BIMSA

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.

2026年03月18日至 05月13日

Weekday	Time	Venue	Online	ID	Password
周三	13:30 - 16:55	Shuangqing	ZOOM B	462 110 5973	BIMSA

Complex analysis, basics of Quantum Mechanics and Quantum Field Theory and some familiarity with perturbation theory

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.

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

Graduate , 博士后 , Researcher

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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.