Virasoro constraints for K3 surfaces and monodromy operators

TL;DR

This paper proves Virasoro constraints for moduli spaces of sheaves on K3 surfaces using monodromy operators, introduces new constraints in rank 0, and constructs a Virasoro algebra representation with central charge 24.

Contribution

It extends Virasoro constraints to K3 surfaces via monodromy operators and introduces new operators and constraints, including a Virasoro algebra representation with central charge 24.

Findings

Virasoro constraints established for K3 surfaces.

New Virasoro operators in negative degree introduced.

Representation of Virasoro algebra with central charge 24 constructed.

Abstract

The Virasoro constraints for moduli spaces of stable torsion free sheaves on a surface with only $(p,p)$-cohomology were recently proved by Bojko-Moreira-Lim. The rank 1 case, which is not restricted to surfaces with only $(p,p)$-cohomology, was established by Moreira. We prove Virasoro constraints for K3 surfaces using Markman monodromy operators, which allow us to reduce to the rank 1 case. We also prove new Virasoro constraints in rank 0. Finally, for K3 surfaces, we introduce new Virasoro operators in negative degree which, together with the previous Virasoro operators, give a representation of Virasoro algebra with central charge $24$.