AdS Geometry, Boundary Problems, and the Dictionary

The AdS/CFT dictionary connects boundary sources and observables to a bulk boundary-value problem. Geometry, admissible falloffs, state preparation, and global data determine which problem is being solved; a bulk mass and a candidate operator dimension alone do not finish the identification.

Enter this chapter

Begin with Anti-de Sitter Geometry and the Conformal Boundary. Readers familiar with the geometry can enter through The GKPW Generating-Functional Dictionary to connect sources with correlators.

Helpful background. Timelike boundaries explains the need for boundary conditions; large-N bulk scaling supplies the semiclassical expansion.

Topics

1. Anti-de Sitter Geometry and the Conformal Boundary — Identify the AdS radius, covering space, and boundary conformal class.
2. Global, Poincaré, and AdS-Rindler Patches — Relate coordinate patches to different boundary domains and time generators.
3. Timelike-Boundary Causality and Boundary-Value Problems — Supply the boundary data needed for deterministic evolution.
4. Asymptotically Locally AdS Boundary Data — Separate prescribed sources from locally fixed and state-dependent coefficients.
5. Radius, Couplings, and the Parameter Map — Track independent curvature, string-coupling, and gravitational scales.
6. Bulk Fields and Boundary Operators — Determine operator weights and admissible quantizations from bulk fields.
7. Spinning Fields, Forms, and Mixed-Symmetry Operators — Include spin, constraints, and representation data in the field map.
8. The GKPW Generating-Functional Dictionary — Generate boundary correlators from a specified bulk source problem.
9. Currents, Stress Tensor, and Bulk Gauge and Metric Fields — Derive current and stress-tensor responses with their Ward identities.
10. Boundary Global Symmetry, Bulk Gauge Symmetry, and Global Form — Recover global gauge data beyond the local current algebra.
11. Extended Operators, Defects, and Brane Charges — Match extended observables to charges and brane boundary conditions.
12. States, Geometries, and Radial Quantization — Relate cylinder energies to AdS excitations.
13. Euclidean Preparation and Lorentzian State Dictionaries — Prepare states and select Lorentzian correlator orderings by gluing contours.
14. Heavy States, Coherent States, and Semiclassical Geometries — Test when small fluctuations justify one semiclassical geometry.
15. Boundary Conditions, Alternate Quantization, and Deformations — Distinguish a change of quantization from a change of source.
16. Dictionary Normalization and Global Data — Check that normalization and global choices describe the same theory.

How the ideas fit together

The generating-functional prescription makes the connection precise: varying boundary sources produces correlators, while bulk gauge and gravitational constraints impose their Ward identities Witten 1998, §§ 2–3. The state and boundary condition determine which bulk solution contributes.

Near-boundary falloffs therefore carry different information. Sources are chosen; locally fixed coefficients follow from the equations; independent response data encode the state after renormalization de Haro, Skenderis, and Solodukhin 2001, §§ 2–5. Exchanging source and response when alternate quantization is admissible changes the boundary theory, rather than merely relabeling one solution.

Claim-domain comparison

Compare the assumptions and limits of these representative conclusions.

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Representative claim domains and validity boundaries for AdS Geometry, Boundary Problems, and the Dictionary

Claim object	State, ensemble, and conventions	Approximation, status, and evidence timing	Uncertainty and counterevidence	Falsifier	Failure condition	Licensed conclusion
scalar mode	Declare mass, patch, and boundary condition; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: AdS patch and conformal boundary → boundary and global conditions → asymptotic field modes → renormalized source and response → operator dictionary. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “normalizability and flux test” check is counterevidence to the promoted claim.	For the declared branch and closed boundary condition, the renormalized norm diverges or forbidden symplectic flux crosses the boundary.	Using a branch outside its normalizability or stability window.	allowed source-response branch
current or stress tensor	Declare gauge group, charges, and counterterms; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: AdS patch and conformal boundary → boundary and global conditions → asymptotic field modes → renormalized source and response → operator dictionary. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “Ward identity and two-point normalization” check is counterevidence to the promoted claim.	With charges and counterterms fixed, the proposed response violates its Ward identity or has the wrong normalized two-point coefficient.	Inferring complete global equivalence from a normalized local operator map.	a normalized operator map
state dictionary	Declare Euclidean cap or Lorentzian initial data; use the volume conventions unless the page states a local replacement.	Dictionary entry or correspondence claim. Control chain: AdS patch and conformal boundary → boundary and global conditions → asymptotic field modes → renormalized source and response → operator dictionary. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “causal and inner-product check” check is counterevidence to the promoted claim.	The proposed state map fails the declared inner-product or causal boundary-data test within its claimed sector and approximation.	Inferring that every heavy state has a smooth geometry from one state construction.	a state in the declared sector

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Review the chapter

Two calculations use the same bulk scalar mass but choose different admissible quantizations. Must their boundary two-point functions describe the same operator?

Solution

No. The mass fixes candidate radial weights, but the quantization selects which falloff is a source and which is a response. The resulting operator dimensions and boundary theories can differ. Agreement requires matching the complete boundary problem, not just the mass.

Where to go next

Continue to String, Brane, and Top-Down Constructions for explicit theory pairs, or Holographic Renormalization and Radial Dynamics to calculate finite responses.

References

• de Haro, Sebastian, Kostas Skenderis, and Sergey N. Solodukhin. “Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence.” Communications in Mathematical Physics 217 (2001): 595–622. arXiv. DOI.
• Witten, Edward. “Anti-de Sitter Space and Holography.” Advances in Theoretical and Mathematical Physics 2 (1998): 253–291. arXiv. DOI.