On the operator content of nilpotent orbifold models

TL;DR

This paper explores the structure of orbifold models in vertex operator algebras with finite nilpotent automorphism groups, establishing a Galois correspondence and linking module categories to twisted quantum doubles.

Contribution

It proves a Galois correspondence between subgroups and subalgebras and relates module categories to twisted quantum doubles via Hochschild 3-cocycles.

Findings

Established Galois correspondence between subgroups and subalgebras.

Proved equivalence of module categories with twisted quantum doubles.

Connected orbifold module categories to Hochschild 3-cocycles.

Abstract

Let $V$ be a simple vertex operator algebra and $G$ be a finite nilpotent group of automorphisms of $V.$ We prove the following in this paper: (1) There is a Galois correspondence between subgroups of $G$ and the vertex operator subalgebras of $V$ which contain $V^G$ given by the map $H\mapsto V^H.$ (2) Assume that for every G\in G$ there is unique simple $g$-twisted $V$-module $M(g).$ Then there exists a Hochschild 3-cocycle $\alpha$ on the integral group $Z[G]$ such that there is an equivalence of categories between $V^G$-module category (whose objects are $V^G$-submodules of direct sums of copies of $\oplus_{g\in G}M(g),$ and whose morphisms are $V^G$-module homomorphisms) and the module category for the twisted quantum double $D_{\alpha}(G)$ associated to $\alpha.$