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.

Let V be a vertex operator algebra and g an automorphism of order T . We construct a sequence of associative algebras A g,n (V ) with n ∈ 1 T Z nonnegative such that A g,n (V ) is a quotient of A g,n+1/T (V ) and a pair of functors between the category of A g,n (V )-modules which are not A g,n−1/T (V )-modules and the category of admissible V -modules. These functors exhibit a bijection between the simple modules in each category. We also show that V is g-rational if and only if all A g,n (V ) are finite-dimensional semisimple algebras.

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Let V be a vertex operator algebra. We construct a sequence of associative Ž . Ž . Ž . Ž . algebras A V n s 0, 1, 2, . . . such that A V is a quotient of A V and a n n n q 1 Ž . Ž . pair of functors between the category of A V -modules which are not A Vn n y 1 modules and the category of admissible V-modules. These functors exhibit a bijection between the simple modules in each category. We also show that V is Ž . rational if and only if all A V are finite-dimensional semisimple algebras.