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

TL;DR

This paper explores the resurgent structure of fermionic spectral traces in topological string theory, revealing exact formulas for Stokes constants and linking asymptotic expansions to number theory, with implications for dual regimes and new asymptotic predictions.

Contribution

It provides the first exact solutions for the resurgent structure of spectral traces in topological string theory, connecting Stokes constants and perturbative coefficients to explicit arithmetic and special functions.

Findings

Exact formulas for Stokes constants as arithmetic functions

Perturbative coefficients as values of known L-functions

Resurgent structure varies with geometry, revealing complex asymptotics

Abstract

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.