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We use the intrinsic one-form and two-form global symmetries of (3+1)$d$bosonic field theories to classify quantum phases enriched by ordinary($0$-form) global symmetry. Different symmetry-enriched phases correspond todifferent ways of coupling the theory to the background gauge field of theordinary symmetry. The input of the classification is the higher-formsymmetries and a permutation action of the $0$-form symmetry on the lines andsurfaces of the theory. From these data we classify the couplings to thebackground gauge field by the 0-form symmetry defects constructed from thehigher-form symmetry defects. For trivial two-form symmetry the classificationcoincides with the classification for symmetry fractionalizations in $(2+1)d$.We also provide a systematic method to obtain the symmetry protectedtopological phases that can be absorbed by the coupling, and we give therelative 't Hooft anomaly for different couplings. We discuss several examplesincluding the gapless pure $U(1)$ gauge theory and the gapped Abelian finitegroup gauge theory. As an application, we discover a tension with a conjecturedduality in $(3+1)d$ for $SU(2)$ gauge theory with two adjoint Weyl fermions.

Symmetry-Enriched Quantum Spin Liquids in $(3+1)d$