[Paper Review] From semi-toric systems to Hamiltonian S^1-spaces

Sonja Hohloch, Silvia Sabatini+1 more|arXiv (Cornell University)|May 30, 2013

Geometric and Algebraic Topology24 references3 citations

TL;DR

This paper establishes a direct correspondence between labeled convex polygons associated with semi-toric systems and Karshon's labeled directed graphs classifying Hamiltonian S¹-spaces. It proves that any such polygon determines the underlying S¹-space's graph up to isomorphism, and characterizes adaptable semi-toric systems—those extendable to Hamiltonian T²-actions—by showing they are precisely those whose associated polygons satisfy the Delzant condition.

ABSTRACT

This paper studies the local and global aspects of semi-toric integrable systems, introduced by Vu Ngoc, using ideas stemming from the theory of Hamiltonian S^1-spaces developed by Karshon. First, we show how any labeled convex polygon associated to a semi-toric system (as defined by Vu Ngoc) determines Karshon's labeled directed graph which classifies the underlying Hamiltonian S^1-space up to isomorphism. Then we characterize adaptable semi-toric systems, i.e. those whose underlying Hamiltonian S^1-action can be extended to an effective Hamiltonian T^2-action, as those which have at least one associated convex polygon which satisfies the Delzant condition.

Motivation & Objective

• To establish a direct correspondence between the labeled convex polygons of semi-toric systems and the labeled directed graphs classifying their underlying Hamiltonian S¹-spaces.
• To characterize which semi-toric systems admit an extension of their S¹-action to an effective Hamiltonian T²-action (adaptable systems).
• To clarify the relationship between symplectic invariants in semi-toric systems and the classification invariants of Hamiltonian S¹-spaces.
• To demonstrate that the labeled directed graph of the underlying S¹-space can be reconstructed solely from the labeled polygon of a semi-toric system.
• To provide a geometric criterion for adaptability based on the combinatorial structure of the associated polygon.

Proposed method

• Use of the Eliasson-Miranda-Zung local normal form to control the geometry near focus-focus singularities.
• Leverage the connectedness of fibers of the semi-toric moment map Φ to ensure global consistency in the construction.
• Adapt Karshon’s graph-theoretic classification of Hamiltonian S¹-spaces to the semi-toric setting via polygon-to-graph mapping.
• Apply piecewise integral affine transformations to relate different polygon representations of the same semi-toric system.
• Analyze vertex smoothness conditions in semi-toric polygons using self-intersection invariants of S¹-orbits.
• Use nodal trades and blow-ups to construct explicit examples of non-adaptable systems with controlled singular point configurations.

Experimental results

Research questions

• RQ1 How can the labeled directed graph of the underlying Hamiltonian S¹-space be reconstructed from the labeled convex polygon of a semi-toric system?
• RQ2 What combinatorial or geometric condition on the labeled polygon ensures that the S¹-action extends to a Hamiltonian T²-action?
• RQ3 What invariants of a semi-toric system determine the isomorphism class of its underlying Hamiltonian S¹-space?
• RQ4 How do the numbers and types of singular points (elliptic-elliptic, focus-focus, and non-free orbits) constrain the structure of the associated polygon?
• RQ5 What characterizes non-adaptable semi-toric systems in terms of their polygonal invariants and vertex smoothness?

Key findings

• Any labeled convex polygon associated to a semi-toric system uniquely determines the labeled directed graph of its underlying Hamiltonian S¹-space.
• A semi-toric system is adaptable (i.e., its S¹-action extends to a Hamiltonian T²-action) if and only if at least one of its associated labeled convex polygons satisfies the Delzant condition.
• Non-adaptable semi-toric systems must have at least one polygon vertex that is not smooth, as characterized by failure of the smoothness conditions in Lemma 4.2.
• For non-adaptable systems, there exists a regular level set J⁻¹(x) containing at least three singular orbits (elliptic-elliptic, focus-focus, or non-free S¹-orbits), with specific combinations of critical point counts as detailed in Corollary 4.14.
• The number of elliptic-elliptic and focus-focus critical points in a fiber J⁻¹(x) is bounded: Ex ≤ 2 and Sx ≤ 2, with stricter constraints in non-adaptable cases.
• Explicit constructions via nodal trades and blow-ups show the existence of semi-toric systems with two elliptic-elliptic points and an arbitrary number of focus-focus points in a single fiber J⁻¹(x₀).