Finite vs. Infinite Decompositions in Conformal Embeddings

Building on work of the first and last author, we prove that an embedding of simple affine vertex algebras Vk(g0) ⊂ Vk(g) , corresponding to an embedding of a maximal equal rank reductive subalgebra g0 into a simple Lie algebra 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 Vk(g) decomposes finitely as a Vk(g0) -module.