We propose an ${ \mathcal N}=3$ nonlinear multiplet coupled to conformal supergravity and use it to formulate the equations of motion for ${ \mathcal N} = 3$ Poincar\'e supergravity. These equations, which are naturally described in a new curved supergeometry with structure group $\mathsf{ SL}(2,\mathbb{ C})$, imply that the ${ \mathcal N} = 3$ super-Bach tensor vanishes, and thus every solution of Poincar\'e supergravity is a solution of conformal supergravity. The aforementioned superspace formulation, which we refer to as $\mathcal N=3$ Einstein superspace, is described in terms of two dimension-$1/2$ superfields: (i) the super-Weyl spinor $W_\alpha$; and (ii) a spinor isospinor $\chi_\alpha^i$.
Categories: hep-th