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

## TL;DR

This paper demonstrates that (3+1)-D topological phases inherently support loop excitations with non-localizable topological charge, called Cheshire charge, which is linked to complex three-loop braiding phenomena.

## Contribution

It establishes that Cheshire charge is a universal feature of all (3+1)-D topological phases, including Abelian ones, and connects these excitations to the mathematical framework of braided fusion 2-categories.

## Key findings

- Cheshire charge exists in all (3+1)-D topological phases.
- Explicit construction of membrane operators in lattice models.
- Connection of loop excitations to braided fusion 2-categories.

## Abstract

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. 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 soluble microscopic lattice models in ${\mathbb{Z}_2}\times{\mathbb{Z}_2}$ 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 $Z(\mathbf{2Vect}_G^{\omega})$, thereby supporting the hypothesis that 2-categories are the correct mathematical framework for (3+1)-dimensional topological phases.