Non-invertible symmetries are the newest expansion of what “symmetry” means in quantum field theory. Unlike ordinary symmetries — which have inverses, so doing and undoing always returns you to the start — non-invertible symmetries transform states in ways that cannot be reversed by any single operation. They are genuinely one-directional at the level of the symmetry algebra.
Putrov and Radhakrishnan prove that in 3+1 dimensions, this non-invertibility dissolves when you look at local operators. If a finite non-invertible symmetry has no topological line operators (a common structural condition), then its action on local operators is necessarily invertible. The global symmetry is non-invertible; its local shadow is not.
The mechanism: a non-invertible symmetry action on local operators decomposes into an invertible operation composed with a gauging interface. The non-invertibility lives in the gauging step — a global operation that cannot be performed locally. Strip away the global structure and what remains is an ordinary, reversible transformation.
This constrains the anomaly structure. Anomaly-free non-invertible symmetries without topological line operators are proven to be “non-intrinsically non-invertible” — their non-invertible character is removable by a choice of presentation, not a deep structural feature.
The structural point: the most exotic symmetries in four-dimensional physics, when examined at the level of what they actually do to local observables, turn out to be invertible in disguise. Non-invertibility is a global phenomenon that locally dissolves. The strangeness is real but delocalized — it lives in the topology of the symmetry action, not in the algebra.