The Schwarzian Theory - Origins

TL;DR

This work deepens the connection between the 1d Schwarzian theory and its 2d Liouville origin by providing a Liouville-path integral derivation that reproduces Schwarzian correlators and clarifies the bulk JT/gravity structure. It extends the Schwarzian paradigm to rational, compact-group models via a 2d BF framework, yielding a 1d particle-on-a-group boundary theory whose correlators admit a clean diagrammatic decomposition built from group representation theory. Explicit analyses of U(1) and SU(2) examples illustrate how bilocal operators arise from dimensional reduction and how fusion coefficients govern correlators, while supersymmetric extensions (N=1, N=2) are mapped to corresponding super-Schwarzian and super-Liouville structures. The results offer a unifying, dimensionally-reduced holographic picture and pave the way for rigorous treatment of rational and supersymmetric generalizations, with potential insights into OTOCs and bulk/boundary dualities in higher-dimensional theories.

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.

Figures (14)

• Figure 1: Scheme of theories and their interrelation.
• Figure 2: Cylindrical surface with ZZ-branes at $\sigma=0,\pi$. The $\tau$-coordinate is chosen periodic with period $T$.
• Figure 3: Left: $\sigma$-dependence of $a$ and $b$ and their behavior at the branes at $\sigma=0$ and $\sigma = \pi$. Right: The doubling trick allows a description in terms of a single function $f(\sigma)$.
• Figure 4: Left: $\sigma$-dependence of $A$ and $B$ and their behavior at the branes at $\sigma=0$ and $\sigma = \pi$. Right: The doubling trick allows a description in terms of a single function $F(\sigma)$.
• Figure 5: Liouville theory in 2d in its different incarnations, and the resulting 1d theory one finds upon taking the double scaling (classical) limit. The redefinition $\dot{f}= e^{\psi}$ utilized by Altland, Bagrets and Kamenev (ABK) altlandBagrets:2017pwq, is the dimensional reduction of the transition from Gerveu-Neveu variables to Bäcklund variables.