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 $(2+1)\mathrm{D}$, 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 $H^{d+2}(B^{p+1}G,U(1))$.

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• Anomalies
• Quantum field theory
• Topological phases of matter