Applications of the factorization theorem of conformal blocks to vertex operator algebras

We apply the factorization and vector bundle propositionerty of the sheaves of conformal blocks on $\overline{\mathscr{M}}_{g,n}$. defined by vertex operator algebras (VOAs) and give geometric proofs of essential results in the representation theory of strongly rational VOAs. We first prove the associativity of the fusion tensor product by investigating four-point conformal blocks on genus zero curves. Then we give a formula to calculate the ranks of VOA-conformal block bundles. Finally, by investigating one-point conformal blocks on genus-one curves, we prove the modular invariance of intertwining operators for VOAs, which is a generalization of Zhu's modular invariance for module vertex operators of strongly rational VOAs, and is a specialization of Huang's recent modular invariance of logarithmic intertwining operators for $C_2$-cofinite VOAs.