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(g) where g=g_0+g_1 is a basic classical simple Lie superalgebra. Let VV_k(g_0) be the subalgebra of V_k(g) generated by g_0. We first classify all levels k for which the embedding VV_k(g_0) in V_k(g) is conformal. Next we prove that, for a large family of such conformal levels, V_k(g) is a completely reducible VV_k(g_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}(sop(2n+8|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}). We also study conformal embeddings gl(n|m) ---> sl(n+1|m) and in most cases we obtain decomposition rules.