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Miguel Moreira: Lectures

January 11, 2023
TITLE: Virasoro constraints: vertex algebras and wall-crossing

ABSTRACT: This talk will be the second one concerning the Virasoro constraints in moduli spaces of sheaves (see Woonam’s abstract), based on joint work with A. Bojko and W. Lim. In this talk, I will focus on the connection between Virasoro constraints and the vertex algebra that D. Joyce recently introduced to study wall-crossing. It turns out that this vertex algebra can be endowed with a conformal element that induces the Virasoro operators that had appeared previously in the literature. In this language, our conjectures/results say that moduli of sheaves define physical/primary states in this vertex operator algebra. From this point of view and Joyce’s theory, we can prove that the Virasoro constraints are compatible with wall-crossing. This is the main new technical tool that allows to prove the constraints for torsion-free sheaves on curves and surfaces by reducing everything to rank 1.