Supergravity coupling of an Abelian vector multiplet ($(A_{\mu },\lambda )$)

An Abelian vector multiplet contains a Maxwell field and a Majorana gaugino. With the doubled gravitino normalization, add $e[-F^2/4-\bar{\lambda }/\nabla \lambda /2-(\kappa /2){\bar{\psi }}_{\mu }{F}_{ab}{\gamma }^{ab}{\gamma }^{\mu }\lambda ]$. The leading transformations are $\delta A_{\mu }=\overline{ϵ}{\gamma }_{\mu }\lambda$ and $\delta \lambda =-F_{ab}{\gamma }^{ab}ϵ/2$. Promoting the rigid parameter to a function produces the current $(F_{ab}/2){\gamma }^{ab}{\gamma }^{\mu }\lambda$ multiplying ${\nabla }_{\mu }\overline{ϵ}$; the gravitino-current term cancels it. The Clifford identity $(F_{ab}{\gamma }^{ab}){\gamma }^{\mu }(F_{cd}{\gamma }^{cd})=8T_{\mathrm{EM}}^{\mu \nu }{\gamma }_{\nu }$ cancels the Maxwell stress variation. Supercovariant field strengths and quartic fermions complete the transformations beyond this order.