{"id":"854f63ec-df11-404a-9733-62b4c399b097","arxiv_id":"2505.09950","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Big quantum D-module mirror symmetry for flag varieties G/P follows from small mirror symmetry via a new unfolding theorem for equivariant F-bundles.","lead":"This paper proves that the big quantum cohomology of flag varieties is mirror symmetric by extending a classical unfolding theorem to torus-equivariant F-bundles. It gives a general way to lift known small-mirror results to big-mirror results in cases where older reconstruction methods fail.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.29's freeness proof is incomplete: Lemma 4.30 is false as stated (C((z)) counterexample) and its Nakayama generation step needs finite generation/separatedness not established for HB,big_R, on which Theorem 4.35 depends.","rationale":"The reader's weakest assumption—freeness of HB,big_R—is exactly the load-bearing point for applying Thm 3.36. I examined Prop. 4.29 more closely. The assertion 'y_jω∈im(∂) iff ω∈im(∂)' is actually correct: ∂ is R[[y]]-linear because the y_j are not coordinates on X∨_P and d~W is linear in y_j; if y_jω=∂η, comparing the coefficient of y_j^1 in η=Σ y_j^k η_k gives ω=∂η_1. So the torsion-freeness part is fine. The gap is in the invocation of Lemma 4.30 to pass from a basis of M/(z)M to a basis of M. Lemma 4.30 is false as stated: C((z)) is a torsion-free C[[z]]-module with M/zM=0 but is not zero. The proof's Nakayama step silently assumes finite generation of M (or at least separatedness), which is precisely what is being proved for HB,big_R. Since Prop. 4.29 is the only place where the B-side big R-linear structure is shown to have finite rank, Thm 4.35 is not established by the written argument. This is not a demonstrated falsity of the theorem—the CP^1 case likely works—so the verdict should remain CONDITIONAL, with the condition being a repaired freeness proof (e.g., adding finite-generation/separatedness hypotheses to Lemma 4.30 or giving a direct argument for HB,big_R). I agree with the reader's identification of freeness as the weakest assumption, though the specific defect is in Lemma 4.30 rather than in the 'iff' assertion.","tokens_in":36199,"tokens_out":32821,"duration_ms":319798,"concrete_test":"Compute HB,big_R explicitly for X=CP^1 (G=SL_2, P=B): with k=C(q), R=C(q)[λ], ~W=z+q/z+y_2 z and equivariant term λ log z, form the cokernel of ∂=u d+d~W∧−λ dz/z∧ over R[[y_1,y_2,u]], and check whether it is free of rank 2 with a basis lifting the reductions at y_2=0. If it fails to be free (or has infinite rank), Theorem 4.35 fails in the base case; if it is free, the freeness statement survives but Lemma 4.30 still needs a finite-generation or separatedness hypothesis to be valid.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim Theorem 4.35 applies the equivariant unfolding theorem (Thm 3.36) to the B-side big F-bundle FB,big. This requires the R-linear (T)-structure HB,big_R to be finite rank over R[[y,u]], which is the content of Prop. 4.29. The proof of Prop. 4.29 is not complete. It states that since y_jω∈im(∂) iff ω∈im(∂), the element y_j is torsion-free, and then invokes Lemma 4.30 to conclude freeness. The 'iff' itself is correct: ∂ is y_j-linear, so comparing coefficients in y_j gives ω=∂η_1 when y_jω=∂η. The problem is Lemma 4.30: as stated it is false. Let R0=C and M=C((z)) as an R0[[z]]-module. Then z is torsion-free, M/zM=0 (empty set is a basis of the reduction), but M is not generated by the empty set over C[[z]]. The proof of Lemma 4.30 uses Nakayama's lemma to pass from generation modulo (z) to generation of M; this is valid only if M is finitely generated over R0[[z]] (or at least separated with a closed-image argument). Finite generation of HB,big_R is exactly what is being proved and is not established a priori; separatedness of the cokernel in the (y_{r+1},...,y_N)-adic topology is also not shown. Thus the freeness claim is not proved by the given argument, and the application of Thm 3.36 to FB,big is unsupported at a load-bearing point.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the notion of an equivariant F-bundle, combining a k-linear F-bundle with a finite-rank R-linear lift of its underlying (T)-structure, and proves an unfolding theorem in the spirit of Hertling-Manin. The main abstract result (Theorem 3.36) asserts existence of maximal unfoldings under (IC), (GC) and freeness of coker μ_v, and uniqueness under (GC'). The authors then apply this to flag varieties G/P: they reformulate the small equivariant quantum D-module mirror symmetry of [13] as an isomorphism of small equivariant F-bundles, construct a big B-model equivariant F-bundle from Rietsch's Landau-Ginzburg mirror via an unfolded superpotential, verify maximality on both sides, and conclude a big equivariant quantum D-module mirror theorem (Theorem 4.35), with a non-equivariant limit in Theorem 4.38. The intended contribution is a mechanism for passing from small to big quantum cohomology mirror symmetry in cases where the small quantum cohomology is neither H2-generated nor semisimple.","tokens_in":36545,"tokens_out":21890,"duration_ms":228372,"significance":"If the main theorem is correct, this is a valuable contribution: it gives a general formal unfolding theorem for equivariant F-bundles and applies it to flag varieties beyond the H2-generated and semisimple cases, such as isotropic Grassmannians of type C. The A-side maximality check (Proposition 4.10) is clean and explicit, the reduction of (GC') to the known generation of localized equivariant quantum cohomology (Lemma 4.6) is economical, and the paper is careful about infinite-rank issues by separating the k-linear F-bundle from its finite-rank R-linear lift. The unfolding proofs are carried out by detailed inductive constructions rather than by black-box analytic arguments. The central application is, however, conditional on a correct proof that the big B-model R-linear lift is a finite-rank free module over R[[y,u]]; the current proof of this fact is not valid.","major_comments":[{"comment":"The exposition of the B-model also asserts 'Since y_jω∈im(∂) iff ω∈im(∂)' without proof. This is true only with the convention that d~W is the vertical part of the differential, i.e. that d does not differentiate the y variables; the authors should state this convention explicitly, since the flatness of the u-direction and the freeness argument both depend on it.","section":"§4.3.1, Proposition 4.29 and Lemma 4.30"}],"minor_comments":[{"comment":"The text reads 'we review the B-side of mirror symmetry for for G/P'; the word 'for' is duplicated.","section":"§4.2, first paragraph"},{"comment":"There is a typo: 'The fist step in our proof' should be 'The first step in our proof'.","section":"§1.2.1"},{"comment":"The displayed equation '∂tiTj = ∂tjTj' should almost certainly read '∂tiTj = ∂tjTi'; as printed it is an identity in only one index and does not express flatness.","section":"§3.2, proof of Lemma 3.16, equation (3.25)"},{"comment":"In the statement of the non-equivariant limit, the source of the isomorphism is written as (HA,big,λ0,∇B,big,λ0); the connection in the first factor should be ∇A,big,λ0, not ∇B,big,λ0.","section":"§4.3.2, Theorem 4.38"},{"comment":"The conclusion of Lemma 4.30 says '{Ω1,...,ΩN}⊂M is an R0[[z]]-basis of M[[z]]'; since the Ωi are elements of M, the intended conclusion is that they form a basis of M as an R0[[z]]-module. The notation should be corrected.","section":"§4.3.1, Lemma 4.30"},{"comment":"In part (2), 'a R[[tI,u]]-module' should be 'an R[[tI,u]]-module'; also, the dependence of the paper on the companion preprint [24] and on the small mirror theorem [13] should be stated more prominently, since several technical results are quoted from these preprints.","section":"§2.4, Definition 2.10"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the authors' companion preprint [24] for framing and extension results, and on [13] for the small equivariant mirror theorem. Both are preprints, and the editors may wish to verify their status and the exact overlap with the present manuscript. This does not by itself affect my recommendation; the decision is driven by the unproved finite freeness of HB,big_R in Proposition 4.29, which is a load-bearing point for Theorem 4.35."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper does two things. First, it proves an equivariant version of Hertling-Manin unfolding for F-bundles, expanding the machinery to modules over a ring with a torus action. That part is solid and interesting. The key idea—that working equivariantly can restore an H2-generation condition even when ordinary small quantum cohomology is neither semisimple nor H2-generated—is genuinely new and worth knowing. Second, it applies the theorem to prove big quantum D-module mirror symmetry for all flag varieties G/P, covering cases like SG(2,2n). If the proof is right, it's a major in-field result.\n\nBut the application has a load-bearing gap. Proposition 4.29 claims the big B-model module HB,big_R is free over R[[y,u]]. The argument rests on Lemma 4.30, and that lemma is false as stated. Take R0 = C, M = C((z)) over C[[z]]. Then z is torsion-free (it's a unit), M/zM = 0, and the empty set is a basis of the reduction, but M is not generated over C[[z]]. Nakayama needs finite generation (or separatedness plus a closed-image argument), and finite generation is exactly what's being proved. The proof of Lemma 4.30 applies Nakayama without warrant, and Proposition 4.29 inherits the problem. Since Theorem 4.35 applies the equivariant unfolding theorem to FB,big, which requires a finite rank R-linear lift, the central mirror symmetry claim is not supported as written.\n\nThe gap may well be fixable. The small B-model module is finite free by [13], and one might prove the big module is a base change of the small one, or use versal deformation theory for isolated singularities to get finite generation. But neither is in the paper. The abstract also overstates the result: the body proves a formal D-module isomorphism, not convergence or a statement about the actual big quantum cohomology ring.\n\nThe citation pattern is mostly fine. The paper leans on [13] (Chow's small mirror theorem) and the authors' own [24] for framing results; that is acceptable if those preprints hold up, but it does make the whole package conditional on works not yet published. The formal unfolding theorem (Theorem 3.28) and the A-side maximality check are the strongest parts. The B-side construction and the freeness of its module are the softest.\n\nVerdict: deserves a serious referee. I would send it out, but the referee should be asked to specifically check Prop 4.29 and Lemma 4.30. If the freeness gap cannot be closed, the flag-variety application falls; the equivariant unfolding theorem itself might still be salvageable.","headline":"A genuinely new equivariant unfolding theorem with a promising application to flag-variety mirror symmetry, but the big B-model freeness proof rests on a false lemma and needs repair before the main theorem is supported.","tokens_in":37122,"tokens_out":5711,"would_cite":false,"duration_ms":59997,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14D15","14M15","14N35","34M56"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the big equivariant quantum D-module of a flag variety $G/P$ is canonically isomorphic to the big Gauss-Manin system of its mirror Landau-Ginzburg model, uniquely extending the small mirror map.","keywords":["equivariant F-bundles","Frobenius manifolds","unfolding theorem","big quantum cohomology","mirror symmetry","flag varieties","Gauss-Manin system","quantum D-modules"],"falsifier":"Compute, for a concrete flag variety such as $\\mathrm{SG}(2,2n)$ with $n=3$, whether the $R[[y,u]]$-module $H_{B,big,R}$ defined by the unfolded superpotential is free: specifically, check if multiplication by any $y_j$ is injective on the module. A single nonzero element annihilated by $y_j$ would violate the freeness assumption in Proposition 4.29 and would invalidate the B-side construction used in the main theorem.","tokens_in":35990,"feed_emoji":"🪞","tokens_out":12751,"duration_ms":104115,"temperature":0.7,"pith_summary":"The paper establishes an unfolding theorem for equivariant F-bundles, a torus-equivariant refinement of Frobenius manifolds, and uses it to prove that big quantum cohomology of flag varieties is mirror-symmetric. Concretely, the big equivariant quantum D-module of $G/P$ is shown to be isomorphic to the big Gauss-Manin system of the mirror Landau-Ginzburg model, with the isomorphism uniquely extending the known small mirror map. The central move is to work equivariantly with respect to the maximal torus: although the small quantum cohomology of many flag varieties is neither generated by divisor classes nor semisimple, its equivariant version becomes divisor-generated after localizing in equivariant parameters, restoring the generation condition needed for unfolding. A sympathetic reader should care because this yields a general small-to-big reconstruction of quantum cohomology for flag varieties of all Lie types, without the restrictive hypotheses that earlier reconstruction theorems required.","feed_headline":"Small-to-big mirror symmetry proved for flag varieties","feed_subtitle":"An equivariant unfolding theorem lifts the small mirror map to the big D-module, without divisor-generation or semisimplicity assumptions.","key_machinery":"The paper's central object is the equivariant F-bundle, defined as a $k$-linear F-bundle $(H,\\nabla)$ over a formal base in infinitely many variables, together with an $R$-linear lift $(H_R,\\nabla_R)$ of its underlying (T)-structure of finite rank over $R = k[\\lambda]$ (the equivariant parameter ring), compatible via a fixed isomorphism $\\alpha$. A (T)-structure is a flat connection in the base directions only, without the $u$-direction. The argument is carried by three pieces of machinery. First, a formal version of the classical unfolding theorem for meromorphic connections (Theorem 3.28) produces maximal unfoldings of finite-rank F-bundles over integral domains under conditions (IC), (GC), and freeness of $\\operatorname{coker}\\mu_v$, with uniqueness from (GC'); second, a characterization (Lemma 3.1) saying an F-bundle is uniquely determined by its underlying (T)-structure and the value of the $u$-direction connection at one point, whenever a framing exists; third, the construction of a maximal unfolding on the B-side by adding deformation terms $y_j f_j$ to the superpotential. The torus action is what makes condition (GC') hold: after localizing in $\\lambda$, the equivariant small quantum cohomology ring is generated by degree-two classes (Lemma 4.6), even though the ordinary small quantum cohomology may be neither divisor-generated nor semisimple.","core_discovery":"On its own terms, the central discovery is Theorem 4.35: there exists a unique isomorphism of equivariant F-bundles $(\\mathrm{mir}^{big}_k, \\Phi^{big}_{mir,k}), (\\mathrm{mir}^{big}, \\Phi^{big}_{mir})$ from the big B-model F-bundle $F_{B,big}$ to the big A-model F-bundle $F_{A,big}$, extending the small equivariant mirror isomorphism of Proposition 4.26. Here $F_{A,big}$ is the equivariant F-bundle built from the equivariant big quantum cohomology of $G/P$, and $F_{B,big}$ is built from the Gauss-Manin system of the unfolded mirror superpotential $W + \\sum_{j=r+1}^N y_j f_j$. The theorem therefore says that the big equivariant quantum D-module of a flag variety and the big Gauss-Manin system of its mirror are the same object, canonically. Taking $\\lambda=0$ gives the non-equivariant big mirror symmetry of Theorem 4.38.","pith_inferences":["The same equivariant-unfolding strategy might generalize to other Fano varieties with a torus action whose localized equivariant cohomology is generated by degree-two classes, potentially yielding big mirror symmetry for broader classes of homogeneous spaces and possibly for some toric or spherical varieties.","If the freeness assumption in Proposition 4.29 could be proved by a general argument about Gauss-Manin systems, the theorem would become unconditional for all $G/P$ without case-checking; conversely, a counterexample for a specific parabolic $P$ would show where the B-side construction breaks over $R$.","The paper leaves two natural upgrades implicit: analytic convergence of the big mirror map around $\\tau=0$, and compatibility with the intersection pairings on both sides; establishing these would promote the formal isomorphism to an isomorphism of genuine Frobenius manifolds.","A direct computational check for a small example such as $\\mathrm{Gr}(3,5)$, comparing the leading quantum-correction terms of the big mirror map with the known small map, would test the uniqueness statement in practice."],"forward_implications":["Big quantum D-module mirror symmetry holds for all flag varieties $G/P$ of simply-connected simple groups, in both equivariant and non-equivariant forms (Theorems 4.35 and 4.38).","All genus-zero Gromov-Witten invariants of $G/P$, not just the small ones, are encoded in the unfolded mirror superpotential; the small mirror map determines them uniquely.","The equivariant unfolding theorem applies where both divisor-generation and semisimplicity fail, for instance the isotropic Grassmannian $\\mathrm{SG}(2,2n)$ in type C.","The formal unfolding theorem is now available over integral domains containing $\\mathbb{Q}$, with existence requiring $\\operatorname{coker}\\mu_v$ free and uniqueness following from (GC') alone, which strengthens the original complex-analytic statement."],"supporting_citations":[{"why":"Supplies the small equivariant quantum D-module mirror isomorphism and the finite freeness of the small B-model module over $R[[y_{\\le r},u]]$.","marker":"[13]"},{"why":"Provides the universal unfolding theorem for meromorphic connections that this paper generalizes to the formal and equivariant setting.","marker":"[23]"},{"why":"Supplies the framing existence and extension results for (T)-structures and the characterization of F-bundles used throughout Section 3.","marker":"[24]"},{"why":"Constructs the mirror Landau-Ginzburg model whose Gauss-Manin system forms the B-side.","marker":"[45]"},{"why":"Supplies Lemma 4.3, the generation of localized equivariant cohomology of $G/P$ by degree-two classes, which underlies the (GC') condition.","marker":"[7]"},{"why":"Provides the explicit small mirror map for the Grassmannian $\\mathrm{Gr}(3,5)$ used as the illustrative example in the paper.","marker":"[36]"}],"fun_headline_variants":["Equivariant unfolding proves big mirror symmetry","From small to big: flag variety mirror symmetry","Unfolding theorem lifts mirror map to big quantum","Big cohomology mirror theorem via equivariant unfolding"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the big B-model module $H_{B,big,R}$ is a finite free module over $R[[y,u]]$; the proof of this rests on the terse assertion that multiplication by $y_j$ has no torsion in the Gauss-Manin module, so if that freeness fails, the B-side cannot be presented as a finite-rank equivariant F-bundle and the unfolding comparison cannot begin.","fun_headline_variants_meta":{"raw":{"variants":["Equivariant unfolding proves big mirror symmetry","From small to big: flag variety mirror symmetry","Unfolding theorem lifts mirror map to big quantum","Big cohomology mirror theorem via equivariant unfolding"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000242,"raw_usage":{"total_tokens":1466,"prompt_tokens":826,"completion_tokens":640,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":442,"completion_tokens_details":{"reasoning_tokens":580}},"tokens_in":442,"tokens_out":640,"duration_ms":6740,"temperature":1.0,"reasoning_tokens":580,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:20:49.826529+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a concrete flag variety such as $\\mathrm{SG}(2,2n)$ with $n=3$, whether the $R[[y,u]]$-module $H_{B,big,R}$ defined by the unfolded superpotential is free: specifically, check if multiplication by any $y_j$ is injective on the module. A single nonzero element annihilated by $y_j$ would violate the freeness assumption in Proposition 4.29 and would invalidate the B-side construction used in the main theorem.","supporting_citations":[{"cited_title":"Unfoldings of meromorphic connections and a construction of Frobenius manifolds","cited_arxiv_id":null,"evidence_quote":"Provides the universal unfolding theorem for meromorphic connections that this paper generalizes to the formal and equivariant setting."},{"cited_title":"A mirror symmetric construction ofqH∗ T (G/P) (q).Adv","cited_arxiv_id":null,"evidence_quote":"Constructs the mirror Landau-Ginzburg model whose Gauss-Manin system forms the B-side."},{"cited_title":"Buch, Pierre-Emmanuel Chaput, Leonardo C","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 4.3, the generation of localized equivariant cohomology of $G/P$ by degree-two classes, which underlies the (GC') condition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the explicit small mirror map for the Grassmannian $\\mathrm{Gr}(3,5)$ used as the illustrative example in the paper."}],"review_version":1}