#### Volume 25, issue 5 (2021)

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

### Chiara Damiolini, Angela Gibney and Nicola Tarasca

Geometry & Topology 25 (2021) 2235–2286
##### 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 $\mathsc{𝒟}$–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.

##### Keywords
vertex algebras, conformal blocks and coinvariants, connections and Atiyah algebras, sheaves on moduli of curves, Chern classes of vector bundles on moduli of curves
##### Mathematical Subject Classification 2010
Primary: 14C17, 14H10, 17B69
Secondary: 16D90, 81R10, 81T40