Supergravity

Definition

Let $M$ be a $D$-dimensional spin manifold. A supergravity theory on $M$ is a classical field theory with the following properties.

1. Its fields include a vielbein $e^a{}_\mu$, which defines a Lorentzian metric $g_{\mu\nu} = \eta_{ab} e^a{}_\mu e^b{}_\nu$, and $N \ge 1$ spin-$3/2$ fields $\psi^I_\mu$ (gravitini), and possibly further bosonic and fermionic fields.
2. Its action is invariant under diffeomorphisms of $M$ and local Lorentz transformations of the vielbein.
3. Its action is invariant under local supersymmetry transformations, whose parameters are spinor fields $\epsilon^I(x)$, under which $\delta e^a{}_\mu$ is proportional to $\bar\epsilon^I \gamma^a \psi^I_\mu$ and $\delta \psi^I_\mu = D_\mu \epsilon^I + \cdots$.