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.
Highlights
Grassmannian formulation extend to the full planar loop integral in four dimensions, prior to integration
After analytically continuing the orthogonal Grassmannian to split signature, we reveal that each on-shell diagram resides in the positive cell of the orthogonal Grassmannian, where all minors are positive
The invariant content of the on-shell diagrams is precisely this stratification, where different on-shell diagrams that belong to the same strata, can be shown to be equivalent though a series of change of variables, whose physical interpretation mounts to the equivalence of distinct BCFW representations
Summary
The scattering amplitudes of ABJM [16] theory will be the focus of our study. It is a ChernSimons matter theory with N = 6 supersymmetry. The most general configuration of the orthogonal Grassmannian, referred to as the “top-cell ”, is k(k − 1)/2-dimensional: Dim(Top Cell)OG(k,2k) = k(k − 1)/2 It was proposed by Sangmin Lee [35] that: A tree-level amplitudes of ABJM theory is given by a sum of the residues of the following integral over a OG(k, 2k) orthogonal Grassmannian Cai. where Ml represent the l-th consecutive minor: Ml ≡ (Cl a1 Cl+1 a2 · · · Cl+k ak ) = (l l + 1 · · · , l + k). We will reverse the previous procedure and consider the top-cell first being partially localized using the (k − 3)(k − 2)/2 number of zeroes in the minors This leaves behind a (2k − 3)-dimensional integral, subject to the constraints of the bosonic delta function δ(C · λ).
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