Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 074, 17 pages      arXiv:1811.03285      https://doi.org/10.3842/SIGMA.2019.074

Combinatorial Expressions for the Tau Functions of $q$-Painlevé V and III Equations

Yuya Matsuhira and Hajime Nagoya
School of Mathematics and Physics, Kanazawa University, Kanazawa, Ishikawa 920-1192, Japan

Received November 24, 2018, in final form September 13, 2019; Published online September 23, 2019

Abstract
We derive series representations for the tau functions of the $q$-Painlevé V, $\mathrm{III_1}$, $\mathrm{III_2}$, and $\mathrm{III_3}$ equations, as degenerations of the tau functions of the $q$-Painlevé VI equation in [Jimbo M., Nagoya H., Sakai H., J. Integrable Syst. 2 (2017), xyx009, 27 pages]. Our tau functions are expressed in terms of $q$-Nekrasov functions. Thus, our series representations for the tau functions have explicit combinatorial structures. We show that general solutions to the $q$-Painlevé V, $\mathrm{III_1}$, $\mathrm{III_2}$, and $\mathrm{III_3}$ equations are written by our tau functions. We also prove that our tau functions for the $q$-Painlevé $\mathrm{III_1}$, $\mathrm{III_2}$, and $\mathrm{III_3}$ equations satisfy the three-term bilinear equations for them.

Key words: $q$-Painlevé equations; tau functions; $q$-Nekrasov functions; bilinear equations.

pdf (414 kb)   tex (20 kb)  

References

  1. Alday L.F., Gaiotto D., Tachikawa Y., Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010), 167-197, arXiv:0906.3219.
  2. Awata H., Feigin B., Shiraishi J., Quantum algebraic approach to refined topological vertex, J. High Energy Phys. 2012 (2012), no. 3, 041, 35 pages, arXiv:1112.6074.
  3. Awata H., Yamada Y., Five-dimensional AGT conjecture and the deformed Virasoro algebra, J. High Energy Phys. 2010 (2010), no. 1, 125, 11 pages, arXiv:0910.4431.
  4. Bershtein M.A., Gavrylenko P.G., Marshakov A.V., Cluster Toda lattices and Nekrasov functions, Theoret. and Math. Phys. 198 (2019), 157-188, arXiv:1804.10145.
  5. Bershtein M.A., Shchechkin A.I., $q$-deformed Painlevé $\tau$ function and $q$-deformed conformal blocks, J. Phys. A: Math. Theor. 50 (2017), 085202, 22 pages, arXiv:1608.02566.
  6. Bershtein M.A., Shchechkin A.I., Painlevé equations from Nakajima-Yoshioka blow-up relations, Lett. Math. Phys., to appear, arXiv:1811.04050.
  7. Bonelli G., Grassi A., Tanzini A., Quantum curves and $q$-deformed Painlevé equations, Lett. Math. Phys. 109 (2019), 1961-2001, arXiv:1710.11603.
  8. Bonelli G., Lisovyy O., Maruyoshi K., Sciarappa A., Tanzini A., On Painlevé/gauge theory correspondence, Lett. Math. Phys. 107 (2017), 2359-2413, arXiv:1612.06235.
  9. Felder G., Müller-Lennert M., Analyticity of Nekrasov partition functions, Comm. Math. Phys. 364 (2018), 683-718, arXiv:1709.05232.
  10. Gamayun O., Iorgov N., Lisovyy O., Conformal field theory of Painlevé VI, J. High Energy Phys. 2012 (2012), no. 10, 038, 25 pages, arXiv:1207.0787.
  11. Gamayun O., Iorgov N., Lisovyy O., How instanton combinatorics solves Painlevé VI, V and IIIs, J. Phys. A: Math. Theor. 46 (2013), 335203, 29 pages, arXiv:1302.1832.
  12. Gambier B., Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est a points critiques fixes, Acta Math. 33 (1910), 1-55.
  13. Grammaticos B., Ramani A., Parameterless discrete Painlevé equations and their Miura relations, J. Nonlinear Math. Phys. 23 (2016), 141-149.
  14. Iorgov N., Lisovyy O., Teschner J., Isomonodromic tau-functions from Liouville conformal blocks, Comm. Math. Phys. 336 (2015), 671-694, arXiv:1401.6104.
  15. Jimbo M., Nagoya H., Sakai H., CFT approach to the $q$-Painlevé VI equation, J. Integrable Syst. 2 (2017), xyx009, 27 pages, arXiv:1706.01940.
  16. Jimbo M., Sakai H., A $q$-analog of the sixth Painlevé equation, Lett. Math. Phys. 38 (1996), 145-154.
  17. Kajiwara K., Noumi M., Yamada Y., Geometric aspects of Painlevé equations, J. Phys. A: Math. Theor. 50 (2017), 073001, 164 pages, arXiv:1509.08186.
  18. Mano T., Asymptotic behaviour around a boundary point of the $q$-Painlevé VI equation and its connection problem, Nonlinearity 23 (2010), 1585-1608.
  19. Murata M., Lax forms of the $q$-Painlevé equations, J. Phys. A: Math. Theor. 42 (2009), 115201, 17 pages, arXiv:0810.0058.
  20. Nagoya H., Irregular conformal blocks, with an application to the fifth and fourth Painlevé equations, J. Math. Phys. 56 (2015), 123505, 24 pages, arXiv:1505.02398.
  21. Nagoya H., Remarks on irregular conformal blocks and Painlevé III and II tau functions, in Proceedings of the Meeting for Study of Number Theory, Hopf Algebras and Related Topics, Yokohama Publ., Yokohama, 2019, 105-124, arXiv:1804.04782.
  22. Painlevé P., Mémoire sur les équations différentielles dont l'intégrale générale est uniforme, Bull. Soc. Math. France 28 (1900), 201-261.
  23. Painlevé P., Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme, Acta Math. 25 (1902), 1-85.
  24. Ramani A., Grammaticos B., Hietarinta J., Discrete versions of the Painlevé equations, Phys. Rev. Lett. 67 (1991), 1829-1832.
  25. Sakai H., Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math. Phys. 220 (2001), 165-229.
  26. Sakai H., Problem: discrete Painlevé equations and their Lax forms, in Algebraic, Analytic and Geometric Aspects of Complex Differential Equations and their Deformations. Painlevé Hierarchies, RIMS Kôkyûroku Bessatsu, Vol. B2, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007, 195-208.
  27. Tachikawa Y., Five-dimensional Chern-Simons terms and Nekrasov's instanton counting, J. High Energy Phys. 2004 (2004), no. 2, 050, 13 pages, arXiv:hep-th/0401184.
  28. Yanagida S., Norm of the Whittaker vector of the deformed Virasoro algebra, arXiv:1411.0462.

Previous article  Next article  Contents of Volume 15 (2019)