The Positive Tropical Grassmannian, the Hypersimplex, and the <i>m</i> = 2 Amplituhedron

Abstract The positive Grassmannian $Gr^{\geq 0}_{k,n}$ is a cell complex consisting of all points in the real whose Plücker coordinates are non-negative. In this paper we consider image and its positroid cells under two different maps: moment map$\mu $ onto hypersimplex [ 31] amplituhedron map$\tilde{Z}$ 6]. For either map, define dissection to be collection images that disjoint cover dense subset image. Positroid dissections interest because they include many matroid subdivisions; meanwhile, can used calculate amplituhedron’s ‘volume’, which turn computes scattering amplitudes $\mathcal{N}=4$ super Yang-Mills. We map call T-duality from 0}_{k+1,n}$ conjecture it induces bijection $\Delta _{k+1,n}$ $\mathcal{A}_{n,k,2}$; prove for (infinite) class BCFW dissections. note particularly striking an $(n-1)$-dimensional polytope while $\mathcal{A}_{n,k,2}$ $2k$-dimensional non-polytopal $Gr_{k,k+2}$. Moreover, tropical secondary fan regular subdivisions hypersimplex, if only 2D faces polytopes. Finally, toward goal generalizing higher $m$, momentum any even $m$.