Fractionalization and Anomalies in Symmetry-Enriched U(1) Gauge Theories*

We classify symmetry fractionalization and anomalies in a 3+1d U(1) gauge theory enriched by a global symmetry group G. We find that, in general, a symmetry-enrichment pattern is specified by 4 pieces of data: ρ, a map from G to the duality symmetry group of this U(1) gauge theory, ν∈H²_ρ[G, U_T(1)], p∈H¹[G, Z_T], and a torsor n over H³_ρ[G, Z]. However, certain choices of (ρ, ν, p, n) are not physically realizable, i.e., they are anomalous. There are two levels of anomalies. The first level of anomalies, deconfinement anomalies, obstruct fractional excitations being deconfined. States with these anomalies can be realized on the boundary of a 4+1d long-range entangled state. In the absence a deconfinement anomaly, there can still be the more familiar 't Hooft anomaly, which forbids certain types of symmetry fractionalization patterns to be implemented in an on-site manner. States with these anomalies can live on the boundary of a 4+1d short-range entangled state. We apply our results to some interesting physical examples.