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

SIGMA 16 (2020), 110, 36 pages      arXiv:1511.01608      https://doi.org/10.3842/SIGMA.2020.110
Contribution to the Special Issue on Primitive Forms and Related Topics in honor of Kyoji Saito for his 77th birthday

Flat Structure on the Space of Isomonodromic Deformations

Mitsuo Kato a, Toshiyuki Mano b and Jiro Sekiguchi c
a) Department of Mathematics, College of Educations, University of the Ryukyus, Japan
b) Department of Mathematical Sciences, Faculty of Science, University of the Ryukyus, Japan
c) Department of Mathematics, Faculty of Engineering, Tokyo University of Agriculture and Technology, Japan

Received March 19, 2020, in final form October 21, 2020; Published online November 03, 2020

Abstract
Flat structure was introduced by K. Saito and his collaborators at the end of 1970's. Independently the WDVV equation arose from the 2D topological field theory. B. Dubrovin unified these two notions as Frobenius manifold structure. In this paper, we study isomonodromic deformations of an Okubo system, which is a special kind of systems of linear differential equations. We show that the space of independent variables of such isomonodromic deformations can be equipped with a Saito structure (without a metric), which was introduced by C. Sabbah as a generalization of Frobenius manifold. As its consequence, we introduce flat basic invariants of well-generated finite complex reflection groups and give explicit descriptions of Saito structures (without metrics) obtained from algebraic solutions to the sixth Painlevé equation.

Key words: flat structure; Frobenius manifold; WDVV equation; complex reflection group; Painlevé equation.

pdf (620 kb)   tex (48 kb)  

References

  1. Arsie A., Lorenzoni P., From the Darboux-Egorov system to bi-flat $F$-manifolds, J. Geom. Phys. 70 (2013), 98-116, arXiv:1205.2468.
  2. Arsie A., Lorenzoni P., Complex reflection groups, logarithmic connections and bi-flat $F$-manifolds, Lett. Math. Phys. 107 (2017), 1919-1961, arXiv:1604.04446.
  3. Arsie A., Lorenzoni P., $F$-manifolds, multi-flat structures and Painlevé transcendents, Asian J. Math. 23 (2019), 877-904, arXiv:1501.06435.
  4. Balser W., Jurkat W.B., Lutz D.A., On the reduction of connection problems for differential equations with an irregular singular point to ones with only regular singularities. I, SIAM J. Math. Anal. 12 (1981), 691-721.
  5. Bessis D., Finite complex reflection arrangements are $K(\pi,1)$, Ann. of Math. 181 (2015), 809-904, arXiv:math.GT/0610777.
  6. Bessis D., Michel J., Explicit presentations for exceptional braid groups, Experiment. Math. 13 (2004), 257-266, arXiv:math.GR/0312191.
  7. Boalch P., Painlevé equations and complex reflections, Ann. Inst. Fourier (Grenoble) 53 (2003), 1009-1022, arXiv:1305.6462.
  8. Boalch P., From Klein to Painlevé via Fourier, Laplace and Jimbo, Proc. London Math. Soc. 90 (2005), 167-208, arXiv:math.AG/0308221.
  9. Boalch P., The fifty-two icosahedral solutions to Painlevé VI, J. Reine Angew. Math. 596 (2006), 183-214, arXiv:math.AG/0406281.
  10. Boalch P., Six results on Painlevé VI, in Théories asymptotiques et équations de Painlevé, Sémin. Congr., Vol. 14, Soc. Math. France, Paris, 2006, 1-20, arXiv:math.AG/0503043.
  11. Boalch P., Higher genus icosahedral Painlevé curves, Funkcial. Ekvac. 50 (2007), 19-32, arXiv:math.AG/0506407.
  12. Boalch P., Some explicit solutions to the Riemann-Hilbert problem, in Differential equations and quantum groups, IRMA Lect. Math. Theor. Phys., Vol. 9, Eur. Math. Soc., Zürich, 2007, 85-112, arXiv:math.DG/0501464.
  13. Boalch P., Towards a non-linear Schwarz's list, in The many facets of geometry, Oxford University Press, Oxford, 2010, 210-236.
  14. David L., Hertling C., Regular $F$-manifolds: initial conditions and Frobenius metrics, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), 1121-1152, arXiv:1411.4553.
  15. Dettweiler M., Reiter S., An algorithm of Katz and its application to the inverse Galois problem, J. Symbolic Comput. 30 (2000), 761-798.
  16. Dettweiler M., Reiter S., Middle convolution of Fuchsian systems and the construction of rigid differential systems, J. Algebra 318 (2007), 1-24.
  17. Dijkgraaf R., Verlinde H., Verlinde E., Notes on topological string theory and $2$D quantum gravity, in String Theory and Quantum Gravity (Trieste, 1990), World Sci. Publ., River Edge, NJ, 1991, 91-156.
  18. 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, arXiv:hep-th/9407018.
  19. Dubrovin B., On almost duality for Frobenius manifolds, in Geometry, Topology, and Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 212, Amer. Math. Soc., Providence, RI, 2004, 75-132, arXiv:math.DG/0307374.
  20. Dubrovin B., Mazzocco M., Monodromy of certain Painlevé-VI transcendents and reflection groups, Invent. Math. 141 (2000), 55-147, arXiv:math.AG/9806056.
  21. Haraoka Y., Linear differential equations on a complex domain, Sugakushobou, 2015 (in Japanese).
  22. Haraoka Y., Kato M., Generating systems for finite irreducible complex reflection groups, Funkcial. Ekvac. 53 (2010), 435-488.
  23. Hertling C., Frobenius manifolds and moduli spaces for singularities, Cambridge Tracts in Mathematics, Vol. 151, Cambridge University Press, Cambridge, 2002.
  24. Hertling C., Manin Yu., Weak Frobenius manifolds, Int. Math. Res. Not. 1999 (1999), 277-286, arXiv:math.QA/9810132.
  25. Hitchin N.J., Poncelet polygons and the Painlevé equations, in Geometry and Analysis (Bombay, 1992), Tata Inst. Fund. Res., Bombay, 1995, 151-185.
  26. Hitchin N.J., A lecture on the octahedron, Bull. London Math. Soc. 35 (2003), 577-600.
  27. Hoge T., Mano T., Röhrle G., Stump C., Freeness of multi-reflection arrangements via primitive vector fields, Adv. Math. 350 (2019), 63-96, arXiv:1703.08980.
  28. Iwasaki K., On algebraic solutions to Painlevé VI, arXiv:0809.1482.
  29. 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.
  30. Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D 2 (1981), 407-448.
  31. Kato M., Mano T., Sekiguchi J., Flat structures without potentials, Rev. Roumaine Math. Pures Appl. 60 (2015), 481-505.
  32. Kato M., Mano T., Sekiguchi J., Flat structures and algebraic solutions to Painlevé VI equation, in Analytic, Algebraic and Geometric Aspects of Differential Equations, Trends Math., Birkhäuser/Springer, Cham, 2017, 383-398.
  33. Kato M., Mano T., Sekiguchi J., Flat structure and potential vector fields related with algebraic solutions to Painlevé VI equation, Opuscula Math. 38 (2018), 201-252.
  34. Kato M., Sekiguchi J., Uniformization systems of equations with singularities along the discriminant sets of complex reflection groups of rank three, Kyushu J. Math. 68 (2014), 181-221.
  35. Katz N.M., Rigid local systems, Annals of Mathematics Studies, Vol. 139, Princeton University Press, Princeton, NJ, 1996.
  36. Kawakami H., Generalized Okubo systems and the middle convolution, Int. Math. Res. Not. 2010 (2010), 3394-3421.
  37. Kawakami H., Mano T., Regular flat structure and generalized Okubo system, Comm. Math. Phys. 369 (2019), 403-431, arXiv:1702.03074.
  38. Kitaev A.V., Quadratic transformations for the sixth Painlevé equation, Lett. Math. Phys. 21 (1991), 105-111.
  39. Kitaev A.V., Grothendieck's dessins d'enfants, their deformations, and algebraic solutions of the sixth Painlevé and Gauss hypergeometric equations, St. Petersburg Math. J. 17 (2006), 169-206.
  40. Konishi Y., Minabe S., Mixed Frobenius structure and local quantum cohomology, Publ. Res. Inst. Math. Sci. 52 (2016), 43-62, arXiv:1405.7476.
  41. Konishi Y., Minabe S., Shiraishi Y., Almost duality for Saito structure and complex reflection groups, J. Integrable Syst. 3 (2018), xyy003, 48 pages, arXiv:1612.03643.
  42. Lisovyy O., Tykhyy Y., Algebraic solutions of the sixth Painlevé equation, J. Geom. Phys. 85 (2014), 124-163, arXiv:0809.4873.
  43. Lorenzoni P., Darboux-Egorov system, bi-flat $F$-manifolds and Painlevé VI, Int. Math. Res. Not. 2014 (2014), 3279-3302, arXiv:1207.5979.
  44. Manin Yu.I., $F$-manifolds with flat structure and Dubrovin's duality, Adv. Math. 198 (2005), 5-26, arXiv:math.DG/0402451.
  45. Maschke H., Aufstellung des vollen Formensystems einer quaternären Gruppe von 51840 linearen Substitutionen, Math. Ann. 33 (1889), 317-344.
  46. Okubo K., Connection problems for systems of linear differential equations, in Japan-United States Seminar on Ordinary Differential and Functional Equations (Kyoto, 1971), Lecture Notes in Math., Vol. 243, Springer, Berlin - Heidelberg, 1971, 238-248.
  47. Orlik P., Basic derivations for unitary reflection groups (with an appendix by Hiroaki Terao and Yoichi Enta), in Singularities (Iowa City, IA, 1986), Contemp. Math., Vol. 90, Amer. Math. Soc., Providence, RI, 1989, 211-228.
  48. Orlik P., Solomon L., Discriminants in the invariant theory of reflection groups, Nagoya Math. J. 109 (1988), 23-45.
  49. Orlik P., Terao H., Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften, Vol. 300, Springer-Verlag, Berlin, 1992.
  50. Oshima T., Fractional calculus of Weyl algebra and Fuchsian differential equations, MSJ Memoirs, Vol. 28, Mathematical Society of Japan, Tokyo, 2012.
  51. Oshima T., Classification of Fuchsian systems and their connection problem, in Exact WKB analysis and microlocal analysis, RIMS Kôkyûroku Bessatsu, Vol. B37, Res. Inst. Math. Sci. (RIMS), Kyoto, 2013, 163-192, arXiv:0811.2916.
  52. Oshima T., Katz's middle convolution and Yokoyama's extending operation, Opuscula Math. 35 (2015), 665-688, arXiv:0812.1135.
  53. Sabbah C., Isomonodromic deformations and Frobenius manifolds. An introduction, Universitext, Springer-Verlag London, Ltd., London, 2007.
  54. Saito K., Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 265-291.
  55. Saito K., On a linear structure of the quotient variety by a finite reflexion group, Publ. Res. Inst. Math. Sci. 29 (1993), 535-579.
  56. 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), 373-408.
  57. Saito M., On the structure of Brieskorn lattice, Ann. Inst. Fourier (Grenoble) 39 (1989), 27-72.
  58. Sekiguchi J., A classification of weighted homogeneous Saito free divisors, J. Math. Soc. Japan 61 (2009), 1071-1095.
  59. Sekiguchi J., Systems of uniformization equations along Saito free divisors and related topics, in The Japanese-Australian Workshop on Real and Complex Singularities - JARCS III, Proc. Centre Math. Appl. Austral. Nat. Univ., Vol. 43, Austral. Nat. Univ., Canberra, 2010, 83-126.
  60. Shephard G.C., Todd J.A., Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274-304.
  61. Terao H., Free arrangements of hyperplanes and unitary reflection groups, Proc. Japan Acad. Ser. A Math. Sci. 56 (1980), 389-392.
  62. Vidūnas R., Kitaev A.V., Quadratic transformations of the sixth Painlevé equation with application to algebraic solutions, Math. Nachr. 280 (2007), 1834-1855, arXiv:math.CA/0511149.
  63. Vidūnas R., Kitaev A.V., Computation of $RS$-pullback transformations for algebraic Painlevé VI solutions, J. Math. Sci. 213 (2016), 706-722, arXiv:0705.2963.
  64. Witten E., On the structure of the topological phase of two-dimensional gravity, Nuclear Phys. B 340 (1990), 281-332.
  65. Yokoyama T., Construction of systems of differential equations of Okubo normal form with rigid monodromy, Math. Nachr. 279 (2006), 327-348.

Previous article  Next article  Contents of Volume 16 (2020)