Local moment maps and the splitting of classical multiplets

TL;DR

This paper introduces local moment maps labeled by the fundamental group, explaining how their branches can be glued via cocycles, and explores how multiply-connected phase spaces lead to multiplet splitting through universal covers.

Contribution

It generalizes global moment maps to local ones using fundamental group labels and cocycles, and analyzes the implications for group actions and state identification in symplectic geometry.

Findings

Proved theorems on liftability of group actions to symplectic covers

Described multiplet splitting via universal covers and non-contractible loops

Connected phase space topology to quantum state identification

Abstract

We generalize the concept of global moment maps to local moment maps, whose different branches are labelled by the elements of the fundamental group of the underlying symplectic manifold. These branches can be smoothly glued together by employing fundamental-group-valued \u Cech cocycles on the phase space. In the course of this work we prove a couple of theorems on the liftability of group actions to symplectic covering spaces, and examine the possible extensions of the original group by the fundamental group of the quotient phase space. It it shown how the splitting of multiplets, this being a consequence of the multiply-connectedness of the quotient phase space, can be described by identification maps on a space of multiplets derived from a symplectic universal covering manifold. The states that are identified in this process are related by certain integrals over non-contractible loops in the quotient phase space.