Higher-Form Anomalies on Lattices

Higher-form symmetry in a tensor product Hilbert space is always emergent: The symmetry generators become genuinely topological only when the Gauss law is energetically enforced at low energies. In this Letter, we present a general method for defining the 't Hooft anomaly of higher-form symmetries in lattice models built on a tensor product Hilbert space. In (2+1)D, for given Gauss-law operators realized by finite-depth circuits that generate a finite 1-form G symmetry, we construct an index representing a cohomology class in H^{4}(B^{2}G,U(1)), which characterizes the corresponding 't Hooft anomaly. This construction generalizes the Else-Nayak characterization of 0-form symmetry anomalies. More broadly, under the assumption of a specified formulation of the p-form G symmetry action and Hilbert space structure in arbitrary d spatial dimensions, we show how to characterize the 't Hooft anomaly of the symmetry action by an index valued in H^{d+2}(B^{p+1}G,U(1)).