Modular invariance for conformal full field algebras

Let $V^{L}$ and $V^{R}$ be simple vertex operator algebras satisfying certain natural uniqueness-of-vacuum, complete reducibility and cofiniteness conditions and let $F$ be a conformal full field algebra over $V^{L}\otimes V^{R}$. We prove that the $q_{\tau}$- $\overline{q_{\tau}}$-traces (natural traces involving $q_{\tau}=e^{2\pi i\tau}$ and $\overline{q_{\tau}}= \overline{e^{2\pi i\tau}}$) of geometrically modified genus-zero correlation functions for $F$ are convergent in suitable regions and can be extended to doubly periodic functions with periods $1$ and $\tau$. We obtain necessary and sufficient conditions for these functions to be modular invariant. In the case that $V^{L}=V^{R}$ and $F$ is one of those constructed by the authors in an earlier paper, we prove that all these functions are modular invariant.