Feigin–Frenkel ODE/IM conjecture on Langlands-dual affine opers

Establish the Feigin–Frenkel conjectural extension of the ODE/IM correspondence by proving that the spectra of quantum KdV systems are encoded by monodromy-free affine opers associated with the Langlands dual affine Lie algebra of the affine Lie algebra attached to the quantum KdV system.

Background

The ODE/IM correspondence links spectral data of certain differential operators to spectra of quantum integrable models such as quantum KdV. Feigin and Frenkel proposed a broad generalization that recasts the correspondence in the language of Langlands duality via affine opers without monodromy.

In this framework, Schrödinger operators are replaced by affine opers for the Langlands dual affine Lie algebra, suggesting a deep duality between spectral problems and representation-theoretic objects.

Despite extensive progress in special cases, the general conjecture remains unresolved and serves as a guiding principle for many developments discussed in the paper, including QQ-systems and Bethe Ansatz relations.