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An Introduction to Resurgence, Trans-Series and Alien Calculus

Daniele Dorigoni

TL;DR

The notes address the problem of extracting non-perturbative physics from perturbative series in quantum theories by developing resurgence theory. They introduce the Borel transform, simple resurgent functions, Stokes automorphisms, and alien derivatives, then build trans-series via bridge equations to encode multi-saddle (instanton) contributions. A median resummation procedure is presented to yield real, unambiguous results, illustrating how Morse theory and Lefschetz thimbles underpin the cancellation of perturbative ambiguities with non-perturbative sectors. The framework provides a principled, transferable approach to relate perturbative data to non-perturbative physics across QM/QFT, with implications for semi-classical analysis and topological string contexts.

Abstract

In these notes we give an overview of different topics in resurgence theory from a physics point of view, but with particular mathematical flavour. After a short review of the standard Borel method for the resummation of asymptotic series, we introduce the class of simple resurgent functions, explaining their importance in physical problems. We define the Stokes automorphism and the alien derivative and discuss these objects in concrete examples using the notion of trans-series expansion. With all the tools introduced, we see how resurgence and alien calculus allow us to extract non-perturbative physics from perturbation theory. To conclude, we apply Morse theory to a toy model path integral to understand why physical observables should be resurgent functions.

An Introduction to Resurgence, Trans-Series and Alien Calculus

TL;DR

The notes address the problem of extracting non-perturbative physics from perturbative series in quantum theories by developing resurgence theory. They introduce the Borel transform, simple resurgent functions, Stokes automorphisms, and alien derivatives, then build trans-series via bridge equations to encode multi-saddle (instanton) contributions. A median resummation procedure is presented to yield real, unambiguous results, illustrating how Morse theory and Lefschetz thimbles underpin the cancellation of perturbative ambiguities with non-perturbative sectors. The framework provides a principled, transferable approach to relate perturbative data to non-perturbative physics across QM/QFT, with implications for semi-classical analysis and topological string contexts.

Abstract

In these notes we give an overview of different topics in resurgence theory from a physics point of view, but with particular mathematical flavour. After a short review of the standard Borel method for the resummation of asymptotic series, we introduce the class of simple resurgent functions, explaining their importance in physical problems. We define the Stokes automorphism and the alien derivative and discuss these objects in concrete examples using the notion of trans-series expansion. With all the tools introduced, we see how resurgence and alien calculus allow us to extract non-perturbative physics from perturbation theory. To conclude, we apply Morse theory to a toy model path integral to understand why physical observables should be resurgent functions.

Paper Structure

This paper contains 9 sections, 2 theorems, 172 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Borel Resummation
  3. The Algebra of Simple Resurgent Functions
  4. Stokes Automorphism and Alien Derivative
  5. Intermezzo on Trans-Series
  6. Trans-series Expansion and Bridge Equation
  7. Median Resummation and Cancellation of Ambiguities
  8. Outlook
  9. Acknowledgment

Key Result

Proposition 1

The Borel transform $\hat{\phi}(\zeta)$ of a formal power series $\tilde{\phi}(z)=\sum_{n=0}^\infty c_n z^{-n-1}$ has a finite radius of convergence if and only if $\tilde{\phi}(z)$ is of Gevrey-1 type

Figures (7)

  • Figure 1: Schematic form of the Borel regularisation procedure.
  • Figure 2: Lateral Borel summation along the direction $\theta$.
  • Figure 3: The difference between left and right resummation along the singular direction $\theta$ as a sum over Hankel contours.
  • Figure 4: To obtain $\Delta_\omega^+$, we have to consider the determination of $\hat{\phi}$ along the path $\gamma_\omega$, issuing from the origin and reaching $\omega$ by avoiding all the singularities from the right.
  • Figure 5: Numerical solutions with asymptotic behaviours of the form $1/(2x)$ (blue), $5/(2x)$ (red), $9/(2x)$ (green).
  • ...and 2 more figures

Theorems & Definitions (26)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Definition 3
  • Example 1
  • Definition 4
  • Definition 5
  • Example 2
  • Example 3
  • Definition 6
  • ...and 16 more