Semitoric integrable systems on symplectic 4-manifolds

TL;DR

This paper introduces new global symplectic invariants for semitoric integrable systems on symplectic 4-manifolds, aiming to classify these systems completely based on the invariants.

Contribution

It develops novel invariants that encode topological, geometric, and analytical data, providing a complete classification of semitoric systems.

Findings

New invariants encode topological and geometric aspects.

Invariants also capture analytical information about singularities.

Semitoric systems are uniquely determined by these invariants.

Abstract

Let M be a symplectic 4-manifold. A semitoric integrable system on M is a pair of real-valued smooth functions J, H on M for which J generates a Hamiltonian S^1-action and the Poisson brackets {J,H} vanish. We shall introduce new global symplectic invariants for these systems; some of these invariants encode topological or geometric aspects, while others encode analytical information about the singularities and how they stand with respect to the system. Our goal is to prove that a semitoric system is completely determined by the invariants we introduce.