A theory of tensor products for module categories for a vertex operator algebra, I

This is the first part in a series of papers developing a tensor product theory for modules for a vertex operator algebra. The goal of this theory is to construct a "vertex tensor category" structure on the category of modules for a suitable vertex operator algebra. The notion of vertex tensor category is essentially a "complex analogue" of the notion of symmetric tensor category, and in fact a vertex tensor category produces a braided tensor category in a natural way. The theory applies in particular to many familiar "rational" vertex operator algebras, including those associated with WZNW models, minimal models and the moonshine module. In this paper (Part I), we introduce the notions of P(z)- and Q(z)-tensor product, where P(z) and Q(z) are two special elements of the moduli space of spheres with punctures and local coordinates, and we present the fundamental properties and constructions of Q(z)-tensor products.