Topological Recursion generates a family of symmetric differential forms (correlators) from some initial data $(Σ,x,y,B)$. We give a functional relation between the correlators of genus $g=0$ generated by the initial data $(Σ,x,y,B)$ and by the initial data $(Σ,y,x,B)$, where $x$ and $y$ are interchanged. The functional relation is derived with the loop insertion operator by computing a functional relation for some intermediate correlators. Additionally, we show that our result is equivalent to the recent result of \cite{Borot:2021thu} in case of $g=0$. Consequently, we are providing a simplified functional relation between generating series of higher order free cumulants and moments in higher order free probability.
Topological recursion for spin enumeration of hypermaps
Center for Geometry and Physics, Institute for Basic Science (IBS)（基础科学研究院几何物理中心）；Hebrew University of Jerusalem（耶路撒冷希伯来大学）；HSE University（高等经济学院）；Skoltech（斯科尔科沃科学技术学院）；NRC “Kurchatov Institute” – ITEP（库尔恰托夫研究所国家研究中心–理论物理与实验物理研究所）；University of Amsterdam（阿姆斯特丹大学）
Highest weight vectors, shifted topological recursion and quantum curves
University of Alberta（阿尔伯塔大学）；University of Saskatchewan（萨斯喀彻温大学）；Syddansk Universitet（南丹麦大学）；Dipartimento di Matematica e Applicazioni, Organization（数学与应用科学系）；INFN（意大利国家核物理研究所）
Painlev'e 2 equation with arbitrary monodromy parameter, topological recursion and determinantal formulas
本文的目的是证明Painlev 'e 2系统的行列式公式与由其谱曲线上的拓扑递归计算出的关联函数是一致的。该结果是建立在两个不同的拉克斯对与Painlev 'e 2系统，即Jimbo-Miwa拉克斯对和Harnad-Tracy-Widom拉克斯对，其光谱曲线不连接任何辛变换。给出了一种不使用插入算子证明拓扑类型性质的新方法。在这个过程中，取时间参数t为无穷大，给出了由Hermite-Weber曲线和Bessel曲线计算的辛不变量F（g）分别等于。这个结果推广了随机矩阵理论在θ = 0的特殊情况下的类似结果。我们相信，这种方法应该适用于所有6个Painlev 'e方程与任意monodromy参数。明确的计算g = 3提供沿文件作为结果的说明。
