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${}_M{q}^{L(0)-c\mathrm{/}24}{\zeta }^{h(0)}($M$a$V$-module)isavector-valuedweakJacobiformofweight0andacertainindex$<h, h >/2$\mathrm{.}Wediscussrefinementsandapplicationsofthisresultwhen$V$ is holomorphic, in particular we prove that if $g=e^{h(0)}$ is a finite order automorphism then Tr${}_V{q}^{L(0)-c\mathrm{/}24}g$ is a modular function of weight 0 on a congruence subgroup of $S{L}_2(Z)$.