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.

Latex file, 31 pages, minor corrections, to appear in Communications in Mathematical Physics

References in corpus (2)

• Field Algebras
• Joseph ideals and lisse minimal W-algebras

Cited by in corpus (9)

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• Conformal embeddings of affine vertex algebras in minimal $W$-algebras I: structural results
• The vertex algebras ${\mathcal{R}}^{(p)}$ and ${\mathcal{V}}^{(p)}$
• Tensor categories of affine Lie algebras beyond admissible levels
• An application of collapsing levels to the representation theory of affine vertex algebras
• Generalized parafermions of orthogonal type
• Invariant subalgebras of the small $\mathcal{N}=4$ superconformal algebra
• On the representation theory of the vertex algebra $L_{-5/2}(sl(4))$
• Kostant's pair of Lie type and conformal embeddings