Symplectic geometry and spectral properties of classical and quantum coupled angular momenta

Floch, Yohann Le; Pelayo, Álvaro

doi:10.48550/arxiv.1607.05419

Cited by 2 publications

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“…then we obtain the coupled spin system. In [12] it is shown that this is indeed a symplectic semitoric manifold and the authors find the two representatives of the polygons associated to the system obtained by Vũ Ngo . c's cutting procedure.…”

Section: Lemma 83 (Classification Of Minimal Words)mentioning

confidence: 94%

“…c [20] from noncompact to compact symplectic semitoric manifolds: if a compact symplectic semitoric manifold with momentum map F = (f 1 , f 2 ) has more than two focus-focus singular points, then f 1 has either a non-unique maximum or a non-unique minimum. The Main Theorem and application can be used to study many integrable systems from classical mechanics such as the spin-orbit system [12,19] (see Section 9.2). It has precisely one focus-focus singularity with monodromy, and is an example of a compact (nontoric) symplectic semitoric manifold.…”

Section: Main Question What Are the Inequivalent Minimal Models Of Co...mentioning

confidence: 99%

“…Since the system has only one focus-focus point, the helix can be obtained as the inwards pointing normal vectors of the semitoric polygon computed in [12], where v 0 is chosen to be the inwards pointing normal vector of an edge adjacent to the fake corner produced by the focus-focus point. This is because in systems with one focus-focus point the same toric momentum map that produces the semitoric polygon also works for the construction of the semitoric helix, as described in Section 5.1 (see Remark 5.4).…”

Section: Lemma 83 (Classification Of Minimal Words)mentioning

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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Section: Lemma 83 (Classification Of Minimal Words)mentioning

confidence: 94%

Section: Main Question What Are the Inequivalent Minimal Models Of Co...mentioning

confidence: 99%

Section: Lemma 83 (Classification Of Minimal Words)mentioning

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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“…Another important example are coupled angular momenta, the classical counterpart of the coupling of two quantum angular momenta as studied by Sadovksii & Zhilinskii [SZ99]. Le Floch & Pelayo have computed some of the symplectic invariants in [LP16] for a particular family parameter, and they also recover the invariants from the joint quantum spectrum. Recently, a compact semitoric system with two focus-focus points has been described by the third author together with Palmer [HP].…”

Section: Introductionmentioning

confidence: 99%

“…The coupled angular momenta are a 4-dimensional semitoric system and can thus be classified using five symplectic invariants. Le Floch & Pelayo computed in [LP16] the number of focus-focus points invariant, the polygon invariant, the height invariant and the linear terms of the Taylor series invariant for t = 1 2 . The Taylor series invariant describes the singular foliation around focus-focus singularities and is particularly difficult to compute explicitly.…”

Section: Introductionmentioning

confidence: 99%

Symplectic classification of coupled angular momenta

2019

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The coupled angular momenta are a family of completely integrable systems that depend on three parameters and have a compact phase space. They correspond to the classical version of the coupling of two quantum angular momenta and they constitute one of the fundamental examples of so-called semitoric systems. Pelayo & Vũ Ngo . c have given a classification of semitoric systems in terms of five symplectic invariants. Three of these invariants have already been partially calculated in the literature for a certain parameter range, together with the linear terms of the so-called Taylor series invariant for a fixed choice of parameter values.In the present paper we complete the classification by calculating the polygon invariant, the height invariant, the twisting-index invariant, and the higher-order terms of the Taylor series invariant for the whole family of systems. We also analyse the explicit dependence of the coefficients of the Taylor series with respect to the three parameters of the system, in particular near the Hopf bifurcation where the focus-focus point becomes degenerate.

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Symplectic geometry and spectral properties of classical and quantum coupled angular momenta

Cited by 2 publications

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Minimal models of compact symplectic semitoric manifolds

Minimal models of compact symplectic semitoric manifolds

Symplectic classification of coupled angular momenta

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