A remark on ${\mathbb Z}_p$-orbifold constructions of the Moonshine vertex operator algebra

TL;DR

This paper demonstrates that certain ${ m Z}_p$-orbifold constructions on the Leech lattice vertex operator algebra produce the Moonshine vertex operator algebra, revealing connections between different orbifold methods and characterizing $V^ atural$ via Ising vectors.

Contribution

It establishes isomorphisms between orbifold constructions of $V_ ext{Leech}$ and $V^ atural$ for specific primes, and characterizes $V^ atural$ using orthogonal Ising vectors.

Findings

${ m Z}_p$-orbifold constructions yield $V^ atural$ for p=3,5,7,13.

Connections between ${ m Z}_p$ and ${ m Z}_2$ orbifold constructions.

Characterization of $V^ atural$ by two orthogonal Ising vectors.

Abstract

For $p = 3,5,7,13$, we consider a ${\mathbb Z}_p$-orbifold construction of the Moonshine vertex operator algebra $V^\natural$. We show that the vertex operator algebra obtained by the ${\mathbb Z}_p$-orbifold construction on the Leech lattice vertex operator algebra $V_\Lambda$ and a lift of a fixed-point-free isometry of order $p$ is isomorphic to the Moonshine vertex operator algebra $V^\natural$. We also describe the relationship between those ${\mathbb Z}_p$-orbifold constructions and the ${\mathbb Z}_2$-orbifold construction in a uniform manner. In Appendix, we give a characterization of the Moonshine vertex operator algebra $V^\natural$ by two mutually orthogonal Ising vectors.