# Calogero-Moser-Krichever system

*Calogero–Moser–Sutherland–Krichever system*

A finite-dimensional Hamiltonian system which is algebraically completely integrable (cf. Completely-integrable differential equation). It admits several generalizations, which will be indicated below. During the enormous activity on mathematical aspects of integrable systems and soliton equations, starting in the late 1960{}s, some prototypes that represent the main features have emerged. For example, the Korteweg–de Vries equation and the non-linear Schrödinger equation for inverse scattering [a8], rank- $ 2 $ perturbations for algebraic complete integrability [a19], [a20], the Kadomtsev–Petviashvili equation (cf. Soliton) for Grassmannians and Schur functions [a30], and the modified Korteweg–de Vries equation for representation theory [a6], [a15].

As a prototype, the Calogero–Moser–Krichever system possesses all the standard features of algebraic complete integrability: 1) a Lax pair, spectral curve, and Abelian integrals; 2) a connection with soliton equations; 3) an $ r $- matrix; 4) a geodesic-motion interpretation. However, it has also more unique features: 5) a link with elliptic functions; 6) a more sophisticated interpretation of the Lax pair, giving a link with the Hitchin system, which is a more powerful source of algebraic complete integrability over moduli spaces of vector bundles; 7) the bispectral property.

A brief illustration of these terms follows.

1) The system has a Hamiltonian function

$$ H = { \frac{1}{2} } \sum _ {i = 1 } ^ { n } p _ {i} ^ {2} + \sum _ {i < j } v ( q _ {i} - q _ {j} ) , $$

where $ ( q _ {1} \dots q _ {n} ; p _ {1} \dots p _ {n} ) $ are position/momentum variables (all quantities are complex numbers) and $ v $ is an even function. If $ L $ and $ M $ are the $ ( n \times n ) $- matrices with entries

$$ L _ {jk } = p _ {j} \delta _ {jk } + \sqrt {- 1 } ( 1 - \delta _ {jk } ) u ( q _ {j} - q _ {k} ) , $$

$$ M _ {jk } = \delta _ {jk } \sum _ {l \neq j } w ( q _ {j} - q _ {l} ) - ( 1 - \delta _ {jk } ) z ( q _ {j} - q _ {k} ) , $$

where $ u $ is odd and $ w $, $ z $ are even, then the equation $ \sqrt {- 1 } {\dot{L} } = [ M,L ] $ implies $ z = - {\dot{u} } $, $ v = u ^ {2} + \textrm{ const } $, as well as a functional equation for $ u $ and $ w $ with solutions

$$ u = { \frac{1}{q} } , { \frac{1}{ \sin q } } , { \frac{1}{ { \mathop{\rm sn} } q } } . $$

These three cases are referred to as rational (Calogero–Moser system), trigonometric (Calogero–Sutherland system) and elliptic (Calogero–Moser–Krichever system), respectively; the first two can be viewed as limits of the third as one or both periods of the Weierstrass function $ { \mathop{\rm sn} } $( cf. Weierstrass elliptic functions) tend to infinity. In each case, the invariants $ F _ {k} = { \mathop{\rm tr} } ( L ^ {k} ) /k $, $ k = 1 \dots n $, are (generically) functionally independent and in involution (cf. Integrals in involution), so that the system is completely integrable. This was the first example [a21] of a Lax pair with as parameter a function over a curve of genus $ > 0 $( generalizations are still (1996) quite rare, cf. [a10] for hyperelliptic parameters).

2) By interpolating an eigenvector of $ L $ into a Baker–Akhiezer function, it was shown in [a21] that the solutions correspond to elliptic (in $ x $) solutions of the Kadomtsev–Petviashvili equation $ u _ {yy } = ( u _ {t} + u _ {xxx } - 6uu _ {x} ) _ {x} $. The first breakthrough in this respect was made in [a1], concerning Korteweg–de Vries solutions and Lamé equations (cf. also Lamé equation); moduli spaces of the corresponding algebro-geometric configurations (tangential covers) were described in [a31] and their density was detected in [a3].

3) Several models of the $ r $- matrix, both dynamic and non-dynamic, recently (1990{}s) became available (for a most complete set of references see [a11]); however a finite-dimensional geodesic motion interpretation 4) has only been achieved for the rational and trigonometric cases [a17].

5) S.P. Novikov posed the following question: Which algebro-geometric Kadomtsev–Petviashvili solutions can be expressed in terms of elliptic functions (cf. also Elliptic function)? These non-trivially overlap with elliptic solitons and a state-of-the-art report can be found in [a22].

6) is the deep theory that ensues if the moving data is viewed as a rank- $ n $ bundle over the elliptic curve, cf. [a4], [a9].

Lastly, 7) is an extra property of the Baker function in the rational case, defined by F.A. Grünbaum [a7] in connection with computerized tomography (roughly stated, the Baker function is both an $ x $- and a $ z $- eigenfunction for a pair of operators with eigenvalues that are functions of $ z $, $ x $, respectively); its manifestation for the matrix $ L $ above (cf. [a1]) is investigated in [a16].

A final word about generalizations: [a27] adapts the geodesic problem 4) to other groups and metrics; [a28] defines a relativistic Calogero–Moser–Krichever system; [a24] and [a25] provide discretized versions of the Calogero–Moser–Krichever and Ruijsenaars system, respectively; an Euler Calogero–Moser–Krichever system is related to a multi-component Kadomtsev–Petviashvili equation in [a23]; and the most recent application of the Calogero–Moser–Krichever system is that conjecturally it provides moduli for solutions to the Seiberg–Witten equations, [a5].

#### References

[a1] | H. Airault, H.P. McKean, J. Moser, "Rational and elliptic solutions of the Korteweg–de Vries equation and a related many-body problem" Commun. Pure Appl. Math. , 30 (1977) pp. 95–148 |

[a2] | E.D. Belokolos, V.Z. Enol'skii, "Algebraically integrable nonlinear equations and Humbert surfaces, plasma theory and nonlinear and turbulent processes in physics" , 1–2 , World Sci. (1988) pp. 20–57 |

[a3] | E. Colombo, G.P. Pirola, E. Previato, "Density of elliptic solitons" J. Reine Angew. Math. , 451 (1994) pp. 161–169 |

[a4] | R. Donagi, E. Markman, "Spectral covers, algebraically completely integrable Hamiltonian systems, and moduli of bundles" , Integrable Systems and Quantum Groups , Lecture Notes in Mathematics , 1620 , Springer (1996) |

[a5] | R. Donagi, E. Witten, "Supersymmetric Yang–Mills systems and integrable systems" hep-th/9510101 (1995) |

[a6] | V.G. Drinfeld, V.V. Sokolov, "Lie algebras and equations of Korteweg–de Vries type" J. Soviet Math. , 30 (1985) pp. 1975–2005 |

[a7] | J.J. Duistermaat, F.A. Grünbaum, "Differential equations in the spectral parameter" Comm. Math. Phys. , 103 (1986) pp. 177–240 |

[a8] | L.D. Faddeev, L.A. Takhtajan, "Hamiltonian methods in the theory of solitons" , Springer (1987) |

[a9] | G. Faltings, "Stable -bundles and projective connections" J. Algebraic Geom. , 2 (1993) pp. 507–568 |

[a10] | Yu.N. Fedorov, "Integrable systems, Lax representations, and confocal quadrics" Amer. Math. Soc. Transl. Ser. 2 , 168 (1995) pp. 173–199 |

[a11] | K. Hasegawa, "Ruijsenaars' commuting difference operators as commuting transfer matrices" q-alg/9512029 (1995) |

[a12] | N. Hitchin, "Stable bundles and integrable systems" Duke Math. J. , 54 (1987) pp. 91–114 |

[a13] | E.L. Ince, "Further investigations into the periodic Lamé function" Proc. Roy. Soc. Edinburgh , 60 (1940) pp. 83–99 |

[a14] | A.R. Its, V.Z. Enol'skii, "Dynamics of the Calogero–Moser system and the reduction of hyperelliptic integrals to elliptic integrals" Funct. Anal. Appl. , 20 (1986) pp. 62–64 |

[a15] | V.G. Kac, J.W. van de Leur, "The -component KP hierarchy and representation theory" , Important Developments in Soliton Theory , Springer (1993) pp. 302–343 |

[a16] | A. Kasman, "Bispectral KP solutions and linearization of Calogero–Moser particle systems" Comm. Math. Phys. , 172 (1995) pp. 427–448 |

[a17] | D. Kazhdan, B. Kostant, S. Sternberg, "Hamiltonian group actions and dynamical systems of Calogero type" Comm. Pure Appl. Math. , 31 (1978) pp. 481–507 |

[a18] | I.M. Krichever, "Rational solutions of the Kadomtsev–Petviashvili equation and integrable systems of particles on a line" Funct. Anal. Appl. , 12 (1978) pp. 59–61 |

[a19] | M. Adler, P. van Moerbeke, "Completely integrable systems, Euclidean Lie algebras, and curves" Adv. in Math. , 38 (1980) pp. 267–317 |

[a20] | M. Adler, P. van Moerbeke, "Linearization of Hamiltonian systems, Jacobi varieties and representation theory" Adv. in Math. , 38 (1980) pp. 318–379 |

[a21] | I.M. Krichever, "Elliptic solutions of the KP equation and integrable systems of particles" Funct. Anal. Appl. , 14 (1980) pp. 282–290 |

[a22] | I.M. (ed.) Krichever, "Special issue on elliptic solitons, dedicated to the memory of J.-L. Verdier" Acta Applic. Math. , 36 : 1–2 (1994) |

[a23] | I.M. Krichever, O. Babelon, E. Billey, M. Talon, "Spin generalizations of the Calogero–Moser system and the matrix KP equation" Amer. Math. Soc. Transl. Ser. 2 , 170 (1995) pp. 83–119 |

[a24] | F.W. Nijhoff, G.-D. Pang, "A time-discretized version of the Calogero–Moser model" Phys. Lett. A , 191 (1994) pp. 101–107 |

[a25] | F.W. Nijhoff, O. Ragnisco, V.B. Kuznetsov, "Integrable time-discretisation of the Ruijsenaars–Schneider model" Comm. Math. Phys. , 176 (1996) pp. 681–700 |

[a26] | S.P. Novikov, "A periodic problem for the KdV equation" Funct. Anal. Appl. , 8 (1974) pp. 236–246 |

[a27] | M.A. Olshanetsky, A.M. Perelomov, "Classical integrable finite-dimensional systems related to Lie algebras" Phys. Rep. , 71 (1981) pp. 313–400 |

[a28] | S.N.M. Ruijsenaars, "Complete integrability of relativistic Calogero–Moser systems and elliptic function identities" Comm. Math. Phys. , 20 (1987) pp. 191–213 |

[a29] | G. Segal, G. Wilson, "Loop groups and equations of KdV type" IHES Publ. Math. , 61 (1985) pp. 5–65 |

[a30] | M. Sato, Y. Sato, "Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold" , Nonlinear Partial Differential Equations in Applied Science , Math. Stud. , 81 , North-Holland (1983) pp. 259–271 |

[a31] | A. Treibich, J.-L. Verdier, "Solitons elliptiques" , The Grothendieck Festschrift , III , Birkhäuser (1990) pp. 437–480 |

[a32] | A.P. Veselov, "Rational solutions of the KP equation and hamiltonian systems" Russian Math. Surveys , 35 : 1 (1980) pp. 239–240 |

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Calogero-Moser-Krichever system.

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