Regularity of fixed-point vertex operator subalgebras

TL;DR

This paper proves that fixed-point subalgebras of regular vertex operator algebras under finite automorphisms are also regular, advancing understanding of symmetry properties and resolving key aspects of the Generalized Moonshine conjecture.

Contribution

It establishes the regularity of fixed-point subalgebras under finite automorphisms and demonstrates $SL_2(\mathbb{Z})$-compatibility of twisted characters, addressing a major conjecture.

Findings

Fixed-point subalgebras are regular under finite automorphisms.

Achieved $SL_2(\mathbb{Z})$-compatibility for twisted characters.

Resolved a principal claim in the Generalized Moonshine conjecture.

Abstract

We show that if $T$ is a simple non-negatively graded regular vertex operator algebra with a nonsingular invariant bilinear form and $\sigma$ is a finite order automorphism of $T$, then the fixed-point vertex operator subalgebra $T^\sigma$ is also regular. This yields regularity for fixed point vertex operator subalgebras under the action of any finite solvable group. As an application, we obtain an $SL_2(\mathbb{Z})$-compatibility between twisted twining characters for commuting finite order automorphisms of holomorphic vertex operator algebras. This resolves one of the principal claims in the Generalized Moonshine conjecture.