[Paper Review] Families of four-dimensional integrable systems with $S^1$-symmetries

Yohann Le Floch, Joseph W. Palmer|arXiv (Cornell University)|Jul 20, 2023

Nonlinear Waves and SolitonsPhysics and Astronomy3 citations

TL;DR

This paper introduces a systematic method to construct explicit four-dimensional integrable systems with $S^1$-symmetries by deforming toric systems via Hamiltonian-Hopf bifurcations, enabling the realization of all strictly minimal semitoric systems—particularly those of types (3a), (3b), and (3c)—through one-parameter families that transition between toric, semitoric, and hypersemitoric types. The key contribution is a constructive approach to realizing every marked semitoric polygon type, including explicit systems on $\mathbb{CP}^2$ and Hirzebruch surfaces, with full classification of invariants such as the height invariant and flap structure.

ABSTRACT

The aim of this paper is to give new insights about families of integrable systems lifting a Hamiltonian $S^1$-space. Specifically, we study one-parameter families $(M^4,ω,F_t=(J,H_t))_{0 \leq t \leq 1}$ of systems with a fixed Hamiltonian $S^1$-space $(M,ω,J)$ and which are semitoric for certain values of the parameter $t$, with a focus on such families in which one singular point undergoes a Hamiltonian-Hopf bifurcation (also called nodal trade in the context of semitoric systems, and more generally almost toric fibrations). Beyond semitoric systems, we also study families containing hypersemitoric systems, and we investigate the local theory of a nodal trade.\\Building on and generalizing the ideas of a previous paper, we show how such families can be used to find explicit semitoric systems with certain desired invariants (bundled in the marked semitoric polygon). This allows us to make progress on the semitoric minimal model program by understanding and coming up with explicit systems for each strictly minimal type (i.e., those not admitting any toric or semitoric type blowdown). In order to obtain these systems, we develop strategies for constructing and understanding explicit examples of semitoric (and hypersemitoric) systems in general. One strategy we make use of is to start from a well-understood system (such as a toric system) and to explicitly induce Hamiltonian-Hopf bifurcations to produce focus-focus singular points. This is an expanded version of the technique used in the aforementioned previous paper, in order to apply it to semitoric systems which include non-trivial isotropy spheres in the underlying $S^1$-space (i.e., $\mathbb{Z}_k$-spheres), which occurs in several of the strictly minimal systems.\\In particular, we give an explicit one-parameter family of systems on $\mathbb{CP}^2$ which transitions between being of toric type, semitoric type, and hypersemitoric type depending on the value of the parameter. We study this system at each stage, computing the marked semitoric polygon of the semitoric system and determining several properties of the hypersemitoric system, including the existence of a unique flap and two parabolic orbits. Furthermore, we study the transitions between these stages.\\We also come up with new explicit semitoric systems on all Hirzebruch surfaces which, together with the previous systems and the systems already contained in the literature, gives an explicit model for every type of strictly minimal system. Moreover, we show how to obtain every strictly minimal system by applying sequences of alternating toric type blowups and blowdowns to simple explicit systems. In particular, we obtain that every strictly minimal semitoric polygon can be obtained from a semitoric system which is part of a family $(M,ω,F_t=(J,H_t))$ which is semitoric for all but a finite number of values of $t$, called a semitoric family.

Motivation & Objective

• To develop a general strategy for constructing explicit semitoric and hypersemitoric systems on familiar manifolds such as $\mathbb{CP}^2$ and Hirzebruch surfaces.
• To resolve the minimal model program for semitoric systems by explicitly realizing all strictly minimal types—those not admitting toric or semitoric blowdowns.
• To understand the local and global behavior of Hamiltonian-Hopf bifurcations (nodal trades) in families of integrable systems with $S^1$-symmetry.
• To compute and classify key invariants—such as the marked semitoric polygon, height invariant, and flap structure—for new families of systems.
• To demonstrate that every strictly minimal semitoric system arises as part of a semitoric family, i.e., a one-parameter family semitoric for all but finitely many $t$.

Proposed method

• Constructing one-parameter families $(M^4, \omega, F_t = (J, H_t))$ starting from toric systems and inducing Hamiltonian-Hopf bifurcations to generate focus-focus singularities.
• Using symplectic reduction on $\mathbb{C}^4$ with a moment map $N = \frac{1}{2}(|z_1|^2 + |z_3|^2 + (n-2)|z_4|^2, |z_2|^2 + |z_4|^2)$ to define the underlying manifold $W_{n-2}(\beta, \beta)$.
• Defining the perturbed Hamiltonian $H_t = \frac{2t-1}{2}|z_3|^2 + 2\gamma t(\mathcal{X} + \delta R^2) - 2\gamma\delta t((n-1)\beta + \alpha)^2$, where $\mathcal{X} = \Re(z_1 z_2 \bar{z}_3^{n-1} z_4)$ and $R = \frac{1}{2}(|z_1|^2 + (n-2)|z_4|^2)$.
• Analyzing the transition between system types by varying $t$: toric for $t < t^-$, semitoric with one focus-focus point for $t^- < t < t^+$, and hypersemitoric for $t$ near $t^+$.
• Computing the marked semitoric polygon via the marked polygon isomorphism, showing explicit polygon types (3a), (3b), and (3c) depending on parameters.
• Using the height invariant and local normal forms to verify the existence of unique flaps and parabolic orbits in hypersemitoric systems.

Experimental results

Research questions

• RQ1 Can all strictly minimal semitoric systems be explicitly constructed using a deformation of toric systems via Hamiltonian-Hopf bifurcations?
• RQ2 What are the precise parameter conditions under which a one-parameter family transitions between toric, semitoric, and hypersemitoric types?
• RQ3 How can the marked semitoric polygon be explicitly computed for systems arising from such bifurcations?
• RQ4 What is the role of $\mathbb{Z}_k$-spheres (non-trivial isotropy spheres) in the construction of strictly minimal systems?
• RQ5 Can every strictly minimal semitoric system be realized as part of a semitoric family, i.e., a continuous family semitoric for all but finitely many $t$?

Key findings

• A one-parameter family on $\mathbb{CP}^2$ is explicitly constructed that transitions from toric to semitoric to hypersemitoric type, with $t^-$ and $t^+$ given by $t^- = \frac{27}{2(27 + 4\sqrt{5})} \approx 0.38$ and $t^+ = \frac{27}{2(27 - 4\sqrt{5})} \approx 0.75$ for $n=4$, $\alpha=2$, $\beta=1$, $\gamma=\frac{1}{90}$, $\delta=15$.
• For $n=4$, the semitoric system in the interval $t \in (t^-, t^+)$ has a marked semitoric polygon of type (3a), as shown in Figure 12(d).
• The hypersemitoric system in the same family features a unique flap and two parabolic orbits, confirming its non-toric, non-simple structure.
• For $n=3$, the height invariant $h_0$ can be made arbitrarily close to $h^+ = \left(1 - \frac{3\sqrt{3}}{4\pi}\right)\beta$ by choosing appropriate $\gamma$, $\delta$, and $t$ in the semitoric regime.
• For $n=4$, the maximal height invariant is $h^+ = \left(1 - \frac{\ln(12 - 8\sqrt{2})}{\pi}\right)\beta$, and this bound is approached for suitable parameter choices.
• The paper constructs explicit semitoric systems of types (3a), (3b), and (3c) on Hirzebruch surfaces and $\mathbb{CP}^2$, completing an explicit model for every strictly minimal type.