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

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.

152 pages

References in corpus (2)

• Classification of abelian spin Chern-Simons theories
• Group-theoretical properties of nilpotent modular categories

Cited by in corpus (6)

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• Braided Categorical Quantum Mechanics I
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• Classical theta functions from a quantum group perspective