Momentum amplituhedron meets kinematic associahedron

Self Cite

In recent years, it has been understood that color-ordered scattering amplitudes can be encoded as logarithmic differential forms on positive geometries. In particular, amplitudes in maximally supersymmetric Yang-Mills theory in spinor helicity space are governed by the momentum amplituhedron. Due to the group-theoretic structure underlying color decompositions, color-ordered amplitudes enjoy various identities which relate different orderings. In this paper, we show how the Kleiss-Kuijf relations arise from the geometry of the momentum amplituhedron. We also show how similar relations can be realised for the kinematic associahedron, which is the positive geometry of bi-adjoint scalar cubic theory.

Section: Discussionmentioning

confidence: 99%

Kleiss-Kuijf relations from momentum amplituhedron geometry

Damgaard

Ferro

Łukowski

et al. 2021

Self Cite

“…What we will study in this paper is a parallel picture for momentum amplituhedron of N = 4 SYM [20], as well as a new positive geometry which we conjecture to be the momentum amplituhedron of ABJM. Although much more involved than kinematic associahedron, both (non-polytopal) geometries can be defined similarly as the intersection of a top-dimensional region in the kinematic space of D = 4 (or D = 3), which requires only positivity but also certain "winding" conditions [27], and a subspace of dimension 2n−4 (or n−3), as we will discuss in order. Again similar to associahedron, we will show that these geometries are images of their moduli space via certain maps in D = 4…”

Section: Jhep02(2022)148mentioning

confidence: 99%

“…Just as the amplituhedron in momentum-twistor space [33], the definition of the momentum amplituhedron implies particular sign patterns [20]. One can show that the brackets Y ii+1 , [ Ỹ ii + 1] The second way for defining the momentum amplituhedron [27] is directly in kinematic space by projecting the kinematic data through Y, Ỹ…”

Section: Jhep02(2022)148mentioning

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.

“…In this picture scattering amplitudes are differential forms living directly in kinematical (momentum twistor) space [2]. This ideas have been later used to study more generally the kinematical differential forms [60,61] and it also lead to the discovery of the Momentum Amplituhedron [62,63] in the spinor helicity space. The immediate big question is if the non-planar N = 4 SYM amplitudes can be also associated with canonical differential forms over geometric space.…”

Section: Introductionmentioning

confidence: 99%

Towards the Gravituhedron: new expressions for NMHV gravity amplitudes

Trnka

2021

In this paper, we present new expressions for n-point NMHV tree-level gravity amplitudes. We introduce a method of factorization diagrams which is a simple graphical representation of R-invariants in Yang-Mills theory. We define the gravity analogues which we call $$ \mathcal{G} $$ G -invariants, and expand the NMHV gravity amplitudes in terms of these objects. We provide explicit formulas of NMHV gravity amplitudes up to eight points in terms of $$ \mathcal{G} $$ G -invariants, and give the general definition for any number of points. We discuss the connection to BCFW representation, special behavior under large momentum shift, the role of momentum twistors and the intricate web of spurious poles cancelation. Because of the close connection between R-invariants and the (tree-level) Amplituhedron for Yang-Mills amplitudes, we speculate that the new expansion for gravity amplitudes should correspond to the triangulation of the putative Gravituhedron geometry.