Semitoric Systems Research Articles

Semitoric systems of non-simple type

Semitoric systems of non-simple type

Constructions of b-semitoric systems

In this article, we introduce b-semitoric systems as a generalization of semitoric systems, specifically tailored for b-symplectic manifolds. The objective of this article is to furnish a collection of examples and investigate the distinctive characteristics of these systems. A b-semitoric system is a four-dimensional b-integrable system that satisfies certain conditions: one of its momentum map components is proper and generates an effective global S1-action and all singular points are non-degenerate and devoid of hyperbolic components. To illustrate this concept, we provide five examples of b-semitoric systems by modifying the coupled spin oscillator and the coupled angular momenta, and we also classify their singular points. Additionally, we describe the dynamics of these systems through the image of their respective momentum maps.

A Family of Semitoric Systems with Four Focus\u2013Focus Singularities and Two Double Pinched Tori

We construct a one-parameter family F_t=(J, H_t)_{0 le t le 1} of integrable systems on a compact 4-dimensional symplectic manifold (M, omega ) that changes smoothly from a toric system F_0 with eight elliptic–elliptic singular points via toric type systems to a semitoric system F_t for t^-< t < t^+. These semitoric systems F_t have precisely four elliptic–elliptic and four focus–focus singular points. Moreover, at t= frac{1}{2}, the system has precisely two focus–focus fibres each of which contains exactly two focus–focus points, giving these fibres the shape of double pinched tori. We exemplarily parametrise one of these fibres explicitly.

The Height Invariant of a Four-Parameter Semitoric System with Two Focus\u2013Focus Singularities

Semitoric systems are a special class of completely integrable systems with two degrees of freedom that have been symplectically classified by Pelayo and Vũ Ngọc about a decade ago in terms of five symplectic invariants. If a semitoric system has several focus–focus singularities, then some of these invariants have multiple components, one for each focus–focus singularity. Their computation is not at all evident, especially in multi-parameter families. In this paper, we consider a four-parameter family of semitoric systems with two focus–focus singularities. In particular, apart from the polygon invariant, we compute the so-called height invariant. Moreover, we show that the two components of this invariant encode the symmetries of the system in an intricate way.

Symplectic invariants of semitoric systems and the inverse problem for quantum systems

Simple semitoric systems were classified about ten years ago in terms of a collection of invariants, essentially given by a convex polygon with some marked points corresponding to focus-focus singularities. Each marked point is endowed with labels which are symplectic invariants of the system. We will review the construction of these invariants, and explain how they have been generalized or applied in different contexts. One of these applications concerns quantum integrable systems and the corresponding inverse problem, which asks how much information of the associated classical system can be found in the spectrum. An approach to this problem has been to try to compute invariants in the spectrum. We will explain how this has been recently achieved for some of the invariants of semitoric systems, and discuss an open question in this direction.

Symplectic classification of coupled angular momenta

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 and Vũ Ngọc 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.

Taylor series and twisting-index invariants of coupled spin–oscillators

About six years ago, semitoric systems on 4-dimensional manifolds were classified by Pelayo & Vũ Ngọc by means of five invariants. A standard example of such a system is the coupled spin–oscillator on S2×R2. Calculations of three of the five semitoric invariants of this system (namely the number of focus–focus singularities, the generalised semitoric polygon, and the height invariant) already appeared in the literature, but the so-called twisting index was not yet computed and, of the so-called Taylor series invariant, only the linear terms were known.In the present paper, we complete the list of invariants for the coupled spin–oscillator by calculating higher order terms of the Taylor series invariant and by computing the twisting index. Moreover, we prove that the Taylor series invariant has certain symmetry properties that make the even powers in one of the variables vanish and allow us to show superintegrability of the coupled spin–oscillator on the zero energy level.

Faithful Semitoric Systems

This paper consists of two parts. The first provides a review of the basic properties of integrable and almost-toric systems, with a particular emphasis on the integral affine structure associated to an integrable system. The second part introduces faithful semitoric systems, a generalization of semitoric systems (introduced by Vu Ngoc and classified by Pelayo and Vu Ngoc) that provides the language to develop surgeries on almost-toric systems in dimension 4. We prove that faithful semitoric systems are natural building blocks of almost-toric systems. Moreover, we show that they enjoy many of the properties that their (proper) semitoric counterparts do.

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

About 6 years ago, semitoric systems were classified by Pelayo &amp; Vũ Ngọc 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.

Integrable systems, symmetries, and quantization

These notes correspond to a mini-course given at the Poisson 2016 conference in Geneva. Starting from classical integrable systems in the sense of Liouville, we explore the notion of non-degenerate singularity and expose recent research in connection with semi-toric systems. The quantum and semiclassical counterpart will also be presented, in the viewpoint of the inverse question: from the quantum mechanical spectrum, can you recover the classical system?

Moduli spaces of semitoric systems

Recently Pelayo–Vũ Ngọc classified simple 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.

Generating Hyperbolic Singularities in Semitoric Systems Via Hopf Bifurcations

Let $(M,\Omega)$ be a connected symplectic 4-manifold and let $F=(J,H) : M \to \mathbb{R}^2$ be a completely integrable system on $M$ with only non-degenerate singularities and for which $J : M \to \mathbb{R}$ is a proper map. Assume that $F$ does not have singularities with hyperbolic blocks and that $p_1,...,p_n$ are the focus-focus singularities of $F$. For each subset $S=\{i_1,...,i_j\}$ we will show how to modify $F$ locally around any $p_i, i \in S$, in order to create a new integrable system $\tilde{F}=(J, \tilde{H}) : M \to \mathbb{R}^2$ such that its classical spectrum $\tilde{F}(M)$ contains $j$ smooth curves of singular values corresponding to non-degenerate transversally hyperbolic singularities of $\tilde{F}$. Moreover the focus-focus singularities of $\tilde{F}$ are precisely $p_i$, $i \in \{1,...,n\} \setminus S$, and each of these $p_i$ is non-degenerate. The proof is based on Eliasson's linearization theorem for non-degenerate singularities, and properties of the Hamiltonian Hopf bifurcation.

Inverse spectral theory for semiclassical Jaynes–Cummings systems

Quantum semitoric systems form a large class of quantum Hamiltonian integrable systems with circular symmetry which has received great attention in the past decade. They include systems of high interest to physicists and mathematicians such as the Jaynes–Cummings model (1963), which describes a two-level atom interacting with a quantized mode of an optical cavity, and more generally the so-called systems of Jaynes–Cummings type. In this paper we consider the joint spectrum of a pair of commuting semiclassical operators forming a quantum integrable system of Jaynes–Cummings type. We prove, assuming the Bohr–Sommerfeld rules hold, that if the joint spectrum of two of these systems coincide up to \(\mathcal {O}(\hbar ^2)\), then the systems are isomorphic.

From compact semi-toric systems to Hamiltonian $S^1$-spaces

We show how any labeled convex polygon associated to a compact semi-toric system, as defined by Vũ ngọc, determinesKarshon's labeled directed graph which classifies the underlyingHamiltonian $S^1$-space up to isomorphism. Then we characterizeadaptable compact semi-toric systems, i.e. those whoseunderlying Hamiltonian $S^1$-action can be extended to an effectiveHamiltonian $\mathbb{T}^2$-action, as those which have at least oneassociated convex polygon which satisfies the Delzant condition.