On the representation theory of the vertex algebra $L_{-5/2}(sl(4))$

Abstract

We study the representation theory of non-admissible simple affine vertex algebra $L_{-5/2}(sl(4))$. We determine an explicit formula for the singular vector of conformal weight four in the universal affine vertex algebra $V^{-5/2}(sl(4))$, and show that it generates the maximal ideal in $V^{-5/2}(sl(4))$. We classify irreducible $L_{-5/2}(sl(4))$--modules in the category $\mathcal{O}$, and determine the fusion rules between irreducible modules in the category of ordinary modules $K{L}_{-5/2}$. It turns out that this fusion algebra is isomorphic to the fusion algebra of $K{L}_{-1}$. We also prove that $K{L}_{-5/2}$ is a semi-simple, rigid braided tensor category. In our proofs we use the notion of collapsing level for the affine $\mathcal{W}$--algebra, and the properties of conformal embedding $gl(4)\hookrightarrow sl(5)$ at level $k=-5/2$ from arXiv:1509.06512. We show that $k=-5/2$ is a collapsing level with respect to the subregular nilpotent element $f_{subreg}$, meaning that the simple quotient of the affine $\mathcal{W}$--algebra $W^{-5/2}(sl(4),f_{subreg})$ is isomorphic to the Heisenberg vertex algebra $M_J(1)$. We prove certain results on vanishing and non-vanishing of cohomology for the quantum Hamiltonian reduction functor $H_{f_{subreg}}$. It turns out that the properties of $H_{f_{subreg}}$ are more subtle than in the case of minimal reducition.

arXiv comment27 pages; final version, to appear in Communications in Contemporary Mathematics

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