The orthogonal momentum amplituhedron and ABJM amplitudes

TL;DR

This paper introduces a geometric object called the momentum space amplituhedron for ABJM theory, revealing its structure and relation to $ ext{N}=4$ super Yang-Mills theory, and verifies it through eight-point amplitude analysis.

Contribution

It defines the momentum space amplituhedron for ABJM amplitudes, connecting it to positive orthogonal Grassmannian and sign flip conditions, and relates it to $ ext{N}=4$ super Yang-Mills theory.

Findings

The amplituhedron's boundaries correspond to Mandelstam variables.

The geometry of ABJM amplitudes is on the boundary of $ ext{N}=4$ sYM.

The canonical form matches the Grassmannian formula for ABJM.

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.