(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.