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

We use higher-form global symmetry to classify the symmetry-enriched phases with ordinary global symmetry in bosonic (3+1)d field theory. Different symmetry-enriched phases correspond to different ways to couple the theory to the background gauge field of the ordinary (0-form) symmetry, which include different symmetry-protected topological (SPT) phases. A general (3+1)d theory has one-form and two-form global symmetries ${\cal A}$ and ${\cal B}$, generated by the symmetry surface and line operators. We assume the two-form symmetry is finite. The two-form symmetry implies the theory has the following symmetry defects: (1) surface defects classified by $H^2({\cal B},U(1))$, they generate a one-form symmetry that acts trivially on all lines. (2) volume defects classified by $H^3({\cal B},U(1))'$, they generate a 0-form symmetry that neither acts on local operators nor permutes the types of non-local operators. The couplings of a (3+1)d bosonic theory to the background of an ordinary unitary symmetry $G$ can be classified by $(\eta_2,\nu_3,\xi)\in H^2_\rho(BG,{\cal A})\times C^3(BG,{\cal B})\times H^1_\sigma(BG, H^3({\cal B},U(1))')$ where $\rho,\sigma$ are fixed $G$-actions induced by permuting the non-local operators. $\delta_\sigma\nu_3$ is subject to a constraint that depends on $\eta_2$ and $\xi$, and $\nu_3$ has an equivalence relation. We determine how the constraint, the classification, and the corresponding 't Hooft anomaly depend on $\xi$. The set of SPT phases of 0-form symmetry that remain inequivalent depends on the couplings $(\eta_2,\nu_3,\xi)$ of the symmetry-enriched phase and can be obtained from the anomaly of the higher-form symmetries. We illustrate our methods with several examples, including the gapless pure $U(1)$ gauge theory and the gapped Abelian finite group gauge theory. We apply the methods to 't Hooft anomaly matching in (3+1)d non-supersymmetric duality.