Les Houches Lectures on Exact WKB Analysis and Painlevé Equations

Abstract: The first part of these lecture notes is devoted to an introduction to the theory of exact WKB analysis for second-order Schrödinger-type ordinary differential equations. It reviews the construction of the WKB solution, Borel summability, connection formulas, and their application to direct monodromy problems. In the second part, we discuss recent developments in applying exact WKB analysis to the study of Painlevé equations. By combining exact WKB analysis with topological recursion, it becomes possible to explicitly compute the monodromy of linear differential equations associated with Painlevé equations, assuming Borel summability and other conditions. Furthermore, by using isomonodromy deformations (integrability of the Painlevé equations), the resurgent structure of the $τ$-function and partition function is analyzed. These lecture notes accompanied a series of lectures at the Les Houches school, ``Quantum Geometry (Mathematical Methods for Gravity, Gauge Theories and Non-Perturbative Physics)'' in Summer 2024.