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.
Highlights
Recent years have brought remarkable progress in our understanding of the mathematical structure of scattering amplitudes, especially in the planar limit of N = 4 super YangMills theory (SYM)
In this paper we study the Grassmannian descriptions of amplitudes in both 4d N = 4 SYM and in 3d ABJM theory
The results are rephrased in the context of on-shell diagrams and the positroid stratification of the Grassmannian, and we show how the ‘boundary operation’ in the Grassmannian integral is related to both the residue theorems and the pole structure
Summary
We briefly review the momentum space spinor helicity formalism in 3d and the associated Grassmannian integral for ABJM amplitudes, which was introduced previously in [11,12,13,14]. It encodes the n=(2k+4)-point NkMHV amplitudes of ABJM theory as contour integrals in the orthogonal Grassmannian OG(k+2, 2k+4). We achieve this goal, which is an important first step towards developing an amplituhedron for the ABJM theory To this end, we first introduce another formulation of the 3d spinor helicity formalism that facilitates the definition of 3d momentum twistors.
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