Some general results on conformal embeddings of affine vertex operator algebras

TL;DR

This paper establishes a general criterion for conformal embeddings of affine Lie algebra vertex operator algebras, constructs new embeddings at various levels, and completes the classification related to Dynkin diagram automorphisms.

Contribution

It introduces a universal criterion for conformal embeddings and constructs all remaining embeddings linked to Dynkin diagram automorphisms.

Findings

Established a criterion for conformal embeddings at arbitrary levels

Constructed new embeddings at rational and negative integer levels

Completed the classification of embeddings related to Dynkin diagram automorphisms

Abstract

We give a general criterion for conformal embeddings of vertex operator algebras associated to affine Lie algebras at arbitrary levels. Using that criterion, we construct new conformal embeddings at admissible rational and negative integer levels. In particular, we construct all remaining conformal embeddings associated to automorphisms of Dynkin diagrams of simple Lie algebras. The semisimplicity of the corresponding decompositions is obtained by using the concept of fusion rules for vertex operator algebras.