Les Houches Lectures on Exact WKB Analysis and Painlevé Equations

TL;DR

The note surveys exact WKB analysis for Schrödinger-type ODEs, detailing WKB solutions, Borel summability, and connection formulas via Voros periods, and then extends these ideas to Painlevé I through isomonodromy, topological recursion, and quantum curves. It constructs τ-functions for PI by coupling TR correlators on an elliptic spectral curve with a discrete Fourier transform to obtain non-perturbative data, while outlining conjectures that connect Stokes data, resurgent structure, and nonperturbative corrections in topological recursion and topological string theory. The work highlights deep links between integrable structures in Painlevé equations, spectral-network concepts, and non-perturbative phenomena, offering a framework for explicit Stokes multipliers and τ-function transformations under monodromy. It also points to broader connections with conformal blocks, BPS structures, and the remodeling program, suggesting avenues for rigorous justification of Borel summability and non-perturbative formulations.

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.

Figures (9)

• Figure 1.1: A path from $\infty_-$ to $\infty_+$. Here and in what follows, the wiggly lines represent branch cuts for $\sqrt{Q(x)}$. The solid part (resp., the dotted part) of the path lie on the first (resp., the second) sheet of $\overline{\Sigma}$.
• Figure 1.2: Example of Stokes graphs.
• Figure 1.3: When there are no saddle connections in the Stokes graph, a path (in red) intersecting with Stokes curves, as shown in the left figure, can be decomposed into several paths that do not intersect with any Stokes curves, as shown in the right figure.
• Figure 1.4: Stokes curve $C$ and adjacent Stokes regions ${\rm I}$, ${\rm II}$.
• Figure 1.5: The upper row figure depicts the Stokes graph of the Airy equation, and the figures (a)--(c) in the lower row represent the Borel planes corresponding to the positions (a)--(c) of $x$ in the upper row figure. The half-line emanating from $-S(x)$ in (a) represents the path of Laplace integration that defines the Borel sum $\Psi_{+}^{\rm Airy, \, I}$, while the integration paths providing the analytic continuation to region ${\rm II}$ with respect to $x$ are shown in Figures (c). The wiggly line represents a branch cut to describe the multivaluedness of $\psi_{+,B}^{\rm Airy}$.

Theorems & Definitions (18)

• Example 1.2
• Remark 1.3
• Definition 1.4
• Definition 1.5
• Proposition 1.6
• Definition 1.7
• Remark 1.8
• Theorem 1.9
• Theorem 1.11: Voros83, Silverstone, KT98, KK12
• Theorem 1.13: Koike2000