{"id":"a24b9814-a12c-4926-a50e-29440dfdef40","arxiv_id":"hep-th/9912012","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The radial evolution equations of five-dimensional supergravity take the form of Callan-Symanzik equations, identifying the classical supergravity action with the quantum effective action of the dual gauge theory.","lead":"This paper shows that the equations governing how fields evolve along the extra dimension in five-dimensional supergravity can be rewritten as the renormalization group equations of the dual four-dimensional gauge theory. It gives theorists a working dictionary between bulk gravity and the running of couplings and correlation functions on the boundary.","discovery_kind":"unclear","skeptic_critique":{"model":"selfhosted/deepseek-v4.1-flash","headline":"The load-bearing soft spot is the term-by-term matching in the derivative expansion: the paper derives β^I twice, from the potential (Eq. 21) and from ∇²φ cancellation (Eq. 25), and never checks the two agree.","rationale":"The paper's main result — that the HJ form of the 5-d constraints yields first-order flow equations for S that take the CS form — is well-motivated and became standard machinery. The reader's verdict (CONDITIONAL, low correctness risk) is appropriate. I agree with the reader that the derivative expansion and term-by-term matching is the load-bearing soft spot, and I sharpen it: the matching yields two expressions for β^I, Eqs. (21) and (25), whose mutual consistency is assumed but not shown. This is the kind of gap that either closes under a short algebraic check or reveals an over-determined ansatz. The paper's external check against Henningson-Skenderis [17] for the anomaly coefficients is genuine support for the fixed-point sector, and the appendix's honest statement that the dimension-4 part of U is undetermined is a feature (the finite effective potential is physical), not a flaw. But the off-fixed-point consistency of the two β expressions is not addressed. I do not think this rises to a rejection; it is a checkable gap in an acknowledged heuristic procedure. Verdict unchanged: CONDITIONAL.","tokens_in":12340,"tokens_out":14067,"duration_ms":246607,"concrete_test":"First re-derive Eq. (21) from the flow equations (17)-(20); the printed factor 6U should be checked, since ȧ = (1/6)Ua and φ̇^I = G^{IJ}∂_J U give β^I = (6/U)G^{IJ}∂_J U. Then check algebraically whether this agrees with Eq. (25), β^I = M^{IJ}∂_J Φ, given the flow equations (23)-(24). Concretely, expand around a fixed point in the G_{IJ} = δ_{IJ} basis of the Appendix, insert U = 6λ + (1/2)λ_I φ² + ... and Φ = Φ_0 + ..., and compare order by order. Agreement to all orders dissolves the concern; a mismatch at some order means the local ansatz (15) is over-determined and Eq. (26) needs correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on the decomposition S = S_loc + Γ (Eq. 14) and matching terms of equal derivative order in the Hamilton-Jacobi constraint (Eq. 13). The paper acknowledges this is systematic only for μ << μ_c, but a sharper issue is that the matching produces two independent expressions for the same β-function: Eq. (21), β^I = 6U G^{IJ}∂_J U, from the leading potential terms, and Eq. (25), β^I = M^{IJ}∂_J Φ, from requiring the ∇²φ coefficient to vanish. For the local ansatz (15) to be consistent these must agree, i.e. M^{IJ}∂_J Φ = 6U G^{IJ}∂_J U must hold given the flow equations (23)-(24). The paper presents (25) as 'another interesting expression' but never verifies the consistency. If the two expressions disagree, the ansatz is over-determined and the derivation of the Callan-Symanzik equation (26), which uses β^I, is not well-defined. The appendix compounds this: it shows the dimension-4 (marginal) part of U is not fixed by V, so the matching leaves the marginal sector free, and it is precisely marginal directions where the two β expressions could differ. This is a genuine gap in an otherwise well-motivated derivation, though it may well close under algebra the paper omits.","agreement_with_reader":"agree"},"referee_report":{"model":"selfhosted/deepseek-v4.1-flash","summary":"The manuscript proposes that the classical Hamilton–Jacobi equations of 5-d supergravity, written in a radial gauge with metric ds² = dr² + g_{μν}dx^μdx^ν, are the holographic image of the renormalization group equations of the dual 4-d large-N gauge theory. Treating the on-shell action S[φ,g] as the Hamilton–Jacobi functional, the authors convert the momentum definitions (8), (10) and flow relations (9), (11) into the constraints (12) and (13), the latter being a functional differential equation for S. Splitting S into a local part S_loc (Eq. 15) and a non-local part Γ (Eq. 14), they match terms of equal derivative order in (13) to obtain the potential relation (16), the beta-function β^I = 6U G^{IJ}∂_J U (Eq. 21), and the relations (23)–(25) between β^I and the local coefficients Φ and M^{IJ} that control the 4-d Newton constant and kinetic terms. Keeping cross terms between U and Γ yields the Callan–Symanzik-type equation (26) and, after varying and taking constant backgrounds, Eq. (28) with anomalous dimensions γ^I_J = ∇_Iβ^J. The trace anomaly (31)–(32) is computed and the fixed-point coefficients are checked against the Henningson–Skenderis result [17]. A final section outlines cutoff removal and the renormalized equation (41), and an appendix analyses how much of U is determined by the 5-d potential V.","tokens_in":12650,"tokens_out":6256,"duration_ms":109658,"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_comments":[{"comment":"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.","section":"Holographic RG-flow and Relations for the local terms, Eqs. (21) and (25)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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.","section":"Callan-Symanzik equation and The conformal anomaly, Eqs. (26), (31), (32)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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.","section":"Removing the UV cut-off, Eqs. (36)-(41)","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"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.","section":"The conformal anomaly, sentence preceding Eq. (32)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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.","section":"The conformal anomaly, Eq. (32)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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.","section":"Abstract and Discussion","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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.","section":"Derivative expansion, Eq. (14)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"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.","section":"The conformal anomaly, footnote 5","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is a short, well-motivated letter whose central derivation follows from standard Hamilton–Jacobi theory and is externally validated at the fixed point against [17]. The main request is an algebraic consistency check between the two beta-function expressions (21) and (25), which I expect closes but which the manuscript should not leave to the reader. The remaining points are presentational. The companion paper [13] is cited as 'to appear' and carries part of the abstract's second claim; the authors should either include that result or adjust the abstract."},"author_rebuttal":null,"desk_editor":{"model":"selfhosted/deepseek-v4.1-flash","letter":"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.","headline":"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.","tokens_in":13276,"tokens_out":2120,"would_cite":true,"duration_ms":35776,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":["11.25.Hf","11.10.Hi","04.50.+h"],"model":"selfhosted/deepseek-v4.1-flash","headline":"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.","keywords":["holographic renormalization group","AdS/CFT correspondence","Hamilton-Jacobi equation","Callan-Symanzik equation","conformal anomaly","beta function","five-dimensional supergravity","holographic c-function"],"falsifier":"Find a five-dimensional potential $V(phi)$ that admits a regular solution filling the ball but for which no $U$ satisfying $(1/3)U^2 - (1/2)(partial U)^2 = V$ exists globally, or for which the resulting $Gamma$ fails the flow equation $(1/sqrt{g})(g_{m n} delta/delta g_{m n} - beta^I delta/delta phi^I) Gamma = 0$ beyond the dropped four-derivative terms; either would break the identification of $S$ with the quantum effective action.","tokens_in":13689,"feed_emoji":"🌀","tokens_out":4229,"duration_ms":80711,"temperature":0.7,"pith_summary":"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.","feed_headline":"Gravity's radial flow obeys the renormalization group","feed_subtitle":"The five-dimensional supergravity action satisfies Callan-Symanzik equations, tying the extra dimension to energy scale.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the prescription identifying the on-shell supergravity action with the generating functional of gauge theory correlators, which the whole identification rests on.","marker":"[2]"},{"why":"Supplies the statement that regularity in the interior of the five-ball fixes a unique classical solution for given boundary data, the premise that makes $S[phi,g]$ well defined.","marker":"[3]"},{"why":"Earlier derivation of the Hamilton-Jacobi relation between the four-dimensional potential and the five-dimensional potential in scalar-gravity domain walls, cited as the closest precedent.","marker":"[12]"},{"why":"Source for the claim that the relation between $U$ and $V$ is a sufficient and possibly necessary condition for stability of domain wall structures.","marker":"[11]"},{"why":"Provides the holographic Weyl anomaly result against which the coefficients $c$ and $d$ are checked at the fixed point.","marker":"[17]"},{"why":"Proposes the quantity built from $V$ as a candidate $c$-function, the interpretation the paper adopts for its anomaly coefficient.","marker":"[7]"},{"why":"Polchinski's exact renormalization group, invoked as the structure the quantum-corrected Hamilton-Jacobi equation would resemble.","marker":"[15]"}],"fun_headline_variants":["Five-dimensional gravity action follows Callan-Symanzik flow","Supergravity action reproduces boundary RG equations","Radial gravity flow maps to gauge theory beta-functions","Hamilton-Jacobi flow yields holographic Callan-Symanzik equations","Beta-functions tie counterterms to cosmological and Newton constants"],"cache_read_input_tokens":0,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Five-dimensional gravity action follows Callan-Symanzik flow","Supergravity action reproduces boundary RG equations","Radial gravity flow maps to gauge theory beta-functions","Hamilton-Jacobi flow yields holographic Callan-Symanzik equations","Beta-functions tie counterterms to cosmological and Newton constants","Trace anomaly matches earlier holographic Weyl computation","Gravity constraint encodes anomalous dimensions of boundary theory","On-shell supergravity action obeys Callan-Symanzik equations","Derivative expansion links five-dimensional gravity to four-dimensional RG","Counterterms affect cosmological and Newton constants through beta-functions"]},"model":"selfhosted/deepseek-v4.1-flash","effort":"low","attempts":1,"cost_usd":0.0,"raw_usage":{"total_tokens":1534,"prompt_tokens":1040,"completion_tokens":494,"prompt_tokens_details":null,"completion_tokens_details":{"reasoning_tokens":366}},"tokens_in":1040,"tokens_out":494,"duration_ms":9943,"image_pages":[],"temperature":1.0,"reasoning_budget":50,"reasoning_tokens":366,"transport_provider":"selfhosted","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-29T18:20:01.778235+00:00","model_set":{"reader":"selfhosted/deepseek-v4.1-flash"},"falsifier":"Find a five-dimensional potential $V(phi)$ that admits a regular solution filling the ball but for which no $U$ satisfying $(1/3)U^2 - (1/2)(partial U)^2 = V$ exists globally, or for which the resulting $Gamma$ fails the flow equation $(1/sqrt{g})(g_{m n} delta/delta g_{m n} - beta^I delta/delta phi^I) Gamma = 0$ beyond the dropped four-derivative terms; either would break the identification of $S$ with the quantum effective action.","supporting_citations":[{"cited_title":"Gauge Theory Correlators from Non-Critical String Theory","cited_arxiv_id":"hep-th/9802109","evidence_quote":"Supplies the prescription identifying the on-shell supergravity action with the generating functional of gauge theory correlators, which the whole identification rests on."},{"cited_title":"Anti De Sitter Space And Holography","cited_arxiv_id":"hep-th/9802150","evidence_quote":"Supplies the statement that regularity in the interior of the five-ball fixes a unique classical solution for given boundary data, the premise that makes $S[phi,g]$ well defined."},{"cited_title":"Modeling the fifth dimension with scalars and gravity","cited_arxiv_id":"hep-th/9909134","evidence_quote":"Earlier derivation of the Hamilton-Jacobi relation between the four-dimensional potential and the five-dimensional potential in scalar-gravity domain walls, cited as the closest precedent."},{"cited_title":"Gravitational Stability and Renormalization-Group Flow","cited_arxiv_id":"hep-th/9909070","evidence_quote":"Source for the claim that the relation between $U$ and $V$ is a sufficient and possibly necessary condition for stability of domain wall structures."},{"cited_title":"The Holographic Weyl anomaly","cited_arxiv_id":"hep-th/9806087","evidence_quote":"Provides the holographic Weyl anomaly result against which the coefficients $c$ and $d$ are checked at the fixed point."},{"cited_title":"Continuous distributions of D3-branes and gauged supergravity","cited_arxiv_id":"hep-th/9906194","evidence_quote":"Proposes the quantity built from $V$ as a candidate $c$-function, the interpretation the paper adopts for its anomaly coefficient."},{"cited_title":"Polchinski, Renormalization and Effective Lagrangians, Nucl","cited_arxiv_id":null,"evidence_quote":"Polchinski's exact renormalization group, invoked as the structure the quantum-corrected Hamilton-Jacobi equation would resemble."}],"review_version":1}