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Mosc. Math. J., 2002, Volume 2, Number 4, Pages 717–752 (Mi mmj70)  

This article is cited in 32 scientific papers (total in 32 papers)

Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations

I. M. Kricheverabc

a Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)
b L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences
c Columbia University

Abstract: We construct the Hamiltonian theory of isomonodromy equations for meromorphic connections with irregular singularities on algebraic curves. We obtain an explicit formula for the symplectic structure on the space of monodromy and Stokes matrices. From these we derive Whitham equations for the isomonodromy equations. It is shown that they provide a flat connection on the space of spectral curves of Hitchin systems.

Key words and phrases: Algebaric curves, meromorphic connections, monodromy matrices, Hamiltonian theory, symplectic form.

Full text: http://www.ams.org/.../abst2-4-2002.html
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MSC: 14, 79
Received: February 18, 2002; in revised form April 20, 2002
Language: English

Citation: I. M. Krichever, “Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations”, Mosc. Math. J., 2:4 (2002), 717–752

Citation in format AMSBIB
\Bibitem{Kri02}
\by I.~M.~Krichever
\paper Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations
\jour Mosc. Math.~J.
\yr 2002
\vol 2
\issue 4
\pages 717--752
\mathnet{http://mi.mathnet.ru/mmj70}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1986088}
\zmath{https://zbmath.org/?q=an:1044.70010}
\elib{http://elibrary.ru/item.asp?id=8379141}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Grava T., “Whitham equations, Bergman kernel and Lax-Levermore minimizer”, Acta Appl. Math., 82:1 (2004), 1–86  crossref  mathscinet  zmath  adsnasa  isi
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    5. I. M. Krichever, O. K. Sheinman, “Lax Operator Algebras”, Funct. Anal. Appl., 41:4 (2007), 284–294  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. Mazzocco M., Mo Man Yue, “The Hamiltonian structure of the second Painlevé hierarchy”, Nonlinearity, 20:12 (2007), 2845–2882  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. Boalch Ph., “Quasi-Hamiltonian geometry of meromorphic connections”, Duke Math. J., 139:2 (2007), 369–405  crossref  mathscinet  zmath  isi
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    12. Kokotov A., Korotkin D., “A new hierarchy of integrable systems associated to Hurwitz spaces”, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 366:1867 (2008), 1055–1088  crossref  mathscinet  zmath  adsnasa  isi
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    15. Heu V., “Stability of rank 2 vector bundles along isomonodromic deformations”, Math. Ann., 344:2 (2009), 463–490  crossref  mathscinet  zmath  isi
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    17. Machu F.-X., “Local Structure of Moduli Spaces”, Internat J Math, 22:12 (2011), 1683–1709  crossref  mathscinet  zmath  isi  elib
    18. Teschner J., “Quantization of the Hitchin Moduli Spaces, Liouville Theory and the Geometric Langlands Correspondence I”, Adv. Theor. Math. Phys., 15:2 (2011), 471–564  crossref  mathscinet  zmath  isi  elib
    19. Yu. P. Bibilo, “Isomonodromic deformations of systems of linear differential equations with irregular singularities”, Sb. Math., 203:6 (2012), 826–843  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    20. D. V. Artamonov, “The Schlesinger system and isomonodromic deformations of bundles with connections on Riemann surfaces”, Theoret. and Math. Phys., 171:3 (2012), 739–753  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    21. Zabrodin A., Zotov A., “Quantum Painlevé-Calogero Correspondence for Painlevé VI”, J. Math. Phys., 53:7 (2012), 073508  crossref  mathscinet  zmath  adsnasa  isi
    22. Wong M.L., “An Interpretation of Some Hitchin Hamiltonians in Terms of Isomonodromic Deformation”, J. Geom. Phys., 62:6 (2012), 1397–1413  crossref  mathscinet  zmath  adsnasa  isi
    23. Artamonov D.V., “Isomonodromic Deformations on Riemann Surfaces”, Painleve Equations and Related Topics (2012), Degruyter Proceedings in Mathematics, eds. Bruno A., Batkhin A., Walter de Gruyter & Co, 2012, 85–89  mathscinet  isi
    24. Wong M.L., “Hecke Modifications, Wonderful Compactifications and Moduli of Principal Bundles”, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 12:2 (2013), 309–367  mathscinet  zmath  isi
    25. A. M. Levin, M. A. Olshanetsky, A. V. Zotov, “Classification of isomonodromy problems on elliptic curves”, Russian Math. Surveys, 69:1 (2014), 35–118  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    26. M. Schlichenmaier, “Multipoint Lax operator algebras: almost-graded structure and central extensions”, Sb. Math., 205:5 (2014), 722–762  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    27. Boalch P.P., “Geometry and Braiding of Stokes Data; Fission and Wild Character Varieties”, Ann. Math., 179:1 (2014), 301–365  crossref  mathscinet  zmath  isi
    28. Boalch Ph., “Poisson Varieties From Riemann Surfaces”, Indag. Math.-New Ser., 25:5, SI (2014), 872–900  crossref  mathscinet  zmath  isi
    29. Schlichenmaier M., “Krichever-Novikov Type Algebras: Theory and Applications”, Krichever-Novikov Type Algebras: Theory and Applications, Degruyter Studies in Mathematics, 53, Walter de Gruyter Gmbh, 2014, 1–360  crossref  mathscinet  isi
    30. Trans. Moscow Math. Soc., 76:2 (2015), 219–236  mathnet  crossref  elib
    31. Aminov G., Arthamonov S., “New -Matrix Linear Problems For the Painlevé Equations III, V”, Constr. Approx., 41:3, SI (2015), 357–383  crossref  mathscinet  zmath  isi  elib
    32. Arata Komyo, “The Moduli Spaces of Parabolic Connections with a Quadratic Differential and Isomonodromic Deformations”, SIGMA, 14 (2018), 111, 22 pp.  mathnet  crossref
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