Cluster algebras and tilings for the m=4 amplituhedron
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| Format: | Preprint |
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2023
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| _version_ | 1866917576527839232 |
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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 $A_{n,k,m}(Z)$ is the image of the positive Grassmannian $Gr_{k,n}^{\geq 0}$ under the map ${Z}: Gr_{k,n}^{\geq 0} \to Gr_{k,k+m}$ induced by a positive linear map $Z:\mathbb{R}^n \to \mathbb{R}^{k+m}$. Motivated by a question of Hodges, Arkani-Hamed and Trnka introduced the amplituhedron as a geometric object whose tilings conjecturally encode the BCFW recursion for computing scattering amplitudes. More specifically, the expectation was that one can compute scattering amplitudes in ${N}=4$ SYM by tiling the $m=4$ amplituhedron $A_{n,k,4}(Z)$ - that is, decomposing the amplituhedron into `tiles' (closures of images of $4k$-dimensional cells of $Gr_{k,n}^{\geq 0}$ on which ${Z}$ is injective) - and summing the `volumes' of the tiles. In this article we prove two major conjectures about the $m=4$ amplituhedron: $i)$ the BCFW tiling conjecture, which says that any way of iterating the BCFW recurrence gives rise to a tiling of the amplituhedron $A_{n,k,4}(Z)$; $ii)$ the cluster adjacency conjecture for BCFW tiles, which says that facets of tiles are cut out by collections of compatible cluster variables for $Gr_{4,n}$. Moreover, we show that each BCFW tile is the subset of $Gr_{k, k+4}$ where certain cluster variables have particular signs. Along the way, we construct many explicit seeds for $Gr_{4,n}$ comprised of high-degree cluster variables, which may be of independent interest in the study of cluster algebras. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_17727 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Cluster algebras and tilings for the m=4 amplituhedron 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 $A_{n,k,m}(Z)$ is the image of the positive Grassmannian $Gr_{k,n}^{\geq 0}$ under the map ${Z}: Gr_{k,n}^{\geq 0} \to Gr_{k,k+m}$ induced by a positive linear map $Z:\mathbb{R}^n \to \mathbb{R}^{k+m}$. Motivated by a question of Hodges, Arkani-Hamed and Trnka introduced the amplituhedron as a geometric object whose tilings conjecturally encode the BCFW recursion for computing scattering amplitudes. More specifically, the expectation was that one can compute scattering amplitudes in ${N}=4$ SYM by tiling the $m=4$ amplituhedron $A_{n,k,4}(Z)$ - that is, decomposing the amplituhedron into `tiles' (closures of images of $4k$-dimensional cells of $Gr_{k,n}^{\geq 0}$ on which ${Z}$ is injective) - and summing the `volumes' of the tiles. In this article we prove two major conjectures about the $m=4$ amplituhedron: $i)$ the BCFW tiling conjecture, which says that any way of iterating the BCFW recurrence gives rise to a tiling of the amplituhedron $A_{n,k,4}(Z)$; $ii)$ the cluster adjacency conjecture for BCFW tiles, which says that facets of tiles are cut out by collections of compatible cluster variables for $Gr_{4,n}$. Moreover, we show that each BCFW tile is the subset of $Gr_{k, k+4}$ where certain cluster variables have particular signs. Along the way, we construct many explicit seeds for $Gr_{4,n}$ comprised of high-degree cluster variables, which may be of independent interest in the study of cluster algebras. |
| title | Cluster algebras and tilings for the m=4 amplituhedron |
| topic | Combinatorics High Energy Physics - Theory Mathematical Physics Algebraic Geometry 05E14, 13F60 |
| url | https://arxiv.org/abs/2310.17727 |