Quantum cohomology of the Hilbert scheme of points on A_n-resolutions

Abstract

We determine the two-point invariants of the equivariant quantum cohomology of the Hilbert scheme of points of surface resolutions associated to type A_n singularities. The operators encoding these invariants are expressed in terms of the action of the affine Lie algebra \hat{gl}(n+1) on its basic representation. Assuming a certain nondegeneracy conjecture, these operators determine the full structure of the quantum cohomology ring. A relationship is proven between the quantum cohomology and Gromov-Witten/Donaldson-Thomas theories of A_n x P^1. We close with a discussion of the monodromy properties of the associated quantum differential equation and a generalization to singularities of type D and E.