Simply-laced isomonodromy systems

Boalch, Philip

doi:10.1007/s10240-012-0044-8

Boalch, Philip ¹

¹ École Normale Supérieure et CNRS 45 rue d’Ulm, 75005, Paris France

Publications Mathématiques de l'IHÉS, Tome 116 (2012), pp. 1-68.

Résumé

A new class of isomonodromy equations will be introduced and shown to admit Kac–Moody Weyl group symmetries. This puts into a general context some results of Okamoto on the 4th, 5th and 6th Painlevé equations, and shows where such Kac–Moody Weyl groups and root systems occur “in nature”. A key point is that one may go beyond the class of affine Kac–Moody root systems. As examples, by considering certain hyperbolic Kac–Moody Dynkin diagrams, we find there is a sequence of higher order Painlevé systems lying over each of the classical Painlevé equations. This leads to a conjecture about the Hilbert scheme of points on some Hitchin systems.

MR Zbl

DOI : 10.1007/s10240-012-0044-8

Affiliations des auteurs :

Boalch, Philip ¹

¹ École Normale Supérieure et CNRS 45 rue d’Ulm, 75005, Paris France

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Boalch, Philip. Simply-laced isomonodromy systems. Publications Mathématiques de l'IHÉS, Tome 116 (2012), pp. 1-68. doi : 10.1007/s10240-012-0044-8. http://archive.numdam.org/articles/10.1007/s10240-012-0044-8/

Bibliographie
Cité par

[1.] Adams, M. R.; Harnad, J.; Hurtubise, J. Dual moment maps into loop algebras, Lett. Math. Phys., Volume 20 (1990), pp. 299-308 | DOI | MR | Zbl

[2.] Adams, M. R.; Harnad, J.; Previato, E. Isospectral Hamiltonian flows in finite and infinite dimensions, I. Generalised Moser systems and moment maps into loop algebras, Commun. Math. Phys., Volume 117 (1988), pp. 451-500 | DOI | MR | Zbl

[3.] Adler, M.; Moerbeke, P. Completely integrable systems, Euclidean Lie algebras, and curves, Adv. Math., Volume 38 (1980), pp. 267-317 | DOI | MR | Zbl

[4.] Balser, W.; Jurkat, W. B.; Lutz, D. A. On the reduction of connection problems for differential equations with an irregular singularity to ones with only regular singularities, I, SIAM J. Math. Anal., Volume 12 (1981), pp. 691-721 | DOI | MR | Zbl

[5.] Biquard, O.; Boalch, P. P. Wild non-Abelian Hodge theory on curves, Compos. Math., Volume 140 (2004), pp. 179-204 | DOI | MR | Zbl

[6.] Boalch, P. P. Symplectic manifolds and isomonodromic deformations, Adv. Math., Volume 163 (2001), pp. 137-205 | DOI | MR | Zbl

[7.] Boalch, P. P. G-bundles, isomonodromy and quantum Weyl groups, Int. Math. Res. Not., Volume 22 (2002), pp. 1129-1166 | DOI | MR | Zbl

[8.] Boalch, P. P. Painlevé equations and complex reflections, Ann. Inst. Fourier, Volume 53 (2003), pp. 1009-1022 (Proceedings of conference in honour of Frédéric Pham, Nice, July 2002) | DOI | Numdam | MR | Zbl

[9.] Boalch, P. P. From Klein to Painlevé via Fourier, Laplace and Jimbo, Proc. Lond. Math. Soc., Volume 90 (2005), pp. 167-208 (math.AG/0308221) | DOI | MR | Zbl

[10.] P. P. Boalch, Irregular connections and Kac–Moody root systems, preprint (June 2008), . | arXiv

[11.] Boalch, P. P. Quivers and difference Painlevé equations, Groups and Symmetries: From the Neolithic Scots to John McKay (CRM Proc. Lecture Notes, 47), AMS, Providence (2009), pp. 25-51 (arXiv:0706.2634) | MR | Zbl

[12.] Boalch, P. P. Towards a non-linear Schwarz’s list, The Many Facets of Geometry: A Tribute to Nigel Hitchin, Oxford Univ. Press, London (2010), pp. 210-236 (arXiv:0707.3375, July 2007) | MR | Zbl

[13.] Cassens, H.; Slodowy, P. On Kleinian singularities and quivers, Singularities, Progress in Mathematics (162), Birkhauser, Basel (1998), pp. 263-288 | DOI | MR | Zbl

[14.] Cosgrove, C. M. Higher-order Painlevé equations in the polynomial class. I. Bureau symbol P2, Stud. Appl. Math., Volume 104 (2000), pp. 1-65 | DOI | MR | Zbl

[15.] Crawley-Boevey, W. Geometry of the moment map for representations of quivers, Compos. Math., Volume 126 (2001), pp. 257-293 | DOI | MR | Zbl

[16.] Crawley-Boevey, W. On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero, Duke Math. J., Volume 118 (2003), pp. 339-352 | DOI | MR | Zbl

[17.] Crawley-Boevey, W.; Shaw, P. Multiplicative preprojective algebras, middle convolution and the Deligne-Simpson problem, Adv. Math., Volume 201 (2006), pp. 180-208 | DOI | MR | Zbl

[18.] Dickey, L. A. Soliton Equations and Hamiltonian Systems, Advanced Series in Mathematical Physics, 26, World Scientific, River Edge, 2003 | DOI | MR | Zbl

[19.] Donagi, R.; Witten, E. Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B, Volume 460 (1996), pp. 299-334 | DOI | MR | Zbl

[20.] R. Y. Donagi, Seiberg-Witten integrable systems. | arXiv | Zbl

[21.] Dubrovin, B. Geometry of 2D topological field theories, Integrable Systems and Quantum Groups (Lect. Notes Math., 1620), Springer, Berlin (1995), pp. 120-348 | DOI | MR | Zbl

[22.] Feingold, A. J.; Kleinschmidt, A.; Nicolai, H. Hyperbolic Weyl groups and the four normed division algebras, J. Algebra, Volume 322 (2009), pp. 1295-1339 | DOI | MR | Zbl

[23.] Garnier, R. Sur une classe de systèmes différentiels Abéliens déduits de la théorie des équations linéaires, Rend. Circ. Mat. Palermo, Volume 43 (1919), pp. 155-191 | DOI | JFM

[24.] Gotay, M. J.; Lashof, R.; Śniatycki, J.; Weinstein, A. Closed forms on symplectic fibre bundles, Comment. Math. Helv., Volume 58 (1983), pp. 617-621 | DOI | MR | Zbl

[25.] Guillemin, V.; Lerman, E.; Sternberg, S. Symplectic Fibrations and Multiplicity Diagrams, Cambridge Univ. Press, Cambridge, 1996 | DOI | MR | Zbl

[26.] Harnad, J. Dual isomonodromic deformations and moment maps to loop algebras, Commun. Math. Phys., Volume 166 (1994), pp. 337-365 | DOI | MR | Zbl

[27.] Jimbo, M.; Miwa, T.; Môri, Y.; Sato, M. Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendant, Physica, Volume 1D (1980), pp. 80-158 | MR | Zbl

[28.] Jimbo, M.; Miwa, T.; Ueno, K. Monodromy preserving deformations of linear differential equations with rational coefficients I, Physica D, Volume 2 (1981), pp. 306-352 | MR | Zbl

[29.] Joshi, N.; Kitaev, A. V.; Treharne, P. A. On the linearization of the Painlevé III-VI equations and reductions of the three-wave resonant system, J. Math. Phys., Volume 48 (2007), p. 42 | DOI | MR | Zbl

[30.] Kac, V. G. Infinite-Dimensional Lie Algebras, Cambridge Univ. Press, Cambridge, 1990 | DOI | MR | Zbl

[31.] Katz, N. M. Rigid Local Systems, Annals of Mathematics Studies, 139, Princeton University Press, Princeton, 1996 | MR | Zbl

[32.] King, A. D. Moduli of representations of finite-dimensional algebras, Q. J. Math. Oxf. Ser. (2), Volume 45 (1994), pp. 515-530 | DOI | MR | Zbl

[33.] Kraft, H.; Procesi, C. Closures of conjugacy classes of matrices are normal, Invent. Math., Volume 53 (1979), pp. 227-247 | DOI | MR | Zbl

[34.] Kronheimer, P. B. The construction of ALE spaces as hyper-Kähler quotients, J. Differ. Geom., Volume 29 (1989), pp. 665-683 | MR | Zbl

[35.] Malgrange, B. Le groupoide de Galois d’un feuilletage, Essays on Geometry and Related Topics, Monogr., 38, Enseignement Math., Geneva, 2001, pp. 465-501 | MR | Zbl

[36.] Moser, J. Geometry of quadrics and spectral theory, Proc. Internat. Sympos., Springer, New York (1980), pp. 147-188 | DOI | MR | Zbl

[37.] H. Nakajima, email correspondence, 10/6/08.

[38.] Nakajima, H. Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J., Volume 76 (1994), pp. 365-416 | DOI | MR | Zbl

[39.] Nakajima, H. Sheaves on ALE spaces and quiver varieties, Mosc. Math. J., Volume 7 (2007), pp. 699-722 | MR | Zbl

[40.] Noumi, M. Affine Weyl group approach to Painlevé equations, Beijing ICM Proc., Volume 3 (2002), p. 497 (math-ph/0304042) | MR | Zbl

[41.] Noumi, M.; Yamada, Y. Affine Weyl groups, discrete dynamical systems and Painlevé equations, Commun. Math. Phys., Volume 199 (1998), pp. 281-295 | DOI | MR | Zbl

[42.] Noumi, M.; Yamada, Y. Higher order Painlevé equations of type $A_{l}^{(1)}$ , Funkc. Ekvacioj, Volume 41 (1998), pp. 483-503 | MR | Zbl

[43.] Noumi, M.; Yamada, Y. A new Lax pair for the sixth Painlevé equation associated with $\hat{𝔰𝔬} (8)$ , Microlocal Analysis and Complex Fourier Analysis, World Scientific, Singapore (2002) | MR | Zbl

[44.] Ogg, A. P. Some special classes of Cartan matrices, Can. J. Math., Volume 36 (1984), pp. 800-819 | DOI | MR | Zbl

[45.] Okamoto, K. Studies on the Painlevé equations. III. Second and fourth Painlevé equations, P _II and P _IV , Math. Ann., Volume 275 (1986), pp. 221-255 | DOI | MR | Zbl

[46.] Okamoto, K. Studies on the Painlevé equations. I. Sixth Painlevé equation P _VI , Ann. Mat. Pura Appl. (4), Volume 146 (1987), pp. 337-381 | DOI | MR | Zbl

[47.] Okamoto, K. Studies on the Painlevé equations. II. Fifth Painlevé equation P _V , Jpn. J. Math. (N.S.), Volume 13 (1987), pp. 47-76 | MR | Zbl

[48.] Okamoto, K. The Painlevé equations and the Dynkin diagrams, Painlevé transcendents (NATO ASI Ser. B, 278), Plenum, New York (1992), pp. 299-313 | MR | Zbl

[49.] T. Oshima, Classification of Fuchsian systems and their connection problem, preprint (October 2008), http://kyokan.ms.u-tokyo.ac.jp/users/preprint/pdf/2008-29.pdf. | MR

[50.] Previato, E. Seventy years of spectral curves, Integrable Systems and Quantum Groups (Lect. Notes Math., 1620), Springer, Berlin (1996), pp. 419-481 | DOI | MR | Zbl

[51.] Sabbah, C. Isomonodromic Deformations and Frobenius Manifolds, Universitext, Springer, London, 2007 | MR | Zbl

[52.] H. Sakai, Isomonodromic deformation and 4-dimensional Painlevé type equations, preprint (October 2010), http://kyokan.ms.u-tokyo.ac.jp/users/preprint/pdf/2010-17.pdf. | MR

[53.] Sanguinetti, G.; Woodhouse, N. M. J. The geometry of dual isomonodromic deformations, J. Geom. Phys., Volume 52 (2004), pp. 44-56 | DOI | MR | Zbl

[54.] Y. Sasano, Four-dimensional Painlevé systems of types $D_{5}^{(1)}$ and $B_{4}^{(1)}$ . | arXiv

[55.] L. Schlesinger, Sur quelques problèmes paramétriques de la théorie des équations différentielles linéaires. Rome ICM talk (1908), pp. 64–68. http://www.mathunion.org/ICM/ICM1908.2/. | JFM

[56.] Szabó, S. Nahm transform for integrable connections on the Riemann sphere, Mém. Soc. Math. Fr. (N.S.), Volume 110 (2007) | Numdam | MR | Zbl

[57.] Umemura, H. Differential Galois theory of infinite dimension, Nagoya Math. J., Volume 144 (1996), pp. 59-135 | MR | Zbl

[58.] Woodhouse, N. M. J. Duality for the general isomonodromy problem, J. Geom. Phys., Volume 57 (2007), pp. 1147-1170 | DOI | MR | Zbl

[59.] Yamakawa, D. Middle convolution and Harnad duality, Math. Ann., Volume 349 (2011), pp. 215-262 | DOI | MR | Zbl

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