Abstract
This review article consists of two parts. In the first part we use the formalism of (exceptional) generalized geometry to derive the scalar field space of SU(2) × SU(2)‐structure compactifications. We show that in contrast to SU(3) × SU(3) structures, there is no dynamical SU(2) × SU(2) structure interpolating between an SU(2) structure and an identity structure. Furthermore, we derive the scalar manifold of the low‐energy effective action for consistent Kaluza‐Klein truncations as expected from 𝒩 = 4 supergravity. In the second part we then determine the general conditions for the existence of stable Minkowski and AdS 𝒩=1 vacua in spontaneously broken gauged 𝒩 = 2 supergravities and construct the general solution under the assumption that two appropriate commuting isometries exist in the hypermultiplet sector. Furthermore, we derive the low‐energy effective action below the scale of partial supersymmetry breaking and show that it satisfies the constraints of 𝒩 = 1 supergravity. We then apply the discussion to special quaternionic‐Kähler geometries which appear in the low‐energy limit of SU(3) × SU(3)‐structure compactifications and construct Killing vectors with the right properties. Finally we discuss the string theory realizations for these solutions.