Logarithmic tensor category theory, IV: Constructions of tensor product bifunctors and the compatibility conditions

TL;DR

This paper develops constructions for tensor product bifunctors in vertex operator algebra module categories, introducing compatibility conditions to ensure their proper behavior within a general tensor category framework.

Contribution

It provides new methods to construct P(z)- and Q(z)-tensor product bifunctors using compatibility conditions in vertex operator algebra tensor categories.

Findings

Constructed P(z)- and Q(z)-tensor product bifunctors

Introduced compatibility conditions for tensor product constructions

Enhanced the theoretical framework for tensor categories in vertex operator algebras

Abstract

This is the fourth part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. In this paper (Part IV), we give constructions of the P(z)- and Q(z)-tensor product bifunctors using what we call "compatibility conditions" and certain other conditions.