Painlevé equations, topological type property and reconstruction by the topological recursion. (English) Zbl 1391.34140
In this paper, the authors study the \(\hbar\)-dependent versions of the six Painlevé equations using the scope of the topological recursion. In particular, following the works of Bergére, Borot and Eynard, they prove that under some assumptions the \(2 \times 2\) Lax pairs associated with the \(\hbar\)-dependent Painlevé equations satisfy a topological type property. The authors also show that for all Painlevé equations the \(\tau\)-functions and the determinantal formulas, defined by the Lax pairs, can be reconstructed by the topological recursion to the corresponding spectral curves.
Reviewer: Tsvetana Stoyanova (Sofia)
MSC:
| 34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |
| 34M56 | Isomonodromic deformations for ordinary differential equations in the complex domain |
| 34M60 | Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent) |
Keywords:
Painlevé equation; \(\tau\)-function; topological recursion; determinantal formula; WKB methodReferences:
| [1] | Bertola, M.; Eynard, B.; Harnad, J., Partition functions for matrix models and isomonodromic tau functions, J. Phys. A, 36, 3067-3083 (2003) · Zbl 1050.37032 |
| [2] | Bertola, M.; Marchal, O., The partition function of the two-matrix model as an isomonodromic tau-function, J. Math. Phys., 50, 013529 (2009) |
| [3] | Forrester, P. J.; Witte, N. S., Random matrix theory and the sixth Painlevé equation, J. Phys. A: Math. Gen., 39 (2006) · Zbl 1119.34070 |
| [4] | Mehta, M. L., (Random Matrices. Random Matrices, Pure and applied mathematics, vol. 142 (2004), Elsevier Academic Press: Elsevier Academic Press Amsterdam) · Zbl 1107.15019 |
| [5] | Tracy, C.; Widom, H., (Introduction To Random Matrices. Introduction To Random Matrices, Springer Lecture Notes in Physics, vol. 424 (1993)), 103-130 · Zbl 0791.15017 |
| [6] | Eynard, B.; Orantin, N., Invariants of algebraic curves and topological expansion, Commun. Number Theory Phys., 1, 347-452 (2007) · Zbl 1161.14026 |
| [7] | Bergère, M.; Borot, G.; Eynard, B., Rational differential systems, loop equations, and application to the \(\text{q}\) th reductions of KP, Ann. Henri Poincaré, 16, 12 (2015) · Zbl 1343.37061 |
| [8] | M. Bergère, B. Eynard, Determinantal formulae and loop equations, 2009. arxiv:0901.3273; M. Bergère, B. Eynard, Determinantal formulae and loop equations, 2009. arxiv:0901.3273 |
| [9] | Jimbo, M.; Miwa, T.; Ueno, K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients I. General theory and \(\tau \)-function, Physica 2D, 2, 306-352 (1981) · Zbl 1194.34167 |
| [10] | Kawai, T.; Takei, Y., WKB analysis of Painlevé transcendents with a large parameter. I, Adv. Math., 118, 1-33 (1996) · Zbl 0848.34005 |
| [11] | Okamoto, K., Polynomial hamiltonians associated with painlevé equations, Proc. Japan Acad., 56, 264-268 (1980) · Zbl 0476.34010 |
| [12] | Iwaki, K.; Marchal, O., Painlevé 2 equation with arbitrary monodromy parameter, topological recursion and determinantal formulas, Ann. Henri Poincaré, 18, 2581-2620 (2017) · Zbl 1385.34067 |
| [13] | Marchal, O.; Eynard, B.; Bergère, M., The sine-law gap probability, Painlevé 5, and asymptotic expansion by the topological recursion, Random Matrices: Theory Appl., 3, 1450013 (2014) · Zbl 1314.15027 |
| [14] | Fokas, A. S.; Its, A. R.; Kapaev, A. A.; Novokshenov, V. Yu., Painlevé transcendents, (The Riemann-Hilbert Approach. The Riemann-Hilbert Approach, Mathematical Surveys and Monographs, vol. 128 (2006), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI) · Zbl 1111.34001 |
| [15] | Jimbo, M.; Miwa, T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients II, Physica 2D, 2, 407-448 (1981) · Zbl 1194.34166 |
| [16] | Jimbo, M.; Miwa, T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients III, Physica 2D, 4, 26-46 (1981) · Zbl 1194.34169 |
| [17] | Kamimoto, S.; Koike, T., On the Borel summability of 0-parameter solutions of nonlinear ordinary differential equations, RIMS Kôkyûroku Bessatsu, B40, 191-212 (2013) · Zbl 1315.34101 |
| [18] | Okamoto, K., Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé, Jpn. J. Math., 5, 1-79 (1979) · Zbl 0426.58017 |
| [19] | Kawai, T.; Takei, Y., (Algebraic Analysis of Singular Perturbation. Algebraic Analysis of Singular Perturbation, Iwanami Series in Modern Mathematics, Translations of Mathematical Monographs, vol. 227 (2005)) · Zbl 1100.34004 |
| [20] | Eynard, B.; Orantin, N., Topological recursion in enumerative geometry and random matrices, J. Phys. A, 42, 293001 (2009) · Zbl 1177.82049 |
| [21] | Iwaki, K.; Saenz, A., Quantum curve and the first painlevé equation, SIGMA, 12 (2016) · Zbl 1339.34096 |
| [22] | Kapaev, A., Quasi-linear Stokes phenomenon for the Painlevé first equation, J. Phys. A: Math. Gen., 37, 46, 11149-11167 (2004) · Zbl 1080.34071 |
| [23] | Gamayun, O.; Iorgov, N.; Lisovyy, O., Conformal field theory of Painlevé VI, J. High Energy Phys., 10, 038 (2012) · Zbl 1397.81307 |
| [24] | Alday, L. F.; Gaiotto, D.; Tachikawa, Y., Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys., 91, 167-197 (2010) · Zbl 1185.81111 |
| [25] | P. Gavrylenko, O. Lisovyy, Pure \(S U ( 2 )\) arxiv:1705.01869; P. Gavrylenko, O. Lisovyy, Pure \(S U ( 2 )\) arxiv:1705.01869 |
| [26] | Iorgov, N.; Lisovyy, O.; Teschner, J., Isomonodromic tau-functions from Liouville conformal blocks, Comm. Math. Phys., 336, 671-694 (2015) · Zbl 1311.30029 |
| [27] | Jimbo, M.; Nagoya, H.; Sakai, H., CFT approach to the q-Painlevé VI equation, J. Integr. Syst., 2 (2017) · Zbl 1400.39008 |
| [28] | Borot, G.; Eynard, B., Geometry of spectral curves and all order dispersive integrable system, SIGMA, 8, 100, 53 (2012) · Zbl 1270.14017 |
| [29] | B. Eynard, The Geometry of integrable systems. Tau functions and homology of Spectral curves. Perturbative definition, 2017. arxiv:1706.04938; B. Eynard, The Geometry of integrable systems. Tau functions and homology of Spectral curves. Perturbative definition, 2017. arxiv:1706.04938 |
| [30] | Bonelli, G.; Lisovyy, O.; Maruyoshi, K.; Sciarappa, A.; Tanzini, A., On Painlevé/gauge theory correspondence, Lett. Math. Phys., 1-55 (2017) |
| [31] | Chekhov, L., Genus one correlation to multi-cut matrix model solutions, Theoret. Math. Phys., 141, 1640-1653 (2004) · Zbl 1178.81217 |
| [32] | Korotkin, D., Solution of matrix Riemann-Hilbert problems with quasi-permutation monodromy matrices, Math. Ann., 329, 2, 335-364 (2004) · Zbl 1059.32002 |
| [33] | Kokotov, A.; Korotkin, D., A new hierarchy of integrable systems associated to Hurwitz spaces, Philos. Trans. R. Soc., 366, 1055-1088 (2007) · Zbl 1153.37414 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
