The quantization of the mirror curve to a toric Calabi–Yau threefold gives rise to quantum-mechanical operators, whose fermionic spectral traces produce factorially divergent power series in the Planck constant. These asymptotic expansions can be promoted to resurgent trans-series. They show infinite towers of periodic singularities in their Borel plane and infinitely many rational Stokes constants, which are encoded in generating functions expressed in closed form in terms of $q$-series. We provide an exact solution to the resurgent structure of the first fermionic spectral trace of the local $\mathbb{P}^2$ geometry in the semiclassical limit of the spectral theory, corresponding to the strongly-coupled regime of topological string theory on the same background in the conjectural TS/ST correspondence. Our approach straightforwardly applies to the dual weakly-coupled limit of the topological string. We present and prove closed formulae for the Stokes constants as explicit arithmetic functions and for the perturbative coefficients as special values of known $L$-functions, while the duality between the two scaling regimes of strong and weak string coupling constant appears in number-theoretic form. A preliminary numerical investigation of the local $\mathbb{F}_0$ geometry unveils a more complicated resurgent structure with logarithmic sub-leading asymptotics. Finally, we obtain a new analytic prediction on the asymptotic behavior of the fermionic spectral traces in an appropriate WKB double-scaling regime, which is captured by the refined topological string in the Nekrasov–Shatashvili limit.

中文翻译：

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拓扑弦理论中的复兴、斯托克斯常数和算术函数

将镜像曲线量子化为复曲面卡拉比-丘三重产生了量子力学算子，其费米子谱迹在普朗克常数中产生阶乘发散的幂级数。这些渐近扩张可以促进复兴的跨系列。它们在 Borel 平面上显示了无限的周期性奇点塔和无限多个有理斯托克斯常数，这些常数被编码在以 $q$ 级数的封闭形式表示的生成函数中。我们提供了谱理论半经典极限下局部 $\mathbb{P}^2$ 几何的第一费米子谱迹的复兴结构的精确解，对应于拓扑弦理论的强耦合机制推测的 TS/ST 对应关系具有相同的背景。我们的方法直接适用于拓扑弦的双弱耦合极限。我们提出并证明了斯托克斯常数作为显式算术函数的封闭公式以及作为已知$L$函数的特殊值的微扰系数的封闭公式，而强弦耦合常数和弱弦耦合常数的两个标度范围之间的对偶性出现在数论中形式。对局部 $\mathbb{F}_0$ 几何的初步数值研究揭示了具有对数次先导渐近的更复杂的复兴结构。最后，我们在适当的 WKB 双尺度范围内获得了费米子谱迹渐近行为的新分析预测，该预测由 Nekrasov-Shatashvili 极限中的精制拓扑弦捕获。