Extremal Black Holes and Nilpotent Orbits

Abstract

Einstein’s equations in general relativity are a set of highly non-linear differential equations. During the 1980’s, Breitenlohner and Maison developed techniques to study stationary solutions to them by exploiting hidden symmetries revealed after dimensional reduction. These methods are applicable in general when seeking solutions allowing for one or more Killing vectors. When reducing a gravity theory down to three dimensions the field content can be dualized into a gravity theory coupled to a non-linear sigma model on a symmetric space G/H. This formulation is manifestly invariant under the Lie groups G and H of global, respectively local, transformations which can be used to generate new solutions from known seed solutions. More recent developments, motivated by supersymmetric string theory, has focused on solution classification through the nilpotent orbits of G as these correspond to certain black hole solutions (so called BPS solutions). This has so far been done for symmetry groups of finite dimensions. This thesis provides a background to the current attempts to generalize this classification in terms of nilpotent orbits to the infinite dimensional affine Kac-Moody algebras, where it is physically expected but not yet understood. These algebras arise from the hidden infinite dimensional symmetries revealed when reducing down to two dimensions and are thus relevant for black hole solutions with two commuting Killing vectors. The thesis covers the basics of dimensional reduction with the solution-generating techniques, nilpotent orbits and their classification, affine Kac-Moody Algebras and includes a Mathematica-package developed to study conjugation in the affine Lie algebras sl+n and g+2 . It aims at providing a pedagogical introduction and thus bridging the gap between master students and current research.