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arxiv: 2510.25688 · v1 · pith:D7AE2UEVnew · submitted 2025-10-29 · ✦ hep-th · gr-qc· quant-ph

Conformal Blocks in 2d Carrollian/Galilean CFTs and Excited State Entanglement Entropy

Peng-Xiang Hao , Shunta Takahashi This is my paper

Pith reviewed 2026-05-21 19:17 UTC · model grok-4.3

classification ✦ hep-th gr-qcquant-ph
keywords Carrollian CFTGalilean CFTentanglement entropyheavy-light conformal blockflat space holographyswing surfaceeigenstate thermalization
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0 comments X

The pith

Entanglement entropy of highly excited states in 2d Carrollian and Galilean CFTs takes a thermal form that matches flat-space holography.

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

The paper derives the heavy-light conformal block in the large central charge limit of 2d Carrollian and Galilean CFTs by absorbing the backreaction of heavy operators into a conformal coordinate transformation. Applying the replica trick to this block shows that the entanglement entropy of highly excited states assumes a thermal expression. This field theory result exactly reproduces the holographic entanglement entropy obtained from the swing surface proposal in 3d Einstein gravity for spinning particles and flat space cosmological solutions. The match yields a dictionary relating the boundary weight Δ and charge ξ to the bulk mass m and angular momentum j, providing a consistency check for the flat space / CFT correspondence.

Core claim

In the large central charge limit, the backreaction of heavy operators is absorbed by a Carrollian or Galilean conformal coordinate transformation, yielding an explicit heavy-light conformal block. The replica trick then produces an entanglement entropy for highly excited states that takes a thermal form. This expression precisely equals the holographic entanglement entropy computed via swing surfaces in 3d Einstein gravity for backgrounds dual to spinning particles and flat space cosmologies, thereby establishing the dictionary between boundary state parameters (Δ, ξ) and bulk parameters (m, j).

What carries the argument

The heavy-light conformal block derived by absorbing heavy operator backreaction through a C/G conformal coordinate transformation in the large central charge limit.

If this is right

  • The eigenstate thermalization hypothesis holds for highly excited states in these theories.
  • A precise bulk-boundary dictionary maps CFT weight Δ and charge ξ to gravity mass m and angular momentum j.
  • The swing surface proposal for holographic entanglement entropy is confirmed for spinning particle and flat space cosmological backgrounds.
  • The replica trick combined with the coordinate-transformed block provides a general method for computing excited-state quantities in C/G CFTs.

Where Pith is reading between the lines

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

  • Similar thermalization patterns may appear in other flat-space holographic models beyond Einstein gravity.
  • The coordinate transformation technique could extend to higher-point correlators or dynamical quantities like out-of-time-order correlators.
  • Explicit checks in known BMS-invariant field theories would test whether the thermal form persists beyond the large central charge limit.

Load-bearing premise

The backreaction of heavy operators can be absorbed by a C/G conformal coordinate transformation in the large central charge limit.

What would settle it

A direct evaluation of the four-point function or entanglement entropy for specific heavy operator weights and charges in a solvable C/G CFT that deviates from the predicted thermal form would falsify the result.

read the original abstract

We advance the study of flat space holography by computing the entanglement entropy of highly excited states in two-dimensional Carrollian/Galilean Conformal Field Theories (C/G CFTs). Our approach is centered on a novel, physically intuitive derivation of the heavy-light conformal block in the large central charge limit, where the backreaction of heavy operators is absorbed by a C/G conformal coordinate transformation. Using this result and the replica trick, we find that the entanglement entropy of highly excited states assumes a thermal form, providing a concrete realization of the Eigenstate Thermalization Hypothesis (ETH). This field-theoretic result perfectly reproduces the holographic entanglement entropy computed via the swing surface proposal in three-dimensional Einstein gravity, for backgrounds corresponding to spinning particles and Flat Space Cosmological solutions. This agreement establishes a precise dictionary relating the weight $\Delta$ and charge $\xi$ of the boundary state to the mass $m$ and angular momentum $j$ of the dual spacetime, offering a powerful consistency check for the Flat/CCFT correspondence.

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

1 major / 1 minor

Summary. The manuscript derives the heavy-light conformal block in 2d Carrollian/Galilean CFTs in the large central charge limit by absorbing the backreaction of heavy operators through a C/G conformal coordinate transformation. It then employs the replica trick to show that the entanglement entropy of highly excited states takes a thermal form, exactly reproducing the holographic entanglement entropy obtained from the swing surface proposal in 3d Einstein gravity for spinning particles and Flat Space Cosmological solutions, and establishes a dictionary mapping boundary weight Δ and charge ξ to bulk mass m and angular momentum j.

Significance. If the central derivation holds, the result supplies a direct field-theoretic realization of the Eigenstate Thermalization Hypothesis in Carrollian/Galilean CFTs and furnishes a non-trivial consistency check for the flat-space/CCFT correspondence. The explicit matching between the CFT entanglement entropy and the swing-surface holographic computation strengthens the proposed duality.

major comments (1)
  1. [abstract and derivation of heavy-light block] The novel derivation of the heavy-light block (abstract and the paragraph describing the coordinate transformation): the assertion that a C/G conformal coordinate change absorbs all backreaction in the large-c limit requires an explicit verification that the transformed Carrollian/Galilean stress tensor (or its null components) contains no residual position-dependent terms. Because the BMS-like or Galilean algebra possesses a different generator structure and central extensions than the Virasoro algebra, it is not immediate that the same coordinate redefinition nullifies the Ward identities as it does in the relativistic case; any surviving non-thermal contribution would alter the replica-trick result away from the claimed thermal form and from the swing-surface holographic value.
minor comments (1)
  1. [abstract] The abstract states that the result 'perfectly reproduces' the holographic entanglement entropy but supplies no explicit equations, error estimates, or numerical checks; including at least the leading-order expressions for the transformed block and the resulting EE would improve verifiability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comment. We address the point regarding the heavy-light block derivation below and have revised the manuscript to include additional explicit verification.

read point-by-point responses

  1. Referee: [abstract and derivation of heavy-light block] The novel derivation of the heavy-light block (abstract and the paragraph describing the coordinate transformation): the assertion that a C/G conformal coordinate change absorbs all backreaction in the large-c limit requires an explicit verification that the transformed Carrollian/Galilean stress tensor (or its null components) contains no residual position-dependent terms. Because the BMS-like or Galilean algebra possesses a different generator structure and central extensions than the Virasoro algebra, it is not immediate that the same coordinate redefinition nullifies the Ward identities as it does in the relativistic case; any surviving non-thermal contribution would alter the replica-trick result away from the claimed thermal form and from the swing-surface holographic value.

    Authors: We appreciate the referee drawing attention to this subtlety. The coordinate transformation employed in our derivation is constructed precisely so that the heavy operators' backreaction is absorbed into a redefinition of the Carrollian/Galilean coordinates, yielding a constant stress tensor in the new frame. Although the BMS-like algebra differs from the Virasoro case, the transformation law for the null components of the stress tensor under C/G conformal maps preserves the central extensions while eliminating position dependence when the map is chosen to match the heavy-state weights and charges. We have verified this explicitly by direct substitution into the transformed stress-tensor expression, confirming that all residual position-dependent terms vanish in the large-central-charge limit; the resulting constant value reproduces the thermal form used in the replica-trick calculation. This verification appears in the main text following the coordinate transformation and is consistent with the subsequent match to the swing-surface holographic entropy. To make the algebra-specific details fully transparent, we will add a short appendix containing the step-by-step transformation of the null stress-tensor components and the check that the Ward identities reduce to the thermal case. revision: yes

Circularity Check

0 steps flagged

Heavy-light block derived via coordinate transformation; thermal EE and holographic match are independent consistency checks

full rationale

The paper's central chain begins with a claimed novel derivation of the heavy-light conformal block by absorbing heavy-operator backreaction into a C/G conformal coordinate transformation in the large-central-charge limit. The replica trick is then applied to this block to obtain the thermal form of entanglement entropy for highly excited states. This field-theoretic result is compared to the swing-surface holographic computation, yielding a dictionary between boundary parameters (Δ, ξ) and bulk parameters (m, j) as a consistency check. No quoted equations or steps reduce the derivation to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation chain; the coordinate transformation and replica-trick steps are presented as independent of the final holographic match. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the large-central-charge approximation and the assumption that a single coordinate transformation fully captures heavy-operator backreaction; no free parameters or new entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption Large central charge limit permits absorption of heavy-operator backreaction by a C/G conformal coordinate transformation
    Invoked to derive the heavy-light conformal block (abstract description of novel derivation).

pith-pipeline@v0.9.0 · 5716 in / 1402 out tokens · 40066 ms · 2026-05-21T19:17:04.853414+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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

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Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages · cited by 1 Pith paper · 27 internal anchors

  1. [1]

    Holographic Derivation of Entanglement Entropy from AdS/CFT

    S. Ryu and T. Takayanagi,Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96(2006) 181602, [hep-th/0603001]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  2. [2]

    Aspects of the BMS/CFT correspondence

    G. Barnich and C. Troessaert,Aspects of the BMS/CFT correspondence, JHEP05(2010) 062, [arXiv:1001.1541]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  3. [3]

    The BMS/GCA correspondence

    A. Bagchi,Correspondence between Asymptotically Flat Spacetimes and Nonrelativistic Conformal Field Theories, Phys. Rev. Lett.105(2010) 171601, [arXiv:1006.3354]. – 28 –

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  4. [4]

    Flat-Space Energy-Momentum Tensor from BMS/GCA Correspondence

    R. Fareghbal and A. Naseh,Flat-Space Energy-Momentum Tensor from BMS/GCA Correspondence, JHEP03(2014) 005, [arXiv:1312.2109]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  5. [5]

    The Carrollian Kaleidoscope

    A. Bagchi, A. Banerjee, P. Dhivakar, S. Mondal, and A. Shukla,The Carrollian Kaleidoscope,arXiv:2506.16164

    work page internal anchor Pith review Pith/arXiv arXiv

  6. [6]

    Bacry and J.-M

    H. Bacry and J.-M. L´ evy-Leblond,Possible kinematics, Journal of Mathematical Physics9 (1968), no. 10 1605–1614

    work page 1968

  7. [7]

    Bergshoeff, J

    E. Bergshoeff, J. Gomis, and G. Longhi,Dynamics of carroll particles, Classical and Quantum Gravity31(2014), no. 20 205009

    work page 2014

  8. [8]

    Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time

    C. Duval, G. W. Gibbons, P. A. Horvathy, and P. M. Zhang,Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time, Class. Quant. Grav.31(2014) 085016, [arXiv:1402.0657]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  9. [9]

    Radiation and Boundary Conditions in the Theory of Gravitation

    A. Trautman,Radiation and Boundary Conditions in the Theory of Gravitation, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys.6(1958), no. 6 407–412, [arXiv:1604.03145]

    work page internal anchor Pith review Pith/arXiv arXiv 1958

  10. [10]

    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]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  11. [11]

    Entropy of three-dimensional asymptotically flat cosmological solutions

    G. Barnich,Entropy of three-dimensional asymptotically flat cosmological solutions, JHEP 10(2012) 095, [arXiv:1208.4371]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  12. [12]

    Holography of 3d Flat Cosmological Horizons

    A. Bagchi, S. Detournay, R. Fareghbal, and J. Sim´ on,Holography of 3D Flat Cosmological Horizons, Phys. Rev. Lett.110(2013), no. 14 141302, [arXiv:1208.4372]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  13. [13]

    Three-dimensional Bondi-Metzner-Sachs invariant two-dimensional field theories as the flat limit of Liouville theory

    G. Barnich, A. Gomberoff, and H. A. Gonz´ alez,Three-dimensional Bondi-Metzner-Sachs invariant two-dimensional field theories as the flat limit of Liouville theory, Phys. Rev. D87 (2013), no. 12 124032, [arXiv:1210.0731]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  14. [14]

    Barnich, H

    G. Barnich, H. A. Gonzalez, A. Maloney, and B. Oblak,One loop partition function of three-dimensional flat gravity, Journal of High Energy Physics2015(2015), no. 4 1–8

    work page 2015

  15. [15]

    Chen and Z

    B. Chen and Z. Hu,Bulk reconstruction in flat holography, JHEP03(2024) 064, [arXiv:2312.13574]

    work page arXiv 2024

  16. [16]

    P.-X. Hao, K. Shinmyo, Y.-k. Suzuki, S. Takahashi, and T. Takayanagi,Bulk Reconstruction of Scalar Excitations in Flat 3/CCFT2 and the Flat Limit from (A)dS 3/CFT2, arXiv:2505.20084

    work page arXiv

  17. [17]

    P.-X. Hao, N. Ogawa, T. Takayanagi, and T. Waki,Flat Space Holography via AdS/BCFT, arXiv:2509.00652

    work page arXiv

  18. [18]

    Entanglement entropy in Galilean conformal field theories and flat holography

    A. Bagchi, R. Basu, D. Grumiller, and M. Riegler,Entanglement entropy in Galilean conformal field theories and flat holography, Phys. Rev. Lett.114(2015), no. 11 111602, [arXiv:1410.4089]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  19. [19]

    Jiang, W

    H. Jiang, W. Song, and Q. Wen,Entanglement Entropy in Flat Holography, JHEP07(2017) 142, [arXiv:1706.07552]

    work page arXiv 2017

  20. [20]

    On Representations and Correlation Functions of Galilean Conformal Algebras

    A. Bagchi and I. Mandal,On Representations and Correlation Functions of Galilean Conformal Algebras, Phys. Lett. B675(2009) 393–397, [arXiv:0903.4524]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  21. [21]

    The BMS Bootstrap

    A. Bagchi, M. Gary, and Zodinmawia,Bondi-Metzner-Sachs bootstrap, Phys. Rev. D96 (2017), no. 2 025007, [arXiv:1612.01730]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  22. [22]

    The nuts and bolts of the BMS Bootstrap

    A. Bagchi, M. Gary, and Zodinmawia,The Nuts and Bolts of the BMS Bootstrap, Class. Quant. Grav.34(2017), no. 17 174002, [arXiv:1705.05890]. – 29 –

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  23. [23]

    Chen, P.-X

    B. Chen, P.-X. Hao, R. Liu, and Z.-F. Yu,On Galilean conformal bootstrap, JHEP06(2021) 112, [arXiv:2011.11092]

    work page arXiv 2021

  24. [24]

    Chen, P.-x

    B. Chen, P.-x. Hao, R. Liu, and Z.-f. Yu,On Galilean conformal bootstrap. Part II.ξ= 0 sector, JHEP12(2022) 019, [arXiv:2207.01474]

    work page arXiv 2022

  25. [25]

    P.-x. Hao, W. Song, X. Xie, and Y. Zhong,A BMS-invariant Free Scalar Model, Phys. Rev. D105(2022), no. 12 125005, [arXiv:2111.04701]

    work page arXiv 2022

  26. [26]

    B. Chen, R. Liu, and Y.-f. Zheng,On higher-dimensional Carrollian and Galilean conformal field theories, SciPost Phys.14(2023), no. 5 088, [arXiv:2112.10514]

    work page arXiv 2023

  27. [27]

    Yu and B

    Z.-f. Yu and B. Chen,Free field realization of the BMS Ising model, JHEP08(2023) 116, [arXiv:2211.06926]

    work page arXiv 2023

  28. [28]

    P.-X. Hao, W. Song, Z. Xiao, and X. Xie,BMS-invariant free fermion models, Phys. Rev. D 109(2024), no. 2 025002, [arXiv:2211.06927]

    work page arXiv 2024

  29. [29]

    Banerjee, S

    A. Banerjee, S. Dutta, and S. Mondal,Carroll fermions in two dimensions, Phys. Rev. D 107(2023), no. 12 125020, [arXiv:2211.11639]

    work page arXiv 2023

  30. [30]

    de Boer, J

    J. de Boer, J. Hartong, N. A. Obers, W. Sybesma, and S. Vandoren,Carroll stories, JHEP 09(2023) 148, [arXiv:2307.06827]

    work page arXiv 2023

  31. [31]

    Hao, W.-X

    P.-X. Hao, W.-X. Lai, W. Song, and Z. Xiao,Modular Hamiltonian and entanglement entropy in the BMS free fermion theory,arXiv:2507.10503

    work page arXiv

  32. [32]

    Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere

    S. Pasterski, S.-H. Shao, and A. Strominger,Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere, Phys. Rev. D96(2017), no. 6 065026, [arXiv:1701.00049]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  33. [33]

    A Conformal Basis for Flat Space Amplitudes

    S. Pasterski and S.-H. Shao,Conformal basis for flat space amplitudes, Phys. Rev. D96 (2017), no. 6 065022, [arXiv:1705.01027]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  34. [34]

    Carrollian Perspective on Celestial Holography

    L. Donnay, A. Fiorucci, Y. Herfray, and R. Ruzziconi,Carrollian Perspective on Celestial Holography, Phys. Rev. Lett.129(2022), no. 7 071602, [arXiv:2202.04702]

    work page internal anchor Pith review arXiv 2022

  35. [35]

    Donnay, A

    L. Donnay, A. Fiorucci, Y. Herfray, and R. Ruzziconi,Bridging Carrollian and celestial holography, Phys. Rev. D107(2023), no. 12 126027, [arXiv:2212.12553]

    work page arXiv 2023

  36. [36]

    Bagchi, S

    A. Bagchi, S. Banerjee, R. Basu, and S. Dutta,Scattering Amplitudes: Celestial and Carrollian, Phys. Rev. Lett.128(2022), no. 24 241601, [arXiv:2202.08438]

    work page arXiv 2022

  37. [37]

    Bagchi, P

    A. Bagchi, P. Dhivakar, and S. Dutta,AdS Witten diagrams to Carrollian correlators, JHEP 04(2023) 135, [arXiv:2303.07388]

    work page arXiv 2023

  38. [38]

    Semi-classical BMS$_3$ blocks and flat holography

    E. Hijano,Semi-classical BMS 3 blocks and flat holography, JHEP10(2018) 044, [arXiv:1805.00949]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  39. [39]

    A. L. Fitzpatrick, J. Kaplan, and M. T. Walters,Virasoro Conformal Blocks and Thermality from Classical Background Fields, JHEP11(2015) 200, [arXiv:1501.05315]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  40. [40]

    Entanglement Entropy at Large Central Charge

    T. Hartman,Entanglement Entropy at Large Central Charge, JHEP09(2013) 145, [arXiv:1303.6955]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  41. [41]

    C. T. Asplund, A. Bernamonti, F. Galli, and T. Hartman,Entanglement Scrambling in 2D Conformal Field Theory, JHEP09(2015) 110, [arXiv:1506.03772]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  42. [42]

    Apolo, H

    L. Apolo, H. Jiang, W. Song, and Y. Zhong,Swing surfaces and holographic entanglement beyond AdS/CFT, JHEP12(2020) 064, [arXiv:2006.10740]

    work page arXiv 2020

  43. [43]

    Entanglement entropy and conformal field theory

    P. Calabrese and J. Cardy,Entanglement entropy and conformal field theory, J. Phys. A42 (2009) 504005, [arXiv:0905.4013]. – 30 –

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  44. [44]

    Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions

    G. Barnich and G. Compere,Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions, Class. Quant. Grav.24(2007) F15–F23, [gr-qc/0610130]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  45. [45]

    Time-dependent Orbifolds and String Cosmology

    L. Cornalba and M. S. Costa,Time dependent orbifolds and string cosmology, Fortsch. Phys. 52(2004) 145–199, [hep-th/0310099]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  46. [46]

    Generalized gravitational entropy

    A. Lewkowycz and J. Maldacena,Generalized gravitational entropy, JHEP08(2013) 090, [arXiv:1304.4926]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  47. [47]

    D. L. Jafferis, A. Lewkowycz, J. Maldacena, and S. J. Suh,Relative entropy equals bulk relative entropy, JHEP06(2016) 004, [arXiv:1512.06431]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  48. [48]

    Apolo, H

    L. Apolo, H. Jiang, W. Song, and Y. Zhong,Modular Hamiltonians in flat holography and (W)AdS/WCFT, JHEP09(2020) 033, [arXiv:2006.10741]. – 31 –

    work page arXiv 2020