Hikita-Nakajima conjecture for the Gieseker variety

TL;DR

This paper proves the Hikita-Nakajima conjecture for Gieseker varieties, explicitly constructing the isomorphism between equivariant cohomology and fixed point algebras, and relates it to rational Cherednik and cyclotomic Hecke algebras.

Contribution

The authors establish the Hikita-Nakajima conjecture for Gieseker varieties and explicitly describe the isomorphism using centers of rational Cherednik and cyclotomic Hecke algebras.

Findings

Proved the conjecture for Gieseker varieties.

Explicitly constructed the algebra isomorphism on generators.

Connected the isomorphism to centers of rational Cherednik and cyclotomic Hecke algebras.

Abstract

Let $\mathfrak{M}_0$ be an affine Nakajima quiver variety, and $\mathcal{M}$ is the corresponding BFN Coulomb branch. Assume that $\mathfrak{M}_0$ can be resolved by the (smooth) Nakajima quiver variety $\mathfrak{M}$. The Hikita-Nakajima conjecture claims that there should be an isomorphism of (graded) algebras $H^*_{S}(\mathfrak{M},\mathbb{C}) \simeq \mathbb{C}[\mathcal{M}_{\mathfrak{s}}^{\mathbb{C}^\times}]$, here $S \curvearrowright \mathfrak{M}_0$ is a torus acting on $\mathfrak{M}_0$ preserving the Poisson structure, $\mathcal{M}_{\mathfrak{s}}$ is the (Poisson) deformation of $\mathcal{M}$ over $\mathfrak{s}=\operatorname{Lie} (S)$, $\mathbb{C}^\times$ is a generic one-dimensional torus acting on $\mathcal{M}$, and $\mathbb{C}[\mathcal{M}_{\mathfrak{s}}^{\mathbb{C}^\times}]$ is the algebra of schematic $\mathbb{C}^\times$-fixed points of $\mathcal{M}_{\mathfrak{s}}$. We prove the Hikita-Nakajima conjecture for $\mathfrak{M}=\mathfrak{M}(n,r)$ Gieseker variety ($ADHM$ space). We produce the isomorphism explicitly on generators. We also describe the Hikita-Nakajima isomorphism above using the realization of $\mathcal{M}_{\mathfrak{s}}$ as the spectrum of the center of rational Cherednik algebra corresponding to $S_n \ltimes (\mathbb{Z}/r\mathbb{Z})^n$ and identify all the algebras that appear in the isomorphism with the center of degenerate cyclotomic Hecke algebra (generalizing some results of Shan, Varagnolo, and Vasserot).