Quantum cohomology of the Springer resolution

TL;DR

This paper establishes a connection between the equivariant quantum differential equation of the cotangent bundle of a flag variety and the affine Knizhnik-Zamolodchikov connection, extending to resolutions of Slodowy slices and discussing generalizations.

Contribution

It identifies the quantum differential equation with the affine KZ connection for cotangent bundles, linking quantum cohomology with integrable systems.

Findings

Equivariant quantum differential equation matches affine KZ connection.

Recovers Kim's quantum cohomology description as a limit.

Extends results to resolutions of Slodowy slices.

Abstract

Let G denote a complex, semisimple, simply-connected group. We identify the equivariant quantum differential equation for the cotangent bundle to the flag variety of G with the affine Knizhnik-Zamolodchikov connection of Cherednik and Matsuo. This recovers Kim's description of quantum cohomology of the flag variety itself as a limiting case. A parallel result is proven for resolutions of the Slodowy slices. Extension to arbitrary symplectic resolutions is discussed.