Quantum Moduli Spaces of Flat Connections

TL;DR

This paper explores the quantization of moduli spaces of flat connections using quantum group gauge theory, highlighting the structure of the resulting algebras and their relation to mapping class group actions.

Contribution

It introduces a framework for constructing quantum algebras of observables for flat connections, emphasizing their structure and mapping class group representations.

Findings

Construction of quantum moduli algebras for flat connections

Explicit formula for mapping class group representations

Insights into the structure of quantum moduli spaces

Abstract

Using the formalism of discrete quantum group gauge theory, one can construct the quantum algebras of observables for the Hamiltonian Chern-Simons model. The resulting moduli algebras provide quantizations of the algebra of functions on the moduli spaces of flat connections on a punctured 2-dimensional surface. In this note we describe some features of these moduli algebras with special emphasis on the natural action of mapping class groups. This leads, in particular, to a closed formula for representations of the mapping class groups on conformal blocks.