Pseudo-differential equations, and the Bethe Ansatz for the classical Lie algebras

Abstract

The correspondence between ordinary differential equations and Bethe ansatz equations for integrable lattice models in their continuum limits is generalised to vertex models related to classical simple Lie algebras. New families of pseudo-differential equations are proposed, and a link between specific generalised eigenvalue problems for these equations and the Bethe ansatz is deduced. The pseudo-differential operators resemble in form the Miura-transformed Lax operators studied in work on generalised KdV equations, classical W-algebras and, more recently, in the context of the geometric Langlands correspondence. Negative-dimension and boundary-condition dualities are also observed.

50 pages, 4 figures. v2: typos corrected, extra numerical results included

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• The ODE/IM Correspondence
• T-systems and Y-systems in integrable systems
• Baxter's Q-operators for supersymmetric spin chains
• T-system on T-hook: Grassmannian Solution and Twisted Quantum Spectral Curve
• Solutions of the T-system and Baxter equations for supersymmetric spin chains
• Baxter's Q-operators and operatorial Backlund flow for quantum (super)-spin chains
• Wronskian Solution for AdS/CFT Y-system
• Bethe Ansatz and the Spectral Theory of affine Lie algebra-valued connections I. The simply-laced case
• Bethe Ansatz and the Spectral Theory of affine Lie algebra--valued connections II. The non simply--laced case
• The full Quantum Spectral Curve for $Ad{S}_4/CF{T}_3$
• Spectra of quantum KdV Hamiltonians, Langlands duality, and affine opers
• Thermodynamic Bethe Ansatz Equations for Minimal Surfaces in AdS_3
• Separation of variables and scalar products at any rank
• Wronskian solutions of the T, Q and Y-systems related to infinite dimensional unitarizable modules of the general linear superalgebra gl(M|N)
• Perturbed Defects and T-Systems in Conformal Field Theory
• Geometric aspects of the ODE/IM correspondence
• ODE/IM correspondence and the Argyres-Douglas theory
• ODE/IM correspondence and modified affine Toda field equations
• TBA equations for the Schrödinger equation with a regular singularity
• The Bethe Ansatz and the Tzitzéica-Bullough-Dodd equation
• A review of the AdS/CFT Quantum Spectral Curve
• Bethe Ansatz equations for the classical $A_n^{(1)}$ affine Toda field theories
• ODE/IM correspondence and Bethe ansatz for affine Toda field equations
• WKB periods for higher order ODE and TBA equations
• Opers for higher states of quantum KdV models
• Polynomial Relations for q-Characters via the ODE/IM Correspondence
• Complex Path Integrals and the Space of Theories
• Spectrum of the reflection operators in different integrable structures
• Wall-crossing of TBA equations and WKB periods for the third order ODE
• Counting monster potentials
• Deforming the ODE/IM correspondence with $\mathrm{T}\bar{\mathrm{T}}$
• ODE/IM correspondence for affine Lie algebras: A numerical approach
• ODE/IM correspondence for modified $B_2^{(1)}$ affine Toda field equation
• Opers for higher states of the quantum Boussinesq model
• Massive ODE/IM Correspondence and Non-linear Integral Equations for $A_r^{(1)}$-type modified Affine Toda Field Equations
• Bethe/Gauge Correspondence for SO/Sp Gauge Theories and Open Spin Chains
• Extended systems of Baxter Q-functions and fused flags I: simply-laced case
• Bukhvostov-Lipatov model and quantum-classical duality
• ODE/IM correspondence and supersymmetric affine Toda field equations
• Elementary functions in Thermodynamic Bethe Ansatz
• On the ODE/IM correspondence for minimal models
• The Generalised Born Oscillator and the Berry-Keating Hamiltonian
• Vacuum energy of the Bukhvostov-Lipatov model
• Bethe/Gauge Correspondence for linear quiver theories with ABCD gauge symmetry and spin chains
• Folding QQ-relations and transfer matrix eigenvalues: towards a unified approach to Bethe ansatz for super spin chains
• WKB analysis of the linear problem for modified affine Toda field equations
• Feigin-Frenkel-Hernandez Opers and the QQ-system
• Integrals of motion in conformal field theory with W-symmetry and the ODE/IM correspondence
• Quasi-exact solvability, resonances and trivial monodromy in ordinary differential equations
• Integrals of motion in $W{E}_6$ CFT and the ODE/IM correspondence
• The ODE/IM Correspondence between $C(2{)}^{(2)}$-type Linear Problems and 2d $\mathcal{N}=1$ SCFT
• Boson-Fermion correspondence, QQ-relations and Wronskian solutions of the T-system