Resonant resurgence and quasi-modularity from QFT

This talk concerns two studies of Feynman integrals in QFT. In a massless case, resurgence in a Dyson-Schwinger equation leads to a trans-series with resonances between instantons producing 6 Stokes constants. In a massive case, periods of modular forms result from quasi-modularity of Lambert series. This led to 20 ways of using complex multiplication to improve Ramanujan's evaluations of zeta values.

Modular properties of brane scattering from integrated correlators

Integrated correlators in N=4 SYM represent a powerful tool to obtain exact results in the coupling constant, and can be used as constraints for dual scattering amplitudes in AdS. In this talk we consider a correlation function in presence of determinant operators, which in the planar limit are heavy operators realizing a giant graviton D3-brane in the dual space. We compute their integrated correlator via supersymmetric localization exactly in the 't Hooft coupling, interpreting such results as the integrated disc and annulus amplitudes from the worldsheet perspective. Next, we derive the SL(2,Z) completion of the result, that nicely matches the holographic expectations for string/brane scattering amplitude. In the end, we discuss the differences with respect to integrated correlators in presence of line defects, which are dual to extended (p,q)-string scattering and transform non-trivially under SL(2,Z). Based on arxiv:2403.17263 and arxiv:2308.16575

Modularity and resurgence in Calabi-Yau compactifications

I'll review some open open problems in determination of the effective action for Calabi-Yau compactifications of type II string theory, and argue that their solution requires a combination of modularity and resurgence.

Strong-weak symmetry and quantum modularity of resurgent topological strings

Quantizing the mirror curve to a toric Calabi-Yau threefold gives rise to quantum operators whose fermionic spectral traces produce factorially divergent series in the Planck constant and its inverse. These are captured by the Nekrasov-Shatashvili and standard topological strings via the TS/ST correspondence. In this talk, I will discuss the resurgence of these dual asymptotic series and present an exact solution for the spectral trace of local P^2. A full-fledged strong-weak symmetry is at play, exchanging the perturbative/nonperturbative contributions to the holomorphic and anti-holomorphic blocks in the factorization of the spectral trace. This relies on a network of relations connecting the dual regimes and building upon the analytic properties of the L-functions with coefficients given by the Stokes constants and the q-series acting as their generating functions. Finally, I will mention how these results fit into a broader paradigm linking resurgence and quantum modularity. This talk is based on arXiv:2212.10606, 2404.10695, and 2404.11550.

Non-perturbative Topological Recursion in Jackiw-Teitelboim Gravity

The connection between matrix models and two-dimensional gravitational theories (including JT gravity) is a very well-established one. For this reason, Topological Recursion has proved to be an invaluable tool for computing observables perturbatively in such theories. In this talk, I will introduce a non-perturbative generalization of Topological Recursion which is particularly well-suited for the study of instanton effects (more specifically, of the ZZ brane type), and apply it to the case of JT gravity. Moreover, by making use of resurgent large order relations, I will show how our results lead to large genus asymptotics of Weil-Petersson volumes.