Topological Representations of $U_q(sl_2)$ on the Torus and the Mapping Class Group

TL;DR

This paper explores the interplay between the mapping class group and quantum group actions on the torus with a puncture, revealing their commutation and explicit vertex form representations in a conformal field theory context.

Contribution

It provides an explicit computation of the mapping class group action on the configuration space of a punctured torus with quantum group coefficients, highlighting their commutation and vertex form structure.

Findings

Mapping class group action commutes with $U_q(sl_2)$ action.

Explicit vertex form of the mapping class group action.

Insights into conformal field theory representations.

Abstract

We compute the mapping class group action on cycles on the configuration space of the torus with one puncture, with coefficients in a local system arising in conformal field theory. This action commutes with the topological action of the quantum group $U_q(sl_2)$, and is given in vertex form.