An Application of Collapsing Levels to the Representation Theory of Affine Vertex Algebras

Authors: Dražen Adamović, Victor G Kac, Pierluigi Möseneder Frajria, Paolo Papi, Ozren Perše
International Mathematics Research Notices, rny237,
Abstract: We discover a large class of simple affine vertex algebras V_k(\mathfrak{g}), associated to basic Lie superalgebras \mathfrak{g} at non-admissible collapsing levels k, having exactly one irreducible \mathfrak{g}-locally finite module in the category \mathcal{O}. In the case when \mathfrak{g} is a Lie algebra, we prove a complete reducibility result for V_k(\mathfrak{g})-modules at an arbitrary collapsing level. We also determine the generators of the maximal ideal in the universal affine vertex algebra V_k(\mathfrak{g}) at certain negative integer levels. Considering some conformal embeddings in the simple affine vertex algebras V_{-\frac{1}{2}}(C_n) and V_{-4}(E_7), we surprisingly obtain the realization of non-simple affine vertex algebras of types B and D having exactly one nontrivial ideal.