ABJM amplitudes and the positive orthogonal grassmannian

TL;DR

This paper establishes a direct link between ABJM scattering amplitudes and the stratification of the positive orthogonal Grassmannian, revealing geometric structures underlying the amplitudes and their singularities.

Contribution

It extends the positive Grassmannian framework from four-dimensional SYM to three-dimensional ABJM theory, connecting on-shell diagrams to the stratification of OG_k and revealing positivity properties.

Findings

On-shell diagrams correspond to stratification of OG_k.

Amplitudes are integrals over the positive orthogonal Grassmannian.

Spurious poles cancel, leaving physical factorization poles.

Abstract

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.