Modular categories and orbifold models

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 "quonent" 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 the category theory analog of the 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 restrictive 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.