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

TL;DR

This paper develops a foundational tensor product theory for modules over vertex operator algebras, aiming to establish a vertex tensor category structure that generalizes symmetric tensor categories.

Contribution

It introduces the notions of P(z)- and Q(z)-tensor products and explores their fundamental properties, laying groundwork for a vertex tensor category framework.

Findings

Defined P(z)- and Q(z)-tensor products for modules

Established fundamental properties of Q(z)-tensor products

Applicable to rational vertex operator algebras such as WZNW and moonshine

Abstract

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.