CSO$_c$ superpotentials

Abstract

Motivated by their application to holographic RG flows and hairy black holes in Einstein-scalar systems, we present a collection of superpotentials driving the dynamics of $\mathcal{N}=2$ and $\mathcal{N}=1$ four-dimensional supergravities. These theories arise as consistent truncations of the electric/magnetic families of $\textrm{CSO}(p,q,r)_{c}$ maximal supergravities, with ${p+q+r=8}$, discovered by Dall'Agata et al. The $\mathcal{N}=2$ and $\mathcal{N}=1$ truncations describe $\textrm{SU}(3)$ and $\mathbb{Z}_{2} \times \textrm{SO}(3)$ invariant sectors, respectively, and contain AdS$_4$ solutions preserving $\mathcal{N}=1,2,3,4$ supersymmetry within the full theories, as well as various gauge symmetries. Realisations in terms of non-geometric type IIB as well as geometric massive type IIA backgrounds are also discussed. The aim of this note is to provide easy to handle superpotentials that facilitate the study of gravitational and gauge aspects of the $\textrm{CSO}(p,q,r)_{c}$ maximal supergravities avoiding the technicalities required in their construction.