Compact automorphism groups of vertex operator algebras

TL;DR

This paper establishes a duality and Galois correspondence relating automorphism groups and modules of simple vertex operator algebras with compact Lie group actions, extending classical symmetry concepts.

Contribution

It introduces a Schur-Weyl type duality and a Galois correspondence for vertex operator algebras with compact automorphism groups, linking algebraic and group-theoretic structures.

Findings

Duality between unitary irreducible modules of G and V^G

Galois correspondence between subalgebras and subgroups for abelian G

Extension of classical symmetry principles to vertex operator algebras

Abstract

Let $V$ be a simple vertex operator algebra which admits the continuous, faithful action of a compact Lie group $G$ of automorphisms. We establish a Schur-Weyl type duality between the unitary, irreducible modules for $G$ and the irreducible modules for $V^G$ which are contained in $V$ where $V^G$ is the space of $G$-invariants of $V.$ We also prove a concomitant Galois correspondence between vertex operator subalgebras of $V$ which contain $V^G$ and closed Lie subgroups of $G$ in the case that $G$ is abelian.