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REVIEW 1 major objections 4 minor 43 references

In eight dimensions, holographic renormalisation of Einstein-AdS gravity is exactly the unique conformal gravity that admits an Einstein sector.

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2026-07-12 00:43 UTC pith:MEQAWCH5

load-bearing objection Solid 8D matching of unique Einstein-sector conformal gravity to holographically renormalised Einstein-AdS; the calculation is explicit and the conjecture is cleanly stated. the 1 major comments →

arxiv 2607.03679 v1 pith:MEQAWCH5 submitted 2026-07-04 hep-th gr-qcmath.DG

Conformal Renormalisation of 8D Einstein Gravity

Giorgos Anastasiou , Ignacio J. Araya , Nicolas Boulanger , Rodrigo Olea , Davide Rovere This is my paper
classification hep-th gr-qcmath.DG
keywords holographic renormalisationconformal gravitytopological regularisationEinstein-AdSWeyl invariantseight dimensionsQ-curvatureFefferman-Graham
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper claims that the standard procedure of holographic renormalisation for eight-dimensional Einstein gravity with a negative cosmological constant is already contained inside a single, unique conformal gravity action that is known to possess Einstein solutions of constant negative curvature. Topological regularisation alone—adding the Euler density with a fixed coefficient so the bulk Lagrangian becomes a polynomial in the AdS curvature—still leaves residual divergences when the conformal boundary is not flat. Those leftover surface terms are precisely cancelled by the extra boundary contributions that arise when the same bulk polynomial is completed to a Weyl-invariant density. On Einstein-AdS manifolds the two actions therefore coincide (up to a universal overall factor). The authors conjecture that the same pattern holds in every even dimension: holographic renormalisation is nothing but the conformal completion of Einstein-Hilbert plus Euler.

Core claim

On Einstein-AdS manifolds the unique eight-dimensional conformal gravity that admits an Einstein sector is proportional to the holographically renormalised Einstein-AdS action obtained by topological regularisation plus three bulk-covariant counterterms: Iren evaluated on Einstein equals −(ℓ⁶/504πG) times that conformal action.

What carries the argument

The unique linear combination of eight-dimensional Weyl invariants (the Boulanger–Rovere Lagrangian L8) that admits an Einstein sector, together with the three total-derivative counterterms □²Φ(2), □Φ(3) and (1/ℓ²)□Φ(2) that cancel the residual Fefferman–Graham divergences left by the Euler-augmented Einstein action.

Load-bearing premise

That the three bulk-covariant total-derivative terms built from the AdS curvature polynomials exhaust all counterterms needed for generic (non-conformally-flat) boundaries; any extra independent Weyl-invariant surface term of the same order would change the matching coefficients.

What would settle it

Compute the on-shell value of the unique 8D conformal gravity on a concrete Einstein-AdS solution whose boundary is not conformally flat (for example a Taub–NUT–AdS instanton) and check whether it exactly equals the holographically renormalised Einstein–Hilbert action; any mismatch would falsify the claimed proportionality.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Holographic renormalisation in every even dimension is conjectured to be equivalent to the conformal completion of Einstein-Hilbert plus the Euler term with the topological-regularisation coefficient.
  • The renormalised volume of an even-dimensional conformally compact Einstein manifold is recovered as the on-shell value of a single conformal gravity action.
  • Codimension-two energy functionals extracted from the same conformal action should automatically be finite for surfaces embedded in asymptotically AdS spacetimes.
  • The residual freedom of adding total-derivative conformal invariants in eight dimensions does not affect the Einstein-sector matching, so the renormalised Einstein action is uniquely fixed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the pattern continues, the obstruction tensor of the Q-curvature in arbitrary even dimension should reproduce the holographic stress tensor of Einstein gravity once the Einstein sector is imposed.
  • The same matching may supply a purely bulk criterion that selects which linear combination of higher-curvature invariants can serve as a holographic dual without ever performing a Fefferman–Graham expansion.
  • Surface functionals obtained by evaluating the 8D conformal action on a conical defect should yield a finite, conformally invariant generalisation of holographic entanglement entropy in seven boundary dimensions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper shows that holographic renormalisation of Einstein-AdS gravity in eight dimensions is encoded in the unique conformal gravity theory that admits an Einstein sector (the Boulanger–Rovere action IBR). Starting from topological regularisation (Einstein–Hilbert plus the Euler density with the coefficient fixed by black-hole thermodynamics), the authors expand the residual divergences in the Fefferman–Graham frame and identify them with boundary Weyl, Cotton and Schouten tensors. They then demonstrate that three bulk-covariant total-derivative counterterms cancel those divergences, and that the resulting finite on-shell density coincides, after a unique matching of six free parameters, with the Einstein-sector restriction of IBR: Iren|E = −(ℓ⁶/504πG) IBR|E. The two independent total-derivative conformal invariants are shown (Appendix C) to vanish on Einstein-AdS manifolds, so they do not affect the equality. The authors conjecture that the same pattern holds in every even dimension.

Significance. If correct, the result supplies a concrete, dimension-by-dimension realisation of the idea that holographic renormalisation is equivalent to a conformal completion of the Einstein–Hilbert action plus the Euler term. The explicit algebraic matching of the six coefficients, the independent FG computation of the residual divergences, and the power-counting argument that the two total-derivative conformal invariants drop out on Einstein-AdS spaces constitute a non-trivial, falsifiable check of that idea in eight dimensions. The work therefore strengthens the existing 4D and 6D evidence and furnishes a clear template for higher even dimensions. The uniqueness of the Einstein-sector conformal action is taken from prior work, but the matching itself is an independent computation that can be verified by direct expansion.

major comments (1)
  1. The claim that the three bulk-covariant operators □²Φ⁽²⁾, □Φ⁽³⁾ and (1/ℓ²)□Φ⁽²⁾ (Eq. 3.17) exhaust the allowed counterterms of the correct asymptotic weight rests on power-counting and on the uniqueness of the Einstein-sector conformal action. While Appendix C and the FG analysis of §§2–3 make this plausible, a short explicit statement that no other independent Weyl-invariant surface density of weight 8 appears at the same order would close the only remaining logical gap in the matching (Eq. 4.20).
minor comments (4)
  1. Several total-derivative terms at the conformal boundary are discarded without an explicit reference to the Stokes theorem or to the fall-off that makes their integrals vanish; a one-sentence justification in §2 and §3 would improve readability.
  2. Notation for the boundary Schouten, Cotton and Bach tensors is introduced gradually; a compact summary table or a pointer to Appendix A at first occurrence would help the non-specialist reader.
  3. Typographical inconsistencies appear in the arXiv identifier (2607.03679) and in a few author e-mail addresses; these should be corrected before publication.
  4. The conjecture for arbitrary even dimension is stated clearly in the abstract and conclusion, but a brief remark on the expected obstruction (growth of the basis of conformal invariants) would place the claim in better context.

Circularity Check

2 steps flagged

Uniqueness of the 8D Einstein-sector conformal gravity is imported from the authors' own prior work [11]; the on-shell matching of free coefficients is an independent algebraic computation, not forced by definition.

specific steps
  1. uniqueness imported from authors [Abstract; §4 opening and Eq. (4.1)–(4.2); conclusion of §4 (Eq. 4.22)]
    "We show that Holographic Renormalisation (HR) in eight dimensions is encoded in the unique conformal gravity theory that admits an Einstein sector with constant negative curvature. [...] The most general conformal gravity action in 8D that admits an Einstein sector was found in Ref. [11], IBR[gµν]=∫M8 d8x√−g L8 [...] Therefore the conclusion is that, for Einstein-AdS manifolds, the renormalised Einstein-AdS action (3.24) is exactly the action (4.1) [...] Iren[gµν]|E=−ℓ6/504πG IBR[gµν]|E."

    The narrative claim that HR is encoded in 'the unique' 8D conformal gravity with an Einstein sector rests on a uniqueness result proved only in the authors' own concurrent paper [11] (Boulanger–Rovere). That uniqueness is not re-derived here; it is imported and then used to single out IBR as the theory that must match the holographically renormalised action. Without that self-cited uniqueness, the algebraic matching still holds for this particular action, but the stronger claim that HR is thereby 'encoded in the unique' such theory would not be justified from within the present paper alone.

  2. self citation load bearing [Introduction, Eqs. (1.9)–(1.14) and (1.18)–(1.20); §2 opening]
    "As a consequence, the following standard thermodynamic relation for Schwarzschild-AdS black holes [...] is recovered if the Euler coupling is fixed by [1] α2n=1/(16πG) (−1)n / [n(2n−2)!] ℓ^{2n−2}. [...] For the appropriate choice (1.14) of coefficient α2n, it is remarkable that the action (1.9) can be written in the form [7] I[gµν]=1/(16πG)∫ √−g ∑k=2^n ak Φ(k)."

    The precise Euler coupling that makes the bulk density a polynomial in the AdS curvature Ω (the starting point of Topological Regularisation) is taken from earlier thermodynamics papers of the same group ([1], [6], [7]). This is load-bearing for the TR side of the comparison, but it is standard setup rather than the new 8D matching; the residual divergences and their cancellation are recomputed independently in §§2–3.

full rationale

The paper's central equality Iren|E = −(ℓ⁶/504πG) IBR|E (Eq. 4.22) is obtained by writing the Boulanger–Rovere conformal Lagrangian (with free overall factor α and free coefficients β,γ of the two total-derivative Weyl invariants) and the topologically regularised Einstein–AdS density plus three bulk-covariant total derivatives (with free c1,c2,c3), then equating coefficients on Einstein-AdS manifolds. That matching uniquely fixes all six parameters and recovers precisely the (c1,c2,c3) already shown in §3 to cancel the residual FG divergences of pure TR. The residual divergences themselves (Eq. 2.14) and the cancellation by □²Φ⁽²⁾, □Φ⁽³⁾, (1/ℓ²)□Φ⁽²⁾ are computed independently from the Fefferman–Graham expansion; Appendix C independently shows the two total-derivative conformal invariants integrate to zero on Einstein-AdS. None of these steps is definitional or a fitted-input-as-prediction. The only load-bearing self-citation is the uniqueness of the Einstein-sector conformal action, taken from the authors' concurrent paper [11] and used to underwrite the claim that HR is encoded in 'the unique' such theory. The topological coefficient α2n is likewise taken from earlier thermodynamics papers of the same group, but that is standard setup for TR, not the new 8D result. Overall mild circularity (score 3): uniqueness is imported, yet the matching computation has independent content and is not forced by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 1 invented entities

The central claim rests on standard differential-geometry identities, the Fefferman-Graham expansion, the classification of 8D conformal invariants, and the uniqueness of the conformal gravity that admits an Einstein sector. No free parameters are fitted to data; the six coefficients that appear are fixed by algebraic matching. The only invented entity is the conjecture for arbitrary even dimension, which is not used in the 8D proof.

axioms (4)
  • domain assumption Fefferman-Graham asymptotic expansion of the metric and of the Weyl tensor components near a conformal boundary (Eqs. 2.6–2.9).
    Standard in AdS/CFT; used throughout §§2–3 to extract divergent terms.
  • domain assumption Uniqueness (up to total derivatives) of the 8D conformal gravity Lagrangian that admits an Einstein sector (IBR, Eq. 4.2).
    Taken from the authors’ prior work [11]; enters as the starting point of §4.
  • standard math Euler density is locally equivalent to the Chern form B7 plus a topological constant (Eq. 3.6).
    Classical Chern-Gauss-Bonnet theorem; used to compare TR with HR counterterms.
  • standard math On Einstein-AdS manifolds the Cotton and Bach tensors vanish and Ω reduces to the Weyl tensor (Eq. A.8).
    Direct consequence of the Einstein condition; used to evaluate all polynomials on-shell.
invented entities (1)
  • Conjecture that HR equals conformal completion of EH+Euler in every even dimension no independent evidence
    purpose: Extrapolates the 4D/6D/8D pattern to a general statement.
    Stated in the abstract and §5; not proved and not required for the 8D result.

pith-pipeline@v1.1.0-grok45 · 24145 in / 2606 out tokens · 18459 ms · 2026-07-12T00:43:22.517311+00:00 · methodology

0 comments
read the original abstract

We show that Holographic Renormalisation (HR) in eight dimensions is encoded in the unique conformal gravity theory that admits an Einstein sector with constant negative curvature. We explicitly relate HR to Topological Regularisation (TR), where the latter prescribes to add the Euler term to the Einstein-Hilbert Lagrangian density, with a precise coefficient so as to ensure that the resulting density is polynomial in the anti de Sitter (AdS) curvature. The polynomial is still asymptotically divergent. We find that the aforementioned, unique conformal gravity action in 8D, reproduces the polynomial, together with extra boundary terms that cancel the divergent terms in the action. We conjecture that, in arbitrary even dimension, HR is equivalent to conformally completing the Einstein-Hilbert action with negative cosmological constant, plus the Euler term with fixed coupling prescribed by TR.

discussion (0)

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Reference graph

Works this paper leans on

43 extracted references · 35 linked inside Pith

  1. [1]

    Olea,Mass, angular momentum and thermodynamics in four-dimensional Kerr-AdS black holes,JHEP06(2005) 023, [hep-th/0504233]

    R. Olea,Mass, angular momentum and thermodynamics in four-dimensional Kerr-AdS black holes,JHEP06(2005) 023, [hep-th/0504233]

    Pith/arXiv arXiv 2005

  2. [2]

    Miskovic and R

    O. Miskovic and R. Olea,Thermodynamics of Einstein-Born-Infeld black holes with negative cosmological constant,Phys. Rev. D77(2008) 124048, [0802.2081]

    Pith/arXiv arXiv 2008

  3. [3]

    R. Aros, M. Contreras, R. Olea, R. Troncoso and J. Zanelli,Conserved charges for gravity with locally AdS asymptotics,Phys. Rev. Lett.84(2000) 1647–1650, [gr-qc/9909015]

    Pith/arXiv arXiv 2000

  4. [4]

    R. Aros, M. Contreras, R. Olea, R. Troncoso and J. Zanelli,Conserved charges for even dimensional asymptotically AdS gravity theories,Phys. Rev. D62(2000) 044002, [hep-th/9912045]

    Pith/arXiv arXiv 2000

  5. [5]

    L. P. Eisenhart,Riemannian geometry, vol. 19. Princeton university press, 1997. – 18 –

    1997

  6. [6]

    Miskovic and R

    O. Miskovic and R. Olea,Topological regularization and self-duality in four-dimensional anti-de Sitter gravity,Phys. Rev. D79(2009) 124020, [0902.2082]

    Pith/arXiv arXiv 2009

  7. [7]

    Miskovic, R

    O. Miskovic, R. Olea and M. Tsoukalas,Renormalized AdS action and Critical Gravity, JHEP08(2014) 108, [1404.5993]

    Pith/arXiv arXiv 2014

  8. [8]

    R. M. Wald,Black hole entropy is the Noether charge,Phys. Rev. D48(1993) R3427–R3431, [gr-qc/9307038]

    Pith/arXiv arXiv 1993

  9. [9]

    Iyer and R

    V. Iyer and R. M. Wald,Some properties of Noether charge and a proposal for dynamical black hole entropy,Phys. Rev. D50(1994) 846–864, [gr-qc/9403028]

    Pith/arXiv arXiv 1994

  10. [10]

    Anastasiou, I

    G. Anastasiou, I. J. Araya and R. Olea,Einstein Gravity from Conformal Gravity in 6D, JHEP01(2021) 134, [2010.15146]

    Pith/arXiv arXiv 2021

  11. [11]

    Boulanger and D

    N. Boulanger and D. Rovere,8D conformal gravity with Einstein sector, and its relation to the Q-curvature,JHEP02(2026) 101, [2511.01368]

    arXiv 2026

  12. [12]

    Anastasiou, I

    G. Anastasiou, I. J. Araya, C. Corral and R. Olea,Noether-Wald charges in six-dimensional Critical Gravity,JHEP07(2021) 156, [2105.02924]

    Pith/arXiv arXiv 2021

  13. [13]

    Anastasiou, I

    G. Anastasiou, I. J. Araya, C. Corral and R. Olea,Conformal Renormalization of topological black holes in AdS 6,JHEP11(2023) 036, [2308.09140]

    Pith/arXiv arXiv 2023

  14. [14]

    de Haro, S

    S. de Haro, S. N. Solodukhin and K. Skenderis,Holographic reconstruction of space-time and renormalization in the AdS / CFT correspondence,Commun. Math. Phys.217(2001) 595–622, [hep-th/0002230]

    Pith/arXiv arXiv 2001

  15. [15]

    C. R. Graham and C. Fefferman,Conformal invariants,Ast´ erisque, Soci´ et´ e Math´ ematique de France131(1985) 95–116

    1985

  16. [16]

    Boulanger and J

    N. Boulanger and J. Erdmenger,A Classification of local Weyl invariants inD= 8,Class. Quant. Grav.21(2004) 4305–4316, [hep-th/0405228]

    Pith/arXiv arXiv 2004

  17. [17]

    Boulanger and J

    N. Boulanger and J. Erdmenger,Corrigendum: A classification of local weyl invariants in d = 8, (2004 class. quantum grav. 21 4305–16),Classical and Quantum Gravity39(jan, 2022) 039501

    2004

  18. [18]

    Boulanger,A Weyl-covariant tensor calculus,J

    N. Boulanger,A Weyl-covariant tensor calculus,J. Math. Phys.46(2005) 053508, [hep-th/0412314]

    Pith/arXiv arXiv 2005

  19. [19]

    C. R. Graham,Volume and area renormalizations for conformally compact Einstein metrics, Rend. Circ. Mat. Palermo S63(2000) 31–42, [math/9909042]

    Pith/arXiv arXiv 2000

  20. [20]

    Ashtekar and A

    A. Ashtekar and A. Magnon,Asymptotically anti-de Sitter space-times,Class. Quant. Grav. 1(1984) L39–L44

    1984

  21. [21]

    Ashtekar and S

    A. Ashtekar and S. Das,Asymptotically Anti-de Sitter space-times: Conserved quantities, Class. Quant. Grav.17(2000) L17–L30, [hep-th/9911230]

    Pith/arXiv arXiv 2000

  22. [22]

    D. P. Jatkar, G. Kofinas, O. Miskovic and R. Olea,Conformal Mass in AdS gravity,Phys. Rev. D89(2014) 124010, [1404.1411]

    Pith/arXiv arXiv 2014

  23. [23]

    Anastasiou, O

    G. Anastasiou, O. Miskovic, R. Olea and I. Papadimitriou,Counterterms, Kounterterms, and the variational problem in AdS gravity,JHEP08(2020) 061, [2003.06425]

    Pith/arXiv arXiv 2020

  24. [24]

    Henningson and K

    M. Henningson and K. Skenderis,The Holographic Weyl anomaly,JHEP07(1998) 023, [hep-th/9806087]. – 19 –

    Pith/arXiv arXiv 1998

  25. [25]

    S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins,Normal forms for tensor polynomials. 1: The Riemann tensor,Class. Quant. Grav.9(1992) 1151–1197

    1992

  26. [26]

    Chen and H

    F.-Y. Chen and H. Lu,Holographic Weyl anomaly in 8D from general higher curvature gravity,Phys. Rev. D111(2025) 086032, [2410.16097]

    Pith/arXiv arXiv 2025

  27. [27]

    J. S. Case, A. Khaitan, Y.-J. Lin, A. J. Tyrrell and W. Yuan,Computing renormalized curvature integrals on Poincar´ e-Einstein manifolds,2404.11319

    Pith/arXiv arXiv

  28. [28]

    A. R. Gover and L. J. Peterson,Conformally invariant powers of the Laplacian, Q-curvature, and tractor calculus,Commun. Math. Phys.235(2003) 339–378, [math-ph/0201030]

    Pith/arXiv arXiv 2003

  29. [29]

    Anastasiou and R

    G. Anastasiou and R. Olea,From conformal to Einstein Gravity,Phys. Rev. D94(2016) 086008, [1608.07826]

    Pith/arXiv arXiv 2016

  30. [30]

    H. L¨ u, Y. Pang and C. N. Pope,Black Holes in Six-dimensional Conformal Gravity,Phys. Rev. D87(2013) 104013, [1301.7083]

    Pith/arXiv arXiv 2013

  31. [31]

    Anastasiou, I

    G. Anastasiou, I. J. Araya, C. Arias and R. Olea,Einstein-AdS action, renormalized volume/area and holographic R´ enyi entropies,JHEP08(2018) 136, [1806.10708]

    Pith/arXiv arXiv 2018

  32. [32]

    C. R. Graham and K. Hirachi,The ambient obstruction tensor and q-curvature,AdS/CFT correspondence: Einstein metrics and their conformal boundaries8(2005) 59–71

    2005

  33. [33]

    Gover and L

    R. Gover and L. Peterson,The ambient obstruction tensor and the conformal deformation complex,Pacific journal of mathematics226(2006) 309–351

    2006

  34. [34]

    Balasubramanian and P

    V. Balasubramanian and P. Kraus,A Stress tensor for Anti-de Sitter gravity,Commun. Math. Phys.208(1999) 413–428, [hep-th/9902121]

    Pith/arXiv arXiv 1999

  35. [35]

    Albin,Renormalizing Curvature Integrals on Poincare-Einstein Manifolds,Adv

    P. Albin,Renormalizing Curvature Integrals on Poincare-Einstein Manifolds,Adv. Math. 221(2009) 140–169, [math/0504161]

    Pith/arXiv arXiv 2009

  36. [36]

    Chang, J

    A. Chang, J. Qing and P. Yang,On the renormalized volumes for conformally compact Einstein manifolds,math/0512376

    Pith/arXiv arXiv

  37. [37]

    Anastasiou, I

    G. Anastasiou, I. J. Araya and R. Olea,Energy functionals from Conformal Gravity,JHEP 10(2022) 123, [2209.02006]

    Pith/arXiv arXiv 2022

  38. [38]

    Anastasiou, I

    G. Anastasiou, I. J. Araya, P. Bueno, J. Moreno, R. Olea and A. Vilar Lopez, Higher-dimensional Willmore energy as holographic entanglement entropy,JHEP01(2025) 081, [2409.19485]

    Pith/arXiv arXiv 2025

  39. [39]

    Lewkowycz and J

    A. Lewkowycz and J. Maldacena,Generalized gravitational entropy,JHEP08(2013) 090, [1304.4926]

    Pith/arXiv arXiv 2013

  40. [40]

    C. R. Graham and E. Witten,Conformal anomaly of submanifold observables in AdS / CFT correspondence,Nucl. Phys. B546(1999) 52–64, [hep-th/9901021]

    Pith/arXiv arXiv 1999

  41. [41]

    Anastasiou, M

    G. Anastasiou, M. Bravo and R. Olea,Asymptotic analysis of energy functionals in anti-de Sitter spacetimes,JHEP09(2025) 093, [2504.06382]

    Pith/arXiv arXiv 2025

  42. [42]

    Boulanger,Algebraic Classification of Weyl Anomalies in Arbitrary Dimensions,Phys

    N. Boulanger,Algebraic Classification of Weyl Anomalies in Arbitrary Dimensions,Phys. Rev. Lett.98(2007) 261302, [0706.0340]

    Pith/arXiv arXiv 2007

  43. [43]

    Boulanger,General solutions of the Wess-Zumino consistency condition for the Weyl anomalies,JHEP07(2007) 069, [0704.2472]

    N. Boulanger,General solutions of the Wess-Zumino consistency condition for the Weyl anomalies,JHEP07(2007) 069, [0704.2472]. – 20 –

    Pith/arXiv arXiv 2007