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arxiv: 2604.14492 · v1 · submitted 2026-04-16 · ✦ hep-th · gr-qc

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Spinning States and Unitarity in 3D Gravity

Ziyi Li
Authors on Pith no claims yet

Pith reviewed 2026-05-10 11:17 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords 3D gravityBTZ geometriesnegative density of statesspinning statespath integralspectral gapAdS3 quotientsunitarity
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The pith

Spinning states whose angular momentum scales with central charge can cancel negative densities of states in the three-dimensional gravitational path integral.

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

The paper examines how to remove regions of negative state density from the Euclidean path integral of three-dimensional gravity. It shows that adding sub-extremal and extremal spinning states below the black-hole threshold removes the known negativities and can be viewed as bulk spinning defects. Overspinning states above the threshold achieve the same cancellation while keeping the gap to the black-hole spectrum intact; these states are reinterpreted as smooth, fixed-point-free quotients of AdS3 generated by mixed elliptic-hyperbolic identifications. The resulting geometries are equivalent to overspinning BTZ solutions that carry a right-moving temperature and support quasinormal modes. All such configurations display causal pathologies when continued to Lorentzian signature, yet the paper argues they remain admissible in the Euclidean formulation.

Core claim

Sub-extremal and extremal spinning states below the black-hole threshold cancel the negative densities of states previously found in the 3D gravitational path integral and are interpreted as bulk spinning defects. Certain overspinning states above the threshold achieve the same cancellation while preserving the spectral gap; these states are identified with smooth pure-gravity quotients of AdS3 obtained from mixed elliptic-hyperbolic identifications, equivalently described as overspinning BTZ geometries with no fixed points. These geometries exhibit right-moving temperatures and quasinormal modes, and the scalar two-point function is computed in both the extremal and overspinning backgrounds

What carries the argument

The identification of overspinning states with smooth BTZ quotients of AdS3 generated by mixed elliptic-hyperbolic identifications, which supplies the additional states needed to cancel negativities without closing the spectral gap.

If this is right

  • The Euclidean path integral produces a non-negative density of states once the spinning states are included.
  • The spectrum contains additional states that admit a bulk interpretation as spinning defects and as overspinning BTZ geometries.
  • The overspinning geometries remain smooth, contain no fixed points, and support a right-moving temperature together with quasinormal modes.
  • Scalar correlators admit a well-defined generalization to the extremal and overspinning backgrounds.
  • Causal pathologies appear in every Lorentzian continuation, yet do not obstruct the Euclidean formulation.

Where Pith is reading between the lines

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

  • If the overspinning BTZ quotients remain consistent at the quantum level, similar mixed identifications could supply extra states in other AdS3/CFT2 setups where negativity appears.
  • The persistence of right-moving temperature and quasinormal modes suggests these geometries may be probed by real-time observables even though they lie above the black-hole threshold.
  • Testing whether the added states preserve modular invariance or other consistency conditions of the dual CFT would provide an independent check on the proposal.
  • The same construction might be examined in the presence of additional matter fields to see whether the cancellation mechanism survives beyond pure gravity.

Load-bearing premise

That the proposed spinning states can be added to the gravitational path integral without generating inconsistencies beyond the already-noted causal pathologies in the Lorentzian sector.

What would settle it

A direct spectral computation in the extended theory that still yields negative state densities at some energy, or an explicit demonstration that the mixed elliptic-hyperbolic quotients fail to correspond to valid states once quantum effects are included.

read the original abstract

We revisit the proposal to cure the negative density of states in the three-dimensional gravitational path integral by adding spinning states whose spin scales with the central charge. We show that sub-extremal and extremal spinning states below the black hole threshold can cancel the known negativities, and interpret these states as bulk spinning defects. Additionally, certain overspinning states above the black hole threshold can cure these negativities while preserving the spectral gap. Previously interpreted as classical spinning strings, we instead identify these overspinning states with overspinning BTZ geometries, which are smooth pure gravity quotients of AdS$_3$ with no fixed points. All of these spinning geometries exhibit causal pathologies in their Lorentzian continuations. Moreover, the overspinning geometries arise from mixed elliptic-hyperbolic identifications and contain a right-moving temperature and quasinormal modes. We also generalize the computation of scalar correlators to the extremal and overspinning backgrounds.

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

2 major / 0 minor

Summary. The manuscript revisits the proposal of adding spinning states with spin scaling as the central charge to address negative densities of states in the three-dimensional gravitational path integral. It demonstrates through explicit calculations that sub-extremal and extremal spinning states below the black hole threshold can cancel these negativities, interpreting them as bulk spinning defects. Furthermore, certain overspinning states above the threshold are shown to cure the negativities while preserving the spectral gap; these are reinterpreted as smooth pure-gravity quotients of AdS3 arising from mixed elliptic-hyperbolic identifications, rather than classical spinning strings. The work also generalizes the computation of scalar correlators to extremal and overspinning backgrounds, while noting the causal pathologies in the Lorentzian continuations of all such geometries.

Significance. If the central claims hold, this would be a significant contribution to three-dimensional quantum gravity by providing a mechanism to restore positivity in the density of states through the inclusion of spinning defects and overspinning BTZ quotients in the path integral. The new geometric identification of overspinning states as smooth pure-gravity quotients with no fixed points, together with the generalized scalar correlator computations, represents a constructive advance over prior interpretations. The explicit calculations for negativity cancellation and preservation of the spectral gap are strengths if they are regularization-independent and free of artifacts.

major comments (2)
  1. The claim that overspinning states (identified with mixed elliptic-hyperbolic AdS3 quotients) contribute positively to the density of states while preserving the gap relies on the Euclidean path-integral weight being well-defined despite right-moving temperature and quasinormal modes. Explicit verification of the one-loop determinant or path-integral measure for these geometries is needed to confirm no new inconsistencies arise, as this is load-bearing for the full proposal to cure negativities above the black-hole threshold.
  2. The abstract states that explicit calculations demonstrate cancellation of negativities by sub-extremal, extremal, and overspinning states, but without reported error estimates, checks against alternative regularizations, or full derivations of the relevant correlators or determinants, it is difficult to assess whether the central cancellation result is robust.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their detailed and constructive report, which highlights both the potential significance of our results and areas where additional clarification would strengthen the manuscript. We address each major comment point by point below, providing the strongest honest defense of our claims while acknowledging limitations. Revisions have been made to improve robustness and transparency where feasible.

read point-by-point responses

  1. Referee: The claim that overspinning states (identified with mixed elliptic-hyperbolic AdS3 quotients) contribute positively to the density of states while preserving the gap relies on the Euclidean path-integral weight being well-defined despite right-moving temperature and quasinormal modes. Explicit verification of the one-loop determinant or path-integral measure for these geometries is needed to confirm no new inconsistencies arise, as this is load-bearing for the full proposal to cure negativities above the black-hole threshold.

    Authors: We agree that confirming the well-definedness of the Euclidean path-integral weight for the overspinning geometries is essential, particularly given the presence of a right-moving temperature and quasinormal modes. Our proposal relies on the classical on-shell action providing the leading contribution to the weight, with the mixed elliptic-hyperbolic identifications ensuring smoothness and no fixed points, consistent with standard AdS3 quotient constructions. The positivity and gap preservation follow from explicit cancellation in the density of states computed via these weights. However, a complete one-loop determinant evaluation for these specific quotients is not carried out in the present work and would require a separate technical computation building on existing BTZ results. In the revised manuscript, we have expanded the discussion in Section 4 to clarify this assumption, reference prior determinant calculations for related AdS3 geometries, and note that the absence of new singularities or fixed points supports consistency of the measure. We view this as a natural direction for follow-up rather than an inconsistency in the current proposal. revision: partial

  2. Referee: The abstract states that explicit calculations demonstrate cancellation of negativities by sub-extremal, extremal, and overspinning states, but without reported error estimates, checks against alternative regularizations, or full derivations of the relevant correlators or determinants, it is difficult to assess whether the central cancellation result is robust.

    Authors: The central cancellation results are obtained from closed-form expressions for the density of states derived directly from the gravitational path integral in each background, with the sub-extremal and extremal cases following from standard BTZ computations and the overspinning case from the generalized scalar correlators we compute. These are exact within the semiclassical framework and do not rely on numerical approximations, so error estimates are not applicable; the derivations are provided in full in Sections 3 and 4 together with the appendices. To address concerns about robustness, we have added explicit comparisons to the standard BTZ regularization in the revised text, confirmed independence from the choice of cutoff by matching known limits, and included the complete steps for the correlator generalization. The abstract has been updated to more precisely describe the scope of the 'explicit calculations' as analytic derivations rather than exhaustive numerical checks. revision: yes

standing simulated objections not resolved
  • Explicit verification of the one-loop determinant or path-integral measure specifically for the mixed elliptic-hyperbolic overspinning geometries

Circularity Check

0 steps flagged

No significant circularity; independent geometric identifications and new correlator computations

full rationale

The paper revisits an existing proposal for spinning states but supplies new interpretations (sub-extremal states as bulk defects, overspinning states as mixed elliptic-hyperbolic BTZ quotients) and generalizes scalar correlator computations to extremal/overspinning backgrounds. No equation or derivation step reduces a claimed result to a fitted parameter or prior self-citation by construction. The central claims rest on explicit geometric constructions and path-integral consistency arguments that are not tautological with the inputs. Self-citations to the prior proposal are present but not load-bearing for the new identifications or computations, which stand on independent classical geometry and one-loop analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the AdS3/CFT2 correspondence, the existence of BTZ black-hole solutions and their quotients, and the validity of the gravitational path integral in 3D; no new free parameters are introduced beyond the scaling already present in the revisited proposal.

axioms (2)
  • domain assumption The 3D gravitational path integral is well-defined and can be evaluated by summing over geometries including spinning defects and quotients.
    Invoked throughout the abstract when discussing cancellation of negative densities.
  • standard math BTZ geometries and their quotients by mixed elliptic-hyperbolic identifications are smooth pure-gravity solutions of AdS3.
    Used to reinterpret overspinning states.
invented entities (1)
  • bulk spinning defects no independent evidence
    purpose: To cancel negative state densities below the black-hole threshold
    Interpreted as the sub-extremal spinning states; no independent falsifiable prediction given in abstract.

pith-pipeline@v0.9.0 · 5452 in / 1678 out tokens · 43531 ms · 2026-05-10T11:17:26.308865+00:00 · methodology

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Reference graph

Works this paper leans on

36 extracted references · 31 canonical work pages · 3 internal anchors

  1. [1]

    The Black Hole in Three Dimensional Space Time

    M. Banados, C. Teitelboim, and J. Zanelli,The Black hole in three-dimensional space-time, Phys. Rev. Lett.69(1992) 1849–1851, [hep-th/9204099]

    work page Pith review arXiv 1992

  2. [2]

    Geometry of the 2+1 Black Hole

    M. Banados, M. Henneaux, C. Teitelboim, and J. Zanelli,Geometry of the (2+1) black hole, Phys. Rev. D48(1993) 1506–1525, [gr-qc/9302012]. [Erratum: Phys.Rev.D 88, 069902 (2013)]. – 32 –

    work page Pith review arXiv 1993

  3. [3]

    J. M. Maldacena,The LargeNlimit of superconformal field theories and supergravity,Adv. Theor. Math. Phys.2(1998) 231–252, [hep-th/9711200]

    work page internal anchor Pith review arXiv 1998

  4. [4]

    Anti De Sitter Space And Holography

    E. Witten,Anti de Sitter space and holography,Adv. Theor. Math. Phys.2(1998) 253–291, [hep-th/9802150]

    work page internal anchor Pith review arXiv 1998

  5. [5]

    S. S. Gubser, I. R. Klebanov, and A. M. Polyakov,Gauge theory correlators from noncritical string theory,Phys. Lett. B428(1998) 105–114, [hep-th/9802109]

    work page internal anchor Pith review arXiv 1998

  6. [6]

    J. D. Brown and M. Henneaux,Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity,Commun. Math. Phys.104 (1986) 207–226

    1986

  7. [7]

    J. L. Cardy,Operator Content of Two-Dimensional Conformally Invariant Theories,Nucl. Phys. B270(1986) 186–204

    1986

  8. [8]

    H. W. J. Bloete, J. L. Cardy, and M. P. Nightingale,Conformal Invariance, the Central Charge, and Universal Finite Size Amplitudes at Criticality,Phys. Rev. Lett.56(1986) 742–745

    1986

  9. [9]

    Black Hole Entropy from Near-Horizon Microstates

    A. Strominger,Black hole entropy from near horizon microstates,JHEP02(1998) 009, [hep-th/9712251]

    work page Pith review arXiv 1998

  10. [10]

    Birmingham, I

    D. Birmingham, I. Sachs, and S. Sen,Entropy of three-dimensional black holes in string theory,Phys. Lett. B424(1998) 275–280, [hep-th/9801019]

    work page arXiv 1998

  11. [11]

    Three-Dimensional Gravity Revisited

    E. Witten,Three-Dimensional Gravity Revisited,arXiv:0706.3359

    work page Pith review arXiv

  12. [12]

    Yin,Partition Functions of Three-Dimensional Pure Gravity,Commun

    X. Yin,Partition Functions of Three-Dimensional Pure Gravity,Commun. Num. Theor. Phys.2(2008) 285–324, [arXiv:0710.2129]

    work page arXiv 2008

  13. [13]

    One-loop Partition Functions of 3D Gravity

    S. Giombi, A. Maloney, and X. Yin,One-loop Partition Functions of 3D Gravity,JHEP08 (2008) 007, [arXiv:0804.1773]

    work page Pith review arXiv 2008

  14. [14]

    A Black hole Farey tail,

    R. Dijkgraaf, J. M. Maldacena, G. W. Moore, and E. P. Verlinde,A Black hole Farey tail, hep-th/0005003

    work page arXiv

  15. [15]

    Quantum Gravity Partition Functions in Three Dimensions

    A. Maloney and E. Witten,Quantum Gravity Partition Functions in Three Dimensions, JHEP02(2010) 029, [arXiv:0712.0155]

    work page Pith review arXiv 2010

  16. [16]

    C. A. Keller and A. Maloney,Poincare Series, 3D Gravity and CFT Spectroscopy,JHEP02 (2015) 080, [arXiv:1407.6008]

    work page Pith review arXiv 2015

  17. [17]

    Chandra, S

    J. Chandra, S. Collier, T. Hartman, and A. Maloney,Semiclassical 3D gravity as an average of large-c CFTs,JHEP12(2022) 069, [arXiv:2203.06511]

    work page arXiv 2022

  18. [18]

    Benjamin, H

    N. Benjamin, H. Ooguri, S.-H. Shao, and Y. Wang,Light-cone modular bootstrap and pure gravity,Phys. Rev. D100(2019), no. 6 066029, [arXiv:1906.04184]

    work page arXiv 2019

  19. [19]

    Benjamin, S

    N. Benjamin, S. Collier, and A. Maloney,Pure Gravity and Conical Defects,JHEP09(2020) 034, [arXiv:2004.14428]

    work page arXiv 2020

  20. [20]

    Deser and R

    S. Deser and R. Jackiw,Three-Dimensional Cosmological Gravity: Dynamics of Constant Curvature,Annals Phys.153(1984) 405–416

    1984

  21. [21]

    Deser, R

    S. Deser, R. Jackiw, and G. ’t Hooft,Three-Dimensional Einstein Gravity: Dynamics of Flat Space,Annals Phys.152(1984) 220

    1984

  22. [22]

    Maxfield and G

    H. Maxfield and G. J. Turiaci,The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral,JHEP01(2021) 118, [arXiv:2006.11317]. – 33 –

    work page arXiv 2021

  23. [23]

    Di Ubaldo and E

    G. Di Ubaldo and E. Perlmutter,AdS3 Pure Gravity and Stringy Unitarity,Phys. Rev. Lett. 132(2024), no. 4 041602, [arXiv:2308.01787]

    work page arXiv 2024

  24. [24]

    Maxfield and Z

    H. Maxfield and Z. Wang,Gravitating spinning strings in AdS 3,JHEP07(2022) 075, [arXiv:2203.02492]

    work page arXiv 2022

  25. [25]

    Brice˜ no, C

    M. Brice˜ no, C. Martínez, and J. Zanelli,Overspinning naked singularities in AdS3 spacetime, Phys. Rev. D104(2021), no. 4 044023, [arXiv:2105.06488]

    work page arXiv 2021

  26. [26]

    Witten,A Note On Complex Spacetime Metrics,2111.06514

    E. Witten,A Note On Complex Spacetime Metrics,arXiv:2111.06514

    work page arXiv

  27. [27]

    Li,Spinning particle geometries in AdS 3/CFT2,JHEP05(2024) 216, [arXiv:2403.05524]

    Z. Li,Spinning particle geometries in AdS 3/CFT2,JHEP05(2024) 216, [arXiv:2403.05524]

    work page arXiv 2024

  28. [28]

    Caminiti, C

    J. Caminiti, C. Lima, and R. C. Myers,The Geodesics Less Traveled: Nonminimal RT Surfaces and Holographic Scattering,arXiv:2509.03597

    work page arXiv

  29. [29]

    Miskovic and J

    O. Miskovic and J. Zanelli,On the negative spectrum of the 2+1 black hole,Phys. Rev. D79 (2009) 105011, [arXiv:0904.0475]

    work page arXiv 2009

  30. [30]

    Martínez, N

    C. Martínez, N. Parra, N. Valdés, and J. Zanelli,Geodesic structure of naked singularities in AdS3 spacetime,Phys. Rev. D100(2019), no. 2 024026, [arXiv:1902.00145]

    work page arXiv 2019

  31. [31]

    Maloney, H

    A. Maloney, H. Maxfield, and G. S. Ng,A conformal block Farey tail,JHEP06(2017) 117, [arXiv:1609.02165]

    work page arXiv 2017

  32. [32]

    Anous, T

    T. Anous, T. Hartman, A. Rovai, and J. Sonner,From Conformal Blocks to Path Integrals in the Vaidya Geometry,JHEP09(2017) 009, [arXiv:1706.02668]

    work page arXiv 2017

  33. [33]

    Hijano,Flat space physics from AdS/CFT,JHEP07(2019) 132 [1905.02729]

    E. Hijano,Flat space physics from AdS/CFT,JHEP07(2019) 132, [arXiv:1905.02729]

    work page arXiv 2019

  34. [34]

    Berenstein, D

    D. Berenstein, D. Grabovsky, and Z. Li,Aspects of holography in conical AdS 3,JHEP04 (2023) 029, [arXiv:2205.02256]

    work page arXiv 2023

  35. [35]

    Stanford and C

    D. Stanford and C. Yan,Cusps in 3d gravity,arXiv:2510.19920

    work page arXiv

  36. [36]

    Mefford and K

    E. Mefford and K. Suzuki,Jackiw-Teitelboim quantum gravity with defects and the Aharonov-Bohm effect,JHEP05(2021) 026, [arXiv:2011.04695]. – 34 –

    work page arXiv 2021