3D Mirror Symmetry for Instanton Moduli Spaces

Abstract

We prove that the Hilbert scheme of k points on {mathbb {C}}^2 (hbox {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}}^times _hbar -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 hbox {Hilb}^k[{mathbb {C}}^2] is obtained via direct limit llongrightarrow 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 hbar -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.