Finite vs infinite decompositions in conformal embeddings

TL;DR

This paper characterizes when affine vertex algebra embeddings are conformal by equalizing central charges and classifies such embeddings, also detailing conditions for finite decompositions as modules.

Contribution

It provides a complete classification of equal rank conformal embeddings and describes conditions for finite decompositions in affine vertex algebras.

Findings

Conformal embeddings occur if and only if central charges are equal.

Classification of all equal rank conformal embeddings.

Conditions for finite module decompositions are identified.

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.