{"id":"e9a5d123-98ce-4621-9d2f-c75f945a0081","arxiv_id":"2411.18062","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Using Wald's formalism and the thermodynamics method, the paper computes conserved charges and dual CFT central charges for WAdS3 black holes in NMG and ENMG, finding results consistent with earlier work.","lead":"This note applies the Wald covariant phase space method to compute the mass, angular momentum, and entropy of warped AdS3 black holes in new massive gravity and its extended version. It then uses a thermodynamics-based conjecture to derive central charges of the dual CFTs. A generalist might read it as a cross-check of whether the holographic dictionary survives higher-derivative deformations in three dimensions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central-charge result Eq. (62) is internally inconsistent: the entropy product quoted in Section V.B does not follow from the entropy formula Eq. (60), and the formulas carry a dimensional mismatch.","rationale":"The reader's verdict is CONDITIONAL, with the weakest assumption identified as the unproven thermodynamics-method conjecture Eq. (1). My stress-test finds a more concrete and more load-bearing internal problem: the entropy product used to derive Eq. (62) is inconsistent with the entropy formula Eq. (60) by an explicit factor, and the displayed formulas have a dimensional mismatch if r± are dimensionless as in Eq. (22). This is checkable algebra rather than a matter of external consensus, so it should be fixed or explained before the central-charge claim is accepted. The issue does not necessarily invalidate the conserved-charge results, which are cross-checked against the literature, so REJECT would be too strong; the paper needs a corrected derivation or an explicit clarification, which keeps the CONDITIONAL verdict. I partially agree with the reader because the dimensional concern was noted in the rationale, but the specific inconsistency between Eq. (60) and the stated S+S- is the decisive point.","tokens_in":17925,"tokens_out":16862,"duration_ms":157990,"concrete_test":"Derive S+S- directly by multiplying the two expressions in Eq. (60) and substituting J from Eq. (52); compare the result with the product asserted in Section V.B. If the ratio is not 1, recompute Eq. (62) from the product that actually follows from Eq. (60): for example, a literal calculation yields 1536ν⁴ℓ²Ξ/G instead of 192ν³ℓΞ/G. Also test dimensions by checking that S±, S+S-, and cL are dimensionless when ℓ and G are treated as lengths; if Eq. (62) changes under restoration of ℓ, the extra ℓ factor is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The core new result is the central charge Eq. (62). Let a=(ν²+3)r+r-. From Eq. (60), S+S- = (16πν²ℓΞ/G)² (2νr+−√a)(2νr−−√a). Using Eq. (52), the bracket equals (4G/Ξ)J, so S+S- = 1024π²ν⁴ℓ²Ξ J/G. The paper instead states S+S- = 2(8π)²ν³ℓΞ J/G = 128π²ν³ℓΞ J/G, which differs by a dimensionful factor 8νℓ. Consequently Eq. (1) gives cL=cR=1536ν⁴ℓ²Ξ/G, not 192ν³ℓΞ/G, unless one of Eqs. (52), (60), or the quoted product is a typo. Independently, in the units implied by Eq. (52) where r± are dimensionless and M~1/ℓ, J is dimensionless, so S± and S+S- must be dimensionless; Eq. (60) and the stated product instead carry factors ℓΞ/G or ℓ²Ξ/G with the dimensions of length or length². Thus Eq. (62) is not supported by the paper's own entropy data as printed; the thermodynamics-method conjecture is a secondary question once this internal inconsistency is resolved.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript derives conserved charges for warped AdS3 black holes in new massive gravity (NMG) and in extended new massive gravity (ENMG) using the Wald covariant phase space formalism in a first-order Chern-Simons-like formulation, and then computes dual CFT central charges by the thermodynamics method. The main new results are the ENMG mass and angular momentum in Eq. (52) and the central charges in Eq. (62). The paper also presents a parallel computation for BTZ black holes and claims consistency with known results in the literature.","tokens_in":18153,"tokens_out":10748,"duration_ms":98253,"significance":"If the results held as stated, the paper would provide the first Wald-formalism derivation of conserved charges for WAdS3 black holes in ENMG and a thermodynamics-method central charge for a higher-derivative, non-maximally-symmetric background. The computations are parameter-free and the supplementary material gives explicit auxiliary-field coefficients, which is a useful resource. However, the central-charge sector as printed is internally inconsistent: the entropy formula, the entropy product, and the central charge do not mutually agree and carry incorrect dimensions. Until that inconsistency is resolved, the main new claim cannot be considered established.","major_comments":[{"comment":"The entropy and central-charge sector is internally inconsistent. With Eq. (52), r± are dimensionless, M has dimension 1/ℓ, and J is dimensionless (G has dimension ℓ in three dimensions), so the entropies S± and the product S+S− must be dimensionless. The prefactor 16πν²ℓΞ/G in Eq. (60) has dimension of length, since Ξ∝ℓ, so Eq. (60) cannot be the entropy. Moreover, substituting Eq. (60) into S+S− and using Eq. (52) gives S+S− = 1024π²ν⁴ℓ²ΞJ/G (or 1024π²ν⁴ΞJ/G if the ℓ in Eq. (60) is removed), whereas the paper states S+S− = 128π²ν³ℓΞJ/G. The two results differ by a factor 8νℓ or 8ν/ℓ, not by a numerical constant. Consequently Eq. (62), c = 192ν³ℓΞ/G, does not follow from the paper's own Eqs. (60) and (52), and it is dimensionful (Ξ∼ℓ) rather than dimensionless. The authors should correct the entropy formula, the product, and the central charge so that they are mutually consistent and dimensionally sound.","section":"Sec. V.B, Eqs. (60)-(62)"},{"comment":"The central derivation of the ENMG charges is not shown. The passage from Eq. (37) and the interior products in Eqs. (48)-(50) to the compact expressions in Eq. (51) is stated as 'substituting the above relations' without displaying the intermediate algebra. The supplementary material lists the Pi and Yi coefficients but not the contractions and simplifications that produce Eq. (51). Since Eq. (52) is used in Section V.B to obtain the central charge, this is a load-bearing step and should be presented in enough detail to be verified by a reader.","section":"Sec. IV.B, Eq. (51)"},{"comment":"The thermodynamics-method formula Eq. (1) is applied to the WAdS3 ENMG background using only the check T+S+ = T−S−. That condition is necessary but not sufficient for Eq. (1). For this higher-derivative, non-maximally-symmetric background, the conjecture should be benchmarked independently. A concrete test would be to compare the corrected central charge with an asymptotic-symmetry-group computation or with the Cardy formula using TL, TR and the energies in Eq. (63).","section":"Sec. V.B, Eq. (1)"}],"minor_comments":[{"comment":"The manuscript contains several typographical errors, including 'taht', 'vinishing', 'balck', 'exppressions', and 'killing'; these should be corrected in a revision.","section":"Throughout"},{"comment":"The condition on ν² is typeset ambiguously as 'ν2 = ˜m2 + 3 /20'; if the intended expression is (m̃²+3)/20, it should be written explicitly to avoid confusion.","section":"Sec. III, Eq. (23)"},{"comment":"The statement 'No new data were created or analysed in this study' is slightly misleading because the supplementary material contains substantial new algebraic expressions; the wording should be adjusted to reflect that the explicit coefficient functions are provided as supplementary material.","section":"Data Availability Statement"},{"comment":"The expression M = (1/2)√(2ν)√(c_L E_L/6) is dimensionally sensitive to the (incorrect) dimensionful c_L given in Eq. (62); after the central charge is corrected, Eq. (63) should be rechecked for dimensional consistency.","section":"Sec. V.B, Eq. (63)"}],"recommendation":"major_revision","confidential_remarks":"The main problem is a load-bearing internal inconsistency in the entropy/central-charge sector, likely a typo, but it affects the headline result and must be fixed. I would ask the authors to provide corrected formulas for S±, S+S−, and c, and ideally an independent check of the corrected central charge against asymptotic symmetry group or Cardy data. The paper also relies on the author's own previous work [44] and [60] for key comparisons; an independent verification would strengthen the manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does a useful thing before it does a wrong thing. It re-derives the conserved charges for WAdS3 black holes in NMG and ENMG using covariant phase space, and the results match earlier work by Clement, Tonni, and Nam-Park-Yi. That is a clean cross-check and worth having on record. The BTZ sector in ENMG is also internally consistent, and the central charge from the thermodynamics method for BTZ (Eq. 56) matches the ASG result. The soft spot is the ENMG WAdS3 central charge. The paper's own Eq. (60) gives S± = 16πν²ℓΞ/G (2νr± − sqrt(...)). Multiplying and using Eq. (52), you get S+S− = 1024π²ν⁴ℓ²Ξ J/G, not the 128π²ν³ℓΞ J/G that appears in Section V.B. The stated product is off by a dimensionful factor 8νℓ. Consequently Eq. (62) is not supported by the paper's own formulas, and the reason is visible: with Ξ having dimension length and G length, both Eq. (60) and Eq. (62) carry length or length² units. Entropy and central charges in 3D should be dimensionless. This is an internal inconsistency, not just a typo. The charge variations (30) and (51) are also presented without derivation, with the auxiliary-field coefficients pushed to supplementary material. That makes verification laborious but not impossible. Novelty is modest. The ENMG conserved charges were in [50], and the same author's ASG central charges [60] overlap with what the thermodynamics method should reproduce. The genuinely new piece is the covariant-phase-space derivation for ENMG and the thermodynamics-method cross-check. If the central charge error is fixed, the paper becomes a useful consistency check. As printed, the main new claim is unsupported. I would send it to a referee anyway, because the error is concrete and fixable and the Wald-formalism portion has value. The referee should be pointed at Eqs. (52), (60), and the entropy product. Without that fix, I would not cite the central charge, and I would not bring it to a reading group.","headline":"Useful Wald-formalism re-derivation, but the ENMG WAdS3 central charge is internally inconsistent and dimensionally wrong as printed.","tokens_in":18721,"tokens_out":8546,"would_cite":false,"duration_ms":71881,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C57","83D05","81T40"],"pacs":["04.70.Bw","04.60.-m","11.25.Hf"],"model":"deepseek-v4-flash","headline":"For warped AdS3 black holes in extended new massive gravity, this paper derives explicit ADM mass, angular momentum, and equal dual CFT central charges $c_L = c_R = 192\\nu^3\\ell\\Xi/G$, using the covariant phase space formalism and the…","keywords":["warped AdS3 black hole","new massive gravity","extended new massive gravity","Noether charge","covariant phase space","central charges","thermodynamics method","entropy product"],"falsifier":"An independent calculation of the asymptotic symmetry group of the ENMG warped AdS3 background would settle the central-charge claim: if the central charges read off from this algebra differ from $192\\nu^3\\ell\\Xi/G$, the thermodynamics-method result is wrong. A cheaper check is to insert the paper's entropies and temperatures into the statistical entropy formula $S = \\pi^2/3\\,(c_L T_L + c_R T_R)$ with $c_L = c_R = 192\\nu^3\\ell\\Xi/G$ and compare with the entropy $S_\\pm$ of Eq. (60) for several values of $r_+$, $r_-$; any discrepancy shows the central charge formula is being misapplied.","tokens_in":17681,"feed_emoji":"🕳️","tokens_out":11963,"duration_ms":97571,"temperature":0.7,"pith_summary":"This paper establishes the full set of conserved charges, entropies, and dual CFT central charges for warped anti-de Sitter black holes in three-dimensional new massive gravity (NMG) and its extended version (ENMG). Using the covariant phase space formalism, it integrates Noether-charge variations to obtain the ADM mass and angular momentum, first for the NMG WAdS3 solution and then for BTZ and WAdS3 black holes in ENMG. It then applies the thermodynamics method, in which equal outer and inner horizon temperatures and entropies imply $c_L=c_R=6\\,d/dN\\,(S_+S_-/4\\pi^2)$, to extract equal left- and right-moving central charges, $c_L=c_R=192\\nu^3\\ell\\Xi/G$ for the ENMG WAdS3 case. A sympathetic reader would care because these explicit formulas are the input needed to check whether the holographic dictionary, matching black hole entropy to a dual CFT through the statistical entropy formula, holds for these higher-derivative backgrounds.","feed_headline":"WAdS3 black hole mass, spin, and CFT charges derived","feed_subtitle":"Covariant phase space yields the charges; the entropy-product thermodynamics method fixes cL = cR.","key_machinery":"The load-bearing object is the Noether charge $(n-2)$-form $Q_\\xi$ associated with the Killing vector $\\xi = \\partial_t + \\Omega_H\\partial_\\varphi$; its integral over a spatial boundary gives the conserved charges and, through $S = (2\\pi/\\kappa)\\oint Q_\\xi$, the entropy. The second device is the thermodynamics method: when outer and inner horizon quantities satisfy $T_+ S_+ = T_- S_-$, the central charges follow from $c_L = c_R = 6\\,d/dN\\,(S_+S_-/4\\pi^2)$ with $N=J$. The paper also relies on a first-order Lagrangian formulation with frame fields and auxiliary fields, which turns the charge variation into an integral over the metric functions $F$, $K$, $N$ and their derivatives.","core_discovery":"The central discovery is that the warped AdS3 black hole, a circle fibration over AdS2 with warping parameter $\\nu$, in extended new massive gravity carries the ADM mass $M = \\Xi/(G\\ell)\\,\\big(\\nu(r_+ + r_-) - \\sqrt{(\\nu^2+3)r_+r_-}\\big)$, angular momentum $J = \\Xi/(4G)\\,\\big[(\\nu(r_+ + r_-) - \\sqrt{(\\nu^2+3)r_+r_-})^2 - \\nu^2(r_+-r_-)^2\\big]$ with $\\Xi$ defined in Eq. (51), and entropies $S_\\pm = 16\\pi\\nu^2\\ell\\Xi/G\\,\\big(2\\nu r_\\pm - \\sqrt{(\\nu^2+3)r_+r_-}\\big)$. These quantities satisfy the first law and Smarr relation on both the outer and inner horizons, and the entropy product $S_+S_-$ is universal, independent of the mass. The paper further claims that the thermodynamics method yields equal dual CFT central charges $c_L = c_R = 192\\nu^3\\ell\\Xi/G$, with an analogous BTZ result in ENMG. The computations reproduce prior results for NMG WAdS3 and for BTZ, which the paper presents as consistency checks.","pith_inferences":["A natural extension the author leaves implicit: the same workflow should assign charges and central charges to any stationary solution of the higher-order Born-Infeld-like extensions, with each curvature order entering only through the prefactor $\\Xi$.","If the central charges in Eq. (62) are confirmed by an independent asymptotic-symmetry computation, it would validate the thermodynamics method on a non-maximally-symmetric higher-derivative background; a mismatch would show the method fails exactly where $T_+S_+=T_-S_-$ is not sufficient.","The explicit charge formulas offer a direct way to test extremality and cosmic censorship bounds for warped black holes in these theories by checking that $r_+ \\ge r_-$ and that the $J/M$ ratios respect the claimed thermodynamics.","Since the paper's computations are background-independent at the level of the covariant phase space, the same Noether-charge integrals could be applied to other warped or Lifshitz-type solutions in NMG and ENMG, where earlier boundary-based methods gave unsatisfactory results."],"forward_implications":["The explicit ENMG WAdS3 mass and angular momentum in Eq. (52), together with the entropy, satisfy the first law $dM = T_\\pm dS_\\pm + \\Omega_\\pm dJ$ and the Smarr relation $M = T_\\pm S_\\pm + 2\\Omega_\\pm J$.","Because $T_+S_+ = T_-S_-$ holds, the entropy product $S_+S_-$ is mass-independent and equals $2(8\\pi)^2\\nu^3\\ell\\Xi\\,J/G$, a universal feature for these diffeomorphism-invariant theories.","The equal central charges $c_L = c_R$ imply the dual CFT is anomaly-free, so the left and right sectors carry the same number of degrees of freedom, unlike theories with gravitational anomaly where $c_L \\neq c_R$.","Using the left and right temperatures, the mass and angular momentum of the WAdS3 black hole can be written in terms of left and right CFT energies as $M = \\tfrac12\\sqrt{2\\nu c_L E_L/6}$ and $J = \\ell^2/(2(\\nu^2+3))\\,(E_L - E_R)$, analogous to TMG and MMG but with ENMG-modified coefficients.","For the BTZ limit in ENMG, the central charges reduce to $c_L=c_R=\\frac{3\\ell}{2G}(1 - \\frac{1}{\\tilde m^2} - \\frac{1}{2\\tilde m^4})$, consistent with earlier asymptotic-symmetry and c-theorem analyses."],"supporting_citations":[{"why":"supplies the Noether-charge formalism on which every conserved-charge computation in the paper is built.","marker":"[25]"},{"why":"provides the covariant phase space derivation of the first law and the entropy formula used in the calculations.","marker":"[26]"},{"why":"gives the BTZ conserved-charge computation in NMG that the paper extends to WAdS3 and ENMG.","marker":"[27]"},{"why":"defines the ENMG action as the higher-derivative extension studied in the paper.","marker":"[14]"},{"why":"supplies the warped AdS3 black hole metric and the left/right temperatures used for the CFT dual analysis.","marker":"[21]"},{"why":"introduces the thermodynamics method for extracting central charges from black hole thermodynamics.","marker":"[38]"},{"why":"provides the specific formula $c_L=c_R=6\\,d/dN\\,(S_+S_-/4\\pi^2)$ used to compute central charges.","marker":"[40]"},{"why":"gives prior ENMG WAdS3 charge results that the paper's Eq. (52) is checked against.","marker":"[50]"}],"fun_headline_variants":["WAdS3 black hole charges via covariant phase space","Universal entropy product for WAdS3 black holes","Equal CFT central charges for WAdS3 black holes","WAdS3 black holes: universal entropy product, equal CFT charges"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on assuming that the formula $c_L=c_R=6\\,d/dN\\,(S_+S_-/4\\pi^2)$, borrowed from earlier work, gives the correct central charges for this higher-derivative, non-maximally-symmetric warped black hole once $T_+S_+=T_-S_-$ holds; the paper checks that condition but does not independently confirm the formula for the ENMG WAdS3 background.","fun_headline_variants_meta":{"raw":{"variants":["WAdS3 black hole charges via covariant phase space","Universal entropy product for WAdS3 black holes","Equal CFT central charges for WAdS3 black holes","WAdS3 black holes: universal entropy product, equal CFT charges"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000862,"raw_usage":{"total_tokens":3725,"prompt_tokens":917,"completion_tokens":2808,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":533,"completion_tokens_details":{"reasoning_tokens":2736}},"tokens_in":533,"tokens_out":2808,"duration_ms":17573,"temperature":1.0,"reasoning_tokens":2736,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:33:23.129060+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"An independent calculation of the asymptotic symmetry group of the ENMG warped AdS3 background would settle the central-charge claim: if the central charges read off from this algebra differ from $192\\nu^3\\ell\\Xi/G$, the thermodynamics-method result is wrong. A cheaper check is to insert the paper's entropies and temperatures into the statistical entropy formula $S = \\pi^2/3\\,(c_L T_L + c_R T_R)$ with $c_L = c_R = 192\\nu^3\\ell\\Xi/G$ and compare with the entropy $S_\\pm$ of Eq. (60) for several values of $r_+$, $r_-$; any discrepancy shows the central charge formula is being misapplied.","supporting_citations":[{"cited_title":"Three-dimensional Chern-Simons black holes","cited_arxiv_id":"0807.4241","evidence_quote":"supplies the Noether-charge formalism on which every conserved-charge computation in the paper is built."},{"cited_title":"Charged Black Holes in New Massive Gravity","cited_arxiv_id":"1007.4323","evidence_quote":"provides the covariant phase space derivation of the first law and the entropy formula used in the calculations."},{"cited_title":"Black Holes in Born-Infeld Extended New Massive Gravity","cited_arxiv_id":"1010.2434","evidence_quote":"gives the BTZ conserved-charge computation in NMG that the paper extends to WAdS3 and ENMG."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces the thermodynamics method for extracting central charges from black hole thermodynamics."},{"cited_title":"Note on Thermodynamics Method of Black Hole/CFT Correspondence","cited_arxiv_id":"1301.0429","evidence_quote":"provides the specific formula $c_L=c_R=6\\,d/dN\\,(S_+S_-/4\\pi^2)$ used to compute central charges."}],"review_version":1}