Resurgence and modularity in string theory

This thesis provides an exploration of the interplay between resurgence analysis and modular invariance in the context of string theory. We focus on two particular applications. Firstly, we analyse a class of modular invariant functions called generalised Eisenstein series that play an important rˆole in string perturbation theory at genus-one, as well as in the low energy effective action for Type IIB supergravity. By extending this space of functions to a broader family, we show how a subtle asymptotic analysis via Cheshire-cat resurgence allows us to recover from perturbative data interesting non-perturbative corrections, which can be interpreted as D-D¯-brane instantons. These results are based on papers [1,2]. Secondly, we consider a related problem in the study of N = 4 maximally supersymmetric SU(N) Yang-Mills theory. By studying certain integrated four-point correlation functions, we show how the large-N expansion at fixed gauge coupling, τ , of such physical quantities yields modular invariant transseries, and we demonstrate the necessity of including non-perturbative, exponentially suppressed terms at large-N, which holographically originate from (p, q)-string world-sheet instantons. These results are based on paper [3]. The thesis furthermore includes a short overview of resurgence analysis, as well as some relevant aspects of the theory of modular functions such as their representation in terms of Poincar´e series and SL(2, Z) spectral theory.