3D Mirror Symmetry for Instanton Moduli Spaces

We prove that the Hilbert scheme of k points on ℂ^2 ( Hilb^k[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 ℂ^× _ħ -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[ℂ^2] is obtained via direct limit l⟶∞ 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 ħ -opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank- N sheaves on ℙ^2 with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual.