{"id":"c98486ac-807f-4664-aeb1-a51316148ec9","arxiv_id":"2501.06643","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Classical type parabolic induction produces twisted Yetter-Drinfeld vertex modules for cohomological Hall algebras, including cases of dimension-zero sheaves on surfaces.","lead":"This paper derives module and comodule structures for cohomological Hall algebras from classical type parabolic induction, where the fixed points of an involution give orthosymplectic objects. It organizes these into a twisted Yetter-Drinfeld vertex module and gives examples for quiver potentials, preprojective algebras, and dimension-zero sheaves on surfaces.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.4.3 rests on an unpublished Artin-stack torus-localization statement and an explicit localized-class identity that is deferred; the shuffle proof covers only a special case.","rationale":"The Reader's weakest_assumption identifies exactly the place where the central argument is least secure: the proof of Theorem 5.4.3 invokes Atiyah-Bott localization for Artin stacks from an unpublished source and leaves the decisive equality of localized classes as a computation. My reading of the proof confirms this. Section 5.4 contains the sentence 'we used Atiyah-Bott torus localisation, see [La3] for details' and then 'It remains to show:, which reduces to proving an equality of localised cohomology classes on M^4 x M^tau'; the subsequent paragraph says the proof finishes by a computation either by hand or via the shuffle case. These are explicit admissions that the theorem is not fully proven within the preprint. The shuffle verification in Section 7.9 is substantial evidence for the quiver family, and the explicit formulas in Section 5.7.5 give a concrete test bed, but neither replaces the missing general geometric computation. Because this concern is precisely the reason for a conditional rather than an accept/reject verdict, I recommend keeping the Reader's verdict unchanged. I do not see an internal contradiction or a more fundamental flaw: the axioms StkCoHAM and StkVA are coherent, the correspondences are constructed with the right fixed-point structure, and the deferred identity is plausible given the Cherednik hexagon relations in Appendix A. The honest assessment is therefore that the paper's central claim is promising but not yet fully demonstrated as written.","tokens_in":72344,"tokens_out":4684,"duration_ms":48020,"concrete_test":"Check the deferred localized-class identity in the base case M = BGL, M^tau = BSp with W = 0, where all correspondences and Euler classes are explicit. Using the formulas of Section 2.3 (normal complexes) and Section 5.7.5 (twists S(z), T(z,w), S_OSp(w1,w2)), compute both sides of the compatibility diagram (48) for low degrees, e.g. n = m = 1, as rational functions in the Chern roots x_i, y_j. If the two sides fail to agree, the deferred computation underlying Theorem 5.4.3 is wrong in the simplest nontrivial case; if they agree, the missing identity is confirmed independently of the unpublished localization reference, leaving [La3] as the sole unresolved foundational condition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem B (Theorem 5.4.3): the critical cohomology of the fixed locus is a twisted Yetter-Drinfeld vertex module, i.e. the compatibility equation (43) between the CoHA action and the Joyce-Liu vertex coaction. The proof in Section 5.4 reduces the compatibility to an equality of localized cohomology classes on M^4 x M^tau, and the text itself flags the two missing ingredients: Atiyah-Bott torus localization for Artin stacks is cited to the unpublished manuscript [La3, Virtual Euler classes for Artin stacks, to appear], and the computation is left as 'The proof now finishes by a computation, which we either do by hand ... as in the shuffle case of section 7.9.' Section 7.9 explicitly carries out the shuffle-algebra verification for quiver examples, but this does not establish the general geometric statement for arbitrary stacks satisfying StkCoHAM and StkVA. Thus Theorem B, and with it Corollaries G and J, is conditional on a foundational reference and a deferred computation. This is not a conflict with consensus; it is an internally flagged gap in the proof as written. If the localization statement or the localized Euler-class identity fails, the twisted Yetter-Drinfeld compatibility is not established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a general framework, for stacks with an involution satisfying axioms StkCoHAM and StkVA, in which the critical cohomology of the fixed locus H^*(M^\\tau,\\phi^\\tau) is simultaneously a module and a vertex comodule for the cohomological Hall algebra of H^*(M,\\phi), and satisfies a twisted Yetter-Drinfeld compatibility condition. The main structural results are Theorem A (module and vertex comodule structures), Theorem B (Theorem 5.4.3, the twisted Yetter-Drinfeld compatibility), Theorem C (factorisation coalgebra structure over configuration spaces), and Theorem D (localised-to-vertex functor). These are applied to quivers with potential, preprojective algebras, and dimension-zero sheaves on a smooth proper surface, with explicit shuffle-algebra formulas and a conjectural relation to AGT in classical types and twisted Yangians.","tokens_in":72587,"tokens_out":5425,"duration_ms":56834,"significance":"If Theorem B is established, the paper would give a uniform and strikingly general mechanism for producing compatible CoHA actions and Joyce-Liu vertex coactions on orthosymplectic fixed loci, covering several important examples at once. The axiomatic formulation and the identification of fixed points with orthosymplectic objects are valuable, and the shuffle formulas in Section 7 provide concrete and checkable statements. The paper contains no fitted parameters, and the target structures are not used as input, so it is not circular in the usual sense. Its main weakness is that several load-bearing steps are delegated to unpublished or to-appear work and to a computation that is only carried out in a special case.","major_comments":[{"comment":"The central compatibility claim is not fully proven in the manuscript. The proof reduces the twisted Yetter-Drinfeld condition to an equality of localised cohomology classes on M^4 x M^tau, and the two key steps are deferred: Atiyah-Bott torus localisation for Artin stacks is cited to the unpublished manuscript [La3], and the final computation is left to be done \"by hand ... as in the shuffle case of section 7.9\". Section 7.9 verifies the compatibility only in the shuffle-algebra setting for quiver examples, and it explicitly assumes that there are no type II orbits and defers the type D case. Thus the statement for arbitrary stacks satisfying StkCoHAM and StkVA is not established as written. Theorem B, and with it the advertised twisted Yetter-Drinfeld statements for the surface and framed preprojective examples, is conditional on [La3] and on a computation that is not included in the main text.","section":"Section 5.4, Theorem 5.4.3, Eq. (43)"},{"comment":"The localised coproduct and orthosymplectic comodule maps use the Euler classes epN_sq and epN^tau_{s,3}, and the proof asserts that the corresponding S and T satisfy the Cherednik hexagon relations. This assertion depends on Corollaries 2.3.7 and 2.3.8 and on Proposition 2.3.10, but the splitting N_s^tau_3 = N_s|_{M^tau x M} oplus K_s in Proposition 2.3.10 is justified only by a short argument. Since the Euler-class factors and the braided cocommutativity of the vertex comodule structure rely on this splitting, this step should be expanded or the precise statement imported from [La2] should be stated in the text.","section":"Section 5.3, Theorem 5.3.2 and Corollary 5.3.4"},{"comment":"The proof that the extension correspondence gives a module structure on the Borel-Moore homology of semistable orthosymplectic perverse coherent sheaves rests on a one-sentence properness argument: because the stability condition lies on the wall where dimension-zero sheaves have the same phase, isotropic extensions by dimension-zero sheaves cannot destabilise. This is plausible but needs a more detailed verification, especially since the semistable open is defined using isotropic subobjects and the wall-crossing behaviour is delicate. The surface example and Corollary G depend on this claim.","section":"Section 6.3, Theorem 6.3.2"}],"minor_comments":[{"comment":"The displayed equation labelled (48) is missing or empty in the text; only the surrounding diagram reference appears. Please restore the equation.","section":"Section 0.1.1, Theorem B statement"},{"comment":"The line \"It remains to show:,\" appears truncated, and the final localised class equality marked by \":\" is not written out explicitly. Please complete this sentence and display the equality.","section":"Section 5.4, proof of Theorem 5.4.3"},{"comment":"The symbol kappa is used both for the orthosymplectic form in Section 1.1 and for the braiding kappa = (tau x id) in Section 0.1.1. Please disambiguate the notation.","section":"Section 1.1, Definition of orthosymplectic form"},{"comment":"The notation rA_1...A_k|B_1,...,B_k,B_ells is introduced, but the asymmetry between the number of A variables and B variables is not explained before it is used in the computation. Adding a short explanation would make the shuffle proof much easier to follow.","section":"Section 7.9.1, shuffle notation"},{"comment":"Several foundational statements are imported from the to-appear references [JKL], [La1], [La2], and [La3]. Please add a sentence or a table indicating which specific results from each reference are used, so that the reader can assess the dependency.","section":"References"},{"comment":"There are numerous typos and OCR-like artifacts, for example \"algberas\" in the abstract, \"compactification\", \"othosymplectic\", \"coputation\", and \"associavity\" in the proof of Proposition 4.2.3. A careful proofreading pass is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper's central theorem is conditional on unpublished work by the authors, especially [La3] on virtual Euler classes for Artin stacks, and on a computation that is only performed in a special shuffle-algebra case. This is not a circularity problem, but it is a verification-dependency problem. I would advise the editor to require that [La3] be made available, or that the relevant localisation statement and the missing localized-class computation be included in full, before the paper is accepted. The breadth of the claimed applications makes this worth pursuing, but the current manuscript does not yet close the proof of its headline theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper is worth a serious referee: it constructs a substantial new framework for orthosymplectic modules of cohomological Hall algebras, with explicit shuffle formulas that are the best part of the work. But the central Theorem B, the twisted Yetter-Drinfeld compatibility, is not proven in the text. Section 5.4 reduces it to an equality of localized classes on M^4 x M^tau, then defers the key computation to Section 7.9, which only handles the shuffle-algebra case, and cites an unpublished manuscript [La3] for the Atiyah-Bott localization on Artin stacks. The text itself flags these gaps; they are not hidden. That does not kill the paper, but it makes the main theorem conditional.\n\nWhat is genuinely new: the idea of taking fixed points of an involution on the moduli stack to produce modules over the CoHA is clean and uniform. It covers quivers with potential, preprojective algebras, and dimension zero sheaves on surfaces, and the orthosymplectic parabolic configuration space is a nice addition. The 'compactification' MOSp of classical type bundles on a surface is a concrete construction that should be of independent interest. The shuffle formulas in Section 7 extend [KS, YZ] to orthosymplectic modules, and they are explicit, checkable, and likely to be useful regardless of Theorem B.\n\nThe soft spots are real but proportionate. The proof of Theorem 5.4.3 leans on [La3] for virtual Euler classes on Artin stacks, and on several to-appear self-citations ([JKL], [La1], [La2]) for foundational factorisation-algebra results. The current preprint is not self-contained. If [La3] and the deferred computation appear and check out, the central claim is plausible; the shuffle examples give strong supporting evidence. The reader's stress test correctly identifies this, and I agree with the conditional verdict.\n\nWho this is for: geometric representation theorists working on CoHAs, vertex algebras, and quiver Yangians. The framework and examples will be valuable. A serious referee should engage with it, but should ask for a complete proof of Theorem B or an explicit statement that it is conditional, and should require that [La3] be available. I would send it to peer review rather than desk reject.","headline":"Substantial new framework and explicit shuffle formulas, but the central twisted Yetter-Drinfeld theorem is explicitly deferred to an unpublished localization paper and a special-case computation.","tokens_in":73130,"tokens_out":3390,"would_cite":true,"duration_ms":33274,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14D23","17B69","14D20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes that for orthosymplectic moduli stacks, parabolic induction gives a critical-cohomology module for the cohomological Hall algebra whose action and vertex coaction are compatible up to a braiding.","keywords":["cohomological Hall algebra","orthosymplectic moduli stack","vertex coalgebra","Yetter-Drinfeld module","quiver with potential","preprojective algebra","dimension zero sheaves","AGT correspondence"],"falsifier":"In the Jordan quiver with zero potential and framing weights $u_1,\\dots,u_r$, compute both sides of identity (48) at dimension vectors $d=d_1=1$ using the shuffle action (85) and the left coaction (70), and compare the rational functions in the variables $x,u,t_1,t_2$; the difference must be exactly the braiding ratio $\\beta_{34}\\beta_{23}\\kappa_4\\beta_{34}$. A second check, independent of the shuffle computation, is whether the deferred equality of localised cohomology classes in Section 5.4 holds when $W=0$ and $M$ is the non-smooth surface stack.","tokens_in":72099,"feed_emoji":"🧶","tokens_out":10841,"duration_ms":191215,"temperature":0.7,"pith_summary":"This paper claims that classical-type parabolic induction, read as the fixed-point data of an involution on a three-step short exact sequence correspondence, turns the critical cohomology of the fixed locus into both a module and a comodule over the cohomological Hall algebra, and that the two structures are compatible: the module is a twisted Yetter-Drinfeld vertex module. The compatibility is encoded by a braided identity in which the action and the coaction commute up to the orthosymplectic braidings $\\beta$ and $\\kappa$, and it is shown to hold in the paper's main examples: quivers with potential, preprojective algebras, and dimension-zero sheaves on smooth proper surfaces. If the main theorem is right, the CoHA action and the vertex coaction that were previously studied separately on such stacks form one coherent quantum-group-like structure, with the type B/C reflection equations governing their commutation. This is of direct interest to the AGT correspondence because the surface case produces an action of the zero-dimensional CoHA, described as the positive modes of a deformed $W$-algebra, on Borel-Moore homology of orthosymplectic instanton moduli.","feed_headline":"Classical-type fixed loci become twisted Yetter-Drinfeld vertex modules","feed_subtitle":"One braided identity governs CoHA actions and vertex coactions for quivers, preprojective algebras, and surfaces.","key_machinery":"The machinery is the orthosymplectic short exact sequence stack $SES^\\tau_3$, the fixed locus of the involution acting on three-step short exact sequences, together with two localised Euler classes $S=\\mathrm{ep}(N_s)$ and $T=\\mathrm{ep}(N^\\tau_{s,3})$ attached to normal complexes. These Euler classes satisfy the Cherednik reflection hexagon relations, and multiplying the naive direct-sum coproduct and parabolic-induction action by them, in a Borcherds twist, produces the coaction and the twisted Yetter-Drinfeld compatibility. Around this core, the paper builds a localised-to-vertex functor $\\Phi$ from factorisation coalgebras on moduli configuration spaces to vertex coalgebras, and an equivalence between $G$-vertex algebras and factorisation algebras over the $G$-Ran space, so the compatibility can be checked on configuration spaces and then imported into vertex-algebra language.","core_discovery":"The central claim is Theorem B (Theorem 5.4.3): for any stack $M$ with involution $\\tau$ satisfying the axioms StkCoHAM and StkVA, the critical cohomology $H(M^\\tau,\\varphi^\\tau)$ is a twisted Yetter-Drinfeld vertex module over the vertex bialgebra $H(M,\\varphi)$, where $\\varphi$ is the vanishing-cycle sheaf of an invariant function. Concretely, there is a CoHA action $m_{\\mathrm{GL-OSp}}: H(M,\\varphi)\\otimes H(M^\\tau,\\varphi^\\tau)\\to H(M^\\tau,\\varphi^\\tau)$ and a compatible localised and vertex coaction $\\Delta_{\\mathrm{GL-OSp}}$, and they satisfy $\\Delta_{\\mathrm{GL-OSp}}\\circ m_{\\mathrm{GL-OSp}}=(m_3\\otimes m)\\circ \\beta_{34}\\beta_{23}\\kappa_4\\beta_{34}\\circ(\\Delta_3\\otimes\\Delta_{\\mathrm{GL-OSp}})$. In words: applying the action and then the coaction is the same as applying the coaction first, then the action, up to the linear and orthosymplectic braidings. The paper proves this at the level of configuration-space factorization coalgebras, transfers it to vertex algebras by the localised-to-vertex functor, and gives a direct shuffle-algebra verification in the quiver case (Theorem 7.9.2). Examples include quivers with potential (Corollary H), preprojective algebras (Corollary J), and dimension-zero sheaves on surfaces (Theorems F and G).","pith_inferences":["If Theorem B is correct, the same braided compatibility should be read as the normal-ordering relation between positive and negative modes of the Drinfeld double of the orthosymplectic CoHA quotient; the paper points toward this route for constructing dual-folded affine quantum groups, but the construction itself is left to future work.","Because the proof reduces everything to a rational-function identity in Chern roots, the shuffle formulas in Section 7 give a concrete and cheap testing ground: verifications at small dimension vectors can serve as a standalone check of the geometric theorem.","For the $S=\\mathbb{A}^2$ framed case, the conjectural Kirwan map to intersection homology of the Uhlenbeck compactification would convert the twisted Yetter-Drinfeld identity into the positive/negative mode commutation relations expected from AGT in classical type; the paper states this as a conjecture, not a theorem."],"forward_implications":["For any quiver with an orientation-reversing involution and an invariant potential, the critical cohomology of the self-dual representations is a $\\tau$-twisted vertex Yetter-Drinfeld module for the quiver CoHA with that potential (Corollary H).","For the preprojective algebra of a quiver, the CoHA acts on Borel-Moore homology of framed orthosymplectic representations, and the action is intertwined with an orthosymplectic shuffle module; the compatible coaction makes the module twisted Yetter-Drinfeld (Corollary J).","For a smooth proper surface, the zero-dimensional CoHA, identified with the positive modes $W^+(S)$ of a deformed $W$-algebra, acts on the Borel-Moore homology of semistable orthosymplectic perverse coherent sheaves, making it a $\\tau$-twisted localised Yetter-Drinfeld module (Theorems F and G).","The semistable orthosymplectic perverse coherent sheaf stack $MOSp$ gives a compactification of the stack of classical-type orthogonal or symplectic bundles on a surface, with closed points matching the Uhlenbeck compactification (Section 6).","The localised-to-vertex functor $\\Phi$ and the $G$-Ran equivalence mean all of the above vertex statements can be re-derived from rational-function data on configuration spaces, so the twisted Yetter-Drinfeld identity is a single localised cohomology-class equality (Theorems 3.4.3 and 4.6.3)."],"supporting_citations":[{"why":"Supplies the Atiyah-Bott torus localisation and virtual Euler classes for Artin stacks used in the crucial equality of localised classes in Theorem 5.4.3; the proof explicitly defers this step to this unpublished manuscript.","marker":"[La3]"},{"why":"Defines the cohomological Hall product on critical cohomology, which is the action side of the module structure under study.","marker":"[KS]"},{"why":"Introduced vertex coalgebra structures on cohomology of moduli stacks, the ordinary vertex coproduct that is twisted in the orthosymplectic setting.","marker":"[Jo]"},{"why":"Provides the vertex coproduct with potential and its compatibility with the CoHA, the structure being extended here to a comodule and twisted Yetter-Drinfeld module.","marker":"[JKL]"},{"why":"Introduced localised coproducts on critical CoHAs, the framework on which the localised coaction side of the theorem is built.","marker":"[Da]"},{"why":"Gives the fixed-point cotangent complex equality used to define virtual classes and normal complexes on the stacks $M^\\tau$ and $SES^\\tau_3$.","marker":"[AKLPR]"},{"why":"Provides the shuffle algebra presentations and affine quantum group relations underlying the shuffle formulas of Section 7.","marker":"[YZ]"},{"why":"Identifies the zero-dimensional CoHA of a surface with the positive modes of a deformed $W$-algebra, making the surface representation concrete.","marker":"[MMSV]"},{"why":"Constructed representations of CoHAs with classical structure groups; the paper's Theorem A recovers and generalises this action when the potential is zero.","marker":"[Yo]"}],"fun_headline_variants":["Classical fixed loci become twisted Yetter-Drinfeld modules","One braided identity governs CoHA actions and vertex coactions","Orthosymplectic braiding yields twisted Yetter-Drinfeld CoHA modules","From quivers to surfaces: twisted modules over CoHAs","CoHA actions on classical loci obey braided Yetter-Drinfeld law"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Theorem 5.4.3 accepts as a premise an unpublished Atiyah-Bott torus localisation statement for virtual Euler classes of Artin stacks, cited as [La3]; if that statement is missing or inapplicable to the fourfold product $M\\times M\\times M\\times M\\times M^\\tau$, the equality of localised classes that produces the twisted Yetter-Drinfeld identity is not established.","fun_headline_variants_meta":{"raw":{"variants":["Classical fixed loci become twisted Yetter-Drinfeld modules","One braided identity governs CoHA actions and vertex coactions","Orthosymplectic braiding yields twisted Yetter-Drinfeld CoHA modules","From quivers to surfaces: twisted modules over CoHAs","CoHA actions on classical loci obey braided Yetter-Drinfeld law"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000431,"raw_usage":{"total_tokens":2246,"prompt_tokens":1034,"completion_tokens":1212,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":650,"completion_tokens_details":{"reasoning_tokens":1118}},"tokens_in":650,"tokens_out":1212,"duration_ms":9440,"temperature":1.0,"reasoning_tokens":1118,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:56:04.384267+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In the Jordan quiver with zero potential and framing weights $u_1,\\dots,u_r$, compute both sides of identity (48) at dimension vectors $d=d_1=1$ using the shuffle action (85) and the left coaction (70), and compare the rational functions in the variables $x,u,t_1,t_2$; the difference must be exactly the braiding ratio $\\beta_{34}\\beta_{23}\\kappa_4\\beta_{34}$. A second check, independent of the shuffle computation, is whether the deferred equality of localised cohomology classes in Section 5.4 holds when $W=0$ and $M$ is the non-smooth surface stack.","supporting_citations":[],"review_version":1}