SYK Correlators from 2D Liouville-de Sitter Gravity

TL;DR

We establish an exact holographic link between the doubled, high-temperature DSSYK model and a 2D Liouville-de Sitter gravity comprising two spacelike Liouville CFTs with total central charge $c_+ + c_- = 26$. By quantizing 3D de Sitter gravity and leveraging Zamolodchikov's analytic continuation of Liouville theory, we map bulk gravitational data to boundary CFT data and derive an exact boundary two-point function that matches the DSSYK correlator for all values of the coupling $ ext{λ} = p^2/N$. The construction hinges on a dual-dS dictionary: the two chiral Liouville sectors have complex central charges summing to 26, the boundary cosmological constant $ ext{μ}_B$ is identified with DSSYK energy, and the boundary two-point function reduces to a theta-function expression mirroring the SYK result. This work provides strong evidence for a unitary, exact holographic dual between DSSYK and de Sitter gravity, with a controlled quantum gravity description of de Sitter space in low dimensions.

Abstract

We introduce and study a candidate gravity dual to the double scaled SYK model in the form of an exactly soluble 2D de Sitter gravity model consisting of two spacelike Liouville CFTs with complex central charge adding up to $c_+ + c_- = 26$. In [1] it was shown that the two-point function of physical operators in a doubled SYK model matches in the semi-classical limit with the Green's function of a massive scalar field in 3D de Sitter space. As further evidence of the duality, we adapt a result from Zamolodchikov to compute the boundary two-point function of the 2D Liouville-de Sitter gravity model on a disk and find that it reproduces the exact DSSYK two-point function to all orders in $λ=p^2/N$. We describe how the 2D Liouville-de Sitter gravity model arises from quantizing 3D de Sitter gravity.

Figures (6)

• Figure 1: The product of two $q$-Gamma functions has the same regular lattice of poles as the inverse power of a theta function.
• Figure 2: Green function of a scalar field attached to the worldline at the center of the static patch.
• Figure 3: Euclidean 3D de Sitter space is obtained by gluing two half three spheres, c.f. Hikida:2022ltr. The observer world line at $\rho=0$ (red) and observer horizon at $\rho=\pi/2$ (green) indicated.
• Figure 4: The 3D functional integral over ${\cal M}{}_{\rm ket}$ equals the 2D disk partition function. The scalar Green's function corresponds to the correlation function of two boundary operators.
• Figure 5: A product of two $\Upsilon$-functions has the same regular lattice of zeros as the theta function.