Cosmic Billiards with Painted Walls in Non-Maximal Supergravities: a worked out example

TL;DR

This paper extends the derivation of smooth cosmic billiard solutions to non-maximal supergravities using Tits Satake projections and introduces the concept of the G-paint group, providing a unified scheme for these theories.

Contribution

It develops a new framework for non-maximal supergravities involving Tits Satake projections and paint groups, with detailed example analysis.

Findings

Derived smooth cosmic billiard solutions for non-maximal supergravities.

Identified the G-paint group as a key symmetry preserved through dimensional reduction.

Established a universal scheme applicable to all homogeneous scalar manifold supergravities.

Abstract

The derivation of smooth cosmic billiard solutions through the compensator method is extended to non maximal supergravities. A new key feature is the non-maximal split nature of the scalar coset manifold. To deal with this, one needs the theory of Tits Satake projections leading to maximal split projected algebras. Interesting exact solutions that display several smooth bounces can thus be derived. From the analysis of the Tits Satake projection emerges a regular scheme for all non maximal supergravities and a challenging so far unobserved structure, that of the paint group G-paint. This latter is preserved through dimensional reduction and provides a powerful tool to codify solutions. It appears that the dynamical walls on which the cosmic ball bounces come actually in painted copies rotated into each other by G-paint. The effective cosmic dynamics is that dictated by the maximal split Tits Satake manifold plus paint. We work out in details the example provided by N=6,D=4 supergravity, whose scalar manifold is the special Kahlerian SO*(12)}/SU(6)xU(1). In D=3 it maps to the quaternionic E_7(-5)/ SO(12) x SO(3). From this example we extract a scheme that holds for all supergravities with homogeneous scalar manifolds and that we plan to generalize to generic special geometries. We also comment on the merging of the Tits-Satake projection with the affine Kac--Moody extensions originating in dimensional reduction to D=2 and D=1.