Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 11 (2015), 068, 21 pages      arXiv:1508.03122      https://doi.org/10.3842/SIGMA.2015.068
Contribution to the Special Issue on Algebraic Methods in Dynamical Systems

Dynamics on Wild Character Varieties

Emmanuel Paul a and Jean-Pierre Ramis b
a) Institut de Mathématiques de Toulouse, CNRS UMR 5219, Équipe Émile Picard, Université Paul Sabatier (Toulouse 3), 118 route de Narbonne, 31062 Toulouse CEDEX 9, France
b) Institut de France (Académie des Sciences) and Institut de Mathématiques de Toulouse, CNRS UMR 5219, Équipe Émile Picard, Université Paul Sabatier (Toulouse 3), 118 route de Narbonne, 31062 Toulouse CEDEX 9, France

Received March 26, 2014, in final form August 05, 2015; Published online August 13, 2015

Abstract
In the present paper, we will first present briefly a general research program about the study of the ''natural dynamics'' on character varieties and wild character varieties. Afterwards, we will illustrate this program in the context of the Painlevé differential equations $P_{\rm VI}$ and $P_{\rm V}$.

Key words: character varieties; wild fundamental groupoid; Painlevé equations.

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References

  1. Babbitt D.G., Varadarajan V.S., Formal reduction theory of meromorphic differential equations: a group theoretic view, Pacific J. Math. 109 (1983), 1-80.
  2. Balser W., Jurkat W.B., Lutz D.A., Birkhoff invariants and Stokes' multipliers for meromorphic linear differential equations, J. Math. Anal. Appl. 71 (1979), 48-94.
  3. Boalch P., Stokes matrices, Poisson Lie groups and Frobenius manifolds, Invent. Math. 146 (2001), 479-506, math.DG/0011062.
  4. Boalch P., Symplectic manifolds and isomonodromic deformations, Adv. Math. 163 (2001), 137-205.
  5. Boalch P., $G$-bundles, isomonodromy, and quantum Weyl groups, Int. Math. Res. Not. 2002 (2002), 1129-1166, math.DG/0108152.
  6. Boalch P., Through the analytic Halo: fission via irregular singularities, Ann. Inst. Fourier (Grenoble) 59 (2009), 2669-2684, arXiv:1305.6465.
  7. Boalch P., Geometry and braiding of Stokes data; fission and wild character varieties, Ann. of Math. 179 (2014), 301-365, arXiv:1111.6228.
  8. Boalch P., Wild character varieties, points on the Riemann sphere and Calabi's examples, arXiv:1501.00930.
  9. Brown R., From groups to groupoids: a brief survey, Bull. London Math. Soc. 19 (1987), 113-134.
  10. Cantat S., Loray F., Dynamics on character varieties and Malgrange irreducibility of Painlevé VI equation, Ann. Inst. Fourier (Grenoble) 59 (2009), 2927-2978.
  11. Casale G., Une preuve galoisienne de l'irréductibilité au sens de Nishioka-Umemura de la première équation de Painlevé, Astérisque (2009), 83-100.
  12. Casale G., Weil J.A., Galoisian methods for testing irreducibility of order two nonlinear differential equations, arXiv:1504.08134.
  13. Dubrovin B., Geometry of $2$D topological field theories, in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348, hep-th/9407018.
  14. Dubrovin B., Mazzocco M., Monodromy of certain Painlevé-VI transcendents and reflection groups, Invent. Math. 141 (2000), 55-147, math.AG/9806056.
  15. Grothendieck A., Esquisse d'un programme, in Geometric Galois Actions, I, London Math. Soc. Lecture Note Ser., Vol. 242, Editors L. Schneps, P. Lochak, Cambridge University Press, Cambridge, 1997, 5-48.
  16. Iwasaki K., A modular group action on cubic surfaces and the monodromy of the Painlevé VI equation, Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), 131-135.
  17. Jimbo M., Miwa T., Ueno K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and $\tau$-function, Phys. D 2 (1981), 306-352.
  18. Kawai T., Koike T., Nishikawa Y., Takei Y., On the complete description of the Stokes geometry for the first Painlevé hierarchy, in Microlocal Analysis and Asymptotic Analysis, RIMS Kôkyûroku Bessatsu, Vol. 1397, Editor T. Kawai, Res. Inst. Math. Sci. (RIMS), Kyoto, 2004, 74-101.
  19. Kawai T., Takei Y., Algebraic analysis of singular perturbation theory, Translations of Mathematical Monographs, Vol. 227, Amer. Math. Soc., Providence, RI, 2005.
  20. Magnus W., Rings of Fricke characters and automorphism groups of free groups, Math. Z. 170 (1980), 91-103.
  21. Malgrange B., Sur les déformations isomonodromiques. I. Singularités régulières, in Mathematics and Physics (Paris, 1979/1982), Progr. Math., Vol. 37, Birkhäuser Boston, Boston, MA, 1983, 401-426.
  22. Malgrange B., Sur les déformations isomonodromiques. II. Singularités irrégulières, in Mathematics and Physics (Paris, 1979/1982), Progr. Math., Vol. 37, Birkhäuser Boston, Boston, MA, 1983, 427-438.
  23. Malgrange B., Le groupoï de de Galois d'un feuilletage, in Essays on Geometry and Related Topics, Monogr. Enseign. Math., Vol. 38, Enseignement Math., Geneva, 2001, 465-501.
  24. Malgrange B., On nonlinear differential Galois theory, Chin. Ann. Math. 23 (2002), 219-226.
  25. Malgrange B., Déformations isomonodromiques, forme de Liouville, fonction $\tau$, Ann. Inst. Fourier (Grenoble) 54 (2004), 1371-1392.
  26. Martinet J., Ramis J.-P., Elementary acceleration and multisummability. I, Ann. Inst. H. Poincaré Phys. Théor. 54 (1991), 331-401.
  27. Ramis J.-P., Confluence et résurgence, J Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989), 703-716.
  28. Ueno K., Monodromy preserving deformation of linear differential equations with irregular singular points, Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), 97-102.
  29. van der Put M., Saito M.H., Moduli spaces for linear differential equations and the Painlevé equations, Ann. Inst. Fourier (Grenoble) 59 (2009), 2611-2667, arXiv:0902.1702.
  30. van der Put M., Singer M.F., Galois theory of linear differential equations, Grundlehren der mathematischen Wissenschaft, Vol. 328, Springer-Verlag, Berlin, 2003.
  31. Witten E., Gauge theory and wild ramification, Anal. Appl. (Singap.) 6 (2008), 429-501, arXiv:0710.0631.


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