3D Mirror Symmetry for Instanton Moduli Spaces

Abstract

AbstractWe prove that the Hilbert scheme of k points on $${\mathbb {C}}^2$$ C 2 ($$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ 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 $${\mathbb {C}}^\times _\hbar $$ C ħ × -action. First, we find a two-parameter family $$X_{k,l}$$ X k , l of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of $$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ Hilb k [ C 2 ] is obtained via direct limit $$l\longrightarrow \infty $$ 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 $$\hbar $$ ħ -opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-N sheaves on $${\mathbb {P}}^2$$ P 2 with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual.