{
  "id": "hep-th/0210297",
  "version": "v2",
  "published": "2002-10-30T23:44:09.000Z",
  "updated": "2003-04-08T13:27:46.000Z",
  "title": "Dirac Operators on Coset Spaces",
  "authors": [
    "A. P. Balachandran",
    "Giorgio Immirzi",
    "Joohan Lee",
    "Peter Presnajder"
  ],
  "comment": "section on Riemannian structure improved, references added",
  "journal": "J.Math.Phys. 44 (2003) 4713-4735",
  "doi": "10.1063/1.1607514",
  "categories": [
    "hep-th",
    "gr-qc",
    "math-ph",
    "math.DG",
    "math.MP"
  ],
  "abstract": "The Dirac operator for a manifold Q, and its chirality operator when Q is even dimensional, have a central role in noncommutative geometry. We systematically develop the theory of this operator when Q=G/H, where G and H are compact connected Lie groups and G is simple. An elementary discussion of the differential geometric and bundle theoretic aspects of G/H, including its projective modules and complex, Kaehler and Riemannian structures, is presented for this purpose. An attractive feature of our approach is that it transparently shows obstructions to spin- and spin_c-structures. When a manifold is spin_c and not spin, U(1) gauge fields have to be introduced in a particular way to define spinors. Likewise, for manifolds like SU(3)/SO(3), which are not even spin_c, we show that SU(2) and higher rank gauge fields have to be introduced to define spinors. This result has potential consequences for string theories if such manifolds occur as D-branes. The spectra and eigenstates of the Dirac operator on spheres S^n=SO(n+1)/SO(n), invariant under SO(n+1), are explicitly found. Aspects of our work overlap with the earlier research of Cahen et al..",
  "revisions": [
    {
      "version": "v2",
      "updated": "2003-04-08T13:27:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dirac operator",
      "coset spaces",
      "define spinors",
      "higher rank gauge fields",
      "compact connected lie groups"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AIP",
      "journal": "J. Math. Phys."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 601010
    }
  }
}