The Heisenberg category of a category

TL;DR

This paper constructs a 2-category that categorifies the Heisenberg algebra associated with a given category, unifying various existing categorical actions and providing a full enhancement of known representations.

Contribution

It introduces a new 2-categorical framework for the Heisenberg algebra, generalizing and unifying previous categorical actions and extending the action on derived categories of symmetric quotient stacks.

Findings

Constructs a k-linear or DG 2-category for the Heisenberg algebra

Defines a 2-categorical analogue of the Fock space representation

Provides a full categorical enhancement of the action on derived categories

Abstract

Starting with a k-linear or DG category admitting a (homotopy) Serre functor, we construct a k-linear or DG 2-category categorifying the Heisenberg algebra of the numerical K-group of the original category. We also define a 2-categorical analogue of the Fock space representation of the Heisenberg algebra. Our construction generalises and unifies various categorical Heisenberg algebra actions appearing in the literature. In particular, we give a full categorical enhancement of the action on derived categories of symmetric quotient stacks introduced by Krug, which itself categorifies a Heisenberg algebra action proposed by Grojnowski.