  We prove an $\text{SL}_2 (\mathbb{Z})$-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 regular VOA containing a regular
subVOA $U$ whose commutant $U^c$ is regular and satisfies $(U^c)^c =U$, then
all simple $U$-modules appear in some simple $V$-module.
