Five Dimensional Minimal Supergravities and Four Dimensional Complex Geometries Available to Purchase

We discuss the relation between solutions admitting Killing spinors of minimal supergravities in five dimensions and four dimensional complex geometries. In the ungauged case (vanishing cosmological constant $Λ = 0$⁠) the solutions are determined in terms of a hyper‐Kähler base space; in the gauged case $(Λ<0)$ the complex geometry is Kähler; in the de Sitter case $(Λ>0)$ the complex geometry is hyper‐Kähler with torsion (HKT). In the latter case some details of the derivation are given. The method for constructing explicit solutions is discussed in each case.