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(Non)displaceability in semitoric systems

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Pith's one-line read This paper proves a rectangle criterion for displacing focus-focus fibers of semitoric systems and classifies fiber (non)displaceability in three example families.

desk verdict Worth refereeing: the semitoric probe idea is novel and the fiber classifications are valuable, but Lemma 3.5's nodal-trade step is a real gap that must be fixed. read the letter →

arxiv 2411.16601 v2 pith:3E5ELJBP submitted 2024-11-25 math.SG math.DSnlin.SI

Sonja Hohloch , Pedro Santos This is my paper · ORCID
classification math.SGmath.DSnlin.SI MSC 53D1257R1770H0653D4053D20
keywords semitoricintegrablesystemsfocus-focusfibersdisplaceabilitymethodofprobespolytopeinvariantnodaltradeLagrangianspheressymplecticquasi-states
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper extends the probe method, a polytope-based technique for proving that certain Lagrangian fibers can be displaced by Hamiltonian diffeomorphisms, from toric systems to semitoric integrable systems. It proves two opposing statements about focus-focus fibers, the singular fibers that distinguish semitoric from toric systems. A focus-focus fiber containing two or more focus-focus points is nondisplaceable, because it contains an embedded Lagrangian sphere. A focus-focus fiber containing a single focus-focus point is displaceable whenever a representative of the polytope invariant contains an affine rectangle $[0,R]\times[0,h]$ with $h

What carries the argument

The central machinery is the polytope invariant of a semitoric system: the momentum image, cut along vertical rays through focus-focus values and straightened by a homeomorphism that is affine away from the cuts. In a representative, a single focus-focus value creates a corner; a nodal trade, which is a surgery operation on almost toric bases, turns that corner into a Delzant corner and makes the system toric nearby. The rectangle $[0,R]\times[0,h]$ in the straightened polytope encodes a symplectic embedding of the product of disks $D(R)\times D(h)$, and the inequality $h<R/2$ is precisely the condition under which a compactly supported Hamiltonian on the big disk displaces every circle of radius up to $h$ (Lemma 2.24). On the nondisplaceability side, the load-bearing object is the focus-focus fiber as a chain of Lagrangian spheres: an embedded Lagrangian sphere has self-intersection $\pm 2$, so it cannot be displaced even topologically. Symplectic quasi-states and pseudoheavy fibers appear as a second, Floer-theoretic tool for identifying the unique nondisplaceable stem fibers in the examples.

What would settle it

Exhibit a semitoric system whose polytope invariant contains the rectangle $[0,R]\times[0,h]$ with $h<R/2$ but whose single focus-focus fiber is provably nondisplaceable, for instance by showing the fiber is superheavy for a symplectic quasi-state or by computing a nonzero Lagrangian Floer obstruction.

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Extended reading notes

Core claim

The central claim is a dichotomy for focus-focus fibers in semitoric systems. If the fiber contains at least two focus-focus points, it is nondisplaceable: topologically it is a closed chain of embedded Lagrangian $2$-spheres, and an embedded Lagrangian sphere in a symplectic $4$-manifold cannot be displaced even by smooth maps homotopic to the inclusion (Proposition 3.3 and Theorem 1.1). If the fiber contains exactly one focus-focus point, displacement is controlled by the polytope invariant: after an integral affine transformation and a nodal trade that replaces the focus-focus corner by a Delzant corner, if the representative contains the affine rectangle $[0,R]\times[0,h]$ with $0<h<R/2$, then the focus-focus fiber lies inside $\pi^{-1}([0,h]\times[0,h])$ for the traded toric fibration and is displaced by a compactly supported Hamiltonian built from the disk lemma (Lemma 2.24, Lemma 3.5, and Theorem 1.2). The paper then checks the rectangle condition in three explicit systems and determines which fibers are stems, which are displaceable, and, in the Kepler and octagon cases, how the answer changes with the height invariant.

Load-bearing premise

The load-bearing premise is that the affine rectangle in a representative of the polytope invariant corresponds to an actual symplectic product of two disks inside the manifold, and that the nodal trade preserves that product embedding while moving the focus-focus fiber into the inner square.

Editorial extensions

If this is right

  • Every focus-focus fiber containing at least two focus-focus points is nondisplaceable, since it contains an embedded Lagrangian sphere.
  • Any semitoric system whose polytope invariant contains the affine rectangle $[0,R]\times[0,h]$ with $h<R/2$ has a displaceable focus-focus fiber, so the displacement question reduces to a check of the polytope invariant.
  • In the coupled spin-oscillator, every fiber, including the focus-focus fiber, is displaceable.
  • For coupled angular momenta with $R_1\neq R_2$, exactly one fiber is nondisplaceable and it is a stem, while the focus-focus fiber is displaceable; for the Kepler problem $R_1=R_2$ the answer changes at a parameter value $t_0$, with the focus-focus fiber displaceable for $t<t_0$, a stem at $t=t_0$, and nondisplaceable together with infinitely many other fibers for $t>t_0$.
  • For the semitoric octagon system, the focus-focus fibers are displaceable when the height invariant satisfies $h<1$ and nondisplaceable when $1\le h\le 3/2$, and at $h=3/2$ the double focus-focus fibers are nondisplaceable by the embedded-sphere argument.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension: the rectangle criterion is purely affine-geometric, so in any concrete semitoric system the (non)displaceability of a single focus-focus fiber could be decided by a finite computation from the polytope invariant and height invariant.
  • The Kepler transition at $t_0$ suggests a general bifurcation pattern: as the height invariant shrinks until the rectangle $[0,R]\times[0,h]$ with $h<R/2$ no longer fits, the focus-focus fiber passes from displaceable to nondisplaceable; checking whether the octagon family exhibits the same transition as $h$ crosses $1$ would test this pattern.
  • The topological argument behind Theorem 1.1 applies to any integrable system whose singular fiber contains an embedded Lagrangian sphere, so the nondisplaceability of multiplicity-two focus-focus fibers likely survives in hypersemitoric and other more general settings.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a generalization of McDuff's method of probes to semitoric integrable systems on four-manifolds. The main theoretical results are Theorem 1.1, asserting that a focus-focus fiber containing at least two focus-focus points is nondisplaceable because it contains embedded Lagrangian spheres, and Theorem 1.2/Lemma 3.5, giving a sufficient condition, expressed through a rectangle in a representative of the polytope invariant, under which a focus-focus fiber of multiplicity one is displaceable after a nodal trade. These tools are then applied to three families: the coupled spin-oscillator, the coupled angular momenta including the Kepler problem, and a semitoric perturbation of the octagon toric system. For each family the paper gives a complete or nearly complete classification of displaceable and nondisplaceable fibers, including stem-type nondisplaceable fibers in the Kepler and octagon examples.

Significance. If the central displacement criterion is correct, the paper introduces a useful and broadly applicable method for semitoric systems, connecting polytope invariants with symplectic rigidity. The nondisplaceability result for multi-pinched focus-focus fibers is simple but elegant and is proved by a clean topological argument. The explicit classifications in Section 4, in particular the stem result at the Kepler parameter t0 and the octagon analysis, would be valuable additions to the symplectic topology of integrable systems. The paper is generally well organized and the appendix computation of the polytope invariant for the octagon family is a concrete contribution. However, the central Lemma 3.5 is proved in a single sentence and relies on an unverified passage from polytope-invariant rectangles to symplectic product embeddings after nodal trade; since all Section 4 applications use this lemma, the central claim needs substantial additional justification.

major comments (4)
  1. [§3.3, Lemma 3.5] The proof of Lemma 3.5 is one sentence: 'After applying a nodal trade ... the semitoric system ... becomes a toric fibration π : (M,ω) → R2 ... F^{-1}(c) is contained in π^{-1}([0,h]×[0,h]).' This is not justified by the cited results. Lemma 2.15 and Theorem 2.16 compare almost toric bases and assert that the associated total spaces are symplectomorphic; they do not directly produce a toric fibration on the original (M,ω) whose preimage contains the given focus-focus fiber. Moreover, the rectangle [0,R]×[0,h] lives in a representative of the polytope invariant, which is obtained by a straightening homeomorphism, and the paper does not show that this affine rectangle corresponds, after the trade, to a symplectic embedding D(R)×D(h) with action-angle coordinates valid across the former singular corner. Because Lemma 3.5 is used in every focus-focus displacement argument in Section 4, this is a load-bearing gap. A detailed proof, or a precise reformulation of the lemma with explicit hypotheses on the nodal trade and the inclusion of the focus-focus fiber, is required.
  2. [§4.2.3, Proposition 4.14] The disjointness estimate in Proposition 4.14 is not satisfied by the stated threshold. The intervals H_t(S_c) have length at most 8(1-t), while consecutive values in A are spaced by 1/(2N); disjointness requires 8(1-t) < 1/(2N), i.e. 1-t < 1/(16N). The hypothesis t > 1 - 1/2^{N+2} only gives 1-t < 1/2^{N+2}, which is insufficient for N=3 and N=4 (for example, when N=3 the maximal interval length is 1/4 while the spacing is 1/6). Since the proposition is stated for every N, the claimed bound is false as written; the correct sufficient bound is t > 1 - 1/(16N). Although Lemma 4.16 later supersedes the counting statement, Proposition 4.14 remains a stated result and must be corrected or removed.
  3. [§4.3.6, Lemmas 4.37 and 4.38] Lemma 4.37, which is essential for the octagon classification when 1 < h ≤ 3/2, is proved only by saying that the proof is analogous to Lemma 4.34 'with the addition that one needs to do certain nodal slides'. The lemma does not specify which nodal slides are performed, why they preserve the fibers being displaced, or how the probe lengths behave after the slides. Lemma 4.38 is likewise dispatched as 'analogous to Lemma 4.29'. These lemmas feed directly into Corollaries 4.40 and 4.41 and Proposition 1.6, so the octagon results for h>1 are not established by the present text. Full proofs, or precise references to the nodal-slide statements used, are needed.
  4. [§4.2.2, Proposition 4.11 and Theorem 4.12] The proof of Proposition 4.11 invokes a nodal slide that 'makes the segment of the eigenline as small as possible while preserving the fiber' and then identifies the resulting torus with the fiber (x,x) of the system for a parameter t̃>t. It is not demonstrated that the given Lagrangian fiber is preserved by the nodal slide, nor that the Hamiltonian value y remains in the range under the identification with the system at t̃. Since Theorem 4.12, the stem result at t=t0, depends on Proposition 4.11, this step needs a precise argument or a citation establishing the invariance of the Chekanov-type torus under the slide and under the parameter change.
minor comments (5)
  1. [§1.1, Theorem 1.2] Theorem 1.2 is stated informally as 'enough space, in the sense of Figure 1.2'; the actual content is in Lemma 3.5. It would be clearer to state a precise theorem with the rectangle hypotheses and the nodal-trade conditions.
  2. [§4.2.3, Proposition 4.14] The phrase 'at least 2N−1+1 nondisplaceable fibers' is confusing: the set A in the proof has 2^N elements, so the intended lower bound is presumably 2^N−1+1 or simply 2^N. Please correct the notation.
  3. [Equation (4.3)] The matrix A is written as belonging to GL(n,Z), but in this four-dimensional setting it should be GL(2,Z).
  4. [§4.2, Proposition 4.4] The claim that Ψ is Hamiltonian because it is the time-1 map of a suitable normalization of G should be written out with the correct sign conventions for the symplectic form ω = −(R1ωS2 ⊕ R2ωS2); this would help the reader verify the Hamiltonian vector field computation.
  5. [§2.13, Theorem 2.33] The statement that 'any integrable system' on a symplectic 4-manifold has a nondisplaceable fiber relies on existence of a partial symplectic quasi-state; the theorem as quoted omits the compactness/rationally-conditioned hypotheses from the cited sources. Adding a short qualification would avoid overstatement.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation found; the central gap in Lemma 3.5 is an unproved inclusion, not a circular reduction.

full rationale

The paper's derivation chain is not circular. The displaceability criterion (Theorem 1.2 / Lemma 3.5) is a sufficient geometric condition in terms of a rectangle inside a representative of the polytope invariant; the rectangle condition is not defined in terms of displaceability, and the displacement mechanism comes from McDuff's probes and the Abreu-Borman-McDuff disk Hamiltonian, both external results. The one-sentence proof of Lemma 3.5, 'After applying a nodal trade... the semitoric system with focus-focus fiber F^{-1}(c) becomes a toric fibration pi : (M, omega) -> R^2. Hereby the set F^{-1}(c) is contained in pi^{-1}([0,h] x [0,h])', asserts an inclusion that is not proved by the cited Symington nodal-trade results, which give symplectomorphisms of total spaces rather than a toric fibration on the original system containing the original fiber. This is a load-bearing proof gap and a correctness risk, but it is not circular: the inclusion is not equivalent to the hypotheses by definition, and no fitted parameter is renamed as a prediction. The cited earlier papers by the same authors (DH21, ADH20) supply parameter-free computations of height invariants and the octagon family; these are independent published inputs, not conclusions derived from the target nondisplaceability statements. Nondisplaceability results use external quasi-state, stem, and Lagrangian-sphere arguments. Hence no significant circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new free parameters fitted to data and no new geometric or physical entities. All free parameters are variables of the systems under study, not fitted constants. The axioms are standard theorems from symplectic geometry or, for the Chekanov tori, imported black boxes whose application is not fully derived.

assumptions (7)
  • standard math Liouville-Arnold-Mineur provides action-angle coordinates near regular fibers (Theorem 2.1 in the paper).
    Used throughout Section 3.1 to write a toric fibration as B x T^2 and to construct the disk embedding.
  • standard math Delzant's classification of toric manifolds by Delzant polytopes.
    Used in Section 2.5 to justify reading fiber structure and probes from the polytope.
  • standard math Vu Ngoc's Theorem 2.18 constructs straightening homeomorphisms and representatives of the semitoric polytope invariant.
    Used throughout Sections 3 and 4 to replace the momentum image by a convex polytope with cuts.
  • standard math Symington's nodal trade and nodal slide theorems (Lemma 2.15, Proposition 2.14, Theorem 2.16) preserve the symplectic manifold.
    Used in Lemma 3.5, Proposition 4.11, and Lemma 4.32 to turn focus-focus fibers into toric fibers or to move nodes.
  • standard math Entov-Polterovich, Usher, and Oh: every symplectic 4-manifold carries a partial symplectic quasi-state, so every integrable system has a nondisplaceable fiber (Theorem 2.33).
    Used in Propositions 4.6 and 4.8 to identify the last nondisplaceable fiber after all others are displaced by probes.
  • standard math Kawasaki-Orita: every integrable system has a pseudoheavy fiber for a partial symplectic quasi-state (Theorem 2.31).
    Used in Lemma 4.28 to produce nondisplaceable fibers intersecting given superheavy fibers.
  • domain assumption Chekanov-type tori in the Kepler problem are nondisplaceable, as imported from Auroux and Tonkonog-Vianna.
    Lemma 4.16 asserts that the fibers with 0 < y <= R - h are Chekanov tori whose potential functions have critical points; the paper does not compute the correspondence.

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Pith. "Pith review of (Non)displaceability in semitoric systems." pith.science (2026). https://pith.science/paper/3E5ELJBP

@misc{pith2026241116601,
  author       = {Pith},
  title        = {Pith review of: (Non)displaceability in semitoric systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3E5ELJBP}},
  note         = {Machine review of arXiv:2411.16601}
}
abstract

We adapt and generalize McDuff's method of probes from toric system to so called semitoric integrable systems and apply it to study the (non)displaceability properties of the fibers of $3$ examples of semitoric integrable systems.

Figures

Figures reproduced from arXiv: 2411.16601 by the authors.

Figure 1.1
Figure 1.1. Straightening homeomorphism f⃗ϵ applied to F(M) to obtain a representative of the polytope invariant f⃗ϵ(F(M)). The focus-focus values of F(M) are c1 and c2. focus-focus points is often seen as a torus with k pinches. Alternatively the fiber F −1 (c) can be seen as a closed chain of Lagrangian k spheres joint at the poles. If k = 1 the sphere is immersed, and if k > 1 the spheres are embedded. Our first result is ab… view at source ↗
Figure 1.2
Figure 1.2. A(f⃗ϵ(F(M))) is the result of applying a suitable integral affine transformation A to the representative of the polytope invariant f⃗ϵ(F(M)). Let 0 < h < a < R 2 . The square [0, a] × [0, a] fits inside the rectangle [0, R] × [0, a]. The rectangle is inside A(f⃗ϵ(F(M))) and does not intersect the cut associated with the focus-focus value c2. The rectangle [0, R]×[0, a] represents the symplectic embedding of the prod… view at source ↗
Figure 3.1
Figure 3.1. Part of the representative of the polytope invariant for ⃗ϵ = −1. The red dot represent the focus-focus value. Consider an embedded Lagrangian sphere i : L → (M, ω) where (M, ω) is a 4- dimensional symplectic manifold. Using Weinstein’s Lagrangian neighborhood the￾orem, we identify a neighborhood of i(L) with the cotangent bundle of S 2 . Therefore the intersection number L · L is such that |L · L| = χ(S 2 ) ̸= 0. N… view at source ↗
Figures from the paper (17 more)
Figure 3.2
Figure 3.2. Figure 3.2: Rectangle [0, R] × [0, h] sitting inside A(∆⃗ϵ). The red dot represents the focus-focus value. Lemma 3.5. The focus-focus fiber of the system (M, ω, F) described above is dis￾placeable. Proof. Recall that for the standard Delzant corner the corner is placed at the or…
Figure 4.1
Figure 4.1. Figure 4.1: These were computed in Pelayo & V˜u Ngo˛c [PV12] and Alonso [Alo19, Section 5.1] [PITH_FULL_IMAGE:figures/full_fig_p022_4_1.png]
Figure 4.2
Figure 4.2. Figure 4.2: A horizontal translation of a representative of the polytope invariant for ⃗ϵ = 1 of the coupled angular momenta system where t − < t < t +. The red segment stands for the cut from above to the focus-focus value along the eigenline. The orange line represents the ver…
Figure 4.3
Figure 4.3. Figure 4.3: Plot of h˜ for t ∈ ]t −, t+[ in blue. Plot of the constant line equal 1 in yellow. Plot of the constant line equal 0 in green. F =(0,0) ( ) ( [PITH_FULL_IMAGE:figures/full_fig_p026_4_3.png]
Figure 4.4
Figure 4.4. Figure 4.4: Representative of the polytope invariant with ⃗ϵ = 1 for the Kepler problem for 1 5 < t < 1. The cut along the eigenline is sketched in red. Proposition 4.8. For 1 5 < t < t0, the Kepler problem (M, ω, Ft) has a unique nondisplaceable fiber. In particular it is a ste…
Figure 4.5
Figure 4.5. Figure 4.5: Image of the nodal trade of the polytope invariant by A ◦ T0,h−2R. The red line represents the image of the cut along the eigenline associated with the focus-focus fiber. Hence we can apply Lemma 3.5 in [PITH_FULL_IMAGE:figures/full_fig_p027_4_5.png]
Figure 4.6
Figure 4.6. Figure 4.6: Representative of the polytope invariant with ⃗ϵ = −1 for the Kepler problem for 1 5 < t < 1. The focus-focus values is represented in red. The cut along the eigenline is sketched in red. T0,h−2R) −1 (c, c))), with c > c0 are displaceable, i.e., the fibers below the …
Figure 4.7
Figure 4.7. Figure 4.7: Representative of the polytope invariant with ⃗ϵ = −1 for the Kepler problem for t0 < t < 1. The cut along the eigenline is sketched in red. The nondisplaceable fibers are identified in green, all the other fibers are displaceable. Lemma 4.16. Let 1 > t > t0 and f⃗ϵ …
Figure 4.8
Figure 4.8. Figure 4.8: The fibers in the octagon that aren’t displaceable by probes are highlighted at red. of (C 8 , ω0). Recall that (M, ω) is obtained as symplectic reduction of (C 8 , ω0) on a level set L˜−1 (0), see Equation (4.5), by a torus action of T 6 , see Equation (4.6). Note t…
Figure 4.9
Figure 4.9. Figure 4.9: (M, ω) as the symplectic reduction of CP2 × CP1 × CP1 × CP1 × CP1 . The dotted lines come in pairs, and each pair corresponds to a reduction by CP1 . The triangle corresponds to CP2 . • Notice that the central fiber in (CP1 , ω0) and the fiber over ( 1 3 , 1 3 ) in (…
Figure 4.10
Figure 4.10. Figure 4.10: Representative of the polytope invariant for the semitoric system when ⃗ϵ = (−1, 1, −1, 1). The dotted lines represent the eigenrays associated with the focus-focus fibers. The red circles represent the 4 dif￾ferent focus-focus values. Recall that for the toric syst…
Figure 4.11
Figure 4.11. Figure 4.11: Representative of the polytope invariant for ⃗ϵ = (−1, 1, 1, 1). The dotted lines represent the eigenrays associated with the focus-focus values. The red dots represent the focus-focus values [PITH_FULL_IMAGE:figures/full_fig_p037_4_11.png]
Figure 4.12
Figure 4.12. Figure 4.12: Image after applying a suitable integral affine transformation to a representative for the polytope invariant with ⃗ϵ = (−1, 1, 1, 1) and h < 1. The dotted lines represent the eigenrays associated with the focus￾focus values. The red dots represent the focus-focus v…
Figure 4.13
Figure 4.13. Figure 4.13: Polytope invariant for ⃗ϵ = (−1, 1, −1, 1) with height invari￾ant h ≥ 1. The dotted horizontal lines indicate how big the eigenrays must be. The dotted vertical lines illustrate the eigenrays associated with the focus-focus values. The red dots represent the focus-f…
Figure 4.14
Figure 4.14. Figure 4.14: Representative of the polytope invariant for ⃗ϵ = (1, 1, 1, 1) and h = 1. The dotted lines are the eigenrays associated with the focus￾focus values. The red dots are the focus-focus values [PITH_FULL_IMAGE:figures/full_fig_p039_4_14.png]
Figure 4.15
Figure 4.15. Figure 4.15: Representative of the polytope invariant for ⃗ϵ = (−1, −1, −1, −1) and h = 1. The dotted lines are the eigenrays associ￾ated with the focus-focus values. The red dots are the focus-focus values. • Applying Lemma 2.22 to [PITH_FULL_IMAGE:figures/full_fig_p039_4_15.png]
Figure 4.16
Figure 4.16. Figure 4.16: Representative for ⃗ϵ = (−1, 1, −1, 1) and h = 1. The dotted lines are the eigenrays associated with the focus-focus values. The red dots are the focus-focus values. Corollary 4.36. Consider the semitoric system (M, ω, Ft) where t is such that h = 1. Then the focus-…

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