Constructing integrable systems of semitoric type

Details Constructing integrable systems of semitoric type Constructing integrable systems of semitoric type

2009-03-19

arxiv.org/abs/0903.3376v1

Mathematics

Symplectic Geometry

28 pages, 4 figures

Scientific paper

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.

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