(3+1)D topological orders with only a ${\mathbb{Z}}_2$-charged particle

Abstract

There is exactly one bosonic (3+1)-dimensional topological order whose only nontrivial particle is an emergent boson: pure ${\mathbb{Z}}_2$ gauge theory. There are exactly two (3+1)-dimensional topological orders whose only nontrivial particle is an emergent fermion: pure "spin-${\mathbb{Z}}_2$" gauge theory, in which the dynamical field is a spin structure; and an anomalous version thereof. I give three proofs of this classification, varying from hands-on to abstract. Along the way, I provide a detailed study of the braided fusion $2$-category ${\mathcal{Z}}_{(1)}(\Sigma \mathbf{SVec})$ of string and particle operators in pure spin-${\mathbb{Z}}_2$ gauge theory.

30 pages

References in corpus (3)

• Generalized Global Symmetries
• Fusion 2-categories and a state-sum invariant for 4-manifolds
• Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions

Cited by in corpus (1)

• Non-invertible topological defects in 4-dimensional ${\mathbb{Z}}_2$ pure lattice gauge theory