The ABJM Amplituhedron

Abstract

In this paper, we take a major step towards the construction and applications of an all-loop, all-multiplicity amplituhedron for three-dimensional planar $\mathcal{N}=6$ Chern-Simons matter theory, or the $\textit{ABJM amplituhedron}$. We show that by simply changing the overall sign of the positive region of the original amplituhedron for four-dimensional planar $\mathcal{N}=4$ super-Yang-Mills (sYM) and performing a symplectic reduction, only three-dimensional kinematics in the middle sector of even-multiplicity survive. The resulting form of the geometry, combined with its parity images, gives the full loop integrand. This simple modification geometrically enforces the vanishing of odd-multiplicity cuts, and manifests the correct soft cuts as well as two-particle unitarity cuts. Furthermore, the so-called ``bipartite structures" of four-point all-loop negative geometries also directly generalize to all multiplicities. We introduce a novel approach for triangulating loop amplituhedra based on the kinematics of the tree region, resulting in local integrands tailored to ``prescriptive unitarity". This construction sheds fascinating new light on the interplay between loop and tree amplituhedra for both ABJM and $\mathcal{N}=4$ sYM: the loop geometry demands that the tree region must be dissected into $\textit{chambers}$, defined by the simultaneous positivity of maximal cuts. The loop geometry is then the ``fibration" of the tree region. Using the new construction, we give explicit results of one-loop integrands up to ten points and two-loop integrands up to eight points by computing the canonical form of ABJM loop amplituhedron.