{"id":"546bc0ed-046d-4d76-a946-16417023a0f1","arxiv_id":"2504.19694","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In CFTs with gravitational anomalies, the imaginary part of timelike entanglement entropy depends only on the right-moving central charge, a chiral asymmetry that may serve as an anomaly detector.","lead":"This paper computes the timelike entanglement entropy for two-dimensional conformal field theories with gravitational anomalies, where left and right moving central charges differ. It finds that the imaginary part of this entropy depends on only one central charge and proposes this chiral asymmetry as a probe of gravitational anomalies.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The c_R dependence of the timelike entanglement entropy is fixed only by an unstated branch choice at (3.3); with the paper's own z=x-t convention and t_12=T>0, the twist correlator (2.4) gives i c_L pi/6 instead.","rationale":"The reader identified the holographic boundary-condition prescription in section 4.3 as the weakest assumption, and I agree that the prescription is underdetermined. However, the more fundamental problem lies one step earlier: the field-theory analytic continuation in section 3.1 is itself ambiguous, and the specific branch chosen in (3.3) is what produces the c_R-dependent imaginary part. A direct continuation of the twist correlator (2.4), which is the object from which (2.5) and hence (2.6) were derived, gives c_L instead under the paper's own z=x-t convention when t_12=T>0. This is not a matter of holographic boundary conditions but a sign/branch issue in the claimed CFT result. Since the central claim is the asymmetric central-charge dependence of the imaginary part, the paper should either justify the +log(-1) branch physically or correct the final formulas to c_L. The qualitative statement that the imaginary part depends on only one central charge may survive either way, but the specific identification with c_R and the detailed holographic matching would change. I therefore keep the verdict conditional, but the condition is stricter than the reader's: the analytic continuation itself must be made unambiguous, not merely the bulk normal-frame prescription. The concern is concrete and testable by direct continuation of (2.4).","tokens_in":19090,"tokens_out":32546,"duration_ms":346367,"concrete_test":"Recompute section 3.1 by applying the analytic continuation to the twist correlator (2.4) itself rather than to the already-integrated expression (2.6). With z=x-t, \\bar z=x+t and t_12=T>0, insert z_12=-T+i0^+, \\bar z_12=T+i0^+ (the i0 prescription that reproduces the known positive imaginary part of the non-anomalous TEE) and take principal logarithms as the paper does. If the replica limit gives Im S_T=c_L pi/6 rather than c_R pi/6, then (3.5b) is one admissible branch and not a consequence of the anomaly; the same check should then be repeated for (3.7) and (3.8), and the holographic normal-frame prescriptions in section 4.3 should be re-examined to see whether they can produce -i pi/(4G mu) instead of +i pi/(4G mu).","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Equations (3.1)-(3.5) do not provide a single-valued analytic continuation. The paper's conventions are z=x-t, \\bar z=x+t, and the future-directed pure timelike interval has z_12=-T, \\bar z_12=T. If one continues the twist-field correlator (2.4) directly, as was used to derive (2.5), the phase from log(-T) sits entirely in the c_L term, yielding S_T=(c_L+c_R)/6 log(T/epsilon)+i c_L pi/6. To obtain the claimed (3.5b), S_T=(c_L+c_R)/6 log(T/epsilon)+i c_R pi/6, one must take \\tanh^{-1}(t_12/x_12)=+i pi/2 at (3.3), i.e. log(-1)=+i pi. But this choice corresponds to R e^{-\\kappa}=+T, whereas the announced interval has x_12-t_12=-T; in other words, the chosen branch reverses the sign of t_12. The final chirality of the imaginary part (c_R versus c_L), which is the central physical claim of the paper, is thus not derived from the gravitational anomaly: it is inserted by the branch convention at (3.3). The holographic construction in section 4.3 is then matched to this same branch, so the bulk agreement does not independently resolve the ambiguity. A corrected or physically justified branch choice could turn (3.5b), (3.7), (3.8), (4.38), (4.51), and (4.57) into c_L-dependent imaginary parts.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies timelike entanglement entropy (TEE) in two-dimensional CFTs with unequal left and right central charges, i.e., with gravitational anomalies. In the field-theoretic part, the authors analytically continue the known spacelike-interval entanglement entropy to a pure timelike interval in three settings: zero temperature, finite temperature with angular potential, and zero temperature with finite angular potential. They obtain complex-valued results whose real part depends on c_L+c_R and whose imaginary part is claimed to depend on a single central charge, specifically c_R. In the holographic part, the paper extends the Castro et al. framework for TMG-AdS3/CFT2 by deriving the Chern-Simons contribution from timelike geodesics as a rotation of the normal frame, prescribing boundary conditions for the normal vectors at null infinity, and matching the resulting holographic TEE with the field-theory answers in the three cases.","tokens_in":19486,"tokens_out":18700,"duration_ms":164628,"significance":"If the central claim held, the chiral sensitivity of the imaginary part of TEE would be a genuinely useful diagnostic: it would show that the timelike entanglement entropy can probe the sign and magnitude of the gravitational anomaly coefficient c_L-c_R. The derivation of the timelike-geodesic Chern-Simons contribution is a useful technical extension of [4], and the paper is careful in several places: the parametrizations (4.27) and (4.33) are explicit, the reduction to the known non-anomalous TEE in the c_L=c_R limit is correctly implemented, and the holographic computation uses the established TMG central charge formula (4.3) without fitting parameters. However, the main physical claim is currently not established as stated because the chirality of the imaginary part is fixed by an unstated branch choice, and there is an internal inconsistency in the finite-temperature holographic equations. For this reason the significance is conditional until those points are resolved.","major_comments":[{"comment":"The claimed imaginary part iπ c_R/6 is fixed by an unstated branch choice rather than by the gravitational anomaly. With the paper's own conventions z=x-t, \\bar z=x+t, and t_{12}=T>0, a pure timelike interval has z_{12}=-T and \\bar z_{12}=T, so the direct continuation of the twist correlator (2.4), equivalently of (2.5), gives S_T^A=(c_L+c_R)/6 log(T/ε)+iπ c_L/6. The value tanh^{-1}(t_{12}/x_{12})=+(1/2)log(-1)=+iπ/2 adopted in footnote 2 corresponds to the opposite sign of log(\\bar z_{12}/z_{12}) and therefore to a branch that effectively reverses the sign of t_{12}. Since the holographic normal-frame prescription in Sec. 4.3 is matched to this same branch, the bulk computations in (4.38), (4.51), and (4.57) do not independently select c_R over c_L. The authors should either derive the branch from a stated physical definition of TEE, such as a specified continuation path in z and \\bar z or an explicit time-orientation convention, or acknowledge that the c_L versus c_R assignment is convention-dependent; in the latter case the proposal to use the imaginary part as a probe of the gravitational anomaly must be re-evaluated.","section":"Sec. 3.1, Eqs. (3.1)-(3.5)"},{"comment":"As printed, Eq. (4.51) does not follow from Eq. (4.50). Eq. (4.50) gives S_anom = (1/(4G_N μ)) log[(sinh(πT/β_R)/β_R)/(sinh(πT/β_L)/β_L)] + iπ/(4G_N μ), which, using 1/(4G_N μ)=-(c_L-c_R)/12, equals -(c_L-c_R)/12 log[(β_L sinh(πT/β_R))/(β_R sinh(πT/β_L))] plus the imaginary term. The second line of Eq. (4.51), however, contains log[(β_R sinh(πT/β_R))/(β_L sinh(πT/β_L))]; the β_L and β_R factors are interchanged. One of the two equations must be corrected. As written, the claimed exact match with the field-theory result (3.7) is not established.","section":"Sec. 4.4, Eqs. (4.50)-(4.51)"},{"comment":"The boundary-condition prescription for the normal frame is a free input. The values S_spacelike_anom=0 in (4.30) and S_timelike_anom=iπ/(4G_N μ) in (4.36) are obtained only for the specific matching rule proposed in this section, and the paper does not derive that rule from the TMG action or from a covariant boundary-value problem. Because the field-theory branch is not fixed by the calculation (see the first major comment), the agreement with (3.5) cannot serve as an independent check of this prescription. The authors should present a derivation of the prescription or, failing that, test its robustness under changes of the regularization (such as the large-β cutoff) and under alternative matching conventions.","section":"Sec. 4.3, Eqs. (4.28)-(4.35)"}],"minor_comments":[{"comment":"The summary states that the TEE receives an additional purely imaginary contribution from the gravitational anomaly, but at finite temperature and angular potential, Eq. (3.7) shows an anomaly-dependent contribution to the real part as well; the wording should be adjusted.","section":"Sec. 5, first paragraph"},{"comment":"The notation tanh^{-1}∞ is imprecise because the function is not defined at infinity and the value depends on the direction of approach; the limit x_{12}→0 with its branch should be written explicitly.","section":"Sec. 3.1, footnote 2"},{"comment":"The stacked fraction in Eq. (4.50) is ambiguous in the printed form; using explicit bracketed ratios would make the ordering of β_L and β_R factors clearer.","section":"Eq. (4.50)"},{"comment":"The caption notes that n_i and n_f are timelike for the spacelike geodesics; a brief sentence on the orientation convention for the triad (v,n,\\tilde n) would help the reader follow the matching rules in Sec. 4.3.","section":"Fig. 2 caption"}],"recommendation":"major_revision","confidential_remarks":"The main risk is the branch ambiguity in Sec. 3.1, which determines the central physical claim (c_R versus c_L in the imaginary part); this must be resolved before the paper can be accepted. The inconsistency between Eqs. (4.50) and (4.51) is a concrete error that also needs correction. I do not see a circularity problem in the sense of fitted parameters: the computation uses standard TMG central charges and the known action from [4], and no parameter is tuned to the target result. The manuscript fits the journal's scope well; the needed revisions are technical rather than conceptual."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The punchline: this is worth reading and refereeing, but the central physical claim — that the imaginary part of TEE in an anomalous CFT is (c_R/6)π — is not actually derived. It is selected by a branch choice at eq. (3.3) that conflicts with the paper's own z=x−t convention. The broader framework, including the new timelike-geodesic Chern-Simons contribution, is solid and useful.\n\nWhat is genuinely new: the derivation of the anomalous contribution from a timelike geodesic (eqs. 4.16–4.21), expressed as a rotation of normal frames rather than a boost, and the holographic construction that assembles spacelike and timelike geodesics into a TEE computation. The field theory sections are clearly organized across zero temperature, finite temperature with angular potential, and extremal cases.\n\nSoft spots, in order. First, the branch issue. Starting from the twist correlator (2.4) with z12=−T, \\bar z12=T, the principal log gives i c_L π/6, not i c_R π/6. To obtain (3.5b) the authors take κ=tanh^{-1}(∞)=+iπ/2 and R=iT, but those choices imply z12=+T, i.e. they reverse the sign of t12 relative to the announced interval. The κ consistent with z12=−T is −iπ/2, yielding i c_L π/6. So the chirality (c_R versus c_L) is an unstated convention, not a prediction of the anomaly. Second, the holographic derivation does not resolve this: the boundary condition prescription in section 4.3 is chosen post hoc to match the same branch. Third, eq. (4.50) has the β_L/β_R ratio inverted relative to what is needed to combine with (4.47) into (4.51); as printed, the algebra gives the opposite sign for the real anomaly term. That is likely a typo, but it should be fixed. On the claimed contradiction in eq. (4.12), I don't find it: with q^2=−1, the ratio in (4.12) gives e^{η_f−η_i}, matching (4.11).\n\nAlso, the paper says the Rindler method [43] reproduces (3.7), and cites [11] without showing the relation. Those connections need discussion, especially since [11] may have overlapping results.\n\nBottom line: the qualitative statement that TEE's imaginary part in an anomalous CFT involves only one central charge is robust; which one is a free convention. The paper deserves a serious referee, but I would not cite the c_R result until the branch is justified. Engage, but require a revision that states the branch convention and its physical basis.","headline":"The paper's new timelike-geodesic Chern-Simons machinery is worth taking seriously, but the central c_R claim is not derived — it is selected by a branch choice at eq. (3.3) that is inconsistent with the paper's own z=x−t convention.","tokens_in":19986,"tokens_out":20728,"would_cite":false,"duration_ms":178452,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"For a timelike interval in a CFT with a gravitational anomaly, the timelike entanglement entropy is $S_T=\\frac{c_L+c_R}{6}\\log(T/\\epsilon)+\\frac{i\\pi c_R}{6}$, and topologically massive gravity in AdS$_3$ reproduces the same formula…","keywords":["timelike entanglement entropy","gravitational anomaly","chiral conformal field theory","topologically massive gravity","AdS3/CFT correspondence","Chern-Simons term","holographic entanglement entropy","BTZ black hole"],"falsifier":"Compute the TEE for the same chiral CFT by a method that does not use the normal-frame boundary prescription, for example a complexified-extremal-surface calculation, a direct pseudo-entropy computation for a free chiral CFT with $c_L\\neq c_R$, or a lattice simulation of 'entanglement in time' on a chiral edge. If the imaginary part is not exactly $\\frac{c_R}{6}\\pi$ at zero temperature, or if the holographic timelike anomaly contribution is not $i\\pi/(4G_N\\mu)$ under an equally natural matching condition, the central claim fails.","tokens_in":18897,"feed_emoji":"⏳","tokens_out":16109,"duration_ms":134732,"temperature":0.7,"pith_summary":"The paper claims that timelike entanglement entropy (TEE) in a two-dimensional conformal field theory with a gravitational anomaly is chiral in its imaginary part: for a pure timelike interval of length $T$, the entropy is $S_T=\\frac{c_L+c_R}{6}\\log(T/\\epsilon)+\\frac{i\\pi c_R}{6}$, so the real part sees both left and right central charges while the imaginary part sees only $c_R$. This asymmetry is new relative to the known non-anomalous TEE and gives a direct way to detect a gravitational anomaly in a chiral CFT. The same result is derived on the gravity side from topologically massive gravity in AdS$_3$, where the Chern-Simons term contributes the rotation of normal frames along a timelike geodesic. The authors obtain exact agreement between field theory and holography at zero temperature, at finite temperature with angular potential, and for the extremal finite-angular-momentum case. If the claim is correct, the imaginary part of TEE becomes a diagnostic of chirality rather than merely a formal continuation artifact.","feed_headline":"Timelike entropy's imaginary part becomes chiral under anomalies","feed_subtitle":"With unequal left and right central charges, only the right central charge enters the imaginary part—a direct gravitational-anomaly signal.","key_machinery":"The load-bearing object is the normal frame attached to the bulk geodesic. In TMG-AdS$_3$ the holographic entanglement entropy is not just a geodesic length: the gravitational Chern-Simons term contributes the on-shell action of a massive spinning particle, which reduces to a boundary term measuring how the normal frame twists or rotates along the curve. For spacelike geodesics this is a Lorentz boost of one timelike and one spacelike normal vector; for timelike geodesics both normals are spacelike and the transport is an $SO(2)$ rotation, giving $S_{\\rm anom}^{\\rm timelike}=\\frac{i}{4G_N\\mu}(\\theta_f-\\theta_i)$. The paper's new step is a boundary-condition prescription (eqs. 4.28--4.35) that fixes the normal vectors at the null-infinity junctions between the spacelike and timelike geodesics using the boundary time direction. That prescription is what makes the spacelike anomaly vanish and the timelike anomaly equal $i\\pi/(4G_N\\mu)$, producing the holographic formula that matches field theory.","core_discovery":"The central discovery is that analytic continuation of the spacelike entanglement entropy to a timelike interval does not just add the universal term $\\frac{c_L+c_R}{12}i\\pi$: with $c_L\\neq c_R$ a second, anomaly term $-\\frac{c_L-c_R}{12}i\\pi$ partially cancels it, leaving $\\frac{c_R}{6}i\\pi$. Holographically, the paper derives the same coefficient by evaluating the on-shell action of a massive spinning particle on the combined spacelike-plus-timelike extremal curves in TMG-AdS$_3$. The Chern-Simons contribution from the spacelike pieces vanishes for a pure timelike interval, while the timelike piece contributes $i\\pi/(4G_N\\mu)$, a finite, interval-length-independent rotation of the normal frame. Summing this with the geodesic lengths and using the TMG central charges $c_L=\\frac{3\\ell}{2G_N}(1-\\frac{1}{\\mu\\ell})$ and $c_R=\\frac{3\\ell}{2G_N}(1+\\frac{1}{\\mu\\ell})$ reproduces the field-theoretic formula in all three thermodynamic settings examined.","pith_inferences":["Beyond the paper: if the imaginary part of TEE is a direct chirality meter, a numerical or condensed-matter probe of timelike correlations in a chiral edge state, using the recently proposed 'entanglement in time' quantity, could test the $c_R/6$ coefficient without invoking holography.","Beyond the paper: the interval-length-independent $i\\pi/(4G_N\\mu)$ from the timelike geodesic has the form of a topological framing term; one might expect it to equal a Berry phase for the normal frame and to be robust under smooth deformations of the timelike curve, a property that could be checked directly in the bulk.","Beyond the paper: replacing the normal-frame matching prescription with another natural condition would change how the anomaly contribution splits between spacelike and timelike geodesics; the paper's agreement with field theory singles out its prescription, so deriving that boundary condition from a variational principle would close the main gap.","Beyond the paper: in backgrounds that are not locally AdS$_3$, such as warped AdS or Lifshitz spacetimes, the timelike anomaly term may acquire $T$-dependence because the normal-frame rotation would no longer be purely topological; the paper's methods give a concrete route to compute it."],"forward_implications":["For a pure timelike interval in an anomalous CFT$_2$, the imaginary part of TEE is $\\frac{c_R}{6}\\pi$, independent of interval length and of $c_L$; measuring this coefficient gives a direct read on the gravitational anomaly.","The holographic TEE in TMG-AdS$_3$ matches the field-theoretic result in three regimes (vacuum, finite temperature with angular potential, and extremal finite angular momentum), validating the normal-frame prescription against analytic continuation.","Setting $c_L=c_R=c$ recovers the standard timelike entanglement entropy $\\frac{c}{3}\\log(T/\\epsilon)+\\frac{i\\pi c}{6}$, so the anomaly result contains the known case as a limit.","The left-moving modes contribute only to the real part of TEE for a pure timelike interval; the imaginary part receives no contribution from $c_L$ at all.","The anomalous imaginary contribution does not depend on $T$; it is a universal constant shift for all timelike intervals in a given theory, unlike the logarithmic real part."],"supporting_citations":[{"why":"Supplies the TMG spinning-particle on-shell action and the spacelike-geodesic anomaly term that the paper extends to timelike geodesics.","marker":"[4]"},{"why":"Introduced timelike entanglement entropy and its holographic description, the baseline that this paper generalizes to gravitational anomalies.","marker":"[22, 23]"},{"why":"Provides the holographic TEE construction using two spacelike geodesics joined by a timelike geodesic, which the paper adapts to TMG-AdS3.","marker":"[28]"},{"why":"Establishes the holographic gravitational-anomaly dictionary for TMG-AdS3 that underlies the unequal left and right central charges.","marker":"[12]"},{"why":"Supplies the replica twist-field method whose spacelike entanglement entropy is analytically continued to timelike intervals.","marker":"[1, 2]"},{"why":"Gives the geodesic-length formula for holographic entanglement entropy that forms the real-part backbone of the holographic TEE.","marker":"[8, 9]"},{"why":"Supply the semi-classical central-charge expressions for TMG-AdS3 used to convert bulk contributions into the boundary central charges.","marker":"[71, 72]"}],"fun_headline_variants":["Imaginary part of timelike entropy exposes gravitational anomalies","Timelike entropy's imaginary part detects chiral central charges","Gravitational anomalies leave mark on timelike entropy imaginary part","Anomaly term in timelike entanglement entropy: only c_R survives","Holographic match: timelike entropy probes anomaly through c_R"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The holographic derivation rests on an underived rule for fixing the normal-frame vectors where the spacelike and timelike geodesics meet at null infinity; if a different natural choice changed the imaginary contribution, the exact match with the field-theory result would break.","fun_headline_variants_meta":{"raw":{"variants":["Imaginary part of timelike entropy exposes gravitational anomalies","Timelike entropy's imaginary part detects chiral central charges","Gravitational anomalies leave mark on timelike entropy imaginary part","Anomaly term in timelike entanglement entropy: only c_R survives","Holographic match: timelike entropy probes anomaly through c_R"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000176,"raw_usage":{"total_tokens":1272,"prompt_tokens":911,"completion_tokens":361,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":527,"completion_tokens_details":{"reasoning_tokens":275}},"tokens_in":527,"tokens_out":361,"duration_ms":3757,"temperature":1.0,"reasoning_tokens":275,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:46:57.451418+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the TEE for the same chiral CFT by a method that does not use the normal-frame boundary prescription, for example a complexified-extremal-surface calculation, a direct pseudo-entropy computation for a free chiral CFT with $c_L\\neq c_R$, or a lattice simulation of 'entanglement in time' on a chiral edge. If the imaginary part is not exactly $\\frac{c_R}{6}\\pi$ at zero temperature, or if the holographic timelike anomaly contribution is not $i\\pi/(4G_N\\mu)$ under an equally natural matching condition, the central claim fails.","supporting_citations":[],"review_version":1}