Hamiltonian systems of Calogero type and two dimensional Yang-Mills theory

TL;DR

The paper establishes a bridge between Calogero-type integrable systems and two-dimensional Yang–Mills theory by constructing exact path-integral representations and performing Hamiltonian reductions from free Lie-algebraic dynamics. It shows how Calogero–Moser models arise both classically (via reduction) and quantum-mechanically (via scaling limits and lattice formulations), and how their wavefunctions correspond to YM data, including representations and monodromies. The work extends to Kac–Moody algebras, connects to matrix-model formulations, and discusses supersymmetric localization and relations to KP/KdV integrable hierarchies, with the large-$k$ limit revealing links to the Generalized Kontsevich Model. Overall, it reveals a unifying framework in which integrable particle systems, 2D YM, and matrix-model techniques mutually inform one another, while outlining several open directions for curved generalizations and vertex-operator interpretations.

Abstract

We obtain integral representations for the wave functions of Calogero-type systems,corresponding to the finite-dimentional Lie algebras,using exact evaluation of path integral.We generalize these systems to the case of the Kac-Moody algebras and observe the connection of them with the two dimensional Yang-Mills theory.We point out that Calogero-Moser model and the models of Calogero type like Sutherland one can be obtained either classically by some reduction from two dimensional Yang-Mills theory with appropriate sources or even at quantum level by taking some scaling limit.We investigate large k limit and observe a relation with Generalized Kontsevich Model.