Logarithmic Intertwining Operators and Genus-One Correlation Functions

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.

References in corpus (3)

• Logarithmic intertwining operators and W(2,2p-1)-algebras
• On the triplet vertex algebra W(p)
• Logarithmic intertwining operators and vertex operators

Cited by in corpus (7)

• Logarithmic conformal field theory, log-modular tensor categories and modular forms
• Convergence of Sewing Conformal Blocks
• Associative algebras and intertwining operators
• On rationality for $C_2$-cofinite vertex operator algebras
• Analytic Conformal Blocks of $C_2$-cofinite Vertex Operator Algebras II: Convergence of Sewing and Higher Genus Pseudo-$q$-traces
• Representation theory of vertex operator algebras and orbifold conformal field theory
• Intertwining operators among twisted modules associated to not-necessarily-commuting automorphisms