Extracting Edge Modes: Reduction of 3D and 2D Gravities

TL;DR

This work analyzes edge modes arising when reducing 3D Einstein gravity and Jackiw–Teitelboim (JT) gravity to edge‑mode theories, showing that the physical degrees of freedom localize as Liouville, Alekseev–Shatashvili (AS), and Schwarzian models under different reductions. It employs a radial Hamiltonian method and a Stueckelberg trick to implement wiggling boundaries, carefully tracking boundary terms and canonical transformations that encode edge dynamics. By comparing CS and metric formulations, and by solving partial bulk equations while keeping edge modes off‑shell, the authors reconcile two historic reductions: Liouville theory (with a potential term depending on background topology) and AS/Schwarzian actions (with temperature‑dependent quadratic terms). A key result is that 3D gravity reduces to two copies of the AS theory or to Liouville theory depending on the decomposition and constraint scheme, while JT gravity reduces to Schwarzian or Liouville‑like edge modes with a potential, clarifying sign ambiguities and the role of boundary conditions. The framework clarifies subtleties in holographic and Hamiltonian reductions and offers a path to extend edge‑mode analyses to higher dimensions and higher‑spin theories. In particular, boundary actions include $S_{\rm Liouville}=\frac{\ell}{32\pi G}\int_{\partial\mathcal{M}} d^{2}x\left(\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi-8 e^{\varphi}\right).$ The Schwarzian edge action arises as $S_{\rm ren}[A,\Phi]\simeq -\frac{\ell}{8\pi G}\int dt\,\varphi_{P}\left(\frac{\dddot{\chi}}{\dot{\chi}}-\frac{3}{2}\Bigl(\frac{\ddot{\chi}}{\dot{\chi}}\Bigr)^2-2\mathcal{L}\,\dot{\chi}^2\right).$ In the JT/BF formulation a Liouville‑like action with a potential also appears: $S_{\rm ren}[A]\simeq \frac{\ell}{16\pi G}\int_{\partial\mathcal{M}} d^{2}x\,[\partial_{\mu}\sigma\partial^{\mu}\sigma-2\mu e^{-\sigma}].$

Abstract

We investigate the boundary reduction of 3D Einstein gravity and JT 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 long-standing ambiguities in the reduction procedure.