Modular invariance for conformal full field algebras

Abstract:

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.