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arxiv: 2606.09346 · v2 · pith:E3VX3EMJnew · submitted 2026-06-08 · ✦ hep-th

Asymptotic Algebras and Holography of Information in CGHS Model

Waheed A. Dar , Nirmalya Kajuri , Rinkesh Panigrahi This is my paper

Pith reviewed 2026-06-27 15:54 UTC · model grok-4.3

classification ✦ hep-th
keywords CGHS modelholography of informationasymptotic algebrasisland formulablack hole information paradoxfuture null infinityright-moving modes
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0 comments X

The pith

The quantum state of right-moving modes can be recovered from an arbitrarily small neighbourhood of the right future null infinity in the CGHS model.

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

This paper studies candidate algebras for the CGHS model to address differences between holography of information and the island formula. It establishes the holography of information by constructing the radiative phase space and asymptotic algebras at future null infinity under standard assumptions. The authors prove that the quantum state of the right-moving modes can be recovered from an arbitrarily small neighbourhood of the right future null infinity. They argue that the existence of an island must violate the commutativity of left- and right-boundary algebras, which highlights a distinction between the two approaches in this model.

Core claim

Making the usual assumptions that are made in HoI literature, the authors establish HoI for the CGHS model coupled to a massless scalar by constructing the radiative phase space and asymptotic algebras at future null infinity. They prove that the quantum state of the right-moving modes can be recovered from an arbitrarily small neighbourhood of the right future null infinity. They then argue that the existence of an island must violate the commutativity of left- and right-boundary algebras and discuss the possible distinction between the HoI and island approaches for the CGHS model.

What carries the argument

The asymptotic algebras at future null infinity, which permit recovery of the right-moving modes quantum state from small neighborhoods.

Load-bearing premise

The usual assumptions that are made in HoI literature when establishing the algebras and recovery.

What would settle it

An explicit construction of an island in the CGHS model that preserves commutativity of the left- and right-boundary algebras would falsify the claimed violation.

read the original abstract

Holography of Information (HoI) and the island formula are two recent approaches to the black hole information loss paradox that yield different Page curves. The difference between the Page curves has been argued to be rooted in the algebra that each approach focuses on. In this paper, we study the candidate algebras for the CGHS model. Making the usual assumptions that are made in HoI literature, we establish HoI for the CGHS model coupled to a massless scalar by constructing the radiative phase space and asymptotic algebras at future null infinity. We prove that the quantum state of the right-moving modes can be recovered from an arbitrarily small neighbourhood of the right future null infinity. We then argue that the existence of an island must violate the commutativity of left- and right-boundary algebras. We discuss the possible distinction between the HoI and island approaches for the CGHS model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript constructs the radiative phase space and asymptotic algebras for the CGHS model coupled to a massless scalar, under the usual assumptions from the Holography of Information (HoI) literature. It proves that the quantum state of the right-moving modes can be recovered from an arbitrarily small neighborhood of the right future null infinity. The paper then argues that the existence of an island must violate the commutativity of left- and right-boundary algebras and discusses possible distinctions between the HoI and island approaches for the CGHS model.

Significance. If the results hold, the work supplies a concrete realization of HoI in the solvable CGHS model, including an explicit algebra construction and a recovery statement from a small neighborhood at null infinity. This could serve as a benchmark for comparing HoI with the island formula in 2D dilaton gravity and may help clarify the origin of differing Page curves. The commutativity-violation argument offers a potential diagnostic for distinguishing the two frameworks.

minor comments (2)
  1. The abstract and introduction invoke 'usual assumptions' from the HoI literature without an explicit enumerated list or dedicated paragraph; adding this (e.g., as a short subsection after the model definition) would make the independence of the recovery proof clearer to readers outside the immediate subfield.
  2. Notation for the left- and right-boundary algebras (introduced when discussing commutativity) should be defined with explicit symbols and commutation relations in the main text rather than relying solely on reference to prior HoI works.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of our results on the radiative phase space, asymptotic algebras, and the recovery statement in the CGHS model, as well as the recommendation for minor revision. No specific major comments are provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under external assumptions

full rationale

The paper states it proceeds by 'making the usual assumptions that are made in HoI literature' to construct the radiative phase space and asymptotic algebras for CGHS + massless scalar. It then proves recovery of the right-moving quantum state from an arbitrarily small neighborhood of right future null infinity and argues that an island would violate left/right algebra commutativity. These steps are presented as following from the algebra definitions rather than reducing to them by construction, renaming, or self-citation chains. No quoted equation or step equates a 'prediction' to a fitted input or imports uniqueness via overlapping-author citations. The result is independent once the standard HoI assumptions (external to this work) are granted, making this a normal non-circular case.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on 'usual assumptions' from HoI literature as the foundation for the algebra construction and recovery; no free parameters, invented entities, or additional axioms are mentioned in the abstract.

axioms (1)
  • domain assumption Usual assumptions made in HoI literature
    Invoked explicitly when establishing the algebras and proving state recovery; these are treated as background without further justification in the abstract.

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discussion (0)

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Reference graph

Works this paper leans on

88 extracted references · 74 canonical work pages · 34 internal anchors

  1. [1]

    S. W. Hawking, Phys. Rev. D14, 2460 (1976)

    1976

  2. [2]

    D. N. Page, Phys. Rev. Lett.71, 1291 (1993), arXiv:gr-qc/9305007

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  3. [3]

    D. N. Page, Phys. Rev. Lett.71, 3743 (1993), arXiv:hep-th/9306083. 17

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  4. [4]

    Laddha, S

    A. Laddha, S. G. Prabhu, S. Raju, and P. Shrivastava, SciPost Phys.10, 041 (2021), arXiv:2002.02448 [hep-th]

    work page arXiv 2021

  5. [5]

    Chowdhury, O

    C. Chowdhury, O. Papadoulaki, and S. Raju, SciPost Phys.10, 106 (2021), arXiv:2008.01740 [hep-th]

    work page arXiv 2021

  6. [6]

    Chowdhury, V

    C. Chowdhury, V. Godet, O. Papadoulaki, and S. Raju, JHEP03, 019 (2022), arXiv:2107.14802 [hep-th]

    work page arXiv 2022

  7. [7]

    Laddha, S

    A. Laddha, S. G. Prabhu, S. Raju, and P. Shrivastava, (2022), arXiv:2207.06406 [hep-th]

    work page arXiv 2022

  8. [8]

    Chakraborty, J

    T. Chakraborty, J. Chakravarty, V. Godet, P. Paul, and S. Raju, JHEP12, 120 (2023), arXiv:2303.16316 [hep-th]

    work page arXiv 2023

  9. [9]

    Jensen, S

    K. Jensen, S. Raju, and A. J. Speranza, JHEP06, 242 (2025), arXiv:2412.21185 [hep-th]

    work page arXiv 2025

  10. [10]

    Gaddam and A

    N. Gaddam and A. H, JHEP06, 034 (2025), arXiv:2410.17316 [hep-th]

    work page arXiv 2025

  11. [11]

    A toy model of black hole complementarity

    S. Banerjee, J.-W. Bryan, K. Papadodimas, and S. Raju, JHEP05, 004 (2016), arXiv:1603.02812 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  12. [12]

    Raju, SciPost Phys.6, 073 (2019), arXiv:1809.10154 [hep-th]

    S. Raju, SciPost Phys.6, 073 (2019), arXiv:1809.10154 [hep-th]

    work page arXiv 2019

  13. [13]

    Raju, Int

    S. Raju, Int. J. Mod. Phys. D28, 1944011 (2019), arXiv:1903.11073 [gr-qc]

    work page arXiv 2019

  14. [14]

    The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole

    A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, JHEP12, 063 (2019), arXiv:1905.08762 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  15. [15]

    Entanglement Wedge Reconstruction and the Information Paradox

    G. Penington, JHEP09, 002 (2020), arXiv:1905.08255 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  16. [16]

    The Page curve of Hawking radiation from semiclassical geometry

    A. Almheiri, R. Mahajan, J. Maldacena, and Y. Zhao, JHEP03, 149 (2020), arXiv:1908.10996 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  17. [17]

    Replica wormholes and the black hole interior

    G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, JHEP03, 205 (2022), arXiv:1911.11977 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  18. [18]

    Replica Wormholes and the Entropy of Hawking Radiation

    A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, JHEP05, 013 (2020), arXiv:1911.12333 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  19. [19]

    Almheiri, T

    A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, Rev. Mod. Phys.93, 035002 (2021), arXiv:2006.06872 [hep-th]

    work page arXiv 2021

  20. [20]

    Almheiri, R

    A. Almheiri, R. Mahajan, and J. Maldacena, (2019), arXiv:1910.11077 [hep-th]

    work page arXiv 2019

  21. [21]

    Almheiri, R

    A. Almheiri, R. Mahajan, and J. E. Santos, SciPost Phys.9, 001 (2020), arXiv:1911.09666 [hep-th]

    work page arXiv 2020

  22. [22]

    Antonini, C.-H

    S. Antonini, C.-H. Chen, H. Maxfield, and G. Penington, JHEP10, 034 (2025), arXiv:2506.04311 [hep-th]

    work page arXiv 2025

  23. [23]

    H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall, and M. Riojas, (2026), arXiv:2602.06543 [hep-th]

    work page arXiv 2026

  24. [24]

    C. G. Callan, Jr., S. B. Giddings, J. A. Harvey, and A. Strominger, Phys. Rev. D45, R1005 (1992), arXiv:hep-th/9111056

    work page internal anchor Pith review Pith/arXiv arXiv 1992

  25. [25]

    S. B. Giddings and W. M. Nelson, Phys. Rev. D46, 2486 (1992), arXiv:hep-th/9204072

    work page internal anchor Pith review Pith/arXiv arXiv 1992

  26. [26]

    S. P. de Alwis, Phys. Rev. D46, 5429 (1992), arXiv:hep-th/9207095

    work page internal anchor Pith review Pith/arXiv arXiv 1992

  27. [27]

    S. B. Giddings and A. Strominger, Phys. Rev. D47, 2454 (1993), arXiv:hep-th/9207034

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  28. [28]

    Liouville Models of Black Hole Evaporation

    A. Bilal and C. G. Callan, Jr., Nucl. Phys. B394, 73 (1993), arXiv:hep-th/9205089

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  29. [29]

    Les Houches Lectures on Black Holes

    A. Strominger, inNATO Advanced Study Institute: Les Houches Summer School, Session 62: Fluctuating Geometries in Statistical Mechanics and Field Theory(1994) arXiv:hep-th/9501071

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  30. [30]

    Observables for Two-Dimensional Black Holes

    J. Gegenberg, G. Kunstatter, and D. Louis-Martinez, Phys. Rev. D51, 1781 (1995), arXiv:gr-qc/9408015

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  31. [31]

    Quantum gravity effects in the CGHS model of collapse to a black hole

    M. Varadarajan, Phys. Rev. D57, 3463 (1998), arXiv:gr-qc/9801058

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  32. [32]

    Varadarajan, J

    M. Varadarajan, J. Phys. Conf. Ser.140, 012007 (2008)

    2008

  33. [33]

    Information is Not Lost in the Evaporation of 2-dimensional Black Holes

    A. Ashtekar, V. Taveras, and M. Varadarajan, Phys. Rev. Lett.100, 211302 (2008), arXiv:0801.1811 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  34. [34]

    Surprises in the Evaporation of 2-Dimensional Black Holes

    A. Ashtekar, F. Pretorius, and F. M. Ramazanoglu, Phys. Rev. Lett.106, 161303 (2011), arXiv:1011.6442 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  35. [35]

    Evaporation of 2-Dimensional Black Holes

    A. Ashtekar, F. Pretorius, and F. M. Ramazanoglu, Phys. Rev. D83, 044040 (2011), arXiv:1012.0077 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  36. [36]

    An Uneventful Horizon in Two Dimensions

    A. Almheiri and J. Sully, JHEP02, 108 (2014), arXiv:1307.8149 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  37. [37]

    J. G. Russo, L. Susskind, and L. Thorlacius, Phys. Rev. D46, 3444 (1992), arXiv:hep-th/9206070

    work page internal anchor Pith review Pith/arXiv arXiv 1992

  38. [38]

    T. M. Fiola, J. Preskill, A. Strominger, and S. P. Trivedi, Phys. Rev. D50, 3987 (1994), arXiv:hep-th/9403137

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  39. [39]

    Hartman, E

    T. Hartman, E. Shaghoulian, and A. Strominger, JHEP07, 022 (2020), arXiv:2004.13857 [hep-th]

    work page arXiv 2020

  40. [40]

    F. F. Gautason, L. Schneiderbauer, W. Sybesma, and L. Thorlacius, JHEP05, 091 (2020), arXiv:2004.00598 [hep-th]

    work page arXiv 2020

  41. [41]

    X. Wang, R. Li, and J. Wang, Phys. Rev. D103, 126026 (2021), arXiv:2104.00224 [hep-th]

    work page arXiv 2021

  42. [42]

    Yu and X.-H

    M.-H. Yu and X.-H. Ge, Phys. Rev. D107, 066020 (2023), arXiv:2208.01943 [hep-th]

    work page arXiv 2023

  43. [43]

    Anegawa and N

    T. Anegawa and N. Iizuka, JHEP07, 036 (2020), arXiv:2004.01601 [hep-th]

    work page arXiv 2020

  44. [44]

    Tian, Symmetry15, 1402 (2023), arXiv:2204.08751 [hep-th]

    J. Tian, Symmetry15, 1402 (2023), arXiv:2204.08751 [hep-th]

    work page arXiv 2023

  45. [45]

    Fitkevich, Phys

    M. Fitkevich, Phys. Rev. D113, 046009 (2026), arXiv:2507.17933 [hep-th]

    work page arXiv 2026

  46. [46]

    K. V. Kuchar, J. D. Romano, and M. Varadarajan, Phys. Rev. D55, 795 (1997), arXiv:gr-qc/9608011

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  47. [47]

    S. B. Giddings, inICTP Summer School in High-energy Physics and Cosmology(1994) pp. 0530–574, arXiv:hep-th/9412138

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  48. [48]

    Crnkovic and E

    C. Crnkovic and E. Witten, (1986)

    1986

  49. [49]

    G. J. Zuckerman, Conf. Proc. C8607214, 259 (1986)

    1986

  50. [50]

    Lee and R

    J. Lee and R. M. Wald, J. Math. Phys.31, 725 (1990)

    1990

  51. [51]

    Barnich, M

    G. Barnich, M. Henneaux, and C. Schomblond, Phys. Rev. D44, R939 (1991)

    1991

  52. [52]

    R. M. Wald, Phys. Rev. D48, R3427 (1993), arXiv:gr-qc/9307038

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  53. [53]

    Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy

    V. Iyer and R. M. Wald, Phys. Rev. D50, 846 (1994), arXiv:gr-qc/9403028

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  54. [54]

    A comparison of Noether charge and Euclidean methods for Computing the Entropy of Stationary Black Holes

    V. Iyer and R. M. Wald, Phys. Rev. D52, 4430 (1995), arXiv:gr-qc/9503052

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  55. [55]

    R. M. Wald and A. Zoupas, Phys. Rev. D61, 084027 (2000), arXiv:gr-qc/9911095

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  56. [56]

    Covariant theory of asymptotic symmetries, conservation laws and central charges

    G. Barnich and F. Brandt, Nucl. Phys. B633, 3 (2002), arXiv:hep-th/0111246

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  57. [57]

    Diffeomorphisms, Noether Charges and Canonical Formalism in 2D Dilaton Gravity

    J. Navarro-Salas, M. Navarro, and C. F. Talavera, Phys. Rev. D52, 6831 (1995), arXiv:hep-th/9411105

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  58. [58]

    Canonical Structure of 2D Black Holes

    J. Navarro-Salas, M. Navarro, and C. F. Talavera, Phys. Lett. B335, 334 (1994), arXiv:hep-th/9405015

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  59. [59]

    W. T. Kim and J. Lee, Int. J. Mod. Phys. A11, 553 (1996), arXiv:hep-th/9502078

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  60. [60]

    Ruzziconi and C

    R. Ruzziconi and C. Zwikel, JHEP04, 034 (2021), arXiv:2012.03961 [hep-th]

    work page arXiv 2021

  61. [61]

    J. F. Pedraza, A. Svesko, W. Sybesma, and M. R. Visser, JHEP12, 134 (2021), arXiv:2107.10358 [hep-th]

    work page arXiv 2021

  62. [62]

    McNees and C

    R. McNees and C. Zwikel, JHEP08, 154 (2023), arXiv:2306.16451 [hep-th]

    work page arXiv 2023

  63. [63]

    Ashtekar, J

    A. Ashtekar, J. Math. Phys.22, 2885 (1981)

    1981

  64. [64]

    Ashtekar and M

    A. Ashtekar and M. Streubel, Proc. Roy. Soc. Lond. A376, 585 (1981)

    1981

  65. [65]

    Ashtekar, Phys

    A. Ashtekar, Phys. Rev. Lett.46, 573 (1981). 18

    1981

  66. [66]

    Ashtekar,ASYMPTOTIC QUANTIZATION: BASED ON 1984 NAPLES LECTURES(1987)

    A. Ashtekar,ASYMPTOTIC QUANTIZATION: BASED ON 1984 NAPLES LECTURES(1987)

    1984

  67. [67]

    Asymptotic symmetries of gravity and soft theorems for massive particles

    M. Campiglia and A. Laddha, JHEP12, 094 (2015), arXiv:1509.01406 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  68. [68]

    On BMS Invariance of Gravitational Scattering

    A. Strominger, JHEP07, 152 (2014), arXiv:1312.2229 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  69. [69]

    Classically this follows from the radiative symplectic form, sinceM B(u) is the Hamiltonian for the step-function vector field Θ(y−u)∂ y

  70. [70]

    However, if we have access to both boundaries, then complete quantum information will be recovered even in this case

    If the quantum gravity effects which resolved the singularity introduces some entanglement between left- and right-moving modes, the entropy would be constant instead of zero. However, if we have access to both boundaries, then complete quantum information will be recovered even in this case

  71. [71]

    thunderpop

    Here is another argument why reflecting boundary conditions cannot explain the right boundary knowing about the left- movers. Recall that the reflecting boundary and the black hole singularity reside at the same locus—the critical curve Ω = Ωcr. Before the black hole forms, this curve is a regular timelike boundary where reflecting conditions can be impos...

  72. [72]

    Arias and D

    R. Arias and D. Fondevila, (2026), arXiv:2603.09763 [hep-th]

    work page arXiv 2026

  73. [73]

    Marolf and H

    D. Marolf and H. Maxfield, JHEP08, 044 (2020), arXiv:2002.08950 [hep-th]

    work page arXiv 2020

  74. [74]

    S. B. Giddings and G. J. Turiaci, JHEP09, 194 (2020), arXiv:2004.02900 [hep-th]

    work page arXiv 2020

  75. [75]

    Marolf and H

    D. Marolf and H. Maxfield, JHEP04, 272 (2021), arXiv:2010.06602 [hep-th]

    work page arXiv 2021

  76. [76]

    Engelhardt, S

    N. Engelhardt, S. Fischetti, and A. Maloney, Phys. Rev. D103, 046021 (2021), arXiv:2007.07444 [hep-th]

    work page arXiv 2021

  77. [77]

    P. Saad, S. H. Shenker, and S. Yao, JHEP10, 076 (2024), arXiv:2107.13130 [hep-th]

    work page arXiv 2024

  78. [78]

    P. Saad, S. H. Shenker, D. Stanford, and S. Yao, JHEP09, 133 (2024), arXiv:2103.16754 [hep-th]

    work page arXiv 2024

  79. [79]

    C. Peng, J. Tian, and J. Yu, (2021), arXiv:2111.14856 [hep-th]

    work page arXiv 2021

  80. [80]

    Blommaert, L

    A. Blommaert, L. V. Iliesiu, and J. Kruthoff, JHEP09, 080 (2022), arXiv:2111.07863 [hep-th]

    work page arXiv 2022

Showing first 80 references.