Minimal models of compact symplectic semitoric manifolds

Kane, Daniel M.; Palmer, Joseph; Pelayo, Álvaro

doi:10.1016/j.geomphys.2017.12.005

Cited by 14 publications

(23 citation statements)

References 24 publications

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“…In this section we introduce the system which is the subject of this paper and prove Theorem 1.1, our main result. This system is minimal in the sense of Kane & Palmer & Pelayo [16], i.e., it is not possible to perform a blowdown of toric type on the system (see Kane [16] and the system discussed in the present paper is minimal of type (2), using the terminology of that paper.…”

Section: A Family Of Systems With Two Focus-focus Pointsmentioning

confidence: 97%

A family of compact semitoric systems with two focus-focus singularities

Hohloch¹,

Palmer²

2018

Journal of Geometric Mechanics

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About 6 years ago, semitoric systems were classified by Pelayo & Vũ Ngo . c by means of five invariants. Standard examples are the coupled spin oscillator on S 2 × R 2 and coupled angular momenta on S 2 × 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 S 2 × 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.where ω S 2 is the standard volume form on the sphere and 0 < R 1 < R 2 are real numbers. For R := (R 1 , R 2 ) and t := (t 1 , t 2 , t 3 , t 4 ) ∈ R 4 define J R , H t : M → R by (1) where (x i , y i , z i ) are Cartesian coordinates on S 2 ⊂ R 3 for i = 1, 2. Then there exist choices of t 1 , t 2 , t 3 , t 4 , R 1 , R 2 such that (M, ω, (J R , H t )) is a semitoric system with exactly two focus-focus points.Theorem 1.1 is restated in more detail in Section 3 as Theorem 3.1. The coupled angular momenta system with coupling parameter t ∈ ]0, 1[ is the special case of Equation (1) with t 1 = t, t 3 = t 4 = 1−t, and t 2 = 0. The coupled angular momenta system describes the rotation of two vectors (with magnitudes R 1 and R 2 ) about the z-axis and has as a second integral a linear combination of the z-component of the first vector and the inner produce of the two vectors, while the system in Equation (1) includes additionally the z-component of the second vector and also

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Section: A Family Of Systems With Two Focus-focus Pointsmentioning

confidence: 97%

A family of compact semitoric systems with two focus-focus singularities

Hohloch¹,

Palmer²

2018

Journal of Geometric Mechanics

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“…Remark 2.23. In [KPP18] the authors define an invariant of a compact semitoric system called the semitoric helix which generalizes the fan of a smooth toric surface. The semitoric helix is a collection of integral vectors which takes into account the effect of the monodromy of the focus-focus points of the semitoric system.…”

Section: Blowups On Delzant Polygonsmentioning

confidence: 99%

Extending compact Hamiltonian $S^1$-spaces to integrable systems with mild degeneracies in dimension four

Hohloch¹,

Palmer²

2021

Preprint

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We show that all compact four-dimensional Hamiltonian S 1 -spaces can be extended to a completely integrable system on the same manifold such that all singularities are non-degenerate, except possibly for a finite number of degenerate orbits of parabolic (also called cuspidal) type -we call such systems hypersemitoric.More precisely, given any compact four dimensional Hamiltonian S 1 -space (M, ω, J) we show that there exists a smooth H : M → R such that (M, ω, (J, H)) is a completely integrable system of hypersemitoric type. Hypersemitoric systems generalize semitoric systems. In addition to elliptic-elliptic, elliptic-regular, and focus-focus singular points which can occur in semitoric systems, hypersemitoric systems may also have hyperbolic-regular and hyperbolic-elliptic singular points (hyperbolic-hyperbolic points cannot appear due to the presence of the global S 1 -action) and moreover degenerate singular points of a relatively tame type called parabolic.Admitting the existence of degenerate points is necessary since there exist compact four-dimensional Hamiltonian S 1 -spaces whose extensions must include degenerate singular points of some kind as we show in the present paper. Parabolic points are among the most common and natural degenerate points, and we show that it is sufficient to only admit these degenerate points in order to extend all Hamiltonian S 1 -spaces. In this sense, hypersemitoric systems are thus the "nicest and smallest" class of systems to which all Hamiltonian S 1 -spaces can be extended. Moreover, we prove several foundational results about these systems, such as the non-existence of loops of hyperbolic-regular points and properties about their fibers.

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“…That being said, the set of standard semitoric fans does not correspond to the set of all minimal semitoric fans, which are those fans which do not admit a reverse corner chop. The problem of classifying all minimal semitoric fans is being addressed in a future paper (which is now available, see [27]).…”

Section: Letmentioning

confidence: 99%

Classifying Toric and Semitoric Fans by Lifting Equations from SL<sub>2</sub>(Z)

Kane¹,

Palmer²,

Pelayo³

2018

SIGMA

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We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group SL 2 (Z) to its preimage in the universal cover of SL 2 (R). With this method we recover the classification of two-dimensional toric fans, and obtain a description of their semitoric analogue. As an application to symplectic geometry of Hamiltonian systems, we give a concise proof of the connectivity of the moduli space of toric integrable systems in dimension four, recovering a known result, and extend it to the case of semitoric integrable systems with a fixed number of focus-focus points and which are in the same twisting index class. In particular, we show that any semitoric system with precisely one focus-focus singular point can be continuously deformed into a system in the same isomorphism class as the Jaynes-Cummings model from optics.

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Minimal models of compact symplectic semitoric manifolds

Cited by 14 publications

References 24 publications

A family of compact semitoric systems with two focus-focus singularities

A family of compact semitoric systems with two focus-focus singularities

Extending compact Hamiltonian $S^1$-spaces to integrable systems with mild degeneracies in dimension four

Classifying Toric and Semitoric Fans by Lifting Equations from SL<sub>2</sub>(Z)

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