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Dictionary Completeness and Global Data

Agreement of local low-point correlators does not specify a complete holographic correspondence. One must also match the global form of symmetry groups, extended operators, charge lattices, anomalies, superselection sectors, boundary conditions, state spaces, and normalization of the observable map. These data distinguish theories that share the same local Lie algebra and perturbative fields.

Required background. Holographic Duality: Claims, Dictionaries, and Regimes defines the theory-pair contract, and Relational, Boundary, and Asymptotic Observables identifies the observables whose completeness is at issue.

Helpful background. Generalized Symmetries, Global Forms, and Anomalies under Duality supplies the underlying QFT distinctions. Additivity, Haag Duality, and Information Completeness gives an operator-algebra perspective.

Local data do not determine the theory

Section titled “Local data do not determine the theory”

Gauge theories with groups SU(N)SU(N) and PSU(N)=SU(N)/ZNPSU(N)=SU(N)/\mathbb Z_N have the same Lie algebra and hence the same perturbative gauge bosons. Their genuine line operators differ. An SU(N)SU(N) Wilson line can carry fundamental NN-ality, whereas the corresponding line is not genuine in the quotient theory without additional surface data Aharony, Seiberg, and Tachikawa 2013.

A proposed correspondence must therefore include at least:

  • the global form of every gauge and global symmetry group;
  • the lattice of genuine electric, magnetic, and dyonic extended operators;
  • ’t Hooft anomalies and higher-form symmetries;
  • allowed bundles and topological terms;
  • superselection sectors and the operations that connect them;
  • boundary conditions and boundary degrees of freedom;
  • the Hilbert space or observable algebra in each sector;
  • state, ensemble, and normalization conventions.

An anomaly match is a powerful necessary test because anomalies are invariant under renormalization-group flow. Higher-form symmetries and their charged extended operators provide further global data Gaiotto et al. 2015. Neither datum is by itself sufficient for equivalence: inequivalent theories can share the same anomaly.

The SU(N) versus PSU(N) application

Section titled “The SU(N) versus PSU(N) application”

Consider two candidate boundary theories that agree on all correlators of local adjoint operators. Their bulk low-energy fields can also agree. Yet changing the global form changes which boundary line operators are genuine and therefore which bulk strings or branes may end at the boundary.

The distinction appears through pairings of electric and magnetic charges. A schematic Dirac pairing obeys

⟨e,m⟩∈Z,\langle e,m\rangle\in\mathbb Z,

but the allowed sublattice depends on the global form and discrete theta data. A complete bulk dictionary must reproduce that lattice, not only the local gauge algebra.

Data comparedWhat agreement establishes
Local stress-tensor and current correlatorsMatching local normalization and dynamics in tested sectors
Genuine line and surface operatorsMatching global form and higher-form symmetry data
AnomaliesA necessary consistency condition across the map
Sector-resolved partition functionsMatching topological sectors and weights
Boundary-condition changesMatching the family of theories or deformations, not just one point

A counterexample to local completeness

Section titled “A counterexample to local completeness”

Suppose dictionaries D1\mathfrak D_1 and D2\mathfrak D_2 agree on every local correlator measured to a fixed order in 1/N1/N, but assign different spectra to genuine Wilson–’t Hooft lines. No local low-point test distinguishes them; an extended-operator experiment does. The local evidence licenses a common local sector, not global equivalence.

Similarly, summing over sectors on one side while fixing a sector on the other can make partition functions agree only after an unannounced averaging operation. The ensemble and sector choice must be part of the equality.

Completeness ceiling

Section titled “Completeness ceiling”

This checklist can refute an incomplete dictionary; passing it does not prove that no unexamined observable exists. Volumes III, IX, and X own the symmetry, CFT, and protected data. Chapter 3 applies the list to AdS boundary conditions, Chapter 25 to quantum-gravity consistency, and Volume XVI to formal completeness questions.

Evidence cutoff. Examples and status statements are fixed to 25 July 2026.

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

Section titled “References”