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A TQFT from quantum Teichmüller theory

Jørgen Ellegaard Andersen, Rinat Kashaev

TL;DR

This paper builds a TQFT from quantum Teichmüller theory by promoting quantum shape-structures on triangulated 3-manifolds to a functor valued in temperate distributions. To handle infinite-dimensionality and triangulation-dependence, it works within the categroid of admissible leveled shaped pseudo 3-manifolds and uses Faddeev's quantum dilogarithm to construct charged tetrahedral operators that satisfy a pentagon identity. The resulting one-parameter family $\{F_\hbar\}$ yields invariants of knots in 3-manifolds and provides a framework that connects hyperbolic geometry with quantum topology via convergence results and conjectural volume asymptotics. The work also develops the necessary analytic machinery, including a symplectic viewpoint on generalized shape structures, gauge properties, and detailed examples illustrating computation of partition functions for classic knots.

Abstract

By using quantum Teichmüller theory, we construct a one parameter family of TQFT's on the categroid of admissible leveled shaped 3-manifolds.

A TQFT from quantum Teichmüller theory

TL;DR

This paper builds a TQFT from quantum Teichmüller theory by promoting quantum shape-structures on triangulated 3-manifolds to a functor valued in temperate distributions. To handle infinite-dimensionality and triangulation-dependence, it works within the categroid of admissible leveled shaped pseudo 3-manifolds and uses Faddeev's quantum dilogarithm to construct charged tetrahedral operators that satisfy a pentagon identity. The resulting one-parameter family yields invariants of knots in 3-manifolds and provides a framework that connects hyperbolic geometry with quantum topology via convergence results and conjectural volume asymptotics. The work also develops the necessary analytic machinery, including a symplectic viewpoint on generalized shape structures, gauge properties, and detailed examples illustrating computation of partition functions for classic knots.

Abstract

By using quantum Teichmüller theory, we construct a one parameter family of TQFT's on the categroid of admissible leveled shaped 3-manifolds.

Paper Structure

This paper contains 38 sections, 20 theorems, 274 equations.

Table of Contents

  1. Introduction
  2. Oriented triangulated pseudo $3$-manifolds
  3. Shaped 3-manifolds
  4. The ${\mathbb Z}{}/3{\mathbb Z}{}$-action on pairs of opposite edges of tetrahedra
  5. Leveled shaped 3-manifolds
  6. The $3-2$ Pachner moves
  7. The categroid of admissible leveled shaped pseudo $3$-manifolds
  8. The TQFT functor
  9. Invariants of knots in 3-manifolds
  10. Future perspectives
  11. The symplectic space of generalized shape structures
  12. The tetrahedral operator of quantum Teichmüller theory
  13. Charged tetrahedral operators
  14. Charged Pentagon identity
  15. Fundamental Lemma
  16. ...and 23 more sections

Key Result

Theorem 1

The map is an affine $H^1(\partial N_0(X), {\mathbb R})$-bundle. The space $\tilde{S}_r(X)$ carries a Poisson structure whose symplectic leaves are the fibers of $\tilde{\Omega}_{X,r}$ and which is identical to the Poisson structure induced by the $H^1(\partial N_0(X), {\mathbb R})$-bundle structure. The na and which maps $S_r(Y)$ to $S_r(X)$. Furthermore, $h$ induces an isomorphism of affine ${

Theorems & Definitions (54)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Theorem 1
  • Definition 8
  • Theorem 2
  • ...and 44 more