(3+1)𝐷 topological orders with only a ℤ₂-charged particle

Source abstract

There is exactly one bosonic (3+1)-dimensional topological order whose only nontrivial particle is an emergent boson: pure Z 2 \mathbb {Z}_2 gauge theory. There are exactly two (3+1)-dimensional topological orders whose only nontrivial particle is an emergent fermion: pure “spin- Z 2 \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 2 -category Z ( 1 ) ( Σ S V e c ) \mathcal {Z}_{(1)}(\Sigma \mathbf {SVec}) of string and particle operators in pure spin- Z 2 \mathbb {Z}_2 gauge theory.