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Braiding phases among topological excitations are key data for physicallycharacterizing topological orders. In this paper, we provide afield-theoretical approach towards a complete list of mutually compatiblebraiding phases of topological orders in (3+1)D spacetime. More concretely,considering a discrete gauge group as input data, topological excitations inthis paper are bosonic \particles carrying gauge charges and loops carryinggauge fluxes. Among these excitations, there are three classes of root braidingprocesses: particle-loop braidings (i.e., the familiar Aharonov-Bohm phase ofwinding an electric charge around a thin magnetic solenoid), multi-loopbraidings [Phys. Rev. Lett. 113, 080403 (2014)], and particle-loop-loopbraidings [i.e., Borromean Rings braiding in Phys. Rev. Lett. 121, 061601(2018)]. A naive way to exhaust all topological orders is to arbitrarilycombine these root braiding processes. Surprisingly, we find that there existillegitimate combinations in which certain braiding phases cannot coexist,i.e., are mutually incompatible. Thus, the resulting topological orders areillegitimate and must be excluded. It is not obvious to identify theseillegitimate combinations. But with the help of the powerful (3+1)D topologicalquantum field theories (TQFTs), we find that illegitimate combinations violategauge invariance. In this way, we are able to obtain all sets of mutuallycompatible braiding phases and all legitimate topological orders. Toillustrate, we work out all details when gauge groups are$\mathbb{Z}_{N_1},\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2},\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times\mathbb{Z}_{N_3}$,and$\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times\mathbb{Z}_{N_3}\times\mathbb{Z}_{N_4}$.Finally, we concisely discuss compatible braidings and TQFTs in (4+1)Dspacetime.

Compatible braidings with Hopf links, multi-loop, and Borromean rings in $(3+1)$-dimensional spacetime