Related papers: Parity duality for the amplituhedron
The (tree) amplituhedron $\mathcal{A}_{n, k, m}$ is introduced by Arkani-Hamed and Trnka in 2013 in the study of $\mathcal{N}=4$ supersymmetric Yang-Mills theory. It is defined in terms of the totally nonnegative Grassmannians. In this…
The (tree) amplituhedron A(n,k,m) is the image in the Grassmannian Gr(k,k+m) of the totally nonnegative part of Gr(k,n), under a (map induced by a) linear map which is totally positive. It was introduced by Arkani-Hamed and Trnka in 2013 in…
In \cite{arkani2018unwinding}, Arkani-Hamed, Thomas and Trnka formulated two conjectural descriptions of the tree amplituhedron $\ampli$ depending on the parity of $m$. When $m$ is even, the description involves the winding number and when…
The (tree) amplituhedron A(n,k,m) is the image in the Grassmannian Gr(k,k+m) of the totally nonnegative part of Gr(k,n), under a (map induced by a) linear map which is totally positive. It was introduced by Arkani-Hamed and Trnka in 2013 in…
The amplituhedron was recently introduced in the study of scattering amplitudes in $N=4$ super Yang-Mills. We compute the cohomology class of a tree amplituhedron subvariety of the Grassmannian to be the truncation of an affine Stanley…
The tree amplituhedra $\mathcal{A}_{n,k}^{(m)}$ are mathematical objects generalising the notion of polytopes into the Grassmannian. Proposed for $m=4$ as a geometric construction encoding tree-level scattering amplitudes in planar…
The hypersimplex $\Delta_{k+1,n}$ is the image of the positive Grassmannian $Gr^{\geq 0}_{k+1,n}$ under the moment map. It is a polytope of dimension $n-1$ in $\mathbb{R}^n$. Meanwhile, the amplituhedron $\mathcal{A}_{n,k,2}(Z)$ is the…
The amplituhedron $A_{n,k,m}(Z)$ is the image of the positive Grassmannian $Gr_{k,n}^{\geq 0}$ under the map ${Z}: Gr_{k,n}^{\geq 0} \to Gr_{k,k+m}$ induced by a positive linear map $Z:\mathbb{R}^n \to \mathbb{R}^{k+m}$. Motivated by a…
The amplituhedron $A_{n,k,m}$ is a geometric object introduced in the context of scattering amplitudes in $N=4$ super Yang Mills. It generalizes the positive Grassmannian (when $n=k+m$), cyclic polytopes (when $k=1$), and the bounded…
The amplituhedron $\mathcal{A}_{n,k,m}$ was introduced by Arkani-Hamed and Trnka (2014) in order to give a geometric basis for calculating scattering amplitudes in planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory. It is a projection…
We investigate the Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory and show that its formulations in twistor and momentum twistor space can be interchanged. In particular we show that the full symmetry can be…
Any totally positive $(k+m)\times n$ matrix induces a map $\pi_+$ from the positive Grassmannian ${\rm Gr}_+(k,n)$ to the Grassmannian ${\rm Gr}(k,k+m)$, whose image is the amplituhedron $\mathcal{A}_{n,k,m}$ and is endowed with a…
In 2005, Britto, Cachazo, Feng and Witten gave a recurrence (now known as the BCFW recurrence) for computing scattering amplitudes in N=4 super Yang Mills theory. Arkani-Hamed and Trnka subsequently introduced the amplituhedron to give a…
Arkani-Hamed, Cachazo, Cheung and Kaplan have proposed a Grassmannian formulation for the S-matrix of N=4 Yang-Mills as an integral over link variables. In parallel work, the connected prescription for computing tree amplitudes in Witten's…
Tree-level scattering amplitudes in planar N=4 super Yang-Mills are known to be Yangian-invariant. It has been shown that integrability allows to obtain a general, explicit method to find such invariants. The uplifting of this result to the…
The amplituhedron is a mathematical object which was introduced to provide a geometric origin of scattering amplitudes in $\mathcal{N}=4$ super Yang Mills theory. It generalizes \emph{cyclic polytopes} and the \emph{positive Grassmannian},…
The amplituhedra arise as images of the totally nonnegative Grassmannians by projections that are induced by linear maps. They were introduced in Physics by Arkani-Hamed \& Trnka (Journal of High Energy Physics, 2014) as model spaces that…
The positive Grassmannian $Gr_{k,n}^{\geq 0}$ is the subset of the real Grassmannian where all Pl\"ucker coordinates are nonnegative. It has a beautiful combinatorial structure as well as connections to statistical physics, integrable…
A dual formulation of the S Matrix for N=4 SYM has recently been presented, where all leading singularities of n-particle N^{k-2}MHV amplitudes are given as an integral over the Grassmannian G(k,n), with cyclic symmetry, parity and…
We present a new formula for all single trace tree amplitudes in four dimensional super Yang-Mills coupled to Einstein supergravity. Like the Cachazo-He-Yuan formula, our expression is supported on solutions of the scattering equations, but…