Conformal embeddings in affine vertex superalgebras

Abstract

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 $V_{-2} (osp(2n +8 \\vert 2n))$ 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 $V_{-2} (D_{n+4})$.