Vertex operator algebras and weak Jacobi forms

TL;DR

This paper proves that trace functions of certain vertex operator algebras form weak Jacobi forms, and explores their modular properties, especially when the algebra is holomorphic and automorphisms are finite order.

Contribution

It establishes the connection between trace functions of strongly regular vertex operator algebras and weak Jacobi forms, extending to modular functions under specific automorphisms.

Findings

Trace functions form vector-valued weak Jacobi forms of weight 0.

Automorphisms of finite order lead to modular functions of weight 0.

Results apply to holomorphic vertex operator algebras.

Abstract

Let $V$ be a strongly regular vertex operator algebra. For a state $h \in V_1$ satisfying appropriate integrality conditions, we prove that the space spanned by the trace functions Tr$_Mq^{L(0)-c/24}\zeta^{h(0)} ($M$ a $V$-module) is a vector-valued weak Jacobi form of weight 0 and a certain index $<h, h >/2$. We discuss refinements and applications of this result when $V$ is holomorphic, in particular we prove that if $g = e^{h(0)}$ is a finite order automorphism then Tr$_V q^{L(0)-c/24}g$ is a modular function of weight 0 on a congruence subgroup of $SL_2(Z)$.