Every m=2 Yangian invariant is labelled by a collection of non-intersecting polygons in an n-gon, yielding an explicit formula whose denominator factors lie in a common Gr(2,n) cluster, thus manifestly satisfying cluster adjacency.
On the Boundaries of the m=2 Amplituhedron
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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 $\mathcal{A}^{(4)}_{n,k}$ 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. This provides an initial step towards proving that the amplituhedron for $m=2$ is homeomorphic to a closed ball.
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Cluster Adjacency for m=2 Yangian Invariants
Every m=2 Yangian invariant is labelled by a collection of non-intersecting polygons in an n-gon, yielding an explicit formula whose denominator factors lie in a common Gr(2,n) cluster, thus manifestly satisfying cluster adjacency.