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REVIEW 3 major objections 5 minor 124 references

The radial flow of 3D gravity is exactly a T-bar-T deformation.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-02 08:11 UTC pith:M6675RSB

load-bearing objection Solid, careful extension of integrable boundary conditions to finite cutoff; the central radial flow is correct on the physical branch, but the 'arbitrary profiles' claim hides a sign restriction that should be fixed. the 3 major comments →

arxiv 2607.06867 v2 pith:M6675RSB submitted 2026-07-07 hep-th gr-qc

On Integrable Structures on Non-compact Boundaries in Three-Dimensional Gravity

Hamed Adami , Kristiansen Lara , Anouchah Latifi , Ren\'e Meyer This is my paper
classification hep-th gr-qc
keywords three-dimensional gravityChern–Simons theoryT-bar-T deformationquasi-local stress tensorintegrable hierarchiesinverse scatteringbi-Hamiltonian structurefinite cutoff
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper studies three-dimensional Einstein gravity with negative cosmological constant, formulated as a Chern–Simons theory, on spatial slices that extend to infinity. Within a diagonal sector of the phase space that reduces the gauge connection to two chiral currents, the author derives an exact, closed radial flow equation for the quasi-local energy density: ∂_r ρ = −(4ℓ²r/(r⁴−ℓ⁴))√(ρ²−J²). This equation has precisely the structure of a T-bar-T deformation and resums all orders in the radial cutoff, giving a non-perturbative holographic renormalization-group flow. The paper also shows that the boundary time evolution is governed by an integrable bi-Hamiltonian hierarchy, while the radial flow itself is not Hamiltonian with respect to the canonical Poisson structure; the two notions of evolution are distinct at finite cutoff. The mechanical core is the exact fluid/gravity dictionary that expresses the energy and momentum densities as algebraic functions of the chiral currents.

Core claim

In the diagonal reduction where the sl(2,R) × sl(2,R) gauge connections are restricted to the Cartan direction (a± = ±A± L0), the flatness condition becomes abelian and the phase space is parametrized by two chiral currents J±(t,x). From the exact reconstruction of the bulk metric, the Brown–York stress tensor on a constant-radius hypersurface is computed in closed form; the energy density ρ and momentum density J are explicit algebraic functions of J± and the radial coordinate. Eliminating the chiral currents yields the exact radial flow equation (63), ∂_r ρ = −(4ℓ²r/(r⁴−ℓ⁴))√(ρ²−J²), which the paper interprets as the realization of the T-bar-T deformation at finite cutoff, valid to all ord

What carries the argument

The load-bearing object is the exact radial flow equation ∂_r ρ = −(4ℓ²r/(r⁴−ℓ⁴))√(ρ²−J²) (Eq. 63), together with the fluid/gravity dictionary (Eqs. 57a–57b) that renders the quasi-local stress tensor an explicit function of the r-independent chiral currents J±. The flow is linearized by X = ρ + √(ρ²−J²), converting it to ∂_r ln X = −4ℓ²r/(r⁴−ℓ⁴), and its integration reproduces the full dictionary. The complementary mechanism is the diagonal ansatz a± = ±A± L0, which abelianizes the flatness condition and splits the boundary dynamics into two independent chiral sectors whose time evolution is a bi-Hamiltonian (Gel'fand–Dikii) hierarchy.

Load-bearing premise

The diagonal reduction of the gauge connections to the Cartan subalgebra (a± = ±A± L0), which abelianizes the flatness condition and reduces the phase space to two chiral u(1) currents; all results—including the radial flow equation and its T-bar-T interpretation—hold only within this subsector.

What would settle it

Compute the quasi-local stress tensor from a generic non-diagonal solution of the Chern–Simons equations (e.g., with a± having off-diagonal components) and test whether the radial derivative of its energy density still satisfies Eq. (63). A violation would show the flow equation is an artifact of the diagonal ansatz rather than a universal property of three-dimensional gravity.

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

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If this is right

  • The quasi-local energy at any finite cutoff is known non-perturbatively from the chiral data: E(r) = (r⁴+ℓ⁴)/(r⁴−ℓ⁴) E_∞ + (4ℓ²r²)/(r⁴−ℓ⁴) E_int, providing a closed-form holographic RG flow.
  • The boundary dynamics is exactly solvable by inverse-scattering methods: the chiral currents are Schrödinger potentials, and the gravitational solution space is parameterized by scattering data (reflection coefficient, discrete eigenvalues, norming constants).
  • The integrable hierarchy and all its conserved charges are common to every cutoff slice; the T-bar-T deformation changes only the dictionary from currents to observables, not the dynamics itself.
  • The radial flow is exactly solvable yet not Hamiltonian with respect to the canonical Poisson structure, showing that solvability and Hamiltonian integrability are distinct once finite-cutoff effects are included.
  • On non-compact boundaries, the finite cutoff produces an effective interaction energy between left- and right-moving solitons that depends on their separation and vanishes in the asymptotic limit.

Where Pith is reading between the lines

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

  • The same variable X = ρ + √(ρ²−J²) that linearizes the radial flow might generalize to higher-dimensional T-bar-T-like flows, where the square root becomes the determinant of the stress tensor; this suggests a broader class of exactly solvable radial evolutions.
  • The no-go theorem indicates that a Hamiltonian description of radial evolution would require an enlarged phase space or a non-commutative deformation of the Poisson structure; such a structure may become relevant when 1/c quantum corrections are included.
  • Because the integrable hierarchy is cutoff-independent, the spectral data (discrete eigenvalues and norming constants) provide universal, scale-invariant labels for gravitational states, potentially offering a sharp characterization across renormalization-group scales.
  • The inverse-scattering description could be used to define a gravitational S-matrix for boundary excitations, with the reflection coefficient playing the role of scattering data; this might connect to flat-space or celestial holography in the appropriate limits.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies AdS3 Einstein gravity in the Chern–Simons formulation, restricted to the diagonal Cartan subsector a± = ±A±L0 (Eq. 13). In this sector the flatness condition is abelian, and the authors derive an exact fluid/gravity dictionary (Eqs. 57a–57b), a closed radial flow equation for the quasi-local energy density (Eq. 63), and interpret it as a holographic T\bar T-type deformation at finite cutoff. They combine this with a bi-Hamiltonian Gel'fand–Dikii hierarchy for the chiral currents, an inverse-scattering description on non-compact boundaries, explicit one-soliton solutions, and an argument that the radial flow is not a Hamiltonian flow on the canonical phase space.

Significance. If the central claims are correct, the paper provides a nonperturbative, exactly solvable radial evolution of the quasi-local Brown–York stress tensor in a nontrivial sector of AdS3, together with a clean gravitational interpretation of inverse-scattering data for KdV-type boundary conditions. The algebraic derivation of Eq. (63) from Eqs. (57) is correct in the physical domain where the induced boundary metric has Lorentzian signature, and the soliton velocities (117) and Hamiltonians (118) are consistent with the Lifshitz scaling structure. These are concrete, checkable results with clear value for finite-cutoff holography and integrable boundary conditions. However, the results are confined to the abelian diagonal ansatz (13), and the domain/sign issue in Eq. (63) discussed below must be addressed before the claims as written are fully supported.

major comments (3)
  1. [Section VII, Eq. (63)] The radial-flow equation is stated without qualification, but it holds only when W in Eq. (35) is positive. Differentiating (57a) and using (57b) gives ∂_r ρ = −(4ℓ² r/(r⁴−ℓ⁴)) sgn(W) √(ρ²−𝒥²). For W<0, e.g. J₊=−J₋ at r>ℓ, the correct sign is opposite to (63). Since W>0 forces J₊J₋≥0 for r>ℓ, the statement in Sec. VII that the equation holds for 'arbitrary boundary profiles' is overbroad. Please either restrict the claim to the physical domain W>0 (and state that restriction explicitly) or include sgn(W) in Eq. (63).
  2. [Abstract; Section II, Eq. (13)] All subsequent results—the fluid dictionary (57), the radial flow (63), the bi-Hamiltonian hierarchy, and the inverse-scattering description—are derived only in the diagonal Cartan subsector (13), where the flatness condition is abelian. The abstract and title present the results as valid for 'three-dimensional gravity' on non-compact boundaries without this qualifier. Although Section II discloses the restriction, the abstract should do so as well; otherwise the paper promises more than it proves.
  3. [Section XII.D] The 'no-go theorem' that the radial flow is not Hamiltonian is essentially a consequence of the radial-gauge choice: J± are r-independent by construction, so the radial flow does not act on the canonical phase space at all. The argument is correct but is more a clarification of the setup than a theorem. Recommend either giving a precise formal statement of the class of Hamiltonian functionals and Poisson brackets considered, or softening the terminology.
minor comments (5)
  1. [Sections VI–VII] The notation is confusing because 𝒥 in Eqs. (57b) and (62) denotes the momentum density while J± in Eq. (14) denote the chiral currents. The radial-flow equation (63) would be much clearer with a distinct symbol, e.g. 𝒥 or Π, for the momentum density.
  2. [Introduction] The text refers to 'the marginal composite operator √(T T̄)'. Finite-cutoff holography is usually associated with the irrelevant T T̄ deformation, while root-T T̄ is marginal. The wording should be checked and clarified to avoid confusion with the standard dictionary.
  3. [Section VII] The statement 'p=ρ, so the Brown–York tensor is traceless' is surprising in the T T̄ context, where the trace is generally nonvanishing. The caveat in Section VII is helpful, but the paper should explicitly state that this trace-free property is a feature of the diagonal fluid dictionary, not of the deformed CFT stress tensor.
  4. [Section X.B.2, Eq. (114)] The sign convention for c± should be stated explicitly. With c±>0 and ν±=c±/(6π), the soliton profiles (108) and Hamiltonians (118) have negative values; this is consistent but nonstandard for readers accustomed to positive-definite KdV charges.
  5. [Section XII.A] The paper notes that explicit trace identities have not been derived. Since the Hamiltonians are claimed to admit an equivalent spectral representation, adding a reference or a brief derivation for the KdV convention used here would strengthen the inverse-scattering interpretation.

Circularity Check

0 steps flagged

No significant circularity; Eq. (63) is derived algebraically from the exact dictionary (57), not fitted or assumed; self-citations are scaffolding only.

full rationale

The central claim, Eq. (63), is derived in Sec. VII by differentiating the exact dictionary (57a) at fixed J± and eliminating J± in favor of ρ and J. This is an algebraic consequence of the stated definitions, not a fitted input or an assumed TTbar flow: no free parameter is tuned to reproduce the radial dependence, and the equation is not imported from the TTbar literature. The TTbar interpretation is explicitly qualified ('By itself, however, this result does not establish an exact equivalence with the universal TTbar flow governing finite-volume energy levels'), so it is not a disguised renaming. The diagonal ansatz (13) is a disclosed restriction, attributed to [15,29,59], and Sec. XIII lists extension beyond it as future work. The bi-Hamiltonian structure and inverse-scattering machinery are either reconstructed from standard Gel'fand-Dikii/GLM formalism or cited to independent spectral literature; the only notable author-overlapping citation, [32] (Adami--Latifi), is used for context and extension (eigenfunction-forced KdV flows), not as proof of a central equation. The paper's own limitations—the 'arbitrary boundary profiles' claim in Sec. VII requires an unstated sign/positivity condition (W>0 in Eq. (35)); Sec. XII.A notes that explicit trace identities have not been derived; Sec. XII.D gives a terse no-go argument—are correctness/omitted-support concerns, not circularity. No step in the derivation reduces to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No numbers are fitted to data; the central charges c± and the consistency-fixed ν± are physical/structural inputs, not fitted parameters. The central new equation (63) is a rearrangement of the paper's own definitions (57) and does not import the target result. The main unproven inputs are structural: the diagonal subsector ansatz and the choice of chemical potentials as Gel'fand-Dikii polynomials. The paper cites prior work for the Gelfand-Dikii bi-Hamiltonian structure and GLM inverse scattering, which are standard. Hence the ledger is short and the burden is correspondingly low.

axioms (5)
  • domain assumption Diagonal subsector ansatz a± = ±A± L0 (Eq. 13): the sl(2,R)×sl(2,R) gauge connections are restricted to the Cartan subalgebra, reducing flatness to an abelian equation.
    All main results (fluid dictionary, radial flow, KdV hierarchy, solitons) are derived in this abelian sector; the paper's title/abstract do not disclose this restriction.
  • ad hoc to paper Chemical potentials are chosen as Gel'fand-Dikii polynomials, μ± = R_{I+1}[J±] (Eqs. 80, 112).
    This is a boundary-condition choice, not derived from the action; it is what makes the boundary dynamics the KdV-type hierarchy. The hierarchy is imposed rather than shown to be the unique consequence of the gravity theory.
  • standard math The second Hamiltonian structure D± = ∂xJ± + 2J±∂x − (c±/24π)∂x³ (Eq. 76) is a Poisson operator compatible with P± = ∂x.
    Classical Virasoro/Gel'fand-Dikii bracket; the paper states compatibility but does not compute the Schouten bracket.
  • domain assumption Short-range, rapidly decaying potentials J± on Σ = R (Section X.A).
    Ensures existence of Jost solutions, Volterra representation, and GLM reconstruction; breaks on compact slices, as the paper notes.
  • domain assumption Vanishing symplectic flux through the asymptotic boundary, Ω = 0 (Eq. 67).
    Used to write μ± = δH/δJ± and define a Hamiltonian boundary theory; standard in covariant phase space but assumed.

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Pith. "Pith review of On Integrable Structures on Non-compact Boundaries in Three-Dimensional Gravity." pith.science (2026). https://pith.science/paper/M6675RSB

@misc{pith2026260706867,
  author       = {Pith},
  title        = {Pith review of: On Integrable Structures on Non-compact Boundaries in Three-Dimensional Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M6675RSB}},
  note         = {Machine review of arXiv:2607.06867}
}
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read the original abstract

We study three-dimensional Einstein gravity with negative cosmological constant on non-compact spatial boundaries within the Chern-Simons formulation. Using an exact fluid/gravity correspondence, we derive a closed radial flow equation for the quasi-local stress tensor and show that it realizes the holographic $T\bar T$ deformation at finite cutoff. We further develop the inverse-scattering description of the boundary dynamics, identifying the gravitational interpretation of the associated spectral data and analyzing the finite-cutoff deformation of soliton solutions. Although the boundary evolution is governed by an integrable bi-Hamiltonian hierarchy, we show that the radial flow itself is not Hamiltonian with respect to the canonical Poisson structure. Our results establish a unified framework connecting integrability, quasi-local gravitational observables, inverse scattering, and finite-cutoff holography on non-compact boundaries.

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Works this paper leans on

124 extracted references · 93 linked inside Pith

  1. [1]

    Af- ter a suitable rescaling of the coordinateξ±, it reduces to the Pöschl–Teller potential with parameterl= 1, a prototypical reflectionless system

    Spectral problem For the single–soliton configuration (108), the Schrödinger equation (89) becomes exactly solvable. Af- ter a suitable rescaling of the coordinateξ±, it reduces to the Pöschl–Teller potential with parameterl= 1, a prototypical reflectionless system. In this case, the spectral problem exhibits a sim- ple structure: there is a single bound ...

  2. [2]

    Fixing the time evolution The dynamics of the hierarchy is determined by the bi- Hamiltonian structure (75). Choosing the Hamiltonian associated with the(I+ 1)-th flow fixes the chemical po- tentials to be µ± =R I+1 [J±].(112) Combining (20) with (109) then yields the evolution equation ∓ π k J± ∂tf± ±∂ tφ± =R I+1 [J±].(113) For generic flows withI≥0, con...

  3. [3]

    T¯T-deformed Gel’fand–Dikii hierarchy

    Hamiltonians The Lifshitz scaling symmetry also determines the functional dependence of the conserved Hamiltonians on the soliton parameters. In the single-soliton sector, the parameterα ± istheonlyindependentdimensionfulquan- tity. Consequently, homogeneity under (85) requires the Hamiltonian generating the(I+ 1)-th flow to scale as H ± I+1 ∝α 2I+3 ± . T...

  4. [4]

    Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity,

    J. D. Brown and M. Henneaux, “Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity,” Commun. Math. Phys.104(1986) 207–226

    1986

  5. [5]

    Asymptotically anti-De Sitter Spaces,

    M. Henneaux and C. Teitelboim, “Asymptotically anti-De Sitter Spaces,”Commun. Math. Phys.98 (1985) 391–424

    1985

  6. [6]

    (2+1)-dimensional gravity as an exactly soluble system,

    E. Witten, “(2+1)-dimensional gravity as an exactly soluble system,”Nucl. Phys.B311(1988) 46

    1988

  7. [7]

    Carlip,Quantum gravity in 2+1 dimensions

    S. Carlip,Quantum gravity in 2+1 dimensions. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 12, 2003

    2003

  8. [8]

    Conformal field theory, (2+1)-dimensional gravity, and the BTZ black hole,

    S. Carlip, “Conformal field theory, (2+1)-dimensional gravity, and the BTZ black hole,”Class. Quant. Grav. 22(2005) R85–R124,gr-qc/0503022

    Pith/arXiv arXiv 2005

  9. [9]

    The Black hole in three-dimensional space-time,

    M. Banados, C. Teitelboim, and J. Zanelli, “The Black hole in three-dimensional space-time,”Phys. Rev. Lett. 69(1992) 1849–1851,hep-th/9204099

    Pith/arXiv arXiv 1992

  10. [10]

    Dimensionally continued black holes,

    M. Banados, C. Teitelboim, and J. Zanelli, “Dimensionally continued black holes,”Phys. Rev. D 49(1994) 975–986,gr-qc/9307033

    Pith/arXiv arXiv 1994

  11. [11]

    A Chern-Simons action for three-dimensional Anti-de Sitter supergravity theories,

    A. Achucarro and P. K. Townsend, “A Chern-Simons action for three-dimensional Anti-de Sitter supergravity theories,”Phys. Lett.B180(1986) 89

    1986

  12. [12]

    Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory,

    S. Elitzur, G. W. Moore, A. Schwimmer, and N. Seiberg, “Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory,”Nucl. Phys. B326(1989) 108–134

    1989

  13. [13]

    The Asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant,

    O. Coussaert, M. Henneaux, and P. van Driel, “The Asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant,” Class.Quant.Grav.12(1995) 2961–2966, gr-qc/9506019

    Pith/arXiv arXiv 1995

  14. [14]

    Global charges in Chern-Simons field theory and the (2+1) black hole,

    M. Bañados, “Global charges in Chern-Simons field theory and the (2+1) black hole,”Phys. Rev.D52 (1995) 5816,hep-th/9405171

    Pith/arXiv arXiv 1995

  15. [15]

    Three-dimensional quantum geometry and black holes,

    M. Bañados, “Three-dimensional quantum geometry and black holes,”hep-th/9901148

    Pith/arXiv arXiv

  16. [16]

    Advanced Lectures on General Relativity,

    G. Compère and A. Fiorucci, “Advanced Lectures on General Relativity,”Lect. Notes Phys.952(2019) 150, 1801.07064

    Pith/arXiv arXiv 2019

  17. [17]

    Boundary conditions for General Relativity on AdS3 and the KdV hierarchy,

    A. Pérez, D. Tempo, and R. Troncoso, “Boundary conditions for General Relativity on AdS3 and the KdV hierarchy,”JHEP06(2016) 103,1605.04490

    Pith/arXiv arXiv 2016

  18. [18]

    Boundary conditions for General Relativity in three-dimensional spacetimes, integrable systems and the KdV/mKdV hierarchies,

    E. Ojeda and A. Pérez, “Boundary conditions for General Relativity in three-dimensional spacetimes, integrable systems and the KdV/mKdV hierarchies,” JHEP08(2019) 079,1906.11226

    Pith/arXiv arXiv 2019

  19. [19]

    Integrable systems and the boundary dynamics of higher spin gravity on AdS3,

    E. Ojeda and A. Pérez, “Integrable systems and the boundary dynamics of higher spin gravity on AdS3,” JHEP11(2020) 089,2009.07829

    Pith/arXiv arXiv 2020

  20. [20]

    Revisiting the asymptotic dynamics of General Relativity on AdS3,

    H. A. González, J. Matulich, M. Pino, and R. Troncoso, “Revisiting the asymptotic dynamics of General Relativity on AdS3,”JHEP12(2018) 115, 1809.02749

    Pith/arXiv arXiv 2018

  21. [21]

    Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields,

    A. Campoleoni, S. Fredenhagen, S. Pfenninger, and S. Theisen, “Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields,”JHEP1011(2010) 007,1008.4744

    Pith/arXiv arXiv 2010

  22. [22]

    Asymptotic W-symmetries in three-dimensional higher-spin gauge theories,

    A. Campoleoni, S. Fredenhagen, and S. Pfenninger, “Asymptotic W-symmetries in three-dimensional higher-spin gauge theories,”JHEP1109(2011) 113, 1107.0290

    Pith/arXiv arXiv 2011

  23. [23]

    Integrable Systems and Spacetime Dynamics,

    M. Cárdenas, F. Correa, K. Lara, and M. Pino, “Integrable Systems and Spacetime Dynamics,”Phys. Rev. Lett.127(2021), no. 16, 161601,2104.09676

    Pith/arXiv arXiv 2021

  24. [24]

    Integrable black hole dynamics in the asymptotic structure of AdS3,

    M. Cárdenas, F. Correa, and M. Pino, “Integrable black hole dynamics in the asymptotic structure of AdS3,”2504.20292

    Pith/arXiv arXiv

  25. [25]

    1/c deformations of AdS3 boundary conditions and the Dym hierarchy,

    K. Lara, M. Pino, and F. Reyes, “1/c deformations of AdS3 boundary conditions and the Dym hierarchy,” JHEP11(2024) 042,2401.12338

    Pith/arXiv arXiv 2024

  26. [26]

    Non-axisymmetric (2+1) black holes with Dym boundary conditions,

    M. Pino and F. Reyes, “Non-axisymmetric (2+1) black holes with Dym boundary conditions,”2511.06567

    arXiv

  27. [27]

    Wsymmetry and integrability of higher spin black holes,

    G. Compère and W. Song, “Wsymmetry and integrability of higher spin black holes,”JHEP1309 (2013) 144,1306.0014

    Pith/arXiv arXiv 2013

  28. [28]

    Lifshitz Scaling, Microstate Counting from Number Theory and Black Hole Entropy,

    D. Melnikov, F. Novaes, A. Pérez, and R. Troncoso, “Lifshitz Scaling, Microstate Counting from Number Theory and Black Hole Entropy,”JHEP06(2019) 054,1808.04034

    Pith/arXiv arXiv 2019

  29. [29]

    Generalized Fefferman-Graham gauge and boundary Weyl structures,

    G. Arenas-Henriquez, F. Diaz, and D. Rivera-Betancour, “Generalized Fefferman-Graham gauge and boundary Weyl structures,”JHEP02 (2025) 007,2411.12513

    Pith/arXiv arXiv 2025

  30. [30]

    Integrable systems with BMS3 Poisson structure and the dynamics of locally flat spacetimes,

    O. Fuentealba, J. Matulich, A. Pérez, M. Pino, P. Rodríguez, D. Tempo, and R. Troncoso, “Integrable systems with BMS3 Poisson structure and the dynamics of locally flat spacetimes,”JHEP01(2018) 148,1711.02646

    Pith/arXiv arXiv 2018

  31. [31]

    Soft Heisenberg hair on black holes in three dimensions,

    H. Afshar, S. Detournay, D. Grumiller, W. Merbis, A. Perez, D. Tempo, and R. Troncoso, “Soft Heisenberg hair on black holes in three dimensions,” Phys. Rev.D93(2016), no. 10, 101503,1603.04824

    Pith/arXiv arXiv 2016

  32. [32]

    Near horizon dynamics of three dimensional black holes,

    D. Grumiller and W. Merbis, “Near horizon dynamics of three dimensional black holes,”SciPost Phys.8 (2020), no. 1, 010,1906.10694

    Pith/arXiv arXiv 2020

  33. [33]

    BTZ black hole with Korteweg–de Vries-type boundary conditions: Thermodynamics revisited,

    C. Erices, M. Riquelme, and P. Rodríguez, “BTZ black hole with Korteweg–de Vries-type boundary conditions: Thermodynamics revisited,”Phys. Rev. D 100(2019), no. 12, 126026,1907.13026

    Pith/arXiv arXiv 2019

  34. [34]

    KdV-charged black holes,

    A. Dymarsky and S. Sugishita, “KdV-charged black holes,”JHEP05(2020) 041,2002.08368

    Pith/arXiv arXiv 2020

  35. [35]

    Integrability in Three-Dimensional Gravity: Eigenfunction-Forced KdV Flows,

    H. Adami and A. Latifi, “Integrability in Three-Dimensional Gravity: Eigenfunction-Forced KdV Flows,”2510.10519

    Pith/arXiv arXiv

  36. [36]

    On the holographic renormalization group,

    J. de Boer, E. P. Verlinde, and H. L. Verlinde, “On the holographic renormalization group,”JHEP08(2000) 003,hep-th/9912012

    Pith/arXiv arXiv 2000

  37. [37]

    Holographic Renormalization,

    M. Bianchi, D. Z. Freedman, and K. Skenderis, “Holographic Renormalization,”Nucl. Phys.B631 (2002) 159–194,hep-th/0112119

    Pith/arXiv arXiv 2002

  38. [38]

    Lecture notes on holographic renormalization,

    K. Skenderis, “Lecture notes on holographic renormalization,”Class. Quant. Grav.19(2002) 5849–5876,hep-th/0209067

    Pith/arXiv arXiv 2002

  39. [39]

    AdS / CFT correspondence and geometry,

    I. Papadimitriou and K. Skenderis, “AdS / CFT correspondence and geometry,”IRMA Lect. Math. Theor. Phys.8(2005) 73–101,hep-th/0404176

    Pith/arXiv arXiv 2005

  40. [40]

    Holographic and Wilsonian Renormalization Groups,

    I. Heemskerk and J. Polchinski, “Holographic and Wilsonian Renormalization Groups,”JHEP06(2011) 031,1010.1264. 24

    Pith/arXiv arXiv 2011

  41. [41]

    Integrating out geometry: Holographic Wilsonian RG and the membrane paradigm,

    T. Faulkner, H. Liu, and M. Rangamani, “Integrating out geometry: Holographic Wilsonian RG and the membrane paradigm,”JHEP08(2011) 051, 1010.4036

    Pith/arXiv arXiv 2011

  42. [42]

    Gravity Is Induced By Renormalization Group Flow,

    H. Adami, M. M. Sheikh-Jabbari, and V. Taghiloo, “Gravity Is Induced By Renormalization Group Flow,” 2508.09633

    Pith/arXiv arXiv

  43. [43]

    Moving the CFT into the bulk withTT,

    L. McGough, M. Mezei, and H. Verlinde, “Moving the CFT into the bulk withTT,”JHEP04(2018) 010, 1611.03470

    Pith/arXiv arXiv 2018

  44. [44]

    Expectation value of composite field T anti-T in two-dimensional quantum field theory,

    A. B. Zamolodchikov, “Expectation value of composite field T anti-T in two-dimensional quantum field theory,”hep-th/0401146

    Pith/arXiv arXiv

  45. [45]

    On space of integrable quantum field theories,

    F. A. Smirnov and A. B. Zamolodchikov, “On space of integrable quantum field theories,”Nucl. Phys. B915 (2017) 363–383,1608.05499

    Pith/arXiv arXiv 2017

  46. [46]

    T ¯T-deformed 2D Quantum Field Theories,

    A. Cavaglià, S. Negro, I. M. Szécsényi, and R. Tateo, “T ¯T-deformed 2D Quantum Field Theories,”JHEP10 (2016) 112,1608.05534

    Pith/arXiv arXiv 2016

  47. [47]

    Holography at finite cutoff with aT2 deformation,

    T. Hartman, J. Kruthoff, E. Shaghoulian, and A. Tajdini, “Holography at finite cutoff with aT2 deformation,”JHEP03(2019) 004,1807.11401

    Pith/arXiv arXiv 2019

  48. [48]

    Modular invariance and uniqueness of T ¯Tdeformed CFT,

    O. Aharony, S. Datta, A. Giveon, Y. Jiang, and D. Kutasov, “Modular invariance and uniqueness of T ¯Tdeformed CFT,”JHEP01(2019) 086,1808.02492

    Pith/arXiv arXiv 2019

  49. [49]

    T¯Tdeformations in general dimensions,

    M. Taylor, “T¯Tdeformations in general dimensions,” Adv. Theor. Math. Phys.27(2023), no. 1, 37–63, 1805.10287

    Pith/arXiv arXiv 2023

  50. [50]

    Cutoff AdS3 versus theT Tdeformation,

    P. Kraus, J. Liu, and D. Marolf, “Cutoff AdS3 versus theT Tdeformation,”JHEP07(2018) 027, 1801.02714

    Pith/arXiv arXiv 2018

  51. [51]

    T¯Tand the mirage of a bulk cutoff,

    M. Guica and R. Monten, “T¯Tand the mirage of a bulk cutoff,”SciPost Phys.10(2021), no. 2, 024, 1906.11251

    Pith/arXiv arXiv 2021

  52. [52]

    An integrable Lorentz-breaking deformation of two-dimensional CFTs,

    M. Guica, “An integrable Lorentz-breaking deformation of two-dimensional CFTs,”SciPost Phys. 5(2018), no. 5, 048,1710.08415

    Pith/arXiv arXiv 2018

  53. [53]

    JT deformed CFT2 and string theory,

    S. Chakraborty, A. Giveon, and D. Kutasov, “JT deformed CFT2 and string theory,”JHEP10(2018) 057,1806.09667

    Pith/arXiv arXiv 2018

  54. [54]

    Strings on warped AdS3 via T¯Jdeformations,

    L. Apolo and W. Song, “Strings on warped AdS3 via T¯Jdeformations,”JHEP10(2018) 165,1806.10127

    Pith/arXiv arXiv 2018

  55. [55]

    Local Fluid Dynamical Entropy from Gravity,

    S. Bhattacharyya, V. E. Hubeny, R. Loganayagam, G. Mandal, S. Minwalla, T. Morita, M. Rangamani, and H. S. Reall, “Local Fluid Dynamical Entropy from Gravity,”JHEP06(2008) 055,0803.2526

    Pith/arXiv arXiv 2008

  56. [56]

    Forced Fluid Dynamics from Gravity,

    S. Bhattacharyya, R. Loganayagam, S. Minwalla, S. Nampuri, S. P. Trivedi, and S. R. Wadia, “Forced Fluid Dynamics from Gravity,”JHEP02(2009) 018, 0806.0006

    Pith/arXiv arXiv 2009

  57. [57]

    Fluid dynamics of R-charged black holes,

    J. Erdmenger, M. Haack, M. Kaminski, and A. Yarom, “Fluid dynamics of R-charged black holes,”JHEP01 (2009) 055,0809.2488

    Pith/arXiv arXiv 2009

  58. [58]

    Hydrodynamics from charged black branes,

    N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Dutta, R. Loganayagam, and P. Surowka, “Hydrodynamics from charged black branes,”JHEP 01(2011) 094,0809.2596

    Pith/arXiv arXiv 2011

  59. [59]

    Gravity and Hydrodynamics: Lectures on the fluid-gravity correspondence,

    M. Rangamani, “Gravity and Hydrodynamics: Lectures on the fluid-gravity correspondence,”Class. Quant. Grav.26(2009) 224003,0905.4352

    Pith/arXiv arXiv 2009

  60. [60]

    The fluid/gravity correspondence,

    V. E. Hubeny, S. Minwalla, and M. Rangamani, “The fluid/gravity correspondence,” inTheoretical Advanced Study Institute in Elementary Particle Physics: String theory and its Applications: From meV to the Planck Scale, pp. 348–383. 2012.1107.5780

    arXiv 2012

  61. [61]

    Most general AdS3 boundary conditions,

    D. Grumiller and M. Riegler, “Most general AdS3 boundary conditions,”JHEP10(2016) 023, 1608.01308

    Pith/arXiv arXiv 2016

  62. [62]

    Near horizon soft hair as microstates of three dimensional black holes,

    H. Afshar, D. Grumiller, and M. M. Sheikh-Jabbari, “Near horizon soft hair as microstates of three dimensional black holes,”Phys. Rev.D96(2017), no. 8, 084032,1607.00009

    Pith/arXiv arXiv 2017

  63. [63]

    Role of surface integrals in the Hamiltonian formulation of general relativity,

    T. Regge and C. Teitelboim, “Role of surface integrals in the Hamiltonian formulation of general relativity,” Ann. Phys.88(1974) 286

    1974

  64. [64]

    Covariant theory of asymptotic symmetries, conservation laws and central charges,

    G. Barnich and F. Brandt, “Covariant theory of asymptotic symmetries, conservation laws and central charges,”Nucl. Phys.B633(2002) 3–82, hep-th/0111246

    Pith/arXiv arXiv 2002

  65. [65]

    Local symmetries and constraints,

    J. Lee and R. M. Wald, “Local symmetries and constraints,”J. Math. Phys.31(1990) 725–743

    1990

  66. [66]

    Some properties of Nöther charge and a proposal for dynamical black hole entropy,

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

    Pith/arXiv arXiv 1994

  67. [67]

    A General definition of ’conserved quantities’ in general relativity and other theories of gravity,

    R. M. Wald and A. Zoupas, “A General definition of ’conserved quantities’ in general relativity and other theories of gravity,”Phys.Rev.D61(2000) 084027, gr-qc/9911095

    Pith/arXiv arXiv 2000

  68. [68]

    Quasilocal energy and conserved charges derived from the gravitational action,

    J. D. Brown and J. W. York, Jr., “Quasilocal energy and conserved charges derived from the gravitational action,”Phys. Rev.D47(1993) 1407–1419

    1993

  69. [69]

    A stress tensor for anti-de Sitter gravity,

    V. Balasubramanian and P. Kraus, “A stress tensor for anti-de Sitter gravity,”Commun. Math. Phys.208 (1999) 413–428,hep-th/9902121

    Pith/arXiv arXiv 1999

  70. [70]

    Black hole entropy from near-horizon microstates,

    A. Strominger, “Black hole entropy from near-horizon microstates,”JHEP02(1998) 009,hep-th/9712251

    Pith/arXiv arXiv 1998

  71. [71]

    The largeNlimit of superconformal field theories and supergravity,

    J. M. Maldacena, “The largeNlimit of superconformal field theories and supergravity,”Adv. Theor. Math. Phys.2(1998) 231–252,hep-th/9711200

    Pith/arXiv arXiv 1998

  72. [72]

    Anti-de Sitter space and holography,

    E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys.2(1998) 253–291, hep-th/9802150

    Pith/arXiv arXiv 1998

  73. [73]

    Gauge theory correlators from non-critical string theory,

    S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, “Gauge theory correlators from non-critical string theory,”Phys. Lett.B428(1998) 105–114, hep-th/9802109

    Pith/arXiv arXiv 1998

  74. [74]

    Emergence of non-linear electrodynamic theories from TT¯-like deformations,

    H. Babaei-Aghbolagh, K. B. Velni, D. M. Yekta, and H. Mohammadzadeh, “Emergence of non-linear electrodynamic theories from TT¯-like deformations,” Phys. Lett. B829(2022) 137079,2202.11156

    Pith/arXiv arXiv 2022

  75. [75]

    Marginal TT¯-like deformation and modified Maxwell theories in two dimensions,

    H. Babaei-Aghbolagh, K. Babaei Velni, D. Mahdavian Yekta, and H. Mohammadzadeh, “Marginal TT¯-like deformation and modified Maxwell theories in two dimensions,”Phys. Rev. D 106(2022), no. 8, 086022,2206.12677

    Pith/arXiv arXiv 2022

  76. [76]

    Metric approach to aT T-like deformation in arbitrary dimensions,

    R. Conti, J. Romano, and R. Tateo, “Metric approach to aT T-like deformation in arbitrary dimensions,” JHEP09(2022) 085,2206.03415

    Pith/arXiv arXiv 2022

  77. [77]

    Root-T¯TDeformations in Two-Dimensional Quantum Field Theories,

    C. Ferko, A. Sfondrini, L. Smith, and G. Tartaglino-Mazzucchelli, “Root-T¯TDeformations in Two-Dimensional Quantum Field Theories,”Phys. Rev. Lett.129(2022), no. 20, 201604,2206.10515

    Pith/arXiv arXiv 2022

  78. [78]

    Geometric Formulation of Generalized Root-TT¯Deformations,

    H. Babaei-Aghbolagh, S. He, T. Morone, H. Ouyang, and R. Tateo, “Geometric Formulation of Generalized Root-TT¯Deformations,”Phys. Rev. Lett.133 (2024), no. 11, 111602,2405.03465. 25

    Pith/arXiv arXiv 2024

  79. [79]

    Root-TT¯deformed boundary conditions in holography,

    S. Ebert, C. Ferko, and Z. Sun, “Root-TT¯deformed boundary conditions in holography,”Phys. Rev. D107 (2023), no. 12, 126022,2304.08723

    Pith/arXiv arXiv 2023

  80. [80]

    Nonlinear automorphism of the conformal algebra in 2D and continuous √ T T deformations,

    D. Tempo and R. Troncoso, “Nonlinear automorphism of the conformal algebra in 2D and continuous √ T T deformations,”JHEP12(2022) 129,2210.00059

    Pith/arXiv arXiv 2022

Showing first 80 references.

This paper was first reviewed by deepseek-v4-flash on August 2, 2026.