A Primer on Resurgent Transseries and Their Asymptotics

Abstract

The computation of observables in general interacting theories, be them quantum mechanical, field, gauge or string theories, is a non-trivial problem which in many cases can only be addressed by resorting to perturbative methods. In most physically interesting problems these perturbative expansions result in asymptotic series with zero radius of convergence. These asymptotic series then require the use of resurgence and transseries in order for the associated observables to become nonperturbatively well-defined. Resurgence encodes the complete large-order asymptotic behaviour of the coefficients from a perturbative expansion, generically in terms of (multi) instanton sectors and for each problem in terms of its Stokes constants. Some observables arise from linear problems, and have a finite number of instanton sectors and associated Stokes constants; some other observables arise from nonlinear problems, and have an infinite number of instanton sectors and Stokes constants. By means of two very explicit examples, and with emphasis on a pedagogical style of presentation, this work aims at serving as a primer on the aforementioned resurgent, large-order asymptotics of general perturbative expansions. This includes discussions of transseries, Stokes phenomena, generalized steepest-descent methods, Borel transforms, nonlinear resonance, and alien calculus. Furthermore, resurgent properties of transseries---usually described mathematically via alien calculus---are recast in equivalent physical languages: either a "statistical mechanical" language, as motions in chains and lattices; or a "conformal field theoretical" language, with underlying Virasoro-like algebraic structures.

192 pages, 76 plots in 41 figures, jheppub-nosort.sty; v2: small changes, corrections & typos, added refs

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Cited by in corpus (135)

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• Supersymmetric phases of 4d N=4 SYM at large N
• Physical Resurgent Extrapolation
• Non-perturbative approaches to the quantum Seiberg-Witten curve
• Hydrodynamic attractors in heavy ion collisions: a review
• The Large D Limit of Einstein's Equations
• The large-$N$ limit of the 4d $\mathcal{N}=1$ superconformal index
• Resurgent Extrapolation: Rebuilding a Function from Asymptotic Data. Painleve I
• Peacock patterns and new integer invariants in topological string theory
• Normalization of D-instanton Amplitudes
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• D-instantons, String Field Theory and Two Dimensional String Theory
• Normalization of Type IIB D-instanton Amplitudes
• Non-Perturbative Completion of Hopf-Algebraic Dyson-Schwinger Equations
• Multi-instantons in minimal string theory and in matrix integrals
• Uniformization and Constructive Analytic Continuation of Taylor Series
• Hydrodynamic attractors in phase space
• Renormalons in integrable field theories
• Nonperturbative effects and resurgence in JT gravity at finite cutoff
• Analytical attractor for Bjorken expansion
• Hydrodynamic Attractors in Ultrarelativistic Nuclear Collisions
• Conjectures about the ground-state energy of the Lieb-Liniger model at weak repulsion
• Resurgence of the large-charge expansion
• Resurgence in the O(4) sigma model
• Exact multi-instantons in topological string theory
• On the Higher Loop Euler-Heisenberg Trans-Series Structure
• New renormalons from analytic trans-series
• On the resurgent structure of quantum periods
• From perturbative to non-perturbative in the O(4) sigma model
• Resonant resurgent asymptotics from quantum field theory
• Painlevé I and exact WKB: Stokes phenomenon for two-parameter transseries
• Backpropagating Hybrid Monte Carlo algorithm for fast Lefschetz thimble calculations
• Testing the Bethe ansatz with large N renormalons
• Exponential Approach to the Hydrodynamic Attractor in Yang-Mills Kinetic Theory
• The resurgent structure of quantum knot invariants
• The full analytic trans-series in integrable field theories
• Towards perturbative renormalization of ${\varphi }^2(i\varphi {)}^{\epsilon }$ quantum field theory
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• Shear flows in far-from-equilibrium strongly coupled fluids
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• Resummation of quantum radiation reaction and induced polarization
• Picard-Lefschetz decomposition and Cheshire Cat resurgence in 3D $\mathcal{N}=2$ field theories
• Effective Theories as Truncated Trans-Series and Scale Separated Compactifications
• Looking through the QCD Conformal Window with Perturbation Theory
• Analytic resurgence in the O(4) model
• Relativistic Hydrodynamics: A Singulant Perspective
• Resurgence and renormalons in the one-dimensional Hubbard model
• Instanton effects vs resurgence in the $O(3)$ sigma model
• Schwinger pair production from Padé-Borel reconstruction
• Effective field theory for dilute Fermi systems at fourth order
• Hydrodynamic Attractor in a Hubble Expansion
• Prescaling relaxation to nonthermal attractors
• 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
• A new renormalon in two dimensions
• Resurgence, Painleve Equations and Conformal Blocks
• Semiclassical instanton theory for reaction rates at any temperature: How a rigorous real-time derivation solves the crossover temperature problem
• Instantons, renormalons and the theta angle in integrable sigma models
• Non-Wilsonian ultraviolet completion via transseries
• Non-perturbative signatures of non-linear Compton scattering
• Resurgence in Lorentzian quantum cosmology: No-boundary saddles and resummation of quantum gravity corrections around tunneling saddle points
• Stokes phenomenon and Hawking Radiation
• Rolling Near the Tachyon Vacuum
• Quantum phase transition and Resurgence: Lessons from 3d $\mathcal{N}=4$ SQED
• Crystals of gauged solitons, force free plasma and resurgence
• Renormalons beyond the Borel plane
• Exact quantization conditions and full transseries structures for $\mathcal{P}\mathcal{T}$ symmetric anharmonic oscillators
• Operator thermalisation in $d>2$: Huygens or resurgence
• The analytic structure of the fixed charge expansion
• Hydrodynamic Attractor in Ultracold Atoms
• Renormalons in quantum mechanics
• Novel features of attractors and transseries in non-conformal Bjorken flows
• Resurgence of the Effective Action in Inhomogeneous Fields
• Generalized Devil's staircase and RG flows
• Running coupling and non-perturbative corrections for O$(N)$ free energy and for disk capacitor
• Statistics of Feynman amplitudes in ${\varphi }^4$-theory
• Transseries for causal diffusive systems
• Three roads to the energy gap
• Summing up perturbation series around superintegrable point
• Resurgence and self-completion in renormalized gauge theories
• The QCD Adler function and the muon $g-2$ anomaly from renormalons
• Exact instanton transseries for quantum mechanics
• Resurgence and semiclassical expansion in two-dimensional large-$N$ sigma models
• Schwinger pair production in spacetime fields: Moiré patterns, Aharonov-Bohm phases and Sturm-Liouville eigenvalues
• Exploring superconformal Yang-Mills theories through matrix Bessel kernels
• Asymptotics in an Asymptotic CFT
• Resurgent representation of the Adler function in the large-${\beta }_0$ approximation of QCD
• Reduction of Order and Transseries Structure of Radiation Reaction
• Deformation of superintegrability in the Miwa-deformed Gaussian matrix model
• Confinement as Analytic Continuation Beyond Infinity
• Resurgence in 2-dimensional Yang-Mills and a genus altering deformation
• Large N instantons from topological strings
• 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
• Introductory Lectures on Resurgence: CERN Summer School 2024
• On the Riemann-Hilbert approach to asymptotics of tronquée solutions of Painlevé I
• All-order Resurgence from Complexified Path Integral in a Quantum Mechanical System with Integrability
• The late to early time behaviour of an expanding plasma: hydrodynamisation from exponential asymptotics
• Constrained extrapolation problem and order-dependent mappings
• Resurgence for the non-conformal Bjorken flow with Fermi-Dirac and Bose-Einstein statistics
• Convergence of Ginzburg-Landau expansions: superconductivity in the Bardeen-Cooper-Schrieffer theory and chiral symmetry breaking in the Nambu-Jona-Lasinio model
• Capturing the cascade: a transseries approach to delayed bifurcations
• Schwinger-Keldysh non-perturbative field theory of open quantum systems beyond the Markovian regime: Application to spin-boson and spin-chain-boson models
• Two-point functions at arbitrary genus and its resurgence structure in a matrix model for 2D type IIA superstrings
• Lefschetz Thimbles and Quantum Phases in Zero-Dimensional Bosonic Models
• Resurgence and $1/N$ Expansion in Integrable Field Theories
• Trans-series from condensates
• Condensates, crystals, and renormalons in the Gross-Neveu model at finite density
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• Averaging method in combinatorics of symmetric polynomials
• Hydrodynamic attractor in periodically driven ultracold quantum gases
• Resurgence in complex Chern-Simons theory at generic levels
• Les Houches lecture notes on moduli spaces of Riemann surfaces
• The factorial growth of topological recursion
• Instantons in ${\varphi }^4$ Theories: Transseries, Virial Theorems and Numerical Aspects
• A saddle-point finder and its application to the spin foam model
• Asymptotics of eigenvalue sums when some turning points are complex
• Gluon Mass Generation from Renormalons and Resurgence
• Recursive Generation of The Semi-Classical Expansion in Arbitrary Dimension
• Resurgence in the Scalar Quantum Electrodynamics Euler-Heisenberg Lagrangian
• Perturbation theory without power series: iterative construction of non-analytic operator spectra
• Smoothing of the higher-order Stokes phenomenon
• Uncovering Hidden Patterns: Approximate Resurgent Resummation from Truncated Series
• Asymptotics of Weil-Petersson volumes and two-dimensional quantum gravities
• Large Orders and Strong-Coupling Limit in Functional Renormalization
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• Instantons and transseries of the Mathieu potential deformed by a $\mathcal{P}\mathcal{T}$-symmetry parameter
• From Divergent Series to Geometry: Resurgence of the Quantum Metric
• Strong coupling expansion of free energy and BPS Wilson loop in $\mathcal{N}=2$ superconformal models with fundamental hypermultiplets
• Matrix models for extremal and integrated correlators of higher rank
• Torsional Hall Viscosity of Massive Chern Insulators: Magnetic Field and Momentum Deformations