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Rabinowitz Floer homology for Legendrian submanifolds in prequantization bundles

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Pith's one-line read An isomorphism equates the Z_d-equivariant Rabinowitz Floer homology of a Legendrian lift to the quantum homology of its Lagrangian base.

desk verdict The paper defines Rabinowitz Floer homology for Legendrian lifts in prequantization bundles and proves an isomorphism to the quantum homology of the base Lagrangian, with explicit ring computations as the main payoff. read the letter →

arxiv 2606.31674 v1 pith:HCSUSAGO submitted 2026-06-30 math.SG

Hanwool Bae , Jungsoo Kang , Sungho Kim This is my paper · ORCID
classification math.SG
keywords RabinowitzFloerhomologyLegendriansubmanifoldsprequantizationbundlesquantummonotoneLagrangiansequivariantLagrangianspheresMaslovnumber
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

The paper defines Rabinowitz Floer homology for Legendrian lifts of closed monotone Lagrangians in prequantization bundles, under the condition that the minimal Maslov number exceeds 2. It proves that the Z_d-equivariant version of this homology is isomorphic as a module to the quantum homology of the projected Lagrangian, where d is the degree of the covering, and that the map is a ring isomorphism when the Maslov number satisfies a stricter bound. The identification is then used to compute explicit quantum homology rings for Lagrangian spheres in quadrics and two-step flag manifolds. Further consequences include vanishing theorems for quantum homology when the base admits a polarization and the Lagrangian is disjoint from a trace set, plus obstructions to topologically simple fillings.

What carries the argument

The Z_d-equivariant Rabinowitz Floer homology of the Legendrian lift, built from the action functional on the covering space and filtered by the covering degree d.

What would settle it

An explicit computation, for any single monotone Lagrangian sphere in a quadric with N_L greater than 2, showing that the rank or ring structure of its Z_d-equivariant Rabinowitz Floer homology differs from the known quantum homology of the sphere.

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Extended reading notes

Core claim

Under the assumption that the minimal Maslov number N_L exceeds 2, the Z_d-equivariant Rabinowitz Floer homology of the Legendrian lift L is isomorphic to the quantum homology of the base Lagrangian L; when N_L is larger still, the isomorphism preserves the ring structures. The construction relies on the prequantization bundle structure and the monotonicity of L to set up the chain complexes and the covering action.

Load-bearing premise

The minimal Maslov number of the Lagrangian must exceed 2 (plus monotonicity) both to define the Rabinowitz Floer homology and to obtain the isomorphism.

Editorial extensions

If this is right

  • The quantum homology ring of Lagrangian spheres in quadrics is computed explicitly via the isomorphism.
  • The quantum homology ring of Lagrangian spheres in two-step flag manifolds is computed explicitly via the isomorphism.
  • Quantum invertibility of the symplectic form implies vanishing of the quantum homology of L.
  • The isomorphism yields obstructions to the existence of topologically simple fillings of the Legendrian L.
  • When the base admits a polarization and L is disjoint from the Lagrangian trace, the quantum homology of L vanishes.

Reading between the lines

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

  • The same isomorphism technique might extend to other contact manifolds that are not necessarily prequantization bundles, provided an analogous covering action can be defined.
  • Computations of quantum homology via this route could be checked against independent algebraic geometry methods for additional classes of monotone Lagrangians.
  • Vanishing results might translate into new constraints on the existence of Lagrangian fillings in higher-dimensional contact manifolds.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 2 minor

Summary. The paper defines Rabinowitz Floer homology for a Legendrian lift ℒ of a closed monotone Lagrangian L in a prequantization bundle Y o (Σ,ω), assuming minimal Maslov number N_L > 2. It establishes an isomorphism between the ℤ_d-equivariant Rabinowitz Floer homology of ℒ and the quantum homology of L (with d the degree of the covering ℒ o L), which is a ring isomorphism under a stricter condition on N_L. The isomorphism is applied to compute the quantum homology ring of Lagrangian spheres in quadrics and two-step flag manifolds, and to obtain vanishing results for quantum homology when (Σ,ω) admits a polarization and L is disjoint from the Lagrangian trace, together with implications for fillings of ℒ.

Significance. If the isomorphism holds, the work supplies a concrete bridge between Rabinowitz Floer homology in the contact setting and quantum homology, permitting explicit computations in both directions and yielding vanishing theorems and filling obstructions as direct consequences. The applications to quadrics and flag manifolds constitute verifiable output that strengthens the result.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'under a more restrictive condition on N_L' is left unspecified; stating the precise numerical threshold would improve immediate readability without altering the theorem statements.
  2. The manuscript would benefit from an explicit comparison table or diagram relating the chain complexes, differentials, and operations used in the Rabinowitz Floer side versus the quantum homology side, to make the isomorphism construction more transparent at a glance.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper defines Rabinowitz Floer homology of the Legendrian lift under the explicit standing hypotheses of monotonicity and N_L > 2 (standard to control disk bubbling), then constructs an isomorphism to the independently studied quantum homology of the base Lagrangian L. No central step reduces by construction to a fitted parameter, self-citation chain, or renamed input; the isomorphism and its ring version under stricter N_L are presented as new results whose consequences (computations on quadrics, vanishing theorems) are derived afterward. The derivation chain is self-contained against external benchmarks in symplectic geometry.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claims rest on standard background in symplectic geometry (Floer theory axioms, quantum homology definitions) and domain assumptions like monotonicity and Maslov number bounds; no free parameters or invented entities are evident from the abstract.

assumptions (2)
  • standard math Standard properties of Floer homology and quantum homology hold in this setting.
    Invoked implicitly when defining the new homology and the isomorphism.
  • domain assumption L is closed monotone Lagrangian with N_L > 2.
    Stated explicitly as the condition under which the definition and isomorphism are established.

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Cite this review

Pith. "Pith review of Rabinowitz Floer homology for Legendrian submanifolds in prequantization bundles." pith.science (2026). https://pith.science/paper/HCSUSAGO

@misc{pith2026260631674,
  author       = {Pith},
  title        = {Pith review of: Rabinowitz Floer homology for Legendrian submanifolds in prequantization bundles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HCSUSAGO}},
  note         = {Machine review of arXiv:2606.31674}
}
abstract

Let $Y$ be a prequantization bundle over an integral symplectic manifold $(\Sigma,\omega)$. Let $L$ be a closed monotone Lagrangian submanifold that admits a Legendrian lift $\mathcal{L}$ in $Y$. Under the assumption that the minimal Maslov number $N_L$ of $L$ is greater than 2, we define the Rabinowitz Floer homology of $\mathcal{L}$. We then establish an isomorphism between the $\mathbb{Z}_d$-equivariant Rabinowitz Floer homology of $\mathcal{L}$ and the quantum homology of $L$, where $d$ is the degree of the covering map $\mathcal{L}\to L$. Under a more restrictive condition on $N_L$, we show that this map is a ring isomorphism. Using this isomorphism, we compute the quantum homology ring of Lagrangian spheres in quadrics and two-step flag manifolds. Furthermore, we investigate the implications of the quantum invertibility of $\omega$ for the vanishing of the quantum homology of $L$ and the obstructions to topologically simple fillings of $\mathcal{L}$. We also show that if $(\Sigma,\omega)$ admits a polarization and $L$ is disjoint from the Lagrangian trace, the quantum homology of $L$ vanishes.

Figures

Figures reproduced from arXiv: 2606.31674 by the authors.

Figure 1
Figure 1. An element of MN (˜ql , p˜k; A; DH). There is a canonical projection map Π : MN (˜ql , p˜k; A; DH) → NN (q, p; A; D), Π(v) := (π ◦ v1, . . . , π ◦ vN ) (3.15) This map restricts to Π : M∗ N (˜ql , p˜k; A; DH) → N ∗ N (q, p; A; D). Proposition 3.11. Let J reg (Y,L) ⊂ JY be the subset of JY whose horizontal part JΣ belongs to J reg (Σ,L) defined in Proposition 2.4. Then for every choice of (˜p, k),(˜q, l) ∈ CritfL × m… view at source ↗
Figure 2
Figure 2. An element of MN(˜rk ′′, p˜k, q˜k ′; A; DP H) There is a canonical projection map Π : MN(˜rk ′′, p˜k, q˜k ′; A; D P H) −→ NN(r, p, q; A; D P ), (3.37) defined by Π(v) := w, where w := (π ◦ v 2H 1 , . . . , π ◦ v H N2+N3 ). We have the same projection map with MN and NN replaced by M∗ N and N ∗ N, respectively. Proposition 3.23. Let J reg,P (Y,L) be the subset of JY consisting of JY whose horizontal part JΣ is an ele… view at source ↗
Figure 3
Figure 3. The limit configuration of vν when N = 0 Since dim L ≥ 3, we may take the underlying simple chains of pearls of Π(v∞,I) and Π(v∞,II). We denote by N ∗ 1 (∅, q; A∗ I ; D) and N ∗ 1 (r, p; A∗ II; D) the moduli spaces containing these under￾lying simple chains of pearls, respectively. They have dim N ∗ 1 (∅, q; A ∗ I ; D) = dim L − indfL (q) + µL(A ∗ I ) ≥ 0, dim N ∗ 1 (r, p; A ∗ II; D) = indfL (r) − indfL (p) + µL(A ∗… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Possible configurations of broken curves [PITH_FULL_IMAGE:figures/full_fig_p069_4.png]
Figure 5
Figure 5. Figure 5: While the SFT-limit of {v˜ν}ν∈N could consist of more than two levels, our argument remains valid as we derive a contradiction by analyzing the topmost component. In the first case of [PITH_FULL_IMAGE:figures/full_fig_p069_5.png]
Figure 5
Figure 5. Figure 5: Possible configurations of SFT-broken curves [PITH_FULL_IMAGE:figures/full_fig_p070_5.png]

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