Logarithmic Intertwining Operators and Genus-One Correlation Functions

TL;DR

This paper develops the theory of genus-one correlation functions for logarithmic intertwining operators in vertex operator algebras, establishing their modular invariance and analytic properties.

Contribution

It introduces a framework for constructing and analyzing genus-one correlation functions for logarithmic intertwining operators, proving their modular invariance and differential equations.

Findings

Formal $q$-traces satisfy differential equations with regular singular points.

Genus-one correlation functions are absolutely convergent and extendable to multivalued analytic functions.

The space of solutions is invariant under the modular group.

Abstract

This is the first of two papers in which we study the modular invariance of pseudotraces of logarithmic intertwining operators. We construct and study genus-one correlation functions for logarithmic intertwining operators among generalized modules over a positive-energy and $C_2$-cofinite vertex operator algebra $V$. We consider grading-restricted generalized $V$-modules which admit a right action of some associative algebra $P$, and intertwining operators among such modules which commute with the action of $P$ ($P$-intertwining operators). We obtain duality properties, i.e., suitable associativity and commutativity properties, for $P$-intertwining operators. Using pseudotraces introduced by Miyamoto and studied by Arike, we define formal $q$-traces of products of $P$-intertwining operators, and obtain certain identities for these formal series. This allows us to show that the formal $q$-traces satisfy a system of differential equations with regular singular points, and therefore are absolutely convergent in a suitable region and can be extended to yield multivalued analytic functions, called genus-one correlation functions. Furthermore, we show that the space of solutions of these differential equations is invariant under the action of the modular group.