  The goal of the present paper is to provide a mathematically rigorous
foundation to certain aspects of rational orbifold 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
numbers of g-twisted sectors for any automorphism g of V of finite order. We
prove that the trace functions and correlations functions associated with such
twisted sectors are holomorphic functions in the upper half-plane and, under
suitable conditions, afford a representations 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.
