Geodesic
Degenerating the elliptic Schlesinger system
Teoretičeskaâ i matematičeskaâ fizika, Tome 174 (2013) no. 1, pp. 3-24 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study various ways of degenerating the Schlesinger system on the elliptic curve with $R$ marked points. We construct a limit procedure based on an infinite shift of the elliptic curve parameter and on shifts of the marked points. We show that using this procedure allows obtaining a nonautonomous Hamiltonian system describing the Toda chain with additional spin $\mathfrak{sl}(N,\mathbb C)$ degrees of freedom.
Keywords: integrable system, Schlesinger system, Toda chain, Inozemtsev limit.
Mots-clés : isomonodromic deformation 
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G. Aminov; S. Arthamonov. Degenerating the elliptic Schlesinger system. Teoretičeskaâ i matematičeskaâ fizika, Tome 174 (2013) no. 1, pp. 3-24. http://geodesic.mathdoc.fr/item/TMF_2013_174_1_a0/

[1] A. M. Levin, M. A. Olshanetsky, Amer. Math. Soc. Transl. Ser. 2, 191 (1999), 223–262 | MR | Zbl

[2] K. Takasaki, Lett. Math. Phys., 44:2 (1998), 143–156 | DOI | MR | Zbl

[3] D. A. Korotkin, “Isomonodromic deformations in genus zero and one: algebro-geometric solutions and Schlesinger transformations”, Integrable Systems: From Classical to Quantum (Montréal, Canada, July 26 – August 6, 1999), CRM Proceedings and Lecture Notes, 26, eds. J. Harnad, G. Sabidussi, P. Winternitz, AMS, Providence, RI, 2000, 87–104 | DOI | MR | Zbl

[4] Yu. Chernyakov, A. Levin, M. Olshanetsky, A. Zotov, J. Phys. A, 39:39 (2006), 12083–12101 | DOI | MR | Zbl

[5] L. Schlesinger, J. für Math., 141 (1912), 96–145 | Zbl

[6] R. Conte, The Painlevé Property. One Century Later, CRM Series in Mathematical Physics, Springer, New York, 1999 | MR | Zbl

[7] A. R. Its, V. Yu. Novokshenov, Isomonodromic Deformation Method in the Theory of Painleve Equations, Lecture Notes in Mathematics, 1191, Springer, Berlin, 1986 | DOI | MR | Zbl

[8] A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov, A. Orlov, Nucl. Phys. B, 357:2–3 (1991), 565–618 | DOI | MR

[9] V. G. Knizhnik, A. B. Zamolodchikov, Nucl. Phys. B, 247:1 (1984), 83–103 | DOI | MR | Zbl

[10] D. Korotkin, H. Samtleben, Internat. J. Modern Phys. A, 12:11 (1997), 2013–2029 | DOI | MR | Zbl

[11] N. Seiberg, E. Witten, Nucl. Phys. B, 426:1 (1994), 19–52, arXiv: hep-th/9407087 | DOI | MR | Zbl

[12] A. Gorsky, S. Gukov, A. Mironov, Nucl. Phys. B, 517:1–3 (1998), 409–461, arXiv: hep-th/9707120 | DOI | MR | Zbl

[13] A. Gorsky, S. Gukov, A. Mironov, Nucl. Phys. B, 518:3 (1998), 689–713, arXiv: hep-th/9710239 | DOI | MR | Zbl

[14] V. I. Inozemtsev, Commun. Math. Phys., 121:4 (1989), 629–638 | DOI | MR | Zbl

[15] A. V. Zotov, Yu. B. Chernyakov, TMF, 129:2 (2001), 258–277 | DOI | DOI | MR | Zbl

[16] A. M. Levin, M. A. Olshanetsky, A. V. Zotov, Commun. Math. Phys., 236:1 (2003), 93–133 | DOI | MR | Zbl

[17] A. V. Smirnov, TMF, 158:3 (2009), 355–369 | DOI | DOI | MR | Zbl

[18] K. Okamoto, J. Fac. Sci. Univ. Tokyo, 33:3 (1986), 575–618 | MR | Zbl

[19] R. Fuchs, C. R. Acad. Sci., 141 (1905), 555–558

[20] E. K. Sklyanin, T. Takebe, Phys. Lett. A, 219:3–4 (1996), 217–225 | DOI | MR | Zbl

[21] B. Enriquez, V. Rubtsov, Math. Res. Lett., 3 (1996), 343–357 | DOI | MR | Zbl

[22] N. Nekrasov, Commun. Math. Phys., 180:3 (1996), 587–603, arXiv: hep-th/9503157 | DOI | MR | Zbl

[23] G. Aminov, S. Arthamonov, J. Phys. A, 44:7 (2011), 075201, 34 pp. | DOI | MR | Zbl

[24] S. B. Artamonov, TMF, 171:2 (2012), 196–207 | DOI | DOI

[25] G. A. Aminov, TMF, 171:2 (2012), 179–195 | DOI | DOI | Zbl

[26] A. A. Belavin, Nucl. Phys. B, 180:2 (1981), 189–200 | DOI | MR | Zbl

[27] A. A. Belavin, V. G. Drinfeld, Funkts. analiz i ego pril., 16:3 (1982), 1–29 | DOI | MR | Zbl

[28] P. P. Kulish, E. K. Sklyanin, Zap. nauchn. sem. LOMI, 95 (1980), 129–160 | DOI | MR | Zbl

[29] V.I. Arnold, Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1974 | MR | Zbl

[30] A. Veil, Ellipticheskie funktsii po Eizenshteinu i Kronekeru, Mir, M., 1978 | MR | Zbl

[31] A. G. Reiman, M. A. Semenov-Tyan-Shanskii, Zap. nauchn. sem. LOMI, 150 (1986), 104–118 | DOI | MR

[32] S. V. Manakov, ZhETF, 67:2 (1974), 543–555 | MR

[33] H. Flashka, Phys. Rev. B, 9:4 (1974), 1924–1925 | DOI | MR

[34] H. Flashka, Progr. Theoret. Phys., 51:3 (1974), 703–716 | DOI | MR

[35] D. Mamford, Lektsii o teta-funktsiyakh, Mir, M., 1988 | MR | Zbl