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

A bstract We study remarkable connections between twistor-string formulas for tree amplitudes in N N = 4 SYM and N N = 6 ABJM, and the corresponding momentum amplituhedron in the kinematic space of D = 4 and D = 3, respectively. Based on the Veronese map to positive Grassmannians, we define a twistor-string map from G + (2, n) to a (2 n− 4) -dimensional subspace of the 4d kinematic space where the momentum amplituhedron of SYM lives. We provide strong evidence that the twistor-string map is a diffeomorphism from G + (2, n) to the interior of momentum amplituhedron; the canonical form of the latter, which is known to give tree amplitudes of SYM, can be obtained as pushforward of that of former. We then move to three dimensions: based on Veronese map to orthogonal positive Grassmannian, we propose a similar twistor-string map from the moduli space M₀, ₍^+ ℳ 0, n + to a (n− 3) -dimensional subspace of 3d kinematic space. The image gives a new positive geometry which conjecturally serves as the momentum amplituhedron for ABJM; its canonical form gives the tree amplitude with reduced supersymmetries in the theory. We also show how boundaries of compactified M₀, ₍^+ ℳ 0, n + map to boundaries of momentum amplituhedra for SYM and ABJM corresponding to factorization channels of amplitudes, and in particular for ABJM case the map beautifully excludes all unwanted channels.