Extracting edge modes: Reduction of 3D and 2D gravities

We investigate the boundary reduction of 3D Einstein gravity and Jackiw-Teitelboim gravity into their respective edge mode theories—namely, the Liouville and Alekseev-Shatashvili, and Schwarzian models. By examining the roles of boundary conditions, canonical transformations, and differing formulations—metric versus Chern-Simons—we clarify how physical degrees of freedom become localized at the boundary and resolve several longstanding ambiguities in the reduction procedure.