Title: Painlevé equations, topological type property and reconstruction by the topological recursion Show Chinese title

Title: Painlevé方程，拓扑类型性质和通过拓扑递归的重构

Abstract: In this article, we prove that we can introduce a small $\hbar$ parameter in the six Painlev\'e equations through their corresponding Lax pairs and Hamiltonian formulations. Moreover, we prove that these $\hbar$-deformed Lax pairs satisfy the Topological Type property proposed by Berg\`ere, Borot and Eynard for any generic choice of the monodromy parameters. Consequently we show that one can reconstruct the formal $\hbar$ series expansion of the tau-function and of the determinantal formulas by applying the so-called topological recursion on the spectral curve attached to the Lax pair in all six Painlev\'e cases. Eventually we illustrate the former results with the explicit computations of the first orders of the six tau-functions. Show Chinese abstract

Abstract: 在本文中，我们证明可以通过相应的Lax对和哈密顿形式，在六个Painlevé方程中引入一个小型$\hbar$参数。 此外，我们证明这些$\hbar$变形的Lax对对于单值化参数的任何一般选择都满足Bergère、Borot和Eynard提出的拓扑类型性质。 因此，我们表明可以通过在所有六个Painlevé情况下对与Lax对相关的谱曲线应用所谓的拓扑递归，重建tau函数和行列式公式的正式$\hbar$级数展开。 最后，我们通过六个tau函数的前几阶的显式计算来说明前面的结果。