  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.
