Modular-invariance of trace functions in orbifold theory and generalized moonshine

The goal of the present paper is to provide a mathematically rigorous foundation to certain aspects of the theory of rational orbifold models in conformal field theory, in other words the theory of rational vertex operator algebras and their automorphisms. Under a certain finiteness condition on a rational Vertex operator algebra V which holds in all known examples, we determine the precise number of g-twisted sectors for any automorphism g of V of finite order. We prove that the trace functions and correlation functions associated with such twisted sectors are holomorphic functions in the upper half-plane and, under suitable conditions, afford a representation of the modular group of the type prescribed in string theory. We establish the rationality of conformal weights and central charge. In addition to conformal field theory itself, where our conclusions are required on physical grounds, there are applications to the generalized Moonshine conjectures of Conway-Norton-Queen and to equivariant elliptic cohomology.