Holographic Renormalization and Radial Dynamics

Holographic renormalization turns divergent near-boundary bulk data into finite boundary observables. The central task is to construct a finite variational problem, so that counterterms, source choices, responses, and Ward identities describe the same theory.

Enter this chapter

Begin with Asymptotically Locally AdS Fields and Fefferman–Graham Expansions to separate source data from state-dependent response.

Helpful background. The GKPW dictionary explains source differentiation; boundary conditions and quantization explains which data are held fixed.

Topics

1. Asymptotically Locally AdS Fields and Fefferman–Graham Expansions — Identify local asymptotic coefficients and independent response data.
2. Radial Cutoffs and Hamilton–Jacobi Flow — Solve for divergent terms through radial Hamilton–Jacobi evolution.
3. Scalar Counterterms and the Renormalized On-Shell Action — Remove scalar-action divergences without confusing contacts and response.
4. Gauge-Field and Differential-Form Counterterms — Renormalize gauge sources consistently with the Gauss constraint.
5. Metric Counterterms and the Boundary Stress Tensor — Obtain the boundary stress tensor from the complete gravitational action.
6. Renormalized One-Point Functions and the Variational Problem — Vary independent sources to compute finite one-point functions.
7. Ward Identities, Weyl Anomalies, and Contact Terms — Derive gauge, diffeomorphism, and trace identities including contacts.
8. Hamiltonian Constraints and Radial Canonical Transformations — Interpret counterterms as transformations of radial canonical data.
9. Holographic RG Flows and Domain-Wall Geometries — Relate suitable domain-wall solutions to boundary RG trajectories.
10. Wilsonian Cutoffs and Integrating-Out Proposals — Distinguish radial factorization from a sharp momentum cutoff.
11. Holographic c-, a-, and F-Theorem Interfaces — State the assumptions behind geometric monotonicity arguments.
12. Finite Counterterms, Schemes, and Multi-Trace Data — Separate scheme changes from deformations of the boundary theory.

How the ideas fit together

The near-boundary expansion determines divergent terms locally in the sources. Counterterms cancel those divergences; varying the resulting finite action then identifies the response. These are successive steps, not interchangeable definitions. Even for a free scalar, the one-point function depends on the normalizable coefficient together with local source terms Skenderis 2002, §§ 5.1–5.4.

A finite local functional of fixed sources changes contact terms. Changing which canonical variable is held fixed, or adding an interaction that changes the boundary condition, can instead change the theory. Ward identities and the variational principle distinguish these operations.

Claim-domain comparison

Compare the assumptions and limits of these representative conclusions.

Scroll horizontally to compare all columns. Use the Left and Right arrow keys when the table region is focused.

Representative claim domains and validity boundaries for Holographic Renormalization and Radial Dynamics

Claim object	State, ensemble, and conventions	Approximation, status, and evidence timing	Uncertainty and counterevidence	Falsifier	Failure condition	Licensed conclusion
divergent on-shell action	Declare radial coordinate and induced fields; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: Fefferman-Graham cutoff data → radial Hamilton-Jacobi recursion → local and finite counterterms → one-point functions and Ward identities → renormalized observable. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “Hamilton-Jacobi cancellation” check is counterevidence to the promoted claim.	After the prescribed counterterms are included, an uncancelled divergence or inadmissible boundary variation remains for the stated sources.	Inferring scheme-independent finite contact terms from divergence cancellation.	finite variational problem
one-point function	Declare source normalization and counterterm action; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: Fefferman-Graham cutoff data → radial Hamilton-Jacobi recursion → local and finite counterterms → one-point functions and Ward identities → renormalized observable. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “Ward and anomaly identities” check is counterevidence to the promoted claim.	With the same source and counterterm action, the proposed response violates the applicable Ward or anomaly identity, including required contact terms.	Identifying a renormalized response with a bare asymptotic bulk coefficient.	renormalized response in one scheme
radial flow	Declare cutoff surface and boundary conditions; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: Fefferman-Graham cutoff data → radial Hamilton-Jacobi recursion → local and finite counterterms → one-point functions and Ward identities → renormalized observable. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “canonical transformation check” check is counterevidence to the promoted claim.	The proposed transformation fails to preserve the declared canonical symplectic structure or radial constraint relations.	Equating radial evolution with ordinary Wilsonian RG without the additional dictionary and qualifications.	radial evolution of data

Download the structured table data (JSON).

Review the chapter

Can a finite local counterterm remove a mismatch between two scalar correlators at separated boundary points?

Solution

A local source functional contributes distributions supported at coincident insertions. It can change contact terms, but cannot erase a genuinely nonlocal mismatch at separated points. One must check the dynamics, normalization, state, or dictionary instead.

Where to go next

Continue to Witten Diagrams and AdS Perturbation Theory to calculate renormalized boundary correlators from bulk interactions.

References

• Skenderis, Kostas. “Lecture Notes on Holographic Renormalization.” Classical and Quantum Gravity 19, 5849–5876 (2002). DOI; arXiv:hep-th/0209067.