Orbifold graph TQFTs

TL;DR

This paper extends orbifold graph TQFTs to include line defects, resulting in a more versatile 3D TQFT framework that simplifies computations and broadens applicability.

Contribution

It introduces a generalized orbifold construction for 3D TQFTs with line defects, associating a ribbon category to the orbifold data and enabling computations on more general skeletons.

Findings

Extended orbifold TQFTs to include line defects.

Associated a ribbon category to the orbifold data.

Simplified computations by allowing more general skeletons.

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.