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

TL;DR

This paper derives a formula for the canonical forms of hypersimplices and explores generalizations of the momentum amplituhedron, discovering new logarithmic forms with similar singularity structures in the spinor helicity space.

Contribution

It provides a universal formula for hypersimplex canonical forms and investigates the properties of the momentum amplituhedron in the case m=2, introducing new logarithmic forms.

Findings

Derived a formula for hypersimplex canonical forms for all n and k.

Found that the existing momentum amplituhedron definition lacks certain properties in m=2.

Discovered momentum amplituhedron-like logarithmic forms with hypersimplex singularity structure in m=2.

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.