A categorification of the boson-fermion correspondence via representation theory of $sl(\infty)

Abstract

In recent years different aspects of categorification of the boson-fermion correspondence have been studied. In this paper we propose a categorification of the boson-fermion correspondence based on the category of tensor modules of the Lie algebra $sl(\infty)$ of finitary infinite matrices. By $\mathbb T^+$ we denote the category of "polynomial" tensor $sl(\infty)$-modules. There is a natural "creation" functor $\mathcal T_N: \mathbb T^+\to \mathbb T^+$, $M\mapsto N\otimes M,\quad M,N\in \mathbb T^+$. The key idea of the paper is to employ the entire category $\mathbb T$ of tensor $sl(\infty)$-modules in order to define the "annihilation" functor $\mathcal D_N: \mathbb T^+\to \mathbb T^+$ corresponding to $\mathcal T_N$. We show that the relations allowing to express fermions via bosons arise from relations in the cohomology of complexes of linear endofunctors on $\mathbb T^+$.