{
  "id": "2207.09952",
  "version": "v1",
  "published": "2022-07-20T14:55:08.000Z",
  "updated": "2022-07-20T14:55:08.000Z",
  "title": "Toledo invariants of Topological Quantum Field Theories",
  "authors": [
    "Bertrand Deroin",
    "Julien Marché"
  ],
  "comment": "57 pages",
  "categories": [
    "math.GT",
    "math.DG"
  ],
  "abstract": "We prove that the Fibonacci quantum representations $\\rho_{g,n}:\\rm{Mod}_{g,n}\\to \\rm{PU}(p,q)$ for $(g,n)\\in\\{(0,4),(0,5),(1,2),(1,3),(2,1)\\}$ are holonomy representations of complex hyperbolic structures on some compactifications of the corresponding moduli spaces $\\mathcal{M}_{g,n}$. As a corollary, the forgetful map between the corresponding compactifications of $\\mathcal M_{1,3}$ and $\\mathcal M_{1,2}$ is a surjective holomorphic map between compact complex hyperbolic orbifolds of different dimensions higher than one, giving an answer to a problem raised by Siu. The proof consists in computing their Toledo invariants: we put this computation in a broader context, replacing the Fibonacci representations with any Hermitian modular functor and extending the Toledo invariant to a full series of cohomological invariants beginning with the signature $p-q$. We prove that these invariants satisfy the axioms of a Cohomological Field Theory and compute the $R$-matrix at first order (hence the usual Toledo invariants) in the case of the $\\rm{SU}_2/\\rm{SO}_3$-quantum representations at any level.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-20T14:55:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "field theory",
      "topological quantum field theories",
      "compact complex hyperbolic orbifolds",
      "complex hyperbolic structures",
      "hermitian modular functor"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 57,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}