Anatomy of AdS black hole on-shell actions

Junho Hong; Sourav Roychowdhury

REVIEW

Anatomy of AdS black hole on-shell actions

Junho Hong, Sourav Roychowdhury

High Energy Physics - Theory · Submitted 10 Nov 2026 · 140 references

Pith's reading: Holographic renormalization confirms quantum statistical relation for general rotating charged AdS black holes in five and six dimensions

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arXiv ID2611.14688 (version 1)

Pith IDKAK3OQ5S, Pith's permanent record of this paper and its review

CitationJunho Hong, Sourav Roychowdhury. "Anatomy of AdS black hole on-shell actions." arXiv:2611.14688 (2026). https://pith.science/paper/KAK3OQ5S

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Major objections (4)

[§2.1, Eq. (2.60) and surrounding text] The paper claims that the holographically renormalized on-shell action satisfies the QSR up to a Casimir term, and that the Casimir term is removed by a finite counterterm with ζ = -1/(96g^3). However, the Casimir term βE_AdS5 in Eq. (2.60) is the on-shell action of global EAdS5 evaluated in the minimal scheme ζ=0. The paper does not explicitly verify that the finite counterterm S_CT-2 in Eq. (2.25) with ζ = -1/(96g^3) indeed cancels this Casimir term for all values of the black hole parameters. The expression for S_CT-2 in Eq. (2.48) is given as 9πβgζ/G_N, which is a constant independent of the black hole parameters. The Casimir term βE_AdS5 = 3πβ/(32g^2 G_N) is also a constant. Setting ζ = -1/(96g^3) gives S_CT-2 = -3πβ/(32g^2 G_N) = -βE_AdS5, so the cancellation is indeed parameter-independent. This is fine. However, the paper should clarify that this cancellation is a scheme choice and that the Casimir term is not 'removed' in a physical sense; it is absorbed into the definition of the finite counterterm. The statement in the abstract and conclusions that the QSR is satisfied 'up to a parameter-independent Casimir term' is accurate, but the paper should be more explicit that the scheme ζ=0 is the minimal scheme and that the scheme ζ=-1/(96g^3) is a specific choice that makes the on-shell action match the QSR exactly. This is already clear in the text, but I raise it as a minor clarification point. Actually, I think this is fine; the paper is clear. I will not make this a major comment.
[§3.1, Eq. (3.9) and §3.4, Eq. (3.34a)] The paper identifies the mismatch between holographic renormalization and background subtraction in five dimensions as consisting of three terms: the Casimir term, a scalar-dependent term ΔS_CT-0, and a geometric term involving J^(2)_00. The scalar term ΔS_CT-0 is given in Eq. (3.10) and vanishes when the three charges are equal. The paper then shows in Eq. (3.16) that using a different radial coordinate (constant ˆr) removes the geometric term but not the scalar term. This demonstrates that background subtraction is ambiguous and depends on the choice of cutoff surface. However, the paper does not provide a proof that no choice of radial coordinate can simultaneously remove both the scalar and geometric terms for general charges. The claim that 'no systematic criterion for the radial cutoff of background subtraction is available' is supported by the examples, but the paper could strengthen this by arguing that the scalar term arises from the finite part of the superpotential counterterm, which is not captured by any background subtraction because the reference has trivial scalars. This is already implicit in the analysis of §3.4. I think this is a valid point, but the paper could be more explicit about the impossibility of a universal background subtraction for non-trivial scalars. I will not raise this as a major comment because the paper is already quite clear. The authors acknowledge this: “It does not provide a criterion for background subtraction itself, namely a necessary and sufficient condition under which background subtraction, with no counterterm added by hand, agrees with holographic renormalization up to the Casimir term and thereby with the on-shell action determined via the QSR, the quantity to be compared with the dual field theory.”
[§2.2, Eq. (2.94)] The paper claims that the holographically renormalized on-shell action for the AdS6 black hole satisfies the QSR exactly, with no Casimir offset. This is verified by using the horizon condition to eliminate m. The verification is presented in a single sentence: 'Using the horizon condition V(r+) = 2mΥ(r+) to eliminate m, we verify that the renormalized on-shell action (2.88) satisfies the QSR exactly.' The paper does not show the steps of this verification, nor does it provide an explicit expression for I^(6) in terms of the thermodynamic quantities. Given the complexity of the expressions, a reader would benefit from at least a sketch of the verification. I consider this a load-bearing point because the central claim for AdS6 rests on this verification. The paper should either provide the explicit algebraic verification in an appendix or cite a source where it is done. Since the paper states 'we verify', it should be possible for the reader to follow the verification. I recommend that the authors include the verification steps, perhaps in an appendix, or at least provide a more detailed outline.
[§4, paragraph on higher-derivative corrections] The paper discusses higher-derivative corrections and states that 'what remains to be confirmed is that the asymptotically global AdS frame renders the finite counterterms and the Casimir term independent of the parameters with these terms included, as it does at two derivatives.' This is a forward-looking statement and not a claim of the paper. However, the paper also cites several works that have already studied higher-derivative corrections using background subtraction or holographic renormalization. The paper should clarify whether the asymptotically global AdS frame has been used in those works, and if not, whether the results of those works are affected by the frame choice. This is a minor point, but it would strengthen the discussion.

Minor objections (10)

[§1, Introduction] The introduction mentions 'the recent holographic renormalization of the AdS7 black hole [60]' as a motivation. It would be helpful to explicitly state that the present paper extends the analysis of [60] to five and six dimensions, which is clear from the abstract but could be stated more directly.
[§2.1, Eq. (2.11)] The expression for Δ_r is very long and involves s_A, s_B, s_C defined in (2.12). It would be helpful to note that Δ_r reduces to the known expression for the equal-charge case (e.g., [83]) when s_1 = s_2 = s_3.
[§2.1, Eq. (2.28)] The coordinate transformation (2.28) is central to the paper. The paper states that it is from [84,85]. It would be helpful to explicitly note that this transformation is valid for any dimension d and that it brings the metric to the asymptotically global AdS form (2.29). This is already clear, but a brief comment would help.
[§2.1, Eq. (2.40)] The expansion of the extrinsic curvature K is given with underbraces indicating the order in ˜r_∞. The notation '/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright' appears to be a LaTeX artifact that did not render correctly. The paper should ensure that the underbraces are properly displayed or replaced with clear text.
[§2.2, Eq. (2.70)] The metric (2.70) is very long and involves many auxiliary functions. It would be helpful to include a reference to the original derivation [98] and to note that the metric is presented in a form that separates the global AdS part from the mass-dependent terms, which is used in the subsequent analysis.
[§3.1, Eq. (3.12)] The definition of ˆr^2 = r^2 + 2ms^2 for the equal-charge case is given. For the general case, the paper defines ˆr^2 = r^2 + (2/3) m s_A in Eq. (3.14). It would be helpful to explain the motivation for this definition, perhaps as the radial coordinate that absorbs the conformal factor (H1H2H3)^{1/3} at large ˆr.
[§3.4, Eq. (3.35)] The exact relation (3.35) between holographic renormalization and background subtraction is a key result. The paper should note that this relation holds for any background, not just the ones considered, as long as both methods are implemented in the same frame. This is implicit but could be stated explicitly.
[§4, Discussion] The discussion of open questions is thorough and honest. The paper acknowledges that holographic renormalization is not yet established for general asymptotically locally AdS black holes and that background subtraction lacks a universal criterion. This is commendable.
[Appendix B.2] The building-block integrals G^{(p)}_{j1 j2} are defined and recurrence relations are given. It would be helpful to note that the integrals are evaluated using the identities (B.8) and that the explicit expressions for G^{(3)}_{20}, G^{(3)}_{02}, and G^{(3)}_{11} are provided. The expressions are quite long; a brief check that they satisfy the recurrence relations would be reassuring.
[General] The paper uses the notation '⋆L' for the Lorentzian Hodge star and '⋆' for the Euclidean one. This is clear, but it would be helpful to remind the reader of the convention in Appendix A, which is already done.

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The paper

The paper claims

Holographic renormalization confirms quantum statistical relation for general rotating charged AdS black holes in five and six dimensions

The reading

This paper establishes that the holographically renormalized Euclidean on-shell action of the most general non-extremal rotating electrically charged AdS black holes in five-dimensional STU gauged supergravity and six-dimensional N=4 SU(2) gauged supergravity satisfies the quantum statistical relation (QSR) of black hole thermodynamics, up to a scheme-dependent Casimir term in odd dimensions and exactly in even dimensions. The key technical move is the use of the asymptotically global AdS frame, a specific radial coordinate in which the cutoff surface is the Einstein static universe $S^{1}$ x $S^{{d-2}}$ for all values of the black hole parameters. This frame makes the counterterms and the Casimir term universal constants, independent of the rotation and charge parameters, so that a single finite counterterm choice removes the Casimir offset in odd dimensions. The work also shows that background subtraction at a constant radial coordinate in the same frame fails to reproduce the QSR, and it identifies the mismatch exactly in terms of the reference action and the finite parts of the holographic counterterms. The results place holographic comparisons of black hole on-shell actions with dual field theory quantities (such as the superconformal index) on a first-principles footing, without relying on the QSR imported from Lorentzian thermodynamics.

What carries the argument

The asymptotically global AdS frame, defined by a coordinate transformation (2.28) from the Boyer-Lindquist-type coordinates to coordinates (τ, ~r, ~μ_i, φ_i) in which the metric asymptotes to global Euclidean AdS_d (2.30) with boundary $S^{1}$ × $S^{{d-2}}$. In this frame the induced metric on the radial cutoff surface is the Einstein static universe for all values of the black hole parameters, making the Casimir energy and the finite counterterms universal constants. The holographic renormalization procedure (2.26) then consists of the bulk on-shell action (2.21)/(2.77), the Gibbons-Hawking-York term (2.22), and the counterterms (2.24) built from the induced metric and the superpotential.

Extended reading notes

Core claim

The paper demonstrates that for the general non-extremal rotating electrically charged AdS5 black hole of the STU model and the AdS6 black hole of N=4 SU(2) gauged supergravity, holographic renormalization performed in the asymptotically global AdS frame yields a renormalized Euclidean on-shell action that satisfies the quantum statistical relation (QSR) of black hole thermodynamics. In five dimensions the relation holds up to a parameter-independent Casimir term that can be removed by a single finite counterterm; in six dimensions it holds exactly with no scheme ambiguity. This is the first direct evaluation of the holographically renormalized on-shell action for these general black holes, confirming that the Euclidean saddle alone suffices for precision holography without importing Lorentzian thermodynamic data via the QSR. The paper also shows that background subtraction in the same frame fails to reproduce the QSR, and it provides an exact decomposition of the mismatch in terms of the renormalized on-shell action of the reference background and the finite parts of the holographic counterterms.

Load-bearing premise

The entire analysis assumes that the asymptotically global AdS frame, defined by the coordinate transformation (2.28), is the correct and universal frame for holographic renormalization of these black holes. The paper acknowledges that holographic renormalization is not yet established for general asymptotically locally AdS black holes, and that the conformally flat boundary $S^{1}$ × $S^{{d-2}}$ is essential for the Casimir term to be a universal constant.

Editorial extensions

If this is right

Holographic comparisons of black hole on-shell actions with the large-N superconformal index — previously reliant on the QSR — are now placed on a first-principles footing, confirmed by direct evaluation of the Euclidean saddle.
The asymptotically global AdS frame provides a canonical radial coordinate for holographic renormalization of rotating charged AdS black holes with conformally flat boundaries, extending the known result from seven dimensions to five and six dimensions.
Background subtraction at constant ~r in the asymptotically global AdS frame is shown to be inadequate for precision holography; a criterion for the cutoff surface remains to be established for this method to be reliable.
The exact identification of the mismatch between holographic renormalization and background subtraction in terms of the reference action and counterterm finite parts provides a diagnostic for future work on background subtraction.
The results confirm that the Euclidean black saddle alone suffices for precision holography, without the need to import Lorentzian thermodynamic data via the QSR.

Reading between the lines

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

The universality of the asymptotically global AdS frame for holographic renormalization suggests that the same frame should be the starting point for extending these results to black holes with non-conformally flat boundaries, such as those with squashed spheres or toroidal boundaries, where the logarithmic counterterms and finite counterterms will depend on the boundary data.
The failure of background subtraction at constant ~r in the asymptotically global AdS frame implies that the Hawking-Horowitz prescription of matching the induced fields on the cutoff surface to 'sufficient order' needs to be made precise; the relative order ~r^{3-d} at which the black hole deviates from the reference may provide a natural criterion.
The parameter-independent Casimir term in odd dimensions, removed by a single finite counterterm, suggests that the scheme ambiguity in odd-dimensional holographic renormalization can be resolved by requiring the QSR to hold, which fixes the finite counterterm uniquely.
The explicit evaluation of the on-shell action for the general AdS6 black hole provides a gravitational check of the entropy function proposal for AdS6 black holes, which was previously inferred from the QSR and the superconformal index.

Original abstract

We revisit the regularized on-shell actions of rotating electrically charged AdS black holes of gauged supergravity via holographic renormalization and background subtraction, with a particular emphasis on the frame in which the radial cutoff is placed. For holographic renormalization, the asymptotically global AdS frame, recently employed in seven dimensions, works equally well in five and six dimensions: the renormalized on-shell action of the AdS$_d$ black holes with $S^1\times S^{d-2}$ conformal boundaries satisfies the quantum statistical relation, up to a parameter-independent Casimir term in odd dimensions, and thereby places the holographic comparisons that so far relied on this relation on a first-principles footing. Background subtraction, in contrast, fails to reproduce this result in the same frame; the mismatch, which we identify exactly in terms of the Casimir term and the counterterms, demands a criterion for the cutoff surface yet to be established. For precision holography, our results settle the choice of frame for holographic renormalization with conformally flat boundaries and delineate the ambiguities that remain in background subtraction and in holographic renormalization for boundaries that are not conformally flat.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-11-13: 1 request, 3,996 input tokens.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.87 · human participants 0.01 · controlled comparison 0.21 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.95 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We revisit the regularized on-shell actions of rotating electrically charged AdS black holes of gauged supergravity via holographic renormalization and background subtraction, with a particular emphasis on the frame in which the radial cutoff is placed. For holographic renormalization, the asymptotically global AdS frame, recently employed in seven dimensions, works equally well in five and six dimensions: the renormalized on-shell action of the AdS$_d$ black holes with $S^1\times S^{d-2}$ conformal boundaries satisfies the quantum statistical relation, up to a parameter-independent Casimir term in odd dimensions, and thereby places the holographic comparisons that so far relied on this relation on a first-principles footing. Background subtraction, in contrast, fails to reproduce this result in the same frame; the mismatch, which we identify exactly in terms of the Casimir term and the counterterms, demands a criterion for the cutoff surface yet to be established. For precision holography, our results settle the choice of frame for holographic renormalization with conformally flat boundaries and delineate the ambiguities that remain in background subtraction and in holographic renormalization for boundaries that are not conformally flat.

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 revisit the regularized on-shell actions of rotating electrically charged AdS black holes of gauged supergravity via holographic renormalization and background subtraction, with a particular emphasis on the frame in which the radial cutoff is placed.	contribution 0.95	0.80	0.06	asserted 0.73
s1	For holographic renormalization, the asymptotically global AdS frame, recently employed in seven dimensions, works equally well in five and six dimensions: the renormalized on-shell action of the AdS$_d$ black holes with $S^1\times S^{d-2}$ conformal boundaries satisfies the quantum statistical relation, up to a parameter-independent Casimir term in odd dimensions, and thereby places the holographic comparisons that so far relied on this relation on a first-principles footing.	result 0.96	0.78	0.36	asserted 1.00
s2	Background subtraction, in contrast, fails to reproduce this result in the same frame; the mismatch, which we identify exactly in terms of the Casimir term and the counterterms, demands a criterion for the cutoff surface yet to be established.	result 0.90	0.90	0.14	asserted 1.00
s3	For precision holography, our results settle the choice of frame for holographic renormalization with conformally flat boundaries and delineate the ambiguities that remain in background subtraction and in holographic renormalization for boundaries that are not conformally flat.	interpretation 0.94	0.88	0.73	asserted 1.00

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

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

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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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