Constructing integrable systems of semitoric type

Abstract

International audienceLet 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

Semiclassical quantization and spectral limits of ħ-pseudodifferential and Berezin-Toeplitz operators

“…If X = R n , then Op (f ) is given by the Weyl quantization. If X is a closed manifold, then Op (f ) is constructed via formula (28).…”

“…be the operator of multiplication by χ j . Then (28) reads Op (f ) = j S j Op j (f )S j , and so Op (f ) is selfadjoint since S j and the Weyl quantization are selfadjoint. Axiom (Q1) follows from the fact that S j Op j (1)S j = S 2 j , and j S 2 j = Id.…”

Asymptotics of action variables near semi-toric singularities

“…There are several global results in our research program (see [PVN11] and [PRVN11] for a description ) that necessitate more conceptual tools than the local models like we used here. In that sense, the aim of articles [Wac14a] and [ZW14] are to establish these result in a fit framework.…”

Open problems, questions and challenges in finite- dimensional integrable systems

“…The classification of semitoric systems in [185,186] (see Section 1) goes a long way to answering the fascinating question "can one hear the shape of a semitoric system"? More precisely, the idea is to recover the classical semitoric invariants from the joint spectrum of a quantum integrable system, as much as possible.…”