On the operator content of nilpotent orbifold models

The present work is in essence a continuation of our paper [DM2]. To describe our results we assume that the reader is familiar with the theory of vertex operator algebras (VOA) and their representations (see for example [B], [FLM] and [FHL]). Suppose that V is a holomorphic VOA and G is a finite (and faithful) group of automorphisms of V. It is then a general conjecture that the fixed vertex operator subalgebra V G of G-invariants is rational, that is, V G has a finite number of simple modules and every module for V G is completely reducible. In fact, following the work of DijkgraafWitten [DW] and Dijkgraaf-Pasquier-Roche [DPR], one can formulate a precise conjecture concerning the category Mod-V G of V -modules. Essentially it says that Mod-V G is equivalent to the category of D(G)-modules, where D(G) is the so-called quantum double of G [Dr] modified by a certain 3-cocycle c ∈ H(G, S). This cocycle itself arises from a quasi-coassociative tensor product on Mod-D(G) which is expected to reflect the algebraic properties of an appropriate notion of tensor product on the category Mod-V . One of the goals of the present paper is to prove a variation on this theme for a broad class of finite groups G, not necessarily abelian, under a suitable hypothesis concerning the so-called twisted sectors for V. Let us explain these results in more detail. For each g ∈ G we have the notion of a g-twisted sector, or g-twisted V -module. It is an important conjecture, invariably assumed in the physics literature, that there is a Supported by NSF grant DMS-9303374 and a research grant from the Committee on Research, UC Santa Cruz. Supported by NSF grant DMS-9122030 and a research grant from the Committee on Research, UC Santa Cruz.