Symmetric quotient stacks and Heisenberg actions

TL;DR

This paper constructs a Heisenberg algebra action on the Grothendieck groups of symmetric quotient stacks for smooth projective varieties, providing a categorification of the algebraic structure.

Contribution

It introduces a novel functorial framework that weakly categorifies the Heisenberg algebra action on symmetric quotient stacks.

Findings

Heisenberg algebra acts on Grothendieck groups of symmetric quotient stacks.

Fock space appears as a subrepresentation within this action.

Provides a functorial categorification of the algebraic structure.

Abstract

For every smooth projective variety, we construct an action of the Heisenberg algebra on the direct sum of the Grothendieck groups of all the symmetric quotient stacks which contains the Fock space as a subrepresentation. The action is induced by functors on the level of the derived categories which form a weak categorification of the action.