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

TL;DR

The work analyzes how BPS extremal states in $D=4$ with $\mathcal{N}=2$ supersymmetry are governed by first-order equations linked to preserved supersymmetries and central charges. It connects rigid and local $\mathcal{N}=2$ theories through Special Kähler geometry, showing Bogomolny-type decompositions of energy and explicit BPS conditions for monopoles, hyperinstanton-like configurations, and extremal black holes, including the horizon-area formula $Area_H = \frac{1}{4\pi} |Z(p,q)|^2$. It also discusses topological twisting to obtain gauged hyperinstanton equations and the interpretation of BPS $p$-branes, and outlines a program to use solvable Lie algebras to obtain a unified, solvable treatment of first-order BPS equations for maximal supergravities. This framework highlights how geometry, topology, and algebraic structures underpin exact, minimal-energy configurations in supersymmetric theories with potential implications for dualities and non-perturbative dynamics.

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.