Gravity, supergravities and integrable systems

Abstract

Around 1968 three wonderful concepts emerged in different places and in seemingly unrelated domains of mathematical physics. They are the Kac-Moody algebras (among them the “affine” Kac-Moody algebras are related to current algebras and to gauge groups over one-dimensional “space-times”), the method of inverse scattering (for nonlinear partial differential equations in two-dimensional space-times), and finally the dual string model which is a two-dimensional field theory describing extended particles moving in a space-time of dimension 26 (10 or 2 if one dresses the string with internal degrees of freedom). In the last two years it was realized that gravity and supergravities provide a three-legged bridge between them and this revived hopes (at least with the author) of breaking the 2-dimensionality constraint for the integrability of interesting nonlinear problems. We shall not here discuss the Yang-Mills self-duality equations for lack of space ; they effectively are reduced to two-dimensions by considering the anti-self-dual null 2-planes. After reviewing the known connections between the 3 concepts listed above, we shall present the table of internal Lie symmetries of the Poincaré (super)- gravities in various numbers of dimensions. Finally, we shall see that a Kac-Moody group (affine type I) plays important roles as a) transformation group of solutions, b) parameter space where fields take their values, c) phase-space.

Based on an invited talk given at the Istanbul Conf. on Group Theoretical Methods in Physics, Aug. 1982.

Laboratoire Propre du CNRS, associé à l'Ecole Normale Supérieure et à l'Université de Paris-Sud.