{
  "id": "1803.06646",
  "version": "v1",
  "published": "2018-03-18T11:46:12.000Z",
  "updated": "2018-03-18T11:46:12.000Z",
  "title": "Toric geometry of $G_2$-manifolds",
  "authors": [
    "Thomas Bruun Madsen",
    "Andrew Swann"
  ],
  "comment": "33 pages",
  "categories": [
    "math.DG",
    "hep-th"
  ],
  "abstract": "We consider $G_2$-manifolds with an effective torus action that is multi-Hamiltonian for one or more of the defining forms. The case of $T^3$-actions is found to be distinguished. For such actions multi-Hamiltonian with respect to both the three- and four-form, we derive a Gibbons-Hawking type ansatz giving the geometry on an open dense set in terms a symmetric $3\\times 3$-matrix of functions. This leads to particularly simple examples of explicit metrics with holonomy equal to $G_2$. We prove that the multi-moment maps exhibit the full orbit space topologically as a smooth four-manifold containing a trivalent graph as the image of the set of special orbits and describe these graphs in some complete examples.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-18T11:46:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C25",
      "53C29",
      "53D20",
      "57R45",
      "70G45"
    ],
    "keywords": [
      "toric geometry",
      "full orbit space",
      "open dense set",
      "torus action",
      "complete examples"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}