The momentum amplituhedron of SYM and ABJM from twistor-string maps

We study remarkable connections between twistor-string formulas for tree amplitudes in ${\cal N}=4$ SYM and N}=6$ ABJM, the corresponding momentum amplituhedron kinematic space of $D=4$ $D=3$, respectively. Based on Veronese map to positive Grassmannians, we define a from $G_{+}(2,n)$ $(2n{-}4)$-dimensional subspace 4d where lives. provide strong evidence that is diffeomorphism $G_+(2,n)$ interior amplituhedron; canonical form latter, which known give SYM, can be obtained as pushforward former. then move three dimensions: based orthogonal Grassmannian, propose similar moduli M}_{0,n}^+$ $(n{-}3)$-dimensional 3d space. The image gives new geometry conjecturally serves ABJM; its amplitude with reduced supersymmetries theory. also show how boundaries compactified amplituhedra ABJM factorization channels amplitudes, particular case beautifully excludes all unwanted channels.