Decompositions of amplituhedra

Abstract

The (tree) amplituhedron An;k;m is the image in the Grassmannian Grk;k+m of the totally nonnegative Grassmannian Gr≥0k;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 planar 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 Gr≥0k;n whose images conjecturally “triangulate” the amplituhedron—that is, their images are ,disjoint and cover a dense subset of An;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 An;k;4 . We also conjecture that for arbitrary even m, there is a decomposition of the amplituhedron An;k;m involving precisely M k; n-k-m;[Formula Presented] top-dimensional cells (of dimension km), where M(a; b; c) is the number of plane partitions contained in an a × b × c box. This agrees with the fact that when m = 4, the number of BCFW cells is the Narayana number Nn 3;k+1=[Formula Presented].