论文标题
最小运动学：全$ k $和$ n $
Minimal Kinematics: An All $k$ and $n$ Peek into ${\rm Trop}^+{\rm G}(k,n)$
论文作者
论文摘要
在本说明中，我们为Cachazo-early-Guevara-Mizera（CEGM）提供了一个公式，用于所有$ k $和$ n $的概括性振幅和我们所谓的最小运动学。我们证明，在最小运动学上，在$ \ Mathbb {cp}^{k-1} $上的$ n $点上的散射方程具有独特的解决方案，并且该解决方案在Veronese嵌入的图像中。最小的运动学是最早的$ k $概括，以$ k = 2 $引入，并使用循环订购的选择。我们猜想了$ m_n^{（k）}的明确公式（\ mathbb {i}，\ mathbb {i}）$，我们已经通过$ n = 10 $分析了所有$ k $。答案是一个简单的理性函数，只有简单的杆子。两极具有循环图的组合结构$ {\ rm c} _n^{（1,2，\ dots，k-2）} $。在广义平面Feynman图方面，还可以使用积极的热带Grassmannian $ {\ rm tr}^+{\ rm g}（k，n）$来评估广义的双节振幅。对于文献中后者已知的所有情况，我们都发现了两个定义之间的完美一致性。特别是，这给出了$ {\ rm tr}^+{\ rm g} $ 90 \，608 $平面阵列的第一个强度一致性检查（4,8）$最近由Cachazo，Guevara，Umbert和Zhang计算出来的。我们还介绍了另一类称为Planar-Basis运动学的特殊运动学类别，该运动学概括了Cachazo，他和Yuan以$ k = 2 $引入的一种，并使用了最近对所有$ k $引入的平面基础。基于所有$ k $的数值计算，我们推测，在平面 - 巴西斯运动学$ m_n^{（k）}（\ Mathbb {i}，\ Mathbb {i}）上，对多维Catalan编号进行了评估，暗示了新颖的组合方法。对于$ k = 2 $，这些是加泰罗尼亚的标准号码。
In this note we present a formula for the Cachazo-Early-Guevara-Mizera (CEGM) generalized biadjoint amplitudes for all $k$ and $n$ on what we call the minimal kinematics. We prove that on the minimal kinematics, the scattering equations on the configuration space of $n$ points on $\mathbb{CP}^{k-1}$ has a unique solution, and that this solution is in the image of a Veronese embedding. The minimal kinematics is an all $k$ generalization of the one recently introduced by Early for $k=2$ and uses a choice of cyclic ordering. We conjecture an explicit formula for $m_n^{(k)}(\mathbb{I},\mathbb{I})$ which we have checked analytically through $n=10$ for all $k$. The answer is a simple rational function which has only simple poles; the poles have the combinatorial structure of the circulant graph ${\rm C}_n^{(1,2,\dots, k-2)}$. Generalized biadjoint amplitudes can also be evaluated using the positive tropical Grassmannian ${\rm Tr}^+{\rm G}(k,n)$ in terms of generalized planar Feynman diagrams. We find perfect agreement between both definitions for all cases where the latter is known in the literature. In particular, this gives the first strong consistency check on the $90\,608$ planar arrays for ${\rm Tr}^+{\rm G}(4,8)$ recently computed by Cachazo, Guevara, Umbert and Zhang. We also introduce another class of special kinematics called planar-basis kinematics which generalizes the one introduced by Cachazo, He and Yuan for $k=2$ and uses the planar basis recently introduced by Early for all $k$. Based on numerical computations through $n=8$ for all $k$, we conjecture that on the planar-basis kinematics $m_n^{(k)}(\mathbb{I},\mathbb{I})$ evaluates to the multidimensional Catalan numbers, suggesting the possibility of novel combinatorial interpretations. For $k=2$ these are the standard Catalan numbers.
