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

摘要

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