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arXiv · 2402.15568

A cluster of results on amplituhedron tiles

Chaim Even-Zohar·Tsviqa Lakrec·Matteo Parisi·Ran Tessler·Melissa Sherman-Bennett·Lauren Williams
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Abstract

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 \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 $m=4$ 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 $m=4$ 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 $m=4$ amplituhedron''.

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Chaim Even-Zohar, Tsviqa Lakrec, Matteo Parisi, Ran Tessler, Melissa Sherman-Bennett, Lauren Williams. 2024-02-23. A cluster of results on amplituhedron tiles. https://doi.org/10.1007/s11005-024-01854-4

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