Hidden symmetries of integrable systems in Yang-Mills theory and Kähler geometry
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1990-1991), Exposé no. 8, 15 p.
@article{SEDP_1990-1991____A8_0,
author = {Takasaki, Kanehisa},
title = {Hidden symmetries of integrable systems in {Yang-Mills} theory and {K\"ahler} geometry},
journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
note = {talk:8},
publisher = {Ecole Polytechnique, Centre de Math\'ematiques},
year = {1990-1991},
zbl = {0746.58039},
mrnumber = {1131581},
language = {en},
url = {http://www.numdam.org/item/SEDP_1990-1991____A8_0/}
}
TY  - JOUR
AU  - Takasaki, Kanehisa
TI  - Hidden symmetries of integrable systems in Yang-Mills theory and Kähler geometry
JO  - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"
N1  - talk:8
PY  - 1990-1991
DA  - 1990-1991///
PB  - Ecole Polytechnique, Centre de Mathématiques
UR  - http://www.numdam.org/item/SEDP_1990-1991____A8_0/
UR  - https://zbmath.org/?q=an%3A0746.58039
UR  - https://www.ams.org/mathscinet-getitem?mr=1131581
LA  - en
ID  - SEDP_1990-1991____A8_0
ER  - 
Takasaki, Kanehisa. Hidden symmetries of integrable systems in Yang-Mills theory and Kähler geometry. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1990-1991), Exposé no. 8, 15 p. http://www.numdam.org/item/SEDP_1990-1991____A8_0/

[Sa-Sa] Sato, M., Sato, Y., Soliton equations as dynamical systems in an infinite dimensional Grassmann manifold, in Nonlinear Partial Differential Equations in Applied Sciences, P.D. Lax, H. Fujita, G. Strang (eds.), North-Holland, 1982 | Zbl 0528.58020

[Se-Wi] Segal, G., and Wilson, G., Loop groups and equations of KdV type, Publ. IHES 61 (1985), 5-65. | Numdam | MR 783348 | Zbl 0592.35112

[Ta1] Takasaki, K., Geometry of universal Grassmannian manifold from algebraic point of view, Reviews in Math. Phys. 1 (1989), 1-46. | MR 1041529 | Zbl 0716.58001

[Ch] Chau, L.-L., Chiral fields, self-dual Yang-Mills fields as integrable systems, and the role of the Kac-Moody algebra, in Nonlinear phenomena, K.B. Wolf (ed.), Lecture Notes in Physics vol. 189, Splinger-Verlag, 1983. | MR 727859 | Zbl 0582.35095

[Bo] Boyer, C.P., The geometry of self-dual Einstein spaces, in Nonlinear Phenomena, K.B. Wolf (ed.), Lecture Notes in Physics vol. 189, Springer, 1983. | MR 727857

[Wal] Ward, R.S., On self-dual gauge fields, Phys. Lett. 61A (1977), 81-82. | MR 443823 | Zbl 0964.81519

[Pe] Penrose, R., Nonlinear gravitons and curved twistor theory, Gen. Rel. Grav. 7 (1976), 31-52. | MR 439004 | Zbl 0354.53025

[Be-Za] Belavin, A.A. and Zakharov, V.E., Yang-Mills equations as inverse scattering problem, Phys. Lett. 73B (1978), 53-57. | MR 484164

[Ch-Pr-Si] Chau, L.-L., Prasad, M.K. and Sinha, A., Some aspects of the linear system for self-dual Yang-Mills fields, Phys. Rev. D24 (1981), 1574-1580. | MR 628859

[Fo-Ho-Pa] Forgács, P., Horváth, Z., and Palla, L., Towards complete integrability of the self-dual equations, Phys. Rev. D23 (1981), 1876- . | MR 610412

[Po] Pohlmeyer, K., On the Lagrangian theory of anti-self-dual fields in four dimensional Euclidean space, Commun. Math. Phys. 72 (1980), 37-47. | MR 573816

[Wa2] Ward, R.S., Completely solvable gauge-field equations in dimension greater than four, Nucl. Phys. B236 (1984), 381-396. | MR 739812

[Co-Go-Ke] Corrigan, E., Goddard, P., and Kent, A., Some comments on the ADHM construction in 4k dimensions, Commun. Math. Phys. 100 (1985), 1-13. | MR 796159 | Zbl 0587.58059

[At-Hi-Dr-Ma] Atiyah, M.F., Hitchin, N.J., Drinfeld, V.G. and Manin, Yu. I., Construction of instantons, Phys. Lett. 65A (1978), 185-187; Atiyah, M.F., Geometry of gauge fields, Scuola Norm. Sup., Pisa, 1979. | MR 598562 | Zbl 0424.14004

[Ya] Yang, C.N., Condition of self-duality for SU(2) gauge fields on Euclidean four-dimensional space, Phys. Rev. Lett. 38 (1977), 1377-1379. | MR 475357

[P1] Plebanski, J.F., Some solutions of complex Einstein equations, J. Math. Phys. 16 (1975), 2395-2402. | MR 386609

[Hi-Ka-Li-Ro] Hitchin, N.J., Kahlhede, A., Lindström, U., and Roček, M., Hyperkähler metrics and supersymmetry, Commun. Math. Phys. 108 (1987), 535-589. | MR 877637 | Zbl 0612.53043

[Tw1] Lerner, D.E., and Sommers, P.D. (ed.), Complex manifold techniques in theoretial physics, Pitman, 1978. | Zbl 0407.00015

[Tw2] Hughston, L.P., and Ward, R.S. (ed.), Advances in twistor theory, Pitman, 1979. | MR 578487 | Zbl 0463.53039

[Tw3] Doebner, H.D., and Palev, T.D. (ed.), Twistor geometry and non-linear systems, Lecture Notes in Mathematics vol. 970, Springer-Verlag, 1982. | MR 699798 | Zbl 0488.00016

[Tw4] Ward, R.S., and Wells, R.O., Twistor geometry and field theory, Cambridge University Press, 1989. | MR 1054377 | Zbl 0714.53059

[Tw5] Mason, L.J., and hughston, L.P. (ed.), Further advances in twistor theory, Pitman, 1990.

[Tw6] Baily, T.N., and Boston, R.J. (ed.), Twistors in mathematics and physics, London Mathematical Society Lecture Note Series vol. 156, Cambridge University Press, 1990. | MR 1089905 | Zbl 0702.53003

[Ka] Kac, V.G., Infinite dimensional Lie algebras, Cambridge Univ. Press, 1985. | MR 823672 | Zbl 0574.17010

[Pr-Se] Pressley, A.N., and Segal, G.B., Loop groups and their representations, Oxford University Press, 1986. | MR 900587 | Zbl 0618.22011

[Da-Ji-Mi-Ka] Date, E., Jimbo, M., Kashiwara, M., Miwa, T., Transformation theory for soliton equations III-VI, J. Phys. Soc. Japan 50 (1982), 3806-3812, 3813-3818; Phyica 4D (1982), 343-365; Publ. RIMS., Kyoto Univ., 18 (1982), 1077-1110.

[Ue-Na] Ueno, K., Nakamura, Y., Transformation theory for anti-self-dual equations and the Riemann-Hilbert problem, Phys. Lett. 109B (1982), 273-278. | MR 647108

(Ch-Ge-WuJ Chau, L.-L., Ge, M.-L., Wu, Y.-S., Kac-Moody algebra in the self-dual Yang-Mills equation, Phys. Rev. D25 (1982), 1086-1094. | MR 646392

[Do] Dolan, L., A new symmetry group of real self-dual Yang-Mills theory, Phys. Lett. 113B (1982), 387-390. | MR 664383

[Bo-P1] Boyer, C.P., Plebanski, J.F., An infinite hierarchy of conservation laws and nonlinear superposition principles for self-dual Einstein spaces, J. Math. Phys. 26 (1985), 229-234. | MR 776490 | Zbl 0555.53044

[Ta2] Takasaki, K., A new approach to the self-dual Yang-Mills equations, Commun. Math. Phys. 94 (1984), 35-59; Hierarchy structure in integrable systems of gauge fields and underlying Lie algebras, Commun. Math. Phys. 127 (1990), 225-238. | MR 763761 | Zbl 0692.58044

[Ta3] Takasaki, K., An infinite number of hidden variables in hyper-Kähler metrics, J. Math. Phys. 30 (1989),1515-1521; Symmetries of hyper-Kähler (or Poisson gauge field) hierarchy, J. Math. Phys. 31 (1990), 1877-1888. | MR 1002256 | Zbl 0683.53017

[Le-Mu] Leznov, A.N., Mukhtarov, M.A., Deformation of algebras and solutions of self-duality equation, J. Math. Phys. 28 (1987), 2574-2578. | MR 913409 | Zbl 0663.35078

[Le-Sa] Leznov, A.N., Saveliev, V.M., Exactly and completely integrable nonlinear dynamical sysmtems, Acta Applicandae Mathematicae 16 (1989), 1-74. | MR 1011902 | Zbl 0683.58025

[Ta4] Takasaki, K., Integrable systems in gauge theory, Kähler geometry and super KP hierarchy - symmetries and algebraic point of view, talk at ICM-90, Kyoto, August, 1990; Kyoto University preprint RIMS-714, September, 1990. | MR 1159305

[Bi-F1-Sa] Biran, B., Floratos, E.G.F., and Savvidy, G.K., The self-dual closed membranes, Phys. Lett. 198B (1987), 329-332. | MR 918701

[F1-Le] Floratos, F.G., and Leontaris, G.K., Integrability of the self-dual membranes in (4+1) dimensions and the Toda lattice, Phys. Lett. 223B (1989), 153-156. | MR 1001109

[Gr-Tz] Grabowski, F., and Tze, C.-H., Generalized self-dual bosonic membranes, vector corss-products and analyticity in higher dimensions, Phys. Lett. 224B (1989), 259-264. | MR 1003764

[Za] Zaikov, R.P., Self-duality in the theory of the bosonic p-branes, Phys. Lett. 211B (1988), 281-284. | MR 958278

[Ar-Sa] Arakelyan, T.A., and Savvidy, G.K., Cocycles of area-preserving diffeomorphisms and anomalies in the theory of relativistic surfaces, Phys. Lett. 214B (1988), 350-356. | MR 969983

[Ba-Po-Se] Bars, I., Pope, C.N., and Sezgin, E., Central extensions of area preserving membrane algebras, Phys. Lett. 210B (1988), 85-91. | MR 959497

[F1-I1] Floratos, F.G., and Iliopoulos, A note on the classical symmetries of the closed bosonic membranes, J., Phys. Lett. 201B (1988), 237-240. | MR 926871

[Ho1] Hoppe, J., DiffAT2, and the curvature of some infinite dimensional manifolds, Phys. Lett. 215B (1988), 706-710. | MR 973588

[Ho2] Hoppe, J., Diffeomorphism groups, quantization, and SU(oo), Int. J. Mod. Phys. A 4 (19) (1989), 5235-5248. | MR 1020144 | Zbl 0707.17018

[Ba1] Bakas, I., The large-N limit of extended conformal symmetries, Phys. Lett. 228B (1989), 57-63. | MR 1015910

[Ba2] Bakas, I., The structure of the W∞ algebra, Maryland University preprint UMD-90-085, November, 1980.

[Oo-Va] Ooguri, H., and Vafa, C., Self-duality and N = 2 string magic, Univ. Chicago preprint, EFI-90-24, April 1990. | MR 1063040 | Zbl 1020.81792

[Ya-Ch] Yamagishi, K., and Chappline, F., Induced 4d self-dual quantum gravity:- W∞ algebraic approach -, Laurence Livermore National Laboratory preprint, April 1990. | Zbl 0736.17040

[Pa1] Park, Q-Han, Self-dual gravity as a large-N limit of the 2D non-linear sigma model, Phys. Lett. 238B (1990), 287-290. | MR 1050734

[Pa2] Park, Q-Han, Extended conformal symmetries in real heavens, Phys. Lett. 236B (1990), 429-432. | MR 1041737

[Wi] Witten, E., Surprises with topological field theorries, Advanced Study Institute preprint IASSNS-HEP-90/37, April, 1990.

[Bo-Fi] Boyer, C., and Finley, J.D., Killing vectors in self-dual, Euclidean Einstein spaces, J. Math. Phys. 23 (1982), 1126-1128. | MR 660020 | Zbl 0484.53051

[Ge-Da] Gegenberg, J.D., and Das, A., Stationary Riemannian space-times with self-dual curvature, Gen. Rel. Grav. 16 (1984), 817-829. | MR 759978 | Zbl 0545.53039

[Hi] Hitchin, N.J., Complex manifolds and Einstein's equations, in Twistor Geometry and Non-linear Systems, H.D. Doebner and T. Weber (eds.), Lecture Notes in Mathematics vol. 970, pp.2-42, Springer-Verlag 1982. | MR 699802 | Zbl 0507.53025

[Jo-To] Jones, P.E., and Tod, K.P., Minitwistor spaces and Einstein-Weyl spaces, Class. Quantum Grav. 2 (1985), 565-577. | MR 795102 | Zbl 0575.53042

[Wa3] Ward, R.S., Winstein-Weyl spaces and SU(∞) Toda fields, Class. Quantum Grav. 7 (1990). L95-L98. | Zbl 0687.53044

[LeBr] Lebrun, C., Explicit self-dual metrics on CP2 # ... # CP2, preprint, 1990.

[Sa-Ve] Saveliev, M.V., and Vershik, A.M., Continual analogues of contragredient Lie algebras, Commun. Math. Phys. 126 (1989), 367-378; New examples of continuum graded Lie algebras, Phys. Lett. 143A (1990), 121-128. | MR 1027502 | Zbl 0691.17012