{"id":"3217e70e-369b-4f0a-865b-286016e192e9","arxiv_id":"2509.07720","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Nodal and smooth genus-0 conformal block dimensions can differ for C2-cofinite non-rational VOAs with non-lowest-generated modules, making conformal block sheaves non-locally-free for N at least 4 and separating the mode transition algebra from the categorical end.","lead":"For certain non-rational vertex operator algebras, spaces of conformal blocks change dimension when a sphere is pinched into a node, breaking a vector bundle property that always holds in the rational case. The paper proves this for C2-cofinite logarithmic VOAs like the triplet algebras and symplectic fermions, with consequences for sheaves of coinvariants and for the Damiolini-Gibney-Krashen mode transition algebra.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main theorem is internally coherent, but its geometric corollaries and the end-mode-algebra consequence depend on two assertions that are imported rather than proved: propagation on non-stable curves (Rem. 2.4) and the identification of the end with smooth blocks (Prop. 2.8).","rationale":"I read the proof of Theorem 2.2 in good faith. The internal chain is coherent: assuming equal dimensions for all X,Y, the representing object D from Eq. (2.6) is forced to be isomorphic to the projective cover P_W by Prop. 2.1, and the surjectivity argument for α plus Cor. 1.19 correctly forces D to be lowest generated, contradicting the choice of W. The DSPS representability step is not obviously wrong: Mod(V) is finite for C2-cofinite V, the external tensor product with P_W^† is exact over C, and Hom(A,−) is left exact, so the functor in Eq. (2.6) is plausibly left exact. However, the paper does not spell out this verification, so I would not call it an established point. The more substantial gap is in Remark 2.4, where propagation to the non-stable curves used for the vector-bundle conclusion is asserted by analogy with [DGT21, Thm. 6.2] and not proved for logarithmic modules. This step is load-bearing for the advertised geometric consequences, because without it the dimension inequality on the two-pointed curves does not transfer to \\overline{M}_{0,N}. A second external dependency appears in Prop. 2.8, where the key identification of the end with smooth conformal blocks is delegated to [GZ25a,GZ25b]. I therefore agree with the reader's CONDITIONAL verdict, while placing the weight slightly differently: the representability concern is real but likely repairable, whereas Rem. 2.4 and Prop. 2.8 contain explicit unproved assertions that a referee should require the author to expand.","tokens_in":15166,"tokens_out":36893,"duration_ms":352155,"concrete_test":"Check the propagation step in Rem. 2.4: start from the 2-pointed nodal block T_B^*(X⊗Y), attach two vacuum insertions to obtain a 4-pointed nodal block on a stable curve in \\overline{M}_{0,4}, and verify that the canonical map T_B^*(X⊗Y) → T_{B'}^*(X⊗Y⊗V^{⊗2}) is an isomorphism (and similarly on the smooth side) by following the Riemann-Roch argument of [DGT21, Thm. 6.2] step by step for the affine curves in (2.13). If the map is not an isomorphism, the vector-bundle corollary fails even if Thm. 2.2 is true.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After checking the internal logic of Thm. 2.2, the contradiction is coherent provided the representing objects A and D in Eqs. (1.8) and (2.6) exist. The use of [DSPS19, Cor. 1.10] is plausible because Mod(V) is finite and the functor U ↦ Hom_{V⊗2}(A, U⊗P_W^†) is left exact, but this verification is not written out. The more exposed soft spot is the step from Thm. 2.2 to the geometric statements: Rem. 2.4 asserts, rather than proves, that propagation of conformal blocks extends to the non-stable affine curves in (2.13) via the Riemann-Roch argument of [DGT21, Thm. 6.2]. If this propagation is not an isomorphism after adding the V-insertions, the dimension inequality for the two-pointed sphere and node (2.3) does not imply non-local-freeness on \\overline{M}_{0,N}. Similarly, Thm. 2.9 depends on Prop. 2.8, whose proof is delegated to the companion preprints [GZ25a, GZ25b]; without that identification the non-isomorphism E ≇ A is not established. These are external dependencies, not internal inconsistencies, and they should be settled before the advertised consequences are treated as fully proved.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies conformal blocks for an N-graded, C_2-cofinite vertex operator algebra V admitting a module that is not generated by its lowest weight subspace. Its main theorem (Theorem 2.2) asserts that, for such V, there exist modules X,Y such that the dimensions of the smooth and nodal two-pointed conformal block spaces differ. From this inequality the author derives that the associated conformal blocks do not form a vector bundle on \\overline{M}_{0,N} for N≥4, that the sheaf of coinvariants is not locally free, and that the mode transition algebra is not isomorphic to the end E (Theorem 2.9). The proof of Theorem 2.2 proceeds by contradiction: if the dimensions always matched, a representing object D for a certain Hom functor would be forced to be isomorphic to a non-lowest-generated projective cover P_W, while an explicit surjection from A⊗P_W to D would force D to be lowest generated.","tokens_in":15389,"tokens_out":11420,"duration_ms":108700,"significance":"If the main theorem and its consequences are fully established, the paper gives a clean negative answer to a natural question raised in [DGK25b] and [DW25]: unlike in the rational case, smooth and nodal conformal block functors are not equivalent for general C_2-cofinite VOAs, and the vector-bundle property fails. The internal contradiction argument for Theorem 2.2 is coherent and the use of lowest generation as the key invariant is elegant. The paper also gives explicit examples (triplet algebras and even symplectic fermion VOAs) where the hypothesis holds. However, several load-bearing inputs are imported from companion preprints or cited without verification, so the advertised consequences are not yet fully proven in this manuscript.","major_comments":[{"comment":"The representing objects A and D are obtained solely by citing [DSPS19, Cor. 1.10]. That corollary applies only to left exact linear functors, but the paper never verifies that W ↦ T_B^*(W^†) and U ↦ Hom_{V⊗2}(A, U⊗P_W^†) are left exact. The second verification is straightforward because U↦U⊗P_W^† is exact as a functor of vector spaces and Hom_{V⊗2}(A,–) is left exact, but the first requires an argument about contragredients and invariance conditions. Without these checks, the existence of D, and hence the contradiction in Theorem 2.2, is not fully established.","section":"§1.4, Eq. (1.8), and §2.2, Eq. (2.6)"},{"comment":"The passage from the two-pointed inequality (2.3) to the vector-bundle and local-freeness statements on \\overline{M}_{0,N} uses propagation of conformal blocks on curves that are explicitly not stable. The remark asserts that the Riemann–Roch proof of [DGT21, Thm. 6.2] still applies because the curves are affine, but it does not provide the argument or a precise statement of the propagation theorem in this setting. Since the curves in (2.13) have components with only two special points, propagation with additional V-insertions is not automatic; this is a load-bearing step for conclusions (a) and (b).","section":"§2.2, Rem. 2.4"},{"comment":"The key identification Hom_{V⊗2}(E, X^†⊗Y^†) ≅ T_N^*(X⊗Y) is delegated to the companion preprints [GZ25a, GZ25b], with only a citation to [FSS20, Cor. 2.9]. Consequently, the advertised non-isomorphism E ≇ A is conditional on results that are not proved or even stated in this manuscript. Either include a proof of Proposition 2.8 or explicitly formulate Theorem 2.9 as depending on [GZ25a, GZ25b].","section":"§2.3, Prop. 2.8 and Thm. 2.9"}],"minor_comments":[{"comment":"There are typos: 'cardinate' should be 'cardinality' and 'indecomposible' should be 'indecomposable'.","section":"§1.1 and Introduction"},{"comment":"The mode transition algebra and the nodal fusion product are both denoted by 'A' in the text; please distinguish them consistently (for example, \\mathcal{A} versus A) in all displayed formulas, especially in Theorem 2.9 and Remark 2.7.","section":"Throughout"},{"comment":"The definition says 'the nodal conformal block functor associated to X', but no X has been introduced in that context; it should say 'associated to Y'.","section":"§1.4, Def. 1.16"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the author's companion preprints [GZ25a, GZ25b] and on unpublished details in [DGK25b]. The editor may wish to require that these references be publicly available or accepted before publication, and that the left-exactness hypotheses for the two representability claims be verified in the text. The core dimension-inequality argument is promising, but the advertised geometric and end-algebra consequences are currently conditional."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper proves something real: for any N-graded C2-cofinite VOA with a module that is not lowest generated, there exist X,Y with dim T_N^*(X⊗Y) ≠ dim T_B^*(X⊗Y) (Thm 2.2). That's a first; it answers [DW25, Q5.6.3] negatively and gives the expected no-vector-bundle conclusion for the triplet and symplectic fermion cases. I read the core proof carefully and it holds together. The representability inputs (1.8) and (2.6) come from [DSPS19] and the nodal fusion product formalism; the contradiction via D ≅ P_W then showing D is lowest generated is sound. The dimension criterion Prop 2.1 is legitimate, and the construction of α with the image argument is self-contained. The examples work: the socle-series weights for W_p and the McRae computation for SF_d^+ do give non-lowest-generated projective covers.\n\nNow the soft spots. The advertised geometric statement — non-local-freeness on \\overline{M}_{0,N} — goes through Rem 2.4, which asserts propagation of blocks for non-stable affine curves in one paragraph rather than proving it. The Riemann-Roch argument in [DGT21, Thm 6.2] likely covers the needed case, but the paper should either include the argument or flag it as an assumption from the companion framework. Similarly, Thm 2.9 (E ≇ A) depends on Prop 2.8, whose proof is delegated to [FSS20] plus the Gui-Zhang preprints. That is a real external dependency, not a flaw in the main theorem, but it makes the end result conditional on the companion papers.\n\nI don't see circularity or fitted parameters. The citation pattern is normal for a paper in this pipeline: the author leans on his own earlier work instrumentally, which is fine because the key inequalities are derived, not assumed.\n\nWho is this for: anyone working on logarithmic conformal blocks, factorization, or the DGK mode transition algebra program. It deserves a serious referee — I would send it out, with the referee asked to check Rem 2.4 and Prop 2.8 against the companion papers. I would not desk reject.\n\nBest.","headline":"A genuinely new dimension-jump result for non-rational conformal blocks; the core proof is sound, but the geometric and end consequences lean on companion-paper results that should be flagged as imports.","tokens_in":15982,"tokens_out":4631,"would_cite":true,"duration_ms":39218,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17B69","81T40","81R10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for every N-graded, C2-cofinite vertex algebra with a non-lowest-generated module, the smooth and nodal conformal block functors are inequivalent, because the dimensions of the relevant spaces on the two-pointed…","keywords":["conformal blocks","nodal curves","vertex operator algebras","logarithmic conformal field theory","C2-cofinite","non-lowest generated modules","mode transition algebra","vector bundle on moduli space"],"falsifier":"Take $\\mathbb{V}=\\mathcal{W}_2$, the smallest triplet algebra, and compute the spaces $T_N^*(X\\otimes Y)$ and $T_B^*(X\\otimes Y)$ for $X$ the projective cover of the non-lowest-generated module and $Y$ its contragredient. If the dimensions turn out equal for every pair, or if the sheaf on $\\overline{\\mathcal{M}}_{0,4}$ is locally free despite the existence of a non-lowest-generated module, the central claim is false. More directly, checking that $\\mathrm{Hom}_{\\mathbb{V}^{\\otimes 2}}(\\mathbb{A},U\\otimes P_W^\\dagger)$ fails to be representable for some $U$ would invalidate the proof of Theorem 2.2.","tokens_in":14911,"feed_emoji":"📐","tokens_out":9017,"duration_ms":72369,"temperature":0.7,"pith_summary":"The paper proves that logarithmic conformal field theories break a property that holds in every rational theory: the spaces of conformal blocks on a nodal curve and on a smooth curve need not have the same dimension. For any N-graded, C2-cofinite vertex operator algebra that has at least one module not generated by its lowest-weight vectors—for instance the triplet algebras $\\mathcal{W}_p$ and the even symplectic fermion algebras $SF_d^+$—there exist modules $X,Y$ for which $\\dim T_N^*(X\\otimes Y)$ differs from $\\dim T_B^*(X\\otimes Y)$. From this single dimension gap the author derives that the sheaf of coinvariants on the moduli space $\\overline{\\mathcal{M}}_{0,N}$ is not locally free for $N\\geq 4$, that the associated conformal-block sheaf is not a vector bundle, and that the mode transition algebra $\\mathfrak{A}$ is not isomorphic to the categorical end $\\mathbb{E}=\\int_{X\\in\\mathrm{Mod}(\\mathbb{V})}X\\otimes X^\\dagger$. The result matters because the vector-bundle property of rational conformal blocks underlies factorization and the geometric interpretation of block spaces; the paper shows that non-semisimple representation theory destroys it.","feed_headline":"Nodal and smooth conformal blocks diverge in logarithmic CFT","feed_subtitle":"For triplet and symplectic fermion algebras, block dimensions differ on nodal vs smooth curves, so coinvariant sheaves are not locally free.","key_machinery":"The argument is carried by three objects and one criterion. The nodal fusion product $\\mathbb{A}$ is the object of $\\mathrm{Mod}(\\mathbb{V}^{\\otimes 2})$ representing the nodal conformal block functor $T_B^*(-)$, i.e. $\\mathrm{Hom}_{\\mathbb{V}^{\\otimes 2}}(\\mathbb{A},W)\\cong T_B^*(W^\\dagger)$; the paper proves (Corollary 1.19) that $\\mathbb{A}$ is lowest generated as a left $\\mathbb{V}$-module. The end $\\mathbb{E}=\\int_{X\\in\\mathrm{Mod}(\\mathbb{V})}X\\otimes X^\\dagger$ is the object representing the smooth conformal block functor and is not lowest generated when non-lowest-generated modules exist. The auxiliary object $D$ represents $\\mathrm{Hom}_{\\mathbb{V}^{\\otimes 2}}(\\mathbb{A},U\\otimes P_W^\\dagger)$ as $\\mathrm{Hom}_{\\mathbb{V}}(D,U)$. The dimension criterion (Proposition 2.1) says that two objects with equal Hom-space dimensions into every module are isomorphic; it converts equal block dimensions into an isomorphism $D\\cong P_W$, producing the contradiction.","core_discovery":"On the paper's own terms, the central discovery is Theorem 2.2: assume there exists a module in $\\mathrm{Mod}(\\mathbb{V})$ that is not lowest generated. Then there exist $X,Y\\in\\mathrm{Mod}(\\mathbb{V})$ such that $\\dim T_N^*(X\\otimes Y)\\neq \\dim T_B^*(X\\otimes Y)$. The proof is a contradiction argument. Assuming equality for all pairs, for an irreducible $W$ whose projective cover $P_W$ is not lowest generated one writes the nodal block space $T_B^*(U^\\dagger\\otimes P_W)$ as $\\mathrm{Hom}_{\\mathbb{V}^{\\otimes 2}}(\\mathbb{A},U\\otimes P_W^\\dagger)$, with $\\mathbb{A}$ the nodal fusion product, and then as $\\mathrm{Hom}_{\\mathbb{V}}(D,U)$ by representability. The smooth block space is $\\mathrm{Hom}_{\\mathbb{V}}(P_W,U)$. If the two dimensions agree universally, Proposition 2.1 forces $D\\cong P_W$. But $D$ is lowest generated: the canonical map $\\alpha:\\mathbb{A}\\otimes P_W\\to D$ is surjective, and $\\mathbb{A}$ is lowest generated as a left $\\mathbb{V}$-module (Corollary 1.19), so $D$ is generated by its lowest-weight subspace. This contradicts the choice of $P_W$. The paper then records the consequences: for $N\\geq 4$ the conformal-block spaces for $X,Y,\\mathbb{V}^{\\otimes(N-2)}$ do not form a vector bundle on $\\overline{\\mathcal{M}}_{0,N}$; the sheaf of coinvariants is not locally free; and in $\\mathrm{Mod}(\\mathbb{V}^{\\otimes 2})$ the end $\\mathbb{E}$ is not isomorphic to the mode transition algebra $\\mathfrak{A}$.","pith_inferences":["A natural next step the paper does not take is to compute the exact dimension difference for the smallest example, the triplet algebra $\\mathcal{W}_2$; the mechanism suggests the gap is governed by the conformal-weight gap between the projective cover $P_W$ and its lowest-weight subspace.","Because the contradiction uses only a single non-lowest-generated projective cover, the same argument should apply to any C2-cofinite VOA whose module category has a projective object whose lowest-weight subspace does not generate it; the class of examples may extend beyond triplets and symplectic fermions.","The paper's closing remark suggests that any restored equality between nodal and smooth blocks would require a nodal-block definition divorced from the Zhu algebra; one could test this by constructing a modified nodal functor on pseudo-modules and checking whether it satisfies factorization."],"forward_implications":["The dimension gap is inherited by higher numbers of marked points: for $N\\geq 4$ the spaces of conformal blocks attached to $X$, $Y$, and $\\mathbb{V}^{\\otimes(N-2)}$ on $\\overline{\\mathcal{M}}_{0,N}$ do not assemble into a vector bundle.","The sheaf of coinvariants for these modules on $\\overline{\\mathcal{M}}_{0,N}$ is not locally free for $N\\geq 4$.","The mode transition algebra $\\mathfrak{A}$ is not isomorphic to the end $\\mathbb{E}$ in $\\mathrm{Mod}(\\mathbb{V}^{\\otimes 2})$; hence the two objects represent different functors in the logarithmic setting.","The smooth and nodal conformal block functors are not equivalent, showing that factorization of conformal blocks cannot hold in the form known from the rational case."],"supporting_citations":[{"why":"Supplies the representability theorem for left exact linear functors on finite C-linear categories that produces the objects $\\mathbb{A}$ and $D$.","marker":"[DSPS19, Cor. 1.10]"},{"why":"Introduces the mode transition algebra $\\mathfrak{A}$ and proves it represents the nodal conformal block functor (cited as Theorem 2.5), the comparison target in Theorem 2.9.","marker":"[DGK25b]"},{"why":"Companion preprint supplying the graphical calculus, the identification of smooth blocks with Hom spaces, and the representability used in Eq. (2.6).","marker":"[GZ25b]"},{"why":"Provides C2-cofiniteness consequences: finite abelian module category, finitely many irreducibles, and existence of projective covers used to find a non-lowest-generated $P_W$.","marker":"[Hua09]"},{"why":"Gives the tensor structure on the triplet $\\mathcal{W}_p$ module category, used to identify the non-lowest-generated projective cover in Corollary 1.13.","marker":"[TW13]"},{"why":"Gives the classification of even symplectic fermion modules and composition factors used in Corollary 1.14.","marker":"[McR23]"},{"why":"Establishes the rational-case factorization and dimension constancy that this paper shows fails generally.","marker":"[DGT24]"},{"why":"Supplies the Eilenberg-Watts-type isomorphism used in Proposition 2.8 to identify Hom spaces of the end $\\mathbb{E}$ with smooth conformal blocks.","marker":"[FSS20]"}],"fun_headline_variants":["Nodal blocks break vector bundle in logarithmic CFT","Block dimensions differ on nodal vs smooth curves","Log CFT: coinvariant sheaf not locally free for N≥4","Mode transition algebra not isomorphic to end in log CFT","Block spaces fail bundle property on M_{0,N} for N≥4"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on a borrowed representability theorem: every left-exact linear functor from the finite module category to vector spaces is represented by an object, applied to the nodal conformal block functor and to the auxiliary functor; if that theorem does not apply to these non-rational module categories, or if the representing object $\\mathbb{A}$ were not generated by its lowest-weight subspace, the contradiction would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Nodal blocks break vector bundle in logarithmic CFT","Block dimensions differ on nodal vs smooth curves","Log CFT: coinvariant sheaf not locally free for N≥4","Mode transition algebra not isomorphic to end in log CFT","Block spaces fail bundle property on M_{0,N} for N≥4"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000967,"raw_usage":{"total_tokens":4233,"prompt_tokens":1179,"completion_tokens":3054,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":795,"completion_tokens_details":{"reasoning_tokens":2969}},"tokens_in":795,"tokens_out":3054,"duration_ms":20099,"temperature":1.0,"reasoning_tokens":2969,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:11:45.369259+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $\\mathbb{V}=\\mathcal{W}_2$, the smallest triplet algebra, and compute the spaces $T_N^*(X\\otimes Y)$ and $T_B^*(X\\otimes Y)$ for $X$ the projective cover of the non-lowest-generated module and $Y$ its contragredient. If the dimensions turn out equal for every pair, or if the sheaf on $\\overline{\\mathcal{M}}_{0,4}$ is locally free despite the existence of a non-lowest-generated module, the central claim is false. More directly, checking that $\\mathrm{Hom}_{\\mathbb{V}^{\\otimes 2}}(\\mathbb{A},U\\otimes P_W^\\dagger)$ fails to be representable for some $U$ would invalidate the proof of Theorem 2.2.","supporting_citations":[],"review_version":2}