The ODE/IM duality, Gaudin models and the Langlands correspondence

The ODE/IM correspondence is a conjectured duality between the spectrum of integrals of motion in quantum integrable systems and spectral properties of certain differential equations. This duality was originally observed to hold in the context of conformal field theories and was later extended to a rather general form that encompasses a wealth of massive quantum integrable field theories.

By contrast, the correspondence is much less explored in the context of quantum spin chains where only very few examples are known.

The mathematical structure underpinning this duality appears to be the renowned Langlands correspondence. Such a statement was proven rigorously in a specific instance, owing to a very general description of integrable systems in terms of Gaudin models. However, this mathematical structure is widely believed to survive a number of possible generalisations, producing a "tesseract of dualities".

This project will explore the structures mentioned above in their various guises, both for field theories and spin chains, starting from some concrete, simple examples and moving from these towards greater generality and mathematical rigour.