Taking the amplituhedron to the limit

TL;DR

This paper investigates the limit of the amplituhedron as the number of particles approaches infinity, revealing its algebraic boundary, stratification, and confirming it as a positive geometry with a simplified residual arrangement.

Contribution

It introduces the concept of the limit amplituhedron, analyzes its algebraic boundary and stratification, and establishes its properties as a positive geometry in the infinite particle limit.

Findings

The limit amplituhedron's algebraic boundary is characterized by Chow hypersurfaces.

Stratification of the boundary involves higher order secants of the rational normal curve.

The residual arrangement of the limit amplituhedron is empty.

Abstract

The amplituhedron is a semialgebraic set given as the image of the non-negative Grassmannian under a linear map subject to a choice of additional parameters. We define the limit amplituhedron as the limit of amplituhedra by sending one of the parameters, namely the number of particles $n$, to infinity. We study this limit amplituhedron for $m = 2$ and any $k$, relating to the number of negative helcity particles. We determine its algebraic boundary in terms of Chow hypersurfaces. This hypersurface in the Grassmannian is stratified by singularities in terms of higher order secants of the rational normal curve. In conclusion, we show that the limit amplituhedron is a positive geometry with a residual arrangement that is empty.