The m=1 amplituhedron and cyclic hyperplane arrangements
The m=1 amplituhedron and cyclic hyperplane arrangements
复制标题
m=1 振幅面体和循环超平面排列
DOI：
10.1093/imrn/rnx140
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
L. Williams
中科院分区：
文献类型：
--
作者：
Steven N. Karp;L. Williams
树型振幅面A（n，k，m）是Grassmannian Gr（k，k+m）中Gr（k，n）的全非负部分在全正线性映射诱导的映射下的像.它是由Arkani-Hamed和Trnka在2013年引入的，目的是为N=4超对称杨-米尔斯理论中散射振幅的计算提供几何基础。当k+m=n时，振幅面同构于完全非负的格拉斯曼，当k=1时，振幅面是循环多面体。虽然m=4的情况与物理学最相关，但对于任何m，振幅面体都是一个有趣的数学对象。本文在m=1的情况下研究它。我们从一个正交的角度出发，定义了一个相关的“B-振幅面体”B（n，k，m），我们证明它与A（n，k，m）同构。我们用这种重新表述来描述的振幅面体的符号变化。然后，我们给出了一个细胞分解的振幅面体A（n，k，1）使用的图像的一个集合的区别细胞的完全非负的格拉斯曼。我们还表明，A（n，k，1）可以识别为一个循环超平面安排的有界面的复形，并描述其细胞如何配合在一起。我们推出A（n，k，1）同胚于一个球.
The (tree) amplituhedron A(n,k,m) is the image in the Grassmannian Gr(k,k+m) of the totally nonnegative part of Gr(k,n), under a (map induced by a) linear map which is totally positive. It was introduced by Arkani-Hamed and Trnka in 2013 in order to give a geometric basis for the computation of scattering amplitudes in N=4 supersymmetric Yang-Mills theory. When k+m=n, the amplituhedron is isomorphic to the totally nonnegative Grassmannian, and when k=1, the amplituhedron is a cyclic polytope. While the case m=4 is most relevant to physics, the amplituhedron is an interesting mathematical object for any m. In this paper we study it in the case m=1. We start by taking an orthogonal point of view and define a related "B-amplituhedron" B(n,k,m), which we show is isomorphic to A(n,k,m). We use this reformulation to describe the amplituhedron in terms of sign variation. We then give a cell decomposition of the amplituhedron A(n,k,1) using the images of a collection of distinguished cells of the totally nonnegative Grassmannian. We also show that A(n,k,1) can be identified with the complex of bounded faces of a cyclic hyperplane arrangement, and describe how its cells fit together. We deduce that A(n,k,1) is homeomorphic to a ball.
影响因子：
1.3
作者：
Federico Ardila;F. Rincón;L. Williams
通讯作者：
Federico Ardila;F. Rincón;L. Williams