Vertex operator algebras and weak Jacobi forms

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$_Mq^{L(0)-c/24}ζ^{h(0)} ($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)$.

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