Resurgence, Stokes constants, and arithmetic functions in topological string theory

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.