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Scattering amplitudes in 4d $\mathcal{N}=4$ super Yang-Mills theory (SYM) canbe described by Grassmannian contour integrals whose form depends on whetherthe external data is encoded in momentum space, twistor space, or momentumtwistor space. After a pedagogical review, we present a new, streamlined proofof the equivalence of the three integral formulations. A similar strategyallows us to derive a new Grassmannian integral for 3d $\mathcal{N}=6$ ABJMtheory amplitudes in momentum twistor space: it is a contour integral in anorthogonal Grassmannian with the novel property that the internal metricdepends on the external data. The result can be viewed as a central steptowards developing an amplituhedron formulation for ABJM amplitudes. Variousproperties of Grassmannian integrals are examined, including boundaryproperties, pole structure, and a homological interpretation of the globalresidue theorems for $\mathcal{N}=4$ SYM.

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