General information
Latest issue
Impact factor

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Mosc. Math. J.:

Personal entry:
Save password
Forgotten password?

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

This article is cited in 34 scientific papers (total in 34 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:
References: PDF file   HTML file

Bibliographic databases:

MSC: 14, 79
Received: February 18, 2002; in revised form April 20, 2002

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
\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

Linking options:

    SHARE: FaceBook Twitter Livejournal

    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    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
    2. Pinchbeck D.J., “Isomonodromic flows for Fuchsian connections on Riemann surfaces”, Int. Math. Res. Not., 2005, no. 40, 2473–2497  crossref  mathscinet  zmath  isi  elib
    3. Teodorescu R., “Relaxation of nonlinear oscillations in BCS superconductivity”, J. Phys. A, 39:33 (2006), 10363–10374  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. Fedorov R.M., “Algebraic and Hamiltonian approaches to isostokes deformations”, Transform. Groups, 11:2 (2006), 137–160  crossref  mathscinet  zmath  isi
    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
    8. Dubrovin B., Mazzocco M., “Canonical structure and symmetries of the Schlesinger equations”, Comm. Math. Phys., 271:2 (2007), 289–373  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. M. Schlichenmaier, O. K. Sheinman, “Central extensions of Lax operator algebras”, Russian Math. Surveys, 63:4 (2008), 727–766  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. O. K. Sheinman, “Lax Operator Algebras and Integrable Hierarchies”, Proc. Steklov Inst. Math., 263 (2008), 204–213  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    11. Hurtubise J., “On the geometry of isomonodromic deformations”, J. Geom. Phys., 58:10 (2008), 1394–1406  crossref  mathscinet  zmath  adsnasa  isi
    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
    13. Schlichenmaier M., “Classification of central extensions of Lax operator algebras”, Geometric Methods in Physics, AIP Conference Proceedings, 1079, 2008, 227–234  crossref  mathscinet  zmath  adsnasa  isi
    14. Dzhamay A., “Factorizations of rational matrix functions with application to discrete isomonodromic transformations and difference Painlevé equations”, J. Phys. A, 42:45 (2009), 454008, 10 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib
    15. Heu V., “Stability of rank 2 vector bundles along isomonodromic deformations”, Math. Ann., 344:2 (2009), 463–490  crossref  mathscinet  zmath  isi
    16. Heu V., “Universal Isomonodromic Deformations of Meromorphic Rank 2 Connections on Curves”, Ann Inst Fourier (Grenoble), 60:2 (2010), 515–549  crossref  mathscinet  zmath  isi
    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
    33. Its A.R., Prokhorov A., “On Some Hamiltonian Properties of the Isomonodromic Tau Functions”, Rev. Math. Phys., 30:7, SI (2018), 1840008  crossref  mathscinet  zmath  isi  scopus
    34. Its A.R., Lisovyy O., Prokhorov A., “Monodromy Dependence and Connection Formulae For Isomonodromic Tau Functions”, Duke Math. J., 167:7 (2018), 1347–1432  crossref  mathscinet  zmath  isi  scopus
  • Moscow Mathematical Journal
    Number of views:
    This page:657

    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2021