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A new formulation of the Teichmüller TQFT

Jørgen Ellegaard Andersen, Rinat Kashaev

TL;DR

The paper develops a novel state-integral formulation of Teichmüller TQFT using the Weil–Gel'fand–Zak transform of Faddeev's quantum dilogarithm, placing edge state variables on $\mathbb{R}$ while making the integrand periodic so the partition function integrates over a compact cube. It proves a pentagon identity for the tetrahedral weights, establishes symmetry and shape-gauge invariance via the WGZ framework, and provides explicit TQFT rules with gluing and conical insertions that yield invariance under $2$-$3$ and $3$-$2$ Pachner moves, together with meromorphic continuation in complex shape parameters. The work integrates a rich set of integral identities for the quantum dilogarithm and a quasi-classical limit, reinforcing the connection to classical dilogarithm topology and enabling potential link invariants within the constructed topological theory. Collectively, the approach yields a robust, topologically invariant Teichmüller TQFT compatible with existing constructions and extending them through a compact, analytically continued state-integral framework.

Abstract

By using the Weil-Gel'fand-Zak transform of Faddeev's quantum dilogarithm, we propose a new state-integral model for the Teichmüller TQFT, where the circle valued state variables live on the edges of oriented leveled shaped triangulations.

A new formulation of the Teichmüller TQFT

TL;DR

The paper develops a novel state-integral formulation of Teichmüller TQFT using the Weil–Gel'fand–Zak transform of Faddeev's quantum dilogarithm, placing edge state variables on while making the integrand periodic so the partition function integrates over a compact cube. It proves a pentagon identity for the tetrahedral weights, establishes symmetry and shape-gauge invariance via the WGZ framework, and provides explicit TQFT rules with gluing and conical insertions that yield invariance under - and - Pachner moves, together with meromorphic continuation in complex shape parameters. The work integrates a rich set of integral identities for the quantum dilogarithm and a quasi-classical limit, reinforcing the connection to classical dilogarithm topology and enabling potential link invariants within the constructed topological theory. Collectively, the approach yields a robust, topologically invariant Teichmüller TQFT compatible with existing constructions and extending them through a compact, analytically continued state-integral framework.

Abstract

By using the Weil-Gel'fand-Zak transform of Faddeev's quantum dilogarithm, we propose a new state-integral model for the Teichmüller TQFT, where the circle valued state variables live on the edges of oriented leveled shaped triangulations.

Paper Structure

This paper contains 12 sections, 7 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}
  3. The Pentagon identity
  4. Symmetries
  5. Analytical continuation
  6. Shape gauge invariance
  7. TQFT rules and tetrahedral symmetries
  8. Appendix A. Faddeev's quantum dilogarithm
  9. Integral analog of $_1\psi_1$-summation formula of Ramanujan
  10. Fourier transformation formulae
  11. Other integral identities
  12. Quasi-classical limit of Faddeev's quantum dilogarithm

Key Result

Theorem 1

Let $X$ be a closed LSTOP 3-manifold. Then, the quantity admits an analytic continuation to a meromorphic function of the complex shapes which is invariant under all shaped $2-3$ and $3-2$ Pachner moves (along balanced edges).

Theorems & Definitions (14)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 2
  • ...and 4 more