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.
Partially supported by the National Science Foundation through the Mathematical Sciences Research Institute.
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References
B. Julia, Kac-Moody Symmetry of Gravitation and Supergravity Theories, in Chicago 1982-AMS-SIAM Proceedings (to appear) (1984).
See for example J.H. Schwarz, Phys. Reports 89 no. 3 (1982).
S.N. Gupta, Phys. Rev. 96 (1954) 1693
S.N. Gupta , Proc. Phys. Soc. (London) A65 (1952) 608.
S. Deser, Gen. Rel. and Grav. 1 (1970) 9.
B. Julia, to appear in the Proceedings of the Conference on Group Theoretical Methods in Physics, College Park (1984) (and work in progress).
J. M. Souriau, Structure des systemes dynamiques (1970) Dunod-Paris.
Helmholtz J. reine und ang. Mathematik 100 (1887) 137.
P. van Nieuwenhiuzen, Phys. Reports 68 (1981) 189.
E. Cremmer, B. Julia and J. Scherk, Phys. Lett. 76 B (1978) 409.
B. de Wit and D. Freedman, Nucl. Phys. B130 (1977) 105.
E. Cremmer and B. Julia, Phys. Lett. 80B (1978) 48
E. Cremmer and B. Julia, Nucl. Phys. B159 (1979) 141.
B. Julia, Application of supergravity to gravitation theory, in “Unified Field Theories of more than 4 dimensions” ed. V. de Sabbata World Scientific 1983.
E. Cremmer, J. Scherk and J.H. Schwarz, Phys. Lett. 84B (1979) 83.
B. Kostant, Adv. in Math. 20 (1976) 179.
See for example M. Jimbo and T. Miwa: Solitons and infinite dimensional Lie algebras (March 1983 R.I.M.S. preprint).
W. Kinnersley and D.M. Chitre, J. Math. Phys. 18 (1977) 1538.
R. Geroch, J. Math. Phys. 13 (1972) 394.
B. Julia, Proceedings of the Johns Hopkins Workshop on Particle Theory (May 81).
B. Julia, Physics Reports, in preparation.
G. Segal and G. Wilson, to be published by I.H.E.S.: Loop groups and equations of KdV type.
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Julia, B. (1985). Supergeometry and Kac-Moody Algebras. In: Lepowsky, J., Mandelstam, S., Singer, I.M. (eds) Vertex Operators in Mathematics and Physics. Mathematical Sciences Research Institute Publications, vol 3. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9550-8_19
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