[Paper Review] The q-Schwarzian and Liouville gravity

Andreas Blommaert, Thomas G. Mertens+1 more|arXiv (Cornell University)|Dec 1, 2023

Black Holes and Theoretical Physics4 citations

TL;DR

This paper establishes an exact holographic duality between the q-Schwarzian quantum mechanics and Liouville gravity (or sinh dilaton gravity), showing that the q-Schwarzian—arising as a deformation of the Schwarzian in double-scaled SYK—precisely describes the boundary dynamics of Liouville gravity on a disk. The duality is proven via a topological Poisson-sigma model formulation, with full matching of thermodynamics, classical two-point functions, and quantum amplitudes.

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.

Motivation & Objective

• To establish a holographic duality between the q-Schwarzian quantum mechanics and Liouville gravity, extending the known duality between Schwarzian mechanics and JT gravity.
• To show that the q-Schwarzian, a deformation of the Schwarzian with parameter $ q = e^{\pi i b^2} $, is dual to sinh dilaton gravity with potential $ V_{\text{qsch}}(\Phi) = \frac{\sinh(2\pi b^2 \Phi)}{\pi b^2} $.
• To prove the duality at the quantum level by reformulating sinh dilaton gravity as a topological gauge theory (Poisson-sigma model), showing equivalence to the q-Schwarzian path integral.
• To match classical observables such as thermodynamics and two-point functions between the q-Schwarzian and Liouville gravity.
• To provide an exact solution of sinh dilaton gravity on disk topology via the exactly solvable q-Schwarzian.

Proposed method

• Formulate sinh dilaton gravity as a topological Poisson-sigma model with a specific Poisson structure related to the quantum group $ \text{SL}_q^+(2,\mathbb{R}) $.
• Derive the boundary action from the Poisson-sigma model by imposing boundary conditions that reduce the bulk dynamics to the q-Schwarzian action.
• Use the group-theoretic structure of the quantum group $ \text{SU}_q(1,1) $ to identify the q-Schwarzian as the boundary quantum mechanics dual to the bulk dilaton gravity.
• Match classical solutions, conserved currents, and symmetry algebras between the q-Schwarzian and Liouville gravity, confirming isomorphism of their classical limits.
• Prove quantum equivalence by showing that the path integral of the Poisson-sigma model yields the same partition function and correlation functions as the q-Schwarzian.
• Leverage the exact solvability of the q-Schwarzian to reproduce known Liouville gravity amplitudes, including the DOZZ three-point function.

Experimental results

Research questions

• RQ1 Is the q-Schwarzian quantum mechanics dual to Liouville gravity (or sinh dilaton gravity) at the quantum level?
• RQ2 Can the classical thermodynamics and two-point functions of the q-Schwarzian match those of Liouville gravity?
• RQ3 Does the Poisson-sigma model formulation of sinh dilaton gravity reproduce the q-Schwarzian as its boundary theory?
• RQ4 How does the q-Schwarzian’s exact solvability relate to the known quantum amplitudes of Liouville gravity?
• RQ5 What is the geometric interpretation of the boundary two-point function $ \langle \mathcal{W}(\varphi) \rangle = e^{-2\Delta \varphi} $ in the context of sinh dilaton gravity?

Key findings

• The q-Schwarzian with $ |q| = 1 $, parameterized as $ q = e^{\pi i b^2} $, is exactly dual to sinh dilaton gravity with potential $ V_{\text{qsch}}(\Phi) = \frac{\sinh(2\pi b^2 \Phi)}{\pi b^2} $, which is equivalent to Liouville gravity.
• Classical thermodynamics and two-point functions of the q-Schwarzian match precisely with those of Liouville gravity, confirming consistency at the classical level.
• The Poisson-sigma model formulation of sinh dilaton gravity reduces to the q-Schwarzian upon imposing appropriate boundary conditions, proving the duality at the path integral level.
• The quantum partition function and correlation functions of the q-Schwarzian reproduce the exact amplitudes of Liouville gravity, including the DOZZ three-point function.
• The q-Schwarzian is shown to be an exact solution of sinh dilaton gravity on disk topology, analogous to how JT gravity is solved by the Schwarzian.
• The boundary two-point function $ \langle e^{-2\Delta \varphi} \rangle $ in the q-Schwarzian matches the semiclassical result in Liouville gravity, despite geometric ambiguities in the bulk interpretation.