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Fredholm Determinant and Nekrasov Sum Representations of Isomonodromic Tau Functions

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Abstract

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.

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References

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Bonelli G., Maruyoshi K., Tanzini A.: Wild quiver gauge theories. J. High Energy Phys. 2012, 31 arXiv:1112.1691 [hep-th] (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MATH  Google Scholar 

  12. Bullimore M.: Defect networks and supersymmetric loop operators. J. High Energy Phys. 2015, 66 arXiv:1312.5001v1 [hep-th] (2015)

    Article  MathSciNet  Google Scholar 

  13. Chekhov L., Mazzocco M.: Colliding holes in Riemann surfaces and quantum cluster algebras. Nonlinearity 31, 54 arXiv:1509.07044 [math-ph] (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Book  MATH  Google Scholar 

  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. 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)

    Article  ADS  MathSciNet  Google Scholar 

  19. Gavrylenko, P.: Isomonodromic \({\tau}\) -functions and WN conformal blocks. J. High Energy Phys. 2015 167 arXiv:1505.00259v1 [hep-th] (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gamayun O., Iorgov N., Lisovyy O.: Conformal field theory of PainlevéI. J. High Energy Phys. 2012, 38 (2012) arXiv:1207.0787 [hep-th]

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  24. Grassi, A., Hatsuda, Y., Marino, M.: Topological strings from quantum mechanics. arXiv:1410.3382 [hep-th]

  25. Harnad J., Its A.R.: Integrable Fredholm operators and dual isomonodromic deformations. Commun. Math. Phys. 226, 497–530 arXiv:solv-int/9706002(2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Book  Google Scholar 

  33. Jimbo M.: Monodromy problem and the boundary condition for some Painlevé equations. Publ. Res. Inst. Math. Sci. 18, 1137–1161 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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. 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. 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)

  40. Mano T.: Asymptotic behaviour around a boundary point of the q-PainlevéVI equation and its connection problem. Nonlinearity 23, 1585–1608 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Nekrasov N.A.: Seiberg–Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864 (2003) arXiv:hep-th/0206161

    Article  MathSciNet  MATH  Google Scholar 

  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. Palmer J.: Determinants of Cauchy–Riemann operators as \({\tau}\) -functions. Acta Appl. Math. 18, 199–223 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  45. Palmer J.: Deformation analysis of matrix models. Physica D 78, 166–185 arXiv:hep-th/9403023v1(1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. Palmer J.: Tau functions for the Dirac operator in the Euclidean plane. Pac. J.Math. 160, 259–342 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  47. Sato M.: Soliton equations as dynamical systems on infinite dimensional Grassmann manifold. N.-Holl. Math. Stud. 81, 259–271 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  48. Sato M., Miwa T., Jimbo M.: Holonomic quantum fields III. Publ. RIMS Kyoto Univ. 15, 577–629 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  49. Sato M., Miwa T., Jimbo M.: Holonomic quantum fields IV. Publ. RIMS Kyoto Univ. 15, 871–972 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  50. Segal G., Wilson G.: Loop groups and equations of KdV type. Publ. Math. IHES 61, 5–65 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. Tracy C.A., Widom H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151–174 (1994) arXiv:hep-th/9211141

    Article  ADS  MathSciNet  MATH  Google Scholar 

  53. Tracy C.A., Widom H.: Fredholm determinants, differential equations and matrix models. Commun. Math. Phys. 163, 33–72 (1994) arXiv:hep-th/9306042

    Article  ADS  MathSciNet  MATH  Google Scholar 

  54. Tsuda T.: UC hierarchy and monodromy preserving deformation. J. Reine Angew. Math. 690, 1–34 (2014) arXiv:1007.3450v2 [math.CA] (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. International Laboratory of Representation Theory and Mathematical Physics, National Research University Higher School of Economics, Moscow, Russian Federation

    P. Gavrylenko

  2. Bogolyubov Institute for Theoretical Physics, Kyiv, 03680, Ukraine

    P. Gavrylenko

  3. Center for Advanced Studies, Skolkovo Institute of Science and Technology, Moscow, Russian Federation, 143026

    P. Gavrylenko

  4. Laboratoire de Mathématiques et Physique Théorique CNRS/UMR 7350, Université de Tours, Parc de Grandmont, 37200, Tours, France

    O. Lisovyy

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Gavrylenko, P., Lisovyy, O. Fredholm Determinant and Nekrasov Sum Representations of Isomonodromic Tau Functions. Commun. Math. Phys. 363, 1–58 (2018). https://doi.org/10.1007/s00220-018-3224-7

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