Scalar quasinormal modes of Kerr--AdS$_{\bf 7}$ via accessory parameter expansions

Juli\'an Barrag\'an Amado

REVIEW

Scalar quasinormal modes of Kerr--AdS$_{\bf 7}$ via accessory parameter expansions

Julián Barragán Amado

General Relativity and Quantum Cosmology · Submitted 2 Sep 2026 · 47 references

Pith's verdict: Technically sound extension of the accessory parameter method to Kerr–AdS7 that delivers new angular eigenvalues, but the QNM computation stalls at the non-rotating real part due to a disclosed order-mixing problem.

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arXiv ID2609.03037 (version 1)

Pith IDDK5JRF7E, Pith's permanent record of this paper and its review

CitationJuli\'an Barrag\'an Amado. "Scalar quasinormal modes of Kerr--AdS$_{\bf 7}$ via accessory parameter expansions." arXiv:2609.03037 (2026). https://pith.science/paper/DK5JRF7E

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Claim paper

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Editor's note

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This paper is a careful, honest piece of work. The new thing is the application of Hill-determinant accessory parameter expansions to scalar perturbations of Kerr–AdS7, whose separated ODEs have five regular singular points. The angular eigenvalues in the slowly rotating limit (Eq. 3.10) are genuinely new and appear correctly derived. The central matching claim—that these expansions coincide with the instanton part of the Nekrasov–Shatashvili function for the SU(2)×SU(2) quiver—is supported by the algebra and consistent with the literature (Refs. 34, 35). The paper earns credit for spelling out the explicit coefficients in the appendices and for flagging its own limitations clearly.

The soft spot is the QNM part. Section 3.2 explains that the Floquet exponent σ cannot be determined order by order because |t| ~ O(1) causes higher-order terms in the accessory expansions to mix with lower orders when re-expressed in black-hole parameters. The paper openly states this. Consequently, the QNM output is restricted to the real part in the non-rotating limit (recovering known Schwarzschild–AdS7 results), and rotation corrections are absent. This is a real limitation, but it is disclosed and it does not undermine the angular eigenvalue results or the matching claim. The paper’s suggestion that rotation corrections first appear at sixth order is plausible but not demonstrated.

Who is this for? Researchers working on the Heun-gravity/ODE-gauge correspondence, higher-dimensional rotating black holes, and thermal correlators from AdS/CFT. The math is sound, the bibliography is relevant, and the presentation is clear. A serious referee should spend time on it; the main thing to check would be the consistency of the angular eigenvalue expansions with symmetry limits and the algebra for the accessory parameter coefficients.

I would not cite this in my own work in the next 12 months (not my subfield), but I would bring it to a reading group focused on black hole perturbation theory or integrability. Yes, it deserves peer review. The authors have produced a respectable, self-aware contribution.

Referee report

Significance

If the results hold, this work provides the first analytic treatment of scalar perturbations in a seven-dimensional rotating black hole with three independent rotation parameters, extending the 'Heun–gravity correspondence' to Fuchsian equations with five regular singular points. The matching of the accessory parameter expansions to the NS instanton function reinforces the AGT correspondence beyond the Heun case. The explicit perturbative angular eigenvalues and the QNM real-part formula are new physical results that could serve as input for holographic computations in the dual $\mathcal{N}=(2,0)$ six-dimensional SCFT. The paper is mathematically careful, openly discusses its limitations (order mixing in $\sigma$, absence of rotation corrections at low order), and provides a self-contained derivation of the accessory parameter expansions via the Hill determinant.

Major objections (2)

[§3.1, Eq. (3.17) and Eq. (3.20)] The derivation of the angular quantization conditions is presented in a highly compressed form. The paper states that enforcing $A_{\rm irr}^{(\sigma)}=0$ yields $\sigma = (|m_1|+|m_2|+2j+1)/2$ and similarly for $\rho$, but the connection coefficients $A_{\rm irr}^{(\sigma)}$ and $A_{\rm irr}^{(\rho)}$ are not explicitly computed in the text; the calculation is merely called 'straightforward'. Given that these conditions are load-bearing for the angular eigenvalue results, the derivation should be expanded or a clear reference to a published derivation of the same structure should be provided (e.g. [43] treats a similar problem for four singular points; the extension to five points is not obvious). Without this, the reader cannot verify that the monodromy argument is correctly applied to the specific ODE with five singular points.
[§3.2, discussion near Eq. (3.31)] The order-mixing problem for the Floquet exponent $\sigma$ is clearly stated, but its consequences for the QNM computation are underappreciated. Because $|t|\sim O(1)$, the expansion of $\sigma$ cannot be determined order by order, and the paper restricts the QNM output to the real part in the non-rotating limit. The claim that 'the first corrections due to rotation appear at sixth total order' (two paragraphs after Eq. (3.47)) is based on the behaviour of $\rho$ alone; since $\sigma$ is undetermined at the relevant orders, this statement is speculative. The paper should explicitly mark this claim as a conjecture or, at a minimum, note that it relies on the assumption that the $\sigma$ expansion does not introduce rotation corrections at lower orders, which remains unproven.

Minor objections (5)

[§2.2.1, Eq. (2.20b)] The definition of $t$ and $q$ in the radial Möbius transformation should be checked for consistency with the subsequent analysis. In particular, the regime $0<|t|<1<|q|<\infty$ used in §3.2 is derived from Eq. (3.27); it would be helpful to state explicitly that this regime corresponds to the small black hole limit, and to note that $t$ is not small in that regime, which is the root of the order-mixing problem.
[§3.1, below Eq. (3.23)] The paper introduces the integer $\ell$ and states $\ell=2(j+k)+|m_1|+|m_2|+|m_3|$ with $j,k\in\mathbb{Z}$. This relation should be derived from the quantization conditions (3.17) and (3.20) rather than stated without comment. Also, the range of the integers $j,k$ (non-negative? any sign?) should be clarified.
[Appendix A.1] The second-order coefficients $\epsilon_{1,1}$, $\epsilon_{2,0}$, etc. are given in full but not used in the main text. The statement 'Further coefficients are available upon request' is reasonable but would be more helpful if the source code or a link to a Mathematica notebook were provided as supplementary material.
[§2.2.1, Eq. (2.21a)] The characteristic exponents at $z=q$ are $\rho^\pm_q = \frac12(3\pm\sqrt{9+\mu^2})$. The notation $\rho$ used here collides with the Floquet exponent $\rho$ introduced later; while the context distinguishes them, a name change (e.g., $\eta_q$) would avoid confusion.
[§3.2, Eq. (3.29)] The expansion of $a_t$ contains a term $O(u_3\gamma^2, u_3^2\gamma, u_3^3, \gamma^3)$, but the subsequent analysis keeps only the leading order. It would be helpful to state explicitly which terms are retained in the perturbative expansions that lead to Eq. (3.47).

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The paper

The paper claims

Accessory expansions match instanton sums for Kerr-AdS7 quasinormal modes.

The reading

This paper studies scalar perturbations of a seven-dimensional Kerr-AdS black hole with three rotation parameters. It shows that for the resulting Fuchsian ordinary differential equations with five regular singular points, the accessory parameter expansions obtained via the Hill determinant method are exactly the same as those computed from the instanton part of the Nekrasov-Shatashvili function of four-dimensional N=2 quiver gauge theories. Using this equivalence, the paper derives perturbative expressions for the angular eigenvalues in the slowly rotating limit and for the real part of the quasinormal mode frequencies in the small black hole limit. If correct, this work extends the gauge-gravity correspondence to higher-dimensional rotating black holes and provides a new analytic tool for computing black hole response in AdS7 backgrounds.

What carries the argument

The central object is the accessory parameter expansion for a Fuchsian ODE in normal form with five singular points at 0, t, q, 1, ∞. The expansions of the two accessory parameters Et and Eq are obtained by imposing that the Hill determinant of a Floquet-type series solution vanishes, yielding double series in the conformal moduli t and q parametrized by the Floquet exponents σ and ρ. These expansions are identified with those coming from the Matone relations of the Nekrasov-Shatashvili function, which relate the accessory parameters to the instanton partition function. This identification carries the argument because it allows the black hole parameters—rotation parameters αi, horizon radius u3, mass parameter µ, and frequency ω—to be expressed in terms of gauge theory data, and the connection problem for the ODE to be solved via the known connection coefficients for semiclassical conformal blocks.

Extended reading notes

Core claim

The central claim is that the accessory parameter expansions for the radial and angular Fuchsian ODEs with five regular singular points, derived via the Hill determinant method, coincide identically with the expansions obtained from the instanton part of the Nekrasov-Shatashvili function of a four-dimensional N=2 superconformal linear quiver gauge theory with gauge group SU(2)×SU(2). By matching these expansions, the author identifies the Floquet exponents σ and ρ with the vevs of the gauge theory scalars, enabling the use of connection formulas from the AGT correspondence to impose boundary conditions. This yields perturbative analytic expressions for the angular separation constants and the real part of the quasinormal mode frequencies, which in the non-rotating limit reduce to known results for Schwarzschild-AdS7.

Load-bearing premise

The double series expansions for the accessory parameters Et and Eq, truncated at low order, correctly capture the leading physical contributions in the perturbative regimes, yet the expansion for one of the Floquet exponents (σ) suffers from order mixing that prevents consistent perturbative determination, limiting the frequency computation to the real part in the non-rotating limit.

Editorial extensions

If this is right

The real part of the quasinormal mode frequencies for Kerr-AdS7 scalar perturbations can now be computed analytically as a series in the small horizon radius and rotation parameters, providing a check for numerical or semiclassical methods.
The equivalence between Hill determinant expansions and Nekrasov-Shatashvili instanton sums strengthens the conjecture that higher-dimensional black hole perturbation theory is dual to lower-dimensional superconformal gauge theories, extending the AdS/CFT dictionary.
The angular eigenvalue expansions obtained in the slowly rotating limit allow one to compute separation constants for arbitrary azimuthal numbers, which are needed for constructing mode solutions and computing Hawking radiation or greybody factors.
The methodology—solving Fuchsian ODEs with five singular points via accessory parameter expansions—can be applied to other higher-dimensional Kerr-NUT-AdS backgrounds with multiple rotation parameters.

Reading between the lines

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

A natural extension would be to compute the imaginary part of the quasinomal frequencies, which requires determinging the Floquert exponent σ to higher order and resolving the order-mixing issue that the paper identifies. A concrete approach may involve resummation or numerical evaluation of the Nekrasov-Shatashvili function.
The agreement between Hill determinant and Nekrasov-Shatashvili expansions suggests that the accessory paramters might be expressible as tau functions of isomonodromic deformations, providing an alternative route to exact solutions.
The paper's observation that rotation corrections first appear at sixth order in the small-parameter expansion indicates that for sufficiently small rotation, the QNM spectrum of Kerr-AdS7 is essentially that of Schwarzschild-AdS7, which could be tested with numerical relativity simulations.

Original abstract

We study scalar perturbations of a seven-dimensional Kerr--AdS black hole with three independent rotation parameters. In this geometry, the Klein--Gordon equation separates into one radial and two angular second-order linear ordinary differential equations, each with five regular singular points. We apply the Hill determinant method to compute analytic expansions of the accessory parameters in two different parameter regimes. The resulting accessory parameter expansions coincide with those obtained from the instanton part of the Nekrasov--Shatashvili function of four-dimensional $\mathcal{N}=2$ quiver gauge theories. These expansions are then used to derive perturbative expressions for the angular eigenvalues in the slowly rotating limit and for the quasinormal mode frequencies in the small black hole limit.

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, 2026-09-23: 1 request, 4,278 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.20 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.86 · 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 study scalar perturbations of a seven-dimensional Kerr--AdS black hole with three independent rotation parameters. In this geometry, the Klein--Gordon equation separates into one radial and two angular second-order linear ordinary differential equations, each with five regular singular points. We apply the Hill determinant method to compute analytic expansions of the accessory parameters in two different parameter regimes. The resulting accessory parameter expansions coincide with those obtained from the instanton part of the Nekrasov--Shatashvili function of four-dimensional $\mathcal{N}=2$ quiver gauge theories. These expansions are then used to derive perturbative expressions for the angular eigenvalues in the slowly rotating limit and for the quasinormal mode frequencies in the small black hole limit.

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 study scalar perturbations of a seven-dimensional Kerr--AdS black hole with three independent rotation parameters.	contribution 0.87	0.91	0.03	asserted 0.98
s1	In this geometry, the Klein--Gordon equation separates into one radial and two angular second-order linear ordinary differential equations, each with five regular singular points.	background 0.91	0.27	0.03	asserted 1.00
s2	We apply the Hill determinant method to compute analytic expansions of the accessory parameters in two different parameter regimes.	method 0.98	0.96	0.06	asserted 1.00
s3	The resulting accessory parameter expansions coincide with those obtained from the instanton part of the Nekrasov--Shatashvili function of four-dimensional $\mathcal{N}=2$ quiver gauge theories.	result 0.97	0.76	0.40	asserted 1.00
s4	These expansions are then used to derive perturbative expressions for the angular eigenvalues in the slowly rotating limit and for the quasinormal mode frequencies in the small black hole limit.	result 0.80	0.94	0.39	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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sources read

arXiv listing abstract v1

not examined

full text
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P abstract

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S sections
T body sentences
C citations
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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.

[1] [43]

F. Novaes, C. Marinho, M. Lencsés and M. Casals,Kerr-de Sitter Quasinormal Modes via Accessory Parameter Expansion,JHEP05(2019) 033 [1811.11912]

View on Pith Pith review arXiv 2019

[2] [1]

R. Emparan and H.S. Reall,Black Holes in Higher Dimensions,Living Rev. Rel.11(2008) 6 [0801.3471]

View on Pith Pith review arXiv 2008

[3] [2]

R. Emparan and R.C. Myers,Instability of ultra-spinning black holes,JHEP09(2003) 025 [hep-th/0308056]

View on Pith Pith review arXiv 2003

[4] [3]

R.C. Myers and M.J. Perry,Black Holes in Higher Dimensional Space-Times,Annals Phys. 172(1986) 304

View on Pith 1986

[5] [4]

S.W. Hawking, C.J. Hunter and M. Taylor,Rotation and the AdS / CFT correspondence, Phys. Rev. D59(1999) 064005 [hep-th/9811056]

View on Pith Pith review arXiv 1999

[6] [5]

G.W. Gibbons, H. Lu, D.N. Page and C.N. Pope,The General Kerr-de Sitter metrics in all dimensions,J. Geom. Phys.53(2005) 49 [hep-th/0404008]

View on Pith Pith review arXiv 2005

[7] [6]

G.W. Gibbons, H. Lu, D.N. Page and C.N. Pope,Rotating black holes in higher dimensions with a cosmological constant,Phys. Rev. Lett.93(2004) 171102 [hep-th/0409155]

View on Pith Pith review arXiv 2004

[8] [7]

W. Chen, H. Lu and C.N. Pope,General Kerr-NUT-AdS metrics in all dimensions,Class. Quant. Grav.23(2006) 5323 [hep-th/0604125]

View on Pith Pith review arXiv 2006

[9] [8]

D.N. Page, D. Kubiznak, M. Vasudevan and P. Krtous,Complete integrability of geodesic motion in general Kerr-NUT-AdS spacetimes,Phys. Rev. Lett.98(2007) 061102 [hep-th/0611083]

View on Pith Pith review arXiv 2007

[10] [9]

V.P. Frolov, P. Krtous and D. Kubiznak,Separability of Hamilton-Jacobi and Klein-Gordon Equations in General Kerr-NUT-AdS Spacetimes,JHEP02(2007) 005 [hep-th/0611245]

View on Pith Pith review arXiv 2007

[11] [10]

T. Oota and Y. Yasui,Separability of Dirac equation in higher dimensional Kerr-NUT-de Sitter spacetime,Phys. Lett. B659(2008) 688 [0711.0078]

View on Pith Pith review arXiv 2008

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