关于m=2振幅面体的边界
On the Boundaries of the m=2 Amplituhedron
摘要：
振幅面体$\mathcal{A}_{n,k}^{(m)}$是现代数学和物理学中非常感兴趣的几何对象：对于数学家来说，它们是基于正性概念的多边形和多面体的组合丰富的概括;对于物理学家来说，振幅面体$\mathcal{A}^{(4)}_{n,k}$编码平面$\mathcal{N}=4$超级杨-米尔斯理论的散射振幅。本文研究了振幅面体$\mathcal{A}_{n,k}^{(2)}$的边界结构。我们对所有维度的所有边界进行分类，并提供其图形枚举。我们发现幅面体的边界偏序集是欧拉的，并证明幅面体的欧拉特征线等于1。这为证明$m=2$的幅面同胚于闭球提供了初步的步骤。
英文摘要：
Amplituhedra $\mathcal{A}_{n,k}^{(m)}$ are geometric objects of great interest in modern mathematics and physics: for mathematicians they are combinatorially rich generalizations of polygons and polytopes, based on the notion of positivity; for physicists, the amplituhedron $\mathcal{A}^{(4)}_{n,k}$ encodes the scattering amplitudes of the planar $\mathcal{N}=4$ super Yang-Mills theory. In this paper we study the structure of boundaries for the amplituhedron $\mathcal{A}_{n,k}^{(2)}$. We classify all boundaries of all dimensions and provide their graphical enumeration. We find that the boundary poset for the amplituhedron is Eulerian and show that the Euler characteristic of the amplituhedron equals one. This provides an initial step towards proving that the amplituhedron for $m=2$ is homeomorphic to a closed ball.
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