Les Houches Lectures on Exact WKB Analysis and Painlev\'e Equations

TL;DR

These lecture notes introduce exact WKB analysis for second-order differential equations and explore its recent applications to Painlevé equations, including monodromy and resurgent structures.

Contribution

The notes combine exact WKB analysis with topological recursion and isomonodromy deformations to advance understanding of Painlevé equations and their associated monodromy.

Findings

Explicit computation of monodromy for Painlevé-related equations

Analysis of the resurgent structure of τ-functions and partition functions

Application of exact WKB to Painlevé equations using topological recursion

Abstract

The first part of these lecture notes is devoted to an introduction to the theory of exact WKB analysis for second-order Schr\"odinger-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 Painlev\'e equations. By combining exact WKB analysis with topological recursion, it becomes possible to explicitly compute the monodromy of linear differential equations associated with Painlev\'e equations, assuming Borel summability and other conditions. Furthermore, by using isomonodromy deformations (integrability of the Painlev\'e equations), the resurgent structure of the $\tau$-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.