论文标题
平面运动学：循环固定点，镜像超电势，k维加泰罗尼亚数字和根系
Planar Kinematics: Cyclic Fixed Points, Mirror Superpotential, k-Dimensional Catalan Numbers, and Root Polytopes
论文作者
论文摘要
在本文中，我们证明了$ n $点的空间中的$ x（k，n）$中的$ \ mathbb {cp}^{k-1} $中的$ n $点的指数，这些配置是在特定环状操作下固定的，这是平面运动学（PK）上普遍散射方程的解决方案。在第一部分中，我们给出了一个建设性的上限：我们表明，这些解决方案将$ \ {1，\ ldots，n \} $的某些上的k-Element子集注入某些上的k-元素子集，因此它们的数量在上面的lyndon单词数量与K One和N-K Zeros的数量相吻合。该证明使用$ g（n-k，n）$的镜像的超电势与$ x（k，n）$的广义chy电位之间的连接有些令人惊讶。我们还检查了最近的猜想，即PK上的概括为$ k $二维的加泰罗尼亚数字，以评估$ k = 3 $ = 3 $和$ n \ leq 40 $和$（k，n）=（6,13）$。然后，我们直接将CEGM广义的BIAD偶会标量振幅直接重新制定为$ {\ rm trop}^+ g（k，n）$上的拉普拉斯变换型积分，我们使用它来评估PK上的幅度，目的是展示GFD如何一起展示GFD的glue。 我们启动了平面运动学点的两个最小晶格多地域社区的研究。其中之一是，在情况下，等级级的根polytope $ \ mathcal {r} _ {k，n} $，在情况下$ k = 2 $，是标准类型A根polytope的投影。另一个表示$π_{k，n} $，在$ k = 2 $的情况下，是AssociaHedron的变性。我们检查并包括$ \ Mathcal {r} _ {3,9} $和$ \ MATHCAL {r} _ {4,9} $，$ \ MATHCAL {r} _ {r} _ {k，n} $的相对体积是多维的catalan数字$ c^{n-- $ m^{（k）}的几何和组合解释（\ mathbb {i} _n，\ mathbb {i} _n）$附近。
In this paper we prove that points in the space $X(k,n)$ of configurations of $n$ points in $\mathbb{CP}^{k-1}$ which are fixed under a certain cyclic action are the solutions to the generalized scattering equations on planar kinematics (PK). In the first part, we give a constructive upper bound: we show that these solutions inject into certain aperiodic k-element subsets of $\{1,\ldots, n\}$, and consequently that their number is bounded above by the number of Lyndon words with k one's and n-k zeros. The proof uses a somewhat surprising connection between the superpotential of the mirror of $G(n-k,n)$ and the generalized CHY potential on $X(k,n)$. We also check the recent conjecture that generalized biadjoint amplitudes evaluate to $k$-dimensional Catalan numbers on PK for several examples including $k=3$ and $n\leq 40$ and $(k,n)=(6,13)$. We then reformulate the CEGM generalized biadjoint scalar amplitude directly as a Laplace transform-type integral over ${\rm Trop}^+ G(k,n)$ and we use it to evaluate the amplitude on PK with the purpose of exhibiting how GFD's glue together. We initiate the study of two minimal lattice polytopal neighborhoods of the planar kinematics point. One of these, the rank-graded root polytope $\mathcal{R}_{k,n}$, in the case $k=2$, is a projection of the standard type A root polytope. The other, denoted $Π_{k,n}$, in the case $k=2$, is a degeneration of the associahedron. We check up to and including $\mathcal{R}_{3,9}$ and $\mathcal{R}_{4,9}$ that the relative volume of $\mathcal{R}_{k,n}$ is the multi-dimensional Catalan number $C^{(k)}_{n-k}$, hinting towards the possibility of deeper geometric and combinatorial interpretations of $m^{(k)}(\mathbb{I}_n,\mathbb{I}_n)$ near the PK point.
