Amplituhedron meets Jeffrey-Kirwan Residue

Abstract

The tree amplituhedra ${\mathcal{A}}_{n,k}^{(m)}$ are mathematical objects generalising the notion of polytopes into the Grassmannian. Proposed for $m=4$ as a geometric construction encoding tree-level scattering amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory, they are mathematically interesting for any $m$. In this paper we strengthen the relation between scattering amplitudes and geometry by linking the amplituhedron to the Jeffrey-Kirwan residue, a powerful concept in symplectic and algebraic geometry. We focus on a particular class of amplituhedra in any dimension, namely cyclic polytopes, and their even-dimensional conjugates. We show how the Jeffrey-Kirwan residue prescription allows to extract the correct amplituhedron volume functions in all these cases. Notably, this also naturally exposes the rich combinatorial and geometric structures of amplituhedra, such as their regular triangulations.

arXiv comment27 pages, 13 figures

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• Cutting Deep Into The Amplituhedron
• Cluster Adjacency for m=2 Yangian Invariants
• An Etude on Recursion Relations and Triangulations
• All-Loop Four-Point Aharony-Bergman-Jafferis-Maldacena Amplitudes from Dimensional Reduction of the Amplituhedron
• Positive Geometries and Differential Forms with Non-Logarithmic Singularities I
• Amplituhedra, and Beyond
• Sign Flip Triangulations of the Amplituhedron
• All-loop cuts from the Amplituhedron
• Non-planar BCFW Grassmannian Geometries
• $\Delta$-Algebra and Scattering Amplitudes
• Triangulations and Canonical Forms of Amplituhedra: a fiber-based approach beyond polytopes
• Triangulation of 2-loop MHV Amplituhedron from Sign Flips
• Poles At Infinity in On-shell Diagrams
• The hypersimplex canonical forms and the momentum amplituhedron-like logarithmic forms
• Positive Geometry for Stringy Scalar Amplitudes