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Modifications of bundles, elliptic integrable systems, and related problems

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We describe a construction of elliptic integrable systems based on bundles with nontrivial characteristic classes, especially attending to the bundle-modification procedure, which relates models corresponding to different characteristic classes. We discuss applications and related problems such as the Knizhnik-Zamolodchikov-Bernard equations, classical and quantum R-matrices, monopoles, spectral duality, Painlevé equations, and the classical-quantum correspondence. For an SL(N,ℂ)-bundle on an elliptic curve with nontrivial characteristic classes, we obtain equations of isomonodromy deformations.

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References

  1. V. I. Arnold, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  2. M. A. Olshanetsky and A. M. Perelomov, Phys. Rep., 71, 313–400 (1981).

    Article  MathSciNet  ADS  Google Scholar 

  3. N. Hitchin, Duke Math. J., 54, 91–114 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Gorsky and N. Nekrasov, “Elliptic Calogero-Moser system from two dimensional current algebra,” arXiv:hepth/9401021v1 (1994).

    Google Scholar 

  5. A. Levin and M. Olshanetsky, Commun. Math. Phys., 188, 449–466 (1997).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. A. Levin, M. Olshanetsky, and A. Zotov, Commun. Math. Phys., 236, 93–133 (2003); arXiv:nlin/0110045v3 (2001).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Yu. Chernyakov, A. Levin, M. Olshanetsky, and A. Zotov, J. Phys. A, 39, 12083–12101 (2006); arXiv:nlin/0602043v1 (2006).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. I. M. Krichever, Russ. Math. Surveys, 32, 185–213 (1977); I. M. Krichever and S. P. Novikov, Russ. Math. Surveys, 35, 53–79 (1980); B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, “Integrable systems I,” Sovrem. Probl. Mat. Fund. Naprav., 4, 210–315 (1985).

    Article  ADS  MATH  Google Scholar 

  9. V. E. Zakharov and A. B. Shabat, Funct. Anal. Appl., 8, No. 3, 226–235 (1974); V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: Method of the Inverse Problem [in Russian], Nauka, Moscow (1980).

    Article  MATH  Google Scholar 

  10. K. Iwasaki, H. Kimura, S. Shimomura, and M. Yoshida, From Gauss to Painlevé, a Modern Theory of Special Functions (Aspects Mathematics, Vol. E16), Friedr. Vieweg and Sohn, Braunschweig (1991).

    Book  MATH  Google Scholar 

  11. R. Conte, ed., The Painlevé Property: One Century Later, Springer, New York (1999).

    MATH  Google Scholar 

  12. R. Garnier, Ann. Sci. École Norm. Sup. 3, 29, 1–126 (1912).

    MathSciNet  MATH  Google Scholar 

  13. M. Jimbo, T. Miwa, and K. Ueno, Phys. D, 2, 306–352 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Jimbo and T. Miwa, Phys. D, 2, 407–448 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Jimbo and T. Miwa, Phys. D, 4, 26–46 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Its and V. Novokshenov, The Isomonodromic Deformation Method in the Theory of Painlevé Equations (Lect. Notes Math., Vol. 1191), Springer, Berlin (1986); A. Fokas, A. Its, A. Kapaev, and V. Novokshenov, Painlevé Transcendents: The Riemann-Hilbert Approach (AMS Math. Surv. Monogr., Vol. 128), Amer. Math. Soc., Providence, R. I. (2006).

    MATH  Google Scholar 

  17. P. Painlevé, C. R. Acad. Sci. (Paris), 143, 1111–01117 (1906).

    Google Scholar 

  18. Yu. Manin, Amer. Math. Soc. Transl. Ser. 2, 186, 131–151 (1998).

    MathSciNet  Google Scholar 

  19. V. I. Inozemtsev, Lett. Math. Phys., 17, 11–17 (1989).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. A. Levin and M. Olshanetsky, “Painlevé-Calogero correspondence,” in: Calogero-Moser-Sutherlend Models (Montréal, QC, 10–15 March 1997, J. F. van Diejen and L. Vinet, eds.), Springer, New York (2000), pp. 313–332.

    Chapter  Google Scholar 

  21. G. Aminov, S. Arthamonov, A. Levin, M. Olshanetsky, and A. Zotov, “Painlevé field theory,” Preprint ITEP-TH-53/12, Inst. Theor. Exp. Phys., Moscow (2012); arXiv:1306.3265v1 [math-ph] (2013); G. Aminov, A. Mironov, A. Morozov, and A. Zotov, “Three-particle integrable systems with elliptic dependence on momenta and theta function identities,” arXiv:1307.1465v1 [hep-th] (2013).

    Google Scholar 

  22. A. M. Levin, M. A. Olshanetsky, A. V. Smirnov, and A. V. Zotov, Commun. Math. Phys., 316, 1–44 (2012); arXiv:1006.0702v4 [math-ph] (2010).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. A. M. Levin, M. A. Olshanetsky, A. V. Smirnov, and A. V. Zotov, J. Geom. Phys., 62, 1810–1850 (2012); arXiv:1007.4127v2 [math-ph] (2010).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. I. M. Krichever, J. Soviet Math., 28, 51–90 (1985); I. M. Krichever, Funct. Anal. Appl., 14, No. 4, 282-290 (1980).

    Article  Google Scholar 

  25. A. G. Reyman and M. A. Semenov-Tian-Shansky, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova, 150, 104–118 (1986).

    MATH  Google Scholar 

  26. N. Nekrasov, Commun. Math. Phys., 180, 587–603 (1996).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. A. V. Zotov and A. M. Levin, Theor. Math. Phys., 146, 45–52 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  28. N. Bourbaki, Lie Groups and Lie Algebras Chapters 4–6, Springer, Berlin (2002).

    Book  MATH  Google Scholar 

  29. N. Nekrasov, A. Rosly, and S. Shatashvili, “Quantization of integrable systems and four dimensional gauge theories,” arXiv:0908.4052v1 [hep-th] Nucl. Phys. B Proc. Suppl., 216, 69–93 (2011); arXiv:1103.3919v1 [hep-th] (2011).

  30. A. Levin and M. Olshanetsky, Commun. Math. Phys., 188, 449–466 (1997).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. M. Atiyah, Proc. London Math. Soc. (3), 7, 414–452 (1957).

    Article  MathSciNet  MATH  Google Scholar 

  32. I. Krichever, O. Babelon, E. Billey, and M. Talon, Amer. Math. Soc. Transl. Ser. 2, 170, 83–119 (1995).

    MathSciNet  Google Scholar 

  33. A. A. Belavin, Nucl. Phys. B, 180, 189–200 (1981); A. A. Belavin and V. G. Drinfel’d, Funct. Anal. Appl., 16, No. 3, 159–180 (1982).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. A. V. Zotov, SIGMA, 1107, 067 (2011); arXiv:1012.1072v2 [math-ph] (2010).

    MathSciNet  Google Scholar 

  35. F. Calogero, J. Math. Phys., 12, 419–436 (1971); Lett. Nuovo Cim., 13, 411 (1975); J. Moser, Advances in Math., 16, 197–220 (1975).

    Article  MathSciNet  ADS  Google Scholar 

  36. E. Markman, Compositio Math., 93, 255–290 (1994).

    MathSciNet  MATH  Google Scholar 

  37. C. Schweigert, Nucl. Phys. B, 492, 743–755 (1997).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  38. R. Friedman and J. Morgan, “Holomorphic principal bundles over elliptic curves,” arXiv:math/9811130v1 (1998).

    Google Scholar 

  39. I. Krichever, O. Babelon, E. Billey, and M. Talon, Amer. Math. Soc. Transl. Ser. 2, 170, 83–119 (1995).

    MathSciNet  Google Scholar 

  40. A. V. Zotov and Yu. B. Chernyakov, Theor. Math. Phys., 129, 1526–1542 (2001); arXiv:hep-th/0102069v3 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  41. L.-C. Li and P. Xu, Commun. Math. Phys., 231, 257–286 (2002).

    Article  ADS  MATH  Google Scholar 

  42. E. Sklyanin, J. Phys. A, 21, 2375–2389 (1988).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  43. H. W. Braden, V. A. Dolgushev, M. A. Olshanetsky, and A. V. Zotov, J. Phys. A, 36, 6979–7000 (2003); arXiv:hep-th/0301121v1 (2003).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  44. A. V. Zotov, A. M. Levin, M. A. Olshanetsky, and Yu. B. Chernyakov, Theor. Math. Phys., 156, 1103–1122 (2008); arXiv:0710.1072v1 [nlin.SI] (2007).

    Article  MATH  Google Scholar 

  45. P. Etingof, O. Schiffmann, and A. Varchenko, Lett. Math. Phys., 62, 143–158 (2002); arXiv:math.QA/0207157v1 (2002); G. Kuroki and T. Takebe, Comm. Math. Phys., 190, 1–56 (1997); arXiv:q-alg/9612033v4 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  46. A. M. Levin, M. A. Olshanetsky, A. V. Smirnov, and A. V. Zotov, SIGMA, 1208, 095 (2012); arXiv:1207.4385v1 [stat.ME] (2012).

    MathSciNet  Google Scholar 

  47. E. K. Sklyanin, L. A. Takhtadzhyan, and L. D. Faddeev, Theor. Math. Phys., 40, 688–706 (1979); E. Sklyanin, “Quantum inverse scattering method: Selected topics,” arXiv:hep-th/9211111v1 (1992); P. Kulish and E. Sklyanin, Phys. Lett. A, 70, 461–463 (1979); A. Izergin and V. Korepin, Commun. Math. Phys., 79, 303–316 (1981); L. D. Faddeev, Lectures on Quantum Inverse Scattering Method, World Scientific, Singapore (1990).

    Article  MathSciNet  Google Scholar 

  48. A. A. Belavin, Nucl. Phys. B, 180, 189–200 (1981); A. A. Belavin and V. G. Drinfeld, Funct. Anal. Appl., 16, No. 3, 159–180 (1982).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  49. G. Felder and C. Wieczerkowski, Commun. Math. Phys., 176, 133–161 (1996); G. Felder, “Conformal field theory and integrable systems associated to elliptic curves,” in: Proc. Intl. Congress of Mathematicians (Zürich, 1994, S. D. Chatterji, ed.), Birkhäuser, Boston (1995), pp. 1247–1255.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  50. E. K. Sklyanin, St. Petersburg Math. J., 6, 397–406 (1995); arXiv:hep-th/9308060v1 (1993).

    MathSciNet  Google Scholar 

  51. E. Billey, J. Avan, and O. Babelon, Phys. Lett. A, 186, 114–118 (1994).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  52. A. M. Levin, M. A. Olshanetsky, A. V. Smirnov, and A. V. Zotov, J. Phys. A, 46, 035201 (2013); arXiv: 1208.5750v1 [math-ph] (2012).

    Article  MathSciNet  ADS  Google Scholar 

  53. A. V. Zotov, Phys. Part. Nucl., 37, 400–443 (2006); A. V. Smirnov, Theor. Math. Phys., 158, 300–312 (2009).

    Article  Google Scholar 

  54. I. Krichever, Commun. Math. Phys., 229, 229–269 (2002).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  55. A. Zotov, Lett. Math. Phys., 67, 153–165 (2004); arXiv:hep-th/0310260v1 (2003).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  56. A. Levin, M. Olshanetsky, and A. Zotov, Commun. Math. Phys., 268, 67–103 (2006); arXiv:math/0508058v2 (2005); A. Levin and A. Zotov, Amer. Math. Soc. Transl. Ser. 2, 221, 173–183 (2007); M. A. Olshanetsky and A. V. Zotov, “Isomonodromic problems on elliptic curve, rigid tops and reflection equations,” in: Elliptic Integrable Systems (Rokko Lect. Math., Vol. 18), Kobe Univ., Kobe, Japan (2005), pp. 149–171.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  57. A. Smirnov, Central Eur. J. Phys., 8, 542–554 (2010); arXiv:0903.1466v1 [math-ph] (2009); G. Aminov and S. Arthamonov, J. Phys. A, 44, 075201 (2011).

    Article  ADS  Google Scholar 

  58. A. Kapustin and E. Witten, “Electric-magnetic duality and the geometric Langlands program,” arXiv:hep-th/ 0604151v3 (2006).

    Google Scholar 

  59. A. M. Levin, M. A. Olshanetsky, and A. V. Zotov, SIGMA, 0905, 065 (2009); arXiv:0811.3056v2 [hep-th] (2008).

    MathSciNet  Google Scholar 

  60. H. Flaschka and A. Newell, Commun. Math. Phys., 76, 65–116 (1980).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  61. I. M. Krichever, Funct. Anal. Appl., 14, No. 4, 282–290 (1980).

    Article  MathSciNet  Google Scholar 

  62. A. M. Levin and M. A. Olshanetsky, Theor. Math. Phys., 123, 609–632 (2000).

    Article  MATH  Google Scholar 

  63. R. Ward, “Integrable systems and twistors,” in: Integrable Systems (Oxford Grad. Texts Math., Vol. 4), Oxford Univ. Press, New York (1999).

    Google Scholar 

  64. A. Weil, Elliptic Functions According to Eisenstein and Kronecker, Springer, Berlin (1976).

    Book  MATH  Google Scholar 

  65. K. Takasaki, J. Math. Phys., 42, 1443–1473 (2001).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  66. A. Zabrodin and A. Zotov, J. Math. Phys., 53, 073507, 073508 (2012); arXiv:1107.5672v2 [math-ph] (2011).

    Article  MathSciNet  ADS  Google Scholar 

  67. B. I. Suleimanov, Differ. Equ., 30, 726–732 (1994).

    MathSciNet  MATH  Google Scholar 

  68. B. I. Suleimanov, Theor. Math. Phys., 156, 1280–1291 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  69. A. Zabrodin and A. Zotov, “Classical-quantum correspondence and functional relations for Painlevé equations,” arXiv:1212.5813v2 [math-ph] (2012).

    Google Scholar 

  70. P. Painlevé, Bull. Soc. Math. Phys. France, 28, 201–261 (1900); Acta Math., 25, 1-85 (1902).

    MATH  Google Scholar 

  71. R. Fuchs, C. R. Acad. Sci. (Paris), 141, 555–558 (1905).

    Google Scholar 

  72. D. Gaiotto, JHEP, 1208, 034 (2012); arXiv:0904.2715v1 [hep-th] (2009); L. F. Alday, D. Gaiotto, and Y. Tachikawa, Lett. Math. Phys., 91, 167–197 (2010); arXiv:0906.3219v2 [hep-th] (2009); A. Mironov and A. Morozov, JHEP, 1004, 040 (2010); arXiv:0910.5670v2 [hep-th] (2009); J. Phys. A, 43, 195401 (2010); arXiv:0911.2396v1 [hep-th] (2009).

    Article  MathSciNet  ADS  Google Scholar 

  73. G. Moore, N. Nekrasov, and S. Shatashvili, Nucl. Phys. B, 534, 549–611 (1998); arXiv:hep-th/9711108v2 (1997); A. Losev, N. Nekrasov, and S. Shatashvili, “Testing Seiberg-Witten solution,” arXiv:hep-th/9801061v1 (1998); Commun. Math. Phys., 209, 77–95 (2000); arXiv:hep-th/9803265v3 (1998); 97–121 (2000); arXiv:hepth/9712241v2 (1997).

    Article  ADS  Google Scholar 

  74. A. Mironov, A. Morozov, Y. Zenkevich, and A. Zotov, JETP Lett., 97, 45–51 (2013); arXiv:1204.0913v1 [hep-th] (2012).

    Article  ADS  Google Scholar 

  75. A. Mironov, A. Morozov, B. Runov, Y. Zenkevich, and A. Zotov, Lett. Math. Phys., 103, 299–329 (2013); arXiv:1307.1502v1 [hep-th] (2013).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  76. M. R. Adams, J. Harnad, and J. Hurtubise, Lett. Math. Phys., 20, 299–308 (1990); J. Harnad, Commun. Math. Phys., 166, 337–365 (1994); arXiv:hep-th/9301076v2 (1993).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  77. D. Mumford, Tata Lectures on Theta I, II (Prog. Math., Vol. 28), Birkhäuser, Boston, Mass. (1983, 1984).

    Book  Google Scholar 

  78. A. Onishchik and E. Vinberg, Seminar on Lie Groups and Algebraic Groups [in Russian], Nauka, Moscow (1988); English transl., Springer, Berlin (1990).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute for Theoretical and Experimental Physics, Moscow, Russia

    A. V. Zotov & A. V. Smirnov

  2. Moscow Institute for Physics and Technology (State University), Dolgoprudnyi, Moscow Oblast, Russia

    A. V. Zotov

  3. Steklov Mathematical Institute, RAS, Moscow, Russia

    A. V. Zotov

  4. Department of Mathematics, Columbia University, New York, USA

    A. V. Smirnov

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  1. A. V. Zotov
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  2. A. V. Smirnov
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Correspondence to A. V. Zotov.

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This paper was written at the request of the Editorial Board.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 177, No. 1, pp. 3–67, October, 2013.

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Zotov, A.V., Smirnov, A.V. Modifications of bundles, elliptic integrable systems, and related problems. Theor Math Phys 177, 1281–1338 (2013). https://doi.org/10.1007/s11232-013-0106-1

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