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

• Published in: arXiv.org 2022-04
• Main Authors: He, Song, Chia-Kai Kuo, Yao-Qi, Zhang
• Format: Article
• Language: English
• Subjects: Amplitudes; Boundaries; Canonical forms; Channels; Isomorphism; Kinematics; Momentum; Strings; Supersymmetry
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2111.02576

We study remarkable connections between twistor-string formulas for tree amplitudes in \({\cal N}=4\) SYM and \({\cal N}=6\) 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 \((2n{-}4)\)-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 \({\cal M}_{0,n}^+\) 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 \({\cal M}_{0,n}^+\) 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.