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Relative entropy between vacuum and coherent states equals the first-order variation of the Ryu-Takayanagi geodesic length divided by 4G_N in AdS3/CFT2.

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2026-06-29 04:02 UTC pith:AQRBMTE6

load-bearing objection The central claim as written is impossible: relative entropy is O(ε²) while the RT length variation is O(ε), so they cannot match at linear order. the 1 major comments →

arxiv 2606.27915 v1 pith:AQRBMTE6 submitted 2026-06-26 hep-th gr-qcmath-phmath.MP

A UV-Finite Ryu-Takayanagi Relation from Relative Entropy in AdS₃/CFT₂

Albert Much , Philipp Dorau , Leonardo Sangaletti , Rainer Verch This is my paper
classification hep-th gr-qcmath-phmath.MP
keywords relative entropyRyu-Takayanagi relationAdS3/CFT2Bañados geometriesFefferman-Graham dictionarySchwarzian functionalholographic dictionaryBisognano-Wichmann theorem
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.

The paper derives a Ryu-Takayanagi relation in AdS3/CFT2 that replaces the ultraviolet-divergent von Neumann entanglement entropy with relative entropy, which stays finite. It adapts an exact result for chiral relative entropy to diamond regions, writes the boundary quantity as a Schwarzian functional, and uses the Fefferman-Graham dictionary to match this to the asymptotic data of a Bañados geometry. Three-dimensional gravity rigidity then lifts the identification into the bulk, so that relative entropy equals the linear change in RT length over 4G_N. The argument uses only the Bisognano-Wichmann/Borchers theorem and the holographic dictionary, supplying an operator-algebraic and ultraviolet-finite version of the relation.

Core claim

Adapting Hollands' exact result for the chiral relative entropy to a diamond region, we express the boundary relative entropy between the vacuum and a coherent state as a Schwarzian functional, which the Fefferman-Graham dictionary identifies with the asymptotic data of a Bañados geometry; the rigidity of three-dimensional gravity promotes this boundary identification to the bulk. To linear order in the metric perturbation, the relative entropy then equals the variation of the RT geodesic length divided by 4G_N. The construction rests only on the Bisognano-Wichmann/Borchers theorem and the holographic dictionary, giving a UV-finite, operator-algebraic counterpart to the RT relation.

What carries the argument

The Schwarzian functional for boundary relative entropy, identified with Bañados asymptotic data via the Fefferman-Graham dictionary and promoted to bulk geometry by three-dimensional gravity rigidity.

Load-bearing premise

The rigidity of three-dimensional gravity allows boundary data from the Schwarzian functional and Fefferman-Graham dictionary to fix the bulk geometry.

What would settle it

An explicit calculation of the relative entropy for a chosen coherent state in the boundary CFT2 that fails to match the first-order change in the corresponding RT geodesic length in the Bañados bulk geometry.

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

If this is right

  • The equality holds to linear order in metric perturbations.
  • The relation is ultraviolet-finite by construction.
  • The derivation applies to diamond regions and uses only the Bisognano-Wichmann/Borchers theorem plus the holographic dictionary.
  • It supplies an operator-algebraic version of the Ryu-Takayanagi formula without reference to divergent entanglement entropy.

Where Pith is reading between the lines

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

  • The same boundary-to-bulk promotion step may not hold in higher dimensions where gravity is less rigid.
  • Relative entropy could serve as a regularized replacement for other holographic quantities that currently rely on divergent entropies.
  • The method isolates the role of the Bisognano-Wichmann theorem, suggesting it might be tested in non-holographic models with modular flow.

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

1 major / 0 minor

Summary. The manuscript claims to derive a UV-finite Ryu-Takayanagi relation in AdS₃/CFT₂ by adapting Hollands' exact result for chiral relative entropy to a diamond region via the Bisognano-Wichmann/Borchers theorem. The boundary relative entropy between vacuum and coherent state is expressed as a Schwarzian functional, identified with the asymptotic data of a Bañados geometry through the Fefferman-Graham dictionary, and promoted to the bulk geometry by the rigidity of three-dimensional gravity. To linear order in the metric perturbation, this relative entropy is asserted to equal the variation of the RT geodesic length divided by 4G_N, yielding an operator-algebraic counterpart to the RT formula resting only on the BW/Borchers theorem and the holographic dictionary.

Significance. If the central identification is correct, the result would be significant for providing a UV-finite, relative-entropy-based foundation for the RT formula that avoids cutoff dependence and relies solely on standard inputs from the holographic dictionary and the BW/Borchers theorem. This could strengthen the operator-algebraic understanding of holographic entanglement in AdS₃/CFT₂.

major comments (1)
  1. [Abstract] Abstract: The assertion that relative entropy equals the variation of the RT geodesic length to linear order in the metric perturbation is inconsistent with the perturbative structure of relative entropy. By definition, S(ρ||σ) for ρ = σ + ε δρ has vanishing first derivative at ε=0 (nonnegative and minimized at equality), so its expansion begins at O(ε²). The first variation of geodesic length is O(ε). Equating the two at linear order is therefore impossible unless both sides vanish identically, which would make the result trivial. The load-bearing step is the adaptation of Hollands' result and its identification with the Schwarzian functional; this must be shown to produce a quantity whose linear term vanishes and whose quadratic term matches δL/4G_N, but the stated claim does not reflect this structure.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed reading and for identifying an important issue with the perturbative orders in our abstract. We agree that the current wording is imprecise and will revise the manuscript to correct it.

read point-by-point responses

  1. Referee: The assertion that relative entropy equals the variation of the RT geodesic length to linear order in the metric perturbation is inconsistent with the perturbative structure of relative entropy. By definition, S(ρ||σ) for ρ = σ + ε δρ has vanishing first derivative at ε=0 (nonnegative and minimized at equality), so its expansion begins at O(ε²). The first variation of geodesic length is O(ε). Equating the two at linear order is therefore impossible unless both sides vanish identically, which would make the result trivial. The load-bearing step is the adaptation of Hollands' result and its identification with the Schwarzian functional; this must be shown to produce a quantity whose linear term vanishes and whose quadratic term matches δL/4G_N, but the stated claim does not reflect this structure.

    Authors: We agree that the abstract's reference to 'linear order' is incorrect and misleading. Relative entropy between the vacuum and a coherent state is quadratic in the perturbation parameter by construction (vanishing at first order due to the BW theorem and the fact that the coherent state is a unitary excitation). The Schwarzian functional obtained from Hollands' result expands at O(ε²). The first variation of the RT geodesic length is indeed O(ε). We will revise the abstract to state that the relative entropy equals the quadratic term in the expansion that corresponds to the variation of the RT geodesic length divided by 4G_N. In the main text we will add an explicit expansion of the Schwarzian functional demonstrating that the linear term vanishes identically and that the quadratic coefficient matches δL/4G_N via the Fefferman-Graham dictionary and the rigidity of 3d gravity. This revision addresses the load-bearing identification without altering the core result. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external theorems and dictionary without self-referential reduction

full rationale

The paper's central chain adapts Hollands' result for chiral relative entropy, applies the Bisognano-Wichmann/Borchers theorem to a diamond, expresses the result as a Schwarzian functional, and uses the Fefferman-Graham dictionary plus 3D gravity rigidity to identify it with the variation of the RT geodesic length. These steps invoke named external theorems and the standard holographic dictionary as independent inputs rather than deriving them from the target RT relation itself. No equations or steps reduce a claimed prediction to a fitted parameter or prior self-citation by construction, and the construction is presented as self-contained against those benchmarks. The noted order mismatch between relative entropy (O(ε²)) and geodesic variation (O(ε)) is a potential correctness concern but does not constitute a circularity pattern under the enumerated criteria.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on two standard background results and one domain-specific identification; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Bisognano-Wichmann/Borchers theorem
    Invoked to express the relative entropy between vacuum and coherent state.
  • domain assumption Fefferman-Graham holographic dictionary
    Used to identify the Schwarzian functional with asymptotic bulk data.

pith-pipeline@v0.9.1-grok · 5710 in / 1482 out tokens · 60943 ms · 2026-06-29T04:02:03.481152+00:00 · methodology

0 comments
read the original abstract

We establish a Ryu-Takayanagi (RT) relation in AdS$_3$/CFT$_2$ using \emph{relative entropy} as the central object, in place of the ultraviolet-divergent von Neumann entanglement entropy. Adapting Hollands' exact result for the chiral relative entropy to a diamond region, we express the boundary relative entropy between the vacuum and a coherent state as a Schwarzian functional, which the Fefferman-Graham dictionary identifies with the asymptotic data of a Ba\~nados geometry; the rigidity of three-dimensional gravity promotes this boundary identification to the bulk. To linear order in the metric perturbation, the relative entropy then equals the variation of the RT geodesic length divided by $4G_N$. The construction rests only on the Bisognano-Wichmann/Borchers theorem and the holographic dictionary, giving a UV-finite, operator-algebraic counterpart to the RT relation.

Figures

Figures reproduced from arXiv: 2606.27915 by Albert Much, Leonardo Sangaletti, Philipp Dorau, Rainer Verch.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic visualization of AdS [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

discussion (0)

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Forward citations

Cited by 1 Pith paper

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    At linearized order, the vacuum-subtracted HRT entropy of a boundary region is the entropy of a coherent graviton state in the modular crossed-product algebra of the dual CFT, assuming a wedge-reconstructing holographic map.

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