{
  "id": "1809.09084",
  "version": "v1",
  "published": "2018-09-24T17:51:53.000Z",
  "updated": "2018-09-24T17:51:53.000Z",
  "title": "Spin$^o$ structures and semilinear (s)pinor bundles",
  "authors": [
    "C. I. Lazaroiu",
    "C. S. Shahbazi"
  ],
  "comment": "49 pages",
  "categories": [
    "math.DG",
    "hep-th"
  ],
  "abstract": "We introduce and explore a new type of extended spinorial structure, based on the groups $\\mathrm{Spin}^o_+(V,h)=\\mathrm{Spin}(V,h)\\cdot\\mathrm{Pin}_{2,0}$ and $\\mathrm{Spin}^o_-(V,h)=\\mathrm{Spin}(V,h)\\cdot \\mathrm{Pin}_{0,2}$, which can be defined on unorientable pseudo-Riemannian manifolds $(M,g)$ of any dimension $d$ and any signature $(p,q)$. In signature $p-q\\equiv_8 3,7$, $(M,g)$ admits a bundle $S$ of irreducible real Clifford modules if and only if the orthonormal coframe bundle of $(M,g)$ admits a reduction to a $\\mathrm{Spin}^{o}_{\\alpha}$ structure with $\\alpha = -1$ if $p-q \\equiv_{8} 3$ or $\\alpha = +1$ if $ p-q \\equiv_{8} 7$. We compute the topological obstruction to existence of such structures, obtaining as a result the topological obstruction for $(M,g)$ to admit a bundle of irreducible real Clifford modules $S$ of complex type. We explicitly show how $\\mathrm{Spin}^o_{\\alpha}$ structures can be used to construct $S$ and give geometric characterizations (in terms of certain associated bundles) of the conditions under which the structure group of $S$ (viewed as a vector bundle) reduces to certain natural subgroups of $\\mathrm{Spin}^o_{\\alpha}$. Finally, we discuss some examples of manifolds admitting $\\mathrm{Spin}^{o}_{\\alpha}$ structures which are constructed using real Grassmannians and certain codimension two submanifolds of spin manifolds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-24T17:51:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "pinor bundles",
      "irreducible real clifford modules",
      "semilinear",
      "orthonormal coframe bundle",
      "topological obstruction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}