Abelian Chern-Simons theory with toral gauge group, modular tensor categories, and group categories
Abstract
Classical and quantum Chern-Simons with gauge group U(1)N were classified by Belov and Moore in belovmoore. 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 frolichkerler,quinn,joyalstreet,eno,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.