Pith. sign in

REVIEW 4 major objections 5 minor 3 cited by

The spectrum of pure dS$_3$ gravity in the static patch

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The static-patch trace in pure dS3 gravity is the future-boundary wavefunction, whose spectral sum gives a dense spectrum with integer degeneracies.

desk verdict The paper's static-patch trace/wavefunction identification is a clearly stated conjecture resting on KSW-violating complex saddles; the entropy check is clean, the spectral reinterpretation is provocative, and it deserves peer review despite the load-bearing weakness. read the letter →

arxiv 2505.06420 v1 pith:BPJLWNLK submitted 2025-05-09 hep-th gr-qc

Joydeep Chakravarty , Alexander Maloney , Keivan Namjou , Simon F. Ross This is my paper · ORCID
classification hep-thgr-qc
keywords deSitterstaticpatchthree-dimensionalgravityLorentzianpathintegralHartle-HawkingwavefunctionPoincarésummodularinvariancespectralformfactorentropy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Three-dimensional Einstein gravity with a positive cosmological constant has no local degrees of freedom, yet the quantum mechanics of its static patch has been unclear because no one knows what the static-patch Hilbert space is. The paper proposes a concrete answer: the Lorentzian path integral for the trace $\operatorname{Tr}(e^{-iHT})$ should be defined by resolving the horizon singularity with a complex contour that connects to future infinity, and the resulting sum of saddle points equals the Hartle-Hawking wavefunction of the torus at $I^+$. Taking that equality as the microscopic definition of the trace, the paper extracts a spin-zero spectrum that is bounded below, discrete but dense, with integer degeneracies, including negative degeneracies that cancel in smooth observables. The same saddle-point sum reproduces the Bekenstein-Hawking entropy $S_{dS}=\pi\ell/(2G)$ at the de Sitter temperature, so the computation connects the entropy to an actual spectrum of states.

What carries the argument

The load-bearing object is a family of complex radial contours in the $u$-coordinate static patch, $ds^2/\ell^2=-\cos^2 u\,dt^2+du^2+\sin^2 u\,d\phi^2$. Starting at the observer at $u=0$, each contour runs to the vicinity of the horizon $u=\pi/2$, then rotates into the future cosmology $u=\pi/2+i\xi$ and reaches $I^+$, thereby resolving the fixed-point singularity of the periodic identification $t\sim t+T$. These contours have the same classical action as the real future cosmology, which matches the exponential factor of $\Psi_{0,1}(\tau)$; their modular images supply the rest of the Poincaré sum. A spectral representation of $\Psi_{HH}$ in terms of Eisenstein series and Maass cusp forms then turns the modular-invariant wavefunction into a $q$-expansion whose coefficients are read as integer degeneracies of static-patch states.

What would settle it

Compute the one-loop determinant of boundary gravitons, or of a free scalar, on the complex contour described in Section 2.1. If the determinant is ill-defined, or if a regulated version of the spectral sum yields non-integer coefficients, then the identification of $\Psi_{HH}$ with the static-patch trace is wrong; if the determinant is finite after a natural regulator such as an observer-induced cutoff at $\beta=2\pi$, the claim survives.

Watch

Extended reading notes

Core claim

The central claim is that the static-patch Hilbert-space trace is exactly the late-time wavefunction of de Sitter gravity on a torus: $$\operatorname{Tr}_{\mathcal{H}}($e^{{-iHT+i\theta J}}$)=\Psi_{HH}(\tau), \qquad \tau=\frac{\$\theta$+iT}{2\pi},$$ where $\Psi_{HH}$ is the Poincaré sum of modular images of the analytically continued thermal-AdS partition function. Each term in that sum is interpreted as a different complex resolution of the cosmological horizon singularity, i.e., a different saddle for the Lorentzian path integral. Reading the wavefunction's spectral expansion as a sum over states gives a $J=0$ spectrum that is bounded below, discrete and dense, with integer degeneracies that can be negative; the negative contributions cancel for smooth observables. The paper takes (1.3) to define what is meant microscopically by the static-patch trace, and verifies the proposal by showing that analytically continuing $T\to i\beta$ makes the $S$-transformed saddle dominate and yield the Bekenstein-Hawking entropy at $\beta=2\pi$.

Load-bearing premise

The load-bearing premise is that complex saddle geometries that fail the Kontsevich-Segal-Witten admissibility criterion are still legitimate contributions to the Lorentzian path integral; the paper shows every contour that resolves the horizon singularity violates that criterion, so the whole trace-wavefunction identification rests on this.

Editorial extensions

If this is right

  • The static-patch entropy at the de Sitter temperature comes from the $S$-transformed saddle $\psi_{1,0}$, not from the naive $\psi_{0,1}$ term, so the full Poincaré sum is essential to the thermodynamics.
  • The spin-zero spectrum is bounded below, discrete but dense, with integer degeneracies; negative degeneracies cancel in smooth observables, explaining why the entropy is finite and why a finite-lifetime observer cannot resolve individual states.
  • Equation (1.3) provides a microscopic definition of $\operatorname{Tr}(e^{-iHT})$, turning de Sitter entropy into a Lorentzian saddle-point computation analogous to the double-cone treatment of AdS black holes.
  • Earlier dS/CFT entropy derivations using an analytically continued Cardy formula are recovered as the semiclassical part of this trace, up to two cancellations of factors of $i$.

Reading between the lines

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

  • Editorial inference: if the identification is exact, then modular invariance of $\Psi_{HH}$ forces a modular-invariant spectral form factor, so the negative degeneracies form a sector that is invisible to $SL(2,\mathbb{Z})$-invariant observables; computing $|Z(T)|^2$ and looking for a ramp-plateau transition would test this.
  • Editorial inference: the KSW violation may be harmless only because pure 3D gravity has no local degrees of freedom; coupling a heavy probe or a conformal matter sector to the same complex contour would make the one-loop determinant sensitive to the contour, giving a sharp test of whether the construction extends beyond pure gravity.
  • Editorial inference: the continuous contribution from Maass cusp forms suggests the full Hilbert space is not a discrete sum; a profitable next step would be to find an arithmetic regularization that includes the continuous density while keeping integer degeneracies, possibly by averaging over cusp forms.
  • Editorial inference: the one-loop divergence at $\beta=2\pi$ is tied to the boundary torus acquiring a null direction; introducing an observer at $r=0$, whose back-reaction shifts the temperature, should cut off that divergence and is a direct extension suggested by the paper's own discussion.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a Lorentzian path-integral definition of the static-patch trace Tr_H(e^{-iHT+iθJ}) in pure three-dimensional de Sitter gravity. It resolves the singularity at the cosmological horizon r=ℓ by complexifying the radial coordinate, connecting the periodically identified static patch to future infinity. The authors identify the full future-boundary Hartle-Hawking wavefunction Ψ_HH(τ), written as a Poincaré sum, with the static-patch trace at the microscopic level. They compute the classical action of the complex saddle, reproduce the Bekenstein-Hawking entropy from the S-transformed saddle, and use Godet's spectral representation to extract a spin-zero spectrum that is bounded, discrete, dense, with integer degeneracies and negative contributions. The paper also discusses one-loop issues, the role of Maass cusp forms, and the failure of the Kontsevich-Segal-Witten condition for the relevant contours.

Significance. If the central conjecture holds, the paper provides a concrete microscopic definition of the static-patch Hilbert space in pure dS3, a long-standing problem. The classical action computation in Sec. 3.1 is clean and the leading entropy check is a genuine result. The derivation of integer spectral degeneracies from the Möbius-function structure is striking and goes beyond previous work. At the same time, the paper is explicit that its central identification is a conjecture rather than a derivation, and several load-bearing technical steps are flagged as open. The significance is high conditional on resolving the validity of the complex saddles and the spectral interchanges.

major comments (4)
  1. [Sec. 2.1, Eq. (2.12)] The paper proves that every complex contour resolving the horizon fixed point violates the KSW bound, and that one cannot reach KSW-satisfying contours from the real axis. Yet the entire saddle-point sum and the trace-wavefunction identification presuppose that these KSW-violating complex geometries are legitimate saddle points of the gravitational path integral. No one-loop determinant around these contours is computed, and no alternative criterion is supplied in its place. The Discussion acknowledges this, but this is not a peripheral technicality: the entropy computation in Sec. 3.3 and the spectral representation in Sec. 4 both pass through the actions of these saddles. As written, the microscopic identification in Sec. 3.2 is a definition rather than a derived result, and the paper needs either a stability test for the complex saddles or a clearly stated alternative criterion for admitting them.
  2. [Sec. 4.1, Eqs. (4.7)-(4.9)] The spectral representation (4.2) relies on an interchange of contour closing and the zeta-function expansion ζ(-2iν)=Σ_n n^{2iν}, which the authors themselves flag as subtle. The expansion converges only for Im ν > 1/2, and the double sum in the second term of (4.2) diverges for fixed n²/m². Since this spectral representation is the main evidence for the discrete spectrum with integer degeneracies, the unregulated divergence means that the claimed spectrum is not yet a demonstrated property of the path integral. The paper adopts the term-by-term order as a working assumption, but a rigorous derivation or a well-defined regulator is needed before this can count as evidence for the central conjecture.
  3. [Sec. 3.3, Eqs. (3.20)-(3.21)] At the de Sitter temperature β=2π, both the numerator and the denominator of K(β) vanish, and the authors note that the ψ_{1,0} dominance approximation breaks down arbitrarily close to β=2π. The suggested resolution by introducing an observer is not quantified. The leading classical entropy match is established, but the advertised agreement with S_dS=πℓ/(2G)+⋯ is only demonstrated at the classical level; the one-loop corrections are not controlled. This should be stated more prominently, since the entropy check is one of the two main pieces of evidence for the conjecture.
  4. [Sec. 4.2, Eq. (4.13)] The Maass cusp form contribution to χ[b] produces a continuous spectral density ρ_j(τ̃) rather than a discrete sum. This means that the full trace Tr_H(e^{-iHT}) is not a discrete sum over states with integer degeneracies, even though the scalar part ψ[Q]+ψ[Q̃] has that form. The paper's headline properties 'integer degeneracies' and 'discrete but dense' therefore apply only to a sector of the wavefunction, and the continuous cusp-form contribution must either be shown to decouple from the static-patch trace or be incorporated into the claimed spectrum. As it stands, the full microscopic content of the trace is not captured by the discrete spectral sum.
minor comments (5)
  1. [Sec. 3.1 and Sec. 3.3] Eq. (3.3) uses the exponent (c_dS-1)/12 while Eq. (3.16) effectively uses (c_dS-13)/12; as written, the classical action matching in Sec. 3.1 agrees with the latter but not with the former. Please reconcile the shift by 12 or clarify the convention.
  2. [Sec. 5] The word 'Maass' is misspelled as 'Mass' in the Discussion; please fix this for consistency with Sec. 4.2.
  3. [Sec. 4.1, Eq. (4.9)] The symbol ψ^{(0)}[Q] is introduced without definition; please define it explicitly before using it in the contour-integral argument.
  4. [Fig. 1] The two contour branches labeled (I) and (II) are not described in the caption; a short explanation of which branch corresponds to which resolution would improve readability.
  5. [Sec. 3.3, after Eq. (3.19)] The argument that terms with c²-d²>1 are subdominant near β=2π is stated without details; a short derivation or a reference would help the reader verify the claim.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction; the trace-wavefunction relation is an explicit conjecture and the spectrum is a consequence, not a hidden input.

full rationale

The derivation chain is not circular. The authors explicitly label the trace-wavefunction equality as a hypothesis or definition: 'Our main hypothesis, motivated by this, is to identify the full wavefunction ΨHH with TrH(e−iHT) . . . we take the expression (1.3) to define what we mean at a microscopic level by the trace over the static patch Hilbert space' (Sec. 3.2). Because this is stated as a proposal rather than derived from the static-patch spectrum, the later reading of Godet's formula as a spectrum (Sec. 4) is a consequence of that conjecture, not an input secretly reintroduced. The Poincaré sum itself comes from independent analytic continuation of the MWK partition function, not from fitting the static-patch trace; the classical action match (Sec. 3.1) is computed independently, and the entropy check (Sec. 3.3) reproduces S_dS from a non-trivial modular image ψ_{1,0} rather than fitting the answer. The admitted KSW obstruction (Sec. 2.1 and Sec. 5) is a correctness and unresolved-premise issue, not a circularity: no quantity is fitted, renamed, or defined in terms of the result it is supposed to predict. Score 2 reflects the heavy reliance on prior constructions involving the same authors and the openly conjectural character of the central identification, but no circular step is present.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted: l/G enters as the physical input that fixes the central charge, and the spectrum labels are derived quantities. The cost of the paper is carried by the six assumptions above, two of which are explicitly flagged by the authors as unsettled.

assumptions (6)
  • ad hoc to paper A KSW-violating complex contour resolving the horizon fixed point is an allowable Lorentzian saddle.
    Sec. 2.1 shows the obstruction and Sec. 5 calls validating this central to the proposal; the paper argues pure 3D gravity may not need KSW.
  • domain assumption The static-patch trace is well-defined and equals the Lorentzian path integral; a Hilbert space with possibly negative-norm states exists.
    Eq. (1.2) and Sec. 3.2; the microscopic trace is defined by the wavefunction.
  • domain assumption Analytic continuation of the AdS MWK partition function gives the dS wavefunction Psi_HH, including the central charge c_dS=13+i3l/(2G).
    Sec. 3, relying on [11,23]; this wavefunction is the object identified with the trace.
  • ad hoc to paper The Eisenstein spectral integrals in Sec. 4.1 can be evaluated by the stated contour closings and by exchanging integration with the zeta-function expansions.
    The paper itself flags this as a subtle step requiring deeper understanding; Eq. (4.2) depends on it.
  • domain assumption The BRST gauge-fixing normalization N^2=1/(y|eta(tau)|^4) fixes the wavefunction normalization.
    Appendix B; needed to match the entropy and the trace normalization.
  • domain assumption Maass cusp forms from [32] with the stated normalization contribute as in Eq. (4.13).
    Sec. 4.2; controls the non-scalar spectrum.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The spectrum of pure dS$_3$ gravity in the static patch." pith.science (2026). https://pith.science/paper/BPJLWNLK

@misc{pith2026250506420,
  author       = {Pith},
  title        = {Pith review of: The spectrum of pure dS$_3$ gravity in the static patch},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BPJLWNLK}},
  note         = {Machine review of arXiv:2505.06420}
}
read the original abstract

We consider the quantum mechanical description of the de Sitter static patch in three-dimensional general relativity. We consider a Lorentzian path integral that conjecturally computes the Fourier transform of the spectrum of the static patch Hamiltonian. We regulate a saddle point for this integral by a complex deformation that connects it to future infinity. Our computation is thus closely connected with the wave function of de Sitter gravity on a torus at future infinity. Motivated by this, we identify an infinite number of saddle points that contribute to our Lorentzian path integral. Their sum gives a surprisingly simple result, which agrees with the expected features of the de Sitter static patch. For example, the thermal entropy, evaluated at the de Sitter temperature, agrees with the Bekenstein-Hawking formula. We also obtain a spectrum in the spin-zero sector, which is bounded, discrete, and has an integer degeneracy of states. It includes a dense spectrum of states, making both positive and negative contributions to the trace, arranged in such a way that negative contributions are invisible in the computation of any smooth observable. Nevertheless, several mysteries remain.

Figures

Figures reproduced from arXiv: 2505.06420 by the authors.

Figure 1
Figure 1. The contour connecting the static patch (in dot) to the spacelike infinity [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Wavefunctions of AdS$_3$ Universes and $T\bar{T}$-deformed Torus Partition Functions

    hep-th 2026-08 reject novelty 6.0 of 10

    The paper's central claim of an invertible bulk-boundary transform for T-Tbar deformed torus partition functions is broken by an incorrect inverse kernel and a factor-of-pi normalization inconsistency.

  2. Imprint of the black hole interior on thermal four-point correlators

    hep-th 2025-12 conditional novelty 6.0 of 10

    Analytically continuing and smearing thermal four-point correlators turns them into flat-space scattering amplitudes at a bulk point inside the black hole.

  3. Microstate counting from defects in de Sitter

    hep-th 2025-11 conditional novelty 6.0 of 10

    Counting defect microstates via Lorentzian wormholes reproduces the de Sitter and Schwarzschild-de Sitter entropy area laws.

Reference graph

Works this paper leans on

38 extracted references · 3 canonical work pages · cited by 3 Pith papers

  1. [1]

    Gibbons and S.W

    G.W. Gibbons and S.W. Hawking,Cosmological Event Horizons, Thermodynamics, and Particle Creation, Phys. Rev. D15 (1977) 2738

    work page 1977

  2. [2]

    Anninos, S.A

    D. Anninos, S.A. Hartnoll and D.M. Hofman,Static Patch Solipsism: Conformal Symmetry of the de Sitter Worldline, Class. Quant. Grav.29 (2012) 075002 [1109.4942]

    arXiv 2012

  3. [3]

    Nakayama,The World-Line Quantum Mechanics Model at Finite Temperature which is Dual to the Static Patch Observer in de Sitter Space, Prog

    R. Nakayama,The World-Line Quantum Mechanics Model at Finite Temperature which is Dual to the Static Patch Observer in de Sitter Space, Prog. Theor. Phys.127 (2012) 393 [1112.1267]

    arXiv 2012

  4. [4]

    Chandrasekaran, R

    V. Chandrasekaran, R. Longo, G. Penington and E. Witten,An algebra of observables for de Sitter space, JHEP 02 (2023) 082 [2206.10780]

    arXiv 2023

  5. [5]

    Loganayagam and O

    R. Loganayagam and O. Shetye,Influence phase of a dS observer. Part I. Scalar exchange, JHEP 01 (2024) 138 [2309.07290]

    arXiv 2024

  6. [6]

    Kolchmeyer and H

    D.K. Kolchmeyer and H. Liu,Chaos and the Emergence of the Cosmological Horizon, 2411.08090

    arXiv

  7. [7]

    Hartle and S.W

    J.B. Hartle and S.W. Hawking,Wave function of the universe, Phys. Rev. D28 (1983) 2960

    work page 1983

  8. [8]

    Strominger,The dS/CFT correspondence, JHEP 10 (2001) 034 [hep-th/0106113]

    A. Strominger,The dS/CFT correspondence, JHEP 10 (2001) 034 [hep-th/0106113]

    arXiv 2001

Show all 38 references
  1. [9]

    Witten,Quantum gravity in de Sitter space, inStrings 2001: International Conference, 6, 2001 [hep-th/0106109]

    E. Witten,Quantum gravity in de Sitter space, inStrings 2001: International Conference, 6, 2001 [hep-th/0106109]

    2001 arXiv

  2. [10]

    Maldacena,Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 05 (2003) 013 [astro-ph/0210603]

    J.M. Maldacena,Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 05 (2003) 013 [astro-ph/0210603]

    2003 arXiv

  3. [11]

    Castro and A

    A. Castro and A. Maloney,The Wave Function of Quantum de Sitter, JHEP 11 (2012) 096 [1209.5757]

    2012 arXiv

  4. [12]

    Chakraborty, J

    T. Chakraborty, J. Chakravarty, V. Godet, P. Paul and S. Raju,The Hilbert space of de Sitter quantum gravity, JHEP 01 (2024) 132 [2303.16315]. 23

    2024 arXiv

  5. [13]

    Anninos, T

    D. Anninos, T. Hartman and A. Strominger,Higher Spin Realization of the dS/CFT Correspondence, Class. Quant. Grav.34 (2017) 015009 [1108.5735]

    2017 arXiv

  6. [14]

    Anninos, F

    D. Anninos, F. Denef, R. Monten and Z. Sun,Higher Spin de Sitter Hilbert Space, JHEP 10 (2019) 071 [1711.10037]

    2019 arXiv

  7. [15]

    Collier, L

    S. Collier, L. Eberhardt and B. Mühlmann,A microscopic realization of dS3, 2501.01486

    arXiv

  8. [16]

    Saad, S.H

    P. Saad, S.H. Shenker and D. Stanford,A semiclassical ramp in SYK and in gravity, 1806.06840

    arXiv

  9. [17]

    Y. Chen, V. Ivo and J. Maldacena,Comments on the double cone wormhole, JHEP 04 (2024) 124 [2310.11617]

    2024 arXiv

  10. [18]

    Chakravarty, A

    J. Chakravarty, A. Maloney, K. Namjou and S.F. Ross,A new observable for holographic cosmology, JHEP 10 (2024) 184 [2407.04781]

    2024 arXiv

  11. [19]

    Kontsevich and G

    M. Kontsevich and G. Segal,Wick rotation and the positivity of energy in quantum field theory, Quart. J. Math. Oxford Ser.72 (2021) 673 [2105.10161]

    2021 arXiv

  12. [20]

    Witten,A note on complex spacetime metrics, 2111.06514

    E. Witten,A note on complex spacetime metrics, 2111.06514

    arXiv

  13. [21]

    Bousso, A

    R. Bousso, A. Maloney and A. Strominger,Conformal vacua and entropy in de Sitter space, Phys. Rev. D65 (2002) 104039 [hep-th/0112218]

    2002 arXiv

  14. [22]

    Balasubramanian, J

    V. Balasubramanian, J. de Boer and D. Minic,Mass, entropy and holography in asymptotically de Sitter spaces, Phys. Rev. D65 (2002) 123508 [hep-th/0110108]

    2002 arXiv

  15. [23]

    Maloney and E

    A. Maloney and E. Witten,Quantum gravity partition functions in three dimensions, JHEP 02 (2010) 029 [0712.0155]

    2010 arXiv

  16. [24]

    Godet,Quantum cosmology as automorphic dynamics, 2405.09833

    V. Godet,Quantum cosmology as automorphic dynamics, 2405.09833

    arXiv

  17. [25]

    Gibbons and S.W

    G.W. Gibbons and S.W. Hawking,Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D15 (1977) 2752

    1977

  18. [26]

    Banihashemi and T

    B. Banihashemi and T. Jacobson,The enigmatic gravitational partition function, in Lemaitre Conference 2024: Black Holes, Gravitational Waves and Space-Time Singularities, 10, 2024 [2411.00267]

    2024 arXiv

  19. [27]

    Maldacena,Comments on the no boundary wavefunction and slow roll inflation, 2403.10510

    J. Maldacena,Comments on the no boundary wavefunction and slow roll inflation, 2403.10510

    arXiv

  20. [28]

    Y. Chen, V. Gorbenko and J. Maldacena,Bra-ket wormholes in gravitationally prepared states, JHEP 02 (2021) 009 [2007.16091]

    2021 arXiv

  21. [29]

    Fumagalli, V

    A. Fumagalli, V. Gorbenko and J. Kames-King,De Sitter Bra-Ket Wormholes, 2408.08351

    arXiv

  22. [30]

    Blacker and S.A

    M.J. Blacker and S.A. Hartnoll,Cosmological quantum states of de Sitter-Schwarzschild are static patch partition functions, JHEP 12 (2023) 025 [2304.06865]

    2023 arXiv

  23. [31]

    Godet,Möbius randomness in the Hartle-Hawking state, 2505.03068

    V. Godet,Möbius randomness in the Hartle-Hawking state, 2505.03068. 24

    arXiv

  24. [32]

    Benjamin, S

    N. Benjamin, S. Collier, A.L. Fitzpatrick, A. Maloney and E. Perlmutter,Harmonic analysis of 2d CFT partition functions, JHEP 09 (2021) 174 [2107.10744]

    2021 arXiv

  25. [33]

    Freidel,Reconstructing AdS/CFT, 0804.0632

    L. Freidel,Reconstructing AdS/CFT, 0804.0632

    arXiv

  26. [34]

    Chakraborty, J

    T. Chakraborty, J. Chakravarty, V. Godet, P. Paul and S. Raju,Holography of information in de Sitter space, JHEP 12 (2023) 120 [2303.16316]

    2023 arXiv

  27. [35]

    Polchinski,String theory

    J. Polchinski,String theory. Vol. 1: An introduction to the bosonic string, Cambridge Monographs on Mathematical Physics, Cambridge University Press (12, 2007), 10.1017/CBO9780511816079

    2007 doi

  28. [36]

    Witten,A note on the canonical formalism for gravity, Adv

    E. Witten,A note on the canonical formalism for gravity, Adv. Theor. Math. Phys.27 (2023) 311 [2212.08270]

    2023 arXiv

  29. [37]

    Carlip,Notes on the (2+1)-dimensional Wheeler-DeWitt equation, Class

    S. Carlip,Notes on the (2+1)-dimensional Wheeler-DeWitt equation, Class. Quant. Grav. 11 (1994) 31 [gr-qc/9309002]

    1994 arXiv

  30. [38]

    Carlip,Lectures on (2+1) dimensional gravity, J

    S. Carlip,Lectures on (2+1) dimensional gravity, J. Korean Phys. Soc.28 (1995) S447 [gr-qc/9503024]. 25

    1995 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.