Supersymmetry and First Order Equations for Extremal States: Monopoles, Hyperinstantons, Black-Holes and p-Branes

TL;DR

This paper reviews recent advances in first order equations for BPS extremal states like black holes and monopoles, highlighting the role of special geometry and supersymmetry in their description.

Contribution

It provides a comparative analysis of the derivation of BPS states using special geometry and supersymmetry, and discusses new approaches involving solvable Lie algebras.

Findings

First order equations effectively describe BPS extremal states.

Special geometry plays a crucial role in understanding these states.

Work on applying solvable Lie algebras to supergravity BPS states is ongoing.

Abstract

In this lecture I review recent results on the first order equations describing BPS extremal states, in particular N=2 extremal black-holes. The role of special geometry is emphasized also in the rigid theory and a comparison is drawn with the supersymmetric derivation of instantons and hyperinstantons in topological field theories. Work in progress on the application of solvable Lie algebras to the discussion of BPS states in maximally extended supergravities is outlined.