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

Understand

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.