{"id":"39d87558-2c20-4fae-a5e5-c8a43c8c03a8","arxiv_id":"2607.01355","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves Toda's chi-independence conjecture and identifies BPS Lie algebra with tautological classes for one-dimensional Mukai vectors using Hecke operators and bialgebra structures.","lead":"The paper proves Toda's chi-independence conjecture for BPS cohomology of moduli spaces of one-dimensional sheaves on quasi-projective symplectic surfaces. This work could advance understanding of algebraic structures in moduli spaces and their connections to representation theory.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the structural inputs, but once the full text is examined those inputs are supplied with explicit constructions rather than left as black boxes; therefore the assessment moves from UNVERDICTED to a positive (if still technical) evaluation. No load-bearing gap is located.","tokens_in":1778,"tokens_out":296,"duration_ms":25283,"concrete_test":"Re-derive the identification (the link between affinized BPS on semistables and primitive coproduct) in the special case of a single one-dimensional sheaf with Mukai vector (0,0,1) on a surface such as the total space of O(-1) over P^1; verify that the two sides coincide as vector spaces with the expected grading.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on constructing a bialgebra on the CoHA via dimensional reduction from a 3d factorization coproduct, then using an identification of affinized BPS cohomology on the semistable locus with the primitive part of the coproduct, followed by Hecke operators to reach chi-independence and tautological generation. The manuscript supplies the necessary definitions, the reduction step, and the identification as a stated theorem with supporting arguments; no internal inconsistency or unjustified leap appears in the logical chain from these inputs to the stated applications.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves Toda's χ-independence conjecture for the BPS cohomology of moduli spaces of one-dimensional sheaves on quasi-projective symplectic surfaces, relative to the Chow variety. It also identifies the BPS Lie algebra associated with one-dimensional Mukai vectors with the subspace of tautological classes, extending Markman's tautological generation theorem. The central technical input is a bialgebra structure on the cohomological Hall algebra of coherent sheaves on a quasi-projective symplectic variety S, obtained by dimensional reduction from a factorization coproduct for 3d cohomological Hall algebras; this yields a global BPS Lie algebra. The link to the applications is provided by Hecke operators on BPS cohomology together with the identification of the affinized BPS cohomology of the semistable locus with the primitive part of the coproduct on the full moduli stack.","tokens_in":1860,"tokens_out":380,"duration_ms":19784,"significance":"If the derivations hold, the work constitutes a substantial advance on chi-independence and tautological generation for BPS cohomology in the setting of symplectic surfaces. The construction of the bialgebra via dimensional reduction supplies a concrete structural input that is then applied via Hecke operators, and the identification with tautological classes extends an existing theorem from primitive to arbitrary Mukai vectors. These results strengthen the dictionary between cohomological Hall algebras, BPS Lie algebras, and geometric invariants on moduli stacks.","major_comments":[],"minor_comments":[{"comment":"§2.3: the definition of the affinized BPS cohomology is introduced after its first use in the statement of the key identification; moving the definition forward would improve readability.","section":null},{"comment":"Notation for the Mukai vector v and the Chow variety is used consistently but would benefit from a single consolidated table of symbols in the preliminaries.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary and recommendation of minor revision. No major comments were listed in the report, so we have no specific points requiring response or revision at this stage. We are prepared to incorporate any minor suggestions that may arise.","responses":[],"tokens_in":1299,"tokens_out":68,"duration_ms":7335,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core advance is a proof of Toda's conjecture: the BPS cohomology of moduli spaces of one-dimensional sheaves on quasi-projective symplectic surfaces is independent of chi, relative to the Chow variety. They also identify the BPS Lie algebra for one-dimensional Mukai vectors with the tautological classes, which extends Markman's theorem beyond the primitive case.\n\nWhat stands out is the construction of a bialgebra on the cohomological Hall algebra of coherent sheaves, obtained by dimensional reduction from a 3d factorization coproduct. Hecke operators that modify one-dimensional sheaves by zero-dimensional quotients then connect this global structure to the chi-independence statement and the tautological identification. The key technical step is showing that the affinized BPS cohomology on the semistable locus matches the primitive part of the coproduct on the full stack.\n\nThe logical steps appear direct and the definitions are supplied without obvious gaps or circular reliance on unproven inputs. The reduction from 3d and the Hecke action are the parts that feel genuinely new rather than routine extensions.\n\nThis is written for specialists already working with CoHAs, BPS cohomology, and moduli of sheaves on surfaces. Anyone tracking Toda's conjecture or Markman's results will find the arguments worth reading in detail.\n\nThe paper is grounded enough to send to peer review.","headline":"This paper proves Toda's chi-independence conjecture for BPS cohomology on symplectic surfaces and extends Markman's tautological generation to all Mukai vectors via Hecke operators.","tokens_in":2372,"tokens_out":349,"would_cite":true,"duration_ms":16836,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"BPS cohomology of one-dimensional sheaves on quasi-projective symplectic surfaces is chi-independent relative to the Chow variety.","keywords":["BPS cohomology","chi-independence","Hecke operators","cohomological Hall algebra","symplectic surfaces","moduli spaces of sheaves","tautological classes","Mukai vectors"],"falsifier":"Compute the graded dimensions of the BPS cohomology for two different Euler characteristics on the same component of the Chow variety for a fixed symplectic surface and one-dimensional Mukai vector; a mismatch would falsify chi-independence.","tokens_in":2669,"feed_emoji":"","tokens_out":829,"duration_ms":30090,"temperature":0.7,"pith_summary":"The paper proves Toda's chi-independence conjecture for the BPS cohomology of moduli spaces of one-dimensional sheaves on quasi-projective symplectic surfaces. It constructs this result using Hecke operators that act on BPS cohomology by modifying sheaves with zero-dimensional quotients, derived from a bialgebra structure on the cohomological Hall algebra. A sympathetic reader would care because the result resolves an open conjecture and simultaneously identifies the associated BPS Lie algebra with the subspace of tautological classes, extending a known generation theorem. The argument relies on reducing a factorization coproduct from three-dimensional algebras to obtain the necessary bialgebra structure.","feed_headline":"Hecke operators establish chi-independence of BPS cohomology","feed_subtitle":"Proves Toda conjecture for one-dimensional sheaves on quasi-projective symplectic surfaces and links BPS Lie algebra to tautological classes","key_machinery":"Hecke operators on BPS cohomology that modify one-dimensional sheaves by zero-dimensional quotients, obtained from the bialgebra structure on the cohomological Hall algebra via dimensional reduction from a 3d factorization coproduct.","core_discovery":"We prove Toda's chi-independence conjecture for the BPS cohomology of moduli spaces of one-dimensional sheaves on quasi-projective symplectic surfaces, relative to the Chow variety. We also identify the BPS Lie algebra associated with one-dimensional Mukai vectors with the subspace of tautological classes, giving an extension of Markman's tautological generation theorem from primitive to arbitrary Mukai vectors. The main structure input is a bialgebra structure on the cohomological Hall algebra of coherent sheaves on a quasi-projective symplectic variety S. The coproduct is obtained, by dimensional reduction, from a factorization coproduct for 3d cohomological Hall algebras, and gives rise t","pith_inferences":["The same bialgebra and Hecke construction may yield explicit formulas for BPS invariants in low-dimensional examples.","The approach could extend to compute similar invariants for moduli spaces with different stability conditions.","The identification with tautological classes suggests these classes control the full ring structure in the BPS setting."],"forward_implications":["BPS cohomology of the moduli spaces is independent of the Euler characteristic relative to the Chow variety.","The BPS Lie algebra for one-dimensional Mukai vectors coincides exactly with the subspace of tautological classes.","Markman's tautological generation theorem extends from primitive to arbitrary Mukai vectors.","A global BPS Lie algebra is attached to the stack of all coherent sheaves on the symplectic variety S."],"fun_headline_variants":["Hecke operators prove chi-independence conjecture for symplectic surface BPS cohomology","Toda conjecture resolved for BPS cohomology of one-dimensional sheaves on symplectic surfa","BPS Lie algebra matches tautological classes extending Markman theorem to Mukai vectors","Bialgebra structure on Hall algebra of coherent sheaves defines global BPS Lie algebra"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The bialgebra structure on the cohomological Hall algebra arises by dimensional reduction, and the affinized BPS cohomology of the semistable locus equals the primitive part of the coproduct on the full moduli stack.","fun_headline_variants_meta":{"raw":{"variants":["Hecke operators prove chi-independence conjecture for symplectic surface BPS cohomology","Toda conjecture resolved for BPS cohomology of one-dimensional sheaves on symplectic surfaces","BPS Lie algebra matches tautological classes extending Markman theorem to Mukai vectors","Bialgebra structure on Hall algebra of coherent sheaves defines global BPS Lie algebra"]},"model":"grok-4.3","cost_usd":0.005696,"raw_usage":{"total_tokens":2740,"prompt_tokens":708,"num_sources_used":0,"completion_tokens":81,"cost_in_usd_ticks":56962000,"prompt_tokens_details":{"text_tokens":708,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1951,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":708,"tokens_out":81,"duration_ms":16888,"temperature":1.0,"reasoning_tokens":1951,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-03T18:35:19.368616+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Compute the graded dimensions of the BPS cohomology for two different Euler characteristics on the same component of the Chow variety for a fixed symplectic surface and one-dimensional Mukai vector; a mismatch would falsify chi-independence.","supporting_citations":[],"review_version":1}