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. We present 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, which is straightforwardly extended to the dual weakly-coupled limit of topological string theory. This leads to proven closed formulae for the Stokes constants as explicit arithmetic functions and for the perturbative coefficients as special values of a known $L$-function, while the duality between the two scaling regimes appears in a concrete number-theoretic form. A preliminary numerical investigation of the local $\mathbb{F}_0$ geometry unveils a more complex resurgent structure with a logarithmic sub-leading asymptotics. Finally, we obtain a new analytic prediction on the asymptotics 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, and it is based on the conjectural Topological Strings/Spectral Theory correspondence.