Exact solutions for Bianchi type cosmological metrics, Weyl orbits of E_{8(8)} subalgebras and p--branes

TL;DR

This paper develops a systematic method to construct and classify time-dependent supergravity solutions using U-duality symmetries, revealing connections to Bianchi cosmologies and providing new exact solutions.

Contribution

It introduces an algorithm for solving supergravity equations via subalgebra embeddings, studies their Weyl group orbits, and links solutions to cosmological models and space-brane configurations.

Findings

Identified a single Weyl orbit of A_r subalgebras for r<6.

Connected solutions to homogeneous Bianchi type 2A cosmologies.

Derived new exact solutions for Bianchi cosmologies with matter.

Abstract

In this paper we pursue further a programme initiated in a previous work and aimed at the construction, classification and property investigation of time dependent solutions of supergravity (superstring backgrounds) through a systematic exploitation of U-duality hidden symmetries. This is done by first reducing to D=3 where the bosonic part of the theory becomes a sigma model on E_{8(8)}/SO(16), solving the equations through an algorithm that produces general integrals for any chosen regular subalgebra G_r of E_{8(8)} and then oxiding back to D=10. Different oxidations and hence different physical interpretations of the same solutions are associated with different embeddings of G_r. We show how such embeddings constitute orbits under the Weyl group and we study the orbit space. This is relevant to associate candidate superstring cosmological backgrounds to space Dp-brane configurations that admit microscopic descriptions. In particular in this paper we show that there is just one Weyl orbit of A_r subalgebras for r < 6$. The orbit of the previously found A_2 solutions, together with space--brane representatives contains a pure metric representative that corresponds to homogeneous Bianchi type 2A cosmologies in D=4 based on the Heisenberg algebra. As a byproduct of our methods we obtain new exact solutions for such cosmologies with and without matter. We present a thorough investigation of their properties.