On the Holographic Renormalization Group

E. Verlinde; H. Verlinde; J. de Boer

REVIEW

On the Holographic Renormalization Group

J. de Boer, E. Verlinde, H. Verlinde

High Energy Physics - Theory · Submitted 2 Dec 1999 · 17 references

Pith's verdict: The Hamilton-Jacobi route from 5-d gravity to the Callan-Symanzik equation holds up; the derivative-expansion matching is the soft part, and the two beta-function expressions are never checked against each other.

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arXiv IDhep-th/9912012 (version 1)

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CitationE. Verlinde, H. Verlinde, J. de Boer. "On the Holographic Renormalization Group." arXiv:hep-th/9912012 (1999). https://pith.science/paper/KL332KMM

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Editor's note

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This is the paper that made 'radial coordinate = RG scale' into an actual derivation rather than an analogy, and the derivation is the part worth your time. The move is to treat the 5-d constraints as a Hamilton-Jacobi problem: the canonical momenta are functional derivatives of the on-shell action, the Hamiltonian constraint becomes a functional differential equation for S, and the leading terms reproduce the Callan-Symanzik equation with the anomaly. That is clean, it is general (no particular dual pair assumed), and it is the reason everyone after 1999 writes the flow equations this way. The relations tying beta functions to the counterterms for the 4-d cosmological and Newton terms (Eqs. 22-25) are the genuinely new content, and the anomaly coefficients c = 3d = lambda^3/8 are checked against Henningson-Skenderis, which is real external validation, not self-agreement. The paper is also more candid about its limits than the abstract suggests: it says the derivative expansion is systematic only for mu << mu_c, it drops the 4-derivative terms in Eq. (26) without writing them, and the appendix shows the dimension-four part of U is not fixed by V. Those are stated, not hidden, and they do not break the main claim. The soft spot is the term-by-term matching itself. The paper derives beta twice, from the potential (Eq. 21) and from requiring the grad-squared-phi coefficient to cancel (Eq. 25), calls the second 'another interesting expression,' and never checks the two agree. That is a fair gap: if they disagreed the local ansatz would be over-determined. My guess is it closes, since both come from the same constraint at different orders and the analogous relation holds in later holographic renormalization treatments, but a student working through this should do the algebra rather than take it on faith. Worth noting the compiled record claims no stated limitations; that is wrong, the text states several. Who should read it: anyone doing holographic RG, anomaly matching, or c-function work, and anyone who wants a compact pedagogical source for the Hamilton-Jacobi method. It deserves a serious referee and I would cite it.

Referee report

Significance

If the correspondence holds, it gives a clean and general derivation of the holographic RG dictionary directly from the 5-d Hamiltonian constraint, without assuming supersymmetry, and it identifies the classical supergravity action with the quantum effective action of the dual theory. The derivation is parameter-free: no new couplings or ad hoc inputs are introduced beyond the standard Hamilton–Jacobi identification of momenta with functional derivatives of S. The fixed-point anomaly coefficients in Eq. (32) are checked against the independent published computation of [17], which is genuine external validation rather than internal consistency. The relations (23)–(25) linking beta functions to the counterterms multiplying R and the scalar kinetic term are a useful and, at the time of writing, novel addition, and the inequality U² ≥ 3V in Eq. (22) gives a monotonicity statement of the c-function type. The paper is also falsifiable in the ordinary sense: the claimed identities between β^I, Φ and M^{IJ} can be checked algebraically against any explicit flow solution.

Major objections (3)

[Holographic RG-flow and Relations for the local terms, Eqs. (21) and (25)] The paper derives the beta function twice: once from the leading potential terms, β^I = 6U G^{IJ}∂_J U (Eq. 21), and once from the requirement that the ∇²φ coefficient in the Hamilton–Jacobi constraint cancel, β^I = M^{IJ}∂_J Φ (Eq. 25). For the local ansatz (15) to be consistent these must agree, i.e. M^{IJ}∂_J Φ = 6U G^{IJ}∂_J U should follow from the flow equations (23)–(24). The manuscript presents (25) only as 'another interesting expression' and never verifies the agreement. This matters because the Callan–Symanzik equation (26) is written in terms of β^I, so if the two expressions differ the local ansatz is over-determined and the flow is not well defined. The appendix sharpens the concern: Eq. (A.9) shows that the dimension-4 part of U is not fixed by V, and marginal directions are precisely where the two expressions could differ. Please add the explicit algebraic check, or state the conditions under which (21) and (25) coincide.
[Callan-Symanzik equation and The conformal anomaly, Eqs. (26), (31), (32)] The treatment of four-derivative terms is not uniform across the paper. In Eq. (26) all four-derivative terms are dropped, with the remark that they 'play no role' for operators at separated points; but in Eq. (31) curvature-squared terms of exactly the same derivative order are retained and assigned definite coefficients in Eq. (32). The reader is not told what principle decides which four-derivative terms survive and which are discarded, nor why the same terms that are negligible in (26) become the anomaly in (31). A short paragraph stating the bookkeeping rule (for example, that only terms built from the local coefficients Φ and U are kept, while terms arising from Γ are dropped) would remove the apparent inconsistency. In addition, the passage from (26) to (28) is summarized as 'some straightforward algebra'; given that this step carries the central claim, the intermediate result before integrating over space should be displayed.
[Removing the UV cut-off, Eqs. (36)-(41)] Equation (37) states that the renormalized action satisfies the flow equation up to unspecified 'local terms', while Eq. (41) for the renormalized correlators contains no such terms. The vanishing of these local contributions in the ǫ → 0 limit is asserted rather than shown, and it is not obvious: the local terms in question are the same counterterms whose divergences are being subtracted in Eq. (36), so their fate depends on the subtraction prescription. Please indicate which local terms survive in (37), why they drop out of (41), and whether the operator renormalization (40) is sufficient to make the anomalous-dimension term in (41) finite.

Minor objections (5)

[The conformal anomaly, sentence preceding Eq. (32)] The symbol Ω appears in 'the coefficients c and d are given in terms of Ω and Φ' but is not defined anywhere in the manuscript; presumably Φ was intended, or a definition is missing.
[The conformal anomaly, Eq. (32)] As printed, d = 2U(Φ² − (3/2)∂_IΦ G^{IJ}∂_JΦ) is dimensionally inconsistent with the stated fixed-point relation c = 3d: at a fixed point the printed form gives d = 2UΦ² while c = 6Φ²/U, which agree only for U = 1. The intended expression is presumably d = (2/U)(Φ² − (3/2)∂_IΦ G^{IJ}∂_JΦ), which does give c = 3d. Please check the typesetting of this equation.
[Abstract and Discussion] The abstract promises 'new relations between the beta-functions and the counterterms that affect the 4-d cosmological and Newton constant'. The Newton-constant relation is Eq. (23), but the cosmological-constant connection is deferred to the companion paper [13], listed as 'to appear'. Since [13] is unpublished, either state the cosmological-constant relation here or soften the abstract so that the claim is supported by the present manuscript.
[Derivative expansion, Eq. (14)] The split S = S_loc + Γ is said to be systematic only when μ << μ_c, with the two parts distinguished by scaling 'up to a redeﬁnition of Γ by ﬁnite local terms'. A sentence quantifying the residual ambiguity, and stating which of the relations (16), (23)–(25) are insensitive to it, would help the reader judge the status of the matching procedure.
[The conformal anomaly, footnote 5] The authors correctly note that a Weyl-tensor-squared term would have to be added by hand to the 5-d Einstein action. It would be worth stating explicitly in the main text that the anomaly match against [17] therefore covers only the R² and R_{μν}R^{μν} sector, so that the agreement is not over-read as a full anomaly match.

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The paper

The paper claims

The paper argues that the radial evolution equations of five-dimensional supergravity are exactly the renormalization group equations of the dual four-dimensional gauge theory, with the classical supergravity action playing the role of the.

The reading

The paper asks whether the classical equations of five-dimensional supergravity, written as evolution equations in the radial direction, are literally the renormalization group equations of the four-dimensional gauge theory living on the boundary. Working in a gauge where the metric reads $ds^2 = dr^2 + g_{m n} dx^m dx^n$, the authors treat the radial coordinate as time and write the supergravity constraints as a Hamilton-Jacobi equation for the classical action $S[phi,g]$. Splitting $S$ into a local part and a non-local remainder $Gamma$, they match terms of the same functional form and read off a relation between the four-dimensional potential and the five-dimensional one, $V = (1/3)U^2 - (1/2) partial_I U G^{I J} partial_J U$, then $\beta$ functions $beta^I = 6 U G^{I J} partial_J U$, and finally a flow equation for $Gamma$ that has the shape of a Callan-Symanzik equation. If this is right, the supergravity action is the quantum effective action of the dual gauge theory and the fifth dimension is an energy scale rather than an ordinary spatial direction. The authors also derive relations fixing the counterterms that shift the four-dimensional cosmological and Newton constants in terms of the $\beta$ functions.

What carries the argument

The engine is Hamilton-Jacobi theory applied to gravity with the radial coordinate as time: the canonical momentum conjugate to a field equals the functional derivative of the on-shell action, $pi_I = (1/sqrt{g}) delta S/delta phi^I$ and $pi^{m n} = (1/sqrt{g}) delta S/delta g_{m n}$, so the Hamiltonian constraint becomes a functional differential equation that fixes the form of $S$ rather than an equation of motion for the boundary fields. Inserting the split into a local piece carrying a potential $U$, a Ricci coupling $Phi$ and a kinetic matrix $M_{I J}$, plus a non-local remainder $Gamma$, and requiring terms of the same functional form to cancel, converts one gravitational constraint into a whole set of renormalization group statements at once. The potential $U$ plays the role of a superpotential and its gradient gives the $\beta$ functions; the coefficients of the local counterterms encode the flow of the couplings.

Extended reading notes

Core claim

The central claim is that the Hamilton constraint of five-dimensional gravity, read as a functional differential equation for the on-shell action $S$, together with a derivative expansion $S = S_{loc} + Gamma$, reproduces the standard renormalization group structure of the boundary theory. Matching the potential terms gives $V = (1/3)U^2 - (1/2) partial_I U G^{I J} partial_J U$, the same relation a superpotential would satisfy, without assuming supersymmetry. Matching the curvature and kinetic terms gives $beta^K partial_K Phi = 2Phi + 6U$ and a corresponding equation for the matrix $M_{I J}$ of two-derivative couplings, and cancelling a leftover $nabla^2 phi$ term gives a further expression for the $\beta$ function in purely four-dimensional terms, $beta^I = M^{I J} partial_J Phi$. Dropping local terms with two or fewer derivatives then leaves a flow equation for the non-local part, $(1/sqrt{g})(g_{m n} delta/delta g_{m n} - beta^I delta/delta phi^I) Gamma = text{4-derivative terms}$, which upon varying and going to a flat background becomes the Callan-Symanzik equation for $n$-point functions with anomalous dimensions $gamma^I{}_J = nabla_I beta^J$. Relaxing flatness gives a trace anomaly $langle T rangle = beta^I partial_I Gamma + c R_{m n}R^{m n} - d R^2$ with $c = 6Phi^2/U$ and $d = 2U(Phi^2 - (3/2) partial_I Phi G^{I J} partial_J Phi)$, which at a fixed point collapses to $c = 3d = (V/3)^{-3/2} = lambda^3/8$, matching the earlier holographic Weyl anomaly computation.

Load-bearing premise

The argument rests on splitting the action into a local piece plus a non-local remainder and matching terms of the same functional form order by order; the authors state this is a systematic expansion only at scales well below the cutoff, and their appendix shows the constraint leaves the dimension-four part of the potential undetermined.

Editorial extensions

If this is right

The radial coordinate of the five-dimensional geometry is an energy scale, and the classical supergravity action evaluated at a finite radius is the generating functional of the dual theory at a finite ultraviolet cutoff.
The counterterms multiplying the four-dimensional cosmological constant and Newton constant are not free parameters: they are determined by the beta functions through $beta^K partial_K Phi = 2Phi + 6U$ and the corresponding equation for $M_{I J}$.
At a fixed point the anomaly coefficients satisfy $c = 3d = lambda^3/8$, reproducing the known holographic Weyl anomaly and giving a candidate monotonic quantity along the flow.
The inequality $U^2 geq 3V$ follows from positivity of the beta-function squares, with equality only at fixed points, so the flow is forced to move monotonically in the potential.
The construction is not tied to a particular dual pair and extends to other dimensions and to any gauge theory obtainable as a relevant or marginal perturbation of a large-$N$ conformal theory.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the identification holds, the Hamilton-Jacobi equation is the leading large-$N$ limit of an exact renormalization group equation; keeping the second-order term the authors mention in their outlook would give a concrete way to test this order by order in $1/N$.
The relations tying beta functions to the Ricci and Newton-constant counterterms suggest that four-dimensional Einstein equations with a cosmological term may be a restatement of the flow equations, which the paper defers to a companion work; a reader could check whether any consistent four-dimensional gravity dual forces the beta functions to obey $beta^I = M^{I J} partial_J Phi$.
Because the appendix shows the constraint fixes only the divergent part of $U$ and leaves dimension-four terms free, those free terms are plausibly the same freedom as scheme choice and local field redefinition on the field theory side, which would make the scheme dependence of the anomaly coefficients explicit.

Original abstract

We propose a direct correspondence between the classical evolution equations of 5-d supergravity and the renormalization group (RG) equations of the dual 4-d large $N$ gauge theory. Using standard Hamilton-Jacobi theory, we derive first order flow equations for the classical supergravity action $S$, that take the usual form of the Callan-Symanzik equations, including the corrections due to the conformal anomaly. This result gives direct support for the identification of $S$ with the quantum effective action of the gauge theory. In addition we find interesting new relations between the beta-functions and the counterterms that affect the 4-d cosmological and Newton constant.

The compiled record

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.05 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.10 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We propose a direct correspondence between the classical evolution equations of 5-d supergravity and the renormalization group (RG) equations of the dual 4-d large $N$ gauge theory. Using standard Hamilton-Jacobi theory, we derive first order flow equations for the classical supergravity action $S$, that take the usual form of the Callan-Symanzik equations, including the corrections due to the conformal anomaly. This result gives direct support for the identification of $S$ with the quantum effective action of the gauge theory. In addition we find interesting new relations between the beta-functions and the counterterms that affect the 4-d cosmological and Newton constant.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We propose a direct correspondence between the classical evolution equations of 5-d supergravity and the renormalization group (RG) equations of the dual 4-d large $N$ gauge theory.	contribution 0.98	0.86	0.69	asserted 0.67
s1	Using standard Hamilton-Jacobi theory, we derive first order flow equations for the classical supergravity action $S$, that take the usual form of the Callan-Symanzik equations, including the corrections due to the conformal anomaly.	method 0.55	0.96	0.16	asserted 1.00
s2	This result gives direct support for the identification of $S$ with the quantum effective action of the gauge theory.	interpretation 0.99	0.78	0.38	asserted 0.76
s3	In addition we find interesting new relations between the beta-functions and the counterterms that affect the 4-d cosmological and Newton constant.	result 0.98	0.95	0.07	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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sources read

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levels run

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References

Works cited (17)

[17]

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E. Witten, Anti-de Sitter Space and Holography , Adv.Theor.Math.Phys.2 (1998) 253, hep-th/9802150; 13

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L. Susskind and E. Witten, The Holographic Bound in Anti- de Sitter Space, hep- th/9805114

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E.T. Akhmedov, A Remark on the AdS/CFT Correspondence and the Renormalizat ion Group Flow , Phys. Lett. B442 (1998) 152, hep-th/9806217

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E. Alvarez and C. G´ omez Geometric Holography, the Renormalization Group and the c-Theorem, Nucl. Phys. B541 (1999) 441, hep-th/9807226

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[7]

D. Z. Freedman, S. S. Gubser, K. Pilch, N. P. Warner, Renormalization Group Flows from Holography–Supersymmetry and a c-Theorem , hep-th/9906194

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L. Girardello, M. Petrini, M. Porrati, A. Zaﬀaroni, The Supergravity Dual of N = 1 Super Yang-Mills Theory , hep-th/9909047

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[9]

L. Girardello, M. Petrini, M. Porrati, A. Zaﬀaroni, Novel Local CFT and Exact Results on Perturbations of N=4 Super Yang Mills from AdS Dynamics , hep-th/9810126; M. Porrati, A. Starinets, RG Fixed Points in Supergravity Duals of 4-d Field Theory and Asymptotically AdS Spaces , Phys.Lett. B454 (1999) 77, hep-th/9903241

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V. Balasubramanian, P. Kraus, Space-time and the Holographic Renormalization Group, Phys.Rev.Lett 83 (1999) 3605, hep-th/9903190,

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K. Skenderis, P.K. Townsend, Gravitational Stability and Renormalization-Group Flow , hep-th/9909070

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O. DeWolfe, D.Z. Freedman, S.S. Gubser, A. Karch, Modelling the ﬁfth dimension with scalars and gravity , hep-th/9909134

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[13]

E. Verlinde and H. Verlinde, RG Flow, gravity, and the cosmological constant , to appear

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K. Behrndt, M. Cvetic, Supersymmetric Domain-Wall World from D=5 Simple Gauged Supergravity, hep-th/9909058; U. Ellwanger, Constraints on a Brane-World from the Vanishing of the Cosmo logical Constant, hep-th/9909103

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J. Polchinski, Renormalization and Eﬀective Lagrangians, Nucl. Phys, B231 (1984) 269

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H. Verlinde, Holography and compactiﬁcation, hep-th/9906182; L. Randall and R. Sun- drum, A Large Mass Hierarchy from a Small Extra Dimension , hep-ph/9905221; An Alternative to Compactiﬁcation , hep-th/9906064

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hep-th 2026-01 conditional novelty 6.0 of 10

The ERG evolution operator is SO(1,d+1)-invariant for any admissible cutoff, with cutoff-dependent conformal generators that reduce to standard AdS isometries for the special AdS-mapping cutoff.
Finite-cutoff Holographic Thermodynamics
hep-th 2025-07 conditional novelty 6.0 of 10

Finite-cutoff holography yields a thermodynamic duality between T^2-deformed CFTs and Schwarzschild-AdS black holes with a Dirichlet wall, including a teardrop coexistence curve with up to three states at one temperature.
Yang-Mills interaction from boundary vector model
hep-th 2025-04 conditional novelty 6.0 of 10

The ERG flow of the USp(2N) singlet sector of a free 3D U(2N) scalar theory yields a bulk AdS4 cubic action that is on-shell equivalent to Yang-Mills plus a field-strength-cubed term.
Gravitational Observatories in AdS$_4$
hep-th 2024-12 conditional novelty 6.0 of 10

With conformal boundary conditions in AdS4, linearized gravity has a new boundary Weyl mode; for a spherical boundary the mode has complex frequencies (exponential growth) at large angular momentum, while for a planar...
Holographic renormalization and the variational problem for mixed boundary conditions via a solution-dependent superpotential-like function
hep-th 2023-11 unverdicted novelty 6.0 of 10

For m²L²=-2 in AdS black holes with integrable mixed boundary conditions, the cubic coefficient in the near-boundary expansion of the solution-dependent W(φ) is fixed by the boundary deformation to ensure a well-posed...
Edge modes of gravity -- II: Corner metric and Lorentz charges
hep-th 2020-07 conditional novelty 6.0 of 10

The corner metric of tetrad gravity satisfies an ultra-local sl(2,R) algebra whose Casimir is the corner area element, and its space-like nature places that Casimir in the unitary discrete series, giving a quantized a...
On the flow of states under $T\overline{T}$
hep-th 2020-06 conditional novelty 6.0 of 10

On the plane, the TTbar deformation of a 2d QFT is a pure Bogoliubov transformation, so suitably dressed operators retain the undeformed symmetry algebra, including conformal symmetry when the seed theory is a CFT.
The boundary dual of the bulk symplectic form
hep-th 2018-06 conditional novelty 6.0 of 10

For holographic field theories, the Kahler form on the space of Euclidean sources, induced by overlaps of path-integral states, equals the bulk gravitational symplectic form integrated over a Cauchy slice.
Exotic RG Flows from Holography
hep-th 2016-11 accept novelty 6.0 of 10

The superpotential equation of Einstein-dilaton gravity admits multi-branched solutions whose dual RG flows bounce, skip fixed points, or run between minima, and these geometries are regular and perturbatively stable.
Ward Identities for Scale and Special Conformal Transformations in Inflation
hep-th 2015-07 accept novelty 6.0 of 10

Single-field inflation obeys Ward identities for scale and special conformal transformations, including a non-linear compensating-reparametrization term, exact to all orders in slow roll.
de Sitter extremal surfaces
hep-th 2015-01 conditional novelty 6.0 of 10

In de Sitter space, real minimal-area extremal surfaces are null past-lightcone wedges with zero area, while complex extremal surfaces have area matching the dS/CFT central charge.
Lifshitz holography: The whole shebang
hep-th 2014-08 conditional novelty 6.0 of 10

A recursive algorithm built on two commuting functional operators generates the full holographic dictionary, including counterterms, Fefferman-Graham expansions, Ward identities and anomalies, for asymptotically local...
Late-time Structure of the Bunch-Davies De Sitter Wavefunction
hep-th 2014-06 accept novelty 6.0 of 10

Perturbative calculation shows the Bunch-Davies wavefunction of light self-interacting scalars in de Sitter acquires terms growing logarithmically in conformal time at tree and loop level, while the pure graviton sect...
A hole-ographic spacetime
hep-th 2013-10 conditional novelty 6.0 of 10

A proposed field-theory measure, differential entropy, equals the area of a spherical hole in AdS3 and the lengths of all closed curves on a spatial slice.
Lifshitz Gravity for Lifshitz Holography
hep-th 2012-11 conditional novelty 6.0 of 10

Lifshitz spacetimes solve vacuum Horava-Lifshitz gravity, and holographic renormalization of that bulk theory yields both central charges of the z=2 anisotropic Weyl anomaly, which the minimal relativistic bulk model ...
Null Geodesics, Local CFT Operators and AdS/CFT for Subregions
hep-th 2012-09 conditional novelty 6.0 of 10

Bulk reconstruction from boundary one-point functions is continuous in global AdS but not in the AdS-Rindler wedge, and a null-geodesic capture criterion is proposed as the diagnostic.
Holographic Geometry of Entanglement Renormalization in Quantum Field Theories
hep-th 2012-08 conditional novelty 6.0 of 10

A quantum-distance definition of the bulk metric in the continuous MERA reproduces anti-de Sitter geometry for conformal free fields and caps off for gapped theories.
Conformal Lifshitz Gravity from Holography
hep-th 2011-12 accept novelty 6.0 of 10

Holographic renormalization of asymptotically Lifshitz spacetimes generates counterterms that reproduce the action of conformal Lifshitz gravity, including the detailed balance condition.
Kerr/CFT, dipole theories and nonrelativistic CFTs
hep-th 2011-08 conditional novelty 6.0 of 10

A class of warped AdS3 supergravity backgrounds is identified as a deformation of the DLCQ of the D1-D5 CFT by a single (1,2) operator, proposing nonrelativistic CFT duals for Kerr/CFT and dipole backgrounds.
Generalized Holographic Quantum Criticality at Finite Density
hep-th 2011-07 conditional novelty 6.0 of 10

Classes of near-extremal Einstein-Maxwell-Dilaton black branes are shown to descend from higher-dimensional AdS, Lifshitz, or dilatonic brane solutions, which explains their scaling behavior and singularity structure.
Holographic Renormalization of general dilaton-axion gravity
hep-th 2011-06 conditional novelty 6.0 of 10

A recursive algorithm solves the radial Hamilton-Jacobi equation in a derivative expansion, yielding explicit boundary counterterms for general dilaton-axion gravity up to five dimensions and generalized Fefferman-Gra...
CFT dual of the AdS Dirichlet problem: Fluid/Gravity on cut-off surfaces
hep-th 2011-06 conditional novelty 6.0 of 10

Fixing a metric on a timelike cut-off surface inside AdS and solving Einstein's equations in the gradient expansion yields a non-conformal hypersurface fluid with the boundary fluid's shear viscosity, whose sound spee...
Static supersymmetric black holes in AdS_4 with spherical symmetry
hep-th 2010-12 conditional novelty 6.0 of 10

Static quarter-BPS asymptotically AdS4 black holes with spherical horizons exist in N=2 Fayet-Iliopoulos gauged supergravity, provided the scalar fields run with radius and are given by harmonic functions.
Holographic Non-Gaussianity
hep-th 2010-11 conditional novelty 6.0 of 10

For a class of holographic inflationary models, the primordial bispectrum is exactly of equilateral shape with f_NL = 5/36, independent of the dual QFT's field content.
Holographic and Wilsonian Renormalization Groups
hep-th 2010-10 conditional novelty 6.0 of 10

The Wilsonian effective action of a holographic field theory is an integral transform of the bulk radial wavefunction and obeys a Hamilton-Jacobi flow equation that necessarily includes multi-trace operators.
Holographic renormalization as a canonical transformation
hep-th 2010-07 accept novelty 6.0 of 10

Holographic renormalization is shown to be a canonical transformation that repairs ill-defined variational problems at infinity, and the same construction is carried out for point particles, flat-space scalars, gravit...
Precision holography for non-conformal branes
hep-th 2008-07 accept novelty 6.0 of 10

Holographic renormalization is worked out in full generality for non-conformal p-branes and fundamental strings, yielding finite one-point functions, Ward identities and two-point functions consistent with the general...
Multi-Trace Deformations in AdS/CFT: Exploring the Vacuum Structure of the Deformed CFT
hep-th 2007-03 conditional novelty 6.0 of 10

Multi-trace deformations of a holographic CFT correspond to mixed boundary conditions in the bulk, and the resulting effective action can be computed in a derivative expansion via Hamilton-Jacobi theory, yielding new ...
Toward a Systematic Holographic QCD: A Braneless Approach
hep-ph 2006-08 conditional novelty 6.0 of 10

Backreaction of the gluon condensate supplies the infrared cutoff in a braneless holographic QCD model, and a dilaton potential reproducing asymptotic freedom yields glueball masses and a finite running coupling.
Dual of Big-bang and Big-crunch
hep-th 2006-03 conditional novelty 6.0 of 10

The Janus supergravity solution and its field theory dual are analytically continued to give a gauge theory dual of a big-bang/big-crunch cosmology, whose spacetime singularities are argued to be artifacts of supergravity.
Thermodynamics of Asymptotically Locally AdS Spacetimes
hep-th 2005-05 conditional novelty 6.0 of 10

Holographic charges equal Noether charges for asymptotically locally AdS spacetimes, so the first law of black hole mechanics holds for these black holes with the holographic mass, angular momentum, and charge as the ...
AdS/CFT correspondence and Geometry
hep-th 2004-04 conditional novelty 6.0 of 10

A Hamiltonian reformulation of holographic renormalization organizes the asymptotic expansion by dilatation weight, yielding universal recursion relations for counterterms and one-point functions valid in all dimensions.

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248 later papers cite it

Evidence: substantial. 247 of the citing papers share no author with it, and 43 of those carry a confident label from Pith's citation classifier.

Pith read 38 of the 386 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

7 later papers by other authors are listed below: 1 built on it and 6 used its method or tool.

Builds on it 1 paper

We think of this as extending the holographic RG methods of [ref] deeper into the bulk.

Brane effective actions and their island rule from $T\overline T$ flows (2026) · Introduction and overview of results · 0.88

Uses its method or tool 6 of 31 papers with this label

First, following [ref] , we expand the flow equations [ref] and [ref] in derivatives.

Gradient Flow and Holography from a Local Wilsonian Cutoff (2021) · The Hamiltonian Constraint · 0.80
Made use of this paper's observation for a theory involving only second derivatives.

Holographic Heat Current as Noether Current (2017) · 0.98 · paraphrased: the licence does not allow quoting
As in this paper, imagined that the boundary action on a radial slice can be evaluated as a function of boundary field values at that scale.

On $AdS_2$ holography from redux, renormalization group flows and $c$-functions (2018) · 0.92 · paraphrased: the licence does not allow quoting
Made use of the Hamilton-Jacobi method first used in the context of AdS/CFT by this paper.

First order flow for non-extremal AdS black holes and mass from holographic renormalization (2014) · 0.93 · paraphrased: the licence does not allow quoting
Used the holographic renormalization, especially the Hamilton-Jacobi approach of this paper, to build up the most general ansatz for the intrinsic counterterms on the boundary.

Deep learning black hole metrics from shear viscosity (2020) · 0.90 · paraphrased: the licence does not allow quoting
By means of the holographic renormalization group [ref] , we can identify this bulk radial cutoff with a IR cutoff in the dual boundary theory.

Magnetic Braneworlds: Cosmology and Wormholes (2024) · The dual description: $𝒩=4$ SYM and confinement · 0.86

How the literature uses it

229 arXiv papers cite this in 355 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

We argue that the classical equations of motion in the SUGRA picture can be interpreted as that of the renormalization group on the SYM side.

A remark on the AdS/CFT correspondence and the renormalization group flow · 0.87

Papers citing it · 195 sentences without a confident label

Builds on it 1 paper

We think of this as extending the holographic RG methods of [ref] deeper into the bulk.

Brane effective actions and their island rule from $T\overline T$ flows · Introduction and overview of results · 0.88

Uses its method or tool 26 papers

Our algorithm for constructing the holographic dictionary hinges upon a certain asymptotic solution of the radial Hamilton-Jacobi (HJ) equation [ref] , subject to asymptotically Lif or hvLf boundary conditions.

Lifshitz holography: The whole shebang · Introduction · 0.91
First, following [ref] , we expand the flow equations [ref] and [ref] in derivatives.

Gradient Flow and Holography from a Local Wilsonian Cutoff · The Hamiltonian Constraint · 0.80
[ref] β(α)=3 α X , where $α(z)$ is defined as [ref] α(z)=e^φ(z) and $X$ is a new dynamical variable [ref] defined as X(z)=φ̇ B/3Ḃ , where $B(z)$ is defined in ([ref]) and $φ$ for the light and heavy quarks model are obtained in ([ref]) and [ref], respectively.

Beta-Functions and RG flows for Holographic QCD with Heavy and Light Quarks: Isotropic case · Beta-Function · 0.81

Compares with it 8 papers

Eq. [ref] coincides with the Hamilton-Jacobi constraint in [ref] .

Gradient Flow and Holography from a Local Wilsonian Cutoff · The Hamiltonian Constraint · 0.98
Furthermore, we will find that the beta function associated to the dimensionless coupling, $g_L = L^d-Δ$, is proportional to the derivative of the prepotential, $W'$, as opposed to the usual holographic beta function $W'/W$ [ref] .

Dimensional regularization for holographic RG flows · Introduction · 0.97
The result we have just presented is very close in spirit to the seminal work of deBoer and Verlinde's [ref] .

Reconstructing AdS/CFT · Solving the Hamilton-Jacobi equation · 0.96

Cites it as support 1 paper

The ideas coming from the AdS/CFT correspondence [ref] where renormalization group flows can be formulated holographically [ref] and scale or Weyl anomalies become geometric [ref] further strengthen our belief that the anomaly can be resolved by our approach.

Unit Invariance as a Unifying Principle of Physics · AdS/CFT Correspondence · 0.98

Cites it as background 102 papers

Second, we reviewed the holographic description of the renormalization group flow which implies Hamiltonian constraints on the action on the boundary of a 5d manifold, following to a large extend [ref] .

Gradient Flow and Holography from a Local Wilsonian Cutoff · Summary and Outlook · 0.95
The question is whether the `traditional' holographic RG flow coming from the classical equation of motion [ref] is the same as the more recently formulated one in [ref] .

Holographic Wilsonian RG Flow and Sliding Membrane Paradigm · Introdution · 0.81
There is a vast literature on the subject of AdS/CFT, however we feel that only a relative small portion of it focuses on the questions asked here : First there is the key original reference of Witten [ref] where the meaning of the correspondence is proposed; the body of work by Skenderis et al. [ref] developing the semi-classical understanding of the correspondence and the holographic renormalisation group; the central claim of Verlinde's and deBoer [ref] that there is a relationship between the Hamilton-Jacobi equation and the holographic renormalisation group, and a Schrödinger picture sketched by Maldacena [ref] in the de Sitter context.

Reconstructing AdS/CFT · Introduction · 0.99

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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L. Susskind and E. Witten, The Holographic Bound in Anti- de Sitter Space, hep- th/9805114

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L. Girardello, M. Petrini, M. Porrati, A. Zaﬀaroni, The Supergravity Dual of N = 1 Super Yang-Mills Theory , hep-th/9909047

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