Modular categories and orbifold models

Abstract

In this paper, we try to answer the following question: given a modular tensor category $\A$ with an action of a compact group $G$, is it possible to describe in a suitable sense the ``quotient'' category $\A/G$? We give a full answer in the case when $\A=\vec$ is the category of vector spaces; in this case, $\vec/G$ turns out to be the category of representation of Drinfeld's double $D(G)$. This should be considered as category theory analog of topological identity ${pt}//G=BG$. This implies a conjecture of Dijkgraaf, Vafa, E. Verlinde and H. Verlinde regarding so-called orbifold conformal field theories: if $\V$ is a vertex operator algebra which has a unique irreducible module, $\V$ itself, and $G$ is a compact group of automorphisms of $\V$, and some not too restricitive technical conditions are satisfied, then $G$ is finite, and the category of representations of the algebra of invariants, $\V^G$, is equivalent as a tensor category to the category of representations of Drinfeld's double $D(G)$. We also get some partial results in the non-holomorphic case, i.e. when $\V$ has more than one simple module.