Orbifolds of Pointed Vertex Operator Algebras I

By a pointed vertex operator algebra (VOA) we mean one whose modules are all simple currents (i.e. invertible), e.g. lattice VOAs. This paper systematically explores the interplay between their orbifolds and tensor category theory. We begin by supplying an elementary proof of the Dijkgraaf-Witten conjecture, which predicts the representation theory of holomorphic VOA orbifolds. We then apply that argument more generally to the situation where the automorphism subgroup fixes all VOA modules, and relate the result to recent work of Mason-Ng and Naidu. Here our results are complete. We then turn to the other extreme, where the automorphisms act fixed-point freely on the modules, and realize any possible nilpotent group as lattice VOA automorphisms. This affords a considerable generalization of the Tambara-Yamagami categories. We conclude by considering some hybrid actions. In this way we use tensor category theory to organize and generalize systematically several isolated examples and special cases scattered in the literature. Conversely, we show how VOA orbifolds can be used to construct broad classes of braided crossed fusion categories and modular tensor categories.