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Resurgence and Transseries

A divergent perturbative series can retain information about exponentially small contributions. Resurgence studies those relations through Borel singularities, Stokes jumps, and transseries. This chapter develops the mechanism in controlled settings while separating ambiguity cancellation from the additional data needed to define a quantum field theory.

Enter this chapter

Section titled “Enter this chapter”

Begin with Large-Order Growth and the Borel Transform to identify factorial growth and construct its Borel transform.

Helpful background. Review Asymptotic Scales, Remainders, Uniformity, and Optimal Truncation and Stokes Jumps, Saddle Dominance, and Contour Dependence for asymptotic remainders and contour-dependent saddle expansions.

Topics

Section titled “Topics”
  1. Large-Order Growth and the Borel Transform — Transforms factorial coefficient growth into analytic information.
  2. Borel Singularities, Lateral Sums, and Stokes Data — Defines contour-dependent sums around Borel singularities.
  3. Transseries Sectors and Parameters — Includes exponential sectors and their independent boundary data.
  4. Instantons and Large-Order Relations — Relates neighboring saddles to late perturbative coefficients.
  5. Renormalons, OPE Ambiguities, and Transseries — Distinguishes momentum-endpoint growth from instanton effects.
  6. Ambiguity Cancellation and Transseries Consistency — Tests cancellation between ambiguous sectors.
  7. Complex Saddles, Quasi-Zero Modes, and Hidden Phases — Tracks phases from complex and correlated saddles.
  8. Constructive Borel Summability and Its Boundaries — States rigorous summability results with their model assumptions.
  9. Evidence and Limits for Resurgence in QFT — Separates established examples from general QFT conjectures.

How the ideas fit together

Section titled “How the ideas fit together”

Two lateral Borel sums can differ by an exponentially small imaginary contribution even when the desired observable is real. A compatible nonperturbative sector must cancel that ambiguity with the same normalization and contour prescription. The cancellation constrains how sectors fit together, but it need not determine every real transseries parameter. Boundary conditions, the observable, and the integration cycle still matter. Therefore, finding a cancellation is a valuable consistency test rather than a universal recipe for constructing an otherwise unspecified QFT. Mariño 2015, §§ 1.9 and 3.2–3.5.

Where to go next

Section titled “Where to go next”

Continue to Integrable Relativistic QFT to distinguish quantum conserved charges from a classical integrability claim.

References

Section titled “References”
  • Mariño, Marcos. Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory. Cambridge University Press, 2015. DOI.