# Perturbation Theory and the WKB method

## Course information

Code: MAT1312F

Instructor: Marco Gualtieri

Class schedule: R2-5 BA6183 [starting Sept. 18]

Evaluation: Exegesis (or Eisegesis) of one of the articles listed
below, or another approved by the instructor. Some additional exercises. Attendance.

Recently there has been a surge of interest in topics such as the Stokes phenomenon, the WKB approximation, the theory of Borel summability of divergent series, and the study of quadratic differentials on a Riemann surface, because of the ways in which these topics illuminate the study of quantum field theories and the meaning of the Feynman diagram series in perturbation theory. The basic aim of this course is to explain perturbation theory and to explore how to go “beyond perturbation theory”.

While we will discuss issues at the forefront of research, we will not assume any background in the area: the only prerequisite is an understanding of basic complex analysis, differential equations and differential geometry, such as is usually covered at the undergraduate level.

## Resources

### Books

The main book we will use is the beautiful book of Kawai and Takei entitled “Algebraic Analysis of Singular Perturbation Theory”, available from the AMS here. Other useful references include:

- Olver’s book “Asympotics and Special Functions”
- Wasow’s book “Asymptotic expansions for ordinary differential equations”
- Mark Holmes: “Introduction to Perturbation Methods”
- Bender and Orszag text: “Advanced Mathematical Methods for Scientists and Engineers”

### Lecture notes

### Articles

- A. Voros,
*The return of the quartic oscillator. The complex WKB method*, Annales de l’I. H. P., section A, tome 39, no 3 (1983), p.211-338 - Bender and Wu “Anharmonic Oscillator” Phys. Rev. 184, 1231 (1969)
- Works of Pham and Delabaere, especially “Resurgence de Voros et periodes des courbes hyperelliptiques”.
- Work of D. Sauzin on resurgence methods, e.g. Mould expansions for the saddle-node and resurgence monomials, summarized here. His work follows the pioneering work by Ecalle, which is also included on this list.
- Zinn-Justin “Perturbation series at large orders in quantum mechanics and field theories: application to the problem of resummation”
- Works of Aoki, Kawai, Koike, Nishikawa, Takei, listed in the main text above. Keyword: “Exact WKB analysis”. To mention one in particular, Takei’s paper “Integral representation for ordinary differential equations of Laplace type and exact WKB analysis”.
- Berk, Nevins and Roberts: New Stokes’ line in WKB theory.
- Kontsevich-Soibelman on Wall crossing, among other works.
- Works of Gaiotto-Moore-Neitzke on Spectral Networks, including http://arxiv.org/abs/0907.3987, http://arxiv.org/abs/1204.4824, http://arxiv.org/abs/1209.0866, http://arxiv.org/abs/1312.2979 (Hollands-Neitzke)
- Witten on Analytic continuation of Chern-Simons Theory and A New Look At The Path Integral Of Quantum Mechanics,