Orbifolds of Reshetikhin-Turaev TQFTs

TL;DR

This paper develops a comprehensive framework for constructing orbifolds of Reshetikhin-Turaev TQFTs using defect TQFT, encompassing group actions, Frobenius algebras, and their relations to Turaev-Viro models.

Contribution

It introduces three new classes of orbifolds for Reshetikhin-Turaev TQFTs, unifies various constructions, and connects them to existing models like Turaev-Viro.

Findings

Orbifolds from spherical fusion categories and Frobenius algebras are constructed.

Turaev-Viro state sum is shown to be an orbifold of the trivial defect TQFT.

Morita equivalence implies isomorphic orbifolds, with applications to Tambara-Yamagami categories.

Abstract

We construct three classes of generalised orbifolds of Reshetikhin-Turaev theory for a modular tensor category $\mathcal{C}$, using the language of defect TQFT from [arXiv:1705.06085]: (i) spherical fusion categories give orbifolds for the "trivial" defect TQFT associated to vect, (ii) $G$-crossed extensions of $\mathcal{C}$ give group orbifolds for any finite group $G$, and (iii) we construct orbifolds from commutative $\Delta$-separable symmetric Frobenius algebras in $\mathcal{C}$. We also explain how the Turaev-Viro state sum construction fits into our framework by proving that it is isomorphic to the orbifold of case (i). Moreover, we treat the cases (ii) and (iii) in the more general setting of ribbon tensor categories. For case (ii) we show how Morita equivalence leads to isomorphic orbifolds, and we discuss Tambara-Yamagami categories as particular examples.