广义Kontsevich图与r-自旋相交数的拓扑递归
Topological recursion for generalised Kontsevich graphs and r-spin intersection numbers
AI总结：
本文定义四类广义Kontsevich图并证明主要类型ciliated maps满足Tutte递归，从而可由拓扑递归计算，进而建立其与r-自旋相交数的恒等式，为组合理解Witten类及探索拓扑递归的辛不变性提供新工具。
摘要：
Kontsevich在证明Witten猜想时引入了某些带边黎曼面模空间组合模型的胞腔分解中的ribbon graphs。本文定义了四种类型的广义Kontsevich图，并找出它们之间的组合关系。我们将主要类型称为ciliated maps，并利用辅助类型证明它们满足一个Tutte递归，进而将其转化为一大类谱曲线的拓扑递归loop equations的组合解释。由此得出，ciliated maps——作为广义Kontsevich矩阵模型（GKM）的Feynman图——可由拓扑递归计算。我们所考虑的GKM特例与r-KdV可积层级相关，而由于后者的string solution编码了Witten的r-自旋类的相交数，我们找到了ciliated maps与r-自旋相交数之间的恒等式，这意味着它们同样受拓扑递归支配。反过来，这为从组合角度理解Witten类铺平了道路。这一关于GKM的新的拓扑递归视角为探索拓扑递归在更广泛谱曲线类上的猜想性辛不变性提供了具体工具。
英文摘要：
Kontsevich introduced certain ribbon graphs as cell decompositions for combinatorial models of moduli spaces of complex curves with boundaries in his proof of Witten's conjecture. In this work, we define four types of generalised Kontsevich graphs and find combinatorial relations among them. We call the main type ciliated maps and use the auxiliary ones to show they satisfy a Tutte recursion that we turn into a combinatorial interpretation of the loop equations of topological recursion for a large class of spectral curves. It follows that ciliated maps, which are Feynman graphs for the Generalised Kontsevich matrix Model (GKM), are computed by topological recursion. Our particular instance of the GKM relates to the r-KdV integrable hierarchy and since the string solution of the latter encodes intersection numbers with Witten's $r$-spin class, we find an identity between ciliated maps and $r$-spin intersection numbers, implying that they are also governed by topological recursion. In turn, this paves the way towards a combinatorial understanding of Witten's class. This new topological recursion perspective on the GKM provides concrete tools to explore the conjectural symplectic invariance property of topological recursion for large classes of spectral curves.
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