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 P2 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 F0 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. ar X iv :2 21 2. 10 60 6v 1 [ he pth ] 2 0 D ec 2 02 2