Resurgence and the Nekrasov-Shatashvili Limit: Connecting Weak and Strong Coupling in the Mathieu and Lam'e Systems

Abstract

The Nekrasov-Shatashvili limit for the low-energy behavior of N=2 and N=2* supersymmetric SU(2) gauge theories is encoded in the spectrum of the Mathieu and Lam'e equations, respectively. This correspondence is usually expressed via an all-orders Bohr-Sommerfeld relation, but this neglects non-perturbative effects, the nature of which is very different in the electric, magnetic and dyonic regions. In the gauge theory dyonic region the spectral expansions are divergent, and indeed are not Borel-summable, so they are more properly described by resurgent trans-series in which perturbative and non-perturbative effects are deeply entwined. In the gauge theory electric region the spectral expansions are convergent, but nevertheless there are non-perturbative effects due to poles in the expansion coefficients, and which we associate with worldline instantons. This provides a concrete analog of a phenomenon found recently by Drukker, Marino and Putrov in the large N expansion of the ABJM matrix model, in which non-perturbative effects are related to complex space-time instantons. In this paper we study how these very different regimes arise from an exact WKB analysis, and join smoothly through the magnetic region. This approach also leads to a simple proof of a resurgence relation found recently by Dunne and Unsal, showing that for these spectral systems all non-perturbative effects are subtly encoded in perturbation theory, and identifies this with the Picard-Fuchs equation for the quantized elliptic curve.

43 pages, 6 figs

References in corpus (15)

• Worldline Instantons II: The Fluctuation Prefactor
• The Stokes Phenomenon and Schwinger Vacuum Pair Production in Time-Dependent Laser Pulses
• Asymptotically free N=2 theories and irregular conformal blocks
• Deforming SW curve
• Non-Perturbative Quantum Geometry III
• RG-Whitham dynamics and complex Hamiltonian systems
• Nonperturbative Effects and the Large-Order Behavior of Matrix Models and Topological Strings
• Modular anomaly equations in N=2* theories and their large-N limit
• Classical irregular block, N=2 pure gauge theory and Mathieu equation
• Resurgence and Holomorphy: From Weak to Strong Coupling
• Non-Perturbative Quantum Geometry II
• Quantum geometry from the toroidal block
• The spectral problem of the ABJ Fermi gas
• Elliptic Faulhaber polynomials and Lamé densities of states
• Resurgent Analysis of Localizable Observables in Supersymmetric Gauge Theories

Cited by in corpus (78)

• A Primer on Resurgent Transseries and Their Asymptotics
• A Monte Carlo algorithm for simulating fermions on Lefschetz thimbles
• Matrix models from operators and topological strings, 2
• Resurgence in Extended Hydrodynamics
• Matrix models from operators and topological strings
• TBA equations and resurgent Quantum Mechanics
• Cheshire Cat resurgence, Self-resurgence and Quasi-Exact Solvable Systems
• Deconstructing zero: resurgence, supersymmetry and complex saddles
• Physical Resurgent Extrapolation
• Non-perturbative approaches to the quantum Seiberg-Witten curve
• Quantum Geometry of Resurgent Perturbative/Nonperturbative Relations
• Exact quantization conditions for the relativistic Toda lattice
• Integrability and cycles of deformed $\mathcal{N}=2$ gauge theory
• Complex Saddles in Two-dimensional Gauge Theory
• Holomorphic Anomaly and Quantum Mechanics
• Resurgence in $\eta$-deformed Principal Chiral Models
• Non-Perturbative Quantum Geometry III
• RG-Whitham dynamics and complex Hamiltonian systems
• A slow review of the AGT correspondence
• The Resurgence of the Cusp Anomalous Dimension
• Remark on the Dunne-Unsal relation in exact semi-classics
• Exact WKB Analysis of N = 2 Gauge Theories
• Analytic (3+1)-dimensional gauged Skyrmions, Heun and Whittaker-Hill equations and resurgence
• Exact Quantization Conditions, Toric Calabi-Yau and Nonperturbative Topological String
• Splitting of surface defect partition functions and integrable systems
• Transmutation of a Trans-series: The Gross-Witten-Wadia Phase Transition
• TBA equations for the Schrödinger equation with a regular singularity
• Gradient flows without blow-up for Lefschetz thimbles
• Accessory parameters in confluent Heun equations and classical irregular conformal blocks
• Exact WKB and the quantum Seiberg-Witten curve for 4d $N=2$ pure $SU(3)$ Yang-Mills, Part I: Abelianization
• Bands and gaps in Nekrasov partition function
• Comments on Exact Quantization Conditions and Non-Perturbative Topological Strings
• Superadiabatic basis in cosmological particle production: application to preheating
• Quantum Seiberg-Witten curve and Universality in Argyres-Douglas theories
• Quantum periods and prepotential in $\mathcal{N}=2$ SU(2) SQCD
• Bethe/Gauge correspondence in odd dimension: modular double, non-perturbative corrections and open topological strings
• Quantum spectral problems and isomonodromic deformations
• Classical limit of irregular blocks and Mathieu functions
• Exact WKB analysis and TBA equations for the Mathieu equation
• Non-perturbative studies of N=2 conformal quiver gauge theories
• Resurgence in the Bi-Yang-Baxter Model
• Quantum periods for $\mathcal{N}=2$ $SU(2)$ SQCD around the superconformal point
• On the large $\Omega$-deformations in the Nekrasov-Shatashvili limit of $\mathcal{N}=2^{\ast }$ SYM
• Pure N=2 Super Yang-Mills and Exact WKB
• Resurgence, Painleve Equations and Conformal Blocks
• Interpolating function and Stokes Phenomena
• Toward the Pole
• Quantum Seiberg-Witten periods for $\mathcal{N}=2$ $SU(N_c)$ SQCD around the superconformal point
• Irregular conformal blocks, Painlevé III and the blow-up equations
• Classical irregular blocks, Hill's equation and PT-symmetric periodic complex potentials
• TBA equations and exact WKB analysis in deformed supersymmetric quantum mechanics
• Crystals of gauged solitons, force free plasma and resurgence
• Quantum periods and TBA equations for $\mathcal{N}=2SU(2)N_f=2$ SQCD with flavor symmetry
• Resurgent transseries $\&$ Dyson-Schwinger equations
• Quantum tunnelling from vacuum in multidimensions
• Solving Heun's equation using conformal blocks
• Exact quantization conditions for the elliptic Ruijsenaars-Schneider model
• Analytic Baby Skyrmions at Finite Density
• Exact instanton transseries for quantum mechanics
• Introductory Lectures on Resurgence: CERN Summer School 2024
• WKB analysis of the linear problem for modified affine Toda field equations
• Unified genus-1 potential and a parametric perturbative/nonperturbative relation
• Resurgence and Dynamics of O(N) and Grassmannian Sigma Models
• Exact Chern-Simons / Topological String duality
• Capturing the cascade: a transseries approach to delayed bifurcations
• Orbital Bifurcations and Shoaling of Cnoidal Waves
• Exact WKB in all sectors I: Potentials with degenerate saddles
• Resurgence of deformed genus-1 curves: A novel perturbative/nonperturbative relation
• TBA equations and quantum periods for D-type Argyres-Douglas theories
• Quantum Hamilton-Jacobi Theory, Spectral Path Integrals and Exact-WKB Analysis
• Mellin-Barnes Representation of the Topological String
• Nonperturbative Formulation of Resonances in Quantum Mechanics Based on Exact WKB Method
• A duality of the classical action yields a reflection symmetry of the quantum energy spectrum
• Solvable spectral problems from 2d CFT and N=2 gauge theories
• Exact-WKB Analysis of Two-level Floquet Systems
• Nonperturbative features in the Lie-algebraic Kähler sigma model with fermions
• Exact WKB in all sectors II: Potentials with non-degenerate saddles
• Pair Production in Real Proper Time and Unitarity Without Borel Ambiguity