Quantum Differential Equation for Slices of the Affine Grassmannian

TL;DR

This paper derives an explicit quantum connection formula for slices of the affine Grassmannian, linking it to the trigonometric KZ equations for the Langlands dual group, advancing understanding of quantum cohomology in geometric representation theory.

Contribution

It provides a new explicit formula for the quantum connection of affine Grassmannian slices using stable envelopes and deformation methods, connecting it to KZ equations.

Findings

Explicit quantum connection formula derived

Identification with trigonometric KZ equations for dual group

Advances in quantum cohomology of affine Grassmannian slices

Abstract

The affine Grassmannian associated to a reductive group $\mathbf{G}$ is an affine analogue of the usual flag varieties. It is a rich source of Poisson varieties and their symplectic resolutions. These spaces are examples of conical symplectic resolutions dual to the Nakajima quiver varieties. In this work, we study their quantum connection. We use the stable envelopes of D. Maulik and A. Okounkov[arXiv:1211.1287] to write an explicit formula for this connection. The classical part of the multiplication comes from [arXiv:2210.09967]. The computation of the purely quantum part is done based on the deformation approach of A. Braverman, D. Maulik and A. Okounkov[arXiv:1001.0056]. For the case of simply-laced $\mathbf{G}$, we identify the quantum connection with the trigonometric Knizhnik-Zamolodchikov equation for the Langlands dual group $\mathbf{G}^\vee$.