Gravitational Edge Mode in Asymptotically AdS$_2$: JT Gravity Revisited

We study the gravitational edge mode of the Jackiw-Teitelboim (JT) gravity and its $sl(2,\mathbb{R})$ BF theory description with the asymptotic AdS$_2$ 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 $PSL(2,\mathbb{R})$ gauging and the path integral measure of the Schwarzian theory. In the $sl(2,\mathbb{R})$ BF theory, we incorporate the gravitational edge mode and derive the Schwarzian theory with $PSL(2,\mathbb{R})$ gauging. We also discuss the path integral measure from the Haar measure in the Iwasawa decomposition of $PSL(2,\mathbb{R})$.