Characters of Crossed Modules and Premodular Categories

Summary

Abstract

A general procedure is presented, associating a premodular category to a finite crossed module, generalizing the representation category of the double of a finite group, and the extent to which the resulting premodular category fails to be modular is explained.

Introduction

Modular Tensor Categories (MTCs for short) have attracted much attention in recent years, which is due to the recognition of their importance in both pure mathematics – 3-dimensional topology, representations of Vertex Operator Algebras (VOAs for short) – and theoretical physics (Rational Conformal Field Theory, Topological Field Theories). They are also closely related to Moonshine: a most interesting (and mysterious) example of a Modular Tensor Category, which is responsible for some of the deeper aspects of Moonshine, is the MTC associated to the Moonshine orbifold, i.e. the fixed point VOA of the Moonshine module under the action of the Monster: note that this MTC is yet to be rigorously constructed.

As in every branch of science, a deeper understanding of Modular Tensor Categories requires a suitable supply of examples. Since the work of Huang, we know that the module category of any rational VOA (satisfying some technical conditions) is modular, but this important result doesn't help us that much, because VOAs are pretty complicated objects usually hard to deal with. This leads to the desire of associating MTCs to simpler and more accessible algebraic objects. There are several such constructions, a most notable case being the one that associates to a finite group the module category of its (Drinfeld) double.