## The geometry of Calogero-Moser systems.(English)Zbl 1135.37023

Summary: We give a geometric construction of the phase space of the elliptic Calogero-Moser system for arbitrary root systems, as a space of Weyl invariant pairs (bundles, Higgs fields) on the $$r$$-th power of the elliptic curve, where $$r$$ is the rank of the root system. The Poisson structure and the Hamiltonians of the integrable system are given natural constructions. We also exhibit a curious duality between the spectral varieties for the system associated to a root system, and the Lagrangian varieties for the integrable system associated to the dual root system. Finally, the construction is shown to reduce to an existing one for the $$A_n$$ root system.

### MSC:

 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) 70G55 Algebraic geometry methods for problems in mechanics 70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics 81R12 Groups and algebras in quantum theory and relations with integrable systems
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### References:

 [1] Comm. Pure Appl. Math., 30, 1, 95-148, (1977) · Zbl 0338.35024 [2] Generalized Calogero-Moser models and universal Lax pair operators, Progr. Theoret. Phys., 102, 3, 499-529, (1999) [3] From solitons to many-body systems · Zbl 1153.37031 [4] Symplectic geometry on moduli spaces of stable pairs, Ann. Sci. École Norm. Sup. (4), 28, 4, 391-433, (1995) · Zbl 0864.14004 [5] Solution of the one-dimensional n-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys., 12, 419-436, (1971) · Zbl 1002.70558 [6] Hilbert schemes, Hecke algebras and the Calogero-Sutherland system [7] Calogero-Moser Lax pairs with spectral parameter for general Lie algebras, 530, 537-610, (1998) · Zbl 0953.37020 [8] Seiberg-Witten integrable systems, 62, Part 2, (1997), Amer. Math. Soc., Providence, RI · Zbl 0896.58057 [9] Symplectic reflection algebras, Calogero-Moser space, and deformed harish-chandra homomorphism, Invent. Math., 147, 2, 243-348, (2002) · Zbl 1061.16032 [10] Principal $$G$$-bundles over elliptic curves, Math. Res. Lett., 5-1, 2, 97-118, (1998) · Zbl 0937.14019 [11] Surfaces and the Sklyanin bracket, Commun. Math. Phys., 230, 485-502, (2002) · Zbl 1041.37034 [12] Liouville integrability of classical Calogero-Moser models, Phys. Lett. A, 279-3, 4, 189-193, (2001) · Zbl 0972.81216 [13] Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles, Funct. Anal. Appl., 14, 282-290, (1980) · Zbl 0473.35071 [14] Root systems and elliptic curves, Inv. Math., 38, 17-32, (1976) · Zbl 0358.17016 [15] Spectral curves and integrable systems, Compositio Math., 93, 255-290, (1994) · Zbl 0824.14013 [16] Three integrable Hamiltonian systems connected with isospectral deformations, Advances in Math., 16, 197-220, (1975) · Zbl 0303.34019 [17] Completely integrable Hamiltonian systems connected with semisimple Lie algebras, Inventiones Math., 37, 93-108, (1976) · Zbl 0342.58017 [18] Exact results for a quantum many-body problem in one-dimension. II, Phys. Rev., A5, 1372-1376, (1972) · Zbl 0277.34049 [19] Collisions of Calogero-Moser particles and an adelic Grassmannian, Inventiones Math., 133, 1-41, (1998) · Zbl 0906.35089
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