[Paper Review] Resurgence of the Kontsevich-Zagier power series

Ovidiu Costin, Stavros Garoufalidis|arXiv (Cornell University)|Sep 21, 2006

Geometric and Algebraic Topology24 references13 citations

TL;DR

This paper establishes the resurgence properties of the Kontsevich-Zagier power series $ f(q) = \sum_{n=0}^\infty (1-q)\cdots(1-q^n) $ by deriving an explicit formula for the Borel transform of its associated formal series $ F(x) = e^{-1/(24x)}f(e^{-1/x}) $. The analytic continuation, singularities, and resurgent structure are fully characterized, enabling right/left and median Borel summation via an integral involving the Dedekind eta function. The key result is a complete resurgence conjecture proven for the trefoil knot and Poincaré homology sphere, with implications for quantum invariants of torus knots and Seifert fibered 3-manifolds.

ABSTRACT

The paper is concerned with the Kontsevich-Zagier formal power series $$ f(q)=\sum_{n=0}^\infty (1-q)... (1-q^n) $$ and its analytic properties. To begin with, we give an explicit formula for the Borel transform of the associated formal power series $F(x)=e^{-1/(24x)}f(e^{-1/x})$ from which its analytic continuation, its singularities and their structure can be manifestly determined. This gives rise to right/left and median summation of the original power series. These sums, which are well-defined in the open right half-plane are expressed by an integral formula involving the Dedekind eta function. The median sum can also be expressed as a series involving the complex error function. Moreover, it is shown using results of Zagier that the limiting values at $-1/(2 πi \a)$ for rational numbers $\a$ coincide with $F(-1/(2 πi \a))$. One motivation for studying the series $f(q)$ is Quantum Topology, which assigns numerical invariants to knotted 3-dimensional objects. Motivated by our results, we formulate a resurgence conjecture for the formal power series of knotted objects, which we prove in the case of the trefoil knot and the Poincare homology sphere, and more generally for torus knots and Seifert fibered 3-manifolds. In a subsequent publication we will study resurgence for a class of power series that includes the quantum invariants of the simplest hyperbolic $4_1$ knot.

Motivation & Objective

• To analyze the analytic and resurgence properties of the Kontsevich-Zagier formal power series $ f(q) = \sum_{n=0}^\infty \prod_{k=1}^n (1-q^k) $.
• To derive an explicit formula for the Borel transform of $ F(x) = e^{-1/(24x)}f(e^{-1/x}) $, enabling full characterization of its analytic continuation and singularities.
• To establish right/left and median Borel summation of the original series using integral representations involving the Dedekind eta function and complex error functions.
• To prove a resurgence conjecture for quantum invariants of knotted 3-manifolds, verified for the trefoil knot and Poincaré homology sphere, and extended to torus knots and Seifert fibered 3-manifolds.
• To lay the foundation for studying resurgence in quantum invariants of hyperbolic 3-manifolds, such as the $4_1$ knot, in future work.

Proposed method

• Derive the Borel transform $ G(p) $ of $ F(x) $, showing it is analytic away from rays $ \lambda \mathbb{N}^+ $, with square-root branch point singularities at $ p = k\lambda $.
• Use the Laplace transform of the averaged Borel transform to recover $ F(x) $, with the median sum expressed via an integral involving the Dedekind eta function.
• Express the median sum as a series involving the complex error function, enabling explicit asymptotic and resurgent expansions.
• Apply results from Zagier on special values to show that the radial limits of $ F(x) $ at $ x = -1/(2\pi i \alpha) $ for rational $ \alpha $ coincide with $ F(-1/(2\pi i \alpha)) $.
• Use transseries expansions and Watson’s lemma to analyze the asymptotic behavior of coefficients and reconstruct $ G(p) $ from its transseries, proving object synthesis.
• Verify the resurgence conjecture for the trefoil knot and Poincaré homology sphere by computing their Borel transforms and showing agreement with the general framework.

Experimental results

Research questions

• RQ1 Can the Borel transform of the Kontsevich-Zagier power series be explicitly computed and its singular structure fully determined?
• RQ2 Does the formal power series admit a well-defined median Borel summation, and can it be expressed via an integral involving the Dedekind eta function?
• RQ3 Do the radial limits of the Borel-summable function $ F(x) $ at $ x = -1/(2\pi i \alpha) $ for rational $ \alpha $ match the values $ F(-1/(2\pi i \alpha)) $?
• RQ4 Can a resurgence conjecture be formulated and proven for quantum invariants of knotted 3-manifolds, such as the trefoil and Poincaré homology sphere?
• RQ5 Can the resurgence structure of quantum invariants be extended to torus knots and Seifert fibered 3-manifolds?

Key findings

• The Borel transform $ G(p) $ of $ F(x) $ is explicitly computed and shown to have square-root branch point singularities along rays $ \lambda \mathbb{N}^+ $, with coefficients determined by periodic functions and algebraic constants.
• The median sum of the original series is given by an integral formula involving the Dedekind eta function, providing a well-defined analytic continuation in the open right half-plane.
• The median sum is also expressed as a series involving the complex error function, enabling explicit asymptotic and resurgent expansions.
• The radial limits of $ F(x) $ at $ x = -1/(2\pi i \alpha) $ for rational $ \alpha $ are shown to coincide with $ F(-1/(2\pi i \alpha)) $, confirming consistency with analytic continuation.
• The resurgence conjecture is proven for the trefoil knot and Poincaré homology sphere, with the Borel transforms of their invariants exhibiting the same resurgent structure as the Kontsevich-Zagier series.
• The framework extends to torus knots and Seifert fibered integer homology 3-spheres, with the Borel transforms of their invariants showing identical resurgence patterns.