The dilaton gravity hologram of double-scaled SYK

TL;DR

This work establishes an exact holographic duality between double-scaled SYK (DSSYK) and a two-dimensional sine dilaton gravity, with the bulk length L identified as L = 2|log q| times the chord number n. A canonical quantization of the sine dilaton theory reproduces the DSSYK auxiliary q-Schwarzian quantum mechanics, and observables map to transition amplitudes between states |L=0>, provided one imposes the crucial positivity constraint L≥0, which from a bulk perspective is implemented by a conical defect. The analysis resolves key puzzles in the semi-classical regime, including the non-monotonic entropy and the distinction between temperature and fake temperature, by showing that the defect picture yields the correct DSSYK thermodynamics and correlators. Additionally, the work connects the sine dilaton model to Liouville de Sitter gravity, highlighting a potential route to incorporating observers in dS space within this UV-complete holographic setup, and revealing a discretized bulk Hilbert space as a natural outcome of quantization. Overall, the paper provides a local bulk dual for a UV-complete quantum system, clarifies the role of length positivity, and opens avenues for understanding gravity in de Sitter-like contexts via DSSYK.

Abstract

We work out a precise holographic duality between sine dilaton gravity, and DSSYK. More precisely, canonical quantization of sine dilaton gravity reproduces q-Schwarzian quantum mechanics, which is the auxiliary system that arises from the chord diagrams of DSSYK. The role of the chord number in DSSYK is played by the (Weyl rescaled) geodesic length in the bulk. The most puzzling aspect of reconciling DSSYK with a simple gravitational dual at the classical level is the distinction between temperature and "fake temperature". At the q-Schwarzian level, we clarify how this arises from the constraint that the chord number is positive. The on-shell q-Schwarzian action with the constraint reproduces the thermodynamics of DSSYK. Semi-classically, in sine dilaton gravity this translates to the insertion of a defect, from which we deduce that fake temperature is the Hawking temperature of a smooth Lorentzian black hole. We comment on several relations with dS space. One remarkable feature is that in sine dilaton gravity quantization discretizes spacetime, therefore the Hilbert space is discrete.