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Turaev-Viro invariant from the modular double of $\mathrm {U}_{q}\mathfrak{sl}(2;\mathbb R)$

Tianyue Liu, Shuang Ming, Xin Sun, Baojun Wu, Tian Yang

TL;DR

This work constructs a family of Turaev-Viro type invariants for hyperbolic 3-manifolds with totally geodesic boundary from the modular double of $U_q\mathfrak{sl}(2;\mathbb{R})$, defined by a state-integral over Kojima triangulations using $b$-6j symbols. It proves convergence and topological invariance of these invariants, and derives their sharp asymptotics as $b\to0$: they decay like $e^{-\mathrm{Vol}(M)/(\pi b^2)}$ with a subleading $1$-loop term given by the adjoint twisted Reidemeister torsion of the double $DM$. The analysis hinges on a detailed saddle-point evaluation of the $b$-6j symbol, the Luo–Yang correspondence between angle structures and hyperbolic polyhedral metrics, and the Murakami–Yano volume formula together with Wong–Yang torsion calculations. The results provide a rigorous bridge between hyperbolic geometry and noncompact quantum invariants, and point toward connections with Teichmüller/Virasoro TQFT and Liouville CFT in a broader 3D–2D correspondence context.

Abstract

We define a family of Turaev-Viro type invariants of hyperbolic $3$-manifolds with totally geodesic boundary from the $6j$-symbols of the modular double of $\mathrm U_{q}\mathfrak{sl}(2;\mathbb R)$, and prove that these invariants decay exponentially with the rate the hyperbolic volume of the manifolds and with the $1$-loop term the adjoint twisted Reidemeister torsion of the double of the manifolds.

Turaev-Viro invariant from the modular double of $\mathrm {U}_{q}\mathfrak{sl}(2;\mathbb R)$

TL;DR

This work constructs a family of Turaev-Viro type invariants for hyperbolic 3-manifolds with totally geodesic boundary from the modular double of , defined by a state-integral over Kojima triangulations using -6j symbols. It proves convergence and topological invariance of these invariants, and derives their sharp asymptotics as : they decay like with a subleading -loop term given by the adjoint twisted Reidemeister torsion of the double . The analysis hinges on a detailed saddle-point evaluation of the -6j symbol, the Luo–Yang correspondence between angle structures and hyperbolic polyhedral metrics, and the Murakami–Yano volume formula together with Wong–Yang torsion calculations. The results provide a rigorous bridge between hyperbolic geometry and noncompact quantum invariants, and point toward connections with Teichmüller/Virasoro TQFT and Liouville CFT in a broader 3D–2D correspondence context.

Abstract

We define a family of Turaev-Viro type invariants of hyperbolic -manifolds with totally geodesic boundary from the -symbols of the modular double of , and prove that these invariants decay exponentially with the rate the hyperbolic volume of the manifolds and with the -loop term the adjoint twisted Reidemeister torsion of the double of the manifolds.

Paper Structure

This paper contains 29 sections, 59 theorems, 465 equations, 25 figures.

Table of Contents

  1. Introduction
  2. Uq(sl(2;R)) 6j-symbol and its asymptotics
  3. Uq(sl(2,R)) Turaev-Viro invariants: state-integral convergence, topological invariance and asymptotics
  4. Outlook and perspectives
  5. Preliminaries
  6. Double sine function
  7. Dilogarithm function
  8. Relationship between double sine function and dilogarithm function
  9. Contour of integration in the $b$-$6j$ symbol
  10. Truncated hyperideal tetrahedra and co-volume function
  11. Hyperbolic polyhedral metrics and co-volume function
  12. Angle structures and generalized hyperbolic polyhedral metrics
  13. Kojima ideal triangulations
  14. Adjoint twisted Reidemeister torsions
  15. Saddle point approximations
  16. ...and 14 more sections

Key Result

Theorem 1.2

Let $(l_1,\dots, l_6)\in{\mathbb R_{>0 }^6}$ be the lengths of the edges of a truncated hyperideal tetrahedron $\Delta.$ Then as $b\to 0,$ In particular,

Figures (25)

  • Figure 1: Contour $\Gamma$ and possible zeros and poles (located in the white rays) of the integrand in \ref{['b-6j']}.
  • Figure 2: Region $H_{\delta,K}$
  • Figure 3: (1) truncated hyperideal tetrahedron, (2) flat tetrahedron
  • Figure 4: Inserted layered flat tetrahedra: the flat tetrahedra respectively with hyperideal vertices $\{p_{ik},p_{jl}, p_1,p_2\},$$\{p_{ik},p_{jl}, p_2,p_3\},$$\{p_{ik},p_{jl}, p_4,p_5\},$$\{p_{ik},p_{jl}, p_5,p_6\}$ and $\{p_{ik},p_{jl}, p_6,p_7\}$ are inserted to connect the ideal triangulations of the hyperideal polygon on the top with hyperideal triangles of vertices $\{p_{ik},p_1,p_2\},$$\{p_{ik},p_2,p_3\},$$\{p_{ik},p_3,p_{jl}\},$$\{p_{ik},p_{jl},p_4\},$$\{p_{ik},p_4,p_5\},$$\{p_{ik},p_5,p_6\}$ and $\{p_{ik},p_6,p_7\}$ and on the bottom with hyperideal triangles of vertices $\{p_{jl},p_3,p_2\},$$\{p_{jl},p_2,p_1\},$$\{p_{jl},p_1,p_{ik}\},$$\{p_{jl},p_{ik},p_7\},$$\{p_{jl},p_7,p_6\},$$\{p_{jl},p_6,p_5\}$ and $\{p_{jl},p_5,p_4\}.$ There are two flat tetrahedra on the right of the edge connecting $p_{ik}$ and $p_{jl}$ (colored in red) and three on the left of it.
  • Figure 5: Region $D_{\delta,c}$
  • ...and 20 more figures

Theorems & Definitions (117)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 107 more