Chern-Simons Theory with Finite Gauge Group

TL;DR

This paper studies Chern-Simons theories with finite gauge groups, focusing on their mathematical structure and implications in 2+1 dimensions, highlighting their relation to quantum groups and modular structures.

Contribution

It provides a detailed mathematical analysis of finite gauge group Chern-Simons theories, emphasizing their modular properties in 2+1 dimensions and connections to quantum groups.

Findings

Path integral reduces to a finite sum, enabling direct mathematical analysis.

Theories exhibit a modular structure in 2+1 dimensions.

Connections to quantum groups suggest broader implications.

Abstract

These theories, which are surely some of the simplest possible quantum field theories, were introduced in a paper of Dijkgraaf and Witten. The path integral reduces to a finite sum, so it is quite amenable to direct mathematical study. Although the theory exisits in arbitrary dimensions, it is most interesting in $2+1$~dimensions, where it has a ``modular structure.'' This is related to quantum groups, and the precise details may give clues as to what happens in other contexts. This paper is written using AMSTeX 2.1, which can be obtained via ftp from the American Mathematical Society (instructions included). 1 encapsulated postscript file was submitted separately in uuencoded tar-compressed format.