The Schwarzian Theory - Origins

TL;DR

This paper explores the 1d Schwarzian theory's connections to 2d Liouville and 3d gravity, extending the theory to models with internal symmetries and identifying its holographic dual as a 2d BF theory.

Contribution

It provides a path-integral derivation of the link between Schwarzian and Liouville theories and generalizes the Schwarzian limit to rational models with internal symmetries.

Findings

Established the link between Schwarzian and Liouville theories

Generalized Schwarzian limit to rational models with symmetries

Identified the holographic dual as a 2d BF theory

Abstract

In this paper we further study the 1d Schwarzian theory, the universal low-energy limit of Sachdev-Ye-Kitaev models, using the link with 2d Liouville theory. We provide a path-integral derivation of the structural link between both theories, and study the relation between 3d gravity, 2d Jackiw-Teitelboim gravity, 2d Liouville and the 1d Schwarzian. We then generalize the Schwarzian double-scaling limit to rational models, relevant for SYK-type models with internal symmetries. We identify the holographic gauge theory as a 2d BF theory and compute correlators of the holographically dual 1d particle-on-a-group action, decomposing these into diagrammatic building blocks, in a manner very similar to the Schwarzian theory.