Quantum integrable systems of particles as gauge theories

We study quantum integrable systems of interacting particles from the point of view proposed by A. Gorsky and N. Nekrasov. We obtain the Sutherland system by a Hamiltonian reduction of an integrable system on the cotangent bundles to an affine $\hat su(N)$ algebra and show that it coincides with the Yang–Mills theory on a cylinder. We point out that there exists a tower of $2d$ quantum field theories. The top of this tower is the gauged $G/G$ WZW model on a cylinder with an inserted Wilson line in an appropriate representation, which in our approach corresponds to Ruijsenaars' relativistic Calogero model. Its degeneration yields the $2d$ Yang–Mills theory, whose small radius limit is the Calogero model itself. We make some comments about the spectra and eigenstates of the models, which one can get from their equivalence with the field theories. Also we point out some possibilities of elliptic deformations of these constructions.