Classifying Toric and Semitoric Fans by Lifting Equations from SL&lt;sub&gt;2&lt;/sub&gt;(Z)

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

doi:10.3842/sigma.2018.016

Cited by 7 publications

(13 citation statements)

References 49 publications

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“…The following statement implies that there exist parameter values for which the system has four nondegenerate rank 0 points, two of them elliptic-elliptic and two focus-focus, and is proved by plugging the values into the criterion in Proposition 3.7. Since nonvanishing and noncoinciding are open conditions, there exist in fact intervals around the parameters (15) where the systems continues to have two focus-focus and two elliptic-elliptic points. Proposition 3.9.…”

Section: 2mentioning

confidence: 99%

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: 2mentioning

confidence: 99%

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

Hohloch¹,

Palmer²

2018

Journal of Geometric Mechanics

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“…Given v, w ∈ Z 2 we denote by [v, w] the 2 × 2 matrix with first column v and second column w and denote by det [10,Proposition 3.7], as in the following diagram:…”

Section: The Algebraic Techniquementioning

confidence: 99%

“…, W descends to a map on G which we also denote W . The map is known as the winding number [10] because if σ ∈ ker(G → SL 2 (Z)) then W (σ) agrees with wind(ρ(σ)) as in Definition 4.10, where ρ is as in Equation 4.2.…”

Section: The Winding Numbermentioning

confidence: 99%

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Minimal models of compact symplectic semitoric manifolds

Kane

Palmer

Pelayo

2018

Journal of Geometry and Physics

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A symplectic semitoric manifold is a symplectic 4-manifold endowed with a Hamiltonian (S 1 ×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 Vũ Ngo . c 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. arXiv:1610.05423v2 [math.SG] 16 Nov 2016 Theorem 1.2 ( W. Fulton 1993). The inequivalent minimal models of symplectic toric manifolds are CP 2 , CP 1 × CP 1 , and a Hirzebruch surface with parameter k = ±1.The Delzant polytopes of the minimal models are: a simplex (M = CP 2 with any multiple of the Fubini-Study form), a rectangle (M = CP 1 × CP 1 with any product form), and a trapezoid (M a Hirzebruch surface, with its standard form). The question is whether Fulton's classification can cover more cases.Main Question. What are the inequivalent minimal models of compact symplectic semitoric manifolds?

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“…In [120] Palmer defined the moduli space of semitoric systems, which is an incomplete metric space, and constructed its completion. In [86] the connectivity properties of this space were studied using SL 2 (Z) equations.…”

Section: Classification Of Hamiltonianmentioning

confidence: 99%

Hamiltonian and symplectic symmetries: An introduction

Pelayo

2017

Bull. Amer. Math. Soc.

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Abstract. Classical mechanical systems are modeled by a symplectic manifold (M, ω), and their symmetries are encoded in the action of a Lie group G on M by diffeomorphisms which preserve ω. These actions, which are called symplectic, have been studied in the past forty years, following the works of Atiyah, Delzant, Duistermaat, Guillemin, Heckman, Kostant, Souriau, and Sternberg in the 1970s and 1980s on symplectic actions of compact Abelian Lie groups that are, in addition, of Hamiltonian type, i.e., they also satisfy Hamilton's equations. Since then a number of connections with combinatorics, finitedimensional integrable Hamiltonian systems, more general symplectic actions, and topology have flourished. In this paper we review classical and recent results on Hamiltonian and non-Hamiltonian symplectic group actions roughly starting from the results of these authors. This paper also serves as a quick introduction to the basics of symplectic geometry.

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

Cited by 7 publications

References 49 publications

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

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

Minimal models of compact symplectic semitoric manifolds

Hamiltonian and symplectic symmetries: An introduction

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