Gravitational Edge Mode in $\mathcal{N}=1$ Jackiw-Teitelboim Supergravity

We study the gravitational edge mode in the $\mathcal{N}=1$ Jackiw-Teitelboim~(JT) supergravity on the disk and it $osp(2|1)$ BF formulation. We revisit the derivation of the finite-temperature Schwarzian action in the conformal gauge of the bosonic JT gravity through wiggling boundary and the frame fluctuation descriptions. Extending our method to $\mathcal{N}=1$ JT supergravity, we derive the finite-temperature super-Schwarzian action for the edge mode from both the wiggling boundary and the superframe field fluctuation. We emphasize the crucial role of the supersymmetric version of the inversion formula in elucidating the relation between the isometry and the $OSp(2|1)$ gauging of the super-Schwarzian action. In $osp(2|1)$ BF formulation, we discuss the asymptotic AdS condition. We employ the Iwasawa-like decomposition of $OSp(2|1)$ group element to derive the super-Schwarzian action at finite temperature. We demonstrate that the $OSp(2|1)$ gauging arises from inherent redundancy in the Iwasawa-like decomposition. We also discuss the path integral measure obtained from the Haar measure of $OSp(2|1)$.