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

The goal of the present paper is to provide a mathematically rigorous found ation to certain aspects of the theory of rational orbifold models in confo rmal field theory, in other words the theory of rational vertex operator al gebras 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-t wisted sectors for any automorphism g of V of finite order. We prove that t he trace functions and correlation functions associated with such twisted s ectors are holomorphic functions in the upper half-plane and, under suitabl e conditions, afford a representation of the modular group of the type pres cribed in string theory. We establish the rationality of conformal weights and central charge. In addition to conformal field theory itself, where our conclusions are req uired on physical grounds, there are applications to the generalized Moonsh ine conjectures of Conway-Norton-Queen and to equivariant elliptic cohomolo gy.