Sign Convention for $A_{\infty}$-Operations in Bott-Morse Case

Kaoru Ono

REVIEW 3 major objections 5 minor 7 references

Sign Convention for $A_{\infty}$-Operations in Bott-Morse Case

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper proves that the filtered $A_{\infty}$-relations hold for the Bott–Morse de Rham model once the sign convention of Definition 3.3 is fixed, and identifies the boundary sign $\kappa$ that makes the cancellation work.

desk verdict Useful sign-bookkeeping for the de Rham Fukaya category, but it silently drops m0/disk-bubble terms, so the filtered A∞ theorem is only proven for unobstructed collections. read the letter →

arxiv 2504.20489 v1 pith:LFM44HFE submitted 2025-04-29 math.SG math.DG

Kaoru Ono This is my paper · ORCID

classification math.SGmath.DG MSC 53D4053D3758A99

keywords filteredA∞-relationBott-MorsedeRhammodelLagrangianFloertheorysignconventionKuranishistructureborderedstablemapsrelativelyspin

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 addresses a sign and orientation problem in Lagrangian Floer theory in the Bott–Morse setting, where Lagrangian submanifolds are allowed to intersect cleanly along positive-dimensional components rather than only at isolated points. The filtered $A_{\infty}$-operations are defined by integrating differential forms along evaluation maps from moduli spaces of holomorphic polygons, and the $A_{\infty}$-relation only holds if the signs are fixed with care. The author defines operations $m_k$ on the de Rham complexes of the clean-intersection components, with coefficients in certain $O(1)$-local systems, and proves (Theorem 4.4) that with the chosen sign $\epsilon$ and boundary sign $\kappa$ they satisfy $\sum_{k'+k''=k+1} m_{k'}\circ \hat{m}_{k''}=0$ for all $k\ge 1$. If correct, this fixes the sign convention needed for a well-defined filtered $A_{\infty}$-category in the de Rham model.

What carries the argument

The central object is the moduli space $\mathcal{M}_{k+1}(B;\mathbf{L};\mathbf{R})$ of bordered stable maps of genus zero with connected boundary and $k+1$ boundary marked points, carrying a Kuranishi structure (a virtual manifold-chart system with obstruction bundles that supports integration along fibers). The argument is carried by sign bookkeeping with orientation bundles of fiber products: the relative orientation $o(\sigma_{\alpha_0};\sigma_{\alpha_1},\ldots,\sigma_{\alpha_k})$ of the evaluation map $ev_0$ is compared with the orientation of the fiber product $\mathcal{M}_{k'+1}\times_{ev}\mathcal{M}_{k''+1}$ appearing in the boundary of $\mathcal{M}_{k+1}$. The factor $(-1)^\kappa$ in Proposition 4.1 is the precise compensation for this comparison, and the factor $(-1)^\epsilon$ in Definition 3.3 compensates the exchange of differential forms and local systems; together they make the Stokes boundary term cancel all composition terms.

What would settle it

Compute the $k=2$ filtered $A_{\infty}$-relation in the simplest nontrivial Bott–Morse example: two relatively spin Lagrangian submanifolds intersecting cleanly in a circle, with a single holomorphic disk class, so that $\mathcal{M}_3$ is a compact oriented manifold with boundary fibering over $S^1$; evaluating the left side of the relation on explicit bump forms supported in the interior of the clean intersection should give zero, and a nonzero result or a sign in disagreement with Proposition 4.1 would falsify the theorem.

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Extended reading notes

Core claim

The central claim is that the naive push-forward definition, without the sign $\epsilon$, would not satisfy the $A_{\infty}$-relations, and the paper identifies exactly what sign is needed. Definition 3.3 sets $m_{k,B}(\xi_1,\ldots,\xi_k)=(-1)^{\epsilon(\xi_1,\ldots,\xi_k)}(ev_0)_!(ev_1^*\xi_1\wedge\cdots\wedge ev_k^*\xi_k)$, where $\epsilon$ records each form's shifted degree and the Maslov indices of the relevant clean-intersection components. Theorem 4.4 states that the resulting operations satisfy the filtered $A_{\infty}$-relation for every $k=1,2,\ldots$. The proof is a sign-counting argument: Proposition 4.1 computes the sign $\kappa$ that compares the orientation of the boundary of the moduli space with the fiber product of two smaller moduli spaces, Lemma 4.3 shows how compositions of operations carry that sign, and the Stokes-type formula of Proposition 2.5 turns the boundary term into the required cancellation.

Load-bearing premise

The load-bearing premise is that extra choices for integration over Kuranishi spaces (perturbations and obstruction-bundle data) do not affect signs, and that the Kuranishi structures from [1,3] exist as used; if either fails, the sign $\kappa$ computed in Proposition 4.1 would not describe the actual moduli spaces and the filtered $A_{\infty}$-relation could break.

Editorial extensions

If this is right

For any finite collection of relatively spin Lagrangian submanifolds with clean intersections, the filtered $A_{\infty}$-category in the de Rham model is well defined with the stated signs.
The same sign convention extends to clean self-intersections and, as the paper notes, to immersed Lagrangian submanifolds with clean self-intersections.
With the coefficient ring replaced by the universal Novikov ring $\Lambda_{0,nov}$, the operations $m_k$ have degree $1$ and hence give a graded filtered $A_{\infty}$-structure, as noted in Remark 3.4.
The $A_{\infty}$-relations imply the usual chain-level identities, in particular that $m_1=d$ together with $m_2$ forms a differential graded algebra up to the specified signs, so the algebraic invariants of the Lagrangian intersection are well defined.

Reading between the lines

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

A natural next step, not taken in the note, would be to carry out the same sign verification for filtered $A_{\infty}$-bimodules and $A_{\infty}$-functors in the de Rham model; the paper notes the bimodule construction only in the singular-chain model.
Because the signs $\epsilon$ and $\kappa$ are given by explicit formulas, the $k=2$ relation could be checked by computer algebra for a moduli space that is a smooth fiber bundle over the clean intersection, providing an independent test of the bookkeeping.
The paper's treatment of signs suggests that the same orientation comparison should govern the obstruction-bundle degrees in more general Kuranishi models; tracking those degrees explicitly could reveal whether the 'ignore CF-perturbations' reduction used here remains valid beyond the present setting.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First thing to know: this note is exactly what it says on the tin—a sign/orientation check for filtered A∞ operations in the de Rham Bott–Morse model—but the version on arXiv proves less than it claims. The operations in Definition 3.3 set m0 ≡ 0, and no m_{0,B} for B ≠ 0 is ever defined. That choice is not harmless. The boundary of M_{k+1}(B) includes strata where one component is a disk with a single marked point (a disk bubble). Those strata are responsible for the m_{k'}(..., m0(B''), ...) terms in the filtered A∞ relation. They are absent from Proposition 4.1 and from the proof of Theorem 4.4. So the theorem only establishes an uncurved A∞ structure, and only under an implicit unobstructedness assumption. For a Lagrangian torus with a nontrivial disk potential, m0 is nonzero and the claim as stated fails. This is the main problem.

What is genuinely useful: the paper writes out the m_{k,B} operations, the sign ε, and the boundary orientation comparison for the polygon–polygon strata with care. Proposition 4.1's κ sign and Lemma 4.3's κ′ are the kind of bookkeeping that is easy to get wrong, and having them on record is valuable for anyone working in the de Rham model. The proof of the relation for the strata that are considered is coherent: equations (4.1)–(4.2) and (4.8) match.

The other caveats are more minor. The paper explicitly says it treats Kuranishi pushforwards as smooth submersions for sign purposes; for a sign paper this deserves more than a sentence, but it is a stated assumption, not a hidden one. And it depends on the unpublished Kuranishi construction [1] for the moduli spaces, which makes the statement conditional on that work. Neither of these is load-bearing if the m0 gap is fixed.

Bottom line: the signs for the m_k, k ≥ 1, are probably right, but the filtered A∞-category statement is not correct as written for general relatively spin Lagrangians. A serious referee should ask the author to either include the m0 terms and the disk-bubble strata, or restrict the theorem to an unobstructed collection and say so. The note could then be a useful reference. As it stands, I would not cite it for the filtered A∞-category claim.

Referee Report

3 major / 5 minor

Summary. This paper addresses sign and orientation conventions in the construction of filtered A∞-operations in the de Rham model of Lagrangian Floer theory in the Bott–Morse (clean intersection) setting. The author defines operations m_{k,B} as integrations along fibers of moduli spaces of bordered stable polygons with Kuranishi structures (Definition 3.3), setting m_{0,0}=0 and m_{1,0}=d, and then states that these operations satisfy the filtered A∞-relations (Theorem 4.4). The proof computes boundary contributions and checks that the sign κ in Proposition 4.1 and the composition sign κ′ in Lemma 4.3 cancel to yield the Stokes boundary term. The paper relies on the Kuranishi structure constructions of [1,3] and on the integration-along-fibers formalism of [6].

Significance. If the sign verification is correct and the underlying Kuranishi constructions are available, the paper would provide a useful de Rham model for the filtered Fukaya category for a collection of relatively spin Lagrangian submanifolds, complementing the singular chain treatment of [4,5]. The explicit sign bookkeeping in Propositions 3.1 and 4.1 is a genuine contribution. However, as it stands the theorem covers only the uncurved case m0=0, and the paper's reduction of Kuranishi integration to the smooth submersion case is asserted rather than proved.

major comments (3)

[Definition 3.3 and Theorem 4.4] The definition sets m_{0,0}=0 and defines no m_{0,B} for B≠0, so m_0≡0. In the filtered A∞-relation written in Section 4, the sum over k′+k″=k+1 includes terms with k″=0 (and k′=0) in which m_0 is inserted; for a general relatively spin collection the disk potential m_0 is generally nonzero (e.g., a Lagrangian torus with nontrivial disk potential in the sense of Fukaya–Oh–Ohta–Ono). The proof of Theorem 4.4 therefore never cancels the boundary strata of the form M_{k′+1}(B′) ×_{ev} M_1(B″), because m_0 was set to zero by fiat. Consequently Theorem 4.4 establishes only an uncurved A∞ structure under an implicit unobstructedness assumption, not the filtered A∞-category relations for general relatively spin Lagrangians. Proposition 3.1(1) prepares orientation data for M_1(B;L) but no operation is defined from it, confirming that the m_0 contribution is omitted. The theorem should either state an explicit unobstructedness hypothesis or include the m_0 terms in the verification.
[Section 2] The paper states: 'For the verification of the sign convention in the filtered A∞-relations, it is sufficient to treat the integration along fibers of a proper weakly submersive strongly smooth map as if the one for proper submersion between smooth manifolds.' This is the basis for suppressing CF-perturbation and obstruction-bundle data in all sign computations, including the orientation comparison in Proposition 4.1 and Lemma 4.2. No proof or detailed reference is given for this reduction. Since the sign κ entering the boundary correspondence depends on orientations of fiber products of Kuranishi spaces, the sufficiency claim is load-bearing for Theorem 4.4. Please either prove the reduction from the formalism of [6, Sections 9 and 10] or state precisely which statements in [6] imply it.
[References [1] and [3]] The moduli spaces M_{k+1}(B;L;R) are assumed to carry tree-like K-systems, and their construction is postponed to [1,3], with [1] listed as 'in preparation'. The verification in Theorem 4.4 is therefore conditional on an unpublished construction. Please clarify whether [3] alone supplies the needed Kuranishi structures for the boundary decomposition used in Proposition 4.1, and update [1] if it is now available.

minor comments (5)

[Abstract] The phrase 'appearing the filtered A∞-formulae' should read 'appearing in the filtered A∞-formulae'.
[Equation (3.4)] The sign ϵ(ξ_1,...,ξ_k) contains a trailing '+1' in its definition; please state explicitly whether this is part of the convention and how it is fixed.
[Remark 3.4] The paper uses the coefficient ring Λ0 rather than Λ_{0,nov}; as stated in Remark 3.4, the operations are not exactly of degree 1 in this ring. Please clarify how the degree property (3.5) is used in the verification of the A∞-relations.
[Definition 3.3] The notation m_{1,0}=d for the de Rham differential may be confused with the operation for B=0 in the same family; consider using a separate symbol such as m_{1}^{dR} for the differential.
[Lemma 4.3] In the display after equation (4.6), the expression ev_{B′}^{∗}∘(ev_{B″}^{0})_! should be explained as composition of pullback with integration along fibers in the correspondence sense; as written it is easy to misread as an operator on a single manifold.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the A∞-relations are derived from Stokes and orientation bookkeeping, not assumed; self-citations are background tools. The m0=0 point is an unobstructedness gap, not a circular step.

full rationale

Definition 3.3 defines m_{k,B} as an explicit integration-along-fibers pushforward (eq. 3.2) with a sign epsilon from eq. (3.4), not from any condition that the target relation hold. The verification in Section 4 starts from Proposition 2.4 (Stokes) and Proposition 2.5, computes m_{k',B'}∘m-hat_{k'',B''} using base change (Proposition 2.3) and the boundary orientation comparison (Proposition 4.1), then cancels the ∂M_{k+1} term against (4.1), (4.2), and Claim 4.5. Each combinatorial sign κ, κ' is computed from orientation-bundle isomorphisms (Φ_B in Proposition 3.1) rather than read off from the filtered A∞ relation. Thus no equation of the theorem is used as an input; there is no fitted parameter or normalized ansatz that makes a 'prediction' equal to its data. The repeated citations to [5,6] concern independent published tools (orientation local systems and Kuranishi integration); the in-preparation [1] supplies K-structure existence and is a dependency, not a circular premise. The choice m_{0,0}=0 in Definition 3.3 means Theorem 4.4 proves an uncurved relation for the stated operations and does not, as written, establish the curved filtered A∞-category for a general relatively spin collection; this is a mathematical gap/overclaim, not a circular reduction, so it does not raise the circularity score.

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

All axioms are standard tools from the FOOO program, mostly from [4,5,6]. The only assumption introduced ad hoc in this paper is the treatment of Kuranishi pushforwards as smooth submersions. No free parameters or new entities are introduced.

assumptions (7)

domain assumption Orientation convention on fiber products in the Kuranishi structure sense as in [5, Section 8.2].
The entire sign bookkeeping of Section 4 uses this convention to order factors in fiber products; a different convention would change kappa.
domain assumption Stokes type formula for strongly smooth maps from K-spaces (Proposition 2.4, from [6, Theorem 9.28]).
Used in equation (4.8) to obtain the boundary term; the proof of Theorem 4.4 depends on it.
domain assumption Base change formula (Proposition 2.3) and composition formula (Proposition 2.6) for integration along fibers, from [6, Chapter 27].
Lemma 4.2 and Lemma 4.3 use the base change and composition formulas to rearrange evaluations; these are imported from [6].
domain assumption Existence of tree-like K-systems (Kuranishi structures) on moduli spaces of bordered stable maps for collections of Lagrangians, deferred to [1] (in preparation) and [3].
The moduli spaces M_{k+1}(B;L;R) and their boundary decompositions are assumed to carry the Kuranishi structure with the stated evaluation maps; the present paper does not construct them.
domain assumption Orientation isomorphism Phi_B (Proposition 3.1) from relative spin structures, after [5, Theorem 8.1.1].
The definition of the relative orientation o(sigma_alpha0; ...) and hence the operations m_{k,B} depend on this isomorphism, imported from [5].
domain assumption The O(1)-local systems Theta_{R_alpha} on clean intersections exist and satisfy Theta^+ tensor O_{R_alpha} tensor Theta^- is canonically trivialized (Remark 3.2).
The local systems are constructed in [5, Proposition 8.1.1]; the m_k operations act on forms with these coefficients.
ad hoc to paper For sign verification, integration along fibers of proper weakly submersive strongly smooth maps from K-spaces can be treated as for proper submersions between smooth manifolds (Section 2).
The paper states this explicitly to suppress CF-perturbation details; if Kuranishi chart signs contributed extra factors, the computed kappa could be incomplete.

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Cite this review

Pith. "Pith review of Sign Convention for $A_{\infty}$-Operations in Bott-Morse Case." pith.science (2026). https://pith.science/paper/LFM44HFE

@misc{pith2026250420489,
  author       = {Pith},
  title        = {Pith review of: Sign Convention for $A_\infty$-Operations in Bott-Morse Case},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LFM44HFE}},
  note         = {Machine review of arXiv:2504.20489}
}

abstract

We describe the sign and orientation issue appearing the filtered $A_{\infty}$-formulae in Lagrangian Floer theory using de Rham model in Bott-Morse setting. After giving the definition of filtered $A_{\infty}$-operations in a Fukaya category, we verify the filtered $A_{\infty}$-formulae.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

7 extracted references · 5 canonical work pages

[1]

Abouzaid M., Fukaya K., Oh Y.G., Ohta H., Ono K., Quantum cohomology and split generation in La- grangian Floer theory, in preparation

work page
[6]

Math., Springer, Singapore, 2020

Fukaya K., Oh Y.G., Ohta H., Ono K., Kuranishi structures and virtual fundamental chains, Springer Monogr. Math., Springer, Singapore, 2020

work page 2020
[3]

Fukaya K., Unobstructed immersed Lagrangian correspondence and filtered A∞-functor, SIGMA 21 (2025), 031, 284 pages, arXiv:1706.02131

arXiv 2025
[2]

Immersed Lagrangian Floer Theory

Akaho M., Joyce D., Immersed Lagrangian Floer theory, J. Differential Geom. 86 (2010), 381–500, arXiv:0803.0717

work page Pith review arXiv 2010
[4]

Part I, AMS/IP Stud

Fukaya K., Oh Y.G., Ohta H., Ono K., Lagrangian intersection Floer theory: Anomaly and obstruction. Part I, AMS/IP Stud. Adv. Math. , Vol. 46, American Mathematical Society, Providence, RI, 2009

work page 2009
[5]

Part II, AMS/IP Stud

Fukaya K., Oh Y.G., Ohta H., Ono K., Lagrangian intersection Floer theory: Anomaly and obstruction. Part II, AMS/IP Stud. Adv. Math. , Vol. 46, American Mathematical Society, Providence, RI, 2009

work page 2009
[7]

Symplectic Geom

Solomon J.P., Tukachinsky S.B., Differential forms, Fukaya A∞ algebras, and Gromov–Witten axioms, J. Symplectic Geom. 20 (2022), 927–994, arXiv:1608.01304

arXiv 2022

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Abouzaid M., Fukaya K., Oh Y.G., Ohta H., Ono K., Quantum cohomology and split generation in La- grangian Floer theory, in preparation

work page

[2] [6]

Math., Springer, Singapore, 2020

Fukaya K., Oh Y.G., Ohta H., Ono K., Kuranishi structures and virtual fundamental chains, Springer Monogr. Math., Springer, Singapore, 2020

work page 2020

[3] [3]

Fukaya K., Unobstructed immersed Lagrangian correspondence and filtered A∞-functor, SIGMA 21 (2025), 031, 284 pages, arXiv:1706.02131

arXiv 2025

[4] [2]

Immersed Lagrangian Floer Theory

Akaho M., Joyce D., Immersed Lagrangian Floer theory, J. Differential Geom. 86 (2010), 381–500, arXiv:0803.0717

work page Pith review arXiv 2010

[5] [4]

Part I, AMS/IP Stud

Fukaya K., Oh Y.G., Ohta H., Ono K., Lagrangian intersection Floer theory: Anomaly and obstruction. Part I, AMS/IP Stud. Adv. Math. , Vol. 46, American Mathematical Society, Providence, RI, 2009

work page 2009

[6] [5]

Part II, AMS/IP Stud

Fukaya K., Oh Y.G., Ohta H., Ono K., Lagrangian intersection Floer theory: Anomaly and obstruction. Part II, AMS/IP Stud. Adv. Math. , Vol. 46, American Mathematical Society, Providence, RI, 2009

work page 2009

[7] [7]

Symplectic Geom

Solomon J.P., Tukachinsky S.B., Differential forms, Fukaya A∞ algebras, and Gromov–Witten axioms, J. Symplectic Geom. 20 (2022), 927–994, arXiv:1608.01304

arXiv 2022