# 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

https://doi.org/10.1016/j.aim.2019.106918

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  where  is a basic classical simple Lie superalgebra. Let  be the subalgebra of  generated by . We first classify all levels  for which the embedding  in  is conformal. Next we prove that, for a large family of such conformal levels,  is a completely reducible–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  as a finite, non simple current extension of . This decomposition uses our previous work [10] on the representation theory of .

We also study conformal embeddings  and in most cases we obtain decomposition rules.