| Summary: | Braiding phases among topological excitations are key data for physically characterizing topological orders. In this paper, we provide a field-theoretical approach toward a complete list of mutually compatible braiding phases of topological orders in (3+1)D spacetime. More concretely, considering a discrete gauge group as input data, topological excitations in this paper are bosonic particles carrying gauge charges and loops carrying gauge fluxes. Among these excitations, there are three classes of root braiding processes: particle-loop braidings (i.e., the familiar Aharonov-Bohm phase of winding an electric charge around a thin magnetic solenoid), multiloop braidings [Wang and Levin, Phys. Rev. Lett. 113, 080403 (2014)PRLTAO0031-900710.1103/PhysRevLett.113.080403], and particle-loop-loop braidings (i.e., Borromean rings braiding in Chan et al. [Phys. Rev. Lett. 121, 061601 (2018)PRLTAO0031-900710.1103/PhysRevLett.121.061601]). A naive way to exhaust all topological orders is to arbitrarily combine these root braiding processes. Surprisingly, we find that there exist illegitimate combinations in which certain braiding phases cannot coexist, i.e., are mutually incompatible. Thus, the resulting topological orders are illegitimate and must be excluded. It is not obvious to identify these illegitimate combinations. But with the help of the powerful (3+1)D topological quantum field theories (TQFTs), we find that illegitimate combinations violate gauge invariance. In this way, we are able to obtain all sets of mutually compatible braiding phases and all legitimate topological orders. To illustrate, we work out all details when gauge groups are Z_{N_{1}},Z_{N_{1}}×Z_{N_{2}},Z_{N_{1}}×Z_{N_{2}}×Z_{N_{3}} and Z_{N_{1}}×Z_{N_{2}}×Z_{N_{3}}×Z_{N_{4}}. Finally, we concisely discuss compatible braidings and TQFTs in (4+1)D spacetime.
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