Decompositions of amplituhedra
Annales de l'Institut Henri Poincare (D) Combinatorics, Physics and their Interactions
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].
National Science Foundation
Amplituhedron, BCFW recursion, Narayana number, Plane partition, Scattering amplitude, Totally nonnegative Grassmannian
Mathematics and Statistics
Steven N. Karp, Lauren K. Williams, and Yan X. Zhang. "Decompositions of amplituhedra" Annales de l'Institut Henri Poincare (D) Combinatorics, Physics and their Interactions (2020): 303-363. https://doi.org/10.4171/AIHPD/87