Grassmannians for scattering amplitudes in 4d $\mathcal{N}=4$ SYM and 3d ABJM

TL;DR

This paper reviews Grassmannian formulations of scattering amplitudes in 4d $ =4$ SYM and introduces a new Grassmannian integral for 3d ABJM theory, advancing the geometric understanding of these amplitudes.

Contribution

It provides a new proof of the equivalence of different Grassmannian integral formulations and introduces a novel Grassmannian integral for 3d ABJM amplitudes with a data-dependent metric.

Findings

Unified Grassmannian formulations for 4d $ =4$ SYM amplitudes.

A new Grassmannian integral for 3d ABJM amplitudes with an orthogonal structure.

Analysis of boundary, pole, and residue properties of the integrals.

Abstract

Scattering amplitudes in 4d $\mathcal{N}=4$ super Yang-Mills theory (SYM) can be described by Grassmannian contour integrals whose form depends on whether the external data is encoded in momentum space, twistor space, or momentum twistor space. After a pedagogical review, we present a new, streamlined proof of the equivalence of the three integral formulations. A similar strategy allows us to derive a new Grassmannian integral for 3d $\mathcal{N}=6$ ABJM theory amplitudes in momentum twistor space: it is a contour integral in an orthogonal Grassmannian with the novel property that the internal metric depends on the external data. The result can be viewed as a central step towards developing an amplituhedron formulation for ABJM amplitudes. Various properties of Grassmannian integrals are examined, including boundary properties, pole structure, and a homological interpretation of the global residue theorems for $\mathcal{N}=4$ SYM.