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An analog of a modular functor from quantized Teichm"uller theory

J. Teschner

TL;DR

This work develops a non-compact analog of a modular functor by quantizing Teichmüller spaces and constructing a tower of representations of the modular groupoid via both Penner/Fock and Fenchel–Nielsen coordinate formalisms. It introduces a constrained-phase-space formulation (Kashaev coordinates) and a quantum Ptolemy-groupoid action, then demonstrates how to decompose representations into irreducible components associated to central charges and boundary data. The results culminate in a proposal for a stable unitary modular functor based on rigged surfaces, outlining the gluing and disjoint-union formalism and connecting to Liouville theory and non-compact quantum group structures. The framework lays the groundwork for explicit matrix coefficients and deeper links to non-compact CFTs, hyperbolic geometry, and potential non-compact topological invariants.

Abstract

It is shown that the quantized Teichm"uller spaces have factorization properties like those required in the definition of a modular functor.

An analog of a modular functor from quantized Teichm"uller theory

TL;DR

This work develops a non-compact analog of a modular functor by quantizing Teichmüller spaces and constructing a tower of representations of the modular groupoid via both Penner/Fock and Fenchel–Nielsen coordinate formalisms. It introduces a constrained-phase-space formulation (Kashaev coordinates) and a quantum Ptolemy-groupoid action, then demonstrates how to decompose representations into irreducible components associated to central charges and boundary data. The results culminate in a proposal for a stable unitary modular functor based on rigged surfaces, outlining the gluing and disjoint-union formalism and connecting to Liouville theory and non-compact quantum group structures. The framework lays the groundwork for explicit matrix coefficients and deeper links to non-compact CFTs, hyperbolic geometry, and potential non-compact topological invariants.

Abstract

It is shown that the quantized Teichm"uller spaces have factorization properties like those required in the definition of a modular functor.

Paper Structure

This paper contains 42 sections, 17 theorems, 82 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Aims
  4. Overview
  5. Outlook
  6. Coordinates for the Teichmüller spaces
  7. The Penner coordinates
  8. Triangulations and fat graphs
  9. Penner coordinates
  10. Fock coordinates
  11. The Ptolemy groupoids
  12. Groupoids vs. complexes
  13. Change of the triangulation
  14. The representation of the mapping class group
  15. Teich-müller space as the phase space of a constrained system
  16. ...and 27 more sections

Key Result

Theorem 1

${}$ Penner P1P2 (a) For any fixed ideal triangulation $\tau$ of $\Sigma$, the function is a homeomorphism. (b) The pull-back of the Weil-Petersson two-form $\omega$ on ${\mathcal{T}}(\Sigma)$ is given by the expression where the summation is extended over the set $\tau_2$ of triangles of $\tau$, and $e_i(t)$, $i=1,2,3$ are the edges bounding the triangle $t$, labelled in the counter-clockwise s

Figures (7)

  • Figure 1: Triangulation of the once-punctured torus.
  • Figure 2: Another representation of the triangulation from Figure \ref{['triang']} and the dual fat graph.
  • Figure 3: Graphical representation of the vertex $v$ dual to a triangle $t$. The marked corner defines a corresponding numbering of the edges that emanate at $v$.
  • Figure 4: Two adjacent triangles and the dual fat graph.
  • Figure 5: Representation of the triangles $t_i$ and $t_j$ in the upper half plane.
  • ...and 2 more figures

Theorems & Definitions (35)

  • Remark 1
  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • Proposition 2
  • ...and 25 more