3d Mirror Symmetry for Instanton Moduli Spaces
Peter Koroteev, Anton M. Zeitlin
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
K\"ahler 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 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 $\hbar$-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.