Kleiss-Kuijf relations from momentum amplituhedron geometry

We consider amplituhedron-like geometries which are defined in a similar way to the intrinsic definition of the amplituhedron but with non-maximal winding number. We propose that for the cases with minimal number of points the canonical form of these geometries corresponds to the product of parity conjugate amplitudes at tree as well as loop level. The product of amplitudes in superspace lifts to a star product in bosonised superspace which we give a precise definition of. We give an alternative definition of amplituhedron-like geometries, analogous to the original amplituhedron definition, and also a characterisation as a sum over pairs of on-shell diagrams that we use to prove the conjecture at tree level. The union of all amplituhedron-like geometries has a very simple definition given by only physical inequalities. Although such a union does not give a positive geometry, a natural extension of the standard definition of canonical form, the globally oriented canonical form, acts on this union and gives the square of the amplitude.

Section: The Union Of Positive Geometriessupporting

confidence: 74%

Amplituhedron-like geometries

Dian

Heslop

2021

“…There are other intriguing features of the geometries and forms, which are made manifest by such maps. While we are limited to a given ordering in momentum twistor space, it is natural to consider relations among amplitudes with different orderings, such as KK (see [40]) and BCJ relations [41]. It is well known that twistor-string formulas (or pushforward) trivialize KK relations, as well as BCJ relations on the support of scattering equations [3].…”

Section: Boundaries Of M 3d (K 2k)mentioning

confidence: 99%

The momentum amplituhedron of SYM and ABJM from twistor-string maps

Kuo

Zhang

2022

We study remarkable connections between twistor-string formulas for tree amplitudes in $$ \mathcal{N} $$ N = 4 SYM and $$ \mathcal{N} $$ 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 $$ {\mathrm{\mathcal{M}}}_{0,n}^{+} $$ 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 $$ {\mathrm{\mathcal{M}}}_{0,n}^{+} $$ 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.

“…The building blocks of on-shell diagrams for ABJM theory are the four-point amplitudes, which in the N = 4 formalism are given by (23) , (23) . 4 One may also consider…”

Section: Jhep01(2022)141mentioning

confidence: 99%

The orthogonal momentum amplituhedron and ABJM amplitudes

Huang

Kojima

Wen

et al. 2022

In this paper, we introduce the momentum space amplituhedron for tree-level scattering amplitudes of ABJM theory. We demonstrate that the scattering amplitude can be identified as the canonical form on the space given by the product of positive orthogonal Grassmannian and the moment curve. The co-dimension one boundaries of this space are simply the odd-particle planar Mandelstam variables, while the even-particle counterparts are “hidden” as higher co-dimension boundaries. Remarkably, this space can be equally defined through a series of “sign flip” requirements of the projected external data, identical to “half” of four-dimensional $$ \mathcal{N} $$ N = 4 super Yang-Mills theory (sYM). Thus in a precise sense the geometry for ABJM lives on the boundary of $$ \mathcal{N} $$ N = 4 sYM. We verify this relation through eight-points by showing that the BCFW triangulation of the amplitude tiles the amplituhedron. The canonical form is naturally derived using the Grassmannian formula for the amplitude in the $$ \mathcal{N} $$ N = 4 formalism for ABJM theory.