Constructing modular categories from orbifold data

TL;DR

The paper develops a general framework for constructing modular fusion categories from orbifold data inside a given modular fusion category, via the category of Wilson lines C_A associated to a simple orbifold datum A. It proves that C_A inherits a modular fusion category structure and provides an exact dimension formula relating Dim C_A to Dim C, φ and ψ, unifying two classical constructions: local A-modules and Drinfeld centers. Through detailed analysis of two key examples, it shows C_A ≅ C_A^loc in the commutative Frobenius case and C_A ≅ Z(S) when C = Vect and S is spherical, thereby connecting orbifold data to Enriched Drinfeld centers and gauging phenomena. The results offer a unified algebraic lens on generalised orbifolds in 3d TQFTs and pave the way for a broader understanding of how stratified TQFT data encode modular categorical structures.

Abstract

In Carqueville et al., arXiv:1809.01483, the notion of an orbifold datum $\mathbb{A}$ in a modular fusion category $\mathcal{C}$ was introduced as part of a generalised orbifold construction for Reshetikhin-Turaev TQFTs. In this paper, given a simple orbifold datum $\mathbb{A}$ in $\mathcal{C}$, we introduce a ribbon category $\mathcal{C}_{\mathbb{A}}$ and show that it is again a modular fusion category. The definition of $\mathcal{C}_{\mathbb{A}}$ is motivated by properties of Wilson lines in the generalised orbifold. We analyse two examples in detail: (i) when $\mathbb{A}$ is given by a simple commutative $Δ$-separable Frobenius algebra $A$ in $\mathcal{C}$; (ii) when $\mathbb{A}$ is an orbifold datum in $\mathcal{C} = \operatorname{Vect}$, built from a spherical fusion category $\mathcal{S}$. We show that in case (i), $\mathcal{C}_{\mathbb{A}}$ is ribbon-equivalent to the category of local modules of $A$, and in case (ii), to the Drinfeld centre of $\mathcal{S}$. The category $\mathcal{C}_{\mathbb{A}}$ thus unifies these two constructions into a single algebraic setting.

Figures (5)

• Figure 1.1: (a) Line defects in a surface defect are labelled by bimodules $M$. (b) $T$-crossings $\tau_1$, $\tau_2$ into adjacent surface defects. (c) Example of a compatibility condition between the $T$-crossings $\tau_i$ and $\alpha$. (d) Example of the $T$-crossing for the tensor product $M \otimes_A N$ of Wilson line defects. (e) Braiding of two Wilson line defects via a surface defect bubble. The various insertions of $\psi$ and $\phi$ are result from the orbifold construction and can be ignored at first.
• Figure 2.1: Identities an orbifold datum has to satisfy. All these string diagrams are drawn in $\mathcal{C}$, but by the comment below \ref{['eq:proj_swallow']} they induce identities also between the appropriate tensor products over $A$ in ${_A \mathcal{C}_A}$. To manipulate expressions involving orbifold data it is best to first ignore all the actions of $\psi$, which is why we draw them in grey.
• Figure 3.1: Identities for an object of $\mathcal{C}_\mathbb{A}$.
• Figure 3.2: Stratifications corresponding to the compositions (a) $H_{12}\circ H_2(M)$ and (b) $H_{21} \circ H_1(M)$.
• Figure 4.1: Left hand side of \ref{['eq:braided_funct_cond']}. Here, in the first equality one uses the natural transformation property of $\psi$ and the half-braidings, in the third we abbreviate $N_{ij}^r = \dim \mathcal{S}(r,ij)$.

Theorems & Definitions (46)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Proposition 2.1
• Definition 2.2
• Definition 3.1
• Example 3.2
• Definition 3.3
• Lemma 3.4
• proof