Logarithmic tensor category theory, VI: Expansion condition, associativity of logarithmic intertwining operators, and the associativity isomorphisms

TL;DR

This paper develops a tensor category framework for vertex operator algebra modules, establishing associativity isomorphisms and a logarithmic operator product expansion theorem, involving complex analytic convergence arguments.

Contribution

It introduces natural associativity isomorphisms for triple tensor products and proves a logarithmic operator product expansion theorem within the tensor category setting.

Findings

Established associativity isomorphisms for tensor products

Proved logarithmic operator product expansion theorem

Addressed convergence and analytic conditions

Abstract

This is the sixth 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 VI), we construct the appropriate natural associativity isomorphisms between triple tensor product functors. In fact, we establish a "logarithmic operator product expansion" theorem for logarithmic intertwining operators. In this part, a great deal of analytic reasoning is needed; the statements of the main theorems themselves involve convergence assertions.