Tits-Satake projections of homogeneous special geometries

TL;DR

This paper classifies homogeneous special geometries in supergravity theories using Tits-Satake projections, organizing them into universality classes that relate to their dynamical solutions and symmetries.

Contribution

It extends Tits-Satake projections to all solvable Lie algebras in homogeneous special geometries, providing a new classification scheme into seven universality classes.

Findings

Most homogeneous special geometries fall into seven universality classes.

The classification commutes with the r- and c-map transformations.

Different classes are characterized by distinct paint groups.

Abstract

We organize the homogeneous special geometries, describing as well the couplings of D=6, 5, 4 and 3 supergravities with 8 supercharges, in a small number of universality classes. This relates manifolds on which similar types of dynamical solutions can exist. The mathematical ingredient is the Tits-Satake projection of real simple Lie algebras, which we extend to all solvable Lie algebras occurring in these homogeneous special geometries. Apart from some exotic cases all the other, 'very special', homogeneous manifolds can be grouped in seven universality classes. The organization of these classes, which capture the essential features of their basic dynamics, commutes with the r- and c-map. Different members are distinguished by different choices of the paint group, a notion discovered in the context of cosmic billiard dynamics of non maximally supersymmetric supergravities. We comment on the usefulness of this organization in universality classes both in relation with cosmic billiard dynamics and with configurations of branes and orbifolds defining special geometry backgrounds.