论文标题
Amplituhedra的三角剖分和规范形式：一种基于纤维的方法
Triangulations and Canonical Forms of Amplituhedra: a fiber-based approach beyond polytopes
论文作者
论文摘要
任何完全积极的$（k+m）\ times n $矩阵都会从正grassmannian $ {\ rm gr} _+（k，n）$诱导地图$π_+$，向格拉斯曼尼亚$ {\ rm gr}（k，rm gr}（k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k，k）$赋予了名为规范形式$ {\bfΩ}（\ Mathcal {a} _ {n，k，m}）$的顶级表格。这种结构是由Arkani-Hamed和Trnka引入的，他们表明$ {\bfΩ}（\ Mathcal {a} _ {n，k，4}）$编码$ \ Mathcal {n} = 4 $ Super Yang-super Yang-Mills理论中的散射幅度。此外，$ {\bfΩ}的计算（\ Mathcal {a} _ {n，k，m}）$减少为查找$ \ Mathcal {a} _ {n，k，k，m} $的三角剖分。但是，尽管多面体的三角剖分被其次级多面体完全捕获，但对多型物体以外的物体的三角剖分的研究仍然不发达。 我们启动了$ \ Mathcal {a} _ {n，k，m} $的细分的几何研究，并提供$π的纤维的具体异常参数化：{\ rm gr}（k，k，k，n）\ dashrightArrow {\ dashrightArrow {\ rm gr}（\ rm gr}（\ rm gr}（k，k，k，k，k，k+m）$。 We then use this to explicitly describe a rational top-degree form $ω_{n,k,m}$ (with simple poles) on the fibers and compute ${\bfΩ}(\mathcal{A}_{n,k,m})$ as a summation of certain residues of $ω_{n,k,m}$.作为我们方法的主要应用，我们开发了一个结构良好的次级Amplituhedra的概念，用于共轭与多面体，即$ n-k-1 = m $（偶数）。我们表明，在这种情况下，$π$的每只光纤都由投影空间进行参数，其音量表格$ω_{n，k，m} $仅具有超平面布置上的杆子。使用这样的线性结构，对于循环多型或共轭的Amplituhedra，我们表明Jeffrey-Kirwan残留物计算$ {\bfΩ}（\ Mathcal {a} _} _ {n，k，m}）$来自$ω__{最后，我们提出了一个更一般的纤维阳性几何形状框架，并分析了诸如纤维多面体和格拉斯曼多印象等示例的新家族。
Any totally positive $(k+m)\times n$ matrix induces a map $π_+$ from the positive Grassmannian ${\rm Gr}_+(k,n)$ to the Grassmannian ${\rm Gr}(k,k+m)$, whose image is the amplituhedron $\mathcal{A}_{n,k,m}$ and is endowed with a top-degree form called the canonical form ${\bfΩ}(\mathcal{A}_{n,k,m})$. This construction was introduced by Arkani-Hamed and Trnka, where they showed that ${\bfΩ}(\mathcal{A}_{n,k,4})$ encodes scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills theory. Moreover, the computation of ${\bfΩ}(\mathcal{A}_{n,k,m})$ is reduced to finding the triangulations of $\mathcal{A}_{n,k,m}$. However, while triangulations of polytopes are fully captured by their secondary polytopes, the study of triangulations of objects beyond polytopes is still underdeveloped. We initiate the geometric study of subdivisions of $\mathcal{A}_{n,k,m}$ and provide a concrete birational parametrization of fibers of $π: {\rm Gr}(k,n)\dashrightarrow {\rm Gr}(k,k+m)$. We then use this to explicitly describe a rational top-degree form $ω_{n,k,m}$ (with simple poles) on the fibers and compute ${\bfΩ}(\mathcal{A}_{n,k,m})$ as a summation of certain residues of $ω_{n,k,m}$. As main application of our approach, we develop a well-structured notion of secondary amplituhedra for conjugate to polytopes, i.e. when $n-k-1=m$ (even). We show that, in this case, each fiber of $π$ is parametrized by a projective space and its volume form $ω_{n,k,m}$ has only poles on a hyperplane arrangement. Using such linear structures, for amplituhedra which are cyclic polytopes or conjugate to polytopes, we show that the Jeffrey-Kirwan residue computes ${\bfΩ}(\mathcal{A}_{n,k,m})$ from $ω_{n,k,m}$. Finally, we propose a more general framework of fiber positive geometries and analyze new families of examples such as fiber polytopes and Grassmann polytopes.
