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Contents 1 Introduction 2 Review of basic concepts 3 Affinizations of vertex operator algebras and the * -operation 4 The notions of P (z)-and Q(z)-tensor product of two modules 5 First construction of Q(z)-tensor product 6 Second construction of Q(z)-tensor product

This is the third 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. In this paper, we focus on a particular element P (z) of a certain moduli space of three-punctured Riemann spheres; in general, every element of this moduli space will give rise to a notion of tensor product, and one must consider all these notions in order to construct a vertex tensor category. Here we present the fundamental properties of the P (z)-tensor product of two modules for a vertex operator algebra. We give two constructions of a P (z)-tensor product, using the results, established in Parts I and II of this series, for a certain other element of the moduli space. The definitions and results in Parts I and II are recalled.

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