Remarks on symplectic circle actions, torsion and loops

TL;DR

This paper investigates loops of symplectic diffeomorphisms on closed symplectic manifolds, showing flux vanishes for contractible orbits, leading to new results on flux groups and conditions for Hamiltonian actions.

Contribution

It establishes that flux vanishes for certain symplectic loops and links fixed points of circle actions to Hamiltonian properties, advancing understanding of symplectic torsion.

Findings

Flux vanishes for loops with contractible orbits

Fixed points imply Hamiltonian circle actions under certain conditions

New results on the structure of the flux group

Abstract

We study loops of symplectic diffeomorphisms of closed symplectic manifolds. Our main result, which is valid for a large class of symplectic manifolds, shows that the flux of a symplectic loop vanishes whenever its orbits are contractible. As a consequence, we obtain a new vanishing result for the flux group and new instances where the presence of a fixed point of a symplectic circle action is a sufficient condition for it to be Hamiltonian. We also obtain applications to symplectic torsion, more precisely, non-trivial elements of $\mathrm{Symp}_{0}(M,\omega)$ that have finite order.