Bohr-Sommerfeld profile surgeries and Disk Potentials
Pith reviewed 2026-05-23 20:11 UTC · model grok-4.3
The pith
Wall-crossing formula describes how disk potentials change under Bohr-Sommerfeld profile surgery.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We construct a new surgery type operation by switching between two exact fillings of Legendrians which we call a BSP surgery. In certain cases, this surgery can preserve monotonicity of Lagrangians. We prove a wall-crossing type formula for the change of the disk-potential under surgery with Bohr-Sommerfeld profiles. As an application, we show that Biran's circle-bundle lifts admit a Bohr-Sommerfeld type surgery. We use the wall-crossing theorem about disk-potentials to construct exotic monotone Lagrangian tori in P^n.
What carries the argument
BSP surgery, defined by switching between two exact fillings of Legendrians, together with the associated wall-crossing formula that tracks the resulting change in disk potential.
If this is right
- The disk potential of a Lagrangian after BSP surgery is related to the original potential by the wall-crossing formula.
- Biran circle-bundle lifts admit Bohr-Sommerfeld type surgery.
- Exotic monotone Lagrangian tori exist in projective space P^n.
- The wall-crossing relation can be used to track invariants across sequences of such surgeries.
Where Pith is reading between the lines
- The formula supplies a systematic way to relate disk potentials of Lagrangians obtained from different fillings of the same Legendrian.
- Similar wall-crossing behavior may appear when the surgery is performed in other symplectic manifolds containing suitable Legendrians.
Load-bearing premise
There exist pairs of exact fillings of the relevant Legendrians that can be switched via the defined BSP surgery while preserving the conditions needed for the wall-crossing formula to apply, particularly for the Biran circle-bundle lifts.
What would settle it
Explicit computation of the disk potential of a specific Biran circle-bundle lift before and after BSP surgery, checking whether the observed change matches the predicted wall-crossing formula.
read the original abstract
We construct a new surgery type operation by switching between two exact fillings of Legendrians which we call a BSP surgery. In certain cases, this surgery can preserve monotonicity of Lagrangians. We prove a wall-crossing type formula for the change of the disk-potential under surgery with Bohr-Sommerfeld profiles. As an application, we show that Biran's circle-bundle lifts admit a Bohr-Sommerfeld type surgery. We use the wall-crossing theorem about disk-potentials to construct exotic monotone Lagrangian tori in $\bP^n$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces Bohr-Sommerfeld profile (BSP) surgery, a new operation obtained by switching between two exact fillings of a Legendrian. It proves a wall-crossing formula describing the change in the disk potential under BSP surgery with Bohr-Sommerfeld profiles. As an application, the authors show that Biran circle-bundle lifts admit BSP surgery and apply the wall-crossing result to produce exotic monotone Lagrangian tori in projective space P^n.
Significance. If the wall-crossing formula is correctly derived and the application to Biran lifts is valid, the work supplies a new constructive technique for modifying monotone Lagrangians while tracking their disk potentials. This could be useful for producing and distinguishing exotic examples in toric symplectic manifolds and for studying wall-crossing phenomena in Lagrangian Floer theory.
major comments (2)
- [§4] §4 (application to Biran lifts): the claim that Biran circle-bundle lifts admit BSP surgery requires explicit verification that there exist pairs of exact fillings of the relevant Legendrians that can be interchanged by the defined surgery while preserving exactness, the existence of suitable bounding cochains, and the monotonicity condition needed for the wall-crossing theorem to apply. The manuscript does not appear to supply a concrete check that these hypotheses remain satisfied after the surgery on the Biran lifts.
- [§3] Theorem on the wall-crossing formula (likely §3): the statement should make clear whether the formula holds for arbitrary exact fillings or only under additional restrictions on the profiles and the choice of fillings; if the latter, the restrictions must be stated explicitly so that the reader can confirm they are met in the Biran-lift application.
minor comments (2)
- [§2] Notation for the disk potential and the surgery parameters should be introduced with a single consistent definition early in the paper rather than piecemeal.
- [§1] The introduction would benefit from a brief comparison table or diagram contrasting BSP surgery with existing Lagrangian surgery operations (e.g., Polterovich surgery).
Simulated Author's Rebuttal
We thank the referee for the careful reading and constructive comments. We address the major comments point by point below.
read point-by-point responses
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Referee: [§4] §4 (application to Biran lifts): the claim that Biran circle-bundle lifts admit BSP surgery requires explicit verification that there exist pairs of exact fillings of the relevant Legendrians that can be interchanged by the defined surgery while preserving exactness, the existence of suitable bounding cochains, and the monotonicity condition needed for the wall-crossing theorem to apply. The manuscript does not appear to supply a concrete check that these hypotheses remain satisfied after the surgery on the Biran lifts.
Authors: We agree that an explicit verification of the hypotheses is needed for the Biran-lift application. In the revised manuscript we will add a dedicated paragraph (or short subsection) in §4 that checks the existence of the required pairs of exact fillings, confirms preservation of exactness and monotonicity, and verifies the existence of suitable bounding cochains for the Legendrians arising from Biran circle-bundle lifts. revision: yes
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Referee: [§3] Theorem on the wall-crossing formula (likely §3): the statement should make clear whether the formula holds for arbitrary exact fillings or only under additional restrictions on the profiles and the choice of fillings; if the latter, the restrictions must be stated explicitly so that the reader can confirm they are met in the Biran-lift application.
Authors: We will revise the statement of the wall-crossing theorem in §3 to explicitly list the restrictions on the Bohr-Sommerfeld profiles and on the choice of exact fillings under which the formula is proved. The revised statement will also note that these restrictions are satisfied by the fillings used in the Biran-lift construction. revision: yes
Circularity Check
Wall-crossing formula and application presented as independent derivation; no reduction to inputs by construction
full rationale
The paper states it constructs BSP surgery, proves a wall-crossing formula for disk potentials, and applies it to Biran lifts to obtain exotic tori. No quoted equations or self-citations in the abstract or described claims reduce the wall-crossing statement or the application to a fitted parameter, self-defined quantity, or load-bearing self-citation chain. The derivation is presented as a theorem with external hypotheses (existence of exact filling pairs preserving monotonicity), which is standard and does not constitute circularity. This matches the default expectation for a proof paper whose central claims remain independent of its own fitted outputs.
discussion (0)
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