Factorization Rules in Quantum Teichm\"uller Theory

TL;DR

This paper explores the representation theory of quantum Teichmüller space, extending Thurston's shear coordinates and analyzing the Weil-Petersson structure, revealing a factorization rule similar to that in conformal field theory.

Contribution

It introduces a new perspective on quantum Teichmüller space representations by extending classical coordinates and connecting to conformal field theory factorization rules.

Findings

Extended Thurston's shear coordinates to augmented Teichmüller space

Analyzed Weil-Petersson Poisson structure in this context

Discovered a factorization rule analogous to conformal field theory

Abstract

We study the representation theory of the quantum Teichmueller space when going to infinity in the classical Teichmueller space. The geometric ingredients are the extension of Thurston's shear coordinates to the augmented Teichmueller space and the study of the Weil-Petersson Poisson structure for this extension. The result is analogous to the factorization rule found in conformal field theory.