ODE/IM correspondence and Bethe ansatz for affine Toda field equations

Abstract

We study the linear problem associated with modified affine Toda field equation for the Langlands dual ${\hat{\mathfrak{g}}}^{\vee }$, where $\hat{\mathfrak{g}}$ is an untwisted affine Lie algebra. The connection coefficients for the asymptotic solutions of the linear problem are found to correspond to the $Q$-functions for $\mathfrak{g}$-type quantum integrable models. The $\psi$-system for the solutions associated with the fundamental representations of $\mathfrak{g}$ leads to Bethe ansatz equations associated with the affine Lie algebra $\hat{\mathfrak{g}}$. We also study the $A_{2r}^{(2)}$ affine Toda field equation in massless limit in detail and find its Bethe ansatz equations as well as T-Q relations.

22 pages

References in corpus (2)

• Pseudo-differential equations, and the Bethe Ansatz for the classical Lie algebras
• Bethe Ansatz and the Spectral Theory of affine Lie algebra-valued connections I. The simply-laced case

Cited by in corpus (14)

• Geometric aspects of the ODE/IM correspondence
• ODE/IM correspondence and the Argyres-Douglas theory
• Lectures on Integrable Structures in Quantum Field Theory and Massive ODE/IM Correspondence
• TBA equations for the Schrödinger equation with a regular singularity
• WKB periods for higher order ODE and TBA equations
• Quantum transfer-matrices for the sausage model
• ODE/IM correspondence for modified $B_2^{(1)}$ affine Toda field equation
• Deforming the ODE/IM correspondence with $\mathrm{T}\bar{\mathrm{T}}$
• ODE/IM correspondence for affine Lie algebras: A numerical approach
• Massive ODE/IM Correspondence and Non-linear Integral Equations for $A_r^{(1)}$-type modified Affine Toda Field Equations
• Bukhvostov-Lipatov model and quantum-classical duality
• Vacuum energy of the Bukhvostov-Lipatov model
• Integrals of motion in $W{E}_6$ CFT and the ODE/IM correspondence
• A short review on TBA equation and scattering amplitude/Wilson loop duality