Decompositions of amplituhedra

Karp, Steven N.; Williams, Lauren K.; Zhang, Yan X

Annales de l’Institut Henri Poincaré D, Tome 7 (2020) no. 3, pp. 303-363.

The (tree) amplituhedron 𝒜 n,k,m is the image in the Grassmannian Gr k,k+m of the totally nonnegative Grassmannian Gr k,n ≥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 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 k,n ≥0, whose images conjecturally "triangulate" the amplituhedron – that is, their images are disjoint and cover a dense subset of 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 𝒜 n,k,4. We also conjecture that for arbitrary even m, there is a decomposition of the amplituhedron 𝒜 n,k,m involving precisely Mk , n - k - m , m 2 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 N n-3,k+1 =1 n-3n-3 k+1n-3 k.

Classification : 05-XX, 14-XX, 15-XX, 81-XX
Mots-clés : Amplituhedron, scattering amplitude, totally nonnegative Grassmannian, BCFW recursion, Narayana number, plane partition