Gravity, supergravities and integrable systems

  • Bernard Julia
Session III — Gravity, Supergravity, Supersymmetry
Part of the Lecture Notes in Physics book series (LNP, volume 180)


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.


Dimensional Reduction Dynkin Diagram Coset Space Loop Algebra Fuchsian System 
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Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Bernard Julia
    • 1
  1. 1.Laboratoire de Physique Théorique de l'Ecole Normale SupérieureParis cedex 05France

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