{
  "id": "2409.08274",
  "version": "v1",
  "published": "2024-09-12T17:59:13.000Z",
  "updated": "2024-09-12T17:59:13.000Z",
  "title": "A functional for Spin(7) forms",
  "authors": [
    "Calin Iuliu Lazaroiu",
    "C. S. Shahbazi"
  ],
  "comment": "22 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "We characterize the set of all conformal Spin(7) forms on an oriented and spin Riemannian eight-manifold $(M,g)$ as solutions to a homogeneous algebraic equation of degree two for the self-dual four-forms of $(M,g)$. When $M$ is compact, we use this result to construct a functional whose self-dual critical set is precisely the set of all Spin(7) structures on $M$. Furthermore, the natural coupling of this potential to the Einstein-Hilbert action gives a functional whose self-dual critical points are conformally Ricci-flat Spin(7) structures. Our proof relies on the computation of the square of an irreducible and chiral real spinor as a section of a bundle of real algebraic varieties sitting inside the K\\\"ahler-Atiyah bundle of $(M,g)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-12T17:59:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "functional",
      "real algebraic varieties sitting inside",
      "spin riemannian eight-manifold",
      "chiral real spinor",
      "algebraic equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}