Constructing integrable systems of semitoric type

TL;DR

This paper presents a comprehensive method to construct and classify semitoric integrable systems on symplectic 4-manifolds using five invariants, unifying geometric, analytic, and combinatorial approaches.

Contribution

It introduces a general construction method for semitoric systems and proves that all such systems can be obtained through this process, completing their classification.

Findings

Provides a construction method from five ingredients

Shows all semitoric systems are obtainable via this method

Classifies systems using five invariants encompassing geometric, analytic, and combinatorial data

Abstract

Let M be a connected, symplectic 4-manifold. A semitoric integrable system on M essentially consists of a pair of independent, real-valued, smooth functions J and H on the manifold M, for which J generates a Hamiltonian circle action under which H is invariant. In this paper we give a general method to construct, starting from a collection of five ingredients, a symplectic 4-manifold equipped a semitoric integrable system. Then we show that every semitoric integrable system on a symplectic 4-manifold is obtained in this fashion. In conjunction with the uniqueness theorem proved recently by the authors (Invent. Math. 2009), this gives a classification of semitoric integrable systems on 4-manifolds, in terms of five invariants. Some of the invariants are geometric, others are analytic and others are combinatorial/group-theoretic.