An $E_8$-approach to the moonshine vertex operator algebra

TL;DR

This paper explores the structure of the moonshine vertex operator algebra using an $E_8$-based approach, employing quadratic spaces and orthogonal groups to analyze automorphisms and subalgebras, revealing new mathematical analogies.

Contribution

It introduces an $E_8$-approach to the moonshine VOA, describing its subalgebras via quadratic spaces and analyzing automorphism group actions, extending previous lattice and code analogies.

Findings

Automorphism group acts transitively on certain subalgebras

Determined stabilizers of specific subalgebras

Established new analogies with Leech lattice and Golay code

Abstract

In this article, we study the moonshine vertex operator algebra starting with the tensor product of three copies of the vertex operator algebra $V_{\sqrt2E_8}^+$, and describe it by the quadratic space over $\F_2$ associated to $V_{\sqrt2E_8}^+$. Using quadratic spaces and orthogonal groups, we show the transitivity of the automorphism group of the moonshine vertex operator algebra on the set of all full vertex operator subalgebras isomorphic to the tensor product of three copies of $V_{\sqrt2E_8}^+$, and determine the stabilizer of such a vertex operator subalgebra. Our approach is a vertex operator algebra analogue of "An $E_8$-approach to the Leech lattice and the Conway group" by Lepowsky and Meurman. Moreover, we find new analogies among the moonshine vertex operator algebra, the Leech lattice and the extended binary Golay code.