Abstract
The Hamiltonian structure of the monodromy preserving deformation equations of Jimboet al [JMMS] is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to the dual spaces of two different loop algebras. The nonautonomous Hamiltonian systems generating the deformations are obtained by pulling back spectral invariants on Poisson subspaces consisting of elements that are rational in the loop parameter and identifying the deformation parameters with those determining the moment maps. This construction is shown to lead to “dual” pairs of matrix differential operators whose monodromy is preserved under the same family of deformations. As illustrative examples, involving discrete and continuous reductions, a higher rank generalization of the Hamiltonian equations governing the correlation functions for an impenetrable Bose gas is obtained, as well as dual pairs of isomonodromy representations for the equations of the Painlevé transcendentsP V and VI .
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References
[AHH1] Adams, M.R., Harnad, J., Hurtubise, J.: Dual Moment Maps to Loop Algebras. Lett. Math. Phys.20, 294–308 (1990)
[AHH2] Adams, M.R., Harnad, J., Hurtubise, J.: Darboux Coordinates and Liouville-Arnold Integration in Loop Algebras. Commun. Math. Phys.155, 385–413 (1993)
[AHP] Adams, M.R., Harnad, J., Previato, E.: Isospectral Hamiltonian Flows in Finite and Infinite Dimensions I. Generalised Moser Systems and Moment Maps into Loop Algebras. Commun. Math. Phys.117, 451–500 (1988)
[HHM] Harnad, J., Hurtubise, J., Marsden, J.E.: Reduction of Hamiltonian Systems with Discrete Symmetries. Preprint CRM (1992)
[HTW] Harnad, J., Tracy, C., Widom, H.: Hamiltonian Structure of Equations Appearing in Random Matrices. In: Low Dimensional Topology and Quantum Field Theory.
[HW] Harnad, J., Wisse, M.-A.: Loop Algebra Moment Maps and Hamiltonian Models for the Painlevé Transcendants. To appear in: Mechanics Day Workshop Proceedings AMS-Fiels Inst. Commun., P.S. Krishnaprasad, T. Ratiu, W.F. Shadwick (eds.) 1994
[IIKS] Its, A.R., Izergin, A.G., Korepin, V.E., Slavnov, N.A.: Differential Equations for quantum correlation functions. Int. J. Mod. Phys.B4, 1003–1037 (1990)
[JMMS] Jimbo, M., Miwa, T., Môri, Y., Sato, M.: Density Matrix of an Impenetrable Bose Gas and the Fifth Painlevé Transcendent. PhysicaID, 80–158 (1980)
[JMU] Jimbo, M., Miwa, T., Ueno, K.: Monodromy Preserving Deformation of Linear Ordinary Differential Equations with Rational Coefficients I. Physica2D, 306–352 (1981)
[JM] Jimbo, M., Miwa, T.: Monodromy Preserving Deformation of Linear Ordinary Differential Equations with Rational Coefficients II, III. Physica2D, 407–448 (1981); ibid., Jimbo, M., Miwa, T.: Monodromy Preserving Deformation of Linear Ordinary Differential Equations with Rational Coefficients II, III. Physica4D, 26–46 (1981)
[M] Moore, G.: Matrix Models of 2D Gravity and Isomonodromic Deformation. Prog. Theor. Phys. Suppl. No.102, 255–285 (1990)
[Ok] Okamoto, K.: The Painlevé equations and the Dynkin Diagrams. In: Painlevé Transcendents. Their Asymptotics and Physical Applications. NATO ASI Series B, Vol.278, 299–313, D. Levi, P. Winternitz (eds.) N.Y.: Plenum Press, 1992
[RS] Reyman, A.G., Semenov-Tian-Shansky, A.: Current algebras and nonlinear partial differential equations. Soviet. Math. Dokl.21, 630–634 (1980); A family of Hamiltonian structures, hierarchy of Hamiltonians, and reduction for first-order matrix differential operators. Funct. Anal. Appl.14, 146–148 (1980)
[S] Semenov-Tian-Shansky, M.A.: What is a Classical R-Matrix. Funct. Anal. Appl.17, 259–272 (1983); Dressing Transformations and Poisson Group Actions. Publ. RIMS Kyoto Univ.21, 1237–1260 (1985)
[TWI] Tracy, C., Widom, H.: Introduction to Random Matrices. UCD preprint ITD 92/92-10 (1992). In: Geometric and Quantum Methods in Integrable Systems. Springer Lecture Notes in Physics, Vol.424, G. Helminck (ed.) Berlin, Heidelberg, Springer, New York: 1993
[TW2] Tracy, C., Widom, H.: Level Spacing Distribution Functions and the Airy Kernel. Commun. Math. Phys.159, 151–174 (1994)
[W] Weinstein, A.: The Local Structure of Poisson Manifolds. J. Diff. Geom.18, 523–557 (1983)
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Communicated by M. Jimbo
Research supported in part by the Natural Sciences and Engineering Research Council of Canada.
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Harnad, J. Dual isomonodromic deformations and moment maps to loop algebras. Commun.Math. Phys. 166, 337–365 (1994). https://doi.org/10.1007/BF02112319
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DOI: https://doi.org/10.1007/BF02112319