Feigin-Frenkel-Hernandez Opers and the QQ-system

This paper represents the completion of our work on the ODE/IM correspondence for the generalised quantum Drinfeld-Sokolov models. We present a unified and general mathematical theory, encompassing all particular cases that we had already addressed, and we fill important analytic and algebraic gaps in the literature on the ODE/IM correspondence. For every affine Lie algebra $\mathfrak{g}$ -- whose Langlands dual $\mathfrak{g}'$ is the untwisted affinisation of a simple Lie algebra -- we study a class of affine twisted parabolic Miura $\mathfrak{g}$-opers, introduced by Feigin, Frenkel and Hernandez. The Feigin-Frenkel-Hernandez opers are defined by fixing the singularity structure at $0$ and $\infty$, and by allowing a finite number of additional singular terms with trivial monodromy. We define the central connection matrix and Stokes matrix for these opers, and prove that the coefficients of the former satisfy the the $QQ$ system of the quantum $\mathfrak{g}'$-Drinfeld-Sokolov (or quantum $\mathfrak{g}'$-KdV) model. If $\mathfrak{g}$ is untwisted, it is known that the trivial monodromy conditions are equivalent to a complete system of algebraic equations for the additional singularities. We prove a suprising negative result in the case $\mathfrak{g}$ is twisted: in this case, the trivial monodromy conditions have no non-trivial solutions.