Saddle Point, Asymptotics, Resurgence & Co.

elwo06
17 mar
Tempo di lettura: 3 min

Decades ago, while investigating finer (non perturbative) effects in quantum physics (instantons and more), I got fashinated by the field.

As I was looking elsewhere, the topic matured, there are even books on the matter nowadays.

I am collecting some of them here.

If you know more or interesting connection, leave a comment. I'll be grateful.

Summary

Broadly speaking, references here are touching on:

Resurgence: Trans-series in QFT, string theory, and integrable systems.
Asymptotics: Hyperasymptotic methods and Stokes phenomena.
Saddle Points: Picard-Lefschetz theory for path integrals and phase transitions.

Resources

With focus on books and reviews/key papers.

I first read about the topic in Quantum Field Theory and Critical Phenomena by Jean Zinn-Justin, immediately captured by resurgence - among the topics.

Resurgence Theory

Key Books & Reviews

Resurgence and Trans-Series in Quantum Theories (2023) By Gerald Dunne and Mithat Ünsal (preprint) A comprehensive review connecting resurgence to quantum mechanics, QFT, and lattice models, emphasizing non-perturbative effects like instantons and renormalons.
Transseries and Resurgence: A New Paradigm in Asymptotics (2023)
By Ovidiu Costin (Cambridge University Press)
Mathematical foundations of trans-series and Écalle’s resurgence theory, with applications to differential equations and quantum mechanics.
Nonperturbative Effects and Resurgence in Quantum Theories (2021)
By Gerald Dunne (Chapter in Handbook of Quantum Field Theory) Focuses on resurgence in gauge theories, including supersymmetric and large-\( N \) systems.

Papers

Resurgence in Quantum Field Theory (2015) By Inês Aniceto, Gökçe Başar, and Ricardo Schiappa (arXiv:1412.3259)
Introduces resurgent trans-series for path integrals and perturbative/non-perturbative relations.
Resurgence and the Conifold (2022)
By Marcos Mariño et al. (arXiv:2204.02989)
Applies resurgence to Calabi-Yau manifolds and topological strings, linking to mirror symmetry.

Asymptotic Expansions

Modern Texts

Hyperasymptotics and Exponential Asymptotics (2023)
By Adrian B. Olde Daalhuis (Cambridge Monographs on Applied Mathematics)
Covers hyperasymptotic methods for extracting exponentially small terms beyond standard asymptotic series.
Asymptotics in Dynamics, Geometry, and PDEs (2021)
Ed. by Bertrand Eynard et al. (Springer)
Explores asymptotics in integrable systems, Painlevé equations, and resurgence.

Applications in Physics

Exponential Asymptotics and Stokes Phenomenon in Hydrodynamics (2022)
By Philippe H. Trinh (J. Fluid Mech.)
Demonstrates hyperasymptotics for fluid instabilities and boundary-layer theory.
Asymptotic Series in String Theory (2020)
By Igor R. Klebanov et al. (arXiv:2008.05268)
Analyzes divergent series in AdS/CFT and matrix models using resurgence.

Saddle Points and Picard-Lefschetz Theory

Books

Instantons and Large N: An Introduction to Non-Perturbative Methods (2022)
By Marcos Mariño (Cambridge University Press)
Covers saddle points, Lefschetz thimbles, and their role in resurgence for path integrals.
Path Integrals and Picard-Lefschetz Theory (2023)
By Aleksandr Cherman and Mithat Ünsal (preprint)
Modern applications to quantum tunneling, sign problems, and lattice QCD.

Key Papers

Lefschetz Thimbles and Quantum Phase Transitions (2021)
By Gerald V. Dunne and Mithat Ünsal (Phys. Rev. D)
Links saddle points to phase transitions in spin systems and gauge theories.
Complex Path Integrals and Resurgence in Quantum Mechanics (2019)
By Ricardo Schiappa and Ricardo Vaz (arXiv:1905.04060)
Combines Picard-Lefschetz theory with resurgent trans-series for double-well potentials.

Interdisciplinary Advances

Resurgence and Holomorphic Floer Theory (2023)
By Tom Bridgeland and Tudor Dimofte (arXiv:2303.10057)
Merges resurgence with symplectic geometry and Floer homology.
Machine Learning and Resurgence (2023)
By Katherine Fraser et al. (arXiv:2304.06984)
Uses neural networks to predict trans-series coefficients and optimize resummation.