Discrete Symmetries of Systems of Isomonodromic Deformations of Second-Order Fuchsian Differential Equations

S. V. Oblezin

Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 2, pp. 38-54.

Résumé

We compute the discrete affine group of Schlesinger transformations for isomonodromic deformations of a Fuchsian system of second-order differential equations. These transformations are treated as isomorphisms between the moduli spaces of logarithmic $sl(2)$-connections with given eigenvalues of the residues on $\mathbb{P}^1$. The discrete structure is computed with the use of the modification technique for bundles with connections. The result generalizes the well-known classical computations of symmetries of the hypergeometric equation, the Heun equation, and the sixth Painlevé equation.

Détail

Keywords: Schlesinger transformations, the Frobenius–Hecke sheaves, Fuchsian systems, the hypergeometric equation, the Heun equation.

Bibliographie

[1] Arinkin D., Lysenko S., “Isomorphisms between moduli of $SL(2)$-bundles with connections on $\mathbb{P}^1\setminus\{x_1,\dots,x_4\}$”, Math. Res. Lett., 4 (1997), 181–190 | DOI | MR | Zbl

[2] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, tt. 1, 3, Nauka, M., 1966 | MR

[3] Birkhoff G. D., Collected mathematical papers, Vol. I, Amer. Math. Soc., 1950 | MR

[4] Bolibrukh A. A., 21-ya problema Gilberta dlya fuksovykh lineinykh sistem, Trudy MIRAN, 206, 1994 | MR | Zbl

[5] Coxeter H. S. M., Moser W. O. J., Generators and relations for discrete groups, Springer-Verlag, 1972 | MR | Zbl

[6] Drinfeld V. G., “Dokazatelstvo globalnoi gipotezy Lenglendsa dlya $GL(2)$ nad funktsionalnym polem”, Funkts. analiz i ego pril., 11:3 (1977), 74–75 | MR

[7] Erdélyi A., “Certain expansions of solutions of the Heun equation”, Quart. J. Math., Oxford Ser., 15 (1944), 62–69 | DOI | MR | Zbl

[8] Enriquez B., Rubtsov V., Hecke–Tyurin parametrization of the Hitchin and KZB systems, arXiv: AG/9911087 | MR

[9] Iwasaki K., Kimura H., Shimomura S., Yoshida M., From Gauss to Painlevé. A modern theory of special functions, Aspects of Mathematics, E16, Friedr. Vieweg Sohn, Braunschweig, 1991 | DOI | MR

[10] Hecke E., Mathematische Werke, Göttingen, 1953 | MR

[11] Heun K., “Zur Theorie Riemann'schen Functionen zweiter Ordnung mit vier Verzweigungspunkten”, Math. Ann., XXXIII (1889) | MR

[12] Hitchin N., “Twistor spaces, Einstein metrics and isomonodromic deformations”, J. Diff. Geom., 3 (1995), 52–134 | MR

[13] Ince E. L., Ordinary differential equations, Oxford Univ. Press, 1953

[14] Krichever I. M., “Integrirovanie nelineinykh uravnenii metodami algebraicheskoi geometrii”, Funkts. analiz i ego pril., 11:1 (1977), 15–31 | MR | Zbl

[15] Jimbo M., Miwa T., “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II”, Physica 2D, 8 (1981), 407–448 | MR

[16] Korotkin D., Manojlović N., Samtleben H., “Schlesinger transformations for elliptic isomonodromic deformations”, J. Math. Phys., 41 (2000), 3125–3141 | DOI | MR | Zbl

[17] Levin A. M., Olshanetsky M. A., Zotov A., “Hitchin systems — symplectic Hecke correspondence and two-dimensional version”, Comm. Math. Phys., 236 (2003), 93–133 | DOI | MR | Zbl

[18] Okamoto K., “Studies in the Painlevé equations I. Sixth Painlevé equation PVI”, Ann. Mat. Pura Appl., 146 (1987), 337–381 | DOI | MR | Zbl

[19] “Über eine Klasse von Differentialsystemen beliebiger Ordnung mit festen kritischen Punkten”, J. Reine Angew. Math., 141 (1912), 96–145 | MR | Zbl

[20] Serr Zh.-P., Algebraicheskie gruppy i polya klassov, Mir, M., 1968 | MR

[21] Sklyanin E. K., “Razdelenie peremennykh v sisteme Godena”, Zap. nauchn. sem. LOMI, 164, 1987, 151–169 | MR

[22] Vakulenko V., Note on the Ruijsenaars–Schneider model, arXiv: QA/9909079 | MR

[23] Weil A., “Généralisation des fonctions abéliennes”, J. Math. Pures Appl., 17 (1938), 47–87