Lectures on Resurgence in Integrable Field Theories

There has been recently considerable progress in understanding the nature of perturbation theory in UV free and gapped $2d$ integrable field theories with renormalon singularities. Thanks to Bethe ansatz and large $N$ techniques, non-perturbative corrections can also be computed and lead to the reconstruction of the trans-series for the free energy in presence of a chemical potential. This is an ideal arena to test resurgence in QFT and determine if and how the exact result can be reconstructed from the knowledge of the perturbative series only. In these notes we give a pedagogical introduction to this subject starting from the basics. In the first lecture we give an overview of applications in QFT of Borel resummations before the advent of resurgence. The second lecture introduces the key concepts of resurgence and finally in the third lecture we discuss a specific application in the context of the principal chiral field model. Extended version of three lectures given at IHES and review talks given at Les Diablerets and Mainz, in 2023.