Geometry of all supersymmetric four-dimensional ${\cal N}=1$ supergravity backgrounds

TL;DR

This paper classifies all supersymmetric solutions in four-dimensional ${ m N}=1$ supergravity, revealing geometric structures and special classes of backgrounds with varying degrees of supersymmetry, including maximally supersymmetric and domain wall solutions.

Contribution

It provides a comprehensive solution to the Killing spinor equations for all ${ m N}=1$ backgrounds, identifying geometric features and classifying solutions with different supersymmetry levels.

Findings

${ m N}=1$ backgrounds admit a null, integrable Killing vector.

${ m N}=2$ backgrounds split into pp-waves and cohomogeneity one solutions.

Maximally supersymmetric backgrounds are locally isometric to $R^{3,1}$ or $AdS_4$.

Abstract

We solve the Killing spinor equations of ${\cal N}=1$ supergravity, with four supercharges, coupled to any number of vector and scalar multiplets in all cases. We find that backgrounds with N=1 supersymmetry admit a null, integrable, Killing vector field. There are two classes of N=2 backgrounds. The spacetime in the first class admits a parallel null vector field and so it is a pp-wave. The spacetime of the other class admits three Killing vector fields, and a vector field that commutes with the three Killing directions. These backgrounds are of cohomogeneity one with homogenous sections either $\bR^{2,1}$ or $AdS_3$ and have an interpretation as domain walls. The N=3 backgrounds are locally maximally supersymmetric. There are N=3 backgrounds which arise as discrete identifications of maximally supersymmetric ones. The maximally supersymmetric backgrounds are locally isometric to either $\bR^{3,1}$ or $AdS_4$.