Decompositions of amplituhedra

Source abstract

The (tree) amplituhedron \mathcal{A}_{n,k,m} is the image in the Grassmannian \mathrm {Gr}_{k,k+m} of the totally nonnegative Grassmannian \mathrm {Gr}_{k,n}^{\geq 0} , 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 planar \mathcal{N}=4 supersymmetric Yang–Mills theory. In the case relevant to physics ( m=4 ), there is a collection of recursively-defined 4k -dimensional BCFW cells in \mathrm {Gr}_{k,n}^{\geq 0} , whose images conjecturally "triangulate" the amplituhedron – that is, their images are disjoint and cover a dense subset of \mathcal{A}_{n,k,4} . In this paper, we approach this problem by first giving an explicit (as opposed to recursive) description of the BCFW cells. We then develop sign-variational tools which we use to prove that when k=2 , the images of these cells are disjoint in \mathcal{A}_{n,k,4} . We also conjecture that for arbitrary even m , there is a decomposition of the amplituhedron \mathcal{A}_{n,k,m} involving precisely M\big(k, n-k-m, \frac{m}{2}\big) top-dimensional cells (of dimension km ), where M(a,b,c) is the number of plane partitions contained in an a \times b \times c box. This agrees with the fact that when m=4 , the number of BCFW cells is the Narayana number N_{n-3,k+1} = \frac{1}{n-3}\binom{n-3}{k+1}\binom{n-3}{k} .