{"id":"4f85361e-77ae-4f93-a77d-e1ad93f952d3","arxiv_id":"2411.10694","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Virasoro-type reduction and inverse Hamiltonian reduction are established for height-two W-algebras in classical Lie types and for the universal W∞-algebra W^{sp}_∞.","lead":"This paper proves that families of W-algebras, symmetry algebras used in conformal field theory, can be reduced to neighboring W-algebras by a simple operation and that the process can be reversed. It extends this reduction to a universal algebra, giving a unified picture for infinitely many examples.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Type BCD main theorem rests on unproved Wakimoto screening formulas (Props. 4.3–4.5); a wrong coefficient would change H0(Wk(g,O)).","rationale":"The reader's weakest assumption already identifies the omitted Wakimoto formulas; I agree that this is the place where the argument is least secure. The main theorems for types B, C, and D are not isolated identities: every displayed screening operator in §5 is derived from Propositions 4.3–4.5, and the proofs of those propositions are not supplied. Because the Virasoro-type reduction H0 is a cohomology of a complex whose differential is built from G+, small changes in the realization can change the kernel; these formulas are exactly the kind of explicit computation where sign or bookkeeping errors occur. I do not see an internal contradiction or a reason to think the claims are false; the type A case is proved in detail, and the BCD cases are plausibly parallel. For that reason the paper should not be rejected, but the omitted computations should be supplied or independently checked before the BCD rows and the W∞ results built on them are accepted. This matches the reader's CONDITIONAL verdict, so I do not change it. A secondary dependency on [21] for the structure, freeness, and simplicity of Wsp∞ affects Theorem C and Corollary 9.2 but is less central than the screening formulas, which underpin the finite-type Theorem A in all BCD cases.","tokens_in":43130,"tokens_out":5264,"duration_ms":58461,"concrete_test":"Independently re-derive Proposition 4.3(i) from the pyramid data in (4.11) by the same exponential-coordinate method used in the proof of Proposition 4.2: compute ρ(e_{α_i}) fully for N=4 and N=5, list all root contributions with Γ(α)>1, and verify the stated P_i^O and ⋆, including P_4^O=Φ_2−γ_3Φ_1 and G+=β_1+β_3+1/2Φ_1^2. Then check the OPE G+(z)G+(w)∼0 and that the intersection of kernels has the low-weight strong generating fields predicted by (3.21) for W_k(so_{2N+1},O[N^2,1]) at a generic level. If any coefficient differs, re-run §5.2.1 with the corrected data and compare H0(Wk(g,O)) to Wk(g,Ohat) for the corresponding orbit in Table 4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Propositions 4.3–4.5 supply the explicit screening operators SO_i = ∫ Y(P_i^O e^{-α_i/(k+h∨)}, z) dz that define Wk(so_{2N+1},O[N^2,1]), Wk(sp_{2N},O[N^2]), and Wk(so_{2N},O[N^2]) inside free fields. Their proofs are explicitly omitted: for example, Proposition 4.3 says 'The computations of ⋆ and the Pi’s are parallel to the proof of Proposition 4.2, which we omit', and Propositions 4.4 and 4.5 have no proof at all. Section 5 then uses these formulas as the unique input: the change of variables (5.19), (5.30), (5.38), (5.49), (5.57), (5.63) is designed to put G+ into a single βγ pair, the cohomology of βγ⋆⊗Φ_{1/2}⊗π is computed, and the resulting kernel is identified with the Wakimoto realization of Wk(g,Ohat) only through the induced coefficients [P_i^O]. Any incorrect constant, sign, missing Φ_{1/2}-term, or wrong zero-graded set ⋆ in these propositions propagates directly into H0(Wk(g,O)) and breaks the isomorphism with Wk(g,Ohat). No independent evidence is supplied for these formulas, such as a small-N check of the strong generating type or characters of the kernel. Thus Theorem A(1) for every row of Table 1 in types B, C, and D, and the special cases feeding Corollary 9.2, hinge on an unverified computation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Virasoro-type quantum Hamiltonian reductions for W-algebras associated with nilpotent orbits of height two in classical types, and their inverse Hamiltonian reductions. The main results are: Theorem A, giving isomorphisms H^0(W_k(g,O)) ≅ W_k(g,\\hat O) for the pairs in Table 1 at generic level and embeddings W_k(g,O) ↪ Π ⊗ W_k(g,\\hat O); Theorem B, a module-level analogue for modules in the Kazhdan–Lusztig category; and Theorem C, computing the Virasoro-type reduction of the universal two-parameter algebra W^{sp}_∞(c,k) as a simple freely generated algebra of type W(2^3,3,4^3,5,6^3,...), with applications to W-algebras of types C and D. The proofs use Wakimoto realizations and screening operators, following and extending previous work on partial reductions in type A.","tokens_in":43494,"tokens_out":7444,"duration_ms":73554,"significance":"If the main theorems are correct, the paper provides a systematic mechanism relating adjacent W-algebras via small, explicit reductions and inverse reductions, including new universal statements for W^{sp}_∞. This would be a valuable contribution to the structure theory of W-algebras, with potential consequences for representation theory and for the program of building W-algebras from fundamental reductions. The manuscript has genuine strengths: the type A proof is explicit, the screening-operator framework is concrete and falsifiable, no fitted parameters are introduced, and the module-level functor isomorphism in Theorem B is a useful extension. The BCD and universal parts, however, currently rest on several omitted computations and on external results, as detailed below; the verdict is therefore conditional.","major_comments":[{"comment":"The Wakimoto realizations for types B, C, and D, including the zero-graded sets ⋆ and all screening coefficients P_i^O, are central inputs for the proof of Theorem 3.6. Proposition 4.3 states that 'the computations of ⋆ and the P_i's are parallel to the proof of Proposition 4.2, which we omit', and Propositions 4.4 and 4.5 have no proof at all. These formulas are used without further verification in Section 5: the change of variables in (5.19), (5.30), (5.38), (5.49), (5.57), and (5.63), and the identification of the reduced screening operators with those of W_k(g,\\hat O), depend directly on every coefficient. I therefore regard the BCD rows of Theorem A(1) and Corollary 9.2 as conditional on unverified computations. Please provide the omitted computations or, failing that, independent checks such as characters or strong-generating-type tests for small N.","section":"§4.2.2–4.2.4, Props. 4.3–4.5"},{"comment":"Theorem 3.6 is stated for generic k, and Remark 3.7 explains that the all-level isomorphism and cohomology vanishing are obtained only when (g,O) is (so_{2N},O_{[N^2]}) or (so_{2N+1},O_{[N^2,1]}) with N even, or (sp_{2N},O_{[N^2]}) with N odd; the other parity cases remain open. The abstract, however, says the reductions are 'established for classical Lie type and nilpotent orbits of height two' without this generic-level qualification. The paper should either soften the abstract and the introductory framing to state the generic-level result, or prove the missing parity cases; as written, the claims exceed the results.","section":"Theorem 3.6, Remark 3.7"},{"comment":"Theorem 6.1(2), the inverse Hamiltonian embedding for all BCD pairs in Table 4, is a main conclusion of the paper, but its proof is one sentence: 'The proof of (2) is similar, we omit it.' Since the embedding is built from the localization trick and an automorphism analogous to (6.12), and since it is claimed for all levels, the omission is load-bearing. Please provide the construction at least for one representative parity case in each type, or restrict the statement to the cases that are actually proved.","section":"§6, Theorem 6.1(2)"},{"comment":"The proof of Theorem 7.1 for general [N,M] is only sketched. After deriving the induced screenings (7.10), the text says 'We show with the same argument as for Theorem 3.5 that this set of screenings can be obtained by [the Wakimoto realization] ... We omit the details.' This is precisely the point where H^0(W_k(sl_{N+M},O_{[N,M]})) is identified with W_k(sl_{N+M},O_{[N+1,M-1]}). Please provide the omitted pyramid/good-pair computation, or explicitly restrict the theorem to the case M=1 proved in Section 5.","section":"§7, after Eq. (7.10)"},{"comment":"The conjugacy claims that identify the formal pairs (f_c, Γ_c) with standard good pairs are not demonstrated. For example, in type B even, f_c in (5.24) contains the term 2E^o_{-N,N-1}, while the pyramid in Figure 12 is said to give f_{\\hat O} containing E^o_{-N+1,N}; the displayed conjugation matrix (5.26) is not accompanied by any computation showing that it preserves the relevant bilinear form and sends f_c to f_{\\hat O}. The same issue occurs in (5.34), (5.44), (5.54), (5.61), and (5.66). Since the good-pair property is required to invoke Theorem 4.1, these checks cannot be omitted.","section":"§5.2–5.4, Eqs. (5.24), (5.34), (5.44), (5.54), (5.61), (5.66)"},{"comment":"The proof of Theorem 9.1 uses, without re-proof, the freeness, complete reducibility over Q(R), and simplicity of W^{sp}_∞(c,k) from [21], as well as the Kazhdan–Lusztig categorical facts in (9.18) from [20,35]. Since [21] is a preprint by one of the authors, the universality part of the paper is conditional on an external result whose status should be made explicit. I recommend either including a short proof of the needed facts or clearly stating this dependency in Theorem 9.1 and Corollary 9.2.","section":"§9, proof of Theorem 9.1"}],"minor_comments":[{"comment":"There are typos in 'vertex algberas' in Proposition 4.4 and 'Propositionn' in §5.3.1; please correct them.","section":"§4.2.4, §5.3.1"},{"comment":"In the paragraph following Eq. (5.58), the cohomology is written as H^p(βγ_{N+1}) but the reduced pair is βγ_N; please fix the index.","section":"§5.4.1"},{"comment":"The text refers to 'Theorem 4.7' after Eq. (7.10), but the only Wakimoto realization theorem in the paper is Theorem 4.1; please correct the cross-reference.","section":"§7"},{"comment":"The tables rely on 'upper row' and 'lower row' to distinguish N even and odd, but this convention is not stated in the captions; please add explicit labels such as 'N even' and 'N odd'.","section":"Tables 1 and 4"},{"comment":"The isomorphisms (5.6) and (7.6) are described as isomorphisms of vertex algebras between βγ-systems, but the domain and codomain indexing is not fully explained; a sentence indicating that these are the standard change-of-basis isomorphisms for βγ vertex algebras would improve readability.","section":"§5.1 and §7"}],"recommendation":"major_revision","confidential_remarks":"The main concern is that several load-bearing computations are omitted and an external preprint [21] supplies key structural facts for the universal part. These are fixable in principle, but the manuscript as submitted does not yet contain complete proofs of Theorems A(1) (BCD cases), A(2), and C as stated. The paper is within the scope of the journal and the overall strategy is promising; I would encourage the editor to request a revision with the missing details."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a real advance—full families of height-two Virasoro-type reductions in classical types, plus a universal version—but the BCD cases rest on omitted Wakimoto computations that a referee should ask to see.\n\nWhat's new: previous work only covered small ranks, notably N=2 in type A. Here they get all N for type A and the height-two rows in types B, C, D, with explicit identification of the target nilpotent orbit. The type A proof is written out in detail, including the change of variables, the cohomology of the free field complex, and the conjugacy argument that identifies the reduced realization with the target Wakimoto realization. Theorem B, extending the result to modules in the Kazhdan–Lusztig category, is a nice bonus. Theorem 9.1, the Virasoro-type reduction of W^sp_infty, is a genuine universal statement and the generating type result W(2^3,3,4^3,5,...) is concrete and checkable. The paper is clearly organized and the overall strategy is coherent.\n\nThe load-bearing gap: Propositions 4.3–4.5, which give the Wakimoto screening operators for types B, C, and D, are stated without proof. Proposition 4.3 explicitly says the computations are parallel to those for type A and are omitted; Propositions 4.4 and 4.5 have no proof at all. These formulas are the unique input to Section 5: every change of variables and every cohomology computation starts from them. An incorrect sign or a missing Φ_{1/2} term would alter the kernel and break the isomorphism with W^k(g, Ohat). So this is not a cosmetic omission. It is probably fillable—the type A computation gives a template—but the referee should ask for the details or at least a small-N check (say N=2 or 3) of the resulting strong generating type or character. Also, Remark 3.7 concedes that the all-level statement holds only in some parity cases; the abstract's 'established' should be read as generic level unless the parity condition is met. Theorem C inherits the structure and simplicity of W^sp_infty from [21], a co-author preprint; that is not disqualifying, but it means the universality result is conditional on [21] being correct. The citation pattern is fine—the self-citations supply the universal algebra and prior small cases, not the conclusions.\n\nWho this is for: specialists in W-algebras and vertex algebra representation theory. It deserves a serious referee; the paper is important enough that the gaps should be fixed rather than desk-rejected. I would recommend sending it to a strong referee with a request to verify or fill in Propositions 4.3–4.5. If those hold up, the paper is solid.","headline":"Strong extension of reduction-by-stages for height-two classical W-algebras, but the BCD main theorem rests on omitted screening computations that a referee should require.","tokens_in":43981,"tokens_out":2267,"would_cite":true,"duration_ms":24042,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17B69","17B67","17B08"],"pacs":[],"model":"deepseek-v4-flash","headline":"The authors prove that a Virasoro-type reduction sends the height-two W-algebras of classical types to the W-algebras of the smallest nilpotent orbit containing them, and that the passage is reversible after tensoring with a free-field…","keywords":["Virasoro-type reduction","W-algebras","quantum Hamiltonian reduction","inverse Hamiltonian reduction","Wakimoto realization","nilpotent orbits","W-infinity algebras","screening operators"],"falsifier":"Work out the rank N=3 case of W_k(sp_6, O_[$3^{2}$]) using the stated Wakimoto realization: if the kernel intersection does not reproduce W_k(g,O) or if the reduced screening operators fail to match the Wakimoto realization of W_k(sp_6, O_[4,2]), the chain of proof breaks.","tokens_in":2143,"feed_emoji":"🔁","tokens_out":2863,"duration_ms":86612,"temperature":0.7,"pith_summary":"This paper establishes that, for a family of W-algebras attached to height-two nilpotent orbits in classical Lie algebras, the Virasoro-type quantum Hamiltonian reduction $H^0$ sends $W_k(g,O)$ isomorphically to $W_k(g,\\hat{O})$, where $\\hat{O}$ is the smallest nilpotent orbit whose closure contains $O$. It also constructs the inverse Hamiltonian reduction: $W_k(g,O)$ embeds into $\\Pi \\otimes W_k(g,\\hat{O})$, with $\\Pi$ the half-lattice vertex algebra. The same statements are promoted to a module level and to the universal two-parameter $W_\\infty$-algebra $W_\\infty^{\\mathrm{sp}}(c,k)$, whose Virasoro-type reduction is shown to be a simple freely generated vertex algebra of type $W(2^3,3,4^3,5,6^3,\\ldots)$. If correct, this gives a systematic reduction and inverse reduction connecting neighboring W-algebras in the closure order of nilpotent orbits, and lifts the pattern to universal objects.","feed_headline":"Virasoro reduction maps height-two W-algebras to orbit neighbors","feed_subtitle":"A Virasoro-style reduction, with inverse embedding, links neighboring W-algebras and lifts to the universal W-infinity.","key_machinery":"The carrying object is the Wakimoto free-field realization of W-algebras at generic levels, which places $W_k(g,O)$ inside $\\beta\\gamma^{\\star} \\otimes \\Phi(\\mathfrak{g}_{1/2}) \\otimes \\pi_{\\mathfrak{h}}^{k+h^\\vee}$ as the common kernel of screening operators $S_i^O = \\int Y(P_i^O e^{-\\alpha_i/(k+h^\\vee)}, z) dz$. The reduction under study is the Virasoro-type BRST complex with differential $d = \\int Y((G^+ + 1)\\phi^*, z) dz$, where $G^+$ is the strong generator playing the role of the positive root of $\\mathfrak{sl}_2$. Gauging the free fields transforms $d$ into a derivative of a single $\\beta$-gamma pair, reducing the cohomology to an intersection of transformed screening operators that is precisely the Wakimoto realization of $W_k(g,\\hat{O})$. The inverse reduction is obtained by localizing that $\\beta$-gamma pair into the half-lattice vertex algebra $\\Pi$, which matches the screening operators on both sides.","core_discovery":"At generic level, each W-algebra in Table 1 is realized inside a free field algebra as the common kernel of screening operators coming from its Wakimoto realization. The paper shows that the strong generator $G^+$, an analogue of the upper nilpotent element of $\\mathfrak{sl}_2$, is a sum of $\\beta$-fields, and that the Virasoro-type BRST differential can be gauged into a differential acting on a single $\\beta$-gamma pair. Computing the cohomology of the Wakimoto modules leaves an intersection of transformed screening operators that exactly matches the Wakimoto realization of $W_k(g,\\hat{O})$. Hence $H^0(W_k(g,O))$ is isomorphic to $W_k(g,\\hat{O})$ with cohomology vanishing, and localizing the relevant $\\beta$-gamma pair into $\\Pi$ identifies the screening operators on both sides, producing the inverse embedding $W_k(g,O) \\hookrightarrow \\Pi \\otimes W_k(g,\\hat{O})$.","pith_inferences":["Iterating the Virasoro-type reduction along chains in the Hesse diagram would produce W-algebras of larger orbits from smaller ones by elementary steps, so the reduction may serve as a building block for the conjectural reduction-by-stages descriptions of W-algebras.","The module-level isomorphism suggests that, at generic level, the Virasoro-type reduction and the inverse localization define an equivalence between weight-module categories of neighboring W-algebras; that equivalence is not constructed in the paper.","The universal theorem motivates searching for analogous two-parameter algebras of type $W(1,2^3,3,4^3,5,\\ldots)$ whose Virasoro-type reduction would be an extension of two even-spin $W_\\infty$-algebras; the explicit two commuting Virasoro vectors and weight-4 primaries in (9.31)-(9.36) give a concrete OPE check.","For the unstarred rows of Table 1, the continuity argument does not settle all levels; a direct attack on the finite-dimensional Slodowy-slice group-action problem the authors identify would decide the remaining cases."],"forward_implications":["For each pair in Table 1, $H^0(W_k(g,O))$ is isomorphic to $W_k(g,\\hat{O})$ at generic level, with cohomology vanishing in all but the unstarred rows of Table 1; type A and the starred rows extend to all levels.","There is an embedding $W_k(g,O) \\hookrightarrow \\Pi \\otimes W_k(g,\\hat{O})$ for all levels, realizing the inverse Hamiltonian reduction.","For $M$ in the Kazhdan–Lusztig category $KL_k(g)$, $H^0(H_O(M))$ is isomorphic to $H_{\\hat{O}}(M)$, so the functors $H^0 \\circ H_O$ and $H_{\\hat{O}}$ are naturally isomorphic.","The Virasoro-type reduction of the universal $W_\\infty^{\\mathrm{sp}}(c,k)$ is a simple freely generated vertex algebra of type $W(2^3,3,4^3,5,6^3,\\ldots)$, free over its base ring, and the W-algebras $W_k(\\mathfrak{sp}_{4n+2}, O_{[2n,2n+2]})$ and $W_k(\\mathfrak{so}_{4n}, O_{[2n-1,2n+1]})$ arise as 1-parameter quotients of it over $\\mathbb{C}(k)$.","The type A statement extends to $W_k(\\mathfrak{sl}_{N+M}, O_{[N,M]})$: $H^0$ gives $W_k(\\mathfrak{sl}_{N+M}, O_{[N+1,M-1]})$ at generic level, with an inverse embedding for all levels."],"supporting_citations":[{"why":"Supplies the Wakimoto realization of W-algebras at generic levels, the exact sequence of screening operators, and the description of W_k(g,O) as an intersection of kernels; every screening-operator computation in Section 5 uses these formulas.","marker":"[37]"},{"why":"Defines the universal two-parameter vertex algebra W_∞^sp(c,k), whose freeness, simplicity, and 1-parameter quotients Theorem 9.1 and Corollary 9.2 assume as input.","marker":"[21]"},{"why":"Established the N=2 case of the Virasoro-type reduction and the inverse-reduction-by-localization technique that Theorems 3.5, 6.1, and 7.1 extend.","marker":"[26]"},{"why":"Introduced the inverse Hamiltonian reduction V_k(sl2) ↪ W_k(sl2,O_[2]) ⊗ Π, the template for the embeddings of Theorems 6.1 and 9.3.","marker":"[53]"},{"why":"Provides the decomposition and cohomology arguments used to prove freeness, vanishing, and the strong generating type of H^0(W_∞^sp(c,k)).","marker":"[43]"},{"why":"Supplies exactness of the Virasoro reduction functor on the relevant category, used to extend the type-A isomorphism from generic levels to all levels.","marker":"[6]"},{"why":"Gives the cohomology vanishing H^{≠0}_O(V_k(g)) = 0 and acyclicity facts that make the Wakimoto exact sequence and the module reductions valid.","marker":"[9]"}],"fun_headline_variants":["Virasoro reduction links W-algebras via inverse embedding","Inverse reduction maps W-algebras to orbit neighbors","BRST reduction yields inverse Hamiltonian embeddings","Height-two W-algebras get inverse reductions","Universal W-infinity from Virasoro reduction"],"cache_read_input_tokens":46080,"weakest_assumption_plain":"The paper relies on the explicit Wakimoto realizations of Propositions 4.3–4.5, whose proofs are omitted, and on the freeness, simplicity, and structure of W_∞^sp(c,k) imported from the authors' preprint [21]; if either input fails, the screening-operator identification and the universal reduction no longer follow.","fun_headline_variants_meta":{"raw":{"variants":["Virasoro reduction links W-algebras via inverse embedding","Inverse reduction maps W-algebras to orbit neighbors","BRST reduction yields inverse Hamiltonian embeddings","Height-two W-algebras get inverse reductions","Universal W-infinity from Virasoro reduction"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000609,"raw_usage":{"total_tokens":2761,"prompt_tokens":798,"completion_tokens":1963,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":414,"completion_tokens_details":{"reasoning_tokens":1892}},"tokens_in":414,"tokens_out":1963,"duration_ms":16039,"temperature":1.0,"reasoning_tokens":1892,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T19:25:24.033322+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Work out the rank N=3 case of W_k(sp_6, O_[$3^{2}$]) using the stated Wakimoto realization: if the kernel intersection does not reproduce W_k(g,O) or if the reduced screening operators fail to match the Wakimoto realization of W_k(sp_6, O_[4,2]), the chain of proof breaks.","supporting_citations":[{"cited_title":"Screening operators and parabolic inductions for a ffine W-algebras","cited_arxiv_id":null,"evidence_quote":"Supplies the Wakimoto realization of W-algebras at generic levels, the exact sequence of screening operators, and the description of W_k(g,O) as an intersection of kernels; every screening-operator computation in Section 5 uses these formulas."},{"cited_title":"Creutzig, V","cited_arxiv_id":null,"evidence_quote":"Defines the universal two-parameter vertex algebra W_∞^sp(c,k), whose freeness, simplicity, and 1-parameter quotients Theorem 9.1 and Corollary 9.2 assume as input."},{"cited_title":"Inverting the Hamiltonian reduction in string th eory","cited_arxiv_id":null,"evidence_quote":"Introduced the inverse Hamiltonian reduction V_k(sl2) ↪ W_k(sl2,O_[2]) ⊗ Π, the template for the embeddings of Theorems 6.1 and 9.3."},{"cited_title":"Quantum Reduction and Representat ion Theory of Supercon- formal Algebras","cited_arxiv_id":null,"evidence_quote":"Provides the decomposition and cohomology arguments used to prove freeness, vanishing, and the strong generating type of H^0(W_∞^sp(c,k))."},{"cited_title":"Representation Theory of Superconformal Alge bras and the Kac-Roan-Wakimoto Conjecture","cited_arxiv_id":null,"evidence_quote":"Supplies exactness of the Virasoro reduction functor on the relevant category, used to extend the type-A isomorphism from generic levels to all levels."},{"cited_title":"Associated Varieties of Modules over Kac-Moody A lgebras and C2-Cofiniteness of W-Algebras","cited_arxiv_id":null,"evidence_quote":"Gives the cohomology vanishing H^{≠0}_O(V_k(g)) = 0 and acyclicity facts that make the Wakimoto exact sequence and the module reductions valid."}],"review_version":1}