Orbifolds of n-dimensional defect TQFTs

TL;DR

This work develops a unified framework for n-dimensional defect TQFTs by introducing generalized orbifolds $\mathcal{Z}_{\mathcal{A}}$ formed by decorating Poincaré duals of triangulated bordisms with defect data and enforcing invariance under $n$-dimensional Pachner moves. It shows how state-sum models and symmetry gauging arise naturally in this setting, with a complete dimensional hierarchy: $2$D orbifolds correspond to $\Delta$-separable symmetric Frobenius algebras in a pivotal bicategory, while $3$D orbifolds correspond to spherical fusion categories internal to Gray categories with duals (via special orbifold data). The paper also introduces Euler completion and point-defect algebras, and provides a higher-categorical formulation of orbifolds in both pivotal bicategories and Gray categories. In its 3D focus, it sets the stage for concrete constructions (e.g., Turaev–Viro and Reshetikhin–Turaev-type orbifolds) in follow-up work, connecting defect TQFTs to topological phases and quantum invariants. The framework offers a systematic pathway to build and classify orbifold theories across dimensions by encoding symmetries and defect data into algebraic structures compatible with Pachner moves.

Abstract

We introduce the notion of $n$-dimensional topological quantum field theory (TQFT) with defects as a symmetric monoidal functor on decorated stratified bordisms of dimension $n$. The familiar closed or open-closed TQFTs are special cases of defect TQFTs, and for $n=2$ and $n=3$ our general definition recovers what had previously been studied in the literature. Our main construction is that of "generalised orbifolds" for any $n$-dimensional defect TQFT: Given a defect TQFT $\mathcal{Z}$, one obtains a new TQFT $\mathcal{Z}_{\mathcal{A}}$ by decorating the Poincaré duals of triangulated bordisms with certain algebraic data $\mathcal{A}$ and then evaluating with $\mathcal{Z}$. The orbifold datum $\mathcal{A}$ is constrained by demanding invariance under $n$-dimensional Pachner moves. This procedure generalises both state sum models and gauging of finite symmetry groups, for any $n$. After developing the general theory, we focus on the case $n=3$.

Theorems & Definitions (71)

• Definition 2.1
• Remark 2.2
• Example 2.3
• Definition 2.4
• Example 2.5
• Definition 2.6
• Lemma 2.7
• proof
• Definition 2.8
• Lemma 2.9