Resurgence, trans-series and non-perturbative effects in Quantum Theory
In general, problems in theoretical physics are not solvable in closed form, and one has to use approximation schemes. Most of these approximations lead to generically divergent formal power series in a small parameter. The course builds from the observation that divergent series are not a failure but carry an exact, complete encoding of non-perturbative physics. Resurgence is the mathematical framework that makes this precise, connecting perturbation theory with non-perturbative effects (instantons, bions, renormalons). Topics of resurgence theory will be reviewed from a physics point of view, with concrete examples (mostly toy models) from Quantum Mechanics and Quantum Field Theory.
Lecturer
Ivan Kostov
Date
18th March ~ 13th May, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday | 13:30 - 16:55 | Shuangqing | ZOOM B | 462 110 5973 | BIMSA |
Prerequisite
Complex analysis, basics of Quantum Mechanics and Quantum Field Theory and some familiarity with perturbation theory
Syllabus
1. Asymptotic series, non-perturbative effects, and differential equations
Asymptotic series and exponentially small corrections. Formal power series and trans-series in ODEs. Classical asymptotics and the Stokes phenomenon. Borel transform and Borel resummation. Non-perturbative effects and large order behavior.
2. Non-perturbative effects in Quantum Mechanics and Quantum Field Theory
Path integral review and saddle-point approximation. The WKB method and quantization conditions. Instantons in double-well potential. The pure quartic oscillator (Balian, Parisi and Voros). Dilute instanton gas approximation. One-loop fluctuation determinant around instanton. Instanton-induced tunneling and level splitting. The instanton–large-order connection (Zinn-Justin, Lipatov). The dispersion relation connecting large orders to non-perturbative effects. Bender-Wu analysis of anharmonic oscillator. The 1/N expansion. Large N instantons. Large N instantons in Matrix Quantum Mechanics.
3. Non-perturbative effects in matrix models
Matrix models at large N: General aspects. The one-cut solution. The multi-cut solution. Large N instantons and eigenvalue tunneling. Large N instantons in the one-cut matrix model. Large N instantons, large order behavior and the spectral curve. Classical asymptotics and the Stokes phenomenon in matrix models.
4. Non-perturbative effects in integrable field theories.
The Bohr-Sommerfeld quantisation: exact vs. perturbative. Thermodynamic Bethe Ansatz (TBA) equations from exact WKB. Relation to spectral determinants and zeta-regularization. The ODE/IQFT correspondence (Dorey-Tateo, Lukyanov-Zamolodchikov). Connection to topological string theory (GV invariants). Trans-series and large N expansion in SU(N) Principal Chiral Field and O(N) Gross-Neveu. N=1 super Yang-Mills theory: resurgence and exact results.
Asymptotic series and exponentially small corrections. Formal power series and trans-series in ODEs. Classical asymptotics and the Stokes phenomenon. Borel transform and Borel resummation. Non-perturbative effects and large order behavior.
2. Non-perturbative effects in Quantum Mechanics and Quantum Field Theory
Path integral review and saddle-point approximation. The WKB method and quantization conditions. Instantons in double-well potential. The pure quartic oscillator (Balian, Parisi and Voros). Dilute instanton gas approximation. One-loop fluctuation determinant around instanton. Instanton-induced tunneling and level splitting. The instanton–large-order connection (Zinn-Justin, Lipatov). The dispersion relation connecting large orders to non-perturbative effects. Bender-Wu analysis of anharmonic oscillator. The 1/N expansion. Large N instantons. Large N instantons in Matrix Quantum Mechanics.
3. Non-perturbative effects in matrix models
Matrix models at large N: General aspects. The one-cut solution. The multi-cut solution. Large N instantons and eigenvalue tunneling. Large N instantons in the one-cut matrix model. Large N instantons, large order behavior and the spectral curve. Classical asymptotics and the Stokes phenomenon in matrix models.
4. Non-perturbative effects in integrable field theories.
The Bohr-Sommerfeld quantisation: exact vs. perturbative. Thermodynamic Bethe Ansatz (TBA) equations from exact WKB. Relation to spectral determinants and zeta-regularization. The ODE/IQFT correspondence (Dorey-Tateo, Lukyanov-Zamolodchikov). Connection to topological string theory (GV invariants). Trans-series and large N expansion in SU(N) Principal Chiral Field and O(N) Gross-Neveu. N=1 super Yang-Mills theory: resurgence and exact results.
Reference
Marcos Mariño, An introduction to resurgence in quantum theory
https://www.marcosmarino.net/uploads/1/3/3/5/133535336/resurgence-course.pdf
Marcos Mariño, Lectures on Non-Perturbative Effects (2012)
https://arxiv.org/pdf/1206.6272
I. Aniceto, G. Basar, and R. Schiappa, A primer on resurgent transseries and their asymptotics.
Physics Reports, 809:1–135, May 2019.
https://arxiv.org/pdf/1802.10441
Resurgence and trans-series in quantum field theory: the CP(N-1) model
G. V. Dunne and M. Ünsal
Journal of High Energy Physics, 2012(11), November 2012
https://arxiv.org/abs/1210.2423
G. V. Dunne. Introductory lectures on resurgence: Cern summer school 2024, 2025.
https://arxiv.org/pdf/2511.15528v1
https://www.marcosmarino.net/uploads/1/3/3/5/133535336/resurgence-course.pdf
Marcos Mariño, Lectures on Non-Perturbative Effects (2012)
https://arxiv.org/pdf/1206.6272
I. Aniceto, G. Basar, and R. Schiappa, A primer on resurgent transseries and their asymptotics.
Physics Reports, 809:1–135, May 2019.
https://arxiv.org/pdf/1802.10441
Resurgence and trans-series in quantum field theory: the CP(N-1) model
G. V. Dunne and M. Ünsal
Journal of High Energy Physics, 2012(11), November 2012
https://arxiv.org/abs/1210.2423
G. V. Dunne. Introductory lectures on resurgence: Cern summer school 2024, 2025.
https://arxiv.org/pdf/2511.15528v1
Audience
Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Ivan Kostov obtained his PhD in 1982 from the Moscow State University, with scientific advisers Vladimir Feinberg and Alexander Migdal. Then he worked in the group of Ivan Todorov at the INRNE Sofia, and since 1990 as a CNRS researcher at the IPhT, CEA-Saclay, France. Currently he is emeritus DR CNRS at IPhT.