On the representation theory of the vertex algebra L-5/2(sl(4))

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 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 W-algebra, and the properties of conformal embedding gl(4) (sic) sl(5) at level k = -5/2 from D. Adamovic et al. [Finite vs infinite decompositions in conformal embeddings, Comm. Math. Phys. 348 (2016) 445-473.]. We show that k = -5/2 is a collapsing level with respect to the subregular nilpotent element fsubreg, meaning that the simple quotient of the affine W-algebra W-5/2(sl(4), fsubreg) 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-fsubreg. It turns out that the properties of H-fsubreg are more subtle than in the case of minimal reduction.