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

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

1995-05-17

arxiv.org/abs/q-alg/9505018v2

J.Pure Appl.Algebra 100 (1995) 141-172

Physics

High Energy Physics

High Energy Physics - Theory

LaTeX file. 40 pages. To appear in the special 100th issue of the Journal of Pure and Applied algebra, dedicated to ``Applicat

Scientific paper

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 The definitions and results in Part I (hep-th/9309076, which has been replaced by a new version with a greatly expanded introduction and updated references) and Part II (hep-th/9309159) are recalled.

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