[Paper Review] Double-scaled SYK, Chords and de Sitter Gravity

Herman Verlinde|arXiv (Cornell University)|Feb 1, 2024

Historical Geography and Cartography5 citations

TL;DR

The paper connects the double-scaled SYK spectrum with 3D de Sitter gravity by identifying the gravity Hamiltonian with a gravitational Wilson line measuring deficit angle, deriving matching chord rules, and computing partition and two-point functions.

ABSTRACT

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.

Motivation & Objective

• Motivate and establish a direct link between the double scaled SYK model and 3D de Sitter gravity.
• Quantize Schwarzschild-de Sitter spacetime and identify a gravitational Wilson line with the SYK Hamiltonian.
• Show that chord rules and energy spectrum arise from the quantum gravity framework.
• Compute the partition function and scalar two-point function in 3D de Sitter gravity using the match with DSSYK.
• Clarify the role of quantum group symmetries and skein relations in the gravity/SYK correspondence.

Proposed method

• Model the SdS spacetime in a first-order SL(2, C) Chern-Simons formulation and identify holonomies L_A and L_Z as phase-space variables.
• Impose a gauge constraint to reduce to the s-wave sector and derive their spectra.
• Derive the Poisson bracket {L_A, L_Z}_PB via skein relations and lift to a quantum skein relation with q = e^{-2π/κ}, where κ = 1/(2G_N).
• Express L_A in terms of Penner-Fock coordinates and use Ptolemy-type relations to obtain an explicit quantum expression for L_A.
• Identify the DSSYK Hamiltonian with L_A divided by sqrt(1−q) and relate spectral angle θ to deficit angle α via 2π α = π − 2θ.
• Demonstrate the matching q-oscillator structure and the resulting spectrum consistent with the double-scaled SYK picture.

Experimental results

Research questions

• RQ1 Can the quantum spectrum and correlation functions of 3D de Sitter gravity be derived from the chord-based DSSYK framework?
• RQ2 How do skein relations and SU(1,1) / SL(2, C) structures encode the same q-deformed oscillator algebra as in DSSYK?
• RQ3 Does the gravitational Wilson line L_A reproduce the DSSYK energy spectrum and its boundedness when interpreted through the q-deformed harmonic oscillator?
• RQ4 What is the precise mapping between de Sitter deficit angle α and DSSYK spectral angle θ, and how does this affect partition functions and two-point functions?

Key findings

• The gravitational Wilson line L_A encodes the SdS deficit angle and, when identified with the DSSYK Hamiltonian as H = L_A/√(1−q), reproduces the same chord rules.
• The holonomies L_A and L_Z form a two-dimensional phase space whose quantum algebra is governed by a real q-deformed structure with q = e^{−2π/κ} = e^{−4π G_N}.
• The skein relations from SL(2, C) Chern-Simons theory generate the same recursive chord relation H|n⟩ = |n+1⟩ + [n]_q |n−1⟩ that arises in DSSYK.
• The mapping fixes the q-oscillator framework on both gravity and DSSYK sides, supporting a holographic link between DSSYK and 3D de Sitter gravity.
• The analysis allows computation of the partition function and scalar two-point function in 3D de Sitter gravity via the established match to DSSYK results.
• The approach emphasizes the role of SU(1,1) invariants and skein/quantum Teichmüller-type structures in realizing the DSSYK–SdS correspondence.