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 ${\mathcal{T}}_N:{\mathbb{T}}^{+}\to {\mathbb{T}}^{+}$, $M\mapsto N\otimes M,M,N\in {\mathbb{T}}^{+}$. 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 ${\mathbb{T}}^{+}$.

17 pages

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Cited by in corpus (4)

• On a categorical Boson-Fermion correspondence
• On Plücker Equations Characterizing Grassmann Cones
• Integrable $sl(\infty )$-modules and Category $\mathcal{O}$ for $\mathfrak{gl}(m|n)$
• Infinite-level Fock spaces, crystal bases, and tensor product of extremal weight modules of type $A_{+\infty }$