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

Abstract

Building on the results of arXiv:2404.11550 and arXiv:2404.10695, 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 arXiv:2404.11550. 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 arXiv:2212.10606 and fully formalized in arXiv:2404.10695 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.