Parity duality for the amplituhedron

Abstract

The (tree) amplituhedron ${\mathcal{A}}_{n,k,m}(Z)$ is a certain subset of the Grassmannian introduced by Arkani-Hamed and Trnka in 2013 in order to study scattering amplitudes in $N=4$ supersymmetric Yang-Mills theory. Confirming a conjecture of the first author, we show that when $m$ is even, a collection of affine permutations yields a triangulation of ${\mathcal{A}}_{n,k,m}(Z)$ for any $Z\in {\mathrm{Gr}}_{>0}(k+m,n)$ if and only if the collection of their inverses yields a triangulation of ${\mathcal{A}}_{n,n-m-k,m}(Z)$ for any $Z\in {\mathrm{Gr}}_{>0}(n-k,n)$. We prove this duality using the twist map of Marsh and Scott. We also show that this map preserves the canonical differential forms associated with the corresponding positroid cells, and hence obtain a parity duality for amplituhedron differential forms.

52 pages, 3 figures

References in corpus (3)

• Unwinding the Amplituhedron in Binary
• The twist for positroid varieties
• The m=1 amplituhedron and cyclic hyperplane arrangements

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• Triangulations and Canonical Forms of Amplituhedra: a fiber-based approach beyond polytopes
• Critical varieties in the Grassmannian
• A cluster of results on amplituhedron tiles
• Gradient flows, adjoint orbits, and the topology of totally nonnegative flag varieties
• The Amplituhedron BCFW Triangulation