On the boundaries of the $m=2$ amplituhedron

Abstract

Amplituhedra ${\mathcal{A}}_{n,k}^{(m)}$ are geometric objects of great interest in modern mathematics and physics: for mathematicians, they are combinatorially rich generalizations of polygons and polytopes, based on the notion of positivity; for physicists, the amplituhedron ${\mathcal{A}}_{n,k}^{(4)}$ encodes the scattering amplitudes of the planar $\mathcal{N}=4$ super Yang–Mills theory. In this paper, we study the structure of boundaries for the amplituhedron ${\mathcal{A}}_{n,k}^{(2)}$. We classify all boundaries of all dimensions and provide their graphical enumeration. We find that the boundary poset for the amplituhedron is Eulerian and show that the Euler characteristic of the amplituhedron equals one. This provides an initial step towards proving that the amplituhedron for $m=2$ is homeomorphic to a closed ball.