The hypersimplex canonical forms and the momentum amplituhedron-like logarithmic forms

Abstract

In this paper we provide a formula for the canonical differential form of the hypersimplex ${\Delta }_{k,n}$ for all $n$ and $k$. We also study the generalization of the momentum amplituhedron ${\mathcal{M}}_{n,k}$ to $m=2$, and we conclude that the existing definition does not possess the desired properties. Nevertheless, we find interesting momentum amplituhedron-like logarithmic differential forms in the $m=2$ version of the spinor helicity space, that have the same singularity structure as the hypersimplex canonical forms.

arXiv comment18 pages, 2 figures

References in corpus (17)

• The Amplituhedron
• Total positivity, Grassmannians, and networks
• Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet
• The EFT-Hedron
• Positive Geometries and Canonical Forms
• The Momentum Amplituhedron
• Cosmological Polytopes and the Wavefunction of the Universe
• Positroid varieties I: juggling and geometry
• On the Positive Geometry of Conformal Field Theory
• Positive configuration space
• Dressians, Tropical Grassmannians, and Their Rays
• Amplituhedron meets Jeffrey-Kirwan Residue
• The positive tropical Grassmannian, the hypersimplex, and the m=2 amplituhedron
• Amplituhedra, and Beyond
• Boundaries of the Amplituhedron with amplituhedronBoundaries
• Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra
• From weakly separated collections to matroid subdivisions