Flat structure and potential vector fields related with algebraic solutions to Painleve VI equation

• Autorzy: Kato M. , Mano T. , Sekiguchi J.

Identyfikatory

DOI

10.7494/OpMath.2018.38.2.201

• Języki publikacji: EN

Abstrakty

EN

A potential vector field is a solution of an extended WDVV equation which is a generalization of a WDVV equation. It is expected that potential vector fields corresponding to algebraic solutions of Painleve VI equation can be written by using polynomials or algebraic functions explicitly. The purpose of this paper is to construct potential vector fields corresponding to more than thirty non-equivalent algebraic solutions.

Słowa kluczowe

EN

flat structure; Painleve VI equation; algebraic solution; potential vector field

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2018
• Tom: Vol. 38, no. 2
• Strony: 201--252
• Opis fizyczny: Bibliogr. 31 poz.

Twórcy

autor

Kato M.

• University of the Ryukyus Colledge of Educations Department of Mathematics Nishihara-cho, Okinawa 903-0213, Japan

autor

Mano T.

• University of the Ryukyus Faculty of Science Department of Mathematical Sciences Nishihara-cho, Okinawa 903-0213, Japan

autor

Sekiguchi J.

• Tokyo University of Agriculture and Technology Faculty of Engineering Department of Mathematics Koganei, Tokyo 184-8588, Japan

Bibliografia

• [1] F.V. Andreev, A.V. Kitaev, Transformations RS%(3) of ranks < 4 and algebraic solutions of the sixth Painleve equation, Comm. Math. Phys. 228 (2002), 151-176.
• [2] P. Boalch, From Klein to Painleve via Fourier, Laplace and Jimbo, Proc. Lond. Math. Soc. (3) 90 (2005), 167-208.
• [3] P. Boalch, The fifty-two icosahedral solutions to Painleve VI, J. Reine Angew. Math. 596 (2006), 183-214.
• [4] P. Boalch, Some explicit solutions to the Riemann-Hilbert problem, IRMA Lect. Math. Theor. Phys. 9 (2007), 85-112.
• [5] P. Boalch, Higher genus icosahedral Painleve curves, Funkcial. Ekvac. 50 (2007), 19-32.
• [6] B. Dubrovin, Geometry of ZD topological field theories, [in:] M. Francoviglia, S. Greco (eds.), Integrable systems and quantum groups, Lecture Notes in Math. 1620, Springer, Cham, 1996, 120-348.
• [7] B. Dubrovin, M. Mazzocco, Monodromy of certain Painleve VI transcendents and reflection groups, Invent. Math. 141 (2000), 55-147.
• [8] C. Hertling, Frobenius Manifolds and Moduli Spaces for Singularities, Cambridge University Press, Cambridge, 2002.
• [9] N.J. Hitchin, Poncelet polygons and the Painleve equation, [in:] Geometry and Analysis, Tata Inst. Fundam. Res. Stud. Math. (1995), 151-185.
• [10] N.J. Hitchin, Lectures on octahedron, Bull. Lond. Math. Soc. 35 (2003), 577-600. [11] K. Iwasaki, On algebraic solutions to Painleve VI, arXiv:0809.1482.
• [12] M. Kato, T. Mano, J. Sekiguchi, Flat structures without potentials, Rev. Roumaine Math. Pures Appl. 60 (2015) 4, 481-505.
• [13] M. Kato, T. Mano, J. Sekiguchi, Flat structure on the space of isomonodromic deformations, arXiv:1511.01608.
• [14] M. Kato, T. Mano, J. Sekiguchi, Flat structures and algebraic solutions to Painleve VI equation, [in:] G. Filipuk, Y. Haraoka, S. Michalik (eds.), Analytic, Algebraic and Geometric Aspects of Differential Equations, Trends Math., Birkhauser/Springer, Cham, 2017, 383-398.
• [15] A.V. Kitaev, Quadratic transformations for the sixth Painleve equation, Lett. Math. Phys. 21 (1991), 105-111.
• [16] A.V. Kitaev, Grothendieck's dessins d'enfants, their deformations and algebraic solutions of the sixth Painleve and Gauss hypergeometric equations, Algebra i Analiz 17 (2005) 1, 224-273.
• [17] A.V. Kitaev, Remarks towards a classification of RSA(i)-transform,ations and algebraic solutions of the sixth Painleve equation, Semin. Congr. 14 (2006), 199-227.
• [18] A.V. Kitaev, R. Vidunas, Quadratic transformations of the sixth Painleve equation with application to algebraic solutions, Math. Nachr. 280 (2007), 1834-1855.
• [19] Y. Konishi, S. Minabe, Local quantum cohomology and mixed Frobenius structure, arXiv: 1405.7476.
• [20] O. Lisovyy, Y. Tykhyy, Algebraic solutions of the sixth Painleve equation, J. Geom. Phys. 85 (2014), 124-163.
• [21] Yu. Manin, F-manifolds with flat structure and Dubrovin's duality, Adv. Math. 198 (2005) 1, 5-26.
• [22] M. Noumi, Y. Yamada, A new Lax pair for the sixth Painleve equation associated with so(8), [in:] T. Kawai, K. Fujita (eds.), Microlocal Analysis and Complex Fourier Analysis, World Scientific, River Edge, 2002, 238-252.
• [23] C. Sabbah, Lsomonodromic Deformations and Frobenius Manifolds. An Introduction, Universitext, Springer-Verlag, London, 2007.
• [24] K. Saito, On the uniformization of complements of discriminant loci, RIMS Kokyuroku, 287 (1977), 117-137.
• [25] K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 27 (1980), 265-291.
• [26] K. Saito, On a linear structure of the quotient variety by a finite reflexion group, Preprint RIMS-288 (1979), Publ. RIMS, Kyoto Univ. 29 (1993), 535-579.
• [27] K. Saito, T. Yano, J. Sekiguchi, On a certain generator system of the ring of invariants of a finite reflection group, Comm. Algebra 8 (1980), 373-408.
• [28] J. Sekiguchi, A classification of weighted homogeneous Saito free divisors, J. Math. Soc. Japan 61 (2009), 1071-1095.
• [29] J. Sekiguchi, Holonomic systems of differential equations of rank two with singularities along Saito free divisors of simple type, [in:] Topics on Real and Complex Singularities, Proceedings of the 4th Japanese-Australian Workshop, 2014, 159-188.
• [30] G.C. Shephard, A.J. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274-304.
• [31] R. Vidunas, A.V. Kitaev, Commutations of RS-pullback transformations for algebraic Painleve VL solutions, J. Math. Sci.(N.Y.) 213 (2016), 706-722.

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).