pith. sign in

arxiv: 2607.01319 · v1 · pith:4GMWLUPDnew · submitted 2026-07-01 · ✦ hep-th · gr-qc

What's the Matter with 3D Gravity?

Robert Bourne , Jackson R. Fliss , Bob Knighton This is my paper

Pith reviewed 2026-07-03 19:40 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords 3D gravityAdS3worldline formalismgeometric quantizationVirasoro conformal blockspath integralWilson spoolthermal AdS
0
0 comments X

The pith

A worldline phase space for matter in 3D gravity, when geometrically quantized, produces Virasoro conformal blocks and a conjecture for the all-orders path integral on thermal AdS3.

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

The paper develops a method to minimally couple a massive scalar field to three-dimensional Einstein gravity with negative cosmological constant by using the worldline formalism. It builds a classical phase space on a spatial slice and applies geometric quantization to obtain a Hilbert space whose states are Virasoro conformal blocks with weights below c/24. The formalism is then used to evaluate the partition function on thermal AdS3 via equivariant localization, matching the known one-loop answer and the AdS3 Wilson spool while offering a candidate expression valid at all orders in the gravitational coupling. A reader would care because three-dimensional gravity with matter remains a tractable arena in which classical and quantum descriptions might be compared directly.

Core claim

By working in the worldline formalism, we construct a classical phase space on an initial time surface Σ, which we quantize using geometric quantization. States in the Hilbert space correspond to Virasoro conformal blocks with operators of conformal weight h<c/24. As an application of our formalism, we compute the partition function on thermal AdS3 through equivariant localization. Our answer reproduces the AdS3 Wilson spool and agrees with the known one-loop result. It further serves as a conjecture for the value of the path integral of gravity minimally coupled to a massive scalar field in thermal AdS3 to all orders in GN.

What carries the argument

The classical phase space of the matter worldline on the initial time surface Σ, quantized geometrically to produce Virasoro conformal blocks as states.

If this is right

  • The partition function on thermal AdS3 equals the AdS3 Wilson spool to all orders in GN.
  • The result agrees with the known one-loop computation from other methods.
  • The Hilbert space of the coupled system consists of Virasoro conformal blocks with conformal weights h below c/24.
  • The same localization procedure supplies a concrete expression for the full non-perturbative path integral of the coupled theory.

Where Pith is reading between the lines

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

  • The method may apply to other matter fields or to different three-dimensional backgrounds once the corresponding worldline phase spaces are identified.
  • Higher-order perturbative checks of the partition function would provide a direct test of whether the localization result holds beyond one loop.
  • The appearance of conformal blocks suggests possible links to Liouville theory or Chern-Simons formulations that remain to be explored within this framework.

Load-bearing premise

The states obtained from geometric quantization of the worldline phase space correctly correspond to Virasoro conformal blocks with h less than c over 24 and capture the Hilbert space of the coupled gravity-matter system.

What would settle it

An explicit two-loop computation of the path integral for gravity coupled to a massive scalar on thermal AdS3 that disagrees with the equivariant localization result would falsify the all-orders conjecture.

read the original abstract

We revisit the problem of minimally coupling matter to Einstein gravity in three dimensions with negative cosmological constant. By working in the worldline formalism, we construct a classical phase space on an initial time surface $\Sigma$, which we quantize using geometric quantization. States in the Hilbert space correspond to Virasoro conformal blocks with operators of conformal weight $h<c/24$. As an application of our formalism, we compute the partition function on thermal $\text{AdS}_3$ through equivariant localization. Our answer reproduces the AdS$_3$ Wilson spool and agrees with the known one-loop result. It further serves as a conjecture for the value of the path integral of gravity minimally coupled to a massive scalar field in thermal $\text{AdS}_3$ to all orders in $G_N$.

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 minimally coupling matter to 3D Einstein gravity with negative cosmological constant via the worldline formalism. It constructs a classical phase space on an initial time surface Σ and quantizes it using geometric quantization, with states identified as Virasoro conformal blocks for operators with h < c/24. The thermal AdS3 partition function is computed through equivariant localization, reproducing the AdS3 Wilson spool and the known one-loop result; this is conjectured to yield the exact path integral of gravity minimally coupled to a massive scalar field in thermal AdS3 to all orders in G_N.

Significance. If the central conjecture holds and the quantization step is independently verified, the work would supply a non-perturbative, all-orders expression for the coupled path integral, extending one-loop and Wilson-spool results and linking worldline geometric quantization to Virasoro blocks in 3D quantum gravity.

major comments (2)
  1. [Abstract] Abstract (paragraph on quantization): the claim that geometric quantization of the worldline phase space on Σ yields states corresponding to Virasoro conformal blocks with h < c/24 and that this Hilbert space is the correct one for the interacting gravity-scalar system is load-bearing for the all-orders conjecture, yet the manuscript provides no explicit verification that the quantized spectrum and inner products match the full set of diffeomorphism-invariant observables of the coupled theory.
  2. [Abstract] Abstract: the statement that equivariant localization reproduces the AdS3 Wilson spool is presented without details establishing whether the localization is independent of prior definitions of that object or reduces to them by construction, which directly affects the strength of the one-loop agreement and the extrapolation to the conjecture.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the two major comments point by point below. Both concerns can be met by clarifying the conjectural status of the Hilbert-space identification and by expanding the localization calculation; no standing objections remain.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on quantization): the claim that geometric quantization of the worldline phase space on Σ yields states corresponding to Virasoro conformal blocks with h < c/24 and that this Hilbert space is the correct one for the interacting gravity-scalar system is load-bearing for the all-orders conjecture, yet the manuscript provides no explicit verification that the quantized spectrum and inner products match the full set of diffeomorphism-invariant observables of the coupled theory.

    Authors: We agree that the manuscript does not supply an exhaustive verification that the quantized spectrum and inner products coincide with every diffeomorphism-invariant observable of the fully coupled theory. The identification follows from applying geometric quantization to the worldline phase space on Σ, which by construction reproduces the known Virasoro-block spectrum for h < c/24. Because the all-orders claim is presented as a conjecture, we will revise the abstract and the relevant discussion sections to state explicitly that the Hilbert-space equivalence is motivated by the quantization procedure but remains conjectural for the interacting system. This change will be accompanied by a short paragraph outlining the scope of the current verification. revision: partial

  2. Referee: [Abstract] Abstract: the statement that equivariant localization reproduces the AdS3 Wilson spool is presented without details establishing whether the localization is independent of prior definitions of that object or reduces to them by construction, which directly affects the strength of the one-loop agreement and the extrapolation to the conjecture.

    Authors: The equivariant localization is carried out directly on the phase space obtained after geometric quantization; the resulting expression is shown to equal the Wilson spool by explicit summation over the fixed-point contributions. The match is therefore derived rather than assumed by construction. We will add the intermediate steps of the localization (including the explicit fixed-point data and the one-loop determinant factors) to the main text, thereby demonstrating the independence of the calculation and strengthening the one-loop agreement section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained with explicit conjecture

full rationale

The paper constructs a classical phase space via the worldline formalism, applies geometric quantization, identifies the resulting states with Virasoro blocks, and evaluates the thermal AdS3 partition function by equivariant localization. This computation is shown to reproduce the known AdS3 Wilson spool and one-loop result, providing an independent consistency check. The extension to all orders in GN is explicitly labeled a conjecture rather than a derived equality, with no self-citations, definitional reductions, or fitted inputs presented as predictions in the abstract or described chain. The load-bearing assumption on the Hilbert space is stated directly but does not reduce the localization result to an input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; the main visible assumption is the applicability of the worldline formalism and geometric quantization to the coupled system. No free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The worldline formalism correctly constructs the classical phase space for minimally coupled matter in 3D Einstein gravity with negative cosmological constant, and geometric quantization of this space yields states corresponding to Virasoro conformal blocks with h < c/24.
    This premise is required for the Hilbert space identification and the subsequent partition-function computation stated in the abstract.

pith-pipeline@v0.9.1-grok · 5659 in / 1402 out tokens · 27610 ms · 2026-07-03T19:40:07.100927+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

80 extracted references · 66 canonical work pages · 28 internal anchors

  1. [1]

    (2+1)-Dimensional Gravity as an Exactly Soluble System,

    E. Witten, “(2+1)-Dimensional Gravity as an Exactly Soluble System,”Nucl. Phys. B311(1988) 46

    1988

  2. [2]

    Three-Dimensional Gravity Revisited

    E. Witten, “Three-Dimensional Gravity Revisited,”arXiv:0706.3359 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  3. [3]

    A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories,

    A. Achucarro and P. K. Townsend, “A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories,”Phys. Lett. B180(1986) 89

    1986

  4. [4]

    Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity,

    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

  5. [5]

    The asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant

    O. Coussaert, M. Henneaux, and P. van Driel, “The Asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant,”Class. Quant. Grav.12(1995) 2961–2966,arXiv:gr-qc/9506019

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  6. [6]

    Geometric actions for three-dimensional gravity

    G. Barnich, H. A. Gonzalez, and P. Salgado-Rebolledo, “Geometric actions for three-dimensional gravity,”Class. Quant. Grav.35no. 1, (2018) 014003, arXiv:1707.08887 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  7. [7]

    A theory of reparameterizations for AdS$_3$ gravity

    J. Cotler and K. Jensen, “A theory of reparameterizations for AdS 3 gravity,”JHEP 02(2019) 079,arXiv:1808.03263 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  8. [8]

    Solving 3d gravity with Virasoro TQFT,

    S. Collier, L. Eberhardt, and M. Zhang, “Solving 3d gravity with Virasoro TQFT,” SciPost Phys.15no. 4, (2023) 151,arXiv:2304.13650 [hep-th]

    work page arXiv 2023

  9. [9]

    3d gravity from Virasoro TQFT: Holography, wormholes and knots,

    S. Collier, L. Eberhardt, and M. Zhang, “3d gravity from Virasoro TQFT: Holography, wormholes and knots,”SciPost Phys.17(2024) 134, arXiv:2401.13900 [hep-th]

    work page arXiv 2024

  10. [10]

    A de Sitter Farey Tail

    A. Castro, N. Lashkari, and A. Maloney, “A de Sitter Farey Tail,”Phys. Rev. D83 (2011) 124027,arXiv:1103.4620 [hep-th]. – 38 –

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  11. [11]

    The Wave Function of Quantum de Sitter

    A. Castro and A. Maloney, “The Wave Function of Quantum de Sitter,”JHEP11 (2012) 096,arXiv:1209.5757 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  12. [12]

    Low-dimensional de Sitter quantum gravity,

    J. Cotler, K. Jensen, and A. Maloney, “Low-dimensional de Sitter quantum gravity,” JHEP06(2020) 048,arXiv:1905.03780 [hep-th]

    work page arXiv 2020

  13. [13]

    Holography in de Sitter Space via Chern-Simons Gauge Theory,

    Y. Hikida, T. Nishioka, T. Takayanagi, and Y. Taki, “Holography in de Sitter Space via Chern-Simons Gauge Theory,”Phys. Rev. Lett.129no. 4, (2022) 041601, arXiv:2110.03197 [hep-th]

    work page arXiv 2022

  14. [14]

    T T + Λ 2 deformed CFT on the stretched dS 3 horizon,

    V. Shyam, “T T + Λ 2 deformed CFT on the stretched dS 3 horizon,”JHEP04 (2022) 052,arXiv:2106.10227 [hep-th]

    work page arXiv 2022

  15. [15]

    De Sitter microstates fromT T+ Λ 2 and the Hawking-Page transition,

    E. Coleman, E. A. Mazenc, V. Shyam, E. Silverstein, R. M. Soni, G. Torroba, and S. Yang, “De Sitter microstates fromT T+ Λ 2 and the Hawking-Page transition,” JHEP07(2022) 140,arXiv:2110.14670 [hep-th]

    work page arXiv 2022

  16. [16]

    Double-scaled SYK and de Sitter holography,

    V. Narovlansky and H. Verlinde, “Double-scaled SYK and de Sitter holography,” JHEP05(2025) 032,arXiv:2310.16994 [hep-th]

    work page arXiv 2025

  17. [17]

    Bulk-local dS 3 holography: the matter withT T+ Λ 2,

    G. Batra, G. B. De Luca, E. Silverstein, G. Torroba, and S. Yang, “Bulk-local dS 3 holography: the matter withT T+ Λ 2,”JHEP10(2024) 072,arXiv:2403.01040 [hep-th]

    work page arXiv 2024

  18. [18]

    A microscopic realization of dS 3,

    S. Collier, L. Eberhardt, and B. M¨ uhlmann, “A microscopic realization of dS 3,” SciPost Phys.18no. 4, (2025) 131,arXiv:2501.01486 [hep-th]

    work page arXiv 2025

  19. [19]

    Quasi-conformal mappings in n -space and the rigidity of hyperbolic space forms,

    G. D. Mostow, “Quasi-conformal mappings in n -space and the rigidity of hyperbolic space forms,”Publ. Math. Inst. Hautes Etudes Sci.34no. 1, (Jan., 1968) 53–104

    1968

  20. [20]

    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 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  21. [21]

    Benjamin, S

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

    work page arXiv 2020

  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 [hep-th]

    work page arXiv 2021

  23. [23]

    Random statistics of OPE coefficients and Euclidean wormholes,

    A. Belin and J. de Boer, “Random statistics of OPE coefficients and Euclidean wormholes,”Class. Quant. Grav.38no. 16, (2021) 164001,arXiv:2006.05499 [hep-th]

    work page arXiv 2021

  24. [24]

    AdS 3 gravity and random CFT,

    J. Cotler and K. Jensen, “AdS 3 gravity and random CFT,”JHEP04(2021) 033, arXiv:2006.08648 [hep-th]

    work page arXiv 2021

  25. [25]

    Semiclassical 3D gravity as an average of large-c CFTs,

    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 [hep-th]

    work page arXiv 2022

  26. [26]

    AdS 3/RMT2 duality,

    G. Di Ubaldo and E. Perlmutter, “AdS 3/RMT2 duality,”JHEP12(2023) 179, arXiv:2307.03707 [hep-th]. – 39 –

    work page arXiv 2023

  27. [27]

    Euclidean wormholes in two-dimensional conformal field theories from quantum chaos and number theory,

    F. M. Haehl, W. Reeves, and M. Rozali, “Euclidean wormholes in two-dimensional conformal field theories from quantum chaos and number theory,”Phys. Rev. D108 no. 10, (2023) L101902,arXiv:2309.02533 [hep-th]

    work page arXiv 2023

  28. [28]

    Approximate CFTs and random tensor models,

    A. Belin, J. de Boer, D. L. Jafferis, P. Nayak, and J. Sonner, “Approximate CFTs and random tensor models,”JHEP09(2024) 163,arXiv:2308.03829 [hep-th]

    work page arXiv 2024

  29. [29]

    A principle of maximum ignorance for semiclassical gravity,

    J. de Boer, D. Liska, B. Post, and M. Sasieta, “A principle of maximum ignorance for semiclassical gravity,”JHEP2024(2024) 003,arXiv:2311.08132 [hep-th]

    work page arXiv 2024

  30. [30]

    3d gravity as a random ensemble,

    D. L. Jafferis, L. Rozenberg, and G. Wong, “3d gravity as a random ensemble,” JHEP02(2025) 208,arXiv:2407.02649 [hep-th]

    work page arXiv 2025

  31. [31]

    Modular-Invariant Random Matrix Theory and AdS3 Wormholes,

    J. Boruch, G. Di Ubaldo, F. M. Haehl, E. Perlmutter, and M. Rozali, “Modular-Invariant Random Matrix Theory and AdS3 Wormholes,”Phys. Rev. Lett.135no. 12, (2025) 121602,arXiv:2503.00101 [hep-th]

    work page arXiv 2025

  32. [32]

    Wormholes in AdS

    J. M. Maldacena and L. Maoz, “Wormholes in AdS,”JHEP02(2004) 053, arXiv:hep-th/0401024

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  33. [33]

    Partition Functions of Three-Dimensional Pure Gravity

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

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  34. [34]

    Di Ubaldo and E

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

    work page arXiv 2024

  35. [35]

    Keeping matter in the loop in dS3 quantum gravity,

    A. Castro, I. Coman, J. R. Fliss, and C. Zukowski, “Keeping matter in the loop in dS3 quantum gravity,”JHEP07(2023) 120,arXiv:2302.12281 [hep-th]. [Erratum: JHEP 09, 004 (2024)]

    work page arXiv 2023

  36. [36]

    Coupling Fields to 3D Quantum Gravity via Chern-Simons Theory,

    A. Castro, I. Coman, J. R. Fliss, and C. Zukowski, “Coupling Fields to 3D Quantum Gravity via Chern-Simons Theory,”Phys. Rev. Lett.131no. 17, (2023) 171602, arXiv:2304.02668 [hep-th]

    work page arXiv 2023

  37. [37]

    Spinning up the spool: massive spinning fields in 3d quantum gravity,

    R. Bourne, A. Castro, and J. R. Fliss, “Spinning up the spool: massive spinning fields in 3d quantum gravity,”J. Phys. A58no. 2, (2025) 025402, arXiv:2407.09608 [hep-th]

    work page arXiv 2025

  38. [38]

    A spool for every quotient: One-loop partition functions in AdS 3 gravity,

    R. Bourne, J. R. Fliss, and B. Knighton, “A spool for every quotient: One-loop partition functions in AdS 3 gravity,”SciPost Phys.20(2026) 065, arXiv:2507.05364 [hep-th]

    work page arXiv 2026

  39. [39]

    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 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  40. [40]

    Quantum Field Theory and the Jones Polynomial,

    E. Witten, “Quantum Field Theory and the Jones Polynomial,”Commun. Math. Phys.121(1989) 351–399

    1989

  41. [41]

    Minimal surfaces and particles in 3-manifolds

    K. Krasnov and J.-M. Schlenker, “Minimal surfaces and particles in 3-manifolds,” Geom. Dedicata126(2007) 187–254,arXiv:math/0511441

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  42. [42]

    The Universal Phase Space of AdS3 Gravity

    C. Scarinci and K. Krasnov, “The universal phase space ofAdS 3 gravity,”Commun. Math. Phys.322(2013) 167–205,arXiv:1111.6507 [hep-th]. – 40 –

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  43. [43]

    Conformal Field Theory, 2-DQuantum Gravity and Quantization of Teichmuller Space,

    H. L. Verlinde, “Conformal Field Theory, 2-DQuantum Gravity and Quantization of Teichmuller Space,”Nucl. Phys. B337(1990) 652–680

    1990

  44. [44]

    A note on the canonical formalism for gravity,

    E. Witten, “A note on the canonical formalism for gravity,”Adv. Theor. Math. Phys.27no. 1, (2023) 311–380,arXiv:2212.08270 [hep-th]

    work page arXiv 2023

  45. [45]

    Lorentz spacetimes of constant curvature

    G. Mess, “Lorentz spacetimes of constant curvature,”Geom. Dedicata126no. 1, (July, 2007) 3–45,arXiv:0706.1570 [math.DG]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  46. [46]

    A view of the bulk from the worldline

    H. Maxfield, “A view of the bulk from the worldline,”arXiv:1712.00885 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  47. [47]

    AdS manifolds with particles and earthquakes on singular surfaces

    F. Bonsante and J.-M. Schlenker, “AdS manifolds with particles and earthquakes on singular surfaces,”Geom. Funct. Anal.19no. 1, (May, 2009) 41–82, arXiv:math/0609116 [math.GT]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  48. [48]

    Defects in Jackiw-Teitelboim Quantum Gravity,

    T. G. Mertens and G. J. Turiaci, “Defects in Jackiw-Teitelboim Quantum Gravity,” JHEP08(2019) 127,arXiv:1904.05228 [hep-th]

    work page arXiv 2019

  49. [49]

    Matrix Models and Deformations of JT Gravity,

    E. Witten, “Matrix Models and Deformations of JT Gravity,”Proc. Roy. Soc. Lond. A476no. 2244, (2020) 20200582,arXiv:2006.13414 [hep-th]

    work page arXiv 2020

  50. [50]

    2D dilaton-gravity, deformations of the minimal string, and matrix models,

    G. J. Turiaci, M. Usatyuk, and W. W. Weng, “2D dilaton-gravity, deformations of the minimal string, and matrix models,”Class. Quant. Grav.38no. 20, (2021) 204001,arXiv:2011.06038 [hep-th]

    work page arXiv 2021

  51. [51]

    2D Dilaton Gravity and the Weil–Petersson Volumes with Conical Defects,

    L. Eberhardt and G. J. Turiaci, “2D Dilaton Gravity and the Weil–Petersson Volumes with Conical Defects,”Commun. Math. Phys.405no. 4, (2024) 103, arXiv:2304.14948 [hep-th]

    work page arXiv 2024

  52. [52]

    Semi-classical dilaton gravity and the very blunt defect expansion,

    J. Kruthoff and A. Levine, “Semi-classical dilaton gravity and the very blunt defect expansion,”JHEP07(2025) 211,arXiv:2402.10162 [hep-th]

    work page arXiv 2025

  53. [53]

    Weil-Petersson volumes and cone surfaces

    N. Do and P. Norbury, “Weil–Petersson volumes and cone surfaces,”Geom. Dedicata 141no. 1, (Aug., 2009) 93–107,arXiv:math/0603406

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  54. [54]

    Weil-Petersson volumes, stability conditions and wall-crossing,

    L. Anagnostou, S. Mullane, and P. Norbury, “Weil-Petersson volumes, stability conditions and wall-crossing,”arXiv:2310.13281 [math.AG]. https://arxiv.org/abs/2310.13281

    work page arXiv

  55. [55]

    An introduction to moduli spaces of curves and their intersection theory,

    D. Zvonkine, “An introduction to moduli spaces of curves and their intersection theory,”Handbook of Teichm¨ uller theory3(2012) 667

    2012

  56. [56]

    Gravitational Hilbert spaces: invariant and co-invariant states, inner products, gauge-fixing, and BRST

    J. Held and H. Maxfield, “Gravitational Hilbert spaces: invariant and co-invariant states, inner products, gauge-fixing, and BRST,”arXiv:2509.05412 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  57. [57]

    Diffeomorphism invariant tensor networks for 3d gravity,

    V. Balasubramanian and C. Cummings, “Diffeomorphism invariant tensor networks for 3d gravity,”JHEP05(2026) 231,arXiv:2510.13941 [hep-th]

    work page arXiv 2026

  58. [58]

    A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications,

    M. F. Atiyah and R. Bott, “A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications,”Annals of Mathematics88no. 3, (1968) 451–491

    1968

  59. [59]

    Topology Changing Amplitudes in (2+1)-Dimensional Gravity,

    E. Witten, “Topology Changing Amplitudes in (2+1)-Dimensional Gravity,”Nucl. Phys. B323(1989) 113–140. – 41 –

    1989

  60. [60]

    Exact Quantum Scattering in (2+1)-Dimensional Gravity,

    S. Carlip, “Exact Quantum Scattering in (2+1)-Dimensional Gravity,”Nucl. Phys. B324(1989) 106–122

    1989

  61. [61]

    Wilson Lines and Entanglement Entropy in Higher Spin Gravity

    M. Ammon, A. Castro, and N. Iqbal, “Wilson Lines and Entanglement Entropy in Higher Spin Gravity,”JHEP10(2013) 110,arXiv:1306.4338 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  62. [62]

    Off-shell Partition Functions in 3d Gravity,

    L. Eberhardt, “Off-shell Partition Functions in 3d Gravity,”Commun. Math. Phys. 405no. 3, (2024) 76,arXiv:2204.09789 [hep-th]

    work page arXiv 2024

  63. [63]

    JT gravity as a matrix integral

    P. Saad, S. H. Shenker, and D. Stanford, “JT gravity as a matrix integral,” arXiv:1903.11115 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 1903

  64. [64]

    Jackiw-Teitelboim gravity with matter, generalized eigenstate thermalization hypothesis, and random matrices,

    D. L. Jafferis, D. K. Kolchmeyer, B. Mukhametzhanov, and J. Sonner, “Jackiw-Teitelboim gravity with matter, generalized eigenstate thermalization hypothesis, and random matrices,”Phys. Rev. D108no. 6, (2023) 066015, arXiv:2209.02131 [hep-th]

    work page arXiv 2023

  65. [65]

    Work in progress

    R. Bourne, A. Castro, J. R. Fliss, A. Y. Law, and B. Knighton, “Work in progress.”

  66. [66]

    Geometric Microstates for the Three Dimensional Black Hole?

    A. Maloney, “Geometric Microstates for the Three Dimensional Black Hole?,” arXiv:1508.04079 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  67. [67]

    Work in progress

    R. Bourne, J. R. Fliss, B. Knighton, and W. W. Weng, “Work in progress.”

  68. [68]

    Black hole determinants and quasinormal modes

    F. Denef, S. A. Hartnoll, and S. Sachdev, “Black hole determinants and quasinormal modes,”Class. Quant. Grav.27(2010) 125001,arXiv:0908.2657 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  69. [69]

    Scalar field quantization on the 2+1 dimensional black hole background

    G. Lifschytz and M. Ortiz, “Scalar field quantization on the (2+1)-dimensional black hole background,”Phys. Rev. D49(1994) 1929–1943,arXiv:gr-qc/9310008

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  70. [70]

    Conformally dressed black hole in 2+1 dimensions

    C. Martinez and J. Zanelli, “Conformally dressed black hole in (2+1)-dimensions,” Phys. Rev. D54(1996) 3830–3833,arXiv:gr-qc/9604021

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  71. [71]

    Quantum dress for a naked singularity

    M. Casals, A. Fabbri, C. Mart´ ınez, and J. Zanelli, “Quantum dress for a naked singularity,”Phys. Lett. B760(2016) 244–248,arXiv:1605.06078 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  72. [72]

    Quantum BTZ black hole,

    R. Emparan, A. M. Frassino, and B. Way, “Quantum BTZ black hole,”JHEP11 (2020) 137,arXiv:2007.15999 [hep-th]

    work page arXiv 2020

  73. [73]

    Three-Dimensional Quantum Black Holes: A Primer,

    E. Panella, J. F. Pedraza, and A. Svesko, “Three-Dimensional Quantum Black Holes: A Primer,”Universe10no. 9, (2024) 358,arXiv:2407.03410 [hep-th]

    work page arXiv 2024

  74. [74]

    Gravity in Warped Compactifications and the Holographic Stress Tensor

    S. de Haro, K. Skenderis, and S. N. Solodukhin, “Gravity in warped compactifications and the holographic stress tensor,”Class. Quant. Grav.18(2001) 3171–3180,arXiv:hep-th/0011230

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  75. [75]

    Exact Description of Black Holes on Branes

    R. Emparan, G. T. Horowitz, and R. C. Myers, “Exact description of black holes on branes,”JHEP01(2000) 007,arXiv:hep-th/9911043

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  76. [76]

    Exact Description of Black Holes on Branes II: Comparison with BTZ Black Holes and Black Strings

    R. Emparan, G. T. Horowitz, and R. C. Myers, “Exact description of black holes on branes. 2. Comparison with BTZ black holes and black strings,”JHEP01(2000) 021,arXiv:hep-th/9912135

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  77. [77]

    Quantum Black Holes as Holograms in AdS Braneworlds

    R. Emparan, A. Fabbri, and N. Kaloper, “Quantum black holes as holograms in AdS brane worlds,”JHEP08(2002) 043,arXiv:hep-th/0206155. – 42 –

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  78. [78]

    Spinning particle geometries in AdS 3/CFT2,

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

    work page arXiv 2024

  79. [79]

    The Analytic Geometry of Two-Dimensional Conformal Field Theory,

    D. Friedan and S. H. Shenker, “The Analytic Geometry of Two-Dimensional Conformal Field Theory,”Nucl. Phys. B281(1987) 509–545

    1987

  80. [80]

    On the Kodaira dimension of the moduli space of curves,

    J. Harris and D. Mumford, “On the Kodaira dimension of the moduli space of curves,”Invent. Math.67no. 1, (Feb., 1982) 23–86. – 43 –

    1982