Supergeometry and Kac-Moody Algebras
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.
KeywordsGauge Theory Dimensional Reduction Vertex Operator Gauge Invariance Supergravity Theory
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- B. Julia, Kac-Moody Symmetry of Gravitation and Supergravity Theories, in Chicago 1982-AMS-SIAM Proceedings (to appear) (1984).Google Scholar
- B. Julia, to appear in the Proceedings of the Conference on Group Theoretical Methods in Physics, College Park (1984) (and work in progress).Google Scholar
- B. Julia, Application of supergravity to gravitation theory, in “Unified Field Theories of more than 4 dimensions” ed. V. de Sabbata World Scientific 1983.Google Scholar
- See for example M. Jimbo and T. Miwa: Solitons and infinite dimensional Lie algebras (March 1983 R.I.M.S. preprint).Google Scholar
- B. Julia, Proceedings of the Johns Hopkins Workshop on Particle Theory (May 81).Google Scholar
- B. Julia, Physics Reports, in preparation.Google Scholar
- G. Segal and G. Wilson, to be published by I.H.E.S.: Loop groups and equations of KdV type.Google Scholar