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REVIEW 2 major objections 5 minor 37 references

The wrapped Fukaya category of the complement, deformed by a Maurer-Cartan element from the divisors, is filtered quasi-equivalent to Seidel's relative Fukaya category.

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2026-07-12 03:55 UTC pith:O4V3D2VO

load-bearing objection Clean proof of the SBEAS24 conjecture identifying the CO(β)-deformed compact wrapped subcategory with Seidel’s relative Fukaya category; standard technology, no new analytic gaps. the 2 major comments →

arxiv 2607.03234 v1 pith:O4V3D2VO submitted 2026-07-03 math.SG

A Deformation of the Compact Fukaya Category via the Relative Fukaya Category

Mohamad Rabah This is my paper
classification math.SG MSC 53D3753D4018G70
keywords relative Fukaya categorywrapped Fukaya categoryMaurer-Cartan elementClosed-Open mapmonotone symplectic manifoldfiltered quasi-equivalenceLagrangian branesNovikov ring
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proves that two constructions of deformed Fukaya categories of a monotone symplectic manifold relative to a divisor are the same up to filtered quasi-equivalence. One construction starts from the wrapped Fukaya category of the complement of the divisor and deforms it by a Maurer-Cartan element coming from the Closed-Open map applied to a class built from the divisor components. The other is Seidel's relative Fukaya category, which counts holomorphic polygons that meet the divisor. The proof shows that the deformation by the simpler Maurer-Cartan element already recovers the relative category, and that this simpler deformation is gauge-equivalent to the more complicated one used in the wrapped setting. A sympathetic reader cares because the equivalence identifies two natural ways of packaging the same geometric data, so calculations or invariants defined in one language can be transferred to the other.

Core claim

The full subcategory of the CO(β)-deformed wrapped Fukaya category whose objects are closed Lagrangian branes in the complement X = M \ D is filtered quasi-equivalent to Seidel's relative Fukaya category F(M, D). The two categories have identical objects and, after a gauge equivalence that replaces β by the elementary Maurer-Cartan element α = sum x_Dj q_j, they have identical structure maps given by the same counts of holomorphic polygons meeting D.

What carries the argument

The Closed-Open L_infty-morphism CO from symplectic cohomology of the complement to the Hochschild cochains of the wrapped Fukaya category, together with the Maurer-Cartan element β obtained by flowing the elementary class α along a homotopy model. Gauge equivalence of α and β, combined with the fact that CO is an L_infty-morphism, produces a zig-zag of filtered quasi-equivalences that identifies the deformed wrapped category with the relative category.

Load-bearing premise

There must exist a single generic choice of almost-complex structures and Hamiltonians, adapted to the divisor, that makes every relevant moduli space for both constructions into a smooth oriented manifold of the expected dimension at the same time.

What would settle it

Exhibit a concrete monotone pair (M, D) and a closed Lagrangian for which the structure constants of the two A_infty-structures (the relative counts versus the CO(α)-deformed wrapped counts) differ by a non-zero power of the Novikov variables, or show that no common regular perturbation data can exist for both moduli problems simultaneously.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Invariants or computations defined via Seidel's relative Fukaya category can be rewritten in the language of the deformed wrapped category of the complement, and vice versa.
  • The elementary Maurer-Cartan element built from the divisor components already produces the full relative deformation; the more complicated gauge-equivalent element β is not needed for the final category.
  • Any future comparison of relative and absolute Fukaya categories that uses either construction can freely switch to the other without changing the quasi-equivalence class.
  • The same zig-zag argument applies whenever a Closed-Open map takes a Maurer-Cartan element to a deformation of a Fukaya category whose coefficient moduli spaces match those of a relative construction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result suggests that many other relative or logarithmic Fukaya constructions may likewise arise as gauge-equivalent deformations of ordinary wrapped categories of the complement.
  • Once the common regular perturbation data are known to exist, numerical or computer-assisted counts of low-degree polygons in simple examples (e.g., projective space with a hyperplane) can be used to check the equality of structure constants directly.
  • The identification may simplify the study of mirror symmetry statements that currently switch between relative and absolute languages, by allowing both sides to be written in a single deformed-wrapped formalism.

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

2 major / 5 minor

Summary. The paper proves Conjecture 1.3 of [SBEAS24]: for a closed monotone symplectic manifold (M,ω) with orthogonal simple normal-crossing divisors D satisfying 2c1(M)=∑λjDj (λj≤2 rational), the full subcategory W(M,D) of the CO(β)-deformed wrapped Fukaya category of X=M∖D, whose objects are closed exact Lagrangian branes, is filtered quasi-equivalent to Seidel’s relative Fukaya category F(M,D). After recalling Liouville structures on X (McLean), domain moduli spaces with Fulton–MacPherson/aligned framings, and the L∞-structure on symplectic cohomology together with the L∞-morphism CO, the proof in §5 proceeds in two steps: (i) gauge equivalence of the Maurer–Cartan elements α=∑xDj qj and β via the homotopy model gΛt yields a zig-zag of filtered quasi-equivalences W(M,D)≃Wα(M,D); (ii) under identical adapted perturbation data the structure constants of the A∞-operations of Wα(M,D) and F(M,D) match, giving a strict isomorphism.

Significance. The result supplies the missing identification between two independently constructed deformations of the compact Fukaya category of the complement of a symplectic divisor. It therefore makes the Maurer–Cartan deformation of wrapped Floer theory constructed in [SBEAS24] available for computations that have previously been performed only with Seidel’s relative Fukaya category, and conversely. The argument is short once the analytic foundations of [BAS24], [SBEAS24] and [PS22] are granted, and it cleanly separates the algebraic gauge-equivalence step from the geometric matching of moduli spaces. The paper does not claim new transversality or compactness theorems; its contribution is the identification itself.

major comments (2)
  1. The opening sentence of the proof of Theorem 5.1 (and the constructions throughout §4) invokes a single generic consistent universal choice of adapted perturbation data that simultaneously renders all moduli spaces appearing in both F(M,D) and the CO(β)-deformed wrapped category smooth oriented manifolds of the expected dimension. While each construction separately asserts such genericity in the cited works, the simultaneous statement is not proved here and is load-bearing for the strict isomorphism of Step 2. A short paragraph confirming that the Baire-category intersection of the two residual sets remains residual (or an explicit reference to a joint transversality result) would close the gap.
  2. Assumption 4.69 (Q-span of the relative Poincaré duals generates H2(M,X;Q) and each class appears at least n+1 times) is used for positivity of intersection (Lemma 4.71) and for the absence of sphere bubbles of codimension 1. The paper treats it as standing, but it is not automatic for every orthogonal simple normal-crossing divisor satisfying (1.1). Either a brief verification that the assumption holds under the hypotheses of Theorem 1.2, or an explicit restriction of the main theorem to divisors satisfying Assumption 4.69, is needed.
minor comments (5)
  1. The abstract and introduction are extremely terse (“We give a proof of Conjecture 1.3 of [SBEAS24]”). A one-sentence statement of the geometric content of the conjecture would help non-specialist readers.
  2. Notation for the two Novikov rings and the two filtrations (P-filtration versus the q-adic filtration) is introduced in several places; a short “Notation” paragraph at the beginning of §4 would reduce the risk of confusion.
  3. In Definition 4.46 the partial L∞-operations are declared zero when all inputs are of the form xDj; a parenthetical remark explaining why this is compatible with the Maurer–Cartan equation would be useful.
  4. Typographical inconsistencies appear (e.g., “adpated”, “campatible”, “equivanlent”, “homoligically”). A careful proof-reading pass is recommended.
  5. The appendix on Gromov’s graph trick is standard; a one-line pointer to the corresponding statements in [MS12] or [AS10] would suffice and free space.

Circularity Check

0 steps flagged

No circularity: the quasi-equivalence is a non-tautological identification of two independently constructed A∞-structures via gauge equivalence of MC elements and matching of structure constants.

full rationale

The paper proves Conjecture 1.3 of [SBEAS24] by showing that the CO(β)-deformed wrapped subcategory W(M,D) is filtered quasi-equivalent to Seidel’s relative Fukaya category F(M,D). The argument proceeds in two steps (Section 5): (i) α and β are gauge-equivalent Maurer–Cartan elements (via the ODE solution β(t) of Proposition 4.47 and the L∞-morphism property of CO from Theorem 4.80), yielding a zig-zag of filtered quasi-equivalences W^α ≃ W through the homotopy model C; (ii) the structure constants of W^α and F coincide once the same adapted perturbation data are used, because both count the same rigid disks with boundary on the closed Lagrangians and interior intersections with D (Lemmas 4.70–4.71 and the explicit description of the moduli spaces). These steps rely on prior constructions of the L∞-structure on symplectic cohomology, the Closed-Open map, and the relative Fukaya category, but those constructions are independent of the identification being proved; the present work supplies the comparison. There is no self-definitional loop (W is defined via CO(β), F via relative counts; they are shown equal, not defined equal), no fitted parameter renamed as a prediction, no uniqueness theorem imported from the same authors to force the result, and no ansatz smuggled via citation. Author lists do not overlap, so the citations to [SBEAS24] and [BAS24] are ordinary external references. The simultaneous-genericity hypothesis for perturbation data is the ordinary transversality assumption of the subfield, already asserted separately for each construction, and does not create a circular reduction. The derivation is therefore self-contained as a comparison theorem and scores 0.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The central claim rests on a large body of standard symplectic geometry (monotone manifolds, Liouville structures on complements of orthogonal divisors, existence of generic almost-complex structures and Hamiltonians, compactness and transversality for moduli of disks with Lagrangian boundary conditions) together with the specific constructions of the relative Fukaya category and of the Maurer-Cartan element β already developed in the cited works. No free parameters are fitted; the only ad-hoc ingredients are the genericity assumptions needed to make all moduli spaces simultaneously regular.

axioms (4)
  • domain assumption M is closed and positively monotone; D is an orthogonal simple transverse collection of symplectic divisors with 2c1(M)=∑λj Dj, 0≤λj≤2.
    Stated in the Introduction and Definition 2.1–2.14; required for the Liouville structure on X and for the degree of the formal variables qj.
  • domain assumption Existence of a cofinal sequence of non-degenerate admissible Hamiltonians and of contact-type almost-complex structures adapted to D that make all Floer and L∞ moduli spaces regular.
    Theorems 4.38, 4.42, 4.44, 4.72, 4.79; used throughout the constructions of SC*, W(X) and the closed-open map.
  • domain assumption The closed-open map CO is an L∞-morphism from the L∞-algebra of symplectic cochains to the Hochschild cochains of the wrapped Fukaya category.
    Theorem 4.80, imported from BAS24; used to transfer the Maurer-Cartan element β into a deformation of W(X).
  • ad hoc to paper Assumption 4.69: the Q-span of the relative Poincaré duals of the Dj generates H2(M,X;Q) and each class appears at least n+1 times.
    Imposed in §4.3.2 to guarantee positivity of intersection and to control sphere bubbles in the relative Fukaya category.

pith-pipeline@v1.1.0-grok45 · 37691 in / 2699 out tokens · 27140 ms · 2026-07-12T03:55:56.400848+00:00 · methodology

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read the original abstract

We give a proof of Conjecture $1.3$ of [SBEAS24].

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Reference graph

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