 Minimal four-dimensional supergravity

Minimal four-dimensional supergravity couples a graviton to one Majorana gravitino. In the normalization $\delta e_{\mu }{}^a=2\kappa \overline{ϵ}{\gamma }^a{\psi }_{\mu }$, $\delta {\psi }_{\mu }={\nabla }_{\mu }ϵ/\kappa$, its action through quadratic fermion order is $\int e[R/(2{\kappa }^2)-2{\bar{\psi }}_{\mu }{\gamma }^{\mu \nu \rho }{\nabla }_{\nu }{\psi }_{\rho }]$. Rescaling the gravitino and parameter by two gives the usual canonical kinetic coefficient $-1/2$. The curvature identity ${\gamma }^{\mu \nu \rho }{\nabla }_{\nu }{\nabla }_{\rho }ϵ=G^{\mu \nu }{\gamma }_{\nu }ϵ/2$ cancels the Einstein-Hilbert variation against the Rarita-Schwinger variation. The omitted cubic-fermion variations require the quartic-fermion completion.