Orbifold graph TQFTs

TL;DR

This work develops a comprehensive framework for generalized orbifolds of 3D defect TQFTs, incorporating line defects, via three core innovations: (i) computation with admissible skeleta that simplify state-sum evaluations beyond triangulations, (ii) construction of a ribbon category W_A of Wilson lines associated to orbifold data, and (iii) a graph TQFT extension that assigns invariants to bordisms with W_A-labeled ribbon graphs. The authors show how special orbifold data enable efficient calculation through admissible skeleta and BLT moves, and they build a robust algebraic backbone (W_A) that mirrors the line-defect structure in the defect TQFT, with a concrete connection to RT-type theories in RT settings. The culmination is an orbifold graph TQFT that pushes ribbon graphs into decorated skeleta and is proven independent of representation choices, establishing a principled pathway to compute orbifold theories in 3D. This work lays the groundwork for applying orbifold techniques to a broad class of 3D TQFTs and sets the stage for a companion paper proving conjectured equivalences with RT-based orbifolds and clarifying the role of Wilson lines in such constructions.

Abstract

A generalised orbifold of a defect TQFT $\mathcal{Z}$ is another TQFT $\mathcal{Z}_{\mathcal{A}}$ obtained by performing a state sum construction internal to $\mathcal{Z}$. As an input it needs a so-called orbifold datum $\mathcal{A}$ which is used to label stratifications coming from duals of triangulations and is subject to conditions encoding the invariance under Pachner moves. In this paper we extend the construction of generalised orbifolds of $3$-dimensional TQFTs to include line defects. The result is a TQFT acting on 3-bordisms with embedded ribbon graphs labelled by a ribbon category $\mathcal{W}_{\mathcal{A}}$ that we canonically associate to $\mathcal{Z}$ and $\mathcal{A}$. We also show that for special orbifold data, the internal state sum construction can be performed on more general skeletons than those dual to triangulations. This makes computations with $\mathcal{Z}_{\mathcal{A}}$ easier to handle in specific examples.

Figures (11)

• Figure 1.1: (a) A local patch of a bordism $M$ with embedded ${\mathcal{W}_{\mathcal{A}}}$-labelled ribbon graph $\mathcal{R}$. (b) A local patch of an admissible $\mathcal{A}$-decorated skeleton $\mathcal{S}$ for $M$ (where all green 2-strata implicitly carry a label $\mathcal{A}_2$) together with two choices to represent $\mathcal{R}$ in $\mathcal{S}$, which are necessarily related by the moves in Figures \ref{['fig:BLTMoves']} and \ref{['fig:omegaMoves']}.
• Figure 2.1: List of allowed local neighbourhoods for skeleta $S$ of a 3-manifold with boundary $M$. As in Section \ref{['sssec:stratMan']}, $S^{(j)}$ denotes the union of all $j$-strata. 2-strata are depicted in green, 1-strata in black, and the boundary of $M$ is grey (colour available online).
• Figure 2.2: BLT moves without orientations. The dotted lines indicate where the 2-strata meet the boundary of the 3-ball in which they are embedded.
• Figure 2.3: Examples of BLT moves with orientations (of a total of 32).
• Figure 2.4: A ribbon graph in $S^3$ (a) and an example of an admissible ribbon diagram representing it (b)

Theorems & Definitions (71)

• Example 2.1
• Definition 2.2
• Remark 2.3
• Remark 2.4
• Definition 2.5
• Definition 2.7
• Remark 2.8
• Definition 2.10
• Lemma 2.11
• proof