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 A and 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(B,U(1)), they generate a one-form symmetry that acts trivially on all lines. (2) volume defects classified by H^3(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 (η_2,ν_3,ξ)∈H^2_ρ(BG,A)×C^3(BG,B)×H^1_σ(BG,H^3(B,U(1))′) where ρ,σ are fixed G-actions induced by permuting the non-local operators. Δ_σν_3 is subject to a constraint that depends on η_2 and ξ, and ν_3 has an equivalence relation. We determine how the classification and the corresponding 't Hooft anomaly depend on ξ. The set of SPT phases of 0-form symmetry that remain inequivalent depends on the couplings (η_2,ν_3,ξ) 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.