On three-dimensional topological field theories constructed from $D^\omega(G)$ for finite group

TL;DR

This paper studies 3d lattice topological field theories based on deformations of quantum doubles of finite groups, exploring their relations to Dijkgraaf-Witten theories and revealing differences depending on the group's structure.

Contribution

It analyzes the Chung-Fukuma-Shapere models constructed from deformed quantum doubles of finite groups, establishing connections to Dijkgraaf-Witten theories for certain groups.

Findings

Chung-Fukuma-Shapere partition function relates to Dijkgraaf-Witten for odd order groups

The relation $\\zcfs = |\zdw|^2$ holds for $G=\Z_{2N+1}$

The relation does not hold for $G=\Z_{2N}$

Abstract

We investigate the 3d lattice topological field theories defined by Chung, Fukuma and Shapere. We concentrate on the model defined by taking a deformation $\D{G}$ of the quantum double of a finite commutative group $G$ as the underlying Hopf algebra. It is suggested that Chung-Fukuma-Shapere partition function is related to that of Dijkgraaf-Witten by $\zcfs = |\zdw|^2$ when $G=\Z_{2N+1}$. For $G=\Z_{2N}$, such a relation does not hold.