Categorical Bernstein operators and the Boson-Fermion correspondence

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 $${\mathcal {H}}$$ 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 $${\mathcal {H}}$$ 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 $${\mathcal {H}}$$ by lifting various Littlewood-Richardson branching isomorphisms to the Karoubian envelope of $${\mathcal {H}}$$ .