Introductory Lectures on Resurgence: CERN Summer School 2024

TL;DR

The paper presents a beginner-friendly survey of resurgence in physics, using physically motivated examples to show how perturbative expansions encode nonperturbative information through transseries and Borel summation. It covers the Airy function, Painlevé II, and the Gross–Witten–Wadia model to illustrate linear and nonlinear Stokes phenomena, followed by the Heisenberg–Euler QED action as a prime QFT example. A core thread is the demonstration that factorial divergence and Borel plane singularities correspond to instanton sectors and phase transitions, with practical summation tools (Padé–Borel, conformal Borel, Richardson acceleration) to extract nonperturbative data. The work emphasizes both analytic structures and numerical methods, highlighting resurgence as a unifying framework across quantum mechanics, quantum field theory, and strong-field QED, and outlines concrete techniques for improved summation and analytic continuation in complex problems.

Abstract

A set of four introductory lectures on Resurgent Asymptotics for Physics (``resurgence") at the CERN Summer School: Continuum Foundations of Lattice Gauge Theories, July 2024. Lecture 1: The Airy function and the Stokes phenomenon. Lecture 2: The nonlinear Stokes phenomenon. Lecture 3: Resurgence in QFT: the Heisenberg-Euler effective action. Lecture 4: Resurgent continuation and summation. The emphasis of these lectures is on physically motivated examples. The lectures include many exercises designed to illustrate some of the key ideas of resurgence.

Figures (12)

• Figure 1: The three basis contours for the Airy function integral in (\ref{['eq:exp-int']}). These curves tend asymptotically to the directions ${\rm arg}(z)=0, \frac{2\pi}{3}, \frac{4\pi}{3}$.
• Figure 2: The integrand function, $e^{x^{3/2}\, S_{\rm real}(v)}=e^{-\frac{2}{3}\, x^{3/2}\, \left(1+\frac{4}{3}v^2\right) \sqrt{1+\frac{1}{3} v^2}}$, from \ref{['eq:res']} and (\ref{['eq:airy-pos']}), plotted along the steepest descent contour \ref{['eq:sd-0']}, with $x=1$ (solid blue), $x=2$ (dashed red) and $x=5$ (dotted black). Each curve has been normalized by dividing by ${\rm Ai}(x)$. The integrand becomes more localized along the steepest descent contour as $x\to\infty$.
• Figure 3: The Airy thimble contours (red solid lines) through the saddle points (black dots), and the stream plot of the vector field $\overline{\vec{\nabla} S}$. The plots are shown for $\theta=0, \frac{99}{100}\frac{2\pi}{3} , \frac{101}{100}\frac{2\pi}{3} , \pi$, from top left to bottom right, where $\theta={\rm arg}(x)$. As $\theta$ changes the saddles move and the thimble contours deform. There is a Stokes jump at $\theta=\frac{2\pi}{3}$, between the 2nd and 3rd plots.
• Figure 4: Plots of the solution to the Painlevé II equation $y"=x\, y+2y^3$. The red curve shows the solution satisfying $y(x)\sim 1.001 {\rm Ai}(x)$ as $x\to +\infty$, and the blue curve shows the solution satisfying $y(x)\sim 0.999 {\rm Ai}(x)$ as $x\to +\infty$. The black curve is the separatrix $\sqrt{-x/2}$. The Hastings-McLeod solution hugs the separatrix as $x\to -\infty$, and $y_{\rm HM}(x)\sim {\rm Ai}(x)$ as $x\to +\infty$. From https://dlmf.nist.gov/32.3.ii.
• Figure 5: The vicinity of the GWW phase transition is probed by a double-scaling limit.