The Positive Tropical Grassmannian, the Hypersimplex, and the m = 2 Amplituhedron

Abstract

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