Regularity of fixed-point vertex operator subalgebras

TL;DR

The work resolves the cyclic orbifold problem for regularity by combining $C_2^0$-cofiniteness, projectivity, and genus-one modular techniques. It proves that fixed-point subalgebras $T^\sigma$ of simple regular VOAs $T$ under finite-order automorphisms are regular, and extends to finite solvable groups, yielding strong structural consequences. A key development is establishing rigidity and projectivity of simple modules via Moore–Seiberg–Huang-type arguments, which, together with modular invariance, leads to regularity and $g$-rationality for twisted sectors. The results have significant impact for holomorphic orbifolds and Generalized Moonshine, providing $SL_2(Z)$-compatibility of twisted-twining characters and strengthening the connections between orbifold theory and moonshine phenomena.

Abstract

We show that if $T$ is a simple non-negatively graded regular vertex operator algebra with a nonsingular invariant bilinear form and $σ$ is a finite order automorphism of $T$, then the fixed-point vertex operator subalgebra $T^σ$ 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.

Theorems & Definitions (82)

• Theorem
• Theorem
• Theorem
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4