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}^{\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} .

Triangulations and Canonical Forms of Amplituhedra: A Fiber-Based Approach Beyond Polytopes

“…More specifically, in the context of scattering amplitudes in N = 4 super Yang-Mills theory [AHT14], where amplitudes can be computed from the geometry of A n,k,4 . Furthermore, the conjectural formula given in [KWZ17] for the number of cells in each triangulation of the amplituhedron is invariant under the operation of swapping the parameters k and , which has motivated further studies [FŁP18,GL20].…”

Positive geometry, local triangulations, and the dual of the Amplituhedron

“…The geometric picture has also been used to obtain a large amount of all-loop data [20,25,26] not currently accessible by any conventional amplitude methods. In parallel, there has also been significant progress trying to understand more formal aspects of the Amplituhedron, including its boundary and combinatorial structure [27][28][29][30][31], the connection to symbol alphabets, properties of final amplitudes (rather than integrands) [32][33][34][35], and many others (see e.g., [36][37][38][39][40]). Despite this progress, there are many open questions, including the determination of the all-loop four-point integrand (see recent progress [20,41,42]).…”

Combinatorics of the geometry of Wilson loop diagrams I: equivalence classes via matroids and polytopes

“…The second paper [3] in this series identifies precisely which positroid cell is associated to a given In recent years, there has been an active program researching the geometry and combinatorics underlying SYM = 4 theory [5,8,13,14,16]. This body of work started with the observation that the on shell (tree level) amplitudes of this theory correspond to the volume of a subspace of a positive Grassmannian, called an Amplituhedron [8].…”

“…Since then, there has been significant work studying the structure of the Amplituhedron both geometrically and combinatorially, for instance [6,7,14,16]. The focus of much of this work has been on understanding the relationship between BCFW diagrams and positroid cells, which are the combinatorial objects from the physics of the system and the corresponding natural objects in the positive Grassmannian respectively.…”