Decompositions of amplituhedra

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 ${\mathrm{Gr}}_{k,n}^{\ge 0}$, 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 ${\mathrm{Gr}}_{k,n}^{\ge 0}$, 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 $M(k,n-k-m,\frac{m}{2})$ 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 $N_{n-3,k+1}=\frac{1}{n-3}\left(\genfrac{}{}{0ex}{}{n-3}{k+1}\right)\left(\genfrac{}{}{0ex}{}{n-3}{k}\right)$.