Modular resurgence, $q$-Pochhammer symbols, and quantum operators from mirror curves

Building on the results of [1,2], we study the resurgence of $q$-Pochhammer symbols and determine their summability and quantum modularity properties. We construct a new, infinite family of pairs of modular resurgent series from the asymptotic expansions of sums of $q$-Pochhammer symbols weighted by suitable Dirichlet characters. These weighted sums fit into the modular resurgence paradigm and provide further evidence supporting our conjectures in [1]. In the context of the topological string/spectral theory correspondence for toric Calabi-Yau threefolds, Kashaev and Mariño proved that the spectral traces of canonical quantum operators associated with local weighted projective planes can be expressed as sums of $q$-Pochhammer symbols. Exploiting this relation, we show that an exact strong-weak resurgent symmetry, first observed by the second author in [3] and fully formalized in [2] for local ${\mathbb{P}}^2$, applies to all local ${\mathbb{P}}^{m,n}$, albeit stripped of some of the underlying number-theoretic properties. Under some assumptions, these properties are restored when considering linear combinations of the spectral traces that reproduce the weighted sums above.