Kontsevich-Zagier幂级数的复苏
Resurgence of the Kontsevich-Zagier power series
摘要：
本文研究Kontsevich-Zagier形式幂级数$$ f(q)=\sum_{n=0}^\infty (1-q)... (1-q^n) $$及其解析性质。首先，我们给出了相应的形式幂级数$F(x)=e^{-1/(24x)}f(e^{-1/x})$的Borel变换的显式公式，由此可以明确地确定它的解析延拓、奇点及其结构。这就产生了原始幂级数的右/左和中值求和。这些和在开右半平面中定义良好，用包含戴德金η函数的积分公式表示。中值和也可以表示为涉及复误差函数的级数。此外，利用Zagier的结果证明了有理数$\a$在$-1/(2 πi \a)$处的极限值与$F(-1/(2 πi \a))$一致.研究系列$f(q)$的一个动机是量子拓扑学，它将数值不变量分配给打结的三维物体。受我们的研究结果的启发，我们制定了一个复苏猜想的正式幂级数的打结对象，我们证明的三叶结和庞加莱同调球的情况下，更一般的环面结和塞弗特纤维3-流形。在随后的出版物中，我们将研究一类幂级数的复苏，其中包括最简单的双曲$4_1$结的量子不变量。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
