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

TL;DR

This paper explores the geometric structures underlying tree amplitudes in ${\

Contribution

It introduces a twistor-string map connecting positive Grassmannians to momentum amplituhedra in 4d and 3d, providing a new geometric perspective on SYM and ABJM amplitudes.

Findings

The twistor-string map is a diffeomorphism to the momentum amplituhedron interior.

The canonical form of the momentum amplituhedron yields tree amplitudes.

Boundaries of moduli spaces correspond to amplitude factorization channels.

Abstract

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.