Global Action-Angle Variables for Non-Commutative Integrable Systems

TL;DR

This paper investigates the challenges to establishing global action-angle variables in non-commutative integrable systems on Poisson manifolds, extending prior symplectic results and introducing new geometric tools.

Contribution

It introduces the concepts of action bundle and action lattice bundle, and analyzes obstructions to global action-angle variables in the broader Poisson setting.

Findings

Global action-angle variables rarely exist beyond local neighborhoods.

Obstructions are often cohomological in nature.

New geometric structures are introduced to understand these obstructions.

Abstract

In this paper we analyze the obstructions to the existence of global action-angle variables for regular non-commutative integrable systems (NCI systems) on Poisson manifolds. In contrast with local action-angle variables, which exist as soon as the fibers of the momentum map of such an integrable system are compact, global action-angle variables rarely exist. This fact was first observed and analyzed by Duistermaat in the case of Liouville integrable systems on symplectic manifolds and later by Dazord-Delzant in the case of non-commutative integrable systems on symplectic manifolds. In our more general case where phase space is an arbitrary Poisson manifold, there are more obstructions, as we will show both abstractly and on concrete examples. Our approach makes use of a few new features which we introduce: the action bundle and the action lattice bundle of the NCI system (these bundles are canonically defined) and three foliations (the action, angle and transverse foliation), whose existence is also subject to obstructions, often of a cohomological nature.