$\mathcal{N}=3$ nonlinear multiplet and supergravity

Abstract

We propose an $\mathcal{N}=3$ nonlinear multiplet coupled to conformal supergravity and use it to formulate the equations of motion for $\mathcal{N}=3$ Poincaré 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 Poincaré 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 $W_{\hat{I}}\pm$; and (ii) a spinor isospinor $\ddot{I}{‡}_{\hat{I}}{\pm }^i$.

37 pages; V2: revised version, comments and reference added