$\mathbb{Z}_2$-orbifold construction associated with $(-1)$-isometry and uniqueness of holomorphic vertex operator algebras of central charge 24

TL;DR

This paper proves the uniqueness of certain holomorphic vertex operator algebras of central charge 24 based on their weight one Lie algebra structure, using a $Z_2$-orbifold construction related to $(-1)$-isometry.

Contribution

It establishes the uniqueness of holomorphic VOAs of central charge 24 for specific Lie algebra types of their weight one space, expanding classification results.

Findings

Uniqueness of VOAs for specified Lie algebra structures.

Application of $Z_2$-orbifold construction with $(-1)$-isometry.

Complete classification for the given Lie algebra types.

Abstract

The vertex operator algebra structure of a strongly regular holomorphic vertex operator algebra $V$ of central charge $24$ is proved to be uniquely determined by the Lie algebra structure of its weight one space $V_1$ if $V_1$ is a Lie algebra of the type $A_{1,4}^{12}$, $B_{2,2}^6$, $B_{3,2}^4$, $B_{4,2}^3$, $B_{6,2}^2$, $B_{12,2}$, $D_{4,2}^2B_{2,1}^4$, $D_{8,2}B_{4,1}^2$, $A_{3,2}^4A_{1,1}^4$, $D_{5,2}^2A_{3,1}^2$, $D_{9,2}A_{7,1}$, $C_{4,1}^4$ or $D_{6,2}B_{3,1}^2C_{4,1}$.