Finite vs. Infinite Decompositions in Conformal Embeddings

Abstract: Building on work of the first and last author, we prove that an embedding of simple affine vertex algebras {V_{\mathbf{k}}({\mathfrak{g}}^0)\subset V_{k}({\mathfrak{g}})}, corresponding to an embedding of a maximal equal rank reductive subalgebra {{\mathfrak{g}}^0}  into a simple Lie algebra {{\mathfrak{g}}}, is conformal if and only if the corresponding central charges are equal. We classify the equal rank conformal embeddings. Furthermore we describe, in almost all cases, when {V_{k}({\mathfrak{g}})} decomposes finitely as a {V_{\mathbf{k}}({\mathfrak{g}}^0)}-module.

Communicated by Y. Kawahigashi. D.A. and O.P. are partially supported by the Croatian Science Foundation under the project 2634 and by the Croatian Scientific Centre of Excellence QuantixLie.

Dražen Adamović, Victor G. Kac, Pierluigi Möseneder Frajria, Paolo Papi, Ozren Perše, Finite vs. Infinite Decompositions in Conformal EmbeddingsCommunications in Mathematical Physics, Volume 348, Issue 2, pp 445–473. doi:10.1007/s00220-016-2672-1.