REVIEW 2 major objections 3 minor 2 cited by
A Brief Note on Complex AdS-Schwarzschild Black Holes
T0 review · 2 major / 3 minor · reviewed 2026-08-04 · deepseek-v4-flash
Pith's one-line read Complex black holes do not control low-temperature AdS thermodynamics; contour arguments show they drop out of the partition function.
desk verdict A genuine new puzzle about complex AdS-Schwarzschild saddles dominating by real action at low T in AdS5+, with a sound mini-superspace contour argument that excludes them, but the reduction to one dimension is assumed, so treat the general claim as conditional. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The key machinery is a mini-superspace integral over the black hole horizon radius r+, Z(beta) = integral from 0 to infinity of dr+ exp[S(r+)-beta E(r+)], together with the steepest-descent analysis of that integral. The real positive r+ axis is identified as (half of) the steepest-descent contour of the thermal AdS saddle (r+=0), because the quadratic term in the action has a negative coefficient. Since the defining contour does not pass through the complex saddles, those saddles do not contribute. For the beta<beta_max case, complexifying Newton's constant G and taking a homology average of steepest-decent contours isolates and cancels the small black hole contribution.
What would settle it
Directly compute the gravitational path integral in a saddle-point approximation without the mini-superspace truncation, at a beta slightly above beta_max, and check whether the result contains a contribution whose entropy scales as the area of a complex horizon; if such a contribution appears, the central claim fails.
Extended reading notes
Core claim
For beta greater than beta_max, the two AdS-Schwarzschild saddles become complex: the horizon radius r+ takes two complex values. A naive comparison of on-shell actions suggests these saddles dominate over thermal AdS at low temperature, threatening the AdS/CFT correspondence. The paper's central claim is that these complex black holes contribute nothing to the partition function, no matter how small their imaginary part. The argument reduces the gravitational path integral to a one-dimensional integral over r+ with an action S(r+)-beta E(r+); because the coefficient of the quadratic term is negative, the steepest-descent contour through the r+=0 (thermal AdS) saddle is the positive real r+
Load-bearing premise
The entire argument relies on the reduction of the full gravitational path integral to a one-dimensional integral over r+ with the positive real axis as the contour; if the true contour in field space is different or includes other degrees of freedom, the complex saddles could contribute.
Editorial extensions
If this is right
- Thermal AdS remains the correct low-temperature saddle in asymptotically AdS gravity in five and higher dimensions, despite the existence of complex black holes with smaller real on-shell action.
- The complex Schwarzschild black holes contribute zero to the thermal partition function for beta>beta_max, even when their imaginary part is arbitrarily small and they pass the Kontsevich-Segal criterion.
- The same contour argument applies in any dimension and rules out even suppressed contributions from the complex black holes, contrary to earlier finite-cutoff AdS4 conclusions.
- In the high-temperature phase, the small black hole saddle does not genuinely contribute: its steepest-descent contribution cancels under median-resummation/homology averaging, leaving thermal AdS and the big black hole as the physical saddles.
Reading between the lines
- The key conceptual lesson is that dominance by on-shell action is not sufficient: a saddle contributes only if the defining contour of the path integral can be deformed onto it. This suggests similar contour analyses could decide whether other complex gravitational saddles (e.g., cosmological or rotating ones) are physical.
- The mini-superspace reduction effectively encodes the 'real energy' condition as the choice of contour along the real r+ axis. A testable extension would be to check whether relaxing that condition—allowing complex energies—leads to contributions from the complex black holes, which would change the low-temperature physics.
- The cancellation of the small black hole via homology averaging hints that 'unstable' saddles in gravitational path integrals should generically be interpreted through median resummation; this may clarify analogous false saddles in higher-dimensional black hole thermodynamics.
- One could test the paper's claim against a direct Lorentzian path-integral computation of the low-temperature partition function, which should show no saddle with black-hole-like entropy at large beta, matching the confined phase of the dual gauge theory.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This note addresses the low-temperature (β>β_max) phase of asymptotically AdS gravitational saddle points. The authors observe that complex AdS-Schwarzschild black holes, obtained by analytically continuing the horizon radius r_+(β), have a real on-shell action smaller than that of thermal AdS in AdS5 (and, they claim, higher dimensions), which would naively make them the dominant saddles and contradict the dual CFT. To resolve this, they propose a mini-superspace path integral over the horizon radius, Eq. (9), with the integration contour taken along the positive real r_+ axis (corresponding to real energy). They show that for β>β_max this contour is (half of) the steepest-descent contour of the r_+=0 thermal AdS saddle and does not pass through the complex saddles, so the latter do not contribute. They also analyze the unstable small black hole for β<β_max by complexifying Newton's constant and taking a homology average, concluding that it does not contribute. The authors contrast their argument with the Kontsevich–Segal criterion and with Ref. [9].
Significance. If the reduction to Eq. (9) is accepted, the paper gives a simple and elegant resolution of a genuine puzzle, showing that a naive on-shell-action comparison is misleading and that steepest-descent/thimble methods provide a more refined criterion. The treatment of the small black hole is also a nice illustration of Stokes phenomena and median resummation. However, the central result is conditional on an unproven mini-superspace reduction; the authors acknowledge this limitation in the text. As it stands, the note is a useful toy-model resolution, but it does not establish the claim for the full gravitational path integral.
major comments (2)
- The central claim—that complex AdS-Schwarzschild saddles do not contribute to Z(β) for β>β_max—is established only inside the one-dimensional model (9). The reduction from the full gravitational path integral to this integral is assumed: the text says 'We approximate the full path integral as just a one-dimensional integral over r+' and footnote 5 drops Jacobian factors. In particular, the integration contour in the full field-space path integral is not derived; the 'reasonable assumption' that it corresponds to the positive real r_+ axis is precisely the assumption that excludes the complex saddles. Without a derivation of the mini-superspace reduction from a Lorentzian/Euclidean definition of the path integral, or an argument that the omitted modes do not affect the thimble decomposition, the statement that these complex black holes 'do not actually contribute' is not established for t
- [Abstract and d=4 specialization] The abstract says that the puzzle arises in AdS5 and higher dimensions and the text later states that the non-contribution holds 'in any dimension,' but the explicit calculation is only for d=4 (AdS5). Equations (6)–(9) are specialized to that case. For d>5 the effective action has different powers of r_+, and the sign of the quadratic coefficient and the contour analysis need to be rechecked. As written, the general claim is an extrapolation. Please either provide the d-dimensional version of Eq. (9) or explicitly state that the detailed analysis is for d=4 and only comment on higher dimensions heuristically.
minor comments (3)
- [Fig. 3] The steepest-descent contours are drawn schematically. It would be helpful to indicate the direction of flow and to clarify the role of the endpoint r_+=0, since the defining contour is half of a thimble rather than a complete thimble.
- [Small black hole discussion] The conclusion that the small black hole does not contribute for β<β_max is based on a homology average over Im G>0 and Im G<0. This is a natural choice, but the statement 'does not contribute' depends on this resummation convention; the text partly acknowledges this, but it would be clearer to state that the conclusion is convention-dependent.
- [References and Eq. (9)] Ref. [29] (Marolf) is cited as a source of the mini-superspace approximation, but the connection between that derivation and Eq. (9) is not explained. A sentence clarifying how Eq. (9) follows from that work, or where it deviates from it, would strengthen the presentation.
Circularity Check
No significant circularity; the central result is a steepest-descent statement within an explicitly assumed mini-superspace integral, and the principal limitation is the assumed real-r+ contour, not a definitional or self-citation circularity.
full rationale
The paper's central claim is derived within an explicit model. Eq. (9) is built from the standard Gibbons-Hawking on-shell action and energy (Eqs. 6-8); no parameter is fitted to force the conclusion. Within that model, the statement that complex black holes do not contribute follows from standard steepest-descent logic: the defining contour is chosen to be the positive real r+ axis, this contour is (half of) the steepest descent contour of the r+=0 thermal AdS saddle, and the complex saddles are off this contour. This is a valid conditional result. The important caveat is that the reduction to the one-dimensional integral over real r+ is assumed, not derived from the full gravitational path integral. The paper flags this explicitly: 'We approximate the full path integral as just a one-dimensional integral over r+.' and 'we are making the reasonable assumption that the contour of integration in the r+ plane is along the real axis.' Thus the exclusion of the complex saddles is contingent on that assumed contour; if the true contour were different, the puzzle would remain. This is a scope limitation and an unsupported premise, but it is not a circularity: the paper does not define the contour in terms of the conclusion, nor does it fit a parameter to the desired answer. The self-citations ([31], [36]) are not load-bearing: [31] is cited only as an example where Kontsevich-Segal-excluded geometries are essential, and [36] is an announced future work. The cited steepest-descent fact [30] is a standard external result. Consequently, the circularity burden is low; the appropriate score is 2 for minor non-load-bearing self-citations.
Assumptions & free parameters
assumptions (4)
- ad hoc to paper The full gravitational path integral may be approximated by the one-dimensional integral over r+ in Eq. (9).
- ad hoc to paper The contour of integration in the r+ plane is along the positive real axis, corresponding to real energy.
- standard math If the defining contour is the steepest descent contour of one saddle and contains no other saddles, only that saddle contributes.
- standard math The on-shell action and energy expressions (7)-(8) for the black hole saddles are correct after holographic renormalization.
Cite this review
Pith. "Pith review of A Brief Note on Complex AdS-Schwarzschild Black Holes." pith.science (2026). https://pith.science/paper/ICI2F65P
@misc{pith2026250908883,
author = {Pith},
title = {Pith review of: A Brief Note on Complex AdS-Schwarzschild Black Holes},
year = {2026},
howpublished = {\url{https://pith.science/paper/ICI2F65P}},
note = {Machine review of arXiv:2509.08883}
}
abstract
In the context of thermodynamics of asymptotically anti-de Sitter spaces, it is often stated that at very low temperatures, there is only one saddle point available-namely, thermal AdS-and hence this sole saddle dictates the low-temperature behavior. However, AdS-Schwarzschild black holes continue to exist at low temperatures as complex saddle points. We point out that the real part of the on-shell action of these complex black holes is smaller than that of thermal AdS at the lowest temperatures, in AdS$_5$ and higher dimensions. So, na\"ively, they should be the "dominant" saddles. This raises a puzzle: if these complex black holes were indeed the relevant saddle points, the physics of the bulk and that of the dual gauge theory would completely disagree at low temperatures. Using a mini-superspace approximation and contour arguments, we argue that these complex black holes do not actually contribute to the gravitational path integral, regardless of the value of their on-shell action. So the standard conclusion that thermal AdS is the correct saddle at the lowest temperatures continues to hold. We also comment on two related matters: whether the Kontsevich-Segal criterion is useful in this setting, and whether the unstable small black hole contributes to the path integral in the high-temperature phase.
Figures
Figures from the paper (1 more)
Forward citations
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Reference graph
Works this paper leans on
-
[9]
Lessons for quantum cosmology from anti–de Sitter black holes,
Alice Di Tucci, Michal P. Heller, and Jean-Luc Lehners, “Lessons for quantum cosmology from anti–de Sitter black holes,” Phys. Rev. D102, 086011 (2020), arXiv:2007.04872 [hep-th]
arXiv 2020
-
[1]
The LargeNlimit of superconformal field theories and supergravity,
Juan Martin Maldacena, “The LargeNlimit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys.2, 231–252 (1998), arXiv:hep-th/9711200
arXiv 1998
-
[2]
Gauge theory correlators from noncritical string theory,
S. S. Gubser, Igor R. Klebanov, and Alexander M. Polyakov, “Gauge theory correlators from noncritical string theory,” Phys. Lett. B428, 105–114 (1998), arXiv:hep-th/9802109
arXiv 1998
-
[3]
Anti de Sitter space and holography,
Edward Witten, “Anti de Sitter space and holography,” Adv. Theor. Math. Phys.2, 253–291 (1998), arXiv:hep-th/9802150
arXiv 1998
-
[4]
Thermodynamics of Black Holes in anti-De Sitter Space,
S. W. Hawking and Don N. Page, “Thermodynamics of Black Holes in anti-De Sitter Space,” Commun. Math. Phys.87, 577 (1983)
1983
-
[5]
Anti-deSitterspace,thermalphasetransition,andconfinementingaugetheories,
EdwardWitten,“Anti-deSitterspace,thermalphasetransition,andconfinementingaugetheories,” Adv. Theor. Math. Phys.2, 505–532 (1998), arXiv:hep-th/9803131
arXiv 1998
-
[6]
The Hagedorn - deconfinement phase transition in weakly coupled large N gauge theories,
Ofer Aharony, Joseph Marsano, Shiraz Minwalla, Kyriakos Papadodimas, and Mark Van Raams- donk, “The Hagedorn - deconfinement phase transition in weakly coupled large N gauge theories,” Adv. Theor. Math. Phys.8, 603–696 (2004), arXiv:hep-th/0310285
arXiv 2004
-
[7]
ActionIntegralsandPartitionFunctionsinQuantumGravity,
G.W.GibbonsandS.W.Hawking,“ActionIntegralsandPartitionFunctionsinQuantumGravity,” Phys. Rev. D15, 2752–2756 (1977)
1977
Show all 37 references
-
[8]
Lecture notes on holographic renormalization,
Kostas Skenderis, “Lecture notes on holographic renormalization,” Class. Quant. Grav.19, 5849– 5876 (2002), arXiv:hep-th/0209067
2002 arXiv
-
[10]
Comments on the no boundary wavefunction and slow roll inflation,
Juan Maldacena, “Comments on the no boundary wavefunction and slow roll inflation,” (2024), arXiv:2403.10510 [hep-th]
2024 arXiv
-
[11]
The no boundary density matrix,
Victor Ivo, Yue-Zhou Li, and Juan Maldacena, “The no boundary density matrix,” JHEP02, 124 (2025), arXiv:2409.14218 [hep-th]
2025 arXiv
-
[12]
Lorentzian Quantum Cosmology,
Job Feldbrugge, Jean-Luc Lehners, and Neil Turok, “Lorentzian Quantum Cosmology,” Phys. Rev. D95, 103508 (2017), arXiv:1703.02076 [hep-th] 8
2017 arXiv
-
[13]
Uses of complex metrics in cosmology,
Caroline Jonas, Jean-Luc Lehners, and Jerome Quintin, “Uses of complex metrics in cosmology,” JHEP08, 284 (2022), arXiv:2205.15332 [hep-th]
2022 arXiv
-
[14]
Holographic No-Boundary Measure,
Thomas Hertog and James Hartle, “Holographic No-Boundary Measure,” JHEP05, 095 (2012), arXiv:1111.6090 [hep-th]
2012 arXiv
-
[15]
Bra-ket wormholes in gravitationally pre- pared states,
Yiming Chen, Victor Gorbenko, and Juan Maldacena, “Bra-ket wormholes in gravitationally pre- pared states,” JHEP02, 009 (2021), arXiv:2007.16091 [hep-th]
2021 arXiv
-
[16]
A semiclassical ramp in SYK and in gravity,
Phil Saad, Stephen H. Shenker, and Douglas Stanford, “A semiclassical ramp in SYK and in gravity,” (2018), arXiv:1806.06840 [hep-th]
2018 arXiv
-
[17]
Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity,
Phil Saad, “Late Time Correlation Functions, Baby Universes, and ETH in JT Gravity,” (2019), arXiv:1910.10311 [hep-th]
2019 arXiv
-
[18]
More quantum noise from wormholes,
Douglas Stanford, “More quantum noise from wormholes,” (2020), arXiv:2008.08570 [hep-th]
2020 arXiv
-
[19]
Comments on the double cone wormhole,
Yiming Chen, Victor Ivo, and Juan Maldacena, “Comments on the double cone wormhole,” JHEP 04, 124 (2024), arXiv:2310.11617 [hep-th]
2024 arXiv
-
[20]
Real-time gauge/gravity duality,
Kostas Skenderis and Balt C. van Rees, “Real-time gauge/gravity duality,” Phys. Rev. Lett.101, 081601 (2008), arXiv:0805.0150 [hep-th]
2008 arXiv
-
[21]
A prescription for holographic Schwinger-Keldysh contour in non-equilibrium systems,
Paolo Glorioso, Michael Crossley, and Hong Liu, “A prescription for holographic Schwinger-Keldysh contour in non-equilibrium systems,” (2018), arXiv:1812.08785 [hep-th]
2018 arXiv
-
[22]
Open quantum systems and Schwinger-Keldysh holograms,
Chandan Jana, R. Loganayagam, and Mukund Rangamani, “Open quantum systems and Schwinger-Keldysh holograms,” JHEP07, 242 (2020), arXiv:2004.02888 [hep-th]
2020 arXiv
-
[23]
An exterior EFT for Hawking radiation,
R. Loganayagam and Godwin Martin, “An exterior EFT for Hawking radiation,” JHEP06, 184 (2025), arXiv:2403.10654 [hep-th]
2025 arXiv
-
[24]
Wick Rotation and the Positivity of Energy in Quantum Field Theory,
Maxim Kontsevich and Graeme Segal, “Wick Rotation and the Positivity of Energy in Quantum Field Theory,” Quart. J. Math. Oxford Ser.72, 673–699 (2021), arXiv:2105.10161 [hep-th]
2021 arXiv
-
[25]
A Note On Complex Spacetime Metrics,
Edward Witten, “A Note On Complex Spacetime Metrics,” (2021), arXiv:2111.06514 [hep-th]
2021 arXiv
-
[26]
Microcanonical Path Integrals and the Holography of small Black Hole Interiors,
Donald Marolf, “Microcanonical Path Integrals and the Holography of small Black Hole Interiors,” JHEP09, 114 (2018), arXiv:1808.00394 [hep-th]
2018 arXiv
-
[27]
Flat entanglement spectra in fixed-area states of quantum gravity,
Xi Dong, Daniel Harlow, and Donald Marolf, “Flat entanglement spectra in fixed-area states of quantum gravity,” JHEP10, 240 (2019), arXiv:1811.05382 [hep-th]
2019 arXiv
-
[28]
Holographic Renyi Entropy from Quantum Error Correction,
Chris Akers and Pratik Rath, “Holographic Renyi Entropy from Quantum Error Correction,” JHEP 05, 052 (2019), arXiv:1811.05171 [hep-th] 9
2019 arXiv
-
[29]
Gravitational thermodynamics without the conformal factor problem: parti- tion functions and Euclidean saddles from Lorentzian path integrals,
Donald Marolf, “Gravitational thermodynamics without the conformal factor problem: parti- tion functions and Euclidean saddles from Lorentzian path integrals,” JHEP07, 108 (2022), arXiv:2203.07421 [hep-th]
2022 arXiv
-
[30]
Analytic Continuation Of Chern-Simons Theory,
Edward Witten, “Analytic Continuation Of Chern-Simons Theory,” AMS/IP Stud. Adv. Math.50, 347–446 (2011), arXiv:1001.2933 [hep-th]
2011 arXiv
-
[31]
Sphere and disk partition functions in Liouville and in matrix integrals,
Raghu Mahajan, Douglas Stanford, and Cynthia Yan, “Sphere and disk partition functions in Liouville and in matrix integrals,” JHEP07, 132 (2022), arXiv:2107.01172 [hep-th]
2022 arXiv
-
[32]
Two dimensional Nearly de Sitter grav- ity,
Juan Maldacena, Gustavo J. Turiaci, and Zhenbin Yang, “Two dimensional Nearly de Sitter grav- ity,” JHEP01, 139 (2021), arXiv:1904.01911 [hep-th]
2021 arXiv
-
[33]
Real observers solving imaginary problems,
Juan Maldacena, “Real observers solving imaginary problems,” (2024), arXiv:2412.14014 [hep-th]
2024
-
[34]
Bulkthimblesdualtotracerelations,
JiHoonLeeandDouglasStanford,“Bulkthimblesdualtotracerelations,” (2024),arXiv:2412.20769 [hep-th]
2024 arXiv
-
[35]
Marcos Mariño,Advanced Topics in Quantum Mechanics(Cambridge University Press, 2021) ISBN 978-1-108-86338-4, 978-1-108-49587-5
2021
-
[36]
Complex Kerr-AdS Black Holes,
Kaustubh Singhi, “Complex Kerr-AdS Black Holes,” to appear
-
[37]
Thermal effective actions from con- formal boundary conditions in gravity,
Batoul Banihashemi, Edgar Shaghoulian, and Sanjit Shashi, “Thermal effective actions from con- formal boundary conditions in gravity,” Class. Quant. Grav.42, 155004 (2025), arXiv:2503.17471 [hep-th] 10
2025 arXiv
Reviewed August 4, 2026 · model on record in the stance chip above.
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