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

TL;DR

This paper computes the equivariant quantum cohomology of Hilbert schemes on A_n surface resolutions, linking it to affine Lie algebra actions and Gromov-Witten theories, and discusses monodromy and generalizations.

Contribution

It explicitly determines two-point invariants and the quantum cohomology ring structure for A_n resolutions, connecting algebraic and geometric theories.

Findings

Operators expressed via affine Lie algebra actions.

Relationship established between quantum cohomology and Gromov-Witten/Donaldson-Thomas theories.

Discussion of monodromy and extensions to D and E singularities.

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.