A quantum group version of quantum gauge theories in two dimensions

TL;DR

This paper introduces a novel approach to quantum gauge theories in two dimensions using quantum groups, specifically $SL_q(2,{f C})$, leveraging Turaev-Viro invariants and extending combinatorial methods.

Contribution

It presents an alternative, combinatorial framework for quantum gauge theories in 2D based on quantum groups and topological invariants, connecting to Witten's and Migdal's approaches.

Findings

Establishes a connection between quantum gauge theories and Turaev-Viro invariants.

Provides a combinatorial method for 2D quantum gauge theories using quantum groups.

Extends topological invariants to arbitrary 3-manifolds in this context.

Abstract

For the special case of the quantum group $SL_q (2,{\bf C})\ (q= \exp \pi i/r,\ r\ge 3)$ we present an alternative approach to quantum gauge theories in two dimensions. We exhibit the similarities to Witten's combinatorial approach which is based on ideas of Migdal. The main ingredient is the Turaev-Viro combinatorial construction of topological invariants of closed, compact 3-manifolds and its extension to arbitrary compact 3-manifolds as given by the authors in collaboration with W. Mueller.