Advertisement

Annales Henri Poincaré

, Volume 15, Issue 2, pp 313–344 | Cite as

Symmetries of Quantum Lax Equations for the Painlevé Equations

  • Hajime NagoyaEmail author
  • Yasuhiko Yamada
Article

Abstract

Based on the fact that the Painlevé equations can be written as Hamiltonian systems with affine Weyl group symmetries, a canonical quantization of the Painlevé equations preserving such symmetries has been studied recently. On the other hand, since the Painlevé equations can also be described as isomonodromic deformations of certain second-order linear differential equations, a quantization of such Lax formalism is also a natural problem. In this paper, we introduce a canonical quantization of Lax equations for the Painlevé equations and study their symmetries. We also show that our quantum Lax equations are derived from Virasoro conformal field theory.

Keywords

Automorphism Group Commutation Relation Weyl Group Dynkin Diagram Gauge Factor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alday, L.F., Gaiotto, D., Tachikawa, Y.: Liouville correlation functions from four-dimensional Gauge Theories. Lett. Math. Phys. 91, 167–197 (2010). arXiv: 0906.3219Google Scholar
  2. 2.
    Awata, H., Fuji, H., Kanno, H., Manabe, M., Yamada, Y.: Localization with a surface operator, irregular conformal blocks and open topological string. Adv. Theor. Math. Phys. 16(3) (2012). arXiv:1008.0574Google Scholar
  3. 3.
    Belavin A.A., Polyakov A.M., Zamolodchikov A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241, 333–380 (1984)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Gamayun, O., Iorgov, N., Lisovyy, O.: Conformal field theory of Painlevé VI. arXiv:1207.0787Google Scholar
  5. 5.
    Harnad, J.: Quantum isomonodromic deformations and the Knizhnik–Zamolodchikov equations. Symmetries and integrability of difference equations. (Estérel, PQ, 1994). CRM Proc. Lecture Notes, vol. 9, pp. 155–161. American Mathematical Society, Providence (1996)Google Scholar
  6. 6.
    Jimbo, M., Miwa, T.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II. Physica 2D, 407–448 (1981)Google Scholar
  7. 7.
    Jimbo, M., Nagoya, H., Sun, J.: Remarks on the confuent KZ equation for \({\mathfrak{sl_2}}\) and quantum Painlevé equations. J. Phys. A Math. Theor. 41, 175205 (2008)Google Scholar
  8. 8.
    Kajiwara, K., Masuda, T., Noumi, M., Ohta, Y., Yamada, Y.: Cubic Pencils and Painlevé Hamiltonians. Funkcialaj Ekvacioj 48, 147–160 (2005). arXiv: nlin/0403009Google Scholar
  9. 9.
    Kawakami H.: Generalized Okubo systems and the middle convolution. Int. Math. Res. Not. 17, 3394–3421 (2010)MathSciNetGoogle Scholar
  10. 10.
    Kuroki, G.: Regularity of quantum τ-functions generated by quantum birational Weyl group actions. arXiv:1206.3419Google Scholar
  11. 11.
    Nekrasov, N.A., Shatashvili, S.L.: Quantization of Integrable Systems and Four Dimensional Gauge Theories. arXiv:0908.4052Google Scholar
  12. 12.
    Nagoya H.: Quantum Painlevé systems of type \({A_l^{(1)}}\). Int. J. Math. 15, 1007–1031 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Nagoya H., Grammaticos B., Ramani A.: Quantum Painlevé equations: from continuous to discrete and back. Regul. Chaotic Dyn. 13(5), 417–423 (2008)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Nagoya H.: A quantization of the sixth Painlevé equation, noncommutativity and singularities. Adv. Stud. Pure Math. 55, 291–298 (2009)MathSciNetGoogle Scholar
  15. 15.
    Nagoya, H., Sun, J.: Confluent primary fields in the conformal field theory. J. Phys. A Math. Theor. 43, 465203 (2010). arXiv:1002.2598Google Scholar
  16. 16.
    Nagoya, H.: Hypergeometric solutions to Schrödinger equations for the quantum Painlevé equations. J. Math. Phys. 52, 083509 (2011)Google Scholar
  17. 17.
    Nagoya H.: Realizations of affine Weyl group symmetries on the quantum Painlevé equations by fractional calculus. Lett. Math. Phys. 102(3), 297–321 (2012)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Novikov D.P.: The 2 × 2 matrix Schlesinger system and the Belavin–Polyakov–Zamolodchikov system. Theor. Math. Phys. 161(2), 1485–1496 (2009)CrossRefzbMATHGoogle Scholar
  19. 19.
    Okamoto K.: Studies on the Painlevé equations, I. Ann. Math. Pura. Appl. 146(4), 337–381 (1987)zbMATHGoogle Scholar
  20. 20.
    Okamoto K.: Studies on the Painlevé equations, II. Jap. J. Math. 13(1), 47–76 (1987)zbMATHGoogle Scholar
  21. 21.
    Okamoto K.: Studies on the Painlevé equations, III. Math. Ann. 275(2), 221–255 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Okamoto K.: Studies on the Painlevé equations, IV. Funkcial. Ekvac. 30(2–3), 305–332 (1987)zbMATHMathSciNetGoogle Scholar
  23. 23.
    Ohyama Y., Kawamuko H., Sakai H., Okamoto K.: Studies on the Painlevé equations, V. J. Math. Sci. Univ. Tokyo 13, 145–204 (2006)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Reshetikhin N.: The Knizhnik–Zamolodchikov system as a deformation of the isomonodromy problem. Lett. Math. Phys. 26, 167–177 (1992)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Suleimanov B.I.: The Hamiltonian structure of Painlevé equations and the method of isomonodromic deformations. Differ. Equ. 30, 726–732 (1994)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Takemura, K.: Integral representation of solutions to Fuchsian system and Heun’s equation. J. Math. Anal. Appl. 342, 52–69 (2008). arXiv:0705.3358Google Scholar
  27. 27.
    Teschner, J.: Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I. Adv. Theor. Math. Phys. 15(2), 471–564 (2011). arXiv:1005.2846Google Scholar
  28. 28.
    Zabrodin, A., Zotov, A.: Quantum Painlevé–Calogero correspondence for Painlevé VI. J. Math. Phys. 53, 073508 (2012). arXiv:1107.5672Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsKobe UniversityKobeJapan

Personalised recommendations