{
  "id": "1009.3179",
  "version": "v1",
  "published": "2010-09-16T13:58:24.000Z",
  "updated": "2010-09-16T13:58:24.000Z",
  "title": "Bergman and Calderón projectors for Dirac operators",
  "authors": [
    "Colin Guillarmou",
    "Sergiu Moroianu",
    "Jinsung Park"
  ],
  "comment": "31 pages",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "For a Dirac operator $D_{\\bar{g}}$ over a spin compact Riemannian manifold with boundary $(\\bar{X},\\bar{g})$, we give a natural construction of the Calder\\'on projector and of the associated Bergman projector on the space of harmonic spinors on $\\bar{X}$, and we analyze their Schwartz kernels. Our approach is based on the conformal covariance of $D_{\\bar{g}}$ and the scattering theory for the Dirac operator associated to the complete conformal metric $g=\\bar{g}/\\rho^2$ where $\\rho$ is a smooth function on $\\bar{X}$ which equals the distance to the boundary near $\\partial\\bar{X}$. We show that $({\\rm Id}+\\tilde{S}(0))/2$ is the orthogonal Calder\\'on projector, where $\\tilde{S}(\\lambda)$ is the holomorphic family in $\\{\\Re(\\lambda)\\geq 0\\}$ of normalized scattering operators constructed in our previous work, which are classical pseudo-differential of order $2\\lambda$. Finally we construct natural conformally covariant odd powers of the Dirac operator on any spin manifold.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-09-16T13:58:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J32",
      "35P25"
    ],
    "keywords": [
      "dirac operator",
      "calderón projectors",
      "spin compact riemannian manifold",
      "calderon projector",
      "construct natural conformally covariant odd"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.3179G"
    }
  }
}