# Conformal blocks from vertex algebras and their connections on $\overline{\mathcal{M}}_{g,n}$

## TL;DR

This paper demonstrates that coinvariants of vertex operator algebra modules form sheaves on moduli spaces of stable pointed curves, extending Verlinde bundles and revealing their D-module structures with projectively flat connections.

## Contribution

It generalizes Verlinde bundles to coinvariants of vertex algebra modules and identifies their twisted logarithmic D-module structure and Atiyah algebra on moduli spaces.

## Key findings

- Sheaves of coinvariants are quasi-coherent on moduli of stable curves.
- These sheaves support a twisted logarithmic D-module structure.
- Identification of the logarithmic Atiyah algebra acting on the sheaves.

## Abstract

We show that coinvariants of modules over vertex operator algebras give rise to quasi-coherent sheaves on moduli of stable pointed curves. These generalize Verlinde bundles or vector bundles of conformal blocks defined using affine Lie algebras studied first by Tsuchiya-Kanie, Tsuchiya-Ueno-Yamada, and extend work of a number of researchers. The sheaves carry a twisted logarithmic D-module structure, and hence support a projectively flat connection. We identify the logarithmic Atiyah algebra acting on them, generalizing work of Tsuchimoto for affine Lie algebras.