论文标题
$ x $ - $ y $ y $ symplectic转换的简单公式
A simple formula for the $x$-$y$ symplectic transformation in topological recursion
论文作者
论文摘要
令$ w_ {g，n} $为某些给定光谱曲线$（x，y）$和$ w^\ vee_ {g，n} $ for $（y，x）$计算的相关器，其中$ x，y $的角色被倒置。这两组相关器$ w_ {g，n} $和$ w^\ vee_ {g，n} $与$ x $ - $ y $ y $ symplectic变换相关。 Bychkov，Dunin-Barkowski，Kazarian和Shadrin计算了两个略有不同的相关器集之间的功能关系。他们与亚历山德罗夫（Alexandrov）一起证明了他们的功能关系确实是拓扑递归中的$ x $ - $ y $ symplectic转换。本文提供了一个相当简单的公式，直接在$ w_ {g，n} $和$ w^\ vee_ {g，n} $之间，该公式由其定理用于Meromorphic $ x $和$ y $，并具有简单而独特的分支点。由于自由概率与完全简单的图与普通图之间的联系，我们结论了时刻和无阶无累积累积物的简化力矩肿瘤关系。
Let $W_{g,n}$ be the correlators computed by Topological Recursion for some given spectral curve $(x,y)$ and $W^\vee_{g,n}$ for $(y,x)$, where the role of $x,y$ is inverted. These two sets of correlators $W_{g,n}$ and $W^\vee_{g,n}$ are related by the $x$-$y$ symplectic transformation. Bychkov, Dunin-Barkowski, Kazarian and Shadrin computed a functional relation between two slightly different sets of correlators. Together with Alexandrov, they proved that their functional relation is indeed the $x$-$y$ symplectic transformation in Topological Recursion. This article provides a fairly simple formula directly between $W_{g,n}$ and $W^\vee_{g,n}$ which holds by their theorem for meromorphic $x$ and $y$ with simple and distinct ramification points. Due to the recent connection between free probability and fully simple vs ordinary maps, we conclude a simplified moment-cumulant relation for moments and higher order free cumulants.
