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Quantum Baxter-Belavin R-matrices and multidimensional lax pairs for Painlevé VI

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Abstract

Quantum elliptic R-matrices satisfy the associative Yang-Baxter equation in Mat(N)⊗2, which can be regarded as a noncommutative analogue of the Fay identity for the scalar Kronecker function. We present a broader list of R-matrix-valued identities for elliptic functions. In particular, we propose an analogue of the Fay identities in Mat(N)⊗2. As an application, we use the ℤ N ×ℤ N elliptic R-matrix to construct R-matrix-valued 2N 2×2N 2 Lax pairs for the Painlevé VI equation (in the elliptic form) with four free constants. More precisely, the case with four free constants corresponds to odd N, and even N corresponds to the case with a single constant in the equation.

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References

  1. A. Levin, M. Olshanetsky, and A. Zotov, JHEP, 1410, 109 (2014); arXiv:1408.6246v3 [hep-th] (2014).

    Article  ADS  MathSciNet  Google Scholar 

  2. A. Polischuk, Adv. Math., 168, 56–95 (2002).

    Article  MathSciNet  Google Scholar 

  3. R. J. Baxter, Ann. Phys., 70, 193–228 (1972).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. A. A. Belavin, Nucl. Phys. B, 180, 189–200 (1981)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. A. A. Belavin and V. G. Drinfeld, Funct. Anal. Appl., 16, 159–180 (1982).

    Article  MathSciNet  Google Scholar 

  6. M. P. Richey and C. A. Tracy, J. Statist. Phys., 42, 311–348 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  7. K. Hasegawa, J. Math. Phys., 35, 6158–6171 (1994).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. V. V. Bazhanov and Y. G. Stroganov, “On connection between the solutions of the quantum and classical triangle equations,” in: Proc. VI Intl. Sem. on High Energy Physics and Quantum Field Theory, Vol. 1, Inst. High Energy Phys., Protvino (1983), pp. 52–2.

    Google Scholar 

  9. L. A. Takhtadzhyan, “Solutions of the triangle equations with Z n ×Z n symmetry as the matrix analogues of the Weierstrass zets and sigma functions [in Russian],” in: Differential Geometry, Lie Groups, and Mechanics: Part VI (Zap. Nauchn. Sem. LOMI, Vol. 133), Nauka, Leningrad. Otdel., Leningrad (1984), pp. 258–2.

    MathSciNet  MATH  Google Scholar 

  10. A. Weil, Elliptic Functions According to Eisenstein and Kronecker (Ergeb. Math. und ihrer Grenzgebiete, Vol. 88), Springer, Berlin (1976).

  11. D. Mumford, Tata Lectures on Theta I (Modern Birkhäuser Classics), Birkhäuser, Basel (2007)

    Book  MATH  Google Scholar 

  12. D. Mumford, Tata Lectures on Theta II: Jacobian Theta Functions and Differential Equations (Modern Birkhäuser Classics), Birkhäuser, Basel (2007).

    Book  Google Scholar 

  13. J. D. Fay, Theta Functions on Riemann Surfaces (Lect. Notes Math., Vol. 352), Springer, Berlin (1973).

  14. A. Levin, M. Olshanetsky, and A. Zotov, JHEP, 1407, 012 (2014); arXiv:1405.7523v3 [hep-th] (2014)

    Article  ADS  Google Scholar 

  15. A. Levin, M. Olshanetsky, and A. Zotov, Nucl. Phys. B, 887, 400–422 (2014); arXiv:1406.2995v2 [math-ph] (2014).

    Article  ADS  MathSciNet  Google Scholar 

  16. M. Aguiar, “nfinitesimal Hopf Algebras,” in: New Trends in Hopf Algebra Theory (Contemp. Math., Vol. 267, N. Andruskiewitsch, W. R. Ferrer Santos, and H.-J. Schneider, eds.), Amer. Math. Soc., Providence, R. I. (2000), pp. 1–2.

    Article  MathSciNet  Google Scholar 

  17. P. Painlevé, C. R. Acad. Sci. Paris, 143, 1111–1117 (1906)

    Google Scholar 

  18. Yu. Manin, “Sixth Painlevé equation, universal elliptic curve, and mirror of P2,” in: Geometry of Differential Equations (Transl. Amer. Math. Soc. Ser. 2, Vol. 186, A. Khovanskij, A. Varchenko, and V. Vassiliev, eds.), Amer. Math. Soc., Providence, R. I. (1998), pp. 131–2.

    MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. N. Joshi, A. Kitaev, and P. Treharne, J. Math. Phys., 48, 103512 (2007); arXiv:0706.1750v3 [math.CA] (2007)

    Article  ADS  MathSciNet  Google Scholar 

  22. M. Noumi and Y. Yamada, “A new Lax pair for the sixth Painlevé equation associated with b so(8),” in: Microlocal Analysis and Complex Fourier Analysis (T. Kawai and K. Fujita, eds.), World Scientific, Singapore (2002), pp. 238–2; arXiv:math-ph/0203029v1 (2002)

    Chapter  Google Scholar 

  23. G. Aminov and S. Arthamonov, “New 2×2-matrix linear problems for Painlevé equations,” arXiv:1112.4688v2 [nlin.SI] (2011).

  24. B. Dubrovin and M. Mazzocco, Invent. Math., 141, 55–147 (2000); arXiv:math/9806056v1 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. P. Boalch, Proc. London Math. Soc. (3), 90, 167–208 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  26. A. Levin and M. Olshanetsky, “Painlevé–Calogero correspondence,” in: Calogero–Moser–Sutherland Models (J. F. van Diejen and L. Vinet, eds.),Springer, eds (2000), pp. 313–2; arXiv:alg-geom/ 9706010v1 (1997)

    Google Scholar 

  27. A. Levin, M. Olshanetsky, and A. Zotov, Commun. Math. Phys., 268, 67–103 (2006); arXiv:math/0508058v2 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. A. Levin and A. Zotov, “On rational and elliptic forms of Painlevé VI equation,” in: Moscow Seminar on Mathematical Physics: II(Transl. Amer. Math. Soc. Ser. 2, Vol. 221, Yu. Neretin, M. A. Olshanetsky, and A. Rosly, eds.), Amer. Math. Soc., Providence, R. I. (2007), pp. 173–2

    MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  30. A. Zabrodin and A. Zotov, J. Math. Phys., 53, 073508 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  31. A. Zabrodin and A. Zotov, J. Math. Phys., Construct. Approx., 41, 385–423 (2015); arXiv:1212.5813v2 [math-ph] (2012)

    Article  MathSciNet  Google Scholar 

  32. A. M. Levin, M. A. Olshanetsky, and A. V. Zotov, Russ. Math. Surveys, 69, 35–118 (2014); arXiv:1311.4498v2 [math-ph] (2013).

    Article  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, National Research, University Higher School of Economics, Moscow, Russia

    A. M. Levin

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

    A. M. Levin & M. A. Olshanetsky

  3. Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Oblast, Russia

    M. A. Olshanetsky

  4. Steklov Mathematical Institute of RAS, Moscow, Russia

    A. V. Zotov

Authors
  1. A. M. Levin
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  2. M. A. Olshanetsky
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  3. A. V. Zotov
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Correspondence to A. M. Levin.

Additional information

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 184, No. 1, pp. 41–56, July, 2015.

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Levin, A.M., Olshanetsky, M.A. & Zotov, A.V. Quantum Baxter-Belavin R-matrices and multidimensional lax pairs for Painlevé VI. Theor Math Phys 184, 924–939 (2015). https://doi.org/10.1007/s11232-015-0306-y

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  • DOI: https://doi.org/10.1007/s11232-015-0306-y

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