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.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
B.K. Harrison, J. Math. Phys. 9 (1968) 1744.
R. Geroch, J. Math. Phys. 12 (1971) 918.
R. Geroch, J. Math. Phys. 13 (1972) 394.
V. Kac, Funct. An. and Appl. 1 (1967) 82, and Math. USSR Izvestija 32 (1968) 1271; R. Moody, Bull. Am. Math. Soc. 73 (1967) 217, and J. of Algebra 10 (1968) 211.
J. Lepowski and R. Wilson, Comm. Math. Phys. 62 (1978) 43.
G. Segal, Comm. Math. Phys. 80 (1981) 301; I. Frenkel and V. Kac, Inventiones 62 (1980) 23.
G. Veneziano, Nuovo Cim. 57A (1968) 190.
W. Nahm, Nucl. Phys. B114 (1976) 174; J. Scherk, Rev. Mod. Phys. 47 (1975) 123.
F. Gliozzi, J. Scherk and D. Olive, Nucl. Phys. B122 (1977) 253.
I. Frenkel, J. Lepowski and A. Meurman, talk at the Chicago SIAM Workshop, July 1982.
E. Cremmer and B. Julia, Nucl. Phys. 8159 (1979) 141.
C.S. Gardner, J.M. Greene, M.D. Kruskal, R. M. Miura, Phys. Rev. Lett. 19 (1967) 1095.
B.A. Dubrovin, V.B. Matveev and S.P. Novikov, Russian Math. Surveys 31 (1976) 59.
For a review see H. Flaschka and A.C. Newell, Comm. Math. Phys. 76 (1980) 65.
V.E. Zakharov and A.B. Shabat, Funct. Anal. and Appl. 13 (1979) 13.
For a review of 2 dimensional problems see A.V. Mikhailov, CERN preprint TH.3194,(1981).
A.M. Polyakov, Phys. Lett. 103B (1981) 207.
J.L. Gervais and A. Neveu, Nucl. Phys. (to appear).
E. Date, M. Jimbo, M. Kashiwara and T. Miwa, RIMS 362 (July 1981); see also G. Segal and G.. Wilson, Oxford preprint (in preparation).
A.G. Reyman and M.A. Semenov-Tian-Shansky, Inventiones Mat. 63 (1981) 423; and V.G. Drinfeld and V.V. Sokolov, Doklady Acad. Nauk. USSR 258 (1981) 457.
M. Adler, Inventiones Mat. 50 (1979) 219.
B. Julia, in Superspace and Supergravity, ed. S. Hawking and M. Rocek, Cambridge 1981, p. 331 (C.U.P.).
B. Julia, “Kac-Moody Symmetry of Gravitation and Supergravity Theories“ (to be published by A.M.S. in Proc. Chicago Meeting, July 1982).
B. Julia, “Infinite Lie Algebras in Physics”, Proc. 5th Johns Hopkins Workshop on Particle Theory, Baltimore, May 1981, p. 23.
D. Maison, J. Math. Phys. 20 (1978) 871; V.A. Belinsky and V.E. Zakharov, Sov. Phys. JETP 48 (1978) 985 and 50 (1979) 1.
W. Kinnersley and D.M. Chitre, J. Math. Phys. 18 (1977) 1538.
I. Hauser and F.J. Ernst. See for example a review by the first author in the Proc. Coyococ 1980 Conference of this series, Lecture Notes in Physics 135, Springer.
B.C. Xanthopoulos, J. Math. Phys. 22 (1981) 1254.
E. Cremmer, see ref. 22).
M. Lüscher and K. Pohlmeyer, Nucl. Phys. B137 (1978) 46.
L. Dolan, Phys. Rev. Lett. 47 (1981) 1371.
Wu Yong-Shi, Nucl. Phys. B211 (1983) 160.
R.P. Zaikov, Dubna preprints E2-80-118, 197 and with B.L. Markowsky E2-80-654; L. Dolan and A. Roos, Phys. Rev. D22 (1980) 2018.
S. Deser and C. Teitelboim, Phys. Rev. D13 (1976) 1592.
K. Ueno and Y. Nakamura, Phys. Lett. 117B (1982) 208; C. Cosgrove, J. Math. Phys. 23 (1982) 615.
A.C. Davies, P.J. Houston, J.M. Leinaas and A.J. Macfarlane, CERN preprint TH 3372.
V.E. Zakharov, in Lecture Notes in Physics 153 (Springer) p. 190.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1983 Springer-Verlag
About this paper
Cite this paper
Julia, B. (1983). Gravity, supergravities and integrable systems. In: Serdaroğlu, M., Ínönü, E. (eds) Group Theoretical Methods in Physics. Lecture Notes in Physics, vol 180. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-12291-5_28
Download citation
DOI: https://doi.org/10.1007/3-540-12291-5_28
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-12291-3
Online ISBN: 978-3-540-39621-5
eBook Packages: Springer Book Archive