The geometry of Calogero-Moser systems
Annales de l'Institut Fourier, Volume 55 (2005) no. 6, p. 2091-2116
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.
Nous donnons une construction géométrique de l’espace de phase du système de Calogero- Moser elliptique, pour des systèmes de racines arbitraires, comme espace de paires (fibrés, champs de Higgs) sur la r-ième puissance de la courbe elliptique, où r est le rang du sytème de racines. La structure de Poisson ainsi que les Hamiltoniens ont alors des constructions géométriques naturelles. Nous exhibons aussi une dualité surprenante entre les variétés spectrales du système de Calogero-Moser associé à un système de racines, et les variétés Lagrangiennes correspondant au système de racines dual. Enfin, nous montrons comment, pour le système A n , notre construction se réduit à une construcion connue.
Classification:  70H06,  14D21
Keywords: Integrable systems, classical mechanics, Calogero-Moser systems, Higgs pairs
     author = {Hurtubise, Jacques and Nevins, Thomas},
     title = {The geometry of Calogero-Moser systems},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {6},
     year = {2005},
     pages = {2091-2116},
     doi = {10.5802/aif.2153},
     zbl = {02230069},
     mrnumber = {2187947},
     language = {en},
     url = {}
Hurtubise, Jacques; Nevins, Thomas. The geometry of Calogero-Moser systems. Annales de l'Institut Fourier, Volume 55 (2005) no. 6, pp. 2091-2116. doi : 10.5802/aif.2153.

[AMM] H. Airault; H. Mckean; J. Moser Comm. Pure Appl. Math. Tome 30 (1977) no. 1, pp. 95-148 | Article | MR 649926 | Zbl 0338.35024

[BCS] A. J. Bordner; E. Corrigan; R. Sasaki Generalized Calogero-Moser models and universal Lax pair operators, Progr. Theoret. Phys., Tome 102 (1999) no. 3, pp. 499-529 | Article | MR 1729871

[BN] D. Ben-Zvi; T. Nevins From solitons to many-body systems (math.AG/0310490,

[Bo] F. Bottacin Symplectic geometry on moduli spaces of stable pairs, Ann. Sci. École Norm. Sup. (4), Tome 28 (1995) no. 4, pp. 391-433 | Numdam | MR 1334607 | Zbl 0864.14004

[Ca] F. Calogero Solution of the one-dimensional n-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys., Tome 12 (1971), pp. 419-436 | Article | MR 280103 | Zbl 1002.70558

[CG] K. Costello; I. Grojnowski Hilbert schemes, Hecke algebras and the Calogero-Sutherland system (math.AG/0310189,

[dHP] E. D'Hoker; D.H. Phong Calogero-Moser Lax pairs with spectral parameter for general Lie algebras (Nuclear Phys. B) Tome 530 (1998), pp. 537-610 | Zbl 0953.37020

[Do] R. Donagi Seiberg-Witten integrable systems, Amer. Math. Soc., Providence, RI, Proc. Sympos. Pure Math., Tome 62, Part 2 (1997) | MR 1492533 | Zbl 0896.58057

[EG] P. Etingof; V. Ginzburg Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math., Tome 147 (2002) no. 2, pp. 243-348 | Article | MR 1881922 | Zbl 1061.16032

[FMW] R. Friedman; J. W. Morgan; E. Witten Principal G-bundles over elliptic curves, Math. Res. Lett., Tome 5-1 (1998) no. 2, pp. 97-118 | MR 1618343 | Zbl 0937.14019

[HuMa] J. Hurtubise; E. Markman Surfaces and the Sklyanin bracket, Commun. Math. Phys., Tome 230 (2002), pp. 485-502 | Article | MR 1937654 | Zbl 1041.37034

[KhS] S. P. Khastgir; R. Sasaki Liouville integrability of classical Calogero-Moser models, Phys. Lett. A, Tome 279-3 (2001) no. 4, pp. 189-193 | MR 1815684 | Zbl 0972.81216

[Kr] I. M. Krichever Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles, Funct. Anal. Appl., Tome 14 (1980), pp. 282-290 | Zbl 0473.35071

[Lo] E. Looijenga Root systems and elliptic curves, Inv. Math., Tome 38 (1976), pp. 17-32 | Article | MR 466134 | Zbl 0358.17016

[Ma] E. Markman Spectral curves and integrable systems, Compositio Math., Tome 93 (1994), pp. 255-290 | Numdam | MR 1300764 | Zbl 0824.14013

[Mo] J. Moser Three integrable Hamiltonian systems connected with isospectral deformations, Advances in Math., Tome 16 (1975), pp. 197-220 | Article | MR 375869 | Zbl 0303.34019

[OP] M. A. Olshanetsky; A. M. Perelomov Completely integrable Hamiltonian systems connected with semisimple Lie algebras, Inventiones Math., Tome 37 (1976), pp. 93-108 | Article | MR 426053 | Zbl 0342.58017

[Su] B. Sutherland Exact results for a quantum many-body problem in one-dimension. II, Phys. Rev., Tome A5 (1972), pp. 1372-1376

[Wi] G. Wilson Collisions of Calogero-Moser particles and an adelic Grassmannian, Inventiones Math., Tome 133 (1998), pp. 1-41 | Article | MR 1626461 | Zbl 0906.35089