Tensor products of modules for a vertex operator algebra and vertex tensor categories

TL;DR

This paper develops a tensor product theory for modules over vertex operator algebras, establishing a vertex tensor category structure that generalizes classical tensor categories and applies to important models like WZNW and moonshine.

Contribution

It introduces the concept of vertex tensor categories and proves that module categories for certain vertex operator algebras naturally form such structures.

Findings

Module categories admit a natural vertex tensor category structure.

Vertex tensor categories produce braided tensor categories.

Applicable to rational vertex operator algebras like WZNW and moonshine.

Abstract

We introduce the main concepts and announce the main results in a theory of tensor products for module categories for a vertex operator algebra. This theory is being developed in a series of papers including hep-th 9309076 and hep-th 9309159. The theory applies in particular to any ``rational'' vertex operator algebra for which products of intertwining operators are known to be convergent in the appropriate regions, including the vertex operator algebras associated with the WZNW models, the minimal models and the moonshine module for the Monster. In this paper, we provide background and motivation; we present the main constructions and properties of the tensor product operation associated with a particular element of a suitable moduli space of spheres with punctures and local coordinates; we introduce the notion of ``vertex tensor category,'' analogous to the notion of tensor category but based on this moduli space; and we announce the results that the category of modules for a vertex operator algebra of the type mentioned above admits a natural vertex tensor category structure, and also that any vertex tensor category naturally produces a braided tensor category structure.