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Double-scaled SYK, Chords and de Sitter Gravity

Herman Verlinde

arXiv:2402.00635v1hep-th

TL;DR

The paper seeks a firmer link between DSSYK and the quantum properties of 3D de Sitter gravity. It matches the gravitational Wilson-line Hamiltonian to DSSYK chord rules, deriving the same recursion and spectrum and using the match to study gravity observables.

  • Problem

    The proposed DSSYK–de Sitter correspondence needs a direct link between DSSYK’s combinatorial chord rules and 3D de Sitter gravity’s quantum properties.

    Source: §1 · p. 3

  • Method

    The paper identifies the de Sitter Hamiltonian with the gravitational Wilson line and matches gravity’s q-oscillator variables to the DSSYK chord algebra.

    Source: §1 · p. 4, §2.4 · p. 16

  • Results

    The gravitational Wilson-line spectrum obeys the DSSYK recursive formula, with the parameter identification uniquely fixed by matching q-oscillator algebras.

    Source: §1 · p. 4

  • Takeaways & Limitations

    The match links the elementary quantities of the two systems, including the SYK Hamiltonian and matter chords with gravitational Wilson lines.

    Source: §3 · p. 22

    The proposed duality restricts 3D de Sitter gravity states to the s-wave sector.

    Source: §3 · p. 22

Abstract

from arXiv · show

We study the partition function of 3D de Sitter gravity defined as the trace over the Hilbert space obtained by quantizing the phase space of non-rotating Schwarzschild-de Sitter spacetime. Motivated by the correspondence with double scaled SYK, we identify the Hamiltonian with the gravitational Wilson-line that measures the conical deficit angle. We express the Hamiltonian in terms of canonical variables and find that it leads to the exact same chord rules and energy spectrum as the double scaled SYK model. We use the obtained match to compute the partition function and scalar two-point function in 3D de Sitter gravity.

1 Introduction

The introduction seeks a direct link between DSSYK chord rules and the quantum properties of 3D de Sitter gravity. It motivates identifying the gravitational Hamiltonian with a Wilson line that measures the Schwarzschild-de Sitter deficit angle.

1 Introduction

  • In the double-scaling limit, DSSYK Hamiltonian moments reduce to chord-diagram counting weighted by q^#intersections, with each chord representing a disorder-average Wick contraction.The limit takes N, p → ∞ with λ = 2p^2/N fixed and q = e^−λ.Source: §1 · p. 3
  • Slicing a chord diagram gives Hilbert states labeled by chord number; Hamiltonian insertions add a nonintersecting chord or close one through intersection-weighted contractions.For n open chords, closing contributions sum powers from q^0 to q^(n−1).Source: §1 · p. 3, Figure 1 · p. 3
  • A shared Uq(sl2) quantum symmetry and real deformation parameter q between 0 and 1 motivate deriving DSSYK’s chord recursion from 3D de Sitter gravity.Pure 3D de Sitter gravity has an SL(2, C) Chern-Simons formulation.Source: §1 · p. 4
  • The authors postulate a dictionary relating DSSYK’s spectral angle θ to the deficit angle 2πα of the bounded 3D Schwarzschild-de Sitter spectrum, with 2πα = π − 2θ.The gravitational Wilson line LA measures the deficit angle and is identified with the de Sitter Hamiltonian.Source: §1 · p. 4, Figure 2 · p. 4
  • The gravitational Wilson-line spectrum follows the DSSYK recursive formula through SL(2, C) skein relations, while matching q-oscillator algebras uniquely fixes the parameter identification.The recursion, generalized to matter chords, also underlies exact DSSYK correlation functions.Source: §1 · pp. 3–4

2 Quantizing 3D Schwarzschild-de Sitter

This section develops the quantization of 3D Schwarzschild–de Sitter gravity using its symmetries, holonomy phase-space variables, quantum algebra, and a Fock-coordinate construction. It relates the gravitational Wilson line and A-cycle holonomy to the Hamiltonian, spectrum, partition function, and scalar two-point functions, including the stated semiclassical limit.

2.1 Symmetries

  • Gauging CPT with the chosen implementation breaks the global SL(2, C) isometry group to SU(1, 1), requiring physical states and operators to be invariant under charge conjugation combined with PT.PT exchanges the north and south static patches while reversing time; the gauged symmetry also requires the masses in the two patches to be identical.Source: §2.1 · p. 6

2.2 Phase space

  • The SdS phase space is two-dimensional: the measurable holonomy trace L_A specifies the geometry up to global isometries, while the time shift z is the second phase-space variable.The deficit angle 2πα is treated as periodic modulo 2π, so a fixed α represents a superposition of geometries whose deficits differ by integer multiples of 2π.Source: §2.2 · pp. 7, 10

2.3 Quantization

  • The first-order formulation identifies 3D gravity with SL(2, C) Chern–Simons theory, whose symplectic structure supplies the brackets used to quantize the gauge-invariant holonomies.The authors use a hybrid approach: derive the holonomy bracket by quantizing first, then construct a Hilbert-space representation of the physical operator algebra.Source: §2.2 · p. 8, §2.3 · pp. 10–11
  • The intersecting holonomies obey a skein relation that lifts to an exact quantum operator relation, with the one-loop shift making it exact to all orders.The framing anomaly can break diffeomorphism invariance for general Wilson-line configurations, but cancels in suitable spinless combinations of the A+ and A− Wilson lines.Source: §2.3 · pp. 11–12
  • The A-cycle holonomy is expressed using Fock coordinates whose commutation and Ptolemy relations provide the main result of the quantization construction.At curvature-singularity punctures, Fock coordinates require regularization using small surrounding geodesics and cross-ratios.Source: §2.3 · pp. 13–15

2.4 Spectrum

  • The Hamiltonian is identified with the A-cycle holonomy, which measures the Schwarzschild-de Sitter deficit angle and energy, and its action matches the double-scaled SYK Hamiltonian.The resulting energy-state density is ρ(θ); the dual variable has a discrete spectrum, and the authors take the match as evidence for a DSSYK-de Sitter correspondence.Source: §2.4 · pp. 16–17

2.5 Partition function

  • The quantized spectrum formally defines a thermal partition function interpreted as an ensemble of non-rotating Schwarzschild-de Sitter spacetimes.The periodic spectral density is interpreted as a sum over semiclassical geometries; the authors propose pure de Sitter corresponds to the first sub-leading saddle, n = 1, while exact functional-integral verification remains future work.Source: §2.5 · pp. 18–20

2.6 Two-point function

  • The authors propose a gravitational Wilson-line expression for the scalar two-point function in 3D de Sitter quantum gravity (c.f. [36]).They present it as an approach to the exact quantum two-point function.Source: §2.6 · pp. 20–21
  • The physical two-Wilson-line combination must be free of framing anomalies, while e−iτH controls the worldline endpoints’ geodesic time difference.The q_n^∆ and q_n^(1−∆) operator matrix elements between energy eigenstates represent DSSYK two-point functions O_∆ and O_(1−∆), respectively.Source: §2.6 · p. 21
  • The gravitational Wilson line produces the two-point function of physical operators in doubled DSSYK, computed using the relation between Pochhammer symbols and Jacobi theta functions.The result further motivates the proposed matter Wilson line.Source: §2.6 · p. 21
  • In the q → 1 limit, the two-point function reduces to the semiclassical massive-scalar result in 3D de Sitter space, expressed in terms of proper-time difference τ.This correspondence may inform future studies of the semiclassical limit of 3D de Sitter quantum gravity and its relationship with DSSYK.Source: §2.6 · p. 21

3 Concluding remarks

The paper establishes a direct match between the 3D de Sitter gravity Hamiltonian and DSSYK chord rules, then situates this link within a broader correspondence involving Liouville gravity.

3 Concluding remarks

  • The gravity Hamiltonian directly matches the combinatorial rules governing the exact DSSYK spectrum and correlation functions.The dictionary identifies the SYK Hamiltonian and matter chords with gravitational Wilson lines measuring the conical deficit and describing matter-particle worldlines.Source: §3 · p. 22
  • The proposed 3D–1D link is restricted to DSSYK observer trajectories and the s-wave sector of topological 3D de Sitter gravity.The paper places this link within a triangle of dual correspondences.Source: §3 · p. 22
  • The broader triangle includes a 3D–2D Chern-Simons/CFT link and a DSSYK–Liouville link whose disk two-point functions match exactly.The companion paper develops the other two links; the 3D gravitational Wilson lines correspond to Verlinde loop operators in Liouville gravity.Source: §3 · p. 22
  • A shared Uq(sl2) symmetry connects DSSYK, 2D Virasoro-Liouville gravity, and 3D de Sitter quantum gravity.In the gravity setting, the symmetry governs fusion and braiding of massive-particle worldlines; deviations from classical generators are of order 1 − q ≃ λ.Source: §3 · p. 23

Scattering and 6j-Symbols

The section connects antipodal communication enabled by de Sitter shockwaves to a SYK out-of-time-ordered correlator, with quantum 6j-symbols encoding the scattering.

Scattering and 6j-Symbols

  • Equal-energy signals sent simultaneously from opposite poles create shockwaves that open a horizon gap, allowing causal communication between antipodal points.The semiclassical setup predicts a four-point-function singularity as the late-time operators approach the signals’ classical arrival time.Source: Scattering and 6j-Symbols · p. 23, Figure 6 · p. 24
  • The SYK four-point function describing antipodal signal detection has the characteristics of an OTOC.The initial operators create particles on opposite sides of the horizon, and later operators detect them across it.Source: Scattering and 6j-Symbols · p. 24
  • Quantum 6j-symbols transform the SYK OTOC into a time-ordered correlator and encode gravitational scattering between the shockwaves.The scattering amplitude equals the expectation value of a tetrahedral configuration of Wilson lines in SL(2, C) Chern-Simons gravity.Source: Scattering and 6j-Symbols · p. 24, Figure 6 · p. 24
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