No-Global-Symmetry and Compactness Results Conditional on AdS/CFT

In an exact AdS/CFT duality with a local boundary QFT and a controlled bulk EFT, a boundary global symmetry acts as a bulk gauge symmetry. Under further completeness and operator-algebra assumptions, an exact internal global symmetry localized in the bulk is excluded and the bulk gauge group is compact. These are conditional holographic conclusions, not theorems about every possible quantum gravity.

Required background. Quantum-Gravity Consistency Claims and Comparison Contract fixes logical status; Boundary Global Symmetry, Bulk Gauge Symmetry, and Global Form supplies the dictionary.

Helpful background. What Is a Symmetry of a QFT? supplies symmetry actions; Completeness and the Operator Basis supplies the state–operator premise.

Evidence cutoff: 25 July 2026.

Boundary current and bulk gauge field

For a continuous compact boundary symmetry $G$, the conserved current obeys

$$\partial_\mu J^\mu_a=0,\qquad Q_a=\int_{\Sigma_{\partial}}d^{d-1}x\,J^0_a .$$

The AdS dictionary maps $J^\mu_a$ to a bulk gauge field $A^a_M$. Boundary charged operators create bulk states carrying the corresponding gauge charge. The transformation can be detected at infinity through a Gauss-law surface charge, so it is not a global symmetry acting only on compactly supported bulk degrees of freedom.

First application: reconstruct the conditional argument

Cover a boundary Cauchy slice by mutually spacelike regions $R_i$. Boundary locality permits the global symmetry operator to be approximated by a product of operators supported in the $R_i$, with buffers. Entanglement-wedge reconstruction maps those factors into bulk wedges near the boundary. If a putative bulk global symmetry acted nontrivially on an operator in a central region spacelike to every wedge representative, the product representation would both commute with and transform that operator—a contradiction. Harlow and Ooguri formalize this reasoning and its compactness refinement Harlow and Ooguri 2021, §§2–6.

Compactness uses additional spectral assumptions: the boundary symmetry group acts faithfully on a discrete operator spectrum with finite-dimensional eigenspaces. A noncompact internal group generally conflicts with those properties. Completeness of charged sectors is related but is not merely the statement that one charged operator exists.

Nonconverses and scope

The result does not say that every gauge symmetry in a bulk EFT has every lattice charge, that global symmetries cannot emerge approximately below a cutoff, or that no higher-form or noninvertible symmetry can appear. Each requires its own operator and defect formulation.

Adversarial control: remove one premise

Without exact duality, reconstruction is evidence rather than an identity. Without boundary locality, the regional factorization step fails. Without operator completeness or a controlled code subspace, a bulk operator need not have the required representatives. Examples compatible with the conclusion cannot repair the missing implication.

The strongest statement is therefore conditional: exact AdS/CFT satisfying the specified locality, spectrum, and reconstruction assumptions forbids an exact bulk internal global symmetry and constrains gauge-group compactness. Extending it beyond AdS or beyond those assumptions remains conjectural.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Harlow, Daniel, and Hirosi Ooguri. “Symmetries in Quantum Field Theory and Quantum Gravity.” Communications in Mathematical Physics 383, 1669–1804 (2021). DOI. Open PDF.