The Positive orthogonal Grassmannian and loop amplitudes of ABJM

TL;DR

This paper explores the combinatorics of the positive orthogonal Grassmannian and its relation to ABJM scattering amplitudes, revealing integrability properties and loop amplitude structures.

Contribution

It introduces a canonical embedding of OG_k into the Grassmannian, analyzes the canonical volume form, and connects combinatorics with integrability and scattering amplitudes.

Findings

Canonical volume form invariant under moves

Reduction of graphs to irreducible forms with dLog forms

Identification of the S-matrix satisfying tetrahedron equations

Abstract

In this paper we study the combinatorics associated with the positive orthogonal Grassmannian OG_k and its connection to ABJM scattering amplitudes. We present a canonical embedding of OG_k into the Grassmannian Gr(k,2k), from which we deduce the canonical volume form that is invariant under equivalence moves. Remarkably the canonical forms of all reducible graphs can be converted into irreducible ones with products of dLog forms. Unlike N=4 super Yang-Mills, here the Jacobian plays a crucial role to ensure the dLog form of the reduced representation. Furthermore, we identify the functional map that arises from the triangle equivalence move as a 3-string scattering S-matrix which satisfies the tetrahedron equations by Zamolodchikov, implying (2+1)-dimensional integrability. We study the solution to the BCFW recursion relation for loop amplitudes, and demonstrate the presence of all physical singularities as well as the absence of all spurious ones. The on-shell diagram solution to the loop recursion relation exhibits manifest two-site cyclic symmetry and reveals that, to all loop, four and six-point amplitudes only have logarithmic singularities.