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arxiv: 2511.15528 · v1 · submitted 2025-11-19 · ✦ hep-th · hep-lat· math-ph· math.MP

Introductory Lectures on Resurgence: CERN Summer School 2024

Gerald V. Dunne This is my paper

Pith reviewed 2026-05-17 20:35 UTC · model grok-4.3

classification ✦ hep-th hep-latmath-phmath.MP
keywords resurgenceasymptoticsStokes phenomenonquantum field theoryHeisenberg-Euler actionAiry functionnon-perturbative effectsBorel summation
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The pith

Resurgence connects perturbative series in quantum field theory to exact non-perturbative results through the Stokes phenomenon.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The lectures introduce resurgent asymptotics as a method to relate divergent perturbative expansions to exact non-perturbative results in physics. They begin with the Airy function to demonstrate the basic Stokes phenomenon, move to nonlinear versions, apply the ideas to the Heisenberg-Euler effective action in QFT, and conclude with resurgent continuation and summation techniques. A reader would care because this framework offers a practical way to extract hidden information from asymptotic series that appear in many physical calculations, turning formal divergences into meaningful predictions.

Core claim

The lectures establish that resurgent asymptotics provides a systematic approach to handling the Stokes phenomenon in both linear and nonlinear differential equations, allowing the resummation of perturbative series in quantum field theory to incorporate non-perturbative contributions, illustrated through the Airy function, the nonlinear Stokes effect, and the Heisenberg-Euler effective action.

What carries the argument

The Stokes phenomenon, which governs the discontinuous changes in asymptotic expansions when crossing certain lines in the complex plane, acting as the key link between perturbative and non-perturbative regimes.

If this is right

  • Resurgent methods applied to the Heisenberg-Euler action reveal non-perturbative strong-field corrections in QED.
  • Resurgent continuation allows perturbative results to be analytically continued across different physical regimes.
  • The included exercises demonstrate how to locate Stokes lines and perform explicit resummations.
  • These techniques extend the reach of perturbative calculations by incorporating instanton and other non-perturbative sectors.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The methods shown could help bridge perturbative and non-perturbative descriptions in lattice gauge theories, consistent with the summer school focus.
  • Similar resurgence tools might address divergent series in statistical mechanics or fluid dynamics beyond the QFT examples.
  • Numerical implementations of resurgent summation could turn these lectures into practical computational aids for high-energy physics.

Load-bearing premise

The target audience has sufficient background in quantum field theory, asymptotics, and complex analysis to follow the lectures and exercises.

What would settle it

A calculation where the Borel resummation of the perturbative series for the Airy function across a Stokes line fails to match the known exact non-perturbative value would challenge the claimed connection.

Figures

Figures reproduced from arXiv: 2511.15528 by Gerald V. Dunne.

Figure 1
Figure 1. Figure 1: The three basis contours for the Airy function integral in ( [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The integrand function, e x 3/2 Sreal(v) = e − 2 3 x 3/2 (1+ 4 3 v 2 ) √ 1+ 1 3 v 2 , from (1.39) and (1.41), plotted along the steepest descent contour (1.37), with x = 1 (solid blue), x = 2 (dashed red) and x = 5 (dotted black). Each curve has been normalized by dividing by Ai(x). The integrand becomes more localized along the steepest descent contour as x → ∞. near v = 0, becoming more and more localize… view at source ↗
Figure 3
Figure 3. Figure 3: The Airy thimble contours (red solid lines) through the saddle points (black dots), [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Plots of the solution to the Painlevé II equation [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The vicinity of the GWW phase transition is probed by a double-scaling limit. [PITH_FULL_IMAGE:figures/full_fig_p028_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Poles of the Painlevé II Hastings-McLeod solution, which are confined to two [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The diagrammatic perturbative expansion of the one loop effective action ( [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A static electric field can tear apart a virtual [PITH_FULL_IMAGE:figures/full_fig_p032_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The form of the particle number N, as a function of longitudinal momentum k, in an electric field (3.53), with carrier phase φ = π 2 . The blue (solid) curve is for spinor QED, and the red (dashed) curve is for scalar QED. See [65, 66, 67]. it is well-known in intense field atomic and molecular physics that the (nonperturbative) ionization spectrum is extremely sensitive to the carrier phase offset paramet… view at source ↗
Figure 10
Figure 10. Figure 10: The Padé and Padé-Conformal approximations to the function [PITH_FULL_IMAGE:figures/full_fig_p054_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The ratio of the approximation to the exact function [PITH_FULL_IMAGE:figures/full_fig_p056_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Padé t-plane poles: N = 50. Suggests a branch point at t = −1 and at t = −∞. To resolve this “hidden" singularity, make a conformal map based on the leading singu￾larity at t = −1: t = 4z (1 − z) 2 ←→ z = √ 1 + t − 1 √ 1 + t + 1 (4.26) Then re-expand about z = 0 to order 50, and make a diagonal Padé approximant in z. This produces Padé poles accumulating to z = −1, which is the conformal map image of t = … view at source ↗
Figure 13
Figure 13. Figure 13: Padé z-plane poles: N = 50. Suggests branch points at z = −1, z = −∞, and at z = ±i. conformal map first, followed by a Padé approximant in the conformal variable z, effectively resolves the hidden singularity at t = −2. This simple method can be applied in many examples where there are multiple collinear singularities. Indeed, in nonlinear problems, one often finds that a leading singularity is re￾peated… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript consists of four introductory lectures on resurgent asymptotics in physics, delivered at the CERN Summer School 2024. Lecture 1 treats the Airy function and the Stokes phenomenon; Lecture 2 covers the nonlinear Stokes phenomenon; Lecture 3 examines resurgence in QFT via the Heisenberg-Euler effective action; Lecture 4 addresses resurgent continuation and summation. The lectures stress physically motivated examples and include exercises.

Significance. As a pedagogical exposition of established material, the lectures could provide a useful entry point for physicists already familiar with QFT, asymptotics, and complex analysis. By focusing on concrete examples such as the Airy function and the Heisenberg-Euler action together with exercises, the work may help disseminate resurgent techniques within the community.

minor comments (3)
  1. [Lecture 1] Lecture 1: the integral representation of the Airy function is introduced without an explicit statement of the contour; adding a short remark on the choice of contour would improve clarity for readers new to the Stokes phenomenon.
  2. [Lecture 3] Lecture 3: the derivation of the Heisenberg-Euler effective action would benefit from a brief citation to the original 1936 paper alongside the modern resurgent references.
  3. Throughout: some exercises ask for explicit numerical checks of asymptotic series; providing a short note on the expected precision or software suggestions would assist readers attempting the exercises independently.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of our introductory lectures on resurgent asymptotics. We appreciate the recommendation to accept the manuscript.

read point-by-point responses

  1. Referee: No major comments raised.

    Authors: We are grateful for the referee's supportive review and the suggestion that the work may help disseminate resurgent techniques. No revisions are necessary. revision: no

Circularity Check

0 steps flagged

Expository lectures with no new derivations or self-referential claims

full rationale

This is a set of four introductory lectures summarizing standard, well-established topics in resurgent asymptotics (Airy function and Stokes phenomenon, nonlinear Stokes, Heisenberg-Euler effective action, resurgent summation) with exercises. The document advances no original theorems, calculations, or predictions that could reduce to fitted inputs or self-citations by construction. All content draws on prior literature without load-bearing steps that equate outputs to inputs via definition or renaming. The derivation chain is absent because the work is pedagogical rather than deductive.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is an introductory review of established techniques; no new free parameters, axioms, or invented entities are introduced.

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