Symplectic Group Actions and Covering Spaces

TL;DR

This paper explores two methods of symplectic reduction for non-Hamiltonian symplectic group actions, establishing conditions under which these methods are related by a symplectic cover and classifying Hamiltonian covers.

Contribution

It introduces a classification of Hamiltonian covers for symplectic group actions and analyzes the properties of lifting actions to covers under specific conditions.

Findings

The natural projection between reduction methods is a symplectic cover under certain conditions.

Classification of all Hamiltonian covers of a given symplectic group action.

Properties of lifted group actions to covers are characterized.

Abstract

For symplectic group actions which are not Hamiltonian there are two ways to define reduction. Firstly using the cylinder-valued momentum map and secondly lifting the action to any Hamiltonian cover (such as the universal cover), and then performing symplectic reduction in the usual way. We show that provided the action is free and proper, and the Hamiltonian holonomy associated to the action is closed, the natural projection from the latter to the former is a symplectic cover. At the same time we give a classification of all Hamiltonian covers of a given symplectic group action. The main properties of the lifting of a group action to a cover are studied.