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Mat. Sb. (N.S.), 1987, Volume 132(174), Number 2, Pages 147–166 (Mi msb1767)  

This article is cited in 49 scientific papers (total in 49 papers)

On Liouville's equation, accessory parameters, and the geometry of Teichmüller space for Riemann surfaces of genus 0

P. G. Zograf, L. A. Takhtadzhyan


Abstract: For the Weil–Petersson metric on the Teichmüller space $T_{0,n}$ of marked Riemann surfaces of genus 0 with $n$ punctures, a potential is constructed in terms of the density of the hyperbolic metric on the corresponding surface (i.e., in terms of a solution of Liouville's equation). It is shown that this potential is a generating function of the accessory parameters of the Fuchsian uniformization of the corresponding Riemann surface. Also, a connection is established between the accessory parameters and the Eichler integrals of Fuchsian groups.
Bibliography: 18 titles.

Full text: PDF file (1020 kB)
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English version:
Mathematics of the USSR-Sbornik, 1988, 60:1, 143–161

Bibliographic databases:

UDC: 517.9
MSC: Primary 30F10, 32G15; Secondary 11F67, 30F35
Received: 01.04.1986

Citation: P. G. Zograf, L. A. Takhtadzhyan, “On Liouville's equation, accessory parameters, and the geometry of Teichmüller space for Riemann surfaces of genus 0”, Mat. Sb. (N.S.), 132(174):2 (1987), 147–166; Math. USSR-Sb., 60:1 (1988), 143–161

Citation in format AMSBIB
\Bibitem{ZogTak87}
\by P.~G.~Zograf, L.~A.~Takhtadzhyan
\paper On~Liouville's equation, accessory parameters, and the geometry of Teichm\"uller space for Riemann surfaces of genus~0
\jour Mat. Sb. (N.S.)
\yr 1987
\vol 132(174)
\issue 2
\pages 147--166
\mathnet{http://mi.mathnet.ru/msb1767}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=882831}
\zmath{https://zbmath.org/?q=an:0655.32022|0642.32010}
\transl
\jour Math. USSR-Sb.
\yr 1988
\vol 60
\issue 1
\pages 143--161
\crossref{https://doi.org/10.1070/SM1988v060n01ABEH003160}


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    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. P. G. Zograf, “Potential of the Weyl–Peterson metric on the Teichmüller space of Riemann surfaces of genus zero with punctures”, Funct. Anal. Appl., 22:4 (1988), 324–325  mathnet  crossref  mathscinet  zmath  isi
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    4. Jian-Min Shen, Zheng-Mao Sheng, Zhong-Hua Wang, “Quantum Liouville theory on the Riemann sphere with n> 3 punctures”, Physics Letters B, 291:1-2 (1992), 53  crossref
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    7. Zheng-Mao Sheng, J Phys A Math Gen, 27:10 (1994), L339  crossref
    8. E. Aldrovandi, L. Bonora, “Liouville and Toda field theories on Riemann surfaces”, Journal of Geometry and Physics, 14:1 (1994), 65  crossref
    9. Takashi Suzuki, “Towards a strong coupling Liouville gravity”, Physics Letters B, 338:2-3 (1994), 175  crossref
    10. Marco Matone, “The Higgs model for anyons and Liouville action”, Journal of Geometry and Physics, 17:1 (1995), 49  crossref
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    12. A. Yu. Orlov, P. Winternitz, “ P ∞ algebra of symmetries of the kadomtsev-petviashvili equation, free fermions, and 2-cocycles in the Lie algebra of pseudo-differential operatorsalgebra of symmetries of the kadomtsev-petviashvili equation, free fermions, and 2-cocycles in the Lie algebra of pseudo-differential operators”, Theor Math Phys, 113:2 (1997), 1393  mathnet  crossref  mathscinet  isi
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    15. Luigi Cantini, Pietro Menotti, Domenico Seminara, Class Quantum Grav, 18:12 (2001), 2253  crossref  mathscinet  zmath
    16. Arkady L. Kholodenko, “Some thoughts about random walks on figure eight”, Physica A: Statistical Mechanics and its Applications, 289:3-4 (2001), 377  crossref
    17. Luigi Cantini, Pietro Menotti, Domenico Seminara, “Proof of Polyakov conjecture for general elliptic singularities”, Physics Letters B, 517:1-2 (2001), 203  crossref
    18. Luigi Cantini, Pietro Menotti, Domenico Seminara, “Liouville theory, accessory parameters and (2+1)-dimensional gravity”, Nuclear Physics B, 638:3 (2002), 351  crossref
    19. Kiselev A., “On the Geometry of Liouville Equation: Symmetries, Conservation Laws, and Backlund Transformations”, Acta Appl. Math., 72:1-2 (2002), 33–49  crossref  mathscinet  zmath  isi
    20. Leszek Hadasz, Zbigniew Jaskólski, “Polyakov conjecture for hyperbolic singularities”, Physics Letters B, 574:1-2 (2003), 129  crossref
    21. A. V. Kiselev, “Methods of geometry of differential equations in analysis of integrable models of field theory”, J. Math. Sci., 136:6 (2006), 4295–4377  mathnet  crossref  mathscinet  zmath  elib  elib
    22. Leszek Hadasz, Zbigniew Jaskólski, “Classical Liouville action on the sphere with three hyperbolic singularities”, Nuclear Physics B, 694:3 (2004), 493  crossref
    23. Aldrovandi E., “On Hermitian-Holomorphic Classes Related to Uniformization, the Dilogarithm, and the Liouville Action”, Commun. Math. Phys., 251:1 (2004), 27–64  crossref  mathscinet  zmath  adsnasa  isi
    24. Seppala M., “Myrberg's Numerical Uniformization of Hyperelliptic Curves”, Ann. Acad. Sci. Fenn. Ser. A1-Math., 29:1 (2004), 3–20  mathscinet  isi
    25. Heng-Yu Chen, N. S. Manton, “The Kähler potential of Abelian Higgs vortices”, J Math Phys (N Y ), 46:5 (2005), 052305  crossref  mathscinet  zmath  adsnasa  isi
    26. V. A. Poberezhnyi, “Special Monodromy Groups and the Riemann–Hilbert Problem for the Riemann Equation”, Math. Notes, 77:5 (2005), 695–707  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    27. Leszek Hadasz, Zbigniew Jaskólski, Marcin Pia̧tek, “Classical geometry from the quantum Liouville theory”, Nuclear Physics B, 724:3 (2005), 529  crossref
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    29. Aldrovandi E., “Hermitian-Holomorphic Deligne Cohomology, Deligne Pairing for Singular Metrics, and Hyperbolic Metrics”, Int. Math. Res. Notices, 2005, no. 17, 1015–1046  crossref  mathscinet  zmath  isi  elib
    30. Leszek Hadasz, Zbigniew Jaskólski, “Liouville theory and uniformization of four-punctured sphere”, J Math Phys (N Y ), 47:8 (2006), 082304  crossref  mathscinet  zmath  isi
    31. Pietro Menotti, “Semiclassical and quantum Liouville theory”, J Phys Conf Ser, 33 (2006), 26  crossref  elib
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    34. G. B. Shabat, V. I. Zolotarskaya, “The Chekhov–Fock parametrization of Teichmüller spaces and dessins d'enfants”, J. Math. Sci., 158:1 (2009), 155–161  mathnet  crossref  mathscinet  elib  elib
    35. V. A. Poberezhnyi, “General Linear Problem of the Isomonodromic Deformation of Fuchsian Systems”, Math. Notes, 81:4 (2007), 529–542  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    36. Yu. V. Brezhnev, “On uniformization of algebraic curves”, Mosc. Math. J., 8:2 (2008), 233–271  mathnet  mathscinet  zmath
    37. Franco Ferrari, Jarosław Paturej, “On a relation between Liouville field theory and a two component scalar field theory passing through the random walk”, Physics Letters B, 664:1-2 (2008), 123  crossref
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    42. Franco Ferrari, Marcin Piatek, “Liouville theory, $ \mathcal{N} = 2 $ gauge theories and accessory parameters”, J. High Energ. Phys, 2012:5 (2012)  crossref
    43. Pietro Menotti, “Accessory parameters for Liouville theory on the torus”, J. High Energ. Phys, 2012:12 (2012)  crossref
    44. A. G. Sergeev, “Lektsii ob universalnom prostranstve Teikhmyullera”, Lekts. kursy NOTs, 21, MIAN, M., 2013, 3–130  mathnet  crossref  zmath  elib
    45. A. Yu. Vasiliev, A. G. Sergeev, “Classical and quantum Teichmüller spaces”, Russian Math. Surveys, 68:3 (2013), 435–502  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    46. Marcin Piatek, “Classical torus conformal block, $ \mathcal{N} $ = 2∗ twisted superpotential and the accessory parameter of Lamé equation”, J. High Energ. Phys, 2014:3 (2014)  crossref
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    48. N. A. Nekrasov, A. A. Roslyi, S. L. Shatashvili, “Darboux coordinates, Yang–Yang functional, and gauge theory”, Theoret. and Math. Phys., 181:1 (2014), 1206–1234  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
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  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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