We study multi-boundary correlators of Witten–Kontsevich topological gravity in two dimensions. We present a method of computing an open string like expansion, which we call the ’t Hooft expansion, of the $n$-boundary correlator for any $n$ up to any order by directly solving the Korteweg–De Vries equation. We first explain how to compute the ’t Hooft expansion of the one-boundary correlator. The algorithm is very similar to that for the genus expansion of the open free energy. We next show that the ’t Hooft expansion of correlators with more than one boundary can be computed algebraically from the correlators with a lower number of boundaries. We explicitly compute the ’t Hooft expansion of the $n$-boundary correlators for $n=1, 2, 3$. Our results reproduce previously obtained results for Jackiw–Teitelboim gravity and also the ’t Hooft expansion of the exact result of the three-boundary correlator which we calculate independently in the Airy case.

中文翻译：

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二维拓扑引力中多边界相关器的't Hooft展开

我们在二维中研究了 Witten-Kontsevich 拓扑引力的多边界相关器。我们提出了一种通过直接求解 Korteweg-De Vries 方程来计算任意阶的任意 $n$ 的 $n$-边界相关器的类似开弦展开的方法，我们称之为 't Hooft 展开。我们首先解释如何计算单边界相关器的 't Hooft 展开。该算法与开放自由能的属扩展算法非常相似。接下来，我们将展示具有多个边界的相关器的 't Hooft 展开可以从具有较少边界数量的相关器进行代数计算。我们显式地计算了 $n=1, 2, 3$ 的 $n$-边界相关器的 't Hooft 展开。