{
  "id": "2310.17727",
  "version": "v1",
  "published": "2023-10-26T18:38:34.000Z",
  "updated": "2023-10-26T18:38:34.000Z",
  "title": "Cluster algebras and tilings for the m=4 amplituhedron",
  "authors": [
    "Chaim Even-Zohar",
    "Tsviqa Lakrec",
    "Matteo Parisi",
    "Ran Tessler",
    "Melissa Sherman-Bennett",
    "Lauren Williams"
  ],
  "categories": [
    "math.CO",
    "hep-th",
    "math-ph",
    "math.AG",
    "math.MP"
  ],
  "abstract": "The amplituhedron $A_{n,k,m}(Z)$ is the image of the positive Grassmannian $Gr_{k,n}^{\\geq 0}$ under the amplituhedron map $Gr_{k,n}^{\\geq 0} \\to Gr_{k,k+m}$ induced by a positive linear map $Z:\\mathbb{R}^n \\to \\mathbb{R}^{k+m}$. It was originally introduced in physics in order to give a geometric interpretation of scattering amplitudes. More specifically, one can compute scattering amplitudes in $N=4$ SYM by decomposing the amplituhedron into 'tiles' (closures of images of $4k$-dimensional cells of $Gr_{k,n}^{\\geq 0}$ on which the amplituhedron map is injective) and summing up the 'volumes' of the tiles. Such a decomposition into tiles is called a tiling. In this article we deepen our understanding of tiles and tilings of the $m=4$ amplituhedron. We prove the cluster adjacency conjecture for BCFW tiles of $A_{n,k,4}(Z)$, which says that facets of BCFW tiles are cut out by collections of compatible cluster variables for $Gr_{4,n}$. We also give an explicit description of each BCFW tile as the subset of $Gr_{k, k+4}$ where certain cluster variables have particular signs. And we prove the BCFW tiling conjecture, which says that any way of iterating the BCFW recurrence gives rise to a tiling of the amplituhedron $A_{n,k,4}(Z)$. Along the way we construct many explicit seeds for the $Gr_{4,n}$ comprised of high-degree cluster variables, which may be of independent interest in the study of cluster algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-26T18:38:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E14",
      "13F60"
    ],
    "keywords": [
      "cluster algebras",
      "bcfw tile",
      "amplituhedron map",
      "high-degree cluster variables",
      "cluster adjacency conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}