Double-scaled SYK, Chords and de Sitter Gravity

TL;DR

This work establishes a quantitative bridge between the double-scaled SYK model and 3D de Sitter gravity by identifying the gravity Hamiltonian with the gravitational Wilson line that measures the conical deficit. Through a SL(2,C) Chern–Simons quantization of non-rotating Schwarzschild–de Sitter space, the authors derive a q-oscillator structure whose spectrum matches the DSSYK recursion, enabling exact partition functions and two-point functions to be computed within the gravity framework. The results reinforce a holographic dictionary where DSSYK chord rules correspond to skein relations of Wilson lines, with a shared U_q(sl2) symmetry governing both spectra and correlators. The approach provides a concrete, solvable testbed for DSSYK–SdS duality and lays groundwork for exploring scattering, 6j-symbols, and higher-dimensional extensions via quantum group methods.

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.

Figures (6)

• Figure 1: Chord diagram produced by contractions between $H$ insertions at successive time steps.
• Figure 2: The gravitational Wilson line $L_A$ wraps the cosmological horizon and measures the deficit angle of the Schwarzschild-de Sitter spacetime. We identify $L_A$ with the de Sitter Hamiltonian.
• Figure 3: 3D de Sitter space with the holonomy variables $L_A$ and $L_Z$ indicated. The left figure shows the Penrose diagram with the two static causal wedges separated by the cosmological horizon, the right figure depicts the spatial constant time slice, which takes the form of a three sphere divided into two hemispheres. In the first order formulation of 3D gravity, the holomies $L_A$ and $L_Z$ are Wilson lines of a flat $SL(2,\mathbb{C})$ connection along two dual cycles with a single intersection point.
• Figure 4: The holonomy variables $L_A$ and $L_Z$ that span the phase space of Schwarzschild-de Sitter (left) and the holonomies $L_Y$ and $L_{\tilde{Y}}$ that arise in the Poisson bracket between $L_A$ and $L_Z$ (right).
• Figure 5: The two triangulations of the two-punctured sphere (left and middle) with the Penner coordinates indicated. The $A$-cycle holonomy $L_A$ is depicted on the right.