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 A_{n} singularities. The operators encoding these invariants are expressed in terms of the action of the the affine Lie algebra g l ^ ( n + 1 ) \widehat {\mathfrak {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 × P 1 A_{n}\times \mathbf {P}^1 . We close with a discussion of the monodromy properties of the associated quantum differential equation and a generalization to singularities of types D D and E E .