WKB analysis of the linear problem for modified affine Toda field equations

Abstract

We study the WKB analysis of the solutions to the linear problem for a modified affine Toda field equation, which is equivalent to the higher-order ordinary differential equation (ODE) studied in the ODE/IM correspondence. After gauge transformation, we diagonalize the flat connection of the linear problem to reduce the latter to a set of independent first-order linear differential equations. We explicitly perform this procedure for classical affine Lie algebras with lower ranks. In particular, we study the WKB solutions of the $D_r^{(1)}$- and $D_{r+1}^{(2)}$-type linear problems, which correspond to the higher-order ODEs with the pseudo-differential operator. The diagonalized connection is obtained from the Riccati equations of the adjoint linear problem and related to the conserved currents of the integrable hierarchy constructed by Drinfeld and Sokolov up to total derivatives.

arXiv comment42 pages

References in corpus (36)

• Supersymmetric Yang-Mills Systems And Integrable Systems
• Integrability and Seiberg-Witten Exact Solution
• Integrable Structure of Conformal Field Theory, Quantum KdV Theory and Thermodynamic Bethe Ansatz
• Integrable systems and supersymmetric gauge theory
• Nekrasov Functions and Exact Bohr-Sommerfeld Integrals
• The ODE/IM Correspondence
• Thermodynamic Bubble Ansatz
• Y-system for Scattering Amplitudes
• T-systems and Y-systems in integrable systems
• Integrable structure of W_3 Conformal Field Theory, Quantum Boussinesq Theory and Boundary Affine Toda Theory
• Resurgence and the Nekrasov-Shatashvili Limit: Connecting Weak and Strong Coupling in the Mathieu and Lam'e Systems
• TBA equations and resurgent Quantum Mechanics
• Pseudo-differential equations, and the Bethe Ansatz for the classical Lie algebras
• Quantum Sine(h)-Gordon Model and Classical Integrable Equations
• Non-perturbative approaches to the quantum Seiberg-Witten curve
• Differential equations for general SU(n) Bethe ansatz systems
• Bethe Ansatz and the Spectral Theory of affine Lie algebra-valued connections I. The simply-laced case
• Integrability and cycles of deformed $\mathcal{N}=2$ gauge theory
• Functional Relations in Stokes Multipliers and Solvable Models related to U_q(A^{(1)}_n)
• Thermodynamic Bethe Ansatz Equations for Minimal Surfaces in AdS_3
• Exact WKB and abelianization for the $T_3$ equation
• ODE/IM correspondence and the Argyres-Douglas theory
• ODE/IM correspondence and modified affine Toda field equations
• The Bethe Ansatz and the Tzitzéica-Bullough-Dodd equation
• Bethe Ansatz equations for the classical $A_n^{(1)}$ affine Toda field theories
• Exact WKB and the quantum Seiberg-Witten curve for 4d $N=2$ pure $SU(3)$ Yang-Mills, Part I: Abelianization
• Opers for higher states of quantum KdV models
• Quantum Seiberg-Witten curve and Universality in Argyres-Douglas theories
• Wall-crossing of TBA equations and WKB periods for the third order ODE
• Deforming the ODE/IM correspondence with $\mathrm{T}\bar{\mathrm{T}}$
• Quantum periods for $\mathcal{N}=2$ $SU(2)$ SQCD around the superconformal point
• ODE/IM correspondence for affine Lie algebras: A numerical approach
• $T$, $Q$ and periods in $SU(3)$ $\mathcal{N}=2$ SYM
• Quantum Seiberg-Witten periods for $\mathcal{N}=2$ $SU(N_c)$ SQCD around the superconformal point
• Massive ODE/IM Correspondence and Non-linear Integral Equations for $A_r^{(1)}$-type modified Affine Toda Field Equations
• ODE/IM correspondence and supersymmetric affine Toda field equations

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