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Calogero-Moser Lax pairs with spectral parameter for general Lie algebras. (English) Zbl 0953.37020

Summary: We construct a Lax pair with spectral parameter for the elliptic Calogero-Moser Hamiltonian systems associated with each of the finite-dimensional Lie algebras, of the classical and of the exceptional type. When the spectral parameter equals one of the three half periods of the elliptic curve, our result for the classical Lie algebras reduces to one of the Lax pairs without spectral parameter that were known previously. These Calogero-Moser systems are invariant under the Weyl group of the associated untwisted affine Lie algebra. For non-simply laced Lie algebras, we introduce new integrable systems, naturally associated with twisted affine Lie algebras, and construct their Lax operators with spectral parameter (except in the case of \(G_2\)).

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
17B81 Applications of Lie (super)algebras to physics, etc.
81T60 Supersymmetric field theories in quantum mechanics
81R12 Groups and algebras in quantum theory and relations with integrable systems
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
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[1] Seiberg, N.; Witten, E.; Seiberg, N.; Witten, E., Monopoles, duality, and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. phys. B, Nucl. phys. B, 431, 494, (1994), hep-th/9410167 · Zbl 1020.81911
[2] Lerche, W.; Lerche, W., Introduction to Seiberg-Witten theory and its stringy origin, (), 83, and references therein · Zbl 0957.81701
[3] Gorski, A.; Krichever, I.M.; Marshakov, A.; Mironov, A.; Morozov, A.; Matone, M.; Nakatsu, T.; Takasaki, K.; Nakatsu, T.; Takasaki, K., Isomonodromic deformations and supersymmetric gauge theories, Phys. lett. B, Phys. lett. B, Mod. phys. lett. A, Int. J. mod. phys. A, 11, 5505-168, (1996), hep-th/9603069 · Zbl 0985.81756
[4] Donagi, R.; Witten, E., Supersymmetric Yang-Mills theory and integrable systems, Nucl. phys. B, 460, 299, (1996), hep-th/9510101 · Zbl 0996.37507
[5] Martinec, E.; Warner, N., Integrable systems and supersymmetric gauge theories, Nucl. phys. B, 459, 97, (1996), hep-th/9509161 · Zbl 0996.37506
[6] E. Martinec, Integrable structures in supersymmetric gauge and string theory, hep-th/9510204.
[7] Sonnenschein, J.; Theisen, S.; Yankielowicz, S., On the relation between the holomorphic prepotential and the quantum moduli in SUSY gauge theories, Phys. lett. B, 367, 145, (1996), hep-th/9510129
[8] T. Eguchi and S.K. Yang, Prepotentials of N = 2 supersymmetric gauge theories and soliton equations, hep-th/9510183;
[9] Itoyama, H.; Morozov, A., Prepotential and the Seiberg-Witten theory, Nucl. phys. B, 491, 529, (1997), hep-th/9512161 · Zbl 0982.32019
[10] Integrability and Seiberg-Witten theory, hep-th/9601168;
[11] I.M. Krichever and D.H. Phong, Symplectic forms in the theory of solitons, hep-th/9708170, to appear in Surveys in Differential Geometry, Vol. III. · Zbl 0931.35148
[12] D’Hoker, E.; Phong, D.H., Calogero-Moser systems in SU(N) Seiberg-Witten theory, Nucl. phys. B, 513, 405, (1998), hep-th/9709053 · Zbl 1052.81624
[13] R. Donagi, Seiberg-Witten integrable systems, alg-geom/9702042;
[14] D.S. Freed, Special Kähler manifolds, hep-th/9712042;
[15] R. Carroll, Prepotentials and Riemann surfaces, hep-th/9802130.
[16] Adler, M.; van Moerbeke, P.; Adler, M.; van Moerbeke, P.; Adler, M.; van Moerbeke, P., The Toda lattice, Dynkin diagrams, singularities and abelian varieties, Advances in math., Advances in math., Invent. math., 103, 223, (1991) · Zbl 0735.14031
[17] Calogero, F.; Moser, J., Integrable systems of non-linear evolution equations, (), 12, 419, (1971)
[18] Krichever, I.M., Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles, Funct. anal. appl., 14, 282, (1980) · Zbl 0473.35071
[19] Hitchin, N., Stable bundles and integrable systems, Duke math. J., 54, 91, (1987) · Zbl 0627.14024
[20] Olshanetsky, M.A.; Perelomov, A.M.; Perelomov, A.M.; Leznov, A.N.; Saveliev, M.V., Group theoretic methods for integration of non-linear dynamical systems, (), 71C, 313, (1992), Birkhäuser Boston, and references therein
[21] D’Hoker, E.; Phong, D.H., Calogero-Moser and Toda systems for twisted and untwisted affine Lie algebras, Nucl. phys. B, 530, 611, (1998), next article in this issue, hep-th/9804125 · Zbl 0953.37019
[22] E. D’Hoker and D.H. Phong, Spectral curves for super-Yang-Mills with adjoint hypermultiplet for general Lie algebras, hep-th/9804126.
[23] Inozemtsev, V.I., The finite Toda lattices, Comm. math. phys., 121, 629, (1989) · Zbl 0677.35086
[24] Inozemtsev, V.I., Lax representation with spectral parameter on a torus for integrable particle systems, Lett. math. phys., 17, 11, (1989) · Zbl 0679.70005
[25] Klemm, A.; Lerche, W.; Theisen, S., Non-perturbative actions of N = 2 supersymmetric gauge theories, Int. J. mod. phys. A, 11, 1929, (1996), hep-th/9505150 · Zbl 1044.81739
[26] D’Hoker, E.; Krichever, I.M.; Phong, D.H.; D’Hoker, E.; Krichever, I.M.; Phong, D.H.; D’Hoker, E.; Krichever, I.M.; Phong, D.H.; D’Hoker, E.; Phong, D.H., Strong coupling expansions in SU(N) Seiberg-Witten theory, Nucl. phys. B, Nucl. phys. B, Nucl. phys. B, Phys. lett. B, 397, 94, (1997), hep-th/9701151 · Zbl 0925.81380
[27] Kac, V., Infinite-dimensional Lie algebras, (1983), Birkhäuser Boston
[28] Goddard, P.; Olive, D., Kac-Moody and Virasoro algebras in relation to quantum physics, Int. J. mod. phys. A, 1, 303, (1986) · Zbl 0631.17012
[29] Krichever, I.M., Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles, Funct. anal. appl., 14, 282, (1980) · Zbl 0473.35071
[30] ()
[31] McKay, W.G.; Patera, J.; Rand, D.W., ()
[32] Kachru, S.; Vafa, C.; Bershadsky, M.; Intrilligator, K.; Kachru, S.; Morrison, D.R.; Sadov, V.; Vafa, C.; Katz, S.; Klemm, A.; Vafa, C.; Katz, S.; Mayr, P.; Vafa, C., Mirror symmetry and exact solutions of 4D N = 2 gauge theories, Nucl. phys. B, Nucl. phys. B, Nucl. phys. B, Adv. theor. math. phys., 1, 53, (1998), hep-th/9706110
[33] Witten, E., Solutions of four-dimensional field theories via M-theory, Nucl. phys. B, 500, 3, (1997), hep-th/9703166 · Zbl 0934.81066
[34] Hanany, A.; Witten, E., Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics, Nucl. phys. B, 492, 152, (1997) · Zbl 0996.58509
[35] Brandhuber, A.; Sonnenschein, J.; Theisen, S.; Yankielowicz, S., M-theory and Seiberg-Witten curves: orthogonal and symplectic groups, Nucl. phys. B, 504, 175, (1997), hep-th/9705232 · Zbl 0934.81050
[36] K. Landsteiner, E. Lopez, New curves from branes, hep-th/9708118; · Zbl 0925.81097
[37] A.M. Uranga, Towards mass deformed N = 4 SO(N) and Sp(K) gauge theories from brane configurations, hep-th/9803054; · Zbl 1031.81655
[38] T. Yokono, Orientifold four plane in brane configurations and N = 4 USp(2N) and SO(2N) theory, hep-th/9803123. · Zbl 1047.81563
[39] Gorskii, A., Branes and integrability in the N = 2 SUSY YM theory, Int. J. mod. phys. A, 12, 1243, (1997), hep-th/9612238
[40] A. Gorskii, S. Gukov and A. Mironov, SUSY field theories, integrable systems and their stringy/brane origin, hep-th/9710239;
[41] S.A. Cherkis and A. Kapustin, Singular monopoles and supersymmetric gauge theories in three dimensions, hep-th/9711145.
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