A Categorification of the Boson–Fermion Correspondence via Representation Theory of sl(∞)

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(∞) of finitary infinite matrices. By ${\mathbb{T}^{+}}$ we denote the category of “polynomial” tensor sl(∞)-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(∞)-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 one to express fermions via bosons arise from relations in the cohomology of complexes of linear endofunctors on ${{\mathbb{T}^{+}}}$.