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Gevrey series in quantum topology. (English) Zbl 1151.57006
A formal power series \(f(x)=\sum_{n\geq0} a_nx^{-n}\) is called Gevrey-\(s\) if there is \(C>0\) such that \(| a_n| \leq C^n(n!)^s\) holds for all \(n\). For example, Gevrey-\(0\) series are exactly the formal power series converging in a neighborhood of \(\infty\). If \(s>0\), Gevrey-\(s\) series are divergent in general. Gevrey series are well-studied in analysis.
The paper under review studies Gevrey properties of several formal power series appearing in quantum topology. For an integral homology \(3\)-sphere \(M\) the authors extract from the Le-Murakami-Ohtsuki invariant \(Z(M)\) of \(M\) a formal power series \(| Z(M)| _x\in\mathbb{Q}[[1/x]]\) whose coefficient of \(1/x^n\) is a certain norm (called Gromov norm by the authors) obtained from the degree \(n\) part of \(Z(M)\), and they show that the power series \(| Z(M)| _x\) is Gevrey-\(1\). This result implies that for each simple Lie algebra \(\mathfrak{g}\) the \(\mathfrak{g}\)-Ohtsuki series of \(M\) is Gevrey-\(1\). The authors also show that the Gromov norm of the Kontsevich integral for a framed link \(L\) in \(S^3\) in a sense similar to the above is Gevrey-\(0\).
The authors also consider Gevrey properties of both the Kashaev invariant of knots in \(S^3\) and the unified \(sl_2\) Witten-Reshetikhin-Turaev invariant of integral homology spheres, taking values in the completion \(\varprojlim_n\mathbb{Z}[q]/((1-q)\cdots(1-q^n))\).

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
17B99 Lie algebras and Lie superalgebras
57M25 Knots and links in the \(3\)-sphere (MSC2010)
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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References:
[1] DOI: 10.1016/0040-9383(95)93237-2 · Zbl 0898.57001 · doi:10.1016/0040-9383(95)93237-2
[2] DOI: 10.1007/s000290050029 · Zbl 0974.16028 · doi:10.1007/s000290050029
[3] DOI: 10.1007/s00029-002-8108-0 · Zbl 1012.57015 · doi:10.1007/s00029-002-8108-0
[4] DOI: 10.2140/gt.2003.7.1 · Zbl 1032.57008 · doi:10.2140/gt.2003.7.1
[5] V., Leningrad Math. J. 2 pp 829– (1991)
[6] E’calle J., Math. Publ. Orsay pp 81– (1981)
[7] Garoufalidis S., Proc. Sympos. Pure Math. AMS 73 pp 173– (2005)
[8] DOI: 10.2140/gt.2005.9.1253 · Zbl 1078.57012 · doi:10.2140/gt.2005.9.1253
[9] Garoufalidis S., Contemp. Math. AMS 416 pp 83– (2006)
[10] DOI: 10.1016/S0040-9383(99)00041-5 · Zbl 0964.57011 · doi:10.1016/S0040-9383(99)00041-5
[11] Geom. Topol. Monogr. 4 pp 55– (2002)
[12] DOI: 10.2977/prims/1145475444 · Zbl 1098.13032 · doi:10.2977/prims/1145475444
[13] DOI: 10.2140/agt.2006.6.1113 · Zbl 1130.57014 · doi:10.2140/agt.2006.6.1113
[14] Huynh V., Fundam. Prikl. Mat. 11 pp 57– (2005)
[15] Modern Phys. Lett. A 39 pp 269– (1997)
[16] Kontsevich M., Adv. Soviet Math. 16 pp 137– (1993)
[17] DOI: 10.1142/S0218216506004737 · Zbl 1114.57013 · doi:10.1142/S0218216506004737
[18] DOI: 10.1215/S0012-7094-00-10224-4 · Zbl 0951.57004 · doi:10.1215/S0012-7094-00-10224-4
[19] DOI: 10.1016/S0166-8641(02)00056-1 · Zbl 1020.57002 · doi:10.1016/S0166-8641(02)00056-1
[20] Lê T. T. Q., Compos. Math. 102 pp 41– (1996)
[21] DOI: 10.1016/S0040-9383(97)00035-9 · Zbl 0897.57017 · doi:10.1016/S0040-9383(97)00035-9
[22] DOI: 10.1142/S0218216505003889 · Zbl 1084.57013 · doi:10.1142/S0218216505003889
[23] DOI: 10.1007/BF02392716 · Zbl 0983.57009 · doi:10.1007/BF02392716
[24] Invent. Math. 123 pp 241– (1996)
[25] J., Bull. Soc. Math. France pp 121– (1993)
[26] DOI: 10.1007/BF01393746 · Zbl 0648.57003 · doi:10.1007/BF01393746
[27] DOI: 10.1007/BF02100618 · Zbl 0739.05007 · doi:10.1007/BF02100618
[28] DOI: 10.1016/S0040-9383(00)00005-7 · Zbl 0989.57009 · doi:10.1016/S0040-9383(00)00005-7
[29] DOI: 10.1016/0377-0427(90)90042-X · Zbl 0738.33001 · doi:10.1016/0377-0427(90)90042-X
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