# 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

## Access Full Article

top## Abstract

top## How to cite

topHurtubise, 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

top- H. Airault, H. McKean, J. Moser, Comm. Pure Appl. Math. 30 (1977), 95-148 Zbl0338.35024MR649926
- A. J. Bordner, E. Corrigan, R. Sasaki, Generalized Calogero-Moser models and universal Lax pair operators, Progr. Theoret. Phys. 102 (1999), 499-529 MR1729871
- D. Ben-Zvi, T. Nevins, From solitons to many-body systems Zbl1153.37031
- F. Bottacin, Symplectic geometry on moduli spaces of stable pairs, Ann. Sci. École Norm. Sup. (4) 28 (1995), 391-433 Zbl0864.14004MR1334607
- F. Calogero, Solution of the one-dimensional n-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971), 419-436 Zbl1002.70558MR280103
- K. Costello, I. Grojnowski, Hilbert schemes, Hecke algebras and the Calogero-Sutherland system
- E. d'Hoker, D.H. Phong, Calogero-Moser Lax pairs with spectral parameter for general Lie algebras, 530 (1998), 537-610 Zbl0953.37020
- R. Donagi, Seiberg-Witten integrable systems, 62, Part 2 (1997), Amer. Math. Soc., Providence, RI Zbl0896.58057MR1492533
- P. Etingof, V. Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243-348 Zbl1061.16032MR1881922
- R. Friedman, J. W. Morgan, E. Witten, Principal $G$-bundles over elliptic curves, Math. Res. Lett. 5-1 (1998), 97-118 Zbl0937.14019MR1618343
- J. Hurtubise, E. Markman, Surfaces and the Sklyanin bracket, Commun. Math. Phys. 230 (2002), 485-502 Zbl1041.37034MR1937654
- S. P. Khastgir, R. Sasaki, Liouville integrability of classical Calogero-Moser models, Phys. Lett. A 279-3 (2001), 189-193 Zbl0972.81216MR1815684
- I. M. Krichever, Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles, Funct. Anal. Appl. 14 (1980), 282-290 Zbl0473.35071
- E. Looijenga, Root systems and elliptic curves, Inv. Math. 38 (1976), 17-32 Zbl0358.17016MR466134
- E. Markman, Spectral curves and integrable systems, Compositio Math. 93 (1994), 255-290 Zbl0824.14013MR1300764
- J. Moser, Three integrable Hamiltonian systems connected with isospectral deformations, Advances in Math. 16 (1975), 197-220 Zbl0303.34019MR375869
- M. A. Olshanetsky, A. M. Perelomov, Completely integrable Hamiltonian systems connected with semisimple Lie algebras, Inventiones Math. 37 (1976), 93-108 Zbl0342.58017MR426053
- B. Sutherland, Exact results for a quantum many-body problem in one-dimension. II, Phys. Rev. A5 (1972), 1372-1376
- G. Wilson, Collisions of Calogero-Moser particles and an adelic Grassmannian, Inventiones Math. 133 (1998), 1-41 Zbl0906.35089MR1626461

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.