The Elliptic Representation of the General Painlevé VI Equation

Communications on Pure and Applied Mathematics

Volumn 55, Issue 10, 2002, Pages 1280-1363

  Guzzetti, Davide ^a  

^a KYOTO UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036350608     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/cpa.10045     Document Type: Article

References (37)

1. Anosov, D. V.; Bolibruch, A. A. The Riemann-Hilbert problem. Aspects of Mathematics, E22. Vieweg and Sohn, Braunschweig, 1994.
2. c. Phys. Rev. Lett. 31 (1973), 1409-1411.
3. Dijkgraaf, R.; Verlinde, H.; Verlinde, E. Topological strings in d < 1. Nuclear Phys. B 352 (1991), no. 1, 59-86.
4. Dubrovin, B. Integrable systems in topological field theory. Nuclear Phys. B 379 (1992), no. 3, 627-689.
5. Dubrovin, B. Geometry of 2D topological field theories. Integrable systems and quantum groups (Montecatini Terme, 1993), 120-348. Lecture Notes in Mathematics, 1620. Springer, Berlin, 1996.
6. Dubrovin, B. Painlevé transcendents in two-dimensional topological field theory. The Painlevé property, 287-412. CRM Series in Mathematical Physics. Springer, New York, 1999.
7. Dubrovin, B. Differential geometry of the space of orbits of a Coxeter group. Surveys in differential geometry: integral systems [integrable systems], 181-211. Surveys in Differential Geometry, 4. International Press, Boston, 1998.
8. Dubrovin, B.; Mazzocco, M. Monodromy of certain Painlevé-VI transcendents and reflection groups. Invent. Math. 141 (2000), no. 1, 55-147.
9. Fuchs, R. Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegene wesentlich singuläre Stellen. Math. Ann. 63 (1907), 301-321.
10. Gambier, B. Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est à points critique fixés. Acta Math. 33 (1910), 1-55.
11. Guzzetti, D. Stokes matrices and monodromy of the quantum cohomology of projective spaces. Comm. Math. Phys. 207 (1999), no. 2, 341-383.
12. Guzzetti, D. Inverse problem and monodromy data for three-dimensional Frobenius manifolds. Math. Phys. Anal. Geom. 4 (2001), no. 3, 245-291.
13. Guzzetti, D. On the critical behavior, the connection problem and the elliptic representation of a Painlevé VI equation. Math. Phys. Anal. Geom. 4 (2001), no. 4, 293-377.
14. Its, A. R.; Novokshenov, V. Y. The isomonodromic deformation method in the theory of Painlevé equations. Lecture Notes in Mathematics, 1191. Springer, Berlin, 1986.
15. Iwasaki, K.; Kimura, H.; Shimomura, S.; Yoshida, M. From Gauss to Painlevé. Aspects of Mathematics, E16. Vieweg and Sohn, Braunschweig, 1991.
16. Jimbo, M. Monodromy problem and the boundary condition for some Painlevé equations. Publ. Res. Inst. Math. Sci. 18 (1982), no. 3, 1137-1161.
17. Jimbo, M.; Miwa, T.; Ueno, K. Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and τ-function. Phys. D 2 (1981), no. 2, 306-352.
18. Jimbo, M.; Miwa, T.; Ueno, K. Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II. Phys. D 2 (1981), no. 3, 407-448.
19. Kontsevich, M.; Manin, Y. Gromov-Witten classes, quantum cohomology, and enumerative geometry. Comm. Math. Phys. 164 (1994), no. 3, 525-562.
20. 2. Geometry of differential equations, 131-151, American Mathematical Society Translations, Series 2, 186. American Mathematical Society, Providence, R.I., 1998.
21. Manin, Y. I. Frobenius manifolds, quantum cohomology, and moduli spaces. American Mathematical Society Colloquium Publications, 47. American Mathematical Society, Providence, R.I., 1999.
22. Mazzocco, M. Picard and Chazy solutions to the Painlevé VI equation. Math. Ann. 321 (2001), no. 1, 157-195.
23. Nörlund, N. E. The logarithmic solutions of the hypergeometric equation. Mat.-Fys. Skr. Danske Vid. Selsk. 2 (1963), no. 5, 58 pp.
24. Painlevé, P. Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme. Acta Math. 21 (1902), 1-85.
25. Painlevé, P. Sur les équations différentielles du second ordre à point critiques fixes. C. R. Acad. Sci. (Paris) 143 (1906), 1111-1117.
26. Picard, É.; Simart, G. Théorie des fonctions algébriques de deux variables indépendantes. Tome I, II. Réimpression corrigée (en un volume) de l'édition en deux volumes de 1897 et 1906. Chelsea, Bronx, N.Y., 1971.
27. Sato, M.; Miwa, T.; Jimbo, M. Holonomic quantum fields. II. The Riemann-Hilbert problem. Publ. Res. Inst. Math. Sci. 15 (1979), no. 1, 201-278.
28. Saito, K. Period mapping associated to a primitive form. Publ. Res. Inst. Math. Sci. 19 (1983), no. 3, 1231-1264.
29. Saito, K.; Yano, T.; Sekiguchi, J. On a certain generator system of the ring of invariants of a finite reflection group. Comm. Algebra 8 (1980), no. 4, 373-408.
30. Sansone, G.; Gerretsen, J. Lectures on the theory of functions of a complex variable. I. Holomorphic functions. Noordhoff, Groningen, 1960.
31. Shimomura, S. Painlevé transcendents in the neighbourhood of fixed singular points. Funkcial. Ekvac. 25 (1982), no. 2, 163-184.
32. Shimomura, S. Series expansions of Painlevé transcendents in the neighbourhood of a fixed singular point. Funkcial. Ekvac. 25 (1982), no. 2, 185-197.
33. Shimomura, S. Supplement to: "Series expansions of Painlevé transcendents in the neighbourhood of a fixed singular point." Funkcial. Ekvac. 25 (1982), no. 3, 363-371.
34. Shimomura, S. A family of solutions of a nonlinear ordinary differential equation and its application to Painlevé equations III, V and VI. J. Math. Soc. Japan 39 (1987), no. 4, 649-662.
35. c. Phys. Rev. Lett. 31 (1973), 1500-1504.
36. Witten, E. On the structure of the topological phase of two-dimensional gravity. Nuclear Phys. B 340 (1990), no. 2-3, 281-332.
37. Wu, T. T.; McCoy, B. M.; Tracy, C. A.; Barouch, E. The spin-spin correlation functions of the two-dimensional Ising model: exact results in the scaling region. Phys. Rev. B 13 (1976), 316-374.