Painlevé 2 equation with arbitrary monodromy parameter, topological recursion and determinantal formulas. (English) Zbl 1385.34067
In this paper, the authors prove that the determinantal formulas and tau-functions associated with two different Painlevé 2 Lax pairs (i.e., the Jimbo-Miwa Lax pair and the Harnad-Tracy-Widom Lax pair) are identical to the correlation functions and symplectic invariants computed by the topological recursion on their corresponding spectral curves in the case of arbitrary nonzero monodromy parameters. They also show that the symplectic invariants of both Lax pairs coincide with the Euler characteristic of the moduli space of Riemann surfaces of genus \(g\) for some limit of the time parameter and, as a consequence, they have succeeded in computing the difference (which appears from constant terms) between the two sets of symplectic invariants explicitly. A key of the proof is to show that the Lax pairs satisfy the topological type property introduced in [M. Bergère et al., ibid. 16, No. 12, 2713–2782 (2015; Zbl 1343.37061)] and [M. Bergère and B. Eynard, “ Determinantal formulae and loop equations”, Preprint, arXiv:0901.3273]. To prove the topological type property the authors propose a new method based on the use of a compatible differential equation in time to bypass the traditional method of insertion operators.
The main results of the paper generalizes similar results of G. Borot and B. Eynard [“Tracy-Widom GUE law and symplectic invariants”, Preprint, arXiv:1011.1418; “The asymptotic expansion of Tracy-Widom GUE law and symplectic invariants”, Preprint, arXiv:1012.2752] obtained from random matrix theory in the special case of vanishing monodromies. This paper clarifies a relation between the integrable structure of the Painlevé 2 equation and the topological recursion.
The main results of the paper generalizes similar results of G. Borot and B. Eynard [“Tracy-Widom GUE law and symplectic invariants”, Preprint, arXiv:1011.1418; “The asymptotic expansion of Tracy-Widom GUE law and symplectic invariants”, Preprint, arXiv:1012.2752] obtained from random matrix theory in the special case of vanishing monodromies. This paper clarifies a relation between the integrable structure of the Painlevé 2 equation and the topological recursion.
Reviewer: Yoshitsugu Takei (Kyoto)
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 |
| 34M25 | Formal solutions and transform techniques for ordinary differential equations in the complex domain |
Keywords:
Painlevé 2 equation; determinantal formula; topological recursion; correlation function; Lax pair; tau function; spectral curve; symplectic invariant; insertion operatorCitations:
Zbl 1343.37061References:
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