ODE/IM correspondence for the Fateev model

The Fateev model is somewhat special among two-dimensional quantum field theories. For different values of the parameters, it can be reduced to a variety of integrable systems. An incomplete list of the reductions includes O(3) and O(4) non-linear sigma models and their continuous deformations (2D and 3D sausages, anisotropic principal chiral field), the Bukhvostov-Lipatov model, the N = 2 supersymmetric sine-Gordon model, as well as the integrable perturbed SU₂(n) ⊗ SU₂(p - 2)/SU₂(n + p - 2) coset CFT. The model possesses a mysterious symmetry structure of the exceptional quantum superalgebras U_q(D̂(2|1;α)). In this work, we propose the ODE/IM correspondence between the Fateev model and a certain generalization of the classical problem of constant mean curvature embedding of a thrice-punctured sphere in AdS₃.