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.