On Generalized Moment Maps for Symplectic Compact Group Actions

TL;DR

This paper introduces a generalized moment map for symplectic actions of compact Lie groups, extending classical concepts and providing new reduction techniques and convexity results, with applications to Hamiltonian actions and surface symmetries.

Contribution

It proposes a new generalized moment map for arbitrary symplectic group actions, extending McDuff's circle-valued maps and establishing key properties and applications.

Findings

Generalized moment maps allow reduction procedures.

In the torus case, they satisfy a convexity theorem.

Application to Hamiltonian actions and symmetry groups of surfaces.

Abstract

A generalized moment map is proposed for arbitrary symplectic actions of compact connected Lie groups on closed symplectic manifolds, in the spirit of the circle -valued maps introduced by D. McDuff in the case of non-Hamiltonian circle actions. We study equivariance properties of generalized moments, show that they allow reduction procedures, and obtain in the torus case a version of the Atiyah-Guillemin-Sternberg convexity theorem. As illustration, we reformulate a proof of M.K. Kim that "complexity one" symplectic torus actions are Hamiltonian, and give a symplectic proof of the finiteness of certain symmetry groups of compact oriented surfaces.