(nonmeromorphic) operator product expansion and the tensor product theory

In [HL1]–[HL5] and [H1], a theory of tensor products of modules for a vertex operator algebra is being developed. To use this theory, one first has to verify that the vertex operator algebra satisfies certain conditions. We show in the present paper that for any vertex operator algebra containing a vertex operator subalgebra isomorphic to a tensor product algebra of minimal Virasoro vertex operator algebras (vertex operator algebras associated to minimal models), the tensor product theory can be applied. In particular, intertwining operators for such a vertex operator algebra satisfy the (nonmeromorphic) commutativity (locality) and the (nonmeromorphic) associativity (operator product expansion). Combined with a result announced in [HL4], the results of the present paper also show that the category of modules for such a vertex operator algebra has a natural structure of a braided tensor category. In particular, for any pair p, q of relatively prime positive integers larger than 1, the category of minimal modules of central charge 1− 6 (p−q) 2 pq for the Virasoro algebra has a natural structure of a braided tensor category. ∗Supported in part by NSF grant DMS-9301020 and by DIMACS, an NSF Science and Technology Center funded under contract STC-88-09648.