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 $ψ$-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 $A^{(2)}_{2r}$ affine Toda field equation in massless limit in detail and find its Bethe ansatz equations as well as T-Q relations.
We study the linear differential system associated with the supersymmetric affine Toda field equations for affine Lie superalgebras, which has a purely odd simple root system. For an affine Lie algebra, the linear problem modified by conformal transformation leads to an ordinary differential equation (ODE) that provides the functional relations in the integrable models. This is known as the ODE/IM correspondence. For the affine Lie superalgebras, the linear equations modified by a superconformal transformation are shown to reduce to a couple of ODEs for each bosonic subalgebra. In particular, …
The massive ODE/IM correspondence is a relation between the linear problem associated with modified affine Toda field equations and two-dimensional massive integrable models. We study the massive ODE/IM correspondence for the $A_r^{(1)}$-type modified affine Toda field equations. Based on the $ψ$-system satisfied by the solutions of the linear problem, we derive the Bethe ansatz equations and determine the asymptotic behavior of the Q-functions for large value of the spectral parameter. We derive the non-linear integral equations for the Q-functions from the Bethe ansatz equations. We compute …
We study the WKB periods for the $(r+1)$-th order ordinary differential equation (ODE) which is obtained by the conformal limit of the linear problem associated with the $A_r^{(1)}$ affine Toda field equation. We compute the quantum corrections by using the Picard-Fuchs operators. The ODE/IM correspondence provides a relation between the Wronskians of the solutions and the Y-functions which satisfy the thermodynamic Bethe ansatz (TBA) equation related to the Lie algebra $A_r$. For the quadratic potential, we propose a formula to show the equivalence between the logarithm of the Y-function and …
We study the WKB periods for the third order ordinary differential equation (ODE) with polynomial potential, which is obtained by the Nekrasov-Shatashvili limit of ($A_2,A_N$) Argyres-Douglas theory in the Omega background. In the minimal chamber of the moduli space, we derive the Y-system and the thermodynamic Bethe ansatz (TBA) equations by using the ODE/IM correspondence. The exact WKB periods are identified with the Y-functions. Varying the moduli parameters of the potential, the wall-crossing of the TBA equations occurs. We study the process of the wall-crossing from the minimal chamber t…
This review describes a link between Lax operators, embedded surfaces and Thermodynamic Bethe Ansatz equations for integrable quantum field theories. This surprising connection between classical and quantum models is undoubtedly one of the most striking discoveries that emerged from the off-critical generalisation of the ODE/IM correspondence, which initially involved only conformal invariant quantum field theories. We will mainly focus of the KdV and sinh-Gordon models. However, various aspects of other interesting systems, such as affine Toda field theories and non-linear sigma models, will …
We study numerically the ODE/IM correspondence for untwisted affine Lie algebras associated with simple Lie algebras including exceptional type. We consider the linear problem obtained from the massless limit of that of the modified affine Toda field equation. We found that the Q-functions in integrable models are expressed as the inner product of the solution of the dual linear problem and the subdominant solution of the linear problem. Using Cheng's algorithm to obtain the solution of the linear problem, we can determine efficiently the zeros of the Q-function, which is known to provide the …
We study the ODE/IM correspondence for all the states of the quantum Boussinesq model. We consider a particular class of third order linear ordinary differential operators and show that the generalised monodromy data of such operators provide solutions to the Bethe Ansatz equations of the Quantum Boussinesq model.
We obtain the exact generalised hydrodynamics for the integrable Toda system. The Toda system can be seen in a dual way, both as a gas and as a chain. In the gas point of view, using the elastic and factorised scattering of Toda particles, we obtain the generalised free energy and exact average currents, and write down the Euler hydrodynamic equations. This is written both as a continuity equation for the density of asymptotic momenta, and in terms of normal modes. This is based on the classical thermodynamic Bethe ansatz (TBA), with a single quasiparticle type of Boltzmann statistics. By expl…
ODE/IM Correspondence at the Free-Fermion Point. Laguerre Wronskians, Shifted Symmetric Functions, and Quantum KdV
Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高等理工学院）；Instituto Superior de Agronomia, Universidade de Lisboa（里斯本大学高等农业学院）；CEMS.UL, Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa（里斯本大学理学院数学系）
From classical Lax ODEs to quantum integrable theories: the moduli
Sezione INFN di Bologna, Dipartimento di Fisica e Astronomia, Università di Bologna（博洛尼亚大学物理与天文学系，意大利国家核物理研究所博洛尼亚分部）；Dipartimento di Fisica dell’Università della Calabria and INFN, Gruppo collegato di Cosenza（卡塔尼亚大学物理系，意大利国家核物理研究所科森扎协作组）
Generalised Gibbs Ensembles from the ODE/IM Correspondence: the Lee--Yang Model
The Institute of Mathematical Sciences（马哈拉施特拉数学科学研究所）；Homi Bhabha National Institute（霍米·巴巴国立研究所）；Dipartimento di Fisica, Università di Torino and INFN, Sezione di Torino（伦敦国王学院数学系）
We derive the Thermodynamic Bethe Ansatz (TBA) equations for the Schrödinger equation with an arbitrary polynomial potential and a regular singular (simple and double pole) term. The TBA equations provide a non-trivial generalization of the ODE/IM correspondence and also give a solution for the Riemann-Hilbert problem in the exact WKB method. We study the TBA equations in detail for the linear and the harmonic oscillator potentials together with inverse and centrifugal terms. As an application, we also compute numerically the Voros spectrum for these potentials using the Bohr-Sommerfeld quanti…
We introduce the notion of a classical dihedral affine Gaudin model, associated with an untwisted affine Kac-Moody algebra $\widetilde{\mathfrak{g}}$ equipped with an action of the dihedral group $D_{2T}$, $T \geq 1$ through (anti-)linear automorphisms. We show that a very broad family of classical integrable field theories can be recast as examples of such classical dihedral affine Gaudin models. Among these are the principal chiral model on an arbitrary real Lie group $G_0$ and the $\mathbb{Z}_T$-graded coset $σ$-model on any coset of $G_0$ defined in terms of an order $T$ automorphism of it…
