  A remarkable connection between perturbative scattering amplitudes of
four-dimensional planar SYM, and the stratification of the positive
grassmannian, was revealed in the seminal work of Arkani-Hamed et. al. Similar
extension for three-dimensional ABJM theory was proposed. Here we establish a
direct connection between planar scattering amplitudes of ABJM theory, and
singularities there of, to the stratification of the positive orthogonal
grassmannian. In particular, scattering processes are constructed through
on-shell diagrams, which are simply iterative gluing of the fundamental
four-point amplitude. Each diagram is then equivalent to the merging of
fundamental OG_2 orthogonal grassmannian to form a larger OG_k, where 2k is the
number of external particles. The invariant information that is encoded in each
diagram is precisely this stratification. This information can be easily read
off via permutation paths of the on-shell diagram, which also can be used to
derive a canonical representation of OG_k that manifests the vanishing of
consecutive minors as the singularity of all on-shell diagrams. Quite
remarkably, for the BCFW recursion representation of the tree-level amplitudes,
the on-shell diagram manifests the presence of all physical factorization
poles, as well as the cancellation of the spurious poles. After analytically
continuing the orthogonal grassmannian to split signature, we reveal that each
on-shell diagram in fact resides in the positive cell of the orthogonal
grassmannian, where all minors are positive. In this language, the amplitudes
of ABJM theory is simply an integral of a product of dlog forms, over the
positive orthogonal grassmannian.

High Energy Physics - Theory