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

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 |$m$|⁠. [ABSTRACT FROM AUTHOR]