The Heisenberg category of a category

In 90s Nakajima and Grojnowski identified the total cohomology of the Hilbert schemes of points on a smooth projective surface with the Fock space representation of the Heisenberg algebra associated to its cohomology lattice. Following a conjecture by Grojnowski, Segal and Wang extended this to any smooth projective variety by replacing Hilbert schemes with symmetric powers and cohomology with equivariant K-theory. Later, Krug lifted this to the level of derived categories.

On the other hand, Khovanov introduced a categorification of the free boson Heisenberg algebra, i.e. the one associated a single point. It is a monoidal category whose morphisms are described by a certain planar diagram calculus which categorifies the Heisenberg relations. A similar categorification was constructed by Cautis and Licata for the Heisenberg algebras of ADE type root lattices.

We show how to associate the Heisenberg 2-category to any smoooth and proper DG category and then define its Fock space 2-representation. This construction can be decategorified via K-theory or via Hochschild homology. It unifies all the results above and extends them to what can be viewed as the generality of arbitrary noncommutative smooth and proper schemes.