Communications in Mathematical Physics

, Volume 242, Issue 1–2, pp 185–219 | Cite as

An Area-Preserving Action of the Modular Group on Cubic Surfaces and the Painlevé VI Equation

  • Katsunori IwasakiEmail author


We construct an area-preserving action of the modular group on a general 4-parameter family of affine cubic surfaces. We present a geometrical background behind this construction, that is, a natural symplectic structure on a moduli space of rank two linear monodromy representations over the 2-dimensional sphere with four punctures, and a natural symplectic action upon it of the braid group on three strings. Studying this action as a discrete dynamical system will be important in discussing the monodromy of the Painlevé VI equation.


Dynamical System Modulus Space Symplectic Structure Braid Group Modular Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Arinkin, D., Lysenko, S.: On the moduli of SL(2)-bundles with connections on ℙ1\{x 1,...,x 4}. Internat. Math. Res. Notices 19, 983–999 (1997)Google Scholar
  2. 2.
    Arinkin, D., Lysenko, S.: Isomorphisms between moduli spaces of SL(2)-bundles with connections on ℙ1\{x 1,...,x 4}. Math. Res. Lett. 4, 181–190 (1997)Google Scholar
  3. 3.
    Boalch, P.P.: Symplectic manifolds and isomonodromic deformations. Adv. Math. 163, 137–205 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Birman, J.S.: Braids, links, and mapping class groups. Ann. Math. Stud., Princeton, NJ: Princeton Univ. Press, 1974Google Scholar
  5. 5.
    Cayley, A.: On the triple tangent planes of surfaces of the third order. Collected Papers I, Cambridge: Cambridge Univ. Press, 1889, pp. 231–326Google Scholar
  6. 6.
    Dubrovin, B.: Painlevé transcendents in two-dimensional topological field theory. In: The Painlevé property, one century later, R. Conte (ed.), New York: Springer-Verlag, 1999Google Scholar
  7. 7.
    Dubrovin, B., Mazzocco, M.: Monodromy of certain Painlevé-VI transcendents and reflection groups. Invent. Math. 141(1), 55–147 (2000)zbMATHGoogle Scholar
  8. 8.
    Goldman, W.M.: The symplectic nature of the fundamental groups of surfaces. Adv. Math. 54, 200–225 (1984)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Guzzetti, D.: On the critical behavior, the connection problem and the elliptic representation of a Painlevé VI equation. J. Math. Phys. Anal. Geom. 4, 293–377 (2001)CrossRefzbMATHGoogle Scholar
  10. 10.
    Guzzetti, D.: The elliptic representation of the general Painlevé VI equation. Comm. Pure Appl. Math. 55, 1280–1363 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Hitchin, N.: Poncelet polygons and the Painlevé equations. In: Geometry and analysis (Bombay, 1992), Bombay: Tata Inst. Fund. Res., 1995, pp. 151–185Google Scholar
  12. 12.
    Hitchin, N.: Twister spaces, Einstein metrics and isomonodromic deformations. J. Diff. Geom. 42, 30–112 (1995)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Hitchin, N.: Frobenius manifolds. In: Gauge theory and symplectic geometry (Montreal, PQ, 1995), NATO Adv. Sci. Inst. Ser. C, Math. Phys. Sci., 488, Dordrecht: Kluwer Acad. Publ., 1997, pp. 69–112Google Scholar
  14. 14.
    Inaba, M., Iwasaki, K., Saito, M.-H.: Bäcklund transformations of the sixth Painlevé equation in terms of Riemann-Hilbert correspondence. To appear in Internat. Math. Res. NoticesGoogle Scholar
  15. 15.
    Inaba, M., Iwasaki, K., Saito, M.-H.: Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. Preprint (2003)Google Scholar
  16. 16.
    Iwasaki, K.: Moduli and deformation for Fuchsian projective connections on a Riemann surface. J. Fac. Sci. Univ. Tokyo, Sect. IA, Math. 38, 431–531 (1991)Google Scholar
  17. 17.
    Iwasaki, K.: Fuchsian moduli on a Riemann surface – its Poisson structure and Poincaré-Lefschetz duality. Pacific J. Math. 155, 319–340 (1992)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Iwasaki, K.: A modular group action on cubic surfaces and the monodromy of the Painlevé VI equation. Proc. Japan Acad. 78, Ser. A, 131–135 (2002)Google Scholar
  19. 19.
    Iwasaki, K., Kimura, H., Shimomura, S., Yoshida, M.: From Gauss to Painlevé. Wiesbaden: Vieweg-Verlag, 1991Google Scholar
  20. 20.
    Jimbo, M.: Monodromy problem and the boundary condition for some Painlevé equation. Publ. Res. Inst. Math. Sci. 18, 1137–1161 (1982)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Jimbo, M., Miwa, T., Ueno, K.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients I – General theory and τ-functions. Physica 2D, 306–352 (1981)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Jimbo, M., Miwa, T.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients II, III. Physica 2D, 407–448 (1981); ibid. 4D, 26–46 (1981)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Kawai, S.: The symplectic nature of the space of projective connections on Riemann surfaces. Math. Ann. 305, 161–182 (1996)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Kawai, S.: Isomonodromic deformation of Fuchsian projective connections on elliptic curves. To appear in Nagoya Math. J. 171 (2003)Google Scholar
  25. 25.
    Manin, Y.I.: Sixth Painlevé equation, universal elliptic curve, and mirror of ℙ2. In: Geometry of differential equations, Amer. Math. Soc. Transl. Ser. 2, 186, Providence, RI: Am. Math. Soc., 1998, pp. 131–151Google Scholar
  26. 26.
    Mazzocco, M.: Picard and Chazy solutions to the Painlevé VI equation. Math. Ann. 321, 157–195 (2001)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Mazzocco, M.: Rational solutions of the Painlevé VI equation. J. Phys. A: Math. Gen. 34, 2281–2294 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Mazzocco, M.: The geometry of the classical solutions of the Garnier systems. Internat. Math. Res. Notices, 2002 (12), 613–646 (2002)Google Scholar
  29. 29.
    Naruki, I.: Cross ratio variety as a moduli space of cubic surfaces. Proc. London Math. Soc. 45(3), 1–30 (1982)zbMATHGoogle Scholar
  30. 30.
    Naruki, I., Sekiguchi, J.: A modification of Cayley’s family of cubic surfaces and birational action of W(E 6) over it. Proc. Japan Acad. 56, Ser. A, 122–125 (1980)Google Scholar
  31. 31.
    Noumi, M., Yamada, Y.: A new Lax pair for the sixth Painlevé equation associated with ŝo(8). In: Microlocal analysis and complex Fourier analysis, Kawai, T. and Fujita, K. eds., NJ: World Scientific, 2002, pp. 238–252Google Scholar
  32. 32.
    Noumi, M., Takano, K., Yamada, Y.: Bäcklund transformations and the manifolds of Painlevé systems. Funkcial. Ekvac. 45, 237–258 (2002)MathSciNetGoogle Scholar
  33. 33.
    Okamoto, K.: Sur les feuilletages associés aux equations du second ordre à points critiques de Painlevé, espace de conditions initiales. Japan. J. Math. 5, 1–79 (1979)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Okamoto, K.: Studies on the Painlevé equations I, sixth Painlevé equation PVI. Annali di Math. Pura, Appl. 146(4), 337–381 (1987)Google Scholar
  35. 35.
    Saito, M.-H., Takebe, T., Terajima, H.: Deformation of Okamoto-Painlevé pairs and Painlevé equations. J. Algebraic Geom. 11, 311–362 (2002)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Saito, M.-H., Terajima, H.: Nodal curves and Riccati solutions of Painlevé equations. math.AG/0201225Google Scholar
  37. 37.
    Sakai, H.: Rational surfaces associated with affine root systems and geometry of the Painlevé equations. Commun. Math. Phys 220, 165–229 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  38. 38.
    Shioda, T., Takano, K.: On some Hamiltonian structures of Painlevé systems, I. Funkcial. Ekvac. 40(2), 271–291 (1997)zbMATHGoogle Scholar
  39. 39.
    Terajima, H.: On the space of monodromy data of Painlevé VI. Preprint, Kobe University, March, 2003Google Scholar
  40. 40.
    Tod, K.P.: Self-dual Einstein metrics from the Painlevé VI equation. Phys. Lett. A 190, 221–224 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  41. 41.
    Yoshida, M.: On the number of apparent singularities of the Riemann-Hilbert problem on Riemann surface. J. Math. Soc. Japan 49, 145–159 (1997)MathSciNetzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Faculty of MathematicsKyushu UniversityHigashi-kuJapan

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