Categorical Bernstein Operators and the Boson-Fermion Correspondence

TL;DR

This paper categorifies the Boson-Fermion correspondence by lifting Bernstein operators to chain complexes in Khovanov's Heisenberg category, establishing categorical Clifford relations and demonstrating the categorical Fock space as a direct summand.

Contribution

It proves conjectures of Cautis and Sussan by constructing categorical analogues of fermionic operators and demonstrating their algebraic relations within the Heisenberg category.

Findings

Categorical fermionic functors satisfy Clifford algebra relations.

Categorical Fock space is a direct summand of the regular representation.

Enhanced graphical calculus with lifted Littlewood-Richardson isomorphisms.

Abstract

We prove a conjecture of Cautis and Sussan providing a categorification of the Boson-Fermion correspondence as formulated by Frenkel and Kac. We lift the Bernstein operators to infinite chain complexes in Khovanov's Heisenberg category H and from them construct categorical analogues of the Kac-Frenkel fermionic vertex operators. These fermionic functors are then shown to satisfy categorical Clifford algebra relations, solving a conjecture of Cautis and Sussan. We also prove another conjecture of Cautis and Sussan demonstrating that the categorical Fock space representation of H is a direct summand of the regular representation by showing that certain infinite chain complexes are categorical Fock space idempotents. In the process, we enhance the graphical calculus of H by lifting various Littlewood-Richardson branching isomorphisms to the Karoubian envelope of H.