Five Dimensional Minimal Supergravities and Four Dimensional Complex Geometries

TL;DR

This paper explores the relationship between five-dimensional minimal supergravity solutions with Killing spinors and four-dimensional complex geometries, classifying solutions based on the cosmological constant and providing methods for explicit construction.

Contribution

It establishes a comprehensive link between supergravity solutions and complex geometries across different cosmological constant regimes, including explicit construction methods.

Findings

In the ungauged case, solutions relate to hyper-Kahler base spaces.

Gauged case solutions correspond to Kahler geometries.

de Sitter case involves hyper-Kahler with torsion (HKT) geometries.

Abstract

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 \Lambda=0) the solutions are determined in terms of a hyper-Kahler base space; in the gauged case (\Lambda<0) the complex geometry is Kahler; in the de Sitter case (\Lambda>0) the complex geometry is hyper-Kahler 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.