Quasimaps to moduli spaces of sheaves

TL;DR

This paper develops a theory of quasimaps to moduli spaces of sheaves on surfaces, establishing their properness, obstruction theory, and connections to threefold sheaf moduli spaces, with applications to wall-crossing formulas relating Gromov-Witten and Donaldson-Thomas theories.

Contribution

It introduces a new framework for quasimaps to sheaf moduli spaces, proving properness and perfect obstruction theory, and relates Gromov-Witten and Donaldson-Thomas theories via wall-crossing formulas.

Findings

Moduli spaces of quasimaps are proper and have a perfect obstruction theory.

Established isomorphism between quasimaps moduli spaces and sheaves on threefolds.

Derived wall-crossing formulas connecting Gromov-Witten and Donaldson-Thomas invariants.

Abstract

We develop a theory of quasimaps to a moduli space of sheaves $M$ on a surface $S$. Under some assumptions, we prove that moduli spaces of quasimaps are proper and carry a perfect obstruction theory. Moreover, they are naturally isomorphic to moduli spaces of sheaves on threefolds $S\times C$, where $C$ is a nodal curve. Using Zhou's theory of entangled tails, we establish a wall-crossing formula which therefore relates the Gromov-Witten theory of $M$ and the Donaldson-Thomas theory of $S\times C$ with relative insertions. We evaluate the wall-crossing formula for Hilbert schemes of points $S^{[n]}$, if $S$ is a del Pezzo surface.