
    +
    5 \cdot V_3
    +
    3 \cdot V_4
    +
    3 \cdot V_5
    +
    2 \cdot V_6
    +
    1 \cdot V_7
  \Big)
  \,,
  \phantom{AAA}
  n \in \mathbb{Z}
$$

whose minimal [[positive number|positive]] [[mass]] (net brane number) is

$$
  \begin{aligned}
    M_{2I}
      & =
    dim(V) / {\vert 2I \vert }
    \\
      & =
    \chi_V([\langle e\rangle]) / {\vert 2I \vert }
    \\
      & =
      \big(
      1 \cdot 1
      +
      4 \cdot 4
      +
      5 \cdot 5
      +
      3 \cdot 6
      +
      3 \cdot 12
      +
      2 \cdot 8
      +
      1 \cdot 8
      \big) / 120
      \\
      &
      =
      120 / 120
      \\
      & = 1
  \end{aligned}
$$

## Related concepts

* [[C-field tadpole cancellation]] in [[M-theory]]

* [[24 branes transverse to K3]] in [[F-theory]] and [[HET-theory]]

## References

### General

The issue was first highlighted in

* {#Sagnotti88} [[Augusto Sagnotti]], _Open strings and their symmetry groups_ in G. Mack et. al. (eds.) Cargese ’87, "Non-perturbative Quantum Field Theory,"  (Pergamon Press, 1988) p. 521 ([arXiv:hep-th/0208020](https://arxiv.org/abs/hep-th/0208020))

The argument is recalled in

* {#GimonPolchinski96} [[Eric Gimon]], [[Joseph Polchinski]], section 3 of _Consistency Conditions for Orientifolds and D-Manifolds_, Phys.Rev.D54:1667-1676, 1996 ([arXiv:hep-th/9601038](https://arxiv.org/abs/hep-th/9601038))

Details are in

* {#Witten12} [[Edward Witten]], section 9.3 of _Superstring Perturbation Theory Revisited_ ([arXiv:1209.5461](https://arxiv.org/abs/1209.5461))

Textbook accounts include

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], section 4.4 of _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge 2012

* {#BlumenhagenLustTheisen13} [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], section 9.4 of _Basic Concepts of String Theory_ Part of the series Theoretical and Mathematical Physics, Springer 2013

Quick illustrations include:

* Marcus Berg, p. 26-36 in _Introduction to Orientifolds_ ([pdf](https://tp.hotell.kau.se/marcus/physics/talks/orienti_short.pdf), [[BergOrientifolds.pdf:file]])

* [slide 18 here](http://www2.kek.jp/theory-center/project/string2higgsflavor/wp-content/uploads/2016/08/part-2.pdf)

Critical outlook in

* {#Moore14} [[Gregory Moore]] p. 21-22 of _[[Physical Mathematics and the Future]]_, talk at [Strings 2014](http://physics.princeton.edu/strings2014/) ([talk slides](http://physics.princeton.edu/strings2014/slides/Moore.pdf), [companion text pdf](http://www.physics.rutgers.edu/~gmoore/PhysicalMathematicsAndFuture.pdf), [[MooreVisionTalk2014.pdf:file]])

The above discussion follows and character tables for [[virtual representation|virtual]] [[permutation representations]] above are taken from

* {#BurtonSatiSchreiber18} [[Simon Burton]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Lift of fractional D-brane charge to equivariant Cohomotopy theory]]_, Journal of Geometry and Physics **161** (2021) 104034 [[arXiv:1812.09679](https://arxiv.org/abs/1812.09679), [Python code](https://arxiv.org/src/1812.09679v1/anc), [doi:10.1016/j.geomphys.2020.104034](https://doi.org/10.1016/j.geomphys.2020.104034)]

* {#SatiSchreiber19} [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Equivariant Cohomotopy implies orientifold tadpole cancellation]]_, Journal of Geometry and Physics **156** (2020) 103775 [[arXiv:1909.12277](https://arxiv.org/abs/1909.12277), [doi:10.1016/j.geomphys.2020.103775](https://doi.org/10.1016/j.geomphys.2020.103775)]

See also

* {#ABIU99} G. Aldazabal, D. Badagnani, [[Luis Ibáñez]], [[Angel Uranga]], _Tadpole versus anomaly cancellation in $D=4,6$ compact IIB orientifolds_, JHEP 9906:031, 1999 ([arXiv:hep-th/9904071](https://arxiv.org/abs/hep-th/9904071))

* {#Uranga00} [[Angel Uranga]], _D-brane probes, RR tadpole cancellation and K-theory charge_, Nucl.Phys.B598:225-246, 2001 ([arXiv:hep-th/0011048](https://arxiv.org/abs/hep-th/0011048))

* Maria E. Angulo, David Bailin, Huan-Xiong Yang, _Tadpole and Anomaly Cancellation Conditions in D-brane Orbifold Models_,  	Int.J.Mod.Phys.A18:3637-3694, 2003 ([arXiv:hep-th/0210150](https://arxiv.org/abs/hep-th/0210150))

* {#Marchesano03} [[Fernando Marchesano]], section 4 of _Intersecting D-brane Models_ ([arXiv:hep-th/0307252](https://arxiv.org/abs/hep-th/0307252))

* {#MarchesanoShiu04} [[Fernando Marchesano]], [[Gary Shiu]], _Building MSSM Flux Vacua_, JHEP0411:041, 2004 ([arXiv:hep-th/0409132](https://arxiv.org/abs/hep-th/0409132))

* {#CKMNW05} C.-M. Chen, G. V. Kraniotis, V. E. Mayes, D. V. Nanopoulos, J. W. Walker, _A K-theory Anomaly Free Supersymmetric Flipped SU(5) Model from Intersecting Branes_, Phys.Lett. B625 (2005) 96-105 ([arXiv:hep-th/0507232](https://arxiv.org/abs/hep-th/0507232))

* {#GarciaUranga05} Inaki Garcia-Etxebarria, [[Angel Uranga]], _From F/M-theory to K-theory and back_, JHEP 0602:008, 2006 ([arXiv:hep-th/0510073](https://arxiv.org/abs/hep-th/0510073))

* {#MaidenShiuStefanski06} John Maiden, [[Gary Shiu]], [[Bogdan Stefanski]], _D-brane Spectrum and K-theory Constraints of D=4, N=1 Orientifolds_, JHEP0604:052,2006 ([arXiv:hep-th/0602038](https://arxiv.org/abs/hep-th/0602038))

* [[Tetsuji Kimura]], Mitsuhisa Ohta, Kei-Jiro Takahashi, _Type IIA orientifolds and orbifolds on non-factorizable tori_, Nucl. Phys. B **798** (2008) 89-123 [[arXiv:0712.2281](https://arxiv.org/abs/0712.2281), [doi:10.1016/j.nuclphysb.2008.01.030](https://doi.org/10.1016/j.nuclphysb.2008.01.030)]

* {#DFM09} [[Jacques Distler]], [[Dan Freed]], [[Greg Moore]], _Orientifold Précis_ in: [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_, Proceedings of Symposia in Pure Mathematics, AMS (2011) ([arXiv:0906.0795](http://arxiv.org/abs/0906.0795), [slides](http://www.ma.utexas.edu/users/dafr/bilbao.pdf))

In view of consistency of [[flux compactifications]]:

* Philip Betzler, Erik Plauschinn, _Type IIB flux vacua and tadpole cancellation_ ([arXiv:1905.08823](https://arxiv.org/abs/1905.08823))

For the [[topological string]]:

* {#Walcher07} [[Johannes Walcher]], _Evidence for Tadpole Cancellation in the Topological String_, Communications in Number Theory and Physics **3** 1 (2009) [[arXiv:0712.2775](https://arxiv.org/abs/0712.2775)]

### Examples and Models

Specifically [[K3]] [[orientifolds]] ($\mathbb{T}^4/G_{ADE}$) in [[type IIB string theory]], hence for [[D9-branes]] and [[D5-branes]]:

* [[Eric Gimon]], [[Joseph Polchinski]], Section 3.2 of: _Consistency Conditions for Orientifolds and D-Manifolds_, Phys. Rev. D54: 1667-1676, 1996 ([arXiv:hep-th/9601038](https://arxiv.org/abs/hep-th/9601038))

* {#GimonJohnson96} [[Eric Gimon]], [[Clifford Johnson]], _K3 Orientifolds_, Nucl. Phys. B477: 715-745, 1996 ([arXiv:hep-th/9604129](https://arxiv.org/abs/hep-th/9604129))

* {#BuchelShiuTye99} Alex Buchel, [[Gary Shiu]], S.-H. Henry Tye, _Anomaly Cancelations in Orientifolds with Quantized B Flux_, Nucl.Phys. B569 (2000) 329-361 ([arXiv:hep-th/9907203](https://arxiv.org/abs/hep-th/9907203))

* P. Anastasopoulos, A. B. Hammou, _A Classification of Toroidal Orientifold Models_, Nucl. Phys. B729:49-78, 2005 ([arXiv:hep-th/0503044](https://arxiv.org/abs/hep-th/0503044))

Specifically [[K3]] [[orientifolds]] ($\mathbb{T}^4/G_{ADE}$) in [[type IIA string theory]], hence for [[D8-branes]] and [[D4-branes]]:

* {#ParkUranga98} J. Park, [[Angel Uranga]], _A Note on Superconformal N=2 theories and Orientifolds_, Nucl. Phys. B542:139-156, 1999 ([arXiv:hep-th/9808161](https://arxiv.org/abs/hep-th/9808161))

* {#AFIRU00a} G. Aldazabal, S. Franco, [[Luis Ibanez]], R. Rabadan, [[Angel Uranga]], _D=4 Chiral String Compactifications from Intersecting Branes_, J. Math. Phys. 42:3103-3126, 2001 ([arXiv:hep-th/0011073](https://arxiv.org/abs/hep-th/0011073))

* {#AFIRU00b} G. Aldazabal, S. Franco, [[Luis Ibanez]], R. Rabadan, [[Angel Uranga]], _Intersecting Brane Worlds_, JHEP 0102:047, 2001 ([arXiv:hep-ph/0011132](https://arxiv.org/abs/hep-ph/0011132))

* {#KataokaShimojo01} H. Kataoka, M. Shimojo, _$SU(3) \times SU(2) \times U(1)$ Chiral Models from Intersecting D4-/D5-branes_, Progress of Theoretical Physics, Volume 107, Issue 6, June 2002, Pages 1291–1296 ([arXiv:hep-th/0112247](https://arxiv.org/abs/hep-th/0112247), [doi:10.1143/PTP.107.1291](https://doi.org/10.1143/PTP.107.1291))

> The $\mathbb{Z}_N$ action with even $N$ contains an order 2 element $[ ...]$  Then there will be D8-branes in the type IIA D4-brane theory. Since the concept of intersecting D-branesinvolves use of the same dimensional D-branes, we restrict ourselves to the case that the order $N$ of $\mathbb{Z}_N$ is odd. ([p. 4](https://arxiv.org/pdf/hep-th/0112247.pdf#page=4))

* {#Honecker01} [[Gabriele Honecker]], _Non-supersymmetric Orientifolds with D-branes at Angles_, Fortsch.Phys. 50 (2002) 896-902 ([arXiv:hep-th/0112174](https://arxiv.org/abs/hep-th/0112174))

* {#Honecker02a} [[Gabriele Honecker]], _Intersecting brane world models from D8-branes on $(T^2 \times T^4/\mathbb{Z}_3)/\Omega\mathcal{R}_1$ type IIA orientifolds_, JHEP 0201 (2002) 025 ([arXiv:hep-th/0201037](https://arxiv.org/abs/hep-th/0201037))

* {#Honecker02b} [[Gabriele Honecker]], _Non-supersymmetric orientifolds and chiral fermions from intersecting D6- and D8-branes_, thesis 2002 ([[HoneckerIntersectingDBraneModels02.pdf:file]])

The [[Witten-Sakai-Sugimoto model]] on [[D4-D8-brane bound states]] for [[QCD]] with [[orthogonal group|orthogonal]] [[gauge groups]] on O-planes:

* {#ImotoSakaiSugimoto09} Toshiya Imoto, [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _$O(N)$ and $USp(N)$ QCD from String Theory_, Prog.Theor.Phys.122:1433-1453, 2010 ([arXiv:0907.2968](https://arxiv.org/abs/0907.2968))

* Hee-Cheol Kim, Sung-Soo Kim, Kimyeong Lee, _5-dim Superconformal Index with Enhanced $E_n$ Global Symmetry_, JHEP 1210 (2012) 142 ([arXiv:1206.6781](https://arxiv.org/abs/1206.6781))

Specifically D5 brane models [[T-duality|T-dual]] to D6/D8 models:

* [[Angel Uranga]], _A New Orientifold of $\mathbb{C}^2/\mathbb{Z}_N$ and Six-dimensional RG Fixed Points_, Nucl. Phys. B577:73-87, 2000 ([arXiv:hep-th/9910155](https://arxiv.org/abs/hep-th/9910155))

* {#FengHeKarchUranga01} Bo Feng, [[Yang-Hui He]], [[Andreas Karch]], [[Angel Uranga]], _Orientifold dual for stuck NS5 branes_, JHEP 0106:065, 2001 ([arXiv:hep-th/0103177](https://arxiv.org/abs/hep-th/0103177))

Specifically for [[D6-branes]]:

* {#IshiharaKataokaSato99} S. Ishihara, H. Kataoka, Hikaru Sato, _$D=4$, $N=1$, Type IIA Orientifolds_, Phys. Rev. D60 (1999) 126005 ([arXiv:hep-th/9908017](https://arxiv.org/abs/hep-th/9908017))

* [[Mirjam Cvetic]], Paul Langacker, Tianjun Li, Tao Liu, _D6-brane Splitting on Type IIA Orientifolds_, Nucl. Phys. B709:241-266, 2005 ([arXiv:hep-th/0407178](https://arxiv.org/abs/hep-th/0407178))

Specifically for [[D3-branes]]/[[D7-branes]]:

* [Feng-He-Karch-Uranga 01](#FengHeKarchUranga01)

Various:

* [[Dieter Lüst]], S. Reffert, E. Scheidegger, S. Stieberger, _Resolved Toroidal Orbifolds and their Orientifolds_, Adv.Theor.Math.Phys.12:67-183, 2008 ([arXiv:hep-th/0609014](https://arxiv.org/abs/hep-th/0609014))

[[!redirects RR-field tadpole cancellations]]

[[!redirects RR-tadpole cancellation]]
[[!redirects RR-tadpole cancellations]]

[[!redirects RR-field tadpole]]
[[!redirects RR-field tadpoles]]

[[!redirects RR-tadpole]]
[[!redirects RR-tadpoles]]

[[!redirects RR-field tadpole anomaly]]


RR-field tadpole cancellation on toroidal orientifolds -- table

**[[RR-field tadpole cancellation]] conditions**
for [[D-branes]] [[wrapped brane|wrapped on]] [[toroidal orbifold|toroidal]] [[orientifolds]]

 in terms of their [[D-brane charge]] $V \in K_G(\ast) = R_G$
in [[equivariant K-theory]] [=](equivariant+K-theory#RelationToRepresentationTheory) [[representation ring]]

 (here $\mathbf{n}_{reg} = k[G/1]$ denotes the [[regular representation]] of [[dimension]] $n = $ [[order of a group|ord(G)]])

| single [[D-brane]] species
 on [[toroidal orbifold|toroidal]] [[orientifold]] |  [local/twisted](RR-tadpole#TwistedTadpoleCancellation)
 [[RR-field tadpole cancellation|tadpole cancellation condition]] |  [global/untwisted](RR-tadpole#UntwistedTadpoleCancellation)
 [[RR-field tadpole cancellation|tadpole cancellation condition]] | comment |
|----|----|----|---|
| [[D5-branes]]
 transv. to $\mathbb{T}^{\mathbf{4}_{\mathbb{H}}}\!\sslash\! \mathbb{Z}_{2}$ | $V = N \cdot \mathbf{2}_{reg}$
 ([Buchel-Shiu-Tye 99 (19)](RR-tadpole#BuchelShiuTye99)) | $V = 16 \cdot \mathbf{2}_{reg}$
 ([Buchel-Shiu-Tye 99 (18)](RR-tadpole#BuchelShiuTye99)) | following
 [Gimon-Polchinski 96](RR-tadpole#GimonPolchinski96),
 [Gimon-Johnson 96](RR-tadpole#GimonJohnson96) |
| [[D5-branes]]
 transv. to $\mathbb{T}^{\mathbf{4}_{\mathbb{H}}} \!\sslash\! \mathbb{Z}_{4}$ | $V = N \cdot \mathbf{4}_{reg}$
 ([Buchel-Shiu-Tye 99 (19)](RR-tadpole#BuchelShiuTye99)) | $V = 8 \cdot \mathbf{4}_{reg}$
 ([Buchel-Shiu-Tye 99 (18)](RR-tadpole#BuchelShiuTye99)) | following
 [Gimon-Polchinski 96](RR-tadpole#GimonPolchinski96),
 [Gimon-Johnson 96](RR-tadpole#GimonJohnson96)  |
| [[D4-branes]]
 transv. to $\mathbb{T}^{\mathbf{1}_{\mathrm{triv}} + \mathbf{4}_{\mathbb{H}}} \!\sslash\! \mathbb{Z}_k$ | $V = N \cdot \mathbf{k}_{reg}$
 ([AFIRU 00a, 4.2.1](RR-tadpole#AFIRU00a)) |   |  |
| [[D4-branes]]
 transv. to $\mathbb{T}^{\mathbf{1}_{\mathrm{triv}} + \mathbf{4}_{\mathbb{H}}} \!\sslash\! \mathbb{Z}_3$ | $V = N \cdot \mathbf{3}_{reg}$
 ([AFIRU 00b, (7.2)](RR-tadpole#AFIRU00b)) | $V =  4 \cdot \mathbf{3}_{reg} + 4 \cdot \mathbf{1}_{triv}$
 ([Kataoka-Shimojo 01, (14)-(17)](RR-tadpole#KataokaShimojo01) |  |
| [[D8-branes]]
 on $\mathbb{T}^{\mathbf{1}_{\mathrm{triv}} + \mathbf{4}_{\mathbb{H}}} \!\sslash\! \mathbb{Z}_3$ | $V = N \cdot \mathbf{3}_{reg}$
  | $V =  4 \cdot \mathbf{3}_{reg} + 4 \cdot \mathbf{1}_{triv}$
 ([Honecker 01, 4](RR-tadpole#Honecker01),
 [Honecker 02a, (25) ∧ (28) ⇔ (29)](RR-tadpole#Honecker02a),
 [Honecker 02b, (3.19)-(3.27)](RR-tadpole#Honecker02b)) | equivalent to D4 case by [[T-duality]]:
 [Honecker 01, p. 2](RR-tadpole#Honecker01),
 [Honecker 02a, 6](RR-tadpole#Honecker02a),
 [Honecker 02b, p. 15](RR-tadpole#Honecker02b)
 review in:
 [Marchesano 03, Sec. 4](RR-tadpole#Marchesano03) |
| [[D6-branes]]
 on $\mathbb{T}^6 \!\sslash\! \mathbb{Z}_4$ |  | $V = 8 \cdot \mathbf{4}_{reg}$
 ([Ishihara-Kataoka-Sato 99, (4.16)](RR-tadpole#IshiharaKataokaSato99)) |   |
| [[D3-branes]]
 on $\mathbb{T}^{\mathbf{4}_{\mathbb{H}}} \!\sslash\! \mathbb{Z}_k$  | $V = N \cdot \mathbf{k}_{reg}$
 ([Feng-He-Karch-Uranga 01, (25)](RR-tadpole#FengHeKarchUranga01)) |  |   |
| [[D7-branes]]
 on $\mathbb{T}^{\mathbf{4}_{\mathbb{H}}} \!\sslash\! \mathbb{Z}_k$  | $V = N \cdot \mathbf{k}_{reg}$
 ([Feng-He-Karch-Uranga 01, (5), (6)](RR-tadpole#FengHeKarchUranga01)) |  |   |


Ruben Minasian

* [webpage](http://string.lpthe.jussieu.fr/members.pl?key=7)

## Selected writings

On [[D-brane charge]] in [[topological K-theory]]:

* [[Ruben Minasian]], [[Gregory Moore]], _K-theory and Ramond-Ramond charge_, JHEP9711:002,1997 ([arXiv:hep-th/9710230](http://arxiv.org/abs/hep-th/9710230))

On [[string-string duality]] and [[duality between heterotic and type II string theory]]:

* [[Michael Duff]], [[Ruben Minasian]], _Putting string-string duality to the test_, Nucl.Phys. B436 (1995) 507-528 ([arXiv:hep-th/9406198](https://arxiv.org/abs/hep-th/9406198))

On [[I8]]:

* {#DuffLiuMinasian95} [[Mike Duff]], [[James Liu]], [[Ruben Minasian]], _Eleven Dimensional Origin of String/String Duality: A One Loop Test_, Nucl.Phys. B452 (1995) 261-282 ([arXiv:hep-th/9506126](https://arxiv.org/abs/hep-th/9506126))

* {#FreedHarveyMinasianMoore98} [[Dan Freed]], [[Jeff Harvey]], [[Ruben Minasian]], [[Greg Moore]], _Gravitational Anomaly Cancellation for M-Theory Fivebranes_, Adv.Theor.Math.Phys.2:601-618, 1998 ([arXiv:hep-th/9803205](https://arxiv.org/abs/hep-th/9803205))

* {#HarveyMinasianMoore98b} [[Jeff Harvey]], [[Ruben Minasian]], [[Greg Moore]], _Non-abelian Tensor-multiplet Anomalies_, JHEP9809:004, 1998 ([arXiv:hep-th/9808060](https://arxiv.org/abs/hep-th/9808060))

On [[higher curvature corrections]] in [[type II supergravity]] and in [[D=11 supergravity]]:

* [[James Liu]], [[Ruben Minasian]], _Higher-derivative couplings in string theory: dualities and the B-field_, Nucl. Phys. B 2013 ([arXiv:1304.3137](https://arxiv.org/abs/1304.3137))

* [[James Liu]], [[Ruben Minasian]], Raffaele Savelli, Andreas Schachner, *Type IIB at eight derivatives: insights from Superstrings, Superfields and Superparticles* $[$[arXiv:2205.11530](https://arxiv.org/abs/2205.11530)$]$

On [[M5-branes]] and their [[quantum anomaly]]-cancellation:

* {#BahBonettiMinasianNardoni18} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], [[Emily Nardoni]], _Class $\mathcal{S}$ Anomalies from M-theory Inflow_, Phys. Rev. D 99, 086020 (2019) ([arXiv:1812.04016](https://arxiv.org/abs/1812.04016))

* {#BBMN19} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], [[Emily Nardoni]], _Anomaly Inflow for M5-branes on Punctured Riemann Surfaces_,  J. High Energ. Phys. 2019, 123 (2019)
([arXiv:1904.07250](https://arxiv.org/abs/1904.07250))

* {#BBMN19b} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], [[Emily Nardoni]], _Anomalies of QFTs from M-theory and Holography_, J. High Energ. Phys. 2020, 125 (2020) ([arXiv:1910.04166](https://arxiv.org/abs/1910.04166))

* {#BahBonettiMinasian20} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], _Discrete and higher-form symmetries in SCFTs from wrapped M5-branes_, J. High Energ. Phys. 2021, 196 (2021) ([arXiv:2007.15003](https://arxiv.org/abs/2007.15003))

On [[type II geometry]] and [[non-geometric vacua]]:

* [[Mariana Graña]], [[Ruben Minasian]], Michela Petrini, [[Daniel Waldram]], _T-duality, generalized geometry and non-geometric backgrounds_,  J. High Energy Phys. 2009, no. 4, 075, 39 pp. [arXiv:0807.4527](http://xxx.lanl.gov/abs/0807.4527) [MR2010i:81323](http://www.ams.org/mathscinet-getitem?mr=2505954) [doi](http://dx.doi.org/10.1088/1126-6708/2009/04/075)

* David Andriot, [[Ruben Minasian]], Michela Petrini, _Flux backgrounds from twists_,  J. High Energy Phys. 2009, no. 12, 028 [arXiv:0903.0633](http://arxiv.org/abs/0903.0633) [MR2011c:81201](http://www.ams.org/mathscinet-getitem?mr=2593014) [doi](http://dx.doi.org/10.1088/1126-6708/2009/12/028)

On [[generalized Calabi-Yau spaces]]:

* [[Mariana Grana]], [[Ruben Minasian]], Michela Petrini, [[Alessandro Tomasiello]], _Generalized structures of $N=1$ vacua_, [hep-th/0505212](http://arxiv.org/abs/hep-th/0505212)

On [[supersymmetric cycles]]:

* [[Marcos Marino]], [[Ruben Minasian]], [[Gregory Moore]], [[Andrew Strominger]], _Nonlinear Instantons from supersymmetric $p$-Branes_, J. High Energy Phys. 2000, no. 1, Paper 5 ([arXiv:hep-th/9911206](https://arxiv.org/abs/hep-th/9911206))

On [[U-duality]] and [[8-dimensional supergravity]]:

* {#LiuMinasian97} [[James Liu]], [[Ruben Minasian]], _U-branes and $T^3$ fibrations_, Nucl.Phys. B510 (1998) 538-554 ([arXiv:hep-th/9707125](http://arxiv.org/abs/hep-th/9707125))

On [[M-theory on Calabi-Yau manifolds]]:

* [[Sergio Ferrara]], [[Ramzi Khuri]],  [[Ruben Minasian]], _M-theory on a Calabi-Yau manifold_, Phys. Lett.B 375:81-88, 1996 ([arXiv:hep-th/9602102](https://arxiv.org/abs/hep-th/9602102))

On [[generalized Calabi-Yau manifolds]]:

* [[Mariana Graña]], [[Ruben Minasian]], Michela Petrini, [[Alessandro Tomasiello]], _Type II Strings and Generalized Calabi-Yau Manifolds_ ([arXiv:hep-th/0409176](http://arxiv.org/abs/hep-th/0409176))

* [[Mariana Grana]], [[Ruben Minasian]], Michela Petrini, [[Alessandro Tomasiello]], _Generalized structures of N=1 vacua_, [hep-th/0505212](http://arxiv.org/abs/hep-th/0505212)

On [[BPS-states]] in [[string theory]]/[[supergravity]] using [[G-structures]]:

* [[Ruben Minasian]], [[Daniël Prins]], [[Hagen Triendl]], _Supersymmetric branes and instantons on curved spaces_, JHEP 10 (2017) 159 ([arXiv:1707.07002](https://arxiv.org/abs/1707.07002))

On [[sphere fiber bundles]] as [[Kaluza-Klein compactifications]] in [[supergravity]] and [[string theory]]:

* [[Federico Bonetti]], [[Ruben Minasian]], Valentí Vall Camell, Peter Weck, *Consistent truncations from the geometry of sphere bundles* [[arXiv:2212.08068](https://arxiv.org/abs/2212.08068)]

On [[D=8 supergravity]]:

* Bing-Xin Lao, [[Ruben Minasian]], *Consistency of eight-dimensional supergravities: Anomalies, Lattices and Counterterms* [[arXiv:2306.07766](https://arxiv.org/abs/2306.07766)]

category: people


Ruben Van Belle

* [institute page](https://www.maths.ed.ac.uk/school-of-mathematics/people/a-z?person=787)

## Selected writings

On the [[Giry monad]] ([[probability]] as a [[codensity monad]]):

* {#VanBelle21} [[Ruben Van Belle]], *Probability monads as codensity monads*  (2021) $[$[arXiv:2111.01250](https://arxiv.org/abs/2111.01250)$]$

category: people


Rudger Kieboom

* [homepage](http://homepages.vub.ac.be/~rkieboom/)

## Selected writings

On the [[Toda bracket]] understood via [[homotopy coherent diagram|homotopy-coherent]] [[pasting diagrams]] in a [[pointed (infinity,1)-category|pointed]] [[homotopy 2-category]]:

* {#HardieKampsKieboom99} [[Keith Hardie]], [[Klaus Heiner Kamps]], [[Rudger  Kieboom]], _Higher homotopy groupoids and Toda brackets_, Homology Homotopy Appl. Volume 1, Number 1 (1999), 117-134 ([euclid:hha/1139840198](https://projecteuclid.org/euclid.hha/1139840198))

On the [[fundamental 2-groupoid]] of [[Hausdorff topological spaces]] as a [[strict 2-groupoid]] ([[Grpd-enriched category]]):

* [[Keith A. Hardie]], [[Klaus H. Kamps]], [[Rudger Kieboom]], *A homotopy 2-groupoid of a Hausdorff space*. Papers in honour of Bernhard Banaschewski (Cape Town, 1996). Appl. Categ. Structures **8** (2000) 209-234 [[doi:10.1023/A:1008758412196](https://doi.org/10.1023/A:1008758412196)]

and as a [[weak 2-groupoid]]:

* [[Keith A. Hardie]], [[Klaus H. Kamps]], [[Rudger Kieboom]], *A Homotopy Bigroupoid of a Topological Space*, Applied Categorical Structures **9** (2001) 311-327 [[doi:10.1023/A:1011270417127](https://doi.org/10.1023/A:1011270417127)]

On the [[canonical model structure]] for [[internal categories]] in [[finitely complete categories]]:

* [[Tomas Everaert]], [[Rudger W. Kieboom]], [[Tim Van der Linden]], _Model structures for homotopy of internal categories_, Theory and Applications of Categories **15** 3 (CT2004) 66-94 [[tac:15-03](http://www.tac.mta.ca/tac/volumes/15/3/15-03abs.html)]

category:people

[[!redirects Rudger W. Kieboom]]

[[!redirects R.W. Kieboom]]

[[!redirects R.W.Kieboom]]
[[!redirects R. Kieboom]]


Rudi Pendavingh

* [Institute page](https://www.tue.nl/en/research/researchers/rudi-pendavingh/)

## Selected writings

On the [[axiomatization]] of [[infinite set|infinite]] [[matroids]]:

* [[Henning Bruhn]], [[Reinhard Diestel]], [[Matthias Kriesell]], [[Rudi Pendavingh]], [[Paul Wollan]], *Axioms for infinite matroids*, Advances in Mathematics, **239** (2013) 18–46 [[arXiv:1003.3919](https://arxiv.org/abs/1003.3919), [doi:10.1016/j.aim.2013.01.011](https://doi.org/10.1016/j.aim.2013.01.011)]

category: people


Rudolf Carnap

* [Stanford Encyclopedia of Philosophy entry](https://plato.stanford.edu/entries/carnap/)

* [Wikipedia entry](http://en.wikipedia.org/wiki/Rudolf_Carnap)

category: people


Rudolf Clausius

* [Wikipedia entry](http://en.wikipedia.org/wiki/Rudolf_Clausius)

category: people

[[!redirects Clausius]]


Rudolf Fritsch

German Wikipedia page

## Selected writings

On [[cell complexes]] ([[CW-complexes]]) in [[topology]] ([[algebraic topology]]):

* {#FritschPiccinini90} [[Rudolf Fritsch]], [[Renzo Piccinini]], _Cellular structures in topology_, Cambridge University Press (1990) ([doi:10.1017/CBO9780511983948](https://doi.org/10.1017/CBO9780511983948), [pdf](https://epub.ub.uni-muenchen.de/4493/1/4493.pdf))

* Rudolf Fritsch, [[Dana May Latch]], _Homotopy inverses for nerve_, Bull. Amer. Math. Soc. (N.S.), Volume 1, Number 1 (1979), 258-262.

## Related $n$Lab entries

* [[CW-complex]]

* [[paracompact topological space]]

* [[CW-complexes are paracompact Hausdorff spaces]]

* [[simplicial approximation theorem]]

category:people


Rudolf Haag

Rudolf Haag was a [[mathematical physicist]] working in the context of [[AQFT]].

* [Wikipedia entry](http://de.wikipedia.org/wiki/Rudolf_Haag)

## Selected writings

On the foundations of [[algebraic quantum field theory]], introducing the [[Haag-Kastler axioms]]:

* {#Haag64} [[Rudolf Haag]], [[Daniel Kastler]], _An algebraic approach to quantum field theory_, Journal of Mathematical Physics, Bd.5, 1964, S.848-861 ([doi:10.1063/1.1704187](https://doi.org/10.1063/1.1704187), [spire:9124](https://inspirehep.net/literature/9124))

following

* {#Haag59} [[Rudolf Haag]], _Discussion des "axiomes" et des propriétés asymptotiques d’une théorie des champs locales avec particules composées, Les Problèmes Mathématiques de la Théorie Quantique des Champs_, Colloque Internationaux du CNRS LXXV (Lille 1957), CNRS Paris (1959), 151.

translated to English as:

* [[Rudolf Haag]], *Discussion of the ‘axioms’ and the asymptotic properties of a local field theory with composite particles*, EPJ H 35, 243–253 (2010) ([doi:10.1140/epjh/e2010-10041-3](https://doi.org/10.1140/epjh/e2010-10041-3))

Textbook account of [[AQFT]]:

* {#Haag} [[Rudolf Haag]], _[[Local Quantum Physics -- Fields, Particles, Algebras]]_ Springer (1992) 2nd., rev. and enlarged ed. Springer (1996) ([ISBN 978-3-642-61458-3] (https://www.springer.com/gp/book/9783540610496))

Introducing the [[Haag–Łopuszański–Sohnius theorem]]:

* {#HaagLopuszanskiSohnius75} [[Rudolf Haag]], [[Jan Łopuszański]], [[Martin Sohnius]], _All possible generators of supersymmetries of the S-matrix_, Nuclear Physics B, 88: 257–274 (1975) (doi:10.1016/0550-3213(75)90279-5, [spire:90850](https://inspirehep.net/literature/90850))

On [[mathematical physics]]:

* _Some people and some problems met in half a century of commitment to mathematical physics_, Personal recollection, The European Physical Journal H November 2010, Volume 35, Issue 3, pp 263-307 ([web](http://link.springer.com/article/10.1140%2Fepjh%2Fe2010-10032-4))

## related $n$Lab entries

* [[Haag-Kastler axioms]], [[AQFT]]

* [[DHR superselection theory]]

* [[Haag–Łopuszański–Sohnius theorem]]

category: people


Rudolf-E. Hoffmann

* [MathGenealogy entry](https://www.mathgenealogy.org/id.php?id=107849)

## Selected writings

Early discussion of the notion of [[center of a category]]:

* {#Hoffmann75} [[Rudolf-E. Hoffmann]], *Über das Zentrum einer Kategorie*, Math. Nachr. **68** (1975) 299-306 [[doi:10.1002/mana.19750680122](https://doi.org/10.1002/mana.19750680122)]

On [[adjoint strings]] of [[functors]]:

* {#Hoffmann80} [[Rudolf-E. Hoffmann]], _Sequences of adjoints for Ens-valued functors_, manuscripta mathematica **32** (1980) [[EuDML;154724](https://eudml.org/doc/154724)]

On the relationship between [[cones]] and [[factorisation systems]]:

* [[Rudolf-E. Hoffmann]], _Factorization of cones_, Mathematische Nachrichten **87** 1 (1979) 221-238. [[doi:10.1002/mana.19790870120](https://doi.org/10.1002/mana.19790870120)]

* [[Rudolf-E. Hoffmann]], _Factorization of cones II, with applications to weak Hausdorff spaces_, in: *Categorical Aspects of Topology and Analysis: Proceedings of an International Conference Held at Carleton University, Ottawa, August 11–15, 1981*, Springer (2006) [[doi:10.1007/BFb0092878](https://doi.org/10.1007/BFb0092878)]

category: people

[[!redirects R.-E. Hoffmann]]


Ruediger Schack

* [institute page](https://cquic.unm.edu/people/ruediger-schack/index.html)

* [GoogleScholar page](https://scholar.google.com/citations?user=tjusX-4AAAAJ&hl=en)

## Selected writings

On [[quantum Bayesianism]]/[[QBism]]:

* {#CFS2001} [[Carlton Caves]], [[Christopher Fuchs]], [[Ruediger Schack]], *Unknown Quantum States: The Quantum de Finetti Representation* (2001) [[arXiv:quant-ph/0104088](http://arxiv.org/abs/quant-ph/0104088)]

* {#FMS2014} [[Christopher Fuchs]], [[David Mermin]], [[Ruediger Schack]], *An Introduction to QBism with an Application to the Locality of Quantum Mechanics*, American Journal of Physics **82**  8 (2014) 749--754 [[arXiv:1311.5253](https://arxiv.org/abs/1311.5253)]

category: people


Ruelle zeta function

> under construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Theta functions
+--{: .hide}
[[!include theta functions - contents]]
=--
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### For dynamical systems

The _Ruelle zeta function_ is defined for a function $f : M \to M$ on a [[compact
manifold]] $M$. Assume that the set $Fix(f^m)$ is finite for all $m \geq 1$. Suppose
that $\phi: M \to \mathbb{C}^{d \times d}$ is a matrix-valued function. The first type of Ruelle zeta function is defined by

$$
\zeta(z) = exp \Bigg\{ \sum_{m \geq 1} \frac{z^m}{m} \sum_{x \in Fix(f^m)} Tr \bigg(\prod^{m-1}_{k=0} \phi(f^k(x)) \bigg) \Bigg\}.
$$

## For compact hyperbolic manifolds

For $X_\Gamma$ a [[hyperbolic manifold]] of odd [[dimension]], and given a [[flat connection|flat]] $GL$-[[principal connection]] $\rho$ on $X_\Gamma$, the _Ruelle zeta function_ is the [[meromorphic function]] $R_\rho$
given for $Re(s)  \gt (dim(X_\gamma)-1)/2 $ by the [[infinite product]] ([[Euler product]]) over the [[characteristic polynomials]] of the [[monodromy]]/[[holonomy]] of the flat connection along [[prime geodesics]]

$$
  R_\rho(s)
   =
  \underset{\gamma \neq e}{\prod} \det
  \left(
    Id - hol_\rho(\gamma) e^{-s l(\gamma)}
  \right)
$$

and defined from there by [[analytic continuation]].

Here

* the product is over [[prime geodesics]] $\gamma$;

* of [[length]] $l(\gamma)$;

* and $hol_\rho(\gamma)$ denotes the [[holonomy]] of $\rho$ along $\gamma$.

## Properties

### Relation to the Selberg zeta function

for the moment see at _[[Selberg zeta function]]_.

### Analogy to the Artin L-function
 {#AnalogyToTheArtinLFunction}

The Ruelle zeta function is [[analogy|analogous]] to the standard definition of an [[Artin L-function]] if one interprets a) a [[Frobenius map]] $Frob_p$ (as discussed there) as an element of the [[arithmetic fundamental group]] of an [[arithmetic curve]] and b) a [[Galois representation]] as a [[flat connection]].

So under this analogy the Ruelle zeta function for hyperbolic 3-manifolds as well as the [[Artin L-function]] for a [[number field]] both are like an [[infinite product]] over primes ([[prime geodesics]] in one case, [[prime ideals]] in the other, see also at _[Spec(Z) -- As a 3-dimensional space containing knots](Spec%28Z%29#As3dSpaceContainingKnots)_) of determinants of monodromies of the given flat connection.

See at _[Artin L-function -- Analogy with Selberg/Ruelle zeta function](Artin%20L-function#AnalogyWithSelbergZeta)_ for more. This analogy has been highlighted in ([Brown 09](#Brown09), [Morishita 12, remark 12.7](#Morishita12)).

### Relation to volume and analytic torsion

The [[special values of L-functions|special value]] of $R_\rho$ at the origin encodes

* the [[Reidemeister torsion]] ([Fried 86, theorem 1](#Fried86));

* the [[volume]] of the hyperbolic manifold ([Mathai 92](#Mathai92), [Lott 92](#Lott92)).

## References

Original articles include

* {#Fried86} [[David Fried]], _Analytic torsion and closed geodesics on hyperbolic manifolds_, Invent. math. 84, 523-540 (1986) ([pdf](http://gdz-lucene.tc.sub.uni-goettingen.de/gcs/gcs?&&action=pdf&metsFile=PPN356556735_0084&divID=LOG_0026&pagesize=original&pdfTitlePage=http://gdz.sub.uni-goettingen.de/dms/load/pdftitle/?metsFile=PPN356556735_0084%7C&targetFileName=PPN356556735_0084_LOG_0026.pdf&))

* {#BunkeOlbrich94a} [[Ulrich Bunke]], [[Martin Olbrich]], _Theta and zeta functions for odd-dimensional locally symmetric spaces of rank one_ ([arXiv:dg-ga/9407012](http://arxiv.org/abs/dg-ga/9407012))

* [[David Ruelle]], _Dynamical zeta functions and transfer operators_, Notices
Amer. Math. Soc. 49 (2002), no. 8, 887895.

Seminar/lecture notes include

* _Selberg and Ruelle zeta functions for compact hyperbolic manifolds_ ([pdf](http://www.math.uni-bonn.de/people/xenia/Oberseminar%202014/Oberseminar%202014.pdf))

* V. Baladi, _Dynamical zeta functions_, arXiv:[1602.05873](http://arxiv.org/abs/1602.05873)

Relation to the [[volume]] of hyperbolic manifolds is discussed in

* {#Mathai92} [[Varghese Mathai]], section 6 of _$L^2$-analytic torsion_, Journal of Functional Analysis Volume 107, Issue 2, 1 August 1992, Pages 369–386

* {#Lott92} [[John Lott]], _Heat kernels on covering spaces and topological invariants_, J. Diff. Geom. 35 no 2 (1992) ([pdf](https://math.berkeley.edu/~lott/covering.pdf))

The [[analogy]] with the [[Artin L-function]] is discussed in

* {#Brown09} Darin Brown, _Lifting properties of prime geodesics_, Rocky Mountain J. Math. Volume 39, Number 2 (2009), 437-454 ([euclid](http://projecteuclid.org/euclid.rmjm/1239113439))

* {#Morishita12} [[Masanori Morishita]], section 12.1 of _Knots and Primes: An Introduction to Arithmetic Topology_, 2012 ([web](https://books.google.co.uk/books?id=DOnkGOTnI78C&pg=PA156#v=onepage&q&f=false))


Rui Fernandes

* [website](http://www.math.ist.utl.pt/~rfern/)

## Selected writings

On [[Lie integration]] in the generality of [[Lie algebroids]] integrating to [[Lie groupoids]] (cf. [[Lie's three theorem]]):

* [[Marius Crainic]], [[Rui Loja Fernandes]], *Integrability of Lie brackets*, Ann. of Math. **157** 2 (2003) 575-620 [[arXiv:math.DG/0105033](http://arxiv.org/abs/math.DG/0105033), [doi:10.4007/annals.2003.157.575](https://doi.org/10.4007/annals.2003.157.575)]

* [[Rui Loja Fernandes]], [[Marius Crainic]], *Lectures on Integrability of Lie Brackets*, Geometry & Topology Monographs **17** (2011) 1–107 [[arxiv:math.DG/0611259](https://arxiv.org/abs/math/0611259), [doi:10.2140/gtm.2011.17.1](http://dx.doi.org/10.2140/gtm.2011.17.1)]

On [[Poisson geometry]]:

* [[Marius Crainic]], [[Rui Loja Fernandes]], *Lectures on Poisson Geometry*, Graduate Studies in Mathematics **217**, Amer. Math. Soc. (2021) [[ISBN:978-1-4704-6430-1](https://bookstore.ams.org/view?ProductCode=GSM/217)]

category: people

[[!redirects Rui Loja Fernandes]]
[[!redirects Rui L. Fernandes]]


Ruizhi Huang

* [personal page](https://sites.google.com/site/hrzsea/)

## Selected writings

On [[gravitino]] [[anomaly cancellation]] in [[D=11 N=1 supergravity]]:

* {#HanHuangLiuZhang20} [[Fei Han]], [[Ruizhi Huang]], [[Kefeng Liu]], [[Weiping Zhang]], *Cubic forms, anomaly cancellation and modularity*, Advances in Mathematics **394** (2022) 108023 [[arXiv:2005.02344](https://arxiv.org/abs/2005.02344), [doi:10.1016/j.aim.2021.108023](https://doi.org/10.1016/j.aim.2021.108023)]

On [[bundle gerbe modules]] with [[higher structure|higher]] [[G-structures]] (including [[String structure]], etc.) and the corresponding [[twisted Chern character]]:

* [[Fei Han]], [[Ruizhi Huang]], [[Varghese Mathai]], *Fractional structures on bundle gerbe modules and fractional classifying spaces* [[arXiv:2203.14439](https://arxiv.org/abs/2203.14439)]

category: people


rule C

## Idea

Rule C is a formal analogue in predicate logic of making a (single) choice when reasoning (in a line of logical proof). Thus, if one has a premise that $(\exists x)\phi(x)$ then one introduces a new
constant $b$ ("chosen $x$") and states $\phi(b)$. Then one
continues proof and finally arrives at a new theorem not containing $b$.

A basic metatheorem is that if a theorem not containing such a $b$ is provable with rule $C$ then it is provable without rule $C$, written $P \vdash_C Q$ then $P\vdash Q$.

## Literature

It seems that the rule is introduced in VI.7 of

* [[J. Barkley Rosser]], _Logic for mathematicians_, 1954

and is now more widely known from 2.6 in

* Elliott Mendelson, _Introduction to mathematical logic_, Chapman and Hall 1964, 1979, 1987, 1997

See also a comment about the proof of the basic metatheorem at

* math.stackexchange:[confusion-about-steps-of-proof-of-rule-c](https://math.stackexchange.com/questions/4115567/confusion-about-steps-of-proof-of-rule-c)

category: logic


Rune Haugseng

* [webpage](http://sites.google.com/site/runehaugseng/)

## Related $n$Lab entries

* [[(infinity,n)-category of correspondences]]

* [[(infinity,1)-Grothendieck construction]]

* [[(infinity,1)-module]], [[Brauer group]]

category: people


Runge-Kutta method

__Runge-Kutta approximation schemes__ are a family of difference schemes used for iterative numerical solution of [[ordinary differential equation]]s.

### Standard Runge-Kutta methods

In engineering it is rather standard to use the 4th order Runge-Kutta difference schemes. General overview is at

* wikipedia [Runge-Kutta methods](http://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods)

* H. Munthe-Kaas, _High order Runge–Kutta methods on manifolds_, Appl. Num. Math., 29, 115-127 (1999)

### For stochastic ODE-s:

* wikipedia [Runge–Kutta method (SDE)](http://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_method_%28SDE%29)

### Relation to algebraic problems, Lie theory and renormalization

* J.C. Butcher, _Coefficients for the study of Runge-Kutta integration processes_, J. Austral. Math. Soc. __3__ (1963), 185-201

In the 1972 Butcher's work Runge-Kutta methods are organized into a group, later called [[Butcher group]]:

* J. C. Butcher, An algebraic theory of integration methods, Math. Comp. 26 (1972), 79–106.

Brouder has shown a relation of Butcher group to Connes-Kreimer Hopf algebra.

* Ch. Brouder, _Runge-Kutta methods and renormalization_,
Europ. Phys. J. C12 (2000) 512–534.

The Butcher-Connes-Kreimer Hopf algebra is a Hopf subalgebra or
Foissy Hopf algebra from work

* L. Foissy, Les algèbres de Hopf des arbres enracinés décorés, I." Bulletin des Sciences Mathematiques 126, no. 3 (2002): 193–239 (arXiv:math/0105212). MR1905177 "Les algèbres de Hopf des arbres enracinés décorés. II." Bulletin des Sciences Mathematiques 126, no. 4 (2002): 249–88 (arXiv:math/0105212). MR1909461

Important family of Lie-algebraic methods for generating integrators are introduced in

* P. E. Crouch, R. Grossman, _Numerical integration of ordinary differential equations on manifolds_, J. Nonlinear Sci. __3__: 1-33 (1993) [doi](https://doi.org/10.1007/BF02429858)

### Connections to symplectic geometry

See also [[symplectic integrator]]s.

* Alberto S. Cattaneo, Benoit Dherin, Giovanni Felder, _Formal symplectic groupoid_, Comm. Math. Phys. __253__ (2005), no. 3, 645–674 [math.SG/0312380](http://arxiv.org/abs/math/0312380) [doi](HTTP://dx.doi.org/10.1007/s00220-004-1199-z)

> The multiplicative structure of the trivial symplectic groupoid over $\mathbb{R}^d$ associated to the zero Poisson structure can be expressed in terms of a generating function. We address the problem of deforming such a generating function in the direction of a non-trivial Poisson structure so that the multiplication remains associative. We prove that such a deformation is unique under some reasonable conditions and we give the explicit formula for it. This formula turns out to be the semi-classical approximation of Kontsevich's deformation formula. For the case of a linear Poisson structure, the deformed generating function reduces exactly to the CBH formula of the associated Lie algebra. The methods used to prove existence are interesting in their own right as they come from an at first sight unrelated domain of mathematics: the Runge--Kutta theory of the numeric integration of ODE's.

* Robert McLachlan, Klas Modin, Olivier Verdier, _Collective Lie--Poisson integrators on $\mathbb{R}^{3}$_, [arxiv/1307.2387](http://arxiv.org/abs/1307.2387)

> We develop Lie--Poisson integrators for general Hamiltonian systems on $\mathbb{R}^{3}$ equipped with the rigid body bracket. The method uses symplectic realisation of $\mathbb{R}^{3}$ on $T^{*}\mathbb{R}^{2}$ and application of __symplectic Runge--Kutta schemes__. As a side product, we obtain simple [[symplectic integrator]]s for general Hamiltonian systems on the sphere $S^{2}$.

* J. M. Sanz-Serra, Runge-Kutta Schemes for Hamiltonian Systems, Bit __28__, pp. 877-883, (1988).

[[!redirects Runge-Kutta approximation scheme]]


Rupert Machleidt

* [webpage](https://www.uidaho.edu/sci/physics/people/faculty/machleid)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Ruprecht_Machleidt)

## Selected writings

On [[meson]] [[effective field theory]] ([[chiral perturbation theory]]) in [[nuclear physics]]:

* [[Rupert Machleidt]], _The Meson Theory of Nuclear Forces and Nuclear Structure_, In: Negele J.W., Vogt E. (eds.) _Advances in Nuclear Physics_ Advances in Nuclear Physics, vol 19. Springer 1989 ([doi:10.1007/978-1-4613-9907-0_2](https://doi.org/10.1007/978-1-4613-9907-0_2))

* {#MachleidtEntem11} [[Rupert Machleidt]], [[David Rodríguez Entem]], _Chiral effective field theory and nuclear forces_, Phys. Rept. 503:1-75, 2011 ([arXiv:1105.2919](https://arxiv.org/abs/1105.2919), [doi:10.1016/j.physrep.2011.02.001](https://doi.org/10.1016/j.physrep.2011.02.001))

category: people


Ruslan Metsaev

* [webpage](http://www.tamm.lpi.ru/staff/metsaev.html)

## related $n$Lab entries

* [[AdS-CFT]]

category: people


Russell O'Connor

* [webpage](http://r6.ca)

## Selected writings

A [[formal proof]] of the [[Gödel incompleteness theorem]] in [[Coq]]:

* [[Russell O'Connor]], *Essential Incompleteness of Arithmetic Verified by Coq*, in: *Theorem Proving in Higher Order Logics. TPHOLs 2005*, Lecture Notes in Computer Science **3603**, Springer (2005) 245-260 [[web](http://r6.ca/Goedel/goedel1.html), [arXiv:cs/0505034](https://arxiv.org/abs/cs/0505034), [doi:10.1007/11541868_16](https://doi.org/10.1007/11541868_16)]

On [[constructive analysis]] (with [[real number]]) in [[type theory]] implemented in [[Coq]] via the [[completion monad]]:

* [[Russell O'Connor]], *A Monadic, Functional Implementation of Real Numbers*, Math. Struc. Comp. Sci. **17** 1 (2007) 129-159 [[arXiv:cs/0605058](https://arxiv.org/abs/cs/0605058), [doi:10.1017/S0960129506005871](https://doi.org/10.1017/S0960129506005871)]

* [[Russell O'Connor]], _Certified exact transcendental real number computation in Coq_, in *Theorem Proving in Higher Order Logics. TPHOLs 2008*,  Lecture Notes in Computer Science **5170** (2008) 246-261 [[arXiv:0805.2438](https://arxiv.org/abs/0805.2438), [doi:10.1007/978-3-540-71067-7_21](https://doi.org/10.1007/978-3-540-71067-7_21)]

* [[Russell O'Connor]], _Incompleteness and Completeness: Formalizing Logic and Analysis in Type Theory_, PhD thesis, Radboud University Nijmegen (2009) [[ubn:2066/76032](https://repository.ubn.ru.nl/handle/2066/76032), [author webpage](http://r6.ca/Thesis/)]

Understanding well-behaved  [[lenses (in computer science)]] as [[coalgebra of a comonad|coalgebras]]  of the [[costate comonad]]:

* {#O’Connor10} [[Russell O’Connor]], *Lenses Are Exactly the Coalgebras for the Store Comonad* (30th Nov 2010) [[r6research:23705](https://r6research.livejournal.com/23705.html)]

* {#O’Connor11} [[Russell O'Connor]], *Functor is to Lens as Applicative is to Biplate: Introducing Multiplate*, contribution to [ICFP '11: ACM SIGPLAN International Conference on Functional Programming](https://dl.acm.org/doi/proceedings/10.1145/2036918) (2011) [[arXiv:1103.2841](https://arxiv.org/abs/1103.2841)]

## Related pages

* [[constructive analysis]]

* [[completion monad]]

category: people

[[!redirects Russel O'Connor]]


Russell O’Connor

* [webpage](http://r6.ca)

## related $n$Lab entries

* [[Gödel incompleteness theorem]]

category: people