An application of collapsing levels to the representation theory of affine vertex algebras

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 $V_{-1/2}(C_n)$ and $V_{-4}(E_7)$, we surprisingly obtain the realization of non-simple affine vertex algebras of types $B$ and $D$ having exactly one non-trivial ideal.

25 pages. In this replacement we correct some typos on page 19 at the end of Section 7, that we learned from T. Arakawa, X. Dai, J. Fasquel. B. Li and A. Moreau