Gravitational edge mode in asymptotically AdS₂: JT gravity revisited

We study the gravitational edge mode of the Jackiw-Teitelboim (JT) gravity and its sl(2, ℝ) BF theory description with the asymptotic AdS₂ boundary condition. We revisit the derivation of the Schwarzian theory from the wiggling boundary as an action for the gravitational edge mode. We present an alternative description for the gravitational edge mode from the metric fluctuation with the fixed boundary, which is often referred as “would-be gauge mode”. We clarify the relation between the wiggling boundary and the would-be gauge mode. We demonstrate a natural top-down derivation of PSL(2, ℝ) gauging and the path integral measure of the Schwarzian theory. In the sl(2, ℝ) BF theory, we incorporate the gravitational edge mode and derive the Schwarzian theory with PSL(2, ℝ) gauging. We also discuss the path integral measure from the Haar measure in the Iwasawa decomposition of PSL(2, ℝ).