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Reshetikhin-Turaev TQFTs close under generalised orbifolds

Nils Carqueville, Vincentas Mulevicius, Ingo Runkel, Gregor Schaumann, Daniel Scherl

Abstract

We specialise the construction of orbifold graph TQFTs introduced in Carqueville et al., arXiv:2101.02482 to Reshetikhin-Turaev defect TQFTs. We explain that the modular fusion category ${\mathcal{C}}_{\mathcal{A}}$ constructed in Mulevičius-Runkel, arXiv:2002.00663 from an orbifold datum $\mathcal{A}$ in a given modular fusion category $\mathcal{C}$ is a special case of the Wilson line ribbon categories introduced as part of the general theory of orbifold graph TQFTs. Using this, we prove that the Reshetikhin-Turaev TQFT obtained from ${\mathcal{C}}_{\mathcal{A}}$ is equivalent to the orbifold of the TQFT for $\mathcal{C}$ with respect to the orbifold datum $\mathcal{A}$.

Reshetikhin-Turaev TQFTs close under generalised orbifolds

Abstract

We specialise the construction of orbifold graph TQFTs introduced in Carqueville et al., arXiv:2101.02482 to Reshetikhin-Turaev defect TQFTs. We explain that the modular fusion category constructed in Mulevičius-Runkel, arXiv:2002.00663 from an orbifold datum in a given modular fusion category is a special case of the Wilson line ribbon categories introduced as part of the general theory of orbifold graph TQFTs. Using this, we prove that the Reshetikhin-Turaev TQFT obtained from is equivalent to the orbifold of the TQFT for with respect to the orbifold datum .

Paper Structure

This paper contains 15 sections, 10 theorems, 96 equations, 15 figures.

Table of Contents

  1. Introduction and summary
  2. Background
  3. Modular fusion categories
  4. Algebras and modules
  5. Orbifold data and associated modular fusion categories
  6. Reshetikhin--Turaev theory
  7. Reshetikhin--Turaev graph TQFT
  8. Reshetikhin--Turaev TQFT with line and surface defects
  9. Orbifold graph TQFT
  10. Proof of isomorphism
  11. Skeleta from surgery
  12. Invariants of closed 3-manifolds
  13. State spaces
  14. Defining identities
  15. Pipe functors

Key Result

Theorem 3.10

Let $([M,\mathcal{R})\colon\Sigma\longrightarrow\Sigma']\in\widehat{\operatorname{Bord}}{}^{\mathrm{rib}}_{3}(\mathcal{C}_\mathcal{A})$ be a $\mathcal{C}$-coloured ribbon bordism and let $\mathcal{S}, \mathcal{T}$ be two $\mathcal{A}$-coloured ribbon diagrams which agree on the boundary components $

Figures (15)

  • Figure 3.1: (a) Part of a defect bordism containing two parallel line defects labelled by $X_1, X_2$. (b) The line defects can be fused to a single line defect labelled by $X_1 \otimes_A X_2$. (c) Evaluating the cylinder containing the projection morphism $\pi\colon X_1\otimes X_2 \longrightarrow X_1 \otimes_A X_2$ with $\mathcal{Z}^{\mathcal{C}}$ provides an isomorphism between the two state spaces. Its inverse is obtained from a similar cylinder containing the inclusion $\imath\colon X_1 \otimes_A X_2 \longrightarrow X_1 \otimes X_2$.
  • Figure 3.2: (a) A cylinder over a defect sphere $S$ with a single $A$-labelled 1-stratum. (b) A choice or ribbonisation of the cylinder. (c) An element of the state space $\mathcal{Z}^{\mathcal{C}}(S)$ is obtained from $f\in\mathcal{C}(A,A)$ by applying the projection $\mathcal{C}(A,A) \longrightarrow {_A \mathcal{C}_A}(A,A)$, which sends $f$ to the string diagram displayed in the picture.
  • Figure 3.3: Upon evaluation with $\mathcal{Z}^{\mathcal{C}}$, contractible $A$-labelled 2-strata $F$ in the interior can be removed by forgetting all $A$-actions on the multimodules $X_i$ which decorate adjacent 1-strata. If $F$ is punctured by a $\gamma$-labelled 0-stratum $p$, then both $F$ and $p$ can be removed at the cost of puncturing one of the 1-strata adjacent to $F$ with a new 0-stratum whose label $\gamma'$ is obtained from $\gamma$ and the $A$-action.
  • Figure 3.4: Orbifold datum $\mathcal{A}=(A,T,\alpha,\overline{\alpha},\psi,\phi)$ as defect labels (for pictures involving the point defects $\psi$ and $\phi$, see Figure \ref{['fig:SpecialOrbifoldDataPhiPsi']}).
  • Figure 3.5: An object $(X,\tau_1,\tau_2,\overline{\tau_{1}},\overline{\tau_{2}})\in\mathcal{C}_\mathcal{A}$ as a set of defect labels.
  • ...and 10 more figures

Theorems & Definitions (26)

  • Definition 2.1
  • Remark 2.3
  • Remark 3.1
  • Remark 3.5
  • Theorem 3.10
  • Example 3.14
  • Example 3.15
  • Theorem 4.1
  • Lemma 4.2
  • proof
  • ...and 16 more