Abelian Chern-Simons theory with toral gauge group, modular tensor categories, and group categories

TL;DR

This paper extends the classification of toral Chern-Simons theories by constructing modular tensor categories from group categories, demonstrating their correspondence with known topological quantum field theories and mapping class group representations.

Contribution

It introduces a method to produce modular tensor categories from group categories that correspond to toral Chern-Simons theories, extending previous classifications.

Findings

Constructed modular tensor categories from group categories for toral Chern-Simons.

Proved the projective representation of the mapping class group matches that of toral Chern-Simons.

Connected algebraic structures with topological quantum field theory classifications.

Abstract

Classical and quantum Chern-Simons with gauge group $\text{U}(1)^N$ were classified by Belov and Moore in \cite{belov_moore}. They studied both ordinary topological quantum field theories as well as spin theories. On the other hand a correspondence is well known between ordinary $(2+1)$-dimensional TQFTs and modular tensor categories. We study group categories and extend them slightly to produce modular tensor categories that correspond to toral Chern-Simons. Group categories have been widely studied in other contexts in the literature \cite{frolich_kerler},\cite{quinn},\cite{joyal_street},\cite{eno},\cite{dgno}. The main result is a proof that the associated projective representation of the mapping class group is isomorphic to the one from toral Chern-Simons. We also remark on an algebraic theorem of Nikulin that is used in this paper.