A Matrix Rank Formula for Vector Bundles of Vertex Operator Algebra Coinvariants and Conformal Blocks

We introduce FA-matrices for computing ranks of vector bundles of coinvariants and conformal blocks associated with modules over vertex operator algebras on the moduli space of stable pointed curves, unifying the notions of fusion and averaging matrices and generalizing Ueno’s work. To illustrate, we compute ranks of vector bundles determined by pointed VOAs and the tensor product of certain VOAs, as well as other examples. As an application, positivity properties of their first Chern classes are analyzed.