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Factorization and Integrable Systems pp 103–129Cite as

Birkhäuser

Matrix Riemann-Hilbert Problems Related to Branched Coverings of ℂℙ1

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Part of the book series: Operator Theory: Advances and Applications ((OT,volume 141))

Abstract

In these notes we solve a class of Riemann-Hilbert (inverse monodromy) problems with an arbitrary quasi-permutation monodromy group. The solution is given in terms of the Szegö kernel on the underlying Riemann surface. In particular, our construction provides a new class of solutions of the Schlesinger system. We present some results on explicit calculation of the corresponding tau-function, and describe the divisor of zeros of the tau-function (the so-called Malgrange divisor) in terms of the theta-divisor on the Jacobi manifold of the Riemann surface. We discuss the relationship of the tau-function to the determinant of the Laplacian operator on the Riemann surface.

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References

  1. Bolibruch, A., The Riemann-Hilbert problem, Russ. Math. Surveys 45 (1990), 1–58.

    Article  Google Scholar 

  2. Zakharov, V.E., Manakov, S.V., Novikov, S.P., Pitaevskii, L.P., Theory of Solitons. The inverse scattering method, Consultants Bureau, New York 1984.

    Google Scholar 

  3. Dubrovin, B., Geometry of 2D topological field theories, In: Integrable systems and quantum groups, pages 120–348, Lecture Notes in Math., 1620, Springer, Berlin 1996.

    Chapter  Google Scholar 

  4. Hitchin, N., Twistor spaces, Einstein metrics and isomonodromic deformations, J. Diff. Geom. 42 (1995), 30–112.

    MathSciNet  MATH  Google Scholar 

  5. Deift, P., Its, A., Zhou, X., A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics, Annals of Math. 146 (1997), 149–235.

    Article  MathSciNet  MATH  Google Scholar 

  6. Belokolos, E., Bobenko, A., Enolski, V., Its, A., Matveev, V., Algebro-geometrical integration of non-linear differential equations, Springer Verlag, Berlin Heidelberg New York 1992.

    Google Scholar 

  7. Malgrange, B., Sur les Déformations Isomonodromiques, In: Mathématique et Physique (E.N.S. Séminaire 1979–1982), pages 401–426, Birkhäuser, Boston 1983.

    Google Scholar 

  8. Jimbo, M., Miwa, T., Ueno, K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients I, Physica 2D (1981), 306–352..

    MathSciNet  Google Scholar 

  9. Okamoto, K., Studies on the Painlevé Equations. I. Sixth Painlevé Equation P VI, Annali Mat. Pura Appl 146 (1987), 337–381.

    Article  MATH  Google Scholar 

  10. Korotkin, D., Finite-gap solutions of stationary axially symmetric Einstein equations in vacuum, Theor.Math.Phys. 77 (1989), 1018–1031.

    Article  MathSciNet  Google Scholar 

  11. Neugebauer, G., Meinel, R., General relativistic gravitational field of the rigidly rotating disk of dust: Solution in terms of ultraelliptic functions, Phys.Rev.Lett. 75 (1995), 3046–3048.

    Article  MathSciNet  MATH  Google Scholar 

  12. Klein, C., Richter, O., Explicit solutions of Riemann-Hilbert problem for the Ernst equation, Phys.Rev.D 57 (1998), 857–862.

    Article  MathSciNet  Google Scholar 

  13. Babich, M., Korotkin, D., Self-dual SU(2)-invariant Einstein manifolds and modular dependence of theta-functions, Lett. Math. Phys. 46 (1998), 323–337.

    Article  MathSciNet  MATH  Google Scholar 

  14. Kitaev, A., Korotkin, D., On solutions of Schlesinger equations in terms of theta-functions, Intern. Math. Res. Notices 17 (1998), 877–905.

    Article  MathSciNet  Google Scholar 

  15. Deift, P., Its, A., Kapaev, A., Zhou, X., On the algebro-geometric integration of the Schlesinger equations, Commun. Math. Phys. 203 (1999), 613–633.

    Article  MathSciNet  MATH  Google Scholar 

  16. Zverovich, E.I., Boundary value problems in the theory of analytic functions in Holder classes on Riemann surfaces, Russ. Math. Surveys 26 (1971), 117–192.

    Article  Google Scholar 

  17. Zamolodchikov, Al.B., Conformal scalar field on the hyperelliptic curve and critical Ashkin-Teller multipoint correlation functions, Nucl. Phys. B285 (1986), 481–503.

    MathSciNet  Google Scholar 

  18. Belavin, A.A., Knizhnik, V.G., Algebraic geometry and the geometry of quantum strings, Phys. Lett. 168B (1986), 201–206.

    MathSciNet  Google Scholar 

  19. Knizhnik, V.G., Analytic fields on Riemann surfaces II, Commun. Math. Phys. 112 (1987), 567–590.

    Article  MathSciNet  MATH  Google Scholar 

  20. Knizhnik, V.G., Multiloop amplitudes in the theory of quantum strings and complex geometry, Sov. Phys. Uspekhi. 32 (1989), 945–971.

    Article  MathSciNet  Google Scholar 

  21. Bershadsky, M., Radul, A., Fermionic fields on ℤN-curves, Commun. Math. Phys. 116 (1988), 689–700.

    Article  MathSciNet  MATH  Google Scholar 

  22. Alvarez-Gaume, L., Moore, G., Vafa, C., Theta-functions, Modular Invariance and Strings, Commun. Math. Phys. 106 (1986), 1–40.

    Article  MathSciNet  MATH  Google Scholar 

  23. Quillen, D., Determinants of Cauchy-Riemann operators over Riemann surface, Funct. Anal. Appl. 19 (1984), 37–41.

    MathSciNet  Google Scholar 

  24. Fay, J., Theta Functions on Riemann Surfaces, Lect. Notes in Math. 352, Springer, Berlin 1973.

    MATH  Google Scholar 

  25. Fay, J., Kernel functions, Analytic torsion and Moduli spaces, Memoirs of the American Mathematical Society 96 (1992), 1–123.

    MathSciNet  Google Scholar 

  26. Takhtajan, L., Semi-classical Lionville theory,complex geometry of moduli spaces, and uniformization of Riemann surfaces, In: New Symmetry Principles in Quantum Field Theory, Eds. Frölich, J., et al Plenum Press, New York 1992.

    Google Scholar 

  27. Korotkin, D., Isomonodromic deformations and Hurwitz spaces, math-ph/0103023, to appear in Isomonodromic deformations and applications, Eds. by Hamad, J. and Its, A., American Mathematical Society 2001.

    Google Scholar 

  28. Grinevich P., Orlov A., Flag Spaces in KP theory and Virasoro action on detD j and Segal-Wilson τ-function, In: Research reports in physics. Problems of modern quantum field theory, pages 86–106, Eds. Belavin, A.A., Klimuk, A.U., Zamolodchikov, A.B., Springer Berlin, Heidelberg 1989.

    Chapter  Google Scholar 

  29. Natanzon, S., Turaev, V., A compactification of the Hurwitz space, Topology 38 (1999), 889–914.

    Article  MathSciNet  MATH  Google Scholar 

  30. Zograf, P., Takhtajan, L., On uniformization of Riemann surfaces and the Weil Petersson metric on Teichmiiller and Schottky spaces, Mat. Sbornik. 132 (1987), 304–327.

    Google Scholar 

  31. Zograf P., The Liouville action on moduli spaces and uniformization of degenerating Riemann surfaces, Algebra i Analiz 1 (1989), 136–160.

    MathSciNet  Google Scholar 

  32. Zograf, P., Takhtajan L., A potential for the Weil-Petersson metric on Torelli space, Zap. Nauch. Sem. LOMI 160 (1987), 110–120.

    MATH  Google Scholar 

  33. Aldrovandi, E., Takhtajan, L., Generating functional in CFT and effective action for two-dimensional quantum gravity on higher-genus Riemann surfaces, Commun. Math. Phys. 188 (1997), 29–67.

    Article  MathSciNet  MATH  Google Scholar 

  34. Mumford, D. Tata Lectures on Theta, I,II, Progress in Mathematics 28, 43 Birkhauser Boston 1983, 84.

    Google Scholar 

  35. Palmer, J., Determinants of Cauchy-Riemann operators as τ-functions, Acta Appl. Mathematicae 18 (1990), 199–223.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Concordia University, 7141 Sherbrook West, Montreal, H4B 1R6, Quebec, Canada

    Dmitry Korotkin

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  1. Dmitry Korotkin
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Editor information

Editors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, IL - Ramat Aviv, 69978, Israel

    Israel Gohberg

  2. Raymond and Beverly Sackler, Tel Aviv University, IL - Ramat Aviv, 69978, Israel

    Israel Gohberg

  3. Faculty of Exact Sciences, Tel Aviv University, IL - Ramat Aviv, 69978, Israel

    Israel Gohberg

  4. A.D. Matematica, FCT, Universidade do Algarve Campus de Gambelas, 8000-117, Faro, Portugal

    Nenad Manojlovic

  5. Departamento de Matemática Instituto Superior Técnico, Ave. Rovisco Pais, 1049-001, Lisboa, Portugal

    António Ferreira dos Santos

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© 2003 Springer Basel AG

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Korotkin, D. (2003). Matrix Riemann-Hilbert Problems Related to Branched Coverings of ℂℙ1 . In: Gohberg, I., Manojlovic, N., dos Santos, A.F. (eds) Factorization and Integrable Systems. Operator Theory: Advances and Applications, vol 141. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8003-9_2

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  • DOI: https://doi.org/10.1007/978-3-0348-8003-9_2

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9400-5

  • Online ISBN: 978-3-0348-8003-9

  • eBook Packages: Springer Book Archive

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