Characters of topological N=2 vertex algebras are Jacobi forms on the moduli space of elliptic supercurves

TL;DR

This paper demonstrates that trace functions from topological N=2 super vertex algebras form Jacobi forms on the moduli space of elliptic supercurves, satisfying specific differential equations and exhibiting modular invariance.

Contribution

It establishes a link between N=2 super vertex algebra characters and Jacobi forms, providing a geometric and differential equation framework for their analysis.

Findings

Trace functions form conformal blocks on elliptic supercurves.

They satisfy linear PDEs with respect to modular parameters.

The associated connection is equivariant under the Jacobi modular group.

Abstract

We show that trace functions on modules of topological N=2 super vertex algebras give rise to conformal blocks on elliptic supercurves. We show that they satisfy a system of linear partial differential equations with respect to the modular parameters of the supercurves. Under some finiteness condition on the vertex algebra these differential equations can be interpreted as a connection on the vector bundle of conformal blocks. We show that this connection is equivariant with respect to a natural action of the Jacobi modular group on the modular parameters and the trace functions. In the appendix we prove the convergence of the trace functions.