The q-Schwarzian and Liouville gravity

TL;DR

This paper establishes a new holographic duality linking the q-Schwarzian quantum mechanics to Liouville gravity, extending the understanding of low-energy holographic dualities and exact solutions in dilaton gravity models.

Contribution

It introduces a novel duality between q-Schwarzian quantum mechanics and Liouville gravity, including a quantum-level proof and exact solutions for sinh dilaton gravity.

Findings

q-Schwarzian is dual to sinh dilaton gravity

The duality is exact at the quantum level

For real q, the duality relates to sine dilaton gravity

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.