Virasoro constraints on moduli of sheaves and vertex algebras

TL;DR

This paper reformulates Virasoro constraints in sheaf theory using vertex algebras, proving their invariance under wall-crossing and establishing conjectural constraints for moduli spaces of sheaves on curves and surfaces.

Contribution

It reinterprets sheaf-theoretic Virasoro constraints via vertex algebras and proves their validity for certain moduli spaces, extending previous conjectures.

Findings

Virasoro constraints are preserved under wall-crossing.

Proved Virasoro constraints for moduli of sheaves on curves.

Reduced complex cases to rank 1 scenarios.

Abstract

In enumerative geometry, Virasoro constraints were first conjectured in Gromov-Witten theory with many new recent developments in the sheaf theoretic context. In this paper, we rephrase the sheaf-theoretic Virasoro constraints in terms of primary states coming from a natural conformal vector in Joyce's vertex algebra. This shows that Virasoro constraints are preserved under wall-crossing. As an application, we prove the conjectural Virasoro constraints for moduli spaces of torsion-free sheaves on any curve and on surfaces with only $(p,p)$ cohomology classes by reducing the statements to the rank 1 case.