Quasimaps to moduli spaces of sheaves

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× 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× C with relative insertions. We evaluate the wall-crossing formula for Hilbert schemes of points S[n] , if S is a del Pezzo surface.