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Holographic Duality: Claims, Dictionaries, and Regimes

A holographic duality claim is meaningful only after both theories, the observables being compared, the parameter map, the state and boundary data, and the limiting procedure have been fixed. The strongest familiar examples support a detailed dictionary and many controlled checks; that is not the same statement as a general theorem of equivalence.

Helpful background. Duality Claims, Dictionaries, Regimes, and Evidence supplies the general QFT vocabulary for duality, while Claim–Evidence Records, Replication, and Retraction Handling explains how distinct evidence should be recorded.

A duality claim is a structured map

Section titled “A duality claim is a structured map”

Let TB{\cal T}_{\mathrm B} denote a boundary theory and Tbulk{\cal T}_{\mathrm{bulk}} a bulk theory. A complete claim must specify a map

D:(HB,AB,HB,ρB)⟶(Hbulk,Abulk,Hbulk,ρbulk),\mathfrak D:\bigl({\cal H}_{\mathrm B},{\cal A}_{\mathrm B},H_{\mathrm B},\rho_{\mathrm B}\bigr) \longrightarrow \bigl({\cal H}_{\mathrm{bulk}},{\cal A}_{\mathrm{bulk}},H_{\mathrm{bulk}},\rho_{\mathrm{bulk}}\bigr),

together with global data, boundary conditions, and a parameter map. Equality of selected generating functionals,

ZB[J]=?Zbulk[ϕ(0)=J],Z_{\mathrm B}[J]\stackrel{?}{=}Z_{\mathrm{bulk}}[\phi_{(0)}=J],

may mean an exact identity in a defined model, an asymptotic equality, or a saddle approximation. The symbol alone does not decide which.

Useful claim classes are:

ClaimWhat has actually been established
Dictionary entryA proposed correspondence between specified observables
Protected matchAgreement insulated from some coupling corrections by symmetry
Perturbative equalityAgreement through a stated order in a controlled expansion
Saddle relationAgreement after selecting a dominant bulk saddle
Numerical testAgreement within stated discretization and statistical errors
Conjectural equivalenceA claim covering a declared complete set of sectors and observables
Definition proposalOne description is taken to define quantities not otherwise constructed

These classes are not rungs on an automatic ladder. A protected equality may be exact yet probe only a small subsector; a numerical comparison may probe unprotected dynamics while remaining finite-precision.

The AdS5/CFT4 example

Section titled “The AdS5/CFT4 example”

The correspondence was introduced by Maldacena 1998; Aharony et al. 2000 reviews its dictionary and parameter regimes.

For the standard normalization tr⁡(TaTb)=12δab\operatorname{tr}(T^aT^b)=\tfrac12\delta^{ab}, the proposed relation between SU(N)SU(N) N=4{\cal N}=4 super-Yang–Mills theory and type-IIB string theory on AdS5×S5AdS_5\times S^5 includes

gYM2=4πgs,λ=gYM2N=L4α′2,G5L3∼1N2.g_{\mathrm{YM}}^2=4\pi g_s, \qquad \lambda=g_{\mathrm{YM}}^2N=\frac{L^4}{\alpha'^2}, \qquad \frac{G_5}{L^3}\sim \frac{1}{N^2}.

The parameter map licenses different approximations in different corners:

  • N→∞N\to\infty suppresses bulk loops when operator normalizations are held fixed appropriately;
  • λ→∞\lambda\to\infty makes L2/α′L^2/\alpha' large and suppresses string-scale curvature corrections;
  • supergravity requires both suppressions, not merely one;
  • finite-NN, finite-λ\lambda equivalence is a broader conjecture than the supergravity calculation.

The observable map is equally important. Local single-trace operators correspond perturbatively to bulk fields, the stress tensor to the bulk metric, conserved currents to gauge fields, and Wilson or ’t Hooft operators to extended bulk objects. The gauge-group global form and spectrum of line operators are therefore part of the claim, even when all local Lie-algebra correlators agree.

An adversarial change of the claim

Section titled “An adversarial change of the claim”

Three changes expose why the slogan “AdS/CFT” is insufficient.

  1. Replacing SU(N)SU(N) by PSU(N)PSU(N) changes genuine line operators and topological sectors while leaving the local Lie algebra unchanged.
  2. Changing an admissible AdS boundary condition changes the boundary deformation or ensemble.
  3. Taking λ→∞\lambda\to\infty before or after N→∞N\to\infty changes which string and loop corrections have been discarded.

After any of these changes, protected local matches may survive, but the original complete dictionary no longer denotes the same pair of theories. The strongest justified conclusion is therefore always indexed by the theories, observables, sectors, and limit order actually tested.

Evidence ceiling and continuation

Section titled “Evidence ceiling and continuation”

The foundational calculations establish a remarkably coherent conditional dictionary and controlled large-NN, strong-coupling limits. They do not constitute a theorem that every observable of two nonperturbatively constructed theories is identical. CFT data are developed in Volume IX, protected inputs in Volume X, and theorem-level reconstruction belongs to Volume XVI. Mutable assessments belong in Research.

Evidence cutoff. Literature-status statements on this page use a cutoff of 25 July 2026; the equations and logical distinctions are not claims that any review or release status has advanced.

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”