Communications in Mathematical Physics

, Volume 363, Issue 1, pp 1–58 | Cite as

Fredholm Determinant and Nekrasov Sum Representations of Isomonodromic Tau Functions

  • P. Gavrylenko
  • O. LisovyyEmail author


We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with n regular singular points on the Riemann sphere and generic monodromy in GL \({(N,\mathbb{C})}\). The corresponding operator acts in the direct sum of N (n − 3) copies of L2 (S1). Its kernel has a block integrable form and is expressed in terms of fundamental solutions of n − 2 elementary 3-point Fuchsian systems whose monodromy is determined by monodromy of the relevant n-point system via a decomposition of the punctured sphere into pairs of pants. For N = 2 these building blocks have hypergeometric representations, the kernel becomes completely explicit and has Cauchy type. In this case Fredholm determinant expansion yields multivariate series representation for the tau function of the Garnier system, obtained earlier via its identification with Fourier transform of Liouville conformal block (or a dual Nekrasov–Okounkov partition function). Further specialization to n = 4 gives a series representation of the general solution to Painlevé VI equation.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alba V.A., Fateev V.A., Litvinov A.V., Tarnopolsky G.M.: On combinatorial expansion of the con-formal blocks arising from AGT conjecture. Lett. Math. Phys. 98, 33–64 (2011) arXiv:1012.1312 [hep-th] (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alday L.F., Gaiotto D., Tachikawa Y.: Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys. 91, 167–197 (2010) arXiv:0906.3219 [hep-th] ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Balogh F.: Discrete matrix models for partial sums of conformal blocks associated to Painlevé transcendents. Nonlinearity 28, 43–56 (2014) arXiv:1405.1871 [math-ph] (2014)ADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Bao L., Mitev V., Pomoni E., Taki M., Yagi F.: Non-lagrangian theories from brane junctions. J. High Energy Phys. 2014, 175 (2014) arXiv:1310.3841 [hep-th]MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bershtein M., Shchechkin A.: Bilinear equations on Painlevé tau functions from CFT. Commun. Math. Phys. 339, 1021–1061 (2015) arXiv:1406.3008v5 [math-ph]ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Bolibrukh A.A.: On Fuchsian systems with given asymptotics and monodromy. Proc. Steklov Inst. Math. 224, 98–106 (1999) (translation from Tr. Mat. Inst. Steklova 224:112–121)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bonelli G., Grassi A., Tanzini A.: Seiberg–Witten theory as a Fermi gas. Lett. Math. Phys. 107, 1–30 (2017) arXiv:1603.01174 [hep-th] (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bonelli G., Maruyoshi K., Tanzini A.: Wild quiver gauge theories. J. High Energy Phys. 2012, 31 arXiv:1112.1691 [hep-th] (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Borodin A., Olshanski G.: Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes. Ann. Math. 161, 1319–1422 (2005) arXiv:math/0109194 [math.RT] (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Borodin, A., Olshanski, G.: Z-measures on partitions, Robinson-Schensted-Knuth correspondence, and \({\beta}\) = 2 random matrix ensembles. In: Bleher, P.M., Its, A.R. (eds.) Random Matrix Models and Their Applications, pp. 71–94. Cambridge University Press, Cambridge arXiv:math/9905189v1 [math.CO] (2001)
  11. 11.
    Borodin A., Deift P.: Fredholm determinants, Jimbo–Miwa–Ueno tau-functions, and representation theory. Commun. Pure Appl. Math. 55, 1160–1230 arXiv:math-ph/0111007(2002) CrossRefzbMATHGoogle Scholar
  12. 12.
    Bullimore M.: Defect networks and supersymmetric loop operators. J. High Energy Phys. 2015, 66 arXiv:1312.5001v1 [hep-th] (2015) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chekhov L., Mazzocco M.: Colliding holes in Riemann surfaces and quantum cluster algebras. Nonlinearity 31, 54 arXiv:1509.07044 [math-ph] (2018)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chekhov L., Mazzocco M., Rubtsov V.: Painlevé monodromy manifolds, decorated character varieties and cluster algebras. Int. Math. Res. Not. 2017, 7639–7691 arXiv:1511.03851v1 [math-ph](2017) Google Scholar
  15. 15.
    Fateev V.A., Litvinov A.V.: Integrable structure, W-symmetry and AGT relation. J. High Energy Phys. 2012, 51 arXiv:1109.4042v2 [hep-th] (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fokas A.S., Its A.R., Kapaev A.A., Novokshenov V.Yu.: Painlevé Transcendents: The Riemann–Hilbert Approach Mathematical Surveys and Monographs, vol. 128. AMS, Providence (2006)CrossRefzbMATHGoogle Scholar
  17. 17.
    Gaiotto, D.: Asymptotically free \({\mathcal{N}}\) = 2 theories and irregular conformal blocks. J. Phys. Conf. Ser. 462, 1 arXiv:0908.0307 [hep-th] (2018)Google Scholar
  18. 18.
    Gaiotto D., Teschner J.: Irregular singularities in Liouville theory and Argyres–Douglas type gauge theories, I. J. High Energy Phys. 2012, 50 arXiv:1203.1052 [hep-th] (2012) ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Gavrylenko, P.: Isomonodromic \({\tau}\) -functions and WN conformal blocks. J. High Energy Phys. 2015 167 arXiv:1505.00259v1 [hep-th] (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gavrylenko P., Marshakov A.: Exact conformal blocks for the W-algebras, twist fields and isomon odromic deformations. J. High Energy Phys. 2016, 181 arXiv:1507.08794 [hep-th] (2016)CrossRefzbMATHGoogle Scholar
  21. 21.
    Gavrylenko P., Marshakov A.: Free fermions, W-algebras and isomonodromic deformations. Theor. Math. Phys. 187, 649–677 (2016) arXiv:1605.04554 [hep-th] (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gamayun O., Iorgov N., Lisovyy O.: Conformal field theory of PainlevéI. J. High Energy Phys. 2012, 38 (2012) arXiv:1207.0787 [hep-th]CrossRefGoogle Scholar
  23. 23.
    Gamayun O., Iorgov N., Lisovyy O.: How instanton combinatorics solves Painlevé VI, V and III’s. J. Phys. A 46, 335203 (2013) arXiv:1302.1832 [hep-th] (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Grassi, A., Hatsuda, Y., Marino, M.: Topological strings from quantum mechanics. arXiv:1410.3382 [hep-th]
  25. 25.
    Harnad J., Its A.R.: Integrable Fredholm operators and dual isomonodromic deformations. Commun. Math. Phys. 226, 497–530 arXiv:solv-int/9706002(2002) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hollands L., Keller C.A., Song J.: Towards a 4d/2d correspondence for Sicilian quivers. J. High Energy Phys. 1110, 100 (2011) arXiv:1107.0973v1 [hep-th]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Iorgov N., Lisovyy O., Teschner J.: Isomonodromic tau-functions from Liouville conformal blocks. Commun. Math. Phys. 336, 671–694 (2015) arXiv:1401.6104 [hep-th] (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Iorgov N., Lisovyy O., Tykhyy Yu.: Painlevé VI connection problem and monodromy of c = 1 conformal blocks. J. High Energy Phys. 2013, 29 (2013) arXiv:1308.4092v1 [hep-th]CrossRefzbMATHGoogle Scholar
  29. 29.
    Its A.R., Izergin A.G., Korepin V.E., Slavnov N.A.: Differential equations for quantum correlation functions. Int. J. Mod. Phys. B 4, 1003–1037 (1990)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Its A.R., Lisovyy O., Prokhorov A.: Monodromy dependence and connection formulae for isomon odromic tau functions. Duke Math. J. 167, 1347–1432 (2018) arXiv:1604.03082 [math-ph]MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Its A., Lisovyy O., Tykhyy Yu.: Connection problem for the sine-Gordon/Painlevé III tau function and irregular conformal blocks. Int. Math. Res. Not. 2015, 8903–8924 (2015) arXiv:1403.1235 [math-ph] (2015)CrossRefzbMATHGoogle Scholar
  32. 32.
    Its, A., Lisovyy, O., Tykhyy, Yu.: Connection problem for the sine-Gordon/Painlevé III tau function and irregular conformal blocks. Int. Math. Res. Not. 2015, 8903–8924 arXiv:1403.1235 [math-ph] (2015)CrossRefGoogle Scholar
  33. 33.
    Jimbo M.: Monodromy problem and the boundary condition for some Painlevé equations. Publ. Res. Inst. Math. Sci. 18, 1137–1161 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Jimbo M., Miwa T., Môri Y., Sato M.: Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Physica 1, 80–158 (1980)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Jimbo M., Miwa T., Ueno K.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients, I. Physica D 2, 306–352 (1981)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Joshi N., Roffelsen P.: Analytic solutions of q-P (A 1) near its critical points. Nonlinear ity 29, 3696 (2016) arXiv:1510.07433 [nlin.SI] (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Korotkin, D.A.: Isomonodromic deformations in genus zero and one: algebrogeometric solutions and Schlesinger transformations. In: Harnad, J., Sabidussi, G., Winternitz, P. (eds.) Integrable Systems: From Classical to Quantum. CRM Proceedings and Lecture Notes. American Mathematical Society. arXiv:math-ph/0003016v1 (2000)
  38. 38.
    Lisovyy, O.: Dyson’s constant for the hypergeometric kernel. In: Feigin B., Jimbo M., Okado M. (eds.) New Trends in Quantum Integrable Systems, pp. 243–267. World Scientific arXiv:0910.1914 [math-ph] (2011)
  39. 39.
    Malgrange, B.: Sur les déformations isomonodromiques, I. Singularités régulières. In: Mathematics and Physics, (Paris, 1979/1982), pp. 401–426; Prog. Math. 37. Birkhäuser, Boston (1983)Google Scholar
  40. 40.
    Mano T.: Asymptotic behaviour around a boundary point of the q-PainlevéVI equation and its connection problem. Nonlinearity 23, 1585–1608 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Nagoya H.: Irregular conformal blocks, with an application to the fifth and fourth Painlevé equations. J. Math. Phys. 56, 123505 (2015) arXiv:1505.02398v3 [math-ph]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Nekrasov N.A.: Seiberg–Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864 (2003) arXiv:hep-th/0206161 MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Nekrasov, N.,Okounkov, A.: Seiberg–Witten theory and randompartitions. In: The Unity of Mathematics, pp. 525–596, Progr. Math. 244. Birkhäuser Boston, Boston. arXiv:hep-th/0306238 (2006)
  44. 44.
    Palmer J.: Determinants of Cauchy–Riemann operators as \({\tau}\) -functions. Acta Appl. Math. 18, 199–223 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Palmer J.: Deformation analysis of matrix models. Physica D 78, 166–185 arXiv:hep-th/9403023v1(1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Palmer J.: Tau functions for the Dirac operator in the Euclidean plane. Pac. J.Math. 160, 259–342 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Sato M.: Soliton equations as dynamical systems on infinite dimensional Grassmann manifold. N.-Holl. Math. Stud. 81, 259–271 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Sato M., Miwa T., Jimbo M.: Holonomic quantum fields III. Publ. RIMS Kyoto Univ. 15, 577–629 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Sato M., Miwa T., Jimbo M.: Holonomic quantum fields IV. Publ. RIMS Kyoto Univ. 15, 871–972 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Segal G., Wilson G.: Loop groups and equations of KdV type. Publ. Math. IHES 61, 5–65 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Shiraishi J., Kubo H., Awata H., Odake S.: A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions. Lett. Math. Phys. 38, 33–51 (1996) arXiv:q-alg/9507034 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Tracy C.A., Widom H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151–174 (1994) arXiv:hep-th/9211141 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Tracy C.A., Widom H.: Fredholm determinants, differential equations and matrix models. Commun. Math. Phys. 163, 33–72 (1994) arXiv:hep-th/9306042 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Tsuda T.: UC hierarchy and monodromy preserving deformation. J. Reine Angew. Math. 690, 1–34 (2014) arXiv:1007.3450v2 [math.CA] (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Wu T.T., McCoy B.M., Tracy C.A., Barouch E.: Spin–spin correlation functions for the two-dimensional Ising model: exact theory in the scaling region. Phys. Rev. B 13, 316–374 (1976)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.International Laboratory of Representation Theory and Mathematical PhysicsNational Research University Higher School of EconomicsMoscowRussian Federation
  2. 2.Bogolyubov Institute for Theoretical PhysicsKyivUkraine
  3. 3.Center for Advanced StudiesSkolkovo Institute of Science and TechnologyMoscowRussian Federation
  4. 4.Laboratoire de Mathématiques et Physique Théorique CNRS/UMR 7350Université de ToursToursFrance

Personalised recommendations