The Witten-Kontsevich theorem states that a certain generating function for intersection numbers on the moduli space of stable curves is a tau-function for the KdV integrable hierarchy. This generating function can be recovered via the topological recursion applied to the Airy curve $x=\frac{1}{2}y^2$. In this paper, we consider the topological recursion applied to the irregular spectral curve $xy^2=\frac{1}{2}$, which we call the Bessel curve. We prove that the associated partition function is also a KdV tau-function, which satisfies Virasoro constraints, a cut-and-join type recursion, and a quantum curve equation. Together, the Airy and Bessel curves govern the local behaviour of all spectral curves with simple branch points.

中文翻译：

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贝塞尔曲线上的拓扑递归

Witten-Kontsevich 定理指出，稳定曲线模空间上交点数的某个生成函数是 KdV 可积层次的 tau 函数。这个生成函数可以通过应用于艾里曲线 $x=\frac{1}{2}y^2$ 的拓扑递归来恢复。在本文中，我们考虑应用于不规则光谱曲线 $xy^2=\frac{1}{2}$ 的拓扑递归，我们称之为贝塞尔曲线。我们证明了相关的配分函数也是一个 KdV tau 函数，它满足 Virasoro 约束、切合型递归和量子曲线方程。艾里曲线和贝塞尔曲线共同控制所有具有简单分支点的光谱曲线的局部行为。