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

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 nontrivial three-loop braiding. We use a dimensional reduction argument to compute the topological degeneracy of loop excitations in the (3 + 1)-dimensional topological phases associated with Dijkgraaf-Witten gauge theories. We explicitly construct membrane operators associated with such excitations in solublemicroscopic lattice models in Z(2) x Z(2) Dijkgraaf Witten phases and generalize this construction to arbitrary membrane-net models. We explain why these loop excitations are the objects in the braided fusion 2-category Z(2Vect(G)(omega)), thereby supporting the hypothesis that 2-categories are the correct mathematical framework for (3 + 1)-dimensional topological phases.