Séminaire Lotharingien de Combinatoire, 78B.20 (2017), 12 pp.

Steven N. Karp and Lauren K. Williams

The m=1 Amplituhedron and Cyclic Hyperplane Arrangements

Abstract. The (tree) amplituhedron An,k,m is the image in the Grassmannian Grk,k+m of the totally nonnegative part of Grk,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" Bn,k,m, which we show is isomorphic to An,k,m. We use this reformulation to describe the amplituhedron in terms of sign variation. We then give a cell decomposition of the amplituhedron An,k,1 using the images of a collection of distinguished cells of the totally nonnegative Grassmannian. We also show that An,k,1 can be identified with the complex of bounded faces of a cyclic hyperplane arrangement. We deduce that An,k,1 is homeomorphic to a ball.

Received: November 14, 2016. Accepted: February 17, 2017. Final version: April 1, 2017.

The following versions are available: