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

TL;DR

This paper introduces FA-matrices to compute ranks of vector bundles related to vertex operator algebras on moduli spaces, unifying fusion and averaging matrices, and explores their positivity properties.

Contribution

It presents a new matrix formula that generalizes previous work, enabling computation of vector bundle ranks and analysis of their geometric properties.

Findings

Computed ranks for bundles from pointed VOAs and tensor products.

Analyzed positivity of first Chern classes of these bundles.

Abstract

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.