One-dimensional Quantum Gravity and the Schwarzian theory

TL;DR

This paper introduces a one-dimensional conformal quantum gravity model that computes the Schwarzian theory partition function, highlighting its simplicity, gauge invariance, and UV finiteness, with implications for understanding quantum gravity at small scales.

Contribution

The paper develops a novel one-dimensional quantum gravity model directly related to the Schwarzian theory, avoiding localization and elucidating gauge invariance and finiteness properties.

Findings

Model computes Schwarzian partition function straightforwardly

The quantum measure is local and does not rely on localization

The model exhibits UV finiteness and Planck-scale emergence

Abstract

We develop a model of one-dimensional (Conformal) Quantum Gravity. By discussing the connection between Goldstone and Gauge theories, we establish that this model effectively computes the partition function of the Schwarzian theory where the $\textrm{SL}(2,\mathbb{R})$ symmetry is realized on the base space. The computation is straightforward, involves a local quantum measure and does not rely on localization arguments. Non-localities in the model are exclusively related to the value of fixed gauge invariant moduli. Furthermore, we study the properties of these models when all degrees of freedom are allowed to fluctuate. We discuss the UV finiteness properties of these systems and the emergence of a Planck's length.