Conformal embeddings in affine vertex superalgebras

Authors: Dražen Adamović, Pierluigi Möseneder Frajria, Paolo Papi, Ozren Perše

Advances in Mathematics, Volume 360, 22 January 2020, 106918

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 superalgebra. 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|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)↪ \hookrightarrow sl(n+1|m) and in most cases we obtain decomposition rules.