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.

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

References in corpus (7)

• Open string amplitudes and large order behavior in topological string theory
• Real-time Feynman path integral with Picard--Lefschetz theory and its applications to quantum tunneling
• 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
• Resurgent Analysis of Localizable Observables in Supersymmetric Gauge Theories

Cited by in corpus (115)

• 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
• Evading the sign problem in the mean-field approximation through Lefschetz-thimble path integral
• The Power of Perturbation Theory
• 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
• Resurgence in the O(4) sigma model
• Resurgence and Holomorphy: From Weak to Strong Coupling
• Modular-invariant large-$N$ completion of an integrated correlator in $\mathcal{N}=4$ supersymmetric Yang-Mills theory
• Resurgence of the Renormalization Group Equation
• On Information Loss in AdS${}_3$/CFT${}_2$
• Conformal mapping of the Borel plane: going beyond perturbative QCD
• From perturbative to non-perturbative in the O(4) sigma model
• Painlevé I and exact WKB: Stokes phenomenon for two-parameter transseries
• 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
• Gravitational catalysis of merons in Einstein-Yang-Mills theory
• Resurgence, Conformal Blocks, and the Sum over Geometries in Quantum Gravity
• The full analytic trans-series in integrable field theories
• Non-local Lagrangians from Renormalons and Analyzable Functions
• 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
• Analytic resurgence in the O(4) model
• Instanton effects vs resurgence in the $O(3)$ sigma model
• An analytic evaluation of gravitational particle production of fermions via Stokes phenomenon
• Hydrodynamic Attractor in a Hubble Expansion
• Existence and Uniqueness of Exact WKB Solutions for Second-Order Singularly Perturbed Linear ODEs
• Effective field theory for dilute Fermi systems at fourth order
• Relative velocities in bidisperse turbulent suspensions
• Prescaling relaxation to nonthermal attractors
• Resurgence in the Bi-Yang-Baxter Model
• Semiclassical Trans-Series from the Perturbative Hopf-Algebraic Dyson-Schwinger Equations: ${\varphi }^3$ QFT in 6 Dimensions
• Resurgence of the QCD Adler function
• Exact WKB analysis for $\mathcal{P}\mathcal{T}$ symmetric quantum mechanics: Study of the Ai-Bender-Sarkar conjecture
• 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
• Quantum phase transition and Resurgence: Lessons from 3d $\mathcal{N}=4$ SQED
• Perturbative/nonperturbative aspects of Bloch electrons in a honeycomb lattice
• $1/\epsilon$ 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
• Exact quantization conditions and full transseries structures for $\mathcal{P}\mathcal{T}$ symmetric anharmonic oscillators
• Resurgence of the Effective Action in Inhomogeneous Fields
• Novel features of attractors and transseries in non-conformal Bjorken flows
• Operator thermalisation in $d>2$: Huygens or resurgence
• Resurgent transseries $\&$ Dyson-Schwinger equations
• Running coupling and non-perturbative corrections for O$(N)$ free energy and for disk capacitor
• Symmetry improvement of 3PI effective actions for O(N) scalar field theory
• Transseries for causal diffusive systems
• A consistent quantum field theory from dimensional reduction
• Conformal mappings in perturbative QCD
• Baryogenesis with the Berry phase
• On the exponentially small corrections to $\mathcal{N}=2$ superconformal correlators at large R-charge
• Two-particle irreducible effective actions versus resummation: analytic properties and self-consistency
• Resurgence and self-completion in renormalized gauge theories
• What if string theory has a de Sitter excited state?
• Resurgent representation of the Adler function in the large-${\beta }_0$ approximation of QCD
• Exact instanton transseries for quantum mechanics
• Exploring superconformal Yang-Mills theories through matrix Bessel kernels
• Reduction of Order and Transseries Structure of Radiation Reaction
• Gravitating Meron-like topological solitons in massive Yang-Mills theory and the Einstein-Skyrme model
• The QCD Adler function and the muon $g-2$ anomaly from renormalons
• Black Hole Entropy, Quantum Corrections and EFT Transitions
• Resurgence and semiclassical expansion in two-dimensional large-$N$ sigma models
• Convergent series for lattice models with polynomial interactions
• 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
• Unified genus-1 potential and a parametric perturbative/nonperturbative relation
• The Hierarchy Problem: From the Fundamentals to the Frontiers
• Introductory Lectures on Resurgence: CERN Summer School 2024
• Resurgence for the non-conformal Bjorken flow with Fermi-Dirac and Bose-Einstein statistics
• Dynamical de Sitter black holes in a quasi-stationary expansion
• Cosmological Weyl-Einsteinian-Cubic Gravity as a Gauge Theory of Gravity
• Lefschetz Thimbles and Quantum Phases in Zero-Dimensional Bosonic Models
• Constrained extrapolation problem and order-dependent mappings
• Two-point functions at arbitrary genus and its resurgence structure in a matrix model for 2D type IIA superstrings
• Resurgence in complex Chern-Simons theory at generic levels
• Perturbative study of large $N$ principal chiral model with twisted reduction
• TBA equations and quantum periods for D-type Argyres-Douglas theories
• Orientation Reversal and the Chern-Simons Natural Boundary
• Resurgence in the Scalar Quantum Electrodynamics Euler-Heisenberg Lagrangian
• Feynman integrals at large loop order and the $\log$-$\Gamma$ distribution
• 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
• Gluon Mass Generation from Renormalons and Resurgence
• Instantons in ${\varphi }^4$ Theories: Transseries, Virial Theorems and Numerical Aspects
• Large Orders and Strong-Coupling Limit in Functional Renormalization
• Uncovering Hidden Patterns: Approximate Resurgent Resummation from Truncated Series
• Resolving Degeneracies in Complex $\mathbb{R}\times S^3$ and $\theta$-KSW
• Nonperturbative features in the Lie-algebraic Kähler sigma model with fermions
• From Divergent Series to Geometry: Resurgence of the Quantum Metric
• Convexity, large charge and the large-N phase diagram of the ${\phi }^4$ theory
• $c_{\mathrm{eff}}$ from Resurgence at the Stokes Line