On the Boundaries of the m=2 Amplituhedron

Understand

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 $\mathcal{A}^{(4)}_{n,k}$ 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.