We prove that the topological recursion formalism can be used to compute the WKB expansion of solutions of second order differential operators obtained by quantization of any hyper-elliptic curve. We express this quantum curve in terms of spectral Darboux coordinates on the moduli space of meromorphic ${\mathfrak{sl}}_2$-connections on ${\mathbb{P}}^1$ and argue that the topological recursion produces a 2g-parameter family of associated tau functions, where 2g is the dimension of the moduli space considered. We apply this procedure to the 6 Painlevé equations which correspond to $g=1$ and consider a $g=2$ example.

中文翻译：

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来自等单系统和拓扑递归的超椭圆曲线的量化

我们证明了拓扑递归形式可以用于计算任何超椭圆曲线量化得到的二阶微分算子解的 WKB 展开。我们用亚纯模空间上的光谱达布坐标来表达这条量子曲线。${\mathfrak{SL}}_2$- 连接 ${\mathbb{磷}}^1$并认为拓扑递归产生了一个 2 g参数族相关的 tau 函数，其中 2 g是所考虑的模空间的维度。我们将此过程应用于 6 个 Painlevé 方程，它们对应于$G=1$ 并考虑一个 $G=2$ 例子。