{"id":"52afe14f-ca56-4ca7-9bd6-f028dafbbff9","arxiv_id":"2502.04253","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A decomposition theorem for cohomology of symmetric stacks yields BPS cohomology, proving cohomological integrality for wide classes of moduli stacks and 3-Calabi-Yau categories.","lead":"This paper proves that the infinite-dimensional cohomology of many moduli stacks can be split into finite-dimensional pieces called BPS cohomology, indexed by the stack's special faces. The decomposition gives a common framework for counting stable objects on Calabi-Yau threefolds and for conjectures in mirror symmetry and Langlands duality.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Almost orthogonality is unverified for general 3-manifold character stacks, and Lemma 4.2.4(ii) gives no virtual criterion for (−1)-shifted symplectic stacks, so Theorem 1.2.12's headline application is not covered.","rationale":"The reader's weakest assumption is exactly the almost orthogonal condition for general 3-manifold character stacks. I agree: this is the single most load-bearing concern because the theorem's proof uses orthogonality (not mere symmetry) in the smallness estimate (Proposition 7.2.1), the cotangent sign representation, and the reduction to the classifying stack via a locally defined quadratic function. The paper honestly flags the open case in §4.3.11, but the abstract and introduction advertise character stacks of 3-manifolds as a central application. Since the theorem is conditional and the condition is not checked for the general motivating class, the scope of the central claim is genuinely limited. I do not see a stronger internal inconsistency: the conditional statement appears coherent, the sign conventions in the (−1)-shifted case are consistent with the omission of sgn_α from Theorem 1.2.12 (by the supercommutativity claim (8.2.8.1)), and the reliance on companion papers is normal for a preprint of this depth. Therefore the reader's CONDITIONAL verdict stands unchanged.","tokens_in":67645,"tokens_out":12770,"duration_ms":121442,"concrete_test":"Choose a compact oriented 3-manifold not covered by Corollaries 4.3.17/4.3.19, such as the mapping torus of a pseudo-Anosov automorphism of a surface or a hyperbolic 3-manifold. For a closed point [K] with positive-dimensional connected stabilizer, compute the representation H^0(T_{Loc_G(M),[K]}) of G^◦_K and determine whether it admits a G^◦_K-invariant non-degenerate symmetric bilinear form. A counterexample would show Theorem 1.2.12 does not apply to that stack; a positive check on a nontrivial example would reduce the gap. Separately, attempting to extend Lemma 4.2.4(ii) to (−1)-shifted symplectic stacks would settle whether virtual orthogonality can ever be promoted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 1.2.12) requires the stack to be almost orthogonal (Definition 4.2.2, Assumption 1.2.5(iii)). For the motivating class of G-character stacks of compact oriented 3-manifolds, the paper proves only almost symmetric (Corollary 4.3.10(ii)), and Lemma 4.2.4(ii) explicitly does not extend the virtual orthogonality criterion to (−1)-shifted symplectic stacks. Since χ(M)=0 for closed 3-manifolds, Lemma 4.3.9 gives [T_{Loc_G(M),[K]}]=0, i.e., virtual orthogonality is automatic, but this does not imply the existence of a G^◦_K-invariant non-degenerate symmetric form on the actual tangent space H^0. The special cases in Corollaries 4.3.17 and 4.3.19 cover only mapping tori of finite-order automorphisms and GL_n/SL_n; the paper states in §4.3.11 that the general case is unknown. Consequently, the decomposition (1.2.12.1) is not established for the general character stacks that motivate the Langlands duality conjecture of §1.2.19, which must be stated conditionally. This is a scope limitation, not an internal inconsistency, but it is load-bearing for the advertised application.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"Bu–Davison–Ibáñez Núñez–Kinjo–Pădurariu prove cohomological integrality and decomposition theorems for three classes of stacks—smooth stacks, 0-shifted symplectic stacks, and oriented (−1)-shifted symplectic stacks—under explicit hypotheses: existence of a good moduli space, affine diagonal, quasi-compact graded points, a global equivariant parameter for special faces, and an (almost) orthogonal tangent-space structure. For a smooth stack U, Theorem 1.2.7 decomposes p_*IC_U into Aut(α)-invariants of terms g_{α,*}IC_{Uα} ⊗ H^*(BG_m^{dim F})^{vir} ⊗ sgn_α; for an oriented (−1)-shifted symplectic stack X, Theorem 1.2.12 gives the analogous decomposition of p_*φ_X into BPS-sheaf summands. The BPS sheaf is defined as the lowest perverse cohomology of the same pushforward, so the theorem is a structural decomposition rather than a definitional tautology. Applications include moduli of G-bundles, twisted Higgs bundles, character stacks of surfaces and certain 3-manifolds, coherent sheaves on Calabi–Yau 3-folds and K3 surfaces, a PBW-type theorem for cohomological Hall algebras of 3-Calabi–Yau categories with commutative orientation data, and conjectural Langlands duality and topological mirror symmetry formulated through BPS cohomology.","tokens_in":67852,"tokens_out":13220,"duration_ms":144500,"significance":"This is a substantial contribution that unifies and generalizes the Meinhardt–Reineke and Davison–Meinhardt theorems and provides a common framework for cohomological integrality across a wide range of moduli stacks. The component-lattice formalism, the notion of special faces, and the cohomological Hall induction are used coherently, and the main theorem is genuinely parameter-free. The paper is also commendably explicit about its hypotheses: almost orthogonality is isolated as an assumption, verified in many examples, and the authors state plainly in §4.3.11 and §1.2.19 that the general character stack of a compact oriented 3-manifold is not known to be almost orthogonal, so the Langlands-duality consequences are conjectural. The proof strategy—reduction to local models via component lattices, purity via Theorem 5.1.3, and the integral isomorphism from [67]—is coherent, and the claimed decompositions are concrete and testable in examples.","major_comments":[],"minor_comments":[{"comment":"In the global-section display following Theorem 1.2.21, the direct sum is indexed by Facesp(X), but the stack being decomposed is Y; the right-hand side H^BM_{-*}(Yα) should presumably be H^BM_{-*}(Y). Please correct the index set and the right-hand side.","section":"§1.2.21"},{"comment":"Equation (1.2.19.2) and the surrounding Langlands-duality discussion are conditional on almost orthogonality of Loc_G(M), which is verified only in the cases of Corollaries 4.3.17 and 4.3.19. The authors do state this, but I recommend moving the caveat into the abstract so that 'a version of Langlands duality' is not misread as a theorem for all compact oriented 3-manifolds.","section":"§1.2.19"},{"comment":"The opening sentence says 'let V be an almost symmetric representation of V'; this should read 'of G'.","section":"§8.2.2"},{"comment":"In the displayed isomorphism (1.2.7.1), an opening parenthesis is missing before 'g_{α,*}IC^◦_{Uα}', making the scope of the Aut(α)-invariants ambiguous.","section":"Theorem 1.2.7"},{"comment":"Given the heavy use of X_α, X_σ^+, g_α, p_α, and the cotangent arrangement throughout §§7–9, a short table or index of the principal maps and their domains would substantially improve readability.","section":"General"}],"recommendation":"minor_revision","confidential_remarks":"This is a long, dense, and high-impact paper. The stress-test concern about general 3-manifold character stacks does not amount to an internal error, because the authors explicitly acknowledge in §4.3.11 and §1.2.19 that almost orthogonality is unverified in general and that the Langlands-duality statements are conjectural. The requested changes are local: fix the displayed formula in §1.2.21 and make the conditional scope of the 3-manifold application visible in the abstract. I do not see a need for further technical revisions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [colleague],\n\nThis is the paper that finally gets cohomological integrality off the quiver-and-Ginzburg train and into general stacks with good moduli spaces. The main theorems (1.2.7, 1.2.12, 1.2.21) decompose cohomology, vanishing cycle cohomology, and Borel-Moore homology under a pointwise almost orthogonal tangent condition. The novelty is real: component lattices and cohomological Hall induction replace quiver-specific arguments, covering moduli of G-bundles, twisted G-Higgs bundles, character stacks of surfaces, and 3-Calabi-Yau categories with commutative orientation data. The smooth case generalizes Meinhardt-Reineke by identifying BPS sheaves with intersection complexes.\n\nWhat is genuinely good: the BPS sheaf is defined as the lowest perverse cohomology of the pushforward, and the paper proves a support lemma and wall-crossing formula. The proof strategy is coherent: reduce to local models via component lattices, prove smallness (Prop 7.2.1), then use CoHI to construct the map and show it is an isomorphism. The paper is honest about what it needs: Theorem 1.2.12 assumes almost orthogonality (Assumption 1.2.5(iii)), and §4.3.11 explicitly says the general 3-manifold case is unknown. Corollaries 4.3.17 and 4.3.19 cover only mapping tori of finite-order automorphisms and GL_n/SL_n. So the Langlands duality conjecture of §1.2.19 is stated conditionally, and rightly so.\n\nThe soft spot is exactly this: the motivating class of 3-manifold character stacks is not covered in full generality, and Lemma 4.2.4(ii) gives no virtual orthogonality criterion for (-1)-shifted symplectic stacks. Since χ(M)=0 for closed 3-manifolds, Lemma 4.3.9 makes the virtual tangent class vanish, but that does not imply existence of a G^◦_K-invariant non-degenerate symmetric form on H^0. The paper knows this; it is a scope gap, not a hole in the proof. There is also a restriction hidden in the 3-Calabi-Yau application: Theorem 1.2.16 needs commutative orientation data (assumption (vi) in §1.2.15), which is not automatic. A second caveat is the reliance on companion papers for component lattices [17] and the integral isomorphism [67]; some statements are only sketched (e.g., Remark 8.1.9). That is typical for a preprint of this scale, but it slows verification.\n\nNo circularity red flag: the BPS sheaf is defined as the lowest piece of the same pushforward that the theorem decomposes, but the proof establishes the isomorphism independently via CoHI and local models. This is standard in the area. No p-hacking or invented entities; the citation pattern is proper.\n\nWho this is for: anyone working on DT theory, cohomological Hall algebras, or the topology of moduli stacks. It deserves a serious referee; conditional acceptance with requests for clarification on the 3-manifold examples would be appropriate. Recommend engaging with it.","headline":"Generalizes cohomological integrality to symmetric stacks; core is solid, but the 3-manifold character-stack application remains conditional on a missing orthogonality check.","tokens_in":68481,"tokens_out":4674,"would_cite":true,"duration_ms":43046,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14D23","14F08","14J32","14C30","55N33"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a broad class of moduli stacks satisfying a pointwise orthogonality condition, the stack's cohomology decomposes canonically into finite-dimensional BPS summands indexed by special faces of a component lattice.","keywords":["cohomological integrality","BPS sheaves","shifted symplectic stacks","monodromic mixed Hodge modules","good moduli spaces","cohomological Hall algebra","3-Calabi-Yau categories","intersection cohomology"],"falsifier":"Compute both sides of the point-level formula (1.2.12.2) for a concrete stack satisfying all four assumptions, for instance the moduli stack of semistable sheaves on a smooth projective Calabi-Yau threefold with generic polarization, using known Donaldson-Thomas invariants; the theorem predicts the multiple-cover identity relating each BPS invariant to the generalized DT invariant, so a single violation of that identity would falsify it. Conversely, no counterexample can be drawn from stacks that fail almost orthogonality, since the theorem is conditional on it.","tokens_in":67400,"feed_emoji":"🧩","tokens_out":6551,"duration_ms":69278,"temperature":0.7,"pith_summary":"The paper claims a single mechanism governs the cohomology of many moduli stacks: smooth stacks, 0-shifted symplectic stacks, and (-1)-shifted symplectic stacks, provided the stack has a good moduli space and its tangent spaces are pointwise almost orthogonal. Under these hypotheses the infinite-dimensional cohomology (or vanishing-cycle cohomology) of the whole stack splits as a direct sum of finite-dimensional BPS cohomologies attached to smaller 'special face' stacks, with Weyl-group-like invariants. A sympathetic reader should care because the decomposition is explicit enough to serve as a universal cohomological integrality theorem: for 3-Calabi-Yau categories it yields a PBW-type theorem for cohomological Hall algebras and recovers the usual BPS invariants, and for smooth stacks it identifies BPS cohomology with intersection cohomology of the good moduli space.","feed_headline":"Symmetry splits stack cohomology into BPS summands","feed_subtitle":"One orthogonality condition yields a universal decomposition; integrality and PBW for 3-Calabi-Yau categories follow.","key_machinery":"The central object is the cohomological Hall induction (CoHI): for each face $(F,\\alpha)$ of the component lattice, the stack of filtered points $U^+_\\sigma$ gives a correspondence $U_\\alpha \\leftarrow U^+_\\sigma \\to U$, and pushing the intersection complex (or DT sheaf) along this correspondence defines a multiplication map from the $\\alpha$-summand to $p_* \\mathrm{IC}_U$. The almost-orthogonal hypothesis makes the relevant map small (Proposition 7.2.1), so the perverse degeneration of CoHI is supercommutative, the cotangent sign representation $\\mathrm{sgn}_\\alpha$ can be extracted, and the map becomes an isomorphism.","core_discovery":"The central claim is Theorem 1.2.12: for an almost orthogonal oriented $(-1)$-shifted symplectic stack $X$ satisfying assumptions (i)-(iv), there is an isomorphism of monodromic mixed Hodge complexes on the good moduli space $\\underline{X}$, $$\\bigoplus_{(F,\\$\\alpha$)\\in \\mathrm{Faces}^{\\mathrm{sp}}(X)} \\left(g_{\\$\\alpha$,*} \\mathrm{BPS}_{X_\\$\\alpha$} \\otimes H^*(B\\mathbb{G}$_m^{{\\dim F}}$)^{\\mathrm{vir}}\\right)^{\\mathrm{Aut}(\\$\\alpha$)} \\simeq p_* \\phi_X,$$ where $\\phi_X$ is the Donaldson-Thomas vanishing-cycle sheaf, $\\mathrm{BPS}_{X_\\alpha}$ is the zeroth perverse cohomology of the localized DT sheaf (the BPS sheaf), and the sum runs over special faces of the component lattice. The smooth-stack analogue replaces BPS sheaves by intersection complexes, expressing $p_* \\mathrm{IC}_U$ as the same kind of direct sum. The paper argues that this decomposition is induced by the cohomological Hall induction, and that the almost-orthogonal condition makes the relevant pushforwards small, so the perverse-degenerate CoHI is symmetric and an isomorphism.","pith_inferences":["Editorial inference: The almost-orthogonal hypothesis is probably stronger than needed for the decomposition itself; the paper's comparison with an algebraic approach suggests a version of cohomological integrality can hold without it, but the identification of BPS sheaves with intersection complexes in the smooth case would then be lost.","Editorial inference: If the theorem extends to all compact oriented 3-manifold character stacks, Langlands duality for the full vanishing-cycle cohomology reduces to a finite-dimensional statement about BPS cohomology, which is a more tractable check.","Editorial inference: A testable extension is to compute the BPS sheaves explicitly for 0-shifted symplectic stacks beyond 2-Calabi-Yau categories, for example for moduli of G-Higgs bundles, and compare with symplectic-duality predictions."],"forward_implications":["For smooth stacks, the theorem gives a closed formula for the cohomology of the stack in terms of intersection cohomology of special-face stacks, generalizing the classical quiver formula to arbitrary reductive groups and non-coprime degrees.","For $(-1)$-shifted symplectic stacks such as moduli stacks of semistable sheaves on Calabi-Yau threefolds, the vanishing-cycle cohomology decomposes into finite-dimensional BPS summands; this is cohomological integrality.","For 3-Calabi-Yau categories with commutative orientation data, the decomposition is a PBW-type theorem for the BPS Lie algebra and cohomological Hall algebra.","For 0-shifted symplectic stacks, Borel-Moore homology decomposes into pure BPS sheaves, yielding purity statements and applications to K3 surfaces.","The finite-dimensional BPS cohomology is proposed as the correct replacement for ordinary cohomology in topological mirror symmetry and Langlands duality for character stacks of 3-manifolds."],"supporting_citations":[{"why":"supplies the component-lattice formalism and constancy/finiteness theorems used to reduce global decompositions to local models","marker":"[17]"},{"why":"establishes the decomposition theorem and weight preservation for good moduli space morphisms, used throughout the proof","marker":"[65]"},{"why":"provides the integral isomorphism and cohomological Hall algebra structure for $(-1)$-shifted symplectic stacks on which CoHI is built","marker":"[67]"},{"why":"supplies the quiver-model cohomological integrality theorem that the smooth-stack result generalizes","marker":"[86]"},{"why":"introduces shifted symplectic structures and the main examples (Higgs bundles, character stacks) the paper treats","marker":"[92]"},{"why":"gives the corresponding integrality theorem for quivers with potentials via vanishing cycles, the model for the $(-1)$-shifted case","marker":"[29]"},{"why":"provides the local structure theorem for stacks with good moduli spaces used in the smallness and etale-local arguments","marker":"[3]"},{"why":"constructs the Donaldson-Thomas monodromic mixed Hodge module $\\phi_X$ whose decomposition is the main object","marker":"[11]"}],"fun_headline_variants":["BPS summands from almost-orthogonal stacks","Cohomology of stacks splits via BPS sheaves","PBW and integrality for 3-Calabi-Yau categories","Topological mirror symmetry via BPS cohomology","One condition splits stack cohomology into BPS parts"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper's central claim collapses if a stack satisfying all other hypotheses but failing almost orthogonality is found: the smallness estimate (Proposition 7.2.1) and the sign identities that make CoHI well-defined both rely on the tangent space being orthogonal for the neutral component of every closed-point stabilizer.","fun_headline_variants_meta":{"raw":{"variants":["BPS summands from almost-orthogonal stacks","Cohomology of stacks splits via BPS sheaves","PBW and integrality for 3-Calabi-Yau categories","Topological mirror symmetry via BPS cohomology","One condition splits stack cohomology into BPS parts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00086,"raw_usage":{"total_tokens":3827,"prompt_tokens":1136,"completion_tokens":2691,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":752,"completion_tokens_details":{"reasoning_tokens":2608}},"tokens_in":752,"tokens_out":2691,"duration_ms":23378,"temperature":1.0,"reasoning_tokens":2608,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T22:59:02.848824+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute both sides of the point-level formula (1.2.12.2) for a concrete stack satisfying all four assumptions, for instance the moduli stack of semistable sheaves on a smooth projective Calabi-Yau threefold with generic polarization, using known Donaldson-Thomas invariants; the theorem predicts the multiple-cover identity relating each BPS invariant to the generalized DT invariant, so a single violation of that identity would falsify it. Conversely, no counterexample can be drawn from stacks that fail almost orthogonality, since the theorem is conditional on it.","supporting_citations":[{"cited_title":"Meinhardt and M","cited_arxiv_id":null,"evidence_quote":"supplies the quiver-model cohomological integrality theorem that the smooth-stack result generalizes"},{"cited_title":"Pantev, B","cited_arxiv_id":null,"evidence_quote":"introduces shifted symplectic structures and the main examples (Higgs bundles, character stacks) the paper treats"}],"review_version":1}