U Duality, Solvable Lie Algebras and Extremal Black-Holes

TL;DR

This paper reviews how Solvable Lie Algebras describe scalar fields in supergravities and their relation to U-duality, focusing on extremal black holes and BPS states, with ongoing extensions to higher supergravities.

Contribution

It introduces the use of Solvable Lie Algebras for scalar sectors in supergravities and explores their role in constructing extremal black holes and BPS states, extending to higher dimensions.

Findings

Solvable Lie Algebras efficiently describe scalar fields in supergravities.

Construction of BPS extremal black holes using differential equations.

Extension of methods to supergravities in dimensions 4 to 11 is underway.

Abstract

In this lecture I review recent results on the use of Solvable Lie Algebras as an efficient description of the scalar field sector of supergravities in relation with their non perturbative structure encoded in the U-duality group. I also review recent results on the construction of BPS saturated states as solution of the first differential equations following from imposing preservation of a fraction of the original supersymmetries. In particular I discuss N=2 extremal black holes that are approximated by a Bertotti Robinson metric near their horizon. The extension of this construction to maximally extended supergravities in all dimensions from 4 to 11 is work in progress where the use of the Solvable Lie algebra approach promises to be of decisive usefulness.