The ODE/IM correspondence is a rich and surprising link between the spectral theory of classical Lax operators and integrable quantum field theories. It dates back to the work of Dorey and Tateo, and Bazhanov, Lukyanov and Zamolodchikov , who proved that the spectral determinants of particular Schroedinger operators coincide with the eigenvalues of the Q operator of the quantum KdV model.

In this talk we consider a class of affine Lie algebra-valued connections (Drinfeld-Sokolov operators) and we construct solutions of the Bethe Ansatz equations of the quantum g-KdV model - for any simple Lie algebra g. The techniques used for the proof include the representation theory of simple Lie algebras and affine Kac-Moody algebras, as well as the asymptotic theory of linear ODEs in the complex plane.

These results were achieved in collaboration with Andrea Raimondo and Daniele Valeri in the recent papers  http://arxiv.org/abs/1501.07421, http://arxiv.org/abs/1511.00895 .