The paper is concerned with the resurgence of the Kontsevich-Zagier series
We give an explicit formula for the Borel transform of the power series when 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 coincide with the values of 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.
L’article porte sur la série de Kontsevich-Zagier
Nous donnons une formule explicite pour sa transformée de Borel lorsque , 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 , coïncident avec les valeurs de 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.
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
Costin, Ovidiu 1; Garoufalidis, Stavros 2
@article{AIF_2011__61_3_1225_0, author = {Costin, Ovidiu and Garoufalidis, Stavros}, title = {Resurgence of the {Kontsevich-Zagier} series}, journal = {Annales de l'Institut Fourier}, pages = {1225--1258}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {61}, number = {3}, year = {2011}, doi = {10.5802/aif.2639}, mrnumber = {2918728}, zbl = {1238.57016}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2639/} }
TY - JOUR AU - Costin, Ovidiu AU - Garoufalidis, Stavros TI - Resurgence of the Kontsevich-Zagier series JO - Annales de l'Institut Fourier PY - 2011 SP - 1225 EP - 1258 VL - 61 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2639/ DO - 10.5802/aif.2639 LA - en ID - AIF_2011__61_3_1225_0 ER -
%0 Journal Article %A Costin, Ovidiu %A Garoufalidis, Stavros %T Resurgence of the Kontsevich-Zagier series %J Annales de l'Institut Fourier %D 2011 %P 1225-1258 %V 61 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2639/ %R 10.5802/aif.2639 %G en %F AIF_2011__61_3_1225_0
Costin, Ovidiu; Garoufalidis, Stavros. Resurgence of the Kontsevich-Zagier series. Annales de l'Institut Fourier, Volume 61 (2011) no. 3, pp. 1225-1258. doi : 10.5802/aif.2639. https://aif.centre-mersenne.org/articles/10.5802/aif.2639/
[1] Approche de la résurgence, Actualités Mathématiques, Hermann, 1993 | MR | Zbl
[2] Premiers pas en calcul étranger, Ann. Inst. Fourier (Grenoble), Volume 43 (1993), pp. 201-224 | DOI | Numdam | MR | Zbl
[3] On Borel summation and Stokes phenomena for rank- nonlinear systems of ordinary differential equations, Duke Math. J., Volume 93 (1998), pp. 289-344 | DOI | MR | Zbl
[4] Resurgence of 1-dimensional series of sum-product type (in preparation)
[5] Resurgence of the Euler-MacLaurin summation formula, Annales de l’Institut Fourier, Volume 58 (2008), pp. 893-914 | DOI | Numdam | MR | Zbl
[6] On optimal truncation of divergent series solutions of nonlinear differential systems, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., Volume 455 (1999) no. 1985, pp. 1931-1956 | DOI | MR | Zbl
[7] Introduction to the Écalle theory, Computer algebra and differential equations, Volume 193, London Math. Soc. Lecture Note Ser., 1994, pp. 59-101 | MR | Zbl
[8] Resurgent methods in semi-classical asymptotics, Ann. Inst. H. Poincaré Phys. Théor., Volume 71 (1999), pp. 1-94 | Numdam | MR | Zbl
[9] Resurgent functions, Vol. I-III, Mathematical Publications of Orsay 81-05, 1981 (ibid 81-06 1981, ibid 85-05 1985) | MR
[10] Well-behaved convolution averages and the non-accumulation theorem for limit-cycles, The Stokes Phenomenon and Hilbert’s 16th problem, World Scientific, 1996, pp. 71-102 | MR | Zbl
[11] Chern-Simons theory, analytic continuation and arithmetic, preprint 2007 arXiv:0711.1716, Acta Math. Vietnam., Volume 33 (2008), pp. 335-362 | MR | Zbl
[12] Gevrey series in quantum topology, J. Reine Angew. Math. (2007), pp. 1-27 (in press) | MR | Zbl
[13] Borel summation and splitting of separatrices for the Hénon map, Ann. Inst. Fourier, Volume 51 (2001), pp. 513-567 | DOI | Numdam | MR | Zbl
[14] On the quantum invariants of knots and integral homology spheres, Geom. Topol. Monogr., Volume 4 (2002), pp. 55-68 | DOI | MR | Zbl
[15] Cyclotomic completions of polynomial rings, Publ. Res. Inst. Math. Sci., Volume 40 (2004), pp. 1127-1146 | DOI | MR | Zbl
[16] Quantum invariant for torus link and modular forms, Comm. Math. Phys., Volume 246 (2004), pp. 403-426 | DOI | MR | Zbl
[17] On the quantum invariant for the Brieskorn homology spheres, Internat. J. Math., Volume 16 (2005), pp. 661-685 | DOI | MR | Zbl
[18] On the Colored Jones Polynomial and the Kashaev invariant, Fundam. Prikl. Mat., Volume 11 (2005), pp. 57-78 | MR | Zbl
[19] Sur les séries de Taylor n’ayant que des singularités algébrico-logarithmiques sur leur cercle de convergence, Comment. Math. Helv., Volume 3 (1931), pp. 266-306 | DOI | MR | Zbl
[20] The hyperbolic volume of knots from the quantum dilogarithm, Modern Phys. Lett. A, Volume 39 (1997), pp. 269-275 | MR | Zbl
[21] Modular forms and quantum invariants of -manifolds, in Sir Michael Atiyah: a great mathematician of the twentieth century, Asian J. Math., Volume 3 (1999), pp. 93-107 | MR | Zbl
[22] Integrality and symmetry of quantum link invariants, Duke Math. J., Volume 102 (2000), pp. 273-306 | DOI | MR | Zbl
[23] A universal quantum invariant of 3-manifolds, Topology, Volume 37 (1998), pp. 539-574 | DOI | MR | Zbl
[24] Special functions and their applications, Dover Publications, Inc., 1972 | MR | Zbl
[25] Introduction aux travaux de J. Écalle, Enseign. Math., Volume 31 (1985), pp. 261-282 | MR | Zbl
[26] Séries lacunaires, Hermann, 1936 (pp. 18) | Zbl
[27] The well-behaved Catalan and Brownian averages and their applications to real resummation, Proceedings of the Symposium on Planar Vector Fields (Lleida, 1996), Publ. Mat., Volume 41 (1997), pp. 209-222 | MR | Zbl
[28] The colored Jones polynomials and the simplicial volume of a knot, Acta Math., Volume 186 (2001), pp. 85-104 | DOI | MR | Zbl
[29] Two examples of resurgence, Analyzable functions and applications (Contemp. Math.), Volume 373 (2005), pp. 355-371 | MR | Zbl
[30] Asymptotics and special functions, Reprint. AKP Classics. A K Peters, Ltd., Wellesley, MA, 1997 | MR | Zbl
[31] Séries divergentes et théories asymptotiques, Panoramas et Syntheses, suppl., 121, Bull. Soc. Math. France, 1993 | MR
[32] Ribbon graphs and their invariants derived from quantum groups, Commun. Math. Phys., Volume 127 (1990), pp. 1-26 | DOI | MR | Zbl
[33] The Yang-Baxter equation and invariants of links, Inventiones Math., Volume 92 (1988), pp. 527-553 | DOI | MR | Zbl
[34] Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics, 18, Walter de Gruyter, Berlin New York, 1994 | MR | Zbl
[35] Quantum field theory and the Jones polynomial, Commun. Math. Physics., Volume 121 (1989), pp. 360-376 | DOI | MR | Zbl
[36] Vassiliev invariants and a strange identity related to the Dedekind eta-function, Commun. Math. Physics., Volume 40 (2001), pp. 945-960 | MR | Zbl
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