The orthogonal momentum amplituhedron and ABJM amplitudes

Abstract

In this paper, we introduce the momentum space amplituhedron for tree-level scattering amplitudes of ABJM theory. We demonstrate that the scattering amplitude can be identified as the canonical form on the space given by the product of positive orthogonal Grassmannian and the moment curve. The co-dimension one boundaries of this space are simply the odd-particle planar Mandelstam variables, while the even-particle counterparts are "hidden" as higher co-dimension boundaries. Remarkably, this space can be equally defined through a series of "sign flip" requirements of the projected external data, identical to "half" of four-dimensional $\mathcal{N}=4$ super Yang-Mills theory (sYM). Thus in a precise sense the geometry for ABJM lives on the boundary of $\mathcal{N}=4$ sYM. We verify this relation through eight-points by showing that the BCFW triangulation of the amplitude tiles the amplituhedron. The canonical form is naturally derived using the Grassmannian formula for the amplitude in the $\mathcal{N}=4$ formalism for ABJM theory.