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

TL;DR

This paper develops an ${ m N}=3$ nonlinear multiplet coupled to conformal supergravity, formulating equations of motion for ${ m N}=3$ Poincaré supergravity within a new curved supergeometry, revealing that solutions satisfy super-Bach tensor conditions.

Contribution

It introduces an ${ m N}=3$ nonlinear multiplet and a novel superspace formulation for ${ m N}=3$ supergravity, connecting conformal and Poincaré supergravity solutions.

Findings

Super-Bach tensor vanishes for solutions of ${ m N}=3$ Poincaré supergravity.

Formulation in terms of super-Weyl spinor and isospinor superfields.

New curved supergeometry with $ m SL(2, ext{C})$ structure group.

Abstract

We propose an ${\mathcal N}=3$ nonlinear multiplet coupled to conformal supergravity and use it to formulate the equations of motion for ${\mathcal N} = 3$ 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 $\chi_\alpha^i$.