Witten Diagrams and AdS Perturbation Theory

Witten diagrams compute boundary correlators by perturbing a specified AdS bulk theory. Propagators and interaction vertices determine the answer only after boundary conditions, source normalization, and Lorentzian ordering are fixed; the result is not yet a flat-space scattering amplitude.

Enter this chapter

Begin with Bulk-to-Boundary and Bulk-to-Bulk Propagators, then use Contact Witten Diagrams for the simplest interacting calculation.

Helpful background. The GKPW dictionary supplies the source prescription; scalar counterterms supplies the finite action.

Topics

1. Bulk-to-Boundary and Bulk-to-Bulk Propagators — Choose normalized Green functions with the required falloffs and prescription.
2. Contact Witten Diagrams — Integrate a local bulk vertex against boundary-sourced solutions.
3. Exchange Witten Diagrams and Conformal Blocks — Separate the exchanged conformal family from accompanying multi-trace data.
4. Spinning and Tensor Witten Diagrams — Impose gauge constraints and boundary tensor structures.
5. Lorentzian Witten Diagrams and Real-Time Orderings — Select causal propagators and analytic continuations for each ordering.
6. AdS Integrals and D-Functions — Reduce scalar contact integrals while retaining normalization and sheets.
7. AdS Unitarity Cuts and Cutting Rules — Express discontinuities through normalizable intermediate states.
8. Loop Witten Diagrams and Bulk EFT Renormalization — Renormalize bulk loops while separating boundary contact terms.
9. Loop Integral Bases and Multi-Trace Mixing — Resolve operator mixing before reading spectral data from a loop result.
10. Bulk Field Redefinitions and Contact Ambiguities — Track induced interactions and boundary terms under field redefinitions.
11. Diagram Normalization and Reproducibility Benchmarks — Check simple analytic fixtures before trusting a complex calculation.

How the ideas fit together

A bulk-to-boundary propagator turns each prescribed source into a bulk solution. Inserting these solutions into a cubic vertex produces a three-point function; its coefficient depends on both the vertex and the external normalization Freedman et al. 1999, § 2. This is why normalizing propagators precedes comparisons of couplings.

Exchange and loop calculations add intermediate states, contact freedom, and mixing. An exchange Witten diagram generally contains multi-trace data as well as the single-trace conformal family of the exchanged field. Lorentzian continuation then selects the discontinuity being tested; changing the ordering changes the observable.

Claim-domain comparison

Compare the assumptions and limits of these representative conclusions.

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Representative claim domains and validity boundaries for Witten Diagrams and AdS Perturbation Theory

Claim object	State, ensemble, and conventions	Approximation, status, and evidence timing	Uncertainty and counterevidence	Falsifier	Failure condition	Licensed conclusion
contact diagram	Declare vertex, measure, and source normalization; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: boundary sources and bulk action → propagators and vertices → AdS diagram integral → OPE and analytic checks → perturbative boundary correlator. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “crossing and short-distance check” check is counterevidence to the promoted claim.	With the required permutations included, the contact contribution violates the stated operator symmetry or its predicted short-distance behavior.	Inferring a unique bulk Lagrangian from a contact contribution.	a contact contribution to a correlator
exchange diagram	Declare field spectrum and propagator branch; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: boundary sources and bulk action → propagators and vertices → AdS diagram integral → OPE and analytic checks → perturbative boundary correlator. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “pole and conformal-block decomposition” check is counterevidence to the promoted claim.	After contact ambiguities are accounted for, the stated exchange pole or conformal-block residue disagrees with the declared spectrum and couplings.	Treating a finite-radius exchange contribution as a flat-space amplitude without taking the required limit.	the stated exchange contribution
loop diagram	Declare regulator, counterterms, and contour; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: boundary sources and bulk action → propagators and vertices → AdS diagram integral → OPE and analytic checks → perturbative boundary correlator. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “unitarity cut and renormalization check” check is counterevidence to the promoted claim.	After all terms required at that order are included, a unitarity cut disagrees or regulator dependence remains that the allowed counterterms cannot remove.	Promoting a fixed-order correction to an exact finite-N correlator.	a fixed-order AdS correction

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Review the chapter

A scalar exchange diagram has the correct single-trace pole. Is that sufficient to identify it with one conformal block?

Solution

No. The pole identifies the exchanged family, but the AdS Green-function integral also produces multi-trace contributions and admits contact ambiguities. Matching a full correlator therefore requires its remaining OPE data and crossing behavior, not only the exchange pole.

Where to go next

Continue to AdS Scattering, Mellin Methods, and Bulk Locality to extract scattering information and test a local bulk interpretation.

References

• Freedman, D. Z., Mathur, S. D., Matusis, A., and Rastelli, L. (1999), “Correlation Functions in the CFT$_d$/AdS$_{d+1}$ Correspondence,” Nuclear Physics B 546, 96–118. arXiv:hep-th/9804058.