Nearly AdS2 Sugra and the Super-Schwarzian

TL;DR

This paper extends nearly $AdS_2$ gravity from the bosonic Jackiw-Teitelboim framework to minimal $\mathcal{N}=1$ supergravity by formulating the boundary in superspace. It derives the super-Schwarzian as the boundary effective action, showing that the supersymmetric Gibbons-Hawking-York term reproduces the correct boundary dynamics when evaluated on the regulated boundary. The analysis yields a boundary action of the form $\int du d\vartheta \Phi_r S[ t,\xi;u,\vartheta]$, with the super-Schwarzian $S$ defined in terms of the bosonic function $f(u)$ and its fermionic partner $\eta(u)$, and confirms that the boundary curvature satisfies $K = 4\epsilon^2 S[\cdot]$ in the supersymmetric setting. These results support a supersymmetric holographic dual of SYK-like models and set the stage for extensions to higher $\mathcal{N}$ and related spectra.

Abstract

In nearly AdS2 gravity the Einstein-Hilbert term is supplemented by the Jackiw-Teitelboim action. Integrating out the bulk metric gives rise to the Schwarzian action for the boundary curve. In the present note, we show how the extension to supergravity leads to the super-Schwarzian action for the superspace boundary.