{
  "id": "math/0512572",
  "version": "v1",
  "published": "2005-12-26T14:44:09.000Z",
  "updated": "2005-12-26T14:44:09.000Z",
  "title": "Multivalued functionals, one-forms and deformed de Rham complex",
  "authors": [
    "Dmitri V. Millionschikov"
  ],
  "categories": [
    "math.AT",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We discuss some applications of the Morse-Novikov theory to some problems in modern physics, where appears a non-exact closed 1-form $\\omega$ (a multi-valued functional). We focus mainly our attention to the cohomology of the de Rham complex of a compact manifold $M^n$ with a deformed differential $d_{\\omega}=d +\\lambda \\omega$. Using Witten's approach to the Morse theory one can estimate the number of critical points of $\\omega$ in terms of the cohomology of deformed de Rham complex with sufficiently large values of $\\lambda$ (torsion-free Novikov's inequalities). We show that for an interesting class of solvmanifolds this cohomology can be computed as the cohomology of the corresponding Lie algebra $\\mathfrak{g}$ associated with the one-dimensional representation $\\rho_{\\lambda \\omega}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-12-26T14:44:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58A12",
      "17B30",
      "17B56",
      "57T15"
    ],
    "keywords": [
      "rham complex",
      "multivalued functionals",
      "cohomology",
      "torsion-free novikovs inequalities",
      "wittens approach"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 701816,
      "adsabs": "2005math.....12572M"
    }
  }
}