Algebraic Geometry Seminar
In a 1996 paper, Ian Grojnowski suggested that the $S^n$-equivariant K-groups of a variety should form a Fock space. About 20 years later, Andreas Krug constructed functors on the corresponding equivariant derived categories which satisfy the relations of a Heisenberg algebra, and which decategorify to an action on Grothendieck groups.
This strongly suggest that there is a monoidal category which categorifies a Heisenberg algebra modeled on the Grothendieck group of a variety, and which acts on derived categories of symmetric quotient stacks. In joint work with Ádám Gyenge and Timothy Logvinenko we go one step further and associate a Heisenberg category to every (smooth and proper dg-) category. Each such Heisenberg category acts on a categorical Fock space. For the derived category of a variety, this Fock category produces an enhancement of Krug's action.