Volume 20, issue 5 (2016)

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Unified quantum invariants for integral homology spheres associated with simple Lie algebras

Kazuo Habiro and Thang T Q Lê

Geometry & Topology 20 (2016) 2687–2835
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 ${\tau }_{M}^{\mathfrak{g}}\left(\xi \right)\in ℂ$ of oriented $3$–manifolds $M$. We construct an invariant ${J}_{M}$ of integral homology spheres $M$, with values in $\stackrel{̂}{ℤ\left[q\right]}$, the cyclotomic completion of the polynomial ring $ℤ\left[q\right]$, 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 Habiro. 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, ${\tau }_{M}\left(\xi \right)$ 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 ${\tau }_{M}\left(\xi \right)$ 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 Lê–Murakami–Ohtsuki invariant. Another consequence is that the WRT quantum invariants ${\tau }_{M}^{\mathfrak{g}}\left(\xi \right)$ 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.

Keywords
quantum invariants, integral homology spheres, quantized enveloping algebras, ring of analytic functions on roots of unity
Primary: 57M27
Secondary: 17B37
Publication
Received: 12 March 2015
Accepted: 25 October 2015
Published: 7 October 2016
Proposed: Vaughan Jones
Seconded: Robion Kirby, Ciprian Manolescu
Authors
 Kazuo Habiro Research Institute for Mathematical Sciences Kyoto University Kitashirakawa-Oiwakecho, Sakyo-ku Kyoto 606-8502 Japan Thang T Q Lê School of Mathematics Georgia Institute of Technology 686 Cherry Street Atlanta, GA 30332-0160 United States