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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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