Supergeometry and Kac-Moody Algebras

Abstract

This exposition is meant to be a sequel to a previous review (to appear in the proceedings of the AMS-SIAM summer seminar on “Applications of Group Theory in Physics and Mathematical Physics” Chicago — July 1982). The main observation and conjectures date back to 1980–81 but further progress will require a concrete realization of affine and hyperbolic Kac-Moody groups and homogeneous spaces beyond the Lie algebraic construction and other formal or algebraic results. The first section summarizes the main features of supersymmetry needed for the construction of a gauge theory thereof. In the next the dimensional reduction of the bosonic part of eleven-dimensional supergravity is brought under the mathematical microscope. We review the impressive computations of General Relativists in section III and the emergence of the loop algebras of so(2,1) and su(2,1). Finally we discuss some links with the main approaches to completely integrable systems. This modest attempt to bring into the same framework such widely different subjects must be superficial, let us hope that the subsequent frustration will generate fruitful work.