论文标题
拓扑递归的符号双重性
Symplectic duality for topological recursion
论文作者
论文摘要
我们考虑加权双旋惠兹的数字，由任意有理函数时间给出的权重为完整周期的指数。这两个特殊的奇异性都是任意的，周期的长度由正式参数控制（两侧最大长度），一方面也有由形式变量程度控制的区别周期。在这些变量中，加权双Hurwitz的数字作为某些差异的扩展系数表示，我们证明这些系数可以满足拓扑递归。 我们的结果部分解决了我们在[ARXIV：2106.08368]中做出的猜想，并基于一个新的显式功能关系系统，用于更通用的$ $（m，n）$ - 相关函数，当两个特殊奇异纤维中正式变量所控制的周期中，这与情况相对应。这些$（m，n）$ - 相关函数是本文的主要主题，而后者的显式功能关系对于加权双Hurwitz数字的组合具有独立的兴趣。我们还将结果放在所谓的“符号双重性”的背景下，这是$ x-y $二元性的概括，这是拓扑递归理论中所知的现象。
We consider weighted double Hurwitz numbers, with the weight given by arbitrary rational function times an exponent of the completed cycles. Both special singularities are arbitrary, with the lengths of cycles controlled by formal parameters (up to some maximal length on both sides), and on one side there are also distinguished cycles controlled by degrees of formal variables. In these variables the weighted double Hurwitz numbers are presented as coefficients of expansions of some differentials that we prove to satisfy topological recursion. Our results partly resolve a conjecture that we made in [arXiv:2106.08368] and are based on a system of new explicit functional relations for the more general $(m,n)$-correlation functions, which correspond to the case when there are distinguished cycles controlled by formal variables in both special singular fibers. These $(m,n)$-correlation functions are the main theme of this paper and the latter explicit functional relations are of independent interest for combinatorics of weighted double Hurwitz numbers. We also put our results in the context of what we call the "symplectic duality", which is a generalization of the $x-y$ duality, a phenomenon known in the theory of topological recursion.
