3d Mirror Symmetry for Instanton Moduli Spaces

Abstract

We prove that the Hilbert scheme of $k$ points on ${\mathbb{C}}^2$ (Hilb${}^k[{\mathbb{C}}^2]$) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the ${\mathbb{C}}_{\hslash }^{\times }$-action. First, we find a two-parameter family $X_{k,l}$ of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of Hilb${}^k[{\mathbb{C}}^2]$ is obtained via direct limit $l\to \infty$ and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted $\hslash$-opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-$N$ sheaves on ${\mathbb{P}}^2$ with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual.

arXiv commentv3: 62 pages, typos corrected, references added, to appear in Commun. Math. Phys

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• Toroidal q-Opers
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• Some Open Questions in Quiver Gauge Theory
• The Zoo of Opers and Dualities
• Tunnels Under Geometries (or Instantons Know Their Algebras)
• q-Opers, QQ-systems, and Bethe Ansatz II: Generalized Minors
• Virtual Coulomb branch and vertex functions
• Opers on the projective line, Wronskian relations, and the Bethe Ansatz
• Superopers revisited