{"id":"507b1e79-d20f-4592-a66f-c8529b6644cc","arxiv_id":"2607.11572","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Under mild assumptions the orbifold Hecke algebra of [X/G] is isomorphic to the degree-0 cohomology of the A∞-endomorphism algebra of a regular cotangent fiber in the bulk-deformed wrapped Fukaya category of T*[X/G].","lead":"The paper proves obstructions that prevent certain nodal orbicurves with ghost components on cyclic quotient singularities from being smoothed. These obstructions are used to identify the orbifold Hecke algebra of [X/G] with the degree-0 endomorphism algebra of a regular cotangent fiber inside the bulk-deformed wrapped Fukaya category of T*[X/G].","discovery_kind":"unification","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The central claim is the existence of the surjective algebra morphism EV:HW0ℏℏℏ(T*[x][X/G])\to Hℏℏℏ|ℏℏℏ0=0([X/G]) that becomes an isomorphism under the topological hypothesis π2(X)⊗Q=0. The analytic engine consists of the obstruction theorems for interior and boundary ghosts (Theorems 5.34 and 5.37) together with the localization that reduces general sequences to the model situation of Propositions 5.32 and 5.36. Both rest on an adapted almost complex structure whose existence is proved, not assumed, for the cotangent-bundle setting of the main theorem. The subsequent algebraic identification of the degree-zero Floer cohomology with the specialized Hecke algebra follows the standard Abouzaid-style evaluation map once the only surviving boundary degenerations are the unobstructed single-stacky-point disks that produce the Hecke relations (6.35). The flatness theorem of Etingof then upgrades surjectivity to isomorphism. No hidden circularity or unproved analytic step remains once the construction of J0 is granted, and that construction is supplied. The reader's assessment of low correctness risk and the ACCEPT verdict are therefore left unchanged.","tokens_in":71621,"tokens_out":660,"duration_ms":7826,"concrete_test":"Verify that the interpolation formula (5.15) for JY on the annular region δY/3≤|(v,w)|<δY produces an ω-tame structure whose C0-distance to Jstd is smaller than the ε of Proposition 5.5; if the resulting J0 fails to remain of contact type or adapted after the final G-equivariant averaging, the jet-evaluation maps of Section 4 lose transversality and the Hecke relations cannot be extracted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption (existence of a contact-type, type-A-adapted G-equivariant J0) is correctly flagged, but Proposition 5.5 constructs it explicitly for cotangent bundles of complex manifolds by interpolating the split almost complex structure JY of (5.8) with the standard structure induced by a G-invariant metric, using a curvature estimate that keeps JY ω-tame near T*Y (Lemma 5.2). The subsequent localization tricks (Lemmas 5.27, 5.30) and jet-vanishing theorems (Theorems 5.34, 5.37) therefore rest on a verified geometric object rather than an unproved existence claim. No other load-bearing gap appears in the derivation of the surjective algebra morphism EV of Theorem 6.6 or in the flatness argument that upgrades it to an isomorphism when π2(X)⊗Q=0.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper develops analytic tools for obstructing smoothings of nodal orbicurves that carry orbighosts mapped into cyclic quotient singularities (Theorems 1.1–1.2, proved via equivariant jet transversality in §4 and two-step gluing with localization in §5). As the main application it constructs the regular bulk-deformed wrapped Fukaya category of the cotangent-bundle orbifold [T*X/G] and proves that the degree-0 cohomology of the endomorphism A∞-algebra of a regular cotangent fibre is related to Etingof’s orbifold Hecke algebra by a surjective algebra morphism EV that becomes an isomorphism whenever π₂(X)⊗ℚ=0 (Theorem 1.6 / Theorem 6.6).","tokens_in":71838,"tokens_out":507,"duration_ms":5358,"significance":"The work supplies one of the first complete computations in orbifold Lagrangian Floer theory that involves a singular Lagrangian (the zero-section). The obstruction package for Am-1-singularities is new and of independent interest for reduced orbifold Gromov–Witten theory. The identification of HW⁰ with the specialized Hecke algebra recovers, in a uniform geometric way, a large class of algebras that appear in representation theory (affine and double-affine Hecke algebras, Broué–Malle–Rouquier algebras, etc.). The constructions are written with full analytic detail and rest on an explicitly constructed adapted almost-complex structure (Proposition 5.5).","major_comments":[],"minor_comments":[{"comment":"Several typographical slips appear in the introduction and abstract (e.g., “orb icurves”, “A∞ -algebra”, “ℏℏℏ”). A careful proof-reading pass would improve readability.","section":null},{"comment":"The orientation package for orbicurves is deferred to Appendix A; a one-sentence pointer in §3.4 to the precise isomorphism det(Du)≅o_x̂ used for signs would help the reader.","section":null},{"comment":"In §7 the examples are listed rather than computed; even a short verification for one classical case (e.g., the A1 double-affine Hecke algebra) would make the geometric origin of the Hecke relation more transparent.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is long and technical, but the central claims are solid and the analytic foundations are carefully laid. It is a natural fit for a top symplectic-geometry journal."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The two things worth knowing: they prove genuine obstruction criteria for smoothing orbicurves with ghosts on cyclic quotient singularities (Theorems 1.1–1.2), upgrading Doan–Walpuski/Ekholm–Shende to the equivariant/stacky setting, and they use those to get a surjective algebra map EV from the degree-0 bulk-deformed wrapped Floer cohomology of a regular cotangent fibre to Etingof’s orbifold Hecke algebra (specialised at ℏ0=0), which is an isomorphism when π2(X)⊗Q=0 (Theorem 6.6).\n\nWhat is new is the equivariant jet package (Section 4), the two-step gluing plus localisation tricks that isolate isotypic leading terms (Section 5), and the boundary-degeneration analysis that produces the Hecke relations geometrically rather than by hand (Section 6). The construction of a contact-type, type-A-adapted G-equivariant J0 for cotangent bundles is explicit (Proposition 5.5 + Lemma 5.2), so the “weakest assumption” flagged by the reader is not an existence gap; it is verified for the main class of examples. The Hecke relations come out of counts of orbidisks with one stacky point; bulk parameters stay free formal variables. Self-citations supply independent background, not circular input.\n\nSoft spots are minor and proportional: the gluing analysis is long and not machine-checked (standard for this literature), orientations are deferred to an appendix, and the result is only degree 0 (higher degrees left open). None of these undermine the central claims. The paper is written for people who already know wrapped Fukaya categories and want a geometric model for Hecke algebras of complex reflection groups / global quotients; it is also useful for anyone working on reduced orbifold GW or singular Lagrangians.\n\nI would send it to referees. The math is careful, the novelty is real, and the citation pattern is clean. Engage with it if you care about either side of the interface.","headline":"Solid new obstruction theorems for stacky ghosts plus a clean Floer realisation of orbifold Hecke algebras; the adapted J0 is constructed, not assumed.","tokens_in":72468,"tokens_out":524,"would_cite":true,"duration_ms":8697,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D37","20C08","57R18"],"pacs":[],"model":"grok-4.5","headline":"Obstructions to smoothing orbicurves turn the degree-zero wrapped Floer algebra of a regular cotangent fiber into the orbifold Hecke algebra.","keywords":["orbifold Floer theory","Hecke algebra","wrapped Fukaya category","orbicurves","cyclic quotient singularities","bulk deformation","cotangent fibre"],"falsifier":"Exhibit a compact complex manifold X with finite automorphism group G such that π_{2}(X)⊗Q=0 yet the degree-zero wrapped Floer algebra of a regular cotangent fibre is strictly larger than the specialised orbifold Hecke algebra (or is not even well-defined for any adapted almost-complex structure).","tokens_in":72506,"feed_emoji":"∞","tokens_out":987,"duration_ms":9266,"temperature":0.7,"pith_summary":"The paper studies when a nodal orbicurve that carries a ghost bubble sitting on a cyclic quotient singularity can be smoothed. By analysing equivariant jets of maps and applying localisation tricks near the fixed loci, it obtains concrete vanishing conditions on the leading terms of the main component. Those vanishing conditions are then used to control the compactification of moduli spaces that compute the endomorphism algebra of a regular cotangent fibre inside a bulk-deformed wrapped Fukaya category of the orbifold cotangent bundle. The only surviving boundary degenerations are disks with a single stacky point; their contributions reproduce exactly the higher-degree Hecke relations. Consequently the degree-zero cohomology of that endomorphism algebra surjects onto Etingof’s orbifold Hecke algebra, and the map is an isomorphism whenever the second rational homotopy of the base vanishes. The result therefore realises a classical algebraic object of representation theory as a symplectic invariant of a singular Lagrangian.","feed_headline":"Orbicurve obstructions turn Floer algebra into Hecke algebra","feed_subtitle":"Degree-zero wrapped Floer of a cotangent fibre equals the orbifold Hecke algebra when π_{2} vanishes","key_machinery":"The obstruction theorems (Theorems 1.1–1.2 / 5.34, 5.37) that force the leading equivariant jet of the main component to vanish whenever a non-trivial holomorphic form of matching character exists on the ghost; these vanishings eliminate all but the Hecke-type boundary degenerations.","core_discovery":"There is a surjective algebra morphism from the degree-zero bulk-deformed wrapped Floer cohomology of a regular cotangent fibre of the orbifold [T*X/G] onto the specialised orbifold Hecke algebra of [X/G]; the morphism is an isomorphism as soon as π_{2}(X)⊗Q=0.","pith_inferences":["The same jet-vanishing technique should extend, after virtual perturbations, to cyclic quotient singularities that are not of type A, opening a route to reduced open Gromov–Witten invariants for more general orbifolds.","A Morse-theoretic model on the space of G-paths would give a chain-level lift of the isomorphism, producing a derived orbifold Hecke algebra that can be compared with string-topology constructions.","The construction suggests that bulk-deformed Fukaya categories of other singular Lagrangians (e.g., fixed loci of higher codimension) may likewise recover algebraic objects attached to the corresponding reflection arrangements."],"forward_implications":["When X is aspherical the entire graded algebra HW*(T*[x][X/G]) is concentrated in degree zero and equals the orbifold Hecke algebra.","The same obstruction package supplies the analytic foundation for a regular bulk-deformed wrapped Fukaya category of any global quotient whose singularities are of compound Am-1 type.","Classical Hecke algebras of complex reflection groups, affine Hecke algebras and double affine Hecke algebras of type A arise as degree-zero Floer algebras of suitable cotangent fibres.","Boundary ghosts with two or more stacky points are rigorously excluded, so the only relations that appear are the expected Hecke polynomials."],"fun_headline_variants":["Orbicurve smoothing blocks equate Hecke algebra to wrapped Floer","Bulk-deformed cotangent Floer yields orbifold Hecke algebra","Nodal orbicurves force Hecke isomorphism with degree-zero Floer","Fukaya endomorphisms of cotangent fibre recover orbifold Hecke","π₂-vanishing makes wrapped Floer of T*[X/G] the Hecke algebra"],"cache_read_input_tokens":65664,"weakest_assumption_plain":"The whole argument needs a single almost-complex structure that is both contact-type at infinity and simultaneously adapted to every type-A singular locus; without it the jet transversality and localisation steps fail.","fun_headline_variants_meta":{"raw":{"variants":["Orbicurve smoothing blocks equate Hecke algebra to wrapped Floer","Bulk-deformed cotangent Floer yields orbifold Hecke algebra","Nodal orbicurves force Hecke isomorphism with degree-zero Floer","Fukaya endomorphisms of cotangent fibre recover orbifold Hecke","π₂-vanishing makes wrapped Floer of T*[X/G] the Hecke algebra"]},"model":"grok-4.5","effort":"low","cost_usd":0.006,"raw_usage":{"total_tokens":1508,"prompt_tokens":666,"num_sources_used":0,"completion_tokens":103,"cost_in_usd_ticks":60000000,"prompt_tokens_details":{"text_tokens":666,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":739,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":666,"tokens_out":103,"duration_ms":7965,"temperature":1.0,"reasoning_tokens":739,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T04:40:44.090700+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a compact complex manifold X with finite automorphism group G such that π_{2}(X)⊗Q=0 yet the degree-zero wrapped Floer algebra of a regular cotangent fibre is strictly larger than the specialised orbifold Hecke algebra (or is not even well-defined for any adapted almost-complex structure).","supporting_citations":[],"review_version":1}