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Integrable systems with symmetries: toric, semitoric, and beyond

Joseph Palmer

TL;DR

This survey synthesizes the theory of integrable systems with torus symmetries, emphasizing toric and semitoric systems and their classification via decorated polygons. It reviews the construction of explicit semitoric examples through one-parameter families and Hamiltonian-Hopf bifurcations, and details the seminal invariants (marked semitoric polygons, Taylor series, twisting index) and corresponding classification theorems (Pelayo–V Ngc, PPT-nonsimple). The text then extends to semitoric families, the semitoric minimal model program, and generalized settings such as hypersemitoric and complexity-one systems, while outlining open problems and future directions in lifting, bifurcation theory, and quantum connections. Collectively, the work provides a coherent framework for navigating from classical toric geometry to richer integrable systems with singularities and symmetry, and offers concrete tools for constructing and classifying next-generation examples. The methodology highlights decorated polygon invariants as a unifying language across toric, semitoric, and beyond, enabling both theoretical classification and explicit realizations with potential applications in symplectic topology and quantum mechanics.

Abstract

This article presents an overview of the theory of integrable systems with symmetries, focusing on toric systems, semitoric systems, and their classifications via decorated polygons. We discuss certain one-parameter families of integrable systems called semitoric families, and explain how deforming systems through controlled bifurcations in such families (and their generalizations) can be used to construct explicit semitoric systems with prescribed invariants. The first part of the paper serves as a quick introduction to integrable systems for newcomers to the field, such as graduate students, while the majority of the exposition surveys recent developments and technical details that will be of interest to experts. It closes with a look at future directions, including hypersemitoric systems and complexity one integrable systems.

Integrable systems with symmetries: toric, semitoric, and beyond

TL;DR

This survey synthesizes the theory of integrable systems with torus symmetries, emphasizing toric and semitoric systems and their classification via decorated polygons. It reviews the construction of explicit semitoric examples through one-parameter families and Hamiltonian-Hopf bifurcations, and details the seminal invariants (marked semitoric polygons, Taylor series, twisting index) and corresponding classification theorems (Pelayo–V Ngc, PPT-nonsimple). The text then extends to semitoric families, the semitoric minimal model program, and generalized settings such as hypersemitoric and complexity-one systems, while outlining open problems and future directions in lifting, bifurcation theory, and quantum connections. Collectively, the work provides a coherent framework for navigating from classical toric geometry to richer integrable systems with singularities and symmetry, and offers concrete tools for constructing and classifying next-generation examples. The methodology highlights decorated polygon invariants as a unifying language across toric, semitoric, and beyond, enabling both theoretical classification and explicit realizations with potential applications in symplectic topology and quantum mechanics.

Abstract

This article presents an overview of the theory of integrable systems with symmetries, focusing on toric systems, semitoric systems, and their classifications via decorated polygons. We discuss certain one-parameter families of integrable systems called semitoric families, and explain how deforming systems through controlled bifurcations in such families (and their generalizations) can be used to construct explicit semitoric systems with prescribed invariants. The first part of the paper serves as a quick introduction to integrable systems for newcomers to the field, such as graduate students, while the majority of the exposition surveys recent developments and technical details that will be of interest to experts. It closes with a look at future directions, including hypersemitoric systems and complexity one integrable systems.

Paper Structure

This paper contains 63 sections, 15 theorems, 73 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments:
  3. Outline
  4. Symplectic geometry, integrable systems, and Hamiltonian group actions
  5. Symplectic geometry
  6. Hamiltonian vector fields
  7. Integrable systems
  8. Motivation
  9. The definition of an integrable system
  10. First consequences
  11. Examples of integrable systems
  12. Hamiltonian group actions
  13. Motivation
  14. Interactions between integrable systems and Hamiltonian group actions
  15. Symplectic toric manifolds
  16. ...and 48 more sections

Key Result

Lemma 2.2

Let $(M,\omega)$ be a symplectic manifold of dimension $2n$, and suppose that are smooth functions which pairwise Poisson commute and are linearly independent almost everywhere. Then $k\leq n$.

Figures (18)

  • Figure 1:
  • Figure 2: Two representatives of the same marked semitoric polygon, related by changing the cut direction on the second marked point. In this example, all vertices are either Delzant or fake (there are no hidden corners in this example).
  • Figure 3: Two representatives of the same marked semitoric polygon, related by a change in the cut direction. Both representatives include a single fake corner and two Delzant corners.
  • Figure 4: Two representatives of the same marked semitoric polygon, related by a change in the cut direction.
  • Figure 5: Four representatives of the same semitoric polygon, which has two focus-focus points. The labels on the vertices are not shown to simplify the figure, but each edge of each polygon has slope either zero, one, or negative one.
  • ...and 13 more figures

Theorems & Definitions (79)

  • Claim 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Claim 2.4
  • proof
  • Corollary 2.5
  • Example 2.6: Harmonic Oscillator
  • Example 2.7
  • ...and 69 more