Relativistic Calogero-Moser model as gauged WZW theory

TL;DR

This work connects quantum integrable particle systems, including Calogero–Moser, Sutherland, and Ruijsenaars models, to two–dimensional topological gauge theories via Hamiltonian reduction of cotangent bundles over affine and loop groups. It explicitly derives the Sutherland model from $T^{*}{\hat{g}}$ and its 2D YM reduction, then generalizes to a Ruijsenaars–type deformation by reducing $T^{*}\hat{G}$ with a central extension, establishing an equivalence to gauged $G/G$ WZW theory with Wilson lines and to a Chern–Simons picture. The spectral data are expressed in Lie–theoretic terms, with wavefunctions tied to representation theory and conjectured Macdonald polynomial structures through the $q,t$ parameters $q=e^{2\pi i/(\kappa+N)}$ and $t=q^{\nu+1}$. This framework links integrable many–body dynamics to topological field theory and quantum groups, suggesting elliptic extensions and deeper connections to affine and double–loop algebras as fruitful future directions.

Abstract

We study quantum intergrable systems of interacting particles from the point of view, proposed in our previous paper. We obtain Calogero-Moser and Sutherland systems as well their Ruijsenaars relativistic generalization by a Hamiltonian reduction of integrable systems on the cotangent bundles over semi-simple Lie algebras, their affine algebras and central extensions of loop groups respectively. The corresponding 2d field theories form a tower of deformations. The top of this tower is gauged G/G WZW model on a cylinder with inserted Wilson line in appropriate representation.