Strong-weak symmetry and quantum modularity of resurgent topological strings on local $\mathbb{P}^2$

Quantizing the mirror curve to a toric Calabi-Yau threefold gives rise to quantum operators whose fermionic spectral traces produce factorially divergent formal power series in the Planck constant and its inverse. These are conjecturally captured by the Nekrasov-Shatashvili and standard topological string free energies, respectively, via the TS/ST correspondence. The resurgent structures of the first fermionic spectral trace of local $\mathbb{P}^2$ in both weak and strong coupling limits were solved exactly by the second author in [1]. Here, we argue that a full-fledged strong-weak resurgent symmetry is at play, exchanging the perturbative/non-perturbative contributions to the holomorphic and anti-holomorphic blocks in the factorization of the spectral trace. This relies on a global net of relations connecting the perturbative series and the discontinuities in the dual regimes, which is built upon the analytic properties of the $L$-functions with coefficients given by the Stokes constants and the $q$-series acting as their generating functions. Then, we show that the latter are holomorphic quantum modular forms for $\Gamma_1(3)$ and are reconstructed by the median resummation of their asymptotic expansions.