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Orbifolds of Reshetikhin-Turaev TQFTs

Nils Carqueville, Ingo Runkel, Gregor Schaumann

TL;DR

This work develops a unified framework to construct generalized orbifolds of Reshetikhin–Turaev TQFTs via 3D defect TQFTs. It shows that (i) Turaev–Viro state sums arise as orbifolds of the trivial theory from spherical fusion categories, (ii) G-crossed extensions yield group orbifolds inside a given modular tensor category, and (iii) commutative Δ-separable Frobenius algebras provide internal orbifold data in ribbon categories. The results include Morita-invariance of orbifold TQFTs, a detailed internal formulation of orbifold data, and explicit constructions and examples such as Tambara–Yamagami categories and ADE-type sl_2 level k theories. The work thereby unifies state-sum models and symmetry gauging within a single defect-TQFT/orbifold framework and extends the construction to nonsemisimple settings via ribbon categories. The key achievement is demonstrating isomorphisms between TV theories and orbifolds of the trivial RT theory, and providing concrete orbifold data from group extensions and Frobenius algebras that yield new RT-type TQFTs.

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 $Δ$-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.

Orbifolds of Reshetikhin-Turaev TQFTs

TL;DR

This work develops a unified framework to construct generalized orbifolds of Reshetikhin–Turaev TQFTs via 3D defect TQFTs. It shows that (i) Turaev–Viro state sums arise as orbifolds of the trivial theory from spherical fusion categories, (ii) G-crossed extensions yield group orbifolds inside a given modular tensor category, and (iii) commutative Δ-separable Frobenius algebras provide internal orbifold data in ribbon categories. The results include Morita-invariance of orbifold TQFTs, a detailed internal formulation of orbifold data, and explicit constructions and examples such as Tambara–Yamagami categories and ADE-type sl_2 level k theories. The work thereby unifies state-sum models and symmetry gauging within a single defect-TQFT/orbifold framework and extends the construction to nonsemisimple settings via ribbon categories. The key achievement is demonstrating isomorphisms between TV theories and orbifolds of the trivial RT theory, and providing concrete orbifold data from group extensions and Frobenius algebras that yield new RT-type TQFTs.

Abstract

We construct three classes of generalised orbifolds of Reshetikhin-Turaev theory for a modular tensor category , 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) -crossed extensions of give group orbifolds for any finite group , and (iii) we construct orbifolds from commutative -separable symmetric Frobenius algebras in . 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.

Paper Structure

This paper contains 14 sections, 14 theorems, 128 equations, 2 figures.

Table of Contents

  1. Introduction and summary
  2. TQFTs with defects and orbifolds
  3. Reshetikhin--Turaev defect TQFTs
  4. Orbifolds of defect TQFTs
  5. Orbifold data for Reshetikhin--Turaev theory
  6. State spaces for spheres
  7. Special orbifold data internal to ribbon fusion categories
  8. Morita equivalent orbifold data
  9. Commutative Frobenius algebras give orbifold data
  10. Turaev--Viro theory
  11. Turaev--Virelizier construction
  12. Orbifold data for the trivial Reshetikhin--Turaev theory
  13. Turaev--Viro theory is an orbifold
  14. Group extensions of modular tensor categories

Key Result

Lemma 3.1

Let $M,N$ be two cyclic multi-modules over a list $(A_{1}, \ldots, A_{n})$ of $\Delta$-separable symmetric Frobenius algebras $A_{i}$. The vector space $\mathcal{Z}^{\mathcal{C}}(S_{M,N})$ associated to the 2-sphere $S_{M,N}$ with $M,N$ on its South and North pole, respectively, is given by the spac

Figures (2)

  • Figure 4.1: Computing the left-hand side of condition \ref{['eq:349n']} for $\mathcal{A}^{\mathcal{S}}$. Note that this is a string diagram in $\operatorname{vect}$, so there is no need to distinguish between over- and under-crossings, but we prefer to keep the notation from Proposition \ref{['prop:internal-orb']}.
  • Figure 4.2: Computing the left-hand side of condition \ref{['eq:350n']} for $\mathcal{A}^{\mathcal{S}}$.

Theorems & Definitions (38)

  • Remark 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.3
  • proof
  • Definition 3.4
  • Proposition 3.5
  • Definition 3.7
  • Proposition 3.8
  • proof
  • ...and 28 more