Conformal embeddings in affine vertex superalgebras

This paper is a natural continuation of our previous work on conformal embeddings of vertex algebras [6], [7], [8]. Here we consider conformal embeddings in simple affine vertex superalgebra $V_k(\mathfrak{g})$ where $\mathfrak{g}={\mathfrak{g}}_{\bar{0}}\oplus {\mathfrak{g}}_{\bar{1}}$ is a basic classical simple Lie superalgebras. Let ${\mathcal{V}}_k({\mathfrak{g}}_{\bar{0}})$ be the subalgebra of $V_k(\mathfrak{g})$ generated by ${\mathfrak{g}}_{\bar{0}}$. We first classify all levels $k$ for which the embedding $\mathcal V_k (\mathfrak g_{\bar 0})$in$V_k(\mathfrak g)$ is conformal. Next we prove that, for a large family of such conformal levels, $V_k(\mathfrak{g})$ is a completely reducible ${\mathcal{V}}_k({\mathfrak{g}}_{\bar{0}})$–module and obtain decomposition rules. Proofs are based on fusion rules arguments and on the representation theory of certain affine vertex algebras. The most interesting case is the decomposition of $V_{-2}(osp(2n+8\mathrm{\mid }2n))$ as a finite, non simple current extension of $V_{-2}(D_{n+4})\otimes V_1(C_n)$. This decomposition uses our previous work [10] on the representation theory of $V_{-2}(D_{n+4})$.