The positive orthogonal Grassmannian and loop amplitudes of ABJM
Journal
Journal of Physics A: Mathematical and Theoretical
Journal Volume
47
Journal Issue
47
Date Issued
2014
Author(s)
Abstract
Herein we study the combinatorics associated with the positive orthogonal Grassmannian OGk and its connection to ABJM scattering amplitudes. We present a canonical embedding of OGk 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 reduced ones with products of dlog forms. Unlike super Yang–Mills, here the Jacobian plays a crucial role to ensure the dlog form of the reduced representation. Furthermore in the positive region, 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 discuss 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.
Type
journal article