Unitary forms for holomorphic vertex operator algebras of central charge $24$

TL;DR

This paper proves that all holomorphic vertex operator algebras of central charge 24 with non-trivial weight one subspaces are unitary, using orbifold constructions from Niemeier lattice VOAs.

Contribution

It establishes the unitarity of a broad class of holomorphic VOAs at central charge 24 via orbifold methods and automorphism group analysis.

Findings

All such VOAs are proven to be unitary.

The unitarity extends from lattice VOAs to orbifolded VOAs.

Method applies orbifold construction and automorphism group properties.

Abstract

We prove that all holomorphic vertex operator algebras of central charge $24$ with non-trivial weight one subspaces are unitary. The main method is to use the orbifold construction of a holomorphic VOA $V$ of central charge $24$ directly from a Niemeier lattice VOA $V_N$. We show that it is possible to extend the unitary form for the lattice VOA $V_N$ to the holomorphic VOA $V$ by using the orbifold construction and some information of the automorphism group $\mathrm{Aut}(V)$.