The geometry of Calogero-Moser systems
[Géométrie des systèmes de Calogero-Moser]
Hurtubise, Jacques1 ; Nevins, Thomas
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, Tome 55 (2005) no. 6, pp. 2091-2116.

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.

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.

DOI : 10.5802/aif.2153
Classification : 70H06, 14D21
Keywords: Integrable systems, classical mechanics, Calogero-Moser systems, Higgs pairs
Mot clés : systémes intégrables, mécanique classique, système de Calogero-Moser, champs de Higgs
Hurtubise, Jacques 1 ; Nevins, Thomas 

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) 
@article{AIF_2005__55_6_2091_0,
     author = {Hurtubise, Jacques and Nevins, Thomas},
     title = {The geometry of {Calogero-Moser} systems},
     journal = {Annales de l'Institut Fourier},
     pages = {2091--2116},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {55},
     number = {6},
     year = {2005},
     doi = {10.5802/aif.2153},
     mrnumber = {2187947},
     zbl = {02230069},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2153/}
}
TY  - JOUR
AU  - Hurtubise, Jacques
AU  - Nevins, Thomas
TI  - The geometry of Calogero-Moser systems
JO  - Annales de l'Institut Fourier
PY  - 2005
SP  - 2091
EP  - 2116
VL  - 55
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2153/
DO  - 10.5802/aif.2153
LA  - en
ID  - AIF_2005__55_6_2091_0
ER  - 
%0 Journal Article
%A Hurtubise, Jacques
%A Nevins, Thomas
%T The geometry of Calogero-Moser systems
%J Annales de l'Institut Fourier
%D 2005
%P 2091-2116
%V 55
%N 6
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2153/
%R 10.5802/aif.2153
%G en
%F AIF_2005__55_6_2091_0
Hurtubise, Jacques; Nevins, Thomas. The geometry of Calogero-Moser systems. Annales de l'Institut Fourier, Tome 55 (2005) no. 6, pp. 2091-2116. doi : 10.5802/aif.2153. http://www.numdam.org/articles/10.5802/aif.2153/

[AMM] H. Airault; H. McKean; J. Moser Comm. Pure Appl. Math., 30 (1977) no. 1, pp. 95-148 | DOI | MR | Zbl

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

[BN] D. Ben-Zvi; T. Nevins From solitons to many-body systems (math.AG/0310490, http://arxiv.org/abs/math.AG/0310490)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cité par Sources :