The functional relation coming from the $x-y$ symplectic transformation of Topological Recursion has a lot of applications; for instance it is the higher order moment-cumulant relation in free probability or can be used to compute intersection numbers on the moduli space of complex curves. We derive the Laplace transform of this functional relation, which has a very nice and compact form as a formal power series in $\hbar$. We apply the Laplace transformed formula to the Airy curve and the Lambert curve which provides simple formulas for $\psi$-class intersections numbers and Hodge integrals on $\overline{\mathcal{M}}_{g,n}$.

中文翻译：

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拓扑递归中$xy$辛变换公式的拉普拉斯变换

拓扑递归的$xy$辛变换得出的函数关系有很多应用；例如，它是自由概率中的高阶矩累积关系，或者可用于计算复杂曲线模空间上的交集数。我们推导出这个函数关系的拉普拉斯变换，它具有非常漂亮和紧凑的形式，作为 $\hbar$ 中的形式幂级数。我们将拉普拉斯变换公式应用于艾里曲线和兰伯特曲线，它为 $\psi$ 类交集数和 $\overline{\mathcal{M}}_{g,n}$ 上的霍奇积分提供了简单的公式。