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3+1 dimensional topological phases can support loop-like excitations inaddition to point-like ones, allowing for non-trivial loop-loop and point-loopbraiding statistics not permitted to point-like excitations alone. Furthermore,these loop-like excitations can be linked together, changing their properties.In particular, this can lead to distinct three-loop braiding, involving twoloops undergoing an exchange process while linked to a third loop. In thiswork, we investigate the loop-like excitations in a 3+1d Hamiltonianrealization of Dijkgraaf-Witten theory through direct construction of theirmembrane operators, for a general finite Abelian group and 4-cocycle twist.Using these membrane operators, we find the braiding relations and fusion rulesfor the loop-like excitations, including those linked to another loop-likeexcitation. Furthermore, we use these membrane operators to constructprojection operators that measure the topological charge and show that thenumber of distinct topological charges measured by the 2-torus matches theground state degeneracy of the model on the 3-torus, explicitly confirming ageneral expectation for topological phases. This direct construction of themembrane operators sheds significant light on the key properties of theloop-like excitations in 3+1 dimensional topological phases.

Twisted Lattice Gauge Theory: Membrane Operators, Three-loop Braiding and Topological Charge