engleski

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)↪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 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_fsubreg. It turns out that the properties of H_fsubreg are more subtle than in the case of minimal reducition.

affine vertex algebras ; vertex operator algebras ; fusion rules ; conformal embeddings ; representation theory

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