Mapping class group actions in Chern-Simons theory with gauge group $G\ltimes\mathfrak{g}^*$

TL;DR

This paper explores how the mapping class group acts on a Poisson algebra in Chern-Simons theory with a semidirect product gauge group, establishing Poisson isomorphisms and quantum representations.

Contribution

It demonstrates the Poisson action of the mapping class group and constructs quantum representations using the quantum double D(G).

Findings

Mapping class group acts via Poisson isomorphisms.

Dehn twists expressed through G-action and ribbon element.

Quantum representations constructed from the algebra.

Abstract

We study the action of the mapping class group of an oriented genus g surface with n punctures and a disc removed on a Poisson algebra which arises in the combinatorial description of Chern-Simons gauge theory when the gauge group is a semidirect product $G\ltimes\mathfrak{g}^*$. We prove that the mapping class group acts on this algebra via Poisson isomorphisms and express the action of Dehn twists in terms of an infinitesimally generated G-action. We construct a mapping class group representation on the representation spaces of the associated quantum algebra and show that Dehn twists can be implemented via the ribbon element of the quantum double D(G) and the exchange of punctures via its universal R-matrix.