Adjoints and Canonical Forms of Tree Amplituhedra

Abstract

We consider semi-algebraic subsets of the Grassmannian of lines in three-space called tree amplituhedra. These arise in the study of scattering amplitudes from particle physics. Our main result states that tree amplituhedra in $\mathrm{Gr}(2,4)$ are positive geometries. The numerator of their canonical form plays the role of the adjoint in Wachspress geometry, and is uniquely determined by explicit interpolation conditions.

27 pages, 5 figures, 3 tables

References in corpus (4)

• Positive Amplitudes In The Amplituhedron
• The positive tropical Grassmannian, the hypersimplex, and the m=2 amplituhedron
• An invitation to positive geometries
• Combinatorics of $m=1$ Grasstopes