Is there a Lagrangian for Super-Gravity? Einsteins field equations can be derived from the Einstein-Hilbert action which only involves the scalar curvature $R$ of the spacetime manifold. This is simply
$$S = \int_M R.$$
The volume form or measure here is left implicit as is quite common when the integral is written over a manifold.
Is there a similar Lagrangian for super-gravity and does it similarly involve a kind of scalar curvature suitably interpreted?
 A: Consider the simple case of $\mathcal{N}=1$ on-shell supergravity in 4 dimensions with no matter content. The full action is made up of the Einstein-Hilbert term and the Rarita-Schwinger action for the gravitino, with
$$
S_{EH}=\frac{1}{2\kappa^2}\int\mathrm d^4 x\det e\ e^\mu_ae^\nu_bR^{ab}_{\mu\nu}(\omega)
$$
where $R_{\mu\nu}^{ab}$ is the field strength of the spin connection $\omega_\mu^{ab}$ and $\det e$ takes the place of the volume element.
Since this SUGRA theory can formally be defined as a gauge theory with gauge group $\text{SO}(1, d-1)$, the commutator of the supercovariant derivatives defines the Riemann tensor above:
$$
[D_\mu, D_\nu]=\frac14R_{\mu\nu}^{ab}\gamma_{ab}
$$
So $e^\mu_ae^\nu_bR^{ab}_{\mu\nu}(\omega)$ is essentially the Ricci scalar in the spin-connection-vielbien formalism. The spin connection here is actually a function of the vielbien, and on introducing supersymmetry, also of the fermionic fields (which you can see when you find the EOM for $\omega$ in the first-order formalism).
The presence of the scalar curvature in the action is a feature of all supergravity theories. There are of course Lagrangian descriptions for things like type IIA SUGRA as well, obtained as the low-energy effective action of type IIA string theory:
$$
S = \frac{1}{2\kappa^2}\int\mathrm d^{10}x\sqrt{-g}e^{-2\phi}\left(R+4\partial_\mu\Phi\partial^\mu\Phi-\frac1{12}|H_3|^2\right)-\frac{1}{4\kappa^2}\int\mathrm d^{10}x\sqrt{-g}\left(|F_2|^2+|\tilde F_4|^2\right)-\frac{1}{4\kappa^2}\int\mathrm d^{10}x\sqrt{-g}\ B_2\wedge F_4\wedge F_4
$$
with the three terms corresponding respectively to the bosonic part, the RR part and the Chern-Simons part. You can remove the ugly $e^{-2\phi}$ in the bosonic action by going to the Einstein frame through a conformal rescaling, whereupon you pick up the familiar Einstein-Hilbert term $\int\mathrm d^n x \sqrt{-g} R$.
