Resurgence and arithmetic of q-series: from quantum operators to quantum modular forms

Perturbative expansions in quantum theory, particularly in quantum field theory and string theory, are typically factorially divergent due to underlying non-perturbative sectors. Resurgence provides a universal toolbox to access the non-perturbative effects hidden within the perturbative series, producing a collection of exponentially small corrections. Under special assumptions, the non-perturbative data extracted via resurgent methods exhibit intrinsic number-theoretic structures that are deeply rooted in the symmetries of the theory. The framework of modular resurgence aims to formalise this observation. In this talk, I will first introduce the systematic, algebraic approach of resurgence to the problem of divergences and describe the emerging bridge between the resurgence of q-series and the analytic and number-theoretic properties of L-functions and quantum modular forms. I will then apply it to the spectral theory of quantum operators associated with toric Calabi-Yau threefolds. Here, a complete realisation of the modular resurgence paradigm is found in the study of the spectral trace of local P^2, where the asymptotics at weak and strong coupling are captured by certain q-series, and is generalised to all local weighted projective planes. This talk is based on arXiv:2212.10606, 2404.10695, 2404.11550, and work to appear soon.