Distinct realizations of generalized global symmetries correspond to distinct phases of matter

The claim was assessed as the general principle articulated by Gaiotto, Kapustin, Seiberg, and Willett (arXiv:1412.5148) and adopted across the subsequent literature, e.g. McGreevy's review of generalized symmetries in condensed matter (arXiv:2204.03045) and the Jena lectures (arXiv:2407.20815), which present phases-as-symmetry-realizations as the standard modern framework.

The supporting argument rests on two subclaims, both seeded as new nodes and both resting on well-established physics. For the Yang-Mills case↗︎, the electric center one-form symmetry of pure SU(N) gauge theory is exact, the Wilson loop is its charged object, and the finite-temperature deconfinement transition coincides with the change of its realization diagnosed by the Polyakov loop; lattice results are consistent with this identification. For the topological-order case↗︎, the identification is explicit in abelian models such as the toric code, where topological order is spontaneous breaking of emergent one-form symmetries, though the general non-abelian statement requires non-invertible symmetry language and the symmetries involved are typically emergent rather than exact. The connecting premise, that the realization of an exact symmetry is a discrete label that cannot change without nonanalyticity, is the standard Landau-type argument extended to generalized symmetries and is not disputed for exact symmetries; it is carried in prose rather than as a node because the discourse does not contest it.

The against argument does not contradict the claim's letter but bounds its force. Fundamental matter explicitly breaks the electric one-form symmetry↗︎ (Wilson lines can end on dynamical charges), and Fradkin-Shenker continuity↗︎ shows Higgs and confining regimes are smoothly connected in lattice gauge-Higgs systems. Current literature confirms this is the live edge: arXiv:2312.16898 states that in gauge theories with fundamental matter there is typically no sharp generalized-symmetry distinction between confining and Higgs regimes, yet finds cases where ordinary-symmetry SPT structure separates them; Cherman, Sen, and Yaffe (Phys. Rev. D 102, 105021) construct sharp Higgs-confinement distinctions via topological vortex order parameters when a global U(1) is broken in both regimes; work on Higgs condensates as SPT phases (arXiv:2211.01376) and on emergent generalized symmetries in ordered phases (SciPost Phys. 17, 080) extends the program rather than refuting it.

Weighing: the claim is a near-consensus organizing principle whose paradigm instances are verified physics, but as stated it is a general correspondence, and its extension beyond exact symmetries is genuinely unsettled, partly as an empirical-theoretical question and partly as a definitional one (what should count as a phase of matter when only emergent symmetries distinguish candidate regimes). That mixture supports SUPPORTED rather than VERIFIED, with credence 0.8 that the claim as stated is true read as the correspondence for exact generalized symmetries with a defensible extension to emergent ones. The verdict would strengthen toward verified if the emergent-symmetry extension were given a settled, generally accepted formulation; it would weaken toward contested if a concrete counterexample showed distinct realizations of an exact generalized symmetry smoothly connected, none is currently known. Marginal yield is modest: a deeper pass through the recent literature on emergent one-form symmetries and finite-temperature subtleties could sharpen the boundary of the claim's validity but is unlikely to change the status.