{"id":"f72a6dce-90fa-4fb4-bb35-0b8d5c3f1a59","arxiv_id":"2507.08417","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A comparison morphism from the K-theoretical Hall algebra to a twisted cohomological Hall algebra is derived from square roots of Todd classes and extended to symmetric quivers with potential.","lead":"The authors construct a cleaner bridge between two algebraic structures that encode representations of a directed graph: the K-theoretical Hall algebra and a twisted cohomological Hall algebra. The bridge also works when the graph carries a potential, which brings two families of quantum-group-like algebras into comparison.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.8 rests on an algebraic-K0 version of vanishing-cycle GRR that is asserted in Remark 4.6 without proof; if the [PT23] topological argument does not carry over, the Todd-twisted Chern character identities fail.","rationale":"I read the paper in good faith: the main new content is the conceptual Todd-square-root comparison and its extension to quivers with potential. For W = 0 the comparison is already known, and the structure of the paper is coherent: the definitions of KHA and CoHA are standard, the Todd class formulas are explicit, and the Zhang-twist mechanism is standard. For the potential case, however, the proof of Theorem 4.8 depends on the vanishing-cycle Grothendieck-Riemann-Roch theorem in algebraic K0. The paper quotes this from a topological K-theory paper and asserts in Remark 4.6 that the same proof works over algebraic K0, without giving the adaptation. This is the most load-bearing unproved assertion in the manuscript: if it fails, the identities used to move Todd classes across pushforwards with vanishing-cycle coefficients are not available, and neither the algebra-map property nor the subsequent finite-length module identification follows. The reader's weakest assumption identifies exactly this point. I do not think the concern warrants rejection: the statement is plausible, the W=0 comparison provides strong independent evidence, and the missing piece is a proof obligation rather than a demonstrated contradiction. I also note an internal label error in Theorem 4.5(1), where W_X and W_Y appear interchanged in the diagram; this should be fixed, but it is not by itself a mathematical objection. The appropriate verdict remains CONDITIONAL, pending a complete proof of the algebraic-K0 GRR and a fully written proof of Theorem 4.8.","tokens_in":9992,"tokens_out":13489,"duration_ms":168614,"concrete_test":"Write out the proof of Theorem 4.5 for algebraic K0 by following the diagram chase in [PT23, Thm 6.8] for the localization D^bPerf(Z_0) → D^b(Z_0) → D_sg(Z_0), explicitly verifying that the boundary map K_1(D_sg(Z_0)) → K_0(Perf(Z_0)) is compatible with the Chern character and contributes nothing to the Todd-twisted pushforward identity. Then re-derive Lemma 3.3 for T_γ-equivariant vanishing cycles using only this algebraic K0-GRR. If the derivation cannot be completed without invoking topological K-theory or without controlling the K_1-boundary, Theorem 4.8 is not proven as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central map v of Theorem 4.8 is built from ch(f) multiplied by square-root Todd classes, and the proof repeatedly moves these Todd classes across the pullbacks and pushforwards p^*, i_*, π_*. This requires the Grothendieck-Riemann-Roch statement Theorem 4.5 for algebraic K0 of matrix-factorization/singularity categories with vanishing-cycle coefficients. The theorem is quoted from [PT23, Thm 6.8], which is proved for topological K-theory, and Remark 4.6 asserts 'the same proof works for algebraic K0' without presenting the adaptation. That is the load-bearing step. The topological proof in loc. cit. combines classical GRR with the localization sequence for D^bPerf(Z_0) ⊂ D^b(Z_0) → D_sg(Z_0), and Remark 4.6 even calls this a short exact sequence; the associated boundary map K_1(D_sg) → K_0(Perf) is not discussed. If that boundary term contributes, or if the argument uses Bott periodicity, Atiyah-Hirzebruch filtration, or the monodromy-invariant splitting of [PT23, (6.1)/(7.2)] in a way that has no algebraic-K0 analogue, then Lemma 3.3's identity ch( i_* p^*(f_1,f_2) ) = Td^{-1}_{M_{γ2,γ1}} m(ch(f_1),ch(f_2)) is unjustified, and Theorem 4.8 does not follow. I am not claiming the theorem is false; I am claiming that the core analytic input for the potential case is currently an omitted proof. The displayed diagram in Theorem 4.5(1) also interchanges the labels W_X and W_Y, which should be corrected before the statement can be used.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper compares the K-theoretical Hall algebra and the cohomological Hall algebra for a symmetric quiver, and extends the comparison to quivers with potential. For W=0, the authors construct a conceptual comparison map v(f)=Td^{1/2} ch(f) into a completed cohomological Hall algebra equipped with a Zhang twist determined by square roots of Todd classes, explaining the ad hoc modification of [LŠVdB24]. For a quiver with potential whose only critical value is 0, they define the analogous map v_sg: R_W -> (\\hat H_W, ◦) and state in Theorem 4.8 that it is an algebra morphism. The proof of the potential case relies on a vanishing-cycle Grothendieck-Riemann-Roch statement (Theorem 4.5) quoted from [PT23] and on compatibility of a cohomology-module action with the maps in the Hall-algebra correspondence.","tokens_in":10346,"tokens_out":11779,"duration_ms":124658,"significance":"If the missing details can be supplied, this is a valuable contribution: it replaces the earlier ad hoc Chern-character twist by a square-root Todd-class construction, extends the comparison to quivers with potential, and yields an identification of finite-length modules over KHA and CoHA. The nonpotential comparison in Section 3 is coherent and the main claim is checked against independently defined Hall algebras; there are no fitted parameters or self-referential normalizations. The potential case follows the same structural pattern but currently rests on two under-proved inputs, so the result is plausible rather than fully established as written.","major_comments":[{"comment":"The potential comparison rests entirely on a Grothendieck-Riemann-Roch statement for algebraic K0 of matrix-factorization/singularity categories with vanishing-cycle coefficients. Theorem 4.5 is quoted from [PT23, Thm 6.8], where the proof is carried out in topological K-theory, and Remark 4.6 asserts 'the same proof works for algebraic K0' without presenting the adaptation. This is load-bearing: the proof of Theorem 4.8 moves square-root Todd classes across the maps p^*, i_*, and π_* precisely by means of Theorem 4.5 (see Lemma 4.7 and the identities (3.1)–(3.4)). The authors should supply the algebraic-K0 argument, in particular the role of the localization sequence D^bPerf(Z0) ↪ D^b(Z0) ↠ D_sg(Z0), the boundary map K_1(D_sg) → K_0(Perf), and any use of Bott periodicity or of the monodromy-invariant splitting of [PT23, (6.1)/(7.2)]. If the topological proof relies on structures with no algebraic-K0 analogue, the identity ch(i_* p^*(f_1,f_2)) = Td^{-1} m(ch(f_1),ch(f_2)) used in the potential comparison is not justified.","section":"§4, Theorem 4.5 and Remark 4.6"},{"comment":"The verification that the module action of H^*_{Tγ}(Mγ,Q) on vanishing-cycle cohomology is compatible with the maps \\tilde p^* and \\tilde i_* is too terse. For \\tilde p^* it is asserted in one sentence, and for \\tilde i_* it is sketched via a diagram and [Dav17, Cor. 2.15]; the T-equivariant analogue of Lemma 3.1 for the potential case is not written out. Since the projection-formula structure of Lemma 3.1 is what makes the twisted multiplication well-defined and associative, this compatibility is a load-bearing part of Theorem 4.8. Please spell out the T-equivariant potential correspondences (including what \\tilde i_* means when the source has torus T_{γ1,γ2} and the target has torus T_γ) and prove the two commutation statements in detail.","section":"§4, proof of Theorem 4.8"}],"minor_comments":[{"comment":"In the statement of Theorem 4.5(1), the labels W_X and W_Y are interchanged: with h: X -> Y and W_X = W_Y ∘ h, the K0 and cohomology groups on the left should be for (X, W_X) and those on the right for (Y, W_Y).","section":"Theorem 4.5(1)"},{"comment":"In Theorem 4.5(2), Td_h is said to lie in bH(X0), but Lemma 4.7 and the pushforward formula Td_h h_* require an element on X (or at least an element whose pullback to X0 equals Td_{h|X0}); please clarify.","section":"Theorem 4.5(2)"},{"comment":"The statement of Lemma 3.2 has a typo: the Chern character maps \\tilde R to b\\tilde H, not to \\tilde R. Also, the proof writes ch(f) = exp(f(x_1,...,x_n)) for a general K-theory class f; this identity holds only for monomial classes. The desired S_n-equivariance follows directly from naturality of the Chern character and should be stated in that way.","section":"Lemma 3.2"},{"comment":"The comparison map is introduced under the name v_sg and then called v in Theorem 4.8; please use one notation.","section":"Section 4.3"},{"comment":"The sentence 'Note that π is the identity in this case' is unclear for the torus quotients; please define \\tilde p, \\tilde i, and the corresponding π precisely.","section":"Section 2.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely correct in substance; the main issue is the omitted algebraic-K0 adaptation of [PT23, Thm 6.8] and the under-specified module-action compatibilities in the proof of Theorem 4.8. The W=0 part of the paper appears sound and is a genuine conceptual improvement over the previous comparison. I would encourage the editor to request a complete proof of the GRR adaptation and a fuller write-up of the T-equivariant potential case before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper gives a conceptual derivation of the comparison morphism from the K-theoretic Hall algebra to a twisted cohomological Hall algebra for symmetric quivers, and extends it to quivers with potential. The conceptual part is the use of square-root Todd classes, which explains the ad hoc Chern character modification in [LŠVdB24]. The extension to the critical CoHA, Theorem 4.8, is genuinely new as far as I can tell, and the overall strategy is sensible: pass to the torus-equivariant setting, prove a twisted multiplicativity statement for a Todd-twisted Chern character, then descend to the G-equivariant Hall algebras.\n\nThe W=0 part looks solid. The formulas in Section 3.1 and the identities (3.1)-(3.4) are explicit and checkable; Lemma 3.3 and Corollary 3.8 follow naturally. I have no serious concerns about that portion.\n\nThe soft spot is exactly where the reader's stress-test lands: Theorem 4.5. The paper quotes a Grothendieck-Riemann-Roch statement for vanishing-cycle cohomology from [PT23], proved there for topological K-theory, and Remark 4.6 asserts without proof that the same argument works for algebraic K0. That assertion is load-bearing for the potential case, because the proof of Theorem 4.8 repeatedly moves Todd classes across pushforwards using this GRR. The stress-test note is right that the exact triangle D^bPerf(Z0) -> D^b(Z0) -> D_sg(Z0) is not literally a short exact sequence of triangulated categories; the associated boundary map and the monodromy-invariant splitting from [PT23] are not discussed. I would not call this a fatal flaw - the topological proof may well carry over - but currently it is an omitted proof, and the paper should either provide the adaptation or state the algebraic-K0 version as a conjecture. The displayed diagram in Theorem 4.5 also has the labels W_X and W_Y interchanged, which is a minor fix but should be corrected.\n\nThe module-action compatibility in the proof of Theorem 4.8 is also sketched rather than written out. The reduction to pullbacks and the claim that the action commutes with i_* via [Dav17, Cor 2.15] is plausible, but the transverse intersection statement and the limit argument deserve more detail.\n\nNet: this is a worthwhile paper that deserves a serious referee. The W=0 comparison is a clean exposition of a known result, and the potential-case theorem is likely true but not yet fully supported. I would accept it for peer review with a request to fill in the algebraic-K0 GRR gap and to expand the compatibility argument. The right comparison point is [LŠVdB24] plus [PT23]; the paper's novelty is the square-root Todd framework, which is a genuine improvement in presentation and a real step toward the potential case.","headline":"A clean Todd-class proof of the known W=0 comparison and a promising potential-case extension that currently hinges on an unproven algebraic-K0 GRR assertion.","tokens_in":10905,"tokens_out":5157,"would_cite":true,"duration_ms":47639,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G20","14C40","14F08","19E08"],"pacs":[],"model":"deepseek-v4-flash","headline":"A Todd-twisted Chern character maps the K-theoretic Hall algebra into the critical cohomological Hall algebra for symmetric quivers with potential.","keywords":["cohomological Hall algebra","K-theoretic Hall algebra","symmetric quiver","potential","Todd class","Chern character","vanishing cycles","Grothendieck-Riemann-Roch"],"falsifier":"Compute the comparison map v explicitly for the Jordan quiver with one vertex and one loop and a generic potential, where both Hall algebras are known (the KHA is related to the spherical Hall algebra and the CoHA to the cohomological Hall algebra of the one-loop quiver). If the twisted multiplication does not match the known Yangian/quantum-affine comparison, or if v fails to preserve the product for a dimension vector (2,2), the theorem would be contradicted.","tokens_in":9768,"feed_emoji":"🧩","tokens_out":2848,"duration_ms":31380,"temperature":0.7,"pith_summary":"The paper proves that for a symmetric quiver with potential whose only critical value is 0, the K-theoretic Hall algebra embeds into a completed, twisted version of the critical cohomological Hall algebra. The comparison map is the Chern character multiplied by a square root of the Todd class, and the twist is implemented by the pullback of these square roots to the critical fiber. This gives a conceptual explanation of an earlier ad-hoc modification used in the zero-potential case and extends the comparison to quivers with potentials. If correct, finite-length modules over the two Hall algebras are identified, aligning K-theoretic and cohomological Hall algebra constructions.","feed_headline":"Hall algebras united by a Todd twist","feed_subtitle":"A square-root Todd class turns the Chern character into an algebra map from K-theoretic to cohomological Hall algebras, extending the…","key_machinery":"The central object is the square-root Todd class $Td^{{1/2}}$_{M_{γ,0}/G_γ}, defined via the unique square root of a power series with constant term 1. The argument carries through a Grothendieck-Riemann-Roch comparison for vanishing-cycle cohomology (Theorem 4.5), which moves Todd classes across the pushforwards i_* and π_*; the twisted product ∘ is a Zhang twist built from the semigroup homomorphism τ ↦ c^γ_τ that factors the Todd-class ratios into separate factors depending on γ_1 and γ_2.","core_discovery":"For a symmetric quiver Q with a potential W such that 0 is the only critical value of tr W, Theorem 4.8 asserts that the map v: R_W → (hat H_W, ∘), f ↦ $Td^{{1/2}}$_{M_{γ,0}/G_γ} ch(f), is a morphism of algebras. Here R_W is the K-theoretic Hall algebra, hat H_W is the completed critical cohomological Hall algebra, and ∘ is the twisted multiplication whose twisting cocycle is given by the pullback ι^* $c^{{γ_1}}$_{γ_2} = ι^* ($b^{{γ_1}}$_{γ_2}/$d^{{γ_1}}$_{γ_2})^{1/2} of square-root Todd-class ratios. The morphism is injective, so the K-theoretic Hall algebra is realized as a subalgebra of a twist of the critical CoHA.","pith_inferences":["If Theorem 4.5 holds in algebraic K-theory, a similar comparison should hold for any symmetric quiver with a generic potential (with multiple critical values) by working fiberwise, though the paper only states the single-critical-value case.","The square-root Todd class formalism suggests a direct bridge to quantum group comparisons: the twist factors c^γ_τ should match the Drinfeld-type twist that relates Yangians and quantum affine algebras, potentially making the comparison canonical rather than ad hoc.","A concrete test would be to compute both sides for the triple quiver with potential, where the KHA is a positive half of a Yangian and the CoHA is a half of a quantum affine algebra, and verify that v recovers the known exponentiation-of-roots map."],"forward_implications":["The K-theoretic Hall algebra of a symmetric quiver with potential is a subalgebra of a completed, twisted critical cohomological Hall algebra.","Finite-length modules over the K-theoretic Hall algebra correspond to finite-length modules over the twisted critical cohomological Hall algebra, extending the identification obtained for zero potential.","The comparison morphism is compatible with the G-equivariant and T-equivariant descriptions, so the shuffle-product presentations of both Hall algebras are matched by the Todd-twisted Chern character.","The construction provides a uniform mechanism, square roots of Todd classes, that explains why the earlier zero-potential comparison required a nontrivial twist."],"supporting_citations":[{"why":"Introduces the cohomological Hall algebra of a quiver with potential, whose multiplication and shuffle formulas are the base object being compared.","marker":"[KS11]"},{"why":"Defines the K-theoretic Hall algebra of a quiver with potential and establishes its associative multiplication via singularity categories.","marker":"[Pad20]"},{"why":"Previous comparison for W=0 using an ad hoc modified Chern character; this paper extends and conceptually explains that result.","marker":"[LŠVdB24]"},{"why":"Supplies the Grothendieck-Riemann-Roch theorem for vanishing-cycle cohomology in topological K-theory, used via Remark 4.6 in the algebraic setting.","marker":"[PT23]"},{"why":"Defines the critical cohomological Hall algebra and the module action of equivariant cohomology on vanishing-cycle cohomology, used in the proof of Theorem 4.8.","marker":"[Dav17]"},{"why":"Provides categorical equivalences between matrix factorizations and singularity categories, used to describe the K-theoretic Hall algebra of a quiver with potential.","marker":"[BFK14]"},{"why":"Establishes the general theory of twisted graded algebras used to define the twisted multiplication ∘.","marker":"[Zha96]"}],"fun_headline_variants":["Square-root Todd class unites Hall algebras","Injective Todd-twisted morphism embeds K-theoretic Hall algebra in CoHA","Square-root Todd cocycle twists CoHA to host K-theoretic Hall subalgebra","Todd twist gives injective algebra map from K-theoretic to cohomological Hall"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof relies on Theorem 4.5, a Grothendieck-Riemann-Roch statement for vanishing-cycle cohomology, holding in algebraic K-theory exactly as it does in topological K-theory, an adaptation that Remark 4.6 asserts without presenting the details.","fun_headline_variants_meta":{"raw":{"variants":["Square-root Todd class unites Hall algebras","Injective Todd-twisted morphism embeds K-theoretic Hall algebra in CoHA","Square-root Todd cocycle twists CoHA to host K-theoretic Hall subalgebra","Todd twist gives injective algebra map from K-theoretic to cohomological Hall"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000573,"raw_usage":{"total_tokens":2608,"prompt_tokens":748,"completion_tokens":1860,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":364,"completion_tokens_details":{"reasoning_tokens":1778}},"tokens_in":364,"tokens_out":1860,"duration_ms":15008,"temperature":1.0,"reasoning_tokens":1778,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:19:13.201702+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the comparison map v explicitly for the Jordan quiver with one vertex and one loop and a generic potential, where both Hall algebras are known (the KHA is related to the spherical Hall algebra and the CoHA to the cohomological Hall algebra of the one-loop quiver). If the twisted multiplication does not match the known Yangian/quantum-affine comparison, or if v fails to preserve the product for a dimension vector (2,2), the theorem would be contradicted.","supporting_citations":[],"review_version":1}