A cluster of results on amplituhedron tiles
arXiv:2402.15568 · doi:10.1007/s11005-024-01854-4
Abstract
The amplituhedron is a mathematical object which was introduced to provide a geometric origin of scattering amplitudes in super Yang Mills theory. It generalizes \emph{cyclic polytopes} and the \emph{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 amplituhedron. Firstly, we provide a full characterization of facets of BCFW tiles in terms of cluster variables for $\mbox{Gr}_{4,n}$. Secondly, we exhibit a tiling of the amplituhedron which involves a tile which does not come from the BCFW recurrence -- the \emph{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 $\mbox{Gr}_{4,n}$. This paper is a companion to our previous paper ``Cluster algebras and tilings for the amplituhedron''.
44 pages, 20 figures. arXiv admin note: text overlap with arXiv:2310.17727