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.

DOI : https://doi.org/10.5802/aif.2153
Classification:  70H06,  14D21
Keywords: Integrable systems, classical mechanics, Calogero-Moser systems, Higgs pairs
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     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 = {http://www.numdam.org/item/AIF_2005__55_6_2091_0}
}
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. http://www.numdam.org/item/AIF_2005__55_6_2091_0/

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