A cluster of results on amplituhedron tiles
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910029426196480 |
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| author | Even-Zohar, Chaim Lakrec, Tsviqa Parisi, Matteo Tessler, Ran Sherman-Bennett, Melissa Williams, Lauren |
| author_facet | Even-Zohar, Chaim Lakrec, Tsviqa Parisi, Matteo Tessler, Ran Sherman-Bennett, Melissa Williams, Lauren |
| contents | 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''. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_15568 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A cluster of results on amplituhedron tiles Even-Zohar, Chaim Lakrec, Tsviqa Parisi, Matteo Tessler, Ran Sherman-Bennett, Melissa Williams, Lauren Combinatorics High Energy Physics - Theory Mathematical Physics Algebraic Geometry 05E14, 13F60 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''. |
| title | A cluster of results on amplituhedron tiles |
| topic | Combinatorics High Energy Physics - Theory Mathematical Physics Algebraic Geometry 05E14, 13F60 |
| url | https://arxiv.org/abs/2402.15568 |