Computing $G$-Crossed Extensions and Orbifolds of Vertex Operator Algebras

TL;DR

This paper develops new tools for computing G-crossed extensions and orbifolds of vertex operator algebras, enabling the determination of their modular tensor categories and advancing the understanding of orbifold constructions.

Contribution

It introduces methods to compute G-crossed extensions and orbifolds of vertex operator algebras, including applications to lattice VOAs and Tambara-Yamagami categories, with new computational techniques.

Findings

Determined the modular tensor category of orbifolds of lattice VOAs under specific involutions.

Established the compatibility of G-crossed extensions with condensations by commutative algebras.

Produced coherence data for generalized Tambara-Yamagami categories.

Abstract

In this article, we develop tools for computing $G$-crossed extensions of braided tensor categories. Their equivariantisations appear as categories of modules of fixed-point subalgebras (or orbifolds) of vertex operator algebras and are often difficult to determine. As the first tool, we show how the seminal work of Etingof, Nikshych and Ostrik on the uniqueness of $G$-crossed extensions can be used to determine the category of modules of orbifold vertex operator algebras. As an application, we determine the modular tensor category of the orbifold of a lattice vertex operator algebra under a lift of $-\mathrm{id}$ for a lattice with odd-order discriminant form. In that case, the de-equivariantisation is of Tambara-Yamagami type. As the second tool, we describe how $G$-crossed extensions and condensations by commutative algebras commute in a suitable sense. This leads to an effective approach to compute new $G$-crossed extensions. As one application, we produce the coherence data that are then used in arXiv:2411.12251 to define a generalisation of the Tambara-Yamagami categories with more than one simple object in the twisted sector. This also yields the modular tensor category of the orbifold of an arbitrary lattice vertex operator algebra under a lift of $-\mathrm{id}$. Finally, we sketch how to categorically approach the general problem of lattice orbifolds under lifts of arbitrary lattice involutions.