Resurgence of the Kontsevich-Zagier series

Ovidiu Costin[1]; Stavros Garoufalidis[2]

  • [1] Ohio State University Department of Mathematics 231 W 18th Avenue Columbus, OH 43210 (USA)
  • [2] School of Mathematics Georgia Institute of Technology Atlanta, GA 30332-0160 (USA)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 3, page 1225-1258
  • ISSN: 0373-0956

Abstract

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The paper is concerned with the resurgence of the Kontsevich-Zagier series f ( q ) = ∑ n = 0 ∞ ( 1 - q ) ⋯ ( 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 / ( 2 π i ) coincide with the values of f ( q ) 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.

How to cite

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Costin, Ovidiu, and Garoufalidis, Stavros. "Resurgence of the Kontsevich-Zagier series." Annales de l’institut Fourier 61.3 (2011): 1225-1258. <http://eudml.org/doc/219786>.

@article{Costin2011,
abstract = {The paper is concerned with the resurgence of the Kontsevich-Zagier series\[ f(q)=\sum \_\{n=0\}^\infty (1-q)\dots (1-q^n) \]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 $1/(2 \pi i)$ coincide with the values of $f(q)$ 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.},
affiliation = {Ohio State University Department of Mathematics 231 W 18th Avenue Columbus, OH 43210 (USA); School of Mathematics Georgia Institute of Technology Atlanta, GA 30332-0160 (USA)},
author = {Costin, Ovidiu, Garoufalidis, Stavros},
journal = {Annales de l’institut Fourier},
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; Poincaré homology sphere},
language = {eng},
number = {3},
pages = {1225-1258},
publisher = {Association des Annales de l’institut Fourier},
title = {Resurgence of the Kontsevich-Zagier series},
url = {http://eudml.org/doc/219786},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Costin, Ovidiu
AU - Garoufalidis, Stavros
TI - Resurgence of the Kontsevich-Zagier series
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 3
SP - 1225
EP - 1258
AB - The paper is concerned with the resurgence of the Kontsevich-Zagier series\[ f(q)=\sum _{n=0}^\infty (1-q)\dots (1-q^n) \]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 $1/(2 \pi i)$ coincide with the values of $f(q)$ 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.
LA - eng
KW - 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; Poincaré homology sphere
UR - http://eudml.org/doc/219786
ER -

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