{
  "id": "math/0204163",
  "version": "v2",
  "published": "2002-04-12T13:16:56.000Z",
  "updated": "2004-07-13T09:40:33.000Z",
  "title": "Adiabatic limits of eta and zeta functions of elliptic operators",
  "authors": [
    "Sergiu Moroianu"
  ],
  "comment": "32 pages, final version",
  "journal": "Math. Z. 246 (2004), 441-471",
  "doi": "10.1007/s00209-003-0578-z",
  "categories": [
    "math.DG"
  ],
  "abstract": "We extend the calculus of adiabatic pseudo-differential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator $\\delta$, constructed from an elliptic family of operators indexed by $S^1$. We show that the regularized values ${\\eta}(\\delta_t,0)$ and $t{\\zeta}(\\delta_t,0)$ are smooth functions of $t$ at $t=0$, and we identify their values at $t=0$ with the holonomy of the determinant bundle, respectively with a residue trace. For invertible families of operators, the functions ${\\eta}(\\delta_t,s)$ and $t{\\zeta}(\\delta_t,s)$ are shown to extend smoothly to $t=0$ for all values of $s$. After normalizing with a Gamma factor, the zeta function satisfies in the adiabatic limit an identity reminiscent of the Riemann zeta function, while the eta function converges to the volume of the Bismut-Freed meromorphic family of connection 1-forms.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-07-13T09:40:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J28",
      "58J52"
    ],
    "keywords": [
      "elliptic operators",
      "adiabatic limit behavior",
      "riemann zeta function",
      "eta function converges",
      "adiabatic pseudo-differential operators"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......4163M"
    }
  }
}