Automorphism groups and uniqueness of holomorphic vertex operator algebras of central charge $24$

TL;DR

This paper characterizes the automorphism groups of all holomorphic vertex operator algebras of central charge 24 with non-trivial weight one Lie algebras and confirms a conjecture related to their classification, reinforcing their uniqueness.

Contribution

It provides a complete description of automorphism groups and verifies a conjecture on the classification of these algebras, establishing their uniqueness.

Findings

Automorphism groups of all such VOAs are described.

A conjecture on the enumeration of these VOAs is confirmed.

The uniqueness of these VOAs with non-trivial weight one Lie algebras is established.

Abstract

We describe the automorphism groups of all holomorphic vertex operator algebras of central charge $24$ with non-trivial weight one Lie algebras by using their constructions as simple current extensions. We also confirm a conjecture of G. H\"ohn on the numbers of holomorphic vertex operator algebras of central charge $24$ obtained as inequivalent simple current extensions of certain vertex operator algebras, which gives another proof of the uniqueness of holomorphic vertex operator algebras of central charge $24$ with non-trivial weight one Lie algebras.