Abstract
We show that a 2-parameter family of \(\tau \)-functions for the first Painlevé equation can be constructed by the discrete Fourier transform of the topological recursion partition function for a family of elliptic curves. We also perform an exact WKB theoretic computation of the Stokes multipliers of associated isomonodromy system assuming certain conjectures.
Similar content being viewed by others
Notes
There are several differences between the usual asymptotic (i.e., \(t \rightarrow \infty \)) and the WKB asymptotics (i.e., \(\hbar \rightarrow 0\)) in general. For Painlevé I, these two asymptotic analysis are equivalent due to homogeneity. See Remark 4.6 below.
References
Alday, L., Gaiotto, D., Tachikawa, Y.: Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys. 91, 167–197 (2010). arXiv:0906.3219 [hep-th]
Aniceto, I., Schiappa, R., Vonk, M.: The resurgence of instantons in string theory. Commub. Number Theor. Phys. 6, 339–496 (2012). arXiv:1106.5922 [hep-th]
Aoki, T., Honda, N., Umeta, Y.: On a construction of general formal solutions for equations of the first Painlevé hierarchy I. Adv. Math. 235, 496–524 (2013)
Aoki, T., Tanda, M.: Borel sums of Voros coefficients of hypergeometric differential equations with a large parameter. RIMS Kôkyûroku 2013, 17–24 (1861)
Aoki, T., Kawai, T., Takei, Y.: WKB analysis of Painlevé transcendents with a large parameter II. In:Structure of Solutions of Differential Equations. World Scientific, pp. 1–49 (1996)
Bassom, A., Clarkson, P., Law, C., McLeod, J.: Application of uniform asymptotics to the second Painlevé transcendent. Arch. Ration. Mech. Anal. 143, 241–271 (1998). arXiv:solv-int/9609005
Bershtein, M., Shchechkin, A.: Bilinear equations on Painleve tau functions from CFT. Commun. Math. Phys. 339, 1021–1061 (2015). arXiv:1406.3008 [math-ph]
Bershtein, M., Shchechkin, A.: Painlevé equations from Nakajima–Yoshioka blow-up relations. Preprint. arXiv:1811.04050 [math-ph]
Bonelli, G., Grassi, A., Tanzini, A.: Quantum curves and \(q\)-deformed Painlevé equations. Lett. Math. Phys. 109, 1961–2001 (2019). arXiv:1710.11603 [hep-th]
Bonelli, G., Lisovyy, O., Maruyoshi, K., Sciarappa, A., Tanzini, A.: On Painlevé/gauge theory correspondence. Lett. Math. Phys. 107, 2359–2413 (2017). arXiv:1612.06235 [hep-th]
Borot, G., Eynard, B.: Geometry of spectral curves and all order dispersive integrable system. SIGMA, 8 (2012), p. 53. arXiv:1110.4936 [math-ph]
Bouchard, V., Chidambaram, N.K., Dauphinee, T.: Quantizing Weierstrass. Commun. Number Theor. Phys. 12, 253–303 (2018). arXiv:1610.00225 [math-ph]
Bouchard, V., Eynard, B.: Reconstructing WKB from topological recursion. Journal de l’Ecole polytechnique - Mathematiques 4, 845–908 (2017). arXiv:1606.04498 [math-ph]
Boutroux, P.: Recherches sur les transcendentes de M. Painlevé et l’étude asymptotique des équations différentielles du seconde ordre. Ann. École Norm. Supér. 30 255–375 (1913)
Bridgeland, T.: Stability conditions on triangulated categories. Ann. Math. 166, 317–345 (2007). arXiv:math.AG/0212237
Bridgeland, T., Smith, I.: Quadratic differentials as stability conditions. I. Publ. Math. IHES 121, 155–278 (2015). arXiv:1302.7030 [math.AG]
Cafasso, M., Gavrylenko, P., Lisovyy, O.: Tau functions as Widom constants. Commun. Math. Phys. 365, 741–772 (2019). arXiv:1712.08546 [math-ph]
Chekhov, L., Eynard, B., Orantin, N.: Free energy topological expansion for the 2-matrix model. JHEP 12, 053 (2006). arXiv:math-ph/0603003
Clarkson, P. A.: Painlevé transcendents. In: Digital Library of Special Functions, Chapter 32. https://dlmf.nist.gov/32
Clarkson, P. A.: Open problems for Painlevé Equations, SIGMA, 15(2019), p. 20. arXiv:1901.10122 [math.CA]
Coman, I., Pomoni, E., Teschner, J.: From quantum curves to topological string partition functions. Preprint. arXiv:1811.01978 [hep-th]
Costin, O.: On Borel summation and Stokes phenomena of nonlinear differential systems. Duke Math. J. 93, 289–344 (1998). arXiv:math/0608408 [math.CA]
Costin, O.: Asymptotics and Borel Summability, Monographs and Surveys in Pure and Applied Mathematics, vol. 141. Chapmann and Hall/CRC, Boca Raton (2008)
David, F.: Non-perturbative effects in matrix models and vacua of two dimensional gravity. Phys. Lett. B. 302, 403–410 (1993). arXiv:hep-th/9212106
Delabaere, E., Dillinger, H., Pham, F.: Résurgence de Voros et périodes des courves hyperelliptique. Annales de l’Institut Fourier 43, 163–199 (1993)
Delabaere, E., Pham, F.: Resurgent methods in semi-classical asymptotics. Annales de l’I.H.P. Physique théorique 71, 1–94 (1999)
Dumitrescu, O., Mulase, M.: Quantum curves for Hitchin fibrations and the Eynard–Orantin theory. Lett. Math. Phys. 104, 635–671 (2014). arXiv:1310.6022 [math.AG]
Dunster, T.M., Lutz, D.A., Schäfke, R.: Convergent Liouville–Green expansions for second-order linear differential equations, with an application to Bessel functions. Proc. R. Soc. Lond. A 440, 37–54 (1993)
Eynard, B., Mariño, M.: A holomorphic and background independent partition function for matrix models and topological strings. J. Geom. Phys. 61, 1181–1202 (2011). arXiv:0810.4273 [hep-th]
Eynard, B., Orantin, N.: Invariants of algebraic curves and topological expansion. Comm. Number Theory Phys. 1, 347–452 (2007). arXiv:math-ph/0702045
Eynard, B., Orantin, N.: Topological recursion in enumerative geometry and random matrices. J. Phys. A Math. Theor., 42, 293001 (117pp) (2009)
Fedoryuk, M.V.: Asymptotic Analysis. Springer, Berlin (1993)
Fokas, A.S., Its, A.R., Kapaev, A.A.: On the asymptotic analysis of the Painlevé equations via the isomonodromy method. Nonlinearity 7, 1291–1325 (1994)
Fokas, A.S., Its, A.R., Kitaev, A.V.: The isomonodromy approach to matric models in 2D quantum gravity. Commun. Math. Phys. 147, 395–430 (1992)
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)
Fuji, K., Suzuki, T.: Drinfeld–Sokolov hierarchies of type A and fourth order Painlevé systems. Funkcial. Ekvac. 53, 143–167 (2010). arXiv:0904.3434 [math-ph]
Gaiotto, D., Moore, G.W., Neitzke, A.: Spectral networks. Ann. Henri Poincaré 14, 1643–1731 (2012). arXiv:1204.4824
Gamayun, O., Iorgov, N., Lisovyy, O.: Conformal field theory of Painlevé VI. JHEP 10, 038 (2012). arXiv:1207.0787 [hep-th]
Gamayun, O., Iorgov, N., Lisovyy, O.: How instanton combinatorics solves Painlevé VI, V and III’s. J. Phys. A Math. Theor. 46, 335203 (2013). arXiv:1302.1832 [hep-th]
Gavrylenko, P.: Isomonodromic \(\tau \)-functions and \(W_N\) conformal blocks. JHEP 2015, 167 (2015). arXiv:1505.00259 [hep-th]
Gavrylenko, P., Iorgov, N., Lisovyy, O.: On solutions of the Fuji–Suzuki–Tsuda system. SIGMA, 14, 123, 27 (2018). arXiv:1806.08650 [math-ph]
Gavrylenko, P., Lisovyy, O.: Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions. Commun. Math. Phys. 363, 1–58 (2018). arXiv:1608.00958 [math-ph]
Gavrylenko, P., Lisovyy, O.: Pure \(SU(2)\) gauge theory partition function and generalized Bessel kernel. Proc. Symp. Pure Math. 18, 181–208 (2018). arXiv:1705.01869 [math-ph]
Garoufalidis, S., Its, A., Kapaev, A., Marino, M.: Asymptotics of the instantons of Painleve I. Int. Math. Res. Not. 2012, 561–606 (2012). arXiv:1002.3634 [math.CA]
Gordoa, P.R., Joshi, N., Pickering, A.: On a Generalized \(2 + 1\) dispersive water wave hierarchy. Publ. RIMS 37, 327–347 (2001)
Grassi, A., Gu, J.: Argyres–Douglas theories, Painlevé II and quantum mechanics, preprint. arXiv:1803.02320 [hep-th]
Hollands, L., Neitzke, A.: Spectral networks and Fenchel–Nielsen coordinates. Lett. Math. Phys. 106, 811–877 (2016). arXiv:1312.2979 [math.GT]
Hone, A.N.W., Zullo, F.: Hirota bilinear equations for Painlevé transcendents. Random Matrices Theory Appl. 7, 1840001 (2018). arXiv:1706.02341 [math.CA]
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]
Its, A.R.: Isomonodromy solutions of equations of zero curvature (Russian). Izv. Akad. Nauk SSSR Ser. Mat., 49 (1985), 530–565; English Transl.: Math. USSR-Izv., 26 (1986), 497–529
Its, A.R., Novokshenov, V.Y.: The isomonodromic deformation method in the theory of Painlevé equations. Lecture Notes in Mathematics, vol. 1191. Springer, Berlin (1986)
Iwaki, K.: Parametric Stokes phenomenon for the second Painlevé equation. Funkcial. Ekvac., 57, 173–243
Iwaki, K.: On WKB theoretic transformations for Painlevé transcendents on degenerate Stokes segments. Publ. Res. Inst. Math. Sci. 51, 1–57 (2015). arXiv:1312.1874 [math.CA]
Iwaki, K.: Topological recursion, quantum curves and the second Painlevé equation. RIMS Kôkyûroku Bessatsu 61, 57–82 (2017)
Iwaki, K., Koike, T., Takei, Y.-M.: Voros coefficients for the hypergeometric differential equations and Eynard–Orantin’s topological recursion— Part I: For the Weber Equation. Preprint; arXiv:1805.10945
Iwaki, K., Koike, T., Takei, Y.-M.: Voros coefficients for the hypergeometric differential equations and Eynard–Orantin’s topological recursion,—part II : for the confluent family of hypergeometric equations. J. Integr. Syst. 4 (2019), xyz004; arXiv:1810.02946
Iwaki, K., Marchal, O.: Painlevé 2 equation with arbitrary monodromy parameter, topological recursion and determinantal formulas. Ann. Henri Poincaré 18, 2581–2620 (2017). arXiv:1411.0875
Iwaki, K., Marchal, O., Saenz, A.: Painlevé equations, topological type property and reconstruction by the topological recursion. J. Geom. Phys. 124, 16–54 (2018). arXiv:1601.02517 [math-ph]
Iwaki, K., Nakanishi, T.: Exact WKB analysis and cluster algebras. J. Phys. A Math. Theor. 47, 474009 (2014). arXiv:1401.7094 [math.CA]
Iwaki, K., Nakanishi, T.: Exact WKB analysis and cluster algebras II: simple poles, orbifold points, and generalized cluster algebras. Int. Math. Res. Not. 2016, 4375–4417 (2016). arXiv:1409.4641 [math.CA]
Iwaki, K., Saenz, A.: Quantum curve and the first Painlevé equation. SIGMA, 12 (2016), 24 pages. arXiv:1507.06557
Jimbo, M., Miwa, T.: Monodromy perserving deformation of linear ordinary differential equations with rational coefficients II. Physica D 2, 407–448 (1981)
Jimbo, M., Miwa, T., Ueno, K.: Monodromy preserving deformations of linear ordinary differential equations with rational coefficients I, general theory tau function. Physica D 2, 306–352 (1981)
Jimbo, M., Nagoya, H., Sakai, H.: CFT approach to the \(q\)-Painlevé VI equation. J. Integr. Syst. 2 (2017), xyx009 arXiv:1706.01940 [math-ph]
Joshi, N., Kruskal, M.D.: An asymptotic approach to the connection problem for the first and the second Painlevé equations. Phys. Lett. A 130, 129–137 (1988)
Kamimoto, S., Koike, T.: On the Borel summability of \(0\)-parameter solutions of nonlinear ordinary differential equations. Preprint of RIMS-1747 (2012)
Kapaev, A.A.: Asymptotics of solutions of the Painlevé equation of the first kind (Russian),Diff. Uravnenija, 24(1988), 1684–1695. English Transl.: Differ. Equ., 24(1989), 1107–1115
Kapaev, A.A.: Quasi-linear Stokes phenomenon for the Painlevé first equation. J. Phys. A. Math. Gen 37, 11149–11167 (2004). arXiv:nlin/0404026 [nlin.SI]
Kapaev, A.A., Kitaev, A.V.: Connection formulae for the first Painlevé transcendent in the complex domain. Lett. Math. Phys. 27, 243–252 (1993)
Kawai, T., Takei, Y.: WKB analysis of Painlevé transcendents with a large parameter. I. Adv. Math. 118, 1–33 (1996)
Kawai, T., Takei, Y.: WKB analysis of Painlevé transcendents with a large parameter. III, Local equivalence of 2-parameter Painleve transcendents. Adv. Math. 134, 178–218 (1998)
English Transl: T. Kawai and Y. Takei. Algebraic Analysis of Singular Perturbation Theory, Translations of Mathematical Monographs, vol 227, AMS (2005)
Kawakami, H., Nakamura, A., Sakai, H.: Degeneration scheme of 4-dimensional Painlevé-type equations. MSJ Memoir 37, 25–111 (2018). arXiv:1209.3836 [math.CA]
Kitaev, A.V.: The justification of the asymptotic formulae obtained by the isomonodromic deformation method (Russian), Zap. Nauchn. Sem. LOMI., 179: 101–109. English Transl: J. Soviet Math. 57(1991), 3131–3135 (1989)
Kitaev, A.V.: The isomonodromy technique and the elliptic asymptotics of the first Painlevé transcendent. Algebra i Analiz 5, 179–211 (1993)
Kitaev, A.V.: Elliptic asymptotics of the first and second Painlevé transcendents (Russian). spekhi Mat. Nauk., 49 (1994), 77–140; English Transl.: Russian Math. Surveys, 49, 81–150 (1994)
Koike, T., Schäfke, R.: On the Borel summability of WKB solutions of Schrödinger equations with rational potentials and its application, in preparation; also Talk given by T. Koike in the RIMS workshop “Exact WKB analysis—Borel summability of WKB solutions” September (2010)
Koike, T., Takei, Y.: On the Voros coefficient for the Whittaker equation with a large parameter—some progress around Sato’s conjecture in exact WKB analysis. Publ. Res. Inst. Math. Sci., Kyoto University 47, 375–395 (2011)
Lisovyy, O., Nagoya, H., Roussillon, J.: Irregular conformal blocks and connection formulae for Painlevé V functions. J. Math. Phys. 59, 091409 (2018). arXiv:1806.08344 [math-ph]
Lisovyy, O., Roussillon, J.: On the connection problem for Painlevé I. J. Phys. A Math. Theor. 50, 255202 (2017). arXiv:1612.08382 [nlin.SI]
Marchal, O., Orantin, N.: Isomonodromic deformations of a rational differential system and reconstruction with the topological recursion: the \({sl}_2\) case. Preprint. arXiv:1901.04344 [math-ph]
Matsuhira, Y., Nagoya, H.: Combinatorial expressions for the tau functions of \(q\)-Painlevé V and III equations. Preprint. arXiv:1811.03285 [math-ph]
Nagoya, H.: Irregular conformal blocks, with an application to the fifth and fourth Painlevé equations. J. Math. Phys. 56, 123505 (2015). arXiv:1505.02398
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 Publications, Yokohama, pp. 105–124 (2019) arXiv:1804.04782
Nakajima, H., Yoshioka, K.: Instanton counting on blowup. I. 4-dimensional pure gauge theory. Inv. math. 162, 313–355 (2005). arXiv:math/0306108
Nakajima, H., Yoshioka, K.: Lectures on Instanton Counting. In lgebraic Structures and Moduli Spaces, CRM Proceedings Lecture Notes 38, AMS, 31–101 (2004) arXiv:math/0311058
Nekrasov, N.A.: Seiberg–Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864 (2004). arXiv:hep-th/0206161
Norbury, P.: Quantum curves and topological recursion. In: Proceedings of Symposia in Pure Mathematics, String-Math 2014(93), 41–65 (2016). arXiv:1502.04394 [math-ph]
Noumi, M., Yamada, Y.: Higher order Painlevé equations of type \(A_\ell ^{(1)}\). Funkcial. Ekvac. 41, 483–503 (1998). arXiv:math/9808003 [math.QA]
Novokshenov, V.Y.: The Boutroux ansatz for the second Painlevé equation in the complex domain (Russian). Izv. Akad. Nauk SSSR Ser. Mat., 54 (1990), no. 6, 1229–1251; English Transl.: Math. USSR-Izv. 37, no. 3, 587–609 (1991)
Okamoto, K.: Polynomial Hamiltonians associated with Painlevé equations I. Proc. Jpn. Acad. Ser. A Math. Sci. 56, 264–268 (1980)
Okamoto, K.: Polynomial Hamiltonians associated with Painlevé equations. II. Differential equations satisfied by polynomial Hamiltonians. Proc. Jpn. Acad. Ser. A Math. Sci. 56, 367–371 (1980)
Painlevé, P.: Sur les équations différentielles du second ordre et d’ordre supérieur dont l’intégrale générale est uniforme. Acta Math. 25, 1–85 (1902)
Ramani, A., Grammaticos, B., Hietarinta, J.: Discrete versions of the Painlevé equations. Phys. Rev. Lett. 67, 1829–1832 (1991)
Sakai, H.: Rational surfaces associated with affine root systems and geometry of the Painlevé equations. Commun. Math. Phys. 220, 165–229 (2001)
750 (1991), 43–51. (M. Sato, T. Aoki, T. Kawai and Y. Takei, Algebraic analysis of singular perturbations (in Japanese; written by A. Kaneko), RIMS Kôkyûroku, 750, 43–51 (1991)
Sauzin, D.: Introduction to 1-summability and resurgence, in Divergent Series, Summability and Resurgence I: Monodromy and Resurgence. Lecture notes in mathematics 2153, (2016). arXiv:1405.0356
Strebel, K.: Quadratic Differentials. Springer, Berlin (1984)
Sutherland, T.: The modular curve as the space of stability conditions of a CY3 algebra. Preprint. arXiv:1111.4184 [math.AG]
Sutherland, T.: Stability conditions for Seiberg–Witten quivers. Ph.D. thesis, University of Sheffield
Takei, Y.: On the connection formula for the first Painlevé equation -from the viewpoint of the exact WKB analysis. Sûrikaisekikenkyûsho Kôkyûroku 931, 70–99 (1995)
Takei, Y.: An explicit description of the connection formula for the first Painlevé equation. In: Toward the Exact WKB Analysis of Differential Equations, Linear or Non-Linear. Kyoto University Press, pp. 271–296 (2000)
Takei, Y.: Sato’s conjecture for the Weber equation and transformation theory for Schrödinger equations with a merging pair of turning points. RIMS Kôkyurôku Bessatsu B10, 205–224 (2008)
van der Put, M., Saito, M.-H.: Moduli spaces for linear differential equations and the Painlevé equations. Ann. Inst. Fourier (Grenoble) 59, 2611–2667 (2009). arXiv:0902.1702 [math.AG]
Voros, A.: The return of the quartic oscillator—the complex WKB method. Ann. Inst. Henri Poincaré 39, 211–338 (1983)
Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge University Press, Cambridge (1902)
Yoshida, S.: 2-Parameter family of solutions for Painlevé equations (I) \(\sim \) (V) at an irregular singular point. Funkcial. Ekvac. 28, 233–248 (1985)
Zabrodin, A., Zotov, A.: Quantum Painlevé–Calogero correspondence. J. Math. Phys. 53, 073507 (2012). arXiv:1107.5672 [math-ph]
Acknowledgements
The author is grateful to Alba Grassi, Akishi Ikeda, Nikolai Iorgov, Michio Jimbo, Akishi Kato, Taro Kimura, Alexander Kitaev, Tatsuya Koike, Oleg Lisovyy, Motohico Mulase, Hajime Nagoya, Hiraku Nakajima, Ryo Ohkawa, Nicolas Orantin, Hidetaka Sakai, Kanehisa Takasaki, Yoshitugu Takei, Yumiko Takei, Yasuhiko Yamada for many valuable comments, suggestions and discussion. He also would like to thank the referees who pointed out the relation between our result and Kitaev’s result [75]. This work was supported by the JSPS KAKENHI Grand Numbers 16K17613, 16H06337, 16K05177, 17H06127, and JSPS and MAEDI under the Japan-France Integrated Action Program (SAKURA). We dedicate the paper to the memory of Tatsuya Koike, who made a lot of important contributions to the theory of the exact WKB analysis and the Painlevé equations. He inspired us by his beautiful papers, talks, and private communications on various occasions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by P. Deift.
To the memory of Tatsuya Koike.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix A. Weierstrass Functions and \(\theta \)-Functions
Here we summarize properties of Weierstrass elliptic functions and \(\theta \)-functions which are relevant in this paper. See [106] for more details.
1.1 Weierstrass functions
1.1.1 Weierstrass \(\wp \)-function
Let
be the periods of smooth elliptic curve \(\Sigma \) defined by \(y^2 = 4x^3 - g_2 x - g_3\). Here A, B are generators of the first homology group \(H_1(\Sigma , {\mathbb {Z}})\). We assume that \(\tau {:}{=} \omega _B/\omega _A\) has a positive imaginary part. The coefficients \(g_2\), \(g_3\) are related to these periods by
Here \(\Lambda = {{\mathbb {Z}}} \cdot \omega _A + {{\mathbb {Z}}} \cdot \omega _B\) be the lattice generated by the two complex numbers \(\omega _A\) and \(\omega _B\).
Under the notations, the Weierstrass elliptic function \(\wp (z) (= \wp (z;g_2,g_3))\) is defined by
It is also constructed as the inverse function of the elliptic integral
and hence \(\wp (z)\) is doubly-periodic function with periods \(\omega _A\) and \(\omega _B\). It has double pole at \(z = m \, \omega _A + n \, \omega _B\) for any \((m,n) \in {\mathbb {Z}}^2\). We also note that \(\wp (z)\) is an even function; that is \(\wp (-z) = \wp (z)\).
The Weierstrass \(\wp \)-function satisfies the following non-linear ODE:
Thus the \(\wp \)-function is used to parametrize elliptic curves. Moreover, differentiating the relation, we also have
1.1.2 Weierstrass \(\zeta \)-function
We also introduce the Weierstrass \(\zeta \)-function
which satisfies \(- \zeta '(z) = \wp (z)\). The function \(\zeta (z)\) is not doubly-periodic, but it satisfies
The constants \(\eta _{A}\) and \(\eta _{B}\) are also expressed as elliptic integral (of the second kind):
The Riemann bilinear identity shows
1.1.3 Weierstrass \(\sigma \)-function
The integral of the Weierstrass zeta function is expressed as the logarithm of the Weierstrass \(\sigma \)-function defined by
This satisfies \(d \log \sigma (z) /dz = \zeta (z)\). \(\sigma \)-function possesses the following quasi-periodicity:
It is also known that the \(\sigma \)-function also satisfies the addition formula:
1.2 \(\theta \)-functions
The Riemann \(\theta \)-function is defined by
This is known to convergent uniformly on \({{\mathbb {C}}} \times {{\mathbb {H}}}\). We also use the \(\theta \)-functions with characteristics:
The parity of these functions are
\(\theta \)-functions with characteristics satisfies various relations. We will use the following identity
in the proof of our main result.
The relation to the Weierstrass \(\sigma \)-function is given as
By taking the logarithm derivative, we have
The last equality follows from the relation
Appendix B. Proof of Theorem 3.7
Here we give a proof of Theorem 3.7 which plays an important role in the proof of our main result. We use the same notation used in Sect. 3.2.3. (For example, the symbol \(z_{[{\hat{j}}]}\) for \(j = 1,\dots , n\) means the \((n-1)\)-tuple of variables \((z_1, \dots , {\hat{z}}_{j}, \dots , z_n)\) without j-th entry.)
Lemma B.1
The function \(F_{g,n}(z_1, \dots , z_n)\) defined in (3.39) satisfies the following equality for \(2g-2+n \ge 1\):
with \(G_{g,n}\) and \(R_{g,n}\) being given as follows:
-
For \(2g-2+n = 1\), we set
$$\begin{aligned}&G_{0,3}(z_1,z_2,z_3) {:}{=} - \sum _{j=2}^{3} \bigl ( P(z_1+z_j) - P(z_1-z_j) \bigr ) \nonumber \\&\qquad \times \biggl ( \frac{1}{2y(z_{1}) \cdot \frac{dx}{dz}(z_{1})} \cdot \frac{\partial F_{0,2}}{\partial z_{1}}(z_{[{\hat{j}}]}) - \frac{1}{2y(z_{j}) \cdot \frac{dx}{dz}(z_{j})} \cdot \frac{\partial F_{0,2}}{\partial z_{j}}(z_{[{\hat{1}}]}) \biggr ) \nonumber \\&\quad +\, \frac{1}{y(z_1) \cdot \frac{dx}{dz}(z_1)} \cdot \frac{\partial F_{0,2}}{\partial z_1}(z_1, z_2) \cdot \frac{\partial F_{0,2}}{\partial z_1}(z_1, z_3), \end{aligned}$$(B.2)$$\begin{aligned}&G_{1,1}(z_1) {:}{=} - \frac{1}{2y(z_{1}) \cdot \frac{dx}{dz}(z_{1})} \cdot \frac{\partial ^2}{\partial u_1 \partial u_2} F_{0,2}(u_1, u_2) \biggl |_{u_{1}=u_{2}=z_{1}}, \end{aligned}$$(B.3)with P(z) being given in (3.13), and for \(2g-2+n \ge 2\), we set
$$\begin{aligned}&G_{g,n}(z_1, \dots , z_n) \nonumber \\&\quad {:}{=} - \sum _{j=2}^{n} \bigl ( P(z_1+z_j) - P(z_1-z_j) \bigr ) \nonumber \\&\qquad \times \biggl ( \frac{1}{2y(z_{1}) \cdot \frac{dx}{dz}(z_{1})} \cdot \frac{\partial F_{g,n-1}}{\partial z_{1}}(z_{[{\hat{j}}]}) - \frac{1}{2y(z_{j}) \cdot \frac{dx}{dz}(z_{j})} \cdot \frac{\partial F_{g,n-1}}{\partial z_{j}}(z_{[{\hat{1}}]}) \biggr ) \nonumber \\&\qquad -\, \frac{1}{2y(z_{1}) \cdot \frac{dx}{dz}(z_{1})} \cdot \frac{\partial ^{2}}{\partial u_{1} \partial u_{2}} \biggl ( F_{g-1,n+1}(u_{1},u_{2},z_{[{\hat{1}}]}) \nonumber \\& + \sum _{\begin{array}{c} g_{1}+g_{2}=g \\ I \sqcup J = [{\hat{1}}] \end{array}}^{\mathrm{stable}} F_{g_{1}, |I|+1}(u_{1},z_{I}) \cdot F_{g_{2}, |J|+1}(u_{2}, z_{J}) \biggr )\Biggr |_{u_{1}=u_{2}=z_{1}}. \end{aligned}$$(B.4) -
\(R_{g,n}(z,z_{2},\dots ,z_{n})\) is a quadratic differential in the variable z (and functions of other variables \(z_2, \dots , z_n\)) defined by
$$\begin{aligned} R_{0,3}(z, z_2, z_3)&{:}{=} \biggl (\int ^{z_{2}}_{0} W_{0,2}(z,z_{2}) \biggr ) \cdot \biggl (\int ^{z_{3}}_{0} W_{0,2}({\bar{z}},z_{3}) \biggr ) \nonumber \\&\qquad +\, \biggl (\int ^{z_{3}}_{0} W_{0,2}(z,z_{3}) \biggr ) \cdot \biggl (\int ^{z_{2}}_{0} W_{0,2}({\bar{z}},z_{2}) \biggr ), \end{aligned}$$(B.5)$$\begin{aligned} R_{1,1}(z)&{:}{=} W_{0,2}(z, {\bar{z}}), \end{aligned}$$(B.6)for \(2g-2+n = 1\), and
$$\begin{aligned}&R_{g,n}(z,z_{2},\dots ,z_{n}) \nonumber \\&{:}{=} \sum _{j=2}^{n} \Biggl [ \biggl (\int ^{z_{j}}_{0} W_{0,2}(z,z_{j}) \biggr ) \cdot \biggl (\int ^{z_{[{\hat{1}}, {\hat{j}}]}}_{0} W_{g,n-1}({\bar{z}}, z_{[{\hat{1}},{\hat{j}}]}) \biggr ) \nonumber \\&\quad +\, \biggl (\int ^{z_{j}}_{0}W_{0,2}({\bar{z}},z_{j}) \biggr ) \cdot \biggl (\int ^{z_{[{\hat{1}}, {\hat{j}}]}}_{0} W_{g,n-1}(z, z_{[{\hat{1}},{\hat{j}}]}) \biggr ) \Biggr ] \nonumber \\&\quad + \int ^{z_{[{\hat{1}}]}}_{0} W_{g-1,n+1}(z,{\bar{z}},z_{[{\hat{1}}]}) \nonumber \\&\quad + \sum _{\begin{array}{c} g_{1}+g_{2}=g \\ I \sqcup J = [{\hat{1}}] \end{array}}^{\text {stable}} \biggl ( \int ^{z_{I}}_{0} W_{g_{1}, |I|+1}(z,z_{I}) \biggr ) \cdot \biggl ( \int ^{z_{J}}_{0} W_{g_{2}, |J|+1}({\bar{z}}, z_{J}) \biggr ) \end{aligned}$$(B.7)for \(2g-2+n \ge 2\). Here, for a set \(L =\{\ell _{1}, \dots , \ell _{k} \} \subset \{1,\dots ,n \}\) of indices, we have used the notation
$$\begin{aligned} \int ^{z_{L}}_{0} W_{g,n}(z_{1},\dots ,z_{n}) {:}{=} \int ^{z_{\ell _{1}}}_{0} \cdots \int ^{z_{\ell _{k}}}_{0} W_{g,n}(z_{1},\dots ,z_{n}). \end{aligned}$$(B.8)
Proof
First we show the claim in the case \(2g-2+n \ge 2\). We employ a similar technique used in the proof of [61, Theorem 3.11].
Integrating the topological recursion relation (3.15) with respect to \(z_{2}, \dots , z_{n}\), we have
Note that, as a differential of z, \(R_{g,n}\) has a simple pole at \(z \equiv z_{1}, {\bar{z}}_{1}, \dots , z_{n}, {\bar{z}}_n\) modulo \(\Lambda \), and no other poles except for the ramification points. Hence, the residue theorem implies
The last term is the integration along the boundary of the fundamental domain \(\Omega \) of the elliptic curve.
The first two lines of the right hand-side of (B.10) coincides with \(G_{g,n}(z_1, \dots , z_n)\) in the desired equality (B.1) (c.f., [27, Theorem 4.7]). On the other hand, since
the integration along \(\partial \Omega \) is computed as follows:
Thus we have proved (B.1) for \(2g-2+n \ge 2\).
The exceptional two cases \((g,n)=(0,3)\) and (1, 1) can be checked similarly by using the identity
which immediately follows from the definition (3.34) of \(F_{0,2}\). \(\quad \square \)
The following formula will be used to relate the A-cycle integral in the right hand-side of (B.1) with the t-derivatives.
Lemma B.2
For \(2g-2+n \ge 1\), we have
where
Proof
Replacing the label \(z_1 \leftrightarrow z_n\) in (B.1) and taking the following residue around \(z_n = 0\), we have
Proposition 3.4 shows that the left hand-side is nothing but the t-derivation of \(F_{g,n-1}\), if we use the x-coordinates. This completes the proof. \(\quad \square \)
Let us set \({\tilde{G}}_{g,n}(z_1,\dots ,z_n) {:}{=} \partial _{z_1} F_{g,n}(z_1,\dots , z_n) - G_{g,n}(z_1,\dots ,z_n)\). By a similar computation in [27, Theorem 6.5], we have
(Recall that \(S_m(x)\)’s are defined in Sect. 3.4.) Here we have used the identity \(y\bigl ( z(x) \bigr ) = \frac{dx}{dz}\bigl ( z(x) \bigr )\).
Next, using Lemmas B.1 and B.2, let us find another expression of the left hand-side of (B.17). First, we note
The first line of the right hand-side is
(C.f., [61, Lemma 4.5].) The notation \([\bullet ]_{\hbar ^{k}}\) means the coefficient of \(\hbar ^k\) in a formal power series \(\bullet \) of \(\hbar \). The second and third lines are also expressed as
and
respectively (c.f., Lemma B.2). Combining (B.17)–(B.21), we have the following recursion relation satisfied by \(S_m\)’s:
Here we used (3.31) and (3.37) to obtain the above expression. Although (B.22) is valid for \(m \ge 1\) a-priori (because it is derived from (B.17) etc.), we can verify that it is also valid for \(m=0\) thanks to the property (3.32). Together with the equation
for the leading term, the recursion relations are summarized into a single PDE
for S given in (3.30). The last equation is equivalent to the PDE (3.40), and hence, we have proved Theorem 3.7.
Rights and permissions
About this article
Cite this article
Iwaki, K. 2-Parameter \(\tau \)-Function for the First Painlevé Equation: Topological Recursion and Direct Monodromy Problem via Exact WKB Analysis. Commun. Math. Phys. 377, 1047–1098 (2020). https://doi.org/10.1007/s00220-020-03769-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-020-03769-2