Differential equations, duality and modular invariance

TL;DR

This paper constructs all chiral genus-one correlation functions from genus-zero functions for certain vertex operator algebras, establishing their fundamental properties including modular invariance, using a novel method different from previous approaches.

Contribution

It introduces a new approach to construct and analyze genus-one correlation functions for vertex operator algebras satisfying specific conditions, extending previous genus-zero results.

Findings

Established modular invariance of genus-one correlation functions.

Derived differential equations satisfied by $q$-traces of intertwining operators.

Solved the problem of constructing chiral genus-one weakly conformal field theories.

Abstract

We solve the problem of constructing all chiral genus-one correlation functions from chiral genus-zero correlation functions associated to a vertex operator algebra satisfying the following conditions: (i) the weight of any nonzero homogeneous elements of V is nonnegative and the weight zero subspace is one-dimensional, (ii) every N-gradable weak V-module is completely reducible and (iii) V is C_2-cofinite. We establish the fundamental properties of these functions, including suitably formulated commutativity, associativity and modular invariance. The method we develop and use here is completely different from the one previously used by Zhu and other people. In particular, we show that the $q$-traces of products of certain geometrically-modified intertwining operators satisfy modular invariant systems of differential equations which, for any fixed modular parameter, reduce to doubly-periodic systems with only regular singular points. Together with the results obtained by the author in the genus-zero case, the results of the present paper solves essentially the problem of constructing chiral genus-one weakly conformal field theories from the representations of a vertex operator algebra satisfying the conditions above.