= Minimal four-dimensional supergravity
{title2=$\mathcal N=1,\quad(e_\mu{}^a,\psi_\mu)$}
Minimal four-dimensional <supergravity> couples a graviton to one Majorana <gravitino>. In the normalization $\delta e_\mu{}^a=2\kappa\bar\epsilon\gamma^a\psi_\mu$, $\delta\psi_\mu=\nabla_\mu\epsilon/\kappa$, 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 $\gamma^{\mu\nu\rho}\nabla_\nu\nabla_\rho\epsilon=G^{\mu\nu}\gamma_\nu\epsilon/2$ cancels the Einstein-Hilbert variation against the Rarita-Schwinger variation. The omitted cubic-fermion variations require the quartic-fermion completion.
Back to article page
