Parity duality for the amplituhedron

TL;DR

This paper proves a duality for the amplituhedron involving triangulations and differential forms, revealing a parity symmetry when the parameter m is even, using the twist map and positroid cell structures.

Contribution

It establishes a parity duality for the amplituhedron's triangulations and differential forms, confirming a conjecture and extending understanding of its geometric structure.

Findings

Triangulations of the amplituhedron are dual under inversion of affine permutations when m is even.

The twist map preserves the canonical differential forms of positroid cells.

A parity duality for amplituhedron differential forms is derived from the triangulation duality.

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 $Z\in \operatorname{Gr}_{>0}(k+m,n)$ if and only if the collection of their inverses yields a triangulation of $\mathcal A_{n,n-m-k,m}(Z)$ for any $Z\in\operatorname{Gr}_{>0}(n-k,n)$. 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.