On generalized gravitational entropy, squashed cones and holography

Aninda Sinha; Arpan Bhattacharyya; Menika Sharma

REVIEW

On generalized gravitational entropy, squashed cones and holography

Arpan Bhattacharyya, Menika Sharma, Aninda Sinha

High Energy Physics - Theory · Submitted 27 Aug 2013 · 40 references

Pith's verdict: A careful technical computation that reproduces known universal terms via the FPS squashed cone plus LM, but the Wald-entropy shortcut rests on a fitted regulator whose 'theory independence' defense is weaker than the paper claims.

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CitationAninda Sinha, Arpan Bhattacharyya, Menika Sharma. "On generalized gravitational entropy, squashed cones and holography." arXiv:1308.5748 (2013). https://pith.science/paper/7UQ2ILBJ

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The useful thing here is the explicit Fefferman-Graham construction: take the FPS squashed cone boundary metrics, expand, and check that the LM generalized gravitational entropy reproduces the known universal log terms in R^2, New Massive Gravity, quasi-topological, and the alpha'^3 IIB correction. That is real work, the answers match the literature (eqs. 6-8, 36-37, 53-56), and the IIB result — no correction to the anomaly — is the right sanity check. Section 5 is also a genuine step forward over the authors' earlier [10]: by taking r -> 0 before n -> 1, they get the Jacobson-Myers surface equation in Gauss-Bonnet without the weak-extrinsic-curvature assumption. That is a cleaner derivation and worth the paper on its own. The Iyer-Wald comparison in 4.5 is suggestive and honestly framed as a conjecture, not a proof. Fine by me at that level of claim. Where I would push back is the Wald-entropy claim in section 4. The regulator b is chosen by hand to land on the expected answer, and the authors say so in 4.1 and again in the Discussion. Their defense is that b is surface-dependent but theory-independent. For the cylinder that defense does less than it looks: eq. 45 factorizes as (1/b^2) times a single lambda-combination, so any b that fixes the Einstein case automatically fixes the lambda_i dependence. The cylinder tests the FG expansion, not the b choice. The sphere (eq. 50) is a real check, since two different lambda-combinations carry different b^2 weights and they do align at beta=1 — but that is one alignment, and the paper does not say why it should survive a third surface. The 2/sqrt(3) discrepancy between the cylinder and sphere values of b is waved through as 'surface dependent'; a local coordinate artifact of the FG expansion near r=0 should not care which surface you picked. Footnote 8, where reproducing dS_n/dn at n=1 proportional to c_T needs d beta/dn tuned at n=1, is the same pattern. So the Wald result is a consistency check, not a prediction, and the paper is more candid about this in the Discussion than in the abstract. Section 3.5's 'mild singularities' is asserted with a stated check rather than shown; minor, but a referee will ask. Who should read it: anyone working on holographic entanglement entropy beyond Einstein gravity, and anyone who wants the FG machinery for squashed cones written down. It deserves referees — the computations are worth checking and the b question is a real open problem the paper is right to flag.

Referee report

Major objections (3)

[§4.1, Eq. (45)] The cylinder case does not independently test the choice of b. Eq. (45) gives A(x,n) proportional to (n²−1) f₀ [(4λ₂+20λ₃−2λ₁)f∞−1]/(b² n⁴ f∞^{3/2}), so the b-dependence (1/b²) and the λ-dependence are multiplicatively separable, and the expected answer has the same single λ-factor up to sign. Any b that reproduces the Einstein (λ=0) result therefore automatically reproduces the correct λ_i dependence. The text presents the cylinder and sphere as parallel checks of the Wald-entropy prescription, but only the sphere (Eq. 50), where two λ-combinations with different b² weighting must align, is a genuine test. This should be stated explicitly rather than left for the reader to infer.
[§4.1, Eqs. (47) and (51)] The required regulator differs between the two surfaces by a factor 2/√3: b_cyl = (2/√3) f₀ √(n²−1) β(n) versus b_sph = f₀ √(n²−1) β(n). The paper labels this 'surface dependent' and moves on, but gives no explanation of why a regulator of the Fefferman-Graham expansion near r=0, where the r,τ structure is common to both boundary metrics, should carry a surface-dependent numerical coefficient. Either a short argument for the 2/√3 factor or an explicit statement that it is unexplained would strengthen the section; as written, the reader cannot tell whether this is a feature of the construction or an accident of the two examples.
[§4.6, footnote 8] The Renyi universality check ∂_n S_n|_{n=1} ∝ c_T requires adjusting ∂_n β(n)|_{n=1} in b, as the footnote acknowledges. This is a second fitted quantity layered on top of the already-fitted b, and it weakens the claim that the approach 'enables us to check some information away from n=1.' The check should be presented as conditional on the choice of β(n), not as an independent confirmation, and the sensitivity of the result to ∂_n β(n)|_{n=1} should be quantified.

Minor objections (5)

[§3.5] The 'mild singularities' claim rests on the assertion that the singular components of the Riemann tensor 'do not enter in the higher derivative actions considered in this paper,' supported only by 'we have explicitly checked.' A single line showing which curvature invariants are finite at r=0 for the R² and quasi-topological actions would make the claim verifiable rather than asserted.
[§3, Eq. (13)] The extraction of the 1/(n−1) factor from the x integral in Eq. (13) is stated but not shown. Since the same step recurs for I₂ and I₃ in Eq. (15), a sentence on the convergence of the x integral and on why the b prefactor combines with the 1/(n−1) to give a finite result would help the reader follow the regularization.
[§5, Eqs. (66)–(68)] The second alternative for handling the r^{2ε} terms (demanding ε/r ∼ 1/ê and r^{2ε} ∼ ê^{1+υ} with υ>0) is stated but not derived. Since this is the step that replaces the weak-extrinsic-curvature assumption of [10, 15], a short derivation or a reference to where it is carried out would make the argument self-contained.
[§4.5] The Noether-charge method is attributed to [40] (Wald's book) in the sentence beginning 'The Noether charge method of [40] needs a bifurcation surface,' but the primary references for the Noether-charge derivation and the Iyer-Wald dynamical-horizon proposal are [17] and [24]. The citation should point to those.
[General] The manuscript contains numerous typographical and spacing artifacts (e.g., 'Ryu-Takayangi' in the introduction, 'logarithimic' in the Discussion, 'the boundary t hat of a regularized squashed cone' in the introduction, and inconsistent spacing in expressions such as 'f 2∞'). These should be cleaned up in the final version.

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The paper

The paper claims

Wald's entropy formula, evaluated in a bulk geometry built from a regularized squashed cone, reproduces the universal anomaly coefficients of entanglement entropy in higher-derivative gravity, provided the cone's regulator is chosen as a特定e

The reading

Holographic entanglement entropy says the entropy of a region in a conformal field theory equals the area of a minimal surface in a higher-dimensional gravity dual. The Lewkowycz–Maldacena derivation of that rule works by analytically continuing a replica index $n$ and taking $n to 1$, but it was originally carried out for Einstein gravity. This paper asks whether the same machinery survives when the bulk action contains curvature-squared, quasi-topological, or string-theoretic $C^4$ corrections, and whether the resulting entropy can be written as Wald's Noether-charge formula rather than as a purpose-built entropy functional. Using a recently proposed regularization of

What carries the argument

The load-bearing device is the squashed-cone boundary metric, in which the transverse sphere or circle is written as $(f_0 + r^n d^{1-n} cos tau)^2 dOmega^2$ and the $r$-$tau$ part as $f(r,b)dr^2 + r^2 dtau^2$. The factor $r^n$ in front of the extrinsic-curvature deformation is what keeps the boundary Ricci scalar finite as $n to 1$; without it the curvature diverges like $(n-1)/r$. Feeding this metric into the Fefferman-Graham expansion, the coefficient $g_{(2)ij} = -(tilde L^2/2)(R^{(0)}_{ij} - g_{(0)ij}R^{(0)}/6)$ controls the $1/rho$ term of the on-shell action, and hence the logarithmic entropy. A second device is the order of limits: send $r to 0$ first and only then $n to 1$, so that $r^{n}$ suppresses the $cos tau$ terms that would otherwise break time-translation symmetry. This is what makes the surface look like a bifurcate horizon and lets Wald's formula apply.

Extended reading notes

Core claim

The authors take the regularized squashed-cone boundary metrics of Fursaev, Patrushev and Solodukhin, extend them into the bulk with a Fefferman-Graham expansion, and evaluate the generalized gravitational entropy. In a general four-derivative theory, in quasi-topological gravity, and in $alpha'^3$ type IIB supergravity, the universal logarithmic terms come out with the expected anomaly coefficients $a$ and $c$ for spherical and cylindrical entangling surfaces; the $C^4$ correction, as expected, leaves them untouched. The central claim is that Wald's entropy functional, evaluated on the codimension-two surface $r=0=tau$ in this Fefferman-Graham metric, gives the same universal terms — even though Wald entropy in pure anti-de Sitter space is known to give the wrong answer. The reason offered is that the regularized cone metric mimics the Iyer-Wald construction for dynamical horizons: taking $r to 0$ before $n to 1$ makes the extrinsic curvatures of the entangling surface vanish, removing the very terms by which the Wald and Jacobson-Myers functionals differ.

Load-bearing premise

The regulator $b$ in the squashed-cone metric is not derived but chosen by hand, as a specific power of $(n-1)$ that depends on the shape of the entangling surface but not on the gravity theory, and the authors state in their Discussion that a justification for this choice still has to be found.

Editorial extensions

If this is right

The universal $a$ and $c$ anomaly coefficients of a four-dimensional CFT are recovered from the generalized gravitational entropy in any of the higher-derivative bulk theories considered, so the holographic entropy rule is not an artifact of Einstein gravity.
Wald's Noether-charge formula, applied in the right background rather than in pure anti-de Sitter space, becomes a practical route to entanglement entropy in an arbitrary higher-derivative theory, bypassing the need to guess a Jacobson-Myers-type entropy functional.
The entangling-surface equation in Gauss-Bonnet gravity follows from the Lewkowycz-Maldacena procedure without the small-extrinsic-curvature assumption used in earlier work, once the $r^n$ regularization and the $r to 0$-then-$n to 1$ ordering are adopted.
The same intermediate expressions yield $partial_n S_n|_{n=1} propto c_T$ for both spherical and cylindrical surfaces, extending a known universality of Renyi entropy near $n=1$ to the cylindrical case where holographic results were not previously available.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the regulator $b$ could be derived from a global regularity condition on the bulk — for instance by demanding the full ten-dimensional or stringy completion be smooth — the Wald computation would turn from a consistency check into a genuine prediction for theories where no independent entropy functional is known.
The parallel between the Fefferman-Graham metric and the Iyer-Wald boost-invariant reconstruction suggests that entanglement entropy in higher-derivative gravity may literally be a Noether charge of an approximate Killing field, which would put black-hole and entanglement entropy on the same footing rather than merely analogous ones.
Because the $C^4$ correction drops out of the universal terms, the construction predicts that $1/lambda$ corrections in $mathcal{N}=4$ super Yang-Mills do not shift the $a$ and $c$ anomalies; a field-theory computation of the anomalous coefficients at finite 't Hooft coupling would test this directly.
The failure of the method away from $n=1$ — the authors note that general Renyi entropies would depend on the detailed form of $f(r,b)$ away from $r=0$ — hints that the regularization is only asymptotically meaningful, and that a proper treatment needs global information such as the temperature of the dual hyperbolic black hole.

Original abstract

We consider generalized gravitational entropy in various higher derivative theories of gravity dual to four dimensional CFTs using the recently proposed regularization of squashed cones. We derive the universal terms in the entanglement entropy for spherical and cylindrical surfaces. This is achieved by constructing the Fefferman-Graham expansion for the leading order metrics for the bulk geometry and evaluating the generalized gravitational entropy. We further show that the Wald entropy evaluated in the bulk geometry constructed for the regularized squashed cones leads to the correct universal parts of the entanglement entropy for both spherical and cylindrical entangling surfaces. We comment on the relation with the Iyer-Wald formula for dynamical horizons relating entropy to a Noether charge. Finally we show how to derive the entangling surface equation in Gauss-Bonnet holography.

The compiled record

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The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

We consider generalized gravitational entropy in various higher derivative theories of gravity dual to four dimensional CFTs using the recently proposed regularization of squashed cones. We derive the universal terms in the entanglement entropy for spherical and cylindrical surfaces. This is achieved by constructing the Fefferman-Graham expansion for the leading order metrics for the bulk geometry and evaluating the generalized gravitational entropy. We further show that the Wald entropy evaluated in the bulk geometry constructed for the regularized squashed cones leads to the correct universal parts of the entanglement entropy for both spherical and cylindrical entangling surfaces. We comment on the relation with the Iyer-Wald formula for dynamical horizons relating entropy to a Noether charge. Finally we show how to derive the entangling surface equation in Gauss-Bonnet holography.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider generalized gravitational entropy in various higher derivative theories of gravity dual to four dimensional CFTs using the recently proposed regularization of squashed cones.	contribution 0.84	0.86	0.07	asserted 0.84
s1	We derive the universal terms in the entanglement entropy for spherical and cylindrical surfaces.	result 0.91	0.96	0.28	asserted 1.00
s2	This is achieved by constructing the Fefferman-Graham expansion for the leading order metrics for the bulk geometry and evaluating the generalized gravitational entropy.	method 1.00	0.94	0.07	asserted 0.99
s3	We further show that the Wald entropy evaluated in the bulk geometry constructed for the regularized squashed cones leads to the correct universal parts of the entanglement entropy for both spherical and cylindrical entangling surfaces.	result 0.99	0.95	0.48	asserted 1.00
s4	We comment on the relation with the Iyer-Wald formula for dynamical horizons relating entropy to a Noether charge.	interpretation 0.90	0.48	0.04	unclear 0.83
s5	Finally we show how to derive the entangling surface equation in Gauss-Bonnet holography.	contribution 0.60	0.93	0.09	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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arXiv listing abstract v3

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full text
figures and tables
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

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Universal Terms of Entanglement Entropy for 6d CFTs
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The universal logarithmic entanglement entropy of 6d CFTs is derived holographically and field-theoretically, with exact agreement on the C^2 and C k^2 terms and agreement on k^4 terms only after assuming a conical-me...
Holographic Entanglement Entropy for the Most General Higher Derivative Gravity
hep-th 2014-11 conditional novelty 6.0 of 10

A generalized Wald entropy plus an anomaly-of-entropy term gives holographic entanglement entropy for gravity actions containing derivatives of the curvature, and the extra term resolves the Hung-Myers-Smolkin mismatch.
On entanglement entropy functionals in higher derivative gravity theories
hep-th 2014-05 conditional novelty 6.0 of 10

For general R² and quasi-topological gravity, the surface equation from the proposed entropy functional does not match the one from the generalized gravitational entropy method.
On short interval expansion of R\'enyi entropy
hep-th 2013-09 conditional novelty 6.0 of 10

Extending the twist-operator OPE to level six yields a new order-1/c contribution to the Rényi entropy of two short disjoint intervals, proposed as the two-loop bulk correction.
Entanglement Entropy for $TT$ deformed CFT in general dimensions
hep-th 2019-04 conditional novelty 4.0 of 10

The exact entanglement entropy of a large-N CFT deformed by a stress-tensor-bilinear operator matches the Ryu-Takayanagi prediction in cutoff AdS for dimensions 2 through 6, up to a renormalization-scheme area term.

How later papers used it

46 later papers cite it

Evidence: substantial. 41 of the citing papers share no author with it, and 7 of those carry a confident label from Pith's citation classifier.

Pith read 12 of the 83 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 paper by the same authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 8 papers with this label · 1 by the same authors

By the same authors

Followed the procedure given in this paper for R^2 theory.

On entanglement entropy functionals in higher derivative gravity theories (2014) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

46 arXiv papers cite this in 83 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 56 sentences without a confident label

Uses its method or tool 8 papers

Nevertheless to compute the entanglement entropy for the present case, based on the results of [ref] , the authors of [ref] have considered the following entropy functional to be minimized to compute the entanglement entropy S_EE=1/4G_N∫dx√(g_ind){1-1/m^2[(R_μνn^μ_in^ν_i- 1/2K_i^2)-3/4R]}, where $i=1,2$ denotes two transverse directions to a co-dimension two hypersurface in the bulk, $n_i^μ$ are two unit mutually orthogonal normal vectors on the co-dimension two hypersurface and $K^(i)$ is trace of the two extrinsic curvature tensors defined by K_μν^(i)=π^σ_ μ π^ρ_ ν∇_ρ(n_i)_σ, with π^σ_ μ=ϵ^σ_ μ+ξ∑_i=1,2(n_i)^σ(n_i)_μ , where $ξ=-1$ for space-like and $ξ=1$ for time-like vectors.

Entanglement Entropy for Logarithmic Conformal Field Theory · Three dimensional log-gravity · 0.96
Wald entropy of the charged black hole in Einstein-Gauss-Bonnet gravity can be obtained from equations ([ref]),([ref]),([ref]) and ([ref]) and is given as: [equation] Holographic entanglement entropy for the action ([ref]) was calculated in [ref] and is given below: [equation] where $h$ is the determinant of the following induced metric on constant $t,r=a$ surface: [equation] Now we are going to simplify equation ([ref]) using Gauss-Codazzi equation [ref] , [equation] From equations ([ref]) and ([ref]) entanglement entropy in Einstein-Gauss-Bonnet gravity simplifies to the following form (which is the gravitational contribution the generalised entropy): [equation] and matter contribution to the entanglement entropy is [ref] [Matter contributions to the entanglement entropy is same as [ref] up to some point which can be obtained using equations ([ref]), ([ref]), ([ref]), ([ref]) and ([ref

Page Curves of Reissner-Nordstr\"om Black Hole in HD Gravity · Page Curves of Charged Black Hole in Einstein-Gauss-Bonnet Gravity · 0.83
Here we will use the gravitational entropy proposed by Dong [ref] and Camps [ref] , which motivated to describe the gravitational dual of the entanglement entropy of CFT in the AdS/CFT setup (see also [ref] ).

Cutoff Scale of Quadratic Gravity from Quantum Focusing Conjecture · Gravitational Entropy · 0.80

Compares with it 1 paper

These results for the universal terms agree with those calculated in Ref. [ref] for the two quasi-topological theories.

On entanglement entropy functionals in higher derivative gravity theories · Universal terms · 0.99

Cites it as support 1 paper

In Ref. [ref] it was shown that this entropy functional leads to the expected universal terms in the entanglement entropy for cylindrical and spherical entangling surfaces, so the FPS functional passes this basic first test.

On entanglement entropy functionals in higher derivative gravity theories · Introduction · 0.98

Cites it as background 14 papers

There were attempts to extend this calculation to find the entangling surface equation in higher curvature theories in [ref] and the corresponding entropy functionals were argued to be different from the Wald entropy [ref] .

Entropy functionals and c-theorems from the second law · Introduction · 0.90
Entanglement entropy for a CFT dual to an asymptotically locally AdS solution in NMG model has been recently studied in [ref] .

Entanglement Entropy for Logarithmic Conformal Field Theory · Three dimensional log-gravity · 0.88
This question has been discussed recently in [ref] .

Free-kick condition for entanglement entropy in higher curvature gravity · Holographic entanglement entropy for NMG · 0.92

What it cites · 55 sentences without a confident label

Contests 3 papers

Recall that the Wald entropy functional in AdS spacetime was not the correct one [ref] .

Holographic Entanglement Entropy in Lovelock Gravities · Connection with Ryu-Takayanagi · contest 0.77
Recall that the Wald entropy functional in AdS spacetime was not the correct one [ref] .

On Holographic Entanglement Entropy and Higher Curvature Gravity · Connection with Ryu-Takayanagi · contest 0.77
We will explain that a modification to the order of limits needs to be done to the use of the LM method in [ref] for deriving the surface equation.

Entanglement entropy in higher derivative holography · Introduction · contest 0.80

Uses the method or tool of 4 papers

Then we computed the generalized gravitational entropy as proposed in [ref] .

Generalized gravitational entropy · Discussion · 1.00
We considered the Fefferman-Graham metric with the regularized metrics in [ref] as the boundary metric.

Distributional Geometry of Squashed Cones · Discussion · 0.94
In order to proceed, we could assume the following as in [ref] that $O() ∼αr/ϵ$ where $α≪1$, then the $ij$, $ir$ components go to zero.

Entanglement entropy in higher derivative holography · Equation for the entangling surface · 0.91

Compares with 5 papers

We will find that the Fefferman-Graham metric, for the cases where the boundary metrics are given by the regularized metrics proposed in [ref] , has some similarities with the Iyer-Wald construction.

Distributional Geometry of Squashed Cones · Introduction · 0.88
The key point in eq.([ref]) as compared to earlier regularizations e.g., [ref] is the introduction of a regulator in the extrinsic curvature terms.

Generalized gravitational entropy · Generalized Gravitational Entropy · 0.80
In particular it coincided with what arises from the Jacobson-Myers (JM) entropy functional [ref] which differs from the Wald entropy functional [ref] in terms quadratic in the extrinsic curvature [ref] .

Holographic Entanglement Entropy in Lovelock Gravities · Introduction · 0.98

Cites as background 11 papers

Recently Lewkowycz and Maldacena (LM) [ref] have proposed a derivation of the Ryu-Takayangi (RT) prescription [ref] for computing entanglement entropy (EE) [ref] in holography [ref] .

Holographic Entanglement Entropy: An Overview · Introduction · 0.99
Recently Lewkowycz and Maldacena (LM) [ref] have proposed a derivation of the Ryu-Takayangi (RT) prescription [ref] for computing entanglement entropy (EE) [ref] in holography [ref] .

Generalized gravitational entropy · Introduction · 0.99
Recently Lewkowycz and Maldacena (LM) [ref] have proposed a derivation of the Ryu-Takayangi (RT) prescription [ref] for computing entanglement entropy (EE) [ref] in holography [ref] .

Entanglement entropy in free quantum field theory · Introduction · 0.99

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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