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 t
he 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.
We study the relative Donaldson-Thomas theory of A_n x P^1, where A_n is the surface resolution of a type A_n singularity. The action of divisor operators in the theory is expressed in terms of operators of the affine algebra hat{gl}(n+1) on Fock spa
ce. Assuming a nondegeneracy conjecture, this gives a complete solution for the theory. The results complete the comparison of this theory with the Gromov-Witten theory of A_n x P^1 and the quantum cohomology of the Hilbert scheme of points on A_n.
