{
  "id": "1309.5746",
  "version": "v3",
  "published": "2013-09-23T09:48:25.000Z",
  "updated": "2014-04-25T20:53:53.000Z",
  "title": "Spectral sections, twisted rho invariants and positive scalar curvature",
  "authors": [
    "Moulay Tahar Benameur",
    "Varghese Mathai"
  ],
  "comment": "25 pages. Minor corrections made, but no changes to the results",
  "categories": [
    "math.DG",
    "hep-th",
    "math-ph",
    "math.KT",
    "math.MP",
    "math.OA"
  ],
  "abstract": "We had previously defined the rho invariant $\\rho_{spin}(Y,E,H, g)$ for the twisted Dirac operator $\\not\\partial^E_H$ on a closed odd dimensional Riemannian spin manifold $(Y, g)$, acting on sections of a flat hermitian vector bundle $E$ over $Y$, where $H = \\sum i^{j+1} H_{2j+1} $ is an odd-degree differential form on $Y$ and $H_{2j+1}$ is a real-valued differential form of degree ${2j+1}$. Here we show that it is a conformal invariant of the pair $(H, g)$. In this paper we express the defect integer $\\rho_{spin}(Y,E,H, g) - \\rho_{spin}(Y,E, g)$ in terms of spectral flows and prove that $\\rho_{spin}(Y,E,H, g)\\in \\mathbb Q$, whenever $g$ is a Riemannian metric of positive scalar curvature. In addition, if the maximal Baum-Connes conjecture holds for $\\pi_1(Y)$ (which is assumed to be torsion-free), then we show that $\\rho_{spin}(Y,E,H, rg) =0$ for all $r\\gg 0$, significantly generalizing our earlier results. These results are proved using the Bismut-Weitzenb\\\"ock formula, a scaling trick, the technique of noncommutative spectral sections, and the Higson-Roe approach.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-04-25T20:53:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "positive scalar curvature",
      "twisted rho invariants",
      "spectral sections",
      "odd dimensional riemannian spin manifold",
      "maximal baum-connes conjecture holds"
    ],
    "publication": {
      "doi": "10.4171/JNCG/209"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1255111,
      "adsabs": "2013arXiv1309.5746T"
    }
  }
}