{
  "id": "1503.03549",
  "version": "v1",
  "published": "2015-03-12T01:28:25.000Z",
  "updated": "2015-03-12T01:28:25.000Z",
  "title": "Unified quantum invariants for integral homology spheres associated with simple Lie algebras",
  "authors": [
    "Kazuo Habiro",
    "Thang T. Q. Lê"
  ],
  "categories": [
    "math.GT",
    "math.QA"
  ],
  "abstract": "For each finite dimensional, simple, complex Lie algebra $\\mathfrak g$ and each root of unity $\\xi$ (with some mild restriction on the order) one can define the Witten-Reshetikhin-Turaev (WRT) quantum invariant $\\tau_M^{\\mathfrak g}(\\xi)\\in \\mathbb C$ of oriented 3-manifolds $M$. In the present paper we construct an invariant $J_M$ of integral homology spheres $M$ with values in the cyclotomic completion $\\widehat {\\mathbb Z [q]}$ of the polynomial ring $\\mathbb Z [q]$, such that the evaluation of $J_M$ at each root of unity gives the WRT quantum invariant of $M$ at that root of unity. This result generalizes the case ${\\mathfrak g}=sl_2$ proved by the first author. It follows that $J_M$ unifies all the quantum invariants of $M$ associated with $\\mathfrak g$, and represents the quantum invariants as a kind of \"analytic function\" defined on the set of roots of unity. For example, $\\tau_M(\\xi)$ for all roots of unity are determined by a \"Taylor expansion\" at any root of unity, and also by the values at infinitely many roots of unity of prime power orders. It follows that WRT quantum invariants $\\tau_M(\\xi)$ for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at $q=1$, and hence by the Le-Murakami-Ohtsuki invariant. Another consequence is that the WRT quantum invariants $\\tau_M^{ \\mathfrak g}(\\xi)$ are algebraic integers. The construction of the invariant $J_M$ is done on the level of quantum group, and does not involve any finite dimensional representation, unlike the definition of the WRT quantum invariant. Thus, our construction gives a unified, \"representation-free\" definition of the quantum invariants of integral homology spheres.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-12T01:28:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M27",
      "17B37"
    ],
    "keywords": [
      "integral homology spheres",
      "simple lie algebras",
      "wrt quantum invariant",
      "unified quantum invariants"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150303549H",
      "inspire": 1351988
    }
  }
}