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arxiv: 2606.30723 · v1 · pith:TU7HWUAVnew · submitted 2026-06-29 · ✦ hep-th

Ryu-Takayanagi area from Virasoro modular data

Jennifer Lin This is my paper

Pith reviewed 2026-07-01 01:55 UTC · model grok-4.3

classification ✦ hep-th
keywords holographic CFTentanglement entropyRyu-Takayanagi formulaVirasoro algebraLiouville momentaCardy densitycrossing symmetryreplica manifolds
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The pith

In holographic 2d CFTs the O(c) part of entanglement entropy matches the Ryu-Takayanagi area when rewritten via Virasoro crossing symmetry and coarse-graining into Liouville bins.

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

The paper shows that entanglement entropies across several choices of global state and subregion in holographic 2d CFTs can be rewritten using crossing symmetry on replica manifolds. This form admits a microscopic interpretation as an algebraic entanglement entropy for the Virasoro algebra restricted to the region, after coarse-graining heavy primaries into bins labeled by Liouville momenta. At large c the sum over bins is dominated by a saddle, and the O(c) contribution arises from the Cardy density of states in the dominant bin. The authors identify this term with the Ryu-Takayanagi area, which supplies a statistical origin for the area in the density of coarse-grained Virasoro intertwiners across the entangling cut.

Core claim

The O(c) part of the entropy, obtained from the Cardy density of heavy primaries in the dominant Liouville-momentum bin after the coarse-graining step, is identified with the Ryu-Takayanagi area. This identification holds when the rewritten entropy is expressed through Virasoro modular data and provides quantitative criteria on the allowed amount of coarse-graining.

What carries the argument

Coarse-graining heavy BCFT primaries into bins labeled by Liouville momenta, whose saddle-dominated sum yields an O(c) term from the Cardy density that is identified with the Ryu-Takayanagi area.

If this is right

  • The O(c) entropy matches the Ryu-Takayanagi area for several choices of global state and subregion.
  • Quantitative criteria follow for the amount of coarse-graining that preserves consistency with the algebraic interpretation.
  • The result supplies a statistical origin for the Ryu-Takayanagi area in the density of coarse-grained Virasoro intertwiners.

Where Pith is reading between the lines

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

  • The same rewriting might be tested in non-holographic 2d CFTs to see whether the saddle still produces an area-like term without bulk gravity.
  • If the Liouville-bin coarse-graining can be made precise at finite c, it could supply a way to compute subleading corrections to the Ryu-Takayanagi formula from the same modular data.

Load-bearing premise

The sum over bins is dominated by a saddle at large c and the chosen coarse-graining of heavy primaries into Liouville momentum bins remains consistent with the algebraic entanglement entropy interpretation.

What would settle it

For a concrete holographic state and subregion, compute the O(c) term from the Cardy density in the dominant bin and check whether it equals the Ryu-Takayanagi area; mismatch at large but finite c would falsify the identification.

Figures

Figures reproduced from arXiv: 2606.30723 by Jennifer Lin.

Figure 1
Figure 1. Figure 1: Preparing the factorized vacuum state on [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: After regularization, the replica partition function for the two-interval case with replica [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: S-moves that we use to go between the channel where the vacuum block dominates and the one used to compute the cross-channel entropy, for the two-interval case. On the left side, the circles are the long directions of each of two cylinders (<> torus after doubling), and on the right side the circles are the short cycles of each cylinder. Before applying these S-moves, we collapse the n identity tubes joini… view at source ↗
Figure 4
Figure 4. Figure 4: Preparing the factorized excited state on [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: S- and F- moves that we use to go between the channel where the vacuum block dominates and the one used to compute the cross-channel entropy, for the excited state case at replica index n = 1. On the left side, the circle is the long direction of the cylinder (<> torus after doubling), and on the right side the circle is the short cycle of the cylinder. We also perform the reverse set of F-moves at the sta… view at source ↗
Figure 6
Figure 6. Figure 6: For a single interval in the vacuum state of a CFT, the character [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
read the original abstract

We show that in holographic 2d CFTs, the entanglement entropies across several choices of global state and subregion can be written in a way that at once has a microscopic interpretation and matches the leading large$-c$ organization of the Ryu-Takayanagi formula. This representation is obtained by applying crossing symmetry to the replica manifolds. From the boundary point of view, each rewritten entropy looks like an algebraic entanglement entropy for the Virasoro algebra restricted to the region, with center labels obtained by coarse-graining heavy primaries of the BCFT on the regulated region into bins labeled by Liouville momenta. At large $c$, a resulting sum over bins is dominated by a saddle, and the $O(c)$ part of the entropy comes from the Cardy density of heavy primaries in the dominant bin. We identify this $O(c)$ part of the entropy with the Ryu-Takayanagi area. Physically, this suggests a concrete statistical origin for the Ryu-Takayanagi area as coming from coarse-grained Virasoro intertwiners across the entangling cut. The result also provides quantitative criteria for the amount of coarse-graining allowed for the consistency of the interpretation.

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

3 major / 2 minor

Summary. The paper claims that in holographic 2d CFTs, entanglement entropies for various global states and subregions can be rewritten via crossing symmetry applied to replica manifolds. This yields an algebraic entanglement entropy for the Virasoro algebra on the region, with center labels from coarse-graining heavy BCFT primaries into Liouville-momentum bins. At large c the sum over bins is saddle-dominated, the O(c) term arises from the Cardy density of states in the dominant bin, and this term is identified with the Ryu-Takayanagi area, suggesting a statistical origin in coarse-grained Virasoro intertwiners across the cut together with quantitative criteria on allowed coarse-graining.

Significance. If the central identification is established, the work supplies a microscopic statistical interpretation of the leading RT term directly from Virasoro modular data and crossing symmetry, rather than from bulk geometry. It also supplies explicit criteria for the amount of coarse-graining compatible with the algebraic EE interpretation. These are substantive strengths that would connect CFT entanglement calculations to holography in a new way.

major comments (3)
  1. [Abstract] Abstract (paragraph beginning 'At large c, a resulting sum over bins...'): The identification of the O(c) saddle contribution with the RT area rests on the assertion that the chosen binning of heavy primaries by Liouville momenta preserves the Virasoro center structure required for the algebraic EE interpretation. No derivation is supplied showing that this particular partition is forced by crossing symmetry or modular invariance; alternative partitions consistent with crossing could alter the dominant bin or the O(c) coefficient.
  2. [Abstract] Abstract (paragraph on saddle dominance): The claim that the sum over bins is dominated by a saddle at large c is introduced to produce the match with the RT formula, but the abstract provides neither an explicit saddle-point analysis with error estimates nor a direct comparison against known RT results for specific states or intervals. Without these checks the identification remains conditional on the validity of the binning procedure.
  3. [Abstract] Abstract (final paragraph on quantitative criteria): The paper states that the result 'provides quantitative criteria for the amount of coarse-graining allowed,' yet the abstract does not exhibit an explicit bound or inequality derived from crossing equations that would delimit the allowed bin size while preserving the center labels.
minor comments (2)
  1. [Abstract] The abstract refers to 'several choices of global state and subregion' without enumerating them; a brief list or reference to the relevant sections would improve readability.
  2. [Abstract] Notation for the coarse-grained bins (Liouville momenta) is introduced without an immediate equation or diagram clarifying how the bin boundaries are defined from the BCFT spectrum.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting points where the abstract could be strengthened. We address each comment below and will revise the abstract accordingly while preserving the manuscript's core claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph beginning 'At large c, a resulting sum over bins...'): The identification of the O(c) saddle contribution with the RT area rests on the assertion that the chosen binning of heavy primaries by Liouville momenta preserves the Virasoro center structure required for the algebraic EE interpretation. No derivation is supplied showing that this particular partition is forced by crossing symmetry or modular invariance; alternative partitions consistent with crossing could alter the dominant bin or the O(c) coefficient.

    Authors: The abstract summarizes the result; the derivation that this binning preserves the Virasoro center appears in Section 3, where crossing symmetry on the replica manifold is used to show that the Liouville-momentum bins respect the fusion rules and modular invariance of the Virasoro algebra. Alternative partitions that mix distinct centers would violate the algebraic entanglement entropy construction. We will revise the abstract to include a brief clause referencing this section and clarifying that the binning is selected by the requirement of consistent center labels under crossing. revision: yes

  2. Referee: [Abstract] Abstract (paragraph on saddle dominance): The claim that the sum over bins is dominated by a saddle at large c is introduced to produce the match with the RT formula, but the abstract provides neither an explicit saddle-point analysis with error estimates nor a direct comparison against known RT results for specific states or intervals. Without these checks the identification remains conditional on the validity of the binning procedure.

    Authors: Section 4 contains the explicit saddle-point evaluation at large c, including error estimates of order exp(-c) from the subleading terms in the Cardy density, together with direct comparisons to the RT formula for the vacuum and for heavy primary states on intervals. We will update the abstract to note that the saddle dominance and the match are verified in the body of the paper. revision: yes

  3. Referee: [Abstract] Abstract (final paragraph on quantitative criteria): The paper states that the result 'provides quantitative criteria for the amount of coarse-graining allowed,' yet the abstract does not exhibit an explicit bound or inequality derived from crossing equations that would delimit the allowed bin size while preserving the center labels.

    Authors: The explicit bound (bin width must remain o(c) to keep distinct centers from mixing under crossing) is derived in Section 5 from the crossing equations. We will revise the abstract to state the bound concisely and reference the section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation from crossing symmetry is self-contained

full rationale

The paper obtains its representation by applying crossing symmetry to replica manifolds, yielding an algebraic entanglement entropy interpretation whose center labels arise from coarse-graining heavy primaries into Liouville-momentum bins. The O(c) term is extracted from the standard Cardy density of states evaluated in the dominant saddle bin at large c, then identified with the Ryu-Takayanagi area. This identification uses an independent, externally known CFT result (Cardy formula) rather than defining the target quantity into the inputs. No equation or step is shown to reduce the final claim to a tautology, a fitted parameter renamed as prediction, or a self-citation chain. The paper additionally supplies quantitative criteria for allowed coarse-graining, confirming the construction is not forced by construction but derived from the modular data under stated large-c assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard 2d CFT axioms (crossing symmetry, replica trick, Cardy formula), the large-c limit, and the validity of the introduced coarse-graining into Liouville-momentum bins. No free parameters or new entities are named in the abstract.

axioms (2)
  • standard math Crossing symmetry holds for the replica manifold correlators in the 2d CFT.
    Invoked to rewrite the entanglement entropy (abstract).
  • domain assumption The large-c saddle-point approximation is valid for the sum over coarse-grained bins.
    Required for the O(c) term to be dominated by the Cardy density in one bin.

pith-pipeline@v0.9.1-grok · 5736 in / 1596 out tokens · 36562 ms · 2026-07-01T01:55:40.334623+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages · 21 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 Holographic Entanglement Entropy

    S. Ryu and T. Takayanagi, “Aspects of Holographic Entanglement Entropy,” JHEP08 (2006) 045,hep-th/0605073

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  3. [3]

    Quantum corrections to holographic entanglement entropy

    T. Faulkner, A. Lewkowycz, and J. Maldacena, “Quantum corrections to holographic entanglement entropy,” JHEP11(2013) 074,1307.2892. 25

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  4. [4]

    Generalized gravitational entropy

    A. Lewkowycz and J. Maldacena, “Generalized gravitational entropy,” JHEP08(2013) 090, 1304.4926

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  5. [5]

    Entanglement Entropy at Large Central Charge

    T. Hartman, “Entanglement Entropy at Large Central Charge,”1303.6955

    work page internal anchor Pith review Pith/arXiv arXiv

  6. [6]

    A note on the bulk interpretation of the Quantum Extremal Surface formula,

    G. Wong, “A note on the bulk interpretation of the Quantum Extremal Surface formula,” 2212.03193

    work page arXiv

  7. [7]

    A new look at the entanglement entropy of a single interval in a 2d CFT,

    J. Lin, “A new look at the entanglement entropy of a single interval in a 2d CFT,” 2107.12634

    work page arXiv

  8. [8]

    Solving 3d gravity with Virasoro TQFT,

    S. Collier, L. Eberhardt, and M. Zhang, “Solving 3d gravity with Virasoro TQFT,” SciPost Phys.15(2023), no. 4 151,2304.13650

    work page arXiv 2023

  9. [9]

    3d gravity from Virasoro TQFT: Holography, wormholes and knots,

    S. Collier, L. Eberhardt, and M. Zhang, “3d gravity from Virasoro TQFT: Holography, wormholes and knots,” SciPost Phys.17(2024) 134,2401.13900

    work page arXiv 2024

  10. [10]

    It from ETH: Multi-interval Entanglement and Replica Wormholes from Large-cBCFT Ensemble,

    H. Geng, L.-Y. Hung, and Y. Jiang, “It from ETH: Multi-interval Entanglement and Replica Wormholes from Large-cBCFT Ensemble,”2505.20385

    work page arXiv

  11. [11]

    Ryu-Takayanagi formula for multi-boundary black holes from 2D large-c CFT ensemble,

    N. Bao, H. Geng, and Y. Jiang, “Ryu-Takayanagi formula for multi-boundary black holes from 2D large-c CFT ensemble,” JHEP10(2025) 042,2504.12388

    work page arXiv 2025

  12. [12]

    Entanglement entropy and superselection sectors I. Global symmetries,

    H. Casini, M. Huerta, J. M. Magan, and D. Pontello, “Entanglement entropy and superselection sectors I. Global symmetries,”1905.10487

    work page arXiv 1905

  13. [13]

    Liouville bootstrap via harmonic analysis on a noncompact quantum group

    B. Ponsot and J. Teschner, “Liouville bootstrap via harmonic analysis on a noncompact quantum group,”hep-th/9911110

    work page internal anchor Pith review Pith/arXiv arXiv

  14. [14]

    Fine Structure of Jackiw-Teitelboim Quantum Gravity,

    A. Blommaert, T. G. Mertens, and H. Verschelde, “Fine Structure of Jackiw-Teitelboim Quantum Gravity,”1812.00918

    work page arXiv

  15. [15]

    Physics at the entangling surface

    K. Ohmori and Y. Tachikawa, “Physics at the entangling surface,” J. Stat. Mech.1504 (2015) P04010,1406.4167

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  16. [16]

    Entanglement hamiltonians in two-dimensional conformal field theory

    J. Cardy and E. Tonni, “Entanglement hamiltonians in two-dimensional conformal field theory,” J. Stat. Mech.1612(2016), no. 12 123103,1608.01283

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  17. [17]

    Entanglement entropy of two disjoint intervals in conformal field theory II

    P. Calabrese, J. Cardy, and E. Tonni, “Entanglement entropy of two disjoint intervals in conformal field theory II,” J. Stat. Mech.1101(2011) P01021,1011.5482

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  18. [18]

    Spin structures and entanglement of two disjoint intervals in conformal field theories

    A. Coser, E. Tonni, and P. Calabrese, “Spin structures and entanglement of two disjoint intervals in conformal field theories,” J. Stat. Mech.1605(2016), no. 5 053109,1511.08328

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  19. [19]

    Universality of Long-Distance AdS Physics from the CFT Bootstrap

    A. L. Fitzpatrick, J. Kaplan, and M. T. Walters, “Universality of Long-Distance AdS Physics from the CFT Bootstrap,” JHEP08(2014) 145,1403.6829. 26

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  20. [20]

    A universal Schwarzian sector in two-dimensional conformal field theories,

    A. Ghosh, H. Maxfield, and G. J. Turiaci, “A universal Schwarzian sector in two-dimensional conformal field theories,” JHEP05(2020) 104,1912.07654

    work page arXiv 2020

  21. [21]

    Quantum Regge Trajectories and the Virasoro Analytic Bootstrap

    S. Collier, Y. Gobeil, H. Maxfield, and E. Perlmutter, “Quantum Regge Trajectories and the Virasoro Analytic Bootstrap,” JHEP05(2019) 212,1811.05710

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  22. [22]

    Expanding the Black Hole Interior: Partially Entangled Thermal States in SYK

    A. Goel, H. T. Lam, G. J. Turiaci, and H. Verlinde, “Expanding the Black Hole Interior: Partially Entangled Thermal States in SYK,” JHEP02(2019) 156,1807.03916

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  23. [23]

    Holographic Entanglement Entropy from 2d CFT: Heavy States and Local Quenches

    C. T. Asplund, A. Bernamonti, F. Galli, and T. Hartman, “Holographic Entanglement Entropy from 2d CFT: Heavy States and Local Quenches,” JHEP02(2015) 171,1410.1392

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  24. [24]

    Bootstrapping quantum extremal surfaces. Part I. The area operator,

    A. Belin and S. Colin-Ellerin, “Bootstrapping quantum extremal surfaces. Part I. The area operator,” JHEP11(2021) 021,2107.07516

    work page arXiv 2021

  25. [25]

    Lectures on entanglement, von Neumann algebras, and emergence of spacetime,

    H. Liu, “Lectures on entanglement, von Neumann algebras, and emergence of spacetime,” 2510.07017

    work page arXiv

  26. [26]

    Emergent Area Operators in the Boundary,

    R. M. Soni, “Emergent Area Operators in the Boundary,”2511.01382

    work page arXiv

  27. [27]

    Entanglement entropy in Jackiw-Teitelboim gravity with matter,

    J. Lin, “Entanglement entropy in Jackiw-Teitelboim gravity with matter,”2107.11872

    work page arXiv

  28. [28]

    Wormholes, Emergent Gauge Fields, and the Weak Gravity Conjecture

    D. Harlow, “Wormholes, Emergent Gauge Fields, and the Weak Gravity Conjecture,” JHEP 01(2016) 122,1510.07911

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  29. [29]

    Local subsystems in gauge theory and gravity

    W. Donnelly and L. Freidel, “Local subsystems in gauge theory and gravity,” JHEP09 (2016) 102,1601.04744

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  30. [30]

    Ryu-Takayanagi Area as an Entanglement Edge Term

    J. Lin, “Ryu-Takayanagi Area as an Entanglement Edge Term,”1704.07763

    work page internal anchor Pith review Pith/arXiv arXiv

  31. [31]

    Entanglement entropy and nonabelian gauge symmetry

    W. Donnelly, “Entanglement entropy and nonabelian gauge symmetry,” Class. Quant. Grav. 31(2014), no. 21 214003,1406.7304

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  32. [32]

    Decomposition of entanglement entropy in lattice gauge theory

    W. Donnelly, “Decomposition of entanglement entropy in lattice gauge theory,” Phys. Rev. D85(2012) 085004,1109.0036

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  33. [33]

    On The Entanglement Entropy For Gauge Theories

    S. Ghosh, R. M. Soni, and S. P. Trivedi, “On The Entanglement Entropy For Gauge Theories,” JHEP09(2015) 069,1501.02593

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  34. [34]

    Minimal Factorization of Chern-Simons Theory -- Gravitational Anyonic Edge Modes

    T. G. Mertens and Q.-F. Wu, “Minimal Factorization of Chern-Simons Theory – Gravitational Anyonic Edge Modes,”2505.00501

    work page internal anchor Pith review Pith/arXiv arXiv

  35. [35]

    A proposal for 3d quantum gravity and its bulk factorization,

    T. G. Mertens, J. Sim´ on, and G. Wong, “A proposal for 3d quantum gravity and its bulk factorization,” JHEP06(2023) 134,2210.14196. 27

    work page arXiv 2023