We study the elliptic and Ruijsenaars–Schneider models which are elliptic generalization of systems given in previous paper by the present authors [Chen et al., J. Math. Phys. 41, 8132 (2000)]. The Lax pairs for these models are constructed by Hamiltonian reduction technology. We show that the spectral curves can be parametrized by the involutive integrals of motion for these models. Taking nonrelativistic limit and scaling limit, we verify that they lead to the systems corresponding to Calogero–Moser and Toda types.
REFERENCES
1.
K.
Hasegawa
, “Ruijsenaars’ commuting difference operators as commuting transfer matrices
,” Commun. Math. Phys.
187
, 289
(1997
).2.
F.W.
Nijhoff
, V.B.
Kuznetsov
, E.K.
Sklyanin
, and O.
Ragnisco
, “Dynamical -matrix for the elliptic Ruijsenaars–Schneider system
,” J. Phys. A
29
, L333
(1996
).3.
A.
Gorsky
and A.
Marshakov
, “Towards effective topological gauge theories on spectral curves
,” Phys. Lett. B
375
, 127
(1996
).4.
5.
H.W.
Braden
, A.
Marshakov
, A.
Mironov
, and A.
Morozov
, “The Ruijsenaars–Schneider model in the context of Seiberg–Witten theory
,” hep-th/9902205;H.W.
Braden
, A.
Marshakov
, A.
Mironov
, and A.
Morozov
, Nucl. Phys. B
558
, 371
(1999
).6.
S.N.M.
Ruijsenaars
, “Complete integrability of relativistic Calogero–Moser systems and elliptic function identities
,” Commun. Math. Phys.
110
, 191
(1987
).7.
S.N.M.
Ruijsenaars
, “Action-angle maps and scattering theory for some finite-dimensional integrable systems
,” Commun. Math. Phys.
115
, 127
(1988
).8.
M.A.
Olshanetsky
and A.M.
Perelomov
, “Classical integrable finite-dimensional systems related to Lie algebras
,” Phys. Rep.
71
, 314
(1981
);A.M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras (Birkhäuser, Boston, 1990).
9.
VI.
Inozemtsev
, “Lax representation with spectral parameter on a torus for integrable particle systems
,” Lett. Math. Phys.
17
, 11
(1989
).10.
E.
D’Hoker
and D.H.
Phong
, “Calogero–Moser Lax pairs with spectral parameter for general Lie algebras
,” hep-th/9804124;11.
A.J.
Bordner
, E.
Corrigan
, and R.
Sasaki
, “Calogero–Moser Models. I. A new formulation
,” hep-th/9805106;A.J.
Bordner
, E.
Corrigan
, and R.
Sasaki
, Prog. Theor. Phys.
100
, 1107
(1998
).12.
A.J.
Bordner
, R.
Sasaki
, and K.
Takasaki
, “Calogero–Moser Models. II. Symmetries and foldings
,” hep-th/9809068;A.J.
Bordner
, R.
Sasaki
, and K.
Takasaki
, Prog. Theor. Phys.
101
, 487
(1999
).13.
A.J.
Bordner
and R.
Sasaki
, “Calogero–Moser Models III: Elliptic potentials and twisting
,” hep-th/9812232;14.
A.J.
Bordner
, E.
Corrigan
, and R.
Sasaki
, “Generalized Calogero–Moser models and universal Lax pair operators
,” hep-th/9905011;A.J.
Bordner
, E.
Corrigan
, and R.
Sasaki
, Prog. Theor. Phys.
102
, 499
(1999
).15.
16.
A.J.
Bordner
, N.S.
Manton
, and R.
Sasaki
, “Calogero–Moser models. V. Supersymmetry and quantum Lax pair
,” hep-th/9910033;A.J.
Bordner
, N.S.
Manton
, and R.
Sasaki
, Prog. Theor. Phys.
103
, 463
(2000
).17.
S.P.
Khastgir
, A.J.
Pocklington
, and R.
Sasaki
, “Quantum Calogero–Moser models: Integrability for all root systems
,” hep-th/0005277.18.
M.A.
Olshanetsky
and A.M.
Perelomov
, “Quantum integrable systems related to Lie algebras
,” Phys. Rep.
94
, 313
(1983
).19.
M.
Bruschi
and F.
Calogero
, “The Lax pair representation for an integrable class of relativistic dynamical systems
,” Commun. Math. Phys.
109
, 481
(1987
).20.
I.
Krichever
and A.
Zabrodin
, “Spin generalization of the Ruijsenaars–Schneider model, the non-Abelian 2D Toda chain, and representations of the Sklyanin algebra
,” Usp. Mat. Nauk
50
, 3
(1995
).21.
Y.B.
Suris
, “Why are the rational and hyperbolic Ruijsenaars–Schneider hierarchies governed by the same -operators as the Calogero–Moser ones?
,” hep-th/9602160.22.
Y.B.
Suris
, “Elliptic Ruijsenaars–Schneider and Calogero–Moser hierarchies are governed by the same-matrix
,” solv-int/9603011;23.
G.E.
Arutyunov
, S.A.
Frolov
, and P.B.
Medvedev
, “Elliptic Ruijsenaars–Schneider model from the cotangent bundle over the two-dimensional current group
,” hep-th/9608013;G.E.
Arutyunov
, S.A.
Frolov
, and P.B.
Medvedev
, J. Math. Phys.
38
, 5682
(1997
).24.
Y.
Komori
and K.
Hikami
, “Conserved operators of the generalized elliptic Ruijsenaars models
,” J. Math. Phys.
39
, 6175
(1998
).25.
Y.
Komori
, “Theta functions associated with the affine root systems and the elliptic Ruijsenaars operators
,” math.QA/9910003.26.
J.F.
van Diejen
, “Integrability of difference Calogero–Moser systems
,” J. Math. Phys.
35
, 2983
(1994
).27.
J.F.
van Diejen
, “Commuting difference operators with polynomial eigenfunctions
,” Compositio. Math.
95
, 183
(1995
).28.
K.
Hasegawa
, T.
Ikeda
, and T.
Kikuchi
, “Commuting difference operators arising from the elliptic -face model
,” J. Math. Phys.
40
, 4549
(1999
).29.
K.
Chen
, B.Y.
Hou
, and W.-L.
Yang
, “The Lax pair for-type Ruijsenaars–Schneider model
,” hep-th/0004006;30.
K.
Chen
, B.Y.
Hou
, and W.-L.
Yang
, “Integrability of the and Ruijsenaars–Schneider models
,” hep-th/0006004;31.
J.
Avan
and G.
Rollet
, “ Ruijsenaars–Schneider models: -matrix structure and Hamiltonians
,” hep-th/0008174.32.
Y.
Ohta
, “Instanton correction of prepotential in Ruijsenaars model associated with SU(2) Seiberg–Witten theory
,” hep-th/9909196;33.
J.
Avan
, “Classical dynamical -matrices for Calogero–Moser systems and their generalizations
,” q-alg/9706024.34.
V.B.
Kuznetsov
, F.W.
Nijhoff
, and E.K.
Sklyanin
, “Separation of variables for the Ruijsenaars system
,” Commun. Math. Phys.
189
, 855
(1997
).35.
S.N.M.
Ruijsenaars
and H.
Schneider
, “A new class of integrable systems and its relation to solitons
,” Ann. Phys. (Leipzig)
170
, 370
(1986
).36.
Paul A.M. Dirac, Lectures on Quantum Physics (Yeshiva University Press, New York, 1964).
37.
38.
39.
E.
D’Hoker
and D.H.
Phong
, “Lectures on supersymmetric Yang–Mills theory and integrable systems
,” hep-th/9912271.40.
A. Marshakov, Seiberg–Witten Theory and Integrable Systems (World Scientific, Singapore, 1998).
41.
E.
D’Hoker
and D.H.
Phong
, “Spectral curves for Super-Yang–Mills with adjoint hypermultiplet for general Lie algebras
,” hep-th/9804126;42.
E.
D’Hoker
and D.H.
Phong
, “Lax pairs and spectral curves for Calogero–Moser and spin Calogero–Moser systems
,” hep-th/9903002.43.
V. I.
Inozemtsev
, “The finite Toda lattices
,” Commun. Math. Phys.
121
, 628
(1989
).44.
E.
D’Hoker
and D.H.
Phong
, “Calogero–Moser and Toda systems for twisted and untwisted affine Lie Algebras
,” hep-th/9804125;45.
S.P.
Khastgir
, R.
Sasaki
, and K.
Takasaki
, “Calogero–Moser Models. IV. Limits to Toda theory
,” hep-th/9907102;S.P.
Khastgir
, R.
Sasaki
, and K.
Takasaki
, Prog. Theor. Phys.
102
, 749
(1999
).46.
I.M.
Krichever
, “Elliptic solutions of the Kadomtsev–Petviashvili equation and integrable systems of particles
,” Funct. Anal. Appl.
14
, 282
(1980
).47.
48.
K.
Takasaki
, “Whitham deformation of Seiberg–Witten curves for classical gauge groups
,” hep-th/9901120;
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