Double-scaled SYK and de Sitter Holography

TL;DR

The paper proposes a low-dimensional de Sitter holography built from a pair of double-scaled SYK models constrained to equal energy. It computes exact two-point functions of dressed, BRST-invariant observables and demonstrates a precise match, in the large-N limit, to the Green's function of a massive scalar in $dS_3$, including antipodal and BD structures and two towers of quasi-normal modes. A candidate gravity dual is presented as a circle-reduced 3D Einstein-de Sitter gravity described by a deformed 2D JT gravity, with matter encoded by a 2D complex scalar whose bulk mass relates to the boundary scaling dimension via $m^2=4\Delta(1-\Delta)$. The work also argues for a finite de Sitter temperature and a bound on observable entropy, offering a concrete framework for de Sitter holography and outlining directions for higher-dimensional generalizations and a fuller holographic dictionary.

Abstract

We propose a new model of low dimensional de Sitter holography in the form of a pair of double-scaled SYK models at infinite temperature coupled via an equal energy constraint $H_L=H_R$. As a test of the duality, we compute the two-point function between two dressed SYK operators ${\cal O}_Δ$ that preserve the constraint. We find that in the large $N$ limit, the two-point function precisely matches with the Green's function of a massive scalar field of mass squared $m^2 = 4Δ(1-Δ)$ in a 3D de Sitter space-time with radius $R_{\text{dS}}/G_N = 4πN/p^2$. In this correspondence, the SYK time is identified with the proper time difference between the two operators. We introduce a candidate gravity dual of the doubled SYK model given by a JT/de Sitter gravity model obtained via a circle reduction from 3D Einstein-de Sitter gravity. We comment on the physical meaning of the finite de Sitter temperature and entropy.

Figures (4)

• Figure 1: The spectral density of the Doubled SYK model looks like a gaussian centered on $E= 0$. The energy eigenstate $|E_0\rangle$ with $E_0=0$ is the state of maximal entropy $S(E_0) = \log \rho(E_0)$.
• Figure 2: Penrose diagram of 2D de Sitter space. The time-like boundaries $UV = \! -\space 1$ denote the location where $e^\Phi =0$.
• Figure 3: Schematic depiction of the two types of de Sitter Green's functions. Based on the explicit answer \ref{['gppone']}-\ref{['gpmtwo']} obtained from the SYK model, we interpret the ${\cal O}^+$ operator as that can emit particles from the north pole and absorb particles from the south pole and vice versa for ${\cal O}^-$.
• Figure 4: 2-point function in 2D de Sitter space for spatially separated points.