Conformal blocks from vertex algebras and their connections on M ¯ g , n

Abstract

We show that coinvariants of modules over vertex operator algebras give rise to quasicoherent 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, Ueno and Yamada, and extend work of others. The sheaves carry a twisted logarithmic $\mathcal{𝒟}$–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.