An internal symmetry in a lattice model is said to be onsiteable if it can be disentangled into an on-site action by introducing ancillas and conjugating with a finite-depth circuit. A standard lore holds that onsiteability is equivalent to being anomaly-free, which is indeed valid for finite 0-form symmetries in $(1+1)\mathrm{D}$. However, for higher-form symmetries, these notions become inequivalent: a symmetry may be on site while still anomalous. In this Letter, we clarify the conditions for onsiteability of higher-form symmetries by proposing an equivalence between onsiteability and the possibility of higher gauging. For a finite 1-form symmetry in $(2+1)\mathrm{D}$, we show that the symmetry is onsiteable if and only if its ‘t Hooft anomaly satisfies a specific algebraic condition that ensures the symmetry can be 1-gauged. We further demonstrate that onsiteable 1-form symmetry in $(2+1)\mathrm{D}$ can always be brought into transversal Pauli operators by ancillas and circuit conjugation. In generic dimensions, we derive necessary conditions for onsiteability using lattice ‘t Hooft anomaly of higher-form symmetry, and conjecture a general equivalence between onsiteability and possibility of higher gauging on lattices. We also identify new “lattice anomalies” intrinsic to higher-form symmetry of lattice models, obstructing on-site realization of symmetry beyond the possibility of higher gauging in continuum.

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• Anyons
• Topological order