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تسجيل مستخدم جديدModuli spaces of semitoric systems
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Joseph Palmer
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Recently Pelayo-V~{u} Ngoc classified semitoric integrable systems in terms of five symplectic invariants. Using this classification we define a family of metrics on the space of semitoric integrable systems. The resulting metric space is incomplete and we construct the completion.
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A symplectic semitoric manifold is a symplectic $4$-manifold endowed with a Hamiltonian $(S^1 times mathbb{R})$-action satisfying certain conditions. The goal of this paper is to construct a new symplectic invariant of symplectic semitoric manifolds,
the helix, and give applications. The helix is a symplectic analogue of the fan of a nonsingular complete toric variety in algebraic geometry, that takes into account the effects of the monodromy near focus-focus singularities. We give two applications of the helix: first, we use it to give a classification of the minimal models of symplectic semitoric manifolds, where minimal is in the sense of not admitting any blowdowns. The second application is an extension to the compact case of a well known result of V~{u} Ngoc about the constraints posed on a symplectic semitoric manifold by the existence of focus-focus singularities. The helix permits to translate a symplectic geometric problem into an algebraic problem, and the paper describes a method to solve this type of algebraic problem.
A semitoric integrable system $F=(J,H)$ on a symplectic $4$-manifold is simple if each fiber of $J$ contains at most one focus-focus critical point. Simple semitoric systems were classified about ten years ago by Pelayo-V~u Ngoc in terms of five inva
riants. In this paper we explain how the simplicity assumption can be removed from the classification by adapting the invariants.
Semitoric systems are a type of four-dimensional integrable system for which one of the integrals generates a global $S^1$-action; these systems were classified by Pelayo and Vu Ngoc in terms of five symplectic invariants. We introduce and study semi
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About 6 years ago, semitoric systems were classified by Pelayo & Vu Ngoc by means of five invariants. Standard examples are the coupled spin oscillator on $mathbb{S}^2 times mathbb{R}^2$ and coupled angular momenta on $mathbb{S}^2 times mathbb{S}^2$,
both having exactly one focus-focus singularity. But so far there were no explicit examples of systems with more than one focus-focus singularity which are semitoric in the sense of that classification. This paper introduces a 6-parameter family of integrable systems on $mathbb{S}^2 times mathbb{S}^2$ and proves that, for certain ranges of the parameters, it is a compact semitoric system with precisely two focus-focus singularities. Since the twisting index (one of the semitoric invariants) is related to the relationship between different focus-focus points, this paper provides systems for the future study of the twisting index.
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