The m=1 Amplituhedron and Cyclic Hyperplane Arrangements

Steven N. Karp ; Lauren K. Williams

Séminaire lotharingien de combinatoire, 78B (2017)

Résumé

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. 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. We deduce that A_n,k,1 is homeomorphic to a ball.