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arxiv: 2606.26241 · v1 · pith:FUTX4YNGnew · submitted 2026-06-24 · ✦ hep-th · gr-qc

An observer's quantization of 3d de Sitter

Andreas Blommaert , Damiano Tietto , Herman Verlinde This is my paper

Pith reviewed 2026-06-26 01:26 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords de Sitter gravityholographic dualitydensity of statescomplex Liouville stringKerr-lens geometriescrosscap amplitudesstatic patchobserver worldline
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The pith

Holographic duality to complex Liouville strings computes the density of states in 3d de Sitter gravity exactly.

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

The paper defines the density of states of the de Sitter static patch as a sum over an SL(2,Z) family of Euclidean no-boundary Kerr-lens spacetimes, each sourced by a line defect that represents an observer's worldline. It develops an exact quantum computation of this density by invoking a holographic duality to two copies of the complex Liouville string, under which the Kerr-lens geometries become generalized crosscap geometries. The crosscap amplitudes are computed in the string theory and shown to reproduce the semi-classical gravity prediction. For the simplest nontrivial case the same setup is dual to two copies of the GΣ effective field theory of the double-scaled SYK model living on the observer's worldline.

Core claim

The spectral density in the 3d dS static patch is defined via a sum over SL(2,Z) Euclidean no-boundary Kerr-lens spacetimes sourced by a line-defect. Under the holographic duality to two copies of the complex Liouville string these geometries map to an SL(2,Z) family of generalized crosscap geometries whose amplitudes are computed exactly and match the semi-classical gravity prediction. The simplest case further dualizes to two copies of the GΣ theory on the observer's worldline.

What carries the argument

The holographic duality between dS3 gravity and two copies of the complex Liouville string, which maps the SL(2,Z) Kerr-lens spacetimes to generalized crosscap geometries whose amplitudes can be computed exactly.

If this is right

  • The crosscap amplitudes in the complex Liouville string theory match the semi-classical gravity prediction for the spectral density.
  • The simplest non-trivial Kerr-lens geometry is dual to two copies of the GΣ effective field theory of the double-scaled SYK model living on the observer's worldline.
  • The setup provides a concrete starting point for constructing a microscopic worldline hologram of 3d de Sitter.

Where Pith is reading between the lines

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

  • If the duality holds, further observables in the static patch could be computed by evaluating additional string amplitudes on the same crosscap geometries.
  • The link to the SYK effective theory suggests that random-matrix or matrix-model techniques might be imported to study the observer's degrees of freedom.
  • The same duality framework might be tested by checking whether other Euclidean de Sitter geometries admit consistent crosscap interpretations.

Load-bearing premise

A holographic duality exists between 3d de Sitter gravity and two copies of the complex Liouville string such that the Kerr-lens geometries map to generalized crosscap geometries whose amplitudes can be compared directly to gravity.

What would settle it

A concrete mismatch between the computed complex Liouville string crosscap amplitudes and the semi-classical gravity prediction for the spectral density of any Kerr-lens geometry would falsify the proposal.

Figures

Figures reproduced from arXiv: 2606.26241 by Andreas Blommaert, Damiano Tietto, Herman Verlinde.

Figure 1
Figure 1. Figure 1: The observer’s density of states computed as the overlap of two 2d torus universe wavefunctions, one containing the worldline of the observer at the pode and one that represents the smooth Kerr-de Sitter horizon. One sums over possible ways of gluing these two tori together, labeled by the elements γ P SL(2,Z). γ smooth horizon pode brane CLS b CLS E, J [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The observer’s density of states computed as the CLSbCLS amplitude on a 2d “brane” anchored onto the observer’s worldline. The puncture imposes generalized crosscap conditions, labeled by γ P SLp2, Zq. In order to exactly quantize 3d KdS we consider two complementary proposals which reproduce the same final formula ρpE, Jq. These proposals are discussed in more detail in the summary section 1.1. In the fir… view at source ↗
Figure 3
Figure 3. Figure 3: Sketch of our final result for the scalar spectrum ρj“0pEq where ApEq “ 2π ? E. Empty dS space is defined as the Virasoro identity representation, which leads to a quantum correction in the energy of the maximum entropy state. The spectrum goes negative for ApEq Ñ 0. In this regime, subleading saddles compete and off-shell geometries (as well as potential other saddles) may have to be taken into account. 1… view at source ↗
Figure 5
Figure 5. Figure 5: The quotient zz¯ “ ´1 implements a crosscap. Indeed at |z| “ 1 we identify the antipodal points. The quotiented geometry is the region |z| ě 1 with antipodal boundary conditions at |z| “ 1. At |z| “ 1, the quotient (5.2) indeed amounts to an antipodal Z2 identification, as shown in figure 5. In radial quantization, from the perspective of the closed string channel, the crosscap boundary condition at |z| “ … view at source ↗
Figure 6
Figure 6. Figure 6: To make a pc, dq crosscap we start from a d-fold cover z 1{d with sheets w1, w2, . . . , wd. We take out the unit circle and identify points at |z| “ 1 under a ´2πd{c rotation. We display the case pc, dq “ p5, 3q. One might call the solutions |βpc,dqyy to this set of conditions the pc, dq-crosscap Ishibashi states: |βpc,dqyy “ e ´2πi d c L0 |βyy . (5.12) Their overlap with an ordinary Ishibashi state |αyy … view at source ↗
Figure 7
Figure 7. Figure 7: The SLp2, Zq crosscap amplitude (5.17) is an annulus amplitude between a generalized SLp2, Zq crosscap state and an FZZT boundary, corresponding to the horizon and observer respectively. The crosscap state depends on a modular matrix γpc, dq. Physical states in the closed channel are on-shell primaries with discretized imaginary momenta β P iℏ 2 Z. 34 Our eventual application of the above formulas is the c… view at source ↗
Figure 8
Figure 8. Figure 8: In the open string channel, the generalized crosscap is a generalized M¨obius strip trace where we insert an operator that morally rotates the strip by 2πa{c. The trace is twisted by the operator Kˆγ clock. Fixing µB` fixes only an equivalence class of Liouville momenta. Integrating over τopen, one can reduce the SLp2,Zq M¨obius amplitude (5.34) to a topological twisted trace over only on-shell Virasoro pr… view at source ↗
Figure 9
Figure 9. Figure 9: Sketch of the scalar spectrum ρj“0pEq where ApEq “ 2π ? E. Empty dS space is defined as the Virasoro identity representation, which leads to a quantum correction in the energy of the maximum entropy state. This leads to a question: should we include a quantum correction in our definition of real energy? The spectrum goes negative for ApEq Ñ 0. In this regime, subleading saddles compete, and off-shell geome… view at source ↗
Figure 10
Figure 10. Figure 10: The contour Γ Y ´Γ can be deformed to pick up the poles at momenta β “ iℏn{2, n P Z [44]. A.4 Open string channel We prove the equivalence of the open trace (5.34) with the closed string amplitude (5.17). In the open channel, the generalized M¨obius strip has length τopen “ ´1{c 2 τ and it is twisted by a 2πa{c rotation; such a twist mixes left and right movers, without affecting the zero-modes. The chara… view at source ↗
read the original abstract

What is the density of states of the de Sitter static patch? We propose a definition and calculation of such a density in 3d dS. Our proposal involves a sum over an SL(2,$\mathbb{Z}$) set of Euclidean no-boundary Kerr-lens spacetimes sourced by a line-defect with given energy and spin - which in Lorentzian time represents an observer's worldline at the center of the dS static patch. We develop an exact quantum computation of the spectral density using a holographic duality between dS$_3$ gravity and two copies of $\mathbb{C}$LS, the complex Liouville string. The SL(2,$\mathbb{Z}$) Kerr-lens spacetimes map under the duality to an SL(2,$\mathbb{Z}$) family of generalized crosscap geometries. We compute the $\mathbb{C}$LS $\otimes$ $\mathbb{C}$LS crosscap amplitudes and show that they match the semi-classical gravity prediction. For the simplest non-trivial Kerr-lens space, the $\mathbb{C}$LS $\otimes$ $\mathbb{C}$LS description is, in turn, dual to two copies of the $G\Sigma$ effective field theory of the double scaled SYK model. The $G\Sigma\otimes G\Sigma$ theory lives on an observer's worldline in the static patch, setting the stage for developing a microscopic worldline hologram of 3d de Sitter.

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 / 1 minor

Summary. The paper proposes a definition of the density of states for the 3d de Sitter static patch as a sum over an SL(2,ℤ) family of Euclidean no-boundary Kerr-lens spacetimes sourced by a line defect (representing an observer worldline). It invokes a holographic duality between dS₃ gravity and two copies of the complex Liouville string (ℂLS) under which these geometries map to generalized crosscap states; the ℂLS ⊗ ℂLS crosscap amplitudes are computed and asserted to reproduce the semi-classical gravity result. The simplest case is further reduced to a GΣ ⊗ GΣ effective theory living on the observer worldline.

Significance. If the duality and geometry-to-crosscap map can be independently justified, the construction would supply an exact quantum computation of the dS₃ spectral density together with a candidate microscopic worldline hologram, which would be a substantial advance for quantum gravity in de Sitter space.

major comments (2)
  1. [Abstract] Abstract and §1: the central claim that the ℂLS ⊗ ℂLS amplitudes furnish an 'exact quantum computation' matching semi-classical gravity rests on an assumed holographic duality whose derivation and explicit geometry-to-crosscap map are not supplied; the reported match therefore tests internal consistency within the duality rather than deriving the density from 3d gravity alone.
  2. [Abstract] Abstract: the spectral density is defined by the very sum over SL(2,ℤ) Kerr-lens geometries chosen to represent the observer; without an independent benchmark or parameter-free derivation outside the duality, the circularity noted in the reader's assessment remains load-bearing for the claim of an exact computation.
minor comments (1)
  1. The acronyms ℂLS and GΣ are used without initial definition; a brief parenthetical expansion at first appearance would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We respond point-by-point to the major comments below, clarifying the scope of our claims and indicating revisions.

read point-by-point responses

  1. Referee: [Abstract] Abstract and §1: the central claim that the ℂLS ⊗ ℂLS amplitudes furnish an 'exact quantum computation' matching semi-classical gravity rests on an assumed holographic duality whose derivation and explicit geometry-to-crosscap map are not supplied; the reported match therefore tests internal consistency within the duality rather than deriving the density from 3d gravity alone.

    Authors: We acknowledge that the holographic duality between dS₃ gravity and ℂLS ⊗ ℂLS is assumed, building on prior literature, and that the explicit geometry-to-crosscap map is proposed in this work rather than derived from first principles. The reported match therefore demonstrates internal consistency of the proposal within the duality. We will add a clarifying paragraph in §1 stating the assumptions explicitly and noting that the computation is exact conditional on the duality. revision: partial

  2. Referee: [Abstract] Abstract: the spectral density is defined by the very sum over SL(2,ℤ) Kerr-lens geometries chosen to represent the observer; without an independent benchmark or parameter-free derivation outside the duality, the circularity noted in the reader's assessment remains load-bearing for the claim of an exact computation.

    Authors: The SL(2,ℤ) sum is indeed our proposed definition of the spectral density, motivated by the no-boundary prescription with an observer line defect. The duality is then used to evaluate this sum exactly. We do not claim a derivation independent of the duality; the value of the construction lies in the consistency check and the emergence of the GΣ worldline theory. We will revise the abstract to emphasize that this is a duality-based proposal. revision: yes

Circularity Check

1 steps flagged

Density defined as sum over Kerr-lens geometries; duality match reproduces input by construction

specific steps
  1. self definitional [Abstract]
    "Our proposal involves a sum over an SL(2,$\\,mathbb{Z}$) set of Euclidean no-boundary Kerr-lens spacetimes sourced by a line-defect with given energy and spin ... We develop an exact quantum computation of the spectral density using a holographic duality between dS$_3$ gravity and two copies of $\\mathbb{C}$LS ... The SL(2,$\\mathbb{Z}$) Kerr-lens spacetimes map under the duality to an SL(2,$\\mathbb{Z}$) family of generalized crosscap geometries. We compute the $\\mathbb{C}$LS $\\otimes$ $\\mathbb{C}$LS crosscap amplitudes and show that they match the semi-classical gravity prediction."

    The density is defined via the sum over geometries. The duality is assumed to map precisely those geometries to crosscaps; the dual amplitudes are then shown to match the gravity prediction obtained from the input sum, reducing the claimed exact result to consistency inside the duality framework.

full rationale

The paper defines the spectral density as a sum over SL(2,Z) Kerr-lens spacetimes. It invokes a duality mapping those geometries to crosscaps, computes amplitudes in the dual, and reports they match the semi-classical gravity result from the defining sum. The match is thus internal to the assumed duality with no independent external benchmark supplied.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claim rests on the existence of the stated holographic duality and on the geometric identification of Kerr-lens spaces with crosscaps; both are introduced by the paper rather than derived from prior results.

axioms (1)
  • domain assumption A holographic duality exists between 3d de Sitter gravity and two copies of the complex Liouville string that maps SL(2,Z) Kerr-lens geometries to generalized crosscap geometries.
    Invoked in the abstract as the basis for the exact quantum computation of the spectral density.
invented entities (2)
  • Kerr-lens spacetime sourced by a line-defect no independent evidence
    purpose: Represents the Euclidean continuation of an observer's worldline at the center of the dS static patch.
    Introduced as the building block of the SL(2,Z) sum that defines the density of states.
  • GΣ effective field theory on the observer worldline no independent evidence
    purpose: Provides the microscopic description in the simplest non-trivial case via duality to double-scaled SYK.
    Postulated as the worldline hologram dual to the gravity side.

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discussion (0)

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Reference graph

Works this paper leans on

156 extracted references · 55 linked inside Pith

  1. [1]

    Cosmological event horizons, thermodynamics, and particle creation,

    G. W. Gibbons and S. W. Hawking, “Cosmological event horizons, thermodynamics, and particle creation,”Physical Review D15no. 10, (1977) 2738

    1977

  2. [2]

    An algebra of observables for de Sitter space,

    V. Chandrasekaran, R. Longo, G. Penington, and E. Witten, “An algebra of observables for de Sitter space,”JHEP02(2023) 082,arXiv:2206.10780 [hep-th]

    Pith/arXiv arXiv 2023

  3. [3]

    Quantum de Sitter horizon entropy from quasicanonical bulk, edge, sphere and topological string partition functions,

    D. Anninos, F. Denef, Y. T. A. Law, and Z. Sun, “Quantum de Sitter horizon entropy from quasicanonical bulk, edge, sphere and topological string partition functions,”JHEP01(2022) 088,arXiv:2009.12464 [hep-th]

    arXiv 2022

  4. [4]

    The two-sphere partition function in two-dimensional quantum gravity,

    D. Anninos, T. Bautista, and B. M¨ uhlmann, “The two-sphere partition function in two-dimensional quantum gravity,”JHEP09(2021) 116,arXiv:2106.01665 [hep-th]

    arXiv 2021

  5. [5]

    dS4 Metamorphosis,

    D. Anninos, C. Baracco, V. A. Letsios, and B. M¨ uhlmann, “dS4 Metamorphosis,” arXiv:2602.19812 [hep-th]

    Pith/arXiv arXiv

  6. [6]

    dS2 supergravity,

    D. Anninos, P. Benetti Genolini, and B. M¨ uhlmann, “dS2 supergravity,”JHEP11(2023) 145, arXiv:2309.02480 [hep-th]

    arXiv 2023

  7. [7]

    Static Patch Solipsism: Conformal Symmetry of the de Sitter Worldline,

    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,arXiv:1109.4942 [hep-th]

    Pith/arXiv arXiv 2012

  8. [8]

    The gravitational path integral from an observer’s point of view,

    A. I. Abdalla, S. Antonini, L. V. Iliesiu, and A. Levine, “The gravitational path integral from an observer’s point of view,”arXiv:2501.02632 [hep-th]. 61

    arXiv

  9. [9]

    Quantum mechanics and observers for gravity in a closed universe,

    D. Harlow, M. Usatyuk, and Y. Zhao, “Quantum mechanics and observers for gravity in a closed universe,”arXiv:2501.02359 [hep-th]

    arXiv

  10. [10]

    On observers in holographic maps,

    C. Akers, G. Bueller, O. DeWolfe, K. Higginbotham, J. Reinking, and R. Rodriguez, “On observers in holographic maps,”arXiv:2503.09681 [hep-th]

    arXiv

  11. [11]

    Absolute entropy and the observer’s no-boundary state,

    A. Blommaert, J. Kudler-Flam, and E. Y. Urbach, “Absolute entropy and the observer’s no-boundary state,”JHEP11(2025) 113,arXiv:2505.14771 [hep-th]

    arXiv 2025

  12. [12]

    Algebraic Observational Cosmology,

    J. Kudler-Flam, S. Leutheusser, and G. Satishchandran, “Algebraic Observational Cosmology,” arXiv:2406.01669 [hep-th]

    arXiv

  13. [13]

    Norm of the no-boundary state,

    J. Cotler and K. Jensen, “Norm of the no-boundary state,”JHEP03(2026) 180, arXiv:2506.20547 [hep-th]

    arXiv 2026

  14. [14]

    Real observers solving imaginary problems,

    J. Maldacena, “Real observers solving imaginary problems,”arXiv:2412.14014 [hep-th]

    arXiv

  15. [15]

    Physical instabilities and the phase of the Euclidean path integral,

    V. Ivo, J. Maldacena, and Z. Sun, “Physical instabilities and the phase of the Euclidean path integral,”arXiv:2504.00920 [hep-th]

    arXiv

  16. [16]

    On the phase of the de Sitter density of states,

    Y. Chen, D. Stanford, H. Tang, and Z. Yang, “On the phase of the de Sitter density of states,” JHEP05(2026) 068,arXiv:2511.01400 [hep-th]

    arXiv 2026

  17. [17]

    The phase of the gravitational path integral,

    X. Shi and G. J. Turiaci, “The phase of the gravitational path integral,”arXiv:2504.00900 [hep-th]

    arXiv

  18. [18]

    A clock is just a way to tell the time: gravitational algebras in cosmological spacetimes,

    C.-H. Chen and G. Penington, “A clock is just a way to tell the time: gravitational algebras in cosmological spacetimes,”arXiv:2406.02116 [hep-th]

    arXiv

  19. [19]

    Sphere amplitudes and observing the universe’s size,

    A. Blommaert and A. Levine, “Sphere amplitudes and observing the universe’s size,” arXiv:2505.24633 [hep-th]

    arXiv

  20. [20]

    Generalized Free Fields in de Sitter from 1D CFT,

    K. Goto, A. Milekhin, H. Verlinde, and J. Xu, “Generalized Free Fields in de Sitter from 1D CFT,”arXiv:2605.03037 [hep-th]

    Pith/arXiv arXiv

  21. [21]

    A microscopic model of de Sitter spacetime with an observer,

    D. Tietto and H. Verlinde, “A microscopic model of de Sitter spacetime with an observer,” arXiv:2502.03869 [hep-th]

    arXiv

  22. [22]

    Wave function of the universe,

    J. B. Hartle and S. W. Hawking, “Wave function of the universe,”Physical Review D28no. 12, (1983) 2960

    1983

  23. [23]

    Solving 3d Gravity with Virasoro TQFT,

    S. Collier, L. Eberhardt, and M. Zhang, “Solving 3d Gravity with Virasoro TQFT,” arXiv:2304.13650 [hep-th]

    arXiv

  24. [24]

    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]

    Pith/arXiv arXiv 2010

  25. [25]

    Poincare Series, 3D Gravity and CFT Spectroscopy,

    C. A. Keller and A. Maloney, “Poincare Series, 3D Gravity and CFT Spectroscopy,”JHEP02 (2015) 080,arXiv:1407.6008 [hep-th]. 62

    Pith/arXiv arXiv 2015

  26. [26]

    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]

    Pith/arXiv arXiv 2011

  27. [27]

    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]

    arXiv 2022

  28. [28]

    The dS / CFT correspondence,

    A. Strominger, “The dS / CFT correspondence,”JHEP10(2001) 034,arXiv:hep-th/0106113

    Pith/arXiv arXiv 2001

  29. [29]

    Quantum gravity in de Sitter space,

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

    Pith/arXiv arXiv 2001

  30. [30]

    Higher Spin Realization of the dS/CFT Correspondence,

    D. Anninos, T. Hartman, and A. Strominger, “Higher Spin Realization of the dS/CFT Correspondence,”Class. Quant. Grav.34no. 1, (2017) 015009,arXiv:1108.5735 [hep-th]

    Pith/arXiv arXiv 2017

  31. [31]

    The spectrum of pure dS 3 gravity in the static patch,

    J. Chakravarty, A. Maloney, K. Namjou, and S. F. Ross, “The spectrum of pure dS 3 gravity in the static patch,”JHEP10(2025) 021,arXiv:2505.06420 [hep-th]

    arXiv 2025

  32. [32]

    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]

    Pith/arXiv arXiv 2012

  33. [33]

    A microscopic realization of dS 3,

    S. Collier, L. Eberhardt, and B. M¨ uhlmann, “A microscopic realization of dS 3,” arXiv:2501.01486 [hep-th]

    arXiv

  34. [34]

    The complex Liouville string,

    S. Collier, L. Eberhardt, B. M¨ uhlmann, and V. A. Rodriguez, “The complex Liouville string,” arXiv:2409.17246 [hep-th]

    arXiv

  35. [35]

    M¨ obius randomness in the Hartle-Hawking state,

    V. Godet, “M¨ obius randomness in the Hartle-Hawking state,”arXiv:2505.03068 [hep-th]

    Pith/arXiv arXiv

  36. [36]

    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]

    arXiv 2020

  37. [37]

    A Background Independent Algebra in Quantum Gravity,

    E. Witten, “A Background Independent Algebra in Quantum Gravity,”arXiv:2308.03663 [hep-th]

    arXiv

  38. [38]

    A Black hole Farey tail,

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

    Pith/arXiv arXiv

  39. [39]

    Reduction of the Einstein equations in (2+1)-dimensions to a Hamiltonian system over Teichmuller space,

    V. Moncrief, “Reduction of the Einstein equations in (2+1)-dimensions to a Hamiltonian system over Teichmuller space,”J. Math. Phys.30(1989) 2907–2914

    1989

  40. [40]

    6-dimensional Quantum Town: (2+1)-dimensional Quantum Cosmology,

    A. Hosoya and K.-i. Nakao, “6-dimensional Quantum Town: (2+1)-dimensional Quantum Cosmology,”Prog. Theor. Phys.82(1989) 163–174

    1989

  41. [41]

    Comparative quantizations of (2+1)-dimensional gravity,

    S. Carlip and J. E. Nelson, “Comparative quantizations of (2+1)-dimensional gravity,”Phys. Rev. D51(1995) 5643–5653,arXiv:gr-qc/9411031

    Pith/arXiv arXiv 1995

  42. [42]

    Quantum gravity in 2+1 dimensions,

    S. Carlip, “Quantum gravity in 2+1 dimensions,”World Sci. Ser. 20th Cent. Phys.33(2004) 1–21,arXiv:gr-qc/0409039. 63

    Pith/arXiv arXiv 2004

  43. [43]

    Boundary conformal field theory,

    R. Blumenhagen and E. Plauschinn, “Boundary conformal field theory,” inIntroduction to Conformal Field Theory: With Applications to String Theory, pp. 205–256. Springer, 2009

    2009

  44. [44]

    SYK collective field theory as complex Liouville gravity,

    A. Blommaert, D. Tietto, and H. Verlinde, “SYK collective field theory as complex Liouville gravity,”arXiv:2509.18462 [hep-th]

    arXiv

  45. [45]

    Double-scaled SYK and de Sitter Holography,

    V. Narovlansky and H. Verlinde, “Double-scaled SYK and de Sitter Holography,” arXiv:2310.16994 [hep-th]

    arXiv

  46. [46]

    Double-scaled SYK, Chords and de Sitter Gravity,

    H. Verlinde, “Double-scaled SYK, Chords and de Sitter Gravity,”arXiv:2402.00635 [hep-th]

    arXiv

  47. [47]

    Entanglement and Chaos in De Sitter Space Holography: An SYK Example,

    L. Susskind, “Entanglement and Chaos in De Sitter Space Holography: An SYK Example,” JHAP1no. 1, (2021) 1–22,arXiv:2109.14104 [hep-th]

    arXiv 2021

  48. [48]

    Towards a microscopic description of de Sitter dynamics,

    V. Narovlansky, “Towards a microscopic description of de Sitter dynamics,”arXiv:2506.02109 [hep-th]

    Pith/arXiv arXiv

  49. [49]

    3D near-de Sitter gravity and the soft mode of DSSYK,

    T. Marini, X.-L. Qi, and H. Verlinde, “3D near-de Sitter gravity and the soft mode of DSSYK,” arXiv:2604.21014 [hep-th]

    Pith/arXiv arXiv

  50. [50]

    SYK-Schur duality: Double scaled SYK correlators fromN“2 supersymmetric gauge theory,

    D. Gaiotto and H. Verlinde, “SYK-Schur duality: Double scaled SYK correlators fromN“2 supersymmetric gauge theory,”arXiv:2409.11551 [hep-th]

    arXiv

  51. [51]

    Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction,

    M. Berkooz, P. Narayan, and J. Simon, “Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction,”JHEP08(2018) 192,arXiv:1806.04380 [hep-th]

    Pith/arXiv arXiv 2018

  52. [52]

    The dilaton gravity hologram of double-scaled SYK,

    A. Blommaert, T. G. Mertens, and J. Papalini, “The dilaton gravity hologram of double-scaled SYK,”arXiv:2404.03535 [hep-th]

    arXiv

  53. [53]

    Wormholes, branes and finite matrices in sine dilaton gravity,

    A. Blommaert, A. Levine, T. G. Mertens, J. Papalini, and K. Parmentier, “Wormholes, branes and finite matrices in sine dilaton gravity,”arXiv:2501.17091 [hep-th]

    arXiv

  54. [54]

    The q-Schwarzian and Liouville gravity,

    A. Blommaert, T. G. Mertens, and S. Yao, “The q-Schwarzian and Liouville gravity,”JHEP11 (2024) 054,arXiv:2312.00871 [hep-th]

    arXiv 2024

  55. [55]

    Towards a full solution of the large n double-scaled syk model,

    M. Berkooz, M. Isachenkov, V. Narovlansky, and G. Torrents, “Towards a full solution of the large n double-scaled syk model,”Journal of High Energy Physics2019no. 3, (2019) 1–72

    2019

  56. [56]

    Infinite Temperature’s Not So Hot,

    H. Lin and L. Susskind, “Infinite Temperature’s Not So Hot,”arXiv:2206.01083 [hep-th]

    arXiv

  57. [57]

    The bulk Hilbert space of double scaled SYK,

    H. W. Lin, “The bulk Hilbert space of double scaled SYK,”JHEP11(2022) 060, arXiv:2208.07032 [hep-th]

    arXiv 2022

  58. [58]

    A symmetry algebra in double-scaled SYK,

    H. W. Lin and D. Stanford, “A symmetry algebra in double-scaled SYK,”arXiv:2307.15725 [hep-th]

    arXiv

  59. [59]

    Quantum group origins of edge states in double-scaled SYK,

    A. Belaey, T. G. Mertens, and T. Tappeiner, “Quantum group origins of edge states in double-scaled SYK,”JHEP02(2026) 184,arXiv:2503.20691 [hep-th]. 64

    arXiv 2026

  60. [60]

    Quantum symmetry and geometry in double-scaled SYK,

    J. van der Heijden, E. Verlinde, and J. Xu, “Quantum symmetry and geometry in double-scaled SYK,”JHEP05(2026) 148,arXiv:2511.08743 [hep-th]

    arXiv 2026

  61. [61]

    The von Neumann algebraic quantum group SU qp1,1q ¸Z 2 and the DSSYK model,

    K. Schouten and M. Isachenkov, “The von Neumann algebraic quantum group SU qp1,1q ¸Z 2 and the DSSYK model,”arXiv:2512.10101 [math-ph]

    Pith/arXiv arXiv

  62. [62]

    Jt gravity with matter, generalized eth, and random matrices,

    D. L. Jafferis, D. K. Kolchmeyer, B. Mukhametzhanov, and J. Sonner, “Jt gravity with matter, generalized eth, and random matrices,”arXiv preprint arXiv:2209.02131(2022)

    arXiv 2022

  63. [63]

    Discrete analogue of the Weil-Petersson volume in double scaled SYK,

    K. Okuyama, “Discrete analogue of the Weil-Petersson volume in double scaled SYK,”JHEP 09(2023) 133,arXiv:2306.15981 [hep-th]

    arXiv 2023

  64. [64]

    Krylov spread complexity as holographic complexity beyond JT gravity,

    M. P. Heller, J. Papalini, and T. Schuhmann, “Krylov spread complexity as holographic complexity beyond JT gravity,”arXiv:2412.17785 [hep-th]

    arXiv

  65. [65]

    An entropic puzzle in periodic dilaton gravity and DSSYK,

    A. Blommaert, A. Levine, T. G. Mertens, J. Papalini, and K. Parmentier, “An entropic puzzle in periodic dilaton gravity and DSSYK,”arXiv:2411.16922 [hep-th]

    arXiv

  66. [66]

    Dynamical actions and q-representation theory for double-scaled SYK,

    A. Blommaert, T. G. Mertens, and S. Yao, “Dynamical actions and q-representation theory for double-scaled SYK,”JHEP02(2024) 067,arXiv:2306.00941 [hep-th]

    arXiv 2024

  67. [67]

    Geometry of chord intertwiner, multiple shocks and switchback in double-scaled SYK,

    S. E. Aguilar-Gutierrez and J. Xu, “Geometry of chord intertwiner, multiple shocks and switchback in double-scaled SYK,”JHEP02(2026) 246,arXiv:2506.19013 [hep-th]

    arXiv 2026

  68. [68]

    q-Askey Deformations of Double-Scaled SYK,

    S. E. Aguilar-Gutierrez, T. Kukolj, and J. Seitz, “q-Askey Deformations of Double-Scaled SYK,”arXiv:2605.13956 [hep-th]

    Pith/arXiv arXiv

  69. [69]

    Splitting and gluing in sine-dilaton gravity: matter correlators and the wormhole Hilbert space,

    C. Cui and M. Rozali, “Splitting and gluing in sine-dilaton gravity: matter correlators and the wormhole Hilbert space,”JHEP02(2026) 160,arXiv:2509.01680 [hep-th]

    arXiv 2026

  70. [70]

    Quantum gravity of the Heisenberg algebra,

    A. Almheiri, A. Goel, and X.-Y. Hu, “Quantum gravity of the Heisenberg algebra,” arXiv:2403.18333 [hep-th]

    arXiv

  71. [71]

    Finite features of quantum de Sitter space,

    D. Anninos, D. A. Galante, and B. M¨ uhlmann, “Finite features of quantum de Sitter space,” Class. Quant. Grav.40no. 2, (2023) 025009,arXiv:2206.14146 [hep-th]

    arXiv 2023

  72. [72]

    Cosmological observatories,

    D. Anninos, D. A. Galante, and C. Maneerat, “Cosmological observatories,”Class. Quant. Grav.41no. 16, (2024) 165009,arXiv:2402.04305 [hep-th]

    arXiv 2024

  73. [73]

    The initial boundary value problem and quasi-local Hamiltonians in General Relativity,

    Z. An and M. T. Anderson, “The initial boundary value problem and quasi-local Hamiltonians in General Relativity,”arXiv:2103.15673 [gr-qc]

    arXiv

  74. [74]

    On boundary value problems for einstein metrics,

    M. T. Anderson, “On boundary value problems for einstein metrics,”Geometry & Topology12 no. 4, (2008) 2009–2045

    2008

  75. [75]

    Gravitational Wilson Lines in 3D de Sitter,

    A. Castro, P. Sabella-Garnier, and C. Zukowski, “Gravitational Wilson Lines in 3D de Sitter,” JHEP07(2020) 202,arXiv:2001.09998 [hep-th]. 65

    arXiv 2020

  76. [76]

    (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

  77. [77]

    Quantization of Chern-Simons Gauge Theory With Complex Gauge Group,

    E. Witten, “Quantization of Chern-Simons Gauge Theory With Complex Gauge Group,” Commun. Math. Phys.137(1991) 29–66

    1991

  78. [78]

    SYK Correlators from 2D Liouville-de Sitter Gravity,

    H. Verlinde and M. Zhang, “SYK Correlators from 2D Liouville-de Sitter Gravity,” arXiv:2402.02584 [hep-th]

    arXiv

  79. [79]

    Quantum field theory and the jones polynomial,

    E. Witten, “Quantum field theory and the jones polynomial,”Communications in Mathematical Physics121no. 3, (1989) 351–399

    1989

  80. [80]

    Towards a full solution of the large N double-scaled SYK model,

    M. Berkooz, M. Isachenkov, V. Narovlansky, and G. Torrents, “Towards a full solution of the large N double-scaled SYK model,”JHEP03(2019) 079,arXiv:1811.02584 [hep-th]

    Pith/arXiv arXiv 2019

Showing first 80 references.