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arxiv: 2605.27122 · v1 · pith:VXL4LXMHnew · submitted 2026-05-26 · 🧮 math.SG

Open Gromov-Witten invariants in genus zero and Lagrangian cobordisms

Amanda Hirschi , Kai Hugtenburg This is my paper

Pith reviewed 2026-06-29 14:29 UTC · model grok-4.3

classification 🧮 math.SG
keywords open Gromov-Witten invariantsgenus zeroLagrangian cobordismsbounding cochainsWDVV relationsHamiltonian isotopysymplectic manifoldsrelatively spin Lagrangians
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The pith

Open Gromov-Witten invariants in genus zero are constructed for arbitrary closed symplectic manifolds and weakly unobstructed embedded Lagrangians.

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

The paper constructs open Gromov-Witten invariants in genus zero. These count holomorphic disks with boundary on a Lagrangian submanifold inside a closed symplectic manifold. The construction applies whenever the Lagrangian is embedded, relatively spin, and made weakly unobstructed by a bounding cochain. It proves that the resulting numbers satisfy the open WDVV relations, do not depend on the choice of almost complex structure, and remain unchanged when the Lagrangian is deformed by a Hamiltonian isotopy. A relation is established between the invariants of two Lagrangians connected by a cobordism.

Core claim

The central claim is that open Gromov-Witten invariants in genus zero can be defined for arbitrary closed symplectic manifolds and embedded relatively spin Lagrangians that are weakly unobstructed by a bounding cochain. The construction relies on prior analytic and algebraic foundations. The invariants satisfy the open WDVV relations, are independent of the almost complex structure, are invariant under Hamiltonian isotopy, and are related for Lagrangians that are cobordant.

What carries the argument

The bounding cochain that renders the Lagrangian weakly unobstructed, which allows the moduli spaces of holomorphic disks to be used to define the invariants via the given algebraic framework.

Load-bearing premise

The Lagrangian admits a bounding cochain that makes it weakly unobstructed and the prior analytic and algebraic foundations are complete.

What would settle it

An explicit computation in a simple example such as the standard sphere showing that the numbers change under a Hamiltonian isotopy or violate the open WDVV relations would falsify the construction.

read the original abstract

We construct open Gromov-Witten invariants in genus zero for arbitrary closed symplectic manifolds and embedded relatively spin Lagrangians, which are weakly unobstructed by a bounding cochain. This uses the foundational work of \cite{HH25,HH26} and the algebraic framework of \cite{ST21}. We prove the open WDVV relations and show that these invariants are independent of the choice of almost complex structure and under Hamiltonian isotopy. We also prove a relation between open Gromov-Witten invariants of cobordant Lagrangians.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs open Gromov-Witten invariants in genus zero for arbitrary closed symplectic manifolds and embedded relatively spin Lagrangians that are weakly unobstructed by a bounding cochain. This uses the foundational analytic setup from HH25 and HH26 together with the algebraic framework of ST21. The authors prove the open WDVV relations, J-independence, invariance under Hamiltonian isotopy, and a relation between the invariants of cobordant Lagrangians.

Significance. If the analytic and algebraic foundations in the cited works hold without gaps, the result would supply a general construction of open GW invariants together with their algebraic relations and invariance properties under cobordisms. This framework could be useful for studying Lagrangian submanifolds in symplectic geometry. The explicit derivation of the WDVV relations and the cobordism invariance are concrete strengths when the prior results are granted.

major comments (2)
  1. [Construction section (post-introduction)] The construction of the invariants (detailed after the introduction) rests entirely on the moduli-space compactness, virtual fundamental classes, and obstruction theory supplied by HH25 and HH26. No independent verification, error estimates, or model-case computation (e.g., Clifford torus in CP^n) is provided to close this dependence, which is load-bearing for the claim that the invariants exist for arbitrary closed symplectic manifolds.
  2. [Sections on WDVV, invariance, and cobordism relations] The proofs of the open WDVV relations, J-independence, Hamiltonian isotopy invariance, and cobordism relation (in the sections following the construction) inherit any analytic gaps in HH25/HH26 without additional checks or self-contained arguments, rendering those theorems conditional on the prior works.
minor comments (2)
  1. [References] The bibliography entries for HH25, HH26, and ST21 should be completed with full publication details once available.
  2. [Notation and setup paragraphs] Notation for the bounding cochain and the precise meaning of 'weakly unobstructed' should be recalled briefly in the main text to aid readers who have not yet consulted the cited foundations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the detailed report. Below we address the major comments point by point, explaining the intended scope of the manuscript.

read point-by-point responses

  1. Referee: [Construction section (post-introduction)] The construction of the invariants (detailed after the introduction) rests entirely on the moduli-space compactness, virtual fundamental classes, and obstruction theory supplied by HH25 and HH26. No independent verification, error estimates, or model-case computation (e.g., Clifford torus in CP^n) is provided to close this dependence, which is load-bearing for the claim that the invariants exist for arbitrary closed symplectic manifolds.

    Authors: The manuscript is explicitly framed as an application of the analytic foundations developed in HH25 and HH26, combined with the algebraic framework of ST21. This dependence is stated in the abstract, the introduction, and the construction section. The contribution consists in defining the open Gromov-Witten invariants in the stated generality and deriving their algebraic and invariance properties. Independent analytic verification or model computations would require substantial new work duplicating or extending the content of the cited papers and therefore lies outside the scope of the present article. revision: no

  2. Referee: [Sections on WDVV, invariance, and cobordism relations] The proofs of the open WDVV relations, J-independence, Hamiltonian isotopy invariance, and cobordism relation (in the sections following the construction) inherit any analytic gaps in HH25/HH26 without additional checks or self-contained arguments, rendering those theorems conditional on the prior works.

    Authors: The proofs of the open WDVV relations, J-independence, Hamiltonian isotopy invariance, and the cobordism relation are algebraic once the virtual fundamental classes are supplied by HH25 and HH26. These arguments apply the framework of ST21 to the open invariants and use standard deformation and cobordism techniques. The manuscript already notes that the results are conditional on the analytic setup of the cited works; no new analytic arguments are claimed or needed for the algebraic derivations. revision: no

Circularity Check

1 steps flagged

Central construction of open GW invariants rests on self-cited analytic foundations HH25/HH26

specific steps
  1. self citation load bearing [Abstract]
    "This uses the foundational work of \cite{HH25,HH26} and the algebraic framework of \cite{ST21}."

    The construction of the invariants (the paper's strongest claim) is justified solely by citing the analytic and algebraic setup from HH25/HH26 by the same authors. The proofs of all subsequent properties (open WDVV, invariance, cobordism relations) therefore reduce to the validity of those prior results, creating a self-citation chain without external or independent closure in this manuscript.

full rationale

The paper's core claim is the construction of open Gromov-Witten invariants for arbitrary closed symplectic manifolds and relatively spin Lagrangians admitting a bounding cochain. This is explicitly achieved by invoking the moduli spaces, virtual classes, and obstruction theory from HH25 and HH26 (same author pair). Subsequent results on WDVV relations, J-independence, and cobordism invariance therefore inherit the analytic setup without providing independent verification or model-case checks. This matches self_citation_load_bearing pattern: the load-bearing premise reduces to prior self-citations whose gap-free status is not re-established here. No other circular steps (e.g., self-definitional equations or fitted predictions) appear in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Ledger entries are inferred from the abstract alone; the central claim rests on the existence of bounding cochains and on the validity of the cited foundational papers.

axioms (2)
  • domain assumption Lagrangians are weakly unobstructed by a bounding cochain
    Stated in the abstract as the condition under which the invariants are defined.
  • domain assumption Foundational results of HH25 and HH26 hold
    Abstract states that the construction uses these works.

pith-pipeline@v0.9.1-grok · 5611 in / 1338 out tokens · 30723 ms · 2026-06-29T14:29:41.523787+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

8 extracted references · 2 canonical work pages · 2 internal anchors

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