Quantum invariants of 3-manifolds: Integrality, splitting, and perturbative expansion
The quantum invariants \(\tau^{\mathfrak g}_M\) of closed \(3\)-manifolds \(M\) are defined for each simple Lie algebra \(\mathfrak g\) which are a function on a set of roots of unity. The author defines an invariant \(\tau^{P\mathfrak g}_M\) for any simple Lie algebra \(\mathfrak g\) which corresponds to the projective group of the simply connected Lie group \(G\) associated to \(\mathfrak g\). The invariant \(\tau^{\mathfrak g}_M\) generalizes the invariants defined first by \textit{R. Kirby} and \textit{P. Melvin} [Invent. Math. 105, 473-545 (1991; Zbl 0745.57006)] for the case \(\mathfrak g=\mathfrak{s}\mathfrak{l}_2\), by \textit{T. Kohno} and \textit{T. Takata} [Bannai, E. (ed.) et al., Progress in algebraic combinatorics. Based on the international symposium on algebraic combinatorics, Fukuoka, Japan, 1993 and the research project on algebraic combinatorics, Kyoto, Japan, April 1994 -- March 1995. Kyoto: Kinokuniya Company Ltd. Adv. Stud. Pure Math. 24, 243-264 (1996; Zbl 0860.57013)] for \({\mathfrak g}=\mathfrak{sl}_n\), and more recently by \textit{S. F. Sawin} [Adv. Math. 165, 1-34 (2002; Zbl 0997.57043)] for any simple Lie algebra \(\mathfrak g \neq D_{2n}\). The decomposition property \(\tau^{\mathfrak g}_M=\tau^{P\mathfrak g}_M\times\tau^G_M\), where \(\tau^G_M\) is a \(3\)-manifold invariant associated to the center of \(G\), which was proved by the above-mentioned researchers for special cases, is obtained for general Lie algebras. The author shows that \(\tau^{P\mathfrak g}_M\) is an algebraic integer if the order of the root of unity is an odd prime, which was first proved by \textit{H. Murakami} [Math. Proc. Camb. Philos. Soc. 117, 237-249 (1995; Zbl 0854.57016)] in the \(\mathfrak{sl}_2\) case. Finally, the author shows that for any simple Lie algebra \(\mathfrak g\), \(\tau^{P\mathfrak g}_M\) seen as an invariant of rational homology spheres \(M\) for roots of unity of prime orders admits a perturbative expansion à la \textit{T. Ohtsuki} [Invent. Math. 123, 241-257 (1996; Zbl 0855.57016)].
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