{
  "id": "2410.14034",
  "version": "v1",
  "published": "2024-10-17T21:14:20.000Z",
  "updated": "2024-10-17T21:14:20.000Z",
  "title": "Fermionic Dyson expansions and stochastic Duistermaat-Heckman localization on loop spaces",
  "authors": [
    "Batu Güneysu",
    "Jonas Miehe"
  ],
  "categories": [
    "math.DG",
    "math.KT",
    "math.PR"
  ],
  "abstract": "Given a self-adjoint operator $H\\geq 0$ and (appropriate) densely defined and closed operators $P_{1},\\dots, P_{n}$ in a Hilbert space $\\mathscr{H}$, we provide a systematic study of bounded operators given by iterated integrals \\begin{align}\\label{oh} \\int_{\\{ 0\\leq s_1\\leq \\dots\\leq s_n\\leq t\\}}\\mathrm{e}^{-s_1H}P_{1}\\mathrm{e}^{-(s_2-s_1)H}P_{2}\\cdots \\mathrm{e}^{-(s_n-s_{n-1})H}P_{n} \\mathrm{e}^{-(t-s_n)H}\\, \\mathrm{d} s_{1} \\ldots \\mathrm{d} s_{n},\\quad t>0. \\end{align} These operators arise naturally in noncommutative geometry and the geometry of loop spaces. Using Fermionic calculus, we give a natural construction of an enlarged Hilbert space $\\mathscr{H}^{(n)}$ and an analytic semigroup $\\mathrm{e}^{-t (H^{(n)}+P^{(n)} )}$ thereon, such that $\\mathrm{e}^{-t (H^{(n)}+P^{(n)} )}$ composed from the left with (essentially) a Fermionic integration gives precisely the above iterated operator integral. This formula allows to establish important regularity results for the latter, and to derive a stochastic representation for it, in case $H$ is a covariant Laplacian and the $P_{j}$'s are first-order differential operators. Finally, with $H$ given as the square of the Dirac operator on a spin manifold, this representation is used to derive a stochastic refinement of the Duistermaat-Heckman localization formula on the loop space of a spin manifold.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-17T21:14:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "loop space",
      "fermionic dyson expansions",
      "stochastic duistermaat-heckman localization",
      "hilbert space",
      "spin manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}