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

We study the representation theory of non-admissible simple affine vertex algebra $L_{-5\mathrm{/}2}(sl(4))$. We determine an explicit formula for the singular vector of conformal weight four in the universal affine vertex algebra $V^{-5\mathrm{/}2}(sl(4))$, and show that it generates the maximal ideal in $V^{-5/2} (sl(4))$\mathrm{.}Weclassifyirreducible$L_{-5/2} (sl(4))$--modulesinthecategory${\mathcal O}$, and determine the fusion rules between irreducible modules in the category of ordinary modules $K{L}_{-5\mathrm{/}2}$. It turns out that this fusion algebra is isomorphic to the fusion algebra of $K{L}_{-1}$. We also prove that $K{L}_{-5\mathrm{/}2}$ 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 $gl(4) \hookrightarrow sl(5)$atlevel$k=-5/2$fromarXiv:\mathrm{1509.06512.}Weshowthat$k=-5/2$isacollapsinglevelwithrespecttothesubregularnilpotentelement$f_{subreg}$,meaningthatthesimplequotientoftheaffine$$\mathcal{W}$$--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.