The q-Schwarzian and Liouville gravity

TL;DR

This work establishes an exact holographic duality between the q-Schwarzian quantum mechanics and Liouville (sinh dilaton) gravity on disk, realized through a Poisson-sigma-model framework that reduces bulk dynamics to boundary q-Schwarzian dynamics. It demonstrates classical thermodynamic and two-point-function matching between the q-Schwarzian and Liouville gravity, and provides a quantum-level equivalence by connecting the boundary theory to the representation theory of SL_q^+(2,R) via the modular double. The results yield a concrete, exact solution for sinh dilaton gravity on the disk and illuminate a gravity-dual for the DSSYK construction in the q → 1 and q < 1 regimes. Together, these insights deepen the understanding of how q-deformations interface with low-dimensional gravity and offer a principled route to exact gravitational amplitudes via boundary quantum mechanics.

Abstract

We present a new holographic duality between q-Schwarzian quantum mechanics and Liouville gravity. The q-Schwarzian is a one parameter deformation of the Schwarzian, which is dual to JT gravity and describes the low energy sector of SYK. We show that the q-Schwarzian in turn is dual to sinh dilaton gravity. This one parameter deformation of JT gravity can be rewritten as Liouville gravity. We match the thermodynamics and classical two point function between q-Schwarzian and Liouville gravity. We further prove the duality on the quantum level by rewriting sinh dilaton gravity as a topological gauge theory, and showing that the latter equals the q-Schwarzian. As the q-Schwarzian can be quantized exactly, this duality can be viewed as an exact solution of sinh dilaton gravity on the disk topology. For real q, this q-Schwarzian corresponds to double-scaled SYK and is dual to a sine dilaton gravity.