Heisenberg categorification and Hilbert schemes

TL;DR

This paper constructs a 2-category related to the Heisenberg algebra and demonstrates its action on derived categories of Hilbert schemes, linking algebraic and geometric structures in representation theory.

Contribution

It introduces a new 2-category H^G associated with a finite subgroup of SL(2,C) and establishes its action on derived categories of Hilbert schemes.

Findings

The Grothendieck group of H^G is isomorphic to an integral form of the Heisenberg algebra.

H^G acts on derived categories of coherent sheaves on Hilbert schemes of points.

Provides a categorification linking algebraic and geometric representation theories.

Abstract

Given a finite subgroup G of SL(2,C) we define an additive 2-category H^G whose Grothendieck group is isomorphic to an integral form of the Heisenberg algebra. We construct an action of H^G on derived categories of coherent sheaves on Hilbert schemes of points on the minimal resolutions of C^2/G.