{"id":"2ba0687f-9bbd-48da-8cdd-7f8e70b70e4f","arxiv_id":"2509.06034","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Chain-level open-closed maps from Hochschild and cyclic homology of a possibly curved Fukaya A-infinity algebra to quantum cohomology, with Gromov-Witten-axiom analogues.","lead":"This paper constructs chain-level open-closed maps from various Hochschild and cyclic homology groups of the Fukaya A-infinity algebra of a Lagrangian submanifold to the quantum cohomology of the ambient symplectic manifold, allowing the algebra to be curved. The maps satisfy analogues of several Gromov-Witten axioms, and are intended to serve as a foundation for gravitational descendants and obstruction theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central theorems rely on an unproven regularity assumption (smooth moduli, proper submersion evaluations) verified only for homogeneous spaces; without the deferred VFC extension, the general statement of Theorems 1–5 is not established.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing point: the paper's theorems are conditional on moduli spaces being smooth orbifolds with proper submersion evaluation maps, and this is verified only for homogeneous spaces. My independent reading confirms that this assumption is used essentially in the proof of Proposition 4.1, which is the geometric engine behind Theorems 1–5. The paper is honest about this limitation in Section 1.2.1, but the abstract and theorem statements do not prominently qualify the scope, so the central claim as a general construction is not fully proven. I do not find an internal contradiction in the algebraic arguments: the Hochschild/cyclic descent proofs in Section 5 are detailed, and the signs appear consistent once the unit and cyclic-symmetry properties are used. The pseudoisotopy section is an additional gap, but it is not needed for the core chain-map theorems. Because the reader already assigned CONDITIONAL with this same concern, my stress-test does not change the verdict. The concrete test I propose would settle whether the regularity assumption genuinely fails outside homogeneous spaces, which is the key unknown for the general claim.","tokens_in":42208,"tokens_out":51838,"duration_ms":566701,"concrete_test":"Take a concrete non-homogeneous pair, e.g., the Clifford torus L ⊂ CP^2 with the standard Kähler form, and examine the disk moduli space M_{k,l+1}(β) for a low-degree class β with Maslov index 2 and small k,l. Explicitly check whether this moduli space is a smooth orbifold with corners and whether the evaluation maps ev^b_1 and ev^i_0 are proper submersions over their images. If any evaluation has non-submersive fibers or the moduli space has singular strata not covered by Props. 4.5/4.7, then the chain-level p^γ of Eq. (3) is not defined by the paper's current formalism and Prop. 4.1—and hence Theorems 1–5 for this target—cannot be obtained without the deferred virtual fundamental class machinery. A single negative example would show the general claim is not covered; a positive example would only support the homogeneous cases already treated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The construction of the p-operators and all descent theorems hinges on the regularity assumption stated in Section 1.2.1: every moduli space M_{k,l+1}(β) is a smooth orbifold with corners and the evaluation maps ev^b and ev^i, when defined, are proper submersions. This is not a peripheral technicality. The operator p^γ in Eq. (3) is defined by pushforward along ev^i_0, and Proposition 4.1 derives the central structure equation by applying Stokes' theorem (Prop. 3.3) to the boundary of these moduli spaces and identifying the boundary strata via the gluing diffeomorphisms in Props. 4.5 and 4.7. If the moduli spaces are not smooth or the evaluations are not proper submersions, the pushforwards are not legitimate currents and the Stokes argument has no meaning. The authors verify the assumption only for homogeneous spaces such as (CP^n, RP^n) and products thereof; for general closed symplectic manifolds they state that the results are 'expected to extend' using virtual fundamental class techniques, citing [6,7,9–11,16–20,2,3,15], but they do not carry out that extension. Thus Theorems 1–5 are proven only under this regularity assumption; the abstract's unqualified claim to construct open-closed maps for a general Lagrangian rests on an unproven VFC extension. The same caveat applies to Section 6: the pseudoisotopy discussion assumes the analogous family regularity and then asserts analogs of Theorems 1–5 without proof (Section 6.3, 'we omit the details here').","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs chain-level open-closed maps p^γ from the Hochschild and cyclic chain complexes of the de Rham model C = A^*(L;R) of a (possibly curved) Fukaya A∞ algebra to the complex of de Rham currents on X, with degree shift n+1. Under a standing regularity assumption on the disk moduli spaces, Theorem 1 proves the structure equation d∘p^γ − (−1)^{n+1}p^γ∘∂_hoch = 0. Theorems 2–4 descend the map to normalized Hochschild, Connes cyclic, and reduced cyclic complexes, the last after quotienting the codomain by the current ζ_L. Theorem 5 extends the construction to extended cyclic homology under the homological triviality assumption i_*([L]) = 0, using the sphere operator q^γ_{∅,1}(η) to cancel the boundary-collapse term. Section 6 sketches pseudoisotopy analogues. The paper also proves GW-like axioms: fundamental class, divisor, and energy zero.","tokens_in":42555,"tokens_out":16586,"duration_ms":191426,"significance":"If the results are accepted under the stated regularity hypotheses, the paper gives a detailed, geometric, chain-level construction of open-closed maps for curved A∞ algebras, with explicit signs and a clean derivation of the structure equation from Stokes' theorem and boundary stratification. The treatment of cyclic and reduced cyclic variants is thorough, and the divisor/fundamental-class/energy-zero properties are exactly the axioms needed for later applications to gravitational descendants and obstruction theory. The paper is not circular: the p-operators are defined geometrically and the chain map property is proved, not assumed. The main weakness is the scope of the regularity assumption: the smoothness of all moduli spaces and the submersion properties of evaluations are verified only for homogeneous spaces and products, while the general symplectic-manifold statement is explicitly deferred to virtual fundamental class techniques. The pseudoisotopy section, moreover, omits the necessary base-dga machinery.","major_comments":[{"comment":"The central construction and the proof of Proposition 4.1 require every moduli space M_{k,l+1}(β) to be a smooth orbifold with corners and the relevant evaluation maps to be proper submersions. Strictly speaking, Section 4.1 only needs ev^i_0 to be a current pushforward, so ev^i_0 itself need not be a submersion; the load-bearing hypotheses are smoothness of the moduli spaces and the submersion property for the evaluation maps entering the closed-open q-operators and the gluing analysis. The paper verifies these hypotheses only for homogeneous spaces and products, and states that the general case is 'expected to extend' via virtual fundamental cycles. This is not a proof. As written, Theorems 1–5 are theorems conditional on a regularity assumption that is verified in a limited class; the unqualified wording of the abstract and of the theorem statements should be corrected, or the VFC ext","section":"§1.2.1, Eq. (3), Proposition 4.1"},{"comment":"The pseudoisotopy section asserts 'analogs of Theorems 1-5' for the maps ˜p, but immediately says 'we omit the details here.' The extension requires (i) Hochschild and cyclic complexes for A∞ algebras over the base dga R = A^*(I;R), for which the only reference is the unpublished '[14] in preparation', and (ii) a proof that the operators ˜q endow C := A^*(I×L;R) with an A∞ structure over this base dga. Neither is supplied. Consequently the pseudoisotopy analogues are not established by this manuscript. The section should either be expanded to a full proof or explicitly labelled as a program/future work rather than a result.","section":"§6.3"},{"comment":"The extension to extended cyclic homology depends on the identity d(p^γ_0(1)) = (−1)^{n+1}p^γ_1(m^γ_0(1)) + q^γ_{∅,1}(i_*1_L), obtained by summing the k=0 case of Proposition 4.1 over l. The displayed derivation in the proof of Theorem 5 is terse: it is easy to misread the notation p^γ_0(1;γ^⊗l), since p^γ_0 has no first input. Please rewrite this computation with the precise definition of p^γ_0(1) and the summation over l, so the sign and the factorial factors are checkable. This is a local issue, but it is load-bearing for Theorem 5.","section":"§5.2, proof of Theorem 5"}],"minor_comments":[{"comment":"The notation in the display defining C^{λ,+}_* mixes the full tensor algebra T(A[1]) with the reduced tensor algebra T(A[1]). Since the distinction between the two is important for the extended cyclic complex, please use different symbols or explicitly say that the equality is after identifying R with the empty tensor factor.","section":"§2.3.2"},{"comment":"The proof says 'For (k+1,l+1,β) ≠ (1,1,β0)' where the statement concerns (k+1,l,β) ≠ (1,0,β0). These are consistent after translating l, but the reader has to pause; please align the indices.","section":"Proposition 4.10, proof"},{"comment":"The forgetful map in the proof is written as π : M_{k,l+2}(β) → M_{k,l+1}(β), but for p_{k,l} the relevant forgetful map should be π : M_{k,l+1}(β) → M_{k,l}(β). Please correct the dimensions.","section":"Proposition 4.15, proof"},{"comment":"There are several typos: 'Hichschild' in §5.2, 'neccesarily' in §2.4, 'quotieting' in §5.2, and 'identify'/'identity' slips. None affect the mathematics, but they should be cleaned up.","section":"Throughout"},{"comment":"Section 3.1 records dζ_S = (−1)^n ζ_L for a chain S with ∂S = L, while Theorem 5 chooses η with dη = −ζ_L. The sign convention is consistent if η is normalized appropriately, but an explicit sentence connecting ζ_S and η would prevent sign confusion in the proof of Theorem 5.","section":"§3.1 and Theorem 5"}],"recommendation":"major_revision","confidential_remarks":"The paper is an honest and technically detailed sequel to the authors' earlier work. The main reason for major revision, rather than acceptance, is scope: the general claims are conditional on a regularity assumption verified only for homogeneous spaces, and the pseudoisotopy section relies on an unpublished reference and omits proofs. I would be willing to accept a version that either proves the missing VFC extension or clearly restricts the main theorems to the case where the regularity assumption is verified, and that either proves or explicitly disclaims the pseudoisotopy analogues."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper delivers what it says: chain-level open-closed maps from Hochschild and several cyclic homology versions of a curved Fukaya A-infinity algebra to de Rham currents, plus Gromov-Witten-type axioms and a pseudoisotopy discussion. The p-operators and their descent to normalized, cyclic, reduced cyclic, and extended cyclic complexes are new relative to the earlier closed-open q-operators, and allowing curvature and working with currents is a real extension. The proofs are explicit and careful, especially Proposition 4.1 and the sign bookkeeping in Theorem 1. I checked several of the algebraic identities and found them consistent; the paper is not circular. The regularity assumption in Section 1.2.1 is stated clearly, and the authors verify it for homogeneous spaces; but the abstract and theorem statements do not carry the caveat, so a reader could reasonably over-read the generality. The stress-test note is on target: the pushforwards and Stokes argument genuinely need the moduli spaces to be smooth orbifolds with proper submersion evaluations, and the promised VFC extension is not carried out. That is a real limitation, but it is a known and openly declared one, not a hidden flaw. The other two soft spots are Section 6.3, where the pseudoisotopy analogues of Theorems 1-5 are asserted without proof, and the reliance on the companion paper [14] for some algebraic technicalities. These are minor relative to the core, because Section 6 is explicitly a discussion and the algebraic constructions in Section 2 are sufficiently concrete. This paper is for specialists in symplectic topology who need open-closed maps with rational curve dependence and cyclic symmetry, especially for gravitational descendants and obstruction theory. It deserves a serious referee: the referee should ask the authors to either prove the VFC extension or state the regularity assumption explicitly in the main theorems, and to either fill in or clearly mark Section 6.3 as conditional. I would send it out.","headline":"A serious, technically detailed paper that builds chain-level open-closed maps for curved Fukaya A-infinity algebras and descends them to cyclic homology; the main theorems are conditional on a regularity assumption verified only for homogeneous spaces, with the general VFC extension deferred.","tokens_in":43071,"tokens_out":1869,"would_cite":true,"duration_ms":25128,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D37","53D45","19D55","58A10","32Q65"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper builds open-closed maps from Hochschild and cyclic homology of a curved Fukaya A∞ algebra to the de Rham current cohomology of the ambient symplectic manifold, and proves they satisfy chain-level Gromov-Witten axioms.","keywords":["open-closed map","Hochschild homology","cyclic homology","Fukaya A-infinity algebra","curved A-infinity algebra","Gromov-Witten axioms","differential forms","currents"],"falsifier":"Find a Lagrangian (X,L) outside the homogeneous class where some moduli space M_{k,l+1}(β) has a non-trivial singular stratum or an evaluation map that is not a proper submersion; then the pushforward defining p^γ is not a current as defined, and the Stokes proof of Proposition 4.1 would need virtual-fundamental corrections not supplied by the paper.","tokens_in":42072,"feed_emoji":"🌀","tokens_out":6306,"duration_ms":67228,"temperature":0.7,"pith_summary":"The paper constructs, on the chain level, open-closed maps from Hochschild and cyclic homologies of the A∞ algebra of differential forms on a Lagrangian submanifold to the de Rham current cohomology of the ambient symplectic manifold. The construction works for possibly curved A∞ algebras and covers several homology variants: ordinary Hochschild, normalized Hochschild, Connes cyclic, reduced cyclic, and extended cyclic homology, with a codomain adjustment or an extra sphere contribution when the Lagrangian is homologically trivial. The authors verify that these operations satisfy analogues of the Gromov-Witten axioms—vanishing on the fundamental class, the divisor equation, and the energy-zero identification with ordinary push-forward of forms. If correct, this yields a differential-form model for open-closed string maps and a foundation intended for gravitational descendants and obstruction theory in open Gromov-Witten theory.","feed_headline":"Disk maps turn Hochschild homology into quantum cohomology","feed_subtitle":"A chain-level differential-form construction that satisfies the Gromov-Witten axioms, curved algebras included.","key_machinery":"The central object is the open-closed map p^γ_{k,l}(α;γ) = (-1)^ε (evi_0)_*(∧_j evi_j^* γ_j ∧ evb_j^* α_j), a current on X obtained by integrating over the moduli space of genus-zero stable disks with k boundary and l+1 interior marked points. The interior marked point 0 is the output; boundary inputs are forms on L and interior inputs are forms on X. The defining structure equation (Proposition 4.1) describes the boundary of these moduli spaces in terms of the closed-open q-operator and sphere contributions, and it is this equation that yields the chain-map property and all the homological descents.","core_discovery":"The central claim is that the operator p^γ = Σ_k p^γ_k, defined by pushing forward, along the interior evaluation map of moduli spaces of stable disks, the exterior product of boundary evaluation pullbacks of differential forms and interior insertions of a closed two-form γ, is a degree n+1 chain map from the Hochschild complex of the curved A∞ algebra C = A*(L;R) to the complex of currents on X. The proof applies Stokes' theorem to the codimension-one boundary of these moduli spaces: the boundary strata produce exactly the A∞-type structure equation that makes d∘p^γ and p^γ∘∂_hoch agree up to the stated sign. The same structure equation, together with unit and cyclic-symmetry properties, yi","pith_inferences":["If the regularity assumption on the moduli spaces could be replaced by virtual fundamental class techniques, the same structure equations and Gromov-Witten axioms should hold for general symplectic targets; the paper states this extension is expected but does not prove it.","The chain-level Gromov-Witten axioms may provide a direct route to open gravitational descendants without first passing to cohomology, which the authors plan in a future work.","The dependence of the extended cyclic map on the choice of the bounding chain η suggests a concrete interplay between open-closed maps and the obstruction-theoretic bounding-chain formalism used for genus-zero open Gromov-Witten invariants."],"forward_implications":["The open-closed map gives chain-level maps from Hochschild and cyclic homology of the Fukaya A∞ algebra to quantum cohomology, so open-string invariants can be compared with closed-string constraints without passing directly to cohomology.","The normalized, reduced, and extended variants make the construction compatible with unit insertions, the empty list, and curved A∞ algebras, widening the class of Lagrangians and bulk deformations to which the method applies.","The Gromov-Witten axioms verified at chain level—fundamental class, divisor, and energy zero—provide the structural input needed for defining open gravitational descendants.","The pseudoisotopy version gives families of such maps as the almost complex structure or the bulk constraint varies, allowing comparison of the resulting invariants across a one-parameter family.","Under homological triviality of the Lagrangian, the extended cyclic map uses sphere contributions weighted by a bounding chain, linking the construction to obstruction theory and the space of bounding chains."],"supporting_citations":[{"why":"Establishes that the cohomology of currents on an orbifold agrees with de Rham cohomology, so the target of p^γ is ordinary cohomology.","marker":"[5]"},{"why":"Supplies the orientation theory for moduli spaces of stable disk maps via relative spin structures and the foundational open-closed map context.","marker":"[8]"},{"why":"Provides the algebraic construction of Hochschild, normalized, cyclic, and extended cyclic complexes for curved A∞ algebras used in Section 2.","marker":"[14]"},{"why":"Gives the Connes cyclic complex formalism and the cyclic identity that the paper generalizes to curved A∞ algebras.","marker":"[21]"},{"why":"Supplies the closed-open operators q, the A∞ structure on C, and the regular-moduli-space assumptions on which Proposition 4.1 relies.","marker":"[28]"},{"why":"Provides the calculus of currents on orbifolds with corners—pushforward, pullback, and Stokes' theorem—that carries the proof of the structure equations.","marker":"[29]"}],"fun_headline_variants":["Disk pushforwards satisfy Gromov-Witten axioms","Differential-form maps satisfy GW axioms","Hochschild-to-current map from Stokes' theorem","Open-closed maps for curved A∞ algebras","Fukaya disk maps verify Gromov-Witten axioms"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"Every moduli space of stable disks and spheres used in the construction is assumed to be a smooth orbifold with corners and to have evaluation maps that are proper submersions, so that Stokes' theorem and pushforwards of currents apply; the paper verifies this assumption only for homogeneous targets such as projective spaces and their products.","fun_headline_variants_meta":{"raw":{"variants":["Disk pushforwards satisfy Gromov-Witten axioms","Differential-form maps satisfy GW axioms","Hochschild-to-current map from Stokes' theorem","Open-closed maps for curved A∞ algebras","Fukaya disk maps verify Gromov-Witten axioms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001321,"raw_usage":{"total_tokens":5142,"prompt_tokens":595,"completion_tokens":4547,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":339,"completion_tokens_details":{"reasoning_tokens":4472}},"tokens_in":339,"tokens_out":4547,"duration_ms":35725,"temperature":1.0,"reasoning_tokens":4472,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T04:35:11.537880+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a Lagrangian (X,L) outside the homogeneous class where some moduli space M_{k,l+1}(β) has a non-trivial singular stratum or an evaluation map that is not a proper submersion; then the pushforward defining p^γ is not a current as defined, and the Stokes proof of Proposition 4.1 would need virtual-fundamental corrections not supplied by the paper.","supporting_citations":[{"cited_title":"de Rham,Differentiable manifolds, Grundlehren der mathematischen Wissenschaften [Funda- mental Principles of Mathematical Sciences], vol","cited_arxiv_id":null,"evidence_quote":"Establishes that the cohomology of currents on an orbifold agrees with de Rham cohomology, so the target of p^γ is ordinary cohomology."},{"cited_title":"Fukaya, Y.-G","cited_arxiv_id":null,"evidence_quote":"Supplies the orientation theory for moduli spaces of stable disk maps via relative spin structures and the foundational open-closed map context."},{"cited_title":"Giterman and J","cited_arxiv_id":null,"evidence_quote":"Provides the algebraic construction of Hochschild, normalized, cyclic, and extended cyclic complexes for curved A∞ algebras used in Section 2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the closed-open operators q, the A∞ structure on C, and the regular-moduli-space assumptions on which Proposition 4.1 relies."}],"review_version":1}