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We prove an $\text{SL}_2 (\mathbb{Z})$-invariance property of multivariabletrace functions on modules for a regular VOA. Applying this result, we providea proof of the inversion transformation formula for Siegel theta series. Asanother application, we show that if $V$ is a regular VOA containing a regularsubVOA $U$ whose commutant $U^c$ is regular and satisfies $(U^c)^c =U$, thenall simple $U$-modules appear in some simple $V$-module.

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