Integrable black hole dynamics in the asymptotic structure of AdS$_{3}$

Francisco Correa; Marcela C\'ardenas; Miguel Pino

REVIEW 3 major objections 4 minor 1 cited by

Integrable black hole dynamics in the asymptotic structure of AdS$_{3}$

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Stationary black holes in the AKNS asymptotic class of AdS3 always have constant left and right temperatures, fixed by the spectral polynomial of the integrable hierarchy, even when the solution is not axisymmetric.

desk verdict A solid extension of the AKNS/AdS3 program with a genuinely new cnoidal black hole, but the 'always constant' temperature claim rests on an unproven diagonal gauge existence. read the letter →

arxiv 2504.20292 v1 pith:UZGJL7UW submitted 2025-04-28 hep-th gr-qcmath-phmath.MPnlin.PSnlin.SI

Marcela Cárdenas , Francisco Correa , Miguel Pino This is my paper · ORCID

classification hep-thgr-qcmath-phmath.MPnlin.PSnlin.SI

keywords AdS3gravityasymptoticsymmetriesAKNShierarchyintegrablesystemsblackholethermodynamicsKdVequationhyperellipticcurveChern-Simons

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper shows that a wide family of asymptotically AdS$_3$ spacetimes, defined by relaxing the standard boundary fall-off, is governed at the boundary by the AKNS integrable hierarchy—an infinite family of nonlinear equations with an infinite tower of conserved charges. The authors construct the field-dependent Killing vectors and canonical charges of this asymptotic structure and show that the charges close into an abelian algebra inherited from the integrable system. Their main thermodynamic result is that every stationary black hole in this class has constant left and right temperatures, with $\beta_\pm = n\pi\ell/\sqrt{(A_\pm)^2+B_\pm C_\pm}$, even when the metric depends on the angular coordinate and is not axisymmetric. The constancy follows because the spectral polynomial $(A_\pm)^2+B_\pm C_\pm$ is constant on-shell, the same hyperelliptic curve that organizes the integrable system. A periodic 'cnoidal' KdV solution realizes the construction explicitly, giving a concrete non-axisymmetric black hole with closed-form mass, entropy, and temperature.

What carries the argument

The load-bearing object is the spectral polynomial of the AKNS hierarchy, $y^2=(A_\pm)^2+B_\pm C_\pm$, which is constant on-shell and defines a hyperelliptic curve whose branch points encode the conserved data of the stationary solution. In the diagonal gauge, the Euclidean connection's temporal component becomes $\eta_\pm=2\sqrt{(A_\pm)^2+B_\pm C_\pm}$, so the holonomy condition for regular black holes fixes $\beta_\pm = n\pi\ell/\sqrt{(A_\pm)^2+B_\pm C_\pm}$. For the KdV reduction this identity factorizes as $16(\lambda^2-E_0)(\lambda^2-E_1)(\lambda^2-E_2)$, connecting the temperature to the band-edge eigenvalues of the associated quantum mechanical problem and making the temperature a purely algebraic function of the curve.

What would settle it

Take any stationary, non-axisymmetric AKNS solution (for example, a nonconstant cnoidal KdV solution with nonzero $p_\pm'$) and evaluate $(A_\pm)^2+B_\pm C_\pm$ at two different angular positions using the explicit fields; if the two values differ on-shell, the spectral polynomial is not constant and the claimed temperature formula collapses.

Watch

Extended reading notes

Core claim

The paper's central claim is that integrability of the boundary dynamics forces the temperature of every stationary black hole in the AKNS asymptotic class to be constant, regardless of axisymmetry. Concretely, after Euclidean continuation and a periodic gauge transformation that diagonalizes the auxiliary connections, the holonomy regularity condition yields $\beta_\pm = n\pi\ell/\sqrt{(A_\pm)^2+B_\pm C_\pm}$ with $n$ odd; the square root is the spectral polynomial of the AKNS system, constant along solutions by the algebro-geometric relation attached to commuting Lax operators. The same logic produces the cnoidal KdV black hole from the $2\pi$-periodic cnoidal solution $p_\pm(\varphi)=\frac{2 n_\pm^2 K_\pm^2}{3\pi^2}[\nu_\pm+1-3\nu_\pm\,\mathrm{sn}^2(\frac{K_\pm\varphi}{\pi n_\pm},\nu_\pm)]$, whose temperature is $\beta_\pm=\frac{\pi\ell}{4\sqrt{(\lambda_\pm^2-E_0^\pm)(\lambda_\pm^2-E_1^\pm)(\lambda_\pm^2-E_2^\pm)}}$, with $E^\pm$ the band-edge energies of the associated quantum mechanical problem. The paper also computes the mass $M=M_+ + M_-$ and the entropy $S=\frac{k}{2}\oint(J_+ + J_-)$ from the Euclidean on-shell action, placing the construction within ordinary black hole thermodynamics.

Load-bearing premise

For every stationary solution, one can find a smooth, angle-periodic change of variables that puts the Euclidean gauge connection into the diagonal form used in the holonomy computation; if such a change does not exist for some stationary AKNS black hole, the constancy proof and the explicit formula for the temperature do not apply.

Editorial extensions

If this is right

Every stationary AKNS black hole, even a non-axisymmetric one with nontrivial angular profile, carries a well-defined constant Hawking temperature fixed only by spectral-curve data.
For the KdV class, the temperature is literally an eigenvalue expression built from band-edge energies of the associated quantum mechanical problem, tying black hole thermodynamics to the band structure of an integrable quantum model.
The conserved charges of the asymptotic symmetry algebra coincide with the infinite tower of AKNS charges, so energy and angular momentum sit inside an abelian algebra; angular momentum is conserved only in the strictly axisymmetric limit.
The cnoidal KdV solution is an explicit non-axisymmetric black hole whose mass and entropy are calculable in closed form, so the framework generates concrete thermodynamic examples beyond the usual stationary axisymmetric solutions.
The same constant-temperature formula should apply to every other member of the AKNS hierarchy, such as mKdV or nonlinear Schrödinger stationary solutions, whenever they admit a periodic stationary configuration.

Reading between the lines

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

A natural next step, not developed in the paper, would be to check whether the constancy of temperature implies a generalized zeroth law for integrable asymptotics, where the spectral curve plays the role usually played by the horizon surface gravity.
The singular values of $\lambda_\pm$ at which the inverse temperature diverges are precisely the branch points of the hyperelliptic curve; this suggests that allowed black hole parameters are classified by the spectral gaps of the associated quantum problem, a selection rule the authors only hint at.
Because the charges are abelian and infinite in number, a generalized Gibbs ensemble built from all AKNS charges may be the correct statistical description of these black holes, potentially refining the entropy count for the cnoidal solution.
One could test the central mechanism by numerically constructing a stationary non-axisymmetric solution of another AKNS member and directly evaluating the angular derivative of $(A_\pm)^2+B_\pm C_\pm$; constancy would confirm that the phenomenon is not an artifact of the KdV example.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Worth reading, but with a pencil. The paper takes the AKNS/AdS3 correspondence from the group's earlier PRL and pushes it into metric-formulation territory: explicit field-dependent Killing vectors, Regge-Teitelboim charges that reproduce the AKNS conserved charges, an abelian algebra proof based on the bi-Hamiltonian recursion, and a cnoidal KdV black hole with temperature tied to the hyperelliptic spectral curve. The cnoidal solution and its thermodynamics are the most concrete new piece; the mass and entropy calculations look careful, and the connection to the Lamé problem is elegant.

The soft spot is the temperature section. The claim that every stationary AKNS black hole has constant left/right temperatures rests on the existence of a periodic gauge transformation that diagonalizes the connection simultaneously on φ and τ. The paper asserts this, gives formulas for the gauge functions, and moves on. The stress-test note is right that the quadratic for f degenerates where C±=0, though in those places it reduces to a linear equation, so f need not blow up. The larger issue is global: a periodic family of commuting sl(2,R) matrices with constant eigenvalues can still have a non-trivial eigenvector bundle, and then the diagonalizing gauge is only quasi-periodic, not periodic. The paper never proves the bundle is trivial on the circle. The same gap underlies the 'always constant' claim for non-axisymmetric solutions. I would not call the claim false; it may well be true, and for the cnoidal example one can check it directly. But it is not established at the level of generality advertised.

Two more minor points. The integrability identities (3.28) are imported from [50] without derivation; they are standard in the AKNS literature, so I don't mind much, but a referee should ask for a self-contained statement. The abelian-algebra proof is terse; it relies on shifting indices in a sum, and the logic is not spelled out. Those are fixable in revision.

Overall: the core construction of charges from the AKNS recursion is solid, and the cnoidal black hole is a genuinely new solution. The temperature story is promising but under-supported. This deserves a proper referee, not a desk rejection. I'd ask the referee to focus on the diagonalization existence condition and the domain of validity of (4.6). For my own work, I'd cite the charge construction and the cnoidal solution, not the 'always constant' theorem until it's tightened.

Referee Report

3 major / 4 minor

Summary. The manuscript studies the relaxed AdS3 boundary conditions introduced in earlier work, in which the Einstein equations reduce to two chiral copies of the AKNS hierarchy. It constructs field-dependent asymptotic Killing vectors, computes the Regge-Teitelboim charges and shows they are integrable and span an abelian algebra, with energy and angular momentum contained in the tower of AKNS charges. The central physical claim is that every stationary black hole in this class has constant left and right temperatures beta_plus/minus = n*pi*l / sqrt((A_plus/minus)^2 + B_plus/minus C_plus/minus), with n odd, because the spectral polynomial (A_plus/minus)^2 + B_plus/minus C_plus/minus is constant on-shell; the paper then constructs an explicit KdV reduction and a cnoidal-wave black hole, computes its mass, temperature and entropy, and identifies the temperature with the band-edge spectrum of the associated Lame problem.

Significance. If the proof gaps are closed, the paper offers a genuine bridge between integrable hierarchies and black hole thermodynamics: the temperature becomes an algebraic function of the spectral curve, the charge algebra is abelian and built from standard AKNS integrals, and the cnoidal KdV solution provides a concrete non-axisymmetric example whose temperature is expressed through hyperelliptic data. The metric-level derivation from (2.1) to (2.4) and the charge calculation are nontrivial and largely cross-checked against the Chern-Simons formulation. These are explicit, falsifiable predictions and useful constructions. The main unresolved issue is the rigor of the universal temperature proof in Section 4.1 and, secondarily, the self-containedness of the charge-integrability relations; these are fixable without changing the overall direction.

major comments (3)

[Section 4.1, Eqs. (4.1)-(4.6)] The proof that beta_plus/minus is constant for all stationary AKNS black holes is not self-contained. First, the existence of a smooth 2pi-periodic gauge transformation g_plus/minus = exp(f_plus/minus L_plus/minus) exp(h_plus/minus L_minus/plus) bringing the connection to the diagonal form (4.2) is asserted without proof; Eq. (4.4) gives f_plus/minus = (A_plus/minus +/- sqrt((A_plus/minus)^2+B_plus/minus C_plus/minus))/C_plus/minus, and for the cnoidal solution (4.10) C_plus/minus = -2p_plus/minus - 4lambda_plus/minus^2 has simple zeros for generic parameters, so neither root is guaranteed to extend to a smooth periodic function on the circle. Second, the holonomy condition (4.1) is imposed on the pure time circle, but for an axisymmetric stationary black hole the thermal cycle is a combination of dtau and dphi; the paper should state when the angular component drops out. Since the advertised universal constancy of the temperature rests on these steps, the proof as written is incomplete. A direct derivation from the constancy of the spectral polynomial and the eigenvalues of a_plus/minus_tau would bypass the gauge-existence question, and I recommend adding it.
[Section 3.2.1, Eqs. (3.28)-(3.31)] The variational identities alpha_plus/minus_n = ((n-1)/2) H_plus/minus_n and beta_plus/minus_(n-1) = delta H_plus/minus_n/delta r_plus/minus, gamma_plus/minus_(n-1) = delta H_plus/minus_n/delta p_plus/minus are imported from [50] without derivation. They are the step that converts the non-integrable variation (3.23) into the integrated charges (3.31), so the identification of the gravitational charges with the AKNS conserved charges depends on them. The notation is also inconsistent: H_n is called both the Hamiltonian density and the conserved charge in the same sentence. Please provide a derivation from the recursion relations (3.26)-(3.27), or at least a precise statement of the normalization and of the Hamiltonian density, and resolve the apparent index shift between delta H_n in (3.29) and H_(n+1) in (3.31).
[Section 3.3.1, Eqs. (3.45)-(3.50)] The proof of the abelian algebra is too terse to be checked. The iteration used to obtain (3.49) assumes that the relation D1 S_(n+1) = D2 S_(n+2) can be applied s times without generating boundary terms, and the lambda powers in the sum are not shown to be invariant under the shift n -> n+s; the conclusion (3.50) then identifies the bracket with its negative, but the intermediate signs are not fully displayed. Please expand the argument so that the vanishing of the bracket can be verified step by step.

minor comments (4)

[Eq. (3.30)] The term gamma_+_n delta p_- appears twice in the integrand; the second occurrence should be gamma_-_n delta p_-.
[Eq. (4.22)] The symbol xi appears in the mass formula without being defined anywhere, which makes the explicit expression for M_plus/minus unusable as written; please define xi or correct the typo.
[References [38] and [41]] References [38] and [41] refer to the same paper (Cardenas, Correa, Lara, Pino, Phys. Rev. Lett. 127 (2021) 161601); please consolidate to a single reference.
[Section 4.2, around Eq. (4.23)] The text refers to 'temperatures (4.23)' before the displayed equation is numbered; please place the equation number in the correct position. Also, the abstract says 'two copies of hyperelliptic curves'; the phrase should be adjusted grammatically to 'two hyperelliptic curves' or similar.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the temperature formula is derived from the Euclidean holonomy condition and the constancy of the AKNS spectral polynomial, not from a fitted input or a self-citation chain.

full rationale

The central derivation is self-contained. Section 2 re-derives the AKNS/gravity dictionary directly from the metric (2.1) and Einstein equations (2.4)-(2.4c), so although the paper builds on the authors' earlier [38]/[41], that citation is not the load-bearing step. Equations (4.1)-(4.6) derive β± = nπℓ/sqrt((A±)^2+B±C±) from the Euclidean holonomy condition exp(β±a±τ) = -I, using the diagonalizing gauge (4.2)-(4.4) and the standard stationary-AKNS fact that the spectral polynomial (2.21) is constant. This is a derived consequence, not an input: no parameter is fitted to the temperatures and then renamed a prediction. The cnoidal KdV temperatures (4.23) and mass (4.20)-(4.22) are computed from the explicit solution parameters, and the entropy (4.25)-(4.26) follows from the horizon boundary term. Self-citations ([38]/[41], [37]) are present but point to prior constructions that the paper reproduces or to standard Miura relations already attributed to [17]. The main caveat is an unproven assumption that a periodic diagonalizing gauge exists (Section 4.1), and the singularities of f± at zeros of C± for the cnoidal example; this is a correctness risk affecting the generality of the 'always constant' claim, not a circular reduction.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

No parameters are fitted to data; the free spectral parameters λ± and solution moduli ν±, n± are choices in the boundary-condition and solution ansatz. The central machinery rests on standard integrable-system theorems, the Chern-Simons description of AdS3 gravity, the Euclidean holonomy criterion for temperature, and two ansatze specific to this paper: the Laurent expansion of the symmetry parameters and the periodic diagonal gauge transformation. No new physical entities are postulated.

free parameters (3)

λ± (spectral parameters)
Constants in the metric (2.1) and in the temperatures (4.6), not fixed by the field equations. Earlier works such as [17,32] set them to zero; here they are kept nonzero and control the temperature.
n (odd integer in temperature formula) = odd integer, n=1 for Hawking temperature
The holonomy condition (4.6) determines β± up to an odd integer n. It is a discrete topological choice, not fitted to data, but it enters the central temperature formula.
ν±, n± (cnoidal solution parameters) = 0 < ν± < 1; n± integer winding
These parameterize the periodic cnoidal solution (4.10) and hence the mass, temperature, and entropy formulas in Section 4.2. They are solution moduli, not ad hoc fits.

assumptions (6)

standard math The AKNS hierarchy is generated by the zero-curvature condition with U and V as 2x2 matrices and Laurent expansions (2.9), and it has an infinite set of commuting conserved charges via its bi-Hamiltonian structure.
Used throughout Sections 2 and 3 to identify Einstein dynamics with AKNS and to construct gravitational charges; cited to [43,46].
standard math Burchnall-Chaundy theorem: commuting Lax operators satisfy a polynomial spectral curve M^2 = R(L), whose right-hand side A^2 + BC is constant on-shell.
Equations (2.20)-(2.21) and (4.14)-(4.15) underlie the constancy of the temperature in Section 4.1.
domain assumption AdS3 gravity is classically equivalent to two sl(2,R) Chern-Simons gauge fields, with the metric given by (2.24).
Standard in the AdS3 literature; used in Sections 2.3 and 4 to pass from the metric to holonomies and boundary terms.
domain assumption The Euclidean black hole is regular if and only if the connection holonomy around the thermal circle is trivial, P exp(β Aτ) = -I, Eq. (4.1).
This is the standard Chern-Simons black hole regularity criterion; the paper relies on it to define the temperature.
ad hoc to paper The state-dependent symmetry parameters α±, β±, γ± have Laurent expansions in powers of λ± with β0 = γ0 = 0, Eq. (3.24).
This ansatz, analogous to the AKNS expansion (2.9), is what makes the charges integrable and abelian. It is not forced by the boundary conditions alone.
ad hoc to paper The diagonal gauge transformation g± = exp(f± L±) exp(h± L∓) with periodic f± and h± exists for the stationary solutions, Eqs. (4.2)-(4.3).
The temperature constancy proof runs through this ansatz; if no such periodic diagonalization exists, Eqs. (4.4)-(4.6) would not follow.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Integrable black hole dynamics in the asymptotic structure of AdS$_{3}$." pith.science (2026). https://pith.science/paper/UZGJL7UW

@misc{pith2026250420292,
  author       = {Pith},
  title        = {Pith review of: Integrable black hole dynamics in the asymptotic structure of AdS$_3$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UZGJL7UW}},
  note         = {Machine review of arXiv:2504.20292}
}

abstract

This work deepens the study of integrable asymptotic symmetries for AdS$_{3}$. They are given by an infinite set of integrable nonlinear equations known as the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, characterized by an also infinite set of abelian conserved charges. We present their field-dependent Killing vectors and the computation of the canonical charges associated to the asymptotic metric, together with their corresponding charge algebra. We study black hole thermodynamics and show that the temperature for stationary black holes falling in the AKNS asymptotics is always constant, even in the case where the solutions are not axisymmetric. This is related to the existence of a hyperelliptic curve, which appears as a fundamental object in many integrable systems. We also present a special solution associated with the Korteweg-de Vries equation, that is a particular case of the AKNS integrable hierarchy. It is presented in the form of a periodic soliton leading to a cnoidal KdV black hole, whose temperature is characterized by two copies of hyperelliptic curves.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On Integrable Structures on Non-compact Boundaries in Three-Dimensional Gravity
hep-th 2026-07 conditional novelty 6.0 of 10

In the diagonal (Cartan) sector of AdS3 gravity, the radial flow of the quasi-local stress tensor satisfies an exact T Tbar-like equation, while the boundary time evolution forms an integrable bi-Hamiltonian hierarchy.

Reference graph

Works this paper leans on

73 extracted references · 31 canonical work pages · cited by 1 Pith paper

[50]

Tu and D

G. Tu and D. Meng, Acta Mathematicae Applicatae Sinica 5 (1989), 89–96

work page 1989
[1]

J. D. Brown and M. Henneaux. Commun. Math. Phys 104 (1986), 207

work page 1986
[2]

Achucarro and P

A. Achucarro and P. K. Townsend, Phys. Lett. B 180 (1986), 89

work page 1986
[3]

Witten, Nucl

E. Witten, Nucl. Phys. B 311 (1988), 46

work page 1988
[4]

Compère, W

G. Compère, W. Song and A. Strominger, JHEP 05 (2013), 152 [arXiv:1303.2662[hep-th]]

arXiv 2013
[5]

Troessaert, JHEP 08 (2013), 044 [arXiv:1303.3296[hep-th]]

C. Troessaert, JHEP 08 (2013), 044 [arXiv:1303.3296[hep-th]]

arXiv 2013
[6]

S. G. A very, R. R. Poojary and N. V. Suryanarayana, JHEP 01 (2014), 144 [arXiv:1304.4252[hep-th]]

arXiv 2014
[7]

Apolo and M

L. Apolo and M. Porrati, JHEP 03 (2014), 116 [arXiv:1401.1197[hep-th]]. 28

arXiv 2014

Show all 73 references

[8]

Compère, P

G. Compère, P. Mao, A. Seraj and M. M. Sheikh-Jabbari, JHE P 01 (2016), 080 [arXiv:1511.06079[hep-th]]

2016 arXiv
[9]

Grumiller, A

D. Grumiller, A. Perez, D. Tempo and R. Troncoso, JHEP 08 (2017), 107 [arXiv:1705.10605[hep-th]]

2017 arXiv
[10]

Henneaux, W

M. Henneaux, W. Merbis and A. Ranjbar, JHEP 03 (2020), 064 [arXiv:1912.09465[hep-th]]

2020 arXiv
[11]

Adami, M

H. Adami, M. M. Sheikh-Jabbari, V. Taghiloo, H. Yavarta noo and C. Zwikel, JHEP 10 (2020), 107 [arXiv:2007.12759[hep-th]]

2020 arXiv
[12]

Geiller, C

M. Geiller, C. Goeller and C. Zwikel, JHEP 09 (2021), 029 [arXiv:2107.01073[hep-th]]

2021 arXiv
[13]

Banados, AIP Conf

M. Banados, AIP Conf. Proc. 484, no.1, 147-169 (1999) [arXiv:hep-th/9901148 [hep-th]]

1999 arXiv
[14]

Carlip and C

S. Carlip and C. Teitelboim, Phys. Rev. D 51, 622-631 (1995) [arXiv:gr-qc/9405070 [gr-qc]]

1995 arXiv
[15]

Banados, T

M. Banados, T. Brotz and M. E. Ortiz, Nucl. Phys. B 545, 340-370 (1999) [arXiv:hep-th/9802076 [hep-th]]

1999 arXiv
[17]

Afshar, S

H. Afshar, S. Detournay, D. Grumiller, W. Merbis, A. Per ez, D. Tempo and R. Tron- coso, Phys. Rev. D 93 (2016) no.10, 101503 [arXiv:1603.04824[hep-th]]

2016 arXiv
[18]

Afshar, D

H. Afshar, D. Grumiller, W. Merbis, A. Perez, D. Tempo an d R. Troncoso, Phys. Rev. D 95 (2017) no.10, 106005 [arXiv:1611.09783[hep-th]]

2017 arXiv
[19]

Grumiller and W

D. Grumiller and W. Merbis, SciPost Phys. 8 (2020) no.1, 010 [arXiv:1906.10694[hep-th]]

2020 arXiv
[20]

Zee, Lect

A. Zee, Lect. Notes Phys. 456 (1995), 99-153 [arXiv:cond-mat/9501022 [cond-mat]]

1995 arXiv
[21]

X. G. Wen, Adv. Phys. 44 (1995) no.5, 405-473

1995
[22]

The quantum Hall ﬂuid and noncommutative Chern-Simons theory

L. Susskind, “The quantum Hall ﬂuid and noncommutative Chern-Simons theory", [arXiv:hep-th/0101029 [hep-th]] . 29

arXiv
[23]

A. P. Polychronakos, JHEP 4904 (2001) 011 [arXiv:hep-th/0103013 [hep-th]]

2001 arXiv
[24]

Witten, Riv

E. Witten, Riv. Nuovo Cim. 39 (2016) no.7, 313-370 [arXiv:1510.07698 [cond-mat.mes-hall]]

2016 arXiv
[25]

Grumiller and M

D. Grumiller and M. Riegler, JHEP 10 (2016), 023 [arXiv:1608.01308[hep-th]]

2016 arXiv
[26]

Coussaert, M

O. Coussaert, M. Henneaux and P. van Driel, Class. Quant . Grav. 12 (1995), 2961- 2966 [arXiv:9506019 [gr-qc]]

1995
[27]

Ciambelli and R

L. Ciambelli and R. G. Leigh, Phys. Rev. D 101 (2020) no. 8, 086020 [arXiv:1905.04339 [hep-th]]

2020 arXiv
[28]

Alessio, G

F. Alessio, G. Barnich, L. Ciambelli, P. Mao and R. Ruzzi coni, Phys. Rev. D 103 (2021) no.4, 046003 [arXiv:2010.15452[hep-th]]

2021 arXiv
[29]

Jia and M

W. Jia and M. Karydas, Phys. Rev. D 104 (2021) no.12, 126031 [arXiv:2109.14014[hep-th]]

2021 arXiv
[30]

Ciambelli, A

L. Ciambelli, A. Delfante, R. Ruzziconi and C. Zwikel, P hys. Rev. D 108 (2023) no.12, 12 [arXiv:2308.15480[hep-th]]

2023 arXiv
[31]

Arenas-Henriquez, F

G. Arenas-Henriquez, F. Diaz and D. Rivera-Betancour, JHEP 02 (2025), 007 [arXiv:2411.12513[hep-th]]

2025 arXiv
[32]

Pérez, D

A. Pérez, D. Tempo and R. Troncoso, JHEP 06 (2016), 103 [arXiv:1605.04490 [hep-th]]

2016 arXiv
[33]

Erices, M

C. Erices, M. Riquelme and P. Rodríguez, Phys. Rev. D 100 (2019) no.12, 126026 [arXiv:1907.13026 [hep-th]]

2019 arXiv
[34]

Dymarsky and S

A. Dymarsky and S. Sugishita, JHEP 05 (2020), 041 [arXiv:2002.08368 [hep-th]]

2020 arXiv
[35]

Ojeda and A

E. Ojeda and A. Pérez, JHEP 08 (2019), 079 [arXiv:1906.11226 [hep-th]]

2019 arXiv
[36]

K. Lara, M. Pino and F. Reyes, JHEP 11 (2024), 042 [arXiv:2401.12338 [hep-th]]

2024 arXiv
[37]

Cárdenas, JHEP 10 (2024), 052 [arXiv:2405.03128 [hep-th]]

M. Cárdenas, JHEP 10 (2024), 052 [arXiv:2405.03128 [hep-th]]

2024 arXiv
[39]

Gravitational solitons ,

V. Belinski and E. Verdaguer, “Gravitational solitons ,” (2001) Cambridge University Press

2001
[40]

Glampedakis, A

K. Glampedakis, A. D. Johnson, and D. Kenneﬁck, Phys. Re v. D 96 (2017), 024036. [arXiv:1702.06459 [gr-qc]]

2017 arXiv
[41]

Cárdenas, F

M. Cárdenas, F. Correa, K. Lara and M. Pino, Phys. Rev. Le tt. 127 (2021), no.16, 161601. [arXiv:2104.09676 [hep-th]]

2021 arXiv
[42]

M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, Phys . Rev. Lett. 31 (1973), 125-127

1973
[43]

M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear E volution Equations and Inverse Scattering (Cambridge University Press, Cambridg e, England, 1991)

1991
[44]

Rogers and W

C. Rogers and W. K. Schief, Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory , Cambridge: Cambridge University Press, 2002

2002
[45]

Sasaki, Nuclear Physics B 154 (1979), 2, 343

R. Sasaki, Nuclear Physics B 154 (1979), 2, 343

1979
[46]

Gesztesy, H

F. Gesztesy, H. Holden, Soliton Equations and Their Algebro-Geometric Solutions (Cambridge Univ. Press, Cambridge, 2003)

2003
[47]

Correa, V

F. Correa, V. Jakubsky, L. M. Nieto and M. S. Plyushchay, Phys. Rev. Lett. 101 (2008), 030403 [arXiv:0801.1671 [hep-th]]

2008 arXiv
[48]

Correa, V

F. Correa, V. Jakubsky and M. S. Plyushchay, J. Phys. A 41 (2008), 485303 [arXiv:0806.1614 [hep-th]]

2008 arXiv
[49]

Regge and C

T. Regge and C. Teitelboim, Annals Phys. 88 (1974), 286

1974
[51]

J. D. Brown and M. Henneaux, J. Math. Phys. 27 (1986), 489-491

1986
[52]

P. G. Drazin and R. S. Johnson, Solitons: An introductio n. Cambridge University Press, Cambridge, England (1996)

1996
[53]

Bunster, M

C. Bunster, M. Henneaux, A. Perez, D. Tempo and R. Tronco so, JHEP 05 (2014), 031 [arXiv:1404.3305 [hep-th]]

2014 arXiv
[54]

Imbimbo, A

C. Imbimbo, A. Schwimmer, S. Theisen and S. Yankielowic z, Class. Quant. Grav. 17 (2000), 1129-1138 [arXiv:hep-th/9910267 [hep-th]] . 31

2000 arXiv
[55]

Penrose, Phys

R. Penrose, Phys. Rev. Lett. 10 (1963), 66-68

1963
[56]

Conformal treatment of inﬁnity,

R. Penrose, “Conformal treatment of inﬁnity,” in Relativity, groups and topology: Les Houches 1963 , B. D. C. DeWitt, ed., pp. 563–584. Gordon and Breach, 1964

1963
[57]

Papadimitriou and K

I. Papadimitriou and K. Skenderis, JHEP 08 (2005), 004 [arXiv:0505190 [hep-th]]

2005
[58]

Compere and D

G. Compere and D. Marolf, Class. Quant. Grav. 25 (2008), 195014 [arXiv:0805.1902 [hep-th]]

2008 arXiv
[59]

Marolf, W

D. Marolf, W. Kelly and S. Fischetti, [arXiv:1211.6347 [gr-qc]]

arXiv
[60]

Rooman and P

M. Rooman and P. Spindel, Nucl. Phys. B 594, (2001) 329–353 [arXiv:0008147 [hep-th]]

2001
[61]

Rooman and P

M. Rooman and P. Spindel, Class. Quant. Grav. 18 (2001) 2117–2124 [arXiv:gr-qc/0011005]

2001 arXiv
[62]

Alessio and G

F. Alessio and G. Esposito, Int. J. Geom. Meth. Mod. Phys . 15 (2018) no. 02, 1830002 [arXiv:1709.05134] [gr-qc]

2018 arXiv
[63]

, Olver, Applications of Lie Groups to Diﬀerential Equations

P.J. , Olver, Applications of Lie Groups to Diﬀerential Equations. Springer-Verlag, (1993) New York

1993
[64]

Maloney, G

A. Maloney, G. S. Ng, S. F. Ross and I. Tsiares, JHEP 03 (2019), 075 [arXiv:1810.11054] [hep-th]

2019 arXiv
[65]

Dymarsky and K

A. Dymarsky and K. Pavlenko, JHEP 01 (2019), 098 [arXiv:1810.11025] [hep-th]

2019 arXiv
[66]

E. M. Brehm and D. Das, Phys. Rev. D 101 (2020) no.8, 086025 [arXiv:1901.10354] [hep-th]

2020 arXiv
[67]

Gauge Theory And Integrab ility, III,

K. Costello and M. Yamazaki, “Gauge Theory And Integrab ility, III,” [arXiv:1908.02289] [hep-th]

1908 arXiv
[68]

Lacroix, J

S. Lacroix, J. Phys. A 55 (2022) no.8, 083001 [arXiv:2109.14278] [hep-th]

2022 arXiv
[69]

Fukuyama and K

T. Fukuyama and K. Kamimura, Phys. Lett. B 160 (1985), 259-262

1985
[70]

Jackiw, Nucl

R. Jackiw, Nucl. Phys. B 252 (1985), 343-356

1985
[71]

Teitelboim, Phys

C. Teitelboim, Phys. Lett. B 126 (1983), 41-45. 32

1983
[72]

Navarro-Salas and P

J. Navarro-Salas and P. Navarro, Nucl. Phys. B 579 (2000), 250-266 [arXiv:hep-th/9910076 [hep-th]

2000 arXiv
[73]

J. Cruz, A. Fabbri, D. J. Navarro and J. Navarro-Salas, P hys. Rev. D 61 (2000), 024011 [arXiv:hep-th/9906187 [hep-th]

2000 arXiv
[74]

A. T. Filippov, Mod. Phys. Lett. A 11 (1996), 1691-1704 [arXiv:hep-th/9605008 [hep-th]

1996 arXiv
[75]

D. J. Navarro and J. Navarro-Salas, Mod. Phys. Lett. A 13 (1998), 2049-2056 [arXiv:hep-th/9807003 [hep-th] . 33

1998 arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [50]

Tu and D

G. Tu and D. Meng, Acta Mathematicae Applicatae Sinica 5 (1989), 89–96

work page 1989

[2] [1]

J. D. Brown and M. Henneaux. Commun. Math. Phys 104 (1986), 207

work page 1986

[3] [2]

Achucarro and P

A. Achucarro and P. K. Townsend, Phys. Lett. B 180 (1986), 89

work page 1986

[4] [3]

Witten, Nucl

E. Witten, Nucl. Phys. B 311 (1988), 46

work page 1988

[5] [4]

Compère, W

G. Compère, W. Song and A. Strominger, JHEP 05 (2013), 152 [arXiv:1303.2662[hep-th]]

arXiv 2013

[6] [5]

Troessaert, JHEP 08 (2013), 044 [arXiv:1303.3296[hep-th]]

C. Troessaert, JHEP 08 (2013), 044 [arXiv:1303.3296[hep-th]]

arXiv 2013

[7] [6]

S. G. A very, R. R. Poojary and N. V. Suryanarayana, JHEP 01 (2014), 144 [arXiv:1304.4252[hep-th]]

arXiv 2014

[8] [7]

Apolo and M

L. Apolo and M. Porrati, JHEP 03 (2014), 116 [arXiv:1401.1197[hep-th]]. 28

arXiv 2014