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Universal topological data of topologically ordered phases can be captured bytopological quantum field theory in continuous space time by taking the limitof low energies and long wavelengths. While previous continuumfield-theoretical studies of topological orders in $3$D real space focus oneither self-statistics, braiding statistics, shrinking rules, fusion rules orquantum dimensions, it is yet to systematically put all topological datatogether in a unified continuum field-theoretical framework. Here, we constructthe topological $BF$ field theory with twisted terms (e.g., $AAdA$ and $AAB$)as well as a $K$-matrix $BB$ term, in order to simultaneously explore all suchtopological data and reach anomaly-free topological orders. Following thespirit of the famous $K$-matrix Chern-Simons theory of $2$D topological orders,we present general formulas and systematically show how the $K$-matrix $BB$term confines topological excitations, and how self-statistics of particles istransmuted between bosonic one and fermionic one. In order to reachanomaly-free topological orders, we explore, within the present continuumfield-theoretical framework, how the principle of gauge invariancefundamentally influences possible realizations of topological data. Moreconcretely, we present the topological actions of (i) particle-loop braidingswith emergent fermions, (ii) multiloop braidings with emergent fermions, and(iii) Borromean-Rings braidings with emergent fermions, and calculate theiruniversal topological data. Together with the previous efforts, our work pavesthe way toward a more systematic and complete continuum field-theoreticalanalysis of exotic topological properties of $3$D topological orders. Severalinteresting future directions are also discussed.

Continuum field theory of 3D topological orders with emergent fermions and braiding statistics