Gravitino-induced torsion ($T_{\mu \nu }{}^a=\frac{{\kappa }^2}{2}{\bar{\psi }}_{\mu }{\gamma }^a{\psi }_{\nu }$)

With canonical Rarita-Schwinger field normalization, one convention gives the displayed torsion tensor. Its contorsion tensor is $K_{\mu ab}={\kappa }^2({\bar{\psi }}_{\mu }{\gamma }_a{\psi }_b-{\bar{\psi }}_{\mu }{\gamma }_b{\psi }_a+{\bar{\psi }}_a{\gamma }_{\mu }{\psi }_b)/4$. The algebraic spin connection equation determines $\tilde{\omega }=\omega (e)+K$; substituting it in the action produces four-fermion terms. This is an instance of Einstein-Cartan theory and is efficiently varied using the 1.5-order formalism. A rescaled gravitino changes the displayed coefficient.