Anyons in discrete gauge theories with Chern-Simons terms

TL;DR

This paper explores how adding a Chern-Simons term to (2+1)-dimensional discrete gauge theories influences the algebraic structure of anyons, linking it to cohomology and topological field theories.

Contribution

It demonstrates that Chern-Simons terms induce non-trivial 3-cocycles on the Hopf algebra D(H), classifying models via cohomology and connecting to Dijkgraaf-Witten theories.

Findings

Chern-Simons terms create a non-trivial 3-cocycle on D(H).

Models are classified by elements of H^3(H,U(1)).

Explicit examples illustrate the theoretical framework.

Abstract

We study the effect of a Chern-Simons term in a theory with discrete gauge group H, which in (2+1)-dimensional space time describes (non-abelian) anyons. As in a previous paper, we emphasize the underlying algebraic structure, namely the Hopf algebra D(H). We argue on physical grounds that the addition of a Chern-Simons term in the action leads to a non-trivial 3-cocycle on D(H). Accordingly, the physically inequivalent models are labelled by the elements of the cohomology group H^3(H,U(1)). It depends periodically on the coefficient of the Chern-Simons term which model is realized. This establishes a relation with the discrete topological field theories of Dijkgraaf and Witten. Some representative examples are worked out explicitly.