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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

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 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.

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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
Received: 5 March 2019
Revised: 19 April 2020
Accepted: 2 October 2020
Published: 3 September 2021
Proposed: Lothar Göttsche
Seconded: Dan Abramovich, Jim Bryan
Chiara Damiolini
Department of Mathematics
Princeton University
Princeton, NJ
United States
Department of Mathematics
Rutgers University
Piscataway, NJ
United States
Angela Gibney
Department of Mathematics
Rutgers University
Piscataway, NJ
United States
Nicola Tarasca
Department of Mathematics and Applied Mathematics
Virginia Commonwealth University
Richmond, VA
United States