3-Loop Braiding and Topological Charge from Membrane Operators in 3+1d Twisted Lattice Gauge Theory

Membrane operators can be used to explicitly construct the loop-like excitations in toy models for topological phases. We discuss the 3+1d twisted lattice gauge theory model, which is a Hamiltonian realization of the Dijkgraaf-Witten topological field theory. By constructing the membrane operators that produce flux loops that are linked to an existing base flux loop, we find the three-loop braiding relations, which describe what happens when two loop-like excitations are exchanged while linked to a third loop. This explicit treatment gives us further insight into how non-Abelian three-loop braiding may arise even for Abelian groups. The membrane operators can also be used to find the conserved topological charges in the model, which are related to the ground state degeneracy.