## Mini-courses

Exact WKB method and Painlevé equation

by K. Iwaki

The exact WKB method, initiated by Voros, is a framework for mathematically rigorous treatment of the classical WKB approximation based on the Borel summation method. This method allows us to describe the monodromy and Stokes matrices of the Schrödinger-type ODEs using period integrals on the classical limit. On the other hand, the Painlevé equations are non-linear differential equations discovered about 100 years ago and are important research subjects that appear in various aspects of mathematical physics. It is known to describe the isomonodromy deformation of linear ODEs. In the first two lectures, I will introduce the theory of exact WKB analysis with details, and I will explain its application to the Painlevé equations (with the viewpoint of topological recursion/quantum curve correspondence) in the remaining two lectures.

Lecture notes, handwritten notes

Non-perturbative Seiberg–Witten geometry

by L. Hollands

In lectures by A. Neitzke last week we have learned about the spectrum of BPS states in 4d 𝒩 = 2 theories, and in particular those of class 𝒮. N. Nekrasov, on the other hand, has taught us about 𝒩 = 2 partition functions that (in some sense) count instanton configurations on a 4d space-time. In particular, he has introduced the Ω-background, and argued that the exact 𝒩 = 2 instanton partition function Z_inst can be computed in this background for an overlapping, but complementary, class of 4d 𝒩 = 2 quiver theories.
In these lectures the goal is to combine these two elements, with the aim of constructing a new partition function that encodes both the instanton partition function as well as the spectrum of BPS states. Spectral networks play a central role in this story, and these lectures owe much to the beautiful works of Gaiotto–Moore–Neitzke. The approach we take is also closely related to various other perspectives and results in the literature, such as the topics of non-perturbative topological string theory, resurgence, isomonodromic tau functions, analytic Langlands, Riemann–Hilbert problems, holomorphic Floer theory, etc. In particular, there is a very close connection to exact WKB analysis, a topic which K. Iwaki will introduce in detail this week.
To make the relation to the exact WKB analysis as clear as possible, I have decided to start these lectures in two dimensions. We will thus start with analysing 2d 𝒩 = (2,2) theories, with emphasis on the class of Landau–Ginzburg models. The latter models are closely related to minimal models, integrable hierarchies of KdV type, and matrix models, so hopefully this will also make a connection to the lectures in earlier weeks in this school.

Lecture notes

Non-perturbative topological strings

by M. Mariño

Topological string theory is at the center of modern mathematical physics, and is deeply related to enumerative geometry, gauge theories, and integrable models, among others. As many other string theories, it is only defined in terms of a factorially divergent perturbative series. In these lectures I will address non-perturbative aspects of this theory. I will first discuss the problem from the point of view of the theory of resurgence. I will then present a non-perturbative definition based on the spectral theory of quantum curves, which provides a precise model of quantum geometry.

Lecture notes, Problem set

## Seminar

Resurgence without transseries: the case of exact WKB in quantum mechanics

by M. Serone

Recovering an exact result from the resummation of a transseries using resurgence is generally a non-trivial task. Whenever the transseries admits an interpretation as a sum over saddle points, by suitable deformations we can get rid of the transseries and replace it with a more tractable single Borel resummable asymptotic series. Within exact WKB, energy eigenvalues in quantum mechanical models with a bounded potential are determined by quantization conditions and generally give rise to transseries in ℏ. We show how suitable deformations allow us to get trivial quantization conditions which lead to Borel resummable series for all energy eigenvalues.