Embeddings of vertex operator algebras associated to orthogonal affine Lie algebras

TL;DR

This paper investigates the embeddings of certain vertex operator algebras associated with affine Lie algebras of types D, B, and F, revealing subalgebra structures and their conformal vectors at specific levels.

Contribution

It demonstrates that vertex operator algebras for affine Lie algebras of types D and B embed into each other at a specific level, and for 4, it shows a subalgebra structure involving three copies of type B within type F.

Findings

$L_{D_{ ext{ell}}}(- ext{ell}+3/2,0)$ embeds into $L_{B_{ ext{ell}}}(- ext{ell}+3/2,0)$

For 4, three copies of $L_{B_{4}}(-5/2,0)$ embed into $L_{F_{4}}(-5/2,0)$

All these VOAs share the same conformal vector

Abstract

Let $L_{D_{\ell}}(-\ell +{3/2},0)$ (resp. $L_{B_{\ell}}(-\ell +{3/2},0)$) be the simple vertex operator algebra associated to affine Lie algebra of type $D_{\ell}^{(1)}$ (resp. $B_{\ell}^{(1)}$) with the lowest admissible half-integer level $-\ell + {3/2}$. We show that $L_{D_{\ell}}(-\ell +{3/2},0)$ is a vertex subalgebra of $L_{B_{\ell}}(-\ell +{3/2},0)$ with the same conformal vector. For $\ell =4$, $L_{D_{4}}(-{5/2},0)$ is a vertex subalgebra of three copies of $L_{B_{4}}(-{5/2},0)$ contained in $L_{F_{4}}(-{5/2},0)$, and all five of these vertex operator algebras have the same conformal vector.