Symplectic quotients and representability: the circle action case

TL;DR

This paper investigates when symplectic quotients from circle actions are diffeomorphic to orbit spaces of proper Lie group actions, showing this only occurs under specific regularity or weight conditions.

Contribution

It establishes necessary conditions for symplectic quotients to be representable as orbit spaces, linking this to orbifold structures and weight configurations.

Findings

Symplectic quotients are diffeomorphic to orbifolds only if the value is regular or weights are restricted.

Effective orbifold structure is a prerequisite for such diffeomorphisms.

At most one positive or negative weight is allowed for the quotient to be representable.

Abstract

Let $S^1$ act on a symplectic manifold in a Hamiltonian fashion with momentum map $\Psi$. Fix a value $a$ of $\Psi$. There is a question of whether the symplectic quotient at $a$ is diffeomorphic to the orbit space of some proper Lie group action. We prove under mild assumptions that this only occurs if the symplectic quotient is diffeomorphic to an effective orbifold. This, in turn, only occurs if $a$ is a regular value, or there is at most one positive weight or at most one negative weight.