{
  "id": "1403.5215",
  "version": "v1",
  "published": "2014-03-20T17:49:04.000Z",
  "updated": "2014-03-20T17:49:04.000Z",
  "title": "A Spectral Perspective on Neumann-Zagier",
  "authors": [
    "Tudor Dimofte",
    "Roland van der Veen"
  ],
  "comment": "53 + 12 pages",
  "categories": [
    "math.GT",
    "hep-th",
    "math.QA"
  ],
  "abstract": "We provide a new topological interpretation of the symplectic properties of gluing equations for triangulations of hyperbolic 3-manifolds, first discovered by Neumann and Zagier. We also extend the symplectic properties to more general gluings of PGL(2,C) flat connections on the boundaries of 3-manifolds with topological ideal triangulations, proving that gluing is a K_2 symplectic reduction of PGL(2,C) moduli spaces. Recently, such symplectic properties have been central in constructing quantum PGL(2,C) invariants of 3-manifolds. Our methods adapt the spectral network construction of Gaiotto-Moore-Neitzke to relate framed flat PGL(2,C) connections on the boundary C of a 3-manifold to flat GL(1,C) connections on a double branched cover S -> C of the boundary. Then moduli spaces of both PGL(2,C) connections on C and GL(1,C) connections on S gain coordinates labelled by the first homology of S, and inherit symplectic properties from the intersection form on homology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-20T17:49:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spectral perspective",
      "connections",
      "neumann-zagier",
      "moduli spaces",
      "spectral network construction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1286653,
      "adsabs": "2014arXiv1403.5215D"
    }
  }
}