Abstract
We study remarkable connections between twistor-string formulas for tree amplitudes in mathcal{N} = 4 SYM and mathcal{N} = 6 ABJM, and the corresponding momentum amplituhedron in the kinematic space of D = 4 and D = 3, respectively. Based on the Veronese map to positive Grassmannians, we define a twistor-string map from G+(2, n) to a (2n−4)-dimensional subspace of the 4d kinematic space where the momentum amplituhedron of SYM lives. We provide strong evidence that the twistor-string map is a diffeomorphism from G+(2, n) to the interior of momentum amplituhedron; the canonical form of the latter, which is known to give tree amplitudes of SYM, can be obtained as pushforward of that of former. We then move to three dimensions: based on Veronese map to orthogonal positive Grassmannian, we propose a similar twistor-string map from the moduli space {mathrm{mathcal{M}}}_{0,n}^{+} to a (n−3)-dimensional subspace of 3d kinematic space. The image gives a new positive geometry which conjecturally serves as the momentum amplituhedron for ABJM; its canonical form gives the tree amplitude with reduced supersymmetries in the theory. We also show how boundaries of compactified {mathrm{mathcal{M}}}_{0,n}^{+} map to boundaries of momentum amplituhedra for SYM and ABJM corresponding to factorization channels of amplitudes, and in particular for ABJM case the map beautifully excludes all unwanted channels.
Highlights
Whose canonical forms [15] encode tree amplitudes and all-loop integrands of the theory; it was defined as the image of positive Grassmannian via a map that was given in terms of bosonized momentum twistor variables [19]
We study remarkable connections between twistor-string formulas for tree amplitudes in N = 4 SYM and N = 6 ABJM, and the corresponding momentum amplituhedron in the kinematic space of D = 4 and D = 3, respectively
We will show that the image of our map gives a (n−3)-dimensional positive geometry which can serve as the ABJM momentum amplituhedron in D = 3 kinematic space; the latter can be defined in a similar fashion as SYM in [20] without referring to the twistor-string map, and its canonical forms give ABJM amplitudes with reduced SUSY.1
Summary
Amplituhedron, the kinematic data is bosonized by introducing 2(n − k) Grassmann-odd variables φαa , α = 1, . Λ An ∈ M + (k + 2, n) where Λ is twisted-positive means that its orthogonal complement is positive Λ⊥ ∈ M +(k−2, n) (see [20]), and we refer to (Λ, Λ) as the kinematic data Given such “positive kinematic data”, there are two ways to define the momentum amplituhedron.. The common way to obtain the volume form is by push-forward the BCFW cell to the momentum amplituhedron and wedge δ4 (P ) d4P to make it become top invariant form. We pushforward canonical forms of BCFW cells ωn(γ,k) to kinematic space [21]: Ω(nγ,k) = This pushforward formula is very similar to the usual Grassmannian integral formula. One can extract the amplitude from the form by multiplying the form Ω(nγ,k) with δ4(P )d4P and make the replacement dλi and dλi to ηi and ηi
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