Modular resurgent structures

Abstract: The theory of resurgence uniquely associates a factorially divergent formal power series with a collection of exponentially small non-perturbative corrections paired with a set of complex numbers known as Stokes constants. When the Borel plane displays a single infinite tower of singularities, the secondary resurgent series are trivial, and the Stokes constants are coefficients of an $L$-function, a rich analytic number-theoretic fabric underlies the resurgent structure of the asymptotic series. We propose a new paradigm of modular resurgence that focuses on the role of the Stokes constants and the interplay of the $q$-series acting as their generating functions with the corresponding $L$-functions. Guided by two pivotal examples arising from topological string theory and the theory of Maass cusp forms, we introduce the notion of modular resurgent series, which we conjecture to have specific summability properties as well as to be related to quantum modular forms.