{
  "id": "1512.08680",
  "version": "v1",
  "published": "2015-12-29T13:43:28.000Z",
  "updated": "2015-12-29T13:43:28.000Z",
  "title": "On the global $2$-holonomy for a $2$-connection on a $2$-bundle",
  "authors": [
    "Wei Wang"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "A crossed module constitutes a strict $2$-groupoid $\\mathcal{G}$ and a $\\mathcal{G}$-valued $2$-cocycle on a manifold defines a $2$-bundle. A $2$-connection on this $2$-bundle is given by a Lie algebra $\\mathfrak g$ valued $1$-form $A $ and a Lie algebra $\\mathfrak h$ valued $2$-form $B $ over each coordinate chart together with $2$-gauge transformations between them, which satisfy the compatibility condition. Locally, the path-ordered integral of $A $ gives us the local $1$-holonomy, and the surface-ordered integral of $(A ,B )$ gives us the local $2$-holonomy. The transformation of local $2$-holonomies from one coordinate chart to another is provided by the transition $2$-arrow, which is constructed from a $2$-gauge transformation. We can use the transition $2$-arrows and the $2$-arrows provided by the $\\mathcal{G}$-valued $2$-cocycle to glue such local $2$-holonomies together to get a global one, which is well defined.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-29T13:43:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "connection",
      "gauge transformation",
      "lie algebra",
      "coordinate chart",
      "manifold defines"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151208680W"
    }
  }
}