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A remarkable connection between perturbative scattering amplitudes offour-dimensional planar SYM, and the stratification of the positivegrassmannian, was revealed in the seminal work of Arkani-Hamed et. al. Similarextension for three-dimensional ABJM theory was proposed. Here we establish adirect connection between planar scattering amplitudes of ABJM theory, andsingularities there of, to the stratification of the positive orthogonalgrassmannian. In particular, scattering processes are constructed throughon-shell diagrams, which are simply iterative gluing of the fundamentalfour-point amplitude. Each diagram is then equivalent to the merging offundamental OG_2 orthogonal grassmannian to form a larger OG_k, where 2k is thenumber of external particles. The invariant information that is encoded in eachdiagram is precisely this stratification. This information can be easily readoff via permutation paths of the on-shell diagram, which also can be used toderive a canonical representation of OG_k that manifests the vanishing ofconsecutive minors as the singularity of all on-shell diagrams. Quiteremarkably, for the BCFW recursion representation of the tree-level amplitudes,the on-shell diagram manifests the presence of all physical factorizationpoles, as well as the cancellation of the spurious poles. After analyticallycontinuing the orthogonal grassmannian to split signature, we reveal that eachon-shell diagram in fact resides in the positive cell of the orthogonalgrassmannian, where all minors are positive. In this language, the amplitudesof ABJM theory is simply an integral of a product of dlog forms, over thepositive orthogonal grassmannian.

ABJM amplitudes and the positive orthogonal grassmannian