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

Title: 潘勒维方程，拓扑类型性质与通过拓扑递归重建

Abstract: In this article we prove that Lax pairs associated with $\hbar$-dependent six Painlev\'e equations 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 $\hbar$-expansion of the isomonodromic $\tau$-function and of the determinantal formulas by applying the so-called topological recursion to the spectral curve attached to the Lax pair in all six Painlev\'e cases. Finally we illustrate the former results with the explicit computations of the first orders of the six $\tau$-functions. Show Chinese abstract

Abstract: 本文中我们证明了与$\hbar$-依赖的六个Painlevé方程相关的Lax对满足Bergère、Borot和Eynard提出的拓扑类型性质，对于任意通用的单值参数选择均成立。由此我们展示了一种方法，即通过将所谓“拓扑递归”应用于与Lax对相关的谱曲线，可以重构等 mono- 稳态$\tau$-函数以及确定性公式的形式$\hbar$-展开式。最后，我们通过计算六个$\tau$-函数的前几阶显式结果来说明上述结果。