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 ⊗ V . We prove that the qτ -qτ -traces (natural traces involving qτ = e2πiτ and qτ = e2πiτ ) 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 τ . 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 [HK], we prove that all these functions are modular invariant.