Logarithmic Intertwining Operators and Genus-One Correlation Functions

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.