𝒩 = 3 Nonlinear Multiplet and Supergravity

We propose an 𝒩 = 3 nonlinear multiplet coupled to conformal supergravity and use it to formulate the equations of motion for 𝒩 = 3 Poincaré supergravity. These equations, which are naturally described in a new curved supergeometry with structure group SL(2, ℂ), imply that the 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 𝒩 = 3 Einstein superspace, is described in terms of two dimension-1/2 superfields: (i) the super-Weyl spinor Wα; and (ii) a spinor isospinor χ_α^i .