A cluster of results on amplituhedron tiles

The amplituhedron is a mathematical object which was introduced to provide a geometric origin of scattering amplitudes in \(\mathcal {N}=4\) super Yang–Mills theory. It generalizes cyclic polytopes and the positive Grassmannian and has a very rich combinatorics with connections to cluster algebras. In this article, we provide a series of results about tiles and tilings of the \(m=4\) amplituhedron. Firstly, we provide a full characterization of facets of BCFW tiles in terms of cluster variables for \(\text{ Gr}_{4,n}\). Secondly, we exhibit a tiling of the \(m=4\) amplituhedron which involves a tile which does not come from the BCFW recurrence—the spurion tile, which also satisfies all cluster properties. Finally, strengthening the connection with cluster algebras, we show that each standard BCFW tile is the positive part of a cluster variety, which allows us to compute the canonical form of each such tile explicitly in terms of cluster variables for \(\text{ Gr}_{4,n}\). This paper is a companion to our previous paper “Cluster algebras and tilings for the \(m=4\) amplituhedron.”