数理解析研究所講究録
第 1866 巻 2013 年 68-81

68

Edge-state integrals on shaped triangulations

Rinat Kashaev
Section de mathematiques, Universit\’e de Gen\‘eve
Abstract

The partition functions of two state integral TQFT models of the Turaev-Viro type suggested in[11, 2] are briefly reviewed. It is also shown that their behavior under adding or removing a vertex
is consistent with the conjectural relationship between two models. This text is partially based on
author’s talk given at the workshop “Intelligence of Low-dimensional Topology” which took place at
RIMS, Kyoto University, on May 22-24, 2013.

1

Introduction

Recently, two state integra13-dimensional TQFT models of the Turaev-Viro type weresuggested in [11, 2]. In this paper, we review these models and analyze their behavior
with respect to the Pachner moves which add or remove vertices. To distinguish between
the models, we will call -model the one of [11], and -model the one of [2]. There are
two main ingredients common to both models: the shaped triangulations and Faddeev’squantum dilogarithm. Let us start by briefly recalling these ingredients.
$R$

1.1

$C$

Shaped triangulations

A triangulation for us is identical to a cellular complex where all cells are simplices.
refined version of triangulation given by A-triangulations in the sense of Hatcher [9] isused in the -model. If $X$ is a triangulation, we denote by
the set of -dimensional
cells of $X$ , and for any $i>j$ , we denote by
the set of pairs $(a, b)$ with(here is itself considered as a triangulation). If
andis the interior of $X,$
we set. For any two sets and $B,$
will denote the set of
all maps from to $B.$
A shaped treangulation is a triangulation $X$ of an oriented 3-dimensional pseudo manifold where each tetrahedron is provided with angles of an ideal hyperbolic tetrahedron.
More formally, a shape structure (or simply a shape) is a map
$A$

$C$

$i$

$\triangle_{i}(X)$$a\in\triangle_{i}(X)$$\Delta_{i,j}(X)$$X^{\circ}$$b\in\Delta_{j}(a)$$a$$B^{A}$$A$$\triangle_{i}(X^{\circ}):=\Delta_{i}(X)\backslash \triangle_{i}(\partial X)$

$A$

(1)

$\alpha:\triangle_{3,1}(X)arrow]0, \pi[$with the condition that the sum of the values at each vertex of each tetrahedron is , or,equivalently, that it takes equal values on opposite edges of each tetrahedron$\pi$$\alpha(T, e)=\alpha(T, e^{op}) , \forall T\in\triangle_{3}(X)$Supported in part by Swlss National Science Foundation.

,

(2)

69
and the sum over all edges of each tetrahedron is$2\pi,$$\sum_{e\in\Delta_{1}(T)}\alpha(T, e)=2\pi, \forall T\in\triangle_{3}(X)$.

(3)

The set of all shape structures on $X$ is denoted $S(X)$ .
By using the orientation of the triangulation, let us choose a cyclic order between threeedges at each vertex in each tetrahedron by saying that follows if by departing fromin the clockwise direction around the common vertex one arrives to . By using such
cyclic order, we define skew-symmetric functions
$f$

$e$

$e$

$f$

$\epsilon_{T}:\triangle_{1}(X)\cross\triangle_{1}(X)arrow\{-1,0,1\}, \forall T\in\triangle_{3}(X)$(4),

with $\epsilon_{T}(e, f)=1$ if and only if both and are edges of which do not form a pairof opposite edges, and follows . The $Neumann-$ Zagier simplectic structure in $S(X)$ isgiven by the following non-degenerate closed two-form [14]:
$f$

$T$

$f$

$e$

$e$

$\omega_{NZ}=\sum_{T\in\triangle_{3}(X)}\sum_{e,f\in\Delta_{1}(T)}\epsilon_{T}(e, f)d(T, e)\wedge d(T, f)$,(5)where any element $(T, e)$ of the set
is considered as a function on $S(X)$ definedby $(T, e)(\alpha):=\alpha(T, e)$ .
Two shape structures on one and the same triangulation $X$ are called gauge equivalent
if they are related by the action of the additive groupinduced by the Poisson
action of the total dihedral angles around the interior edges of the triangulationl.
formal characterization is as follows. Two shape structures and on a triangulation $X$
are gauge equivalent if and only if there exists a function$\triangle_{3,1}(X)$$\mathbb{R}^{\triangle_{1}(\mathring{X})}$$A$$\alpha$$\beta$(6)$g:\triangle_{1}(X)arrow \mathbb{R}$such that$g|_{\triangle_{1}(\partial X)}=0$

(7)

,

and
$\beta(T, e)=\alpha(T, e)+\sum_{f\in\triangle_{1}(T)}\epsilon_{T}(e, f)g(f) , \forall(T, e)\in\triangle_{3,1}(X)$.

(8)

We say that two shaped triangulations are related by a shaped 2–3 Pachner move ifthey differ only in the interior of a three dimensional ball as is shown in this picture$rightarrow$

where it is assumed that the total dihedral angles on the corresponding boundary edges are
pairwise equal. This implies that the only interior edge in the right hand side carries thelIn [1] this equivalence is called based gauge equivalence, while the (non-based) gauge equivalence is defined differently

70
total dihedral angle . In that case the edge is called balanced.
shaped triangulation
where all edges are interior and balanced is known as (strict) angle structure introduced
by Casson, Rivin and Lackenby as a linearized version of the hyperbolic structure [15, 13].
(quantum) state of a triangulation $X$ is an assignment of real numbers to all edges,
i.e. a map$A$$2\pi$$a$$A$$x:\Delta_{1}(X)arrow \mathbb{R}$

1.2

.

(9)

Faddeev’s quantum dilogarithm

Following [7], Faddeev’s quantum dilogartthm function [6] is defined by the formula$\Phi_{\hslash}(z):=\exp(\int_{\mathbb{R}+i\epsilon}\frac{e^{-i2xz}dx}{4\sinh(xb)\sinh(xb^{-1})x})$where$\hslash\in \mathbb{C}\backslash \mathbb{R}_{\leq 0},$

$b$

(10)

,

is any root of the equation$(b+b^{-1})^{-2}=\hslash$

,

(11)

and is an arbitrary sufficiently small positive real. From now on, we choose the uniquesuch that $0\leq\arg b<\pi/2$ , andwill be defined to be $(b+b^{-1})^{-1}$ . This implies that$\epsilon$$b$$\sqrt{\hslash}$$\Re\sqrt{\hslash}>0.$

The integral in (10) is absolutely convergent only in the strip
satisfies the functional equations$| \Im z|<|\Re(\frac{1}{2\sqrt{\hslash}})|$(12)$\Phi_{\hslash}(z-ib^{\pm 1}/2)=(1+e^{2\pi b^{\pm 1}z})\Phi_{\hslash}(z+ib^{\pm 1}/2)$

which can be used for extending the domain of definition ofplane. One has the inversion relation$\Phi_{\hslash}(z)\Phi_{\hslash}(-z)=e^{i\pi z^{2}}e^{-\pi i(2+\hslash^{-1})/12}$

and in the case when $\arg b>0$$( i.e. \hslash\not\in]0, \frac{1}{4}])$$\Phi_{\hslash}(z)$where itto the whole complex

,

(13)

one can show that$\Phi_{\hslash}(z)=\frac{(-qe^{2\pi bz};q)_{\infty}}{(-\overline{q}e^{2\pi b^{-1}z};\overline{q})_{\infty}}, q:=e^{2\pi ib^{2}} \overline{q}:=e^{-2\pi ib^{-2}}$

,

(14)

where we use the common notation in the theory of basic hypergeometric series$(x;y)_{\infty}:= \prod_{n=0}^{\infty}(1-xy^{n})$

.

(15)

Formula (14) and the continuity argument permit to extract the complete information
about the analytic properties of Faddeev’s quantum dilogarithm. Namely, it is a meromorphic function on with the set of poles$\mathbb{C}$$P_{\hslash}=ib( \frac{1}{2}+\mathbb{Z}_{\geq 0})+ib^{-1}(\frac{1}{2}+\mathbb{Z}_{\geq 0})$the set of zeros-$P_{\hslash}$

,

(16)

, i.e.
$(\Phi_{\hslash}(z))^{\pm 1}=0\Leftrightarrow z\in\mp P_{\hslash}$

,

(17)

71
and the asymptotic behavior at infinity:$\Phi_{\hslash}(z)|_{|z|arrow\infty}=\Phi_{\hslash}^{(\infty)}(z)+o(1)$

,

(18)

where
$\Phi_{\hslash}^{(\infty)}(z):=\{\begin{array}{ll}1 if |\arg z|>\frac{\pi}{2}+\arg be^{i\pi z^{2}}e^{-\pi i(2+\hslash^{-1})/12} if |\arg z|<\frac{\pi}{2}-\arg b\frac{(\overline{q}_{)}\cdot\overline{q})_{\infty}}{\frac{}{}\Theta(ibz;b^{2})\Theta(ib^{-1}z;-b^{-2}),(q,q)_{\infty}} if |\arg z-\frac{\pi}{2}|<\arg b\end{array}$if $| \arg z+\frac{\pi}{2}|<\arg b$

with the Jacobi theta-function$\Theta(z;\tau):=\sum_{n\in \mathbb{Z}}e^{\pi i\tau n^{2}+2\pi izn}, \Im\tau>0$

In the unitary case, corresponding toproperty$\hslash\in \mathbb{R}_{>0}$

(19)

.

, we have the complex conjugation(20)$\overline{\Phi_{\hslash}(z)}\Phi_{\hslash}(\overline{z})=1$

which implies that for any selfadjoint operator in a Hilbert space, the operator
is unitary. Moreover, if and are selfadjoint operators satisfying Heisenberg’s commutation relation$pq-qp=(2\pi i)^{-1}$ ,(21)
for example, the operators in
defined by
$A$

$p$

$\Phi_{\hslash}(A)$$q$$L^{2}(\mathbb{R})$$pf(x):=\frac{1}{2\pi i}\frac{\partial f(x)}{\partial x}, qf(x):=xf(x)$

,

(22)

then one has Faddeev’s quantum five term identity [5](23)$\Phi_{\hslash}(p)\Phi_{\hslash}(q)=\Phi_{\hslash}(q)\Phi_{\hslash}(p+q)\Phi_{\hslash}(p)$which can be shown to be equivalent to the integral version of Ramanujan’s
formula:$1\psi_{1}$$\int_{\mathbb{R}+i\epsilon}\frac{\Phi_{\hslash}(x+u)}{\Phi_{\hslash}(x-\frac{i}{2\sqrt{\hslash}})}e^{-2\pi iwx}dx=\frac{\Phi_{\hslash}(u)\Phi_{\hslash}(\frac{i}{2\sqrt{\hslash}}-w)}{\Phi_{\hslash}(u-w)}e^{\pi i(1+\hslash^{-1})/12}$

summation(24)

where the integral is absolutely convergent under the conditions$0<\Im w<\Im u<\Re\perp$(25)$2\sqrt{\hslash}.$As a particular limiting case of (24), we have the following Fourier transformation formula.

$\int_{\mathbb{R}+i\epsilon}\Phi_{\hslash}(x)e^{-2\pi iwx}dx=\Phi_{\hslash}(\frac{i}{2\sqrt{\hslash}}-w)e^{-\pi iw^{2}}e^{\pi i(1+\hslash^{-1})/12}$(26)

In what follows, for simplicity of presentation, we will always assume the unitary case,i.e. that
Remark 1. Faddeev’s quantum dilogarithm is closely related to Shintani’s double sinefunction [16, 12, 3], but the operator identity (23) seems not to be known before [5].
$\hslash\in \mathbb{R}_{>0}.$

72
2

The

2.1

Weight functions$R$-modelFor a tetrahedron$T$with shape$\alpha$and in state , one assigns a weight function$x$$W_{\hslash}(T, \alpha, x)=\prod_{j=1}^{3}\Psi_{\hslash}(x_{j+1}+x_{j+1}’-x_{j-1}-x_{j-1}’+\frac{i}{\sqrt{\hslash}}(\frac{1}{2}-\frac{\alpha_{j}}{\pi}))$(27)

where all indices are considered modulo 3, the notation for the shape and state variables
corresponds to this picture

while the function$\Psi_{\hslash}(x)$is a renormalized version of Faddeev’s quantum dilogarithm:(28)$\Psi_{\hslash}(x)=\frac{\Phi_{\hslash}(x)}{\Phi_{\hslash}(0)}e^{-i\pi x^{2}/2}$

which corresponds to the simplest version of the inversion relation (13):$\Psi_{\hslash}(x)\Psi_{\hslash}(-x)=1$

.

(29)

Remark 2. For calculational purposes, it is useful to write the integral Ramanujan formula (24) in the form$\varphi(x, y):=\int_{\mathbb{R}}\frac{\Phi_{\hslash}(t+\frac{x}{2})}{\Phi_{\hslash}(t-\frac{x}{2})}e^{2\pi ity}dt$$= \Psi_{\hslash}(x-\frac{i}{2\sqrt{\hslash}})\Psi_{\hslash}(y+\frac{i}{2\sqrt{\hslash}})\Psi_{\hslash}(-x-y+\frac{i}{2\sqrt{\hslash}})$(30)

so that, denoting$x*j \alpha :=x_{j+1}+x_{j+1}’-x_{j-1}-x_{j-1}’-\frac{i\alpha_{j}}{\pi\sqrt{\hslash}}, \forall j\in\{1,2,3\}$

,

(31)

the tetrahedral weight function (27) can be written$W_{\hslash}(T, \alpha, x)=\varphi(\frac{i}{\sqrt{\hslash}}+x*j\alpha, x*k\alpha)$

,

(32)

where $\{j, k\}$ is any 2-element subset of {1, 2, 3}.
To a triangulation $X$ in state and with shape , we associate the weight function
given by the product of tetrahedral weight functions for all tetrahedra of $X$ with induced
states and shapes:
.
(33)
$x$

$\alpha$$W_{\hslash}(X, \alpha, x)=\prod_{T\in\triangle_{3}(X)}W_{\hslash}(T, x|_{T}, \alpha|_{T})$

73
2.2

Partition functions

A state gauge transformation on a triangulation $X$ is an elementin the space of states according to the map$\mathbb{R}^{\Delta_{1}(X)}\cross \mathbb{R}^{\triangle o(X)}arrow \mathbb{R}^{\triangle_{1}(X)}, (x, g)\mapsto x^{g}$$g\in \mathbb{R}^{\triangle_{0}(X)}$

which acts
(34),

defined by the formula$x^{g}(e)=x(e)+g(v_{1})+g(v_{2}) , \partial e=\{v_{1}, v_{2}\}$

.

(35)

For any subset, an element
will be thought of as a state gaugetransformation vanishing on the complement of . One can easily verify the state gauge
invariance of the weight function (33):$S\subset\Delta_{0}(X)$$g\in \mathbb{R}^{S}$$S$$W_{\hslash}(X, \alpha, x)=W_{\hslash}(X, \alpha, x^{g}) , \forall g\in \mathbb{R}^{\triangle o(X)}$

.

(36)

A state gauge fixing on a triangulation $X$ is an assignment to each interior vertex of $X$of a real valued linear formon the vector space of statessuch that$\lambda$$\mathbb{R}^{\triangle_{1}(X)}$$\lambda_{v}$$\langle\lambda_{v}, x^{g}\rangle=\langle\lambda_{v}, x\rangle+g(v) , \forall g\in \mathbb{R}^{\triangle o(\mathring{X})}$

To a triangulation $X$ of shape$\alpha$partition function

and boundary state.

$s\in \mathbb{R}^{\triangle_{1}(\partial X)}$(37)
, we associate the$R_{\hslash}(X, \alpha, s):=\int_{x\in \mathbb{R}^{\triangle_{1}(X)}}W_{\hslash}(X, \alpha, x)\delta(s-x|_{\partial X})\delta(\langle\lambda, x\rangle)dx$

where$\lambda$.

(38)

is a state gauge fixing on $X,$$\delta(s-x|_{\partial X}):=\prod_{e\in\triangle_{1}(\partial X)}\delta(s(e)-x(e))$$\delta(\langle\lambda, x\rangle):=\prod_{v\in\triangle o(\mathring{X})}\delta(\langle\lambda_{v}, x\rangle)$$dx:= \prod_{e\in\triangle_{1}(X)}dx(e)$

,

(39)

(40)

.

(41)

The main result of [11] is the following theorem.

Theorem 1 ([11]). The partition function
is an absolutely convergent integral
independent of the choice of the state gauge fixing and invariant under the shaped 2-3
Pachner moves and the shape gauge tmnsformations.
$R_{\hslash}(X, \alpha, s)$$\lambda$

3

The

3.1

Leveled shaped triangulations$C$-modelSimilarly to known TQFT models of the Reshetikhin-Turaev type, there is a phase ambiguity in the Teichm\"uller TQFT of [1, 2], which is taken int account by assigning a real$0$

74
number called level to a shaped triangulation and extending the shaped gauge transformations and the shaped 2–3 Pachner moves in a certain specific way. More precisely, a, and two leveled shapes
leveled shape on a triangulation $X$ is a pairandon a triangulation $X$ are gauge equivalent if and only if and aregauge equivalent through a function in equations (6) $-(8)$ and$(\alpha, u)\in S(X)\cross \mathbb{R}$$(\alpha, u)$$(\beta, v)$$\alpha$$\beta$$g$$v=u+ \frac{1}{4\pi}\sum_{(T,e)\in\triangle_{31}(X)},g(e)(\alpha(T, e)-\frac{\pi}{3})$

.

(42)

We also say that leveled shaped triangulations $(X, \alpha, u)$ and $(Y, \beta, v)$ are related by aleveled shaped 2–3 Pachner move if the underlying shaped triangulations are related bya shaped 2–3 Pachner move, where, for example, is obtained from $X$ by removing an
edge
and
$Y$

$e\in\triangle_{1}(X)$$v=u+ \frac{1}{48}\sum_{(T,f)\in\Delta_{31}(X)},\epsilon_{T}(e, f)\alpha(T, f)$

3.2
Let

.

(43)

Weight functions$T\in \mathbb{R}^{3}$be an Euclidean tetrahedron with vertex ordering mapping$v:\{0,1,2,3\}\ni i\mapsto v_{i}\in\triangle_{0}(T)$

.

(44)

We say that is positive (respectively negative) if the vectors
form a$\alpha\inS(T)$right-handed (respectively left-handed) frame in . For any shapeand state, we associate a weight function$T$$v_{0}v_{1},$$v_{0}v_{2},$$v_{0}v_{3}$$\mathbb{R}^{3}$$x\in \mathbb{R}^{\triangle_{1}(T)}$$B_{\hslash}(T, \alpha, x)=g_{\alpha_{1},\alpha}3(x_{02}+x_{13}-x_{03}-x_{12}, x_{02}+x_{13}-x_{01}-x_{23})$

if

$T$

is positive, and the same but complex conjugate expression if$x_{ij} :=x(v_{i}v_{j}) , \alpha_{i} :=\alpha_{T}(v_{0}v_{i})/2\pi$$\tilde{\psi}_{a,c}’(s):=e^{-\pi is^{2}}\tilde{\psi}_{a,c}(s)$

(46)

,

(47)

,

$\tilde{\psi}_{a,c}(s):=\int_{\mathbb{R}}\psi_{a,c}(t)e^{-2\pi ist}dt$

is negative. Here(45),$g_{a,c}(s, t):= \sum_{m\in \mathbb{Z}}\tilde{\psi}_{a,c}’(s+m)e^{\pi it(s+2m)}$

$T$

(48)

,

and
$\psi_{a,c}(t):=\frac{e^{2\pi at/\sqrt{\hslash}}e^{\pi i(-a(a+c)+(4(a-c)+1)/24)/\hslash}}{\Phi_{\hslash}(t-i(a+c)/\sqrt{\hslash})}$

.

(49)

Remark 3. In formula (46), we use a transformation which associates to functions on. This transform by itself has a number of remarkablequasi-periodic functions on
properties, and in the literature, it can be traced back to the works of Weil [17], Gel’fand[8], and Zak [18].
$\mathbb{R}$$\mathbb{R}^{2}$

75
Let $X$ be a -triangulation of an oriented pseudo 3-manifold, where we distinguishbetween positive and negative tetrahedra. For any shape $\alpha\in S(X)$ and state
we define the weight function$\triangle$$x\in \mathbb{R}^{\triangle_{1}(X)},$$B_{\hslash}(X, \alpha, x):=\prod_{T\in\Delta_{3}(X)}B_{\hslash}(T, \alpha|_{\triangle_{3,1}(T)}, x|_{\triangle_{1}(T)})$

3.3

(50)

.

Partition functionsLet
be a shaped -triangulation in a fixed boundary statea partition function by the formula$(X, \alpha)$$\triangle$$s\in \mathbb{R}^{\triangle_{1}(\partial X)}$

. We define(51)$C_{\hslash}(X, \alpha, s):=\int_{x\in[0,1]^{\triangle_{1}(\dot{X})}}\int_{x\in\mathbb{R}^{\triangle_{1}(\partial X)}}B_{\hslash}(X, \alpha, x)\delta(\dot{x}-s)dx$

where$\mathring{x}:=x|_{\triangle_{1}(\mathring{X})}, \dot{x}:=x|_{\Delta_{1}(\partial X)}$

,

(52)

$\delta(\dot{x}-s):=\prod_{e\in\triangle_{1}(\partial X)}\delta(x(e)-s(e)) , dx:=\prod_{e\in\triangle_{1}(X)}dx(e)$

.

(53)

Theorem 2 ([2]). The partition function
is absolutely convergent integral
which admits complex analytic continuation to a meromorphic function of its arguments(complex shapes and boundary states) and, for any leveled shape, thequantity
is invariant under the leveled shaped 2–3 Pachner moves andthe leveled shape gauge tmnsformations.
$C_{\hslash}(X, \alpha, s)$$(\alpha, u)\in S(X)\cross \mathbb{R}$$e^{-iu/\hslash}C_{\hslash}(X, \alpha, s)$4

Calculations with two tetrahedra

It is expected thatand -models are related according to the following conjecture.
Conjecture 1 ([11, 2]). For any shaped -triangulationof a closed 3-manifold, one
has the equality(54)
Here we give an additional justification for this conjecture by calculating and comparing
the two partition functions for triangular pillowcases triangulated into two tetrahedra with
one interior vertex.
It is convenient to encode -triangulations graphically by associating to each tetrahedron $T$ an element as in this picture
$R$-

$C$

$\triangle$$(X, \alpha)$$R_{\hslash}(X, \alpha)=2|C_{\hslash}(X, \alpha)|^{2}$$\triangle$$T=\ovalbox{\tt\small REJECT}$

where the vertical segments, ordered from left to right, correspond to the faces
for$i\in\{0,1,2,3\}$ , and by joining the corresponding segments for identified faces. Let us
consider the diagram$\partial_{i}T$

(55)

76
which represents acommon faces:$\triangle$-triangulation composed of two tetrahedra$\partial_{i}T^{-}\sim\partial_{i}T^{+}, i\in\{1,2,3\}$

glued along three$T^{\pm}$(56),$i\in\{0,1,2,3\}$ , where the vertex
isso that the -skeleton consists of four vertices
in the interior, and all others are on the boundary. Thus, is a triangulation of a 3-ballwith one interior vertex. Topologically, it can be conveniently visualized as a cellularwith all four vertices in the plane
decomposition of the standard unit ball inat the positions$0$$v_{0}$$v_{i},$$Y$$\mathbb{R}^{3}\simeq \mathbb{C}\cross \mathbb{R}$$\mathbb{C}\cross\{0\}$$v_{0}=(0,0) , v_{1}=(1,0) , v_{2}=(e^{2\pi i/3},0) , v_{3}=(e^{-2\pi i/3},0)$

(57)

,

with the six edges being given by three arcs of the equator and three straight radii of theequatorial unit disc.
by the formula
Define a meromorphic function on$\mathbb{C}^{3}$$f_{\hslash}(z):= \prod_{j=1}^{3}\Psi_{\hslash}(\frac{i}{2\sqrt{\hslash}}(1-\frac{z_{j}}{\pi})) , Z=(z_{1}, z_{2}, z_{3})$and a mapof :$w:S(Y)arrow \mathbb{R}_{>0}^{3}\subset \mathbb{C}^{3}$

(58)

,

which associates total dihedral angles to interior edges$Y$$w_{j}(\alpha) :=\alpha(T^{+}, v_{0}v_{j})+\alpha(T^{-}, v_{0}v_{j}) , \forall j\in\{1,2,3\}$

(59)

.

Theorem 3. The following equalities hold true:(60)$R_{\hslash}(Y, \alpha, s)=R_{\hslash}(Y, \alpha) :=|f_{\hslash}(w(\alpha))|^{2}$$C_{\hslash}(Y, \alpha, s)=C_{\hslash}(Y, \alpha) :=f_{\hslash}(w(\alpha))e^{\frac{i}{4\pi\hslash}\chi(\alpha)}$

(61)

,

where
$\chi(\alpha):=c_{1}^{-}c_{3}^{+}-c_{3}^{-}c_{1}^{+}, c_{j}^{\pm}:=\alpha(T^{\pm}, v_{0}v_{j})-\frac{\pi}{3}, \forall j\in\{1,3\}$

.

(62)

. After performing the (trivial) integration over boundary edge states in (38), the partition function becomes the following triple integral:

Proof. Calculation of$R_{\hslash}(Y, \alpha, s)$$R_{\hslash}(Y, \alpha, s)$$= \int_{\mathbb{R}^{3}}\varphi(2c_{\hslash}+x_{2}+s_{2}-x_{3}-s_{3}-2c_{\hslash}\alpha_{1}^{+}, x_{3}+s_{3}-x_{1}-s_{1}-2c_{\hslash}\alpha_{2}^{+})$$\cross\varphi(2c_{\hslash}+x_{3}+s_{3}-x_{2}-s_{2}-2c_{\hslash}\alpha_{1}^{-}, x_{1}+s_{1}-x_{3}-\mathcal{S}_{3}-2c_{\Gamma_{1}}\alpha_{2}^{-})\delta(x_{3})dx_{1}dx_{2}dx_{3}(63)$

where
(64)

$x_{j};=x(v_{0}v_{j}) , s_{j}:=s(v_{j+1}v_{j-1}) , j\in\{1,2,3\}$ ,$c_{\hslash}:= \frac{i}{2\sqrt{\hslash}}, \alpha_{j}^{\pm}:=\alpha(T^{\pm}, v_{0}v_{j})/\pi, j\in\{1,2,3\}$is chosen to be
and the state gauge fixing at the only interior vertex
Integrating overand shifting two other variables as follows:$v_{0}$(65),$\langle\lambda_{v_{0}},$$x\rangle=x_{3}.$$x_{3}$$x_{1}\mapsto x_{1}+s_{3}-s_{1}, X_{2}\mapsto x_{2}+s_{3}-s_{2}$

,

(66)

77
we observe that the integral is manifestly independent of the boundary state:r.h. of (63)$s$.

$= \int_{\mathbb{R}^{2}}\varphi(2c_{\hslash}+x_{2}-2c_{\hslash}\alpha_{1}^{+}, -x_{1}-2c_{\hslash}\alpha_{2}^{+})\varphi(2c_{\hslash}-x_{2}-2c_{\Gamma_{1}}\alpha_{1}^{-}, x_{1}-2c_{\Gamma_{l}}\alpha_{2}^{-})dx_{1}dx_{2}$(67)Futhermore, by using the analyticity of the integrand, we can make complex shifts(68)$x_{1}\mapsto x_{1}+c_{\hslash}(\alpha_{2}^{-}-\alpha_{2}^{+}) , x_{2}\mapsto x_{2}+c_{\hslash}(\alpha_{1}^{+}-\alpha_{1}^{-})$

without changing the value of the integral so that, introducing the normalized total dihedral angles$\omega_{i}:=c_{\hslash}(\alpha_{i}^{+}+\alpha_{i}^{-}) , i\in\{1,2,3\}, \sum_{i=1}^{3}\omega_{i}=2c_{\hslash}$

(69)

,

we arrive at a manifestly shape gauge invariant form
r.h.s of (67).

$= \int_{\mathbb{R}^{2}}\varphi(2c_{\hslash}+x_{2}-\omega_{1}, -x_{1}-\omega_{2})\varphi(2c_{\Gamma_{1}}-x_{2}-\omega_{1}, x_{1}-\omega_{2})dx_{1}dx_{2}$(70)Now, if we use the integral representation (30), we obtainr.h. of (70)$s$$= \int_{\mathbb{R}^{4}}\frac{\Phi_{\hslash}(t_{1}+c_{\hslash}+\frac{1}{2}(x_{2}-\omega_{1}))\Phi_{\hslash}(t_{2}+c_{\hslash}-\frac{1}{2}(x_{2}+\omega_{1}))}{\Phi_{\hslash}(t_{1}-c_{\hslash}-\frac{1}{2}(x_{2}-\omega_{1}))\Phi_{\hslash}(t_{2}-c_{\hslash}+\frac{1}{2}(x_{2}+\omega_{1}))}$$\cross e^{2\pi i(t_{1}(-x_{1}-\omega_{2})+t_{2}(x_{1}-\omega_{2}))}dt_{1}dt_{2}dx_{1}dx_{2}$

(71)

.

Integration over produces a delta-fUnction which removes one of the -variables bringing
us back to a 2-dimensional integral:$t$$x_{1}$

r.h. of (71)$s$,$= \int_{\mathbb{R}^{2}}\frac{\Phi_{\hslash}(t+c_{\hslash}+\frac{1}{2}(x-\omega_{1}))\Phi_{T\iota}(t+c_{\hslash}-\frac{1}{2}(x+\omega_{1}))}{\Phi_{\hslash}(t-c_{\hslash}-\frac{1}{2}(x-\omega_{1}))\Phi_{\hslash}(t-c_{\hslash}+\frac{1}{2}(x+\omega_{1}))}e^{-4\pi it\omega_{2}}dtdx$(72)

where we have removed the indices of the integration variables. Now, we make twoconsecutive shifts of integrations variables,
followed by $x\mapsto x-t$ :$t \mapsto t+\frac{1}{2}x$r.h. of (72)$s$$= \int_{\mathbb{R}^{2}}\frac{\Phi_{\hslash}(t+c_{\hslash}+x-\omega_{1})\Phi_{\hslash}(t+c_{\hslash}-\frac{1}{2}\omega_{1})}{\Phi_{\hslash}(t-c_{\hslash}+\frac{1}{2}\omega_{1}\Phi_{\hslash}(t-c_{\hslash}+x+\frac{1}{2}\omega_{1})}e^{-4\pi i(t+\frac{1}{2}x)\omega_{2}}dtdx$$= \int_{\mathbb{R}^{2}}\frac{\Phi_{\hslash}(c_{\hslash}+x-\frac{1}{2}\omega_{1})\Phi_{\hslash}(t+c_{\hslash}-\frac{1}{2}\omega_{1})}{\Phi_{\hslash}(t-c_{\hslash}+\frac{1}{2}\omega_{1})\Phi_{\hslash}(-c_{\hslash}+x+\frac{1}{2}\omega_{1})}e^{-2\pi i(t+x)\omega_{2}}dtdx$$=( \int_{\mathbb{R}}\frac{\Phi_{\hslash}(c_{\hslash}+x-\frac{1}{2}\omega_{1})}{\Phi_{\hslash}(-c_{\hslash}+x+\frac{1}{2}\omega_{1})}e^{-2\pi ix\omega_{2}}dx)^{2}=(\varphi(2c_{\Gamma\iota}-\omega_{1}, -\omega_{2}))^{2}$$=( \prod_{i=1}^{3}\Psi_{\hslash}(c_{\hslash}-\omega_{i}))^{2}=(f_{\hslash}(w(\alpha)))^{2}=|f_{\hslash}(w(\alpha))|^{2}$

where the last equality is valid because the quantities$\Psi_{\hslash}(c_{\Gamma_{l}}-\omega_{i})$,are real numbers.

(73)

78
. After performing the (trivial) integration over boundary
Calculation ofedge states in (51), the partition function is the following triple integral:$C_{\hslash}(Y, \alpha, s)$$C_{\hslash}(Y, \alpha, s)=\int_{[0,1]^{3}}g_{a^{+},c^{+}}(x_{2}+s_{2}-x_{3}-s_{3}, x_{2}+s_{2}-x_{1}-s_{1})$$\cross\overline{g}_{a^{-},c^{-}}(x_{2}+s_{2}-x_{3}-s_{3}, x_{2}+s_{2}-x_{1}-s_{1})dx_{1}x_{2}dx_{3}$

,

(74)

where the notation in (64) is used and$a^{\pm}:= \frac{\alpha(T^{\pm},v_{0}v_{1})}{2\pi}, c^{\pm}:=\frac{\alpha(T^{\pm},v_{0}v_{3})}{2\pi}$

.

(75)

After the change of the integration variables(76)$(x_{1}, x_{3})=(-y-s_{1}+x_{2}+s_{2}, -x-s_{3}+x_{2}+s_{2})$the integrand becomes independent of the boundary states
integration overgives the multiplicative factor 1:$s_{i}$and the variable$x_{2}$

so that$x_{2}$

r.h. of (74)$s$$= \int_{[0,1]^{2}}g_{a+,c+}(x, y)\cross\overline{g}_{a^{-},c^{-}}(x, y)dxdy$

.

(77)

Substituting the definition (46), we haver.h. of (77)$s$$= \int_{[0,1]^{2}}\sum_{m,n\in \mathbb{Z}}\tilde{\psi}_{a\dagger,c+}’(x+m)e^{\pi iy(2m+x)}\overline{\tilde{\psi}’}_{a^{-},c^{-}}(x+n)e^{-\pi iy(2n+x)}dxdy$$= \int_{[0,1]}\sum_{m,n\in \mathbb{Z}}\tilde{\psi}_{++}’a,c(x+m)\overline{\tilde{\psi}’}_{a^{-},c^{-}}(x+n)\delta_{m,n}dx$$= \int_{[0,1]}\sum_{m\in \mathbb{Z}}\tilde{\psi}_{a^{+},c^{+}}’(x+m)\overline{\tilde{\psi}’}_{a^{-},c^{-}}(x+m)dx$$= \int_{\mathbb{R}}\tilde{\psi}_{a^{+},c^{+}}’(x)\overline{\tilde{\psi}’}_{a^{-},c^{-}}(x)dx=\langle\tilde{\psi}_{a+,c^{+}}’|\tilde{\psi}_{a^{-},c^{-\rangle=}}’\langle\psi_{a+,c^{+}}|\psi_{a^{-},c}-\rangle$

,

(78)

where in the last two equalities we use Dirac’s bra-ket notation for the scalar product in theHilbert space
and unitarity of the transformations in (47) and (48) with respect tothis scalar product. Substituting now (49) and, by using analyticity of Faddeev’s quantum
dilogarithm, shifting the integration variable in imaginary direction, we finally obtain$L^{2}(\mathbb{R})$$\langle\psi_{a+,c^{+}}|\psi_{a^{-},c}-\rangle e^{\pi i(a^{-}+c^{+}-a^{+}-c^{-})/6\hslash}$$=e^{\pi i(a^{-}(a^{-}+c^{-})-a^{+}(a^{+}+c^{+}))/r_{1}} \int_{\mathbb{R}\Phi_{\hslash}(x-\frac{i}{\sqrt{\hslash}}(a^{+}+C^{+}))^{e^{2\pi(a^{+}+a^{-})x/\sqrt{\hslash}}}}^{\Phi_{\hslash}(x+\frac{i}{\sqrt{\hslash}}(a^{-}+c^{-}))}dx$$=e^{\pi i(a^{-}c^{+}-a^{+}c^{-})/\hslash} \int_{\mathbb{R}}\frac{\Phi_{\hslash}(x+\frac{i}{2\sqrt{\hslash}}(a^{+}+a^{-}+c^{+}+c^{-}))}{\Phi_{\hslash}(x-\frac{i}{2\sqrt{\hslash}}(a^{+}+a^{-}+\mathcal{C}^{+}+c^{-}))}e^{2\pi(a^{+}+a^{-})x/\sqrt{\hslash}}dx$79$=e^{\pi i(a^{-}c^{+}-a^{+}c^{-})/\hslash} \varphi(\frac{i}{\sqrt{\hslash}}(a^{+}+a^{-}+c^{+}+c^{-}),$$=e^{\pi i(a^{-}c^{+}-a^{+}c^{-})}f_{\hslash}(w(\alpha))$$- \frac{i}{\sqrt{\hslash}}(a^{+}+a^{-}))$.

(79)$\square$Remark 4. Let
be a shaped -triangulation containing as a sub-triangulation,and let $F(X, \alpha, s)$ be any one of the two partition functions. Then, from Theorem 3 it
easily follows that one has the following factorization$(X, \alpha)$$Y$$\triangle$$F(X, \alpha, s)=F(Y, \alpha)F(X’, \alpha’, s)$

,

(80)

where
is obtained fromby applying a shaped 2–0 Pachner move: $X’$ isobtained from $X$ by removing the interior part of and identifying the remaining two
triangular faces with each other, and$(X\prime, \alpha’)$$(X, \alpha)$$Y$$\alpha’=\alpha|_{\Delta_{3,1}(X\backslash \mathring{Y})}$As Theorem 3 also implies thatwith Conjecture 1.
4.1.

$R_{\hslash}(Y, \alpha)=|C_{\hslash}(Y, \alpha)|^{2}$(81), factorization (80) is consistentApplication to knot invariants

An $H$-triangulation is a pair $(X, H)$ where $X$ is a triangulation of a closed compact oriented 3-manifold $M$ and $H$ is a Hamiltonian 1-dimensional sub-complex of $X$ representing
a link $L\subset M$ . Pachner moves of types 2–3, 0–2 and their inverses can be extended torelative Pachner moves of $H$-triangulations as follows. Let $(X, H)$ be an $H$-triangulation.
All moves which do not involve the edges of $H$ are applicable and stay the same as in thecase of usual triangulations. The moves which do not add or remove vertices and which
involve edges from $H$ are forbidden.
2–0 move which removes a vertex necessarily
involves at least two edges of $H$ which form two sides of a unique triangle , and the moveis applicable if and only if it does not involve any other edge from $H$ , and upon applicationof the move one replaces those two edges of $H$ by the third side of triangle . Topologically,
this move corresponds to an elementary isotopy of a polygonal link. Originally, relative
Pachner moves of $H$-triangulations were introduced in [10] and it was conjectured that
their finite sequences can relate any two topologically equivalent $H$ -triangulations. The
conjecture subsequently was proven in [4].
The shaped version of relative Pachner moves is straightforward, but under the moves
which add or remove edges, initially balanced edges can become unbalanced so that topological significance of such moves is yet not clear. However, in the case of generalized
shapes where the angles can be any real numbers, positive or negative, shapes becomea very convenient means to encode Hamiltonian paths: one imposes the condition thatthe total dihedral angle around an edge is zero if the edge is from the Hamiltonian pathandotherwise. In that case, all shaped relative Pacher moves preserve this class ofshapes. This scheme was used in [1] for defining renormalized partition functions of 1vertex $H$-triangulations. Theorem 3 allows us to extend this definition to the case of$H$ -triangulations
with arbitrary number of vertices as follows.
$A$

$t$

$t$

$2\pi$

80
Let $(X, H, \alpha)$ be a shaped$H-\Delta$-triangulation. Let$\omega_{\alpha}:\triangle_{1}(X)arrow \mathbb{R}_{>0}, e\mapsto\sum_{T\in\Delta_{3}(X)}\alpha(T, e)$,be the map which associates to each edge the total dihedral angle around it.
. Defineoreither the partition function(82)$L\dot{e}tF$

be$|C_{\hslash}|$$R_{\hslash}$$v_{F}(x):=| \Psi_{\hslash}(\frac{i}{2\sqrt{\hslash}}(1-\frac{x}{\pi}))|^{n_{F}}$

(83)

where
$n_{F}:=\{\begin{array}{l}1 if F=|C_{\hslash}|2 if F=R_{\hslash}\end{array}$

Let
that

$\alpha_{H}$(84)be a generalized (i.e. possibly non-positive real valued) shape structure on $X$ such$\omega_{\alpha_{H}}(e)=\{\begin{array}{l}0 if e\in\triangle_{1}(H)2\pi otherwise\end{array}$(85)

We define the renormalized partition function by the following formula:$\tilde{F}(X, H) :=\lim_{\alphaarrow\alpha_{H}}\frac{F(X,\alpha)}{\prod_{e\in\Delta_{1}(H)}\nu_{F}(\omega_{\alpha}(e))}$(86)

Theorem 4. Under the condition that the limit in (86) exists, the renormalized partition
and is invariant under shaped
does not depend on the choice offunction
relative Pachner moves of the pair $(X, H)$ .
$\tilde{F}(X, H)$$\alpha_{H}$Proof. The proof follows from a straightforward verification of the shaped relative Pachnermoves and shape gauge invariance.
Acknowledgement$\square$$s$

Ellegaard Andersen, Feng Luo and Grigory
This work is based on joint works with
Vartanov, and I would like thank them for collaboration and fruitful discussions. This
work was completed during my visit to MSC of Tsignhua University, and it is a great
pleasure to thank people of MSC for their hospitality. would also like to thank Tomotada
Ohtsuki for inviting me to write this article and his hospitality at RIMS, Kyoto University.
This work is supported in part by Swiss National Science Foundation.
$J\emptyset rgen$$I$

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Ellegaard Andersen and Rinat Kashaev. A new formulation of the Teichm\"uller
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Suisse-mail address: Rinat. Kashaev@unige. ch$E$