Differential equations and logarithmic intertwining operators for strongly graded vertex algebras

TL;DR

This paper derives differential equations for matrix elements of logarithmic intertwining operators in strongly graded vertex algebras, enabling verification of key properties in logarithmic tensor category theory.

Contribution

It introduces systems of differential equations for intertwining operators in strongly graded vertex algebras, advancing the mathematical framework for logarithmic tensor categories.

Findings

Derived differential equations for matrix elements of intertwining operators.

Verified convergence and extension properties in logarithmic tensor category theory.

Supported the development of a rigorous mathematical foundation for strongly graded vertex algebras.

Abstract

We derive certain systems of differential equations for matrix elements of products and iterates of logarithmic intertwining operators among strongly graded generalized modules for a strongly graded conformal vertex algebra under suitable assumptions. Using these systems of differential equations, we verify the convergence and extension property needed in the logarithmic tensor category theory for such strongly graded generalized modules developed by Huang, Lepowsky and Zhang.