Local representations of the quantum Teichmuller space

TL;DR

This paper introduces local algebraic representations of the quantum Teichmuller space, classifies them via classical geometric data, and constructs a fiber bundle over Teichmuller space to study surface diffeomorphism invariants.

Contribution

It provides a new classification of quantum Teichmuller space representations and develops a fiber bundle framework for analyzing surface diffeomorphisms.

Findings

Classified local representations by classical geometric data.

Constructed a fiber bundle over Teichmuller space.

Developed invariants for surface diffeomorphisms.

Abstract

We introduce a certain type of representations for the quantum Teichmuller space of a punctured surface, which we call local representations. We show that, up to finitely many choices, these purely algebraic representations are classified by classical geometric data. We also investigate the family of intertwining operators associated to such a representations. In particular, we use these intertwiners to construct a natural fiber bundle over the Teichmuller space and its quotient under the action of the mapping class group. This construction also offers a convenient framework to exhibit invariants of surface diffeomorphisms.