Cheshire charge in (3+1)-D topological phases

Understand

We show that (3+1)-dimensional topological phases of matter generically support loop excitations with topological degeneracy.

• The loops carry "Cheshire charge": topological charge that is not the integral of a locally-defined topological charge density.
• Cheshire charge has previously been discussed in non-Abelian gauge theories, but we show that it is a generic feature of all (3+1)-D topological phases (even those constructed from an Abelian gauge group).
• Indeed, Cheshire charge is closely related to non-trivial three-loop braiding.