The geometry of Calogero-Moser systems

Jacques Hurtubise[1]; Thomas Nevins

  • [1] McGill University, department of mathematics, Montréal QC H3A 2K6 (Canada), Université de Montréal, centre de recherches mathématiques, QC H3P 3J7 (Canada), University of Massachusetts, department of mathematics, Amherst (USA), University of Illinois at Urbana-Champaign, department of mathematics, Urbana IL 61801 (USA)

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 6, page 2091-2116
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Hurtubise, Jacques, and Nevins, Thomas. "The geometry of Calogero-Moser systems." Annales de l’institut Fourier 55.6 (2005): 2091-2116. <http://eudml.org/doc/116245>.

@article{Hurtubise2005,
abstract = {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.},
affiliation = {McGill University, department of mathematics, Montréal QC H3A 2K6 (Canada), Université de Montréal, centre de recherches mathématiques, QC H3P 3J7 (Canada), University of Massachusetts, department of mathematics, Amherst (USA), University of Illinois at Urbana-Champaign, department of mathematics, Urbana IL 61801 (USA)},
author = {Hurtubise, Jacques, Nevins, Thomas},
journal = {Annales de l’institut Fourier},
keywords = {Integrable systems; classical mechanics; Calogero-Moser systems; Higgs pairs; integrable systems},
language = {eng},
number = {6},
pages = {2091-2116},
publisher = {Association des Annales de l'Institut Fourier},
title = {The geometry of Calogero-Moser systems},
url = {http://eudml.org/doc/116245},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Hurtubise, Jacques
AU - Nevins, Thomas
TI - The geometry of Calogero-Moser systems
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 6
SP - 2091
EP - 2116
AB - 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.
LA - eng
KW - Integrable systems; classical mechanics; Calogero-Moser systems; Higgs pairs; integrable systems
UR - http://eudml.org/doc/116245
ER -

References

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  13. I. M. Krichever, Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles, Funct. Anal. Appl. 14 (1980), 282-290 Zbl0473.35071
  14. E. Looijenga, Root systems and elliptic curves, Inv. Math. 38 (1976), 17-32 Zbl0358.17016MR466134
  15. E. Markman, Spectral curves and integrable systems, Compositio Math. 93 (1994), 255-290 Zbl0824.14013MR1300764
  16. J. Moser, Three integrable Hamiltonian systems connected with isospectral deformations, Advances in Math. 16 (1975), 197-220 Zbl0303.34019MR375869
  17. M. A. Olshanetsky, A. M. Perelomov, Completely integrable Hamiltonian systems connected with semisimple Lie algebras, Inventiones Math. 37 (1976), 93-108 Zbl0342.58017MR426053
  18. B. Sutherland, Exact results for a quantum many-body problem in one-dimension. II, Phys. Rev. A5 (1972), 1372-1376 
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