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

TL;DR

This paper investigates the representation theory of the non-admissible affine vertex algebra $L_{-5/2}(sl(4))$, providing explicit formulas, classifying modules, and analyzing fusion rules and tensor categories.

Contribution

It offers the first explicit singular vector formula, classifies irreducible modules, and establishes the fusion algebra and tensor category properties for $L_{-5/2}(sl(4))$.

Findings

Explicit singular vector formula in $V^{-5/2}(sl(4))$

Fusion algebra isomorphic to that of $KL_{-1}$

Proves $KL_{-5/2}$ is semi-simple and rigid

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 $KL_{-5/2}$. It turns out that this fusion algebra is isomorphic to the fusion algebra of $KL_{-1}$. We also prove that $KL_{-5/2}$ 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.