CSO$_c$ superpotentials

TL;DR

This paper provides explicit superpotentials for certain four-dimensional supergravity theories derived from maximal supergravities, aiding the study of holographic RG flows, black holes, and related gravitational phenomena.

Contribution

It introduces manageable superpotentials for $ extrm{CSO}(p,q,r)_{c}$ supergravities, simplifying their analysis and applications in holography and black hole physics.

Findings

Superpotentials for $ extrm{CSO}(p,q,r)_{c}$ supergravities are constructed.

AdS$_4$ solutions with various supersymmetries are identified.

Connections to non-geometric and geometric string backgrounds are discussed.

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.