Resurgence of the Kontsevich-Zagier series

Costin, Ovidiu; Garoufalidis, Stavros

Annales de l'Institut Fourier, Tome 61 (2011) no. 3, pp. 1225-1258.

L’article porte sur la série de Kontsevich-Zagier

$$f\left(q\right)=\sum _{n=0}^{\infty }(1-q)\cdots (1-q^n)$$

Nous donnons une formule explicite pour sa transformée de Borel lorsque $q=e^{1/x}$, d’où son prolongement analytique, ses singularités (toutes sur l’axe des réels positifs) et la monodromie locale peuvent être déterminés. Nous donnons également deux formules (l’une impliquant la fonction éta de Dedekind, et l’autre impliquant la fonction d’erreur complexe) pour la sommation à droite, à gauche et médiane de la transformée de Borel. Nous démontrons aussi que les valeurs limites de la somme médiane, aux multiples rationnels de $1/\left(2i\pi \right)$, coïncident avec les valeurs de $f\left(q\right)$ aux racines complexes de l’unité. Notre théorème s’étend plus généralement à la série entière des noeuds du tore et les 3-variétés fibrées de Seifert associées par la topologie quantique.

The paper is concerned with the resurgence of the Kontsevich-Zagier series

$$f\left(q\right)=\sum _{n=0}^{\infty }(1-q)\cdots (1-q^n)$$

We give an explicit formula for the Borel transform of the power series when $q=e^{1/x}$ from which its analytic continuation, its singularities (all on the positive real axis) and the local monodromy can be manifestly determined. We also give two formulas (one involving the Dedekind eta function, and another involving the complex error function) for the right, left and median summation of the Borel transform. We also prove that the limiting values of the median sum at rational multiples of $1/\left(2\pi i\right)$ coincide with the values of $f\left(q\right)$ at the corresponding complex roots of unity. Our resurgence theorem extends more generally to the power series of torus knots and Seifert fibered 3-manifolds associated by Quantum Topology.

Classification : 57M27, 40D99
Keywords: resurgence, analytic continuation, Borel summability, analyzability, asymptotic expansions, transseries, Zagier-Kontsevich power series, strange identity, trefoil, Poincare homology sphere, Habiro ring, Laplace transform, Borel transform, knots, 3-manifolds, quantum topology, TQFT, perturbative quantum field theory, Gevrey series, resummation.
Mot clés : résurgence, prolongement analytique, Kontsevich-Zagier séries, transformée de Laplace, transformée de Borel, noeuds, TQFT, séries de Gevrey

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