Heisenberg categorification and Hilbert schemes

Abstract

Given a finite subgroup $\Gamma \subset {\mathrm{SL}}_2\left(\mathbb{C}\right)$ we define an additive $2$-category ${\mathcal{H}}_{\Gamma }$ whose Grothendieck group is isomorphic to an integral form ${\mathfrak{h}}_{\Gamma }$ of the Heisenberg algebra. We construct an action of ${\mathcal{H}}_{\Gamma }$ on derived categories of coherent sheaves on Hilbert schemes of points on the minimal resolutions $\widehat{{ℂ}^2/\Gamma }$.