Only flat spacetime is full BPS in four dimensional $$\mathcal{N}=3$$ and $$\mathcal{N}=4$$ supergravity

A bstract We investigate the fully supersymmetric solutions in a class of $$\mathcal{N}=3$$ and $$\mathcal{N}=4$$ higher derivative Poincaré supergravity theories. These class of theories are obtained within the framework of conformal supergravity using the standard Weyl multiplet and contains a set of terms related to the Weyl square term by supersymmetry. We work in the superconformal formalism and show that flat spacetime is the only fully supersymmetric solution in these theories. However, in $$\mathcal{N}=2$$ Poincaré supergravity theories that falls into the same class, two fully supersymmetric stationary solutions exist: Bertotti-Robinson geometry ( AdS 2 × S 2 ) and flat spacetime, which are known in the literature. We discuss the reason behind the richer vacua in $$\mathcal{N}=2$$ supergravity compared to its $$\mathcal{N}=3$$ and $$\mathcal{N}=4$$ cousins.