Symplectic manifolds and isomonodromic deformations. (English) Zbl 1001.53059

Boalch, Philip

Adv. Math. 163, No. 2, 137-205 (2001).

The author studies the moduli spaces of meromorphic connections with poles of arbitrary order over Riemann surfaces together with the corresponding spaces of monodromy data involving Stokes matrices. Generalizing the Atiyah-Bott approach [M. F. Atiyah and R. Bott, Phil. Trans. R. Soc. Lond. 308, 523-615 (1983; Zbl 0509.14014)], natural symplectic structures on extended spaces of singular connections are found and described both explicitly and from an infinite dimensional viewpoint. By the symplectic space construction an intrinsic symplectic description of the isomonodromic deformation equations of Jimbo, Miwa and Ueno is given and the existing results for the six Painlevé equations and Schlesinger’s equations are put into a uniform framework.

Reviewer: Stanislaw Janeczko (Warszawa)

MSC:

53D20	Momentum maps; symplectic reduction
34M55	Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
53D30	Symplectic structures of moduli spaces

Keywords:

symplectic manifold; Kähler structures; meromorphic connections; monodromy map

Citations:

Zbl 0509.14014

References:

[1]	Ablowitz, M. J.; Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering (1991), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0762.35001
[2]	Adams, M. R.; Harnad, J.; Hurtubise, J., Isospectral Hamiltonian flows in finite and infinite dimensions, II. Integration of the flows, Comm. Math. Phys., 134, 555-585 (1990) · Zbl 0717.58051
[3]	Alekseev, A.; Malkin, A.; Meinrenken, E., Lie group valued moment maps, J. Differential Geom., 48, 445-495 (1998) · Zbl 0948.53045
[4]	Andersen, J. E.; Mattes, J.; Reshetikhin, N., The Poisson structure on the moduli space of flat connections and chord diagrams, Topology, 35, 1069-1083 (1996) · Zbl 0857.58009
[5]	Atiyah, M. F., The Geometry and Physics of Knots (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0696.57002
[6]	Atiyah, M. F.; Bott, R., The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. London, 308, 523-615 (1982) · Zbl 0509.14014
[7]	Audin, M., Lectures on integrable systems and gauge theory, (Hurtubise, J.; Lalonde, F., Gauge Theory and Sympletic Geoemtry. Gauge Theory and Sympletic Geoemtry, NATO ASI Series C: Math & Phys. (1995), Kluwer: Kluwer Dordrecht/Boston/London) · Zbl 0873.58016
[8]	Babbitt, D. G.; Varadarajan, V. S., Formal reduction theory of meromorphic differential equations: a group theoretic view, Pacific J. Math., 109, 1-80 (1983) · Zbl 0533.34010
[9]	Babbitt, D. G.; Varadarajan, V. S., Local moduli for meromorphic differential equations, Astérisque, 169-170, 1-217 (1989) · Zbl 0683.34003
[10]	Balser, W.; Braaksma, B. J.L.; Ramis, J. P.; Sibuya, Y., Multisummability of formal power series solutions of linear ordinary differential equations, Asymptotic Anal., 2, 27-45 (1991) · Zbl 0754.34057
[11]	Balser, W.; Jurkat, W. B.; Lutz, D. A., Birkhoff in variants and Stokes’ multipliers for meromorphic linear differential equations, J. Math. Anal. Appl., 71, 48-94 (1979) · Zbl 0415.34008
[12]	Beauville, A., Jacobiennes des courbes spectrales et systèmes Hamiltoniens complètement intégrables, Acta Math., 164, 211-235 (1990) · Zbl 0712.58031
[13]	Biswas, I.; Guruprasad, K., Principal bundles on open surfaces and in variant functions on Lie groups, Internat. J. Math., 4, 535-544 (1993) · Zbl 0789.58017
[14]	Boalch, P. P., Symplectic Geometry and Isomonodromic Deformations (1999), Oxford University
[15]	Borel, A., Linear Algebraic Groups (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0726.20030
[16]	Bott, R.; Tu, L. W., Differential Forms in Algebraic Topology. Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, 82 (1995), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0496.55001
[17]	Boutet de Monvel, L.; Douady, A.; Verdier, J.-L., Séminaire E.N.S. Mathématique et Physique. Séminaire E.N.S. Mathématique et Physique, Progress in Math., 37 (1983), Birkhäuser: Birkhäuser Boston
[18]	Donagi, R.; Markman, E., Spectral covers, algebraically integrable Hamiltonian systems and moduli of bundles, (Francaviglia, M.; Greco, S., Integrable Systems and Quantum Groups. Integrable Systems and Quantum Groups, Lecture Notes in Mathematics, 1620 (1995), Springer-Verlag: Springer-Verlag Berlin), 1-119 · Zbl 0853.35100
[19]	Dubrovin, B., Geometry of 2D topological field theories, (Francaviglia, M.; Greco, S., Integrable Systems and Quantum Groups. Integrable Systems and Quantum Groups, Lecture Notes in Mathematics, 1620 (1995), Springer-Verlag: Springer-Verlag Berlin), 120-348 · Zbl 0841.58065
[20]	Faddeev, L. D.; Takhtajan, L. A., Hamiltonian Methods in the Theory of Solitons (1987), Springer-Verlag: Springer-Verlag Berlin · Zbl 1327.39013
[21]	Fock, V. V.; Rosly, A. A., Flat connections and Polyubles, Teoret. Mat. Fiz., 95, 228-238 (1993) · Zbl 0849.58030
[22]	Goldman, W. M., The symplectic nature of fundamental groups of surfaces, Adv. in Math., 54, 200-225 (1984) · Zbl 0574.32032
[23]	Griffiths, P.; Harris, J., Principles of Algebraic Geometry (1994), Wiley Interscience: Wiley Interscience New York · Zbl 0836.14001
[24]	Guillemin, V.; Sternberg, S., Symplectic Techniques in Physics (1984), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0576.58012
[25]	Guruprasad, K.; Huebschmann, J.; Jeffrey, L.; Weinstein, A., Group systems, groupoids and moduli spaces of parabolic bundles, Duke Math. J., 89, 377-412 (1997) · Zbl 0885.58011
[26]	Hamilton, R. S., The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (New Ser.), 7, 65-222 (1982) · Zbl 0499.58003
[27]	Harnad, J., Quantum isomonodromic deformations and the Knizhnik-Zamolodchikov equations, Symmetries and Integrability of Difference Equations Estérel, PQ, 1994 (1996), Amer. Math. Soc: Amer. Math. Soc Providence, p. 155-161 · Zbl 0860.35123
[28]	Harnad, J.; Wisse, M. A., Loop algebra moment maps and Hamiltonian models for the Painlevé transcendants, Fields Inst. Commun., 7, 155-169 (1993) · Zbl 0871.58049
[29]	Hitchin, N. J., Frobenius manifolds (notes by D. Calderbank), (Hurtubise, J.; Lalonde, F., Gauge Theory and Sympletic Geometry. Gauge Theory and Sympletic Geometry, NATO ASI Series C: Maths and Phys., 488 (1995), Kluwer: Kluwer Dordrecht/Boston/London) · Zbl 0867.53027
[30]	Hitchin, N. J., Twistor spaces, Einstein metrics and isomonodromic deformations, J. Differential Geom., 42, 30-112 (1995) · Zbl 0861.53049
[31]	Hörmander, L., The Analysis of Linear Partial Differential Operators. I (1983), Springer-Verlag: Springer-Verlag Berlin · Zbl 0521.35001
[32]	Hsieh, P.; Sibuya, Y., Note on regular perturbations of linear ordinary differential equations at irregular singular points, Funkcial. Ekvac., 8, 99-108 (1966) · Zbl 0145.10404
[33]	Hurtubise, J.; Lalonde, F., Gauge Theory and Symplectic Geometry. Gauge Theory and Symplectic Geometry, NATO ASI Series C: Maths Phys., 488 (1995), Kluwer: Kluwer Dordrecht/Boston/London
[34]	Iwasaki, K., Moduli and deformation for Fuchsian projective connections on a Riemann surface, J . Fac. Sci. Univ. Tokyo, Sect. 1A, Math., 38, 431-531 (1991) · Zbl 0754.30032
[35]	Iwasaki, K., Fuchsian moduli on a Riemann surface - its Poisson structure and Poincaré Lefschetz duality, Pacific J. Math., 155, 319-340 (1992) · Zbl 0770.32012
[36]	Iwasaki, K.; Kimura, H.; Shimomura, S.; Yoshida, M., From Gauss to Painlevé: A Modern Theory of Special Functions (1991), Aspects of Mathematics: Aspects of Mathematics Vieweg · Zbl 0743.34014
[37]	Jeffrey, L., Extended moduli spaces of flat connections on Riemann surfaces, Math. Ann., 298, 667-692 (1994) · Zbl 0794.53017
[38]	Jimbo, M.; Miwa, T., Monodromy preserving deformations of linear differential equations with rational coefficients II, Physica 2D, 407-448 (1981) · Zbl 1194.34166
[39]	Jimbo, M.; Miwa, T.; Mori, Y.; Sato, M., Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendant, Physica 1D, 80-158 (1980) · Zbl 1194.82007
[40]	Jimbo, M.; Miwa, T.; Ueno, Kimio, Monodromy preserving deformations of linear differential equations with rational coefficients I, Physica 2D, 306-352 (1981) · Zbl 1194.34167
[41]	Karshon, Y., An algebraic proof for the symplectic structure of moduli space, Proc. Amer. Math. Soc., 116, 591-605 (1992) · Zbl 0790.14012
[42]	Kobayashi, S., Differential Geometry of Complex Vector Bundles. Differential Geometry of Complex Vector Bundles, Kanô Memorial Lectures, 5 (1987), Iwanami Shoten/Princeton University Press: Iwanami Shoten/Princeton University Press Tokyo/Princeton · Zbl 0708.53002
[43]	Loday-Richaud, M., Stokes phenomenon, multisummability and differential Galois groups, Ann. Inst. Fourier, 44, 849-906 (1994) · Zbl 0812.34004
[44]	Malgrange, B., Ideals of Differentiable Functions (1966), Bombay/Oxford University PressTATA Institute of fundamental Research: Bombay/Oxford University PressTATA Institute of fundamental Research Oxford · Zbl 0177.17902
[45]	Malgrange, B., Sur les déformations isomonodromiques, I, singularités regulières, (Boutet de Monvel, L.; Douady, A.; Verdier, J.-L., Seminaire E.N.S. Mathématique et Physique. Seminaire E.N.S. Mathématique et Physique, Progress in Math., 37 (1983), Birkhäuser: Birkhäuser Boston), 401-426 · Zbl 0528.32017
[46]	Malgrange, B., Sur les déformations isomonodromiques, II, singularités irregulières, (Boutet de Monvel, L.; Douady, A.; Verdier, J.-L., Seminaire E.N.S. Mathématique et Physique. Seminaire E.N.S. Mathématique et Physique, Progress in Math., 37 (1983), Birkhäuser: Birkhäuser Boston), 427-438 · Zbl 0528.32018
[47]	Malgrange, B., Sommation des séries divergentes, Exposition. Math., 13, 163-222 (1995) · Zbl 0836.40004
[48]	Malgrange, B.; Ramis, J.-P., Fonctions multisommables, Ann. Inst. Fourier (Grenoble), 42, 353-368 (1992) · Zbl 0759.34007
[49]	Martinet, J., Singularities of Smooth Functions and Maps (1982), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0522.58006
[50]	Martinet, J.; Ramis, J. P., Elementary acceleration and multisummability, Ann. inst. H. Poincaré, Phys. Théorique, 54, 331-401 (1991) · Zbl 0748.12005
[51]	Mason, L. J.; Woodhouse, N. M.J., Integrability, Self-Duality and Twistor Theory (1996), Oxford Univ. Press: Oxford Univ. Press Oxford · Zbl 0856.58002
[52]	McCoy, B. M.; Tracy, C. A.; Wu, T. T., Painlevé functions of the third kind, J. Math. Phys., 18, 1058-1092 (1977) · Zbl 0353.33008
[53]	Francaviglia, M.; Greco, S., Integrable Systems and Quantum Groups. Integrable Systems and Quantum Groups, Lecture Notes in Mathematics, 1620 (1995), Springer: Springer Berlin
[54]	Milnor, J., Remarks on infinite-dimensional Lie groups, Relativity, Groups and Topology, II Les Houches, 1983 (1984), North-Holland: North-Holland Amsterdam, p. 1007-1057 · Zbl 0594.22009
[55]	Miwa, T., Painlevé property of monodromy preserving deformation equations and the analyticity of \(τ\) functions, Publ. RIMS, Kyoto Univ., 17, 703-721 (1981) · Zbl 0605.34005
[56]	Narasimhan, M. S.; Seshadri, C. S., Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math., 82, 540-567 (1965) · Zbl 0171.04803
[57]	Okamoto, K., On the \(τ\) function of the Painlevé equations, Physica 2D, 525-535 (1981) · Zbl 1194.34171
[58]	Range, R. M., Holomorphic Functions and Integral Representations in Several Complex Variables. Holomorphic Functions and Integral Representations in Several Complex Variables, Graduate Texts in Mathematics, 108 (1986), Springer-Verlag: Springer-Verlag Berlin · Zbl 0591.32002
[59]	Reshetikhin, N., The Knizhnik-Zamolodchikov system as a deformation of the isomonodromy problem, Lett. Math. Phys., 26, 167-177 (1992) · Zbl 0776.17016
[60]	Sato, M.; Miwa, T.; Jimbo, M., Holonomic quantum fields I-V, Publ. RIMS Kyoto Univ., 14, 223-267 (1978) · Zbl 0383.35066
[61]	Schlesinger, L., Über eine Klasse von Differentialsystemen beliebiger Ordnung mit feten kritischen Punkten, J. Math., 141, 96-145 (1912) · JFM 43.0385.01
[62]	Sibuya, Y., Simplification of a system of linear ordinary differential equations about a singular point, Funkcial. Ekvac., 4, 29-56 (1962) · Zbl 0145.32305
[63]	Sibuya, Y., Perturbation of linear ordinary differential equations at irregular singular points, Funkcial. Ekvac., 11, 235-246 (1968) · Zbl 0228.34036
[64]	C. T . Simpson, Moduli of representations of the fundamental group of a smooth projective variety, I, II, Publ. Math. I.H.E.S.791994, 47-129; 80; C. T . Simpson, Moduli of representations of the fundamental group of a smooth projective variety, I, II, Publ. Math. I.H.E.S.791994, 47-129; 80 · Zbl 0891.14005
[65]	Tod, K. P., Self-dual Einstein metrics from the Painlevé VI equation, Phys. Lett. A, 190, 221-224 (1994) · Zbl 0960.83505
[66]	Treves, F., Topological Vector Spaces, Distributions and Kernels. Topological Vector Spaces, Distributions and Kernels, Pure and Applied Mathematics Monographs, 25 (1967), Acadamic Press: Acadamic Press New York, London · Zbl 0171.10402
[67]	Ugaglia, M., On a Poisson structure on the space of Stokes matrices, Internat. Math. Res. Notices, 9, 473-493 (1999) · Zbl 0939.34073
[68]	Umemura, H., Irreducibility of Painlevé transcendental functions, Sugaku Expositions, 2, 231-252 (1989) · Zbl 0807.12005
[69]	Umemura, H., Painlevé equations and classical functions, Sugaku Expositions, 11, 77-100 (1998) · Zbl 1011.34075
[70]	Varadarajan, V. S., Linear meromorphic differential equations: a modern point of view, Bull. AMS, 33, 1-42 (1996) · Zbl 0862.34004
[71]	Vergne, M., La structure de Poisson sur l’algèbre symétrique d’une algèbre de Lie nilpotente, Bull. Soc. Math. France, 100, 301-335 (1972) · Zbl 0256.17002
[72]	Wasow, W., Asymptotic Expansions for Ordinary Differential Equations (1976), Wiley Interscience: Wiley Interscience New York · Zbl 0169.10903
[73]	Wu, T. T.; McCoy, B. M.; Tracy, C. A.; Barouch, E., Spin-spin correlation functions for the two-dimensional Ising model: exact theory in the scaling region, Phys. Rev. B, 13, 316-374 (1976)