VERTEX OPERATOR ALGEBRAS AND WEAK JACOBI FORMS

Let V be a strongly regular vertex operator algebra. For a state h is an element of V-1 satisfying appropriate integrality conditions, we prove that the space spanned by the trace functions Tr-M q(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 SL2(Z).