A modular invariance property of multivariable trace functions for regular vertex operator algebras

We prove an SL2(Z)SL2(Z)-invariance property of multivariable trace functions on modules for a regular VOA. Applying this result, we provide a proof of the inversion transformation formula for Siegel theta series. As another application, we show that if V is a simple regular VOA containing a simple regular subVOA U   whose commutant UcUc is simple, regular, and satisfies (Uc)c=U(Uc)c=U, then all simple U-modules appear in some simple V-module.