The Bethe Ansatz equations were initially conceived as a
method to solve some particular Quantum Integrable Models (IM), but
are nowadays a central tool of investigation in a variety physical and
mathematical theory such as string theory, super-symmetric Gauge
theories, and Donaldson-Thomas invariants. Surprisingly, it has been
observed, in several examples, that the solutions of the same Bethe
Ansatz equations are provided by the monodromy data of some ordinary
differential operators with an irregular singularity (ODE/IM
correspondence).

In this talk I will present the results of my investigation on the
ODE/IM correspondence quantum g-KdV models, where
g is an untwisted affine Kac-Moody algebra. I will construct solutions
of the corresponding Bethe Ansatz equations, as
the (irregular) monodromy data of a meromorphic L(g)-oper, where L(g)
is the Langlands dual algebra of g.

The talk is based on:

[1] D Masoero, A Raimondo, D Valeri, Bethe Ansatz and the Spectral
Theory of affine Lie algebra-valued connections I. The simply-laced
case. Comm. Math. Phys. (2016)

[2] D Masoero, A Raimondo, D Valeri, Bethe Ansatz and the Spectral
Theory of affine Lie algebra-valued connections II: The non
simply-laced case. Comm. Math. Phys. (2017)

[3] D Masoero, A Raimondo, Opers corresponding to Higher States of the
g-Quantum KdV model. arXiv 2018.