Differential equations and intertwining operators

TL;DR

This paper demonstrates that under certain finiteness conditions, matrix elements of intertwining operators in vertex operator algebras satisfy differential equations, leading to a braided tensor category structure on modules.

Contribution

It establishes the connection between finiteness conditions and differential equations for intertwining operators, enabling the construction of tensor categories for vertex operator algebras.

Findings

Matrix elements satisfy differential equations with regular singular points.

Finiteness of fusion rules follows from these differential systems.

Products of intertwining operators satisfy convergence and extension properties.

Abstract

We show that if every module W for a vertex operator algebra V satisfies the condition that the dimension of W/C_1(W) is less than infinity, where C_1(W) is the subspace of W spanned by elements of the form u_{-1}w for u in V of positive weight and w in W, then matrix elements of products and iterates of intertwining operators satisfy certain systems of differential equations. Moreover, for prescribed singular points, there exist such systems of differential equations such that the prescribed singular points are regular. The finiteness of the fusion rules is an immediate consequence of a result used to establish the existence of such systems. Using these systems of differential equations and some additional reducibility conditions, we prove that products of intertwining operators for V satisfy the convergence and extension property needed in the tensor product theory for V-modules. Consequently, when a vertex operator algebra V satisfies all the conditions mentioned above, we obtain a natural structure of vertex tensor category (consequently braided tensor category) on the category of V-modules and a natural structure of intertwining operator algebra on the direct sum of all (inequivalent) irreducible V-modules.