On the integrality of the Witten–Reshetikhin–Turaev 3-manifold invariants

Abstract

We prove that the SU(2) and SO(3) Witten-Reshetikhin-Turaev invariants of any 3-manifold with any odd colored link inside at any root of unity are algebraic integers. In the late 80s, Witten (Wi) and Reshetikhin-Turaev (RT) associated with any closed oriented 3-manifold M (possibly with a colored link inside), any root of unityand any compact Lie group G a complex numberG M (�), called the quantum or WRT invariant of M. The case G = SU(2) is most studied. In this case, for roots of unity of odd order, there is a projective or G = SO(3) version introduced by Kirby and Melvin (KM). This projective version, when defined, i.e. when the order ofis odd, determines the SU(2) version. Since more than 20 years, the problem of integrality of the WRT invariants has been intensively studied. The interest to this problem was drawn by the theory of perturbative 3-manifold invariants generalizing those of Casson and Walker (O), by the construction of Integral Topological Quantum Field Theories (G, GM) and their topological applications and more recently, by attempts to categorify the WRT invariants (Kho). In this paper we completely solve the integrality problem for both SO(3) and SU(2) versions of the WRT invariants. Before stating our results, let us give a brief introduction into the history of this subject. In 1995 Murakami (Mu) established the integrality of the WRT SO(3)-invariant for rational homology 3-spheres at roots of unity of prime orders. The proof was extended to all 3-manifolds by Masbaum and Roberts (MR). The integrality at roots of prime orders was the main starting point of the construction of perturbative 3-manifold invariants by Ohtsuki (O) and integral Topological Quantum Field Theories by Gilmer and Masbaum (GM). Masbaum and Wenzl (MW), and independently Takata and Yokota (TY), proved the integrality of the projective WRT SU(n)-invariant for all 3-manifolds, always under the assumption that the order of the root of unity is prime. Finally the third author (Le1) established the integrality of the projective WRT invariant associated with any compact simple Lie group, again at roots of unity of prime orders. The case when the order of the root of unity is not prime is more complicated. The first integrality result for all roots of unity was obtained by Habiro (Ha) in the case of SU(2) and integral homology 3-spheres. Habiro's proof was based on the existence of the unified invariant, a kind of generating function for WRT invariants at all roots of unity. Habiro and the third author (HL) subsequently defined the unified WRT invariant for all simple