Cubic Pencils and Painlev\'e Hamiltonians

Preprint English OPEN
Kajiwara, Kenji; Masuda, Tetsu; Noumi, Masatoshi; Ohta, Yasuhiro; Yamada, Yasuhiko;
(2004)
  • Subject: Nonlinear Sciences - Exactly Solvable and Integrable Systems | High Energy Physics - Theory | Mathematics - Algebraic Geometry
    arxiv: Mathematics::Algebraic Geometry | Mathematics::Symplectic Geometry
We present a simple heuristic method to derive the Painlev\'e differential equations from the corresponding geometry of rational surafces. We also give a direct relationship between the cubic pencils and Seiberg-Witten curves.
  • References (10)

    [1] K.Okamoto, Sur les feuilletages associ´es aux ´equation du second ordre `a points critiques fixes de P. Painlev´e, Japan. J. Math. 5 (1979) 1-79.

    [2] T.Sioda and K.Takano, On Some Hamiltonian Structures of Painlev´e Systems,I, Funkcial. Ekvac, 40 (1997) 271-291.

    [3] H.Sakai, Rational surfaces with affine root systems and geometry of the Painlev´e equations, Commun. Math. Phys. 220 (2001) 165-221.

    [4] M.Saito, T.Takebe and H.Terajima, Deformation of Okamoto-Painlev´e pairs and Painlev´e equations, J. Algebraic Geom. 11 (2002) 311-362. H.Terajima, Local cohomology of generalized Okamoto-Painlev´e pairs and Painlev´e equations, Japan. J. Math. (N.S.) 28 (2002) 137-162.

    [5] K.Kajiwara, T.Masuda, M.Noumi, Y.Ohta and Y.Yamada, 10E9 solutions to the elliptic Painlev´e equation, J. Phys. A:Math.Gen. 36 (2003) L263-L272.

    [6] N.Seiberg and E.Witten, Monopoles, Duality and Chiral Symmetry Breaking in N = 2 Supersymmetric QCD, Nuclear Phys. B431 (1994) 484-550.

    [7] P.C.Argyres, M.R.Plesser, N.Seiberg and E.Witten, New N = 2 Superconformal Field Theories in Four Dimensions, Nuclear Phys. B461 (1996) 71-84.

    [8] O.J.Ganor, D.R.Morrison and N.Seiberg, Branes, Calabi-Yau Spaces, and Troidal Compactification of the N = 1 Six-Dimensional E8 Theory, Nuclear Phys. B487 (1997) 93-127.

    [9] S.Mizoguchi and Y.Yamada, W (E10) Symmetry, M -theory and Painlev´e equations, Phys. Lett. B537 (2002) 130-140.

    [10] T.Eguchi and K.Sakai, Seiberg-Witten Curve for E-String Theory Revisited, Adv. Theor. Math. Phys. 7 (2003) 421-457.

  • Metrics
Share - Bookmark