An Introduction to Resurgence, Trans-Series and Alien Calculus

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.

arXiv comment48 pages, 7 figures; v2 typos fixed, references added, minor corrections

References in corpus (26)

• Renormalons
• High density QCD on a Lefschetz thimble?
• Analytic Continuation of Liouville Theory
• Lectures on non-perturbative effects in large N gauge theories, matrix models and strings
• Resurgence and Trans-series in Quantum Field Theory: The CP(N-1) Model
• Open string amplitudes and large order behavior in topological string theory
• Hydrodynamic gradient expansion in gauge theory plasmas
• The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects
• Uniform WKB, Multi-instantons, and Resurgent Trans-Series
• Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c=1 Matrix Models
• Generating Non-perturbative Physics from Perturbation Theory
• A semiclassical realization of infrared renormalons
• Nonperturbative Ambiguities and the Reality of Resurgent Transseries
• Floer homology and the heat flow
• Resurgence in QFT: Unitons, Fractons and Renormalons in the Principal Chiral Model
• Resurgence theory, ghost-instantons, and analytic continuation of path integrals
• Asymptotics of the instantons of Painleve I
• Real-time Feynman path integral with Picard--Lefschetz theory and its applications to quantum tunneling
• The Resurgence of Instantons: Multi-Cut Stokes Phases and the Painleve II Equation
• Instantons in Quantum Mechanics and Resurgent Expansions
• Introduction to 1-summability and resurgence
• A New Look At The Path Integral Of Quantum Mechanics
• Resurgence and Holomorphy: From Weak to Strong Coupling
• Real-Time Feynman Path Integral Realization of Instantons
• Resonances in Extreme Mass-Ratio Inspirals: Asymptotic and Hyperasymptotic Analysis
• Resurgent Analysis of Localizable Observables in Supersymmetric Gauge Theories

Cited by in corpus (116)

• Advances in QED with intense background fields
• Hydrodynamics Beyond the Gradient Expansion: Resurgence and Resummation
• Minimally subtracted six loop renormalization of $O(n)$-symmetric ${\varphi }^4$ theory and critical exponents
• QCD thermalization: Ab initio approaches and interdisciplinary connections
• Resurgence and the Nekrasov-Shatashvili Limit: Connecting Weak and Strong Coupling in the Mathieu and Lam'e Systems
• RG Flows and Bifurcations
• Lefschetz-thimble analysis of the sign problem in one-site fermion model
• Structure of Lefschetz thimbles in simple fermionic systems
• Resurgence in Extended Hydrodynamics
• Hydrodynamics, resurgence and trans-asymptotics
• Exact properties of an integrated correlator in $\mathcal{N}=4$ $SU(N)$ SYM
• Cheshire Cat resurgence, Self-resurgence and Quasi-Exact Solvable Systems
• Hydrodynamic attractors in heavy ion collisions: a review
• Resurgence in complex Chern-Simons theory
• The Power of Perturbation Theory
• Evading the sign problem in the mean-field approximation through Lefschetz-thimble path integral
• Harnessing S-Duality in $\mathcal{N}=4$ SYM & Supergravity as $SL(2,\mathbb{Z})$-Averaged Strings
• Lefschetz-thimble techniques for path integral of zero-dimensional $O(n)$ sigma models
• Non-perturbative rheological behavior of a far-from-equilibrium expanding plasma
• Complex saddle points and the sign problem in complex Langevin simulation
• Resurgence in $\eta$-deformed Principal Chiral Models
• Spacetime foam: a review
• Remark on the Dunne-Unsal relation in exact semi-classics
• Nonperturbative effects and resurgence in JT gravity at finite cutoff
• Multi-flavor massless QED${}_2$ at finite densities via Lefschetz thimbles
• Resurgence of the large-charge expansion
• Mathematical Methods of Theoretical Physics
• Resurgence and Holomorphy: From Weak to Strong Coupling
• Resurgence in the O(4) sigma model
• Modular-invariant large-$N$ completion of an integrated correlator in $\mathcal{N}=4$ supersymmetric Yang-Mills theory
• Resurgence of the Renormalization Group Equation
• Conformal mapping of the Borel plane: going beyond perturbative QCD
• On Information Loss in AdS${}_3$/CFT${}_2$
• From perturbative to non-perturbative in the O(4) sigma model
• Exact-WKB analysis for SUSY and quantum deformed potentials: Quantum mechanics with Grassmann fields and Wess-Zumino terms
• Resonant resurgent asymptotics from quantum field theory
• Painlevé I and exact WKB: Stokes phenomenon for two-parameter transseries
• Gravitational catalysis of merons in Einstein-Yang-Mills theory
• The full analytic trans-series in integrable field theories
• Resurgence, Conformal Blocks, and the Sum over Geometries in Quantum Gravity
• Non-local Lagrangians from Renormalons and Analyzable Functions
• Analytic resurgence in the O(4) model
• Effective Theories as Truncated Trans-Series and Scale Separated Compactifications
• Picard-Lefschetz decomposition and Cheshire Cat resurgence in 3D $\mathcal{N}=2$ field theories
• Quantum oscillations in the zeroth Landau Level and the serpentine Landau fan
• An analytic evaluation of gravitational particle production of fermions via Stokes phenomenon
• Instanton effects vs resurgence in the $O(3)$ sigma model
• Existence and Uniqueness of Exact WKB Solutions for Second-Order Singularly Perturbed Linear ODEs
• Relative velocities in bidisperse turbulent suspensions
• Effective field theory for dilute Fermi systems at fourth order
• Hydrodynamic Attractor in a Hubble Expansion
• Prescaling relaxation to nonthermal attractors
• Semiclassical Trans-Series from the Perturbative Hopf-Algebraic Dyson-Schwinger Equations: ${\varphi }^3$ QFT in 6 Dimensions
• Resurgence in the Bi-Yang-Baxter Model
• Exact WKB analysis for $\mathcal{P}\mathcal{T}$ symmetric quantum mechanics: Study of the Ai-Bender-Sarkar conjecture
• Resurgence of the QCD Adler function
• Resurgence of a de Sitter Glauber-Sudarshan State: Nodal Diagrams and Borel Resummation
• To the cusp and back: Resurgent analysis for modular graph functions
• Resurgence in Lorentzian quantum cosmology: No-boundary saddles and resummation of quantum gravity corrections around tunneling saddle points
• Perturbative/nonperturbative aspects of Bloch electrons in a honeycomb lattice
• Quantum phase transition and Resurgence: Lessons from 3d $\mathcal{N}=4$ SQED
• $1/\epsilon$ problem in resurgence
• Crystals of gauged solitons, force free plasma and resurgence
• Novel Screening with Two Bodies: Summing the ladder in disformal scalar-tensor theories
• The analytic structure of the fixed charge expansion
• Resurgent transseries $\&$ Dyson-Schwinger equations
• Exact quantization conditions and full transseries structures for $\mathcal{P}\mathcal{T}$ symmetric anharmonic oscillators
• Operator thermalisation in $d>2$: Huygens or resurgence
• Novel features of attractors and transseries in non-conformal Bjorken flows
• Resurgence of the Effective Action in Inhomogeneous Fields
• Symmetry improvement of 3PI effective actions for O(N) scalar field theory
• Transseries for causal diffusive systems
• Running coupling and non-perturbative corrections for O$(N)$ free energy and for disk capacitor
• A consistent quantum field theory from dimensional reduction
• Two-particle irreducible effective actions versus resummation: analytic properties and self-consistency
• Baryogenesis with the Berry phase
• On the exponentially small corrections to $\mathcal{N}=2$ superconformal correlators at large R-charge
• Conformal mappings in perturbative QCD
• Resurgence and self-completion in renormalized gauge theories
• The QCD Adler function and the muon $g-2$ anomaly from renormalons
• Reduction of Order and Transseries Structure of Radiation Reaction
• Exact instanton transseries for quantum mechanics
• Resurgence and semiclassical expansion in two-dimensional large-$N$ sigma models
• What if string theory has a de Sitter excited state?
• Exploring superconformal Yang-Mills theories through matrix Bessel kernels
• Gravitating Meron-like topological solitons in massive Yang-Mills theory and the Einstein-Skyrme model
• Black Hole Entropy, Quantum Corrections and EFT Transitions
• Resurgent representation of the Adler function in the large-${\beta }_0$ approximation of QCD
• Introductory Lectures on Resurgence: CERN Summer School 2024
• The Hierarchy Problem: From the Fundamentals to the Frontiers
• All-order Resurgence from Complexified Path Integral in a Quantum Mechanical System with Integrability
• The small-$N$ series in the zero-dimensional $O(N)$ model: constructive expansions and transseries
• Convergent series for lattice models with polynomial interactions
• Unified genus-1 potential and a parametric perturbative/nonperturbative relation
• Lefschetz Thimbles and Quantum Phases in Zero-Dimensional Bosonic Models
• Dynamical de Sitter black holes in a quasi-stationary expansion
• Cosmological Weyl-Einsteinian-Cubic Gravity as a Gauge Theory of Gravity
• Constrained extrapolation problem and order-dependent mappings
• Resurgence for the non-conformal Bjorken flow with Fermi-Dirac and Bose-Einstein statistics
• Two-point functions at arbitrary genus and its resurgence structure in a matrix model for 2D type IIA superstrings
• Perturbative study of large $N$ principal chiral model with twisted reduction
• Resurgence in complex Chern-Simons theory at generic levels
• Orientation Reversal and the Chern-Simons Natural Boundary
• TBA equations and quantum periods for D-type Argyres-Douglas theories
• A Computational Companion to Transient de Sitter and Quasi de Sitter States in SO(32) and E_8 X E_8 Heterotic String Theories I: Formalisms
• Resurgence in the Scalar Quantum Electrodynamics Euler-Heisenberg Lagrangian
• Instantons in ${\varphi }^4$ Theories: Transseries, Virial Theorems and Numerical Aspects
• Gluon Mass Generation from Renormalons and Resurgence
• Feynman integrals at large loop order and the $\log$-$\Gamma$ distribution
• Nonperturbative features in the Lie-algebraic Kähler sigma model with fermions
• Convexity, large charge and the large-N phase diagram of the ${\phi }^4$ theory
• Resolving Degeneracies in Complex $\mathbb{R}\times S^3$ and $\theta$-KSW
• $c_{\mathrm{eff}}$ from Resurgence at the Stokes Line
• Large Orders and Strong-Coupling Limit in Functional Renormalization
• From Divergent Series to Geometry: Resurgence of the Quantum Metric
• Uncovering Hidden Patterns: Approximate Resurgent Resummation from Truncated Series