Vertex Operator Algebra Analogue of Embedding $D_8$ into $E_8$

TL;DR

This paper constructs an explicit embedding of the vertex operator algebra associated with $D_8$ into that of $E_8$, showing the latter as an extension of the former by a simple module, with implications for conformal field theory.

Contribution

It provides an explicit construction of the embedding of $L_{D_8}(1, 0)$ into $L_{E_8}(1, 0)$ and describes the module structure as an extension, advancing vertex operator algebra theory.

Findings

Explicit embedding constructed

$L_{E_8}(1, 0)$ is an extension of $L_{D_8}(1, 0)$

Module structure clarified for conformal field theory

Abstract

Let $L_{D_8}(1, 0)$ and $L_{E_8}(1, 0)$ be the simple vertex operator algebras associated to untwisted affine Lie algebra $\widehat{{\mathbf g}}_{D_{8}}$ and $\widehat{{\mathbf g}}_{E_8}$ with level 1 respectively. In the 1980s by I. Frenkel, Lepowsky and Meurman as one of the many important preliminary steps toward their construction of the moonshine module vertex operator algebra, they use roots lattice showing that $L_{D_8}(1, 0)$ can embed into $L_{E_8}(1, 0)$ as a vertex operator subalgebra(\cite{5, 6, 8}). Their construct is a base of vertex operator theory. But the embedding they gave using the fact $L_{\mathbf g}(1,0)$ is isomorphic to its root lattice vertex operator algebra $V_L$. In this paper, we give an explicitly construction of the embedding and show that as an $L_{D_8}(1, 0)$-module, $L_{E_8}(1, 0)$ is isomorphic to the extension of $L_{D_8}(1, 0)$ by its simple module $L_{D_8}(1, \overline{\omega}_8)$. It may be convenient to be used for conformal field theory.