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.
Categories: hep-th math-ph math.GT math.MP math.QA