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Tau Functions as Widom Constants

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Abstract

We define a tau function for a generic Riemann–Hilbert problem posed on a union of non-intersecting smooth closed curves with jump matrices analytic in their neighborhood. The tau function depends on parameters of the jumps and is expressed as the Fredholm determinant of an integral operator with block integrable kernel constructed in terms of elementary parametrices. Its logarithmic derivatives with respect to parameters are given by contour integrals involving these parametrices and the solution of the Riemann–Hilbert problem. In the case of one circle, the tau function coincides with Widom’s determinant arising in the asymptotics of block Toeplitz matrices. Our construction gives the Jimbo–Miwa–Ueno tau function for Riemann–Hilbert problems of isomonodromic origin (Painlevé VI, V, III, Garnier system, etc) and the Sato–Segal–Wilson tau function for integrable hierarchies such as Gelfand–Dickey and Drinfeld–Sokolov.

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References

  1. Adler, M., Cafasso, M., van Moerbeke, P.: Nonlinear PDEs for Fredholm determinants arising from string equations. In: Algebraic and Geometric Aspects of Integrable Systems and Random Matrices. AMS Contemporary Mathematics 593 (2013)

  2. Balogh F., Yang D.: Geometric interpretation of Zhou’s explicit formula for the Witten–Kontsevich tau function. Lett. Math. Phys. 107, 1837–1857 (2017) arXiv:1412.4419 [math-ph]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Basor E.L., Widom H.: On a Toeplitz determinant identity of Borodin and Okounkov. Integr. Equ. Oper. Theory 37, 397–401 (2000) arXiv:math/9909010v3 [math.FA]

    Article  MathSciNet  MATH  Google Scholar 

  4. Bershtein M., Shchechkin A.: q-deformed Painlevé tau function and q-deformed conformal blocks. J. Phys. A 50, 085202 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Bertola M.: The dependence on the monodromy data of the isomonodromic tau function. Commun. Math. Phys. 294, 539–579 (2010) arXiv:0902.4716 [nlin.SI]; corrigendum: arXiv:1601.04790 [math-ph]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Bertola M.: The Malgrange form and Fredholm determinants. SIGMA 13, 046 (2017) arXiv:1703.00046 [math-ph]

    MathSciNet  MATH  Google Scholar 

  7. Bonelli G., Lisovyy O., Maruyoshi K., Sciarappa A., Tanzini A.: On Painlevé/gauge theory correspondence. Lett. Math. Phys. 107, 2359–2413 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Borodin A., Okounkov A.: A Fredholm determinant formula for Toeplitz determinants. Integr. Equ. Oper. Theory 37, 386–396 (2000) arXiv:math/9907165 [math.CA]

    Article  MathSciNet  MATH  Google Scholar 

  9. Cafasso M.: Block Toeplitz determinants, constrained KP and Gelfand–Dickey hierarchies. Math. Phys. Anal. Geom. 11, 11–51 (2008) arXiv:0711.2248 [math.FA]

    Article  MathSciNet  MATH  Google Scholar 

  10. Cafasso, M., Du Crest de Villeneuve, A., Yang, D.: Drinfeld–Sokolov hierarchies, tau functions, and generalized Schur polynomials, (2017). arXiv:1709.07309

  11. Cafasso M., Wu C.-Z.: Tau functions and the limit of block Toeplitz determinants.. Int. Math. Res. Not. 2015, 10339–10366 (2015) arXiv:1404.5149 [math-ph]

    Article  MathSciNet  MATH  Google Scholar 

  12. Cafasso, M., Wu, C.-Z.: Borodin-Okounkov formula, string equation and topological solutions of Drinfeld–Sokolov hierarchies (2015). arXiv:1505.00556

  13. Chekhov L., Mazzocco M.: Colliding holes in Riemann surfaces and quantum cluster algebras. Nonlinearity 31(1), 54–107 (2017) arXiv:1509.07044 [math-ph]

    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., rnw219 (2016). arXiv:1511.03851 [math-ph]

  15. Deift P., Its A., Krasovsky I.: Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model: some history and some recent results. Commun. Pure Appl. Math. 66, 1360–1438 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Drinfeld V.G., Sokolov V.V.: Lie algebras and equations of Korteweg–de Vries type. J. Math. Sci. 30, 1975–2036 (1985)

    Article  MATH  Google Scholar 

  17. Dubrovin B., Matveev V., Novikov S.: Non-linear equations of Korteweg–de Vries type, finite-zone linear operators, and Abelian varieties. Russ. Math. Surv. 31, 59–146 (1976)

    Article  MATH  Google Scholar 

  18. Dubrovin, B., Zhang, Y.: Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov–Witten invariants (2001). arXiv:math/0108160 [hep-th]

  19. Enolski V., Harnad J.: Schur function expansions of KP \({\tau}\)-functions associated to algebraic curves. Russ. Math. Surv. 66, 767–807 (2011) arXiv:1012.3152 [math-ph]

    Article  Google Scholar 

  20. Eynard, B.: The Geometry of integrable systems. Tau functions and homology of spectral curves. Perturbative definition arXiv:1706.04938

  21. Fokas, A.S., Its, A.R., Kapaev, A.A., Novokshenov, V.Y.: Painlevé Transcendents: The Riemann–Hilbert Approach. Mathematical Surveys and Monographs, 128. AMS, Providence (2006)

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

    Article  MATH  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]

    Article  MathSciNet  MATH  Google Scholar 

  24. Gavrylenko, P., Lisovyy, O.: Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions (2016). arXiv:1608.00958 [math-ph]

  25. Gavrylenko P., Lisovyy O.: Pure SU(2) gauge theory partition function and generalized Bessel kernel. Proc. Symp. Pure Math. 98(2018), 181–206 (2017) arXiv:1705.01869 [math-ph]

    MathSciNet  Google Scholar 

  26. Its A.R., Jin B.-Q., Korepin V.E.: Entropy of XY spin chain and block Toeplitz determinants, in “Universality and renormalization”. Fields Inst. Commun. 50, 151–183 (2007)

    MATH  Google Scholar 

  27. Its, A. R., Lisovyy, O., Prokhorov, A.: Monodromy dependence and connection formulae for isomonodromic tau functions. Duke Math. J. 167(7), 1347–1432 (2018). arXiv:1604.03082 [math-ph]

  28. Its A.R., Mezzadri F., Mo M.Y.: Entanglement entropy in quantum spin chains with finite range interaction. Commun. Math. Phys. 284, 117–185 (2008) arXiv:0708.0161 [math-ph]

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. 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 

  30. 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 

  31. Jimbo M., Nagoya H., Sakai H.: CFT approach to the q-Painlevé VI equation. J. Int. Syst. 2, xyx009 (2017) arXiv:1706.01940 [math-ph]

    MATH  Google Scholar 

  32. Kac V.: Infinite-Dimensional Lie Algebras. Cambridge Univ. Press, Cambridge (1994)

    Google Scholar 

  33. Kac V., Schwarz A.: Geometric interpretation of the partition function of 2D gravity. Phys. Lett. B 257, 329–334 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  34. Kontsevich M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147, 1–23 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. Kowalevskaya S.V.: Sur une propriété du système d’équations différentielles qui définit la rotation d’un corps solide autour d’un point fixe. Acta Math. 14, 81–93 (1890)

    Article  MathSciNet  Google Scholar 

  36. Macdonald I.G.: Symmetric Functions and Hall Polynomials. Oxford Univ. Press, Oxford (1998)

    MATH  Google Scholar 

  37. Malgrange, B.: Sur les déformations isomonodromiques, I. Singularités régulières, in “Mathematics and Physics”, (Paris, 1979/1982); Prog. Math. 37, 401–426 (1983)

  38. Matveev V.: 30 years of finite-gap integration theory. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 366(1867), 837–875 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Moore G.: Geometry of the string equations. Commun. Math. Phys. 133(2), 261–304 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. 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 

  41. 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 

  42. Nekrasov, N., Okounkov, A.: Seiberg–Witten theory and random partitions. In: The Unity of Mathematics, pp. 525–596, Progr. Math. 244 (2006). arXiv:hep-th/0306238

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

    Article  MathSciNet  MATH  Google Scholar 

  44. Plemelj J.: Problems in the Sense of Riemann and Klein. Wiley, Hoboken (1964)

    MATH  Google Scholar 

  45. Riemann B.: Theorie der Abel’schen functionen. J. Reine Angew. Math. 54, 101–155 (1857)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  48. Szegő G.: On certain Hermitian forms associated with the Fourier series of a positive function. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952, 228–238 (1952)

    MathSciNet  MATH  Google Scholar 

  49. Tracy C. A., Widom H.: Fredholm determinants, differential equations and matrix models. Commun. Math. Phys. 163, 33–72 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. Widom H.: Asymptotic behavior of block Toeplitz matrices and determinants. Adv. Math. 13, 284–322 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  51. Widom H.: Asymptotic behavior of block Toeplitz matrices and determinants. II. Adv. Math. 21, 1–29 (1976)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. Wilson G.: Collisions of Calogero–Moser particles and an adelic Grassmannian. Invent. Math. 133(1), 1–41 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank M. Bertola, T. Grava, Y. Haraoka, N. Iorgov, A. Its, K. Iwaki, H. Nagoya, A. Prokhorov, and V. Roubtsov for useful discussions. The present work was supported by the PHC Sakura Grant No. 36175WA and CNRS/PICS project, “Isomonodromic deformations and conformal field theory”. The work of P.G. was partially supported the Russian Academic Excellence Project ‘5-100’ and by the RSF Grant No. 16-11-10160. In particular, results of Subsection 3.1 have been obtained using support of Russian Science Foundation. P.G. is a Young Russian Mathematics award winner and would like to thank its sponsors and jury. M.C. acknowledges the support of the project IPaDEGAN (H2020-MSCA-RISE-2017), Grant No. 778010.

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Authors and Affiliations

  1. LAREMA, Université d’Angers, 2 bd Lavoisier, 49045, Angers, France

    M. Cafasso

  2. Center for Advanced Studies, Skolkovo Institute of Science and Technology, Moscow, Russia, 143026

    P. Gavrylenko

  3. Department of Mathematics and International Laboratory of Representation Theory and Mathematical Physics, National Research University Higher School of Economics, Moscow, Russia, 119048

    P. Gavrylenko

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

    P. Gavrylenko

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

    O. Lisovyy

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  1. M. Cafasso
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Correspondence to O. Lisovyy.

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Communicated by P. Deift

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Cafasso, M., Gavrylenko, P. & Lisovyy, O. Tau Functions as Widom Constants. Commun. Math. Phys. 365, 741–772 (2019). https://doi.org/10.1007/s00220-018-3230-9

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