Semitoric Systems Overview

• Semitoric systems are two-degrees-of-freedom integrable models on 4D symplectic manifolds defined by a momentum map F=(J,H) with J generating an effective S¹-action and exhibiting only nondegenerate singularities.
• They feature characteristic focus–focus singularities that manifest as multipinched tori, leading to nontrivial monodromy and necessitating a vertical cutting procedure to regularize the affine structure.
• The constructed affine invariant Δ generalizes the standard toric momentum polygon by encoding shear jumps and decomposing the image into distinct regions, thus enabling complete symplectic classification even when J is non-proper.

A semitoric system is a two-degrees-of-freedom integrable system on a $4$-dimensional symplectic manifold $(M, \omega)$, specified by a momentum map $F=(J, H): M \to \mathbb{R}^2$ such that $J$ generates an effective Hamiltonian $S^1$-action and all singularities are nondegenerate in the sense of Williamson, with no hyperbolic blocks. Generalized semitoric systems relax the requirement that $J$ be proper, instead only assuming $F$ itself is proper and all regular fibers are connected. The presence of focus-focus singularities—fibers that are topologically multipinched tori—is a characteristic feature. The image $F(M)$ is a singular integral affine manifold whose structure encodes both the global monodromy and the local action geometry of the integrable system. To address the global complexity introduced by focus-focus points and possibly non-proper $J$, the construction of an affine invariant generalizes Vũ Ngọc’s vertical cutting procedure, yielding a set $\Delta$ in $\mathbb{R}^2$ that serves as a complete symplectic invariant for the system (Pelayo et al., 2013).

1. Structure and Properties of Generalized Semitoric Systems

A generalized semitoric system is defined by a proper map $F=(J,H): M \to \mathbb{R}^2$ on a connected, oriented, $4$-dimensional symplectic manifold, where:

• $J$ is the momentum map for an effective Hamiltonian $S^1$-action; $J$ itself may not be proper.
• The singularities are nondegenerate (no hyperbolic blocks), corresponding to Williamson types elliptic–elliptic, elliptic–regular, or focus–focus.
• F admits properness (so $F^{-1}$(compact) is compact), but individual $J$ fibers may be noncompact.
• The presence of isolated focus–focus singularities is allowed.

This framework models key classical mechanical systems, such as the spherical pendulum, which are non-toric due to the non-properness of $J$. The systems preserve connectedness of regular fibers and exclude Hamiltonian flows with hyperbolic singularities.

2. Focus–Focus Singularities and Affine Monodromy

Focus–focus singularities are distinguished by their linearization: complex conjugate eigenvalues with nonzero real and imaginary parts in the linearized flows at these critical points. Locally, Eliasson’s normal form applies: in canonical coordinates $(x_1, x_2, \xi_1, \xi_2)$, the integrals near a focus–focus point are given by $q_1 = x_1\xi_2 - x_2\xi_1$ and $q_2 = x_1\xi_1 + x_2\xi_2$. The preimage $F^{-1}(c_i)$ over a focus–focus value $c_i$ is a multipinched torus, and the existence of such points is associated with nontrivial monodromy—i.e., the failure of global action–angle coordinates to trivialize the affine structure globally.

In the image $F(M)$, each focus–focus value $c_i = (x_i, y_i)$ appears as an isolated interior point. The monodromy is topologically and geometrically encoded by the conjunction of these singular fibers and global affine data.

3. Construction of the Vertical Cutting and Affine Invariant

To regularize the global affine structure in the presence of focus–focus monodromy, the construction of the invariant $\Delta$ involves:

• Vertical Cutting Procedure: For each focus–focus value $c_i = (x_i, y_i)$ and chosen sign $\varepsilon_i = \pm1$, define the vertical half-line (cut)

$$\ell_i^{(\varepsilon_i)} = \{ (x_i, y) \in \mathbb{R}^2 \mid \varepsilon_i(y - y_i) \geq 0 \}$$

and perform a "cut" along this set in $F(M)$.

• Homeomorphism to Affine Model: There exists a homeomorphism

$f: F(M) \to f(F(M)) \subset \mathbb{R}^2,$

with $f(x, y) = (x, f^{(2)}(x, y))$, such that: - $f$ is an affine diffeomorphism away from the union of cuts. - Across a cut, the derivative of $f$ jumps according to

$$\lim_{(x,y) \to c^-} d f(x, y) = T^{k(c)} \cdot \lim_{(x,y)\to c^+} d f(x, y),$$

where $$T = \begin{pmatrix} 1 & 0 \ 1 & 1 \end{pmatrix} \in SL(2, \mathbb{Z})$$ is a shear matrix, and $k(c) = \sum_{i \in I_c} \varepsilon_i k_i$ combines the contributions of intersecting cuts.

• Affine Invariant Set: The symplectic invariant is

$\Delta := f(F(M)) \subset \mathbb{R}^2$

defined up to the natural action of a subgroup of $Aff(2, \mathbb{Z})$. $\Delta$ encodes the corrected affine structure of the system, rectifying for monodromy.

4. Geometry of the Invariant: Decomposition into Region Types

The invariant $\Delta$ admits a canonical decomposition into a union of four types of planar regions:

Region Type	Description	Structural Formula
I	Convex polygon slices	$$\mathcal{R} = \Delta \cap \{ (x, y) \mid x \in I \}$$
II	Between convex, piecewise-linear lower and lower-semicontinuous upper bounds	$$\mathcal{R} = \{ (x, y) \mid x \in I,\, f(x) < y < g(x) \}$$, $f$ convex/continuous, $g$ lower semicontinuous
III	Reversed roles (upper semicontinuous lower bound, concave upper)	Analogous, with bounds reversed
IV	Bounded between upper/lower semicontinuous functions	Both bounds possibly only semicontinuous

This stratification captures both topological and geometric complexity arising from the focus–focus monodromy and potential lack of properness of $J$, distinguishing the generalized case from the toric one.

5. Comparison to Toric Systems

In the toric setting, the momentum map $\mu: M \to \mathbb{R}^2$ is proper, both commuting flows generate an effective $T^2$-action, and the momentum image $\mu(M)$ is a convex polytope (the Delzant polygon, by the Atiyah–Guillemin–Sternberg theorem). There are no focus–focus singularities, thus no monodromy:

• The vertical cutting procedure is unnecessary; $f$ is the identity.
• The affine invariant $\Delta$ coincides with the standard convex momentum polygon.

Generalized semitoric systems must be distinguished from their toric counterparts primarily by the emergence of nontrivial monodromy, the greater richness of the affine structure, and the relaxation of properness constraints on the generating component.

6. Classification, Applications, and Broader Context

The construction of the affine invariant $\Delta$ enables:

• Extension of symplectic classification from toric to a broader class including classical mechanical systems with nonproper $J$ (e.g., the spherical pendulum).
• Encoding of both the number and multiplicity of focus–focus singularities by the affine monodromy jumps (“shear indices”) and the geometry of $\Delta$, allowing fine symplectic distinction between systems.
• Fundamental tools for studying Lagrangian torus fibrations, applications in mirror symmetry, wall-crossing phenomena, and toric degenerations in algebraic geometry. The refined affine structure is directly linked to the global properties of action–angle variables and provides a precise measure of when these cannot be trivialized globally due to singular monodromy.
• Influence on recent developments in the theory of almost-toric manifolds, where generalized semitoric systems serve as local models for charting more singular Lagrangian fibrations.

The vertical cutting technique and the associated homeomorphism $f$ serve as a template for generalizing affine invariants to broader classes of integrable systems with singular Lagrangian fibrations, forming a bridge between classical toric geometry and the contemporary landscape of integrable systems with more intricate singularities.

7. Summary Table: Generalized Semitoric versus Toric Systems

Feature	Generalized Semitoric	Toric
Properness	$F$ proper; $J$ may not be proper	Both components proper
S¹-action	Present (generated by $J$)	Present (full $T^2$)
Allowed Singularities	Nondegenerate, no hyperbolic	Elliptic, no focus–focus
Focus–focus Points	Allowed; finite number	None
Monodromy	Possible (nontrivial)	None
Affine Invariant ($\Delta$)	Image via vertical cuts; union of region types I–IV	Delzant polytope

These properties underscore how the generalized semitoric theory expands the range of integrable systems covered by convexity and classification theorems and provides a robust symplectic invariant, $\Delta$, incorporating monodromy data that goes beyond the classical toric context (Pelayo et al., 2013).