{
  "id": "2407.13614",
  "version": "v1",
  "published": "2024-07-18T15:52:54.000Z",
  "updated": "2024-07-18T15:52:54.000Z",
  "title": "The integration problem for principal connections",
  "authors": [
    "Javier Fernández",
    "Francisco Kordon"
  ],
  "categories": [
    "math.DG",
    "math-ph",
    "math.DS",
    "math.MP"
  ],
  "abstract": "In this paper we introduce the Integration Problem for principal connections. Just as a principal connection on a principal bundle $\\phi:Q\\rightarrow M$ may be used to split $TQ$ into horizontal and vertical subbundles, a discrete connection may be used to split $Q\\times Q$ into horizontal and vertical submanifolds. All discrete connections induce a connection on the same principal bundle via a process known as the Lie or derivative functor. The Integration Problem consists of describing, for a principal connection $\\mathcal{A}$, the set of all discrete connections whose associated connection is $\\mathcal{A}$. Our first result is that for \\emph{flat} principal connections, the Integration Problem has a unique solution among the \\emph{flat} discrete connections. More broadly, under a fairly mild condition on the structure group $G$ of the principal bundle $\\phi$, we prove that the existence part of the Integration Problem has a solution that needs not be unique. Last, we see that, when $G$ is abelian, given compatible continuous and discrete curvatures the Integration Problem has a unique solution constrained by those curvatures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-18T15:52:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53B15",
      "53C05",
      "37J06",
      "70G45"
    ],
    "keywords": [
      "principal connection",
      "principal bundle",
      "unique solution",
      "discrete connections induce",
      "integration problem consists"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}