No exact global symmetries in quantum gravity

Fix a standalone ultraviolet-complete quantum-gravity theory in spacetime dimension $d\ge 3$, its boundary conditions and its physical Hilbert space ${\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$. Standalone here means that the full quantum state space, observables and dynamics are defined without coupling the putative gravitational system to an additional higher-dimensional gravitational host. A localized gravitational subsystem whose consistency requires such a host is a relative theory; the conjecture is tested on its full completion, not on that subsystem in isolation.

For the ordinary internal-symmetry version, supply algebras $\mathcal{A}(R)\subseteq \mathcal{B}({\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}})$ of bounded physical bulk operators assigned to bounded regions $R$, with a specified relational localization and gravitational dressing prescription wherever needed. A candidate exact internal global symmetry is a strongly continuous unitary representation $U:G\to \mathcal{U}({\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}})$ of a topological group $G$, commuting with the specified time evolution and acting trivially on spacetime coordinates. Conjugation ${\alpha }_g(A):=U(g)AU(g{)}^{-1}$ must preserve every $\mathcal{A}(R)$, preserve all dynamics and correlation functions, and act faithfully on their union: for each $g\mathrm{\ne }e$ some $R$ and $A\in \mathcal{A}(R)$ satisfy ${\alpha }_g(A)\mathrm{\ne }A$. Where a local stress tensor is part of the effective description it is neutral under this action. The ordinary conjecture is that no nontrivial $G$ with these properties exists.

A gauge redundancy acts trivially on physical states and operators. A physical asymptotic gauge transformation can instead act on charged sectors or on operators with gauge Wilson lines to the boundary while commuting with all bounded-region gauge-invariant bulk algebras; it is not a global symmetry by the faithful bulk-action criterion above. The operator algebras used in that criterion exclude boundary-anchored gauge-charge insertions; the specified neutral gravitational dressing is distinguished from such gauge dressing. Spacetime isometries are not internal symmetries and are not prohibited by this version. For higher-form or more general versions, the additional open task is to specify the charged extended operators, their support, topological-defect action and boundary conditions, then formulate the corresponding prohibition; an arbitrary nontrivial action on the total state space is not an adequate definition. Approximate low-energy symmetries are not exact symmetries of the full theory. Arbitrary $d<3$ worldsheet-gravity models are outside this target-space formulation.

The universal conjecture is distinct from conditional holographic theorems and from charge-completeness and cobordism proposals. The assumed existence of a full ultraviolet completion is a scope condition, not an existence theorem for any proposed effective theory.

Scalar, gauge and theory-family data for swampland conjectures

Arkani-Hamed et al., pp. 1–4

Harlow–Ooguri, pp. 1–6

McNamara–Vafa, pp. 1–3, 7–9

Status review dated 2025-09-05: the bounded record is Arkani-Hamed–Motl–Nicolis–Vafa, PDF pp. 1–4; Harlow–Ooguri, PDF pp. 1–6; McNamara–Vafa, PDF pp. 1–3 and 7–9; and Lehnert, §II.J, PDF pp. 65–68. Lehnert states that its reviewed literature was published before 2025 (PDF p. 2, note 8) and says that the no-global-symmetry literature is too extensive for its review to cover exhaustively (PDF pp. 67–68). Harlow–Ooguri's argument uses an asymptotically anti-de Sitter/conformal-field-theory dual pair, entanglement-wedge reconstruction, and splittability, and treats internal symmetries in this letter; it does not establish the universal claim on every quantum-gravity background. The bounded sources leave the universal version conjectural; higher-form/generalized extensions and approximate low-energy symmetries need their own formulations. This item makes no status claim about literature after that cutoff and does not claim an exhaustive search. The cobordism proposal is related but is not used as proof.

Selective formulation supplement dated 2026-11-05: Harlow–Ooguri, §§III–V, distinguish faithful bulk-local internal actions from physical asymptotic gauge actions; their discussion includes the need for relational gravitational localization. Anastasi et al., introduction, distinguish relative localized gravity from the complete system with its gravitational host. These scope checks do not establish a universal theorem or independently certify the models in that paper. The historical status review above remains bounded to its listed record.