Exact WKB and the quantum Seiberg-Witten curve for 4d $N=2$ pure $SU(3)$ Yang-Mills, Part I: Abelianization

Abstract

We investigate the exact WKB method for the quantum Seiberg-Witten curve of 4d $N=2$ pure $SU(3)$ Yang-Mills, in the language of abelianization. The relevant differential equation is a third-order equation on ${\mathbb{CP}}^1$ with two irregular singularities. We employ the exact WKB method to study solutions to such a third-order equation and the associated Stokes phenomena. We also investigate the exact quantization condition for a certain spectral problem. Moreover, exact WKB analysis leads us to consider new Darboux coordinates on a moduli space of flat SL(3,$\mathbb{C}$)-connections. In particular, in the weak coupling region we encounter coordinates of higher length-twist type generalizing Fenchel-Nielsen coordinates. The Darboux coordinates are conjectured to admit asymptotic expansions given by the formal quantum periods series; we perform numerical analysis supporting this conjecture.

arXiv comment28 pages, 5 figures, 6 tables

References in corpus (70)

• Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory
• Monopoles, Duality and Chiral Symmetry Breaking in N=2 Supersymmetric QCD
• Liouville Correlation Functions from Four-dimensional Gauge Theories
• N=2 dualities
• Geometric Engineering of Quantum Field Theories
• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• Simple Singularities and N=2 Supersymmetric Yang-Mills Theory
• Four-dimensional wall-crossing via three-dimensional field theory
• The Vacuum Structure and Spectrum of N=2 Supersymmetric SU(N) Gauge Theory
• Loop and surface operators in N=2 gauge theory and Liouville modular geometry
• Supersymmetric vacua and Bethe ansatz
• Instantons and recursion relations in N=2 Susy gauge theory
• Nekrasov Functions and Exact Bohr-Sommerfeld Integrals
• Quantization of Integrable Systems and Four Dimensional Gauge Theories
• Quantum integrability and supersymmetric vacua
• The Omega Deformation, Branes, Integrability, and Liouville Theory
• On AGT relation in the case of U(3)
• A Primer on Resurgent Transseries and Their Asymptotics
• Darboux coordinates, Yang-Yang functional, and gauge theory
• Nonperturbative Effective Actions of N=2 Supersymmetric Gauge Theories
• Spectral networks
• Quantum Geometry of Refined Topological Strings
• Multi-Instantons and Exact Results I: Conjectures, WKB Expansions, and Instanton Interactions
• On non-conformal limit of the AGT relations
• Uniform WKB, Multi-instantons, and Resurgent Trans-Series
• BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters
• Exact solution of Kerr black hole perturbations via CFT${}_2$ and instanton counting. Greybody factor, Quasinormal modes and Love numbers
• Defects and Quantum Seiberg-Witten Geometry
• Generating Non-perturbative Physics from Perturbation Theory
• Deformed Prepotential, Quantum Integrable System and Liouville Field Theory
• Resurgence theory, ghost-instantons, and analytic continuation of path integrals
• Nekrasov Functions from Exact BS Periods: the Case of SU(N)
• Multi-Instantons and Exact Results II: Specific Cases, Higher-Order Effects, and Numerical Calculations
• Matone's Relation in the Presence of Gravitational Couplings
• Resurgence and the Nekrasov-Shatashvili Limit: Connecting Weak and Strong Coupling in the Mathieu and Lam'e Systems
• Deforming SW curve
• Multi-Instantons and Exact Results III: Unified Description of the Resonances of Even and Odd Anharmonic Oscillators
• QNMs of branes, BHs and fuzzballs from Quantum SW geometries
• TBA equations and resurgent Quantum Mechanics
• Toward Picard-Lefschetz Theory of Path Integrals, Complex Saddles and Resurgence
• Quantum Geometry of Resurgent Perturbative/Nonperturbative Relations
• Non-perturbative approaches to the quantum Seiberg-Witten curve
• Wild Wall Crossing and BPS Giants
• Resurgence in sine-Gordon quantum mechanics: Exact agreement between multi-instantons and uniform WKB
• Opers, surface defects, and Yang-Yang functional
• Nekrasov prepotential with fundamental matter from the quantum spin chain
• Exact Resurgent Trans-series and Multi-Bion Contributions to All Orders
• On Gauge Theory and Topological String in Nekrasov-Shatashvili Limit
• Holomorphic Anomaly and Quantum Mechanics
• On exact-WKB analysis, resurgent structure, and quantization conditions
• Wilson Loops and Chiral Correlators on Squashed Sphere
• Exact WKB and abelianization for the $T_3$ equation
• Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on $S^1$
• Exact WKB methods in $SU(2)$ $N_f=1$
• Exact WKB Analysis of N = 2 Gauge Theories
• Intersecting defects in gauge theory, quantum spin chains, and Knizhnik-Zamolodchikov equations
• Riemann-Hilbert correspondence and blown up surface defects
• Mathieu equation and Elliptic curve
• ODE/IM correspondence and the Argyres-Douglas theory
• Quantum Spin Systems and Supersymmetric Gauge Theories, I
• TBA equations for the Schrödinger equation with a regular singularity
• Voros symbols as cluster coordinates
• Virtual turning points and bifurcation of Stokes curves for higher order ordinary differential equations
• Quantum periods and prepotential in $\mathcal{N}=2$ SU(2) SQCD
• On the Quantization of Seiberg-Witten Geometry
• ODE/IM correspondence for affine Lie algebras: A numerical approach
• $T$, $Q$ and periods in $SU(3)$ $\mathcal{N}=2$ SYM
• $1/\epsilon$ problem in resurgence
• Quantum periods and TBA equations for $\mathcal{N}=2SU(2)N_f=2$ SQCD with flavor symmetry
• Abelianisation of Logarithmic ${\mathfrak{sl}}_2$-Connections

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