3d Mirror Symmetry for Instanton Moduli Spaces

Understand

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.